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notation.pdf", "pages": 3, "bytes": 96523, "sha256": "4079cd45b3f2abbaa241fd9de55faad09860dcd13367d611ae3d1e7b9930654a" }, { "name": "14 Sigma notation sol.pdf", "pages": 10, "bytes": 306125, "sha256": "ee94f9dcb04cd9e64fa746c7c58299552b3b07dd67cfb59cc0648799e69c8d75" }, { "name": "15 complex numbers.pdf", "pages": 5, "bytes": 173750, "sha256": "0faffc4848db79887e3ad9d04c35b9c91b89bf4f835dc4bab19fd0580a27c685" }, { "name": "15 complex numbers sol.pdf", "pages": 23, "bytes": 771224, "sha256": "ae9c70fae706ce76355a80acd9f8e8b692a7a89198526d78b1a8157809dc384e" } ], "topics": [ { "id": "01", "title": "Graphs & transformations", "questionFile": "01 curve sketching and transformation.pdf", "solutionFile": "01 Curve sketching and transformation sol.pdf", "questionPages": 14, "solutionPages": 28, "count": 20, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 20, "reviewed": 20 }, { "id": "02", "title": "Linear systems", "questionFile": "02 System of linear eqn.pdf", "solutionFile": "02 System of linear eqn sol.pdf", "questionPages": 5, "solutionPages": 12, "count": 15, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 15, "reviewed": 15 }, { "id": "03", "title": "Inequalities", "questionFile": "03 Inequalities.pdf", "solutionFile": "03 Inequalities sol.pdf", "questionPages": 4, "solutionPages": 16, "count": 14, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 14, "reviewed": 14 }, { "id": "04", "title": "Functions", "questionFile": "04 Functions.pdf", "solutionFile": "04 Functions sol.pdf", "questionPages": 9, "solutionPages": 28, "count": 17, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 17, "reviewed": 17 }, { "id": "05", "title": "Differentiation", "questionFile": "05 Differentiation (Gradient function).pdf", "solutionFile": "05 Differentiation sol.pdf", "questionPages": 4, "solutionPages": 7, "count": 7, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 7, "reviewed": 7 }, { "id": "06", "title": "Applications of differentiation", "questionFile": "06 App of diff.pdf", "solutionFile": "06 App of diff sol.pdf", "questionPages": 9, "solutionPages": 29, "count": 18, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 18, "reviewed": 18 }, { "id": "07", "title": "Integration techniques", "questionFile": "07 Integration tech.pdf", "solutionFile": "07 Integration tech sol.pdf", "questionPages": 5, "solutionPages": 17, "count": 14, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 14, "reviewed": 14 }, { "id": "08", "title": "Applications of integration", "questionFile": "08 App of integration.pdf", "solutionFile": "08 App of integration sol.pdf", "questionPages": 8, "solutionPages": 22, "count": 15, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 15, "reviewed": 15 }, { "id": "09", "title": "Series expansion", "questionFile": "09 series expansion.pdf", "solutionFile": "09 seriex expansion sol.pdf", "questionPages": 27, "solutionPages": 27, "count": 23, "identical": true, "inventoryStatus": "visually checked question groups and subparts", "authored": 23, "reviewed": 23 }, { "id": "10", "title": "Differential equations", "questionFile": "10 DE.pdf", "solutionFile": "10 DE sol.pdf", "questionPages": 9, "solutionPages": 33, "count": 18, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 18, "reviewed": 18 }, { "id": "11", "title": "Vectors", "questionFile": "11 vectors.pdf", "solutionFile": "11 vectors sol.pdf", "questionPages": 12, "solutionPages": 44, "count": 23, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 23, "reviewed": 23 }, { "id": "12", "title": "Arithmetic & geometric progressions", "questionFile": "12 APGP.pdf", "solutionFile": "12 APGP sol.pdf", "questionPages": 8, "solutionPages": 32, "count": 20, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 20, "reviewed": 20 }, { "id": "13", "title": "Recurrence relations", "questionFile": "13 Recurrence relations.pdf", "solutionFile": "13 Recurrence relation sol.pdf", "questionPages": 5, "solutionPages": 13, "count": 13, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 13, "reviewed": 13 }, { "id": "14", "title": "Sigma notation", "questionFile": "14 Sigma notation.pdf", "solutionFile": "14 Sigma notation sol.pdf", "questionPages": 3, "solutionPages": 10, "count": 12, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 12, "reviewed": 12 }, { "id": "15", "title": "Complex numbers", "questionFile": "15 complex numbers.pdf", "solutionFile": "15 complex numbers sol.pdf", "questionPages": 5, "solutionPages": 23, "count": 19, "identical": false, "inventoryStatus": "visually checked question groups and subparts", "authored": 19, "reviewed": 19 } ], "questions": [ { "id": "01-01", "topic": "01", "number": 1, "questionPages": [ 1 ], "solutionPages": [ 1, 2 ], "attribution": "2010/SRJC/Prelim/II/3", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "1 y A(4, 3)\n\n\n 1\n\n x\n 0 2 C(3,0)\n\n B(1, – 1)\n\n\n\n\n The diagram above shows the graph of y  f  x  . On separate diagrams, sketch the graphs\n of\n (i) y  2f  x   1 , [3]\n (ii) y  f  x  , [2]\n (iii) y  f 1  2 x  . [3]\n showing in each case, the coordinates of the points corresponding to A, B, C and the\n equations of the asymptotes.\n [2010/SRJC/Prelim/II/3]", "methods": [ "Coordinate mapping", "Absolute-input reflection" ], "prerequisites": [ "Input versus output transformations", "Asymptotes" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 1, solution PDF pages 1,2. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "01-02", "topic": "01", "number": 2, "questionPages": [ 1 ], "solutionPages": [ 3, 4 ], "attribution": "modified 2010/RVHS/Prelim/II/1", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed-with-qualification", "difficulty": "not yet assessed", "extractedText": "2 The diagram shows the graph of y = f(x). The curve passes through the origin O, the points\n A(–4a, 0) and C(–2a, 0). The points B(3a,2) and D(a,2) are stationary points and the\n lines y = 0 and x = a are asymptotes of the curve.\n\n\n\n\n B\n\n\n a x\n A C 0\n\n D\n\n\n Sketch on separate clearly labelled diagrams, the graphs of\n (i) y  1  f ( x) ; [3]\n (ii) y  f ( x ) ; [2]\n 1\n (iii) y  , [3]\n f( x)\n indicating clearly the points corresponding to A, B, C and D, and the asymptotes, where\n applicable.\n [modified 2010/RVHS/Prelim/II/1]\n\n\n\n 1\n\f2025 TJC H2 Math (9758) JC2 MSM: GRAPHS AND CURVE SKETCHING", "methods": [ "Input/output absolute values", "Reciprocal graph" ], "prerequisites": [ "Domains", "Zeros and poles" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 1, solution PDF pages 3,4. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "An asymptote’s reciprocal limit can produce a hole; it does not automatically restore the excluded input. Source qualification: solution PDF p.4 labels the reciprocal limit at x=a as an intercept. Under the original function domain it is a hole; an intercept requires an explicitly defined continuous extension." }, { "id": "01-03", "topic": "01", "number": 3, "questionPages": [ 2 ], "solutionPages": [ 5 ], "attribution": "2021/ASRJC/Prelim/I/Q7(i)", "candidatePartLabels": [], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed-with-qualification", "difficulty": "not yet assessed", "extractedText": "3  x  2\n 2\n\n The curve C with equation y =\n 1 is transformed onto the curve C2 with equation\n x 1\n y  f ( x) . The curve C1 has a minimum turning point (0, 4) which corresponds to the point\n with coordinates (a, b) on the curve C2, where a, b > 0. Given that f(x) has the form\n p2 x2\n  q , where p, q are positive constants, express p and q in terms of a and b. [4]\n px  1\n [2021/ASRJC/Prelim/I/Q7(i)]", "methods": [ "Variable rescaling", "Derivative stationary points" ], "prerequisites": [ "Rational differentiation", "Extremum classification" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 2, solution PDF pages 5. Independent computations and qualifications are documented in that report.", "parts": [], "warning": "The given q>0 implies b>4; retain this feasibility condition." }, { "id": "01-04", "topic": "01", "number": 4, "questionPages": [ 2 ], "solutionPages": [ 6 ], "attribution": "modified 2010/ACJC/Prelim/I/3", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed-with-qualification", "difficulty": "not yet assessed", "extractedText": "4 The diagram below shows the graph of y  f ( x ) . The curve cuts the axes at (1,0) and\n (0,3). It has a maximum point at  3,5 , and asymptotes x  2 and y  2 .\n y\n\n\n\n\n x\n\n\n\n\n Sketch on separate diagrams, the graphs of\n (i) y  1 , [3]\n f(x)\n (ii) y  f(x)  4 , [3]\n showing clearly asymptotes, intercepts and coordinates of turning points, where available.\n [modified 2010/ACJC/Prelim/I/3]", "methods": [ "Coordinate mapping", "Reciprocal graph", "Absolute-output folding" ], "prerequisites": [ "Asymptote limits", "Stationary points" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 2, solution PDF pages 6. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii" ], "warning": "Source solution PDF p.6 labels the reciprocal limit (2,0) as an intercept, although the original input remains excluded; the authored version marks a hole unless a continuous extension is intended. The same page labels the absolute-output image of (−1,0) as (−1,−4); its correct height is |0−4|=4." }, { "id": "01-05", "topic": "01", "number": 5, "questionPages": [ 2 ], "solutionPages": [ 7 ], "attribution": "2021/CJC/Prelim/I/1", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "5 A curve has equation y  f ( x), where\n  8 x  1 for 0  x  3,\n f x  \n  11  2 x for 3  x  5,\n and f ( x)  f ( x  5) for all real values of x.\n (i) Sketch the curve for 2  x  8. [2]\n 1\n (ii) On a separate diagram, sketch the curve with equation y  f  x  1  , for\n 2 \n 2  x  8. [2]\n [2021/CJC/Prelim/I/1]\n\n\n\n\n 2\n\f2025 TJC H2 Math (9758) JC2 MSM: GRAPHS AND CURVE SKETCHING", "methods": [ "Period reduction", "Input transformation" ], "prerequisites": [ "Square-root graph", "Piecewise functions" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 2, solution PDF pages 7. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii" ], "warning": "" }, { "id": "01-06", "topic": "01", "number": 6, "questionPages": [ 3 ], "solutionPages": [ 9, 10 ], "attribution": "modified 2015/RVHS/Prelim/I/10", "candidatePartLabels": [ "a", "b.i", "b.ii", "b.iii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed-with-qualification", "difficulty": "not yet assessed", "extractedText": "6 (a) A curve y = f(x) is transformed by a stretch with scale factor 2 parallel to the y-axis,\n followed by a reflection about the x-axis. The equation of the resultant curve is\n  1 \n y  ln 2\n , where x  a and a  1. Find f(x) and sketch the graph of y  f ( x ).\n  ( x  a) \n [4]\n\n (b) The diagram shows the graph of y  g( x  1). The graph passes through the origin and\n has a turning point A(3, 4). The asymptotes of the graph are x  2 and y  2 x  4.\n y\n\n\n y = 2x – 4\n\n\n ●\n A\n\n x\n 0\n\n\n\n\n x=2\n On separate diagram, sketch the following graphs indicating the points corresponding\n to the axial intercepts, turning point and asymptotes, where necessary.\n (i) y  g( x) [2]\n (ii) y  g'( x) [3]\n 1\n (iii) y  [3]\n g( x)\n [modified 2015/RVHS/Prelim/I/10]\n\n\n\n\n 3\n\f2025 TJC H2 Math (9758) JC2 MSM: GRAPHS AND CURVE SKETCHING", "methods": [ "Undo transformations", "Derivative from monotonicity", "Reciprocal graph" ], "prerequisites": [ "Log laws", "Derivative signs", "Excluded inputs" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 3, solution PDF pages 9,10. Independent computations and qualifications are documented in that report.", "parts": [ "a", "b.i", "b.ii", "b.iii" ], "warning": "Detailed derivative limits follow the supplied qualitative sketch; no exact formula for g or numerical g′(0) is given. Source qualification: solution PDF p.10 labels the reciprocal limit at x=3 as an intercept. Under the original function domain it is a hole; an intercept requires an explicitly defined continuous extension." }, { "id": "01-07", "topic": "01", "number": 7, "questionPages": [ 4 ], "solutionPages": [ 11, 12 ], "attribution": "2018/NJC/Prelim/I/8", "candidatePartLabels": [ "a", "b" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "7 (a) Describe a sequence of exactly three transformations that will transform the curve with\n 1 2x  a\n equation y  onto the curve with equation y  , where a is a positive\n x3 3 x\n constant. [3]\n Given instead that a < -6, use a non-graphical method to determine the range of values\n 2x  a\n of x where the graph of y  is concave upwards. [2]\n 3 x\n\n (b) The diagrams below show the graphs of y  f  x  and y  f  x  , where the\n equations of the asymptotes and the coordinates of the turning points are given. The\n 3\n gradient of the graph with equation y  f  x  at the origin is  .\n 4\n y y\n\n\n\n\n O\n x O x\n\n\n\n\n On separate diagrams, sketch the graph of y  f ( x) including the coordinates of the points\n where the graphs cross the x- and y- axes and the equations of the asymptotes, if any. [2]\n [2018/NJC/Prelim/I/8]", "methods": [ "Rational decomposition", "Second derivative", "Sign reconstruction" ], "prerequisites": [ "Absolute-value graphs", "Concavity" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 4, solution PDF pages 11,12. Independent computations and qualifications are documented in that report.", "parts": [ "a", "b" ], "warning": "" }, { "id": "01-08", "topic": "01", "number": 8, "questionPages": [ 4 ], "solutionPages": [ 12 ], "attribution": "2015/SAJC/Prelim/I/5", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "8 x2 y 2\n The ellipse C1 with the equation   1 is transformed by a stretch with scale factor\n 62 32\n k parallel to the x-axis, followed by a translation of m units in the positive x-direction,\n followed by a translation of n units in the negative y-direction, to form C2 .\n (i) Find the equation of C2 in terms of k, m, and n. [3]\n (ii) Given that C2 is a circle centred at  4, 7  , find the values of k, m and n. [2]\n [2015/SAJC/Prelim/I/5]", "methods": [ "Coordinate substitution", "Semiaxis comparison" ], "prerequisites": [ "Ellipse standard form", "Translations" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 4, solution PDF pages 12. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii" ], "warning": "" }, { "id": "01-09", "topic": "01", "number": 9, "questionPages": [ 4 ], "solutionPages": [ 13, 14 ], "attribution": "2011/PJC/Promo/9", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "9 The curve C has equation\n ax  b\n y ,\n x2  c\n where a, b and c are constants. It is given that x  2 is an asymptote of C.\n (i) State the value of c. [1]\n  1\n (ii) Given also that C has a turning point at  6,  , find the values of a and b. [4]\n  4\n (iii) Sketch C, stating the equations of the asymptotes and the coordinates of the turning\n points. [You need not indicate the coordinates of intersections with the axes.] [3]\n (iv) Hence find the range of values of k for which the equation kx 2  3x  10  4k  0 has\n no real root. [2]\n [2011/PJC/Promo/9]\n 4\n\f2025 TJC H2 Math (9758) JC2 MSM: GRAPHS AND CURVE SKETCHING", "methods": [ "Value/derivative system", "Horizontal-line range", "Discriminant" ], "prerequisites": [ "Quotient rule", "Quadratic roots" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 4, solution PDF pages 13,14. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "" }, { "id": "01-10", "topic": "01", "number": 10, "questionPages": [ 5 ], "solutionPages": [ 15 ], "attribution": "No exam attribution printed with this question", "candidatePartLabels": [], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "10 Sketch the graphs of y  ln 1  x  and y   x  1 on the same diagram, showing clearly\n the equations of any asymptotes and any intersections with the axes.\n Hence, solve the inequality ln 1  x   x  1  0 .\n\n x  a  a2", "methods": [ "Domain split", "Monotonicity", "Numerical intersections" ], "prerequisites": [ "Logarithm domain", "Absolute values" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 5, solution PDF pages 15. Independent computations and qualifications are documented in that report.", "parts": [], "warning": "" }, { "id": "01-11", "topic": "01", "number": 11, "questionPages": [ 5 ], "solutionPages": [ 15, 16 ], "attribution": "2012/NJC/Promo/11(a)", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "11 The curve C has equation given by y = , where a is a negative constant.\n a( x  a)\n (i) Write down the equations of the asymptotes of C. [2]\n (ii) Sketch the graph of C, labelling clearly the asymptotes and axial intercepts. [2]\n x  a  a2\n (iii) Deduce the range of values of k for which the equation = k has one\n a( x  a)\n negative real root. [2]\n (iv) State the range of values of x for which the curve is concave downwards. [1]\n [2012/NJC/Promo/11(a)]", "methods": [ "Polynomial division", "Inverse level equation", "Second derivative" ], "prerequisites": [ "Rational graphs", "Sign analysis" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 5, solution PDF pages 15,16. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "" }, { "id": "01-12", "topic": "01", "number": 12, "questionPages": [ 5 ], "solutionPages": [ 16 ], "attribution": "2009/SRJC/Prelim/I/7", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "12 x2  b\n The curve C has equation y  ,where a > 0 and b > 0.\n xa\n (i) State the coordinates of the intersection(s) of the curve C with the axes in terms of a\n and b. [1]\n (ii) Find the equation(s) of the asymptote(s). [2]\n (iii) Draw a sketch of the curve C, labeling the equation(s) of its asymptote(s) and\n coordinates of any intersection with the axes. [2]\n (iv) Hence find the range of values of k, where k is a positive constant, for which the\n equation x2 + b = (x + a)(kx – a) has no real root. [2]\n [2009/SRJC/Prelim/I/7]\n ax 2  bx  c", "methods": [ "Polynomial division", "Line comparison", "Degenerate quadratic analysis" ], "prerequisites": [ "Discriminant", "Rational asymptotes" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 5, solution PDF pages 16. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "" }, { "id": "01-13", "topic": "01", "number": 13, "questionPages": [ 5 ], "solutionPages": [ 17, 18 ], "attribution": "modified 2010/RVHS/Prelim/I/7", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "13 The diagram below shows a sketch of the graph of y  where a, b, c and d\n xd\n are constants. The asymptotes, also shown in the diagram, are x  2 and y  x  4.\n y\n\n\n\n −2 4\n O x\n −4\n\n\n\n\n (i) Write down the value of d and find the values of a and b. [4]\n (ii) Given that the curve passes through the point (0, 4.5) , find the value of c. [1]\n (iii) By drawing a suitable curve on the same diagram shown above, determine the range\n 2\n 2  ax 2  bx  c  2\n of values of k such that the equation  x  4      k , where a, b, c\n  xd \n and d are the values found above, has at least one negative root. [3]\n [modified 2010/RVHS/Prelim/I/7]\n\n\n 5\n\f2025 TJC H2 Math (9758) JC2 MSM: GRAPHS AND CURVE SKETCHING", "methods": [ "Asymptote coefficient comparison", "Circle intersection", "Distance monotonicity" ], "prerequisites": [ "Polynomial division", "Circle equation", "Derivative signs" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 5, solution PDF pages 17,18. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "01-14", "topic": "01", "number": 14, "questionPages": [ 6 ], "solutionPages": [ 19, 20 ], "attribution": "2011/NJC/Prelim/I/12b", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed-with-qualification", "difficulty": "not yet assessed", "extractedText": "14 ax 2  x  4\n The curve C has equation y  , where a is a constant.\n x2\n dy\n (i) Find in terms of a, and deduce the range of values of a if C has two distinct\n dx\n stationary points. [4]\n (ii) Given that the asymptotes of C intersect at the point (−2, −7), show that a = 2. [2]\n (iii) Sketch C, indicating clearly the asymptotes and stationary points. [2]\n (iv) For r  0 , find the range of values of r, correct to 3 decimal places, such that the\n x2\n equation ln( x  r )  2 has exactly one real root. [2]\n 2x  x  4\n [2011/NJC/Prelim/I/12b]", "methods": [ "Quotient rule", "Discriminant", "Asymptote intersection", "Monotone root counting" ], "prerequisites": [ "Rational poles", "Logarithm domain", "Limits" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 6, solution PDF pages 19,20. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "Source solution PDF p.20 labels A=(−1.186,0) in one sentence, although its graph and r≥1.186 conclusion use the positive root. Authored working keeps the exact positive threshold separately." }, { "id": "01-15", "topic": "01", "number": 15, "questionPages": [ 6 ], "solutionPages": [ 21, 22 ], "attribution": "2018/AJC/Prelim/II/3", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased; graphical features retained", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "15 2(b  2a )\n The curve C has equation y  ax  b  2a  , where a and b are real constants such\n x2\n that a > 0, b  2a and x  2.\n (i) By using differentiation, find the range of values of b in terms of a such that C has\n no stationary points. [4]\n (ii) Given that b = 3a, sketch C, stating the equations of any asymptotes and the\n coordinates of the points where C crosses the axes. [3]\n 2\n  y\n (iii) On the same diagram as C, sketch the graph of    ( x  1)2  4, showing clearly\n a\n the equations of the asymptotes and turning points. Hence solve the inequality\n 2\n x 1   ( x  1) 2  4. [4]\n x2\n [2018/AJC/Prelim/II/3]", "methods": [ "Derivative positivity", "Hyperbola comparison", "Sign before squaring" ], "prerequisites": [ "Rational graphs", "Square-root inequalities" ], "classificationStatus": "reviewed against rendered question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-01.md; visually audited question PDF pages 6, solution PDF pages 21,22. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "Squaring without first requiring the rational side to be nonnegative would incorrectly include 20. The inclusion of 2a in the inverse domain awaits clarification." }, { "id": "04-16", "topic": "04", "number": 16, "questionPages": [ 6 ], "solutionPages": [ 25, 26 ], "attribution": "[2011/RI/Promo/13]", "candidatePartLabels": [ "a", "b.i", "b.ii", "b.iii", "b.iv" ], "partInventoryStatus": "visually checked against Functions question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "16 The function f is defined by\n f:x ( x − 7)( x + 1), x  0.\n\n x2 − 7 x\n (a) Use an algebraic method to find the exact solution of f ( x )  . [5]\n 2− x\n\n (b) (i) Determine if the inverse function of f exists. [2]\n It is given that g is a one-one function, whose domain and range are given by the\n intervals ( 0,  ) and ( 4,  ) respectively.\n (ii) Find the range of fg . [2]\n (iii) A function h is one-one if, for any  and  in the domain of h\n such that h ( ) = h (  ) , then  =  . Show that fg is one-one. [2]\n\n If g( x) = 4e x for x > 0, find the exact value of ( fg ) ( 20 ) .\n −1\n (iv) [4]\n\n [2011/RI/Promo/13]\n\n\n\n\n 29\n\f2025 TJC H2 Math (9758) JC2 MSM: FUNCTIONS", "methods": [ "Rational sign chart", "Restricted injectivity", "Factorization", "Inverse value by equation" ], "prerequisites": [ "Inequality signs", "Quadratic turning points", "Exponentials" ], "classificationStatus": "reviewed against Functions question pages", "reviewEvidence": "Functions question PDF pp.6, solution PDF pp.25,26 visually checked. Working independently rearranged with domain/branch/endpoints retained; targeted numeric identities in tests/core-reviewed-maths.test.mjs.", "parts": [ "a", "b.i", "b.ii", "b.iii", "b.iv" ], "warning": "" }, { "id": "04-17", "topic": "04", "number": 17, "questionPages": [ 7 ], "solutionPages": [ 27, 28 ], "attribution": "[2019/TJC/Promo/9]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against Functions question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; source wording paraphrased", "mathematicalReview": "reviewed-with-qualification", "difficulty": "not yet assessed", "extractedText": "17 Functions f and g are defined by\n\n x2 − 6 x − 6\n f :x , x  , x  −1 ,\n x +1\n ax + 1\n g:x , x , x  −b ,\n x+b\n where a and b are constants.\n (i) Sketch the graph of y = f ( x) , giving the coordinates of the turning points and the\n equation of the asymptotes. Write down the range of f. [3]\n\n (ii) Find the value of a and the range of values of b such that both composite functions\n fg and gf exist. [4]\n\n (iii) Find g−1(x). Given that g−1(x) = g(x) for all real x, x  −b , find b in terms of a. [3]\n [2019/TJC/Promo/9]\n\n\n\n\n 30\n\f2025 TJC H2 Math (9758) JC2 MSM: FUNCTIONS", "methods": [ "Polynomial division", "Derivative turning points", "Range inclusion", "Degenerate-case analysis", "Rational inverse" ], "prerequisites": [ "Rational asymptotes", "Set complements", "Parameters and injectivity" ], "classificationStatus": "reviewed against Functions question pages", "reviewEvidence": "Functions question PDF pp.7, solution PDF pp.27,28 visually checked. Working independently rearranged with domain/branch/endpoints retained; targeted numeric identities in tests/core-reviewed-maths.test.mjs.", "parts": [ "i", "ii", "iii" ], "warning": "Source qualification: solution PDF p.28 assumes ab≠1 in part (ii). Under that part’s literal wording, a constant-map family also works. Part (iii) requires an inverse, which rules out constant maps; if that assumption is intended throughout, the printed part (ii) answer applies." }, { "id": "05-01", "topic": "05", "number": 1, "questionPages": [ 1 ], "solutionPages": [ 1 ], "attribution": "2009/DHS/Prelim/I/4", "candidatePartLabels": [ "a", "b" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "1 (a) Suppose the following facts are known about the function g and its derivative:\n g(5) = 1, g '(5) = −3.\n\n If f ( x) = tan ( eg( x ) ) , find the value of f '(5) . [3]\n\n x2\n (b) Find, by differentiation, the range of values of x for which the function y =\n 2x − 1\n increases as x increases. [3]\n [2009/DHS/Prelim/I/4]\n\n d  −1  2 x   −2", "methods": [ "Chain rule", "Quotient rule", "Derivative sign chart" ], "prerequisites": [ "Trigonometric derivatives in radians", "Exponentials", "Rational inequalities" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-05.md; rendered original question PDF pages 1 and solution PDF pages 1; independently worked first-pass mathematical review", "parts": [ "a", "b" ], "warning": "Do not omit the derivative of g. A squared denominator is positive only where nonzero." }, { "id": "05-02", "topic": "05", "number": 2, "questionPages": [ 1 ], "solutionPages": [ 2 ], "attribution": "2009/YJC/Prelim/I/5(a)", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed with endpoint qualification", "difficulty": "not yet assessed", "extractedText": "2 (i) Show that  cos  2 \n = for −1  x  1 .\n 1+ x  1+ x\n 2\n dx \n Explain, in your working, how you use the condition −1  x  1 to conclude your\n answer. [3]\n\n (ii) Hence, determine the constants A and B satisfying the equation\n  2x \n cos −1  2 \n + A tan −1 x = B , for −1  x  1 . [3]\n  1 + x \n [2009/YJC/Prelim/I/5(a)]", "methods": [ "Chain rule", "Quotient rule", "Absolute-value simplification", "Constant derivative", "Double-angle identity" ], "prerequisites": [ "Principal inverse trigonometric branches", "Radians", "One-sided derivatives" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-05.md; rendered original question PDF pages 1 and solution PDF pages 2; independently worked first-pass mathematical review", "parts": [ "i", "ii" ], "warning": "The printed derivative includes endpoints without saying one-sided. The identity itself holds on the closed interval." }, { "id": "05-03", "topic": "05", "number": 3, "questionPages": [ 1 ], "solutionPages": [ 3 ], "attribution": "modified 2010/RVHS/Prelim/II/1", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "3 The diagram shows the graph of y = f(x). The curve passes through the origin O, the points\n A(–4a, 0) and C(–2a, 0). The points B(−3a,2) and D(−a,−2) are stationary points and the\n lines y = 0 and x = a are asymptotes of the curve.\n\n\n\n\n B\n\n\n a x\n A C 0\n\n D\n\n\n\n Sketch the graph of y = f '( x), indicating clearly the points corresponding to A, B, C and\n D, and the asymptotes, where applicable. [2]\n [modified 2010/RVHS/Prelim/II/1]\n\n\n\n\n 33\n\f2025 TJC H2 Math (9758) JC2 MSM: DIFFERENTIATION (GRADIENT FUNCTION)", "methods": [ "Qualitative gradient graph", "Stationary-point mapping", "Concavity inspection" ], "prerequisites": [ "Slope signs", "Asymptotes", "Stationary points" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-05.md; rendered original question PDF pages 1 and solution PDF pages 3; independently worked first-pass mathematical review", "parts": [], "warning": "Roots of f need not be roots of its derivative." }, { "id": "05-04", "topic": "05", "number": 4, "questionPages": [ 2 ], "solutionPages": [ 4 ], "attribution": "modified 2010/ACJC/Prelim/I/3", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "4 The diagram below shows the graph of y = f ( x) . The curve cuts the axes at (−1,0) and\n (0,3). It has a maximum point at ( 3,5 ) , and asymptotes x = −2 and y = 2 .\n y\n\n\n\n\n x\n\n\n\n\n Sketch the graph of y = f '(x), showing clearly asymptotes, intercepts and coordinates of\n turning points, where available. [3]\n [modified 2010/ACJC/Prelim/I/3]", "methods": [ "Qualitative gradient graph", "Concavity inspection", "Asymptotic slope" ], "prerequisites": [ "Gradient versus height", "Turning points", "Asymptotes" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-05.md; rendered original question PDF pages 2 and solution PDF pages 4; independently worked first-pass mathematical review", "parts": [], "warning": "A value f(x)=2 at one point does not imply zero derivative. Read the actual slope behaviour." }, { "id": "05-05", "topic": "05", "number": 5, "questionPages": [ 2 ], "solutionPages": [ 5, 6 ], "attribution": "2008/NJC/Prelim/I/11(i)", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "5\n\n\n\n\n 3\n\n\n\n\n The diagram above shows the graph of the function y = f ( x) where\n ax 2 + bx + c + 1 3\n f ( x) = , a, b, c, d  . The oblique asymptote intersects the axes at x =\n x+d 2\n and y = 3 . Find the values of a, b and d.\n dy\n By considering , show that c > −1. [5]\n dx\n Sketch, on a separate diagram, the graph of y = f ( x) . [3]\n [2008/NJC/Prelim/I/11(i)]\n 34\n\f2025 TJC H2 Math (9758) JC2 MSM: DIFFERENTIATION (GRADIENT FUNCTION)", "methods": [ "Polynomial division", "Coefficient comparison", "Derivative sign", "Degenerate-case exclusion" ], "prerequisites": [ "Oblique asymptotes", "Rational derivatives", "Quadratic equations" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-05.md; rendered original question PDF pages 2 and solution PDF pages 5,6; independently worked first-pass mathematical review", "parts": [], "warning": "No stationary points alone gives c≥−1; the stated vertical asymptote excludes equality." }, { "id": "05-06", "topic": "05", "number": 6, "questionPages": [ 3 ], "solutionPages": [ 6 ], "attribution": "2009/HCI/Prelim/I/3", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "6 The diagram below shows the graph of y = f '( x) . The curve passes through the origin and\n has turning points at (−3, 0) and (−1.5, −1). The x-axis and x = 1 are the two asymptotes of\n the curve.\n y\n\n\n\n\n x\n −3 O\n (−1.5, −1)\n\n x=1\n\n (i) Find the range of values of x for which the graph of y = f ( x) is strictly increasing and\n concave downwards. [1]\n (ii) State the x-coordinates of all the stationary points of the graph of y = f ( x) and\n determine the nature of each point. [2]\n [2009/HCI/Prelim/I/3]", "methods": [ "Derivative sign", "Second-derivative sign", "First-derivative stationary test" ], "prerequisites": [ "Concavity", "Sign changes", "Stationary inflection" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-05.md; rendered original question PDF pages 3 and solution PDF pages 6; independently worked first-pass mathematical review", "parts": [ "i", "ii" ], "warning": "A derivative zero alone does not establish a maximum or minimum." }, { "id": "05-07", "topic": "05", "number": 7, "questionPages": [ 3 ], "solutionPages": [ 7 ], "attribution": "2021/DHS/Promo/7", "candidatePartLabels": [ "i", "ii.a", "ii.b", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed with endpoint convention qualification", "difficulty": "not yet assessed", "extractedText": "7 The graphs of y = f '( x) and y = f ( x) are shown below.\n y y\n\n\n\n\n (3, 3)\n\n\n\n x x\n O (3, 0) O\n\n\n\n x=2\n x=2\n\n\n (i) State the nature of all stationary points of the graph of y = f ( x). [2]\n\n (ii) State the range of values of x where\n (a) f is decreasing, [2]\n (b) f is concave upwards. [1]\n\n (iii) Sketch the graph of y = f ( x) indicating clearly the asymptotes, coordinates of the\n stationary points and the intersections with the axes. [3]\n [2021/DHS/Promo/7]\n\n\n 35\n\f2025 TJC H2 Math (9758) JC2 MSM: DIFFERENTIATION (GRADIENT FUNCTION)", "methods": [ "Derivative sign", "Concavity", "Absolute-value branch reconstruction" ], "prerequisites": [ "Derivative graphs", "Reflection in x-axis", "Asymptotes" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-05.md; rendered original question PDF pages 3 and solution PDF pages 7; independently worked first-pass mathematical review", "parts": [ "i", "ii.a", "ii.b", "iii" ], "warning": "A minimum of |f| need not be a minimum of f. Closed endpoints describe full branches, not strict derivative inequalities." }, { "id": "06-01", "topic": "06", "number": 1, "questionPages": [ 1 ], "solutionPages": [ 1 ], "attribution": "2012/NYJC/Prelim/I/3", "candidatePartLabels": [ "derivative", "i", "ii", "geometry" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "1 The equation of a curve is given by 4 x3 + 3x 2 y = y 3 − 2 . Find in terms of x and y,\n dx\n simplifying your answer. [2]\n The curve meets the line y = − x at point P. Find\n (i) the coordinates of P and [2]\n (ii) the equation of the tangent at P. [2]\n The tangent to the curve at P cuts the x-axis at R and the normal to the curve at P cuts the\n y-axis at Q. If O denotes the origin, use the results above to give a geometrical description\n of the quadrilateral OQPR. [1]\n [2012/NYJC/Prelim/I/3]\n The equation of a closed curve is given by ( nx + y ) + 4 ( nx − y ) = 20 , where n is a\n 2 2", "methods": [ "Implicit differentiation", "Substitution", "Implicit tangent line", "Perpendicular normal" ], "prerequisites": [ "Product rule", "Curve-line intersection", "Coordinate geometry" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 1 and solution PDF pages 1; independently worked first-pass mathematical review", "parts": [ "derivative", "i", "ii", "geometry" ], "warning": "An undefined slope is not automatically a vertical tangent at every singularity. Here the nonzero implicit gradient verifies it." }, { "id": "06-02", "topic": "06", "number": 2, "questionPages": [ 1 ], "solutionPages": [ 2, 3 ], "attribution": "2009/IJC/Prelim/I/11", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed with nonzero-parameter qualification", "difficulty": "not yet assessed", "extractedText": "2\n constant. Find\n (i) the gradient of the curve at the point ( x, y ) , [2]\n (ii) the coordinates of the points on the curve at which the tangent is parallel to the\n y−axis, [4]\n (iii) the equations of the normal at the points where the curve cuts the x-axis. [4]\n [2009/IJC/Prelim/I/11]", "methods": [ "Implicit differentiation", "Vertical tangent condition", "Normal equations" ], "prerequisites": [ "Chain rule", "Quadratic substitution", "Negative reciprocal slope" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 1 and solution PDF pages 2,3; independently worked first-pass mathematical review", "parts": [ "i", "ii", "iii" ], "warning": "The answer list prints only one sign in part (iii); both x-axis intersections require a normal. Closedness supplies the unstated n≠0 condition." }, { "id": "06-03", "topic": "06", "number": 3, "questionPages": [ 1 ], "solutionPages": [ 3, 4, 5 ], "attribution": "2023/MI/Prelim/II/4", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "3 MI Prelim 9758/2023/02/Q4\n The curve C has equation 2 x 2 + y 2 = 20 .\n (i) Sketch curve C, giving the exact coordinates of any points where the curve meets\n the axes. [2]\n dy 2x\n (ii) Show that =− . [1]\n dx y\n (iii) It is given that point P (a, b) is on C. The normal to C at P passes through the point\n (1, 0). Find the possible coordinates of P. [5]\n [2023/MI/Prelim/II/4]\n 2", "methods": [ "Implicit differentiation", "Normal incidence", "Special-case analysis", "Implicit gradient vector" ], "prerequisites": [ "Ellipse intercepts", "Normal versus tangent", "Factoring equations" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 1 and solution PDF pages 3,4,5; independently worked first-pass mathematical review", "parts": [ "i", "ii", "iii" ], "warning": "Dividing by b would lose the two horizontal normals." }, { "id": "06-04", "topic": "06", "number": 4, "questionPages": [ 1 ], "solutionPages": [ 5, 6 ], "attribution": "2022/HCI/Prelim/I/9", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "4 A curve C is given by the equation e x y = ln x 2 , where x  0 .\n dy 2\n Show that 2 x ln x 2 + e x = .\n 2\n (i) [2]\n dx x\n (ii) Find the equation of the tangent to C at the point P where x = 1 . Given that the\n tangent at P meets C again at the point Q, find the coordinates of Q, leaving your\n answer correct to 5 significant figures. [4]\n\n (iii) Determine the acute angle, in degrees, between the tangents to C at the points P\n and Q. [3]\n [2022/HCI/Prelim/I/9]\n\n\n\n\n 37\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF DIFFERENTIATION", "methods": [ "Implicit product rule", "Numerical intersection", "Angle between lines" ], "prerequisites": [ "Natural logarithms of x²", "Chain rule", "Radians and degree conversion" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 1 and solution PDF pages 5,6; independently worked first-pass mathematical review", "parts": [ "i", "ii", "iii" ], "warning": "ln(x²) is defined for negative x too; replacing it by 2ln x discards the branch containing Q." }, { "id": "06-05", "topic": "06", "number": 5, "questionPages": [ 2 ], "solutionPages": [ 7, 8 ], "attribution": "modified 2011/SRJC/Prelim/II/3", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "5 A curve C is defined by the parametric equations\n x =  − cos  , y =  + cos  , 0    2\n\n (i) Find the coordinate of the point(s) where the tangent is parallel to x-axis. [3]\n (ii) Find the equation of tangent to the curve C that is parallel to the y-axis. [2]\n \n (iii) Find the exact equation of the tangent to the curve at the point P where  =\n . [3]\n 6\n (iv) Hence, determine whether the tangent at point P will meet the curve C again. [3]\n [modified 2011/SRJC/Prelim/II/3]", "methods": [ "Parametric differentiation", "Tangent equation", "Intersection sign analysis" ], "prerequisites": [ "Trigonometric derivatives", "Intermediate value theorem", "Parameter domains" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 2 and solution PDF pages 7,8; independently worked first-pass mathematical review", "parts": [ "i", "ii", "iii", "iv" ], "warning": "A tangent can touch at one parameter and intersect again elsewhere; do not infer the second meeting from tangency alone." }, { "id": "06-06", "topic": "06", "number": 6, "questionPages": [ 2 ], "solutionPages": [ 9, 10 ], "attribution": "2011/NYJC/Prelim/I/9", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed with source-domain discrepancy", "difficulty": "not yet assessed", "extractedText": "6 The diagram shows a sketch of the curve y = xe− x .\n y\n\n\n\n\n x\n\n\n\n\n (i) By differentiation, find the range of values of x for which the graph of y = xe− x is\n decreasing. [3]\n\n (ii) Determine the range of values of x for which the graph of y = xe− x is decreasing and\n concave downwards. [3]\n\n (iii) The tangent to the curve at P ( x, y ) meets the y-axis at R ( 0, h ) . Express h in terms\n of x , and find the greatest possible value of h. [6]\n [2011/NYJC/Prelim/I/9]", "methods": [ "Product rule", "Derivative sign", "Tangent intercept", "Constrained optimisation", "Counterexample" ], "prerequisites": [ "Exponentials", "Concavity", "Global versus local maxima" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 2 and solution PDF pages 9,10; independently worked first-pass mathematical review", "parts": [ "i", "ii", "iii" ], "warning": "Printed greatest value assumes an unstated domain restriction. Keep the local answer and unrestricted conclusion distinct." }, { "id": "06-07", "topic": "06", "number": 7, "questionPages": [ 2 ], "solutionPages": [ 10, 11, 12 ], "attribution": "2022/NYJC/Prelim/I/7", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "7 A curve is defined parametrically by\n at at 2\n x= ,y= ,\n t +1 t +1\n where a is a non-zero constant and t  , t  −1 .\n (i) Show that the equation of the normal to the curve at the point T with parameter t is\n given by\n t (t + 1)(t + 2) y + (t + 1) x = at (t 3 + 2t 2 + 1) . [4]\n  1 \n (ii) The normal to the curve at the point P  − a, a  meets the curve again at point Q.\n  2 \n Find the exact coordinates of Q in terms of a. [4]\n (iii) Let M be the mid-point of PQ. Find a cartesian equation of the curve traced by M\n if a varies. [2]\n [2022/NYJC/Prelim/I/7]\n\n 38\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF DIFFERENTIATION", "methods": [ "Parametric differentiation", "Normal equation", "Quadratic factorisation", "Parameter elimination" ], "prerequisites": [ "Quotient rule", "Midpoint formula", "Locus restrictions" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 2 and solution PDF pages 10,11,12; independently worked first-pass mathematical review", "parts": [ "i", "ii", "iii" ], "warning": "The Cartesian equation alone includes an unattained origin; retain a≠0. Do not divide by t(t+2) when deriving a formula valid at horizontal tangents." }, { "id": "06-08", "topic": "06", "number": 8, "questionPages": [ 3 ], "solutionPages": [ 12, 13 ], "attribution": "2023/ACJC/Promo/7", "candidatePartLabels": [ "a", "b", "c", "d" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "8 The curve C has parametric equations\n x = 4cos  , y = 3sin  , 0     .\n (a) Point Q lies on C and is moving along C, such that its x coordinate is increasing at\n the constant rate of 15 units per second. Determine the exact rate of change of  at\n  3 3\n the instant when the coordinates of Q is  2, . [3]\n  2 \n \n It is given that point P also lies on C and has parameter p where 0  p  .\n 2\n (b) Show that the equation of the tangent to C at the point P, is given by\n  3 \n y =  − cot p  x + 3 cosec p. [3]\n  4 \n (c) The tangent to the curve at P meets the x-axis at the point A and the y-axis at the\n point B. Find the area of the triangle OAB. [2]\n (d) Let M be the mid-point of AB. Find a cartesian equation of the curve traced by M\n as p varies. [2]\n [2023/ACJC/Promo/7]", "methods": [ "Parametric rates", "Tangent equation", "Intercept triangle area", "Trigonometric elimination" ], "prerequisites": [ "Chain rule", "Half base times height", "Positive parameter branch" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 3 and solution PDF pages 12,13; independently worked first-pass mathematical review", "parts": [ "a", "b", "c", "d" ], "warning": "The midpoint locus is not an ordinary ellipse in x,y: reciprocal coordinates satisfy the ellipse identity." }, { "id": "06-09", "topic": "06", "number": 9, "questionPages": [ 3 ], "solutionPages": [ 14, 15 ], "attribution": "2017/SRJC/Prelim/I/10", "candidatePartLabels": [ "i", "ii", "iii", "iv", "v" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "9 A particle is moving along a curve, C , such that its position at time t seconds after it is\n set into motion is given by the parametric equations\n x = t + e−2t , y = t − e−2t .\n (i) State the coordinates of the initial position of the article. [1]\n (ii) Explain what would happen to the path of the particle after a long time. [1]\n\n At the time of 2 seconds after the particle was set into motion, an external force struck\n the particle resulting in the particle moving in a straight line along the normal to the path\n at the point of collision.\n (iii) Find an equation for the normal to the curve C at the point t = 2 , leaving your\n answer correct to 3 decimal places. [3]\n\n After T seconds, where T > 2, the particle reaches point A, which lies on the x-axis, and\n stops moving.\n (iv) Find the coordinates of the point A. Hence, give a sketch of the path traced by the\n particle, indicating the coordinates of any axial intercepts. [4]\n (v) Find the total area bounded by the path of the particle in the first T seconds and the\n positive x-axis. [4]\n [2017/SRJC/Prelim/I/10]\n\n\n\n\n 39\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF DIFFERENTIATION", "methods": [ "Parametric differentiation", "Normal line", "Numerical root", "Parametric integration", "Area decomposition" ], "prerequisites": [ "Velocity components", "Exponentials", "Triangle area", "Definite integral substitution" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 3 and solution PDF pages 14,15; independently worked first-pass mathematical review", "parts": [ "i", "ii", "iii", "iv", "v" ], "warning": "Do not integrate from t=0 as if the entire path and axis enclosed one positive region; the early curve doubles back in x." }, { "id": "06-10", "topic": "06", "number": 10, "questionPages": [ 4 ], "solutionPages": [ 16, 17 ], "attribution": "Not printed in supplied question", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "10 y Cannonball (x, y)\n\n\n\n O Ground\n x\n The diagram shows the trajectory of a cannonball fired off from an origin O with an initial\n speed of v ms −1 and at an angle of   above the ground. At time t seconds, the position\n of the cannonball can be modelled by the parametric equations\n x = (v cos ) t , y = (v sin  ) t − 5t 2 ,\n where x m is the horizontal distance of the cannonball with respect to O and y m is the\n vertical distance of the cannonball with respect to ground level.\n (i) Find the horizontal distance, d m, that a cannonball would have travelled by the time\n it hits the ground. Leave your answer in terms of v and  . [4]\n Use v = 200 to answer the remaining parts of the question.\n An approaching target is travelling at a constant speed of 10 ms-1 along the ground. A\n cannonball is fired towards the target when it is 3000 m away. You may assume the height\n of the moving target is negligible.\n (ii) Show that in order to hit the target, the possible angles at which the cannonball\n should be fired are 22.7 and 69.5 . [2]\n (iii) Explain at which angle the cannonball should be fired in order to hit the target earlier.\n [2]\n (iv) Given that  = 22.7, find the angle that the tangent to the trajectory makes with the\n horizontal when x = 370. [4]\n\n\n\n\n 40\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF DIFFERENTIATION", "methods": [ "Parametric projectile motion", "Time elimination", "Numerical trigonometric equation", "Tangent slope" ], "prerequisites": [ "Ground intercepts", "Constant-speed distance", "Trigonometric angles in degrees" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 4 and solution PDF pages 16,17; independently worked first-pass mathematical review", "parts": [ "i", "ii", "iii", "iv" ], "warning": "Use degree mode for this model. The approaching target travels towards the origin, so its distance decreases." }, { "id": "06-11", "topic": "06", "number": 11, "questionPages": [ 5 ], "solutionPages": [ 18 ], "attribution": "2009/MJC/Prelim/I/11", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "11 A manufacturer wants to design a lidless container with a square base and a capacity of\n 1000 cm3 . Given that the uniform thickness of the container is 1 cm, using differentiation,\n find the internal length of the tank, x cm, such that the volume of the material used is\n minimised. [6]\n 1 cm\n 1 cm\n\n\n\n\n x cm\n\n\n 1 cm\n\n\n 1 cm\n\n [2009/MJC/Prelim/I/11]", "methods": [ "Volume constraint", "Outer-minus-inner volume", "Differentiation", "Factorisation", "Global minimum" ], "prerequisites": [ "Cuboid volumes", "Uniform wall thickness", "Positive dimensions" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 5 and solution PDF pages 18; independently worked first-pass mathematical review", "parts": [], "warning": "Using x+1 for the outer base side misses one wall; using h+2 adds a lid the design does not have." }, { "id": "06-12", "topic": "06", "number": 12, "questionPages": [ 5 ], "solutionPages": [ 19, 20 ], "attribution": "2022/NYJC/Prelim/I/Q2", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "12 [It is given that a right circular cone with base radius r and height h has volume 1 πr 2 h\n 3\n and curved surface area  rl , where l is the slanted height of the cone.]\n\n\n 8x cm\n 15x\n\n\n w cm\n\n\n\n\n The model of a house is made up of three parts.\n • The roof is modelled by the curved surface of a right circular cone with base radius\n 15x cm and height 8x cm.\n • The walls are modelled by the curved surface of a cylinder of radius 15x cm and\n height w cm.\n • The floor is modelled by a circular disc of radius 15x cm.\n The three parts are joined together as shown in the diagram. The model is made of\n material of negligible thickness. It is given that the volume of the model is a fixed value\n k cm3, and the external surface area is a minimum. Use differentiation to find the exact\n values of x and w in terms of k. Simplify your answers. [8]\n [2022/NYJC/Prelim/I/Q2]\n\n\n 41\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF DIFFERENTIATION", "methods": [ "Pythagoras", "Fixed-volume elimination", "Differentiation", "Convex minimum" ], "prerequisites": [ "Cone and cylinder volumes", "Curved surface area", "Positive dimensions" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 5 and solution PDF pages 19,20; independently worked first-pass mathematical review", "parts": [], "warning": "Do not count the joined circular cone base as exposed area, and do not omit the floor." }, { "id": "06-13", "topic": "06", "number": 13, "questionPages": [ 6 ], "solutionPages": [ 20, 21 ], "attribution": "2010/NYJC/Prelim/II/Q2", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "13 A circular cylinder is inscribed in a right circular cone of base radius 2c and vertical\n height c. One of the circular ends of the cylinder lies on the base of the cone and the other\n end is in contact with the inner surface of the cone. If the cylinder has base radius x, find\n expressions for the volume V and the total surface area S of the cylinder in terms of c and\n x. If c is fixed and x varies, find the maximum value of V. [8]\n Prove that S cannot exceed 8πc2. [2]\n [2010/NYJC/Prelim/II/Q2]", "methods": [ "Similar triangles", "Volume optimisation", "Derivative sign", "Surface-area bound" ], "prerequisites": [ "Cone geometry", "Cylinder total area", "Feasible radius" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 6 and solution PDF pages 20,21; independently worked first-pass mathematical review", "parts": [], "warning": "The surface-area bound is a limiting value for nonzero cylinder height; distinguish a supremum from an attained maximum." }, { "id": "06-14", "topic": "06", "number": 14, "questionPages": [ 6 ], "solutionPages": [ 22, 23 ], "attribution": "2008/RJC/Prelim/I/11(b)", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "14 A movie theatre screen which is 5 m high, has its lower edge 1 m above an observer’s\n eye. The visual angle  of the observer seated x m away is as shown in the diagram\n below.\n\n\n\n\n 5\n\n\n\n\n 1\n\n\n\n\n 6 1\n (i) Show that  = tan −1   − tan −1   . [1]\n x x\n (ii) Find the exact distance the observer should sit to obtain the largest visual\n angle.\n [You need not establish that the distance gives the largest visual angle.] [4]\n (iii) Suppose that the observer is situated between 2 m and 15 m from the screen. Find,\n to the nearest degree, the smallest visual angle. [2]\n [2008/RJC/Prelim/I/11(b)]", "methods": [ "Right-triangle angles", "Inverse-trigonometric differentiation", "Endpoint comparison" ], "prerequisites": [ "Radians in calculus", "Distance positivity", "Degree conversion" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 6 and solution PDF pages 22,23; independently worked first-pass mathematical review", "parts": [ "i", "ii", "iii" ], "warning": "Do not mistake the interior stationary maximum for the minimum on a closed interval." }, { "id": "06-15", "topic": "06", "number": 15, "questionPages": [ 6 ], "solutionPages": [ 23, 24 ], "attribution": "2023/EJC/Prelim/II/4", "candidatePartLabels": [ "a", "b", "c" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "15 A curve C has equation\n x3 + y 3 − xy = A\n\n where A is a non-zero constant.\n\n (a) Show that any stationary points on C lie on\n y = 3x 2 . [2]\n\n (b) Find, in terms of A, the x-coordinates of the stationary points, and hence determine\n the range of values of A for which C has two distinct stationary points. [4]\n (c) Suppose that C has two stationary points. Determine the nature of each of the\n stationary points found in part (b). [3]\n [2023/EJC/Prelim/II/4]\n\n 42\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF DIFFERENTIATION", "methods": [ "Implicit differentiation", "Quadratic substitution", "Discriminant", "Second-derivative test", "Regularity check" ], "prerequisites": [ "Real cube roots", "Product rule", "Nonzero parameter" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 6 and solution PDF pages 23,24; independently worked first-pass mathematical review", "parts": [ "a", "b", "c" ], "warning": "The printed range A>−1/27 must be read alongside its original A≠0 assumption. At the discriminant boundary the derivative is 0/0." }, { "id": "06-16", "topic": "06", "number": 16, "questionPages": [ 7 ], "solutionPages": [ 25 ], "attribution": "2009/SRJC/Prelim/II/1", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "16 The diagram shows a vertical cross-section of a container in the form of an inverted cone\n of height 24cm and base radius 9cm. The container is initially empty. Water is poured\n into the container at a constant rate of 30 cm3s −1 and the height of water level in the\n container at time t is h cm.\n 9cm\n\n\n\n\n 24cm\n\n\n\n\n 3\n (i) Show that the volume of water in the container is given by V =  h3 . [2]\n 64\n (ii) Find the value of h at the instant when the rate of increase in the depth of water is\n 5.65 cms−1, leaving your answer in 2 decimal places. [3]\n\n [2009/SRJC/Prelim/II/1]\n\n 1", "methods": [ "Similar triangles", "Volume substitution", "Related rates" ], "prerequisites": [ "Cone volume", "Chain rule", "Positive square root" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 7 and solution PDF pages 25; independently worked first-pass mathematical review", "parts": [ "i", "ii" ], "warning": "Radius changes as the water rises; it is not the constant full-container radius 9 cm." }, { "id": "06-17", "topic": "06", "number": 17, "questionPages": [ 7 ], "solutionPages": [ 26, 27, 28 ], "attribution": "2017/ACJC/Prelim/I/8", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "17 The curve C and the line L have equations y = x 2 and y = x − 2 respectively.\n 2\n (i) The point A on C and the point B on L are such that they have the same\n x-coordinate. Find the coordinates of A and B that gives the shortest distance AB.\n [3]\n (ii) The point P on C and the point Q on L are such that they have the same\n y-coordinate. Find the coordinates of P and Q that gives the shortest distance PQ.\n [3]\n (iii) Find the exact area of the polygon formed by joining the points found in (i) and (ii).\n [2]\n (iv) A variable point on the curve C with coordinates ( s, s 2 ) starts from the origin O and\n moves along the curve with s increasing at a rate of 2 units/s. Find the rate of change\n of the area bounded by the curve, the y-axis and the line y = s 2 , at the instant when\n s= 2. [4]\n [2017/ACJC/Prelim/I/8]\n\n\n\n 43\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF DIFFERENTIATION", "methods": [ "Distance optimisation", "Completing the square", "Branch comparison", "Triangle area", "Integration and related rates" ], "prerequisites": [ "Quadratic derivatives", "Square roots", "Coordinate geometry", "Chain rule" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 7 and solution PDF pages 26,27,28; independently worked first-pass mathematical review", "parts": [ "i", "ii", "iii", "iv" ], "warning": "The source line has x=2y+4, not 2y+2; one intermediate printed line contains the latter typo but its final Q is correct." }, { "id": "06-18", "topic": "06", "number": 18, "questionPages": [ 8 ], "solutionPages": [ 28, 29 ], "attribution": "2008/AJC/Prelim/I/7", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original diagrams preserved in source pages", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "18 A capsule made of two similar hemispheres of radius r cm, one at each end of a cylinder,\n is inscribed into a fixed cone of height 12 cm and base radius 9 cm. The two figures have\n a common axis of symmetry and the cross-section is shown below.\n\n\n\n\n (i) Show that the height of the cylinder of the capsule, x cm, is related to the radius of\n 8\n the hemisphere through the equation, x = 12 − r . [3]\n 3\n (ii) Given that, when r = 2 , x is decreasing at a rate of 0.1 cms-1, find the rate of\n change at this instant of the volume of the capsule. [4]\n 4\n [Given: Volume of sphere =  r 3 ]\n 3\n [2008/AJC/Prelim/I/7]", "methods": [ "Similar right triangles", "Tangent-radius geometry", "Volume elimination", "Related rates" ], "prerequisites": [ "Pythagoras", "Sphere and cylinder volumes", "Signs of rates" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "evidence/review-06.md; rendered original question PDF pages 8 and solution PDF pages 28,29; independently worked first-pass mathematical review", "parts": [ "i", "ii" ], "warning": "The shrinking cylinder height makes the radius increase; keep dx/dt negative." }, { "id": "07-01", "topic": "07", "number": 1, "questionPages": [ 1 ], "solutionPages": [ 1, 2, 3 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a", "b", "c", "d", "e", "f", "g", "h", "i", "j" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "1 Evaluate\n 1 1\n  2 x ( ln x ) dx  x + 2 x + 5 dx\n 5\n (a) (f) 2\n\n\n x2 − x\n  x 2 − 16 dx\n 2\n 4− x\n (b)  xe dx (g)\n\n x2 − 6 x + 1\n  ( ln x ) dx  ( 3x + 1) x 2 + 3 dx\n 2\n (c) (h)\n ( )\n 4 e 2 x +1\n (d)  1 + cos 4 d (i)  dx\n ( )\n 3\n e2 x + 1\n\n 1 x +1\n  sin 2 xe  1 − x2 dx\n cot x\n (e) dx (j)", "methods": [ "Reverse chain rule", "Integration by parts", "Partial fractions", "Completing the square", "Trigonometric identities" ], "prerequisites": [ "Chain rule", "Logarithms and domains", "Rational algebra", "Inverse trigonometric derivatives" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a", "b", "c", "d", "e", "f", "g", "h", "i", "j" ], "warning": "" }, { "id": "07-02", "topic": "07", "number": 2, "questionPages": [ 1 ], "solutionPages": [ 3, 4 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "main" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "2 Without the use of a calculator, show that\n 2 x−2\n −2 x 2 − 2 x + 4 dx = p ln 3 + q π ,\n where p and q are exact real constants to be determined. [6]\n [2023/HCI/Prelim/I/4]\n 1", "methods": [ "Absolute-value sign analysis", "Completing the square", "Reverse chain rule" ], "prerequisites": [ "Logarithmic integrals", "Arctangent integral", "Exact angles" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "main" ], "warning": "" }, { "id": "07-03", "topic": "07", "number": 3, "questionPages": [ 1 ], "solutionPages": [ 4, 5 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a", "b" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "3 (a) Find  3 − 4t 2 dt . [3]\n\n (b) Use the substitution u = 5 x to find  5 x cos 2 5 x dx . ( ) [5]\n\n [2010/HCI/Prelim/I/6]", "methods": [ "Partial fractions", "Exponential substitution", "Power-reduction identity" ], "prerequisites": [ "Logarithmic antiderivatives", "Chain rule", "Double-angle cosine" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a", "b" ], "warning": "Do not replace only 5ˣ by u while leaving dx unchanged. The factor $1/\\ln5$ is essential." }, { "id": "07-04", "topic": "07", "number": 4, "questionPages": [ 1 ], "solutionPages": [ 5 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a.derivative", "a.integral", "b" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "4 (a) Write down\n d\n dx\n ( )\n tan( x 2 ) . Hence find  x3 sec2 x 2 dx.( ) [4]\n\n 4\n (b) Evaluate  x 2 x − 3 dx without the use of the graphic calculator. [3]\n 0\n [2010/DHS/Prelim/I/6 (part)]\n\n  x cos x dx.\n 2", "methods": [ "Integration by parts", "Reverse chain rule", "Piecewise integration" ], "prerequisites": [ "Chain rule", "Trigonometric derivatives", "Absolute value" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a.derivative", "a.integral", "b" ], "warning": "" }, { "id": "07-05", "topic": "07", "number": 5, "questionPages": [ 1 ], "solutionPages": [ 6 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a", "b.indefinite", "b.definite" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "5 (a) Find [1]\n\n (b) Use integration by parts to find  x cos 2 x dx. [3]\n \n\n  x cos x dx.\n 2 2\n Hence or otherwise find \n [3]\n 4\n\n [2020/RI/Prelim/I/5]\n\n\n  1− (2x −1) dx", "methods": [ "Reverse chain rule", "Integration by parts", "Power-reduction identity" ], "prerequisites": [ "Chain rule", "Trigonometric identities", "Exact definite integrals" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a", "b.indefinite", "b.definite" ], "warning": "" }, { "id": "07-06", "topic": "07", "number": 6, "questionPages": [ 1 ], "solutionPages": [ 7 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "6 x\n (i) Use the substitution u = 2x – 1 to find . [4]\n 2\n\n\n\n\n  sin (2x −1) dx .\n −1\n (ii) Hence find [2]\n\n [2020/ACJC/Prelim/I/1]\n 46\n\f2025 TJC H2 Math (9758) JC2 MSM: TECHNIQUES OF INTEGRATION", "methods": [ "Linear substitution", "Integration by parts" ], "prerequisites": [ "Inverse sine derivative", "Square-root domains", "Reverse chain rule" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "i", "ii" ], "warning": "" }, { "id": "07-07", "topic": "07", "number": 7, "questionPages": [ 2 ], "solutionPages": [ 8 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "main" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "7 Find  e2 x cos3x dx. [5]\n [2023/EJC/Prelim/I/5b]", "methods": [ "Repeated integration by parts" ], "prerequisites": [ "Exponential derivatives", "Trigonometric derivatives", "Linear algebra" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "main" ], "warning": "The short answer key on question PDF p.4 (printed49) omits the factor $1/13$ and includes an unrelated part (a). The full solution PDF p.8 has the correct factor; the original key is preserved." }, { "id": "07-08", "topic": "07", "number": 8, "questionPages": [ 2 ], "solutionPages": [ 8, 9 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a.constants", "a.integral", "b.i", "b.ii" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "8 (a) Given that\n A(sin  + cos ) + B(cos  – sin )  4 sin ,\n find the values of the constants A and B. [2]\n\n 4sin \n 1\n \n Hence find the exact value of  4 d . [3]\n 0 sin  + cos \n\n\n 2t\n (b) (i) Express in partial fractions. [2]\n (t + 1) 2\n 5 1\n (ii) Hence, find the exact value of  dx , using the substitution\n 1 x+ 2x −1\n t= 2x −1 . [6]\n [2010/SAJC/Prelim/I/11]\n\n ln 2 e3 x", "methods": [ "Coefficient comparison", "Logarithmic derivative", "Radical substitution", "Partial fractions" ], "prerequisites": [ "Trigonometric identities", "Chain rule", "Definite-integral bounds" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a.constants", "a.integral", "b.i", "b.ii" ], "warning": "" }, { "id": "07-09", "topic": "07", "number": 9, "questionPages": [ 2 ], "solutionPages": [ 10 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a", "b.sketch", "b.integral" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "9 (a) Use the substitution u = e x + 2 to find the exact value of  dx . [4]\n 0 ex + 2\n\n (b) Sketch the graph of y = e x − 2 for 0 ≤ x ≤ 2. State the exact value of x where the\n graph crosses the x-axis. [2]\n Hence, find the exact value of ln k such that\n 2 e k 1\n 0 e x − 2 dx = \n e x ln x\n dx . [5]\n\n [2011/TPJC/Prelim/I/9]", "methods": [ "Exponential substitution", "Polynomial division", "Piecewise integration", "Logarithmic derivative" ], "prerequisites": [ "Exponential graph", "Area versus signed integral", "Changed bounds" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a", "b.sketch", "b.integral" ], "warning": "" }, { "id": "07-10", "topic": "07", "number": 10, "questionPages": [ 2 ], "solutionPages": [ 11, 12 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a", "b.i", "b.ii" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "10 (a) Use the substitution x = 2 tan  to find the exact value of\n\n \n 2 1\n 2\n dx . [6]\n 3 x( x 2 + 4)\n\n\n (b) (i) By writing x = A(1 − x) + B , where A and B are constants, prove that\n 3\n x 2 1\n\n  dx = (1 − x ) 2 − 2 (1 − x ) 2 + C , where C is an arbitrary constant. [2]\n 1− x 3\n\n (ii) Hence, find  (\n x sin −1 x dx .) [3]\n\n [2009/SRJC/Prelim/I/9]\n\n\n\n\n 47\n\f2025 TJC H2 Math (9758) JC2 MSM: TECHNIQUES OF INTEGRATION\n\n\n  2− x", "methods": [ "Trigonometric substitution", "Linear decomposition", "Integration by parts" ], "prerequisites": [ "Positive square roots", "Inverse sine derivative", "Exact trigonometric values" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a", "b.i", "b.ii" ], "warning": "" }, { "id": "07-11", "topic": "07", "number": 11, "questionPages": [ 3 ], "solutionPages": [ 13, 14 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a.i", "a.ii", "a.iii", "b.i", "b.ii" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "11 (a) (i) Find  dx . [2]\n  ( 4 x − x2 − 1 )\n  1\n (ii) Find  dx . [2]\n  ( 4 x − x2 − 1 )\n  x\n (iii) Using the results of part (i) and (ii), find  dx . [2]\n  ( 4 x − x2 − 1)\n  x\n (b) (i) By using the substitution u = x , find  dx . [4]\n  1+ x\n\n 0 ln (1 + x ) dx , leaving your answer in the form\n 9\n (ii) Find the exact value of\n c\n a ln b − , where a, b and c are integers to be determined. [4]\n b\n [2016/VJC/Promo/11]\n\n\n  x x + 1 dx .\n π 1", "methods": [ "Reverse chain rule", "Completing the square", "Linear combination", "Radical substitution", "Integration by parts" ], "prerequisites": [ "Inverse sine integral", "Polynomial division", "Definite integral" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a.i", "a.ii", "a.iii", "b.i", "b.ii" ], "warning": "" }, { "id": "07-12", "topic": "07", "number": 12, "questionPages": [ 3 ], "solutionPages": [ 14, 15 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a", "b.i", "b.ii" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "12 (a) Use the substitution x = tan t , where 0  t  , to find [4]\n 2 2 2\n\n\n\n (b) (i) Write down\n d\n dx\n ( cos x3 ) . [1]\n\n (ii) Hence, find  x5 sin x3 dx. [4]\n\n [2023/SAJC/Prelim/II/3]", "methods": [ "Trigonometric substitution", "Integration by parts", "Reverse chain rule" ], "prerequisites": [ "Secant identity", "Chain rule", "Inverse substitutions" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a", "b.i", "b.ii" ], "warning": "" }, { "id": "07-13", "topic": "07", "number": 13, "questionPages": [ 3 ], "solutionPages": [ 15, 16 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "13 \n (i) Verify, without the use of a calculator, that  = is a root of the equation\n 4\n tan 2  − 2 cos 2  = 0 . [1]\n\n x x\n (ii) By sketching the graphs of y = tan 2   and y = 2 cos 2   , solve exactly the\n 2 2\n\n x x\n inequality tan 2    2 cos 2   for −  x   . [2]\n 2 2\n 2\n\n\n \n 3  x  x\n (iii) Hence evaluate exactly tan 2   − 2 cos 2   dx . [6]\n 0 2 2\n\n [2011/IJC/Prelim/I/11]\n\n\n\n\n 48\n\f2025 TJC H2 Math (9758) JC2 MSM: TECHNIQUES OF INTEGRATION", "methods": [ "Graphical sign analysis", "Trigonometric identities", "Piecewise integration" ], "prerequisites": [ "Even functions", "Exact angles", "Absolute value" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "07-14", "topic": "07", "number": 14, "questionPages": [ 4 ], "solutionPages": [ 16, 17 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-07.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed with source inconsistency", "difficulty": "not yet assessed", "extractedText": "14 (i) Find\n d\n dx\n (\n sin −1 x 2 . ) [1]\n\n   x \n \n 5\n (ii) Evaluate the integral ln  tan −1    dx numerically. [1]\n 1   10  \n k\n  −1  x  \n \n 5\n (iii) Given that 0  k  1 and\n 0\n x 2 sin −1 x 2 dx =\n  ln  tan  10   dx ,\n 1\n\n\n\n  1− x\n k\n x4\n show that dx = ak 3 sin −1 k 2 + b , where a and b are constants to be\n 0 4\n\n determined. [4]\n [2010/RVHS/Prelim/I/4]", "methods": [ "Chain rule", "Numerical quadrature", "Integration by parts", "Sign feasibility" ], "prerequisites": [ "Inverse sine domain", "Improper-integral limits", "Logarithms" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-07.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "i", "ii", "iii" ], "warning": "Confirmed source inconsistency: the equality in (iii) is impossible for the stated k. The printed constants are retained and explained separately; no replacement question or answer is invented. Question PDF p.4 (printed49), solution PDF pp.16–17." }, { "id": "08-01", "topic": "08", "number": 1, "questionPages": [ 1 ], "solutionPages": [ 1 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "1 (i) Sketch the graphs of y = and y = 1 on the same diagram, showing\n 1 + ( x − 2)2\n clearly the coordinates of the stationary point and points of intersection between the\n curves. [2]\n (ii) Find the exact area of region R enclosed by the two curves, the lines x = 1 and\n x = 3. [3]\n (iii) Find the volume generated when the region R is rotated 3600 about the y-axis. [4]\n [2011/MI/Prelim/II/2]", "methods": [ "Curve sketching", "Definite area", "Washers", "Cylindrical shells" ], "prerequisites": [ "Stationary points", "Arctangent integral", "Inverse branches" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "08-02", "topic": "08", "number": 2, "questionPages": [ 1 ], "solutionPages": [ 2 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a", "b" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "2 1 1\n Curve C1 has equation y = and curve C2 has equation y = . Region Q is\n x 2 − x2\n 1\n bounded by line x = and both curves C1 and C2 .\n 2\n (a) Find the exact area of region Q. [3]\n (b) The region bounded by the line y = 5 and the curves C1 and C2 in the first quadrant\n is rotated about the x–axis by 2 radians. Find the volume of the solid formed. [3]\n [2021/ASRJC/Prelim/I/4]", "methods": [ "Curve intersection", "Definite area", "Piecewise washers" ], "prerequisites": [ "Square-root domains", "Inverse sine integral", "Rational integrals" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a", "b" ], "warning": "Do not use one inner-radius formula across the whole interval. The curves exchange roles at x=1." }, { "id": "08-03", "topic": "08", "number": 3, "questionPages": [ 1 ], "solutionPages": [ 3, 4 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "3 10\n The curve C has equation y = 3 − 2 .\n x − 2x + 4\n (i) Without using a calculator, determine the exact coordinates of the stationary point\n of C. [2]\n (ii) Sketch C, stating clearly the equations of any asymptotes. [2]\n\n The region R is bounded by C, the line 3 y + 16 x = 15 and the y-axis.\n (iii) Find the exact area of the region R. [3]\n (iv) Find the volume generated when R is rotated through 2 radians about the y-axis.\n [3]\n [2021/EJC/Prelim/II/4]\n x2 + 5x + 3", "methods": [ "Stationary points", "Curve sketching", "Definite area", "Washers", "Cylindrical shells" ], "prerequisites": [ "Completing the square", "Inverse branches", "Arctangent integral" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "" }, { "id": "08-04", "topic": "08", "number": 4, "questionPages": [ 1 ], "solutionPages": [ 5 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "main" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "4 The curve C has equation y = .\n x +1\n\n 9\n Region S is bounded by C, the y-axis, and the line y = . Find the volume of the solid\n 2\n formed when region S is rotated about the x-axis completely. [3]\n [2019/EJC/Prelim/I/6(iii)]\n x −1", "methods": [ "Polynomial division", "Curve intersection", "Washers" ], "prerequisites": [ "Rational algebra", "Logarithmic integration", "Volume of revolution" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "main" ], "warning": "" }, { "id": "08-05", "topic": "08", "number": 5, "questionPages": [ 1 ], "solutionPages": [ 6 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "main" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "5 Find the volume generated when the area bounded by the curve y = , the x-axis,\n x +1\n and the lines x = 2 and x = 3 is rotated completely about the y-axis. Give your answer\n correct to 3 significant figures. [4]\n [2011/SAJC/Prelim/II/2(a)]\n\n\n\n 51\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF INTEGRATION\n\n\n x2 y 2 y x", "methods": [ "Cylindrical shells", "Washers", "Rational division" ], "prerequisites": [ "Inverse rational function", "Piecewise radii", "Definite integration" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "main" ], "warning": "" }, { "id": "08-06", "topic": "08", "number": 6, "questionPages": [ 2 ], "solutionPages": [ 7, 8 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "6 (i) The region R is bounded by the ellipse 2\n + 2 = 1 and the line − = 1 as\n a b b a\n shown in the diagram below.\n y\n\n\n b\n R\n x\n −a O a\n\n\n −b\n\n 1 0 b\n Show that the area of R is − ab +  a 2 − x 2 dx .\n 2 − a a\n Hence, by substituting x = a sin  , find the exact area of R in terms of a and b. [8]\n\n (ii) Find the volume of revolution formed when R is rotated completely about the\n y-axis. Give your answer in the form ka 2b , where k is a constant to be determined.\n [4]\n [2010/MJC/Prelim/II/5]", "methods": [ "Trigonometric substitution", "Area decomposition", "Washers" ], "prerequisites": [ "Ellipse equations", "Positive roots", "Cone volume" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "i", "ii" ], "warning": "" }, { "id": "08-07", "topic": "08", "number": 7, "questionPages": [ 2 ], "solutionPages": [ 9 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a", "b", "c" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "7 It is given that\n   π \n  tan  4a x  for − a  x  a ,\n \n f ( x) = \n sin  π x  for a  x  3a ,\n   2a \n\n and that f ( x) = f ( x − 4a) for all real values of x , where a is a positive real constant.\n (a) Sketch the graph of y = f ( x) for − 4a  x  7a. [3]\n\n (b) By using the sketch in part (a), write down an equation relating f ( x) and f (− x) .\n [1]\n 4a ka\n (c) Show that  f ( x ) dx = (1 + ln m) , where k and m are constants to be\n −2 a π\n determined. [5]\n [2023/HCI/Prelim/I/7]\n\n\n\n\n 52\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF INTEGRATION", "methods": [ "Periodic extension", "Odd symmetry", "Piecewise integration" ], "prerequisites": [ "Tangent and sine graphs", "Logarithmic secant integral", "Signed area" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a", "b", "c" ], "warning": "This asks for signed integration, not total shaded area. Positive and negative lobes cancel." }, { "id": "08-08", "topic": "08", "number": 8, "questionPages": [ 3 ], "solutionPages": [ 10, 11 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed with source branch qualification", "difficulty": "not yet assessed", "extractedText": "8 The curve C is defined by the parametric equations\n x = t − sin t , y = sin 2 t , 0  t  π .\n \n The point A on C has parameter and is a maximum point. The line l passes through the\n 2\n origin and the point A. The region R is bounded by C and l as shown in the diagram below.\n y\n\n A\n\n R\n\n\n\n O\n x\n\n\n (i) Find the coordinates of A. [1]\n (ii) Using an algebraic method, find the exact area of R. [5]\n (iii) Show that the Cartesian equation of C is given by x = sin −1 y − y . [1]\n (iv) Find the volume of the solid formed when R is rotated by 2π radians about the y-\n axis. [3]\n [2023/ACJC/Prelim/I/8]", "methods": [ "Parametric area", "Inverse sine branch", "Washers" ], "prerequisites": [ "Chain rule", "Power-reduction identity", "Triangle area" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "Confirmed branch qualification: the printed Cartesian expression is valid only for 0≤t≤π/2. Counterexample t=π gives (x,y)=(π,0), while the printed expression gives x=0. Question PDF p.3 (printed53); solution PDF p.11 explicitly adds the rising-arc restriction." }, { "id": "08-09", "topic": "08", "number": 9, "questionPages": [ 3 ], "solutionPages": [ 11 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "main" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed with depicted-region qualification", "difficulty": "not yet assessed", "extractedText": "9 The curve C is defined by the parametric equations\n x = 3t 2 , y = t 3 + 3t\n where t is a real number.\n 1\n A line l cuts C at two points P and Q with parameter t = 1 and t = − respectively.\n 2\n y\n l\n C\n P\n\n x\n Q\n\n\n\n\n Find the area of the region bounded by C and the line l, giving your answer correct to\n 3 significant figures. [5]\n [2011/TPJC/Prelim/I/1]\n\n\n\n\n 53\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF INTEGRATION", "methods": [ "Parametric area", "Trapezium subtraction" ], "prerequisites": [ "Parameter derivatives", "Line equation", "Definite polynomial integration" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "main" ], "warning": "The full real-parameter curve has a third intersection at t=7, outside the depicted arc. The article explicitly selects the region shown in the source diagram; the original wording “at two points” is preserved in the PDF." }, { "id": "08-10", "topic": "08", "number": 10, "questionPages": [ 4 ], "solutionPages": [ 12, 13 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "limits", "area" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "10 A curve C is defined by the parametric equations\n 1 1\n x=− , y= , where −2  t  0 .\n t 4 − t2\n 1 1\n The region S is bounded by C, the lines x = , x= and the x-axis.\n 3 2\n b\n Show that the area of S can be expressed as \n 1\n dt , where a and b are\n a t 2 4 − t 2\n constants to be determined in exact form.\n By using the substitution t = 2sin  , find the exact area of S. [6]\n [2011/SAJC/Prelim/II/2(b)]", "methods": [ "Parametric area", "Trigonometric substitution" ], "prerequisites": [ "Chain rule", "Positive roots", "Cotangent integral" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "limits", "area" ], "warning": "" }, { "id": "08-11", "topic": "08", "number": 11, "questionPages": [ 4 ], "solutionPages": [ 14 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a", "b" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "11 (a) A curve is defined by the parametric equations\n x = 2t 2 , y = sin ( −t ) , for t  0 .\n 2\n Find the exact area bounded by the curve, the x-axis and the line x = . [5]\n 8\n  \n (b) The curve C has equation y = x 2 sec x tan 2 x, where − x .\n 2 2\n 2\n The region R is bounded by the curve C and the line y =  2 . Find the volume\n 3\n obtained when R is rotated through 2π radians about the x-axis, giving your\n answer correct to 2 decimal places. [4]\n [2011/MJC/Prelim/II/2]\n\n\n\n\n 54\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF INTEGRATION\n\n\n  6 3x \n ", "methods": [ "Parametric area", "Integration by parts", "Symmetric washers", "Numerical quadrature" ], "prerequisites": [ "Sign of sine", "Even functions", "Volume of revolution" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a", "b" ], "warning": "" }, { "id": "08-12", "topic": "08", "number": 12, "questionPages": [ 5 ], "solutionPages": [ 15, 16 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a", "b", "c.i", "c.ii" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "12 x\n  cot 2 x + cot 2 x − sin sin  dx .\n 4\n (a) Find [3]\n  2 2 \n\n (b) The diagram shows the graphs of y = e 2 x and y = ( x + 1) .\n 2\n\n\n y\n\n\n\n 1\n\n\n\n x\n\n R is the finite region bounded by the two curves y = e 2 x and y = ( x + 1) . Find the\n 2\n\n\n\n volume of the solid formed when R is rotated through 2 radians about the y-axis,\n giving your answer correct to 4 decimal places. [3]\n\n (c) A curve C has parametric equations\n  t t t t 3 \n x = 2  sin − cos  , y = sin + cos , for − t − .\n  2 2 2 2 2 2\n The line L with equation y = −2 x − 5 meets the curve C at the point ( −2, −1) . S is\n the region enclosed by line L, curve C and the x-axis as shown in the diagram below.\n y\n\n x\n C S L\n\n\n\n\n (i) Show that the x-intercept of curve C is −2 2 . [2]\n (ii) Find the exact area of S. [6]\n [2020/ASRJC/Prelim/II/4]\n\n\n\n\n 55\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF INTEGRATION", "methods": [ "Trigonometric identities", "Reverse chain rule", "Cylindrical shells", "Parametric area" ], "prerequisites": [ "Curve intersections", "Product-to-sum identity", "Negative signed area" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a", "b", "c.i", "c.ii" ], "warning": "" }, { "id": "08-13", "topic": "08", "number": 13, "questionPages": [ 6 ], "solutionPages": [ 18 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "a.i", "a.ii", "a.iii", "b" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed with notation qualification", "difficulty": "not yet assessed", "extractedText": "13 (a)\n y\n\n\n 1\n y=1\n\n\n\n\n R\n y=0 x\n O\n\n e− x\n The diagram shows the curve C with equation y = . The region R is\n 1 + e− x\n bounded by C, the positive x-axis and the positive y-axis.\n n e− x\n (i) Obtain a formula for  dx in terms of n, where n  0 . [2]\n 0 1 + e− x\n\n (ii) Hence find the exact area of R. [2]\n (iii) Denoting the answer you have obtained in part (i) by q, write down, in terms\n of q, the area of R if C is scaled by factor 2 parallel to the y-axis. [1]\n y\n (b)\n\n\n\n\n x\n O\n\n A girl intends to design a bowl by rotating a section AB of the curve y = ln x\n completely about the y-axis. She wants the bowl to hold 300 cm3 of fluid.\n Given that the diameter of the rim is twice that of the base, write down, in terms\n of a, the coordinates of the point B. Hence find the exact value of a. [5]\n [2010/VJC/Prelim/I/11 (modified)]\n\n\n\n\n 56\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF INTEGRATION", "methods": [ "Improper integration", "Vertical scaling", "Discs" ], "prerequisites": [ "Logarithmic derivative", "Limits", "Inverse exponential" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "a.i", "a.ii", "a.iii", "b" ], "warning": "Notation qualification in (a)(iii): q is defined in the question as a finite-n integral, while R extends to infinity. The printed “2q” is recorded separately from the total area 2ln2. Question PDF p.6 (printed56), solution p.18." }, { "id": "08-14", "topic": "08", "number": 14, "questionPages": [ 7 ], "solutionPages": [ 19 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "sum", "inequality" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "14 The graph of y = 2x for 0  x  1 , is shown in the diagram below. Rectangles, each of\n 1\n width , where n is an integer are drawn under the curve.\n n\n\n\n\n\n O\n\n 1\n\n ( )\n Show that the total area of all the n rectangles is 1\n . [2]\n n 2 −1 n\n\n\n\n\n(ii) By considering the area of the region bounded by the curve y = 2 x , x = 1 and the axes,\n\n ( 1\n briefly explain why ln 2  n 2 n − 1 . ) [3]\n\n [2008/NJC/Prelim/I/3]", "methods": [ "Geometric progression", "Lower Riemann sum" ], "prerequisites": [ "Increasing functions", "Area under a curve", "Exponential integration" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "sum", "inequality" ], "warning": "" }, { "id": "08-15", "topic": "08", "number": 15, "questionPages": [ 7 ], "solutionPages": [ 20, 21, 22 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "i", "ii", "iii", "translation", "volume" ], "partInventoryStatus": "all printed tasks visually checked; see evidence/review-08.md", "status": "authored", "transcription": "transcribed from rendered question pages; wording paraphrased, notation preserved", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "15\n (i) Prove that\n d\n dx\n ( tan −1 x ) =\n 1\n 1 + x2\n . [2]\n\n 1\n The diagram shows a sketch of the graph of y = .\n 1 + x2\n y\n\n\n\n\n x\n O\n\n (ii) By considering the shaded rectangle and the area of the region between the graph\n and the x-axis for n − 1  x  n , where n  1 , show that\n 1\n  tan −1 ( n ) − tan −1 ( n − 1) . [2]\n 1+ n 2\n\n\n 1 1 1 1\n (iii) Deduce that + + + +  tan −1 ( n ) . [2]\n 2 5 10 1+ n 2\n\n\n\n 1 1\n The region R is bounded by the curve y = and the lines x = 3 and y = . Write\n 1+ x 2\n 2\n 1\n down the equation of the curve obtained when the curve y = is translated by\n 1 + x2\n 3 units in the negative x-direction. Hence, or otherwise, find the volume of the solid\n formed when R is rotated completely about the line x = 3 . [4]\n [2015/NYJC/Prelim/II/4 (modified)]\n 57\n\f2025 TJC H2 Math (9758) JC2 MSM: APPLICATIONS OF INTEGRATION", "methods": [ "Implicit differentiation", "Lower Riemann sum", "Telescoping sum", "Axis translation", "Discs", "Cylindrical shells" ], "prerequisites": [ "Inverse tangent", "Decreasing functions", "Inverse branches" ], "classificationStatus": "reviewed against rendered source and worked solution", "reviewEvidence": "evidence/review-08.md: exact rendered source page audit, worked algebra and independent numerical verification. Source printed statements are retained separately where qualified.", "parts": [ "i", "ii", "iii", "translation", "volume" ], "warning": "" }, { "id": "09-01", "topic": "09", "number": 1, "questionPages": [ 1 ], "solutionPages": [ 1 ], "attribution": "[2022/EJC/Prelim/II/1]", "candidatePartLabels": [ "a", "b", "c" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "1 It is given that f ( x) = 4 − 3x .\n (a) Find the binomial expansion for f ( x), up to and including the term in x 2 . Give the\n coefficients as exact fractions in their simplest form, and state the range of values of\n x for which this expansion is valid. [4]\n 1 1847\n (b) By taking x = , show that 13  . [3]\n 4 512\n 1\n (c) We may also substitute x = to obtain an approximation of 13. Witout further\n 13\n calculation, explain why this leads to a better approximation of 13 than the value\n shown in (b). [1]\n [2022/EJC/Prelim/II/1]\n1 [Solution]\n 1 Need to factorise out 4\n(a) f ( x ) = ( 4 − 3x ) 2 to make the constant\n 1 term 1.\n  3x \n 1\n 2\n = 4 1 − \n 2\n Use the law of indices\n  4 \n (AB )n = AnB n\n   1  1  \n   1  3 x   2  − 2   3 x  2  Then use binomial\n = 2 1 +   −  +     −\n   + . ..  theorem\n   2  4  2!  4  \n   (1 + x )n\n  \n n(n − 1) 2\n 3 9 = 1 + nx + x +\n = 2 − x − x 2 + ... 2!\n 4 64\n where n = ½ and replace\n 3 4 4 4 3x\n Validity range: − x  1  x   −  x  x by − .\n 4 3 3 3 4\n 1\n(b) Substitute x = into the expansion: The binomial expansion\n 4\n of (1 + f(x ) ) , where n\n 2 n\n 1 31 9 1\n 4 − 3   = 2 −   −   + ...\n 4 4  4  64  4  is a negative integer or\n 3 9 fraction, is valid for the\n = 2− − + ...\n 16 1024 range of x given by\n\n 13 2048 − 3  64 − 9 1847 f(x )  1.\n  =\n 4 1024 1024\n 1847 1847\n 13  2  = (shown)\n 1024 512\n(c) 1\n The value obtained by substituting x = will be a better approximation for 13\n 13\n 1 1\n because x = is closer to 0 than x = .\n 13 4\n\n\n\n 1\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n 1", "methods": [ "Maclaurin expansion", "Coefficient comparison" ], "prerequisites": [ "Binomial theorem", "Power-series order" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 1, solution PDF pages 1. Independent computations and qualifications are documented in that report.", "parts": [ "a", "b", "c" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-02", "topic": "09", "number": 2, "questionPages": [ 2 ], "solutionPages": [ 2 ], "attribution": "[2009/IJC/Prelim/I/7]", "candidatePartLabels": [ "expansion", "comparison", "approximation" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "2\n  1− x 3\n Find the expansion of   in ascending powers of x, up to and including the term\n  1+ 2x \n in x 2 . State the range of values of x for which the expansion is valid. [5]\n 1\n Without performing any calculations, explain why putting x = into the result gives a\n 6\n 4\n better approximation to 3 5 than putting x = − . [1]\n 11\n 1\n Hence by putting x = , find an approximation for 3 5 , expressing your answer as a\n 6\n fraction in its lowest terms. [2]\n [2009/IJC/Prelim/I/7]\n2 [Solution]\n\n (1 − x ) 3 = 1 − x +   −  ( − x ) + ......\n 1\n 1 1 1 2 2\n\n 3 2! 3  3 \n 1 1\n = 1 − x − x 2 + .....\n 3 9\n\n (1 + 2 x ) 3 = 1 − (2 x) +   − \n 1\n 1 1 1 4\n  −  ( 2 x ) + ......\n − 2\n\n 3 2!  3  3 \n 2 8\n = 1 − x + x 2 + .....\n 3 9\n 1\n Since both the expansions\n  1 − x 3\n 1 1\n −\n   = (1 − x) (1 − 2 x)\n 3 3\n of\n  1 + 2 x  1 −1\n  1 1  2 8  (1 − x ) 3 and (1 + 2x ) 3\n = 1 − x − x 2 + 1 − x + x 2 + \n  3 9  3 9  must be valid at the same\n = 1− x + x +\n 2 time, we take the\n intersection of the 2\n For expansion to be valid,\n regions.\n 1\n − x  1 and 2 x  1  x  1 and x\n 2\n  −1  x  1\n 2 2 –1/2 1/2\n Both values of x are within the range of validity of x.\n 1 4 1\n However, since 0   − , x = will give a better\n 6 11 6\n approximation as it is closer to zero.\n 1 1\n  1 3  5 3\n  1−  6\n 1\n 1 6  5 3\n Substituting x = ,   =  = \n 6  2  8  8\n 1+ \n  6 6\n 1\n 2\n  5 3 1 1\n Thus,   = 1 −   +   + .....\n 8 6 6\n\n ( 5) 3  2 1 − +  =\n 1\n 1 1 31\n \n  6 36  18\n 2\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)", "methods": [ "Maclaurin expansion", "Coefficient comparison" ], "prerequisites": [ "Binomial theorem", "Power-series order" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 2, solution PDF pages 2. Independent computations and qualifications are documented in that report.", "parts": [ "expansion", "comparison", "approximation" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers. Source p.2 has a product-line typo (1−2x) in place of (1+2x); the actual binomial expansion and result use the plus sign." }, { "id": "09-03", "topic": "09", "number": 3, "questionPages": [ 3 ], "solutionPages": [ 3 ], "attribution": "[2008/HCI/Prelim/I/6]", "candidatePartLabels": [ "partial fractions", "expansion", "coefficient" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "3 2x2 + 3x – 1\n Express in partial fractions.\n (x – 1)(x2 + 1)\n 2x2 + 3x – 1\n Hence find the series expansion of in ascending powers of x, up to and\n (x – 1)(x2 + 1)\n including the term in x4. Find the coefficient of x2008. [6]\n [2008/HCI/Prelim/I/6]\n3 [Solution]\n 2x2 + 3x – 1 A Bx + C For expansion of\n Let = + 2\n (x – 1)(x + 1) x – 1 x + 1\n 2\n (1 − x )−1 ,\n 2x + 3x – 1 = A(x2 + 1) + (Bx + C)(x – 1)\n 2\n Coefficient of x2008\n Let x = 1: 4 = 2A A=2 =-2(1) = -2\n Let x = 0: –1 = 2 – C  C = 3 For expansion of\n Compare coefficients of x2 : 2 = 2 + B  B = 0 (1 + x 2 ) −1\n 2x2 + 3x – 1 2 3 = 1 + ( −1)x 2\n = + 2\n (x – 1)(x + 1) x – 1 x + 1\n 2\n + ( −1)2 (x 2 )2\n = –2(1 – x)–1 + 3(1 + x2)–1\n + ( −1)3 (x 2 )3 +\n = –2(1 + x + x2 + x3 + x4 + ...) + 3(1 – x2 + x4 – ...)\n Coefficient of x2008\n = 1 – 2x – 5x2 – 2x3 + x4 +...\n = 3(-1)1004 = 3\n Coefficient of x2008 = x2(1004) is –2 + 3 = 1", "methods": [ "Partial fractions", "Geometric series" ], "prerequisites": [ "Polynomial identity", "Even powers" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 3, solution PDF pages 3. Independent computations and qualifications are documented in that report.", "parts": [ "partial fractions", "expansion", "coefficient" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-04", "topic": "09", "number": 4, "questionPages": [ 3 ], "solutionPages": [ 3, 4 ], "attribution": "[2020/HCI/Prelim/I/2]", "candidatePartLabels": [ "coefficient equation" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "4 1\n The coefficient of x 3 in the expansion of is 5 times the coefficient of x 2 in the\n ( 2 + ax )\n 5\n\n\n\n expansion of 4 − ax where a is a non-zero constant. Find the exact value of a . [5]\n [2020/HCI/Prelim/I/2]\n4 1\n = ( 2 + ax )\n −5\n\n ( 2 + ax )\n 5\n\n\n −5\n  ax \n −5\n = 2 1 + \n  2\n 1  5ax ( −5 )( −6 )  ax  ( −5 )( −6 )( −7 )  ax  \n 2 3\n\n =  1 − +   +   + ...\n 32  2 2!  2  3!  2  \n\n 1 ( −5 )( −6 )( −7 )  a \n 3\n 35 3\n Coefficient of x =  3\n   =− a\n 32 3! 2 256\n 1\n  ax  2\n 4 − ax = 2 1 − \n  4 \n  1  ax  ( 12 )( − 12 )  ax 2 \n = 2 1 +  −  +  −  + ...\n  2  4  2!  4  \n\n  ( 12 )( − 12 )  a  2  1 2\n Coefficient of x = 2  −   = − a\n 2\n\n  2!  4   64\n\n\n 3\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n 35 3  1 \n − a = 5  − a2 \n 256  64 \n 35a 3 = 20a 2\n 7 a = 4, since a  0\n 4\n a=\n 7\n It is given that the first 3 terms in the expansion of ( 2 − ax )", "methods": [ "Maclaurin expansion", "Coefficient comparison" ], "prerequisites": [ "Binomial theorem", "Power-series order" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 3, solution PDF pages 3,4. Independent computations and qualifications are documented in that report.", "parts": [ "coefficient equation" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-05", "topic": "09", "number": 5, "questionPages": [ 4 ], "solutionPages": [ 4 ], "attribution": "[ACJC/2009/Prelim/II/2]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "5 −b\n in ascending powers of x\n 1 9\n are + x + cx 2 + ... , where a, b and c are positive constants.\n 8 16\n (i) Find the values of a, b and c. [7]\n (ii) Show that the coefficient of x is in the form p ( r + 1)( r + 2 ) q , where p and q\n r r\n\n\n are constants to be determined. [4]\n [ACJC/2009/Prelim/II/2]\n 5 [Solution]\n −b\n(i)  ax  Need to factorise out 2\n ( 2 − ax )\n −b\n = 2−b  1 − \n  2  to make the constant\n term 1.\n −b\n   ax  (−b)(−b − 1)  ax \n 2\n \n = 2 1 + (−b)  −  +  −  + ...  Use the law of indices\n   2  2!  2  \n   (AB )n = AnB n\n  ab a 2b(b + 1) 2  Then use binomial\n = 2−b 1 + x+ x + ... \n  2 8  theorem\n\n Comparing constant term: 2− b =\n 1\n  b=3 (1 + x )n\n 8 n(n − 1) 2\n = 1 + nx + x +\n 1  ab  9 2!\n Comparing coefficient of x:  =  a=3\n 8  2  16 where n = -b and replace\n 1  a b(b + 1)  27\n 2\n x by −\n ax\n .\n Comparing coefficient of x2: c=  =\n 8 2\n 8  16\n(ii) Coefficient of x r\n − b ( −b)( −b − 1)...( −b − ( r − 1))  a \n r\n In the coefficient of xr,\n =2 − \n r!  2 ( −b − 0)( −b − 1)...( −b − ( r − 1 ))\n r!\n 1 (−3)(−4)...(−r − 2)  3 \n r\n\n = −  there are r factors in\n 8 r!  2\n the numerator for r! in\n 1 (−1) r (3)(4)...(r + 2)\n r\n 3 the denominator.\n = (−1) r  \n 8 r! 2\n (3)(4)...(r + 2)  3 \n r\n 1\n = (−1) 2 r  \n 8 r! 2\n 1 (r + 1)(r + 2)  3 \n r\n\n =  \n 8 2 2\n r\n 1 3\n = (r + 1)(r + 2)  \n 16 2\n 1 3\n Thus, p = and q = .\n 16 2\n 4\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)", "methods": [ "Maclaurin expansion", "Coefficient comparison" ], "prerequisites": [ "Binomial theorem", "Power-series order" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 4, solution PDF pages 4. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-06", "topic": "09", "number": 6, "questionPages": [ 5 ], "solutionPages": [ 5 ], "attribution": "[modified DHS/2008/Prelim/I/5]", "candidatePartLabels": [ "expansion", "validity", "tangent" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "6 1 + x2\n Find the first three terms in the expansion of in ascending powers of x.\n 3 − 2x\n State the range of values of x for which the expansion is valid. [3]\n 2 + 2 x2\n Deduce that the equation of the tangent to the curve y = at the point where\n 3 − 2x\n x = 0, is given by 9 y = 2 3x + 6 3 . [2]\n [modified DHS/2008/Prelim/I/5]\n6 [Solution]\n 1 + x2 −1 Need to factorise out\n = (1 + x 2 )(3 − 2 x) 2\n 3 − 2x 3 to make the\n − 12 constant term 1.\n (\n = (1 + x 2 )  3 1 − 2 x \n  3  ) Use the law of\n indices\n ( )\n − 1\n = 1 (1 + x 2 ) 1 − 2 x 2 (AB )n = AnB n\n 3 3\n\n ( ) ( ) − 2 x +  theorem\n Then use binomial\n  −1 −3\n 2 \n ( )( ) ( 3 ) \n 2\n = 1 (1 + x ) 1 + − 1 − x + 2\n 2 2\n 3  2 3 2!\n   (1 + x )n\n n(n − 1) 2\n  1 1  = 1 + nx + x +\n = 1 (1 + x 2 ) 1 + x + x 2 +  2!\n 3  3 6 \n where n = -½ and\n  1 1 \n = 1 1 + x + x 2 + x 2 +  2x\n 3 3 6  replace x by − .\n 3\n 1 1 7 2\n = + x+ x +\n 3 3 3 6 3\n\n 2 3 3 3\n Validity range: − x  1  x   −  x \n 3 2 2 2\n\n 2 + 2 x2  1 + x2  dy 2 14\n y= = 2  = + (2x ) +\n 3 − 2x dx 3 3 6 3\n  3 − 2x \n At x = 0,\n  1 1 7 2 \n = 2 + x+ x +  2 dy 2\n  3 3 3 6 3  y= and =\n 3 dx 3 3\n 2 2 14 2\n = + x+ x + Thus\n 3 3 3 6 3 equation of tangent at\n x = 0 is\n Thus, equation of tangent at x = 0 is\n 2 2\n 2 2 y− = (x − 0)\n y= + x 3 3 3\n 3 3 3 2 2\n y= + x\n 3 3 3\n  3 3y = 6 + 2x\n which are the 1st 2\n  9 y = 6 3 + 2 3x (Shown)\n terms (constant &\n term in x) of expansion\n of y.\n\n 5\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)", "methods": [ "Binomial expansion", "Tangent from series" ], "prerequisites": [ "Negative fractional exponent", "Derivative at zero" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 5, solution PDF pages 5. Independent computations and qualifications are documented in that report.", "parts": [ "expansion", "validity", "tangent" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-07", "topic": "09", "number": 7, "questionPages": [ 6 ], "solutionPages": [ 6 ], "attribution": "[2009/SAJC/JCT/8]", "candidatePartLabels": [ "partial fractions", "a", "b" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "7 9 + 4 x2\n Express in partial fractions. [3]\n (1 − 2 x) 2 (2 + x)\n\n Hence find in its simplest form,\n 9 + 4 x2\n (a) the first four terms in the expansion of in ascending powers of x\n (1 − 2 x) 2 (2 + x)\n and state the set of values of x for which the expansion is valid.\n (b) the coefficient of x r in the expansion. [6]\n [2009/SAJC/JCT/8]\n7 [Solution]\n 9 + 4 x2 = A + B C\n +\n (1 − 2 x) (2 + x)\n 2\n 1 − 2 x (1 − 2 x ) 2\n 2+ x\n\n  9 + 4 x2 = A(1 − 2 x)(2 + x) + B(2 + x) + C (1 − 2 x) 2\n 1 5\n Substitute x = : 9 +1 = B  B=4\n 2 2\n Substitute x = −2 : 9 + 16 = 25C  C =1\n Comparing constant term: 9 = 2 A + 2(4) + 1  A = 0\n 9 + 4 x2 4 1\n Thus, = + .\n (1 − 2 x) (2 + x) (1 − 2 x) 2 + x\n 2 2\n\n\n\n 9 + 4 x2\n = 4 (1 − 2 x ) + ( 2 + x )\n −2 −1\n(a)\n (1 − 2 x) (2 + x)\n 2\n\n −1\n\n = 4 (1 − 2 x )−2 + 1 + \n 1 x\n 2 2\n  (−2)(−3) (−2)(−3)(−4) \n = 4 1 + (−2)(−2 x) + ( −2 x ) 2 + ( −2 x)3 +  Since both the\n  2 ! 3 !  expansions of\n 1  x  (−1)(−2)  x  ( −1)(−2)(−3)  x  \n 2 3\n −1\n + 1 + (−1)   +   +   +  −2  x\n 2  (1 − 2x ) & 1 + \n 2 2!  2  3! 2   2\n\n = ( 4 + 16 x + 48 x 2 + 128 x 3 + ) + 12 1 − 12 x + 14 x − 81 x +\n 2 3 \n \n \n must be valid at the\n same time, we take\n the intersection of\n = 9 + 63 x + 385 x 2 + 2047 x3 +\n 2 4 8 16 the 2 regions.\n x 1\n Taking intersection of  1 and |2x|< 1, we have |x| < .\n 2 2\n  1 1 –1/2 1/2 2\n Solution set for which expansion is valid is  x  : −  x   .\n  2 2\n 1  (1 − X )\n r −2\n(b) r1\n  \n Coefficient of x = 4 (r + 1)2  + (−1)   \n r r\n 2   2   = 1 + 2X + 3X 2 +\n\n r +1\n r +2 r1 (1 + X )−1\n = (r + 1)2 + (−1)  \n 2 = 1 + ( −1)(X ) + ( −1)2 X 2 +\n\n 6\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)", "methods": [ "Partial fractions", "Geometric-series differentiation" ], "prerequisites": [ "Repeated factors", "General coefficient" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 6, solution PDF pages 6. Independent computations and qualifications are documented in that report.", "parts": [ "partial fractions", "a", "b" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-08", "topic": "09", "number": 8, "questionPages": [ 7 ], "solutionPages": [ 7 ], "attribution": "[2018/DHS/Prelim/I/2]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "8 (i) Using appropriate expansions from the List of Formulae (MF26), show that the series\n ( )\n expansion of ln 1 + e 2 x in ascending powers of x, up to and including the term in\n 2\n x\n x 2 , is ln 2 + x + . [3]\n 2\n (ii) Find the set of values of x such that the value for the above expansion for ln 1 + e 2 x ( )\n is within  0.3 of the value of ln 1 + e 2 x . ( ) [2]\n 2\n (iii) Deduce an expansion for up to and including the term in x. [2]\n 1 + e−2 x\n [2018/DHS/Prelim/I/2]\n 8 [Solution] For e2x, use MF27:\n(i)  (2 x) 2  x2 x3\n ln (1 + e 2 x ) = ln 1 + 1 + 2 x + + ...  ex = 1 + x + + +\n 2! 3!\n  2! \n = ln ( 2 + 2 x + 2 x + ...)\n 2 where x is replaced by 2x.\n Do not expand as follows:\n = ln ( 2 ) (1 + x + x 2 + ...)  (\n ln 1 + e 2 x )\n = ln 2 + ln 1 + ( x + x + ...) \n 2 \n = ln  1 + 1 + 2x +\n (2x )2 \n + ... \n  2! \n ( x + x + ...) + ...  \n 2 2\n\n = ln 2 + ( x + x + ...) − (1 + 2x + ...) +\n 2\n 2\n = (1 + 2x + ... ) −\n 2 2\n x2 as each bracketed term contains a\n = ln 2 + x ++... (shown)\n 2 constant and terms in x, x2, x3, …\n(ii)  x2 \n ( )\n −0.3  ln 1 + e2 x −  ln 2 + x +   0.3\n  2\n Using GC, {x  : −1.55  x  1.55}\n\n\n\n\n(iii) x2\n ln (1 + e 2 x ) = ln 2 + x +\n + ...\n 2\n Differentiate both sides wrt x :\n 2e 2 x\n = 1 + x + ...\n 1 + e2 x\n 2e 2 x / e 2 x\n = 1 + x + ... (  by e 2 x in numerator & denominator)\n (1 + e ) / e\n 2x 2x\n\n\n 2\n  = 1 + x + ...\n 1+e −2 x\n 7\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)", "methods": [ "Composition of series", "Numerical error inequality", "Termwise differentiation" ], "prerequisites": [ "Logarithm expansion", "Symmetry", "Monotonicity" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 7, solution PDF pages 7. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-09", "topic": "09", "number": 9, "questionPages": [ 8 ], "solutionPages": [ 8, 9 ], "attribution": "[2020/MI/Prelim/I/6]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "9 It is given that y = e x cos x .\n dy\n (i) Show that 2 y = y 2 − e x sin x . [2]\n dx\n (ii) By further differentiation of the result in part (i), find the Maclaurin series for y, up\n to and including the term in x2. [4]\n (iii) Using standard series from the List of Formulae (MF26), expand e x cos x as far as\n the term in x2 and verify that the same result is obtained in part (ii). [3]\n [2020/MI/Prelim/I/6]\n9 [Solution]\n(i)\n y = e x cos x\n\n Method 1: Implicit differentiation\n y 2 = e x cos x --- (1) Square the equation so\n that the resulting eqn\n = ( e x ) ( − sin x ) + ( cos x ) ( e x )\n dy\n 2y (1) is easier to handle\n dx\n dy for repeated\n 2y = y 2 − e x sin x. (shown) differentiation using\n dx\n implicit differentiation\n Method 2: Direct differentiation and product rule.\n 1\n y = ( e x cos x ) 2\n\n 1\n = ( e cos x ) 2 ( e x ) ( − sin x ) + ( cos x ) ( e x ) \n dy 1 x −\n\n dx 2\n dy 1  1  2 x\n =   ( y − e sin x )\n dx 2  y \n dy\n 2y = y 2 − e x sin x (shown)\n dx\n\n\n\n\n 8\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n dy d  dy \n9(ii) 2y = y 2 − e x sin x ... (1) y \n dx dx  d x \n d 2 y dy  dy  d  dy  dy d\n 2 y 2 +  2  = 2 y − ( e x ) ( cos x ) + ( sin x ) ( e x ) \n dy =y  + \n dx  dx  dx dx\n (y )\n dx dx  dx  dx\n d2y dy dy\n 2 =y + \n d2 y  dy \n 2 y 2 + 2   = 2 y − ( y 2 + e x sin x ) ... (2) d x 2 d x dx\n dy\n dx  dx  dx\n When x = 0, y = 1,\n When x = 0, y = e0 cos 0 = 1. to obtain the values of\n dy dy 1 dy d2y\n From (1): 2(1) = (1)2 − e0 sin 0  = . , , subst. them\n dx dx 2 dx dx 2\n 2\n into their respective eqns.\n d2 y 1 1\n From (2): 2(1) 2 + 2   = 2(1)   − (12 + e 0 sin 0 ) No Need to make them the\n dx 2 2 subject first.\n\n d2 y 1 d2 y 1 Subst the values of\n 2 + = 0  =− .\n dx 2\n 2 dx 2\n 4 dy d2y\n y, , when x = 0\n dx dx 2\n 1  1\n   −  into Maclaurin’s Theorem:\n Hence, y = 1 +  2 1  4 2\n x + x + ... , f(0) 2\n f(x ) = f(0) + f(0)x + x\n 1! 2! 2!\n 1 1 +\n i.e. y = 1 + x − x 2 + ... where y = f(x).\n 2 8\n 9 Method 1:\n 1 1\n ( cos x ) 2 From MF27,\n x\n(iii) y = e cos x = e\n x 2\n\n 1 Maclaurin expansion for\n   1  1  1 2   x2 2 x2 x 4\n = 1 +  x  +  x  + ... 1 − + ...  cos x = 1 − + − .\n   2  2!  2    2  2! 4!\n\n  1  x  1x\n 1\n 2\n 1 2 For e 2 , replace x by x\n = 1 + x + x + ...  1 − + ...  2\n  2 8  4 \n in the standard series\n x2 1 1 1 1\n = 1 − + x + x 2 + ... = 1 + x − x 2 + ... (verified) x x\n 2\n 4 2 8 2 8 e =1+x + + .\n 2!\n\n Method 2:\n 1\n y = e x cos x = ( e x cos x ) 2\n 1\n  1  x\n 2\n  2\n = 1 + x + x 2 + ...  1 − + ...  \n  2!  2 \n 1\n  x2 1 2\n = 1 − + x + x 2 + ... \n  2 2 \n 1\n = (1 + x + ...) 2\n 1 1\n − \n 1 2 2 2 1 1\n = 1+ x + x + ... = 1 + x − x 2 + ... (verified)\n 2 2! 2 8\n 9\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)", "methods": [ "Implicit differentiation", "Product of standard series" ], "prerequisites": [ "Product rule", "Square-root branch" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 8, solution PDF pages 8,9. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-10", "topic": "09", "number": 10, "questionPages": [ 10 ], "solutionPages": [ 10, 11 ], "attribution": "[2023/DHS/Prelim/P1/6]", "candidatePartLabels": [ "a", "b", "c", "d" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "10 In this question, you may use expansions from the List of Formulae (MF26).\n\n (a) Given that f ( x) = esin 3 x , find the Maclaurin expansion of f ( x) in ascending powers\n of x, up to and including the term x2. Show that this expansion is independent of x3.\n [4]\n\n (b) Use your expansion from part (a) and integration to find an approximate expression\n esin 3 x 0.2  2 \n 2\n for  2\n dx. Hence obtain an approximate value for    e\n sin 3 x\n dx, correct\n x 0.1\n x\n to 4 decimal places. [3]\n 2\n 0.2  2 \n Use your calculator to find  esin 3 x dx, correct to 4 decimal places.\n 0.1  x \n (c) [1]\n  \n\n (d) A student compares the answers to parts (b) and (c) and concludes that the\n approximation is accurate. Give a reason to support his conclusion. [1]\n [2023/DHS/Prelim/P1/6]\n10 [Solution]\n (3 x)3 9 x3 From MF27,\n(a) sin 3 x = (3 x) − + ... = 3 x − + ...\n 3! 2 Replace x by 3x in the\n (sin 3 x) 2 (sin 3 x)3 standard Maclaurin’s\n f ( x) = e sin 3 x\n = 1 + (sin 3 x) + + + ...\n 2! 3! series\n 9 x3 1 9 x3 2 1 9 x3 x\n 3\n\n = 1 + (3 x − + ...) + (3 x − + ...) + (3 x − + ...)3 + ... sin x = x − +\n 2 2 2 6 2 3!\n\n 9 x3 1 1 to obtain expansion for\n = 1 + 3x − + [(3 x)2 ] + [(3x)3 ] + ... sin 3x.\n 2 2 6\n 9 2\n  1 + 3x + x + 0 x3 (independent of x3)\n 2 For e sin 3x ,\n Alternative (by differentiation) replace x by\n let y = esin 3 x 9x 3\n sin 3x = x − + ... in\n dy 2\n = 3cos 3 x  esin 3 x = 3cos 3 x  y the standard series\n dx\n d2 y x2 x3\n dy ex = 1 + x + + +\n 2\n = 3cos 3 x − 9sin 3x  y 2! 3!\n dx dx\n .\n d3 y d2 y dy dy\n 3\n = 3cos 3 x 2\n − 9sin 3 x − 9sin 3 x  − 27 cos 3 x  y\n dx dx dx dx\n When x = 0,\n dy d2 y d3 y\n y = 1, = 3, = 9, =0\n dx dx 2 dx3\n 9\n  y = 1 + 3x + x 2 + 0 x3 +\n 2\n\n\n\n\n 10\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n10 esin 3 x 9\n  x 2 dx   ( x + 3x + 2 ) dx\n −2 −1\n(b)\n 9\n = − x −1 + 3ln x + x + C where C is an arbitrary constant\n 2\n 2\n 0.2  2  0.2 4esin 3 x\n  dx =  0.1\n sin 3 x\n   e\n 0.1  x \n dx\n x2\n 0.2\n  9 \n = 4  − x −1 + 3ln x + x \n  2  0.1\n = 30.1178 (4 d.p.)\n 2\n(c) 0.2  2 \n Using GC,    e\n sin 3 x\n dx = 29.9995 (4 d.p.)\n 0.1  x \n\n(d) The approximation is accurate as the values of x (between 0.1 and 0.2) are close to 0\n for the magnitude of x4 and higher powers of x to be neglected.\n Alternative:\n | 30.1178 − 29.9995 |\n % error = 100 = 0.3943%\n 29.9995\n Since percentage error is small, approximation is accurate.", "methods": [ "Series composition", "Termwise integration", "Numerical quadrature" ], "prerequisites": [ "Sine and exponential expansions", "Logarithmic integral" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 10, solution PDF pages 10,11. Independent computations and qualifications are documented in that report.", "parts": [ "a", "b", "c", "d" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-11", "topic": "09", "number": 11, "questionPages": [ 11 ], "solutionPages": [ 11, 12 ], "attribution": "[2020/NYJC/Prelim/I/9]", "candidatePartLabels": [ "i", "ii", "iii", "iv", "v" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "11 (i) Using standard series from the List of Formula (MF26), expand ln(k + 2 x)n in\n ascending powers of x, as far as the term in x3 , where k and n are real constants,\n k\n k  0 , n  0 and x  − . [2]\n 2\n (ii) State the range of values of x for which the expansion in part (i) is valid. [1]\n −2\n (iii) Find the exact value of  ln (1 + 2 x ) d x .\n 1\n [3]\n 0\n −2\n (iv) Use your series from part (i) to estimate  ln (1 + 2 x ) d x .\n 1\n [2]\n 0\n\n (v) Calculate the percentage error of the estimate in part (iv), and comment on the\n accuracy of your approximation using your answer from part (ii). [2]\n [2020/NYJC/Prelim/I/9]\n11 [Solution]\n Need to factorise out k\n(i) ln(k + 2 x) n = n ln ( k + 2 x ) to make the constant\n   2 x  term 1.\n = n ln  k  1 +  \n   k  Apply rule of logarithms\n   2 x  ln (ab) = ln a + ln b .\n = n ln k + ln  1 +  \n   k  From MF27, use standard\n  2x 1  2x  1  2x \n 2 3  series\n = n ln k + −   +   − \n  k 2 k  3 k   x2 x3\n ln(1 + x ) = x − + −\n 2 3\n  2 x 2 x 2 8 x3 \n = n ln k + − 2 + 3 −  up to term in x3 where −1  x  1 ,\n  k k 3k  2x\n replacing x by .\n(ii) 2x k k k\n Range of validity: −1  1  −  x \n k 2 2\n 11\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n 11 dv\n u = ln (1 + 2 x ) =1 Use Integration by Parts:\n(iii) dx\n du 2  u dv dx = uv −  v du dx\n = v=x  \n dx 1 + 2 x  dx  dx\n\n\n  ln (1 + 2 x ) dx = −2 ln (1 + 2 x ) dx\n 1 −2 1\n\n 0 0\n\n  1 2x \n = −2   x ln (1 + 2 x )  0 − \n 1\n dx \n 0 1+ 2x\n  \n  1 1 \n = −2  ln 3 −  1 − dx \n  0 1+ 2x \n   1  \n 1\n\n = −2  ln 3 −  x − ln (1 + 2 x )  \n   2  0 \n \n   1 \n = −2  ln 3 − 1 − ln 3 \n   2 \n 3 \n = −2  ln 3 − 1\n 2 \n = 2 − 3ln 3\n(iv) Taking n = −2 and k = 1 ,\n  8 \n    2 x − 2 x + 3 x  dx\n 1 1\n ln (1 + 2 x ) d x = −2\n −2 2 3\n 0 0\n\n 1\n  2 2 \n = −2  x 2 − x3 + x 4 \n  3 3 0\n = −2\n(v) −2 − ( 2 − 3ln 3) Percentage error\n % error =  100  54.3%\n 2 − 3ln 3 approx value − actual value\n =  100\n actual value\n Since the series expansion in part (ii) is valid for\n 1 1\n −  x  , while the limits of the integral is for 0  x  1\n 2 2\n (outside the range of validity), the use of the series in part\n (i) to estimate the integral is wrong (not valid). This results\n in a large error.\n dy", "methods": [ "Logarithm expansion", "Integration by parts", "Error calculation" ], "prerequisites": [ "Logarithm laws", "Convergence endpoints" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 11, solution PDF pages 11,12. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv", "v" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-12", "topic": "09", "number": 12, "questionPages": [ 12 ], "solutionPages": [ 13 ], "attribution": "[2008/RJC/Prelim /II/5(c)]", "candidatePartLabels": [ "derivative", "series" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "12 The function y = f ( x) satisfies the differential equation 2 y = y 2 − 1 and its graph\n dx\n passes through the point (0,3) .\n d 2 y 20\n Show that when x = 0 , = . [3]\n dx 2 27\n Find the Maclaurin’s series of y in ascending powers of x, up to and including the term\n in x3 . [3]\n [2008/RJC/Prelim /II/5(c)]\n 12\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n12 [Solution]\n dy\n 2 y = y 2 − 1 ----- (1)\n dx\n Differentiate (1) w.r.t. x, we get Use implicit differentiation\n d2 y  dy \n 2\n dy and product rule.\n 2y + 2  = 2y\n dx 2\n  dx  dx d  dy \n y \n 2 dx  d x \n d 2 y  dy  dy\n y 2 +  = y d  dy  dy d\n dx  dx  dx\n ----- (2) =y  + \n dx  dx  dx dx\n (y )\n d2y dy dy\n dy 4 =y + \n When x = 0 , y = 3 , = using (1) d x 2 d x dx\n dx 3\n Subst into (2),\n 2\n d2 y  4  4\n 3 +   = 3 \n dx  3 \n 2\n 3\n d 2 y 1  16  20\n = 4−  = (Shown)\n dx 2 3  9  27\n\n Differentiate (2) w.r.t. x , we get Let y = f(x).\n d y  dy   d y   dy   d y \n 2\n 3 2\n d y  dy \n 2 2\n To find the Maclaurin’s\n y +   2  + 2  2  = y 2 +  \n dx  dx   dx \n 3\n  dx   dx  dx  dx  series for y, up to and\n including the term in x3,\n  dy   d y \n 2\n d3 y 2\n d 2 y  dy \n y 3 + 3   2  = y 2 +   differentiate up to the 3rd\n dx  dx   dx  dx  dx \n d3y\n derivative .\n dy 4 d 2 y 20 d 3 y 28 dx 3\n When x = 0 , y = 3 , = , = , =\n dx 3 dx 2 27 dx 3 81 Subst the values of\n\n By Maclaurin’s Theorem, dy d2y d3y\n y, , , when x = 0\n dx dx 2 dx 3\n  20  x  28  x\n 2 3\n 4\n y = 3 + x +   +   + ... into Maclaurin’s Theorem:\n 3  27  2!  81  3!\n f(0) 2\n 4 10 2 14 3 f(x ) = f(0) + f(0)x + x\n = 3+ x + x + x + ... 2!\n 3 27 243 f(0) 3\n + x +\n 3!\n 1", "methods": [ "Implicit differentiation", "Maclaurin coefficients" ], "prerequisites": [ "Product rule", "Factorials" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 12, solution PDF pages 13. Independent computations and qualifications are documented in that report.", "parts": [ "derivative", "series" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-13", "topic": "09", "number": 13, "questionPages": [ 13 ], "solutionPages": [ 13, 14 ], "attribution": "[2008/AJC/Prelim/I/4]", "candidatePartLabels": [ "identity", "series", "approximation" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "13\n It is given that y = ( cos x ) 2 .\n 2\n d2 y  dy \n Show that 2 y 2 + 2   + y 2 = 0 . [2]\n dx  dx \n Find Maclaurin’s series for y in ascending powers of x, up to and including the term in x2.\n [3]\n 1\n By choosing a suitable value for x, deduce the approximate relation 4  1 + k , where\n 2\n\n 2\n k is a constant to be determined. [2]\n [2008/AJC/Prelim/I/4]\n\n 13\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n13 [Solution] Square the eqn so that the\n 1\n y = ( cos x )  y 2 = cos x --- (1)\n 2 resulting eqn (1) is easier to\n handle for repeated\n dy\n 2 y = − sin x differentiation using implicit\n dx diffn and product rule.\n 2 2\n d2 y  dy  d2 y  dy  Note:\n 2 y 2 + 2   = − cos x  2 y 2 + 2   = − y 2\n dx  dx  dx  dx  d  dy  d  dy  dy d\n 2\n y =y\n dx  dx \n  + \n dx  dx  d x dx\n (y )\n d2 y  dy \n 2y + 2   + y2 = 0 d2y dy dy\n dx 2\n  dx  =y + \n dx 2 dx dx\n\n When x = 0, y = 1 To find the Maclaurin’s series\n for y, up to and including the\n dy dy term in x2, differentiate up to\n 2 =0 =0\n dx dx d2y\n the 2nd derivative .\n d2 y d2 y 1 dx 2\n 2 2 + 0 +1 = 0  2 = −\n dx dx 2 Subst the values of\n dy d2y\n y, , when x = 0 (y =f(x))\n  1 dx dx 2\n − \n y = 1+ \n 2 2 1 into Maclaurin’s Theorem:\n Maclaurin’s series: x + ...  1 − x 2\n 2 4 f(0) 2\n f(x ) = f(0) + f(0)x + x +\n 2!\n 1\n 1 1 \n ( cos x ) 2 = 4  cos x = x=\n 2 2 4\n \n Sub x = :\n 4\n 1 \n 2\n 1 1\n  1−   = 1−  2\n 4\n 2 4 4  64\n 1\n Thus, k = − .\n 64", "methods": [ "Implicit differentiation", "Series substitution" ], "prerequisites": [ "Even functions", "Trigonometric exact values" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 13, solution PDF pages 13,14. Independent computations and qualifications are documented in that report.", "parts": [ "identity", "series", "approximation" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-14", "topic": "09", "number": 14, "questionPages": [ 14 ], "solutionPages": [ 14, 15 ], "attribution": "[2010/MI/Prelim/I/7]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "14 (i) Given that f ( x ) = ln ( 2 x + 1) , find f ( 0 ) , f  ( 0 ) , f  ( 0 ) and f  ( 0 ) . Hence obtain\n the first three non-zero terms in the Maclaurin’s series for f ( x ) . [5]\n\n (ii) Hence, or otherwise, show that the first three non-zero terms in the expansion of\n ln ( 2 x 2 + 3x + 1) are ax + bx 2 + ax3 , where a and b are constants to be found. [3]\n [2010/MI/Prelim/I/7]\n14 [Solution]\n d g'(x )\n f ( x ) = ln ( 2 x + 1) 1) ln ( g(x ) )  =\n dx   g(x )\n 2 where g(x ) = 2x + 1\n = 2 ( 2 x + 1)\n −1\n f ( x) =\n 2x + 1\n f ( x) = −2 ( 2 x + 1) (2) = −4 ( 2 x + 1) d \n −2 −2\n ( g(x ) )  = n ( g(x ) ) g'(x )\n n n −1\n 2)\n \n dx  \n \n f ( x) = −4(−2) ( 2 x + 1) (2) = 16 ( 2 x + 1)\n −3 −3\n\n\n\n f (0) = 0, f (0) = 2, f (0) = −4, f (0) = 16\n 14\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n 1 2 1\n f ( x) = f (0) + xf (0) + x f (0) + x3f (0) + ...\n 2 3!\n 1 1\n = 0 + 2 x + x 2 (−4) + x3 (16) + ...\n 2 6\n 8\n  f ( x) = 2 x − 2 x 2 + x 3 + ...\n 3\n14 ln ( 2 x 2 + 3x + 1) = ln(2 x + 1)( x + 1) Since question states “hence\n(ii)\n = ln(2 x + 1) + ln( x + 1) or …”, so can use the result for\n ln(2x + 1). In order to do so,\n 8 1 1\n = 2 x − 2 x 2 + x 3 + x − x 2 + x 3 + ... factorise 2x 2 + 3x + 1 and\n 3 2 3\n 5 (\n re-express ln 2x 2 + 3x + 1 in )\n = 3x − x 2 + 3x 3 + ...\n 2 terms ln(2x + 1) by using the\n 5 rule of logarithms\n Thus, a = 3, b = −\n 2 ln(AB) = ln A + ln B.", "methods": [ "Differentiation at zero", "Factorization", "Logarithm expansion" ], "prerequisites": [ "Chain rule", "Logarithm product law" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 14, solution PDF pages 14,15. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-15", "topic": "09", "number": 15, "questionPages": [ 15 ], "solutionPages": [ 15, 16 ], "attribution": "[2022/DHS/Prelim/I/6]", "candidatePartLabels": [ "a", "b", "c" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "15 Let y = cos  ln(1 + x) for x  1.\n d2 y dy\n (a) Show that (1 + x) 2 2\n + (1 + x) + y = 0. By further differentiation of this result,\n dx dx\n show that the Maclaurin series for y is given by\n 1 − 12 x 2 + 12 x 3 − 12\n 5 x 4 + ... . [6]\n (b) By using the result in part (a), deduce the Maclaurin series for sin  ln(1 + x)  in\n ascending powers of x up to and including x3. [2]\n (c) Hence state the tangent to the curve y = sin  ln(1 + x)  at x = 0. [1]\n [2022/DHS/Prelim/I/6]\n15 [Solution] Let u = 1+x, v = ln(1+x) = lnu\n(a) dy sin  ln(1 + x)  Use Chain Rule:\n =− d\n dx 1+ x cos ln(1 + x ) \n dx\n d Differentiate\n  (1 + x)\n dy\n = − sin  ln(1 + x) =\n dx\n ( cos v )\n from outside\n dx d dv\n = ( cos v )  dx to inside:\n dv →→\n Differentiating with respect to x, d\n = − sin v ( )\n ln u\n d 2 y dy cos  ln(1 + x)  y dx\n (1 + x) + =− =− = − sin v\n d\n (ln u )  ddxu\n dx 2\n dx 1+ x 1+ x du\n 1 d\n  (1 + x) 2\n d2 y dy\n + (1 + x) + y = 0 (shown)\n = − sin v  \n u dx\n (1 + x )\n 2\n dx dx 1\n = − sin (ln(1 + x ) )  1\n 1+ x\n Differentiating with respect to x, Use implicit diffn & product rule.\n d3 y d2 y d 2 y dy dy d  d2y \n (1 + x) 2 3 + 2(1 + x) 2 + (1 + x) 2 + + =0  (1 + x )2 \n dx  dx 2 \n dx dx dx dx dx\n d  d2y  d2y d\n  (1 + x)\n 3\n 2 d y d2 y dy\n + 3(1 + x) 2 + 2 = 0\n = (1 + x )2  + \n dx  dx 2  dx 2 dx\n (1 + x )2 ( )\n 3\n dx dx dx d3y d2y\n = (1 + x )2 3\n +  2(1 + x )\n dx dx 2\n 15\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n Differentiating again with respect to x,\n d4 y d3 y d3 y d2 y d2 y\n (1 + x) 2 4 + 2(1 + x) 3 + 3(1 + x) 3 + 3 2 + 2 2 = 0\n dx dx dx dx dx\n 4 3 2\n d y d y d y\n  (1 + x) 2 4 + 5(1 + x) 3 + 5 2 = 0\n dx dx dx\n\n dy d2 y d3 y d4 y\n When x = 0, y = 1, = 0, 2 = −1, 3 = 3, 4 = −10\n dx dx dx dx\n\n  y = 1 − 2!\n 1 x 2 + 3 x 3 − 10 x 4 + \n 3! 4!\n = 1 − 1 x 2 + 1 x3 − 5 x 4 + (shown)\n 2 2 12\n dy\n(b) sin  ln(1 + x)  = −(1 + x)\n dx\n ( )\n = −(1 + x) − x + 32 x 2 − 53 x 3 + \n = x − 12 x 2 + 16 x3 +  --- (*)\n\n For y = sin [ln(1+x)],\n(c)  Equation of tangent to curve at x = 0 is y = x.\n dy\n when x = 0, y = 0, =1.\n dx\n Equation of tangent at x = 0:\n y − 0 = 1(x − 0)\n y = x (const. & term in x in (*))", "methods": [ "Differential identity", "Termwise differentiation" ], "prerequisites": [ "Chain rule", "Product rule", "Series order" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 15, solution PDF pages 15,16. Independent computations and qualifications are documented in that report.", "parts": [ "a", "b", "c" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-16", "topic": "09", "number": 16, "questionPages": [ 16 ], "solutionPages": [ 16, 17, 18 ], "attribution": "[2022/ACJC/Prelim/I/7]", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "16 It is given that y = ln(2 + sin 2 x) .\n 2\n d2 y  dy \n (i) Show that e y + e y   = −4sin 2 x . [2]\n  dx \n 2\n dx\n (ii) By further differentiation of the above results, find the Maclaurin series for y, up to\n and including the term in x3 . [3]\n (iii) Verify that the series for ln(2 + sin 2 x) is the same as the result obtained in part (ii),\n if the standard series from the List of Formulae (MF26) are used. [3]\n ln(2 + sin 2 x)\n (iv) Hence, deduce the series expansion for , up to and including the term\n 1− x\n in x 2 . [2]\n [2022/ACJC/Prelim/I/7]\n16 [Solution]\n(i) y = ln(2 + sin 2 x)\n  e y = 2 + 2sin 2 x\n Differentiate w.r.t. x:\n dy\n ey = 2cos 2 x\n dx\n 2\n d2 y  dy \n ey + e y   = −4sin 2 x (shown)\n  dx \n 2\n dx\n\n\n 16\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n16 Differentiate again w.r.t. x: dy\n(ii) Let u =\n y  dy d y   y dy  dy \n 2\n d y  y dy  d y\n 3 2 2\n dx\n ey +e  2 + e 2  2 +e   = −2 cos 2 x Using Chain Rule:\n dx  dx  dx\n 3\n  dx dx   dx   dx \n 2\n d3 y d 2 y dy  dy \n 3 d  dy \n e y 3 +3e y 2 + e y   = −2 cos 2 x  \n dx  dx \n dx dx dx  dx \n d\n ( u)\n 2\n =\n dx\n 2 3\n dy d y d y\n When x = 0, y = ln 2, = 1, 2 = −1, 3 = −2\n d du\n ( u) \n dx dx dx 2\n =\n du dx\n Maclaurin’s series for y :\n du\n ( −1) x 2 + ( −2 ) x3 + ... = 2u\n  y = ln 2 + x + dx\n 2! 3! dy d  dy \n =2   \n 1 1 dx dx  dx \n = ln 2 + x − x 2 − x3 + ...\n 2 3\n dy  d2y \n =2  \n dx  dx 2 \n\n\n\n\n(iii) y = ln ( 2 + sin 2 x ) From MF27,\n Replace x by 2x in\n  ( 2x) \n 3\n\n  ln  2 + 2 x −  the standard\n  6  Maclaurin’s series\n  \n 3\n x\n   2  sin x = x − +\n = ln  2  1 + x − x 3   3!\n   3 \n to obtain expansion\n  2  for sin 2x.\n = ln 2 + ln  1 + x − x 3 \n  3 \n 2 3\n  2   2  Factorise out 2 to\n  x − x3   x − x3  make the constant\n  2 \n = ln 2 +  x − x 3  − \n 3   3 \n + + ... term 1.\n  3  2 3\n 2 1 1 Apply rule of\n  ln 2 + x − x 3 − x 2 + x 3 logarithms\n 3 2 3\n ln (ab) = ln a + ln b .\n 1 2 1 3\n = ln 2 + x − x − x + ... (verified) From MF27, use\n 2 3\n standard series\n ln(1 + x )\n x2 x3\n =x− + −\n 2 3\n replacing x by\n 2\n x − x3.\n 3\n\n\n\n\n 17\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n\n 16 ln(2 + sin 2 x)\n(iv) 1− x\n 1 2 1 3\n ln 2 + x − x − x Use result from\n  2 3 (iii).\n 1− x\n Use binomial\n  1 1  1\n =  ln 2 + x − x 2 − x 3  (1 − x ) 2\n −\n expansion for\n  2 3  −1\n (1 − x ) 2 .\n   1  3  \n   −  −  \n  1 \n   ln 2 + x − x 2  1 + x + \n 1 2  2 \n (−x) \n 2\n\n  2  2 2! \n  \n  \n  1  1 3 \n =  ln 2 + x − x 2  1 + x + x 2 \n  2  2 8 \n 1 2 1  1 3 \n  ln 2 + x − x +  ln 2  x + x 2 +  ln 2  x 2\n 2 2  2 8 \n  1  3 \n = ln 2 + 1 + ln 2  x +  ln 2  x 2\n  2  8 ", "methods": [ "Implicit differentiation", "Series composition", "Binomial multiplication" ], "prerequisites": [ "Exponential chain rule", "Order tracking" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 16, solution PDF pages 16,17,18. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-17", "topic": "09", "number": 17, "questionPages": [ 18 ], "solutionPages": [ 18, 19 ], "attribution": "[2022/ASRJC/Prelim/I/4]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "17 (i) Show that the first two non-zero terms of the Maclaurin series for tan  is given by\n 1\n  +  3 . You may use the standard results given in the List of Formulae (MF26).\n 3\n [2]\n C\n\n\n 1\n A B\n O 1\n x\n\n In the right-angle triangle OBC shown above, point A lies on OB such that OA = 1 , OB\n \n = x , where x > 1 and OC = 1 . It is given that angle COB is radians and that angle ACB\n 2\n is  radians (see diagram).\n 2 tan \n (ii) Show that AB = . [2]\n 1 − tan \n (iii) Given that  is a sufficiently small angle, show that\n AB  a + b 2 + c 3\n for exact real constants a, b and c to be determined. [3]\n [2022/ASRJC/Prelim/I/4]\n\n\n\n\n 18\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n\n\n\n17 [Solution]\n sin \n(i) tan  =\n cos \n Maclaurin Expansion for\n = sin  ( cos  )\n −1\n\n sin  and cos can be found\n −1\n     \n 3 2\n in MF27.\n    − 1 −  (since  is small)\n  3!  2! \n Apply binomial expansion for\n      3 2\n (1 − x ) −1 = 1 + x + x 2 + x 3 + ,\n    − 1 + \n  3!  2! \n 2\n replacing x by .\n  3\n  3\n 2\n  + −\n 2! 3!\n 1\n = + 3\n 3\n \n tan + tan  From MF27,\n   \n(ii) tan  +   = 4\n 4  1 − tan  tan  under Trigonometry,\n 4 tan A + tan B\n tan(A + B) = .\n 1 + tan  1 − tan A tan B\n x = \n 1 − tan  Replace A by and B by  .\n 4\n 1 + tan \n AB = −1\n 1 − tan \n 2 tan \n AB =\n 1 − tan \n AB = 2 tan  (1 − tan  )\n −1\n 17\n(iii) Use result from (i) for\n −1\n   3    3  tan , then apply binomial\n  2   +  1 −   +  \n  3   3  expansion for\n (1 − x )−1 = 1 + x + x 2 + ,\n  3    3    3  \n 2\n \n  2   +  1 +   +  +   +   3\n  3   3  3  replacing x by  + .\n   3\n  2 3 \n   2 +  (1 +  +  )\n 2\n\n  3 \n 8 3\n  2 + 2 2 +\n 3\n 8\n a = 2, b = 2, c =\n 3\n\n\n\n\n 19\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n 2 ", "methods": [ "Trigonometric geometry", "Tangent addition", "Geometric series" ], "prerequisites": [ "Radians", "Right-triangle ratios" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 18, solution PDF pages 18,19. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-18", "topic": "09", "number": 18, "questionPages": [ 20 ], "solutionPages": [ 20, 21 ], "attribution": "[2023/SAJC/Prelim/P2/2]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "18 In triangle PQR, PRQ = , RPQ = +  , and PR = a units, where a is a positive\n 3 6\n real constant.\n 3a\n (i) Show that PQ = . [4]\n cos  − 3 sin \n (ii) Given that  is sufficiently small, show that PQ  3 a (1 + b  + c  2 ) , where b\n and c are real constants. [3]\n (iii) Given that   0 , find the range of values of  such that the percentage error of\n the approximation in (ii) is less than 5%. [2]\n [2023/SAJC/Prelim/P2/2]\n18 R\n(i)\n 2\n a 3\n\n  +\n P 6 Q\n\n 2   \n PQR =  − −  + \n 3 6 \n \n = −\n 6\n Using Sine Rule,\n PQ a From MF27,\n =\n 2   under Trigonometry,\n sin sin  −  \n 3 6  sin (A – B)\n 2 = sin A cos B − cos A sin B\n a sin\n PQ = 3 replacing\n   \n sin cos  − cos sin  A by and B by  .\n 6 6 6\n  3\n a \n  2 \n =\n 1 3\n cos  − sin \n 2 2\n 3a\n =\n cos  − 3 sin \n\n\n\n\n 20\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n\n\n18 3a From MF27,\n PQ =\n(ii) cos  − 3 sin  use Maclaurin expansion for\n cos  and sin .\n 3a\n \n 2 Then apply binomial\n 1− − 3\n 2 expansion for\n −1\n   (1 − x ) −1 = 1 + x + x 2 +\n  2 \n = 3 a 1 −  3  +   2\n   2  replacing x by 3 +\n 2\n .\n\n   2   2 \n 2\n \n = 3 a 1+  3  +  +  3  +  +\n  \n   2   2  \n  \n   2  \n = 3 a 1 +  3  +  + ( 3 2 + )+ \n   2  \n  7 \n  3 a 1 + 3  +  2 \n  2 \n\n(iii)  7  3a Percentage error\n 3 a 1 + 3  +  2  −\n  2  cos  − 3 sin  approx value − actual value\n Let f ( ) = =\n actual value\n  100\n 3a\n = 100 f( )\n cos  − 3 sin \n  7 2 1\n 1 + 3  +   −\n 2  cos  − 3 sin \n = \n 1\n cos  − 3 sin \n\n  f ( )  0.05\n\n From GC,\n\n y = f ( )\n\n\n\n\n y = 0.05\n\n 0 0.194\n\n 0    0.194 (3sf )\n\n\n\n\n 21\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)", "methods": [ "Sine rule", "Reciprocal series", "Numerical error inequality" ], "prerequisites": [ "Angle sum", "Radians", "Relative error" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 20, solution PDF pages 20,21. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-19", "topic": "09", "number": 19, "questionPages": [ 22 ], "solutionPages": [ 22 ], "attribution": "[2018/AJC/Prelim/I/2]", "candidatePartLabels": [ "perimeter" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "19\n P\n 5 cm\n\n\n \n O\n Q R\n 7 cm\n\n\n The diagram shows a triangle OPQ with OP = 5 cm , OQ = 7 cm and POQ =  radians .\n Given that OP and OR are radii of a circle with centre O and  is a sufficiently small\n angle, show that the perimeter of PQR can be approximated by a + b + c 2 where a, b\n and c are constants to be determined. [5]\n [2018/AJC/Prelim/I/2]\n19 [Solution]\n PQ 2 = 52 + 7 2 − 2 ( 5 )( 7 ) cos  Use Cosine Rule.\n\n  PQ = 74 − 70 cos \n\n Perimeter of PQR = PQ + QR + arc length PR\n = 74 − 70cos  + 2 + 5\n 1\n     2 2\n   74 − 70 1 −   + 2 + 5\n   2 \n 1 Factorise out 4 to make\n ( )\n = 4 + 35 2 2 + 2 + 5 the constant term 1.\n 1 Use the law of indices\n  35  2\n = 2  1 +  2  + 2 + 5 (AB )n = AnB n\n  4 \n Then use binomial\n  35 \n  2 1 +  2  + 2 + 5 theorem\n  8 \n 1 1\n 35 2 (1 + x ) 2 = 1 + x + ...\n = 4 + 5 +  2\n 4\n 35 2\n 35 replacing x by  .\n  a = 4, b = 5, c = 4\n 4\n It is given that y = 1 + e\n tan x", "methods": [ "Cosine rule", "Binomial expansion", "Arc length" ], "prerequisites": [ "Radians", "Boundary identification" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 22, solution PDF pages 22. Independent computations and qualifications are documented in that report.", "parts": [ "perimeter" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-20", "topic": "09", "number": 20, "questionPages": [ 22 ], "solutionPages": [ 22, 23 ], "attribution": "[2009/MI/PU2 Promo/I/7]", "candidatePartLabels": [ "identity", "i", "ii", "integral" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "20 .\n d2 y 3 2\n = ( y 2 − 1) sec2 x , and hence show that when x = 0 , 2 =\n dy\n Show that 2 y . [5]\n dx dx 16\n Find\n (i) the first three terms of the Maclaurin’s series for y, [1]\n dy\n (ii) the first two terms of the Maclaurin’s series for , [2]\n dx\n giving your answers in surd form.\n 1\n Use the series to obtain an approximate value for  y dx , giving your answer in surd\n −1\n\n form. [2]\n [2009/MI/PU2 Promo/I/7]\n 22\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n\n\n\n20 [Solution]\n Square the equation so that\n y = 1 + e tan x the resulting eqn is easier to\n y = 1+ e\n 2 tan x handle for repeated\n differentiation using implicit\n Differentiating wrt x,\n differentiation and product\n dy\n (\n 2 y = sec2 x e tan x\n dx\n ) rule.\n\n\n  2y\n dy\n dx\n (\n = sec2 x y 2 − 1 )( ) (Shown) Note:\n d  dy \n 1) y \n Differentiating wrt x, d x  dx \n d  d y  dy d\n  dy \n 2 =y  +  (y )\n d2 y\n ( dy\n )\n 2 y 2 + 2   = y 2 − 1 2sec 2 x.tan x + 2 y sec 2 x\n dx  dx  dx\n dx  d x  dx dx\n d2y d y dy\n =y + \n dx 2 dx d x\n When x = 0 , y = 2\n d\n dy\n =\n 2 2)\n dx\n (\n e f( x ) )\n dx 4\n = e f( x )  f'(x )\n 2\n d2 y  2  2\n 2 2 2\n + 2   = 0 + 2 2   .1 = 1 d\n dx  4   4  3) (sec2 x )\n dx\n d2 y 3 1 2 3 = 2sec x  (sec x tan x )\n  2 =   = 2 (Shown)\n dx 4 2 2 2 16\n By Maclaurin’s Theorem, Subst the values of\n(i) dy d2y\n 2 3  x2  y, , when x = 0 (y =f(x))\n y= 2+ x+ 2   + ... dx dx 2\n 4 16  2!  into Maclaurin’s Theorem:\n f(0) 2\n 2 3 2 2 f(x ) = f(0) + f(0)x + x +\n y= 2+ x+ x + ... 2!\n 4 32\n 2 3 2 2\n(ii) y= 2+ x+ x + ...\n 4 32\n Differentiating wrt x,\n dy 2 3 2\n = + ( 2 x ) + ...\n dx 4 32\n dy 2\n  ( 4 + 3x ) (Surd form)\n dx 16\n  3 2 2\n  \n 1 1\n 2\n y dx   2 + x+ x  dx\n −1 −1\n  4 32 \n 1\n  x 2 x3 \n = 2 x + + \n  8 32  −1\n  37 29  33\n = 2 + = 2 (Surd form)\n  32 32  16\n\n 23\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n\n x 2", "methods": [ "Maclaurin expansion", "Coefficient comparison" ], "prerequisites": [ "Binomial theorem", "Power-series order" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 22, solution PDF pages 22,23. Independent computations and qualifications are documented in that report.", "parts": [ "identity", "i", "ii", "integral" ], "warning": "The rendered question requests two derivative-series terms, not three. The earlier compact transcription misread this request; it has been corrected during the complete native audit. The printed two-term solution is not missing a requested term. The quadratic integral estimate is a truncation, without a small-error guarantee over the full interval." }, { "id": "09-21", "topic": "09", "number": 21, "questionPages": [ 24 ], "solutionPages": [ 24 ], "attribution": "[2008/HCI/Prelim P2/1]", "candidatePartLabels": [ "identity", "series", "reciprocal" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "21 If y = 2 tan −1   , show that ( x + 2 x + 3) ddyx = 2.\n 2\n [1]\n  x+3\n By further differentiation of the above result, find the Maclaurin’s series expansion for y\n in ascending powers of x up to and including the term in x3 .\n 1\n Hence find the first three non-zero terms in the expansion of 2 . [7]\n x + 2x + 3\n [2008/HCI/Prelim P2/1]\n21 [Solution]\n x 2\n y = 2 tan −1 \n  x + 3 \n  \n  \n d\n dy\n \n = 2\n 1\n \n   2( x + 3) − x 2  dx\n (\n tan −1 f(x ) )\n  2   ( x + 3) 2\n \n dx\n  1 +\n 2x\n    1\n 2  =  f'(x )\n  ( x + 3)  1 + ( f(x ) )\n 2\n\n\n  3 2  x 2\n = 2\n  ( x + 3) + 2 x \n 2 2 where f(x ) =\n x +3\n .\n  \n 2 Use quotient rule to find\n =\n x + 2x + 3\n 2\n f’(x).\n\n (\n  x2 + 2 x + 3 ) ddyx = 2 (Shown)\n Differentiate w.r.t. x,\n\n ( x2 + 2 x + 3 ) d2 y\n dx 2\n + 2( x + 1)\n dy\n dx\n =0 Let y = f(x).\n To find the Maclaurin’s\n Differentiate again w.r.t. x, series for y, up to and\n including the term in x3,\n ( ) dx\n 3 2 2\n d y d y d y dy\n x2 + 2 x + 3 + 2( x + 1) + 2( x + 1) +2 =0\n 3\n dx 2\n dx 2\n dx differentiate up to the 3rd\n d3y\n (\n  x + 2x + 32d3 y\n dx 3\n + 4( x + )\n 1)\n d2 y\n dx 2\n +2\n dy\n dx\n =0 derivative\n dx 3\n .\n\n When x = 0, Subst the values of\n dy 2 2\n d y 4 3\n d y 4 dy d2y d3y\n y = 0, = , =− , = y, , , when x = 0\n dx 3 dx 2\n 9 dx3 27 dx dx 2 dx 3\n\n into Maclaurin’s Theorem:\n  Maclaurin’s series for y is\n f(0) 2\n 2 2 2 f(x ) = f(0) + f(0)x + x\n y  x − x 2 + x3 (up term in x 3 ) 2!\n 3 9 81 f(0) 3\n + x +\n 3!\n 1 1 dy 1  2 4 2 2\n =   − x+\n 27 \n x\n x + 2x + 3\n 2\n 2 dx 2  3 9\n 1 2 1 2\n = − x+ x\n 3 9 27\n\n\n 24\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n\n   dy", "methods": [ "Inverse-tangent differentiation", "Differential recurrence" ], "prerequisites": [ "Quotient rule", "Maclaurin factorials" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 24, solution PDF pages 24. Independent computations and qualifications are documented in that report.", "parts": [ "identity", "series", "reciprocal" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-22", "topic": "09", "number": 22, "questionPages": [ 25 ], "solutionPages": [ 25, 26 ], "attribution": "[modified 2009/ACJC/Prelim/I/11]", "candidatePartLabels": [ "i", "ii", "iii", "iv", "v" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "22 (i) Given that y = cot  2 x +  , find an equation relating and y . [2]\n  4 dx\n d3 y  d 2 y  dy 2 \n Hence show that = k  y 2 +    , where k is a constant to be found. [2]\n dx3  dx  dx  \n (ii) Find the Maclaurin’s series for y up to and including the term in x3 if x is sufficiently\n small for powers of x higher than x3 to be neglected. [2]\n  13 \n (iii) Using the Maclaurin’s expansion for y, estimate the value of cos ec 2   , giving\n  50 \n your answer in the form a + b + c 2 , where a, b and c are constants to be\n determined. [4]\n (iv) Using your result from part (ii), state the equation of the tangent to the curve\n  \n y = cot  2 x +  at x = 0. [1]\n  4\n (v) Explain why a Maclaurin series for y = cot 2x cannot be found. [1]\n [modified 2009/ACJC/Prelim/I/11]\n22 [Solution]\n(i)  \n y = cot  2 x + \n  4\n dy     \n  = −2cosec 2  2 x +  = − 2 1 + co t 2  2 x +  \n dx  4   4\n\n = −2 (1 + y 2 )\n dy\n \n dx\n Differentiate wrt x,\n d2 y dy\n 2\n = −4 y\n dx dx\n Differentiate wrt x,\n 2\n d3 y  dy  d2 y\n = − 4   − 4 y\n dx 3  dx  dx 2\n d3 y  d 2 y  dy  2 \n  = −4 y 2 +   \n dx 3  dx  dx  \n Therefore, k = −4\n22 dy d2 y d3 y Subst the values of\n When x = 0, y = 1, = −4 , = 16 , = −128\n(ii) dx dx 2 dx3 dy d2y d3y\n y, , , when x = 0\n  Maclaurin’s series is dx dx 2 dx 3\n into Maclaurin’s Theorem:\n   16 128 3\n cot  2 x +  = 1 − 4 x + x 2 − x + ... f(x ) = f(0) + f(0)x +\n f(0) 2\n x\n  4 2! 3! 2!\n 64 f(0) 3\n  1 − 4 x + 8 x 2 − x3 + x +\n 3!\n 3\n where y = f(x).\n\n\n 25\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n 22   64\n cot  2 x +   1 − 4 x + 8 x 2 − x 3\n(iii)  4 3\n Differentiating wrt x,\n  \n −2cosec 2  2 x +   −4 + 16 x − 64 x 2\n  4\n  \n Therefore, cos ec 2  2 x +   2 − 8 x + 32 x 2\n  4\n  13 \n Let 2 x + = . Then x =\n 4 50 200\n Hence,\n  13    2\n cosec2   2− +\n  50  25 1250\n 1 1\n where a = 2, b = − , c =\n 25 1250\n   1st 2 terms (constant and\n(iv) Equation of tangent to curve of y = cot  2 x +  at x = 0\n  4 term in x) in the\n is y = 1 − 4 x. Maclaurin’s series for y.\n\n(v) When x = 0, cot 2x is undefined. Hence Maclaurin’s\n expansion for y = cot 2 x cannot be found.\n −1", "methods": [ "Differential recurrence", "Series differentiation", "Angle substitution" ], "prerequisites": [ "Trigonometric identities", "Analyticity at expansion point" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 25, solution PDF pages 25,26. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv", "v" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "09-23", "topic": "09", "number": 23, "questionPages": [ 26 ], "solutionPages": [ 26, 27 ], "attribution": "[2010/SAJC/Prelim P1/9]", "candidatePartLabels": [ "i", "ii", "iii", "iv.a", "iv.b" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "23 dy e tan x\n It is given that = , where tan −1 x denotes the principal value.\n dx 1 + x 2\n (i) Find an expression for y in terms of x given that y = 1 when x = 0. [2]\n\n ( ) ddxy + (2 x −1) ddyx = 0 .\n 2\n (ii) Show that 1 + x 2 2\n [2]\n\n (iii) By further differentiation of the result in (ii), find the Maclaurin’s series for y up\n to and including the term in x3. [4]\n (iv) Hence, deduce the series expansion for\n −1\n e tan x −1 x\n (a) (b) e2 x+ tan , up to and including the term in x2. [4]\n (1 + x)2\n [2010/SAJC/Prelim P1/9]\n23 [Solution]\n −1\n(i) dy e tan x\n =\n dx 1 + x 2\n −1\n dy e tan x\n  dx dx =  1 + x2 dx\n −1\n  y = e tan x + C\n When x = 0, y = 1  1 = e0 + C  C = 0\n −1\n  y = e tan x\n\n\n\n 26\n\f2025 TJC H2 Math (9758) JC2 MSM: MACLAURIN’S SERIES (includes Binomial Expansion)\n\n −1\n23 y = e tan x\n(ii) −1\n dy e tan x y\n = =\n dx 1 + x 2 1 + x 2\n\n (\n  1 + x2\n dy\n dx\n =y )\n Differentiating w.r.t. x,\n\n (\n 1+ x\n d2 y\n dx 2 )\n dy dy\n + 2x =\n 2\n dx dx\n\n ( 1 + x2 ) d2 y\n dx 2\n + (2 x − 1)\n dy\n dx\n = 0 (shown)\n\n(iii)\n (\n 1+ x\n d2 y\n dx 2 ) dy\n + (2 x − 1) = 0\n 2\n dx\n Differentiating w.r.t. x,\n\n (\n 1 + x2\n d3 y\n dx 3\n + 2)x\n d2 y\n dx 2\n + (2 x − 1)\n d2 y\n dx 2\n dy\n +2 =0\n dx\n\n (\n 1 + x2\n d3 y\n dx 3\n + )\n (4 x − 1)\n d2 y\n dx 2\n dy\n +2 =0\n dx\n When x = 0, y = 1 (given)\n dy d2 y d3 y\n = 1, = 1, = −1\n dx dx 2 dx3\n 1 2 1 3\n Thus the Maclaurin's Series is y = 1 + x + x − x +\n 2 6\n −1\n(iv) e tan x −1\n = e tan x (1 + x) −2\n(a) (1 + x) 2 Use binomial expansion\n\n   for (1 + x ) −2 .\n 1 1\n (\n = 1 + x + x 2 − x 3 + ...  1 − 2 x + 3 x 2 + ...\n  2 6 \n )\n 3\n = 1 − x + x 2 + ...\n 2\n −1 −1\n(iv) e 2 x + tan x = e 2x e tan x From MF27,\n(b)\n  (2 x) 2\n  1 1  Use standard\n = 1 + 2 x + + ...  1 + x + x 2 − x3 + ... \n  2!  2 6  Maclaurin’s series\n 9 x2\n = 1 + 3x + x 2 + ... ex = 1 + x + + ,\n 2!\n 2\n replacing x by 2x.\n\n\n\n\n 27", "methods": [ "Chain-rule integration", "Differential recurrence", "Series multiplication" ], "prerequisites": [ "Principal inverse tangent", "Binomial series" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-09.md; visually audited question PDF pages 26, solution PDF pages 26,27. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv.a", "iv.b" ], "warning": "Both Topic 09 PDFs are identical and include full solutions beside the questions; opening source pages reveals answers." }, { "id": "10-01", "topic": "10", "number": 1, "questionPages": [ 1 ], "solutionPages": [ 1 ], "attribution": "[2010/MI/Prelim/I/11(b) (modified)]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "1 (i) Find the general solution of the differential equation   = 2. [2]\n  x  dx\n (ii) Sketch the particular solution of the differential equation in (i) for which y = 0 and\n x = 0. [2]\n (iii) Describe the gradient of every solution curve as x → + . [1]\n [2010/MI/Prelim/I/11(b) (modified)]", "methods": [ "Separation of variables", "Initial-value solution" ], "prerequisites": [ "Integration", "Exponential and logarithmic algebra" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 1, solution PDF pages 1. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "The original equation is undefined at x=0. The requested initial condition uses the smooth extension y′=2x/(1+x²); this interpretation follows the printed working rather than treating the displayed equation as defined there." }, { "id": "10-02", "topic": "10", "number": 2, "questionPages": [ 1 ], "solutionPages": [ 1 ], "attribution": "[modified 2011/RVHS/Promo/12ai]", "candidatePartLabels": [ "general solution" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "2 Find the general solution of the differential equation\n d2 y\n = sec x tan x . [2]\n dx 2\n [modified 2011/RVHS/Promo/12ai]", "methods": [ "Direct integration twice" ], "prerequisites": [ "Secant antiderivative", "Connected domains" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 1, solution PDF pages 1. Independent computations and qualifications are documented in that report.", "parts": [ "general solution" ], "warning": "" }, { "id": "10-03", "topic": "10", "number": 3, "questionPages": [ 1 ], "solutionPages": [ 2 ], "attribution": "[2012/RVHS/Prelim/II/1 (modified)]", "candidatePartLabels": [ "substitution", "general solution", "i", "ii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "3 Use the substitution u = xy, where u is a function of x, to reduce the differential equation\n dy du\n x + (1 − x3 ) y − x 2 = 0 to = (u + 1) x 2 .\n dx dx\n Hence show that the general solution to the differential equation is of the form\n Aef ( x ) − 1\n y= , where f is a function in x. [5]\n x\n Sketch, on separate diagrams, the particular solution curve for\n (i) A = 2 and (ii) A = −3 . [2]\n [2012/RVHS/Prelim/II/1 (modified)]", "methods": [ "Product substitution", "Separation of variables", "Curve analysis" ], "prerequisites": [ "Product rule", "Exponential limits", "Excluded zero" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 1, solution PDF pages 2. Independent computations and qualifications are documented in that report.", "parts": [ "substitution", "general solution", "i", "ii" ], "warning": "" }, { "id": "10-04", "topic": "10", "number": 4, "questionPages": [ 1 ], "solutionPages": [ 3 ], "attribution": "[2012/NJC/Prelim/I/4]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "4 It is known that a particular type of bacteria grows very well under certain controlled\n conditions in a specially prepared Petri dish. The researcher believes that the growth rate of\n dx\n such bacteria can be modelled by t 2 − 2 xt = − x 2 , where x milligrams is the amount of\n dt\n bacteria grown in the dish after t hours.\n (i) Using the substitution x = ut 2 , show that the differential equation can be reduced to\n du\n = −u 2 . [2]\n dt\n (ii) Find x in terms of t, given that there was 0.2 milligrams of bacteria after 15 minutes.\n Hence find the amount of this particular type of bacteria after 4 hours. [4]\n (iii) Explain if this mathematical model is a realistic one. [2]\n [2012/NJC/Prelim/I/4]", "methods": [ "Product substitution", "Separation of variables", "Model assessment" ], "prerequisites": [ "Unit conversion", "Long-term limits" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 1, solution PDF pages 3. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "10-05", "topic": "10", "number": 5, "questionPages": [ 1 ], "solutionPages": [ 4 ], "attribution": "[2019/NYJC/Prelim/I/7]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "5 A spherical tank with negligible thickness and internal radius a cm contains water. At time\n t s, the water surface is at a height x cm above the lowest point of the tank and the volume\n 1\n of water in the tank, V cm3, is given by V =  x 2 ( 3a − x ) . Water flows from the tank,\n 3\n through an outlet at its lowest point, at a rate  k x cm3s−1 , where k is a positive constant.\n\n Show that ( 2ax − x 2 ) = −k x .\n dx\n (i) [2]\n dt\n (ii) Find the general solution for t in terms of x , a and k . [3]\n (iii) Find the ratio T1 : T2 , where T1 is the time taken to empty the tank when initially it is\n completely full, and T2 is the time taken to empty the tank when initially it is half\n full. [4]\n [2019/NYJC/Prelim/I/7]\n 68\n\f2025 TJC H2 Math (9758) JC2 MSM: DIFFERENTIAL EQUATIONS", "methods": [ "Related rates", "Separation of variables", "Definite integration" ], "prerequisites": [ "Sphere-cap volume", "Fractional-power integration" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 1, solution PDF pages 4. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "10-06", "topic": "10", "number": 6, "questionPages": [ 2 ], "solutionPages": [ 5, 6, 7, 8 ], "attribution": "[2024/TJC/Prelim/I/11]", "candidatePartLabels": [ "a.i", "a.ii", "b.i", "b.ii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "6 A metal ball is released from the surface of the liquid in a tall cylinder. The ball falls\n vertically through the liquid and the distance, x cm, that the ball has fallen in time t seconds\n is measured. The speed of the ball at time t seconds is v cms −1 . The ball is released in a\n manner such that x = 0 and v = 0 when t = 0 .\n (a) The motion of the ball is modelled by the differential equation\n d 2 x 1 dx\n + − 10 = 0 .\n dt 2 2 dt\n dx\n It is given that v = .\n dt\n (i) Show that the differential equation can be written as\n dv 1\n = 10 − v. [1]\n dt 2\n (ii) Using the differential equation in (a)(i), find v in terms of t. Hence find x in terms\n of t. [6]\n (b) The metal ball is now released in another tall cylinder filled with a different liquid.\n However, for this liquid, the motion of the ball is modelled by the differential equation\n dv\n = 10 − k 2v 2 , where k is a positive constant.\n dt\n It is given that x = 0 and v = 0 when t = 0 .\n (i) Find v in terms of t and k. [5]\n (ii) When the ball falls through this liquid, its speed will approach its “terminal speed”\n which is the speed it will attain after a long time. Find the ball’s terminal speed in\n terms of k. You must show sufficient working to justify your answer. [2]\n [2024/TJC/Prelim/I/11]", "methods": [ "Separation of variables", "Partial fractions", "Integration of velocity" ], "prerequisites": [ "Velocity and acceleration", "Exponential limits" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 2, solution PDF pages 5,6,7,8. Independent computations and qualifications are documented in that report.", "parts": [ "a.i", "a.ii", "b.i", "b.ii" ], "warning": "" }, { "id": "10-07", "topic": "10", "number": 7, "questionPages": [ 2 ], "solutionPages": [ 8, 9 ], "attribution": "[modified 2015/YJC/Promo/9]", "candidatePartLabels": [ "DE", "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "7 A certain radioactive substance is known to decay at a rate proportional to the mass present.\n A block of this substance has an initial mass of 200 grams. After 40 hours, its mass is\n reduced to 180 grams. Given that x grams is the mass of the substance present at time t\n hours, write down a differential equation relating x and t. [1]\n  1 \n  ln 0.9  t\n (i) Show that x = 200e  40  . [4]\n (ii) Find the half-life of the radioactive substance, that is, the time it takes for the substance\n to decay to half of its mass, giving your answer correct to the nearest hour. [2]\n (iii) Sketch the curve that shows the mass of the block of radioactive substance over time.\n Hence explain what happens to the radioactive substance eventually. [2]\n [modified 2015/YJC/Promo/9]", "methods": [ "Exponential decay", "Parameter calibration" ], "prerequisites": [ "Logarithms", "Half-life", "Nonnegative time" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 2, solution PDF pages 8,9. Independent computations and qualifications are documented in that report.", "parts": [ "DE", "i", "ii", "iii" ], "warning": "" }, { "id": "10-08", "topic": "10", "number": 8, "questionPages": [ 2 ], "solutionPages": [ 9, 10 ], "attribution": "[2023/RI/Prelim/I/6]", "candidatePartLabels": [ "a", "b", "c" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "8 Alice bakes a large pie and removes it from the oven at 1 pm. The temperature reading of\n the pie using a food thermometer is 200 C at that instant.\n She left the pie on the kitchen countertop to cool down. The temperature of the kitchen is\n kept at a constant 32 C. The temperature of the pie t minutes after it is removed from the\n oven is  C. This temperature decreases at a rate proportional to the difference between\n the temperature of the pie and the temperature of the kitchen.\n At 1.15 pm, Alice checks the temperature of the pie again and it is 180 C.\n (a) Find an exact expression for  in terms of t, simplifying your answer. [5]\n (b) Sketch a graph of  against t. [2]\n (c) Food safety experts caution that bacteria grow fastest in food within the temperature\n range of 5 C to 60 C, known as the temperature danger zone. Find the latest time to\n the nearest minute to safely store the pie to avoid this danger zone. [2]\n [2023/RI/Prelim/I/6]\n 69\n\f2025 TJC H2 Math (9758) JC2 MSM: DIFFERENTIAL EQUATIONS", "methods": [ "Newton cooling", "Exponential calibration", "Threshold equation" ], "prerequisites": [ "Ambient temperature", "Logarithms", "Clock-time conversion" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 2, solution PDF pages 9,10. Independent computations and qualifications are documented in that report.", "parts": [ "a", "b", "c" ], "warning": "" }, { "id": "10-09", "topic": "10", "number": 9, "questionPages": [ 3 ], "solutionPages": [ 11, 12 ], "attribution": "[2022/RI/Prelim/I/10]", "candidatePartLabels": [ "i", "ii", "iii", "iv", "v" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "9 Glucose in the blood stream is reduced at a rate proportional to the amount of glucose\n present in the blood stream.\n Let G milligrams (mg) be the amount of glucose in 1 decilitre (dL) of blood stream at time\n t minutes and let λ denote the positive constant of proportionality.\n (i) Write down a differential equation relating G, λ and t. Solve this differential equation\n to find an expression of G in terms of λ and t. [3]\n Through extensive research, doctors recommended that a healthy glucose level in the blood\n stream of an adult should be between 70 mg/dL to 100 mg/dL.\n An adult patient, Neo, has 80 mg/dL of glucose in his blood stream at time t = 0 minutes.\n Due to a medical condition, he is not able to extract glucose from the food he eats. As he\n is not able to replenish the glucose in his blood stream, he is thus losing his glucose in the\n blood stream at a rate (in mg/dL per minute) corresponding to  = 0.005 per minute.\n (ii) Find the approximate time it would take for Neo’s glucose level in the blood stream\n to fall below the healthy range if there is no intake of glucose. [2]\n In order to maintain Neo’s glucose level in the blood stream in the healthy range, glucose\n is injected into his blood stream intravenously at a constant rate of μ mg/dL per minute.\n (iii) Write down a differential equation relating G, μ and t to model Neo’s situation. Solve\n this differential equation to find an expression of G in terms of μ and t. [5]\n (iv) Explain whether it is recommended to keep injecting glucose at a rate of 0.7 mg/dL\n per minute into Neo’s blood stream. [2]\n (v) Find the maximum range of values of μ so that Neo’s glucose level in the blood\n stream will always be in the healthy range. [1]\n [2022/RI/Prelim/I/10]", "methods": [ "Constant-input exponential model", "Equilibrium bounds" ], "prerequisites": [ "Linear first-order equations", "Limits", "Interval endpoints" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 3, solution PDF pages 11,12. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv", "v" ], "warning": "Printed solution p.12 / question answer p.8 uses 0.35<μ<0.5. Inclusive healthy bounds permit both endpoints. Counterexamples μ=0.35 and μ=0.5 remain within the interval for every finite t; exclude them only if the limiting equilibrium must also be strictly interior." }, { "id": "10-10", "topic": "10", "number": 10, "questionPages": [ 3 ], "solutionPages": [ 13 ], "attribution": "[2012/HCI/I/8]", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "10 Jesse leans a 6 metre rod XY in the position as shown below. At time t seconds, the two\n ends X and Y are x and y metres from O respectively.\n\n Y\n\n\n\n\n X\n O\n (i) State an equation relating x and y. [1]\n As the rod slips, Jesse proposes that y decreases at a rate proportional to y.\n (ii) Based on Jesse’s proposal, show that\n dx k (36 − x 2 )\n = , where k is a constant. [3]\n dt x\n Initially, X is 4 metres from O. Take k = 2.\n (iii) Find an expression for x in terms of t and find the time taken for Y to be 3 metres\n from O from the instance Jesse lets go of the rod, giving your answer to 1 decimal\n place. [5]\n (iv) Sketch the graph of x against t. Hence, comment on whether Jesse’s proposed model\n is appropriate. Justify your answer. [3]\n [2012/HCI/I/8]\n 70\n\f2025 TJC H2 Math (9758) JC2 MSM: DIFFERENTIAL EQUATIONS", "methods": [ "Related rates", "Separation of variables" ], "prerequisites": [ "Pythagoras", "Implicit differentiation", "Positive roots" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 3, solution PDF pages 13. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "" }, { "id": "10-11", "topic": "10", "number": 11, "questionPages": [ 4 ], "solutionPages": [ 14, 15 ], "attribution": "[2019/RVHS/Prelim/I/10]", "candidatePartLabels": [ "i", "ii", "iii", "iv", "v" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "11 Since the 1990s, a group of scientists has started conservation work to prevent the\n extinction of a rare species of flying fox in the wild Western Australia forests. The\n number of such species, in thousands, observed in the forests at time t years after the\n start of the conservation is denoted by N. It is known that the death rate of the flying\n foxes is proportional to the number of flying foxes present and that the birth rate of the\n flying foxes has been maintained constant at 2000 per year. There were only 1000 flying\n foxes in the forests at the start of the conservation and the rate of increase of the number\n of flying foxes was 1500 per year when there were 3000 flying foxes.\n (i) Form a differential equation relating N and t and show that it can be reduced to\n dN\n 6 = 12 − N . [2]\n dt\n (ii) Solve the differential equation to find N in terms of t. [5]\n (iii) Deduce the minimum number of years in integer, needed for the number of flying\n foxes to exceed 6000. [2]\n (iv) Sketch a graph to show how the number of flying foxes in the forests varies with\n time and suggest the long term behavior of the flying fox population. [3]\n (v) Suggest a possible limitation of the model in part (i). [1]\n [2019/RVHS/Prelim/I/10]", "methods": [ "Constant-input exponential model", "Integer threshold" ], "prerequisites": [ "Thousands scaling", "Logarithmic inequalities" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 4, solution PDF pages 14,15. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv", "v" ], "warning": "" }, { "id": "10-12", "topic": "10", "number": 12, "questionPages": [ 4 ], "solutionPages": [ 16, 17 ], "attribution": "[2020/CJC/Prelim/II/4]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "12 Botanists are studying how excessive logging is affecting the population of forest trees in\n a country in South-East Asia. The population of trees in a forest after t years is denoted by\n x, in thousands. One simple model proposed is that the population increases at a rate\n proportional to its population size. It is known that trees are chopped off at a constant rate\n of 3 thousand trees per year and the population stays constant when x = p .\n (i) Find a differential equation involving x and t. [3]\n (ii) The initial population of trees in a forest is 5 thousand trees. Solve the differential\n 3t\n equation in part (i) and show that x = (5 − p)e p + p. [5]\n (iii) Given that the population of tree decreases in a finite time, state an inequality satisfied\n by p.\n Sketch the graph of x against t for this case, showing clearly the coordinates of the\n graph’s point of intersection with the horizontal axis. Hence explain the meaning of\n this axial intercept in the context of the question. [4]\n [2020/CJC/Prelim/II/4]\n\n\n\n\n 71\n\f2025 TJC H2 Math (9758) JC2 MSM: DIFFERENTIAL EQUATIONS", "methods": [ "Equilibrium calibration", "Separation of variables", "Model domain" ], "prerequisites": [ "Proportional growth", "Exponential sign", "Extinction threshold" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 4, solution PDF pages 16,17. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "10-13", "topic": "10", "number": 13, "questionPages": [ 5 ], "solutionPages": [ 18, 19 ], "attribution": "[2023/RVHS/Prelim/I/12]", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "13 In a tank at an aquarium, fish pollute the water at a constant rate. A filter is installed into a\n new clean tank to remove the pollutant. The rate at which it removes the pollutant from the\n tank is proportional to the square root of the amount of pollutant present in the tank at that\n instant.\n Immediately after being cleaned of all pollutant, the amount of pollutant in the tank is\n increasing at a rate of 5 units per day. It is known that eventually, with the filter installed\n in a new clean tank, the amount of pollutant in the tank would stabilize at 75 units.\n dQ Q\n (i) Show that = 5− , where Q is the amount of pollutant present at time t. [3]\n dt 3\n Q\n (ii) Using the substitution x = , show that the differential equation in part (i) can be\n 3\n dx\n written as 6 x = 5− x . [2]\n dt\n (iii) Hence or otherwise, show that the particular solution of the differential equation in\n b\n  Q\n Q\n part (i) is  a −  e 3\n = m e pt\n  3\n \n where a, b, p and m are constants to be determined. [6]\n The water in the tank becomes toxic if the pollutant level gets too high. To avoid this, the\n tank is cleaned when the pollutant level reaches 48 units.\n (iv) By comparing the time taken for the pollutant level to reach 48 units with and without\n the filter, comment on the effectiveness of the filter. [2]\n [2023/RVHS/Prelim/I/12]", "methods": [ "Square-root substitution", "Separation of variables", "Implicit solution" ], "prerequisites": [ "Equilibrium calibration", "Logarithms", "Physical branch" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 5, solution PDF pages 18,19. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "" }, { "id": "10-14", "topic": "10", "number": 14, "questionPages": [ 5 ], "solutionPages": [ 20, 21, 22 ], "attribution": "[2017/RVHS/Promo/10]", "candidatePartLabels": [ "a.substitution", "a.solution", "b.DE", "b.solution" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "14 (a) By means of the substitution z = xy , show that the differential equation\n dy 1\n 2x = − y\n dx x\n can be reduced to the form\n 2 dz 1\n = . [2]\n 1 + z dx x\n Hence, solve the differential equation, giving your answer in the form f ( x, y ) = Ax,\n where f ( x, y ) is a function in x and y, and A is an arbitrary constant. [3]\n\n (b) A small town has a population of 1500 cats. During an epidemic, 50 cats are initially\n infected with this disease and these infected cats in the town spread the disease to\n other cats at a rate proportional to the product of the number of infected cats and the\n number of cats that are not infected.\n It is also given that the number of infected cats increase at a rate of 15 cats per day\n when the number of infected cats is 100.\n Let x denote the number of infected cats (in hundreds) in t days. Show that the rate\n dx\n of increase of infected cats, can be expressed as\n dt\n dx\n = kx (15 − x ) ,\n dt\n where k is an exact constant to be determined. [2]\n Find x in terms of t. [4]\n [2017/RVHS/Promo/10]\n 72\n\f2025 TJC H2 Math (9758) JC2 MSM: DIFFERENTIAL EQUATIONS", "methods": [ "Product substitution", "Logistic separation", "Partial fractions" ], "prerequisites": [ "Implicit branches", "Hundreds scaling", "Logarithmic odds" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 5, solution PDF pages 20,21,22. Independent computations and qualifications are documented in that report.", "parts": [ "a.substitution", "a.solution", "b.DE", "b.solution" ], "warning": "" }, { "id": "10-15", "topic": "10", "number": 15, "questionPages": [ 6 ], "solutionPages": [ 22, 23 ], "attribution": "[2020/ACJC/Prelim/II/2]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "15 A scientist and his student are investigating the growth of bacteria in a petri dish. At time\n t minutes after the bacteria is first introduced into the dish, the number of bacteria in the\n dish is denoted by x, in hundreds. Initially, 100 bacteria were introduced into the dish. In\n 10 minutes, the number of bacteria became 400.\n (i) The student claims that the growth rate of the bacteria in the dish is inversely\n proportional to the square root of the number of bacteria in the dish. Write down a\n differential equation for this situation and solve it to get x as a function of t. [4]\n (ii) The scientist claims that\n = m (100 x − x 2 ) ,\n dx\n dt\n where m is a constant.\n H\n Show that x = , where H is a whole number to be determined. (You need\n 1 + Ae− Hmt\n not find the values of A and m. [4]\n (iii) Whose model do you think is better, the student’s or the scientist’s? Justify your\n answer. [1]\n [2020/ACJC/Prelim/II/2]", "methods": [ "Power-law separation", "Logistic separation", "Model comparison" ], "prerequisites": [ "Fractional powers", "Partial fractions", "Population units" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 6, solution PDF pages 22,23. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "10-16", "topic": "10", "number": 16, "questionPages": [ 6 ], "solutionPages": [ 23, 24, 25, 26 ], "attribution": "[2021/TJC/Prelim/I/10]", "candidatePartLabels": [ "a.i", "a.ii", "b.i", "b.ii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "16 On a mangrove swamp, scientists are investigating the population of mudskippers and\n crabs. Initially, there are 800 mudskippers and 800 crabs on the mangrove swamp. At time\n t years, the number of mudskippers and crabs are M and C respectively.\n (a) For the mudskippers, the scientists discovers that every year, the growth rate is 0.4%\n of the population size and the death rate is 5 mudskippers per year.\n (i) Write down a differential equation relating M and t. [1]\n (ii) Solve the differential equation in part (i) and determine what happens to the\n mudskipper population in the future. [4]\n (b) For the crabs, the scientists propose that C and t are related by the differential equation\n dC\n = 8C − 0.005C 2 .\n dt\n (i) Find the number of crabs in the mangrove swamp when the rate of change of C\n is a maximum. [2]\n (ii) Find C in terms of t. Sketch a graph of C against t. [5]\n [2021/TJC/Prelim/I/10]", "methods": [ "Linear exponential model", "Completing the square", "Logistic separation" ], "prerequisites": [ "Percentages", "Quadratic maximum", "Physical extinction cutoff" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 6, solution PDF pages 23,24,25,26. Independent computations and qualifications are documented in that report.", "parts": [ "a.i", "a.ii", "b.i", "b.ii" ], "warning": "" }, { "id": "10-17", "topic": "10", "number": 17, "questionPages": [ 6 ], "solutionPages": [ 27, 28 ], "attribution": "[2022/ASRJC/Prelim/I/11]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "17 The cylindrical tank in a research laboratory has a cross-sectional area of 4 m2 . To cool\n the tank, water is pumped in and out of the tank simultaneously. The volume and height of\n the water in the tank at any time t minutes is given by V (m3) and h (metres) respectively.\n Clean water is pumped into the tank at a rate that is proportional to h2 and the water is\n pumped out from the tank at a rate that is proportional to h.\n (i) Assume that the water does not overflow and that there is no change to the height of\n dh kh(h − 10)\n the water when h is 10, show that = where k is a real constant. [4]\n dt 4\n\n The tank was initially filled with clean water to a height of 2 metres. When the height of\n the water is 5 metres, the volume of water is increasing at a rate of 5.5 m3 per minute.\n (ii) Find the exact value of k. Hence find h in terms of t. [5]\n (iii) Sketch a graph of h against t. Hence write down the minimum height of the\n cylindrical tank that will not result in the overflow of the water. [3]\n [2022/ASRJC/Prelim/I/11]\n 73\n\f2025 TJC H2 Math (9758) JC2 MSM: DIFFERENTIAL EQUATIONS", "methods": [ "Rate balance", "Logistic separation", "Model consistency check" ], "prerequisites": [ "Tank volume", "Sign of physical rates", "Initial-value calibration" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 6, solution PDF pages 27,28. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "Question PDF p.6 (printed73), solution pp.27–28: the printed k=−11/50 is algebraically consistent with the given derivative datum but incompatible with positive inflow ∝h² and positive outflow ∝h. The original wording and solution are preserved; the logistic interpretation is explicitly conditional." }, { "id": "10-18", "topic": "10", "number": 18, "questionPages": [ 7 ], "solutionPages": [ 29, 30, 31, 32, 33 ], "attribution": "[2021/HCI/Prelim/I/10]", "candidatePartLabels": [ "i", "ii", "iii", "iv", "v" ], "partInventoryStatus": "visually checked against rendered question and solution PDFs", "status": "authored", "transcription": "transcribed and paraphrased from rendered PDF, maths independently checked", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "18 A flu virus is spreading in a community which has a fixed population of N people.\n Scientists discover that at time t weeks from the beginning, the rate of change of the\n number of people who are infected is proportional to the product of the number of people\n infected, I , and the number of people who are not infected.\n (i) Write down a differential equation relating I and t. [1]\n It is noted that at the beginning, 1% of the people in the community is infected with the\n virus. 12 weeks later, the proportion of people infected in the community has risen to 25%.\n N\n (ii) By solving the differential equation obtained in part (i), show that I = t\n .\n −\n 99(33 12\n ) +1\n [6]\n (iii) Find the least number of weeks needed for the proportion of people infected to\n exceed 50%, giving your answer to the nearest integer. [2]\n (iv) Sketch the graph of I against t. Explain, with justification, what happens to the\n number of people infected if this situation continues indefinitely. [2]\n (v) State a possible limitation of the model in part (i). [1]\n [2021/HCI/Prelim/I/10]\n\n\n\n\n 74\n\f2025 TJC H2 Math (9758) JC2 MSM: DIFFERENTIAL EQUATIONS", "methods": [ "Logistic separation", "Odds calibration", "Threshold inequality" ], "prerequisites": [ "Partial fractions", "Population proportions", "Exponential limits" ], "classificationStatus": "reviewed against rendered question and solution pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-10.md; visually audited question PDF pages 7, solution PDF pages 29,30,31,32,33. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv", "v" ], "warning": "" }, { "id": "11-01", "topic": "11", "number": 1, "questionPages": [ 1 ], "solutionPages": [ 1, 2 ], "attribution": "[modified 2017/DHS/Promo/6]", "candidatePartLabels": [ "collinearity", "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "1 The position vectors of the points A, B and C relative to the origin O are −i + 5 j + k ,\n i − j + 5k and pi − 4 j + 7k respectively, where p is a constant.\n Given that A, B and C are collinear, show that p = 2. [2]\n (i) The point N divides AC in the ratio AN : NC =  :1 − . Find the value of  given\n that ON is perpendicular to AC. [4]\n (ii) Hence find the exact area of triangle OAC. [2]\n (iii) Find the position vector of the point D such that OADC is parallelogram. [2]\n (iv) Find the coordinates of the point E such that OAEC is a kite. [2]\n [modified 2017/DHS/Promo/6]", "methods": [ "Collinearity", "Dot-product perpendicular foot", "Cross-product area", "Reflection" ], "prerequisites": [ "Position vectors", "Section ratio", "Parallelogram diagonals" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 1, solution PDF pages 1,2. Independent computations and qualifications are documented in that report.", "parts": [ "collinearity", "i", "ii", "iii", "iv" ], "warning": "" }, { "id": "11-02", "topic": "11", "number": 2, "questionPages": [ 1 ], "solutionPages": [ 3 ], "attribution": "[2018/EJC/Prelim/I/5]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "2 Referred to the origin O, the points P and Q have position vectors p and q where p and\n q are non-parallel, non-zero vectors. Point R is on PQ produced such that\n PQ : QR = 1:  . Point M is the mid-point of OR.\n (i) Find the position vector of R in terms of  , p and q . [1]\n F is a point on OQ such that F, P and M are collinear.\n (ii) Find the ratio OF : FQ, in terms of  . [4]\n [2018/EJC/Prelim/I/5]", "methods": [ "External section vector", "Compare independent coefficients" ], "prerequisites": [ "Linear independence", "Midpoint" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 1, solution PDF pages 3. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii" ], "warning": "" }, { "id": "11-03", "topic": "11", "number": 3, "questionPages": [ 1 ], "solutionPages": [ 4 ], "attribution": "[modified 2009/NYJC/Prelim/I/3]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "3 1  1 1 7\n     \n Lines l1 and l2 are given by r =  0  +   1  and r =  2  +   5  respectively.\n \n 1  −2   −1  1\n        \n (i) Show that l1 and l2 are skew lines. [3]\n (ii) Find the position vector of the foot of perpendicular from the point A(1, 2, −1) to line\n l1 . [3]\n [modified 2009/NYJC/Prelim/I/3]", "methods": [ "Simultaneous coordinate equations", "Dot-product projection" ], "prerequisites": [ "Skew versus parallel lines", "Line parameters" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 1, solution PDF pages 4. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii" ], "warning": "" }, { "id": "11-04", "topic": "11", "number": 4, "questionPages": [ 1 ], "solutionPages": [ 5, 6 ], "attribution": "[2010/ACJC/Prelim/I/9]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "4 Relative to the origin O, two points A and B have position vectors given by\n a = 3i + j + 3k and b = 5i − 4 j + 3k respectively.\n (i) Find the length of the projection of OA on OB . [2]\n (ii) Hence, or otherwise, find the position vector of the point C on OB such that AC is\n perpendicular to OB. [2]\n (iii) Find a vector equation of the reflection of the line AB in the line AC. [3]\n [2010/ACJC/Prelim/I/9]", "methods": [ "Scalar projection", "Vector projection", "Reflection about a line" ], "prerequisites": [ "Dot product", "Line equations" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 1, solution PDF pages 5,6. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "11-05", "topic": "11", "number": 5, "questionPages": [ 1 ], "solutionPages": [ 6, 7 ], "attribution": "[2010/NJC/Prelim/I/3]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "5 The points A and B have position vectors a and b respectively, relative to the origin O\n such that a = b . The point P with position vector p lies on AB such that b  p = a  p .\n (i) Show that AB is perpendicular to OP. [2]\n (ii) Determine the position vector of the point D in terms of a and b, where D is the\n reflection of O about the line AB. [2]\n (iii) Give the geometrical meaning of a  b . [1]\n [2010/NJC/Prelim/I/3]\n\n\n 77\n\f2025 TJC H2 Math (9758) JC2 MSM: VECTORS", "methods": [ "Dot-product orthogonality", "Midpoint reflection", "Cross-product area" ], "prerequisites": [ "Equal lengths", "Perpendicular bisector" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 1, solution PDF pages 6,7. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "11-06", "topic": "11", "number": 6, "questionPages": [ 2 ], "solutionPages": [ 7 ], "attribution": "[2018/DHS/Prelim/II/1]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed with source qualification; see warning and topic review evidence", "difficulty": "not yet assessed", "extractedText": "6 With reference to the origin O, the position vectors of point A and B are a and b, where\n a and b are non-zero and non-parallel vectors.\n (i) State the shortest distance from B to the line passing through O and A. [1]\n (ii) Given that c = a + b, state the geometrical meaning of b  c and show that\n bc = ab . [2]\n (iii) Given that a  2b = d  3a, find a linear relationship between a, b and d. [2]\n [2018/DHS/Prelim/II/1]", "methods": [ "Cross-product distance", "Bilinearity", "Zero cross-product criterion" ], "prerequisites": [ "Cross product", "Parallel vectors" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 2, solution PDF pages 7. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "Printed source relation needs a qualification for the valid case 2b+3d=0; original answer is preserved in the source panel." }, { "id": "11-07", "topic": "11", "number": 7, "questionPages": [ 2 ], "solutionPages": [ 8 ], "attribution": "[modified 2008/NYJC/Promo/5]", "candidatePartLabels": [ "parameter", "distance" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "7 The line l passes through the points A and B with coordinates (1,2, −1) and (2, a,1)\n respectively. Given that the line l cuts the x-axis, show that a = −2 . [3]\n Find the shortest distance of the origin from l. [5]\n [modified 2008/NYJC/Promo/5]", "methods": [ "Coordinate constraints", "Foot projection", "Cross-product distance" ], "prerequisites": [ "Line parametrisation", "Perpendicular distance" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 2, solution PDF pages 8. Independent computations and qualifications are documented in that report.", "parts": [ "parameter", "distance" ], "warning": "" }, { "id": "11-08", "topic": "11", "number": 8, "questionPages": [ 2 ], "solutionPages": [ 9, 10 ], "attribution": "[2017/ACJC/Prelim/I/9]", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed with source qualification; see warning and topic review evidence", "difficulty": "not yet assessed", "extractedText": "8 The position vectors of A, B and C with respect to the origin O are a, b and c\n respectively. It is given that AC = 4CB and a + b = a + b .\n 2 2 2\n\n\n\n (i) By considering (a + b).(a + b) , show that a and b are perpendicular. [2]\n (ii) Find the length of the projection of c on a in terms of a . [3]\n (iii) Given that F is the foot of the perpendicular from C to OA and f denotes the\n position vector OF , state the geometrical meaning of c  f . [1]\n (iv) Two points X and Y move along line segments OA and AB respectively such that\n 1\n OX = (cos 3t )i + (sin 3t ) j + k ,\n 2\n OY = (sin t )i + (cos t ) j − 2k ,\n where t is a real parameter, 0  t  2π . By expressing the scalar product of OX\n and OY in the form of p sin(qt ) + r where p, q and r are real values to be\n determined, find the greatest value of the angle XOY. [5]\n [2017/ACJC/Prelim/I/9]", "methods": [ "Expand squared norm", "Section formula", "Cross-product area", "Trigonometric angle optimisation" ], "prerequisites": [ "Dot product", "Sine addition formula", "Monotonicity of arccos" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 2, solution PDF pages 9,10. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "The printed motion description names line segments, but the prescribed coordinate paths are circular. This qualification does not alter the computed angle." }, { "id": "11-09", "topic": "11", "number": 9, "questionPages": [ 2 ], "solutionPages": [ 11 ], "attribution": "[2017/VJC/Prelim/II/2]", "candidatePartLabels": [ "i", "ii.collinear", "ii.area" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "9 Referred to the origin O, the points A, B and D are such that OA = a , OB = b and\n OD = d. The point C is such that OACB is a parallelogram and angle OAC is\n 2π\n radians.\n 3\n (i) Given that a is a unit vector and b = 4 , find the length of projection of OC onto\n OA . [3]\n (ii) Given that  a +  b + d = 0 and  +  + 1 = 0 , show that A, B and D are collinear.\n [3]\n If  = 4 , find the area of triangle OBD, leaving your answer in the form k a  b\n where k is a constant to be determined. [3]\n [2017/VJC/Prelim/II/2]\n\n 78\n\f2025 TJC H2 Math (9758) JC2 MSM: VECTORS", "methods": [ "Parallelogram addition", "Angle orientation", "Affine combination", "Cross-product area" ], "prerequisites": [ "Dot product", "Triangle area" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 2, solution PDF pages 11. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii.collinear", "ii.area" ], "warning": "" }, { "id": "11-10", "topic": "11", "number": 10, "questionPages": [ 3 ], "solutionPages": [ 12, 13 ], "attribution": "[2010/DHS/Prelim/II/4]", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed with source qualification; see warning and topic review evidence", "difficulty": "not yet assessed", "extractedText": "10 The planes Π1 and Π2 are defined by\n  2 1\n  \n Π1 : r  4 = 10, Π2 : r   3  = 8.\n    \n 1 1\n (i) Find the acute angle between the two planes. [3]\n (ii) Obtain a vector equation of l1 , the line of intersection of the two planes. [4]\n x−2 7−z\n The Cartesian equation of another line, l2 , is given by = , y = m, where m is\n 6 3\n a real constant.\n (iii) If the plane Π1 and line l2 intersect at the point (6, m, 5), find the value of m. [2]\n (iv) Show that the lines l1 and l2 are perpendicular for all values of m. [2]\n [2010/DHS/Prelim/II/4]", "methods": [ "Normal-vector angle", "Solve plane pair", "Direction dot product" ], "prerequisites": [ "Plane normals", "Skew lines" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 3, solution PDF pages 12,13. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "The source uses “perpendicular” for orthogonal directions. Actual intersection occurs only at m=−2/5." }, { "id": "11-11", "topic": "11", "number": 11, "questionPages": [ 3 ], "solutionPages": [ 13, 14 ], "attribution": "[2010/JJC/Prelim/I/7]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "11 The line l passes through the points A and B with coordinates (−1, 2, 3) and\n (5, 14, 11) respectively. The plane p has equation 2 x + 3 y − 6 z = −7.\n (i) Show that the line l is parallel to but not contained on the plane p. [3]\n (ii) Find the distance of the line l from the plane p. [3]\n (iii) Find a cartesian equation of the plane which contains l and is perpendicular to p.\n [2]\n [2010/JJC/Prelim/I/7]", "methods": [ "Normal-direction dot product", "Point-plane distance", "Cross-product normal" ], "prerequisites": [ "Line direction", "Plane equation" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 3, solution PDF pages 13,14. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "11-12", "topic": "11", "number": 12, "questionPages": [ 3 ], "solutionPages": [ 15, 16 ], "attribution": "[2010/AJC/Prelim/I/13]", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "12 The equations of a plane 1 and a line l are\n  2\n x −1 y + 4\n 1 : r   1  = 6 and l: = = z − 5 respectively.\n   3 2\n  −2 \n The point A has position vector 3i − j + 4k .\n (i) Find the shortest distance between point A and the plane 1 . [2]\n (ii) B is another point such that AB = −5 j − 2k . Find the length of projection of AB\n onto the plane 1 . [2]\n (iii) Using your answers in (i) and (ii), find the area of triangle ABC, where C is the\n reflection of A in the plane 1 . [2]\n (iv) Find the cartesian equation of the plane 2 which contains the line l and the origin.\n Hence, find the line of intersection between the planes, 1 and 2 . [4]\n [2010/AJC/Prelim/I/13]\n\n\n\n\n 79\n\f2025 TJC H2 Math (9758) JC2 MSM: VECTORS", "methods": [ "Point-plane distance", "Tangential projection", "Reflection", "Cross-product plane normal" ], "prerequisites": [ "Pythagoras", "Triangle area", "Plane intersection" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 3, solution PDF pages 15,16. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "" }, { "id": "11-13", "topic": "11", "number": 13, "questionPages": [ 4 ], "solutionPages": [ 17, 18, 19 ], "attribution": "[modified 2010/IJC/Prelim/I/12]", "candidatePartLabels": [ "i.intersection", "i.distance", "ii", "iii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "13 2  3  4\n    \n A plane  1 has equation r  −1 = 9 and a line l1 has equation r = 0 +   1  .\n      \n 3 0  −2 \n (i) Find the coordinates of P, the point of intersection of l1 and  1 . [4]\n Hence, or otherwise, find the shortest distance from point A(3, 0, 0) to  1 . [2]\n The equations of planes  2 and  3 are given as\n 1  2   2 \n  2 : r = 1 + s  0  + t  2  , and\n      \n 1  −3   1 \n  3 :  x + y − z =  , where  ,   .\n (ii) Find an equation of plane  2 in the form rn = d . Explain why the planes  1 and\n  2 intersect. [4]\n (iii) The line of intersection of planes  1 and  2 is l2 . The line l2 has equation\n  3 2\n  \n r = 3 + 1 .\n    \n  2  −1\n Given that l2 and  3 do not have any points in common, find the conditions\n satisfied by  and  . [3]\n [modified 2010/IJC/Prelim/I/12]", "methods": [ "Line-plane substitution", "Cross-product normal", "Parallel excluded line" ], "prerequisites": [ "Point-plane distance", "Normal proportionality" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 4, solution PDF pages 17,18,19. Independent computations and qualifications are documented in that report.", "parts": [ "i.intersection", "i.distance", "ii", "iii" ], "warning": "" }, { "id": "11-14", "topic": "11", "number": 14, "questionPages": [ 4 ], "solutionPages": [ 19, 20 ], "attribution": "[2009/IJC/Prelim/I/13]", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "14 S\n R\n\n P Q\n D C\n\n k j\n A i B\n The diagram above shows a cuboid with horizontal rectangular base ABCD, where\n AB = 5 units, AD = 3 units and AP = 2 units. The edges AP, BQ, CR and DS are\n vertical, and PQRS is the top of the cuboid. The point A is taken as the origin and\n vectors i, j and k, each of length 1 unit, are taken along AB, AD and AP respectively.\n (i) Show that the line DQ has equation\n r = 3j +  (5i − 3j + 2k ) , where  is a parameter.\n Find an equation of the line BS. [4]\n (ii) Hence calculate the acute angle between the lines DQ and BS, giving your\n answer to the nearest 0.1. [3]\n 2\n The point X lies on RC such that RX = RC , and the point M is the midpoint of BD.\n 3\n The plane p contains the points R, B and M.\n (iii) Find, in scalar product form, an equation of the plane p. [3]\n (iv) Find the exact value of the shortest distance from X to the plane p. [3]\n [2009/IJC/Prelim/I/13]\n 80\n\f2025 TJC H2 Math (9758) JC2 MSM: VECTORS", "methods": [ "Coordinate geometry", "Direction angle", "Cross-product plane", "Point-plane distance" ], "prerequisites": [ "Cuboid vertices", "Section ratio" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 4, solution PDF pages 19,20. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "" }, { "id": "11-15", "topic": "11", "number": 15, "questionPages": [ 5 ], "solutionPages": [ 21 ], "attribution": "[modified 2010/NYJC/Prelim/II/4]", "candidatePartLabels": [ "parameters", "distance", "third-plane" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "15 The plane p1 has equation x + y − 2 z = 4. A plane p2 with equation 2 x + ay + bz = −4\n (where a and b are constants) is parallel to p1 , find the values of a and b and also the\n exact distance between p1 and p2 . [3]\n The plane p3 is parallel to p1 and p2 . Find the equation of p3 such that p2 is exactly\n in between p1 and p3 . [2]\n [modified 2010/NYJC/Prelim/II/4]", "methods": [ "Proportional normals", "Normalise plane equations", "Equal signed spacing" ], "prerequisites": [ "Point-plane distance" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 5, solution PDF pages 21. Independent computations and qualifications are documented in that report.", "parts": [ "parameters", "distance", "third-plane" ], "warning": "" }, { "id": "11-16", "topic": "11", "number": 16, "questionPages": [ 5 ], "solutionPages": [ 22, 23, 24 ], "attribution": "[2010/NJC/Prelim/I/9]", "candidatePartLabels": [ "i", "ii", "iii.foot", "iii.projection", "iv" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "16 0\n Relative to the origin O, the point A has position vector given by OA =  1  and A lies\n 0\n  \n 1\n on the plane 1 with equation defined by r   3  = 3 . Another plane  2 has equation\n  2\n  \n y = x . The planes 1 and  2 intersect at line l.\n (i) Find a vector equation of the line l. [1]\n 7\n (ii) Show that the cosine of the acute angle between the planes 1 and  2 is . [2]\n 7\n (iii) Find the position vector of the foot of perpendicular, OF , from point A to the line\n l. Hence, find the exact length of projection of AF onto the plane  2 . [5]\n (iv) Another plane  3 has equation px + qy = 1 , where p and q are real constants. Find\n the condition in which p and q must satisfy such that l intersects  3 at exactly one\n point. [2]\n [2010/NJC/Prelim/I/9]", "methods": [ "Plane intersection", "Normal angle", "Orthogonal line projection", "Tangential vector projection" ], "prerequisites": [ "Dot product", "Pythagoras" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 5, solution PDF pages 22,23,24. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii.foot", "iii.projection", "iv" ], "warning": "" }, { "id": "11-17", "topic": "11", "number": 17, "questionPages": [ 5 ], "solutionPages": [ 24, 25, 26 ], "attribution": "[2010/CJC/Prelim/II/2]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "17 The equations of the line l1 and the plane p1 are respectively,\n 1  −1  1  2 \n        \n r =  0  +   1  and r = s 1 + t  −1 , where , s, t  .\n 0 1  1  1 \n        \n (i) Find the acute angle between l1 and p1 . [3]\n  \n  \n (ii) A second plane p2 has equation r   1  = 1 . Given that the two planes p1 and p2\n  \n  \n 4 x − 15 5\n intersect at the line l2 : = y, z = , find the values of α and β. [3]\n −2 2\n (iii) The plane p3 with equation 2 x + by + z = 1 is parallel to l2 . Find the value of b.\n Hence find the distance between l2 and p3 . [4]\n [2010/CJC/Prelim/II/2]\n\n\n\n 81\n\f2025 TJC H2 Math (9758) JC2 MSM: VECTORS", "methods": [ "Cross-product normal", "Line-plane angle", "Coefficient identity", "Parallel line distance" ], "prerequisites": [ "Sine angle formula", "Point-plane distance" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 5, solution PDF pages 24,25,26. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "11-18", "topic": "11", "number": 18, "questionPages": [ 6 ], "solutionPages": [ 26, 27, 28, 29 ], "attribution": "2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "18 The points A and B have position vectors 3i + j and 3i + 3j respectively. The line l1\n and the planes 1 and  2 have equations as follows:\n 2 1 0\n  \n l1 : r = OA +   1  , 1 : x + 2 z = 3 ,  2 : r =   1  +   1  ,\n  \n  −1  0 1\n      \n where  ,  and   .\n It is given that the planes 1 and  2 intersect in the line l2 .\n (i) Find a vector equation of the line l2 and show that the line l2 is parallel to the line\n l1 . Hence, find the shortest distance between the lines l1 and l2 .\n (ii) The points C and D are on the lines l1 and l2 respectively such that ABC = 90\n and ABDC is a parallelogram. Find the position vectors of the points C and D .\n Hence, find the area of the parallelogram ABDC .\n (iii) The plane  3 is parallel to the plane  2 and is equidistant to both point A and the\n plane  2 . Show that the equation of the plane  3 is given by r  (i − j + k ) = 1 .\n (iv) Find the position vector of the foot of the perpendicular from the point A to the\n plane  3 .", "methods": [ "Plane intersection", "Parallel line distance", "Parallelogram addition", "Midplane", "Normal projection" ], "prerequisites": [ "Cross product", "Dot product", "Plane distance" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 6, solution PDF pages 26,27,28,29. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "" }, { "id": "11-19", "topic": "11", "number": 19, "questionPages": [ 6 ], "solutionPages": [ 30, 31, 32, 33 ], "attribution": "[2023/TMJC/Prelim/I/9]", "candidatePartLabels": [ "a", "b", "c", "d", "e" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "19 A home interior designer is looking to design a pull-down wardrobe lift. To test his\n concept, the designer built a small-scale model of it, in which points ( x, y, z ) are\n defined relative to a corner of the model at O ( 0, 0, 0 ) . The rods used in the model can\n be modelled by equations of straight lines.\n 5 1\n The main rod can be modelled by the line l1 with equation r =  − 5  +  1,   .\n    \n 8 0\n Another rod, line l2 , passes through the points A ( 5, − 5, 8 ) and B ( 8, − 4, 3) .\n (a) A supporting wire connects B to a point F on l1 such that BF is perpendicular to\n l1 . Find the position vector of F. [3]\n (b) A retractable rod, line l R , is placed such that it can be modelled by reflecting l2\n about l1 . Find a vector equation of l R . [3]\n The panels of the small-scale model can be modelled by equations of planes. The main\n panel, plane p1 , contains the main rod l1 and an attachment point C ( 0, 0, 6 ) . A side\n panel, plane p2 , can be represented by the equation p2 : −17 x − 37 y + 4 z = 24 .\n (c) Show that p1 can be represented by the cartesian equation − x + y + 5 z = 30. [3]\n (d) A structural rod, line lS , needs to be placed at the intersection between p1 and p2 .\n Find a vector equation of lS . [2]\n (e) A decorative plank, plane p3 , has equation p3 : −3x +  y + 15z =  ,\n where  ,   . If the distance between p1 and p3 is exactly 3 3 units, find the\n possible values of  and  . [3]\n [2023/TMJC/Prelim/I/9]\n 82\n\f2025 TJC H2 Math (9758) JC2 MSM: VECTORS", "methods": [ "Line projection", "Reflection", "Plane through line and point", "Plane intersection", "Parallel-plane distance" ], "prerequisites": [ "Dot and cross products", "Line equations" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 6, solution PDF pages 30,31,32,33. Independent computations and qualifications are documented in that report.", "parts": [ "a", "b", "c", "d", "e" ], "warning": "" }, { "id": "11-20", "topic": "11", "number": 20, "questionPages": [ 7 ], "solutionPages": [ 34, 35, 36 ], "attribution": "[2017/YJC/Prelim/I/12]", "candidatePartLabels": [ "i", "ii", "sensor" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "20 Laser (Light Amplification by Stimulated Emission of Radiation) has many applications\n including medicine, data storage, military and industrial uses. It has the property of\n spatial coherence, which allows the laser beam to stay narrow over long distances.\n When a laser beam is projected onto a mirror at an angle, it reflects off the mirror at the\n same angle.\n An engineer is designing a device that does industrial cutting using a laser beam. To\n make the device compact, the device has a mirror to reflect the beam before it leaves the\n device. The laser beam source is located at the origin O. It projects an incident beam\n with direction vector i + 2 j + k . The beam hits the mirror at the point P with angle  .\n The mirror has an equation − x + 2 y + 3z = 12 .\n\n\n O\n Incident Reflected\n beam beam\n\n\n  \n P mirror\n\n\n\n\n O'\n (i) Find the acute angle  that the beam makes with the mirror. [2]\n (ii) By finding O', the image of O in the mirror, find a vector equation of the line\n that the reflected beam is on. [7]\n The engineer plans to install a sensor at (3,1, 0) to monitor the heat produced by the\n laser. For the sensor to work properly, the sensor must be less than 2 units\n away from either the incident or the reflected beam. Determine if the sensor will\n work properly. [4]\n [2017/YJC/Prelim/I/12]\n\n\n\n\n 83\n\f2025 TJC H2 Math (9758) JC2 MSM: VECTORS", "methods": [ "Line-plane angle", "Plane reflection", "Line projection", "Ray admissibility" ], "prerequisites": [ "Normal vectors", "Perpendicular distance" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 7, solution PDF pages 34,35,36. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "sensor" ], "warning": "For beam distances, check the foot lies on the actual ray or incident segment; an infinite-line distance alone may be insufficient." }, { "id": "11-21", "topic": "11", "number": 21, "questionPages": [ 8 ], "solutionPages": [ 36, 37, 38 ], "attribution": "[2017/JJC/Prelim/II/3]", "candidatePartLabels": [ "i", "ii", "iii", "iv", "v" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "21 Peter is using equations of planes to model two hillsides that meet along a river. The\n river is modelled by the line where the two planes meet.\n\n\n H1 H2\n\n\n\n\n One of the hillsides, H 1 , contains the points A, B and C with coordinates (3, 0, 2) ,\n (1, 0,3) and (2, −3,5) respectively. The point A is on the river. The other hillside H 2 has\n equation 2 x − y + kz = 14, where k is a constant.\n (i) Find a vector equation of H 1 in scalar product form. [4]\n (ii) Show that k = 4 and deduce that point B is also on the river. [3]\n (iii) Write down a cartesian equation of the river. [1]\n (iv) Show that B is the point on the river that is nearest to C. Hence find the exact\n shortest distance from C to the river. [3]\n (v) Find the acute angle between BC and H 2 . [2]\n [2017/JJC/Prelim/II/3]\n\n\n\n\n 84\n\f2025 TJC H2 Math (9758) JC2 MSM: VECTORS", "methods": [ "Plane from three points", "Plane intersection", "Perpendicular foot", "Line-plane angle" ], "prerequisites": [ "Dot and cross products", "Normal vectors" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 8, solution PDF pages 36,37,38. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv", "v" ], "warning": "" }, { "id": "11-22", "topic": "11", "number": 22, "questionPages": [ 9 ], "solutionPages": [ 39, 40, 41, 42 ], "attribution": "[2018/ACJC/Prelim/I/11]", "candidatePartLabels": [ "i", "ii", "iii", "iv", "v" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "22\n H T\n\n\n\n\n C\n V\n H Q S\n H\n R\n Figure 1. H Figure 2.\n\n\n Figure 1 shows a methane molecule consisting of a carbon atom with four hydrogen\n atoms symmetrically placed around it. Figure 2 shows the tetrahedron structure of the\n methane molecule with the centres of the hydrogen atoms represented by points Q, R, S\n and T and the centre of the carbon atom represented by point V.\n The points Q, R and S has coordinates (8, 1, 8), (8, 7, 2) and (2, 1, 2) respectively and\n form an equilateral triangle.\n (i) Find a cartesian equation of the plane p which passes through the points Q, R and\n S. [4]\n (ii) Find a cartesian equation of the plane p1 which passes through the midpoint of\n QR and is perpendicular to QR. [2]\n Plane p2 which passes through the midpoint of RS and is perpendicular to RS has\n equation x + y = 9.\n (iii) Find a vector equation of the line l where p1 and p2 meet. [1]\n (iv) The point T is on the line l such that QRST is a regular tetrahedron with QR = QT.\n Show that the possible coordinates for point T is (2, 7, 8). Hence, or otherwise,\n find the coordinates of a point on plane p that is closest to point T. [5]\n 3\n (v) Given that TV is of the vertical height of tetrahedron QRST. Find the\n 4\n coordinates of point V and hence show that the bonding angle TVQ of the methane\n molecule is 109.5 (correct to 1 decimal place). [4]\n [2018/ACJC/Prelim/I/11]\n\n\n\n\n 85\n\f2025 TJC H2 Math (9758) JC2 MSM: VECTORS", "methods": [ "Perpendicular bisector planes", "Equidistance", "Point-plane projection", "Vector angle" ], "prerequisites": [ "Regular tetrahedron", "Dot product", "Plane equations" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 9, solution PDF pages 39,40,41,42. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv", "v" ], "warning": "There are two possible tetrahedron vertices on opposite sides of the base; the source asks for the displayed one." }, { "id": "11-23", "topic": "11", "number": 23, "questionPages": [ 10 ], "solutionPages": [ 43, 44 ], "attribution": "[2010/PJC/Prelim/II/3]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independent vector calculation compared with rendered question and answer pages", "difficulty": "not yet assessed", "extractedText": "23\n D C\n\n\n 5m\n 6m 6m\n A 10 m\n B\n H G\n 3m k j 3m\n 4m\n O i\n E 10 m F\n\n The diagram above shows a partial design of the roof of the gallery of an\n amphitheatre. ABCD is an inclined rectangular roof, where AB = 10 m and BC = 5 m.\n EFGH is a rectangle on a horizontal ground, where EF = 10 m and EH = 4 m.\n The points A and B are 3 m directly above E and F respectively. The points C and D\n are 6 m directly above G and H respectively.\n Point O is the centre of EFGH and is taken as the origin. Perpendicular unit vectors i,\n j, k are in the direction of EF , EH and EA respectively.\n (i) Find a vector equation of the line passing through A and C. [2]\n (ii) A spot light is fixed at the top of a vertical pillar erected at point Q with\n coordinates (15,6,0) . Find the height of the pillar if the top of the pillar is\n collinear with A and C. [3]\n (iii) The main electrical supply to the gallery runs along the diagonal AC. Find the\n position vector of point X on AC such that the length of the electric cable that is\n required to provide electricity from the main electrical supply to an amplifier\n mounted at D, will be the least. [3]\n [2010/PJC/Prelim/II/3]\n\n\n\n\n 86\n\f2025 TJC H2 Math (9758) JC2 MSM: VECTORS", "methods": [ "Coordinate model", "Line parametrisation", "Constrained perpendicular foot" ], "prerequisites": [ "Centred coordinates", "Dot-product projection" ], "classificationStatus": "reviewed against Vectors question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-11.md; visually audited question PDF pages 10, solution PDF pages 43,44. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "12-01", "topic": "12", "number": 1, "questionPages": [ 1 ], "solutionPages": [ 1 ], "attribution": "[2020/DHS/Prelim/I/1]", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "1 3x − 2 (3 x − 2) 2 (3 x − 2)3\n The geometric series is given by + + + ... where x  0.\n 4x 4x 4x\n Explain why the sum to infinity exists for 13  x  1, and find its value in terms of x. [5]\n [2020/DHS/Prelim/I/1]", "methods": [ "Identify GP ratio", "Convergence inequality", "Infinite sum" ], "prerequisites": [ "Geometric series", "Rational simplification" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [], "warning": "" }, { "id": "12-02", "topic": "12", "number": 2, "questionPages": [ 1 ], "solutionPages": [ 1, 2 ], "attribution": "[2022/RVHS/Promo/3(ii)]", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "2 The difference between the sum of the first n terms of the geometric progression and its\n 3\n sum to infinity is less than 0.4% of the sum to infinity. Assuming the value of r is , find\n 4\n the least value of n. [3]\n [2022/RVHS/Promo/3(ii)]", "methods": [ "Tail fraction", "Logarithmic inequality" ], "prerequisites": [ "GP partial sums", "Logarithms", "Integer rounding" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [], "warning": "The percentage statement implicitly concerns a nonzero positive total; a zero GP cannot have a strict error less than zero." }, { "id": "12-03", "topic": "12", "number": 3, "questionPages": [ 1 ], "solutionPages": [ 2, 3, 4 ], "attribution": "[2022/YIJC/Promo/8]", "candidatePartLabels": [ "a", "b", "c", "d" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "3 An arithmetic series has first term a and common difference d, where a  0 and d is non-\n zero. The first, eighth and thirteenth terms of the arithmetic series are equal to the first three\n terms of a geometric series respectively.\n 49\n (a) Show that a = − d . [2]\n 2\n (b) Show that the geometric series is convergent and find its sum to infinity, S, in terms of\n a. [3]\n (c) Given that the sum of the first n terms of the arithmetic series is 25 times the third term\n of the geometric series, find the value of n. [3]\n (d) Find the least value of k such that the sum of the first k terms of the geometric series is\n more than 98% of S. [3]\n [2022/YIJC/Promo/8]", "methods": [ "Equal GP ratios", "AP sum", "Relative tail" ], "prerequisites": [ "Nth terms", "Finite sums", "Logarithms" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "a", "b", "c", "d" ], "warning": "" }, { "id": "12-04", "topic": "12", "number": 4, "questionPages": [ 1 ], "solutionPages": [ 4, 5, 6 ], "attribution": "[2023/ACJC/Promo/8]", "candidatePartLabels": [ "a.i", "a.ii", "a.iii", "b.i", "b.ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "4 (a) (i) The sum of the first n terms of a sequence u1 , u2 , u3 ,... is given by Sn = e2 n2 − 3n .\n Show that un = e2 ( 2n − 1) − 3 . [2]\n (ii) Hence show that the sequence is an arithmetic progression. [2]\n (iii) State the first 3 odd-numbered terms, u1 , u3 and u5 . Hence find the least value of n\n such that the sum of the first n odd-numbered terms is more than 500. [3]\n\n (b) (i) Find the range of values of x for which the following geometric series converges,\n 1 2 3\n +1 +1 +1\n e − e x + e x − e x + ... . [2]\n (ii) Given that x = −1 , find the sum of all the terms of this geometric series after, and\n including, the 8th term. [3]\n [2023/ACJC/Promo/8]", "methods": [ "Difference of partial sums", "Subsequence AP", "GP tail" ], "prerequisites": [ "Exponentials", "AP sums", "Convergence" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "a.i", "a.ii", "a.iii", "b.i", "b.ii" ], "warning": "" }, { "id": "12-05", "topic": "12", "number": 5, "questionPages": [ 1 ], "solutionPages": [ 6, 7 ], "attribution": "TJC JC2 2025 H2 Maths MSM; no separate exam attribution printed for Q5", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "5 The fifth, tenth and twentieth terms of a convergent geometric progression, G, are\n the first three consecutive terms of an arithmetic progression, A.\n (i) Determine the common ratio of G. [3]\n Given that the first term of G is 2,\n (ii) evaluate the sum to infinity for G, [2]\n (iii) find the sum of the first 10 odd-numbered terms in A. [3]\n\n\n\n\n 89\n\f2025 TJC H2 Math (9758) JC2 MSM: ARITHMETIC & GEOMETRIC PROGRESSIONS", "methods": [ "AP equal differences", "Polynomial substitution", "Subsequence sum" ], "prerequisites": [ "GP nth term", "Quadratic equations" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "i", "ii", "iii" ], "warning": "The printed solution divides by r^4 and thereby omits the zero-ratio GP. The displayed main answer follows its intended nondegenerate branch." }, { "id": "12-06", "topic": "12", "number": 6, "questionPages": [ 2 ], "solutionPages": [ 8, 9 ], "attribution": "[2009/NYJC/Prelim/I/9]", "candidatePartLabels": [ "a", "b" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "6 (a) In a convergent geometric progression, the sum of the first n terms is equal to the\n 1\n sum of the remaining terms. Given also that the ( n + 1) th term is , determine the\n 2\n value of the ( 2n + 1) th term. [4]\n (b) An arithmetic progression has first term a and common difference d. The eighth,\n third and second term of the progression are successive terms of an infinite geometric\n progression.\n If the first term of the geometric progression is 10, find the sum of the even-numbered\n terms of the progression. [4]\n A sequence is formed in which the n th term is given by ln unun+1 , where un is the\n n th term of the geometric progression. Show that the sequence forms an arithmetic\n progression and state the value of the common difference. [3]\n [2009/NYJC/Prelim/I/9]", "methods": [ "Tail equality", "Equal ratios", "Even-indexed GP", "Logarithmic sequence" ], "prerequisites": [ "GP tails", "Logarithm laws" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "a", "b" ], "warning": "An infinite GP need not have a finite sum. The printed solution supplies a convergence assumption absent from part (b) wording." }, { "id": "12-07", "topic": "12", "number": 7, "questionPages": [ 2 ], "solutionPages": [ 9, 10 ], "attribution": "[2024/NJC/Prelim/II/2]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "7 An arithmetic series has first term a and common difference d, where a and d are non-\n zero. A convergent geometric series has first term b and common ratio r, where b is\n positive and r is non-zero. It is given that the fourth and ninth terms of the arithmetic\n series are equal to the sixth and ninth terms of the geometric series respectively and the\n eleventh term of the arithmetic series is br 6 less than the fourteenth term of the geometric\n series.\n (i) Show that r satisfies the equation 5r 8 − 7r 3 − 5r + 2 = 0 and solve this equation,\n giving your answer correct to 4 decimal places. [4]\n (ii) Using this value of r, deduce that for any positive integer n, the sum of the terms of\n 3\n the geometric series after, but not including, the nth term is less than b. [2]\n 5\n [2024/NJC/Prelim/II/2]", "methods": [ "Simultaneous equations", "Polynomial root", "Tail bound" ], "prerequisites": [ "AP and GP nth terms", "Numerical roots" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "i", "ii" ], "warning": "" }, { "id": "12-08", "topic": "12", "number": 8, "questionPages": [ 2 ], "solutionPages": [ 10, 11 ], "attribution": "[2009/IJC/Prelim/I/12]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "8 A geometric series G is given by e − e1+ + e1+ 2 − e1+3 + , where  is a negative real\n number.\n (i) Show that G is convergent. [2]\n (ii) The sum to infinity of G is S. Given that the sum to infinity of the even numbered\n 1\n terms of G, i.e. the second, fourth, sixth, … terms, is − S , find the exact value of\n 3\n . [4]\n (iii) In another series H, each term is the modulus of the corresponding term of G. Given\n that  = −0.03 , find the greatest value of n such that the product of the first n terms\n of H exceeds 15 000. [4]\n [2009/IJC/Prelim/I/12]\n\n\n\n\n 90\n\f2025 TJC H2 Math (9758) JC2 MSM: ARITHMETIC & GEOMETRIC PROGRESSIONS", "methods": [ "GP convergence", "Even-indexed GP", "Logarithm of product" ], "prerequisites": [ "Exponent laws", "Quadratic inequalities" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "12-09", "topic": "12", "number": 9, "questionPages": [ 3 ], "solutionPages": [ 12, 13 ], "attribution": "[modified 2011/AJC/Prelim/I/12]", "candidatePartLabels": [ "intro", "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "9 The sum of the first n terms of a series, S n , is given by\n 1\n Sn = 1 − ( a − 1)  , where a is a constant and a  1, n  + .\n n\n\n a \n Obtain an expression for the nth term of the series, Tn and prove that S n is a geometric\n series. [3]\n If the sequence Tn  is now grouped as follows:\n ( T1 ), ( T2 , T3 , T4 ), ( T5 , T6 , T7 , T8 , T9 ), …\n where each subsequent bracket has 2 terms more than the previous bracket, find\n (i) the total number of terms in the first n brackets. [2]\n (ii) the middle term of the 11th bracket in terms of a. [2]\n (iii) the range of values of a for the sum to infinity of the series to exist.\n Find the least value of n for the sum of all the terms in the first n brackets to be\n 39\n within 0.1% of the sum to infinity of the series when a = . [4]\n 20\n [modified 2011/AJC/Prelim/I/12]", "methods": [ "Difference of sums", "Odd-number counting", "Tail inequality" ], "prerequisites": [ "GP terms", "Sigma bounds", "Logarithms" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "intro", "i", "ii", "iii" ], "warning": "The source excludes a=1 but omits the necessary a≠0 restriction; its convergence answer also omits the zero-series endpoint a=2." }, { "id": "12-10", "topic": "12", "number": 10, "questionPages": [ 3 ], "solutionPages": [ 14, 15 ], "attribution": "[2009/DHS/Prelim/I/9]", "candidatePartLabels": [ "a", "b.i", "b.ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "10 (a) The first 2 terms of a geometric progression are p and q, where q < p and p ≠ 0. If\n the sum of the first n terms is equal to the sum to infinity of the remaining terms,\n prove that p n = 2q n . [3]\n (b) The arithmetic progression 2, 4, 6, 8, 10, 12, ... is arranged in rows in the following\n way:\n 1st Row: 2\n 2nd Row: 4, 6\n rd\n 3 Row: 8, 10, 12\n th\n 4 Row: 14, 16, 18, 20\n …\n (i) Show that the first term in the nth row is n 2 − n + 2. [2]\n (ii) Find the sum of all the terms from the 1 row to the (n – 1)th row.\n st\n [3]\n [2009/DHS/Prelim/I/9]", "methods": [ "GP tail equality", "Triangular indexing", "AP sum" ], "prerequisites": [ "Finite GP sums", "Triangular numbers" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "a", "b.i", "b.ii" ], "warning": "" }, { "id": "12-11", "topic": "12", "number": 11, "questionPages": [ 3 ], "solutionPages": [ 15, 16 ], "attribution": "[2013/CJC/Prelim/II/1]", "candidatePartLabels": [ "i", "ii.a", "ii.b" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "11 (i) A certain machine A is pumping a liquid into an empty container of total volume\n 850 m3.\n The first pumping action fills the container with a volume of 4 m3. Each subsequent\n pumping action fills the container with 0.5 m3 more than the previous pumping\n action. Pumping continues until the container is completely filled with the liquid.\n Find the volume of liquid that overflows from the container at the final pumping\n action. [3]\n (ii) Suppose that, after the 50th pumping action by machine A, a different machine B is\n used instead to fill the remaining volume of the container. Using machine B, the first\n pumping action fills the container with a volume of 5 m3. Each subsequent pumping\n 5\n action fills the container with the amount filled in the previous pumping action.\n 6\n (a) Find the total volume of the container filled after the 10th pumping action by\n B. [3]\n (b) Explain whether the container would be completely filled eventually. [2]\n [2013/CJC/Prelim/II/1]\n 91\n\f2025 TJC H2 Math (9758) JC2 MSM: ARITHMETIC & GEOMETRIC PROGRESSIONS", "methods": [ "AP cumulative sum", "Integer threshold", "GP limit" ], "prerequisites": [ "AP and GP sums", "Volumes" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "i", "ii.a", "ii.b" ], "warning": "" }, { "id": "12-12", "topic": "12", "number": 12, "questionPages": [ 4 ], "solutionPages": [ 17 ], "attribution": "[2022/NJC/Prelim/I/7]", "candidatePartLabels": [ "a", "b", "c" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "12 Amy cuts off pieces of ribbon from a long roll of ribbon. The first piece of ribbon she cuts\n off is 160 cm long and each successive piece is 8 cm shorter than the preceding piece.\n (a) What is the maximum number of pieces of ribbon that Amy can cut off? [2]\n After Amy has cut off the number of pieces found in part (a), her friend, Bala, continues\n to cut off more pieces from the same remaining long roll of ribbon, such that the first piece\n is 160 cm long and each successive piece is p times as long as the preceding piece, where\n p is a constant such that 0  p  1.\n (b) Given that the total length of ribbon that Bala cuts off can never be greater than 10\n m regardless of the number of pieces he cuts off, find the largest value of p. Taking p\n to be this value and by solving an appropriate inequality, find the maximum number\n of pieces that can be cut off before the total length of ribbon that Bala cuts off exceeds\n 9.5 m. [4]\n (c) After Amy has cut off the number of pieces found in part (a) and Bala has cut off the\n number of pieces found in part (b), there is 4.5 cm of ribbon left in the roll of ribbon.\n Determine the length of the original long roll of ribbon, leaving your answer to the\n nearest cm. [2]\n [2022/NJC/Prelim/I/7]", "methods": [ "Positive AP terms", "GP ceiling", "Logarithmic threshold" ], "prerequisites": [ "Unit conversion", "AP sums", "GP sums" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "a", "b", "c" ], "warning": "A zero-length twenty-first piece is not a ribbon piece; convert metres to centimetres before combining totals." }, { "id": "12-13", "topic": "12", "number": 13, "questionPages": [ 4 ], "solutionPages": [ 18, 19 ], "attribution": "[2023/JPJC/Prelim/I/8]", "candidatePartLabels": [ "i", "ii", "iii", "iv", "v" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "13 A marathon runner and a sprinter are running around in laps on a track. Each lap is\n 400m. The sprinter runs his first lap in 56 seconds, and the time he takes for each\n subsequent lap is 6 seconds more than the previous lap. The marathon runner runs his first\n lap in 1 minute, and the time he takes for each subsequent lap is 4% more than the time\n taken for the previous lap.\n (i) Show that the time the sprinter takes to complete n laps is 3n2 + 53n. [2]\n (ii) Find the time the marathon runner takes to complete n laps, giving your answer in\n terms of n . [2]\n (iii) How many complete laps does it take for the marathon runner to first exceed a total\n time of 40 minutes? [3]\n The 2 runners decide to do a 12km race with the same starting point.\n (iv) Determine which runner will be the first to complete the race. [3]\n (v) Determine which lap the winner is on when he first overtakes the other runner. [3]\n [2023/JPJC/Prelim/I/8]", "methods": [ "AP elapsed time", "GP elapsed time", "Integer comparison" ], "prerequisites": [ "Finite sums", "Unit conversion" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "i", "ii", "iii", "iv", "v" ], "warning": "Lap totals establish a lead change between laps 3 and 4. The precise within-lap crossing time needs a speed model; the source gives only lap durations." }, { "id": "12-14", "topic": "12", "number": 14, "questionPages": [ 4 ], "solutionPages": [ 20, 21 ], "attribution": "[2014/NYJC/JC2/mid-year/P2/4]", "candidatePartLabels": [ "a", "b.i", "b.ii", "b.iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "14 (a) Find the sum of all the integers in the set 1, 2, 3, , 2014 which are not multiples\n of 3. [4]\n\n (b) An elastic ball released from a height h falls vertically onto a horizontal floor and\n rebounds to a height ah where a is a constant such that 0 < a < 1. The ball is initially\n released from a height H above the horizontal floor.\n (i) Write down an expression, in terms of H, a and n, for the height to which the\n th\n ball rises after the n bounce. [1]\n 3\n Given that a = ,\n 4\n (ii) find the number of times the ball has bounced so that the height risen is less\n H\n than for the first time, [3]\n 20\n (iii) calculate, in terms of H, the total distance covered by the ball just before it\n comes to a complete rest. [2]\n [2014/NYJC/JC2/mid-year/P2/4]\n 92\n\f2025 TJC H2 Math (9758) JC2 MSM: ARITHMETIC & GEOMETRIC PROGRESSIONS", "methods": [ "Subtract AP sums", "GP height", "Infinite distance" ], "prerequisites": [ "Integer multiples", "Logarithmic inequalities" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "a", "b.i", "b.ii", "b.iii" ], "warning": "" }, { "id": "12-15", "topic": "12", "number": 15, "questionPages": [ 5 ], "solutionPages": [ 21, 22, 23 ], "attribution": "[2012/PJC/Prelim/I/9]", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "15 A hamster farm has 500 hamsters to sell. The farmer sells k hamsters at the end of every\n week, where k is a constant factor of 500 (e.g. 5, 100, etc.). The selling price of a hamster\n is $10 in the first week and it drops by 5% in each subsequent week. It is assumed that\n there is neither birth nor death of hamsters in the farm.\n (i) State the total number of weeks for the farmer to sell all his hamsters in terms of k.\n [1]\n (ii) Show that the total proceeds from selling all the hamsters is\n  500 \n $ 200k 1 − 0.95 k  . [3]\n  \n  \n (iii) Given that the cost of rearing a hamster is $0.50 a week, find the total cost incurred\n in rearing the hamsters, when the farmer has sold all his hamsters. [3]\n (iv) If all other costs are negligible, find the least value of k for the farmer to make a\n profit. [2]\n [2012/PJC/Prelim/I/9]", "methods": [ "GP revenue", "AP holding costs", "Divisor enumeration" ], "prerequisites": [ "Factors", "Finite sums", "Profit" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "i", "ii", "iii", "iv" ], "warning": "Round a numerical threshold only after imposing k divides 500; an arbitrary integer such as 22 is not allowed." }, { "id": "12-16", "topic": "12", "number": 16, "questionPages": [ 5 ], "solutionPages": [ 23, 24 ], "attribution": "[2022/ASRJC/Prelim/I/9]", "candidatePartLabels": [ "a.i", "a.ii", "b.i", "b.ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "16 (a) An arithmetic progression has first term a and common difference d, where a > 0 and\n d  0. The eighth, third and second term of the progression are the first three terms of\n an infinite geometric progression.\n (i) Find the common ratio of the geometric progression. [3]\n (ii) Find the exact sum of the odd-numbered terms of the geometric progression in\n terms of a. [3]\n (b) A programmer coded a program involving a rabbit-fox chase along a straight path to\n model the actual hunt for a rabbit by a fox.\n The rabbit first hop is 1.75 m. In each subsequent hop, the distance covered is 1%\n less than its previous hop. The fox first leaps 3 m. In each subsequent leap, the\n distance covered is 0.02 m less than its previous leap. Initially the rabbit is 60 m ahead\n of the fox and assume that the rabbit and the fox start and end each hop and leap at\n the same time.\n (i) By finding the total distance travelled by the fox and the rabbit after n leaps and\n hops respectively, find the minimum number of hops and leaps for the fox to\n catch up with the rabbit. [4]\n (ii) Find the number of leaps the fox takes before it comes to a stop. Hence, find the\n minimum starting distance, in metre, between the fox and the rabbit such that\n the fox will never catch up with the rabbit. Leave your answer to the nearest\n integer. [2]\n [2022/ASRJC/Prelim/I/9]", "methods": [ "Equal GP ratios", "Odd-index sum", "AP–GP distance", "Discrete maximisation" ], "prerequisites": [ "Finite sums", "Integer thresholds" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "a.i", "a.ii", "b.i", "b.ii" ], "warning": "The model specifies synchronised endpoints. Within-hop motion is unspecified; the distance calculation follows the source endpoint/constant-progress model. A strict real lower bound has an infimum rather than an attained minimum." }, { "id": "12-17", "topic": "12", "number": 17, "questionPages": [ 5, 6 ], "solutionPages": [ 25, 26 ], "attribution": "[2020/TMJC/Prelim/I/10]", "candidatePartLabels": [ "a", "b.i", "b.ii", "b.iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "17 Chloe plans to invest $k on the first day of every month with the bank, starting on\n 1 January 2021. The bank offers two different investment plans:\n Plan A: Each $k invested earns a fixed bonus of $5 at the end of every month for which\n it has been in the account. This bonus is added to the account. The accumulated bonuses\n themselves do not earn any further bonus.\n (a) If Chloe takes up Plan A, find the value of k such that the total value of all the\n investments, including bonuses, is worth $22 200 at the end of 31 December 2022.\n [3]\n 93\n\f2025 TJC H2 Math (9758) JC2 MSM: ARITHMETIC & GEOMETRIC PROGRESSIONS\n\n Plan B: The interest rate is 0.2% per month, so that on the last day of each month, the\n amount in the account on that day is increased by 0.2%.\n (b) Assume that Chloe takes up Plan B for the rest of the question.\n (i) Find, in terms of k, the amount of interest earned from the first $k invested on\n 1 January 2021 at the end of 31 December 2022. [1]\n (ii) Find the minimum value of k (to the nearest integer) so that the total amount in\n the account is worth at least $22 200 at the end of 31 December 2022. [3]\n (iii) Let k = 900 . Find the month in which the total in the account will first exceed\n $30000. Explain whether this occurs on the first or last day of the month. [5]\n [2020/TMJC/Prelim/I/10]", "methods": [ "AP bonus sum", "Annuity-due GP", "Event-time comparison" ], "prerequisites": [ "Monthly indexing", "Compounding", "Ceilings" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "a", "b.i", "b.ii", "b.iii" ], "warning": "A month-end formula alone can give the right month but the wrong day; check the deposit at the start of that month." }, { "id": "12-18", "topic": "12", "number": 18, "questionPages": [ 6 ], "solutionPages": [ 26, 27 ], "attribution": "[2024/JPJC/Prelim/I/8]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "18 Selena decides to borrow $200 000 from the bank to fund her home purchase on 1 January\n 2025. On the first day of each month, 0.5% interest is added to the amount owed with the\n first interest amount added on 1 January 2025. On the last day of each month, she makes\n a repayment of $x to the bank, starting from 31 January 2025.\n (i) Show that the amount of money Selena owes the bank at the end of n months is\n 200 000(1.005)n − 200 x(1.005n − 1). [3]\n (ii) If she repays the bank $1500 a month on the last day of each month, on which date\n will she fully repay the loan? What is the amount of the last repayment? [4]\n (iii) If she wants to fully repay the loan in 10 years, how much will she need to repay the\n bank every month? [2]\n [2024/JPJC/Prelim/I/8]", "methods": [ "Affine recurrence", "GP accumulation", "Integer month", "Directed currency rounding" ], "prerequisites": [ "Compound interest", "Dates", "Ceiling" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "12-19", "topic": "12", "number": 19, "questionPages": [ 6 ], "solutionPages": [ 28, 29 ], "attribution": "[2014/CJC/JC1/mid-year/7]", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "19 A fisherman purchased a fishing boat for $1000 at the beginning of year 2011. The annual\n operational cost is $120 in 2011 and increases by $40 every year thereafter.\n (i) Show that the total cost, inclusive of the purchasing cost and operational costs, up to\n the nth year is 20n 2 + 100n + 1000. [1]\n The annual income from fishing is $500 in 2011 but will decrease by 10% every year\n thereafter due to overfishing.\n (ii) Show that the total income from fishing will never exceed $5000, no matter how\n many years have passed. [2]\n The boat can be sold for $600 in 2011 and the sale price decreases by 10% every year\n thereafter. The boat is sold at the end of the nth year after income from the fishing is\n collected.\n (iii) Find the total income, from both fishing and the sale of the boat, in terms of n. [3]\n The fisherman will sell the boat in the final year when the total income, from both fishing\n and the sale of the boat, is more than the total cost.\n (iv) Hence, find the year in which the fisherman will sell the boat. [2]\n [2014/CJC/JC1/mid-year/7]\n\n\n\n\n 94\n\f2025 TJC H2 Math (9758) JC2 MSM: ARITHMETIC & GEOMETRIC PROGRESSIONS", "methods": [ "AP cost", "GP revenue", "Profit interval" ], "prerequisites": [ "Finite sums", "Year indexing" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "i", "ii", "iii", "iv" ], "warning": "" }, { "id": "12-20", "topic": "12", "number": 20, "questionPages": [ 7 ], "solutionPages": [ 30, 31, 32 ], "attribution": "[2022/SAJC/Prelim/I/9]", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "20 The coach of Besto running club designed a training programme such that runners begin\n with a 400 m run on the first training session. On each subsequent session, the distance\n covered is 250 m more than the distance covered on the previous session.\n (i) Find the minimum number of sessions required for runners from Besto on the training\n programme to run at least 20 km in a training session. [3]\n For another group of runners in Besto, a circuit training exercise was designed to build\n up their stamina.\n In this exercise, this group of runners from Besto run from a starting point O to and from\n a series of points, P1 , P2 , P3 , , increasingly far away in a straight line. In the exercise,\n they start at O and run stage 1 from O to P1 , and back to O, then stage 2 from O to P2 ,\n and back to O, and so on.\n\n\n O 50 m 100 m 300 m 900 m\n Fig. 1\n\n The distances between the points are such that OP1 = 50 m , P1 P2 = 100 m , P2 P3 = 300 m\n and Pn Pn +1 = 3Pn −1 Pn (see Fig. 1).\n (ii) Find an expression for the distance run by a runner from Besto who completes n stages\n of the circuit training exercise. [3]\n (iii) Hence, find the distance from O and the direction of travel, of a runner from Besto\n undergoing the circuit training exercise after he has run exactly 42 km. [3]\n Another running club, Choco, designed a different training programme. The runners in\n Choco began with running 400 m on the 1st session. On each subsequent session, the\n distance covered was increased by 10% of the distance covered on the previous session.\n From the 11th session onwards and for all subsequent sessions, the distance covered was\n increased by r% of the distance covered on the previous session.\n (iv) Given that the runners from Choco club covered at least 20 km on the 70th training\n session, find the range of values of r. [4]\n [2022/SAJC/Prelim/I/9]\n\n\n\n\n 95\n\f2025 TJC H2 Math (9758) JC2 MSM: ARITHMETIC & GEOMETRIC PROGRESSIONS", "methods": [ "AP threshold", "Nested GP distances", "Stage remainder", "Piecewise GP" ], "prerequisites": [ "Finite GP sum", "Percent changes", "Units" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-12.md", "parts": [ "i", "ii", "iii", "iv" ], "warning": "Nine increases reach session 10; sixty further increases reach session 70. Rounded lower bounds are approximate and should not replace the exact inequality in calculations." }, { "id": "13-01", "topic": "13", "number": 1, "questionPages": [ 1 ], "solutionPages": [ 1 ], "attribution": "[2011/ACJC/Promo/13b]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "1 The sum, S n , of the first n terms of a sequence U1 , U 2 , U 3 ,… is given by\n n\n Sn = ( c − 7n ) ,\n 2\n where c is a constant.\n (i) Find U n in terms of c and n. [2]\n (ii) Find a recurrence relation of the form U n +1 = f (U n ) . [2]\n [2011/ACJC/Promo/13b]\n 3", "methods": [ "Difference of partial sums", "AP recurrence" ], "prerequisites": [ "Partial sums", "Arithmetic progressions" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-13.md", "parts": [ "i", "ii" ], "warning": "" }, { "id": "13-02", "topic": "13", "number": 2, "questionPages": [ 1 ], "solutionPages": [ 1 ], "attribution": "[2011/DHS/Prelim/II/4b(i),(ii)]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "2 A sequence u1, u2 , u3 , ... is such that u1 = and\n 2\n 1 +\n un +1 = un + , for all n  .\n (n + 1)(n + 2)\n (i) Find the values of u2 , u3 and u4 . [2]\n (ii) Form a conjecture for un in terms of n. [1]\n [2011/DHS/Prelim/II/4b(i),(ii)]", "methods": [ "Iteration", "Telescoping differences" ], "prerequisites": [ "Fractions", "Partial fractions" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-13.md", "parts": [ "i", "ii" ], "warning": "" }, { "id": "13-03", "topic": "13", "number": 3, "questionPages": [ 1 ], "solutionPages": [ 2, 3 ], "attribution": "[2023/DHS/Prelim/II/2]", "candidatePartLabels": [ "a.i.A", "a.i.B", "a.ii", "b.i", "b.ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "3 (a) A sequence of real numbers u1 , u2 , u3 . . . is such that\n un+1 = 3 − 0.5un , for n  1.\n It is given that u1 = c where c is a constant.\n (i) Describe how the sequence behaves when\n (A) c = 5, [1]\n (B) c = 2. [1]\n\n (ii) Find the value of c for which 2u3 = −5u2 . [3]\n\n (b) Another sequence is defined by v1 = p, v2 = 2, where p is a constant, and\n vn+2 = 2vn + vn+1 − 1, for n  1.\n (i) If v3 + 1 = 2u2 − u1 , find a relationship between c and p. [2]\n (ii) Find the value of p for which v5 = 77. [2]\n [2023/DHS/Prelim/II/2]", "methods": [ "Fixed-point shift", "Iteration", "Simultaneous equations" ], "prerequisites": [ "Linear equations", "Geometric sequences" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-13.md", "parts": [ "a.i.A", "a.i.B", "a.ii", "b.i", "b.ii" ], "warning": "" }, { "id": "13-04", "topic": "13", "number": 4, "questionPages": [ 1 ], "solutionPages": [ 3, 4 ], "attribution": "[2011/TJC/Prelim/II/1)]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "4 The nth term of a sequence is denoted by an . The sequence is defined by\n an+1 = pan + q for all n  1 , where p and q are constants.\n The first three terms of the sequence are given by a1 = 0, a2 = 2 and a3 = 7 .\n (i) Find the values of p and q. [2]\n (ii) Express an in terms of n, for n  3 . [4]\n [2011/TJC/Prelim/II/1)]\n\n\n\n\n 97\n\f2025 TJC H2 Math (9758) JC2 MSM: RECURRENCE RELATIONS", "methods": [ "Fit recurrence coefficients", "Fixed-point shift" ], "prerequisites": [ "Linear equations", "GP nth term" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-13.md", "parts": [ "i", "ii" ], "warning": "" }, { "id": "13-05", "topic": "13", "number": 5, "questionPages": [ 2 ], "solutionPages": [ 4, 5 ], "attribution": "[2015/TJC/Promo/6]", "candidatePartLabels": [ "a", "b.i", "b.ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "5 A sequence {un} is defined by the recurrence relation\n 2\n un +1 = for n  + and u1  0,1 or 2 .\n 2 − un\n (a) Prove that the sequence does not converge. [3]\n\n (b) Given that u1 = 4 ,\n (i) write down the next 4 terms of the sequence, [1]\n (ii) prove algebraically that un + 4 = un . [3]\n [2015/TJC/Promo/6]", "methods": [ "Limit contradiction", "Iterated rational map" ], "prerequisites": [ "Real quadratic equations", "Composition", "Convergence" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-13.md", "parts": [ "a", "b.i", "b.ii" ], "warning": "The limit substitution must first exclude the pole L=2; dividing by a denominator tending to zero is invalid." }, { "id": "13-06", "topic": "13", "number": 6, "questionPages": [ 2 ], "solutionPages": [ 5 ], "attribution": "[2015/AJC/Promo/11(b)]", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "6 A sequence of real numbers u1 , u2 , u3 satisfies the recurrence relation\n u n = u n−1+ n + 2 n for n  2 .\n (n + a)(n + b)\n Given that u 1= 4 , show that u n = + 2 n+1 , where a, b are integers to be\n 2\n determined. Hence determine, with reasons, the behaviour of the sequence. [5]\n [2015/AJC/Promo/11(b)]", "methods": [ "Unroll recurrence", "AP sum", "GP sum" ], "prerequisites": [ "Finite sums", "Exponential growth" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-13.md", "parts": [], "warning": "" }, { "id": "13-07", "topic": "13", "number": 7, "questionPages": [ 2 ], "solutionPages": [ 6 ], "attribution": "[2012/MI/Prelim/10]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "7 The diagram below shows that graph of y = ln ( x − 1) − x + 4 .\n The roots of the equation ln ( x − 1) − x + 4 = 0 are denoted by α and β, where α < β.\n y\n\n\n\n\n x\n 1 α β\n\n\n\n\n (i) Find the values of α and β, each correct to 3 decimal places. [2]\n\n A sequence of real numbers x1 , x2 , x3 , ... satisfies the recurrence relation\n xn +1 = ln ( xn − 1) + 4 for n  1 .\n (ii) Prove algebraically that, if the sequence converges, then it converges to either α or β.\n [3]\n (iii) By considering the maximum point of y = ln ( x − 1) − x + 4 , show that xn +1  xn + 2 .\n [3]\n [2012/MI/Prelim/10]\n\n\n\n\n 98\n\f2025 TJC H2 Math (9758) JC2 MSM: RECURRENCE RELATIONS", "methods": [ "Numerical root finding", "Limit equation", "Derivative maximum" ], "prerequisites": [ "Logarithm domain", "Differentiation", "Convergence" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-13.md", "parts": [ "i", "ii", "iii" ], "warning": "Finding possible limits does not prove convergence; every logarithm requires $x_n$>1." }, { "id": "13-08", "topic": "13", "number": 8, "questionPages": [ 3 ], "solutionPages": [ 7 ], "attribution": "[2012/IJC/Prelim/II/3]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "8 A sequence of positive numbers x1 , x2 , x3 , ... satisfies the recurrence relation\n xn +1 = 10 − 3 xn , for n = 1, 2, 3, ... .\n (i) Given that the sequence converges to l, find the value of l. [2]\n (ii) Prove that ( xn+1 ) − l 2 = 3 ( l − xn ) .\n 2\n [2]\n (iii) Use the result in part (ii) to show that if xn  l , then xn +1  l . [2]\n [2012/IJC/Prelim/II/3]", "methods": [ "Fixed-point equation", "Difference of squares", "Sign analysis" ], "prerequisites": [ "Square roots", "Quadratic equations" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-13.md", "parts": [ "i", "ii", "iii" ], "warning": "The source has a square root over $10-3x_n$; plain text extraction loses it. The recurrence is assumed defined and positive at every step." }, { "id": "13-09", "topic": "13", "number": 9, "questionPages": [ 3 ], "solutionPages": [ 7, 8 ], "attribution": "[2011/TPJC/Prelim/I/2]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "9 A sequence u1, u2, u3, … is such that u1 = 1 and\n 5u + 4\n un+1 = n , for all n ≥ 1.\n 2un + 3\n As n → , un → l.\n (i) Find the exact value of l. [2]\n 5x + 4\n (ii) By considering the graph of y = − x , show that un+1 < un, if un > l. [3]\n 2x + 3\n (iii) State the behavior of un when u1 = 0. [1]\n [2011/TPJC/Prelim/I/2]", "methods": [ "Fixed-point equation", "Factored increment", "Invariant interval" ], "prerequisites": [ "Rational inequalities", "Monotone convergence" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-13.md", "parts": [ "i", "ii", "iii" ], "warning": "" }, { "id": "13-10", "topic": "13", "number": 10, "questionPages": [ 3 ], "solutionPages": [ 8, 9 ], "attribution": "[2011/TJC/Promo/14]", "candidatePartLabels": [ "a.i", "a.ii", "a.iii", "b.i", "b.ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "10 A sequence of real numbers {un} satisfies the recurrence relation\n 1\n un +1 = un + 1 for n  + .\n 2\n (a) (i) Given that {un} converges to l, find the value of l. [2]\n (ii) If un < l , show that un < un+1 < l. [3]\n (iii) If u1  l , describe the behaviour of {un}. [2]\n +\n (b) Another sequence {vn} is such that vn = un − 2 for n  .\n (i) Show that {vn} is a geometric progression. [3]\n n −1\n 1\n (ii) If u1 = 3 , show that un = 2 +   . [2]\n 2\n [2011/TJC/Promo/14]", "methods": [ "Fixed-point shift", "Invariant interval", "GP formula" ], "prerequisites": [ "Linear inequalities", "GP convergence" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-13.md", "parts": [ "a.i", "a.ii", "a.iii", "b.i", "b.ii" ], "warning": "Use $v_{n+1}=v_n/2$ rather than dividing by $v_n$, which could be zero." }, { "id": "13-11", "topic": "13", "number": 11, "questionPages": [ 3 ], "solutionPages": [ 9, 10 ], "attribution": "[2011/CJC/Prelim/I/2]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "11 A sequence of positive real numbers x1 , x2 , x3 , ... satisfies the relation\n 3 − xn\n xn +1 = for n  1.\n 2 xn + 1\n (i) Given that the sequence converges to l, find the exact value of l. [3]\n 3− x\n (ii) By using a sketch of the graph of y = − x , and considering xn +1 − xn ,\n 2x +1\n prove that\n xn+1  xn if 0  xn  l ,\n xn+1  xn if xn  l . [3]\n [2011/CJC/Prelim/I/2]\n\n\n\n\n 99\n\f2025 TJC H2 Math (9758) JC2 MSM: RECURRENCE RELATIONS", "methods": [ "Fixed-point quadratic", "Factored increment", "Iterated map" ], "prerequisites": [ "Rational functions", "Sign analysis" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-13.md", "parts": [ "i", "ii" ], "warning": "The printed question assumes convergence only in (i). Do not infer convergence of every sequence from the one-step inequalities." }, { "id": "13-12", "topic": "13", "number": 12, "questionPages": [ 4 ], "solutionPages": [ 11, 12 ], "attribution": "[2011/YJC/Prelim/I/1]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "12\n\n\n\n\n x 3 x 3\n The diagram shows the graph of y = x\n − x . The two roots of the equation x − x = 0\n 2 4 2 4\n are 0 and  .\n (i) Find the exact value of  . [2]\n\n A sequence of positive real numbers x1 , x2 , x3 ,  satisfies the recurrence relation\n x 1\n xn+1 = xn + xn , for n  1 .\n 2 n\n 4\n (ii) By considering xn +1 − xn , show that xn+1  xn if 0  xn   . [2]\n (iii) For the case 0  x1   , use the result in (i) and (ii) to find the exact value that the\n sequence converges to. [3]\n [2011/YJC/Prelim/I/1]", "methods": [ "Fixed-point factorisation", "Invariant interval", "Monotone convergence" ], "prerequisites": [ "Exponentials", "Logarithms", "Differentiation" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-13.md", "parts": [ "i", "ii", "iii" ], "warning": "The denominator is 2 raised to $x_n$. Establishing an upper bound is necessary; increasing alone does not imply convergence." }, { "id": "13-13", "topic": "13", "number": 13, "questionPages": [ 4 ], "solutionPages": [ 12, 13 ], "attribution": "[2014/TJC/Promo/7]", "candidatePartLabels": [ "i", "ii", "iii.a", "iii.b", "iv" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "13 The diagram shows the graph of y = x5 + 2 x − 8 . The root of the equation x5 +2x − 8= 0 is\n denoted by .\n\n\n\n\n (i) Find the value of  correct to 3 decimal places. [1]\n 1\n The real numbers xn satisfy the recurrence relation xn+1 = (8 − 2 xn )5 for n  1 .\n (ii) Using the result in (i), show that if the sequence converges, it will converge to . [2]\n 1\n (iii) By considering the graphs of y = x and y = (8 − 2 x) 5 on the same diagram, or\n otherwise, prove that if   xn  4 , then\n (a) 0  xn+1   ,\n (b)   xn+2  xn . [3]\n (iv) It is given that x1 = 3. Use the results in part (iii) to obtain an inequality relating 0, ,\n x1, x2 and x3. With the help of a graphic calculator, describe the behaviour of the\n sequence. [2]\n [2014/TJC/Promo/7]\n 100\n\f2025 TJC H2 Math (9758) JC2 MSM: RECURRENCE RELATIONS", "methods": [ "Numerical root", "Fixed-point equation", "Two-step bound" ], "prerequisites": [ "Real fifth roots", "Monotonicity", "Convergence" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-13.md", "parts": [ "i", "ii", "iii.a", "iii.b", "iv" ], "warning": "" }, { "id": "14-01", "topic": "14", "number": 1, "questionPages": [ 1 ], "solutionPages": [ 1 ], "attribution": "[2015/TJC/MCT/1]", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "1 Find the set of values of a for which  ( 2r + a )  a 2 . [5]\n r =3\n [2015/TJC/MCT/1]\n\n n ( n + 1)( 2n + 1)\n n", "methods": [ "AP summation", "Quadratic inequality" ], "prerequisites": [ "Sigma bounds", "Integer restrictions" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-14.md", "parts": [], "warning": "Do not keep the algebraic branch a<−2: it is incompatible with the stated finite-sum indexing." }, { "id": "14-02", "topic": "14", "number": 2, "questionPages": [ 1 ], "solutionPages": [ 1, 2 ], "attribution": "[2024/MI/Prelim/I/9b]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "2 It is given that r =\n r =1\n 2\n 6\n .\n\n n\n (\n n 4n 2 − 1 ).\n (i) Show that  ( 2r −1) =\n r =1\n 2\n 3\n [3]\n\n\n (ii) Hence find an expression for 132 + 152 + ... + ( 4m − 1) in terms of m.\n 2\n [4]\n [2024/MI/Prelim/I/9b]\n\n ( )\n n n\n n n\n Given that  r = (n + 1)(2n + 1), show that  (2r + 1)(2r + 3) = 4n + 18n + 23 .\n 2 2", "methods": [ "Expand polynomial", "Partial-sum subtraction" ], "prerequisites": [ "Sum of integers", "Sum of squares" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-14.md", "parts": [ "i", "ii" ], "warning": "" }, { "id": "14-03", "topic": "14", "number": 3, "questionPages": [ 1 ], "solutionPages": [ 3 ], "attribution": "[2023/DHS/Prelim/II/2]", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "3\n r =1 6 r =1 3\n Hence find the exact sum of the 25th to the 75th term of the series. [5]\n [2023/DHS/Prelim/II/2]\n\n  1 r +1\n k ", "methods": [ "Polynomial expansion", "Partial-sum subtraction" ], "prerequisites": [ "Standard finite sums" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-14.md", "parts": [], "warning": "The lower endpoint is included: subtract S_24, not S_25." }, { "id": "14-04", "topic": "14", "number": 4, "questionPages": [ 1 ], "solutionPages": [ 3, 4 ], "attribution": "[2022/HCI/Prelim/II/4a]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "4 (i) Find   −  + ln ( r + 1)  in terms of k. Simplify your answer. [4]\n  2 \n r =1  \n   r +1\n \n  1\n (ii) Hence determine if   −  + ln ( r + 1)  exists. [1]\n  2 \n r =1  \n [2022/HCI/Prelim/II/4a]\n n sin ( 2n + 1)  − sin \n Given that  cos 2r =", "methods": [ "GP summation", "Logarithm product", "Divergence" ], "prerequisites": [ "Factorials", "Convergence" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-14.md", "parts": [ "i", "ii" ], "warning": "" }, { "id": "14-05", "topic": "14", "number": 5, "questionPages": [ 1 ], "solutionPages": [ 4 ], "attribution": "[modified 2024/RI/Prelim/I/5]", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "5\n , where 0     ,\n r =1 2sin \n show that the sum of the series\n sin 2 10 + sin 2 11 + sin 2 12 + ... + sin 2 20 , for 0    \n sin(41 ) − sin(19 )\n is k − , where k is a constant to be determined. [4]\n 4sin \n [modified 2024/RI/Prelim/I/5]\n  n −1 ", "methods": [ "Double-angle identity", "Partial-sum subtraction" ], "prerequisites": [ "Trigonometric identities", "Sigma indexing" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-14.md", "parts": [], "warning": "" }, { "id": "14-06", "topic": "14", "number": 6, "questionPages": [ 1 ], "solutionPages": [ 5, 6 ], "attribution": "[2023/MI/Prelim/II/3]", "candidatePartLabels": [ "a", "b.i", "b.ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "6 (a) A sequence u2 , u3 , u4 ,... is such that un = ln   where n  2 .\n  n +1\n Find in terms of N, an expression for S N , where S N = u2 + u3 + u4 ... + uN −1 + uN ,\n leaving your answer in the form ln a − f ( N ) , where a is a positive constant. [3]\n\n  n\n (b) (i) Find the set of values of x for which the series  \n 3x \n  has a finite sum. [3]\n n =1  x + 1 \n\n  n\n (ii) Given that x = 0.25, find the value of  \n 3x \n  . [2]\n n =1  x + 1 \n\n [2023/MI/Prelim/II/3]\n\n 102\n\f2025 TJC H2 Math (9758) JC2 MSM: SIGMA NOTATION\n\n n\n r+2 1 1 \n r+2", "methods": [ "Log product cancellation", "GP convergence", "Rational inequality" ], "prerequisites": [ "Log laws", "Quadratic inequalities" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-14.md", "parts": [ "a", "b.i", "b.ii" ], "warning": "" }, { "id": "14-07", "topic": "14", "number": 7, "questionPages": [ 2 ], "solutionPages": [ 6, 7 ], "attribution": "[modified 2008/TPJC/Prelim/I/7]", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "7 Given that \n r =1 2\n r +1\n = −\n r (r + 1) 2 (n + 1)2 n +1\n , evaluate \n r =5 2\n r +1\n r (r + 1)\n , leaving your\n\n p\n answer in the form of where p and q are real. [4]\n q\n [modified 2008/TPJC/Prelim/I/7]\n r + 3r + 1 3\n n 2\n 1 1", "methods": [ "Tail subtraction", "Limit of partial sums" ], "prerequisites": [ "Geometric decay", "Sigma bounds" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-14.md", "parts": [], "warning": "" }, { "id": "14-08", "topic": "14", "number": 8, "questionPages": [ 2 ], "solutionPages": [ 7 ], "attribution": "[modified 2024/EJC/Prelim/I/6]", "candidatePartLabels": [ "a", "b" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "8 It is given that  = − − .\n r =1 ( r + 2)! 2 ( n + 1)! (n + 2)!\n \n r 2 + 3r + 1\n (a) Explain why  is convergent, and state the limit. [2]\n r =1 ( r + 2)!\n \n r 2 − r −1\n (b) Find the exact value of  . [3]\n r =6 r!\n [modified 2024/EJC/Prelim/I/6]\n n\n 4 2 2", "methods": [ "Factorial limit", "Index shift", "Tail subtraction" ], "prerequisites": [ "Factorials", "Partial sums" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-14.md", "parts": [ "a", "b" ], "warning": "" }, { "id": "14-09", "topic": "14", "number": 9, "questionPages": [ 2 ], "solutionPages": [ 8 ], "attribution": "[modified 2020/ASRJC/Prelim/I/5]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "9 The sum  is denoted by Sn. Given that Sn = − ,\n r =1 (2r + 3)(2r + 5) 5 2n + 5\n (i) find the smallest value of n for which Sn is within 10−3 of the sum to infinity, [3]\n 2n\n 2\n (ii) find  in terms of n. [3]\n r =10 4r\n 2\n −1\n [modified 2020/ASRJC/Prelim/I/5]\n N \n 3n  1   1  3N +1 ", "methods": [ "Tail-error inequality", "Reindexing", "Telescoping fractions" ], "prerequisites": [ "Absolute error", "Integer thresholds" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-14.md", "parts": [ "i", "ii" ], "warning": "" }, { "id": "14-10", "topic": "14", "number": 10, "questionPages": [ 2 ], "solutionPages": [ 8, 9 ], "attribution": "[modified 2020/ACJC/Prelim/I/5]", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "10 Given that    1 −   =  − 3  ,\n n =1  n + 1  2n   2  N + 1 \n N\n 3n (2n + 1)  3N + 2 \n deduce  , leaving your answer in the form a  − b  , where a and b\n n =1 ( n + 1)( n + 2)  N +2 \n are constants to be determined. [3]\n [modified 2020/ACJC/Prelim/I/5]\n 2500\n 1 10001 − 1", "methods": [ "Reindexing", "Partial-sum subtraction" ], "prerequisites": [ "Exponent rules", "Sigma bounds" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-14.md", "parts": [], "warning": "The bases are powers 3^n, not the product 3n; extracted text loses superscripts." }, { "id": "14-11", "topic": "14", "number": 11, "questionPages": [ 2 ], "solutionPages": [ 9 ], "attribution": "[modified 2012/ACJC/Prelim/I/10]", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "11 Given that  4r − 3 + 4r + 1 =\n r =1 4\n ,\n\n 2500\n 1\n hence show, without using a calculator, that   24 . [3]\n r =0 4r + 1 + 4r + 5\n [modified 2012/ACJC/Prelim/I/10]\n ", "methods": [ "Index shift", "Lower bound", "Rationalisation" ], "prerequisites": [ "Square roots", "Inequalities" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-14.md", "parts": [], "warning": "There are 2501 terms in the requested sum because its lower index is zero." }, { "id": "14-12", "topic": "14", "number": 12, "questionPages": [ 2 ], "solutionPages": [ 10 ], "attribution": "[2023/ACJC/Prelim/II/4b]", "candidatePartLabels": [], "partInventoryStatus": "visually checked against rendered question PDF", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased; original attribution and pages retained in source provenance", "mathematicalReview": "reviewed", "difficulty": "not yet assessed", "extractedText": "12 The root test is a test for convergence of infinite series of the general form  ar . The\n r =0\n\n test states that the series converges when lim n an  1 , and diverges when\n n→\n\n lim n an  1 . When lim n an = 1 , the test is inconclusive. Use the test to explain why\n n→ n→\n r\n  \n −3r + rx  3\n the series    converges for all values of x. [3]\n r =0  5r + 7 \n 3\n\n [2023/ACJC/Prelim/II/4b]\n\n\n\n 103\n\f2025 TJC H2 Math (9758) JC2 MSM: SIGMA NOTATION", "methods": [ "Root test", "Dominant powers" ], "prerequisites": [ "Absolute values", "Limits of rational expressions" ], "classificationStatus": "reviewed against rendered question pages and independently checked working", "reviewEvidence": "evidence/review-14.md", "parts": [], "warning": "Keep the absolute value in the root test: the underlying ratio tends to −3/5, but the root-test limit is positive 3/5." }, { "id": "15-01", "topic": "15", "number": 1, "questionPages": [ 1 ], "solutionPages": [ 1 ], "attribution": "[2017/MJC/Prelim/I/3]", "candidatePartLabels": [ "main" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "1 Do not use a calculator in answering this question.\n Showing your working, find the complex numbers z and w which satisfy the simultaneous\n equations\n 4iz − 3w = 1 + 5i and\n 2 z + (1 + i ) w = 2 + 6i. [5]\n [2017/MJC/Prelim/I/3]", "methods": [ "Complex elimination", "Conjugate rationalisation" ], "prerequisites": [ "Complex arithmetic", "Linear equations" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 1, solution PDF pages 1. Independent computations and qualifications are documented in that report.", "parts": [ "main" ], "warning": "" }, { "id": "15-02", "topic": "15", "number": 2, "questionPages": [ 1 ], "solutionPages": [ 1, 2 ], "attribution": "[2024/ACJC/Prelim/I/1]", "candidatePartLabels": [ "main" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "2 The complex numbers z and w satisfy the following equations.\n iw2 + 2 wz = 2i\n z + iz = 2 + iw\n Find z and w, giving your answers in the form of a + ib where a and b are real numbers.\n [4]\n [2024/ACJC/Prelim/I/1]", "methods": [ "Substitution", "Complex quadratic formula" ], "prerequisites": [ "Complex arithmetic", "Quadratic formula" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 1, solution PDF pages 1,2. Independent computations and qualifications are documented in that report.", "parts": [ "main" ], "warning": "" }, { "id": "15-03", "topic": "15", "number": 3, "questionPages": [ 1 ], "solutionPages": [ 2, 3 ], "attribution": "[2015/JJC/Prelim/II/1]", "candidatePartLabels": [ "a.i", "a.ii", "b" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "3 (a) Given that arg ( a + ib ) =  , where a and b are real and positive, find in terms of \n and/or  , the value of\n (i) arg ( a − ib ) ,\n (ii) arg ( b + ia ) . [3]\n (b) Solve the simultaneous equations\n iz + 2w = 0 and z − w* = 3,\n giving z and w in the form a + ib, where a and b are real. [5]\n [2015/JJC/Prelim/II/1]", "methods": [ "Argand reflection", "Compare real and imaginary parts" ], "prerequisites": [ "Conjugation", "Quadrants" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 1, solution PDF pages 2,3. Independent computations and qualifications are documented in that report.", "parts": [ "a.i", "a.ii", "b" ], "warning": "" }, { "id": "15-04", "topic": "15", "number": 4, "questionPages": [ 1 ], "solutionPages": [ 3 ], "attribution": "[modified 2023/HCI/Prelim/II/3a]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "4 A complex number z is given by z = a + (a − 3)i, where a is a positive real constant. It\n π\n is given that arg z =  , where −   0.\n 2\n Find, leaving your answers in terms of ,\n (i) arg(−2 z ), [2]\n (ii) arg( z − 2a). [2]\n [modified 2023/HCI/Prelim/II/3a]", "methods": [ "Rotation by a negative scalar", "Quadrant correction" ], "prerequisites": [ "Principal arguments", "Argand reflection" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 1, solution PDF pages 3. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii" ], "warning": "An inverse tangent alone does not identify the quadrant." }, { "id": "15-05", "topic": "15", "number": 5, "questionPages": [ 1 ], "solutionPages": [ 4 ], "attribution": "[2024/RVHS/Prelim/I/5b]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "5 (i) The roots of the equation w = 4i are w1 and w2 . Find w1 and w2 in cartesian form\n 2\n\n\n x + iy , showing your working. [3]\n (ii) Hence, or otherwise, find in exact cartesian form the roots v1 and v2 of the equation\n v 2 − 10v + ( 25 − i ) = 0 . [3]\n [2024/RVHS/Prelim/I/5b]", "methods": [ "Compare real and imaginary parts", "Completing the square" ], "prerequisites": [ "Complex multiplication", "Quadratic roots" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 1, solution PDF pages 4. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii" ], "warning": "" }, { "id": "15-06", "topic": "15", "number": 6, "questionPages": [ 1 ], "solutionPages": [ 5, 6 ], "attribution": "[2023/CJC/Prelim/I/1]", "candidatePartLabels": [ "main" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "6 The complex numbers z and w satisfy the following equations.\n 2z +1 = w\n 2w − z = 4 + 24i\n Find z and w, giving your answers in the form a + ib where a and b are real numbers.\n [4]\n [2023/CJC/Prelim/I/1]\n\n\n 105\n\f2025 TJC H2 Math (9758) JC2 MSM: COMPLEX NUMBERS", "methods": [ "Modulus equation", "Component comparison", "Reject extraneous squared root" ], "prerequisites": [ "Modulus", "Real and imaginary parts" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 1, solution PDF pages 5,6. Independent computations and qualifications are documented in that report.", "parts": [ "main" ], "warning": "The printed first equation contains modulus bars; plain extraction omits them. Squaring creates a rejected candidate." }, { "id": "15-07", "topic": "15", "number": 7, "questionPages": [ 2 ], "solutionPages": [ 7, 8 ], "attribution": "[2022/VJC/Promo/9]", "candidatePartLabels": [ "i", "ii", "iii", "iv" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "7 The complex number z satisfies the equation\n z 2 + (2 − 4i) z = c,\n where c is a complex constant.\n (i) Given that z1 and z2 are two distinct complex roots of the above equation, show that\n z1 + z2\n = −1 + 2i. [2]\n 2\n It is given that c = 4i and lm ( z1 )  lm ( z2 ) .\n (ii) Find z1 and z2 in the form x + iy, where x and y are real numbers. Leave the values\n of x and y in exact form. [3]\n (iii) On an Argand diagram, indicate clearly the points P1 and P2 which represent z1 and\n z2 respectively. [2]\n (iv) Point O and Point Q represent complex number 0 and w respectively. It is given that\n OPP1 2Q is a parallelogram. Using diagram in part (iii), find w. [2]\n [2022/VJC/Promo/9]", "methods": [ "Vieta root sum", "Complex quadratic formula", "Vector addition" ], "prerequisites": [ "Quadratic coefficients", "Argand coordinates" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 2, solution PDF pages 7,8. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii", "iv" ], "warning": "" }, { "id": "15-08", "topic": "15", "number": 8, "questionPages": [ 2 ], "solutionPages": [ 8, 9 ], "attribution": "[2023/CJC/Prelim/I/4 modified]", "candidatePartLabels": [ "a", "b", "c", "d.i", "d.ii" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "8 It is given that f ( x) is a cubic polynomial with real coefficients. The diagram shows the\n curve with equation y = f ( x).\n y\n\n y = f ( x)\n\n\n x\n O\n\n\n\n\n (a) What can be said about all the roots of the equation f ( x) = 0? [2]\n\n Suppose f ( x) = 2 x3 − 7 x 2 + 16 x + c , where c is a real number.\n (b) Show that c = −15 , if the equation f ( x) = 0 has a root x = 1 − 2i. [1]\n (c) Without using a calculator, determine the other roots of the equation f ( x) = 0. [3]\n (d) Hence, find the roots of the equation\n (i) −15w3 + 16w2 − 7 w + 2 = 0, [3]\n (ii) 2iw − 7 w − 16iw + 15 = 0.\n 3 2\n [2]\n [2023/CJC/Prelim/I/4 modified]", "methods": [ "Conjugate root theorem", "Factorisation", "Reciprocal substitution", "Rotation substitution" ], "prerequisites": [ "Polynomial roots", "Complex conjugates" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 2, solution PDF pages 8,9. Independent computations and qualifications are documented in that report.", "parts": [ "a", "b", "c", "d.i", "d.ii" ], "warning": "" }, { "id": "15-09", "topic": "15", "number": 9, "questionPages": [ 2 ], "solutionPages": [ 10, 11 ], "attribution": "[2020/VJC/Prelim/I/5]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "9 (i) One of the roots of the equation x − 3x + c = 0 , where c is real, is 2 − 3i .\n 3\n\n Without the use of a calculator, find the value of c and the other roots of the\n equation. [5]\n (ii) If c is a non-zero purely imaginary number, explain if it is possible for x − 3x + c = 0\n 3\n\n to have real roots. [1]\n (iii) It is given instead that c is a real number. By considering the graph of y = x − 3x ,\n 3\n\n\n find the range of values of c for which the equation x3 − 3x + c = 0 has only real\n roots. [2]\n [2020/VJC/Prelim/I/5]\n 106\n\f2025 TJC H2 Math (9758) JC2 MSM: COMPLEX NUMBERS", "methods": [ "Conjugate pair factorisation", "Imaginary-part argument", "Turning points" ], "prerequisites": [ "Cubic roots", "Differentiation", "Repeated roots" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 2, solution PDF pages 10,11. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "Three distinct real roots require strict inequalities; “only real roots” includes repeated roots." }, { "id": "15-10", "topic": "15", "number": 10, "questionPages": [ 3 ], "solutionPages": [ 11, 12, 13 ], "attribution": "[2017/IJC/Prelim/I/4]", "candidatePartLabels": [ "a", "b" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "10 A graphic calculator is not to be used in answering this question.\n (a) The equation w3 + pw2 + qw + 30 = 0 , where p and q are real constants, has a root\n w = 2 − i . Find the values of p and q, showing your working. [3]\n\n (b) The equation z 2 + (−5 + 2i) z + (21 − i) = 0 has a root z = 3 + ui , where u is real\n constant. Find the value of u and hence find the second root of the equation in\n cartesian form, a + bi , showing your working. [5]\n [2017/IJC/Prelim/I/4]", "methods": [ "Conjugate root factorisation", "Component comparison", "Vieta sum" ], "prerequisites": [ "Real-coefficient conjugacy", "Complex multiplication" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 3, solution PDF pages 11,12,13. Independent computations and qualifications are documented in that report.", "parts": [ "a", "b" ], "warning": "The conjugate of a root need not be a root of a polynomial with complex coefficients." }, { "id": "15-11", "topic": "15", "number": 11, "questionPages": [ 3 ], "solutionPages": [ 13, 14 ], "attribution": "[2018/RI/Promo/11a]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "11 (i) Solve the equation z 2 − (1 + 2i ) z + 1 + 7i = 0. [2]\n (ii) It is given that one of the roots of the equation\n\n z 3 − (1 + 4i ) z 2 − ( 3 − 9i ) z + (14 − 2i ) = 0\n\n is purely imaginary. Find all the roots of the equation, giving your answers in\n cartesian form a + ib . [6]\n [2018/RI/Promo/11a]", "methods": [ "Complex quadratic factorisation", "Pure imaginary substitution", "Factor theorem" ], "prerequisites": [ "Polynomial division", "Component equations" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 3, solution PDF pages 13,14. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii" ], "warning": "" }, { "id": "15-12", "topic": "15", "number": 12, "questionPages": [ 3 ], "solutionPages": [ 14, 15, 16 ], "attribution": "[2024/YIJC/Prelim/I/3]", "candidatePartLabels": [ "a", "b" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "12 Do not use a calculator in answering this question.\n (a) Given that f ( x) is a polynomial of degree 4 with real coefficients, explain whether\n it is possible for f ( x) = 0 to have 3 non-real roots and 1 real root. [1]\n (b) One of the roots of the equation 2 x4 − 15x3 + ax2 − 63x + b = 0 , where a and b are\n real, is 3 − 2i . Find the other roots of the equation and the values of a and b. [6]\n [2024/YIJC/Prelim/I/3]", "methods": [ "Conjugate root theorem", "Coefficient comparison", "Quadratic formula" ], "prerequisites": [ "Polynomial expansion", "Conjugate pairs" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 3, solution PDF pages 14,15,16. Independent computations and qualifications are documented in that report.", "parts": [ "a", "b" ], "warning": "" }, { "id": "15-13", "topic": "15", "number": 13, "questionPages": [ 3 ], "solutionPages": [ 17 ], "attribution": "[2024/DHS/Prelim/I/7]", "candidatePartLabels": [ "a", "b", "c" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed with source qualification; see warning and topic review evidence", "difficulty": "not yet assessed", "extractedText": "13 The points A, B and C represent the complex numbers a, b and c respectively, such that\n a = 0, b = 3 and c = −2 + i. The three complex numbers are roots to the equation\n f( z ) = 0 where f(z) is a quartic polynomial with real coefficients and z is a complex\n variable.\n (a) Express f(z) as a product of two quadratic factors with real coefficients. [3]\n (b) Sketch an Argand diagram showing the roots of the equation f( z ) = 0 . [2]\n (c) The point W represents the complex number w, such that c = i w. Find the value of\n AC AW and the area of the triangle APW where P represents the complex number\n c − b. [4]\n [2024/DHS/Prelim/I/7]\n π", "methods": [ "Conjugate factorisation", "Rotation by i", "Planar determinant area" ], "prerequisites": [ "Complex conjugation", "Dot product", "Triangle area" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 3, solution PDF pages 17. Independent computations and qualifications are documented in that report.", "parts": [ "a", "b", "c" ], "warning": "The printed polynomial assumes a monic quartic; the literal root data permit any nonzero real scalar multiple." }, { "id": "15-14", "topic": "15", "number": 14, "questionPages": [ 3 ], "solutionPages": [ 18 ], "attribution": "[modified 2019/NJC/Prelim/I/1]", "candidatePartLabels": [ "main" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "14 The complex number z has modulus r and argument  , where 0  r  1 and    π .\n 2\n The complex number w is such that w = z and arg( w) + arg( z ) = π .\n\n Let the points A, B, C, D and E represent the complex numbers z, w, z − w, z* and −iz\n respectively, where z* denotes the conjugate of z.\n On a single Argand diagram, illustrate these five points. [5]\n [modified 2019/NJC/Prelim/I/1]\n\n\n\n 107\n\f2025 TJC H2 Math (9758) JC2 MSM: COMPLEX NUMBERS", "methods": [ "Polar coordinates", "Reflection", "Rotation", "Vector difference" ], "prerequisites": [ "Sine and cosine", "Complex conjugation" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 3, solution PDF pages 18. Independent computations and qualifications are documented in that report.", "parts": [ "main" ], "warning": "" }, { "id": "15-15", "topic": "15", "number": 15, "questionPages": [ 4 ], "solutionPages": [ 18 ], "attribution": "[2020/VJC/Prelim/II/3a]", "candidatePartLabels": [ "main" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "15 The complex numbers w and z are such that w = −1 − i 3 and z * + z  0 . The points P\n and Q in the Argand diagram represent the complex numbers w and z * + z respectively.\n Find arg ( w ) , leaving your answer in exact form. Hence, state the possible values of angle\n POQ, where O is the origin. [3]\n [2020/VJC/Prelim/II/3a]", "methods": [ "Quadrant correction", "Conjugate sum" ], "prerequisites": [ "Principal argument", "Real axis" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 4, solution PDF pages 18. Independent computations and qualifications are documented in that report.", "parts": [ "main" ], "warning": "" }, { "id": "15-16", "topic": "15", "number": 16, "questionPages": [ 4 ], "solutionPages": [ 19 ], "attribution": "[2016/VJC/Prelim/II/3]", "candidatePartLabels": [ "main" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "16 The points A, B, C and D represent the complex numbers −2 + 5i , z1 , 4 + i and z2\n respectively. Given that ABCD is a square, labelled in an anti-clockwise direction, show\n that z1 = −1 . Find z2 . [4]\n [2016/VJC/Prelim/II/3]\n ", "methods": [ "Quarter-turn multiplication", "Diagonal midpoint" ], "prerequisites": [ "Complex rotation", "Square diagonals" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 4, solution PDF pages 19. Independent computations and qualifications are documented in that report.", "parts": [ "main" ], "warning": "" }, { "id": "15-17", "topic": "15", "number": 17, "questionPages": [ 4 ], "solutionPages": [ 19, 20 ], "attribution": "[2020/DHS/Prelim/I/3 modified]", "candidatePartLabels": [ "i", "ii" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "17 The complex number z is such that arg( z ) = − and the complex number w has modulus\n 3\n  \n of 2 and an argument of . It is also known that arg( w − z ) =\n . Let W and Z be the\n 4 3\n points representing the complex numbers w and z respectively.\n (i) By using an Argand diagram with reference from the origin O, show that\n \n angle OWZ = radians. [2]\n 12\n (ii) Hence or otherwise, find the modulus of z. [2]\n [2020/DHS/Prelim/I/3 modified]", "methods": [ "Directed arguments", "Sine rule", "Cartesian verification" ], "prerequisites": [ "Triangle angles", "Polar coordinates" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 4, solution PDF pages 19,20. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii" ], "warning": "The angle of w−z belongs to the vector ZW, not WZ." }, { "id": "15-18", "topic": "15", "number": 18, "questionPages": [ 4 ], "solutionPages": [ 20, 21 ], "attribution": "[2010/MJC/Prelim/I/11]", "candidatePartLabels": [ "shape", "identity" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "18 The complex numbers z and w are given by z = 3 + i and w = −1 + i 3 .\n Sketch an Argand diagram, with origin O, showing the points Z, W and P representing\n the complex numbers z, w and z + w respectively. Describe the geometrical shape of\n OWPZ. [3]\n 5\n By considering the argument of z + w , deduce that tan = 2+ 3 . [3]\n 12\n [2010/MJC/Prelim/I/11]", "methods": [ "Complex rotation", "Parallelogram addition", "Rationalise a tangent ratio" ], "prerequisites": [ "Argand coordinates", "Square diagonal" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 4, solution PDF pages 20,21. Independent computations and qualifications are documented in that report.", "parts": [ "shape", "identity" ], "warning": "" }, { "id": "15-19", "topic": "15", "number": 19, "questionPages": [ 4 ], "solutionPages": [ 22, 23 ], "attribution": "[modified 2013/TJC/Prelim/I/5]", "candidatePartLabels": [ "i", "ii", "iii" ], "partInventoryStatus": "visually checked against question PDF pages 1–4", "status": "authored", "transcription": "transcribed from rendered question PDF; wording paraphrased", "mathematicalReview": "reviewed: independently derived and compared with rendered question/answer pages", "difficulty": "not yet assessed", "extractedText": "19 A graphic calculator is not to be used in answering this question.\n p\n Two complex numbers p and q are given by p = 1 − i and q = −1 + 3i , and z = .\n q\n (i) Express z in the form x + yi, where x and y are exact real values to be determined.\n [2]\n z1 z z \n (ii) It is known that = 1 and arg  1  = arg z1 − arg z2 .\n z2 z2  z2 \n By considering the moduli and arguments of p and q, find the exact values of |z| and\n arg z, where −  arg z   . [4]\n\n  11  6− 2\n (iii) Hence, show that sin    = . [3]\n  12  4\n [modified 2013/TJC/Prelim/I/5]\n\n\n\n\n 108\n\f2025 TJC H2 Math (9758) JC2 MSM: COMPLEX NUMBERS", "methods": [ "Conjugate rationalisation", "Polar quotient", "Cartesian-to-polar comparison" ], "prerequisites": [ "Complex division", "Principal arguments", "Sine symmetry" ], "classificationStatus": "reviewed against Complex Numbers question pages", "reviewEvidence": "Full source/mathematical review recorded in evidence/review-15.md; visually audited question PDF pages 4, solution PDF pages 22,23. Independent computations and qualifications are documented in that report.", "parts": [ "i", "ii", "iii" ], "warning": "Argument subtraction may need reduction into the principal interval; here −11π/12 already lies there." } ], "counts": { "groups": 248, "authored": 248, "reviewed": 248, "subparts": 684, "pending": 0 } }