9709 Pure Mathematics 3 Oct/Nov 2022 Paper 32

21 questions and parts, 75 marks in total. Marks come from the published mark scheme.

1. Question 1

   4 marks · 4/75

   This assesses the use of logarithm or index laws to solve an exponential equation and present the result in a specified logarithmic form.

   Examiner report: Many candidates knew the relevant logarithm rules, but brackets around an exponent were often omitted. Other frequent problems were mishandling the product involving 5 and 3, or making sign errors while collecting terms. The index-based route was also sometimes left unsimplified.

   Question paper p.2 · Mark scheme p.6

2. Question 2 (a)

   2 marks · 6/75

   This assesses the factor theorem for finding an unknown constant in a polynomial.

   Examiner report: Most candidates used the factor theorem successfully, although substitution signs and arithmetic caused avoidable losses. Some used the reciprocal of the required root instead of the root itself.

   Question paper p.3 · Mark scheme p.7

3. Question 2 (b)

   4 marks · 10/75

   This assesses factorising a cubic using a known factor and solving a polynomial inequality by sign analysis.

   Examiner report: A substantial number tried to apply a quadratic formula directly to the cubic or used an invalid rearrangement. Even after finding the quadratic factor, some assumed it had real zeros and did not justify the sign of the product.

   Question paper p.3 · Mark scheme p.8

4. Question 3

   6 marks · 16/75

   This assesses differentiation of a trigonometric product, use of double-angle identities, and numerical solution of the stationary-point condition.

   Examiner report: Recasting the curve before differentiating often led to a shorter solution. Product-rule and double-angle mistakes were common, followed by algebraic slips when solving the resulting equation. A few candidates reported an angle in degrees despite the radian interval.

   Question paper p.4 · Mark scheme p.9

5. Question 4 (a)

   3 marks · 19/75

   This assesses the representation of a linear combination of sine and cosine as a single shifted cosine.

   Examiner report: Many responses were correct, but errors often came from expanding the shifted angle with the wrong sign. Some treated a coefficient itself as the cosine of the shift or supplied an incorrectly rounded angle.

   Question paper p.5 · Mark scheme p.10

6. Question 4 (b)

   5 marks · 24/75

   This assesses the use of a compound-angle form to solve a trigonometric equation over a restricted degree interval.

   Examiner report: Most candidates made effective use of the preceding part. The main difficulty was tracking that the argument is 2x, rather than x, and incorporating the shift correctly; some also gave answers with more precision than their working supported.

   Question paper p.5 · Mark scheme p.10

7. Question 5 (a)

   3 marks · 27/75

   This assesses solving a quadratic equation whose variable is complex and expressing both roots in rectangular form.

   Examiner report: Rewriting the quadratic as a square was generally the most efficient approach. Errors included treating the imaginary coefficient as an ordinary real number and incorrectly assuming that the roots had to be conjugates.

   Question paper p.6 · Mark scheme p.11

8. Question 5 (b)

   1 mark · 28/75

   This assesses plotting complex numbers on an Argand diagram with a consistent scale.

   Examiner report: There were many accurate sketches, but some lacked a usable linear scale or used irregular numerical markings. A few diagrams failed to place the two points symmetrically in accordance with the roots found earlier.

   Question paper p.6 · Mark scheme p.11

9. Question 5 (c)

   3 marks · 31/75

   This assesses finding exact moduli and arguments of complex numbers while accounting for their quadrants.

   Examiner report: The modulus was usually handled well, whereas the pair of arguments was less reliable. In particular, candidates sometimes used the same acute reference angle for both points without adjusting for their different positions.

   Question paper p.7 · Mark scheme p.12

10. Question 5 (d)

    1 mark · 32/75

    This assesses geometric justification that the three plotted points form an equilateral triangle.

    Examiner report: Showing three equal sides was the clearest successful route. Unsupported claims based on only two equal sides were common, and some candidates asserted equal angles without demonstrating them.

    Question paper p.7 · Mark scheme p.12

11. Question 6 (a)

    4 marks · 36/75

    This assesses forming side vectors from position vectors and applying the scalar-product formula to an angle at a vertex.

    Examiner report: Many candidates knew the scalar-product method, but incorrect vector pairs were frequent. Some used position vectors directly or a vector involving BC, while others calculated the angle but did not explicitly state its cosine.

    Question paper p.8 · Mark scheme p.13

12. Question 6 (b)

    4 marks · 40/75

    This assesses obtaining an exact sine from a known cosine and using it in the area formula for a triangle.

    Examiner report: Although the standard area formula was often identified, many answers stopped at a decimal despite the requirement for exactness. The identity needed to convert the known cosine into an exact sine was not recalled by everyone.

    Question paper p.9 · Mark scheme p.14

13. Question 7 (a)

    1 mark · 41/75

    This assesses differentiating a squared trigonometric function by the chain rule.

    Examiner report: Some candidates unnecessarily began solving the full differential equation. The usual issue in direct attempts was losing the multiplier from differentiating the square, while quotient-rule solutions sometimes contained avoidable algebra errors.

    Question paper p.10 · Mark scheme p.15

14. Question 7 (b)

    7 marks · 48/75

    This assesses separation and integration of a first-order differential equation followed by use of an initial condition.

    Examiner report: Candidates who separated the variables correctly generally made useful progress. Sign errors occurred when integrating the cotangent-related term, and recognising the required forms involving secant and powers of sine was difficult for some. The initial condition had to be used before evaluating at the later angle.

    Question paper p.10 · Mark scheme p.16

15. Question 8 (a)

    1 mark · 49/75

    This assesses reading the positive nonzero x-intercept of a curve involving sin√x from the diagram and trigonometric zeros.

    Examiner report: The origin was often selected even though the marked intercept was elsewhere. Other mistakes arose from stopping after finding a value for the square root or mishandling the squaring step.

    Question paper p.12 · Mark scheme p.17

16. Question 8 (b)

    7 marks · 56/75

    This assesses integration by substitution followed by integration by parts for an exact area.

    Examiner report: The substitution method was generally understood, but the differential was frequently transformed incorrectly, producing an extra or missing factor of u. Some otherwise successful solutions gave decimal rather than exact results.

    Question paper p.12 · Mark scheme p.17

17. Question 9 (a)

    4 marks · 60/75

    This assesses deriving a circular-segment area equation from sector and triangle geometry.

    Examiner report: Recognising the central angle was the key geometric step. Some candidates reversed the subtraction or struggled to simplify the sine of the supplementary angle, leading to an incorrect form of the required equation.

    Question paper p.14 · Mark scheme p.18

18. Question 9 (b)

    2 marks · 62/75

    This assesses numerical verification that a root lies between two stated bounds.

    Examiner report: Both endpoints had to be tested and the numerical comparisons made explicit. Errors commonly came from using degrees, entering the formula incorrectly, or expecting the fixed-point form to produce a sign change when that is not necessary.

    Question paper p.15 · Mark scheme p.18

19. Question 9 (c)

    3 marks · 65/75

    This assesses fixed-point iteration and control of numerical accuracy.

    Examiner report: This was one of the stronger numerical parts of the paper. The main losses came from too few iterations, insufficient displayed precision, or omitting a clear concluding statement.

    Question paper p.15 · Mark scheme p.19

20. Question 10 (a)

    5 marks · 70/75

    This assesses decomposing a rational function into partial fractions with a linear factor and an irreducible quadratic factor.

    Examiner report: Many candidates chose the correct decomposition structure, but missing brackets changed the intended algebra in several scripts. Arithmetic mistakes then prevented some otherwise sound coefficient comparisons from being completed.

    Question paper p.16 · Mark scheme p.20

21. Question 10 (b)

    5 marks · 75/75

    This assesses integrating a partial-fraction decomposition and combining exact logarithmic terms.

    Examiner report: The first logarithmic integral was usually obtained, while the quadratic term caused more difficulty. Some candidates ignored its numerator or left an x-dependent factor attached to the logarithm; others made errors when combining the endpoint contributions.

    Question paper p.17 · Mark scheme p.20