9709 Pure Mathematics 3 Feb/Mar 2025 Paper 32

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

1. Question 1

   3 marks · 3/75

   Question paper p.3 · Mark scheme p.9

2. Question 2

   5 marks · 8/75

   Finding a gradient from an implicitly defined curve by differentiating products and composite logarithmic expressions.

   Examiner report: Candidates commonly handled the product term better than the logarithm, whose chain-rule derivative was a frequent source of error. Further marks were also lost when isolating dy/dx.

   Question paper p.4 · Mark scheme p.10

3. Question 3 (a)

   3 marks · 11/75

   Translating a shaded Argand-plane region bounded by a horizontal line and a circle into inequalities involving a complex number.

   Examiner report: This was difficult for many candidates: some found the circular boundary but omitted the second condition, used an equation, or confused the imaginary part with an imaginary number.

   Question paper p.6 · Mark scheme p.11

4. Question 3 (b)

   3 marks · 14/75

   Maximising the modulus of a complex number over a bounded region in the Argand plane.

   Examiner report: Only a small proportion obtained complete solutions. Successful approaches generally located the appropriate extreme point first and then evaluated its distance, while many made no meaningful attempt.

   Question paper p.6 · Mark scheme p.11

5. Question 4

   6 marks · 20/75

   Solving a trigonometric equation in a specified degree interval by converting it into a quadratic in tan x.

   Examiner report: Most candidates used the intended quadratic method, although some began with an incorrect tangent expansion or reciprocal identity. Others lost the final mark by giving an incorrect or excessive set of angles.

   Question paper p.7 · Mark scheme p.12

6. Question 5

   5 marks · 25/75

   Finding exact Cartesian square roots of a complex number by equating real and imaginary parts after squaring.

   Examiner report: This was generally well answered. The main incomplete responses either supplied only one root or paired the signs of the real and imaginary parts incorrectly.

   Question paper p.8 · Mark scheme p.13

7. Question 6

   7 marks · 32/75

   Solving a first-order separable differential equation involving sin² of a multiple angle and an initial condition.

   Examiner report: Candidates who knew how to integrate the squared trigonometric factor usually completed the problem. Others separated the variables successfully but could not integrate that factor, and some failed to simplify their final expression fully.

   Question paper p.9 · Mark scheme p.14

8. Question 7 (a)

   3 marks · 35/75

   Deriving a fixed-point equation for the stationary point of a product involving a power of x and cos 2x.

   Examiner report: Most candidates differentiated correctly, but a few substituted p into the curve before differentiating. The final rearrangement needed a clear conversion to an equation involving tan 2x.

   Question paper p.10 · Mark scheme p.15

9. Question 7 (b)

   2 marks · 37/75

   Establishing that a root lies between two decimal bounds by numerical sign or endpoint comparison.

   Examiner report: This part was usually handled effectively. Candidates used either opposite signs of a residual or direct comparisons with the iteration function; both approaches were acceptable when the numerical evidence supported the conclusion.

   Question paper p.11 · Mark scheme p.15

10. Question 7 (c)

    3 marks · 40/75

    Using fixed-point iteration to approximate the stationary-point parameter to a specified accuracy.

    Examiner report: This was one of the strongest parts of the paper. The main problems were using an incorrect iteration formula, working in degrees instead of radians, or failing to display the required precision.

    Question paper p.11 · Mark scheme p.16

11. Question 8 (a)

    5 marks · 45/75

    Demonstrating that two three-dimensional parametric lines are skew rather than intersecting or being parallel.

    Examiner report: Many candidates knew both conditions required, but arithmetic mistakes weakened intersection checks. Explanations of non-parallelism were often incomplete because the direction vectors were not explicitly identified or compared as scalar multiples.

    Question paper p.12 · Mark scheme p.17

12. Question 8 (b)

    3 marks · 48/75

    Question paper p.13 · Mark scheme p.18

13. Question 9 (a)

    5 marks · 53/75

    Determining unknown polynomial coefficients from a known factor and a specified remainder on division of the derivative.

    Examiner report: The approach was usually understood, but arithmetic slips were common. Direct substitution into the polynomial and its derivative tended to be more reliable than using remainder arguments through polynomial division.

    Question paper p.14 · Mark scheme p.18

14. Question 9 (b)

    3 marks · 56/75

    Completing the factorisation of a cubic polynomial after one linear factor and its coefficients are known.

    Examiner report: Some candidates appeared to rely on a calculator and gave unsupported roots. Others found the quadratic factor but stopped before resolving it into linear factors.

    Question paper p.15 · Mark scheme p.19

15. Question 9 (c)

    2 marks · 58/75

    Solving a cubic polynomial inequality from its linear factorisation and the sign changes at its roots.

    Examiner report: Fully correct responses were uncommon, partly because this depended on the preceding factorisation. A sign diagram or a sketch of the cubic would have helped candidates select the correct intervals.

    Question paper p.15 · Mark scheme p.19

16. Question 10 (a)

    5 marks · 63/75

    Decomposing a rational function with one linear and one irreducible quadratic denominator into partial fractions.

    Examiner report: The standard decomposition was familiar to most candidates, but coefficient arithmetic caused repeated errors. Those who chose the correct numerator form generally progressed well.

    Question paper p.16 · Mark scheme p.20

17. Question 10 (b)

    6 marks · 69/75

    Evaluating a definite rational-function integral exactly after partial-fraction decomposition.

    Examiner report: The linear-denominator term was usually integrated correctly, whereas the quadratic term was often not split into two manageable integrals. Some candidates also left an added logarithm instead of rewriting the result as a subtraction, or combined logarithms incorrectly.

    Question paper p.17 · Mark scheme p.21

18. Question 11

    6 marks · 75/75

    Question paper p.18 · Mark scheme p.22