9709 Pure Mathematics 3 May/June 2025 Paper 32

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

1. Question 1

   5 marks · 5/75

   Solving an exponential equation by converting it into a quadratic in a suitable exponential expression and applying a numerical accuracy requirement.

   Question paper p.3 · Mark scheme p.9

2. Question 2 (a)

   4 marks · 9/75

   Using the binomial expansion for a fractional negative power and multiplying the resulting series by a linear factor.

   Question paper p.4 · Mark scheme p.10

3. Question 2 (b)

   1 mark · 10/75

   Identifying the interval of convergence for a binomial series.

   Question paper p.4 · Mark scheme p.10

4. Question 3

   5 marks · 15/75

   Representing simultaneous modulus and argument inequalities on an Argand diagram.

   Question paper p.5 · Mark scheme p.11

5. Question 4

   6 marks · 21/75

   Question paper p.6 · Mark scheme p.12

6. Question 5

   5 marks · 26/75

   Finding complex square roots in Cartesian form by equating real and imaginary parts.

   Question paper p.7 · Mark scheme p.13

7. Question 6 (a)

   2 marks · 28/75

   Using graphical comparison to establish the number of roots of an equation involving a modulus and a sine function.

   Question paper p.8 · Mark scheme p.14

8. Question 6 (b)

   2 marks · 30/75

   Bracketing a root numerically using a sign change or equivalent comparison at two endpoints.

   Question paper p.8 · Mark scheme p.15

9. Question 6 (c)

   3 marks · 33/75

   Applying fixed-point iteration to approximate a root to a prescribed decimal accuracy.

   Question paper p.9 · Mark scheme p.16

10. Question 7 (a)

    3 marks · 36/75

    Rewriting a linear combination of sine and cosine as a single shifted cosine.

    Question paper p.10 · Mark scheme p.17

11. Question 7 (b)

    4 marks · 40/75

    Solving a trigonometric equation after applying an amplitude-phase transformation and accounting for all inverse-cosine branches.

    Question paper p.11 · Mark scheme p.17

12. Question 8

    7 marks · 47/75

    Solving a first-order separable differential equation using an initial condition.

    Question paper p.12 · Mark scheme p.18

13. Question 9 (a)

    2 marks · 49/75

    Forming a vector equation for the line through two points in three-dimensional space.

    Question paper p.14 · Mark scheme p.19

14. Question 9 (b)

    4 marks · 53/75

    Finding the cosine of a three-dimensional angle using the scalar-product formula.

    Question paper p.14 · Mark scheme p.19

15. Question 9 (c)

    3 marks · 56/75

    Question paper p.15 · Mark scheme p.20

16. Question 10 (a)

    2 marks · 58/75

    Dividing one polynomial by another and identifying the quotient and remainder.

    Question paper p.16 · Mark scheme p.21

17. Question 10 (b)

    6 marks · 64/75

    Evaluating a definite integral involving an inverse tangent by integration by parts and a rational integral.

    Question paper p.16 · Mark scheme p.22

18. Question 11 (a)

    6 marks · 70/75

    Finding the exact horizontal coordinate of a maximum of a trigonometric curve by differentiation.

    Question paper p.18 · Mark scheme p.23

19. Question 11 (b)

    5 marks · 75/75

    Finding a trigonometric area by making the specified substitution in a definite integral.

    Question paper p.19 · Mark scheme p.25