9709 Pure Mathematics 3 Oct/Nov 2025 Paper 32

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

1. Question 1 (a)

   1 mark · 1/75

   Sketching a translated absolute-value graph using its vertex and linear branches.

   Question paper p.2 · Mark scheme p.9

2. Question 1 (b)

   2 marks · 3/75

   Solving an inequality involving an absolute value by considering the relevant sign cases.

   Question paper p.2 · Mark scheme p.9

3. Question 2

   4 marks · 7/75

   Solving an exponential equation by applying index and logarithm laws.

   Question paper p.3 · Mark scheme p.10

4. Question 3 (a)

   2 marks · 9/75

   Using an Argand diagram and the equation of a circle to find an exact complex intersection point.

   Question paper p.4 · Mark scheme p.11

5. Question 3 (b)

   3 marks · 12/75

   Finding the largest argument of points in a circular region on an Argand diagram.

   Question paper p.4 · Mark scheme p.12

6. Question 4

   6 marks · 18/75

   Question paper p.5 · Mark scheme p.13

7. Question 5 (a)

   2 marks · 20/75

   Showing that a repeated polynomial factor remains present after differentiation.

   Question paper p.6 · Mark scheme p.14

8. Question 5 (b)

   5 marks · 25/75

   Determining unknown coefficients when a polynomial has a repeated linear factor.

   Question paper p.6 · Mark scheme p.15

9. Question 6 (a)

   2 marks · 27/75

   Using graphs to establish the number of roots of a trigonometric equation on a specified interval.

   Question paper p.7 · Mark scheme p.18

10. Question 6 (b)

    2 marks · 29/75

    Locating a trigonometric root within a narrower interval by checking signs.

    Question paper p.7 · Mark scheme p.18

11. Question 6 (c)

    3 marks · 32/75

    Using fixed-point iteration to approximate a trigonometric root.

    Question paper p.7 · Mark scheme p.19

12. Question 7 (a)

    4 marks · 36/75

    Implicitly differentiating an equation containing powers and products of x and y.

    Question paper p.8 · Mark scheme p.20

13. Question 7 (b)

    4 marks · 40/75

    Question paper p.9 · Mark scheme p.20

14. Question 8 (a)

    3 marks · 43/75

    Question paper p.10 · Mark scheme p.21

15. Question 8 (b)

    5 marks · 48/75

    Question paper p.11 · Mark scheme p.22

16. Question 9 (a)

    5 marks · 53/75

    Decomposing a rational function into partial fractions with parameter-dependent coefficients.

    Question paper p.12 · Mark scheme p.23

17. Question 9 (b)

    4 marks · 57/75

    Integrating a partial-fraction decomposition and simplifying the exact result over parameter-dependent limits.

    Question paper p.13 · Mark scheme p.24

18. Question 10 (a)

    4 marks · 61/75

    Forming and simplifying a differential equation from volume flow rates and a measured rate of change.

    Question paper p.14 · Mark scheme p.25

19. Question 10 (b)

    5 marks · 66/75

    Solving a separable differential equation for the time needed to reach a specified depth.

    Question paper p.15 · Mark scheme p.26

20. Question 11 (a)

    2 marks · 68/75

    Finding vectors from the midpoint of a line segment to two three-dimensional points.

    Question paper p.16 · Mark scheme p.27

21. Question 11 (b)

    3 marks · 71/75

    Finding the exact cosine of a three-dimensional angle using vectors.

    Question paper p.16 · Mark scheme p.27

22. Question 11 (c)

    4 marks · 75/75

    Question paper p.17 · Mark scheme p.29