9709 Pure Mathematics 3 Oct/Nov 2025 Paper 31

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

1. Question 1

   4 marks · 4/75

   This assesses definite integration by parts when the integrand contains a logarithm.

   Question paper p.2 · Mark scheme p.9

2. Question 2 (a)

   3 marks · 7/75

   This assesses the use of logarithm laws to transform a logarithmic equation into a quadratic in the variable.

   Question paper p.3 · Mark scheme p.9

3. Question 2 (b)

   2 marks · 9/75

   This assesses solving the quadratic obtained from a logarithmic equation and selecting a value within the logarithms' domains.

   Question paper p.3 · Mark scheme p.9

4. Question 3 (a)

   4 marks · 13/75

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

   Question paper p.4 · Mark scheme p.10

5. Question 3 (b)

   4 marks · 17/75

   Question paper p.5 · Mark scheme p.10

6. Question 4

   5 marks · 22/75

   This assesses differentiation of a product involving exponential and trigonometric factors to locate a maximum.

   Question paper p.6 · Mark scheme p.11

7. Question 5 (a)

   3 marks · 25/75

   This assesses translating boundaries on an Argand diagram into inequalities involving a complex number.

   Question paper p.9 · Mark scheme p.12

8. Question 5 (b)

   3 marks · 28/75

   This assesses minimising the argument of points in a region bounded by a circle and a vertical line.

   Question paper p.9 · Mark scheme p.12

9. Question 6

   5 marks · 33/75

   This assesses solving a quadratic equation with complex coefficients and presenting both roots in Cartesian complex form.

   Question paper p.10 · Mark scheme p.13

10. Question 7

    5 marks · 38/75

    This assesses differentiation of parametrically defined variables.

    Question paper p.11 · Mark scheme p.15

11. Question 8

    8 marks · 46/75

    This assesses separation and integration of a first-order differential equation followed by use of two conditions to determine constants.

    Question paper p.12 · Mark scheme p.16

12. Question 9 (a)

    2 marks · 48/75

    This assesses using graphs to establish the number of roots of a trigonometric-exponential equation on a restricted interval.

    Question paper p.14 · Mark scheme p.17

13. Question 9 (b)

    2 marks · 50/75

    This assesses numerical bracketing of a root by testing the equation at two nearby endpoints.

    Question paper p.14 · Mark scheme p.17

14. Question 9 (c)

    1 mark · 51/75

    This assesses recognising that a convergent iteration must have a fixed point and relating that fixed point to the original equation.

    Question paper p.15 · Mark scheme p.18

15. Question 9 (d)

    3 marks · 54/75

    This assesses numerical fixed-point iteration to obtain a root to a specified accuracy.

    Question paper p.15 · Mark scheme p.18

16. Question 10 (a)

    6 marks · 60/75

    This assesses decomposition of a rational function into partial fractions with one linear and one irreducible quadratic denominator.

    Question paper p.16 · Mark scheme p.19

17. Question 10 (b)

    5 marks · 65/75

    This assesses combining partial fractions with binomial or geometric expansions to obtain a power series near zero.

    Question paper p.17 · Mark scheme p.20

18. Question 11 (a)

    2 marks · 67/75

    This assesses forming a vector equation for a line through two points in three-dimensional space.

    Question paper p.18 · Mark scheme p.21

19. Question 11 (b)

    4 marks · 71/75

    This assesses finding the intersection of two three-dimensional lines using component equations.

    Question paper p.18 · Mark scheme p.21

20. Question 11 (c)

    4 marks · 75/75

    Question paper p.19 · Mark scheme p.22