9709 Pure Mathematics 3 May/June 2025 Paper 33

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

1. Question 1 (a)

   1 mark · 1/75

   This tests sketching the graph of an absolute-value function formed from a linear expression.

   Question paper p.2 · Mark scheme p.9

2. Question 1 (b)

   3 marks · 4/75

   This tests solving an absolute-value inequality by considering the relevant linear cases.

   Question paper p.2 · Mark scheme p.10

3. Question 2

   4 marks · 8/75

   Question paper p.3 · Mark scheme p.11

4. Question 3

   4 marks · 12/75

   This tests exact integration of a squared trigonometric function over specified limits.

   Question paper p.4 · Mark scheme p.11

5. Question 4 (a)

   3 marks · 15/75

   This tests the relationship between complex multiplication in polar form and complex conjugation.

   Question paper p.5 · Mark scheme p.12

6. Question 4 (b)

   3 marks · 18/75

   This tests the use of conjugate roots and the root-sum and root-product relationships for a polynomial with real coefficients.

   Question paper p.5 · Mark scheme p.12

7. Question 5 (a)

   4 marks · 22/75

   This tests implicit differentiation of a curve containing both a product and an exponential factor.

   Question paper p.6 · Mark scheme p.13

8. Question 5 (b)

   2 marks · 24/75

   This tests finding tangent gradients at points where the curve has a specified x-coordinate.

   Question paper p.6 · Mark scheme p.14

9. Question 6

   6 marks · 30/75

   This tests combining a complex-ratio condition with a modulus condition to determine complex numbers.

   Question paper p.7 · Mark scheme p.14

10. Question 7 (a)

    3 marks · 33/75

    This tests decomposition of a rational function into partial fractions with linear denominators.

    Question paper p.8 · Mark scheme p.17

11. Question 7 (b)

    4 marks · 37/75

    This tests using geometric-series expansions to obtain a power series for a rational function.

    Question paper p.9 · Mark scheme p.17

12. Question 7 (c)

    1 mark · 38/75

    Question paper p.9 · Mark scheme p.17

13. Question 8 (a)

    4 marks · 42/75

    Question paper p.10 · Mark scheme p.18

14. Question 8 (b)

    4 marks · 46/75

    This tests solving a trigonometric equation by applying an identity and selecting solutions in a restricted interval.

    Question paper p.11 · Mark scheme p.20

15. Question 9 (a)

    2 marks · 48/75

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

    Question paper p.12 · Mark scheme p.20

16. Question 9 (b)

    4 marks · 52/75

    This tests locating the foot of a perpendicular from a point to a line using a scalar-product condition.

    Question paper p.12 · Mark scheme p.21

17. Question 9 (c)

    2 marks · 54/75

    Question paper p.13 · Mark scheme p.21

18. Question 10 (a)

    8 marks · 62/75

    This tests solving a first-order differential equation by separation, trigonometric simplification, integration by parts, and an initial condition.

    Question paper p.14 · Mark scheme p.22

19. Question 10 (b)

    2 marks · 64/75

    Question paper p.15 · Mark scheme p.22

20. Question 11 (a)

    4 marks · 68/75

    This tests differentiating a function involving a square root and a trigonometric factor, then using a stationary-point condition.

    Question paper p.16 · Mark scheme p.23

21. Question 11 (b)

    2 marks · 70/75

    This tests locating a root numerically by demonstrating a sign change over a stated interval.

    Question paper p.16 · Mark scheme p.23

22. Question 11 (c)

    2 marks · 72/75

    This tests showing that a convergent fixed-point iteration has the required limiting value.

    Question paper p.17 · Mark scheme p.23

23. Question 11 (d)

    3 marks · 75/75

    This tests numerical fixed-point iteration to obtain a specified decimal accuracy.

    Question paper p.17 · Mark scheme p.24