9709 Pure Mathematics 3 Oct/Nov 2025 Paper 33

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

1. Question 1

   4 marks · 4/75

   Question paper p.2 · Mark scheme p.9

2. Question 2

   3 marks · 7/75

   Dividing a polynomial by a linear factor and identifying both the quotient and the remainder.

   Question paper p.3 · Mark scheme p.10

3. Question 3

   4 marks · 11/75

   Solving an exponential equation by taking logarithms and expressing the result as a quotient of logarithms.

   Question paper p.4 · Mark scheme p.11

4. Question 4

   5 marks · 16/75

   Interpreting complex-number modulus inequalities as geometric loci on an Argand diagram.

   Question paper p.5 · Mark scheme p.12

5. Question 5 (a)

   4 marks · 20/75

   Question paper p.6 · Mark scheme p.13

6. Question 5 (b)

   4 marks · 24/75

   Solving a trigonometric equation using a previously established identity over a specified interval in degrees.

   Question paper p.7 · Mark scheme p.14

7. Question 6

   6 marks · 30/75

   Question paper p.8 · Mark scheme p.15

8. Question 7

   6 marks · 36/75

   Solving an equation involving a complex variable and its conjugate by separating real and imaginary parts.

   Question paper p.10 · Mark scheme p.16

9. Question 8 (a)

   3 marks · 39/75

   Finding a stationary point of a product involving an exponential and a logarithm, then deriving the equation satisfied by its x-coordinate.

   Question paper p.12 · Mark scheme p.17

10. Question 8 (b)

    2 marks · 41/75

    Using sketches of two functions to establish that a nonlinear equation has exactly one positive root.

    Question paper p.12 · Mark scheme p.18

11. Question 8 (c)

    2 marks · 43/75

    Bracketing a root numerically by evaluating a continuous function at two specified endpoints.

    Question paper p.13 · Mark scheme p.19

12. Question 8 (d)

    3 marks · 46/75

    Applying fixed-point iteration to approximate the root of a rearranged equation.

    Question paper p.13 · Mark scheme p.19

13. Question 9 (a)

    3 marks · 49/75

    Writing vector equations of lines in three dimensions from point and direction information.

    Question paper p.14 · Mark scheme p.20

14. Question 9 (b)

    3 marks · 52/75

    Finding the acute angle between two three-dimensional lines using their direction vectors.

    Question paper p.14 · Mark scheme p.20

15. Question 9 (c)

    3 marks · 55/75

    Question paper p.15 · Mark scheme p.21

16. Question 10 (a)

    2 marks · 57/75

    Decomposing a rational function into partial fractions with linear denominators.

    Question paper p.16 · Mark scheme p.21

17. Question 10 (b)

    6 marks · 63/75

    Solving a separable differential equation using partial fractions, logarithmic integration, an initial condition, and algebraic rearrangement.

    Question paper p.16 · Mark scheme p.22

18. Question 11 (a)

    6 marks · 69/75

    Locating the minimum of a product involving secant squared and the square root of a tangent expression by differentiation.

    Question paper p.18 · Mark scheme p.23

19. Question 11 (b)

    6 marks · 75/75

    Evaluating an exact area integral by making the prescribed tangent substitution.

    Question paper p.19 · Mark scheme p.24