9709 Pure Mathematics 3 May/June 2025 Paper 35

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

1. Question 1

   4 marks · 4/75

   Solving an exponential equation by applying logarithms and handling powers and products correctly.

   Question paper p.3 · Mark scheme p.9

2. Question 2

   5 marks · 9/75

   Question paper p.4 · Mark scheme p.10

3. Question 3 (a)

   2 marks · 11/75

   Dividing complex numbers in polar or exponential form and expressing the result with a principal argument.

   Question paper p.5 · Mark scheme p.11

4. Question 3 (b)

   2 marks · 13/75

   Interpreting complex division as a combination of a scaling transformation and a rotation on an Argand diagram.

   Question paper p.5 · Mark scheme p.11

5. Question 4

   5 marks · 18/75

   Finding a stationary point when the curve contains a product involving a logarithm of a power.

   Question paper p.6 · Mark scheme p.12

6. Question 5 (a)

   3 marks · 21/75

   Using the geometry of a circle in the complex plane to find the extreme distances from the origin.

   Question paper p.7 · Mark scheme p.13

7. Question 5 (b)

   3 marks · 24/75

   Finding the least possible argument of points on a circle using a tangent from the origin.

   Question paper p.7 · Mark scheme p.14

8. Question 6 (a)

   5 marks · 29/75

   Finding dy/dx for a parametric curve and simplifying it into the requested cosecant form.

   Question paper p.8 · Mark scheme p.16

9. Question 6 (b)

   4 marks · 33/75

   Finding the equation of a normal to a parametrically defined curve at a specified parameter value.

   Question paper p.9 · Mark scheme p.18

10. Question 7 (a)

    3 marks · 36/75

    Differentiating an inverse tangent function and solving for points with a prescribed gradient.

    Question paper p.10 · Mark scheme p.19

11. Question 7 (b)

    5 marks · 41/75

    Evaluating a definite integral involving arctan by integration by parts.

    Question paper p.11 · Mark scheme p.20

12. Question 8 (a)

    2 marks · 43/75

    Using graph sketches to establish that a trigonometric equation has exactly one root on an interval.

    Question paper p.12 · Mark scheme p.22

13. Question 8 (b)

    2 marks · 45/75

    Bracketing a root numerically by evaluating an appropriate function at two endpoints.

    Question paper p.12 · Mark scheme p.23

14. Question 8 (c)

    2 marks · 47/75

    Showing that an inverse-trigonometric iteration is a fixed-point rearrangement of the original equation.

    Question paper p.13 · Mark scheme p.24

15. Question 8 (d)

    3 marks · 50/75

    Using fixed-point iteration to approximate a root to a specified accuracy.

    Question paper p.13 · Mark scheme p.25

16. Question 9 (a)

    5 marks · 55/75

    Decomposing a rational function into partial fractions with two linear denominators.

    Question paper p.14 · Mark scheme p.26

17. Question 9 (b)

    4 marks · 59/75

    Obtaining a Maclaurin-style expansion of a partial-fraction expression through the quadratic term.

    Question paper p.15 · Mark scheme p.27

18. Question 10 (a)

    3 marks · 62/75

    Using equal vector lengths to determine an unknown coordinate.

    Question paper p.16 · Mark scheme p.28

19. Question 10 (b)

    2 marks · 64/75

    Using the parallelogram properties of a rhombus to find a missing position vector.

    Question paper p.17 · Mark scheme p.29

20. Question 10 (c)

    3 marks · 67/75

    Calculating an angle between two three-dimensional vectors using the scalar product.

    Question paper p.17 · Mark scheme p.30

21. Question 11

    8 marks · 75/75

    Solving a first-order separable differential equation using an initial condition and then evaluating the solution at a specified x-value.

    Question paper p.18 · Mark scheme p.31