9709 Pure Mathematics 3 May/June 2025 Paper 31

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

1. Question 1 (a)

   1 mark · 1/75

   This assesses sketching a modulus-based graph accurately.

   Question paper p.2 · Mark scheme p.9

2. Question 1 (b)

   2 marks · 3/75

   This assesses solving an inequality involving an absolute value.

   Question paper p.2 · Mark scheme p.9

3. Question 2

   3 marks · 6/75

   This assesses using logarithm laws and exponentiation to rearrange an equation involving logarithms.

   Question paper p.3 · Mark scheme p.10

4. Question 3

   6 marks · 12/75

   This assesses finding complex numbers subject to a real quotient condition and a modulus constraint.

   Question paper p.4 · Mark scheme p.11

5. Question 4

   6 marks · 18/75

   This assesses differentiating a parametrically defined curve and finding a tangent at a specified point.

   Question paper p.6 · Mark scheme p.12

6. Question 5

   5 marks · 23/75

   This assesses applying the factor/remainder theorem to determine unknown polynomial coefficients.

   Question paper p.8 · Mark scheme p.13

7. Question 6 (a)

   2 marks · 25/75

   This assesses multiplication of complex numbers written in polar exponential form.

   Question paper p.9 · Mark scheme p.14

8. Question 6 (b)

   2 marks · 27/75

   This assesses representing complex numbers and their transformed images on an Argand diagram.

   Question paper p.9 · Mark scheme p.14

9. Question 6 (c)

   2 marks · 29/75

   This assesses interpreting multiplication by a complex number as geometric transformations.

   Question paper p.9 · Mark scheme p.14

10. Question 7 (a)

    4 marks · 33/75

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

    Question paper p.10 · Mark scheme p.15

11. Question 7 (b)

    4 marks · 37/75

    This assesses solving a trigonometric equation after applying a compound-angle transformation.

    Question paper p.11 · Mark scheme p.15

12. Question 8 (a)

    2 marks · 39/75

    This assesses forming a vector equation for a line through two points in three dimensions.

    Question paper p.12 · Mark scheme p.16

13. Question 8 (b)

    4 marks · 43/75

    This assesses testing whether two three-dimensional vector lines intersect.

    Question paper p.12 · Mark scheme p.16

14. Question 8 (c)

    3 marks · 46/75

    This assesses finding the acute angle between two three-dimensional directions using a scalar product.

    Question paper p.13 · Mark scheme p.16

15. Question 9 (a)

    5 marks · 51/75

    This assesses evaluating a logarithmic definite integral and rearranging the resulting equation into an exponential fixed-point form.

    Question paper p.14 · Mark scheme p.17

16. Question 9 (b)

    2 marks · 53/75

    This assesses verifying a numerical bracket for the solution of an equation involving a definite integral.

    Question paper p.15 · Mark scheme p.17

17. Question 9 (c)

    3 marks · 56/75

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

    Question paper p.15 · Mark scheme p.18

18. Question 10 (a)

    3 marks · 59/75

    This assesses polynomial division by a quadratic expression.

    Question paper p.16 · Mark scheme p.19

19. Question 10 (b)

    5 marks · 64/75

    This assesses solving a first-order differential equation by separation and integration with an initial condition.

    Question paper p.16 · Mark scheme p.20

20. Question 11 (a)

    6 marks · 70/75

    This assesses locating the maximum of a product involving trigonometric functions and a square root.

    Question paper p.18 · Mark scheme p.21

21. Question 11 (b)

    5 marks · 75/75

    This assesses finding a volume of revolution by integrating the square of a trigonometric curve.

    Question paper p.19 · Mark scheme p.21