9709 Pure Mathematics 1 May/June 2025 Paper 12

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

1. Question 1

   4 marks · 4/75

   This assesses recognition and sequencing of graph transformations from the displayed graphs.

   Question paper p.2 · Mark scheme p.9

2. Question 2

   4 marks · 8/75

   This assesses solving a simultaneous nonlinear equation and a linear equation by elimination.

   Question paper p.3 · Mark scheme p.10

3. Question 3

   4 marks · 12/75

   This assesses selection and evaluation of a binomial term with a specified power of x.

   Question paper p.4 · Mark scheme p.11

4. Question 4 (a)

   3 marks · 15/75

   This assesses related rates using differentiation and the chain rule.

   Question paper p.5 · Mark scheme p.12

5. Question 4 (b)

   2 marks · 17/75

   This assesses using the stationary-point condition to determine an unknown parameter.

   Question paper p.5 · Mark scheme p.12

6. Question 5 (a)

   2 marks · 19/75

   This assesses determining the range of a transformed cosine function.

   Question paper p.6 · Mark scheme p.13

7. Question 5 (b)

   2 marks · 21/75

   This assesses sketching a trigonometric graph over a specified interval.

   Question paper p.6 · Mark scheme p.13

8. Question 5 (c)

   1 mark · 22/75

   Question paper p.6 · Mark scheme p.13

9. Question 6

   6 marks · 28/75

   This assesses finding a composite area using a curve integral and elementary geometric area.

   Question paper p.7 · Mark scheme p.14

10. Question 7 (a)

    3 marks · 31/75

    Question paper p.8 · Mark scheme p.15

11. Question 7 (b)

    4 marks · 35/75

    This assesses solving a trigonometric equation by reducing it to a quadratic in tan theta.

    Question paper p.9 · Mark scheme p.15

12. Question 8 (a)

    3 marks · 38/75

    This assesses extracting a circle's centre and its intersections with a horizontal line.

    Question paper p.10 · Mark scheme p.16

13. Question 8 (b)

    2 marks · 40/75

    This assesses finding a central angle in an isosceles triangle formed by two radii and a chord.

    Question paper p.11 · Mark scheme p.16

14. Question 8 (c)

    4 marks · 44/75

    This assesses calculating a circular segment area from sector and triangle areas.

    Question paper p.11 · Mark scheme p.17

15. Question 9 (a)

    3 marks · 47/75

    This assesses integrating a second derivative and using a stationary-point condition to find a constant.

    Question paper p.12 · Mark scheme p.17

16. Question 9 (b)

    2 marks · 49/75

    This assesses locating a second stationary point and classifying it with the second derivative.

    Question paper p.12 · Mark scheme p.18

17. Question 9 (c)

    3 marks · 52/75

    This assesses integrating a first derivative to obtain the curve and using a point to determine the remaining constant.

    Question paper p.13 · Mark scheme p.18

18. Question 9 (d)

    4 marks · 56/75

    This assesses finding a normal equation from a prescribed tangent gradient and expressing the result linearly.

    Question paper p.13 · Mark scheme p.19

19. Question 10 (a)(i)

    2 marks · 58/75

    This assesses using equal differences in an arithmetic progression to determine a parameter.

    Question paper p.14 · Mark scheme p.19

20. Question 10 (a)(ii)

    3 marks · 61/75

    This assesses finding an arithmetic-series sum from its first term and common difference.

    Question paper p.14 · Mark scheme p.20

21. Question 10 (b)

    5 marks · 66/75

    This assesses determining a geometric progression and its infinite sum from non-consecutive terms.

    Question paper p.15 · Mark scheme p.20

22. Question 11 (a)

    2 marks · 68/75

    This assesses completing the square for a quadratic expression.

    Question paper p.16 · Mark scheme p.21

23. Question 11 (b)(i)

    3 marks · 71/75

    This assesses finding the inverse of a restricted quadratic function.

    Question paper p.16 · Mark scheme p.21

24. Question 11 (b)(ii)

    4 marks · 75/75

    This assesses finding the inverse of a composite function involving a restricted quadratic.

    Question paper p.16 · Mark scheme p.22