9709 Pure Mathematics 1 May/June 2025 Paper 15

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

1. Question 1

   4 marks · 4/75

   This assesses finding a curve from its derivative using integration and a given point.

   Question paper p.2 · Mark scheme p.9

2. Question 2

   5 marks · 9/75

   This assesses extracting selected coefficients from binomial expansions and solving the resulting equation in a parameter.

   Question paper p.3 · Mark scheme p.10

3. Question 3 (a)

   2 marks · 11/75

   This assesses solving a quadratic exactly by rewriting it in completed-square form.

   Question paper p.4 · Mark scheme p.11

4. Question 3 (b)

   3 marks · 14/75

   This assesses applying the quadratic roots to a trigonometric equation by treating the tangent as the unknown.

   Question paper p.4 · Mark scheme p.11

5. Question 4

   5 marks · 19/75

   This assesses finding a volume of revolution using integration of the square of a curve’s ordinate.

   Question paper p.5 · Mark scheme p.12

6. Question 5 (a)

   4 marks · 23/75

   This assesses using circle geometry to determine missing lengths and an arc length before finding a perimeter.

   Question paper p.6 · Mark scheme p.13

7. Question 5 (b)

   2 marks · 25/75

   Question paper p.7 · Mark scheme p.13

8. Question 6 (a)

   3 marks · 28/75

   This assesses comparing terms from an arithmetic progression and a geometric progression.

   Question paper p.8 · Mark scheme p.14

9. Question 6 (b)

   5 marks · 33/75

   Question paper p.9 · Mark scheme p.14

10. Question 7 (a)

    2 marks · 35/75

    This assesses using the gradient formula and a known line gradient to determine an unknown coordinate.

    Question paper p.10 · Mark scheme p.15

11. Question 7 (b)

    2 marks · 37/75

    This assesses finding a missing vertex of a parallelogram from coordinate vectors or equivalent coordinate relationships.

    Question paper p.10 · Mark scheme p.15

12. Question 7 (c)

    5 marks · 42/75

    This assesses constructing a perpendicular bisector from coordinates and using its axis intercepts to find a triangle’s area.

    Question paper p.11 · Mark scheme p.15

13. Question 8 (a)

    5 marks · 47/75

    This assesses using the point and stationary-point conditions to determine two coefficients in a cubic.

    Question paper p.12 · Mark scheme p.16

14. Question 8 (b)

    3 marks · 50/75

    This assesses locating the second stationary point of a cubic and finding its full coordinates.

    Question paper p.13 · Mark scheme p.16

15. Question 8 (c)

    3 marks · 53/75

    This assesses using related rates and the chain rule to connect the rates of change of two coordinates.

    Question paper p.13 · Mark scheme p.17

16. Question 9 (a)

    4 marks · 57/75

    This assesses describing the combined transformations that map one trigonometric graph onto another.

    Question paper p.14 · Mark scheme p.18

17. Question 9 (b)

    4 marks · 61/75

    This assesses sketching two restricted trigonometric graphs after applying the stated transformations.

    Question paper p.14 · Mark scheme p.19

18. Question 9 (c)

    4 marks · 65/75

    This assesses finding a specified value of the inverse of the transformed cosine function.

    Question paper p.15 · Mark scheme p.19

19. Question 9 (d)

    1 mark · 66/75

    This assesses checking the range-domain compatibility required for forming a composite function.

    Question paper p.15 · Mark scheme p.20

20. Question 10 (a)

    4 marks · 70/75

    This assesses finding the tangent to a circle from its centre and the radius direction.

    Question paper p.16 · Mark scheme p.21

21. Question 10 (b)

    5 marks · 75/75

    This assesses using the non-intersection condition between a line and a circle to derive an inequality in a parameter.

    Question paper p.17 · Mark scheme p.23