9709 Pure Mathematics 1 May/June 2025 Paper 13

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

1. Question 1

   4 marks · 4/75

   This assesses differentiation of a curve followed by finding the tangent line at a specified point.

   Question paper p.2 · Mark scheme p.9

2. Question 2

   4 marks · 8/75

   This assesses the use of geometric-progression formulae, including the condition for an infinite sum.

   Question paper p.3 · Mark scheme p.10

3. Question 3

   4 marks · 12/75

   This assesses definite integration involving a reciprocal squared linear expression and determination of an unknown constant.

   Question paper p.4 · Mark scheme p.11

4. Question 4 (a)

   3 marks · 15/75

   This assesses the first terms of a binomial expansion in ascending powers of x.

   Question paper p.5 · Mark scheme p.12

5. Question 4 (b)

   3 marks · 18/75

   This assesses using a truncated binomial expansion to approximate a numerical power.

   Question paper p.5 · Mark scheme p.12

6. Question 5

   6 marks · 24/75

   Question paper p.6 · Mark scheme p.13

7. Question 6

   6 marks · 30/75

   This assesses simultaneous use of the nth-term and sum formulae for an arithmetic progression.

   Question paper p.7 · Mark scheme p.14

8. Question 7 (a)

   3 marks · 33/75

   This assesses determining where a curve is decreasing from the sign of its derivative.

   Question paper p.8 · Mark scheme p.15

9. Question 7 (b)

   4 marks · 37/75

   This assesses recovering a curve from its derivative using information about a maximum point.

   Question paper p.8 · Mark scheme p.15

10. Question 8 (a)

    2 marks · 39/75

    This assesses finding a central angle from the geometry of a square and the associated right triangles.

    Question paper p.9 · Mark scheme p.16

11. Question 8 (b)

    3 marks · 42/75

    This assesses finding the perimeter of a region bounded partly by a circular arc.

    Question paper p.9 · Mark scheme p.17

12. Question 8 (c)

    3 marks · 45/75

    This assesses finding the area enclosed by a circular arc and straight line segments.

    Question paper p.9 · Mark scheme p.17

13. Question 9 (a)

    4 marks · 49/75

    This assesses using perpendicular gradients, or an equivalent distance argument, to determine an unknown coordinate.

    Question paper p.10 · Mark scheme p.18

14. Question 9 (b)

    5 marks · 54/75

    This assesses finding a circle’s tangent using the midpoint of a diameter and perpendicular gradients.

    Question paper p.11 · Mark scheme p.20

15. Question 10 (a)

    4 marks · 58/75

    This assesses locating stationary points by differentiating a rational-plus-linear curve.

    Question paper p.12 · Mark scheme p.22

16. Question 10 (b)

    3 marks · 61/75

    This assesses using the second derivative to classify stationary points.

    Question paper p.12 · Mark scheme p.22

17. Question 10 (c)(i)

    1 mark · 62/75

    Question paper p.13 · Mark scheme p.23

18. Question 10 (c)(ii)

    3 marks · 65/75

    This assesses transforming a curve equation under a translation followed by reflection in the x-axis.

    Question paper p.13 · Mark scheme p.23

19. Question 11 (a)

    4 marks · 69/75

    This assesses finding parameter values from the minimum value of a quadratic function.

    Question paper p.14 · Mark scheme p.24

20. Question 11 (b)

    6 marks · 75/75

    Question paper p.15 · Mark scheme p.26