9709 Pure Mathematics 1 Oct/Nov 2025 Paper 13

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

1. Question 1 (a)

   3 marks · 3/75

   This assesses use of the binomial expansion to obtain successive powers of x.

   Question paper p.2 · Mark scheme p.9

2. Question 1 (b)

   2 marks · 5/75

   This assesses extraction of a particular coefficient after multiplying two polynomials.

   Question paper p.2 · Mark scheme p.9

3. Question 2 (a)

   2 marks · 7/75

   Question paper p.3 · Mark scheme p.10

4. Question 2 (b)

   4 marks · 11/75

   Question paper p.3 · Mark scheme p.10

5. Question 3 (a)

   1 mark · 12/75

   Question paper p.4 · Mark scheme p.11

6. Question 3 (b)

   1 mark · 13/75

   Question paper p.4 · Mark scheme p.11

7. Question 3 (c)

   1 mark · 14/75

   This assesses interpreting chord gradients as estimates of the derivative at a point.

   Question paper p.4 · Mark scheme p.11

8. Question 4 (a)

   2 marks · 16/75

   This assesses finding a specified term of an arithmetic progression from its initial terms.

   Question paper p.5 · Mark scheme p.12

9. Question 4 (b)

   4 marks · 20/75

   This assesses finding the sum to infinity of a geometric progression whose first terms contain an unknown.

   Question paper p.5 · Mark scheme p.12

10. Question 5 (a)

    2 marks · 22/75

    This assesses finding a central angle from a chord length and the radius of a circle.

    Question paper p.6 · Mark scheme p.13

11. Question 5 (b)

    3 marks · 25/75

    This assesses finding the exact area of a circular segment using sector and triangle areas.

    Question paper p.6 · Mark scheme p.13

12. Question 6 (a)

    5 marks · 30/75

    This assesses describing a composite transformation of a cubic graph in the correct order.

    Question paper p.7 · Mark scheme p.14

13. Question 6 (b)

    3 marks · 33/75

    This assesses reading the parameters of a transformed cubic from its graph or transformation description.

    Question paper p.7 · Mark scheme p.15

14. Question 7

    4 marks · 37/75

    This assesses finding the inverse of a restricted rational function and checking when the function is self-inverse.

    Question paper p.8 · Mark scheme p.16

15. Question 8 (a)

    3 marks · 40/75

    This assesses finding the centre and radius of a circle from the endpoints of a diameter.

    Question paper p.9 · Mark scheme p.17

16. Question 8 (b)

    5 marks · 45/75

    This assesses finding points of contact between a circle and tangents of a prescribed gradient.

    Question paper p.9 · Mark scheme p.18

17. Question 9

    8 marks · 53/75

    This assesses finding a volume of revolution by integrating the squared radius about the x-axis.

    Question paper p.10 · Mark scheme p.20

18. Question 10 (a)(i)

    3 marks · 56/75

    This assesses completing the square for a quadratic function.

    Question paper p.12 · Mark scheme p.22

19. Question 10 (a)(ii)

    1 mark · 57/75

    This assesses determining the range of a quadratic from its completed-square form.

    Question paper p.12 · Mark scheme p.22

20. Question 10 (b)

    5 marks · 62/75

    This assesses using known roots of a quadratic to determine two unknown constants.

    Question paper p.13 · Mark scheme p.23

21. Question 11 (a)

    6 marks · 68/75

    This assesses finding a tangent to a rational curve and its intersection with a straight line.

    Question paper p.14 · Mark scheme p.25

22. Question 11 (b)(i)

    3 marks · 71/75

    This assesses locating stationary points by solving the equation obtained from setting the first derivative to zero.

    Question paper p.15 · Mark scheme p.26

23. Question 11 (b)(ii)

    4 marks · 75/75

    This assesses using the second derivative to classify stationary points.

    Question paper p.15 · Mark scheme p.26