9709 Pure Mathematics 1 Oct/Nov 2025 Paper 11

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

1. Question 1

   4 marks · 4/75

   Finding the parameter values that make a quadratic have two different real solutions using its discriminant.

   Question paper p.2 · Mark scheme p.9

2. Question 2 (a)

   3 marks · 7/75

   Using the terms of a geometric progression to determine an angle from its common ratio.

   Question paper p.3 · Mark scheme p.9

3. Question 2 (b)

   2 marks · 9/75

   Finding the exact sum to infinity of a geometric progression.

   Question paper p.3 · Mark scheme p.9

4. Question 3

   5 marks · 14/75

   Selecting relevant terms in two binomial expansions and using their coefficients to determine a positive parameter.

   Question paper p.4 · Mark scheme p.10

5. Question 4 (a)

   3 marks · 17/75

   Question paper p.5 · Mark scheme p.11

6. Question 4 (b)

   3 marks · 20/75

   Describing a reflection and translation that transform one parabola into another.

   Question paper p.5 · Mark scheme p.11

7. Question 5 (a)

   3 marks · 23/75

   Proving a trigonometric identity by rewriting powers of tangent in terms of sine and cosine.

   Question paper p.6 · Mark scheme p.12

8. Question 5 (b)

   4 marks · 27/75

   Solving a trigonometric equation by applying the identity from part (a) and considering every angle in the stated interval.

   Question paper p.6 · Mark scheme p.13

9. Question 6 (a)

   1 mark · 28/75

   Determining the range of a quadratic function subject to a restricted domain.

   Question paper p.7 · Mark scheme p.14

10. Question 6 (b)

    2 marks · 30/75

    Finding the inverse of a restricted quadratic function.

    Question paper p.7 · Mark scheme p.14

11. Question 6 (c)

    4 marks · 34/75

    Solving an equation involving the composition of two functions.

    Question paper p.7 · Mark scheme p.14

12. Question 7 (a)

    2 marks · 36/75

    Finding the area of a circular segment from a sector and an isosceles triangle.

    Question paper p.8 · Mark scheme p.15

13. Question 7 (b)(i)

    3 marks · 39/75

    Using related rates to find how the area of a circular segment changes as its radius increases.

    Question paper p.9 · Mark scheme p.15

14. Question 7 (b)(ii)

    3 marks · 42/75

    Finding the rate of change of an arc length when the radius is varying.

    Question paper p.9 · Mark scheme p.15

15. Question 8 (a)

    3 marks · 45/75

    Finding the area between a square-root curve, a horizontal line, and the vertical axis.

    Question paper p.10 · Mark scheme p.16

16. Question 8 (b)

    4 marks · 49/75

    Finding a volume of revolution about the y-axis using integration with respect to y.

    Question paper p.11 · Mark scheme p.16

17. Question 9 (a)

    4 marks · 53/75

    Using arithmetic-series sums and the condition on three terms to determine a common difference.

    Question paper p.12 · Mark scheme p.17

18. Question 9 (b)

    4 marks · 57/75

    Question paper p.13 · Mark scheme p.17

19. Question 10 (a)

    4 marks · 61/75

    Finding the intersection coordinates of a circle and a straight line by simultaneous solution.

    Question paper p.14 · Mark scheme p.18

20. Question 10 (b)

    3 marks · 64/75

    Finding the area of a triangle formed by the circle's centre and the two line–circle intersection points.

    Question paper p.15 · Mark scheme p.18

21. Question 11 (a)

    3 marks · 67/75

    Finding the equation of a normal to a curve from its derivative and a point on the curve.

    Question paper p.16 · Mark scheme p.19

22. Question 11 (b)

    3 marks · 70/75

    Finding the rate at which the gradient of a curve changes by differentiating its derivative.

    Question paper p.16 · Mark scheme p.19

23. Question 11 (c)

    5 marks · 75/75

    Question paper p.17 · Mark scheme p.19