9709 Pure Mathematics 1 Oct/Nov 2025 Paper 15

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

1. Question 1 (a)

   2 marks · 2/75

   This assesses forming the equation of a circle from its centre and radius.

   Question paper p.3 · Mark scheme p.7

2. Question 1 (b)

   3 marks · 5/75

   This assesses using a circle equation to determine possible coordinates of a point.

   Question paper p.3 · Mark scheme p.7

3. Question 2 (a)

   3 marks · 8/75

   This assesses finding and simplifying the common multiplier of a geometric progression involving surds.

   Question paper p.4 · Mark scheme p.8

4. Question 2 (b)

   2 marks · 10/75

   This assesses applying the sum-to-infinity formula for a convergent geometric progression.

   Question paper p.4 · Mark scheme p.8

5. Question 3 (a)

   2 marks · 12/75

   This assesses completing the square for a quadratic expression.

   Question paper p.5 · Mark scheme p.9

6. Question 3 (b)

   1 mark · 13/75

   This assesses interpreting the single intersection of a quadratic curve with a horizontal line.

   Question paper p.5 · Mark scheme p.9

7. Question 4 (a)

   2 marks · 15/75

   This assesses finding the perimeter of a composite design containing a circular arc and part of a rhombus.

   Question paper p.6 · Mark scheme p.10

8. Question 4 (b)

   5 marks · 20/75

   This assesses finding the area of a composite sector-and-rhombus design after using a perimeter condition.

   Question paper p.6 · Mark scheme p.10

9. Question 5

   3 marks · 23/75

   Question paper p.7 · Mark scheme p.11

10. Question 6 (a)

    3 marks · 26/75

    This assesses transforming a trigonometric equation using reciprocal identities and the Pythagorean identity.

    Question paper p.8 · Mark scheme p.11

11. Question 6 (b)

    4 marks · 30/75

    This assesses solving a trigonometric quadratic and listing every angle in a full revolution.

    Question paper p.9 · Mark scheme p.11

12. Question 7 (a)

    3 marks · 33/75

    This assesses modelling the volume of an open cylinder subject to a fixed surface-area constraint.

    Question paper p.10 · Mark scheme p.12

13. Question 7 (b)

    3 marks · 36/75

    This assesses locating the radius that maximises a one-variable volume function using differentiation.

    Question paper p.11 · Mark scheme p.12

14. Question 7 (c)

    2 marks · 38/75

    This assesses evaluating the maximum volume after finding the optimising radius.

    Question paper p.11 · Mark scheme p.12

15. Question 8 (a)

    4 marks · 42/75

    This assesses using integration to obtain the volume generated by rotating a region about the x-axis and then applying an inequality.

    Question paper p.12 · Mark scheme p.13

16. Question 8 (b)

    3 marks · 45/75

    This assesses solving a quadratic inequality subject to a positive-variable restriction.

    Question paper p.12 · Mark scheme p.13

17. Question 9 (a)

    3 marks · 48/75

    This assesses comparing binomial coefficients in an expansion.

    Question paper p.13 · Mark scheme p.14

18. Question 9 (b)

    4 marks · 52/75

    This assesses combining a coefficient condition with a previously found ratio to determine two positive constants.

    Question paper p.13 · Mark scheme p.14

19. Question 10 (a)

    2 marks · 54/75

    This assesses solving a simple equation involving a function defined by a rational expression.

    Question paper p.14 · Mark scheme p.15

20. Question 10 (b)

    4 marks · 58/75

    This assesses finding an inverse function algebraically and determining its domain from the original range.

    Question paper p.14 · Mark scheme p.15

21. Question 10 (c)

    3 marks · 61/75

    This assesses simplifying a composite rational function and identifying a constant multiplier.

    Question paper p.15 · Mark scheme p.16

22. Question 11 (a)

    5 marks · 66/75

    This assesses finding a tangent line to a cubic curve using differentiation.

    Question paper p.16 · Mark scheme p.17

23. Question 11 (b)

    6 marks · 72/75

    This assesses finding an exact area bounded by a curve, a tangent, and the x-axis.

    Question paper p.17 · Mark scheme p.18

24. Question 11 (c)(i)

    1 mark · 73/75

    This assesses transforming a curve under a horizontal stretch and expressing the result as a polynomial.

    Question paper p.17 · Mark scheme p.20

25. Question 11 (c)(ii)

    2 marks · 75/75

    This assesses transforming a point and an area under a horizontal scaling.

    Question paper p.17 · Mark scheme p.20