9709 Pure Mathematics 1 Oct/Nov 2025 Paper 12

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

1. Question 1 (a)

   2 marks · 2/75

   This assesses completing the square for a quadratic expression.

   Question paper p.2 · Mark scheme p.9

2. Question 1 (b)

   1 mark · 3/75

   Question paper p.2 · Mark scheme p.9

3. Question 1 (c)

   2 marks · 5/75

   This assesses solving a quadratic exactly using its completed-square form.

   Question paper p.2 · Mark scheme p.9

4. Question 2

   3 marks · 8/75

   This assesses selecting and evaluating the constant term in a binomial expansion containing positive and negative powers of x.

   Question paper p.3 · Mark scheme p.10

5. Question 3 (a)

   3 marks · 11/75

   This assesses interpreting transformations of a function when the input is multiplied and a constant is added to the output.

   Question paper p.4 · Mark scheme p.11

6. Question 3 (b)

   2 marks · 13/75

   This assesses expressing successive vertical scaling and reflection in function notation.

   Question paper p.4 · Mark scheme p.11

7. Question 4 (a)

   1 mark · 14/75

   Question paper p.5 · Mark scheme p.12

8. Question 4 (b)

   5 marks · 19/75

   This assesses integrating a derivative and using a point on the curve to determine the integration constant.

   Question paper p.5 · Mark scheme p.12

9. Question 5 (a)

   3 marks · 22/75

   This assesses finding the x-coordinate of a maximum point using differentiation.

   Question paper p.6 · Mark scheme p.13

10. Question 5 (b)

    5 marks · 27/75

    Question paper p.7 · Mark scheme p.14

11. Question 6 (a)

    2 marks · 29/75

    This assesses sketching a vertically shifted and scaled sine curve over one complete standard interval.

    Question paper p.8 · Mark scheme p.15

12. Question 6 (b)(i)

    1 mark · 30/75

    This assesses counting intersections between a sine curve and a straight-line graph on a closed interval.

    Question paper p.8 · Mark scheme p.15

13. Question 6 (b)(ii)

    1 mark · 31/75

    Question paper p.8 · Mark scheme p.15

14. Question 6 (c)

    5 marks · 36/75

    Question paper p.9 · Mark scheme p.16

15. Question 7 (a)

    4 marks · 40/75

    This assesses finding the equation of a circle from the endpoints of a diameter.

    Question paper p.10 · Mark scheme p.17

16. Question 7 (b)

    3 marks · 43/75

    This assesses finding a tangent equation using the perpendicular relationship between a radius and a tangent.

    Question paper p.10 · Mark scheme p.18

17. Question 7 (c)

    4 marks · 47/75

    Question paper p.11 · Mark scheme p.19

18. Question 8 (a)

    3 marks · 50/75

    This assesses combining the common-difference and common-ratio conditions of arithmetic and geometric progressions.

    Question paper p.12 · Mark scheme p.20

19. Question 8 (b)(i)

    5 marks · 55/75

    This assesses finding the sum to infinity of a geometric progression after resolving the possible parameter values.

    Question paper p.12 · Mark scheme p.21

20. Question 8 (b)(ii)

    3 marks · 58/75

    This assesses finding a finite arithmetic-series sum from the progression information.

    Question paper p.12 · Mark scheme p.22

21. Question 9 (a)

    4 marks · 62/75

    This assesses differentiating a rational function and using the derivative to classify its monotonic behaviour.

    Question paper p.14 · Mark scheme p.22

22. Question 9 (b)

    1 mark · 63/75

    This assesses deciding whether a function has an inverse from its one-to-one behaviour.

    Question paper p.14 · Mark scheme p.22

23. Question 9 (c)

    1 mark · 64/75

    This assesses finding the range of a linear function with a strict lower domain bound.

    Question paper p.15 · Mark scheme p.23

24. Question 9 (d)

    2 marks · 66/75

    This assesses determining when a composite function is defined by matching the inner function’s range to the outer function’s domain.

    Question paper p.15 · Mark scheme p.23

25. Question 10 (a)

    2 marks · 68/75

    This assesses using circle geometry and an exact triangle calculation to relate the two radii.

    Question paper p.16 · Mark scheme p.24

26. Question 10 (b)

    7 marks · 75/75

    This assesses finding an exact shaded area by combining sectors and triangles from two related circles.

    Question paper p.17 · Mark scheme p.25