9709 Pure Mathematics 1 May/June 2025 Paper 11

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

1. Question 1

   4 marks · 4/75

   Question paper p.3 · Mark scheme p.9

2. Question 2 (a)

   2 marks · 6/75

   Finding the gradient of a normal from the given derivative at a specified point.

   Question paper p.4 · Mark scheme p.10

3. Question 2 (b)

   4 marks · 10/75

   Recovering a curve equation by integrating a derivative and using a point on the curve to determine the constant.

   Question paper p.4 · Mark scheme p.10

4. Question 3 (a)

   5 marks · 15/75

   Using geometric-progression information to determine a term when the common ratio is negative.

   Question paper p.5 · Mark scheme p.11

5. Question 3 (b)

   2 marks · 17/75

   Finding the exact sum to infinity of a geometric progression.

   Question paper p.5 · Mark scheme p.12

6. Question 4

   5 marks · 22/75

   Finding the area enclosed between two intersecting curves using definite integration.

   Question paper p.6 · Mark scheme p.13

7. Question 5 (a)(i)

   2 marks · 24/75

   Finding the first three terms of a binomial expansion in ascending powers of x.

   Question paper p.7 · Mark scheme p.15

8. Question 5 (a)(ii)

   2 marks · 26/75

   Expanding a binomial with a fractional linear term to obtain its first three powers of x.

   Question paper p.7 · Mark scheme p.15

9. Question 5 (b)

   3 marks · 29/75

   Using selected terms from two binomial expansions to determine an unknown parameter from a product coefficient.

   Question paper p.7 · Mark scheme p.15

10. Question 6 (a)

    4 marks · 33/75

    Finding intersections of a curve and a straight line by substitution and solving the resulting quadratic.

    Question paper p.8 · Mark scheme p.16

11. Question 6 (b)

    5 marks · 38/75

    Using the discriminant to identify parameter values for which a curve and line have no real intersections.

    Question paper p.9 · Mark scheme p.17

12. Question 7 (a)

    4 marks · 42/75

    Applying implicit differentiation and related rates to connect the rates of change of x and y.

    Question paper p.10 · Mark scheme p.18

13. Question 7 (b)

    5 marks · 47/75

    Question paper p.11 · Mark scheme p.19

14. Question 8

    8 marks · 55/75

    Using circle geometry and tangent equations to find the area of a triangle formed by two tangents and a vertical chord line.

    Question paper p.12 · Mark scheme p.20

15. Question 9 (a)

    4 marks · 59/75

    Using sector and triangle area formulae to find the area of a circular segment.

    Question paper p.14 · Mark scheme p.21

16. Question 9 (b)

    4 marks · 63/75

    Combining the cosine rule with sector geometry to calculate a segment area numerically.

    Question paper p.15 · Mark scheme p.21

17. Question 10 (a)

    5 marks · 68/75

    Describing a sequence of graph transformations, including their order, for a square-root function.

    Question paper p.16 · Mark scheme p.22

18. Question 10 (b)

    2 marks · 70/75

    Sketching the inverse of a function by reflecting its graph in the line of symmetry between a function and its inverse.

    Question paper p.16 · Mark scheme p.23

19. Question 10 (c)

    2 marks · 72/75

    Finding an inverse function algebraically from a given function rule.

    Question paper p.17 · Mark scheme p.23

20. Question 10 (d)

    1 mark · 73/75

    Determining the range of an inverse function from the domain of the original function.

    Question paper p.17 · Mark scheme p.23

21. Question 10 (e)

    1 mark · 74/75

    Evaluating a composition involving a separately defined function and an inverse function.

    Question paper p.17 · Mark scheme p.24

22. Question 10 (f)

    1 mark · 75/75

    Checking whether a composite function is defined by comparing the first function's range with the second function's domain.

    Question paper p.17 · Mark scheme p.24