9709 Pure Mathematics 3 Oct/Nov 2025 Paper 35

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

1. Question 1 (a)

   1 mark · 1/75

   This assesses sketching a graph involving the modulus of a linear expression.

   Question paper p.2 · Mark scheme p.9

2. Question 1 (b)

   3 marks · 4/75

   This assesses solving an inequality containing an absolute value.

   Question paper p.2 · Mark scheme p.9

3. Question 2

   4 marks · 8/75

   This assesses representing modulus inequalities as loci on an Argand diagram and identifying their intersection.

   Question paper p.3 · Mark scheme p.10

4. Question 3 (a)

   2 marks · 10/75

   This assesses linearising an exponential relationship by taking logarithms.

   Question paper p.4 · Mark scheme p.10

5. Question 3 (b)

   3 marks · 13/75

   This assesses determining the constants in an exponential model from two observations.

   Question paper p.4 · Mark scheme p.10

6. Question 4

   5 marks · 18/75

   This assesses implicit differentiation of an equation containing products, logarithms and composite functions.

   Question paper p.5 · Mark scheme p.11

7. Question 5

   5 marks · 23/75

   This assesses solving a complex equation involving a conjugate by separating real and imaginary parts.

   Question paper p.6 · Mark scheme p.12

8. Question 6 (a)

   4 marks · 27/75

   This assesses differentiating a trigonometric quotient and rewriting the result using only sine.

   Question paper p.7 · Mark scheme p.13

9. Question 6 (b)

   2 marks · 29/75

   This assesses using the derivative to prove that a curve has no stationary points.

   Question paper p.7 · Mark scheme p.14

10. Question 7 (a)

    4 marks · 33/75

    This assesses changing variables in a definite integral using a stated substitution.

    Question paper p.8 · Mark scheme p.15

11. Question 7 (b)

    4 marks · 37/75

    This assesses evaluating the transformed definite integral exactly.

    Question paper p.9 · Mark scheme p.15

12. Question 8 (a)

    5 marks · 42/75

    This assesses expressing a sum of shifted sine terms as a single amplitude-phase sinusoid.

    Question paper p.10 · Mark scheme p.16

13. Question 8 (b)

    4 marks · 46/75

    This assesses solving a trigonometric equation after applying an amplitude-phase transformation.

    Question paper p.11 · Mark scheme p.16

14. Question 9 (a)

    2 marks · 48/75

    This assesses using direction vectors to impose perpendicularity on two lines.

    Question paper p.12 · Mark scheme p.17

15. Question 9 (b)

    4 marks · 52/75

    This assesses finding when two three-dimensional parametric lines intersect.

    Question paper p.12 · Mark scheme p.17

16. Question 9 (c)

    4 marks · 56/75

    This assesses finding parameter values from the acute angle between two lines in three dimensions.

    Question paper p.13 · Mark scheme p.18

17. Question 10 (a)

    5 marks · 61/75

    This assesses evaluating a parameter-dependent exponential integral by integration by parts.

    Question paper p.14 · Mark scheme p.18

18. Question 10 (b)

    2 marks · 63/75

    This assesses numerically verifying that a root lies within a specified interval.

    Question paper p.15 · Mark scheme p.19

19. Question 10 (c)

    3 marks · 66/75

    This assesses using fixed-point iteration to approximate a root to a specified accuracy.

    Question paper p.15 · Mark scheme p.20

20. Question 11 (a)

    1 mark · 67/75

    This assesses translating a proportional growth description into a differential equation.

    Question paper p.16 · Mark scheme p.20

21. Question 11 (b)

    8 marks · 75/75

    This assesses solving a logistic differential equation using separation of variables and initial conditions.

    Question paper p.16 · Mark scheme p.21