9709 Pure Mathematics 3 Oct/Nov 2024 Paper 31

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

1. Question 1

   5 marks · 5/75

   This assesses the factor theorem together with the remainder theorem for a cubic polynomial.

   Question paper p.3 · Mark scheme p.6

2. Question 2

   5 marks · 10/75

   This assesses integration by parts involving a logarithmic factor and evaluation of an exact definite integral.

   Question paper p.4 · Mark scheme p.7

3. Question 3

   4 marks · 14/75

   This assesses implicit differentiation of an equation containing a logarithm and a product of variables.

   Question paper p.5 · Mark scheme p.8

4. Question 4 (a)

   3 marks · 17/75

   Question paper p.6 · Mark scheme p.9

5. Question 4 (b)

   5 marks · 22/75

   This assesses solving a trigonometric equation by applying an identity and accounting for all solutions in a specified interval.

   Question paper p.7 · Mark scheme p.9

6. Question 5 (a)

   2 marks · 24/75

   This assesses using graphs and their intersection properties to establish uniqueness of a numerical root.

   Question paper p.8 · Mark scheme p.10

7. Question 5 (b)

   2 marks · 26/75

   This assesses bracketing a root numerically by comparing the two sides of an equation at specified endpoints.

   Question paper p.8 · Mark scheme p.10

8. Question 5 (c)

   3 marks · 29/75

   This assesses fixed-point iteration for finding a root to a stated decimal accuracy.

   Question paper p.9 · Mark scheme p.10

9. Question 6 (a)

   4 marks · 33/75

   This assesses finding the exact coordinates of a specified minimum point by differentiating a trigonometric product.

   Question paper p.10 · Mark scheme p.11

10. Question 6 (b)

    4 marks · 37/75

    Question paper p.11 · Mark scheme p.11

11. Question 7 (a)

    5 marks · 42/75

    This assesses decomposition of a rational function into partial fractions with a linear and an irreducible quadratic denominator.

    Question paper p.12 · Mark scheme p.12

12. Question 7 (b)

    4 marks · 46/75

    Question paper p.13 · Mark scheme p.12

13. Question 8 (a)

    2 marks · 48/75

    This assesses rationalising the denominator of a complex reciprocal and separating real and imaginary parts.

    Question paper p.14 · Mark scheme p.13

14. Question 8 (b)

    3 marks · 51/75

    This assesses verifying that the complex numbers from part (a) satisfy a circle equation in the Argand plane.

    Question paper p.14 · Mark scheme p.13

15. Question 8 (c)

    3 marks · 54/75

    This assesses interpreting equations involving real parts and moduli as loci on an Argand diagram.

    Question paper p.15 · Mark scheme p.13

16. Question 8 (d)

    1 mark · 55/75

    This assesses identifying the geometric centre and radius represented by a complex-number locus.

    Question paper p.15 · Mark scheme p.13

17. Question 9 (a)

    2 marks · 57/75

    This assesses forming a three-dimensional vector equation for a line from a point and a parallel direction.

    Question paper p.16 · Mark scheme p.14

18. Question 9 (b)

    3 marks · 60/75

    This assesses finding the parameter at which a point with a given position vector lies on a vector line.

    Question paper p.16 · Mark scheme p.14

19. Question 9 (c)

    5 marks · 65/75

    Question paper p.17 · Mark scheme p.14

20. Question 10 (a)

    4 marks · 69/75

    This assesses modelling a cylindrical tank with a differential equation using related rates and a proportional outflow law.

    Question paper p.18 · Mark scheme p.15

21. Question 10 (b)

    6 marks · 75/75

    This assesses solving a separable differential equation and using two depth–time conditions to predict when the tank becomes empty.

    Question paper p.19 · Mark scheme p.15