9709 Pure Mathematics 3 Oct/Nov 2023 Paper 31

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

1. Question 1

   5 marks · 5/75

   This assesses differentiation of a rational function and the use of a specified tangent gradient to locate points on a curve.

   Examiner report: Most candidates selected an appropriate differentiation rule, and those who formed the correct quadratic usually completed the question successfully. The main losses came from algebraic processing errors.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   4 marks · 9/75

   This assesses the interpretation and combination of distance and argument loci on an Argand diagram.

   Examiner report: A considerable number of candidates made no attempt. Although the two fixed points were often placed correctly, several drew an incorrect boundary; candidates with both loci correct generally identified the required region.

   Question paper p.3 · Mark scheme p.7

3. Question 3

   4 marks · 13/75

   This assesses the logarithmic linearisation of an exponential model and the recovery of its constants from two plotted points.

   Examiner report: Many candidates misread the vertical coordinates as values of y rather than ln y, leading them to substitute directly into the original exponential model. Very few used the intended straight-line logarithmic equation.

   Question paper p.4 · Mark scheme p.8

4. Question 4 (a)

   3 marks · 16/75

   This assesses rationalisation of a complex denominator and conversion to Cartesian form.

   Examiner report: Those attempting this part generally knew that conjugate multiplication was required. Arithmetic slips were frequent, and some candidates mishandled the square of the imaginary unit.

   Question paper p.5 · Mark scheme p.9

5. Question 4 (b)

   2 marks · 18/75

   This assesses the use of an argument condition to relate the real and imaginary components of a complex number.

   Examiner report: Nearly half of the candidates left this part blank. Among the attempts, a common difficulty was translating the stated argument into a condition on the components.

   Question paper p.5 · Mark scheme p.9

6. Question 5 (a)

   4 marks · 22/75

   Question paper p.6 · Mark scheme p.10

7. Question 5 (b)

   2 marks · 24/75

   This assesses the solution of a trigonometric equation over a specified interval using the tangent result and its periodicity.

   Examiner report: Candidates who found the correct tangent value generally obtained both required angles. Even with an incorrect tangent, some gained credit by applying the period of the tangent function correctly.

   Question paper p.7 · Mark scheme p.10

8. Question 6 (a)

   3 marks · 27/75

   This assesses differentiation of parametric equations and formation of dy/dx by the chain rule.

   Examiner report: Many candidates were comfortable with the standard parametric differentiation procedure, and a number of complete solutions were seen.

   Question paper p.8 · Mark scheme p.11

9. Question 6 (b)

   3 marks · 30/75

   This assesses the relationship between the gradients of perpendicular tangent and normal lines in a parametric curve.

   Examiner report: This was more demanding than part (a). A recurring error was to set the tangent gradient equal to the stated normal gradient instead of using the negative reciprocal relationship.

   Question paper p.8 · Mark scheme p.11

10. Question 7

    7 marks · 37/75

    This assesses separation and integration of a differential equation subject to an initial condition.

    Examiner report: Most candidates recognised that the variables should be separated and many integrated at least one side correctly. Notation was sometimes incomplete, and errors often appeared when converting the logarithmic relation back to an expression for x².

    Question paper p.9 · Mark scheme p.12

11. Question 8 (a)

    2 marks · 39/75

    This assesses the use of graph shapes and intersections to establish uniqueness of a root.

    Examiner report: Some sketches did not have the correct intercepts or shapes. Even when both curves were recognisable, candidates sometimes omitted a clear intersection or failed to explain how the drawing established uniqueness.

    Question paper p.10 · Mark scheme p.13

12. Question 8 (b)

    2 marks · 41/75

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

    Examiner report: Several candidates used an unsuitable sign-change argument rather than comparing the two functions. Some also failed to identify which functions their numerical values represented, making the reasoning unclear.

    Question paper p.10 · Mark scheme p.13

13. Question 8 (c)

    1 mark · 42/75

    This assesses recognition of a fixed-point iteration as a rearrangement of the original root equation.

    Examiner report: Many candidates did not attempt this part. The essential step was to connect the logarithmic iteration formula explicitly with an equivalent rearrangement of the initial equation.

    Question paper p.11 · Mark scheme p.13

14. Question 8 (d)

    3 marks · 45/75

    This assesses numerical fixed-point iteration and control of the requested accuracy.

    Examiner report: Those who attempted the calculation usually reached the correct result. The usual problems were insufficient decimal accuracy or stopping before the displayed iterates supported the requested rounding.

    Question paper p.11 · Mark scheme p.14

15. Question 9 (a)

    4 marks · 49/75

    This assesses differentiation of a product involving an exponential function and identification of its maximum point.

    Examiner report: Most candidates used the product rule, but the chain-rule factor from differentiating the exponential was often omitted.

    Question paper p.12 · Mark scheme p.14

16. Question 9 (b)

    5 marks · 54/75

    This assesses definite integration after a substitution involving the square of x.

    Examiner report: The general substitution method was understood by many candidates, but sign mistakes and missing numerical factors were common.

    Question paper p.13 · Mark scheme p.15

17. Question 10 (a)

    5 marks · 59/75

    This assesses decomposition of a rational function into partial fractions with one repeated linear factor.

    Examiner report: Many candidates handled the partial fractions confidently. The main difficulties were the repeated denominator factor and occasional arithmetic errors.

    Question paper p.14 · Mark scheme p.16

18. Question 10 (b)

    5 marks · 64/75

    This assesses the use of geometric and binomial expansions to obtain a Maclaurin expansion from partial fractions.

    Examiner report: Only a minority handled the binomial expansions fully successfully. Errors were especially common when extracting factors of two from the shifted denominators and when managing signs.

    Question paper p.15 · Mark scheme p.17

19. Question 10 (c)

    1 mark · 65/75

    Question paper p.15 · Mark scheme p.18

20. Question 11 (a)

    1 mark · 66/75

    This assesses the reading of a cuboid diagram and the construction of a midpoint position vector.

    Examiner report: Candidates who understood the vector construction found this part straightforward.

    Question paper p.16 · Mark scheme p.18

21. Question 11 (b)

    4 marks · 70/75

    This assesses the calculation of an angle between two three-dimensional vectors using the scalar product.

    Examiner report: Several candidates showed a sound grasp of the scalar-product method. Some used incompatible vector directions, while others obtained the dot product correctly but divided by an incorrect product of magnitudes.

    Question paper p.16 · Mark scheme p.19

22. Question 11 (c)

    5 marks · 75/75

    This assesses the shortest distance from a point to a line in three dimensions.

    Examiner report: The few candidates who began with a general point on the line often completed the problem effectively. Most unsuccessful attempts did not establish a valid method for finding the minimum distance.

    Question paper p.17 · Mark scheme p.19