9709 Pure Mathematics 3 Oct/Nov 2023 Paper 32

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

1. Question 1 (a)

   1 mark · 1/75

   This assesses recognition and sketching of the piecewise-linear graph defined by an absolute value.

   Examiner report: Responses varied considerably: some graphs had the wrong number or shape of branches, misplaced the corner, or failed to show the required extent of the graph.

   Question paper p.3 · Mark scheme p.6

2. Question 1 (b)

   4 marks · 5/75

   This assesses solving an inequality containing an absolute value by considering the relevant linear cases.

   Examiner report: The boundary calculations were usually attempted successfully, but sign errors were frequent and some candidates discarded one valid interval or included an incorrect bounded interval.

   Question paper p.3 · Mark scheme p.7

3. Question 2

   4 marks · 9/75

   This assesses differentiation of parametric equations and use of the parametric gradient formula at a specified parameter value.

   Examiner report: Many candidates knew the required quotient of derivatives, but omitted a logarithmic factor, mishandled the exponential derivative, or failed to complete the substitution at the requested parameter.

   Question paper p.4 · Mark scheme p.8

4. Question 3

   5 marks · 14/75

   This assesses application of the factor theorem and the remainder theorem to determine unknown polynomial coefficients.

   Examiner report: A large proportion reached the correct pair, although long division often introduced algebraic mistakes; some treated the divisor associated with the remainder as though it were a factor.

   Question paper p.5 · Mark scheme p.9

5. Question 4 (a)

   4 marks · 18/75

   This assesses interpreting modulus and real-part conditions as loci on an Argand diagram.

   Examiner report: Most diagrams identified the circular region, but some confused a real-part restriction with an imaginary-part restriction or shaded several possible regions without making the intended one clear.

   Question paper p.6 · Mark scheme p.10

6. Question 4 (b)

   2 marks · 20/75

   This assesses maximising the argument of a complex number over a restricted planar region.

   Examiner report: Many attempts found a length or modulus rather than an angle. Only a minority set up both geometric contributions correctly, while coordinate estimates from the sketch were not sufficiently precise.

   Question paper p.6 · Mark scheme p.11

7. Question 5

   6 marks · 26/75

   This assesses integration of a rational function after algebraic division and decomposition into standard forms.

   Examiner report: A substantial number of candidates tried to integrate before simplifying the fraction, and some used invalid product or integration-by-parts arguments. Successful solutions generally began with the necessary algebraic split.

   Question paper p.7 · Mark scheme p.12

8. Question 6 (a)

   2 marks · 28/75

   This assesses using graphs to establish the existence and uniqueness of a root of a trigonometric equation on a stated interval.

   Examiner report: Many sketches did not resemble the required trigonometric curves. Even when both curves were acceptable, candidates often omitted any indication of where the intersection supporting the conclusion occurred.

   Question paper p.8 · Mark scheme p.14

9. Question 6 (b)

   2 marks · 30/75

   This assesses bracketing a root numerically by evaluating a suitable function at two endpoints.

   Examiner report: The numerical sign-change approach was common, but some candidates stopped after calculating values without drawing the bracketing conclusion; degree-mode calculations also caused errors.

   Question paper p.8 · Mark scheme p.14

10. Question 6 (c)

    3 marks · 33/75

    This assesses fixed-point iteration and appropriate rounding of a numerically determined trigonometric root.

    Examiner report: Most candidates used the recurrence competently, though some showed too few iterations to support the rounding or reported a conclusion inconsistent with their displayed sequence.

    Question paper p.9 · Mark scheme p.15

11. Question 7 (a)

    3 marks · 36/75

    This assesses proving a triple-angle cosine identity from the addition formula and double-angle identities.

    Examiner report: The prescribed starting formula was generally used, but algebraic slips involving signs, powers, and double-angle substitutions prevented some otherwise sound derivations from reaching the identity.

    Question paper p.10 · Mark scheme p.15

12. Question 7 (b)

    5 marks · 41/75

    This assesses solving a trigonometric equation by using a proven identity and reducing it to an algebraic equation in cosine.

    Examiner report: Some candidates incorrectly expanded the cosine of a sum or lost the product term. Others divided by cosine and thereby omitted a solution, or added values outside the permitted interval.

    Question paper p.11 · Mark scheme p.16

13. Question 8 (a)

    4 marks · 45/75

    This assesses manipulation of a complex equation and comparison of real and imaginary components.

    Examiner report: Expansion was often completed correctly, but many candidates did not proceed to compare components. Those who did sometimes created unnecessary algebraic complications while eliminating the scale factor.

    Question paper p.12 · Mark scheme p.17

14. Question 8 (b)

    3 marks · 48/75

    This assesses solving the quadratic parameter condition and recovering the corresponding real multiplier in a complex equation.

    Examiner report: Candidates who had a usable relation for the multiplier usually completed this part efficiently; others found the parameter values but did not determine or pair the associated multipliers.

    Question paper p.13 · Mark scheme p.17

15. Question 9 (a)

    5 marks · 53/75

    This assesses finding a stationary maximum of a product of trigonometric functions and reporting its position numerically.

    Examiner report: The product-rule start was common, but the chain-rule factor and sign in differentiating the composite cosine were frequent sources of error. Some correct results were then reported in the wrong units or to unsuitable precision.

    Question paper p.14 · Mark scheme p.18

16. Question 9 (b)

    4 marks · 57/75

    This assesses finding an exact area by integrating a trigonometric curve over the portions bounded by the axis.

    Examiner report: Many candidates spent effort finding limits but made no progress with the antiderivative. Incorrect power-integration rules were common, whereas substitution or a suitable product-to-sum route led to the successful solutions.

    Question paper p.15 · Mark scheme p.19

17. Question 10 (a)

    4 marks · 61/75

    This assesses finding an unknown component of a three-dimensional direction vector from the angle between two lines.

    Examiner report: A frequent unproductive approach was to seek an intersection of the lines. Among scalar-product attempts, errors mainly arose from miscalculating the vector length or using the angle itself instead of its cosine.

    Question paper p.16 · Mark scheme p.20

18. Question 10 (b)

    5 marks · 66/75

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

    Examiner report: Many candidates could construct the joining vector but did not know how to impose the perpendicular condition. Clear scalar-product work generally led to the full distance calculation, whereas omitted magnitude steps left unsupported final results.

    Question paper p.17 · Mark scheme p.21

19. Question 11 (a)

    8 marks · 74/75

    This assesses solving a first-order differential equation by separation, partial fractions, use of an initial condition, and algebraic rearrangement.

    Examiner report: This was difficult for many candidates from the separation stage onward. Common problems included losing the minus sign, using an invalid partial-fraction split, or stopping after obtaining a relation involving the quotient of the dependent-variable terms.

    Question paper p.18 · Mark scheme p.23

20. Question 11 (b)

    1 mark · 75/75

    This assesses determining the limiting value of the dependent variable from the explicit solution as the independent variable grows without bound.

    Examiner report: Only a small proportion gave a correct limit; many responses remained expressions involving the independent variable or did not attempt this short concluding step.

    Question paper p.19 · Mark scheme p.24