9709 Pure Mathematics 3 May/June 2023 Paper 32

20 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.6

2. Question 2

   3 marks · 7/75

   This assesses the use of logarithm laws, domain restrictions and exact algebra when solving a logarithmic equation.

   Examiner report: Many candidates reached the positive solution but failed to discard the inadmissible negative possibility; answers given only as decimals also missed the exactness requirement.

   Question paper p.4 · Mark scheme p.7

3. Question 3 (a)

   2 marks · 9/75

   This assesses interpreting a complex modulus equation as a geometric locus on an Argand diagram.

   Examiner report: Most diagrams were accurate, although some circles were displaced into the wrong region or were shaded internally rather than representing the circumference.

   Question paper p.5 · Mark scheme p.7

4. Question 3 (b)

   2 marks · 11/75

   This assesses finding the minimum modulus of points lying on a circular complex locus.

   Examiner report: A recurring misunderstanding was to give an angle instead of a distance, while some candidates found the centre-to-origin distance but did not account for the radius.

   Question paper p.5 · Mark scheme p.8

5. Question 4

   5 marks · 16/75

   Question paper p.6 · Mark scheme p.9

6. Question 5 (a)

   3 marks · 19/75

   This assesses expansion and simplification of a polynomial in a complex number whose imaginary component is symbolic.

   Examiner report: Although many candidates had the right overall approach, sign and arithmetic errors were frequent, especially when handling the square of the imaginary term; incomplete expansion also made unsupported calculator-like answers difficult to credit.

   Question paper p.7 · Mark scheme p.10

7. Question 5 (b)

   3 marks · 22/75

   This assesses using the real part of a complex expression to determine a parameter and then finding the argument in the correct quadrant.

   Examiner report: Some candidates rejected the required negative parameter, while others correctly found the parameter but then calculated the argument of the polynomial value instead of the original complex number.

   Question paper p.7 · Mark scheme p.10

8. Question 6 (a)

   2 marks · 24/75

   This assesses demonstrating that a root lies in a specified interval by comparing endpoint values.

   Examiner report: Several otherwise suitable solutions lost the final credit because the numerical comparison was not followed by a clear statement that the root had been bracketed.

   Question paper p.8 · Mark scheme p.11

9. Question 6 (b)

   2 marks · 26/75

   This assesses proving that a convergent iteration has a limit satisfying the original trigonometric equation.

   Examiner report: A number of candidates iterated numerically even though an algebraic argument was required; confusion between inverse tangent and cotangent also prevented otherwise promising rearrangements from succeeding.

   Question paper p.8 · Mark scheme p.11

10. Question 6 (c)

    3 marks · 29/75

    This assesses carrying out numerical fixed-point iteration to the requested accuracy.

    Examiner report: This was generally a strong part, though degree-mode calculations received no credit and some candidates rounded the final estimate upward incorrectly.

    Question paper p.9 · Mark scheme p.12

11. Question 7 (a)

    4 marks · 33/75

    This assesses implicit differentiation of a relation containing x, y and a product xy.

    Examiner report: Many candidates completed the differentiation successfully, but misplaced minus signs and unclear derivative notation caused avoidable losses.

    Question paper p.10 · Mark scheme p.13

12. Question 7 (b)

    5 marks · 38/75

    This assesses using a gradient condition with an implicitly defined curve to obtain exact coordinates.

    Examiner report: The usual successful route was followed by many candidates, but sign errors in the linear relation were common and decimal coordinates did not satisfy the exact-answer requirement.

    Question paper p.11 · Mark scheme p.14

13. Question 8 (a)

    7 marks · 45/75

    This assesses solving a separable first-order differential equation using an initial condition.

    Examiner report: Frequent difficulties involved interpreting the exponential denominator, integrating its negative exponent, and obtaining the correct coefficient for the inverse-tangent term; some inappropriate logarithmic integrations were attempted.

    Question paper p.12 · Mark scheme p.15

14. Question 8 (b)

    1 mark · 46/75

    This assesses evaluating the limiting behaviour of the explicit differential-equation solution.

    Examiner report: Candidates with a sound expression from part (a) usually succeeded, whereas unsupported guesses such as a zero or infinite limit were seen among weaker responses.

    Question paper p.13 · Mark scheme p.15

15. Question 9 (a)

    5 marks · 51/75

    This assesses decomposing a rational function into partial fractions with a repeated linear factor.

    Examiner report: This was generally handled confidently, although some candidates treated the repeated factor as though it required a quadratic numerator, and coefficient-comparison work was vulnerable to sign and arithmetic slips.

    Question paper p.14 · Mark scheme p.16

16. Question 9 (b)

    5 marks · 56/75

    This assesses integrating a partial-fraction decomposition over definite limits and simplifying the exact result.

    Examiner report: Sign and coefficient mistakes were common, particularly in the repeated-factor integral; some candidates also omitted sufficient intermediate work or tried to force the given result instead of checking their integration.

    Question paper p.15 · Mark scheme p.17

17. Question 10 (a)

    5 marks · 61/75

    This assesses locating the maximum of a product involving a linear factor and a square root.

    Examiner report: The product rule was usually recognised, but chain-rule errors with the square root were widespread, and some candidates stopped after finding the x-coordinate or gave the y-coordinate only approximately.

    Question paper p.16 · Mark scheme p.18

18. Question 10 (b)

    5 marks · 66/75

    This assesses evaluating a bounded area by transforming the integral with a specified substitution.

    Examiner report: Many candidates found the endpoint values but then placed the transformed limits in the wrong order; further losses came from incomplete substitution and algebraic errors while simplifying the new integrand.

    Question paper p.17 · Mark scheme p.19

19. Question 11 (a)

    5 marks · 71/75

    This assesses testing whether two three-dimensional lines intersect by comparing their parametric components.

    Examiner report: Some candidates calculated only the direction vector AB rather than the full line equation, while others found parameter values but failed to state clearly why the unmatched third component ruled out an intersection.

    Question paper p.18 · Mark scheme p.20

20. Question 11 (b)

    4 marks · 75/75

    This assesses finding the foot of a perpendicular from a point to a line in three dimensions.

    Examiner report: Successful solutions were generally concise, but many attempts used an incorrect vector in the scalar product or did not know how to begin; early vector errors often prevented the parameter and final point from being found.

    Question paper p.19 · Mark scheme p.21