9709 Pure Mathematics 3 May/June 2023 Paper 31

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

1. Question 1

   3 marks · 3/75

   Solving an exponential equation by converting it into a quadratic in an exponential expression and then using logarithms.

   Examiner report: The report found this accessible when treated as a quadratic, but errors arose from applying logarithm rules incorrectly and accepting a negative exponential value.

   Question paper p.2 · Mark scheme p.6

2. Question 2 (a)

   1 mark · 4/75

   Sketching the graph of a modulus function.

   Examiner report: A number of sketches were curved or placed the vertex on the wrong axis; the meeting point and the straight-line shape were frequent weaknesses.

   Question paper p.3 · Mark scheme p.6

3. Question 2 (b)

   3 marks · 7/75

   Solving a linear inequality containing an absolute value.

   Examiner report: Although squaring was possible, it often produced an additional irrelevant boundary that was wrongly included in the final solution.

   Question paper p.3 · Mark scheme p.7

4. Question 3

   4 marks · 11/75

   Question paper p.4 · Mark scheme p.8

5. Question 4 (a)

   2 marks · 13/75

   Using double-angle identities to transform a trigonometric equation.

   Examiner report: The identities were often used correctly, but some solutions did not explicitly complete the rearrangement into an equation in the requested form.

   Question paper p.5 · Mark scheme p.8

6. Question 4 (b)

   4 marks · 17/75

   Solving a trigonometric equation over a specified interval after algebraic factorisation.

   Examiner report: The factorised route was successful for some candidates, whereas incorrect trigonometric identities led others to equations that could not be solved reliably.

   Question paper p.5 · Mark scheme p.9

7. Question 5 (a)

   4 marks · 21/75

   Implicit differentiation of an equation containing a product of x and y.

   Examiner report: Some candidates differentiated the constant as though it varied, or treated the squared-y term as a product involving the constant.

   Question paper p.6 · Mark scheme p.10

8. Question 5 (b)

   4 marks · 25/75

   Finding points on an implicitly defined curve where the tangent is vertical.

   Examiner report: There was confusion between vertical and horizontal tangents, and some candidates substituted an inappropriate value for y or kept a solution that contradicted the resulting conditions.

   Question paper p.7 · Mark scheme p.10

9. Question 6 (a)

   3 marks · 28/75

   Using position vectors and the ordered vertices of a parallelogram to locate a missing vertex.

   Examiner report: Many solutions used the wrong directed side relationship, suggesting that the sequential naming of the vertices had not been taken into account; explanations were also often unclear.

   Question paper p.8 · Mark scheme p.11

10. Question 6 (b)

    3 marks · 31/75

    Finding the cosine of an angle between two three-dimensional vectors using the scalar product.

    Examiner report: The scalar-product method was generally understood, but some candidates stopped at an unsimplified radical expression or gave a decimal despite the demand for an exact value.

    Question paper p.9 · Mark scheme p.11

11. Question 6 (c)

    4 marks · 35/75

    Finding the area of a parallelogram from two side lengths and the angle between them.

    Examiner report: Only a small proportion used the straightforward relationship between the parallelogram and the two equal triangles; this part was frequently left blank.

    Question paper p.9 · Mark scheme p.12

12. Question 7

    8 marks · 43/75

    Solving a separable first-order differential equation using trigonometric integration and an initial condition.

    Examiner report: Separation was often achieved, but coefficient errors in the trigonometric integral and incomplete treatment of the x-integral prevented many solutions from reaching the numerical result.

    Question paper p.10 · Mark scheme p.13

13. Question 8 (a)

    5 marks · 48/75

    Decomposing a rational function into partial fractions with a repeated linear factor.

    Examiner report: The partial-fraction structure was usually recognised, although arithmetic slips affected the constants; the alternative two-term setup was valid but less convenient later.

    Question paper p.12 · Mark scheme p.14

14. Question 8 (b)

    5 marks · 53/75

    Integrating a partial-fraction decomposition over finite limits and presenting an exact result.

    Examiner report: Sign and coefficient errors in logarithms were common, and the reciprocal-square term was sometimes incorrectly integrated or turned into an unnecessary extra logarithm.

    Question paper p.13 · Mark scheme p.14

15. Question 9 (a)

    5 marks · 58/75

    Using integration by parts twice to derive an equation for an unknown limit.

    Examiner report: The integration process was generally familiar, but applying the limits and completing the algebraic rearrangement caused substantial difficulty.

    Question paper p.14 · Mark scheme p.15

16. Question 9 (b)

    2 marks · 60/75

    Verifying a numerical bracket for a root using endpoint calculations.

    Examiner report: Some responses asserted the interval without numerical support, while others left the part unanswered despite several acceptable ways of checking it.

    Question paper p.15 · Mark scheme p.15

17. Question 9 (c)

    3 marks · 63/75

    Applying fixed-point iteration to obtain a specified numerical accuracy.

    Examiner report: The iteration was often set up correctly, but insufficient working accuracy or an incorrect final rounding decision led to lost marks.

    Question paper p.15 · Mark scheme p.15

18. Question 10 (a)

    2 marks · 65/75

    Applying the factor theorem to a polynomial.

    Examiner report: This was one of the more commonly completed parts; obtaining the quadratic quotient by division was also a useful route for the later work.

    Question paper p.16 · Mark scheme p.16

19. Question 10 (b)

    3 marks · 68/75

    Verifying a specified complex root of a polynomial.

    Examiner report: Using the quadratic factor was usually simpler than direct cubic expansion, but many candidates either omitted the verification or relied on calculator-generated roots without demonstrating substitution.

    Question paper p.16 · Mark scheme p.17

20. Question 10 (c)

    7 marks · 75/75

    Finding all complex square roots of the roots of a polynomial.

    Examiner report: The connection with the preceding parts was often missed; attempts commonly began with an unproductive direct squaring of one earlier root, and many responses were absent.

    Question paper p.17 · Mark scheme p.18