9709 Pure Mathematics 3 Feb/Mar 2024 Paper 32

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

1. Question 1

   3 marks · 3/75

   This assesses polynomial division by a quadratic and identification of the resulting quotient and remainder.

   Examiner report: Most candidates handled this confidently. The usual losses came from arithmetic or sign slips, often caused by untidy alignment during long division.

   Question paper p.3 · Mark scheme p.6

2. Question 2 (a)

   4 marks · 7/75

   This assesses use of the binomial expansion to obtain a specified coefficient after multiplication by a linear factor.

   Examiner report: A common difficulty was putting the square root into the correct factorised form before applying the expansion. Multiplication by the linear factor was generally more successful, even when the earlier expansion was imperfect.

   Question paper p.4 · Mark scheme p.7

3. Question 2 (b)

   1 mark · 8/75

   Question paper p.4 · Mark scheme p.7

4. Question 3 (a)

   3 marks · 11/75

   This assesses conversion of a squared complex number into principal modulus-argument form.

   Examiner report: Expanding the square first tended to prevent quadrant mistakes. Those who doubled an argument without checking its branch sometimes selected the wrong principal value.

   Question paper p.5 · Mark scheme p.8

5. Question 3 (b)

   3 marks · 14/75

   This assesses simultaneous use of modulus and argument conditions for a product and quotient of complex numbers.

   Examiner report: Many responses used only the modulus information. The second condition requires considering the relevant multiples of a straight angle, which produces two possibilities.

   Question paper p.5 · Mark scheme p.8

6. Question 4

   4 marks · 18/75

   This assesses logarithm laws and the solution of a pair of linear equations in logarithms.

   Examiner report: The initial logarithmic rearrangements were often correct, but some candidates did not complete the simultaneous-equation step or the final linear combination.

   Question paper p.6 · Mark scheme p.9

7. Question 5 (a)

   4 marks · 22/75

   This assesses interpretation of complex-modulus inequalities as loci on an Argand diagram and the shading of their intersection.

   Examiner report: Many circles were drawn correctly, but diagrams often lacked a clear centre, radius, scale, or correctly placed vertical boundary. A ruler and compass would have improved the construction.

   Question paper p.7 · Mark scheme p.10

8. Question 5 (b)

   2 marks · 24/75

   This assesses geometric maximisation of the argument of points in a specified Argand-plane region.

   Examiner report: Some candidates recognised the relevant extremal geometry from a correct sketch and completed the trigonometry. Others either left the part blank or used the wrong portion of the region.

   Question paper p.7 · Mark scheme p.10

9. Question 6 (a)

   4 marks · 28/75

   This assesses implicit differentiation of a relation containing products and a squared dependent variable.

   Examiner report: Most candidates knew the method, but notation errors occurred when derivatives were attached indiscriminately to terms. Inconsistent use of the equation being differentiated also prevented some otherwise correct solutions from earning all marks.

   Question paper p.8 · Mark scheme p.11

10. Question 6 (b)

    3 marks · 31/75

    This assesses using the derivative condition for a horizontal tangent together with the original curve equation.

    Examiner report: A frequent error was setting the denominator rather than the numerator to zero. Others formed the correct quadratic but asserted the absence of real solutions without providing a supporting calculation.

    Question paper p.9 · Mark scheme p.11

11. Question 7 (a)

    3 marks · 34/75

    This assesses differentiation of a product involving an exponential, followed by rearrangement of the stationary-point condition.

    Examiner report: Most candidates differentiated and rearranged successfully. The main weakness was failing to use the product rule or stopping before the required logarithmic form.

    Question paper p.10 · Mark scheme p.12

12. Question 7 (b)

    2 marks · 36/75

    This assesses numerical verification that a root of a fixed-point equation lies inside a stated interval.

    Examiner report: Solutions were generally strong. A few calculated endpoint values but did not explicitly connect them to the required conclusion.

    Question paper p.11 · Mark scheme p.12

13. Question 7 (c)

    3 marks · 39/75

    This assesses fixed-point iteration and justified rounding of a numerical root.

    Examiner report: Most candidates completed this well. Some stated a rounded value without showing that the iteration had settled sufficiently to justify it.

    Question paper p.11 · Mark scheme p.12

14. Question 8 (a)

    4 marks · 43/75

    This assesses expressing a linear combination of sine and cosine as a single shifted sine function.

    Examiner report: Responses were often either fully correct or made little progress. Some used the coefficients in the original expression as though they were already the amplitude and phase data, and a few worked in degrees.

    Question paper p.12 · Mark scheme p.13

15. Question 8 (b)

    5 marks · 48/75

    This assesses solving a trigonometric equation using the amplitude-phase form and listing every solution in a bounded interval.

    Examiner report: The main difficulty was completeness: many candidates found one or two roots but missed further periodic solutions or retained values outside the interval. The result from part (a) was often used effectively even when that part was not fully correct.

    Question paper p.13 · Mark scheme p.13

16. Question 9 (a)

    4 marks · 52/75

    This assesses vector verification that a three-dimensional quadrilateral is a rectangle.

    Examiner report: Many candidates established parallel or equal opposite sides but stopped at proving only a parallelogram. A perpendicularity check was still needed to establish the stronger classification.

    Question paper p.14 · Mark scheme p.14

17. Question 9 (b)

    4 marks · 56/75

    This assesses finding the acute angle between two spatial diagonals using the scalar-product formula.

    Examiner report: The instruction to use a scalar product was important; non-vector trigonometric approaches received no credit. Some candidates calculated an angle between a side and a diagonal instead of between the two diagonals.

    Question paper p.15 · Mark scheme p.15

18. Question 10 (a)

    5 marks · 61/75

    This assesses decomposition of a rational function into partial fractions with three linear factors.

    Examiner report: This part was generally well answered. Where marks were lost, the coefficient calculations were often longer than necessary rather than conceptually wrong.

    Question paper p.16 · Mark scheme p.16

19. Question 10 (b)

    5 marks · 66/75

    This assesses integrating a partial-fraction expression over finite limits and simplifying logarithms to the required exact form.

    Examiner report: Many candidates earned the integration marks but struggled with the endpoint substitutions and simplification. Substituting an arbitrary value for the parameter could give early progress but did not meet the exact-form requirement.

    Question paper p.17 · Mark scheme p.17

20. Question 11

    9 marks · 75/75

    This assesses solving a separable differential equation involving integration by parts and a double-angle trigonometric integral.

    Examiner report: Separation was commonly recognised, while the repeated integration on the left caused sign and method errors. On the right, some candidates did not convert the double-angle expression into a standard secant-squared integral. Exact rather than decimal presentation was required at the end.

    Question paper p.18 · Mark scheme p.18