9709 Pure Mathematics 3 Oct/Nov 2022 Paper 33

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

1. Question 1

   4 marks · 4/75

   This assesses the use of logarithm laws to form and solve an algebraic equation while respecting logarithm domains and numerical accuracy.

   Examiner report: Most candidates handled the logarithmic manipulation and quadratic solution, but many retained an inadmissible root or failed to give the result in the required decimal form.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   5 marks · 9/75

   This assesses the use of binomial expansions and multiplication of series to obtain terms through a specified power of x.

   Examiner report: The product method was generally effective, although some candidates added rather than multiplied the series or supplied only the requested coefficient instead of the whole expansion.

   Question paper p.3 · Mark scheme p.7

3. Question 3

   5 marks · 14/75

   Question paper p.4 · Mark scheme p.8

4. Question 4

   5 marks · 19/75

   This assesses parametric differentiation, including the chain rule and trigonometric simplification of dy/dx.

   Examiner report: Differentiating x was commonly successful, but many candidates omitted the factor arising from the inner angle in y; clear intermediate work was needed to justify the final simplification.

   Question paper p.5 · Mark scheme p.8

5. Question 5 (a)

   4 marks · 23/75

   This assesses the interpretation of modulus inequalities as geometric regions in an Argand diagram.

   Examiner report: Most candidates recognised the required circle and horizontal boundary, but the translated centre was often misplaced even when the radius was correct.

   Question paper p.6 · Mark scheme p.9

6. Question 5 (b)

   2 marks · 25/75

   This assesses the geometric maximisation of the argument of a complex number within a restricted Argand region.

   Examiner report: This part was difficult for many candidates; successful solutions generally used the relevant intersection on the boundary, whereas weaker diagrams did not make the chosen angle or point clear.

   Question paper p.6 · Mark scheme p.9

7. Question 6

   6 marks · 31/75

   This assesses solving a quadratic with complex coefficients and expressing the resulting roots in rectangular complex form.

   Examiner report: Many candidates used a calculator instead of displaying the required complex-number work; errors in identifying the coefficient signs were also common, while only a small group completed the real-and-imaginary-parts alternative successfully.

   Question paper p.7 · Mark scheme p.10

8. Question 7 (a)

   4 marks · 35/75

   This assesses the conversion of a linear combination of sine and cosine into a single shifted cosine expression.

   Examiner report: The rearrangement was usually recognised, but sign errors in matching the sine coefficient led to incorrect values of the phase angle.

   Question paper p.8 · Mark scheme p.10

9. Question 7 (b)

   4 marks · 39/75

   This assesses applying a previously derived trigonometric form to solve an equation involving a multiple angle over a restricted degree interval.

   Examiner report: Candidates who completed part (a) generally progressed well, though some solved for x instead of 2x or omitted the second admissible angle.

   Question paper p.9 · Mark scheme p.11

10. Question 8 (a)

    3 marks · 42/75

    This assesses differentiation of a quotient involving an exponential function and use of the stationary-point condition.

    Examiner report: The quotient-rule approach was common and usually successful; errors mainly involved reversed numerator terms, a missing denominator, or incomplete algebra after setting the derivative to zero.

    Question paper p.10 · Mark scheme p.11

11. Question 8 (b)

    2 marks · 44/75

    This assesses numerical verification of the location of a root by demonstrating a sign change across a stated interval.

    Examiner report: Several candidates performed the numerical checks but did not finish with a statement explaining what the sign change establishes.

    Question paper p.11 · Mark scheme p.11

12. Question 8 (c)

    3 marks · 47/75

    This assesses fixed-point iteration and the presentation of iterates to justify a rounded numerical root.

    Examiner report: This was answered well overall, but some candidates stopped recording values at three decimal places, which was insufficient for the specified justification.

    Question paper p.11 · Mark scheme p.12

13. Question 9 (a)

    3 marks · 50/75

    This assesses finding position vectors using midpoint and internal-division properties in three-dimensional vector geometry.

    Examiner report: Frequent errors came from confusing a position vector with a vector from an endpoint, reversing the division ratio, or misunderstanding the role of the origin.

    Question paper p.12 · Mark scheme p.12

14. Question 9 (b)

    2 marks · 52/75

    This assesses forming a vector equation for a line from two known points in three-dimensional space.

    Examiner report: The method was widely understood, but omissions of the parameter or reliance on incorrect vectors from part (a) lost marks.

    Question paper p.12 · Mark scheme p.12

15. Question 9 (c)

    4 marks · 56/75

    This assesses finding the intersection of two vector lines by equating their components.

    Examiner report: Many candidates formed the second line correctly and solved the simultaneous equations, but earlier vector errors and the use of one common parameter for both lines often prevented full credit.

    Question paper p.13 · Mark scheme p.13

16. Question 10 (a)

    1 mark · 57/75

    Question paper p.14 · Mark scheme p.13

17. Question 10 (b)

    6 marks · 63/75

    This assesses solving a separable differential equation with an initial condition and using the solution to find a specified time.

    Examiner report: The main obstacles were incorrect separation or integration signs, failure to determine the constant from the initial state, leaving the result implicit, and reporting an exact expression rather than the requested contextual time.

    Question paper p.14 · Mark scheme p.14

18. Question 10 (c)

    2 marks · 65/75

    This assesses rearranging the differential-equation solution to make volume the subject and interpreting its long-term behaviour.

    Examiner report: Only a minority obtained the explicit volume formula; some nevertheless gained the final conclusion by correctly identifying the long-term tendency.

    Question paper p.15 · Mark scheme p.14

19. Question 11 (a)

    5 marks · 70/75

    This assesses decomposition of a rational function into partial fractions with one linear and one irreducible quadratic denominator.

    Examiner report: Nearly all candidates knew the appropriate structure; the few errors generally came from using an incomplete numerator over the quadratic factor or making arithmetic slips when finding the constants.

    Question paper p.16 · Mark scheme p.15

20. Question 11 (b)

    5 marks · 75/75

    This assesses exact integration of a partial-fraction decomposition involving logarithmic and inverse-tangent terms.

    Examiner report: The logarithmic contribution was usually handled correctly, but many candidates placed the scaling factor incorrectly in the inverse-tangent term or treated the quadratic reciprocal as though it integrated to a logarithm.

    Question paper p.17 · Mark scheme p.15