9709 Pure Mathematics 3 Oct/Nov 2020 Paper 33

17 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

   4 marks · 8/75

   This assesses interpreting modulus inequalities as loci on an Argand diagram and identifying their common region.

   Examiner report: Errors commonly involved treating one or both loci incorrectly, especially placing the smaller circle wrongly or omitting the overlap; equal scaling and explicit shading would have improved clarity.

   Question paper p.4 · Mark scheme p.7

3. Question 3

   5 marks · 13/75

   This assesses differentiation of parametric equations and simplification using trigonometric identities.

   Examiner report: Most candidates handled this well, but some omitted chain-rule factors or used the trigonometric identities incompletely when simplifying the derivative quotient.

   Question paper p.5 · Mark scheme p.8

4. Question 4

   6 marks · 19/75

   This assesses solving a logarithmic equation while respecting the stated base and the required numerical accuracy.

   Examiner report: Weak responses often confused common logarithms with natural logarithms, skipped the log-law step, discarded a valid negative value, or used the wrong rounding convention.

   Question paper p.6 · Mark scheme p.9

5. Question 5 (a)

   2 marks · 21/75

   This assesses using standard-function sketches to establish the number of solutions of a trigonometric-exponential equation.

   Examiner report: This was one of the less successful parts: many sketches missed the cosecant shape or the exponential curve’s limiting behaviour, so the claimed number of crossings was not convincingly shown.

   Question paper p.7 · Mark scheme p.10

6. Question 5 (b)

   3 marks · 24/75

   This assesses applying a fixed-point iteration accurately in radians and judging convergence to the requested precision.

   Examiner report: Most candidates obtained the limit, but some used degree mode or failed to show enough correctly rounded iterations to support the final accuracy.

   Question paper p.7 · Mark scheme p.10

7. Question 6 (a)

   3 marks · 27/75

   This assesses rewriting a linear combination of sine and cosine in amplitude-phase form.

   Examiner report: A recurring weakness was giving a decimal approximation for the amplitude or quoting inconsistent sine, cosine, and tangent relationships instead of satisfying the requested exact-and-approximate format.

   Question paper p.8 · Mark scheme p.11

8. Question 6 (b)

   4 marks · 31/75

   This assesses using the amplitude-phase result to solve a trigonometric equation over a restricted degree interval.

   Examiner report: The first solution was usually found, whereas the second was often omitted; some candidates also failed to make the substitution involving one-third of the variable or rounded outside the accepted tolerance.

   Question paper p.9 · Mark scheme p.11

9. Question 7 (a)

   3 marks · 34/75

   This assesses direct verification of a complex root of a polynomial.

   Examiner report: A substitution left in unexpanded form was not persuasive enough; candidates needed to show the complex arithmetic explicitly, without relying on a calculator.

   Question paper p.10 · Mark scheme p.12

10. Question 7 (b)

    4 marks · 38/75

    This assesses finding the remaining roots of a cubic with real coefficients after one complex root is known.

    Examiner report: Common mistakes included ignoring the leading coefficient and reporting a linear factor as though it were the associated root; the conjugate root also needed to be stated as one of the requested roots.

    Question paper p.11 · Mark scheme p.12

11. Question 8

    6 marks · 44/75

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

    Examiner report: Most candidates recognised the separable structure, though some mishandled the split rational expression or incorrectly treated the separate antiderivatives as factors rather than as terms in one integrated relation.

    Question paper p.12 · Mark scheme p.14

12. Question 9 (a)

    5 marks · 49/75

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

    Examiner report: The standard decomposition was generally successful; errors arose mainly from arithmetic during coefficient determination or from multiplying through by an inappropriate power of the repeated denominator.

    Question paper p.14 · Mark scheme p.15

13. Question 9 (b)

    5 marks · 54/75

    This assesses obtaining a Maclaurin expansion from partial fractions and combining terms through the quadratic term.

    Examiner report: Sign slips and incorrect powers of the denominator’s constant factor were frequent, and some otherwise correct component expansions were spoiled when the final coefficients were combined.

    Question paper p.15 · Mark scheme p.16

14. Question 10 (a)

    5 marks · 59/75

    This assesses locating a stationary minimum by differentiating a product involving an exponential function.

    Examiner report: Many difficulties came from differentiating the exponential factor incorrectly, especially mishandling its half factor; taking logarithms was an unnecessary route and often complicated the algebra.

    Question paper p.16 · Mark scheme p.17

15. Question 10 (b)

    5 marks · 64/75

    This assesses finding an area by integrating an exponential-product curve between its intercepts with the coordinate axes.

    Examiner report: The repeated integration-by-parts process caused sign and missing-term errors, particularly when exponential integration was mishandled; substituting limits before completing the antiderivative also led to avoidable mistakes.

    Question paper p.17 · Mark scheme p.18

16. Question 11 (a)

    5 marks · 69/75

    This assesses finding the intersection of two three-dimensional lines given in vector form.

    Examiner report: The method was usually understood, but arithmetic slips affected otherwise sound work; checking the resulting point in both line equations would have exposed many of these errors.

    Question paper p.18 · Mark scheme p.19

17. Question 11 (b)

    6 marks · 75/75

    This assesses using the scalar-product formula and vector magnitudes to determine a parameter from an acute angle between two lines.

    Examiner report: Some candidates used position vectors or forgot the absolute/sign alternative, while others simplified a vector length incorrectly; these errors often hid an otherwise correct angle method.

    Question paper p.19 · Mark scheme p.20