9709 Pure Mathematics 3 Oct/Nov 2020 Paper 31

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 circular loci on an Argand diagram and identifying their common region.

   Examiner report: Some responses treated the loci as something other than circles or placed the smaller circle at the wrong point. The scale need not be exact, but the centres, overlap and shading should be unambiguous.

   Question paper p.4 · Mark scheme p.7

3. Question 3

   5 marks · 13/75

   This assesses differentiating parametric equations and simplifying the quotient of the two derivatives with double-angle identities.

   Examiner report: The main errors were omitting chain-rule factors and failing to simplify both numerator and denominator with suitable trigonometric identities. The standard parametric derivative formula was otherwise well understood.

   Question paper p.5 · Mark scheme p.8

4. Question 4

   6 marks · 19/75

   This assesses applying base-10 logarithm laws and solving the resulting quadratic to the required decimal precision.

   Examiner report: Weak solutions often used the wrong exponential base, skipped the logarithm laws, or rounded to three significant figures instead of three decimal places. Some also discarded the negative solution without checking the original logarithmic domain.

   Question paper p.6 · Mark scheme p.9

5. Question 5 (a)

   2 marks · 21/75

   This assesses using qualitative graph features to establish that a trigonometric–exponential equation has exactly two intersections in the stated interval.

   Examiner report: This was one of the less successful parts. Frequent omissions included the U-shaped cosecant branch, its asymptotes, the correct initial value of the exponential curve and its approach towards its horizontal asymptote.

   Question paper p.7 · Mark scheme p.10

6. Question 5 (b)

   3 marks · 24/75

   This assesses carrying out fixed-point iteration accurately and using successive approximations to round a root.

   Examiner report: Most candidates handled the iteration effectively, but some used degree mode or failed to display every intermediate value to the requested precision. Rechecking the calculator sequence would help detect such slips.

   Question paper p.7 · Mark scheme p.10

7. Question 6 (a)

   3 marks · 27/75

   This assesses converting a linear combination of sine and cosine into a single shifted cosine and determining exact amplitude with a rounded phase angle.

   Examiner report: The reliable route was to compare coefficients after expansion. Errors commonly came from interchanging the sine and cosine relationships, giving a decimal rather than exact amplitude, or reporting the angle to the wrong accuracy.

   Question paper p.8 · Mark scheme p.11

8. Question 6 (b)

   4 marks · 31/75

   This assesses solving a trigonometric equation by using the amplitude–phase form from the preceding part and applying the required interval to the transformed variable.

   Examiner report: Most candidates connected this part correctly to the previous result, but some forgot to transform the angle argument. The second solution was often missed, and premature rounding sometimes moved a value outside the accepted range.

   Question paper p.9 · Mark scheme p.11

9. Question 7 (a)

   3 marks · 34/75

   This assesses verifying a complex root of a polynomial by explicit algebraic substitution.

   Examiner report: A substitution left as an unevaluated polynomial expression was not convincing evidence of verification. Candidates were expected to reduce all powers of the imaginary unit and show the real and imaginary parts cancelling.

   Question paper p.10 · Mark scheme p.12

10. Question 7 (b)

    4 marks · 38/75

    This assesses using conjugate roots and polynomial factorisation to determine the remaining roots of a polynomial with real coefficients.

    Examiner report: A common omission was failing to state the conjugate root explicitly. Other errors arose from ignoring the leading coefficient of the cubic or confusing a linear factor with the value of its zero; calculator-only answers did not show the required method.

    Question paper p.11 · Mark scheme p.12

11. Question 8

    6 marks · 44/75

    This assesses solving a first-order separable differential equation and applying an initial condition to determine the integration constant.

    Examiner report: Most candidates separated the variables and integrated successfully. Some mishandled the split rational expression or tried to multiply separately integrated results, while modulus notation was generally unnecessary for the conditions in this question.

    Question paper p.12 · Mark scheme p.14

12. Question 9 (a)

    5 marks · 49/75

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

    Examiner report: The majority obtained the decomposition, though arithmetic mistakes were fairly common. A frequent structural error was multiplying by an unnecessary extra power of the repeated factor, and coefficient comparison sometimes led to avoidable algebraic slips.

    Question paper p.14 · Mark scheme p.15

13. Question 9 (b)

    5 marks · 54/75

    This assesses using binomial-series expansions for the partial-fraction terms and combining them through the quadratic term.

    Examiner report: The initial binomial expansions were often correct, but signs and factors from the constant denominator were frequently mishandled when combining them. Another source of error was failing to distribute a numerator factor across the expansion.

    Question paper p.15 · Mark scheme p.16

14. Question 10 (a)

    5 marks · 59/75

    This assesses differentiating a product involving an exponential function and locating the stationary minimum of the curve.

    Examiner report: The main difficulty was differentiating the exponential factor correctly, especially retaining the divisor from its exponent. Some candidates knew to set the derivative to zero but took an unnecessary logarithm instead of cancelling the non-zero exponential factor.

    Question paper p.16 · Mark scheme p.17

15. Question 10 (b)

    5 marks · 64/75

    This assesses finding the bounded area by integrating the curve, using integration by parts repeatedly and applying the axis intercepts as limits.

    Examiner report: Many candidates found this manageable but lost terms or signs when performing the integrations separately. Substituting limits before completing and simplifying the antiderivative also led to errors.

    Question paper p.17 · Mark scheme p.18

16. Question 11 (a)

    5 marks · 69/75

    This assesses finding the intersection of two vector lines by equating corresponding components and then determining the unknown parameter in a direction vector.

    Examiner report: Many solutions were excellent, but isolated arithmetic errors reduced otherwise correct work. Checking the resulting point in both line equations would have exposed several of these mistakes.

    Question paper p.18 · Mark scheme p.19

17. Question 11 (b)

    6 marks · 75/75

    This assesses finding a parameter from the acute angle between two line directions using the scalar-product formula and vector magnitudes.

    Examiner report: Some candidates used position vectors instead of direction vectors or omitted the absolute-value/sign issue associated with an acute angle between lines. Algebraic mistakes in simplifying a vector length were also common, and the final mark was often lost when only one parameter value was found.

    Question paper p.19 · Mark scheme p.20