9709 Pure Mathematics 3 Oct/Nov 2021 Paper 31

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

1. Question 1

   4 marks · 4/75

   Question paper p.2 · Mark scheme p.6

2. Question 2 (a)

   3 marks · 7/75

   This tests rewriting a linear combination of sine and cosine as a single shifted sine function.

   Examiner report: Some candidates handled the angle calculation but then failed to give the amplitude in exact form, particularly when they found the angle before determining R.

   Question paper p.3 · Mark scheme p.7

3. Question 2 (b)

   2 marks · 9/75

   This assesses using the range of a sinusoidal expression to find the extrema of its square.

   Examiner report: Only a small proportion used the range of a shifted sine correctly; many responses were absent or did not address the effect of squaring.

   Question paper p.3 · Mark scheme p.7

4. Question 3 (a)

   4 marks · 13/75

   This tests finding a stationary point by differentiating a product involving an exponential function.

   Examiner report: There were several reasonable uses of the product rule, but errors in the subsequent algebra prevented some candidates from obtaining the full coordinate pair.

   Question paper p.4 · Mark scheme p.8

5. Question 3 (b)

   2 marks · 15/75

   This assesses classifying a stationary point as a maximum or minimum.

   Examiner report: Several methods were available, but candidates who calculated values without identifying whether they represented heights or gradients often failed to justify the classification.

   Question paper p.4 · Mark scheme p.8

6. Question 4

   6 marks · 21/75

   Question paper p.5 · Mark scheme p.9

7. Question 5 (a)

   3 marks · 24/75

   This assesses converting cotangent identities into a quadratic equation in tan θ.

   Examiner report: Most attempted responses showed awareness of the required trigonometric identities, although rearrangement and arithmetic mistakes frequently spoiled the final quadratic.

   Question paper p.6 · Mark scheme p.10

8. Question 5 (b)

   3 marks · 27/75

   This tests solving the trigonometric equation through the quadratic in tan θ and selecting all angles in the stated interval.

   Examiner report: A correct quadratic often led to one valid angle, but some candidates did not use tangent's periodic behavior to recover the second obtuse solution.

   Question paper p.6 · Mark scheme p.10

9. Question 6

   6 marks · 33/75

   This assesses using a binomial expansion and coefficient comparison to determine unknown constants.

   Examiner report: Many candidates did not start the expansion of the square root. Those who did generally formed suitable equations in a and b and could then solve them.

   Question paper p.7 · Mark scheme p.11

10. Question 7 (a)

    1 mark · 34/75

    This tests differentiating a nested logarithm by the chain rule.

    Examiner report: Because the derivative was supplied, marks depended on showing a convincing chain-rule calculation rather than simply repeating the result.

    Question paper p.8 · Mark scheme p.12

11. Question 7 (b)

    7 marks · 41/75

    This assesses solving a separable differential equation using an initial condition.

    Examiner report: A number of candidates inserted the initial condition before separating the variables, so they missed the role of the logarithmic derivative from part (a). Only a small group progressed through the integration successfully.

    Question paper p.8 · Mark scheme p.12

12. Question 7 (c)

    1 mark · 42/75

    This tests interpreting the limiting behavior of the solution of a differential equation.

    Examiner report: Many responses to this part were missing or based on an incorrect form of the solution, so the required interpretation could not be made.

    Question paper p.9 · Mark scheme p.12

13. Question 8 (a)

    5 marks · 47/75

    This assesses integration by parts followed by rearrangement into a fixed-point equation.

    Examiner report: Only a limited number recognised the integration-by-parts structure, and coefficient errors were common even when the overall method was appropriate.

    Question paper p.10 · Mark scheme p.13

14. Question 8 (b)

    2 marks · 49/75

    This tests verifying that a solution of an implicit equation lies within a specified interval by numerical comparison.

    Examiner report: The strongest responses supplied numerical endpoint comparisons and drew the interval conclusion. Merely saying that such a comparison should be made was not sufficient evidence.

    Question paper p.11 · Mark scheme p.13

15. Question 8 (c)

    3 marks · 52/75

    This assesses using fixed-point iteration to approximate a solution to a specified accuracy.

    Examiner report: Many candidates did not attempt iteration, while some appeared to use a calculator-generated solution as their starting value. Insufficient decimal accuracy in the displayed iterations also lost marks.

    Question paper p.11 · Mark scheme p.13

16. Question 9 (a)

    2 marks · 54/75

    This tests using the scalar product of direction vectors to establish that two lines are perpendicular.

    Examiner report: Candidates who selected the direction vectors and evaluated their scalar product correctly generally obtained both marks.

    Question paper p.12 · Mark scheme p.14

17. Question 9 (b)

    5 marks · 59/75

    This assesses finding and verifying the intersection of two three-dimensional parametric lines.

    Examiner report: Two recurring omissions were failing to check the third component after finding parameter values and proving intersection without actually giving the intersection vector.

    Question paper p.12 · Mark scheme p.14

18. Question 9 (c)

    4 marks · 63/75

    This tests finding the shortest distance from the origin to a line in three dimensions.

    Examiner report: Very few candidates began with a general point on the line and the perpendicularity condition, which was the most direct route to progress in this part.

    Question paper p.13 · Mark scheme p.15

19. Question 10 (a)

    4 marks · 67/75

    This assesses using a known non-real root of a polynomial with real coefficients to determine its real parameters.

    Examiner report: The minority who attempted this usually computed the square and cube of the given complex number before substitution. Comparing the two components was then needed to finish the parameter calculation.

    Question paper p.14 · Mark scheme p.16

20. Question 10 (b)

    1 mark · 68/75

    Question paper p.14 · Mark scheme p.16

21. Question 10 (c)

    2 marks · 70/75

    This assesses factoring a real-coefficient polynomial using a conjugate pair of complex roots.

    Examiner report: Very few candidates showed how the conjugate pair produces a quadratic with real coefficients, which was the key structural idea in this part.

    Question paper p.15 · Mark scheme p.16

22. Question 10 (d)(i)

    4 marks · 74/75

    This tests representing simultaneous modulus and argument conditions on an Argand diagram.

    Examiner report: Candidates familiar with geometric interpretations of modulus and argument generally produced clear diagrams; others struggled to combine the circular and angular restrictions into one shaded region.

    Question paper p.15 · Mark scheme p.16

23. Question 10 (d)(ii)

    1 mark · 75/75

    Question paper p.15 · Mark scheme p.16