9709 Pure Mathematics 3 Oct/Nov 2021 Paper 33

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

1. Question 1

   3 marks · 3/75

   Polynomial division by a quadratic divisor, including identification of both the quotient and the remainder.

   Examiner report: Long division was the most common approach, but errors frequently occurred in the constant part of the quotient or in one of the remainder terms.

   Question paper p.2 · Mark scheme p.6

2. Question 2 (a)

   1 mark · 4/75

   Sketching the graph of an absolute-value linear function.

   Examiner report: Some sketches were asymmetric or were incorrectly stopped at the vertical axis, even though the graph continues for negative x-values.

   Question paper p.3 · Mark scheme p.6

3. Question 2 (b)

   3 marks · 7/75

   Solving an inequality involving an absolute value and selecting the valid region.

   Examiner report: The graph from part (a) could have helped select the correct branch and solution region; solving all possible linear cases often introduced an irrelevant extra boundary.

   Question paper p.3 · Mark scheme p.7

4. Question 3

   4 marks · 11/75

   Using index laws and logarithms to solve an exponential equation to a specified numerical accuracy.

   Examiner report: A common difficulty was treating the equation as a logarithm problem too early instead of first applying the laws of indices; some solutions also lacked enough intermediate work.

   Question paper p.4 · Mark scheme p.8

5. Question 4

   5 marks · 16/75

   Integration by parts involving a product of x and a trigonometric function, followed by exact evaluation of definite limits.

   Examiner report: Sign mistakes and incorrect factors from integrating the half-angle terms were frequent, and some candidates did not show enough detail when evaluating the trigonometric limits or gave a decimal despite the request for an exact result.

   Question paper p.5 · Mark scheme p.8

6. Question 5

   5 marks · 21/75

   Question paper p.6 · Mark scheme p.9

7. Question 6 (a)

   5 marks · 26/75

   Expressing a linear combination of sine and cosine as a single shifted cosine.

   Examiner report: Most work was successful; the main recurring issue was presenting the amplitude as a decimal instead of in exact form.

   Question paper p.7 · Mark scheme p.9

8. Question 6 (b)

   2 marks · 28/75

   Using the phase-shifted cosine form to locate its minimum over a specified degree interval.

   Examiner report: Many candidates looked for a minimum when the cosine argument was zero or at a right angle rather than at its negative extreme.

   Question paper p.7 · Mark scheme p.9

9. Question 7 (a)

   4 marks · 32/75

   Implicit differentiation of a logarithmic relation using the chain rule.

   Examiner report: The chain-rule step for the logarithm was a significant source of difficulty; rearranging the original equation before differentiating sometimes made errors more likely.

   Question paper p.8 · Mark scheme p.10

10. Question 7 (b)

    3 marks · 35/75

    Finding a horizontal tangent on an implicitly defined curve.

    Examiner report: This part was generally handled well, with most candidates using the zero-gradient condition and then returning to the original equation.

    Question paper p.9 · Mark scheme p.10

11. Question 8 (a)

    5 marks · 40/75

    Constructing a three-dimensional vector equation of a line from points defined by a pyramid's geometry.

    Examiner report: Some candidates gave only the direction vector rather than a full line equation, and others failed to use a vector symbol for the variable position on the line.

    Question paper p.10 · Mark scheme p.10

12. Question 8 (b)

    4 marks · 44/75

    Finding the shortest distance from the origin to a line in three dimensions.

    Examiner report: The scalar-product condition was the most widely used successful method, followed by calculating the length of the perpendicular position vector.

    Question paper p.11 · Mark scheme p.11

13. Question 9 (a)

    4 marks · 48/75

    Differentiating a rational function containing a square-root power and locating its stationary point.

    Examiner report: Sign errors were common in product- and quotient-rule work, especially in the middle term; some quotient-rule solutions also failed to square the denominator.

    Question paper p.12 · Mark scheme p.12

14. Question 9 (b)

    6 marks · 54/75

    Evaluating a radical integral by substituting u equal to the square root of x.

    Examiner report: The substitution itself was often correct, but mistakes followed when changing the integrand or limits; several solutions mixed x and u in the antiderivative or mishandled logarithms of expressions requiring absolute-value consideration.

    Question paper p.13 · Mark scheme p.12

15. Question 10 (a)

    2 marks · 56/75

    Forming a differential equation from a proportional-rate model and using initial data to determine its constant.

    Examiner report: The main obstacle was converting the verbal proportionality statement into an equation with an unknown constant; once that was done, the supplied data were usually used correctly.

    Question paper p.14 · Mark scheme p.13

16. Question 10 (b)

    5 marks · 61/75

    Solving a separable differential equation and applying an initial condition to reach a fixed-point equation.

    Examiner report: Although separation was usually achieved, integration and the treatment of the repeated factor of 19 caused confusion. Some candidates also omitted the integration constant or stopped before converting the logarithmic relation into the required exponential equation.

    Question paper p.14 · Mark scheme p.13

17. Question 10 (c)

    3 marks · 64/75

    Question paper p.15 · Mark scheme p.13

18. Question 10 (d)

    1 mark · 65/75

    Question paper p.15 · Mark scheme p.13

19. Question 11 (a)

    2 marks · 67/75

    Writing a complex number in exponential polar form with the principal argument.

    Examiner report: The modulus was usually found correctly, but the argument was often assigned to the wrong quadrant; an Argand sketch would have helped resolve the ambiguity.

    Question paper p.16 · Mark scheme p.14

20. Question 11 (b)

    2 marks · 69/75

    Applying polar powers to a complex number and establishing that the sixth power is real.

    Examiner report: The instruction to use the earlier polar form was important; restarting with a binomial expansion did not address the intended method.

    Question paper p.16 · Mark scheme p.14

21. Question 11 (c)(i)

    4 marks · 73/75

    Representing simultaneous argument and real-part inequalities on an Argand diagram.

    Examiner report: Correct diagrams needed equal real and imaginary scales, a clearly identified point u, and the vertical boundary with its relevant intersection; omissions of these features led to incomplete loci.

    Question paper p.17 · Mark scheme p.14

22. Question 11 (c)(ii)

    2 marks · 75/75

    Maximising the modulus of a complex number over a bounded region shown on an Argand diagram.

    Examiner report: Some candidates identified the appropriate triangular region but forgot that the imaginary coordinates were measured from the height of u, producing a modulus based on an incorrect vertical displacement.

    Question paper p.17 · Mark scheme p.14