9709 Pure Mathematics 3 Feb/Mar 2021 Paper 32

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

1. Question 1

   3 marks · 3/75

   Solving a logarithmic equation by applying logarithm laws and then solving the resulting algebraic equation.

   Examiner report: The logarithmic manipulation was generally handled well, but some responses used three decimal places instead of the required three significant figures.

   Question paper p.3 · Mark scheme p.6

2. Question 2

   5 marks · 8/75

   Using the factor and remainder theorems to determine two unknown coefficients in a polynomial.

   Examiner report: The remainder theorem was usually an effective approach; polynomial division was valid but appeared to create more opportunities for arithmetic mistakes.

   Question paper p.4 · Mark scheme p.6

3. Question 3

   6 marks · 14/75

   Converting a trigonometric equation into a quadratic in tan x and finding every solution in a specified angular interval.

   Examiner report: Typical errors involved treating cotangent incorrectly, dropping factors while clearing a denominator, and rejecting the second-quadrant solution because its tangent arose from a negative value.

   Question paper p.5 · Mark scheme p.7

4. Question 4 (a)

   6 marks · 20/75

   Solving a first-order separable differential equation using integration and an initial condition.

   Examiner report: Separation was usually successful, although sign errors in the trigonometric integral and an incorrect value for cos π sometimes derailed the constant-finding step.

   Question paper p.6 · Mark scheme p.7

5. Question 4 (b)

   1 mark · 21/75

   Sketching a trigonometric solution curve over one complete interval using its key geometric features.

   Examiner report: This sketch was a common difficulty: many drawings missed one or more of the endpoint intercepts or lacked the required symmetry.

   Question paper p.7 · Mark scheme p.8

6. Question 5 (a)

   3 marks · 24/75

   Rewriting a linear combination of sine and cosine as a single phase-shifted sine function.

   Examiner report: The main weaknesses were failure to give the amplitude exactly, insufficient angular accuracy, and confusing which coefficient determines the phase angle.

   Question paper p.8 · Mark scheme p.8

7. Question 5 (b)

   5 marks · 29/75

   Solving a transformed compound-angle trigonometric equation over a restricted interval.

   Examiner report: Some candidates did not make the double-angle substitution and others discarded a negative intermediate angle instead of converting it to a valid angle in the required range.

   Question paper p.9 · Mark scheme p.8

8. Question 6 (a)

   3 marks · 32/75

   Decomposing a rational function with two linear factors into partial fractions.

   Examiner report: The parameter a made the constants less familiar; choosing arbitrary numerical values for x often caused trouble unless the dependence on a was restored correctly.

   Question paper p.10 · Mark scheme p.9

9. Question 6 (b)

   4 marks · 36/75

   Evaluating a definite integral after partial-fraction decomposition.

   Examiner report: Although logarithmic antiderivatives were recognised, the sign on the term involving the decreasing linear factor was often wrong; absolute-value notation was also needed in some alternative approaches.

   Question paper p.11 · Mark scheme p.9

10. Question 7 (a)

    5 marks · 41/75

    Determining whether two three-dimensional lines intersect and whether their directions are parallel, in order to establish that they are skew.

    Examiner report: A frequent conceptual error was testing the position vectors rather than the direction vectors when deciding whether the lines were parallel; parallel directions need only be proportional, not identical.

    Question paper p.12 · Mark scheme p.10

11. Question 7 (b)

    3 marks · 44/75

    Finding the acute angle between two lines from their direction vectors.

    Examiner report: Most solutions were successful, but some used position or line vectors, or paired a dot product with magnitudes calculated from different vectors.

    Question paper p.13 · Mark scheme p.10

12. Question 8 (a)

    3 marks · 47/75

    Dividing complex numbers and expressing the result in Cartesian form.

    Examiner report: Working was generally shown, as required, but arithmetic slips in the conjugate multiplication were common; a calculator could not replace the required complex-number method.

    Question paper p.14 · Mark scheme p.11

13. Question 8 (b)

    2 marks · 49/75

    Converting a complex number from Cartesian form to exact polar-exponential form.

    Examiner report: The modulus was usually found correctly, whereas the argument was often given in degrees or as a decimal rather than as an exact radian value; an Argand sketch would have helped with the quadrant.

    Question paper p.14 · Mark scheme p.11

14. Question 8 (c)

    2 marks · 51/75

    Interpreting complex-number addition and scalar multiplication as vector geometry in an Argand diagram.

    Examiner report: Many responses identified parallelism but stopped before stating how the length and direction of BC compare with OA.

    Question paper p.15 · Mark scheme p.11

15. Question 8 (d)

    2 marks · 53/75

    Using complex arguments to determine the angle between two directed lines in an Argand diagram.

    Examiner report: Because the target value was supplied, full credit depended on showing the connection between the geometric angle and the quotient's argument; merely subtracting separate inverse-tangent expressions was not sufficient.

    Question paper p.15 · Mark scheme p.12

16. Question 9 (a)

    2 marks · 55/75

    Verifying that a fixed point of a function lies inside a given interval by comparing the function with the identity function.

    Examiner report: Some candidates calculated several endpoint values but failed to compare each function value with the corresponding x-value, so the sign-change argument was incomplete.

    Question paper p.16 · Mark scheme p.13

17. Question 9 (b)

    3 marks · 58/75

    Using fixed-point iteration to approximate a root and reporting the convergence to the requested accuracy.

    Examiner report: The iteration was generally carried out successfully, though a small number of final values were rounded incorrectly.

    Question paper p.16 · Mark scheme p.13

18. Question 9 (c)

    6 marks · 64/75

    Differentiating an exponential quotient and solving the resulting equation exactly for a prescribed derivative value.

    Examiner report: The quotient-rule differentiation was usually sound, but errors arose in forming the quadratic; some solutions retained an invalid root or gave a decimal despite the request for an exact form.

    Question paper p.17 · Mark scheme p.14

19. Question 10 (a)

    5 marks · 69/75

    Finding the exact area under a trigonometric curve by substitution and standard identities.

    Examiner report: Once the integrand had been expressed correctly in the new variable, the integration was usually completed; common failures came from mishandling the identities or retaining the original x-limits.

    Question paper p.18 · Mark scheme p.15

20. Question 10 (b)

    6 marks · 75/75

    Locating the maximum of a product of trigonometric functions using differentiation, the chain rule, and trigonometric identities.

    Examiner report: The principal difficulty was retaining the chain-rule factors when differentiating the multiple-angle terms, which prevented some candidates from obtaining a solvable trigonometric condition.

    Question paper p.19 · Mark scheme p.15