9709 Pure Mathematics 3 May/June 2021 Paper 32

18 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

   4 marks · 8/75

   Representing simultaneous modulus and argument conditions on an Argand diagram.

   Examiner report: The circle was generally handled better than the half-line; some diagrams placed the ray through the circle’s centre or used inconsistent axis scales. Clearer construction of the axes and loci would have helped.

   Question paper p.3 · Mark scheme p.7

3. Question 3 (a)

   3 marks · 11/75

   Using logarithms to identify a straight-line relationship and its gradient.

   Examiner report: Logarithmic manipulation was often successful, but some responses did not justify linearity, used an unsuitable logarithm base without changing it properly, or gave a decimal instead of an exact gradient.

   Question paper p.4 · Mark scheme p.7

4. Question 3 (b)

   2 marks · 13/75

   Using a graph intercept to determine a parameter in a logarithmic model.

   Examiner report: A common misunderstanding was to set x itself to zero and attempt to use ln 0. Only a minority correctly interpreted the intercept as the point where the horizontal coordinate is ln x equal to zero.

   Question paper p.4 · Mark scheme p.8

5. Question 4

   5 marks · 18/75

   Evaluating an inverse-tangent integral by integration by parts.

   Examiner report: This was one of the harder questions. Some candidates confused inverse tangent with cotangent, while others struggled to create the rational integral or mishandled coefficients and trigonometric values at the limits.

   Question paper p.5 · Mark scheme p.8

6. Question 5

   5 marks · 23/75

   Finding exact square roots of a complex number in Cartesian form.

   Examiner report: Successful solutions usually compared coefficients after squaring a general complex number. Errors arose from incorrect signs, misuse of i², and failure to match the possible values of the two components into valid pairs.

   Question paper p.6 · Mark scheme p.10

7. Question 6 (a)

   3 marks · 26/75

   Proving a trigonometric identity using reciprocal and double-angle formulae.

   Examiner report: The route using a common denominator was usually the clearest. Lost marks often followed missing brackets or an incorrect reciprocal identity; an alternative tangent-based route was possible but tended to create more opportunities for error.

   Question paper p.7 · Mark scheme p.11

8. Question 6 (b)

   4 marks · 30/75

   Using a proved trigonometric identity to evaluate a definite integral exactly.

   Examiner report: Most unsuccessful attempts involved a sign error in the antiderivative or inaccurate trigonometric values. Decimal approximations were particularly unhelpful where an exact result was required.

   Question paper p.7 · Mark scheme p.12

9. Question 7

   7 marks · 37/75

   Solving a first-order separable differential equation with two point conditions.

   Examiner report: Candidates who separated variables correctly generally progressed well. Some assumed the proportionality constant without justification, used the points too early, or stopped after finding a logarithmic relation instead of making y the subject.

   Question paper p.8 · Mark scheme p.13

10. Question 8

    8 marks · 45/75

    Finding stationary points of an exponential-trigonometric product.

    Examiner report: The product rule was often known, but differentiating tan²x and handling signs caused difficulty. Some divided by tan x and lost the stationary point at zero; others retained extraneous negative solutions or stopped before solving the trigonometric condition.

    Question paper p.10 · Mark scheme p.14

11. Question 9 (a)

    5 marks · 50/75

    Decomposing a rational function into partial fractions with a linear and an irreducible quadratic denominator.

    Examiner report: The decomposition structure and coefficient method were widely understood. Bracket omissions in the numerator over the quadratic factor nevertheless led to avoidable algebraic errors.

    Question paper p.12 · Mark scheme p.15

12. Question 9 (b)

    5 marks · 55/75

    Expanding a partial-fraction expression in ascending powers of x.

    Examiner report: Many candidates knew the expansion technique, but transcription between the two parts was a recurring problem, especially when signs and brackets in the denominator were copied inaccurately.

    Question paper p.13 · Mark scheme p.15

13. Question 10 (a)

    3 marks · 58/75

    Translating a geometric area condition into a trigonometric equation.

    Examiner report: This was a challenging part. Finding the missing parallel side was the main hurdle; those who obtained it correctly usually succeeded provided the area equation was written carefully rather than reverse-engineered from the target.

    Question paper p.14 · Mark scheme p.16

14. Question 10 (b)

    2 marks · 60/75

    Using numerical endpoint checks to bracket a root.

    Examiner report: Most candidates could calculate suitable endpoint values. Some interpreted the signs incorrectly, particularly when working with the original function rather than a difference from zero, and therefore failed to justify the reversal of inequalities.

    Question paper p.14 · Mark scheme p.16

15. Question 10 (c)

    2 marks · 62/75

    Showing that a fixed-point iteration has the required root as its limit.

    Examiner report: There were many correct arguments, although some responses retained iteration subscripts unnecessarily or carried out numerical iterations here instead of proving the fixed-point implication.

    Question paper p.15 · Mark scheme p.16

16. Question 10 (d)

    3 marks · 65/75

    Applying a trigonometric fixed-point iteration to approximate a root.

    Examiner report: Those using radians generally performed well. Common weaknesses were choosing an unrelated starting value or stopping after too few iterations, especially when starting from an endpoint of the bracket.

    Question paper p.15 · Mark scheme p.17

17. Question 11 (a)

    4 marks · 69/75

    Using vector lengths and the scalar-product formula to find an angle.

    Examiner report: Many candidates could perform the length and dot-product calculations, but some did not explicitly establish the equal lengths or reported the acute supplementary angle despite obtaining a negative cosine.

    Question paper p.16 · Mark scheme p.17

18. Question 11 (b)

    6 marks · 75/75

    Finding points on a line from a midpoint and a distance ratio.

    Examiner report: This was among the least successful parts. A frequent misconception was to identify the midpoint vector with half of the vector AB; others found the line but did not form the AP length equation or connect it correctly to the stated ratio.

    Question paper p.17 · Mark scheme p.18