9709 Pure Mathematics 3 May/June 2021 Paper 33

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

1. Question 1

   4 marks · 4/75

   This assesses the use of the binomial expansion for a fractional index, including the correct ordering and simplification of terms.

   Examiner report: Responses were usually successful, but some candidates altered the coefficient inside the bracket, arranged the powers in the wrong order, or multiplied the whole expansion to avoid fractions.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   5 marks · 9/75

   This assesses solving an exponential equation by converting it into a quadratic in a suitable power of the base.

   Examiner report: A number of candidates took logarithms of the original equation and misused logarithm laws. The more effective approach was to substitute for the exponential term and solve the resulting quadratic.

   Question paper p.3 · Mark scheme p.6

3. Question 3 (a)

   5 marks · 14/75

   This assesses differentiation of parametric equations and simplification of the quotient of the two derivatives.

   Examiner report: Some candidates left exponential factors insufficiently simplified, which prevented the final accuracy mark. Other errors came from omitting the derivative of the standalone t or mishandling the product rule.

   Question paper p.4 · Mark scheme p.7

4. Question 3 (b)

   2 marks · 16/75

   This assesses locating a stationary point on a parametrically defined curve and finding its exact ordinate.

   Examiner report: The factorised derivative made this part short for many candidates. Some retained a parameter value excluded by the domain, while others gave a decimal instead of the requested exact form.

   Question paper p.4 · Mark scheme p.7

5. Question 4 (a)

   3 marks · 19/75

   This assesses decomposition of a rational function into partial fractions with linear denominators.

   Examiner report: Most candidates handled this well; the main observed problem was assigning the two constants to the wrong denominators.

   Question paper p.5 · Mark scheme p.8

6. Question 4 (b)

   4 marks · 23/75

   This assesses integrating partial fractions and applying logarithm laws to a definite integral.

   Examiner report: The logarithmic integrations were a source of errors, particularly the factors arising from the inner linear functions. Some also struggled to combine the resulting logarithms into the required format.

   Question paper p.5 · Mark scheme p.8

7. Question 5 (a)

   4 marks · 27/75

   This assesses repeated use of the tangent double-angle identity to derive an algebraic equation in tan θ.

   Examiner report: The first two identity applications were generally secure. Errors commonly arose when squaring the intermediate expression or rearranging; dividing by a tangent factor earlier would have simplified the algebra for many candidates.

   Question paper p.6 · Mark scheme p.8

8. Question 5 (b)

   3 marks · 30/75

   This assesses solving a trigonometric equation from an algebraic equation in tan θ while respecting a degree interval.

   Examiner report: Many candidates found the acute solution but failed to use the negative tangent value to obtain the second angle in the required domain. Unsupported numerical answers could not always receive the method credit.

   Question paper p.7 · Mark scheme p.9

9. Question 6 (a)

   2 marks · 32/75

   This assesses using graph sketches and their intersection behaviour to establish the number of roots of a transcendental equation.

   Examiner report: Several sketches missed important transformation features: the exponential curve should approach its horizontal asymptote and pass through its correct intercept, while the cotangent graph needs the correct endpoint, curvature and behaviour near zero. The intersection corresponding to the root also needed to be indicated.

   Question paper p.8 · Mark scheme p.9

10. Question 6 (b)

    2 marks · 34/75

    This assesses bracketing a numerical root by evaluating an appropriate continuous difference or pair of functions at two endpoints.

    Examiner report: This was generally done well. Correct arithmetic and use of radians were important, and calculator mode was a source of occasional numerical discrepancies.

    Question paper p.8 · Mark scheme p.9

11. Question 6 (c)

    3 marks · 37/75

    This assesses fixed-point iteration and reporting a numerical root to the requested precision.

    Examiner report: Nearly all candidates obtained the required numerical result. The main requirement was to show sufficient iterations, rather than simply presenting an unsupported final value.

    Question paper p.9 · Mark scheme p.9

12. Question 7 (a)(i)

    1 mark · 38/75

    This assesses the relationship between the gradients of a tangent and a normal using the geometry of the diagram.

    Examiner report: Very few candidates supplied a genuine justification; many merely copied the relation to be shown. A short argument involving tangent and normal gradients or corresponding right-angled triangles was needed.

    Question paper p.10 · Mark scheme p.10

13. Question 7 (a)(ii)

    2 marks · 40/75

    This assesses translating the area of a right-angled triangle into a differential equation using the result from part (i).

    Examiner report: Full credit required explicit statements for the triangle area and the substitutions for its two perpendicular sides. Omitting these links made otherwise correct algebra insufficiently justified.

    Question paper p.10 · Mark scheme p.10

14. Question 7 (b)

    6 marks · 46/75

    This assesses solving a separable first-order differential equation and applying an initial condition.

    Examiner report: Many solutions were clear and complete. A recurring issue was assigning the wrong sign to the integral of the tangent function, with some candidates incorrectly treating it as an integral of a squared secant.

    Question paper p.11 · Mark scheme p.10

15. Question 8 (a)

    4 marks · 50/75

    This assesses differentiation of a logarithmic power function to locate a maximum and give exact coordinates.

    Examiner report: Most solutions were strong. Errors in the quotient rule included incorrect signs, reversed numerator terms and missing powers in the denominator; the logarithmic equation also had to be solved exactly rather than numerically.

    Question paper p.12 · Mark scheme p.11

16. Question 8 (b)

    6 marks · 56/75

    This assesses integration by parts and the use of the resulting expression to prove a strict upper bound for an improper-looking family of finite integrals.

    Examiner report: The integration by parts was often carried out competently, but the final inequality was harder. Errors included incorrect signs or constants, choosing a particular numerical a, and not making the comparison needed for all permitted values of a.

    Question paper p.13 · Mark scheme p.11

17. Question 9 (a)

    3 marks · 59/75

    This assesses vector subtraction and the use of a parallel-side relation to determine an unknown position vector.

    Examiner report: Most candidates were successful, with occasional arithmetic slips and confusion between the directions of DC and CD.

    Question paper p.14 · Mark scheme p.12

18. Question 9 (b)

    1 mark · 60/75

    This assesses writing a vector equation for a line through two given points.

    Examiner report: Many candidates knew the direction vector but did not present a complete vector equation, particularly failing to identify the right-hand side as a general position vector.

    Question paper p.14 · Mark scheme p.12

19. Question 9 (c)

    5 marks · 65/75

    This assesses finding the perpendicular separation of parallel vector lines and then applying the trapezium area formula.

    Examiner report: A wide range of valid vector and geometric methods was seen. The main omission was completing only one of the two requested tasks, namely finding the separation but not the area, or vice versa.

    Question paper p.15 · Mark scheme p.12

20. Question 10 (a)

    3 marks · 68/75

    This assesses direct verification of a complex root in a polynomial equation without relying on a calculator.

    Examiner report: Most candidates verified the root correctly. Some unnecessarily attempted factorisation; the expected evidence was a careful direct expansion, with enough intermediate terms to show that the arithmetic was not simply calculator output.

    Question paper p.16 · Mark scheme p.13

21. Question 10 (b)

    7 marks · 75/75

    This assesses factorisation of a quartic over the complex numbers and solution of the resulting quadratic factors.

    Examiner report: Correct roots without supporting algebra received limited credit because working was required. The successful approach usually formed the factor from the conjugate pair before finding and solving the second quadratic; sign errors and incomplete treatment of the remaining complex roots were common.

    Question paper p.16 · Mark scheme p.13