9709 Pure Mathematics 3 May/June 2024 Paper 33

16 questions and parts. Marks come from the published mark scheme.

Source note: published mark scheme is missing question 11.

1. Question 1

   4 marks

   This assesses solving an exponential equation by applying logarithm laws and handling the unknown in the indices.

   Examiner report: Most candidates used a valid logarithmic approach, although some introduced errors while simplifying powers before taking logs. Both logarithm bases and alternative preliminary index rearrangements were acceptable.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   5 marks

   This assesses differentiating a product involving an exponential and a trigonometric function, then locating a stationary point in a specified interval.

   Examiner report: The main difficulty was unnecessary expansion of the double-angle terms, which made the subsequent work harder. Some responses also included an extra value of the variable or failed to provide the ordinate.

   Question paper p.3 · Mark scheme p.8

3. Question 3

   5 marks

   This assesses finding complex square roots in Cartesian form by equating real and imaginary parts and forming the required quartic.

   Examiner report: Most candidates formed the two required component equations successfully. Weak responses either squared the wrong expression at the outset or moved to a polar approach without producing the requested quartic; numerical methods were also sometimes used before the necessary algebra had been shown.

   Question paper p.4 · Mark scheme p.9

4. Question 4

   4 marks

   This assesses linearising an exponential relationship and using two points on a logarithmic graph to determine its constants.

   Examiner report: A frequent mistake was confusing the intercept in the transformed line with the constant already named in the question. Rounding the intercept too early and sign errors also commonly led to an incorrect value for the multiplicative constant.

   Question paper p.5 · Mark scheme p.10

5. Question 5

   5 marks

   This assesses decomposition of a rational expression into a constant term and proper partial fractions with two distinct linear factors.

   Examiner report: Many candidates selected a form that could not produce the quadratic numerator, so they could earn little beyond an initial method mark. Dividing first, or including the constant term from the beginning, avoided this problem.

   Question paper p.6 · Mark scheme p.11

6. Question 6 (a)

   5 marks

   This assesses representing simultaneous modulus and argument inequalities on an Argand diagram.

   Examiner report: Responses varied widely in quality. Common graphical errors included using a full line instead of a ray, placing the ray on the wrong side of the circle’s centre, and shading the wrong part of the circular region.

   Question paper p.7 · Mark scheme p.12

7. Question 6 (b)

   2 marks

   This assesses maximising the argument of points in an Argand region using tangent geometry and trigonometry.

   Examiner report: Only a small proportion obtained both marks. The successful method generally used the angle to the circle’s centre together with the angle between the centre line and a tangent; some candidates selected incorrect sides in their trigonometric ratios or relied on an unjustified graphical approximation.

   Question paper p.7 · Mark scheme p.12

8. Question 7 (a)

   1 mark

   This assesses verifying that a proposed linear factor divides a cubic polynomial.

   Examiner report: This part was handled very well, with most candidates showing the substitution and its arithmetic clearly.

   Question paper p.8 · Mark scheme p.13

9. Question 7 (b)

   2 marks

   This assesses polynomial division or coefficient comparison to obtain the quadratic quotient after removing a known linear factor.

   Examiner report: Most candidates earned the available marks, usually by long division. A factorisation method was less common but equally valid when completed correctly.

   Question paper p.8 · Mark scheme p.13

10. Question 7 (c)

    3 marks

    Question paper p.9 · Mark scheme p.14

11. Question 8 (a)

    3 marks

    This assesses rewriting a linear combination of sine and cosine as a single shifted cosine and determining its exact amplitude and phase shift.

    Examiner report: The amplitude was generally found correctly. Errors mainly involved choosing a negative inverse-tangent value or assigning incompatible values to the sine and cosine of the phase angle.

    Question paper p.10 · Mark scheme p.14

12. Question 8 (b)

    5 marks

    This assesses using the result from part (a) to integrate a squared reciprocal trigonometric expression.

    Examiner report: Candidates who connected this part to the preceding representation usually made good progress. A common loss of marks came from omitting the scaling factor introduced when integrating a function whose argument contains a multiple of the variable.

    Question paper p.11 · Mark scheme p.15

13. Question 9 (a)

    5 marks

    This assesses deriving a differential equation from a volume formula, an inverse-time rate condition, and related rates.

    Examiner report: Many candidates did not introduce a proportionality constant or used the wrong sign for a decreasing volume. Others substituted the numerical data without first displaying the general differential equation, which prevented them from obtaining the later marks.

    Question paper p.12 · Mark scheme p.16

14. Question 9 (b)

    6 marks

    Question paper p.13 · Mark scheme p.17

15. Question 10 (a)

    6 marks

    This assesses finding possible parameters from the angle between two three-dimensional line directions using scalar products and vector magnitudes.

    Examiner report: The initial dot-product and modulus calculations were usually sound. A common error was replacing the square root of a quadratic expression by a linear expression, while others used a calculator on the unsquared equation instead of first deriving the required quadratic.

    Question paper p.14 · Mark scheme p.18

16. Question 10 (b)

    5 marks

    This assesses finding the intersection of two parameterised lines in three dimensions and determining the associated parameter and position vector.

    Examiner report: This was generally regarded as accessible, although some candidates omitted the requested position vector. Checking the resulting point in both line equations would have helped detect inconsistent simultaneous-equation solutions.

    Question paper p.15 · Mark scheme p.19