9709 Pure Mathematics 3 May/June 2024 Paper 32

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

1. Question 1 (a)

   1 mark · 1/75

   This assesses sketching a modulus graph with the correct vertex, symmetry and scale.

   Examiner report: Many responses recognised the V-shape but lost credit through incorrect placement, missing axis labels or assigning an arbitrary value to the parameter.

   Question paper p.3 · Mark scheme p.6

2. Question 1 (b)

   2 marks · 3/75

   This assesses solving an inequality involving an absolute value by identifying the relevant cases or critical points.

   Examiner report: Squaring was a common approach, but algebraic slips involving the parameter were frequent, and an unwanted critical value was often retained instead of rejected.

   Question paper p.3 · Mark scheme p.6

3. Question 2

   5 marks · 8/75

   This assesses decomposition of a rational expression into partial fractions when the numerator and denominator have equal degree.

   Examiner report: Errors often arose because the denominator was unfactored initially and because some candidates omitted the constant term required by the equal degrees of numerator and denominator.

   Question paper p.4 · Mark scheme p.7

4. Question 3 (a)

   3 marks · 11/75

   This assesses using logarithms to convert an exponential relation into the equation of a straight line.

   Examiner report: Although logarithms were usually taken correctly, missing brackets and unjustified cancellation caused problems; some candidates also stopped without explaining why the rearranged equation represented a line.

   Question paper p.5 · Mark scheme p.9

5. Question 3 (b)

   2 marks · 13/75

   This assesses specialising the logarithmic line equation using a power relationship between the two bases.

   Examiner report: This part was generally successful, though some responses left logarithmic ratios unsimplified rather than reducing the gradient and intercept fully.

   Question paper p.5 · Mark scheme p.9

6. Question 4

   6 marks · 19/75

   This assesses implicit differentiation of a curve containing products of y and exponential functions, followed by evaluation at a specified y-value.

   Examiner report: The differentiation was often handled well, but finding the corresponding x-coordinate caused difficulty. Incorrect logarithmic manipulation, failure to discard an invalid root and sign errors during rearrangement were common.

   Question paper p.6 · Mark scheme p.10

7. Question 5 (a)

   2 marks · 21/75

   This assesses locating a root numerically by comparing the two sides of an equation at the ends of a stated interval.

   Examiner report: Numerical endpoint work was often correct, but some candidates did not state the resulting conclusion clearly, and degree mode produced many wrong evaluations.

   Question paper p.7 · Mark scheme p.12

8. Question 5 (b)

   1 mark · 22/75

   This assesses showing that a convergent fixed-point iteration has a limit satisfying the original equation.

   Examiner report: Most candidates established the relationship, but those working backwards sometimes stopped before restoring the indexed iteration and completing the link with the original equation.

   Question paper p.7 · Mark scheme p.12

9. Question 5 (c)

   3 marks · 25/75

   This assesses using fixed-point iteration to obtain a root to a specified accuracy.

   Examiner report: Many calculations were successful; the principal failures came from degree-mode calculations and from stopping before enough decimal places had been established.

   Question paper p.7 · Mark scheme p.12

10. Question 6 (a)

    4 marks · 29/75

    This assesses differentiating a product involving an exponential and locating the maximum point of the curve.

    Examiner report: The x-coordinate was often found, but mistakes with the factor and sign in differentiating the exponential, or failure to calculate the corresponding y-coordinate, prevented full credit.

    Question paper p.8 · Mark scheme p.13

11. Question 6 (b)

    5 marks · 34/75

    Question paper p.9 · Mark scheme p.13

12. Question 7 (a)

    3 marks · 37/75

    Question paper p.10 · Mark scheme p.14

13. Question 7 (b)

    6 marks · 43/75

    This assesses applying a proved identity and double-angle formulae to evaluate an exact definite trigonometric integral.

    Examiner report: Candidates who identified the connection with part (a) often made a good start, but many struggled to transform the squared trigonometric term or made errors with antiderivatives and endpoint brackets.

    Question paper p.11 · Mark scheme p.15

14. Question 8 (a)

    3 marks · 46/75

    This assesses forming a vector equation for a line through two points in three-dimensional space.

    Examiner report: Most candidates knew the required structure, but a common presentation error was assigning an equation to the line label rather than to the vector r; some also used the unrelated point A.

    Question paper p.12 · Mark scheme p.16

15. Question 8 (b)

    4 marks · 50/75

    This assesses finding the coordinates of the intersection of two three-dimensional vector lines.

    Examiner report: A number of responses only demonstrated consistency of the two line equations rather than extracting the intersection coordinates. Sign and addition mistakes in the simultaneous equations were also common.

    Question paper p.12 · Mark scheme p.16

16. Question 8 (c)

    5 marks · 55/75

    This assesses using an equality of three-dimensional distances to locate a point on a parametrised line.

    Examiner report: Many candidates did not interpret the notation as a distance condition. Those who did often formed one of the required vectors correctly but failed to complete the resulting equation; vector notation in column form also caused some presentation issues.

    Question paper p.13 · Mark scheme p.17

17. Question 9 (a)

    1 mark · 56/75

    This assesses multiplying two complex numbers and collecting the real and imaginary components in exact surd form.

    Examiner report: Most candidates earned the mark, with lost credit mainly caused by arithmetic slips or failure to display the imaginary coefficient unambiguously.

    Question paper p.14 · Mark scheme p.17

18. Question 9 (b)

    4 marks · 60/75

    This assesses converting complex numbers from Cartesian form to polar-exponential form using exact moduli and principal arguments.

    Examiner report: The moduli were usually obtained correctly, whereas quadrant selection for the arguments was more troublesome; some candidates calculated the polar form of the product instead of the two requested numbers.

    Question paper p.14 · Mark scheme p.18

19. Question 9 (c)

    2 marks · 62/75

    This assesses proving geometric properties of an Argand-diagram triangle from complex-number moduli and arguments.

    Examiner report: A proof needed separate statements for the isosceles property and the right angle. Merely listing lengths without the associated deductions, or assuming the right angle in the argument, was insufficient.

    Question paper p.14 · Mark scheme p.18

20. Question 9 (d)

    3 marks · 65/75

    This assesses using arguments of products and the rectangular form of a complex number to prove an exact tangent identity.

    Examiner report: Few solutions completed the proof. Many did not state that the product-argument rule was being used or failed to connect the calculated tangent with the correct point on the Argand diagram.

    Question paper p.15 · Mark scheme p.19

21. Question 10 (a)

    2 marks · 67/75

    This assesses differentiating a reciprocal power of cosine by the chain rule and expressing the result in secant notation.

    Examiner report: The intended derivative required visible working. Some candidates confused the independent variable or wrote a derivative with respect to x, while others lost a minus sign in the chain rule.

    Question paper p.16 · Mark scheme p.19

22. Question 10 (b)

    8 marks · 75/75

    This assesses solving a separable differential equation using the result from part (a), standard rational integration and an initial condition.

    Examiner report: Separation was commonly done correctly, but progress often stopped when the rational expression was not split before integration. Even complete symbolic work was sometimes spoiled by calculator errors or an impossible cosine value.

    Question paper p.16 · Mark scheme p.20