9709 Pure Mathematics 3 May/June 2024 Paper 31

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

1. Question 1

   4 marks · 4/75

   This assesses use of the binomial expansion followed by multiplication and collection of terms through degree two.

   Examiner report: Many candidates made sign or simplification errors, especially by expanding with x instead of 2x. Some stopped after finding only the quadratic term, while others tried to expand the simple linear factor unnecessarily.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   4 marks · 8/75

   This tests solving a logarithmic equation by combining logarithms, removing the logarithm, and selecting a valid numerical root.

   Examiner report: Combining the logarithms was generally the most effective entry point. Although many then solved the quadratic successfully, some retained an inadmissible negative solution or used an invalid logarithm identity.

   Question paper p.3 · Mark scheme p.6

3. Question 3

   4 marks · 12/75

   This assesses linearising an exponential model and using the gradient and intercept of a graph against ln x to determine its constants.

   Examiner report: Only a small proportion began with the necessary logarithmic transformation. A frequent mistake was treating the displayed ln x coordinates as ordinary x-values.

   Question paper p.4 · Mark scheme p.7

4. Question 4 (a)

   2 marks · 14/75

   This tests conversion of a complex number from Cartesian form to polar form with an argument in the prescribed principal range.

   Examiner report: The modulus was usually found by candidates familiar with polar form, but the argument was often placed in the wrong quadrant.

   Question paper p.5 · Mark scheme p.7

5. Question 4 (b)

   2 marks · 16/75

   This assesses division of complex numbers written in polar form, including manipulation of moduli and arguments.

   Examiner report: Relatively few candidates used the standard polar-division method confidently. Finding the modulus was more common than handling the argument correctly.

   Question paper p.5 · Mark scheme p.7

6. Question 5

   7 marks · 23/75

   This tests differentiation of a quotient containing composite functions and finding every stationary point in a specified interval.

   Examiner report: Most candidates recognised the quotient structure, but errors in differentiating either the exponential factor or the squared cosine factor were common. Even with such errors, some obtained credit for forming and solving the trigonometric quadratic.

   Question paper p.6 · Mark scheme p.8

7. Question 6 (a)

   2 marks · 25/75

   This assesses graphical comparison of an exponential function and a cosecant function to establish the number of roots on an interval.

   Examiner report: The exponential curve was often recognisable, but many sketches treated the cosecant incorrectly instead of viewing it as the reciprocal of a sine function.

   Question paper p.8 · Mark scheme p.9

8. Question 6 (b)

   2 marks · 27/75

   This tests numerical verification that the root lies between two specified bounds, using the original equation in radians.

   Examiner report: Correct endpoint checks were seen regularly, but degree-mode calculations and incorrect evaluations of the cosecant expression caused many failures.

   Question paper p.8 · Mark scheme p.9

9. Question 6 (c)

   1 mark · 28/75

   This assesses identifying the fixed-point form of the equation and connecting convergence of the iteration with the original root.

   Examiner report: Those who attempted this part generally succeeded. The main difficulty was failing to reverse the rearrangement fully to obtain the stated iteration with its subscripts.

   Question paper p.9 · Mark scheme p.9

10. Question 6 (d)

    3 marks · 31/75

    This tests numerical fixed-point iteration from a given starting value and appropriate checking of the requested accuracy.

    Examiner report: Fully successful iterations were obtained by some candidates, while others used degrees, evaluated cosecant inaccurately, or did not maintain the required decimal precision.

    Question paper p.9 · Mark scheme p.9

11. Question 6 (e)

    1 mark · 32/75

    Question paper p.9 · Mark scheme p.10

12. Question 7 (a)

    4 marks · 36/75

    This tests interpretation of complex-number loci as a circle and a perpendicular bisector on an Argand diagram.

    Examiner report: The circle was often drawn with the correct radius, but its centre was sometimes reflected into the wrong position. The second locus was frequently missed or not recognised as a perpendicular bisector.

    Question paper p.11 · Mark scheme p.10

13. Question 7 (b)

    2 marks · 38/75

    This assesses finding the shortest separation between a circle and a straight-line locus.

    Examiner report: Only a small number of candidates identified that the required distance was the line-to-centre distance reduced by the radius; many did not establish a workable geometric method.

    Question paper p.11 · Mark scheme p.10

14. Question 8

    7 marks · 45/75

    This tests definite integration by the suggested substitution, including handling trigonometric identities, transformed limits, and fractional powers.

    Examiner report: Only a minority demonstrated a clear substitution method. A crucial missing step in many attempts was the differential relationship arising from differentiating 1 − sin x.

    Question paper p.12 · Mark scheme p.11

15. Question 9 (a)

    1 mark · 46/75

    This assesses using the scalar product of direction vectors to impose perpendicularity of two lines.

    Examiner report: Only a minority evaluated the direction-vector dot product successfully enough to establish the given parameter value.

    Question paper p.14 · Mark scheme p.12

16. Question 9 (b)

    4 marks · 50/75

    This tests finding the intersection of two three-dimensional lines by comparing their component parametrisations.

    Examiner report: Candidates who understood the parameter-matching process often reached the intersection correctly; others did not set up the component equations effectively.

    Question paper p.14 · Mark scheme p.12

17. Question 9 (c)

    2 marks · 52/75

    This assesses checking whether a given point belongs to a vector line by finding a consistent parameter value.

    Examiner report: Very few responses successfully established membership of the given point on the line.

    Question paper p.15 · Mark scheme p.12

18. Question 9 (d)

    2 marks · 54/75

    This tests reflection of a point in a line using the known intersection and the perpendicular relationship established earlier.

    Examiner report: Correct answers were rare. Although the preceding parts supplied the necessary intersection and perpendicular geometry, many candidates did not use that information to construct the reflected point.

    Question paper p.15 · Mark scheme p.12

19. Question 10 (a)

    3 marks · 57/75

    This assesses implicit differentiation of a relation involving x and tan y, followed by substitution of the original relation.

    Examiner report: Only a few candidates carried out the intended implicit-differentiation route. Rewriting the relation with an inverse tangent and quoting a formula did not demonstrate the required method.

    Question paper p.16 · Mark scheme p.13

20. Question 10 (b)

    7 marks · 64/75

    This tests integration by parts together with use of the derivative from part (a) and integration of a rational function.

    Examiner report: Some candidates recognised integration by parts, but relatively few connected the calculation with the result from part (a). Many stopped at the remaining rational integral.

    Question paper p.16 · Mark scheme p.13

21. Question 11 (a)

    2 marks · 66/75

    This assesses forming a differential equation from a proportional-rate model and the supplied initial rate information.

    Examiner report: A common approach was to assume the displayed differential equation and merely check it against the data. That verifies consistency but does not establish the equation, so it earned no credit.

    Question paper p.18 · Mark scheme p.14

22. Question 11 (b)

    9 marks · 75/75

    This tests separation of variables, partial fractions, logarithmic integration, use of an initial condition, and consolidation into one logarithm.

    Examiner report: Partial-fraction work often earned some credit even when separation was incomplete. Sign errors in the term containing the uninfected population were frequent, and some otherwise correct solutions failed to combine the logarithms at the end.

    Question paper p.18 · Mark scheme p.15