9709 Pure Mathematics 1 May/June 2024 Paper 13

22 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 theorem together with multiplication of expansions to extract a specified power of x.

   Examiner report: Most responses showed a sound command of the binomial expansion and successfully combined the relevant products.

   Question paper p.3 · Mark scheme p.6

2. Question 2 (a)

   3 marks · 7/75

   This tests the interpretation of a translated cosine graph, including the locations of a zero and a minimum.

   Examiner report: This was a weaker part, with some candidates leaving it incomplete. Reading the graph as a shifted basic cosine curve was effective, although algebraic and calculus approaches also worked when fully developed.

   Question paper p.4 · Mark scheme p.6

3. Question 2 (b)

   2 marks · 9/75

   This assesses exact manipulation of inverse trigonometric functions and the solution of an equation involving them.

   Examiner report: Responses were generally successful. Both radian and degree methods appeared, but consistency of angular units was important.

   Question paper p.4 · Mark scheme p.6

4. Question 3 (a)

   4 marks · 13/75

   This tests the modelling of a reflex circular sector using perimeter and area formulae, followed by simultaneous algebraic solution.

   Examiner report: The standard arc-length and sector-area relationships were usually identified correctly. Full credit depended on showing the solution of the resulting quadratic or cubic rather than merely quoting its roots.

   Question paper p.5 · Mark scheme p.7

5. Question 3 (b)

   2 marks · 15/75

   This assesses the use of the sine area formula when the included angle is the reflex angle of a sector.

   Examiner report: The crucial issue was using the angle belonging to the triangle rather than the sector parameter directly. Confusion between radians and degrees was particularly damaging.

   Question paper p.5 · Mark scheme p.7

6. Question 4 (a)

   3 marks · 18/75

   This tests the conversion of a trigonometric equation into a quadratic in sin θ using identities and algebraic simplification.

   Examiner report: Most candidates handled this reduction effectively, usually by rewriting the tangent and then applying a standard trigonometric identity.

   Question paper p.6 · Mark scheme p.7

7. Question 4 (b)

   3 marks · 21/75

   This assesses solving a trigonometric equation after replacing θ by 2x, with careful attention to the stated angular interval.

   Examiner report: Incomplete solutions often failed to account for the doubled angle. Errors also arose from using the wrong supplementary relationship or rounding the angle for 2x too early.

   Question paper p.6 · Mark scheme p.8

8. Question 5 (a)

   3 marks · 24/75

   This tests differentiation and the method for finding the coordinates of a stationary point.

   Examiner report: The general procedure was familiar, but differentiating the reciprocal term incorrectly was a recurring issue. Some otherwise correct solutions also omitted the algebra needed to solve the stationary-point equation.

   Question paper p.7 · Mark scheme p.8

9. Question 5 (b)

   2 marks · 26/75

   This assesses classification of a stationary point using the second derivative or an equivalent change-of-gradient argument.

   Examiner report: Most candidates used the second derivative successfully, though stating the classification without demonstrating the sign was insufficient. The break in the domain at zero made a first-derivative sign-change approach less commonly used.

   Question paper p.7 · Mark scheme p.9

10. Question 5 (c)

    2 marks · 28/75

    This tests the use of the derivative to determine monotonic behaviour over a specified positive domain.

    Examiner report: Many solutions correctly recognised the sign of the derivative. A substantial number, however, supported the conclusion only with isolated substitutions, which does not establish the result throughout the interval.

    Question paper p.7 · Mark scheme p.9

11. Question 6 (a)

    4 marks · 32/75

    This assesses integration of a rational derivative involving a linear inner expression and determination of the integration constant from a point on the curve.

    Examiner report: This was handled well overall, with many candidates showing a clear understanding of both integration and the use of the given point. Misreading the negative power as a positive power led to serious loss of credit, and treating the curve as a straight line limited the available marks.

    Question paper p.8 · Mark scheme p.10

12. Question 6 (b)

    3 marks · 35/75

    This tests the effect of successive graph transformations on the equation of a curve.

    Examiner report: This was considerably more challenging than the preceding integration part. Common difficulties included using the wrong horizontal substitution, applying the translations in the wrong order, and altering only part of the expression.

    Question paper p.8 · Mark scheme p.10

13. Question 7 (a)

    2 marks · 37/75

    Question paper p.9 · Mark scheme p.11

14. Question 7 (b)

    5 marks · 42/75

    This tests locating the first and last terms of an arithmetic progression within bounds and summing the resulting finite block.

    Examiner report: Only a smaller proportion of candidates completed all stages correctly. Frequent issues were choosing the wrong lower endpoint in a difference of sums and failing to convert a non-integer position into the appropriate integer term.

    Question paper p.9 · Mark scheme p.11

15. Question 8

    8 marks · 50/75

    Question paper p.10 · Mark scheme p.12

16. Question 9 (a)

    5 marks · 55/75

    This tests differentiation of a square-root composite function and the construction of a tangent in a prescribed integer-coefficient form.

    Examiner report: The tangent-line procedure was broadly understood, but differentiating the radical caused difficulty for some. A recurring error was to use the normal gradient after correctly finding the derivative.

    Question paper p.12 · Mark scheme p.14

17. Question 9 (b)

    3 marks · 58/75

    This assesses the disc method for finding a volume of revolution about the x-axis, including squaring the curve and evaluating a definite integral.

    Examiner report: Many candidates selected the correct method. Omitting the square on the radius prevented progress, while giving only a final numerical value without displaying the integration earned very limited credit.

    Question paper p.13 · Mark scheme p.14

18. Question 10 (a)

    3 marks · 61/75

    This tests forming and solving an equation for the common ratio of a geometric progression from specified terms.

    Examiner report: The term formula was generally applied effectively. Showing the solution of the quadratic, rather than simply presenting a ratio, was important for full marks.

    Question paper p.14 · Mark scheme p.15

19. Question 10 (b)

    2 marks · 63/75

    This assesses application of the finite geometric-series formula and appropriate numerical rounding.

    Examiner report: Most candidates used the finite-sum formula correctly and obtained an appropriately rounded result.

    Question paper p.14 · Mark scheme p.15

20. Question 10 (c)

    3 marks · 66/75

    Question paper p.15 · Mark scheme p.15

21. Question 11 (a)

    3 marks · 69/75

    This assesses rewriting a quadratic function as a shifted square and reading its range from that form.

    Examiner report: Many correct forms were seen, but sign changes caused errors for some candidates. A correct range obtained without the requested square-form method received only limited credit.

    Question paper p.16 · Mark scheme p.16

22. Question 11 (b)

    6 marks · 75/75

    This assesses finding an inverse linear function, composing it with a quadratic, and using the single-intersection condition to determine the parameter and point.

    Examiner report: The inverse-function route was the most common successful approach, although some candidates confused the order of composition. Method marks were often lost when the repeated-root quadratic was solved or its root stated without supporting algebra; an alternative route using a double composition also worked.

    Question paper p.17 · Mark scheme p.17