9709 Pure Mathematics 1 May/June 2024 Paper 11

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

1. Question 1 (a)

   2 marks · 2/75

   Question paper p.2 · Mark scheme p.6

2. Question 1 (b)

   3 marks · 5/75

   This assesses solving a quartic by treating it as a quadratic in x² and selecting the real solutions.

   Examiner report: The usual successful approach introduced x² before factorising. Lost method credit often resulted from omitting the substitution or failing to retain the common numerical factor during factorisation.

   Question paper p.2 · Mark scheme p.6

3. Question 2 (a)

   4 marks · 9/75

   This assesses describing successive graph transformations in the correct order.

   Examiner report: The order was important: successful responses stretched first and translated second. Descriptions that merely said ‘down 2’ were sometimes judged incomplete because the direction needed to be unambiguous.

   Question paper p.3 · Mark scheme p.7

4. Question 2 (b)

   2 marks · 11/75

   This assesses forming the equation of a transformed sine curve from its graph transformations.

   Examiner report: Many candidates used a general form involving a multiple of sin x and a constant; a substantial number then obtained the required expression.

   Question paper p.3 · Mark scheme p.7

5. Question 3 (a)

   2 marks · 13/75

   This assesses extracting a specified binomial coefficient and using it to determine a parameter.

   Examiner report: The standard successful calculation used the x³ term directly. Errors commonly came from treating the cube of ax as though only x, rather than both a and x, were raised to the third power.

   Question paper p.4 · Mark scheme p.7

6. Question 3 (b)

   3 marks · 16/75

   This assesses finding a coefficient in a product of a binomial expansion and a linear factor.

   Examiner report: Candidates who identified the x² term generally progressed well. Some made the same index error as in part (a), while others still earned method credit by combining the two relevant contributions correctly.

   Question paper p.4 · Mark scheme p.8

7. Question 4 (a)

   1 mark · 17/75

   Question paper p.5 · Mark scheme p.8

8. Question 4 (b)

   1 mark · 18/75

   Question paper p.5 · Mark scheme p.8

9. Question 4 (c)

   1 mark · 19/75

   Question paper p.5 · Mark scheme p.8

10. Question 5 (a)

    3 marks · 22/75

    This assesses proving a trigonometric identity by applying a fundamental identity and simplifying algebraically.

    Examiner report: A substitution followed by factorisation was a common successful route; working on both sides was also acceptable when the equivalence was clearly demonstrated.

    Question paper p.6 · Mark scheme p.9

11. Question 5 (b)

    3 marks · 25/75

    This assesses using a proven identity to solve a trigonometric equation over a complete degree interval.

    Examiner report: Some candidates introduced an extra angle by solving an unnecessary quadratic. Others used an incorrect cosine value, so no valid trigonometric solutions could be obtained.

    Question paper p.6 · Mark scheme p.9

12. Question 6 (a)

    2 marks · 27/75

    This assesses sketching the inverse of a restricted rational function.

    Examiner report: A fourth-quadrant curve was often drawn, but the line y=x and the correct relative placement of the asymptotes were frequently missing.

    Question paper p.7 · Mark scheme p.9

13. Question 6 (b)

    3 marks · 30/75

    This assesses finding an inverse function algebraically while respecting the original domain and range.

    Examiner report: Many candidates obtained most of the algebraic rearrangement marks. A common issue was choosing the sign without considering the range of the inverse.

    Question paper p.7 · Mark scheme p.10

14. Question 6 (c)

    1 mark · 31/75

    Question paper p.7 · Mark scheme p.10

15. Question 6 (d)

    1 mark · 32/75

    This assesses explaining why a function and its inverse cannot have a common point in this case.

    Examiner report: The clearest accepted explanations referred either to the absence of an intersection or to the incompatible signs of the relevant function values.

    Question paper p.7 · Mark scheme p.10

16. Question 7 (a)

    1 mark · 33/75

    This assesses using right-triangle geometry and inverse trigonometry to determine an angle in radians.

    Examiner report: The most common successful route related the angle to the inverse sine or to a right angle minus an inverse cosine. Degree-mode calculations had to be converted back to radians.

    Question paper p.8 · Mark scheme p.10

17. Question 7 (b)

    7 marks · 40/75

    This assesses finding the perimeter and area of a composite shape containing straight sides, circular arcs and sectors.

    Examiner report: Many candidates applied the radian formulae correctly and identified the required pieces. Some stopped after finding just one of the two requested quantities.

    Question paper p.8 · Mark scheme p.11

18. Question 8 (a)

    4 marks · 44/75

    This assesses using equal differences in an arithmetic progression to determine a term.

    Examiner report: Both a simultaneous-equation method and a single equation in p were effective. Errors included confusing p with the common difference, sign slips, and using a sum formula instead of a term formula.

    Question paper p.10 · Mark scheme p.12

19. Question 8 (b)

    4 marks · 48/75

    This assesses determining a geometric ratio from three algebraic terms and applying the sum-to-infinity formula.

    Examiner report: Candidates were generally familiar with forming the quadratic, but some did not show enough working to justify its solution. The extraneous algebraic root also needed to be rejected.

    Question paper p.11 · Mark scheme p.13

20. Question 9

    6 marks · 54/75

    This assesses finding a volume of revolution using washers and combining it with the volume of a cylindrical part of the region.

    Examiner report: The strongest solutions set up an integral of the square of the curve. A number of candidates calculated area rather than volume, and relatively few included the cylindrical contribution.

    Question paper p.12 · Mark scheme p.14

21. Question 10

    8 marks · 62/75

    This assesses finding tangents from an external point to a circle using a repeated-intersection condition.

    Examiner report: The substitution method was the most successful. Candidates who reached the quadratic condition in the slope usually found both contact points, whereas some incorrectly treated the intercept or slope condition.

    Question paper p.14 · Mark scheme p.15

22. Question 11 (a)

    5 marks · 67/75

    This assesses differentiating a rational function and using a sign analysis to find where it decreases.

    Examiner report: Differentiation and the stationary values were often found correctly, but only a small proportion gave a fully correct interval description. The excluded point at zero and the sign changes on either side required particular care.

    Question paper p.16 · Mark scheme p.17

23. Question 11 (b)

    8 marks · 75/75

    This assesses constructing tangent and normal lines, finding their intersection, and calculating the area of a coordinate triangle.

    Examiner report: Successful candidates formed both line equations and equated them to locate the intersection. Others made limited progress, often failing to connect the line equations with the required triangular area.

    Question paper p.17 · Mark scheme p.18