9709 Pure Mathematics 1 May/June 2024 Paper 12

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

1. Question 1

   3 marks · 3/75

   This tests binomial expansion, coefficient comparison, and solving for a positive parameter.

   Examiner report: Most candidates handled this well, but some lost accuracy by dropping the square on the parameter or applying the multiplier to the wrong coefficient.

   Question paper p.3 · Mark scheme p.6

2. Question 2

   5 marks · 8/75

   This tests the description and ordering of combined transformations of a quadratic graph.

   Examiner report: Clear answers used precise transformation terminology and made the sequence explicit; imprecise descriptions of movements made otherwise correct work harder to credit.

   Question paper p.4 · Mark scheme p.7

3. Question 3 (a)

   3 marks · 11/75

   This tests the use of trigonometric identities to convert an equation involving tangent and cosine into a quadratic in sine.

   Examiner report: Many candidates completed the algebra successfully, although some omitted the zero during the intermediate rearrangement and only attached it at the end.

   Question paper p.5 · Mark scheme p.9

4. Question 3 (b)

   3 marks · 14/75

   This tests solving a trigonometric quadratic and selecting all sine solutions in a full revolution.

   Examiner report: Unshown calculator work frequently led to a missing branch or an angle outside the permitted range; the report emphasised that a numerical quadratic solution needs supporting working.

   Question paper p.5 · Mark scheme p.9

5. Question 4 (a)

   1 mark · 15/75

   Question paper p.6 · Mark scheme p.10

6. Question 4 (b)

   2 marks · 17/75

   This tests identifying the range of a reciprocal quadratic function and deciding whether it has an inverse function.

   Examiner report: A common weakness was giving only one boundary of the range or being unclear about whether the original function or its inverse was being described.

   Question paper p.6 · Mark scheme p.10

7. Question 4 (c)

   4 marks · 21/75

   This tests composition of functions, solving a surd equation, and applying domain restrictions.

   Examiner report: Although many candidates set up the composite equation correctly, a substantial number did not complete the surd algebra or retained a root that was outside the allowed input range.

   Question paper p.7 · Mark scheme p.11

8. Question 5 (a)

   4 marks · 25/75

   This tests arithmetic-progression notation and the exact use of the finite-sum formula.

   Examiner report: The method was often known, but decimal approximations were sometimes given where the question required an exact result, with early rounding also causing errors.

   Question paper p.8 · Mark scheme p.12

9. Question 5 (b)(i)

   2 marks · 27/75

   This tests finding the common ratio and applying the infinite geometric-series formula.

   Examiner report: Most candidates found the series sum, but some presented an algebraically equivalent nested fraction that did not meet the required form.

   Question paper p.9 · Mark scheme p.12

10. Question 5 (b)(ii)

    3 marks · 30/75

    This tests evaluating a finite geometric sum numerically for a specified angle.

    Examiner report: This part was generally successful, though some candidates calculated only the tenth term instead of adding the first ten terms.

    Question paper p.9 · Mark scheme p.13

11. Question 6 (a)

    4 marks · 34/75

    This tests differentiation to locate a minimum and solving for a positive-axis intercept.

    Examiner report: Most candidates used differentiation effectively; some who tried completing the square misidentified the stationary x-coordinate.

    Question paper p.10 · Mark scheme p.14

12. Question 6 (b)

    5 marks · 39/75

    This tests finding the area between a curve and a straight line using definite integration.

    Examiner report: Some candidates repeated work from the previous part instead of using its coordinates, while clearer solutions usually evaluated the two areas separately before taking their difference.

    Question paper p.11 · Mark scheme p.15

13. Question 7 (a)

    5 marks · 44/75

    This tests the tangent condition for a circle using substitution and the discriminant.

    Examiner report: The substitution-and-discriminant approach was the most reliable; expansion mistakes and omission of the circle’s constant term were frequent sources of lost marks.

    Question paper p.12 · Mark scheme p.18

14. Question 7 (b)

    3 marks · 47/75

    This tests using the radius direction to find a diameter perpendicular to a tangent.

    Examiner report: Many candidates spent time finding the tangent’s contact point even though the centre alone was sufficient; some also appeared unsure that the requested line must pass through the centre.

    Question paper p.13 · Mark scheme p.18

15. Question 8 (a)(i)

    2 marks · 49/75

    This tests right-triangle trigonometry in a symmetric sector diagram.

    Examiner report: A notable number of candidates tried to use arc length for this straight-line measurement; the successful method used the right-triangle geometry.

    Question paper p.14 · Mark scheme p.19

16. Question 8 (a)(ii)

    4 marks · 53/75

    This tests decomposing a composite plate into elementary regions and evaluating its area.

    Examiner report: Sector areas were often found correctly, but the perpendicular height of the central region caused difficulty; labelling separate component areas would have made partial methods clearer.

    Question paper p.14 · Mark scheme p.20

17. Question 8 (b)

    4 marks · 57/75

    This tests forming a perimeter equation when the sector radius is no longer fixed.

    Examiner report: Many candidates carried over the length from part (a) despite the changed radius, rather than rebuilding that boundary length before forming the perimeter equation.

    Question paper p.15 · Mark scheme p.21

18. Question 9 (a)

    4 marks · 61/75

    This tests determining where a derivative is negative by solving a quadratic inequality.

    Examiner report: Some candidates analysed only the squared term and missed the additional linear contribution; others gave separate inequalities without making their logical connection clear.

    Question paper p.16 · Mark scheme p.22

19. Question 9 (b)

    4 marks · 65/75

    Question paper p.17 · Mark scheme p.23

20. Question 10 (a)

    4 marks · 69/75

    This tests implicit rate relationships for a moving point on a curve.

    Examiner report: Differentiation was usually the successful starting point, but many candidates did not find the x-coordinate associated with the specified y-value and therefore could not continue.

    Question paper p.18 · Mark scheme p.24

21. Question 10 (b)

    6 marks · 75/75

    This tests combining a curve gradient, coordinate geometry, midpoint calculation, and the equation of a perpendicular bisector.

    Examiner report: Common problems included failing to reuse the earlier curve information, confusing a tangent gradient with the perpendicular gradient, and stopping before converting the final line into the requested standard form.

    Question paper p.19 · Mark scheme p.25