9709 Pure Mathematics 1 May/June 2021 Paper 13

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

1. Question 1

   3 marks · 3/75

   This assesses integration of a derivative and use of a known point to determine a curve equation.

   Examiner report: This was generally handled well: candidates usually integrated appropriately and then used the point to determine the missing constant.

   Question paper p.3 · Mark scheme p.6

2. Question 2

   4 marks · 7/75

   This assesses differentiation and the use of gradient conditions to determine the largest interval on which a function decreases.

   Examiner report: Most candidates recognised the need to differentiate; the main omission was failing to justify explicitly why the derivative must be negative over the specified interval.

   Question paper p.4 · Mark scheme p.6

3. Question 3

   6 marks · 13/75

   This assesses the condition for a line to be tangent to a quadratic curve and the recovery of the corresponding contact points.

   Examiner report: The discriminant approach was common and often logically presented, but some candidates relied on calculator outputs without showing the algebra. Differentiation provided a valid alternative route.

   Question paper p.5 · Mark scheme p.7

4. Question 4 (a)

   2 marks · 15/75

   This assesses manipulation of trigonometric ratios to transform an equation into the stated cosine form.

   Examiner report: The identity was usually understood, although some solutions skipped the cancellation step and therefore did not make the reduction sufficiently visible.

   Question paper p.6 · Mark scheme p.8

5. Question 4 (b)

   2 marks · 17/75

   This assesses rearrangement of an equation to make cosine the subject.

   Examiner report: Most candidates changed the subject successfully; a recurring algebraic slip was failing to collect both cosine terms before factorising.

   Question paper p.6 · Mark scheme p.8

6. Question 4 (c)

   2 marks · 19/75

   This assesses solving a trigonometric equation using the cosine relation and restricting solutions to a specified interval.

   Examiner report: Many candidates found one valid angle but missed its companion or included extraneous values; degree answers were also seen despite the interval being given in radians.

   Question paper p.6 · Mark scheme p.8

7. Question 5 (a)

   2 marks · 21/75

   This assesses application of the sector-area formula with an angle measured in radians.

   Examiner report: The sector formula was almost universally selected correctly; occasional responses converted the angle to degrees instead of leaving it in radians.

   Question paper p.7 · Mark scheme p.9

8. Question 5 (b)

   4 marks · 25/75

   This assesses combining arc length, right-triangle trigonometry and the cosine rule to find a composite perimeter.

   Examiner report: Arc length was generally used correctly and many complete perimeters were obtained. Rounding intermediate quantities too early sometimes spoiled the final accuracy.

   Question paper p.7 · Mark scheme p.9

9. Question 6 (a)

   4 marks · 29/75

   This assesses completing the square and comparing two quadratic functions to express one as a shifted version of the other.

   Examiner report: Completing the square was familiar to most candidates, but some did not show both requested forms, making the comparison less clear.

   Question paper p.8 · Mark scheme p.9

10. Question 6 (b)

    2 marks · 31/75

    This assesses describing a graph transformation from the algebraic relationship between two functions.

    Examiner report: A number of responses named a translation without specifying its direction precisely; vector notation or clearly separated horizontal and vertical movements was more reliable.

    Question paper p.8 · Mark scheme p.10

11. Question 7 (a)

    2 marks · 33/75

    This assesses use of the binomial theorem to obtain the initial terms in ascending powers of the variable.

    Examiner report: The expansion was generally well known. Errors mainly involved signs or arranging the terms in the wrong order.

    Question paper p.9 · Mark scheme p.11

12. Question 7 (b)

    5 marks · 38/75

    This assesses extraction of a selected coefficient from a product and solution of the resulting equation in a parameter.

    Examiner report: Many candidates selected the relevant terms successfully. Full credit depended on solving the resulting equation completely, including the negative possibilities, and on factorisation or another displayed algebraic method.

    Question paper p.9 · Mark scheme p.11

13. Question 8 (a)

    4 marks · 42/75

    Question paper p.10 · Mark scheme p.12

14. Question 8 (b)

    3 marks · 45/75

    Question paper p.11 · Mark scheme p.13

15. Question 9 (a)

    3 marks · 48/75

    This assesses use of the infinite geometric-series formula together with a percentage condition.

    Examiner report: The governing relationship was usually identified, though supporting algebra was often abbreviated. Candidates needed to show enough work to justify the quadratic and its factorisation or solution.

    Question paper p.12 · Mark scheme p.13

16. Question 9 (b)

    6 marks · 54/75

    This assesses modelling terms and finite sums of two arithmetic progressions and solving the resulting simultaneous equations.

    Examiner report: Most candidates used the arithmetic-progression formulae appropriately. Errors commonly arose from misreading the first term or common difference of the second progression when simplifying the ratios.

    Question paper p.13 · Mark scheme p.14

17. Question 10 (a)

    2 marks · 56/75

    This assesses proving that three coordinate points form a right angle using analytic geometry.

    Examiner report: Several valid methods appeared, with perpendicular gradients being especially common. Clear justification was needed; merely asserting the angle was right was insufficient.

    Question paper p.14 · Mark scheme p.14

18. Question 10 (b)

    1 mark · 57/75

    Question paper p.14 · Mark scheme p.14

19. Question 10 (c)

    2 marks · 59/75

    This assesses constructing a circle equation from its centre and radius.

    Examiner report: The standard form was well understood. Some longer approaches using all three points introduced unnecessary algebra, and unsimplified radius terms sometimes prevented a fully acceptable final form.

    Question paper p.14 · Mark scheme p.15

20. Question 10 (d)

    5 marks · 64/75

    This assesses finding the endpoint of a diameter and using perpendicularity to obtain the tangent equation.

    Examiner report: The midpoint or vector approach was efficient, but this was the part most often left incomplete. Successful solutions generally combined the contact point with a perpendicular-gradient line equation.

    Question paper p.15 · Mark scheme p.15

21. Question 11 (a)

    4 marks · 68/75

    This assesses differentiation of a function with fractional powers and identification of its stationary minimum.

    Examiner report: Differentiation was usually completed competently, but errors often occurred while solving the fractional-power equation or while finding the associated y-coordinate.

    Question paper p.16 · Mark scheme p.16

22. Question 11 (b)

    4 marks · 72/75

    This assesses finding a tangent line at a parameter-dependent point and its intercept on the y-axis.

    Examiner report: Most candidates could evaluate both the gradient and point and then form the tangent; the main difficulties were algebraic simplification and carrying the parameter powers consistently.

    Question paper p.17 · Mark scheme p.16

23. Question 11 (c)

    3 marks · 75/75

    This assesses integration of the curve between two parameter-dependent vertical boundaries.

    Examiner report: The integration method was recognised by many candidates. A frequent source of error was mishandling the three-halves power, particularly its coefficient and the parameter factor.

    Question paper p.17 · Mark scheme p.17