9709 Pure Mathematics 1 May/June 2021 Paper 11

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

1. Question 1

   4 marks · 4/75

   Integrating a derivative and using a known point to determine the constant in the equation of a curve.

   Examiner report: Most candidates could reverse the differentiation, but some omitted the integration constant or differentiated instead of integrating.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   5 marks · 9/75

   Using arithmetic-progression sum formulae to recover the first term and common difference, then finding a specified term.

   Examiner report: The main difficulty was sometimes answering with a sum of terms instead of the requested single term; elimination was generally less error-prone than lengthy substitution.

   Question paper p.3 · Mark scheme p.6

3. Question 3 (a)

   3 marks · 12/75

   Question paper p.4 · Mark scheme p.7

4. Question 3 (b)

   3 marks · 15/75

   Finding a coefficient in a product of polynomial expansions by matching combinations of powers.

   Examiner report: Strong solutions isolated the relevant product terms. Errors often arose when only two terms of the quadratic were written or when the full product was expanded unnecessarily.

   Question paper p.4 · Mark scheme p.7

5. Question 4

   3 marks · 18/75

   Question paper p.5 · Mark scheme p.7

6. Question 5

   4 marks · 22/75

   Using consecutive terms of a geometric progression to determine its ratio and then its infinite sum.

   Examiner report: Some candidates treated the sequence as arithmetic. Among geometric-progression methods, sign mistakes in the ratio, first term or final summation were frequent.

   Question paper p.6 · Mark scheme p.8

7. Question 6

   5 marks · 27/75

   Using the condition for a line to be tangent to a quadratic curve to determine a parameter.

   Examiner report: Most candidates began by equating the two equations, but algebraic slips in forming the quadratic or discriminant often prevented completion. A differentiation route was attempted by relatively few candidates.

   Question paper p.7 · Mark scheme p.9

8. Question 7 (a)

   2 marks · 29/75

   Question paper p.8 · Mark scheme p.10

9. Question 7 (b)

   3 marks · 32/75

   Solving a trigonometric equation involving a squared tangent over a specified degree interval.

   Examiner report: Many candidates reached the quadratic but stopped after finding tan²θ, rather than continuing to tanθ. Others missed solutions within the stated interval or did not check them in the original equation.

   Question paper p.9 · Mark scheme p.10

10. Question 8 (a)

    2 marks · 34/75

    Using the geometry of equal radii and a chord, together with symmetry, to establish a central angle.

    Examiner report: An equilateral-triangle observation gave a concise route, while splitting the triangle also worked. Since the angle was supplied in the question, unsupported assertion was not enough.

    Question paper p.10 · Mark scheme p.11

11. Question 8 (b)

    3 marks · 37/75

    Calculating an exact perimeter made from circular arcs and semicircular boundaries.

    Examiner report: A frequent omission was the curved length contributed by the semicircles. Decimal approximations were unsuitable when an exact perimeter was requested.

    Question paper p.10 · Mark scheme p.11

12. Question 8 (c)

    5 marks · 42/75

    Decomposing a composite circular region into sectors, segments and semicircles to find its exact area.

    Examiner report: This was difficult because it required an overall plan rather than one standard formula. Many attempts contained an individual sector, triangle or semicircle calculation without completing the full subtraction.

    Question paper p.11 · Mark scheme p.12

13. Question 9 (a)

    1 mark · 43/75

    Identifying the range of a restricted quadratic function.

    Examiner report: Some candidates used the wrong variable when describing the range or represented the endpoint and inequality incorrectly.

    Question paper p.12 · Mark scheme p.13

14. Question 9 (b)

    2 marks · 45/75

    Finding the inverse of a one-to-one restricted quadratic function.

    Examiner report: The main pitfalls were retaining an unnecessary ± sign, mishandling the constant inside the square root, and confusing inverse-function notation with differentiation notation.

    Question paper p.12 · Mark scheme p.13

15. Question 9 (c)

    3 marks · 48/75

    Solving an intersection equation between the restricted quadratic function and a given linear function.

    Examiner report: Although many candidates formed the quadratic correctly, the domain restriction was often not applied when selecting the final solution.

    Question paper p.12 · Mark scheme p.13

16. Question 9 (d)

    5 marks · 53/75

    Evaluating a nested composition involving an inverse function and determining an unknown coefficient.

    Examiner report: Misreading the inverse notation and composing the functions in the wrong order were common. Expansion errors also caused otherwise suitable methods to fail.

    Question paper p.13 · Mark scheme p.14

17. Question 10 (a)

    2 marks · 55/75

    Finding where a circle meets the x-axis by solving the resulting quadratic equation.

    Examiner report: Some candidates did not first impose the x-axis condition. Once it was used, most obtained the required pair of intersections without difficulty.

    Question paper p.14 · Mark scheme p.14

18. Question 10 (b)

    6 marks · 61/75

    Finding the intersection of two circle tangents using the centre–radius geometry and perpendicular gradients.

    Examiner report: Both coordinate geometry and implicit differentiation were viable. Sign errors in the centre, radius slopes or perpendicular gradients were common, as were omissions of dy/dx when differentiating implicitly.

    Question paper p.14 · Mark scheme p.15

19. Question 11 (a)

    5 marks · 66/75

    Finding a normal line to a curve at a specified point using differentiation and the reciprocal-gradient relationship.

    Examiner report: The coefficient from the chain rule was often lost, and some candidates found the tangent gradient but did not convert it to the normal gradient. A few substituted into the curve without differentiating.

    Question paper p.16 · Mark scheme p.16

20. Question 11 (b)

    3 marks · 69/75

    Locating a stationary point by setting the first derivative equal to zero.

    Examiner report: Candidates with a sound derivative generally completed this part successfully. Omitting the linear term in the derivative often made the stationary-point equation unsolvable.

    Question paper p.16 · Mark scheme p.16

21. Question 11 (c)

    2 marks · 71/75

    Classifying a stationary point using the second-derivative test or a justified sign-change test.

    Examiner report: The second-derivative approach was usually clearest. Sign-chart approaches were sometimes stated without enough numerical evidence, while chain-rule errors were commonly carried into the second derivative.

    Question paper p.17 · Mark scheme p.16

22. Question 11 (d)

    4 marks · 75/75

    Finding the exact area under a curve over a stated interval by definite integration.

    Examiner report: Incorrect handling of the inner linear expression was widespread. Some candidates treated the lower endpoint as making the whole expression vanish, and calculator-only results lacked sufficient supporting work.

    Question paper p.17 · Mark scheme p.17