9709 Pure Mathematics 1 May/June 2022 Paper 11

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

1. Question 1 (a)

   2 marks · 2/75

   Completing the square in a quadratic expression and identifying the resulting constants.

   Examiner report: Most candidates used the expected method successfully, although sign slips and arithmetic mistakes affected some responses.

   Question paper p.2 · Mark scheme p.6

2. Question 1 (b)

   2 marks · 4/75

   Solving a quadratic exactly by using a completed-square result.

   Examiner report: Although the wording encouraged use of the preceding result, many candidates instead applied the quadratic formula.

   Question paper p.2 · Mark scheme p.6

3. Question 2

   5 marks · 9/75

   Using arithmetic-progression formulae to determine unknown terms and then a later partial sum.

   Examiner report: The main difficulty was using the sum formula correctly: treating its final term as an independent number, rather than expressing it through the first term and common difference, prevented further progress.

   Question paper p.3 · Mark scheme p.6

4. Question 3 (a)

   3 marks · 12/75

   Selecting specified terms in binomial expansions and extracting their coefficients in terms of a parameter.

   Examiner report: Many candidates selected the wrong terms or mishandled powers, while others lost the parameter factors by giving only numerical coefficients.

   Question paper p.4 · Mark scheme p.7

5. Question 3 (b)

   3 marks · 15/75

   Question paper p.5 · Mark scheme p.7

6. Question 4 (a)

   4 marks · 19/75

   Question paper p.6 · Mark scheme p.8

7. Question 4 (b)

   2 marks · 21/75

   Solving a trigonometric equation by applying a previously proved identity and respecting an open interval.

   Examiner report: Very few complete solutions were seen. Dividing by the squared tangent discarded the valid case, and some otherwise correct solutions also retained an endpoint excluded by the interval.

   Question paper p.7 · Mark scheme p.9

8. Question 5 (a)

   3 marks · 24/75

   Finding an exact circular-segment area by subtracting a right-triangle area from a sector area.

   Examiner report: The sector calculation was generally secure, but finding and subtracting the triangle area caused more difficulty; decimal forms could not satisfy the exact-answer requirement.

   Question paper p.8 · Mark scheme p.10

9. Question 5 (b)

   4 marks · 28/75

   Using a perpendicular chord-related length to determine the angle and then calculating the perimeter of a circular segment.

   Examiner report: Only a small proportion noticed that the changed length implied a different central angle. Continuing with the angle from part (a) allowed only limited method credit.

   Question paper p.9 · Mark scheme p.11

10. Question 6 (a)

    3 marks · 31/75

    Finding the inverse of a rational function with a restricted original domain.

    Examiner report: Many solutions were sound, but some failed while isolating the squared variable or attached an unnecessary plus-or-minus sign to the inverse.

    Question paper p.10 · Mark scheme p.12

11. Question 6 (b)

    4 marks · 35/75

    Verifying an algebraic rearrangement of a rational function and determining its range from domain restrictions.

    Examiner report: The algebraic verification was usually done correctly, but relatively few candidates identified both ends of the range.

    Question paper p.11 · Mark scheme p.13

12. Question 6 (c)

    1 mark · 36/75

    Deciding whether a composite function is defined from the relationship between the range of one function and the domain required for the next application.

    Examiner report: The report noted that a satisfactory explanation had to compare outputs with the permitted inputs; merely referring to the range without this connection was insufficient.

    Question paper p.11 · Mark scheme p.13

13. Question 7 (a)

    4 marks · 40/75

    Finding curve-line intersections involving a square-root function and a straight line.

    Examiner report: The initial equation and squaring step were often correct, but a frequent error was squaring the two terms of the line separately, producing incorrect points.

    Question paper p.12 · Mark scheme p.13

14. Question 7 (b)

    5 marks · 45/75

    Calculating the area between a square-root curve and a line using definite integration.

    Examiner report: Errors commonly arose when integrating the square-root term and when evaluating the endpoints. Calculator-only results could not receive full justification.

    Question paper p.13 · Mark scheme p.14

15. Question 8 (a)

    5 marks · 50/75

    Describing a composite sequence of graph transformations involving horizontal and vertical changes and a translation.

    Examiner report: Most responses contained at least one error in the horizontal transformation. Typical issues were reversing the scale factor, using the wrong axis, or choosing a translation vector suited to a different order.

    Question paper p.14 · Mark scheme p.16

16. Question 8 (b)

    3 marks · 53/75

    Solving a sine equation in degrees over a bounded interval using both sine branches.

    Examiner report: Some candidates struggled to isolate the sine value or applied the operations to the inner angle in the wrong order; omitting the second branch was also common.

    Question paper p.15 · Mark scheme p.16

17. Question 9 (a)

    4 marks · 57/75

    Putting a circle equation into centre-radius form and locating its lowest point.

    Examiner report: Sign and radius-squared errors were frequent, and some candidates found the centre and radius but did not complete the request for the lowest point.

    Question paper p.16 · Mark scheme p.17

18. Question 9 (b)

    6 marks · 63/75

    Finding the slopes for which a line meets a circle twice by using a discriminant condition.

    Examiner report: Although both direct substitution and a rearranged circle equation were viable, expansion and collection errors meant that relatively few candidates reached the correct inequality.

    Question paper p.17 · Mark scheme p.17

19. Question 10 (a)

    1 mark · 64/75

    Classifying a stationary point with the second-derivative test.

    Examiner report: Correct responses usually required only the substitution and classification; sign mistakes at the negative x-value caused the main losses.

    Question paper p.18 · Mark scheme p.18

20. Question 10 (b)

    5 marks · 69/75

    Recovering a curve from its second derivative using two integrations and two point conditions.

    Examiner report: This was one of the more demanding parts. Common omissions included integration constants, the second integration, and the fact that the derivative at a stationary point is zero.

    Question paper p.18 · Mark scheme p.18

21. Question 10 (c)

    3 marks · 72/75

    Showing uniqueness of a stationary point by solving the first-derivative equation.

    Examiner report: Many candidates formed the derivative equation but did not solve it fully or explain why no other solution existed; some incorrectly used the second derivative instead.

    Question paper p.19 · Mark scheme p.18

22. Question 10 (d)

    3 marks · 75/75

    Using related rates and the chain rule to connect the rates of change of x and y along a curve.

    Examiner report: The required substitution into the derivative was often made, but progress then depended on applying the chain rule with the rate factors in the correct order.

    Question paper p.19 · Mark scheme p.19