9709 Pure Mathematics 1 May/June 2022 Paper 13

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

1. Question 1

   4 marks · 4/75

   This assesses selection of a binomial-expansion term and formation of an equation from its coefficient.

   Examiner report: This was usually handled successfully using the binomial formula, but some candidates reported only one sign when two values were required.

   Question paper p.3 · Mark scheme p.6

2. Question 2 (a)

   1 mark · 5/75

   Question paper p.4 · Mark scheme p.6

3. Question 2 (b)

   1 mark · 6/75

   Question paper p.4 · Mark scheme p.6

4. Question 2 (c)

   1 mark · 7/75

   Question paper p.4 · Mark scheme p.6

5. Question 3 (a)

   1 mark · 8/75

   This assesses use of the arithmetic-progression sum formula and algebraic rearrangement.

   Examiner report: Successful responses normally showed an intermediate sum formula rather than jumping straight to the displayed result.

   Question paper p.5 · Mark scheme p.7

6. Question 3 (b)

   4 marks · 12/75

   This tests simultaneous use of the nth-term formula and the sum relation for an arithmetic progression.

   Examiner report: Most candidates used the preceding result effectively and then found the common difference with the correct progression formula.

   Question paper p.5 · Mark scheme p.7

7. Question 4 (a)

   3 marks · 15/75

   This assesses the effect of a two-dimensional translation on a quadratic graph and subsequent expansion.

   Examiner report: The main error was reversing the direction of the horizontal movement; completing the square often helped candidates avoid this.

   Question paper p.6 · Mark scheme p.7

8. Question 4 (b)

   2 marks · 17/75

   This tests recognition and description of a single horizontal stretch or compression from two quadratic equations.

   Examiner report: Many descriptions incorrectly combined a stretch with a translation, and some omitted either the direction or the scale factor.

   Question paper p.6 · Mark scheme p.8

9. Question 5 (a)

   4 marks · 21/75

   This assesses solving an equation containing both a square root and its reciprocal by substitution.

   Examiner report: Calculator-generated roots without supporting algebra did not demonstrate the required method; factorisation was a particularly reliable approach.

   Question paper p.7 · Mark scheme p.8

10. Question 5 (b)

    3 marks · 24/75

    This tests transferring the permissible values from a radical equation to a trigonometric equation and finding all angles in a full revolution.

    Examiner report: Candidates who had the valid values from part (a) generally found the four angles, although quadrant selection and rounding still caused some errors.

    Question paper p.7 · Mark scheme p.8

11. Question 6 (a)

    2 marks · 26/75

    This assesses completing the square for a quadratic function.

    Examiner report: Most candidates completed the square correctly with little difficulty.

    Question paper p.8 · Mark scheme p.9

12. Question 6 (b)

    1 mark · 27/75

    This tests finding the range of a quadratic when its domain is restricted.

    Examiner report: The crucial point was applying the restricted domain rather than using the range of the unrestricted quadratic.

    Question paper p.8 · Mark scheme p.9

13. Question 6 (c)

    3 marks · 30/75

    This assesses finding the inverse of a restricted quadratic function.

    Examiner report: The algebra was generally sound; the deciding issue was choosing the branch compatible with the restriction on the original function.

    Question paper p.9 · Mark scheme p.9

14. Question 6 (d)

    2 marks · 32/75

    This tests composition of two functions followed by algebraic simplification.

    Examiner report: Both substitution into the completed-square form and direct use of the original definition produced many correct solutions.

    Question paper p.9 · Mark scheme p.9

15. Question 7 (a)

    2 marks · 34/75

    This assesses finding the equation of a line from a circle’s centre and another point.

    Examiner report: This was a straightforward part for most candidates, with many obtaining the line equation correctly.

    Question paper p.10 · Mark scheme p.10

16. Question 7 (b)

    5 marks · 39/75

    This tests solving for the intersection of a line and circle and selecting the geometrically relevant point.

    Examiner report: The simultaneous algebra was usually completed well, but some candidates kept both intersections instead of using the diagram to choose the relevant one.

    Question paper p.11 · Mark scheme p.10

17. Question 8 (a)

    4 marks · 43/75

    This assesses differentiation of fractional powers and use of a point-slope equation for a tangent.

    Examiner report: Fractional indices were handled competently by most candidates, though some confused the tangent gradient with that of the normal.

    Question paper p.12 · Mark scheme p.10

18. Question 8 (b)

    4 marks · 47/75

    This tests finding a bounded area between a curve and a horizontal line using definite integration.

    Examiner report: Integration was generally successful; the common omission was stopping at the area under the curve instead of taking the difference from the line’s area.

    Question paper p.13 · Mark scheme p.11

19. Question 9 (a)

    5 marks · 52/75

    This assesses combined use of triangle geometry, radians, and arc length to calculate a composite perimeter.

    Examiner report: Candidates used several valid routes for the triangle measurements, and the arc-length formula was usually applied correctly when the angle was kept in a consistent unit.

    Question paper p.14 · Mark scheme p.11

20. Question 9 (b)

    3 marks · 55/75

    This tests finding a composite shaded area by combining a sector and a triangle.

    Examiner report: The sector calculation was generally secure, and Heron’s formula as well as the included-angle formula were accepted for the triangle.

    Question paper p.15 · Mark scheme p.12

21. Question 10 (a)

    4 marks · 59/75

    Question paper p.16 · Mark scheme p.12

22. Question 10 (b)

    6 marks · 65/75

    This tests implicit use of the chain rule with related rates for motion along a curve.

    Examiner report: Differentiation was familiar to most candidates, but errors often arose when translating the two coordinate rates into the gradient condition or when handling the resulting power equation.

    Question paper p.17 · Mark scheme p.12

23. Question 11 (a)

    6 marks · 71/75

    This assesses the tangent condition for a line and a reciprocal curve, including the restriction to a unique contact point.

    Examiner report: A frequent issue was retaining algebraic possibilities that conflict with the positive-constant conditions or with the requirement that there is only one contact point.

    Question paper p.18 · Mark scheme p.13

24. Question 11 (b)

    4 marks · 75/75

    This tests finding a normal to a reciprocal curve and locating its second intersection with the curve.

    Examiner report: This was one of the more demanding parts; only a small proportion of candidates completed the algebraic factorisation and isolated the second intersection successfully.

    Question paper p.19 · Mark scheme p.14