9709 Pure Mathematics 1 Oct/Nov 2020 Paper 11

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

1. Question 1

   3 marks · 3/75

   This tests the use of a discriminant to determine when a line and a quadratic curve have no common point.

   Examiner report: The successful approach was generally to remove y first and then analyse the resulting quadratic. Attempts based on removing x tended not to lead to a workable condition.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   4 marks · 7/75

   This assesses integration of a derivative and use of a point on a curve to determine the integration constant.

   Examiner report: Candidates who treated the given derivative as something to integrate usually made good progress, especially once they used the supplied point to determine the constant.

   Question paper p.3 · Mark scheme p.6

3. Question 3

   3 marks · 10/75

   This tests related rates for a sphere, including linking the rates of change of volume and radius.

   Examiner report: Very few responses made the necessary chain-rule connection between the two rates.

   Question paper p.4 · Mark scheme p.7

4. Question 4

   3 marks · 13/75

   Question paper p.5 · Mark scheme p.7

5. Question 5 (a)

   4 marks · 17/75

   This tests selection of terms in a binomial expansion and comparison of their coefficients.

   Examiner report: Many candidates expanded far more of the binomial than was needed. The key difficulty was retaining the correct powers of both 2 and a in the two selected terms.

   Question paper p.6 · Mark scheme p.7

6. Question 5 (b)

   1 mark · 18/75

   Question paper p.6 · Mark scheme p.8

7. Question 6

   5 marks · 23/75

   This assesses differentiation of a composite square-root function and selection of a valid point with a specified gradient.

   Examiner report: The differentiation was usually manageable, but many candidates retained an extraneous value created when the equation was squared and failed to test it in the original condition.

   Question paper p.7 · Mark scheme p.8

8. Question 7 (a)

   3 marks · 26/75

   Question paper p.8 · Mark scheme p.9

9. Question 7 (b)

   3 marks · 29/75

   This tests solving a trigonometric equation using an identity and listing all solutions in a specified interval.

   Examiner report: The positive tangent case was usually handled, but the solution associated with negative tangent was often omitted.

   Question paper p.9 · Mark scheme p.9

10. Question 8 (a)

    3 marks · 32/75

    This assesses the sum-to-infinity formula for geometric progressions and elimination of common quantities.

    Examiner report: Most candidates knew the infinite geometric-sum formula and could begin the elimination, although some did not complete the algebra linking the two ratios.

    Question paper p.10 · Mark scheme p.9

11. Question 8 (b)

    4 marks · 36/75

    This tests combining a term relationship in geometric progressions with an earlier ratio relation to find a sum in terms of the first term.

    Examiner report: The second part was often not attempted unless the first ratio relation had been obtained. The essential step was eliminating one ratio before determining the required sum.

    Question paper p.11 · Mark scheme p.10

12. Question 9 (a)

    2 marks · 38/75

    This tests finding a tangent to a circle from the coordinates of its centre and a point of contact.

    Examiner report: This was generally an accessible part, with many candidates correctly using perpendicular gradients to connect the diameter and tangent.

    Question paper p.12 · Mark scheme p.11

13. Question 9 (b)

    3 marks · 41/75

    This assesses finding a circle equation from its centre and a point on its circumference.

    Examiner report: The standard centre-radius form led to more reliable work than the general circle equation, where sign errors were frequent.

    Question paper p.13 · Mark scheme p.11

14. Question 9 (c)

    3 marks · 44/75

    This tests finding intersections of a line and circle by substitution and solving a quadratic.

    Examiner report: Candidates who had correct equations from the earlier parts generally completed this by eliminating y and solving the resulting quadratic.

    Question paper p.13 · Mark scheme p.11

15. Question 10 (a)

    3 marks · 47/75

    This tests the use of right-triangle trigonometry and the tangent-radius property in a sector configuration.

    Examiner report: The geometric setup was the main obstacle; those who recognised the right triangle and related the radius to the tangent generally proceeded successfully.

    Question paper p.14 · Mark scheme p.12

16. Question 10 (b)

    3 marks · 50/75

    This assesses sector perimeter calculations using an arc-length formula and the radius obtained in part (a).

    Examiner report: The arc-length formula was usually known, but some candidates incorrectly used the small circle's radius instead of the sector radius for the arc and straight sides.

    Question paper p.15 · Mark scheme p.12

17. Question 10 (c)

    4 marks · 54/75

    This tests area calculation in a composite sector diagram involving a triangle or kite and a circular sector.

    Examiner report: Successful solutions separated the kite-shaped contribution from the sector contribution. The sector-area formula was often available, but combining the two geometrical regions correctly was less common.

    Question paper p.15 · Mark scheme p.13

18. Question 11 (a)

    1 mark · 55/75

    Question paper p.16 · Mark scheme p.14

19. Question 11 (b)

    4 marks · 59/75

    This tests inversion of a composite function and identification of the inverse function's domain.

    Examiner report: Working from the unexpanded composite made the inverse more straightforward. The domain was sometimes missed, although it could be obtained directly from the original range or the inverse expression.

    Question paper p.16 · Mark scheme p.14

20. Question 11 (c)

    4 marks · 63/75

    Question paper p.17 · Mark scheme p.14

21. Question 12 (a)

    4 marks · 67/75

    This tests finding intersections between a square-root curve and a straight line by reducing the equation to a quadratic.

    Examiner report: Using √x as the quadratic variable was usually more dependable than isolating the radical and squaring, which sometimes introduced errors.

    Question paper p.18 · Mark scheme p.15

22. Question 12 (b)

    2 marks · 69/75

    This tests demonstrating that a point on a curve is stationary by evaluating its derivative there.

    Examiner report: A direct derivative evaluation at the known coordinate was sufficient, and most candidates who attempted this part could complete it.

    Question paper p.18 · Mark scheme p.15

23. Question 12 (c)

    6 marks · 75/75

    Question paper p.19 · Mark scheme p.16