9709 Pure Mathematics 1 Oct/Nov 2021 Paper 11

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

1. Question 1 (a)

   1 mark · 1/75

   This tests expansion and simplification of a squared binomial containing a reciprocal term.

   Examiner report: Errors commonly occurred in the middle term, especially through mishandling the power of x.

   Question paper p.3 · Mark scheme p.6

2. Question 1 (b)

   2 marks · 3/75

   This tests use of the binomial theorem to obtain an expansion in ascending powers of x.

   Examiner report: Some responses lost marks through incorrect coefficients or by leaving out the constant term.

   Question paper p.3 · Mark scheme p.6

3. Question 1 (c)

   2 marks · 5/75

   This tests selection and combination of products that contribute to a specified power in a product of expansions.

   Examiner report: The clearest solutions considered only the relevant products; multiplying everything out often created avoidable errors.

   Question paper p.3 · Mark scheme p.6

4. Question 2

   5 marks · 10/75

   This tests using the discriminant to find parameter values for which a curve and a line have no common point.

   Examiner report: Sign mistakes in forming the quadratic were frequent, and some candidates found only the boundary values rather than the required intervals.

   Question paper p.4 · Mark scheme p.7

5. Question 3

   4 marks · 14/75

   This tests factorisation of a trigonometric equation followed by solving the resulting equations over a stated interval.

   Examiner report: Replacing tangent by sine divided by cosine made the algebra longer and often prevented candidates from reaching both valid solutions.

   Question paper p.5 · Mark scheme p.8

6. Question 4 (a)

   4 marks · 18/75

   This tests simultaneous use of the infinite geometric-series sum and the finite arithmetic-series sum.

   Examiner report: A common mistake was to use the formula for an arithmetic term instead of its sum; sign and arithmetic slips were also seen.

   Question paper p.6 · Mark scheme p.9

7. Question 4 (b)

   2 marks · 20/75

   This tests finding the position of a zero term in an arithmetic progression.

   Examiner report: Even when part (a) was unsuccessful, a correct general-term equation using the candidate’s value of a could still demonstrate the required method.

   Question paper p.6 · Mark scheme p.9

8. Question 5 (a)

   3 marks · 23/75

   Question paper p.7 · Mark scheme p.9

9. Question 5 (b)(i)

   1 mark · 24/75

   Question paper p.7 · Mark scheme p.10

10. Question 5 (b)(ii)

    1 mark · 25/75

    Question paper p.7 · Mark scheme p.10

11. Question 6 (a)

    3 marks · 28/75

    This tests finding arc lengths and combining them with a straight side to obtain the perimeter of a composite circular shape.

    Examiner report: Some candidates could not extract the required sector angle from the diagram, preventing progress with the arc calculation.

    Question paper p.8 · Mark scheme p.10

12. Question 6 (b)

    4 marks · 32/75

    This tests finding the exact area of a shape made from circular sectors, segments, and a triangle.

    Examiner report: Several approaches were accepted, but errors included halving the triangle area incorrectly and using an unsuitable triangle-area formula.

    Question paper p.9 · Mark scheme p.11

13. Question 7 (a)

    2 marks · 34/75

    This tests finding a circle’s radius from its centre and a point on the circumference, then writing its equation.

    Examiner report: Frequent errors included using the radius rather than its square and assigning an incorrect sign in the circle equation.

    Question paper p.10 · Mark scheme p.12

14. Question 7 (b)

    7 marks · 41/75

    This tests finding the intersections of a line and circle and then calculating the exact chord length.

    Examiner report: The main difficulties were eliminating the variable accurately, solving the resulting quadratic, and completing the final distance calculation when coordinates were available.

    Question paper p.10 · Mark scheme p.12

15. Question 8 (a)

    2 marks · 43/75

    This tests completing the square for a quadratic expression.

    Examiner report: Most candidates found the horizontal shift, but fewer obtained the correct constant term as well.

    Question paper p.12 · Mark scheme p.13

16. Question 8 (b)

    1 mark · 44/75

    Question paper p.12 · Mark scheme p.13

17. Question 8 (c)

    1 mark · 45/75

    This tests finding the range of a restricted quadratic function.

    Examiner report: Incorrect inequality directions and expressions describing the domain rather than the range were common.

    Question paper p.12 · Mark scheme p.14

18. Question 8 (d)

    3 marks · 48/75

    This tests finding an inverse function algebraically while respecting the original restricted domain.

    Examiner report: The algebra was familiar to many candidates, but the branch inconsistent with the original domain was often retained.

    Question paper p.13 · Mark scheme p.14

19. Question 8 (e)

    3 marks · 51/75

    This tests applying a two-dimensional translation to a quadratic graph and expanding the result.

    Examiner report: Correct solutions used either the original or completed-square form, while others made errors in modifying the equation.

    Question paper p.13 · Mark scheme p.15

20. Question 9 (a)

    4 marks · 55/75

    This tests integrating a derivative and using a known function value to determine the constant of integration.

    Examiner report: Many candidates integrated successfully, but omitting the integration constant prevented completion of the question.

    Question paper p.14 · Mark scheme p.15

21. Question 9 (b)

    5 marks · 60/75

    This tests finding stationary points by solving a disguised quadratic equation in x² and evaluating the original function.

    Examiner report: A suitable factorisation or substitution was needed for the method marks; arithmetic mistakes in the y-coordinates were particularly common.

    Question paper p.15 · Mark scheme p.16

22. Question 9 (c)

    1 mark · 61/75

    Question paper p.15 · Mark scheme p.16

23. Question 9 (d)

    2 marks · 63/75

    This tests classifying stationary points using the second-derivative test.

    Examiner report: The intended follow-through was to use the second derivative at the values found earlier; unsupported labels without evaluated signs were not sufficient.

    Question paper p.15 · Mark scheme p.17

24. Question 10 (a)

    4 marks · 67/75

    This tests evaluating an improper integral involving a shifted power of a linear expression.

    Examiner report: Index manipulation caused difficulty, particularly losing the factor introduced by differentiating the linear expression or failing to evaluate the limiting term.

    Question paper p.16 · Mark scheme p.18

25. Question 10 (b)

    4 marks · 71/75

    This tests finding a volume of revolution by integrating the square of a curve’s ordinate.

    Examiner report: The most frequent omission was the squaring required by the volume formula; some otherwise sound attempts then had only a coefficient error.

    Question paper p.16 · Mark scheme p.18

26. Question 10 (c)

    4 marks · 75/75

    This tests finding a normal’s y-intercept from the derivative at a specified point.

    Examiner report: Some candidates found the derivative and tangent gradient but did not complete the separate step of using the normal line to obtain the intercept.

    Question paper p.17 · Mark scheme p.19