9709 Pure Mathematics 1 Feb/Mar 2023 Paper 12

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

1. Question 1

   4 marks · 4/75

   This assesses finding when a line and a quadratic curve intersect for every value of a parameter.

   Examiner report: Most candidates knew to create a quadratic and use its discriminant, but many did not explain what the simplified discriminant implied; algebraic sign mistakes were also common.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   4 marks · 8/75

   This assesses applying successive graph transformations algebraically.

   Examiner report: The reflection was the transformation most often mishandled. Other errors included incorrect expansion after the horizontal stretch, scaling only selected terms, and failing to present the final expression in the specified form.

   Question paper p.3 · Mark scheme p.6

3. Question 3

   4 marks · 12/75

   This assesses differentiation by the chain rule and interpreting equal rates of change.

   Examiner report: The differentiation itself was generally secure, but some candidates did not translate the wording about rates into the condition on the gradient; others incorrectly treated the derivative as zero.

   Question paper p.4 · Mark scheme p.7

4. Question 4 (a)

   2 marks · 14/75

   This assesses using the general term of an arithmetic progression with correct term counting.

   Examiner report: A common mistake was to use the second measurement as the first term without adjusting the term number, so many candidates lost the accuracy mark despite using the right progression formula.

   Question paper p.5 · Mark scheme p.7

5. Question 4 (b)

   3 marks · 17/75

   This assesses modelling repeated growth with a geometric progression.

   Examiner report: Many candidates found the ratio correctly but assigned the wrong number of repetitions or terms to the calculation.

   Question paper p.5 · Mark scheme p.7

6. Question 5

   6 marks · 23/75

   This assesses using a circle equation and a line intersection to determine a second point on a circle.

   Examiner report: The circle-and-line method was the usual successful approach, although some candidates used the wrong point when finding the radius. A sketch could have helped candidates who could not identify a starting method.

   Question paper p.6 · Mark scheme p.8

7. Question 6

   6 marks · 29/75

   This assesses extracting selected terms from a binomial expansion and comparing their coefficients.

   Examiner report: Most candidates located the relevant terms, but errors in handling powers were frequent and the negative possibility was often discarded when taking a square root.

   Question paper p.7 · Mark scheme p.9

8. Question 7 (a)

   5 marks · 34/75

   This assesses solving a trigonometric equation by reducing it to a quadratic in cosine.

   Examiner report: The trigonometric identities were generally used well and many solutions were correct, but unsupported calculator roots did not demonstrate the required quadratic method. The valid solutions were usually located in the correct quadrants.

   Question paper p.8 · Mark scheme p.10

9. Question 7 (b)

   3 marks · 37/75

   Question paper p.9 · Mark scheme p.10

10. Question 8 (a)

    5 marks · 42/75

    This assesses combining right-triangle geometry, central angles and arc length to find a perimeter.

    Examiner report: Most candidates found the required length and angle successfully using elementary trigonometry or Pythagoras. Premature rounding affected final values, and a few applications of Pythagoras were made to triangles that were not right-angled.

    Question paper p.10 · Mark scheme p.11

11. Question 8 (b)

    3 marks · 45/75

    This assesses finding a circular-sector area and removing the triangular part to obtain a shaded area.

    Examiner report: The sector calculation was usually understood, and the triangle-subtraction method was the common successful approach. Rounding the angle or lengths too early often led to an inaccurate final result.

    Question paper p.11 · Mark scheme p.11

12. Question 9 (a)

    1 mark · 46/75

    This assesses determining the range of a restricted quadratic function.

    Examiner report: Most responses gave an acceptable description of the outputs, but some wrote the domain restriction instead of the range.

    Question paper p.12 · Mark scheme p.12

13. Question 9 (b)

    3 marks · 49/75

    This assesses finding an inverse function when the original quadratic has a restricted domain.

    Examiner report: Most candidates reached the square-root stage, but the domain restriction was often not used to select the correct branch. Some notation also made it unclear whether the radical covered the entire fraction.

    Question paper p.12 · Mark scheme p.12

14. Question 9 (c)

    5 marks · 54/75

    Question paper p.13 · Mark scheme p.12

15. Question 10 (a)

    1 mark · 55/75

    This assesses using a known gradient at a point to determine an unknown constant in a derivative formula.

    Examiner report: Most candidates recognised that the derivative had to be evaluated at the specified point, but full credit required showing the intermediate evaluation rather than simply asserting the given result.

    Question paper p.14 · Mark scheme p.13

16. Question 10 (b)

    4 marks · 59/75

    This assesses integrating a derivative and using a point on the curve to determine the integration constant.

    Examiner report: The integration was handled well by many candidates. Frequent omissions were the integration constant and the correct treatment of division by the fractional power appearing in the derivative.

    Question paper p.14 · Mark scheme p.13

17. Question 10 (c)

    3 marks · 62/75

    This assesses locating a stationary point from a derivative and then finding both of its coordinates.

    Examiner report: Most candidates found the stationary x-coordinate correctly, but some stopped before calculating the associated y-coordinate.

    Question paper p.15 · Mark scheme p.13

18. Question 10 (d)

    2 marks · 64/75

    This assesses classifying a stationary point with the second-derivative test.

    Examiner report: This was generally answered successfully by evaluating the second derivative at the stationary coordinate; a few candidates instead tried to infer the classification from nearby gradient values.

    Question paper p.15 · Mark scheme p.14

19. Question 11 (a)

    2 marks · 66/75

    This assesses finding a straight-line equation from two points.

    Examiner report: Nearly all candidates attempted this successfully. The main errors were reversing the gradient ratio or interchanging coordinates when applying the line formula.

    Question paper p.16 · Mark scheme p.14

20. Question 11 (b)

    9 marks · 75/75

    This assesses finding a volume of revolution about the y-axis using integration with respect to y.

    Examiner report: A substantial number of responses did not account for the specified axis of rotation, failed to square the radius functions, or mixed x-limits with y-integration. Full credit depended on showing both integrations, their limit substitutions, and the final subtraction rather than relying on an unsupported calculator result.

    Question paper p.16 · Mark scheme p.15