9709 Pure Mathematics 1 Oct/Nov 2021 Paper 13

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

1. Question 1

   4 marks · 4/75

   This assesses the interpretation of a composite graph transformation and the importance of applying transformations in the stated order.

   Examiner report: Most candidates described the transformations successfully when they used the reflection first and then shifted the graph upwards; reverse ordering commonly led to an incorrect shift, while vectors would have made the descriptions clearer.

   Question paper p.2 · Mark scheme p.6

2. Question 2 (a)

   1 mark · 5/75

   Question paper p.3 · Mark scheme p.7

3. Question 2 (b)

   4 marks · 9/75

   This assesses the extraction of a coefficient from a product of expansions and the solution of the resulting quadratic equation.

   Examiner report: Most candidates recognised that two products contribute to the degree-two term, but a fully justified expansion and solution was required rather than unsupported roots.

   Question paper p.3 · Mark scheme p.7

4. Question 3 (a)

   2 marks · 11/75

   Question paper p.4 · Mark scheme p.8

5. Question 3 (b)

   3 marks · 14/75

   This assesses the use of differentiation and positivity to decide whether a polynomial function is increasing throughout its domain.

   Examiner report: Differentiation was usually done correctly, but weaker solutions did not justify positivity for all real inputs; checking only a few values is insufficient.

   Question paper p.4 · Mark scheme p.8

6. Question 4 (a)

   2 marks · 16/75

   This assesses the nth-term formula for an arithmetic progression together with an inequality or boundary check.

   Examiner report: Trial methods and the standard nth-term approach both produced correct results when the boundary condition was handled accurately.

   Question paper p.5 · Mark scheme p.8

7. Question 4 (b)

   3 marks · 19/75

   This assesses the formula for the sum of an arithmetic progression and the formation and solution of an equation involving two sums.

   Examiner report: Most candidates formed the two series sums correctly; errors mainly arose from using an incorrect number of terms or mishandling the common difference.

   Question paper p.5 · Mark scheme p.8

8. Question 5 (a)

   1 mark · 20/75

   This assesses the use of right-triangle trigonometry to determine an angle in radians from the geometry of the diagram.

   Examiner report: This was answered well overall, with only a small number of candidates unnecessarily converting through degrees.

   Question paper p.6 · Mark scheme p.9

9. Question 5 (b)

   6 marks · 26/75

   Question paper p.7 · Mark scheme p.9

10. Question 6 (a)

    1 mark · 27/75

    This assesses the graphical interpretation of an inverse function.

    Examiner report: Although many candidates recognised that reflection was involved, relatively few used the line y=x rather than another axis or line.

    Question paper p.8 · Mark scheme p.9

11. Question 6 (b)

    4 marks · 31/75

    This assesses finding an inverse function algebraically when the original relation contains a square root in the denominator.

    Examiner report: The main difficulty was the algebra after changing the subject, especially errors from squaring the negative term and choosing the correct final sign.

    Question paper p.8 · Mark scheme p.10

12. Question 6 (c)

    1 mark · 32/75

    This assesses the domain condition needed for a composite function to be defined.

    Examiner report: Candidates who focused on the parameter controlling the interval generally handled this part well; giving a value of x instead did not answer the question.

    Question paper p.9 · Mark scheme p.10

13. Question 6 (d)

    2 marks · 34/75

    This assesses evaluation and simplification of a composite function in the correct order.

    Examiner report: Substitution into the outer function was usually correct, while factorising and cancelling the denominator separated stronger solutions from weaker ones.

    Question paper p.9 · Mark scheme p.10

14. Question 7 (a)

    4 marks · 38/75

    This assesses a full trigonometric-algebraic derivation involving the tangent identity and the Pythagorean identity.

    Examiner report: A complete chain of algebra was required; incomplete solutions often omitted one of the identities or jumped to the final expression without showing the rearrangement.

    Question paper p.10 · Mark scheme p.11

15. Question 7 (b)

    4 marks · 42/75

    This assesses solving a trigonometric equation by reducing it to a quadratic in sin x and selecting all angles in a specified interval.

    Examiner report: Most candidates used the result from the preceding part, but full credit depended on showing how the quadratic was solved and including both valid angles.

    Question paper p.11 · Mark scheme p.11

16. Question 8 (a)

    6 marks · 48/75

    This assesses finding the area between two intersecting curves using definite integration with fractional powers.

    Examiner report: The integration was generally accessible; common issues included unclear limit substitution and treating the problem as a volume rather than a planar area.

    Question paper p.12 · Mark scheme p.12

17. Question 8 (b)

    4 marks · 52/75

    This assesses differentiation of a fractional power, conversion from tangent gradient to normal gradient, and use of a line equation.

    Examiner report: Most candidates found the derivative and then proceeded correctly to the normal equation and intercept.

    Question paper p.13 · Mark scheme p.12

18. Question 9 (a)

    7 marks · 59/75

    This assesses finding circle-line intersections in exact surd form and calculating a chord length from the resulting coordinates.

    Examiner report: The intersection method was well understood, but exact surd coordinates required a proper quadratic solution rather than decimal approximations; the distance formula was then usually applied correctly.

    Question paper p.14 · Mark scheme p.13

19. Question 9 (b)

    5 marks · 64/75

    This assesses finding tangent gradients to a circle from a fixed external point using either a repeated-root condition or geometric trigonometry.

    Examiner report: This part was frequently omitted, and a sketch would have helped reveal the two possible tangents; the geometric approach was considerably shorter than the full discriminant algebra.

    Question paper p.15 · Mark scheme p.14

20. Question 10 (a)

    3 marks · 67/75

    This assesses differentiating a first-derivative expression and using the second-derivative test at a minimum to restrict a parameter.

    Examiner report: Most candidates knew to use a second derivative, but some incorrectly differentiated the constant term as though it depended on x.

    Question paper p.16 · Mark scheme p.15

21. Question 10 (b)

    4 marks · 71/75

    This assesses integration of a parameterised derivative and determination of the integration constant from a known point on the curve.

    Examiner report: This was generally handled well; the main recurring error was failing to integrate the constant contribution with respect to x.

    Question paper p.16 · Mark scheme p.15

22. Question 10 (c)

    4 marks · 75/75

    This assesses locating the second stationary point and classifying it with the second derivative.

    Examiner report: Many successful solutions found both stationary x-values, but some stopped before calculating the corresponding coordinate or did not identify whether the point was a maximum or minimum.

    Question paper p.17 · Mark scheme p.15