9709 Pure Mathematics 1 Oct/Nov 2021 Paper 12

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

1. Question 1

   4 marks · 4/75

   Question paper p.2 · Mark scheme p.6

2. Question 2 (a)

   3 marks · 7/75

   This assesses identifying the separate transformations represented by replacing x with 2x and then subtracting 3 from the function value.

   Examiner report: Transformation descriptions were often successful, although some candidates confused the terminology or used the reciprocal scale factor incorrectly and others failed to specify the direction.

   Question paper p.3 · Mark scheme p.7

3. Question 2 (b)

   2 marks · 9/75

   This assesses reversing a combined graph transformation to map a point on the transformed curve back to the original curve.

   Examiner report: This was more difficult for many candidates, who applied the transformations forwards to the supplied point instead of reversing them.

   Question paper p.3 · Mark scheme p.7

4. Question 3 (a)

   2 marks · 11/75

   This assesses evaluating a composite function by applying the function twice to the given input.

   Examiner report: The part was generally handled well; a less efficient approach seen was to derive a full expression for the composite function before substituting the numerical input.

   Question paper p.4 · Mark scheme p.7

5. Question 3 (b)

   3 marks · 14/75

   This assesses finding the inverse of a fractional linear function by rearranging its equation.

   Examiner report: Most candidates knew the usual rearrangement method for this type of inverse function.

   Question paper p.4 · Mark scheme p.8

6. Question 4

   4 marks · 18/75

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

   Examiner report: This was usually answered successfully, though some weaker responses treated the derivative as describing a straight line rather than integrating it; omission of the necessary divisors also occurred.

   Question paper p.5 · Mark scheme p.8

7. Question 5 (a)

   3 marks · 21/75

   This assesses using equal successive differences in an arithmetic progression whose terms are given trigonometrically.

   Examiner report: Many candidates found the trigonometric form demanding; some used only two of the three terms or treated the two-term gaps as single common differences.

   Question paper p.6 · Mark scheme p.9

8. Question 5 (b)

   3 marks · 24/75

   This assesses finding an arithmetic progression’s common difference and applying the finite-sum formula to 25 terms.

   Examiner report: Some candidates found a term rather than a sum, while others lost accuracy by using an incorrect gap for the common difference or omitted the part after an unsuccessful first section.

   Question paper p.6 · Mark scheme p.9

9. Question 6

   5 marks · 29/75

   This assesses solving a geometric progression problem from a specified term and an infinite sum, with a restriction on the common ratio.

   Examiner report: Many complete solutions were seen, but some candidates used the inequality as though it supplied the ratio itself, did not show how the quadratic was solved, or gave a decimal despite the request for an exact form.

   Question paper p.7 · Mark scheme p.10

10. Question 7 (a)

    2 marks · 31/75

    This assesses combining right-triangle geometry and trigonometry to determine an angle in radians to a specified accuracy.

    Examiner report: Several approaches were valid, but some solutions lacked enough accurate intermediate figures to justify the requested three-significant-figure result.

    Question paper p.8 · Mark scheme p.11

11. Question 7 (b)

    4 marks · 35/75

    This assesses finding a curved-region area by combining a circular sector with a right-triangle subtraction.

    Examiner report: Candidates generally recognised the sector, but weaker attempts used one of the supplied sides as its radius or lost accuracy when subtracting the triangle.

    Question paper p.9 · Mark scheme p.11

12. Question 8 (a)

    4 marks · 39/75

    This assesses extracting the x² coefficient from a product involving a binomial expansion and solving the resulting quadratic in a.

    Examiner report: Most candidates identified the relevant expansion terms, although some included only one contribution or made sign errors when handling the negative binomial term.

    Question paper p.10 · Mark scheme p.12

13. Question 8 (b)

    4 marks · 43/75

    This assesses using the repeated-root condition to find when the coefficient expression has only one possible value of a.

    Examiner report: The wording indicating a new condition was missed by some candidates, who reused the alternatives from part (a); successful solutions usually formed the quadratic and applied the discriminant condition.

    Question paper p.11 · Mark scheme p.13

14. Question 9 (a)

    3 marks · 46/75

    This assesses related rates for the radius of a mound when its volume is given as a function of radius.

    Examiner report: Differentiation was often correct, but the chain-rule relationship was a frequent source of error; a few candidates incorrectly introduced sphere or hemisphere volume formulae instead of using the model supplied.

    Question paper p.12 · Mark scheme p.15

15. Question 9 (b)

    3 marks · 49/75

    This assesses reversing the related-rate relationship to find the mound’s radius and then its volume.

    Examiner report: Many candidates found this part difficult or left it blank, and some incorrectly carried the numerical result from part (a) into this separate situation.

    Question paper p.13 · Mark scheme p.15

16. Question 10 (a)

    3 marks · 52/75

    This assesses finding a parameter from the condition that a curve has a stationary point at a specified x-value.

    Examiner report: Most candidates recognised the need to differentiate and set the result to zero, although a few instead set the original function equal to zero.

    Question paper p.14 · Mark scheme p.16

17. Question 10 (b)

    2 marks · 54/75

    This assesses classifying the stationary point using the second derivative.

    Examiner report: The second-derivative method was widely used, but a small number of candidates failed to evaluate it at the required x-value.

    Question paper p.15 · Mark scheme p.16

18. Question 10 (c)

    2 marks · 56/75

    This assesses determining the range of a function from the nature and uniqueness of its stationary point.

    Examiner report: This was less secure: some responses gave only one function value or used the wrong inequality direction, rather than interpreting the unique minimum correctly.

    Question paper p.15 · Mark scheme p.16

19. Question 11 (a)

    5 marks · 61/75

    This assesses finding where a normal to a curve crosses the x-axis using differentiation and the negative-reciprocal gradient rule.

    Examiner report: Part (a) was generally strong, though some candidates differentiated correctly but did not substitute the point’s x-coordinate before finding the normal.

    Question paper p.16 · Mark scheme p.17

20. Question 11 (b)

    6 marks · 67/75

    This assesses calculating a composite shaded area using integration for the curved section and elementary geometry for the triangular section.

    Examiner report: Integration was usually understood, but the upper limit was sometimes confused with the intercept from part (a); other common omissions were the triangular component or an unnecessary attempt to integrate the normal.

    Question paper p.17 · Mark scheme p.18

21. Question 12 (a)

    5 marks · 72/75

    This assesses finding the tangent to a circle and using its axis intercepts to calculate a triangle’s area.

    Examiner report: Many candidates used the centre-and-radius method successfully, while weaker solutions used the origin instead of the circle’s centre or stopped after finding the intercepts without calculating the area.

    Question paper p.18 · Mark scheme p.19

22. Question 12 (b)

    3 marks · 75/75

    This assesses using the geometry of an equilateral triangle inscribed in a circle to determine its exact area.

    Examiner report: This was challenging for many candidates; those who found the radius often still needed to infer the relevant central geometry from the diagram, and almost half of the candidates did not attempt the part.

    Question paper p.19 · Mark scheme p.21