9709 Pure Mathematics 1 May/June 2021 Paper 12

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

1. Question 1 (a)

   2 marks · 2/75

   Completing the square for a quadratic expression in a prescribed form.

   Examiner report: This was generally handled well; comparing coefficients after expansion was more reliable than factoring out a coefficient and trying to restore it later.

   Question paper p.2 · Mark scheme p.6

2. Question 1 (b)

   2 marks · 4/75

   Using the repeated-root condition for a quadratic equation, together with a completed-square representation.

   Examiner report: Some responses found the parameter but stopped before finding the requested root. Others used the discriminant successfully, although the link with part (a) gave a shorter route.

   Question paper p.2 · Mark scheme p.6

3. Question 2 (a)

   3 marks · 7/75

   Describing combined graph transformations using precise transformation terminology.

   Examiner report: Transformation language caused difficulty. Informal descriptions involving movement were not sufficiently precise, and the required vector, direction and scale-factor terminology was often missing.

   Question paper p.3 · Mark scheme p.7

4. Question 2 (b)

   2 marks · 9/75

   Applying a reflection and a horizontal stretch to the equation of a curve.

   Examiner report: The reflection was often dealt with correctly, but the subsequent horizontal stretch was frequently applied to the argument incorrectly.

   Question paper p.3 · Mark scheme p.7

5. Question 3 (a)

   1 mark · 10/75

   Evaluating a curve equation at a specified coordinate to the requested accuracy.

   Examiner report: Several answers were numerically correct but did not follow the specified four-decimal-place format.

   Question paper p.4 · Mark scheme p.8

6. Question 3 (b)

   1 mark · 11/75

   Finding the gradient of a chord from two points on a curve.

   Examiner report: Accuracy requirements again caused avoidable losses, with some otherwise correct calculations not being presented to the required precision.

   Question paper p.4 · Mark scheme p.8

7. Question 3 (c)

   2 marks · 13/75

   Interpreting secant gradients as an estimate of the tangent gradient by considering a limiting process.

   Examiner report: Many responses noticed the numerical trend but did not connect it explicitly to chord gradients approaching a limiting tangent gradient.

   Question paper p.4 · Mark scheme p.8

8. Question 4

   5 marks · 18/75

   Selecting terms in binomial expansions and equating coefficients involving positive and negative powers.

   Examiner report: The coefficient work was often successful, but a noticeable number of solutions discarded one of the two permissible parameter values.

   Question paper p.5 · Mark scheme p.8

9. Question 5 (a)

   2 marks · 20/75

   Composing a function with itself and simplifying the resulting polynomial.

   Examiner report: The main error was failing to square the entire inner expression when applying the outer function.

   Question paper p.6 · Mark scheme p.9

10. Question 5 (b)

    4 marks · 24/75

    Solving an equation involving a composite polynomial by reducing it to a quadratic in x².

    Examiner report: Answers obtained solely from calculator equation-solving were not supported by enough algebra. Incorrect or non-expanding factors were also common, as was forgetting to take square roots after solving for x².

    Question paper p.6 · Mark scheme p.9

11. Question 6

    4 marks · 28/75

    Using midpoint and perpendicular-gradient properties to determine unknown coordinates in coordinate geometry.

    Examiner report: A sketch often prevented the mistaken assumption that the given point on the bisector was B. Successful methods used both the perpendicular-gradient condition and the midpoint condition.

    Question paper p.7 · Mark scheme p.10

12. Question 7 (a)

    2 marks · 30/75

    Proving that a line is tangent to a circle using the radius, point-of-contact and perpendicularity criteria.

    Examiner report: Many solutions established only one of the two necessary facts. A complete justification required both membership of A on the circle and perpendicularity of the radius and tangent.

    Question paper p.8 · Mark scheme p.12

13. Question 7 (b)

    3 marks · 33/75

    Finding a circle with a prescribed tangent at a known point and a fixed radius.

    Examiner report: Some responses reproduced the original circle or placed the centre at A. The successful approach used the opposite displacement from A to locate the second centre.

    Question paper p.8 · Mark scheme p.12

14. Question 8 (a)

    5 marks · 38/75

    Combining arithmetic- and geometric-progression relationships to determine unknown terms.

    Examiner report: The strongest work treated the two progressions separately before linking them. Weaker attempts mixed their rules or gave insufficient algebra when solving the resulting quadratic.

    Question paper p.9 · Mark scheme p.13

15. Question 8 (b)

    3 marks · 41/75

    Finding the sum of a specified number of terms of an arithmetic progression.

    Examiner report: This part was usually more secure once the progression parameters from part (a) had been found.

    Question paper p.9 · Mark scheme p.14

16. Question 9

    6 marks · 47/75

    Finding a volume of revolution using washers about the x-axis.

    Examiner report: Typical errors included using an incorrect intersection, integrating the wrong quantity, or omitting the volume corresponding to the lower boundary. Both a washer integral and an equivalent cylinder-subtraction method were viable.

    Question paper p.10 · Mark scheme p.14

17. Question 10 (a)

    4 marks · 51/75

    Question paper p.12 · Mark scheme p.16

18. Question 10 (b)

    3 marks · 54/75

    Solving a trigonometric equation by using a proved identity and respecting a restricted interval.

    Examiner report: Cancelling the tangent factor caused the zero-tangent solution to disappear in many scripts. A few responses also equated the expression with the wrong right-hand side.

    Question paper p.13 · Mark scheme p.17

19. Question 11 (a)

    2 marks · 56/75

    Using the stationary-point condition to determine an unknown constant in a derivative.

    Examiner report: Most candidates recognised that a stationary point requires a zero gradient, although some incorrectly used the point's y-coordinate or differentiated the supplied derivative.

    Question paper p.14 · Mark scheme p.17

20. Question 11 (b)

    4 marks · 60/75

    Recovering a curve equation by integration and using a point to determine the constant of integration.

    Examiner report: Integration was generally understood. The most frequent omission was the constant of integration, while a few solutions substituted the point without showing how the constant was obtained.

    Question paper p.14 · Mark scheme p.17

21. Question 11 (c)

    2 marks · 62/75

    Differentiating a composite derivative to obtain the second derivative.

    Examiner report: A common technical error was omitting the extra factor from differentiating the linear expression inside the power.

    Question paper p.15 · Mark scheme p.18

22. Question 11 (d)

    2 marks · 64/75

    Classifying a stationary point using the second-derivative test.

    Examiner report: Substitution into the second derivative was usually completed correctly. A sign-based argument using nearby first-derivative values would also have needed the locations and values to be stated explicitly.

    Question paper p.15 · Mark scheme p.18

23. Question 12 (a)

    2 marks · 66/75

    Using rotational symmetry and circle geometry to determine an angle between adjacent rope tangencies.

    Examiner report: Assuming the small triangle was equilateral without justification was a common weakness. The cleanest reasoning used the sixfold symmetry of the arrangement.

    Question paper p.16 · Mark scheme p.19

24. Question 12 (b)

    4 marks · 70/75

    Calculating the total length of a rope made from straight tangents and circular arcs.

    Examiner report: This part was more accessible when the angle from part (a) was reused. The straight pieces were diameters, while the curved pieces required arc length rather than a sector area or circumference used indiscriminately.

    Question paper p.16 · Mark scheme p.20

25. Question 12 (c)

    2 marks · 72/75

    Finding the area of a regular hexagon formed by the centres of six equal outer circles.

    Examiner report: Several decompositions worked, including triangles and trapezia. Incorrect answers often failed to reflect the required symmetry or used a sector-area formula for the hexagon.

    Question paper p.17 · Mark scheme p.21

26. Question 12 (d)

    3 marks · 75/75

    Decomposing a compound planar region into polygonal, rectangular and sector components.

    Examiner report: The successful solutions built the region from six rectangles, six sectors and the hexagon. Treating the arrangement as several complete circles did not account for the uncovered connecting portions.

    Question paper p.17 · Mark scheme p.22