9709 Pure Mathematics 1 Feb/Mar 2021 Paper 12

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

1. Question 1 (a)

   1 mark · 1/75

   Question paper p.3 · Mark scheme p.6

2. Question 1 (b)

   2 marks · 3/75

   This tests applying the binomial expansion when the quantity substituted for x is negative and contains a factor of 2.

   Examiner report: The negative multiple inside the bracket caused errors for some candidates, but many obtained the expansion correctly.

   Question paper p.3 · Mark scheme p.6

3. Question 1 (c)

   2 marks · 5/75

   This tests finding a specified coefficient by multiplying two truncated polynomial expansions and collecting terms of the required degree.

   Examiner report: Candidates who had obtained both preceding expansions generally selected the relevant products successfully.

   Question paper p.3 · Mark scheme p.6

4. Question 2

   4 marks · 9/75

   This tests reducing an equation containing a repeated squared expression to a polynomial equation through substitution, then solving and reversing the substitution.

   Examiner report: This was usually accessible when a substitution was made. Unsupported calculator solutions did not demonstrate the required algebraic method.

   Question paper p.4 · Mark scheme p.6

5. Question 3

   4 marks · 13/75

   Question paper p.5 · Mark scheme p.7

6. Question 4

   5 marks · 18/75

   This tests finding when a line and a parabola meet at two distinct points by forming a quadratic in x and applying the discriminant condition.

   Examiner report: The elimination and discriminant approach was widely understood, but calculator-only solutions for the final quadratic in k could lose method credit.

   Question paper p.6 · Mark scheme p.7

7. Question 5 (a)

   4 marks · 22/75

   This tests reading a graph to identify and describe the vertical stretch and horizontal translation that form a combined transformation.

   Examiner report: The two changes were often recognised, but descriptions were sometimes too vague or assigned the increased vertical displacement to a translation instead of a vertical stretch.

   Question paper p.7 · Mark scheme p.8

8. Question 5 (b)

   2 marks · 24/75

   This tests expressing a graph obtained by combining a vertical scale change and a horizontal shift in function notation.

   Examiner report: Those who correctly interpreted the graph usually transferred the two changes into function notation; some obtained the expression independently even without completing the previous part.

   Question paper p.7 · Mark scheme p.8

9. Question 6 (a)

   3 marks · 27/75

   This tests using related rates and the chain rule to connect the rate of change of y with respect to time to the rate of change of x with respect to time.

   Examiner report: The chain-rule approach was common, and identifying the requested x-with-time rate usually led to the correct result.

   Question paper p.8 · Mark scheme p.8

10. Question 6 (b)

    4 marks · 31/75

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

    Examiner report: Most candidates recognised the need to integrate and often found the constant correctly; some stopped before explicitly giving the final equation of the curve.

    Question paper p.9 · Mark scheme p.9

11. Question 7 (a)

    3 marks · 34/75

    This tests completing the square for a restricted quadratic function and using its domain to determine the range.

    Examiner report: The completed-square form was generally found, but the domain restriction led some candidates to choose the wrong side of the range.

    Question paper p.10 · Mark scheme p.9

12. Question 7 (b)

    2 marks · 36/75

    This tests finding the inverse of a one-to-one restricted quadratic function by interchanging variables and selecting the branch consistent with the original domain.

    Examiner report: The algebraic rearrangement was usually understood, but the correct branch was chosen only by candidates who used the original domain properly.

    Question paper p.11 · Mark scheme p.9

13. Question 7 (c)

    3 marks · 39/75

    This tests composing two functions, forming a quadratic equation, and rejecting solutions that do not lie in the domain of the inner function.

    Examiner report: Most candidates formed the quadratic, but some gave insufficient work for solving it or retained a root outside the permitted domain.

    Question paper p.11 · Mark scheme p.10

14. Question 8 (a)

    5 marks · 44/75

    This tests determining a circle from three points by recognising its centre and radius from the coordinate geometry.

    Examiner report: The geometrical observation involving the three points commonly led to a correct circle. Direct substitution into a general circle equation was less reliable because of algebraic slips.

    Question paper p.12 · Mark scheme p.10

15. Question 8 (b)

    2 marks · 46/75

    This tests finding a tangent to a circle by using the perpendicular relationship between the tangent and the radius at the point of contact.

    Examiner report: This part was often left blank. Candidates who used the centre from the earlier part generally proceeded correctly, whereas treating AC as parallel to the tangent was invalid.

    Question paper p.13 · Mark scheme p.10

16. Question 9 (a)(i)

    3 marks · 49/75

    This tests using the sum to infinity of a geometric progression to determine its common ratio and hence its second term.

    Examiner report: Most candidates used the infinite-sum formula effectively and obtained the second term, often helped by the form of the stated sum.

    Question paper p.14 · Mark scheme p.11

17. Question 9 (a)(ii)

    2 marks · 51/75

    This tests evaluating a finite geometric sum for twelve terms after substituting a specified angle.

    Examiner report: The formula was generally known, but some solutions did not make the angle substitution or subsequent numerical work sufficiently explicit.

    Question paper p.14 · Mark scheme p.11

18. Question 9 (b)

    4 marks · 55/75

    This tests finding an arithmetic progression term from its first two terms and then applying the nth-term formula at a specified angle.

    Examiner report: The common difference was often found correctly. Errors mainly occurred when substituting the angle or when a sum formula was used instead of the nth-term relation.

    Question paper p.15 · Mark scheme p.11

19. Question 10 (a)

    5 marks · 60/75

    This tests modelling equal areas in a sector by equating the sector area with twice the area of the isosceles triangle formed by the two marked points.

    Examiner report: Most candidates identified the equal-area relationship and used the sector formula correctly. A minority struggled with radians or with attempts based on a triangle’s height and base.

    Question paper p.16 · Mark scheme p.12

20. Question 10 (b)

    3 marks · 63/75

    This tests generalising the equal-area model to an angle theta and solving the resulting inequality for a length ratio constrained to be below one.

    Examiner report: This was frequently omitted. Even when the general area equation was obtained, some candidates included negative values that are impossible for the positive length ratio.

    Question paper p.17 · Mark scheme p.12

21. Question 11 (a)

    2 marks · 65/75

    This tests finding an x-axis intercept by setting the curve’s y-coordinate to zero and solving an equation involving negative fractional powers.

    Examiner report: Most candidates began by setting y to zero and often reached the correct intercept, but unsupported numerical answers received no credit.

    Question paper p.18 · Mark scheme p.13

22. Question 11 (b)

    4 marks · 69/75

    This tests differentiating a curve, evaluating its gradient at an intercept, and forming the tangent equation there.

    Examiner report: Correct derivative expressions were common, though a small number of candidates found a normal instead of the requested tangent.

    Question paper p.18 · Mark scheme p.13

23. Question 11 (c)

    2 marks · 71/75

    This tests locating a stationary point by setting the derivative equal to zero and solving for x.

    Examiner report: Most attempts used the correct stationary-point condition and produced valid working; as in part (a), unsupported answers were not sufficient.

    Question paper p.19 · Mark scheme p.13

24. Question 11 (d)

    4 marks · 75/75

    This tests finding a bounded area by integrating the curve between its x-axis intercept and the specified vertical line.

    Examiner report: Many candidates used the intercept as the lower limit and showed the endpoint substitution. The area calculation was generally successful when the integration was correct.

    Question paper p.19 · Mark scheme p.14