9709 Pure Mathematics 1 May/June 2023 Paper 12

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

1. Question 1

   3 marks · 3/75

   This tests integration of a derivative involving a shifted negative power and the use of a point to determine the constant.

   Examiner report: Most candidates chose integration, though some treated the relation as linear or differentiated it instead. A few lost credit through mishandling the negative coefficient or omitting the dependent variable from the final equation.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   4 marks · 7/75

   This tests extracting binomial coefficients from two expansions and solving the resulting equation for a parameter.

   Examiner report: The binomial work was generally strong. Common slips included failing to square the parameter in one coefficient and rejecting one of the two possible signs at the end.

   Question paper p.3 · Mark scheme p.6

3. Question 3 (a)

   2 marks · 9/75

   This tests rewriting a quadratic by completing the square while retaining a parameter in the constant term.

   Examiner report: Most candidates knew the required method, but some did not preserve the leading factor when forming the square.

   Question paper p.4 · Mark scheme p.7

4. Question 3 (b)

   1 mark · 10/75

   This tests using a completed-square form or discriminant to determine when a quadratic has no real zero.

   Examiner report: Only a minority made direct use of the result from part (a). Discriminant methods often reached the right intermediate inequality but were then simplified incorrectly.

   Question paper p.4 · Mark scheme p.7

5. Question 4

   3 marks · 13/75

   Question paper p.5 · Mark scheme p.7

6. Question 5

   4 marks · 17/75

   This tests finding the area under a curve by definite integration between its intercepts.

   Examiner report: This was the strongest question overall. The main difficulties were integrating the fractional coefficient and failing to display the endpoint substitutions clearly enough.

   Question paper p.6 · Mark scheme p.8

7. Question 6 (a)

   2 marks · 19/75

   This tests combining triangle geometry with the arc-length formula to establish the radius of the sector.

   Examiner report: Many used the sine or cosine rule successfully, but some assigned the wrong angle in the auxiliary triangle. A complete derivation was needed even though the target expression was supplied.

   Question paper p.7 · Mark scheme p.9

8. Question 6 (b)

   4 marks · 23/75

   This tests finding a circular-segment area by subtracting the area of a triangle from the sector area.

   Examiner report: The sector method was generally understood, although some substituted the arc length instead of the radius. Triangle areas were obtained either by splitting the figure or by using the included angle.

   Question paper p.7 · Mark scheme p.9

9. Question 7 (a)(i)

   3 marks · 26/75

   This tests expanding a squared trigonometric expression and solving the resulting simple trigonometric equation over a specified interval.

   Examiner report: Most followed the requested expansion, but the cross-product term was sometimes lost. Some responses supplied only two of the required values.

   Question paper p.8 · Mark scheme p.10

10. Question 7 (a)(ii)

    2 marks · 28/75

    This tests checking which candidates from a squared equation satisfy the original unsquared equation.

    Examiner report: Many candidates made this verification harder than necessary, while a sizeable group left it blank. Direct substitution into the original equation was sufficient.

    Question paper p.8 · Mark scheme p.10

11. Question 7 (b)

    3 marks · 31/75

    This tests proving a trigonometric identity by combining rational terms over a common denominator and simplifying with the Pythagorean identity.

    Examiner report: This was one of the more familiar parts and produced many successful proofs. Credit depended on showing enough algebra rather than merely asserting the identity.

    Question paper p.9 · Mark scheme p.11

12. Question 7 (c)

    3 marks · 34/75

    This tests using a proved identity to solve a trigonometric equation while handling a cancellable factor without losing solutions.

    Examiner report: The frequent error was cancelling the common factor without retaining its zero case, which removed valid solutions. A substantial number of candidates did not attempt this part.

    Question paper p.9 · Mark scheme p.12

13. Question 8 (a)

    2 marks · 36/75

    This tests sketching the inverse of a one-to-one function by reflecting its graph in the line y = x.

    Examiner report: Many knew that reflection was required but plotted the diagonal inaccurately or reflected in a horizontal line through an intercept instead.

    Question paper p.10 · Mark scheme p.13

14. Question 8 (b)

    2 marks · 38/75

    This tests finding an inverse function algebraically for a restricted sine function.

    Examiner report: The rearrangement was often successful, but some attempted to divide by an inverse-sine expression instead of applying the inverse function.

    Question paper p.10 · Mark scheme p.14

15. Question 8 (c)

    2 marks · 40/75

    This tests completing a periodic graph and deciding whether the extended function is one-to-one.

    Examiner report: The missing portion was often drawn without the required flattening near the endpoint. Some candidates also failed to connect the shape of their graph with the existence of an inverse.

    Question paper p.11 · Mark scheme p.14

16. Question 8 (d)

    6 marks · 46/75

    This tests describing a specified sequence of graph transformations, including scale factors, directions, invariant axes, and order.

    Examiner report: Fully correct descriptions were seen, but many candidates found the new transformations topic difficult. In particular, the listed transformations were sometimes right but applied in the wrong sequence.

    Question paper p.11 · Mark scheme p.15

17. Question 9 (a)

    4 marks · 50/75

    This tests finding possible first terms of a geometric progression from a specified second term and an infinite sum.

    Examiner report: Most candidates formed the necessary equations and completed the algebra. Unsupported calculator output was a recurring reason for losing method credit.

    Question paper p.12 · Mark scheme p.16

18. Question 9 (b)

    4 marks · 54/75

    This tests comparing the nth terms of two geometric progressions and proving a multiplicative relationship between them.

    Examiner report: Many did not recognise that the results from part (a) were intended for immediate use. Algebra involving powers of the two common ratios was the main source of incorrect proofs.

    Question paper p.13 · Mark scheme p.17

19. Question 10 (a)

    5 marks · 59/75

    This tests using the tangency condition between a line and a circle to determine the possible horizontal position of the centre.

    Examiner report: Most candidates substituted the line correctly and used a discriminant, although a few replaced the unknown parameter prematurely. Calculator-based solutions without displayed algebra were not fully credited.

    Question paper p.14 · Mark scheme p.18

20. Question 10 (b)

    4 marks · 63/75

    This tests finding the normal to a circle at the contact point using the radius and the centre.

    Examiner report: Candidates generally recognised that the normal passes through the centre and has the perpendicular gradient. Some unnecessarily calculated the contact point, although a direct centre-based method was available.

    Question paper p.15 · Mark scheme p.19

21. Question 10 (c)

    4 marks · 67/75

    This tests finding the two tangents to a circle that have a prescribed direction.

    Examiner report: This was frequently omitted or replaced by a repetition of part (b). Successful solutions used either a diameter parallel to the original tangent or a discriminant applied to a general parallel line.

    Question paper p.15 · Mark scheme p.20

22. Question 11 (a)

    2 marks · 69/75

    This tests differentiating a square-root composite function using the chain rule.

    Examiner report: Most candidates knew the method. Typical slips were forgetting the inner derivative factor or incorrectly multiplying the constant derivative by that factor.

    Question paper p.16 · Mark scheme p.21

23. Question 11 (b)

    2 marks · 71/75

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

    Examiner report: The stationary-point condition was usually recognised, but manipulating the negative power and retaining the parameter in the correct place caused errors for some candidates.

    Question paper p.16 · Mark scheme p.21

24. Question 11 (c)

    4 marks · 75/75

    This tests using a tangent angle to obtain a gradient, locating the corresponding point, and forming the normal equation.

    Examiner report: Many candidates differentiated confidently but did not connect the tangent angle with its gradient, sometimes setting the derivative to zero instead. Those who used the correct gradient condition could usually complete the coordinate and normal steps.

    Question paper p.17 · Mark scheme p.22