9709 Pure Mathematics 1 Oct/Nov 2022 Paper 12

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

1. Question 1 (a)

   3 marks · 3/75

   Finding the equation of a perpendicular bisector from two points.

   Examiner report: Most candidates could find the midpoint and gradients, but common errors included using an endpoint instead of the midpoint or not reversing and negating the original gradient.

   Question paper p.3 · Mark scheme p.6

2. Question 1 (b)

   3 marks · 6/75

   Forming the equation of a circle from its centre and a point on its circumference.

   Examiner report: The centre-radius form led to more success than expanding the equation, while some candidates confused the distance with its square.

   Question paper p.3 · Mark scheme p.6

3. Question 2

   5 marks · 11/75

   Using consecutive-term differences and the arithmetic-series sum formula.

   Examiner report: A noticeable minority treated the progression as geometric or stopped with the sum still depending on the parameter instead of determining it.

   Question paper p.4 · Mark scheme p.7

4. Question 3 (a)

   2 marks · 13/75

   Using the discriminant condition for a quadratic to have no real roots.

   Examiner report: Although the discriminant was usually identified, some candidates selected the outside intervals, kept only positive parameter values, or included the zero-discriminant boundary.

   Question paper p.5 · Mark scheme p.7

5. Question 3 (b)

   3 marks · 16/75

   Solving a trigonometric quadratic by substituting a trigonometric expression and restricting the angles.

   Examiner report: Most attempts used factorisation successfully, but one valid cosine solution was sometimes rejected and unsupported calculator outputs did not receive credit.

   Question paper p.5 · Mark scheme p.8

6. Question 4

   4 marks · 20/75

   Determining a geometric progression from term information and finding a later term.

   Examiner report: Efficient solutions first extracted a product involving the first term and ratio; weaker attempts used arithmetic-progression methods, mishandled the given sum, or found a sum instead of a single term.

   Question paper p.6 · Mark scheme p.8

7. Question 5 (a)

   3 marks · 23/75

   Sketching the result of a horizontal stretch followed by a vertical translation.

   Examiner report: This was difficult for many candidates: the horizontal scaling was often reversed, and some showed only the upward shift.

   Question paper p.7 · Mark scheme p.9

8. Question 5 (b)

   2 marks · 25/75

   Expressing a transformed function in terms of the original function.

   Examiner report: The upward constant was frequently included, but the altered input of the original function was much more often missed.

   Question paper p.7 · Mark scheme p.9

9. Question 6 (a)

   3 marks · 28/75

   Completing the square for a quadratic expression.

   Examiner report: This part was generally handled well, with many candidates obtaining all available credit.

   Question paper p.8 · Mark scheme p.9

10. Question 6 (b)

    3 marks · 31/75

    Solving a quadratic using a completed-square representation.

    Examiner report: Some candidates ignored the requested connection with part (a) and restarted from the expanded quadratic, which made this part less successful.

    Question paper p.8 · Mark scheme p.10

11. Question 6 (c)

    3 marks · 34/75

    Sketching an upward-opening quadratic and locating its stationary point from completed-square information.

    Examiner report: Frequent issues were omitting the sketch, drawing a maximum, or confusing the vertex with an axis intercept; some also differentiated unnecessarily instead of using part (a).

    Question paper p.9 · Mark scheme p.10

12. Question 7 (a)

    3 marks · 37/75

    Question paper p.10 · Mark scheme p.11

13. Question 7 (b)

    4 marks · 41/75

    Solving a trigonometric equation using a proven identity over a specified interval.

    Examiner report: Candidates who used part (a) generally progressed well, though the negative tangent branch was often overlooked.

    Question paper p.11 · Mark scheme p.12

14. Question 8 (a)

    4 marks · 45/75

    Integrating a derivative and using a point on the curve to determine the constant.

    Examiner report: Integration and use of the given point were usually understood, but some candidates applied an incorrect coefficient-and-power procedure rather than the power rule.

    Question paper p.12 · Mark scheme p.13

15. Question 8 (b)

    2 marks · 47/75

    Finding a stationary point's x-coordinate from a derivative.

    Examiner report: Setting the derivative to zero was widely recognised, but solving the resulting fractional-power equation caused difficulty for some candidates.

    Question paper p.13 · Mark scheme p.13

16. Question 8 (c)

    1 mark · 48/75

    Question paper p.13 · Mark scheme p.13

17. Question 9 (a)

    1 mark · 49/75

    Question paper p.14 · Mark scheme p.13

18. Question 9 (b)

    2 marks · 51/75

    Using a known composite-function value to determine an unknown constant.

    Examiner report: This part was among the strongest on the paper and was usually completed correctly.

    Question paper p.14 · Mark scheme p.14

19. Question 9 (c)

    1 mark · 52/75

    Deciding whether a function has an inverse by considering one-to-one behaviour.

    Examiner report: Many recognised the relevant issue, but merely mentioning a graphical test without explaining the failure of one-to-one behaviour was not enough.

    Question paper p.14 · Mark scheme p.14

20. Question 9 (d)

    3 marks · 55/75

    Finding the inverse of a linear function and composing it with another function.

    Examiner report: The inverse process was often understood, but algebraic simplification was a common weakness; some candidates instead inverted the wrong composite function.

    Question paper p.15 · Mark scheme p.14

21. Question 9 (e)

    1 mark · 56/75

    Checking whether the range of one function is contained in the domain required for composition.

    Examiner report: This was one of the least successful explanations on the paper, with only the strongest responses clearly relating the range of the inner function to the outer function's domain.

    Question paper p.15 · Mark scheme p.14

22. Question 10 (a)

    3 marks · 59/75

    Calculating the perimeter formed by circular arcs using radian measure.

    Examiner report: Many candidates knew that two arcs had to be added, but a common error was using the minor angle for the outer arc; radian-based working tended to be more reliable than degree conversion.

    Question paper p.16 · Mark scheme p.15

23. Question 10 (b)

    2 marks · 61/75

    Finding the difference between two triangle areas from two sides and their included angle.

    Examiner report: Some candidates did not interpret “difference” as a subtraction, while the included-angle area formula generally produced more accurate work than separate base-height calculations.

    Question paper p.17 · Mark scheme p.15

24. Question 10 (c)

    3 marks · 64/75

    Combining sector areas with a triangular-area difference to find a composite cross-sectional area.

    Examiner report: The main difficulty was recognising that the result from part (b) must be combined with both sector areas; requested rounding was also sometimes ignored.

    Question paper p.17 · Mark scheme p.15

25. Question 11 (a)

    3 marks · 67/75

    Finding the coordinates of the minimum point of a quadratic curve.

    Examiner report: Differentiation was the more dependable approach, whereas some candidates gave the correct coordinates without sufficient supporting work and consequently lost credit.

    Question paper p.18 · Mark scheme p.16

26. Question 11 (b)

    5 marks · 72/75

    Finding the area between two curves over their given intersection interval.

    Examiner report: Many complete solutions were seen, but some unnecessarily tried to calculate the intersections again instead of using those supplied in the question.

    Question paper p.18 · Mark scheme p.17

27. Question 11 (c)

    3 marks · 75/75

    Applying the chain rule to a related-rates problem involving a moving point on a curve.

    Examiner report: Some candidates used the derivative from an earlier part or divided the two rates instead of multiplying them; correct chain-rule applications were otherwise common.

    Question paper p.19 · Mark scheme p.18