9709 Pure Mathematics 1 Oct/Nov 2022 Paper 13

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

1. Question 1

   3 marks · 3/75

   Question paper p.2 · Mark scheme p.6

2. Question 2 (a)

   2 marks · 5/75

   Question paper p.3 · Mark scheme p.6

3. Question 2 (b)

   1 mark · 6/75

   Finding the range of a quadratic function subject to a restricted domain.

   Examiner report: A common weakness was to use only the vertex value and overlook the restriction on the domain, which makes the relevant boundary unattainable.

   Question paper p.3 · Mark scheme p.6

4. Question 2 (c)

   3 marks · 9/75

   Finding the inverse of a restricted quadratic function.

   Examiner report: The rearrangement was usually sound, but signs were often mishandled under the square root. The final branch choice also needed to reflect the original restriction on the quadratic's input.

   Question paper p.3 · Mark scheme p.6

5. Question 3 (a)

   2 marks · 11/75

   Using the binomial theorem to obtain the first three terms of an expansion.

   Examiner report: Most candidates knew the required expansion and obtained the requested terms accurately.

   Question paper p.4 · Mark scheme p.7

6. Question 3 (b)

   2 marks · 13/75

   Expanding a negative binomial using the binomial theorem.

   Examiner report: This part was usually correct, particularly when the preceding binomial work had been done reliably.

   Question paper p.4 · Mark scheme p.7

7. Question 3 (c)

   2 marks · 15/75

   Finding a coefficient in a product of two truncated binomial expansions.

   Examiner report: The intended use of the earlier expansions was evident in nearly all successful responses; errors generally followed from an earlier expansion mistake.

   Question paper p.4 · Mark scheme p.7

8. Question 4

   5 marks · 20/75

   Using related rates and the chain rule to convert a volume rate into a depth rate with changed units.

   Examiner report: The main difficulty was remembering the derivative of the inner expression. Stronger solutions found the depth rate first and converted units afterwards; others either omitted the conversion or attempted it too early.

   Question paper p.5 · Mark scheme p.8

9. Question 5 (a)

   1 mark · 21/75

   Question paper p.6 · Mark scheme p.8

10. Question 5 (b)

    1 mark · 22/75

    Question paper p.6 · Mark scheme p.8

11. Question 5 (c)

    2 marks · 24/75

    Finding the centre after a translation but before a subsequent vertical stretch.

    Examiner report: Many candidates obtained the horizontal coordinate but failed to undo the vertical scaling. The centre of the final ellipse alone was not sufficient for this part.

    Question paper p.6 · Mark scheme p.8

12. Question 5 (d)

    2 marks · 26/75

    Reversing a translation to recover the centre of the original circle.

    Examiner report: This was often completed successfully, including by following through from an imperfect centre in the preceding part.

    Question paper p.6 · Mark scheme p.9

13. Question 6

    5 marks · 31/75

    Evaluating an exact trigonometric expression from an inverse-cosine definition without calculator trigonometry.

    Examiner report: The report noted that unsupported calculator values could not demonstrate the required exact method. Successful candidates justified the missing ratio algebraically or through a right triangle before combining the fractions.

    Question paper p.7 · Mark scheme p.9

14. Question 7 (a)

    4 marks · 35/75

    Finding points where a given derivative has a specified tangent gradient.

    Examiner report: Forming the equation was usually manageable, but expanding it into a quartic often led to failure. The efficient route was to take positive and negative fourth roots and retain both possibilities.

    Question paper p.8 · Mark scheme p.10

15. Question 7 (b)

    3 marks · 38/75

    Integrating a derivative and using a point condition to determine the constant of integration.

    Examiner report: Integration and use of the point condition were generally well understood, and most candidates presented the resulting function as their final statement.

    Question paper p.9 · Mark scheme p.10

16. Question 8 (a)

    4 marks · 42/75

    Using circle geometry and arc length to find the perimeter of a symmetric lens-shaped region.

    Examiner report: Many candidates found the required angle and assembled the repeated arcs correctly. Marks were sometimes lost through leaving an inverse-trigonometric form or rounding the angle too early.

    Question paper p.10 · Mark scheme p.11

17. Question 8 (b)

    3 marks · 45/75

    Finding the area of a lens-shaped region by combining circular sectors and triangles.

    Examiner report: Several valid approaches were seen, but some solutions paired a sector with the wrong triangle or did not make clear which pieces were being combined, making the geometry difficult to verify.

    Question paper p.11 · Mark scheme p.11

18. Question 9 (a)

    3 marks · 48/75

    Finding the sum to infinity of a geometric progression from two of its terms.

    Examiner report: This topic was generally well mastered, with candidates usually identifying the ratio correctly and applying the convergence condition appropriately.

    Question paper p.12 · Mark scheme p.11

19. Question 9 (b)

    6 marks · 54/75

    Relating terms of a geometric progression to terms of an arithmetic progression and summing the arithmetic progression.

    Examiner report: Successful solutions kept the two progressions distinct. A serious error was to assume that their first terms were the same merely because selected later terms were linked.

    Question paper p.13 · Mark scheme p.12

20. Question 10 (a)

    4 marks · 58/75

    Finding the intersection point of a circle and a straight line by substitution.

    Examiner report: Most candidates formed the correct quadratic and gained substantial credit. Some lost the method mark by writing calculator roots without showing factorisation, the quadratic formula with substitution, or an equivalent process.

    Question paper p.14 · Mark scheme p.12

21. Question 10 (b)

    4 marks · 62/75

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

    Examiner report: The presence of the straight line caused some candidates to choose the wrong region or integrand. Those who used the circle's squared ordinate and the correct limits generally completed the calculation, although the required exact format was sometimes missed.

    Question paper p.15 · Mark scheme p.13

22. Question 10 (c)

    2 marks · 64/75

    Finding the exact perimeter of a region bounded partly by a circular arc.

    Examiner report: This was one of the more challenging parts. Common problems included treating the line gradient as the required angle, using degrees in an exact arc calculation, and converting to decimals before completing the exact form.

    Question paper p.15 · Mark scheme p.13

23. Question 11 (a)

    3 marks · 67/75

    Using equality of distances between coordinate points to determine an unknown parameter.

    Examiner report: The distance formula was used effectively by most candidates. An alternative perpendicular-bisector method was less common but also valid.

    Question paper p.16 · Mark scheme p.13

24. Question 11 (b)(i)

    4 marks · 71/75

    Using gradients to impose perpendicularity on two lines containing a variable point.

    Examiner report: Responses varied considerably. Many formed the two gradients correctly but then equated them or changed only one sign instead of applying the perpendicular-gradient condition. Some also omitted the required method for solving the quadratic.

    Question paper p.16 · Mark scheme p.14

25. Question 11 (b)(ii)

    4 marks · 75/75

    Finding the equation of a circle through three points when two radii are perpendicular.

    Examiner report: The most efficient solutions used the diameter consequence of the right angle. Attempts based on intersecting perpendicular bisectors were more prone to algebraic errors, and some candidates confused the radius with the diameter.

    Question paper p.16 · Mark scheme p.14