9709 Pure Mathematics 1 Feb/Mar 2020 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 assesses the use of differentiation to determine whether a function is monotonic on a stated domain.

   Examiner report: Most candidates found the derivative successfully, but some did not connect its sign over the whole domain to the required conclusion.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   4 marks · 7/75

   This assesses recognition and description of combined graph transformations from a transformed function formula.

   Examiner report: Descriptions based only on vague movements were not sufficient; each operation needed a precise transformation name and an axis-based specification.

   Question paper p.3 · Mark scheme p.6

3. Question 3

   4 marks · 11/75

   This assesses the use of integration to find a volume of revolution about the y-axis.

   Examiner report: The rotation was generally interpreted correctly, though a few responses omitted the circular-area factor or failed to show the limit substitution.

   Question paper p.4 · Mark scheme p.7

4. Question 4

   4 marks · 15/75

   This assesses related rates using implicit differentiation and the given rates of change of x and y.

   Examiner report: The chain-rule approach was commonly used correctly, and most candidates handled the subsequent algebra efficiently.

   Question paper p.5 · Mark scheme p.7

5. Question 5

   5 marks · 20/75

   Question paper p.6 · Mark scheme p.8

6. Question 6 (a)

   3 marks · 23/75

   This assesses selection of a particular binomial term and the use of its coefficient and power of x.

   Examiner report: Selecting the relevant term was often done without expanding everything, but some candidates reported only one of the possible constant values.

   Question paper p.7 · Mark scheme p.8

7. Question 6 (b)

   2 marks · 25/75

   This assesses selection of a different term in a binomial expansion using a value found in the preceding part.

   Examiner report: Finding this term was less secure than in part (a); including the associated power of x sometimes helped reveal whether the correct term had been chosen.

   Question paper p.7 · Mark scheme p.8

8. Question 7

   6 marks · 31/75

   This assesses calculation of a composite shaded area using a sector, a right triangle and trigonometry in radians.

   Examiner report: The geometry and radian measures were usually handled well, but premature rounding in intermediate trigonometric calculations caused avoidable errors in the final value.

   Question paper p.8 · Mark scheme p.9

9. Question 8 (a)

   1 mark · 32/75

   Question paper p.9 · Mark scheme p.9

10. Question 8 (b)

    5 marks · 37/75

    This assesses the use of arithmetic and geometric progressions in a percentage comparison over time.

    Examiner report: The strongest responses identified both progression types correctly and kept extra precision during the calculation before rounding the final percentage.

    Question paper p.9 · Mark scheme p.9

11. Question 9 (a)

    2 marks · 39/75

    This assesses rewriting a quadratic in completed-square form.

    Examiner report: Most candidates were comfortable with the completing-square process and earned the available marks.

    Question paper p.10 · Mark scheme p.10

12. Question 9 (b)

    3 marks · 42/75

    This assesses finding an inverse function for a restricted quadratic and stating its domain.

    Examiner report: The algebraic inversion was generally sound, but the original restriction was sometimes overlooked, leading to the wrong root or an omitted domain.

    Question paper p.10 · Mark scheme p.10

13. Question 9 (c)

    2 marks · 44/75

    This assesses evaluation and solution of a composite function subject to a domain restriction.

    Examiner report: Using the completed-square representation simplified the composition; the main difficulty was discarding a mathematically obtained root that violated the original restriction.

    Question paper p.11 · Mark scheme p.10

14. Question 9 (d)

    1 mark · 45/75

    This assesses determining the largest domain bound that makes a composition well defined.

    Examiner report: A useful approach was to impose the outer-function boundary on the linear expression at the endpoint of its domain.

    Question paper p.11 · Mark scheme p.10

15. Question 10 (a)

    3 marks · 48/75

    This assesses finding a stationary point from a supplied gradient formula.

    Examiner report: Most candidates recognised the stationary-point condition, but full credit depended on showing the algebra and identifying the result specifically as a rather than merely as a possible x-value.

    Question paper p.12 · Mark scheme p.11

16. Question 10 (b)

    3 marks · 51/75

    This assesses classification of a stationary point using the second derivative.

    Examiner report: The second-derivative method was almost universal; errors in differentiating the fractional power were the main source of lost marks.

    Question paper p.12 · Mark scheme p.11

17. Question 10 (c)

    4 marks · 55/75

    This assesses recovering a curve equation by integrating its gradient and using a known point.

    Examiner report: The general integration strategy was well understood, although mistakes occurred in integrating either the shifted power term or the linear term.

    Question paper p.13 · Mark scheme p.11

18. Question 11 (a)

    4 marks · 59/75

    This assesses solving a quadratic trigonometric equation by treating tangent as the variable and then finding all angles in an interval.

    Examiner report: Many solutions missed the angle associated with the negative tangent value, despite the interval allowing it.

    Question paper p.14 · Mark scheme p.12

19. Question 11 (b)

    2 marks · 61/75

    This assesses use of the discriminant to determine when a quadratic in tan x has no real solutions.

    Examiner report: The discriminant criterion was commonly selected correctly, with sign handling in the terms involving k causing most remaining errors.

    Question paper p.14 · Mark scheme p.12

20. Question 11 (c)

    3 marks · 64/75

    This assesses identifying a parameter that produces three trigonometric solutions and then listing those solutions in the required interval.

    Examiner report: Candidates who understood the tangent graph saw that the extra solution had to arise from tangent equal to zero; the parameter and all three angles were then required.

    Question paper p.15 · Mark scheme p.12

21. Question 12 (a)

    3 marks · 67/75

    This assesses finding a circle equation from the endpoints of a diameter.

    Examiner report: The midpoint-and-half-diameter method was the most dependable; confusing the full diameter length with the radius still allowed some follow-through credit later.

    Question paper p.16 · Mark scheme p.13

22. Question 12 (b)

    2 marks · 69/75

    This assesses translating a circle and forming its new equation.

    Examiner report: The centre translation was generally straightforward, and candidates could preserve credit when an earlier radius error was used consistently.

    Question paper p.16 · Mark scheme p.13

23. Question 12 (c)

    4 marks · 73/75

    This assesses finding the common chord of two circles by using either their radical axis or the perpendicular bisector of their centres.

    Examiner report: Both approaches appeared regularly; because the question required a demonstration, incomplete intermediate algebra prevented some otherwise correct final lines from receiving full credit.

    Question paper p.17 · Mark scheme p.14

24. Question 12 (d)

    2 marks · 75/75

    This assesses substituting a known intersection line into a circle equation and simplifying to a quadratic in x.

    Examiner report: Substitution into one circle was usually the quickest route, while eliminating y from both circle equations demanded considerably more careful algebra.

    Question paper p.17 · Mark scheme p.14