9709 Pure Mathematics 3 Feb/Mar 2022 Paper 32

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

1. Question 1

   4 marks · 4/75

   Question paper p.3 · Mark scheme p.6

2. Question 2

   4 marks · 8/75

   This tests representing simultaneous modulus and argument conditions as loci on an Argand diagram.

   Examiner report: Many diagrams placed the circle's centre incorrectly or used a full line instead of the required ray. Incorrect side-of-ray shading and unequal axis scales also caused problems.

   Question paper p.4 · Mark scheme p.7

3. Question 3

   5 marks · 13/75

   This tests converting a power relationship into a straight-line logarithmic model and extracting its constants from plotted log data.

   Examiner report: A frequent mistake was treating the displayed logarithmic coordinates as ordinary x and y values, or applying another logarithm to them, instead of using them directly in the linear relation.

   Question paper p.5 · Mark scheme p.8

4. Question 4

   5 marks · 18/75

   This tests differentiation of parametric equations using the chain rule and trigonometric simplification.

   Examiner report: The report highlights confusion between the parameter and x, together with omitted chain-rule factors. Some candidates also gave a large simplification step without enough intermediate work to support it.

   Question paper p.6 · Mark scheme p.9

5. Question 5

   6 marks · 24/75

   This tests solving simultaneous trigonometric conditions involving a tangent addition formula and angle restrictions.

   Examiner report: Although many candidates handled the main algebra well, arithmetic slips were common and some solutions missed the obtuse pair of angles.

   Question paper p.7 · Mark scheme p.10

6. Question 6

   6 marks · 30/75

   This tests solving an equation involving a complex number and its conjugate by separating real and imaginary components.

   Examiner report: Introducing real components was generally effective, but some candidates treated the imaginary part of the real right-hand side incorrectly. Others lost a solution by cancelling a factor instead of considering the factor being zero.

   Question paper p.8 · Mark scheme p.11

7. Question 7 (a)

   2 marks · 32/75

   This tests using the intersection of two sketches to establish the number of roots on a specified interval.

   Examiner report: The report distinguishes a genuine curve sketch from plotting a few isolated points. Candidates needed to show the main shapes over most of the interval and indicate how the single crossing represents the required root.

   Question paper p.9 · Mark scheme p.12

8. Question 7 (b)

   2 marks · 34/75

   This tests numerically bracketing a root by comparing the two sides of an equation at the stated endpoints.

   Examiner report: Some candidates evaluated the trigonometric functions in degrees, which invalidated the endpoint comparison. The report stresses comparing both functions at each endpoint rather than giving unsupported numerical claims.

   Question paper p.9 · Mark scheme p.12

9. Question 7 (c)

   3 marks · 37/75

   This tests applying a fixed-point iteration accurately enough to obtain a rounded numerical root.

   Examiner report: Degree-mode calculations were again a major source of incorrect results. The displayed iterates also needed enough consistency to support the final rounding.

   Question paper p.9 · Mark scheme p.13

10. Question 8 (a)

    3 marks · 40/75

    This tests polynomial division when the divisor is quadratic and the dividend is cubic.

    Examiner report: This part was generally handled well, though some candidates made arithmetic errors during the final stages of the division.

    Question paper p.10 · Mark scheme p.13

11. Question 8 (b)

    5 marks · 45/75

    Question paper p.11 · Mark scheme p.14

12. Question 9

    9 marks · 54/75

    This tests solving a separable first-order differential equation using partial fractions and an exact initial condition.

    Examiner report: The separation step was often successful, but the coefficient attached to one partial-fraction term caused difficulty. Decimal approximations were also used where the question required an exact logarithmic or exponential simplification.

    Question paper p.12 · Mark scheme p.15

13. Question 10 (a)

    3 marks · 57/75

    This tests constructing a vector equation for the line through two points in three-dimensional space.

    Examiner report: Many candidates found the direction correctly but stopped at the vector joining the points rather than writing the equation of the whole line. The left side was also sometimes labelled with a point or line name instead of a position vector.

    Question paper p.14 · Mark scheme p.16

14. Question 10 (b)

    3 marks · 60/75

    This tests finding the acute angle between two three-dimensional line directions using a scalar product.

    Examiner report: Correct numerical angles were usually obtained when the directions had first been identified correctly. Common wrong choices were a position vector on the second line or a vector from the origin to one of the points.

    Question paper p.14 · Mark scheme p.16

15. Question 10 (c)

    4 marks · 64/75

    This tests checking whether two three-dimensional lines have a common point by comparing their component equations.

    Examiner report: Most candidates formed component equations and solved for a parameter, but some did not complete the final consistency check or compared the resulting points incorrectly.

    Question paper p.15 · Mark scheme p.17

16. Question 11 (a)

    6 marks · 70/75

    This tests locating the maximum of a product of trigonometric functions by differentiation and solving the resulting trigonometric equation.

    Examiner report: The main difficulty was retaining the factor generated when differentiating the inner angle in the cosine term. An algebraic substitution can simplify the work, but the report notes that it sometimes introduced further errors.

    Question paper p.16 · Mark scheme p.18

17. Question 11 (b)

    5 marks · 75/75

    This tests evaluating a trigonometric area integral by substitution and exact limit conversion.

    Examiner report: Errors commonly arose from an incorrect double-angle identity or from confusing the original x-limits with the transformed limits. Exact endpoint trigonometric values were required rather than calculator decimals.

    Question paper p.17 · Mark scheme p.19