9709 Pure Mathematics 3 May/June 2022 Paper 32

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

1. Question 1

   4 marks · 4/75

   This tests solving a logarithmic equation by converting it into an exponential equation and applying the required numerical accuracy.

   Examiner report: Most candidates handled this routine opening question successfully. Errors mainly came from mishandling the initial logarithm transformation, confusing the placement of the exponent, or rounding incorrectly.

   Question paper p.3 · Mark scheme p.6

2. Question 2

   5 marks · 9/75

   Question paper p.4 · Mark scheme p.7

3. Question 3

   5 marks · 14/75

   This tests using the factor and remainder theorems to form simultaneous equations for unknown polynomial coefficients.

   Examiner report: Substitution was generally more reliable than polynomial division. A recurring mistake was using the wrong value for the remainder condition, together with occasional sign or arithmetic errors when solving for the coefficients.

   Question paper p.5 · Mark scheme p.8

4. Question 4

   6 marks · 20/75

   This tests differentiating a product involving a power of cosine and a square root of sine, then finding a stationary point using trigonometric algebra.

   Examiner report: The product rule was often attempted, but the chain-rule factors were frequently lost or given incorrect coefficients. Further errors arose while removing negative powers and reducing the equation to one trigonometric function; some final values were also insufficiently accurate.

   Question paper p.6 · Mark scheme p.9

5. Question 5 (a)

   2 marks · 22/75

   This tests establishing the existence and uniqueness of a root by comparing the graphs of the two sides of an equation.

   Examiner report: Many sketches missed essential features: the logarithmic curve was not shown approaching a vertical asymptote, and the quadratic was sometimes misplaced or given the wrong orientation. A correct pair of curves still needed an explicit indication that the intersection represented the root.

   Question paper p.7 · Mark scheme p.10

6. Question 5 (b)

   2 marks · 24/75

   This tests numerical bracketing of a root by checking the signs or relative sizes of the two sides at the endpoints.

   Examiner report: This part was often completed correctly, but some candidates gave calculations without a clear final inference. When four separate values were used, the comparison between them was not always made explicit.

   Question paper p.7 · Mark scheme p.10

7. Question 5 (c)

   3 marks · 27/75

   This tests using fixed-point iteration to approximate a root and judging when the approximation is accurate enough.

   Examiner report: Most candidates iterated successfully from a starting value in the bracket. Weaknesses included stopping too soon, failing to retain the requested decimal places, and not giving the final answer in the specified two-decimal format.

   Question paper p.7 · Mark scheme p.11

8. Question 6 (a)

   7 marks · 34/75

   This tests solving a first-order separable differential equation, including integration by parts, use of an initial condition, and rearrangement for y.

   Examiner report: Candidates who separated the variables generally progressed well, but common difficulties included not splitting the exponential into x and y factors, sign errors in integration, and using an unsuitable constant after taking logarithms. Some incorrectly differentiated a product instead of integrating by parts.

   Question paper p.8 · Mark scheme p.12

9. Question 6 (b)

   1 mark · 35/75

   This tests substituting a specified x-value into the solution of the differential equation and presenting the result in the requested exact structure.

   Examiner report: This was usually answered correctly when part (a) had produced a valid solution.

   Question paper p.9 · Mark scheme p.12

10. Question 7 (a)

    4 marks · 39/75

    This tests implicit differentiation of an algebraic curve and rearrangement to obtain dy/dx.

    Examiner report: Many solutions were successful, but some candidates left an extraneous dy/dx statement in the working or failed to carry it consistently through the rearrangement. The supplied target form helped expose sign mistakes, but any incorrect differentiation still needed to be corrected.

    Question paper p.10 · Mark scheme p.13

11. Question 7 (b)

    5 marks · 44/75

    This tests locating horizontal tangents from an implicit derivative and then imposing the original curve equation to find every corresponding point.

    Examiner report: Most candidates recognised that a horizontal tangent requires a zero numerator. Errors included mishandling the factor involving x, incorrect powers during substitution, and omitting the additional point associated with that factor.

    Question paper p.11 · Mark scheme p.13

12. Question 8 (a)

    5 marks · 49/75

    This tests decomposing a rational function into partial fractions with a linear factor and an irreducible quadratic factor.

    Examiner report: The decomposition was often treated efficiently by choosing convenient values of x. Some candidates assumed the quadratic numerator was constant without justification, while others found that the coefficient of x was zero but did not complete the deduction.

    Question paper p.12 · Mark scheme p.14

13. Question 8 (b)

    5 marks · 54/75

    Question paper p.13 · Mark scheme p.14

14. Question 9 (a)

    4 marks · 58/75

    This tests finding the condition for two three-dimensional lines to intersect by equating their component coordinates and eliminating the parameters.

    Examiner report: This unfamiliar elimination problem caused considerable difficulty. Candidates who formed component equations and systematically removed the parameters were most successful; repeated attempts often led to algebraic or arithmetic errors.

    Question paper p.14 · Mark scheme p.15

15. Question 9 (b)

    4 marks · 62/75

    This tests using the perpendicularity condition for two lines to obtain a second equation and solve simultaneously with the intersection condition.

    Examiner report: Those who connected perpendicular lines with a zero scalar product generally completed this part. The main failures came from arithmetic and algebra, or from not using the direction vectors in the perpendicularity test.

    Question paper p.15 · Mark scheme p.15

16. Question 9 (c)

    2 marks · 64/75

    This tests finding the intersection point of two vector lines after their parameters or direction constants have been determined.

    Examiner report: This was accessible once the preceding constants had been found. Candidates commonly used an already available parameter equation to obtain the point directly.

    Question paper p.15 · Mark scheme p.16

17. Question 10 (a)

    3 marks · 67/75

    This tests substituting a complex root into a cubic, expanding powers of a complex number, and determining a real coefficient.

    Examiner report: Direct substitution was the most common approach, but the expansion needed to be shown rather than delegated to a calculator. Binomial expansions sometimes introduced arithmetic errors; an approach based on finding a quadratic factor also worked when presented clearly.

    Question paper p.16 · Mark scheme p.16

18. Question 10 (b)

    4 marks · 71/75

    This tests using conjugate roots of a polynomial with real coefficients, forming a quadratic factor, and finding the remaining root.

    Examiner report: Most candidates knew to use the conjugate root and then divide by the resulting quadratic. Confusion between a root and a factor, sign errors in the quadratic, and attempts to take square roots of the given complex number were seen.

    Question paper p.16 · Mark scheme p.17

19. Question 10 (c)

    2 marks · 73/75

    This tests interpreting a complex modulus equation as a circle on an Argand diagram.

    Examiner report: The centre and radius were usually identified correctly. Some diagrams used unequal or non-linear axis scales, which made the intended circle appear elliptical.

    Question paper p.17 · Mark scheme p.18

20. Question 10 (d)

    2 marks · 75/75

    This tests finding the greatest argument of a point on a circle by using the tangent from the origin and expressing the resulting angle in radians.

    Examiner report: Successful solutions generally relied on a clearly marked diagram and supporting calculations. The required unit was radians, and incomplete geometric justification or degree answers lost accuracy credit.

    Question paper p.17 · Mark scheme p.18