9709 Pure Mathematics 3 May/June 2022 Paper 33

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

1. Question 1

   4 marks · 4/75

   This assesses solving a strict inequality involving absolute values when the coefficients include a positive parameter.

   Examiner report: Most candidates used the efficient quadratic approach and selected the region between the boundary values; unsupported final responses were only successful when the result itself was correct.

   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 (a)

   3 marks · 12/75

   This assesses applying logarithm laws to convert a logarithmic equation into a quadratic polynomial equation.

   Examiner report: Some candidates handled the logarithm coefficient correctly but combined the constant and squared factor as a sum rather than as a product.

   Question paper p.5 · Mark scheme p.9

4. Question 3 (b)

   2 marks · 14/75

   This assesses transferring a derived logarithmic-quadratic method to a changed variable and reporting a valid numerical root.

   Examiner report: Errors included solving for the substituted variable instead of y, retaining an inadmissible root, and failing to give the required decimal accuracy.

   Question paper p.5 · Mark scheme p.9

5. Question 4 (a)

   4 marks · 18/75

   This assesses differentiating a product involving an exponential and tangent factor, then transforming the result with trigonometric identities.

   Examiner report: Some responses appeared to choose the constants without justification, while others differentiated only one factor or obtained the wrong coefficient for the double-angle term.

   Question paper p.6 · Mark scheme p.10

6. Question 4 (b)

   3 marks · 21/75

   This assesses finding all stationary points by solving the trigonometric condition obtained from a derivative.

   Examiner report: The main omissions were finding only one stationary point and working in degrees rather than radians.

   Question paper p.7 · Mark scheme p.11

7. Question 5 (a)

   3 marks · 24/75

   This assesses representing conjugate-related complex numbers geometrically and recognising the resulting quadrilateral from its side relationships.

   Examiner report: Some diagrams had no usable scale, and although many candidates recognised the parallel-side property, they did not give the required quadrilateral classification.

   Question paper p.8 · Mark scheme p.12

8. Question 5 (b)

   3 marks · 27/75

   This assesses division of complex numbers by rationalising the denominator and separating real and imaginary parts.

   Examiner report: The report stressed that a displayed method was necessary; simply entering the calculation into a calculator and recording the result did not demonstrate the required complex-number working.

   Question paper p.8 · Mark scheme p.13

9. Question 5 (c)

   2 marks · 29/75

   This assesses using complex arguments or the tangent double-angle formula to prove an inverse-tangent identity.

   Examiner report: The strongest solutions explicitly accounted for the signs of both arguments; missing this information prevented some otherwise promising arguments from being complete.

   Question paper p.9 · Mark scheme p.14

10. Question 6 (a)

    5 marks · 34/75

    This assesses finding the derivative of a parametrically defined curve by differentiating both coordinates with respect to the parameter.

    Examiner report: A recurrent notation problem was replacing the parameter by x, which made the chain-rule argument incomplete even when the underlying differentiation was understood.

    Question paper p.10 · Mark scheme p.15

11. Question 6 (b)

    3 marks · 37/75

    This assesses forming a tangent equation to a parametric curve at a point specified by its y-coordinate.

    Examiner report: Some candidates interchanged x and the parameter in the tangent calculation, while others used the normal gradient instead of the tangent gradient.

    Question paper p.11 · Mark scheme p.16

12. Question 7 (a)

    5 marks · 42/75

    This assesses decomposing a rational function into partial fractions with a linear numerator over the irreducible quadratic factor.

    Examiner report: Most candidates chose the right structure, but mistakes arose when a constant rather than a linear numerator was used over the quadratic factor or when the resulting simultaneous equations were solved incorrectly.

    Question paper p.12 · Mark scheme p.16

13. Question 7 (b)

    5 marks · 47/75

    This assesses combining partial fractions with geometric or binomial expansions to obtain a power series through the quadratic term.

    Examiner report: Many errors came from mishandling the signs and factors when converting the two denominators into expansion forms, especially placing extracted constants on the wrong side of a fraction.

    Question paper p.13 · Mark scheme p.17

14. Question 8 (a)

    5 marks · 52/75

    This assesses solving a separable differential equation with a power of the population variable and a trigonometric time factor.

    Examiner report: Although many candidates separated the variables successfully, sign and coefficient errors were common, particularly when integrating the negative fractional power of N and the cosine containing 0.02t.

    Question paper p.14 · Mark scheme p.18

15. Question 8 (b)

    2 marks · 54/75

    This assesses using a second population observation to determine the constant in a solved differential equation.

    Examiner report: The principal difficulties were algebraic rearrangement and treating the trigonometric argument as degrees rather than radians.

    Question paper p.15 · Mark scheme p.18

16. Question 8 (c)

    2 marks · 56/75

    This assesses rearranging the population relation into an explicit model and optimising it over time.

    Examiner report: Many candidates struggled to make N the subject after substituting the constant, so only a small proportion completed the maximum-value calculation correctly.

    Question paper p.15 · Mark scheme p.19

17. Question 9 (a)

    3 marks · 59/75

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

    Examiner report: Some candidates paired a dot product with magnitudes belonging to different vectors, producing an inconsistent angle calculation.

    Question paper p.16 · Mark scheme p.20

18. Question 9 (b)

    4 marks · 63/75

    This assesses locating the perpendicular projection of a point onto a vector line in three dimensions.

    Examiner report: A common conceptual error was making the position vector of the foot perpendicular to the line direction; a diagram would have helped candidates impose perpendicularity on the segment from A instead.

    Question paper p.16 · Mark scheme p.20

19. Question 9 (c)

    2 marks · 65/75

    Question paper p.17 · Mark scheme p.21

20. Question 10 (a)

    5 marks · 70/75

    This assesses integration by parts followed by applying limits and rearranging an equation into a fixed-point form.

    Examiner report: This part was generally handled very well, with most candidates completing the integration by parts and showing enough algebra to connect the definite integral to the required rearrangement.

    Question paper p.18 · Mark scheme p.22

21. Question 10 (b)

    2 marks · 72/75

    This assesses verifying that a root lies within a numerical interval by comparing the two sides of its defining equation at the endpoints.

    Examiner report: A few candidates discussed a sign change without actually comparing the variable with the fixed-point expression, which does not directly establish the required endpoint crossover.

    Question paper p.19 · Mark scheme p.22

22. Question 10 (c)

    3 marks · 75/75

    This assesses applying fixed-point iteration and judging convergence to the requested decimal accuracy.

    Examiner report: Successful solutions included sufficiently precise intermediate iterates and enough repetitions to demonstrate that the displayed two-decimal result had stabilised.

    Question paper p.19 · Mark scheme p.23