9709 Pure Mathematics 3 Oct/Nov 2022 Paper 31

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

1. Question 1 (a)

   1 mark · 1/75

   Sketching a modulus graph by recognising the two linear branches of an absolute-value function.

   Examiner report: The main graphical error was putting the vertex on the vertical axis; a few candidates also drew curved rather than linear branches.

   Question paper p.3 · Mark scheme p.6

2. Question 1 (b)

   3 marks · 4/75

   Solving an inequality involving an absolute value, using the relevant branch of the graph or algebraic cases.

   Examiner report: Only a smaller proportion used the sketch to identify the appropriate region. Squaring was common, but it introduced two critical values that were often both retained incorrectly.

   Question paper p.3 · Mark scheme p.7

3. Question 2

   4 marks · 8/75

   Representing simultaneous modulus, real-part and argument restrictions on an Argand diagram.

   Examiner report: Some diagrams misplaced the circle, while others treated the constant-real-part boundary as horizontal instead of vertical.

   Question paper p.4 · Mark scheme p.8

4. Question 3

   4 marks · 12/75

   Solving an equation with exponential terms by applying logarithm or index laws and expressing the result in the requested logarithmic form.

   Examiner report: Logarithm rules were generally handled well by candidates who knew them, but missing brackets and incorrect treatment of the product involving the base 3 caused frequent errors.

   Question paper p.5 · Mark scheme p.9

5. Question 4

   5 marks · 17/75

   Question paper p.6 · Mark scheme p.10

6. Question 5 (a)

   3 marks · 20/75

   Multiplying and dividing complex numbers in exponential polar form.

   Examiner report: This polar-form manipulation was difficult for many candidates; those who knew how to combine the modulus and argument usually completed the part successfully.

   Question paper p.7 · Mark scheme p.10

7. Question 5 (b)

   1 mark · 21/75

   Question paper p.7 · Mark scheme p.10

8. Question 6 (a)

   4 marks · 25/75

   Question paper p.8 · Mark scheme p.11

9. Question 6 (b)

   3 marks · 28/75

   Using a previously proved identity to solve a trigonometric equation within a degree interval.

   Examiner report: Only a minority made effective use of the preceding identity. Candidates who reduced the equation correctly generally obtained the permitted angles.

   Question paper p.9 · Mark scheme p.11

10. Question 7 (a)

    3 marks · 31/75

    Differentiating a quotient involving a trigonometric denominator and using a prescribed tangent gradient.

    Examiner report: There were many attempts using the appropriate differentiation rule, but incorrect signs or numerical factors often stopped the derivation from reaching the required form.

    Question paper p.10 · Mark scheme p.12

11. Question 7 (b)

    2 marks · 33/75

    Verifying a numerical bracket for a root by evaluating a suitable continuous expression at both endpoints.

    Examiner report: Some candidates omitted this verification. Others used a fixed-point equation without clearly showing the endpoint evidence needed to establish the interval.

    Question paper p.11 · Mark scheme p.12

12. Question 7 (c)

    3 marks · 36/75

    Applying fixed-point iteration to approximate a solution to a specified decimal accuracy.

    Examiner report: A number of candidates left the iteration blank. Those who used radians generally progressed well, but the required intermediate precision was not always shown.

    Question paper p.11 · Mark scheme p.12

13. Question 8 (a)

    5 marks · 41/75

    Solving a separable first-order differential equation with an initial condition.

    Examiner report: Correct solutions usually began with separation and integrated the x-term successfully. Mistakes in the coefficient of the time integral and failure to use the initial condition were frequent.

    Question paper p.12 · Mark scheme p.13

14. Question 8 (b)

    3 marks · 44/75

    Using a further data point to determine a parameter and finding the long-term limiting amount from a differential-equation solution.

    Examiner report: This continuation was often not attempted, particularly when part (a) had not been completed. Exact intermediate work was acceptable, but the final limiting quantity needed to be numerical.

    Question paper p.13 · Mark scheme p.13

15. Question 9 (a)

    5 marks · 49/75

    Finding the stationary minimum of a product involving a linear factor and an exponential factor.

    Examiner report: Most attempts used the product rule, though sign and coefficient errors were common. Some candidates stopped after finding the stationary condition or failed to provide an exact ordinate.

    Question paper p.14 · Mark scheme p.14

16. Question 9 (b)

    5 marks · 54/75

    Finding an exact area under an exponential curve using integration by parts and axis boundaries.

    Examiner report: Integration by parts was widely recognised, but coefficients were often lost during the process. Candidates who completed the antiderivative generally used the limits correctly.

    Question paper p.15 · Mark scheme p.14

17. Question 10 (a)

    5 marks · 59/75

    Decomposing a rational function with a repeated linear factor into partial fractions.

    Examiner report: Most attempted solutions adopted the right partial-fraction structure. Arithmetic and algebra slips, particularly missing brackets, accounted for many incorrect constants.

    Question paper p.16 · Mark scheme p.15

18. Question 10 (b)

    5 marks · 64/75

    Combining binomial-series expansions of partial fractions to obtain a power series through the quadratic term.

    Examiner report: This part was difficult for many candidates and was often left unanswered. Errors frequently occurred when extracting factors of 2 before applying the binomial expansion, followed by mistakes when collecting terms.

    Question paper p.17 · Mark scheme p.15

19. Question 11 (a)

    4 marks · 68/75

    Using a three-dimensional diagram and section ratios to express position and displacement vectors in component form.

    Examiner report: Some candidates correctly interpreted enough of the diagram to make progress, but the spatial relationships and ratio construction caused difficulty.

    Question paper p.18 · Mark scheme p.16

20. Question 11 (b)

    3 marks · 71/75

    Calculating the angle between two three-dimensional vectors using the scalar product.

    Examiner report: This part depended on having usable vectors from part (a), although correct scalar-product and magnitude work with an incorrect pair of vectors could still receive method credit.

    Question paper p.19 · Mark scheme p.16

21. Question 11 (c)

    4 marks · 75/75

    Finding the shortest distance from a point to a line in three dimensions.

    Examiner report: A large number of candidates did not attempt this part. The few successful approaches showed that the key step was locating the foot of the perpendicular on the line before evaluating its length.

    Question paper p.19 · Mark scheme p.17