9709 Pure Mathematics 3 Oct/Nov 2020 Paper 32

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

1. Question 1

   3 marks · 3/75

   This tests solving an exponential equation after removing a natural logarithm.

   Examiner report: Most candidates could form the equation without logarithms, but some stopped before isolating the exponential or missed the required accuracy. A frequent error was incorrectly distributing the logarithm over the sum inside its argument.

   Question paper p.3 · Mark scheme p.6

2. Question 2 (a)

   4 marks · 7/75

   This tests using the binomial expansion for a fractional power and simplifying its first four terms.

   Examiner report: Many responses were complete, though arithmetic simplification caused some losses. Common mistakes involved expanding in x instead of the required multiple of x or selecting the wrong number of terms.

   Question paper p.4 · Mark scheme p.6

3. Question 2 (b)

   1 mark · 8/75

   This tests identifying the interval of convergence of a binomial expansion.

   Examiner report: This part was often left blank, and some candidates considered only one side of the required restriction or repeated part of their expansion instead of giving a validity condition.

   Question paper p.4 · Mark scheme p.6

4. Question 3 (a)

   3 marks · 11/75

   This tests converting an exponential relation into a linear equation and reading its exact gradient.

   Examiner report: Many candidates began correctly but lost the meaning of the exponent by omitting brackets. Others did not explain why the resulting equation represented a straight line or gave a decimal or incorrectly signed gradient.

   Question paper p.5 · Mark scheme p.7

5. Question 3 (b)

   2 marks · 13/75

   This tests finding an exact intersection coordinate by combining a logarithmic line equation with another straight-line equation.

   Examiner report: Candidates who kept the logarithmic form generally made good progress, whereas decimal coefficients prevented an exact response. Some also failed to simplify the resulting logarithmic denominator or made rearrangement errors.

   Question paper p.5 · Mark scheme p.7

6. Question 4 (a)

   3 marks · 16/75

   This tests applying the tangent addition formula and converting a trigonometric equation into a quadratic in tan θ.

   Examiner report: Concise correct solutions were common. Errors included treating the tangent of a sum as a sum of tangents and mishandling cot θ by using an incorrect reciprocal factor.

   Question paper p.6 · Mark scheme p.8

7. Question 4 (b)

   3 marks · 19/75

   This tests solving a trigonometric equation through its quadratic in tan θ and selecting all angles in a specified interval.

   Examiner report: Premature rounding and including extra roots were common sources of lost marks. Some candidates wrongly rejected the negative tangent root instead of finding its corresponding angle in the prescribed interval.

   Question paper p.7 · Mark scheme p.8

8. Question 5 (a)

   3 marks · 22/75

   This tests finding the gradient of a parametrically defined curve using a ratio of derivatives.

   Examiner report: Most candidates knew the parametric-gradient method, but differentiating cos²θ was a recurring difficulty. Some produced a sign or power error, while others complicated the work by changing to a double-angle form unnecessarily.

   Question paper p.8 · Mark scheme p.9

9. Question 5 (b)

   4 marks · 26/75

   This tests locating the maximum value of a parametric gradient by differentiating the gradient with respect to its parameter.

   Examiner report: Only the strongest responses distinguished a maximum gradient from a point where the gradient itself is zero. Several candidates selected the wrong sign for the coordinate despite the position of the point in the diagram.

   Question paper p.9 · Mark scheme p.10

10. Question 6 (a)

    3 marks · 29/75

    This tests division of complex numbers and conversion to Cartesian form.

    Examiner report: The standard conjugate method was widely recognised, but arithmetic involving i² caused some slips. A smaller group used the wrong conjugate or did not finish by comparing real and imaginary components.

    Question paper p.10 · Mark scheme p.11

11. Question 6 (b)

    2 marks · 31/75

    This tests representing complex numbers as points on an Argand diagram.

    Examiner report: Many diagrams were suitably accurate, although uneven scales weakened some representations. A few candidates interchanged the axes or left the diagram unfinished.

    Question paper p.10 · Mark scheme p.11

12. Question 6 (c)

    3 marks · 34/75

    This tests using the argument rule for division of complex numbers to establish an exact inverse-tangent identity.

    Examiner report: Because the target relation was supplied, a clear argument equation was still needed to justify it. A common misconception was to add the arguments during division, and some candidates assigned the denominator the wrong sign of argument.

    Question paper p.11 · Mark scheme p.12

13. Question 7 (a)

    7 marks · 41/75

    This tests solving a separable differential equation involving a squared trigonometric function and an exponential factor.

    Examiner report: Separation was usually identified, but sign and factor errors were frequent in the exponential integral. The secant-square integral and its factor of 2 were particularly troublesome; successful candidates generally completed the calculation correctly.

    Question paper p.12 · Mark scheme p.13

14. Question 7 (b)

    1 mark · 42/75

    This tests finding the long-term behaviour of a solution from its explicit expression.

    Examiner report: Some responses merely said that x increased or decreased, without evaluating the limiting process. Stronger solutions explicitly considered the vanishing exponential contribution.

    Question paper p.13 · Mark scheme p.13

15. Question 8 (a)

    3 marks · 45/75

    This tests finding displacement vectors and comparing the lengths of two three-dimensional line segments.

    Examiner report: Most candidates formed the displacement vectors correctly, but fewer went on to compare their magnitudes as the question required. Numerical slips in the length calculations were also common.

    Question paper p.14 · Mark scheme p.14

16. Question 8 (b)

    3 marks · 48/75

    This tests finding the angle between two three-dimensional direction vectors using the scalar product.

    Examiner report: The main method was generally understood, although many candidates expected an acute result and overlooked that the negative scalar product indicates an obtuse angle.

    Question paper p.14 · Mark scheme p.14

17. Question 8 (c)

    4 marks · 52/75

    This tests deciding whether two three-dimensional lines intersect by comparing their vector equations.

    Examiner report: Many candidates knew how to construct the two line equations, but arithmetic errors and the use of one shared parameter caused difficulties. Merely displaying unequal parameter results without explaining the resulting non-intersection was not enough.

    Question paper p.15 · Mark scheme p.15

18. Question 9 (a)

    5 marks · 57/75

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

    Examiner report: The usual partial-fraction structure was often used successfully. Mistakes included treating the quadratic as though it factored into real linear terms, using an incomplete numerator over it, and copying the wrong constant from the original numerator.

    Question paper p.16 · Mark scheme p.16

19. Question 9 (b)

    6 marks · 63/75

    Question paper p.17 · Mark scheme p.16

20. Question 10 (a)

    3 marks · 66/75

    This tests differentiating a product and using the stationary-point condition to derive an equation for the minimum location.

    Examiner report: Most candidates used the product rule correctly. Some stopped before rearranging into the requested form, while others produced unnecessarily long solutions or made algebraic sign errors.

    Question paper p.18 · Mark scheme p.17

21. Question 10 (b)

    3 marks · 69/75

    This tests numerical fixed-point iteration and reporting a converged value to specified accuracy.

    Examiner report: The iterative method was familiar to most candidates, but some ignored the given starting value or worked in degrees. Other responses listed figures without showing how the iterations were generated or did not provide enough precision to support the final rounding.

    Question paper p.18 · Mark scheme p.17

22. Question 10 (c)

    6 marks · 75/75

    This tests finding a volume of revolution by integrating the square of a curve and evaluating a trigonometric integral by parts.

    Examiner report: Only a minority set up the volume integral correctly, and integration of the squared cosine factor caused further errors. Some attempted to integrate x cos x directly, used an incorrect antiderivative for cos²x, or selected the later endpoint rather than the boundary of the shaded region.

    Question paper p.19 · Mark scheme p.18