9709 Pure Mathematics 3 May/June 2020 Paper 31

22 questions and parts. Marks come from the published mark scheme.

1. Question 1

   4 marks

   This assesses the use of logarithms to solve an exponential inequality exactly.

   Question paper p.2 · Mark scheme p.6

2. Question 2 (a)

   4 marks

   This assesses the binomial expansion of a negative power up to a specified degree.

   Question paper p.3 · Mark scheme p.6

3. Question 2 (b)

   1 mark

   Question paper p.3 · Mark scheme p.6

4. Question 3

   6 marks

   This assesses the use of compound-angle tangent identities and the selection of solutions in a specified angular interval.

   Question paper p.4 · Mark scheme p.7

5. Question 4 (a)

   4 marks

   This assesses differentiation of a product involving an exponential and trigonometric factor, followed by numerical root-finding.

   Question paper p.5 · Mark scheme p.7

6. Question 4 (b)

   2 marks

   This assesses classification of a stationary point using the behaviour of the derivative nearby.

   Question paper p.5 · Mark scheme p.7

7. Question 5 (a)

   3 marks

   This assesses polynomial division when the divisor is quadratic.

   Question paper p.6 · Mark scheme p.8

8. Question 5 (b)

   5 marks

   This assesses integration after algebraic division, including logarithmic and inverse-tangent antiderivatives.

   Question paper p.7 · Mark scheme p.8

9. Question 6 (a)

   3 marks

   This assesses the use of tangent geometry, sector areas, and triangle areas to derive a trigonometric equation.

   Question paper p.8 · Mark scheme p.9

10. Question 6 (b)

    2 marks

    This assesses numerical verification that a continuous equation has a root inside a given interval.

    Question paper p.9 · Mark scheme p.9

11. Question 6 (c)

    3 marks

    This assesses fixed-point iteration for a trigonometric equation and appropriate rounding of the converged root.

    Question paper p.9 · Mark scheme p.9

12. Question 7 (a)

    4 marks

    This assesses differentiation of a trigonometric quotient and use of trigonometric identities to establish a sign.

    Question paper p.10 · Mark scheme p.9

13. Question 7 (b)

    4 marks

    This assesses recognition of an exact logarithmic antiderivative and evaluation of a definite integral.

    Question paper p.11 · Mark scheme p.10

14. Question 8 (a)

    8 marks

    This assesses solving a first-order separable differential equation with a proportionality constant determined by two points.

    Question paper p.12 · Mark scheme p.10

15. Question 8 (b)

    1 mark

    This assesses interpreting the long-term behaviour of the function obtained from a differential equation.

    Question paper p.13 · Mark scheme p.11

16. Question 9 (a)

    3 marks

    This assesses the use of position vectors and the dot product to prove perpendicularity at a vertex.

    Question paper p.14 · Mark scheme p.11

17. Question 9 (b)

    2 marks

    This assesses finding vector lengths and using them to establish that a triangle has two equal sides.

    Question paper p.14 · Mark scheme p.11

18. Question 9 (c)

    4 marks

    This assesses finding the shortest distance from the origin to a line in three-dimensional vector form.

    Question paper p.15 · Mark scheme p.11

19. Question 10 (a)(i)

    3 marks

    This assesses rationalising a complex denominator and separating a complex number into real and imaginary parts.

    Question paper p.16 · Mark scheme p.12

20. Question 10 (a)(ii)

    3 marks

    This assesses using conjugation and argument information to determine a real parameter in a complex expression.

    Question paper p.16 · Mark scheme p.12

21. Question 10 (b)(i)

    4 marks

    This assesses translating modulus inequalities into geometric boundaries on an Argand diagram.

    Question paper p.17 · Mark scheme p.13

22. Question 10 (b)(ii)

    2 marks

    This assesses finding the smallest argument of points in a geometrically defined region.

    Question paper p.17 · Mark scheme p.13