9709 Pure Mathematics 3 May/June 2020 Paper 32

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

1. Question 1

   4 marks

   This assesses polynomial long division involving a quadratic divisor.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   5 marks

   This assesses the use of logarithms to linearise an exponential model and determine its constants from two points.

   Question paper p.3 · Mark scheme p.6

3. Question 3

   5 marks

   This assesses definite integration by parts for an integrand containing a power of x and a logarithm.

   Question paper p.4 · Mark scheme p.7

4. Question 4

   6 marks

   This assesses finding a stationary point of a product of trigonometric functions and solving the resulting trigonometric equation numerically.

   Question paper p.5 · Mark scheme p.7

5. Question 5 (a)

   3 marks

   This assesses rewriting a linear combination of sine and cosine as a single shifted cosine.

   Question paper p.6 · Mark scheme p.8

6. Question 5 (b)

   4 marks

   This assesses solving a trigonometric equation by using a previously obtained amplitude-phase form over a specified degree interval.

   Question paper p.7 · Mark scheme p.8

7. Question 6 (a)

   4 marks

   This assesses locating the maximum of a rational function on the non-negative x-axis using differentiation.

   Question paper p.8 · Mark scheme p.8

8. Question 6 (b)

   5 marks

   This assesses evaluating a bounded area by a substitution that converts the integrand to an inverse-tangent form.

   Question paper p.9 · Mark scheme p.9

9. Question 7

   9 marks

   This assesses solving a separable first-order differential equation with an initial condition and partial fractions.

   Question paper p.10 · Mark scheme p.10

10. Question 8 (a)

    4 marks

    This assesses solving a complex equation involving a number and its conjugate by separating real and imaginary parts.

    Question paper p.12 · Mark scheme p.11

11. Question 8 (b)(i)

    4 marks

    This assesses interpreting modulus and argument inequalities geometrically on an Argand diagram.

    Question paper p.13 · Mark scheme p.11

12. Question 8 (b)(ii)

    2 marks

    This assesses extracting the smallest imaginary coordinate from a geometrically defined complex-number region.

    Question paper p.13 · Mark scheme p.11

13. Question 9 (a)

    5 marks

    This assesses using tangency to relate two intersecting curves and eliminate an unknown parameter.

    Question paper p.14 · Mark scheme p.12

14. Question 9 (b)

    3 marks

    This assesses applying a fixed-point iteration to obtain a root to a stated decimal accuracy.

    Question paper p.15 · Mark scheme p.12

15. Question 9 (c)

    2 marks

    This assesses finding a parameter by substituting the numerically determined contact point into one of the curve equations.

    Question paper p.15 · Mark scheme p.12

16. Question 10 (a)

    5 marks

    This assesses vector geometry involving a midpoint, an internal division point, and the equation of a line in three dimensions.

    Question paper p.16 · Mark scheme p.13

17. Question 10 (b)

    3 marks

    This assesses finding the intersection of two three-dimensional vector lines.

    Question paper p.17 · Mark scheme p.13

18. Question 10 (c)

    3 marks

    This assesses calculating an angle between two lines using the scalar-product formula.

    Question paper p.17 · Mark scheme p.13