9709 Pure Mathematics 3 May/June 2020 Paper 33

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

1. Question 1

   4 marks

   Question paper p.2 · Mark scheme p.6

2. Question 2

   5 marks

   This assesses exact integration of a linear function multiplied by an exponential function over specified limits.

   Question paper p.3 · Mark scheme p.7

3. Question 3 (a)

   2 marks

   This assesses transforming a logarithmic equation into a quadratic equation using an exponential substitution.

   Question paper p.4 · Mark scheme p.8

4. Question 3 (b)

   4 marks

   This assesses solving the transformed quadratic equation and converting its admissible exponential root back to a value of the original variable.

   Question paper p.4 · Mark scheme p.8

5. Question 4 (a)

   3 marks

   This assesses differentiation of a product involving an inverse tangent function and an inner linear expression.

   Question paper p.5 · Mark scheme p.9

6. Question 4 (b)

   3 marks

   This assesses using a derivative and a point on a curve to determine the intercept of its tangent.

   Question paper p.5 · Mark scheme p.9

7. Question 5

   6 marks

   This assesses solving a trigonometric equation by rewriting all terms in one variable and applying a quadratic method.

   Question paper p.6 · Mark scheme p.10

8. Question 6 (a)

   2 marks

   This assesses demonstrating uniqueness of a real solution through the intersection of two graphs.

   Question paper p.8 · Mark scheme p.11

9. Question 6 (b)

   2 marks

   This assesses identifying the fixed-point equation associated with a proposed iterative formula.

   Question paper p.8 · Mark scheme p.11

10. Question 6 (c)

    3 marks

    Question paper p.9 · Mark scheme p.11

11. Question 7 (a)

    2 marks

    This assesses decomposing a rational function with two linear factors into partial fractions.

    Question paper p.10 · Mark scheme p.12

12. Question 7 (b)

    2 marks

    This assesses using a partial-fraction representation to expand the square of a rational function.

    Question paper p.10 · Mark scheme p.12

13. Question 7 (c)

    5 marks

    This assesses integrating a squared rational function after partial-fraction decomposition and evaluating definite limits.

    Question paper p.11 · Mark scheme p.12

14. Question 8 (a)

    5 marks

    This assesses finding a missing vertex of a parallelogram with vectors and testing whether adjacent sides have equal length.

    Question paper p.12 · Mark scheme p.13

15. Question 8 (b)

    3 marks

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

    Question paper p.13 · Mark scheme p.13

16. Question 8 (c)

    2 marks

    This assesses calculating the area of a parallelogram from two side vectors and their included angle.

    Question paper p.13 · Mark scheme p.14

17. Question 9 (a)

    5 marks

    This assesses solving simultaneous relations involving complex numbers by elimination and a complex quadratic.

    Question paper p.14 · Mark scheme p.14

18. Question 9 (b)

    5 marks

    This assesses representing combined modulus, argument, and real-part restrictions as loci on an Argand diagram.

    Question paper p.15 · Mark scheme p.14

19. Question 10 (a)

    4 marks

    This assesses forming a related-rates differential equation for the changing depth of water in a hemispherical tank.

    Question paper p.16 · Mark scheme p.15

20. Question 10 (b)

    8 marks

    This assesses solving a separable differential equation and using two physical conditions to determine its constants.

    Question paper p.17 · Mark scheme p.15