9709 Pure Mathematics 3 Feb/Mar 2020 Paper 32

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

1. Question 1 (a)

   1 mark · 1/75

   This assesses recognition and accurate sketching of a graph involving an absolute value.

   Examiner report: Some responses gave only one straight line or produced an incomplete graph. The corner and the correct intercept information were often missing, and several sketches were drawn over too small a range.

   Question paper p.2 · Mark scheme p.6

2. Question 1 (b)

   3 marks · 4/75

   This assesses solving an inequality containing a modulus and a linear expression.

   Examiner report: Although many candidates located the boundary value, fewer stated the resulting interval correctly. Errors also arose when candidates squared only one side or expanded the squared linear expression incorrectly.

   Question paper p.2 · Mark scheme p.6

3. Question 2

   4 marks · 8/75

   This assesses the use of logarithm laws, domain restrictions and quadratic solving in an exact-form equation.

   Examiner report: Typical difficulties included treating a coefficient as an added term when applying a logarithm law, retaining an inadmissible root, and giving a decimal rather than the requested exact result.

   Question paper p.3 · Mark scheme p.7

4. Question 3 (a)

   2 marks · 10/75

   This assesses using graph intersections to establish the existence and uniqueness of a trigonometric equation's root.

   Examiner report: The reciprocal-cosine graph was frequently inaccurate, especially near its asymptote or at the end of the interval. Some line sketches also crossed the horizontal axis at an incorrect location.

   Question paper p.4 · Mark scheme p.7

5. Question 3 (b)

   2 marks · 12/75

   This assesses numerical verification that a root lies within a stated interval.

   Examiner report: This was generally handled successfully by candidates.

   Question paper p.4 · Mark scheme p.7

6. Question 3 (c)

   3 marks · 15/75

   This assesses applying a fixed-point iteration in radians and reporting a root to the required accuracy.

   Examiner report: Most candidates made good progress here; a recurring mistake was leaving the calculator in degree mode instead of using radians.

   Question paper p.4 · Mark scheme p.7

7. Question 4

   7 marks · 22/75

   This assesses integration by parts followed by exact evaluation of a definite trigonometric integral.

   Examiner report: Candidates often completed the first integration step but then lost marks through an incorrect logarithm, mishandled limits, decimal working, or sign errors when removing brackets.

   Question paper p.5 · Mark scheme p.8

8. Question 5 (a)

   4 marks · 26/75

   Question paper p.6 · Mark scheme p.8

9. Question 5 (b)

   3 marks · 29/75

   This assesses solving a trigonometric equation using an identity and restricting the solutions to a specified interval.

   Examiner report: Most candidates used the preceding identity and found at least one solution. Common problems were missing the second solution, using degrees, or mishandling the algebra after applying the double-angle relation.

   Question paper p.7 · Mark scheme p.9

10. Question 6 (a)

    7 marks · 36/75

    This assesses separation of variables, integration involving an inverse tangent, and use of an initial condition in a differential equation.

    Examiner report: Separation was often achieved, but many attempts stalled at the integrations. The coefficient required inside the inverse-tangent expression was a particularly common source of error.

    Question paper p.8 · Mark scheme p.9

11. Question 6 (b)

    1 mark · 37/75

    This assesses finding the limiting behaviour of a function obtained from a differential equation.

    Examiner report: This depended on having an expression from part (a) containing the decaying exponential term. Candidates who had that structure were often able to obtain the follow-through mark.

    Question paper p.9 · Mark scheme p.10

12. Question 7 (a)

    4 marks · 41/75

    This assesses implicit differentiation of a curve containing products and powers of both variables.

    Examiner report: This was one of the stronger parts of the paper. The main slips were sign errors and failing to show the factorisation needed before rearranging the derivative.

    Question paper p.10 · Mark scheme p.10

13. Question 7 (b)

    5 marks · 46/75

    This assesses identifying vertical tangents from an implicit derivative and finding the corresponding points on the curve.

    Examiner report: Some candidates interpreted a vertical tangent as requiring a derivative of zero or one. Others found the proportional branch but overlooked the additional branch arising when the other factor is zero.

    Question paper p.11 · Mark scheme p.10

14. Question 8 (a)

    3 marks · 49/75

    This assesses constructing three-dimensional displacement vectors from cuboid geometry, a division ratio and a midpoint.

    Examiner report: The main errors were incorrect signs in the displacement to the midpoint and failure to use the actual edge lengths when applying the ratios.

    Question paper p.12 · Mark scheme p.11

15. Question 8 (b)

    2 marks · 51/75

    This assesses forming a vector equation for a line through two points in three-dimensional space.

    Examiner report: Some candidates attached the parameter to a position vector instead of the line direction. Others described the line without writing the required vector equation with the position-vector symbol on the left.

    Question paper p.12 · Mark scheme p.11

16. Question 8 (c)

    4 marks · 55/75

    This assesses finding the perpendicular projection of a point onto a line using vectors and a scalar product.

    Examiner report: This was considered one of the more difficult parts. Many candidates used only position vectors from the origin, chose the wrong vector for the perpendicularity condition, or gave the displacement from the given point instead of the foot's position vector.

    Question paper p.13 · Mark scheme p.11

17. Question 9 (a)

    5 marks · 60/75

    This assesses decomposition of a rational function into partial fractions with three distinct linear factors.

    Examiner report: This question was answered well overall. Even candidates using the more laborious simultaneous-equations method often succeeded when their algebra was organised carefully.

    Question paper p.14 · Mark scheme p.12

18. Question 9 (b)

    5 marks · 65/75

    This assesses obtaining a power-series expansion by expanding each partial-fraction term and combining coefficients.

    Examiner report: Most candidates earned substantial credit. The most frequent issue was mishandling the factor involving the denominator with a constant term other than one before applying the series formula.

    Question paper p.15 · Mark scheme p.12

19. Question 10 (a)

    6 marks · 71/75

    This assesses solving simultaneous linear equations whose unknowns are complex numbers.

    Examiner report: This was challenging for many candidates. A common misconception was to treat a complex unknown as real when comparing real and imaginary parts; insufficient written working also made otherwise promising calculator-based attempts unsupported.

    Question paper p.16 · Mark scheme p.13

20. Question 10 (b)(i)

    2 marks · 73/75

    This assesses interpreting a complex modulus equation as a locus on an Argand diagram.

    Examiner report: The circle was often misplaced because one or both signs in its centre were read incorrectly. Unequal axis scales also made the radius and shape impossible to verify reliably.

    Question paper p.17 · Mark scheme p.13

21. Question 10 (b)(ii)

    2 marks · 75/75

    This assesses finding the smallest argument of points on a circular locus using tangent geometry.

    Examiner report: Many candidates stopped after finding the argument of the centre. Others used the wrong side of the tangent triangle, confusing the relevant sine relationship with a tangent calculation.

    Question paper p.17 · Mark scheme p.13