9709 Pure Mathematics 3 May/June 2021 Paper 31

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

1. Question 1

   4 marks · 4/75

   Question paper p.2 · Mark scheme p.6

2. Question 2

   5 marks · 9/75

   This tests substitution of an exponential expression into a quadratic, selection of the admissible root, and numerical solution with a uniqueness justification.

   Examiner report: Many responses began by clearing the denominator, but few then treated the exponential as the quadratic variable. An invalid approach was to split the logarithm of a sum. The rejected quadratic root needed a mathematical explanation rather than being labelled simply as an error.

   Question paper p.3 · Mark scheme p.7

3. Question 3 (a)

   4 marks · 13/75

   This assesses expansion of compound-angle sine and cosine expressions and exact manipulation to obtain a formula for the tangent of the unknown angle.

   Examiner report: Some candidates used an incorrect compound-angle identity, especially for the cosine term. Others reached an equation in sine and cosine but could not convert it into a statement about the tangent.

   Question paper p.4 · Mark scheme p.8

4. Question 3 (b)

   2 marks · 15/75

   This tests using a previously obtained tangent value to find every solution of the trigonometric equation in a specified degree interval.

   Examiner report: Several candidates stopped at an equivalent negative-angle result or gave only one solution. Starting again from the original equation generally did not help, whereas linking directly to part (a) was the effective route.

   Question paper p.4 · Mark scheme p.8

5. Question 4 (a)

   2 marks · 17/75

   This assesses proof of a trigonometric identity using double-angle formulae.

   Examiner report: The shortest successful proofs used the two standard double-angle rearrangements. Missing brackets or incorrect signs in those substitutions caused otherwise promising solutions to fail.

   Question paper p.5 · Mark scheme p.8

6. Question 4 (b)

   4 marks · 21/75

   This tests applying a proved identity to a definite integral and evaluating the resulting exact trigonometric antiderivative.

   Examiner report: The definite integral proved difficult for many candidates, particularly when the identity from part (a) was not used. Even after choosing the tangent-square route, only some handled the antiderivative and endpoint substitution accurately.

   Question paper p.5 · Mark scheme p.9

7. Question 5 (a)

   2 marks · 23/75

   This assesses solving a quadratic with complex coefficients or roots by applying the quadratic formula and simplifying powers of i.

   Examiner report: Many candidates left the whole question blank. Among those attempting this part, the main difficulty was simplifying the expression under the square root without sign or factor errors.

   Question paper p.6 · Mark scheme p.9

8. Question 5 (b)

   2 marks · 25/75

   This tests interpreting the locations of complex roots on the Argand diagram and translating that condition into a discriminant restriction.

   Examiner report: Only a small number of candidates recognised that roots on the imaginary axis require a negative discriminant; this geometric condition was often missed entirely.

   Question paper p.6 · Mark scheme p.9

9. Question 5 (c)

   3 marks · 28/75

   This assesses using the geometry of an equilateral triangle in the Argand plane to relate the parameters in the complex roots.

   Examiner report: Some candidates stated a general equilateral-triangle condition but did not connect it correctly with the particular root coordinates. Very few carried that geometric condition through to a usable algebraic relation.

   Question paper p.7 · Mark scheme p.10

10. Question 6 (a)

    5 marks · 33/75

    This assesses differentiation of parametric equations and analysis of the domain to establish the sign of the curve's gradient.

    Examiner report: Most candidates knew to find both parameter derivatives. A recurrent slip was losing the factor 3 in dx/dt, while stronger solutions also used the restriction imposed by the logarithm to justify the sign.

    Question paper p.8 · Mark scheme p.11

11. Question 6 (b)

    3 marks · 36/75

    This tests finding a tangent to a parametric curve at its intersection with the y-axis.

    Examiner report: Candidates who began by imposing x equal to zero generally made progress through the parameter, point, and gradient calculations. Others were unsure how to begin the tangent construction.

    Question paper p.9 · Mark scheme p.12

12. Question 7 (a)

    4 marks · 40/75

    This assesses differentiating a function involving an inverse tangent and a fractional power, then using the stationary-point condition to obtain a fixed-point equation.

    Examiner report: The stronger solutions differentiated immediately from the maximum-point condition, often using the quotient rule successfully. Later algebraic rearrangement was the main source of slips.

    Question paper p.10 · Mark scheme p.12

13. Question 7 (b)

    2 marks · 42/75

    This tests numerical verification that a root of the fixed-point equation lies between two specified bounds.

    Examiner report: Degree-mode calculations prevented some otherwise valid attempts from working. There was also confusion between checking a fixed-point difference and checking a zero of a rearranged function, so numerical values were sometimes presented without a coherent conclusion.

    Question paper p.11 · Mark scheme p.13

14. Question 7 (c)

    3 marks · 45/75

    This assesses fixed-point iteration and reporting a numerical root to the requested precision.

    Examiner report: The iterative procedure was generally understood, and candidates working in radians commonly reached the correct rounded result.

    Question paper p.11 · Mark scheme p.13

15. Question 8 (a)

    4 marks · 49/75

    This tests finding the angle between a line segment and a line direction in three dimensions using the scalar-product formula.

    Examiner report: The scalar-product method was familiar to candidates who attempted it, although some selected the wrong vector for AB. The line's direction vector was usually identified correctly.

    Question paper p.12 · Mark scheme p.13

16. Question 8 (b)

    5 marks · 54/75

    This assesses using a parameterised line and equal-distance geometry to locate a point equidistant from two given points.

    Examiner report: This was rarely completed. A few candidates formed one of the required displacement vectors, but it was uncommon to see the two squared distances equated and solved.

    Question paper p.13 · Mark scheme p.14

17. Question 9 (a)

    5 marks · 59/75

    This assesses product-rule differentiation of a power times a logarithm and exact determination of the curve's stationary point.

    Examiner report: Many candidates applied the product rule successfully and found the x-coordinate, but index manipulation caused errors. Finding the associated y-coordinate was a further stumbling block.

    Question paper p.14 · Mark scheme p.14

18. Question 9 (b)

    5 marks · 64/75

    Question paper p.15 · Mark scheme p.15

19. Question 10

    11 marks · 75/75

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

    Examiner report: The instruction about partial fractions helped many candidates begin the algebra, but some supplied coefficients without showing how they arose. The reciprocal-square integral was a frequent obstacle; those who integrated correctly usually used the initial condition successfully.

    Question paper p.16 · Mark scheme p.16