9709 Pure Mathematics 3 Oct/Nov 2023 Paper 33

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

1. Question 1

   4 marks · 4/75

   Solving an exponential inequality involving an absolute value and presenting the interval to the required accuracy.

   Examiner report: A direct two-sided inequality was usually more efficient than squaring. Common faults were mishandling the two signs, using invalid logarithm rules, leaving the bounds as separate statements, and failing to apply the requested precision.

   Question paper p.3 · Mark scheme p.6

2. Question 2

   5 marks · 9/75

   Interpreting complex-number inequalities as geometric loci and shading their common region on an Argand diagram.

   Examiner report: Some diagrams lacked a clear scale or were too small, and several candidates treated the distance-comparison locus as a circle. The required side of the perpendicular bisector was also frequently selected incorrectly; accurate construction and clear boundary labelling would have helped.

   Question paper p.4 · Mark scheme p.7

3. Question 3

   5 marks · 14/75

   Using polynomial evaluations at divisor roots to determine two unknown coefficients from given remainders.

   Examiner report: The intended evaluation method generally worked well, although sign and arithmetic slips were common. Some candidates used unnecessarily complicated division or misunderstood how the stated remainder is obtained from the corresponding root.

   Question paper p.5 · Mark scheme p.8

4. Question 4

   5 marks · 19/75

   Solving a quadratic equation with complex coefficients and expressing both roots in Cartesian complex form.

   Examiner report: Choosing the quadratic formula was usually the most successful route. Errors often arose from copying the constant coefficient, handling the minus sign in the formula, or expanding complex products; some otherwise correct solutions failed to present the roots in the requested format.

   Question paper p.6 · Mark scheme p.10

5. Question 5

   6 marks · 25/75

   Finding all stationary points of a function defined as an exponential quotient and giving their coordinates exactly.

   Examiner report: The quotient method was common and was often set up correctly, but forgetting the denominator square or mishandling the exponential derivative caused losses. Cancelling a factor of x removed one stationary point for some candidates, while others gave decimal rather than exact coordinates.

   Question paper p.7 · Mark scheme p.12

6. Question 6 (a)

   3 marks · 28/75

   Rewriting a trigonometric equation in terms of powers of sine and reducing it to the specified polynomial equation.

   Examiner report: This part was handled successfully by most candidates. The clearest solutions made the identity substitutions explicitly and avoided large unexplained jumps between lines.

   Question paper p.8 · Mark scheme p.13

7. Question 6 (b)

   3 marks · 31/75

   Solving a trigonometric equation over a full angular cycle after reducing it to a quadratic in sin²θ.

   Examiner report: Most candidates found the admissible squared-sine value, but some stopped after the first two angles or omitted the angles associated with negative sine. The complete set requires considering all four quadrants where appropriate.

   Question paper p.9 · Mark scheme p.14

8. Question 7 (a)

   3 marks · 34/75

   Implicit differentiation of an equation containing both x and y to obtain a formula for dy/dx.

   Examiner report: Most candidates recognised the need for implicit differentiation. A recurring presentation problem was deleting the constant term without showing that its derivative is zero, rather than demonstrating why it disappears.

   Question paper p.10 · Mark scheme p.14

9. Question 7 (b)

   5 marks · 39/75

   Locating points on an implicitly defined curve where the tangent is horizontal.

   Examiner report: The main omission was losing the x=0 case. Some candidates also imposed a zero denominator as an additional condition, or rejected valid zero and negative y-values instead of retaining all points produced by the curve equation.

   Question paper p.11 · Mark scheme p.15

10. Question 8

    7 marks · 46/75

    Solving a separable first-order differential equation using an initial condition.

    Examiner report: Recognition that the reciprocal cosine squared becomes a secant-squared expression was important. Frequent coefficient errors involved the factors from integrating with respect to 3y or 4x, and a few candidates used the wrong tangent argument or omitted the integration constant.

    Question paper p.12 · Mark scheme p.16

11. Question 9 (a)

    5 marks · 51/75

    Decomposing a rational function into partial fractions with a linear numerator over the quadratic factor.

    Examiner report: Most candidates obtained full credit, but some paired numerators with the wrong factors or used a constant where a linear numerator was required. Correct structure was essential before the coefficient calculations could earn full credit.

    Question paper p.14 · Mark scheme p.17

12. Question 9 (b)

    5 marks · 56/75

    Obtaining a Maclaurin expansion of a rational function by expanding its partial-fraction components and combining terms through degree three.

    Examiner report: This part exposed many coefficient and sign errors. Typical problems included using the wrong scaled variable, losing factors of 2, mishandling the cubic sign, and making mistakes when multiplying the component series by their numerators.

    Question paper p.15 · Mark scheme p.18

13. Question 9 (c)

    1 mark · 57/75

    Determining the exact interval in which the power-series expansion is valid.

    Examiner report: This was difficult for many candidates. Some stopped with an absolute-value condition instead of converting it into the single required inequality.

    Question paper p.15 · Mark scheme p.19

14. Question 10 (a)

    4 marks · 61/75

    Finding the equation of a tangent to a trigonometric product curve at a specified point.

    Examiner report: The product rule was generally understood, although a few candidates differentiated the factors incorrectly. Others found only the gradient or wrote the normal instead of completing the tangent equation.

    Question paper p.16 · Mark scheme p.19

15. Question 10 (b)

    5 marks · 66/75

    Evaluating exactly the area bounded by a curve and the x-axis using integration by parts.

    Examiner report: Many solutions were sound and used the correct bounds. Marks were sometimes lost when the integration-by-parts working was omitted or when an exact result was replaced by a decimal.

    Question paper p.17 · Mark scheme p.20

16. Question 11 (a)

    2 marks · 68/75

    Normalising a three-dimensional direction vector to produce a unit vector.

    Examiner report: This short part caused disproportionate difficulty. Some candidates found the magnitude but did not use it to normalise the direction vector, while others wrote a general point instead of a unit direction.

    Question paper p.18 · Mark scheme p.20

17. Question 11 (b)

    2 marks · 70/75

    Constructing a vector equation for the line through two given points in three dimensions.

    Examiner report: A valid direction was often found, but the final equation was not always written as a position-vector equation. Confusion between a point, a direction vector, and a general line parameter caused many lost accuracy marks.

    Question paper p.18 · Mark scheme p.21

18. Question 11 (c)

    5 marks · 75/75

    Classifying two three-dimensional lines by testing their directions and checking whether they have a common point.

    Examiner report: Candidates were more comfortable with the component equations here, but parallelism was often not addressed or was rejected using an invalid justification. Sign errors in the direction vectors frequently led to incorrect parameter values and conclusions.

    Question paper p.19 · Mark scheme p.21