9709 Pure Mathematics 3 May/June 2023 Paper 33

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

1. Question 1

   4 marks · 4/75

   This assesses solving a logarithmic equation by applying logarithm laws and then obtaining a numerical root.

   Examiner report: Most candidates manipulated the logarithms successfully. The usual lost mark came from failing to present the final numerical value with the stipulated rounding.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   3 marks · 7/75

   This assesses polynomial division and the identification of both quotient and remainder.

   Examiner report: Long division was the common approach and was generally successful. Some candidates made a sign error in the linear remainder after subtraction.

   Question paper p.3 · Mark scheme p.7

3. Question 3

   4 marks · 11/75

   This assesses interpreting modulus inequalities geometrically on an Argand diagram and shading their common region.

   Examiner report: The main weaknesses were incomplete axis scaling, replacing the straight-line boundary by a circle, and shading the wrong side of the boundary. Accurate diagrams with suitable scales usually gained all the available marks.

   Question paper p.4 · Mark scheme p.8

4. Question 4

   5 marks · 16/75

   This assesses differentiation of parametric equations and simplification of the ratio of the two parameter derivatives.

   Examiner report: Most candidates handled the differentiation well, but some failed to differentiate the parameter term or changed the parameter to another symbol during the work. A complete derivation was important because the question required a demonstration rather than only the final result.

   Question paper p.5 · Mark scheme p.9

5. Question 5 (a)

   3 marks · 19/75

   This assesses applying the product rule to a trigonometric product and using the stationary-point condition at the maximum.

   Examiner report: The product rule was usually applied correctly. Errors arose from mishandling the factor introduced by differentiating the multiple angle, omitting the zero-derivative condition, or confusing an inverse tangent with a reciprocal.

   Question paper p.6 · Mark scheme p.10

6. Question 5 (b)

   3 marks · 22/75

   This assesses fixed-point iteration for a nonlinear equation and judging convergence to a specified accuracy.

   Examiner report: This was generally answered successfully. The principal procedural mistake was evaluating the trigonometric functions in degrees rather than radians; many candidates displayed more iterations than strictly necessary.

   Question paper p.7 · Mark scheme p.10

7. Question 6 (a)

   4 marks · 26/75

   This assesses rewriting a sum of shifted cosine functions as a single cosine with amplitude and phase shift.

   Examiner report: Most candidates reached the correct cosine and sine coefficients. Frequent slips involved the sign of the sine term, the value of the sine of the angle used in the expansion, and leaving the cosine of the standard angle unevaluated.

   Question paper p.8 · Mark scheme p.11

8. Question 6 (b)

   4 marks · 30/75

   This assesses solving a trigonometric equation using the amplitude-phase form and finding every solution in a restricted interval.

   Examiner report: A common omission was the second admissible solution. Other errors came from applying the phase shift to the wrong quantity, treating the doubled angle as the original angle, or mixing degree and radian measures.

   Question paper p.9 · Mark scheme p.12

9. Question 7 (a)

   4 marks · 34/75

   Question paper p.10 · Mark scheme p.13

10. Question 7 (b)

    4 marks · 38/75

    Question paper p.11 · Mark scheme p.14

11. Question 8

    8 marks · 46/75

    This assesses solving a separable differential equation with a rational expression in the dependent variable and applying an initial condition.

    Examiner report: Many complete solutions were seen, but some stopped after separating variables or integrating only the reciprocal of the independent variable. The arctangent component was sometimes misidentified, and a few otherwise sound solutions failed to simplify the final logarithmic equation sufficiently.

    Question paper p.12 · Mark scheme p.15

12. Question 9 (a)

    4 marks · 50/75

    This assesses using vector direction conditions and a known point to determine unknown constants in a line equation.

    Examiner report: Many solutions were efficient and correct. A recurring error was to assume that the two lines intersected and equate them, which imposes a condition not supplied in this part and can lead to using a different point on the line.

    Question paper p.14 · Mark scheme p.17

13. Question 9 (b)

    6 marks · 56/75

    This assesses finding the foot of a perpendicular from a point to a line and using a directed section ratio to locate another point.

    Examiner report: This was substantially more difficult for candidates. Typical mistakes involved reversing vectors, interchanging the section ratios, mishandling squared lengths, and calculating a vector to the required point without converting it into the position vector from the origin.

    Question paper p.15 · Mark scheme p.18

14. Question 10 (a)

    5 marks · 61/75

    This assesses setting up and evaluating a partial-fraction decomposition with one linear factor and a repeated linear factor.

    Examiner report: The majority obtained the decomposition correctly. Errors included using an unsuitable fraction structure, multiplying by an incorrect power of a denominator, and attempting to invert the factors term by term instead of clearing the common denominator.

    Question paper p.16 · Mark scheme p.20

15. Question 10 (b)

    5 marks · 66/75

    This assesses expanding partial fractions by standard series and combining terms through the quadratic power of the variable.

    Examiner report: Most candidates could begin the relevant expansions. The main difficulties were extracting the correct powers of the constant denominator and simplifying the squared terms without changing their coefficients.

    Question paper p.17 · Mark scheme p.21

16. Question 11 (a)

    6 marks · 72/75

    This assesses rationalising a complex quotient, using an argument condition to determine an integer parameter, and expressing the result in Cartesian form.

    Examiner report: Many candidates recognised the need for Cartesian form, but mistakes in the conjugate multiplication produced incorrect denominator or numerator terms. Some retained an inadmissible second parameter value or used the wrong sign when interpreting the argument condition.

    Question paper p.18 · Mark scheme p.24

17. Question 11 (b)

    3 marks · 75/75

    Question paper p.19 · Mark scheme p.26