9709 Pure Mathematics 3 Feb/Mar 2023 Paper 32

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

1. Question 1

   3 marks · 3/75

   This tests the use of logarithm laws and algebraic rearrangement to make a variable the subject of an equation.

   Examiner report: Most candidates handled this successfully, but some incorrectly treated the logarithm of a sum or difference as separate logarithms.

   Question paper p.2 · Mark scheme p.6

2. Question 2 (a)

   3 marks · 6/75

   This tests the representation of argument inequalities and Cartesian restrictions on an Argand diagram.

   Examiner report: Many diagrams had incorrect origins, full lines instead of rays, wrong symmetry, or an incorrectly placed vertical boundary; unequal or missing scales were also frequent problems.

   Question paper p.3 · Mark scheme p.6

3. Question 2 (b)

   2 marks · 8/75

   This tests the geometric determination of an extreme argument from a region on an Argand diagram.

   Examiner report: Very few candidates made progress here, usually because the required point on the lower boundary and the vertical edge had not been identified correctly in part (a).

   Question paper p.3 · Mark scheme p.7

4. Question 3

   5 marks · 13/75

   This tests polynomial division or coefficient comparison when a polynomial remainder is prescribed.

   Examiner report: Unknown coefficients made long division difficult for many candidates. Common errors included multiplying the divisor by the remainder and using the real zero of the remainder instead of a zero of the divisor.

   Question paper p.4 · Mark scheme p.7

5. Question 4

   5 marks · 18/75

   This tests solving a complex equation involving a conjugate by separating its real and imaginary components.

   Examiner report: Sign slips, omitted terms, and failure to set both components to zero prevented many candidates from reaching the final solutions.

   Question paper p.5 · Mark scheme p.9

6. Question 5 (a)

   3 marks · 21/75

   This tests differentiation of parametric equations using the product rule and the chain rule relationship between derivatives.

   Examiner report: Many candidates were successful, although some differentiated only the exponential factor in x or did not apply the quotient of parametric derivatives correctly.

   Question paper p.6 · Mark scheme p.10

7. Question 5 (b)

   3 marks · 24/75

   This tests finding a normal to a parametric curve at a specified parameter value and verifying a point on that normal.

   Examiner report: Candidates often found the curve point but forgot to invert and change the sign of the tangent gradient when obtaining the normal.

   Question paper p.7 · Mark scheme p.10

8. Question 6 (a)

   3 marks · 27/75

   This tests the conversion of a linear combination of sine and cosine into a single shifted cosine.

   Examiner report: Some candidates quoted a formula without applying the coefficient matching correctly, while others gave decimal rather than exact values or interchanged the relevant trigonometric ratios.

   Question paper p.8 · Mark scheme p.11

9. Question 6 (b)

   4 marks · 31/75

   This tests solving a trigonometric equation after applying an auxiliary-angle form and selecting every solution in a specified interval.

   Examiner report: A frequent omission was the second solution. Others solved for x while still treating the angle as though it were x rather than 2x.

   Question paper p.9 · Mark scheme p.11

10. Question 7 (a)

    4 marks · 35/75

    This tests modelling sector and segment areas and simplifying the resulting trigonometric equation.

    Examiner report: Candidates who knew how to begin generally completed the argument, but others lost factors of one-half, used the wrong total angle for the major sector, or failed to simplify the triangle area appropriately.

    Question paper p.10 · Mark scheme p.12

11. Question 7 (b)

    2 marks · 37/75

    This tests locating a root numerically by demonstrating a change of sign or equivalent endpoint inequalities.

    Examiner report: This was generally answered well, though a small number of candidates left it blank or did not complete the numerical sign-change argument.

    Question paper p.11 · Mark scheme p.12

12. Question 7 (c)

    3 marks · 40/75

    This tests fixed-point iteration and the presentation of sufficient numerical evidence for rounding a root.

    Examiner report: Most candidates performed well, but some stopped too early and claimed the required accuracy before their iterates supported it.

    Question paper p.11 · Mark scheme p.13

13. Question 8 (a)

    4 marks · 44/75

    This tests differentiation of a product involving a logarithm and the exact determination of a stationary minimum.

    Examiner report: The differentiation was usually done correctly. The main difficulties were abandoning exact form and mishandling the negative logarithmic y-coordinate.

    Question paper p.12 · Mark scheme p.13

14. Question 8 (b)

    5 marks · 49/75

    This tests finding an exact area under a logarithmic curve using integration by parts and definite limits.

    Examiner report: Integration by parts was often well executed, but many candidates failed to account for the fact that the graph lies below the axis over part of the bounded region.

    Question paper p.13 · Mark scheme p.14

15. Question 9

    7 marks · 56/75

    This tests separation of variables, integration of an exponential in y and a squared trigonometric function in x, and use of an initial condition.

    Examiner report: Separation itself was usually understood, but many candidates could not correctly convert the exponential and squared-sine factors into integrable forms, with sign errors in the trigonometric identity also appearing.

    Question paper p.14 · Mark scheme p.15

16. Question 10 (a)

    3 marks · 59/75

    This tests the scalar-product formula for the obtuse angle between two three-dimensional vectors.

    Examiner report: Many candidates gained full credit, although some made elementary errors with magnitudes or selected the acute rather than the obtuse angle.

    Question paper p.16 · Mark scheme p.16

17. Question 10 (b)

    2 marks · 61/75

    This tests constructing a vector equation for a line through two points in three dimensions.

    Examiner report: The direction vector was usually understood, but arithmetic slips and incorrect notation for the vector on the left-hand side were common.

    Question paper p.16 · Mark scheme p.16

18. Question 10 (c)

    4 marks · 65/75

    This tests finding the intersection of two three-dimensional lines given in vector form.

    Examiner report: Most candidates knew the procedure, but copying and arithmetic errors were frequent. Checking both parameters and the resulting point would have exposed many of these mistakes.

    Question paper p.17 · Mark scheme p.16

19. Question 11 (a)

    5 marks · 70/75

    This tests decomposition of a rational function into partial fractions with a quadratic and a linear factor in the denominator.

    Examiner report: This part was generally handled confidently, though some candidates used an inappropriate constant numerator over the quadratic factor or an unnecessary linear numerator over the linear factor.

    Question paper p.18 · Mark scheme p.17

20. Question 11 (b)

    5 marks · 75/75

    Question paper p.19 · Mark scheme p.17