9709 Pure Mathematics 3 Oct/Nov 2021 Paper 32

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

1. Question 1

   4 marks · 4/75

   This assesses the use of logarithms to solve an exponential equation and present the result in a specified logarithmic form.

   Examiner report: Many candidates knew that logarithms were needed, but some made an invalid manipulation of the powers. Others stopped after forming a logarithmic equation or incorrectly cancelled logarithmic terms instead of completing the algebra.

   Question paper p.3 · Mark scheme p.6

2. Question 2

   4 marks · 8/75

   This assesses solving a strict inequality involving two modulus expressions and a positive parameter.

   Examiner report: Squaring was the usual approach, but errors often came from mishandling the factor 2 or expanding brackets. Some candidates found the quadratic but then investigated its discriminant instead of solving for the variable, while others rejected a valid boundary because the parameter was positive.

   Question paper p.4 · Mark scheme p.7

3. Question 3 (a)

   2 marks · 10/75

   This assesses a direct proof that complex conjugation preserves addition.

   Examiner report: Many responses showed uncertainty about the conjugate notation or made unsupported assertions. A common weakness was failing to work out the two sides independently before comparing them.

   Question paper p.5 · Mark scheme p.8

4. Question 3 (b)

   4 marks · 14/75

   This assesses solving a complex equation containing a conjugate by using real and imaginary components.

   Examiner report: Only a minority clearly used the representation in real and imaginary parts and connected it with the preceding result. Some equated the components with the original variables incorrectly, while others tried to manipulate the equation directly in the complex unknown.

   Question paper p.5 · Mark scheme p.8

5. Question 4

   5 marks · 19/75

   This assesses partial-fraction decomposition when the rational expression first requires polynomial division.

   Examiner report: Those who noticed the fraction was not proper often succeeded. A frequent error was to omit the polynomial part and begin with an unsuitable two-fraction form; some completed the division but did not decompose the remaining fraction.

   Question paper p.6 · Mark scheme p.9

6. Question 5 (a)

   4 marks · 23/75

   This assesses interpreting modulus and imaginary-part inequalities as loci on an Argand diagram.

   Examiner report: Fully correct diagrams were seen, but inconsistent axis scales sometimes distorted the circle. Other errors included placing the circle in the wrong quadrant or drawing the wrong horizontal boundary, which consequently led to incorrect shading.

   Question paper p.7 · Mark scheme p.10

7. Question 5 (b)

   3 marks · 26/75

   This assesses finding the greatest polar argument of points in a shaded Argand region using tangent geometry.

   Examiner report: This part was frequently left blank. Some otherwise promising attempts used the wrong inverse-trigonometric ratio, and answers in radians could not receive the final accuracy mark.

   Question paper p.7 · Mark scheme p.10

8. Question 6 (a)

   3 marks · 29/75

   This assesses deriving a product-to-sum identity from the sine addition and subtraction formulae.

   Examiner report: Most candidates began with a suitable addition expansion and followed the intended route. A notable minority tried to relate the two sine expressions by an invalid equality instead of applying the standard formulae.

   Question paper p.8 · Mark scheme p.11

9. Question 6 (b)

   3 marks · 32/75

   This assesses using a previously established trigonometric identity to evaluate a definite integral exactly.

   Examiner report: Many candidates did not exploit the identity supplied in the preceding part and instead attempted integration by parts without success. Other losses arose from incorrect coefficients, reversed or mishandled limits, and omitting enough intermediate work to support a given result.

   Question paper p.9 · Mark scheme p.11

10. Question 7

    7 marks · 39/75

    This assesses solving a first-order separable differential equation with an initial condition, including an integration by parts step.

    Examiner report: Some candidates applied the initial condition before obtaining a usable general relation and then could not progress. Common technical problems were the integral involving the dependent variable, numerical factors, the sign in the exponential, and taking the final reciprocal incorrectly.

    Question paper p.10 · Mark scheme p.12

11. Question 8 (a)

    3 marks · 42/75

    This assesses proving a fourth-power trigonometric identity by expanding a squared Pythagorean relation.

    Examiner report: Many candidates did not use the fact that the squared sum equals one and instead set it equal to zero. Others reached the double-angle stage through incomplete or incorrect algebra.

    Question paper p.12 · Mark scheme p.13

12. Question 8 (b)

    4 marks · 46/75

    This assesses solving a trigonometric equation obtained from the preceding identity over a specified degree interval.

    Examiner report: Although most attempted to use the earlier result, errors commonly arose when the double angle was reduced to a single angle or when the square on the sine was misinterpreted. Many found one or two values but did not list all four, and rounding was sometimes inaccurate.

    Question paper p.13 · Mark scheme p.13

13. Question 9 (a)

    4 marks · 50/75

    This assesses implicit differentiation of an equation containing products of functions of both variables.

    Examiner report: The intended method was generally recognised, but signs and brackets were frequent sources of error. Some candidates wrote the supplied result without showing a valid derivation, while weaker responses showed little familiarity with implicit differentiation.

    Question paper p.14 · Mark scheme p.14

14. Question 9 (b)

    4 marks · 54/75

    This assesses locating a point where an implicitly defined curve has a vertical tangent.

    Examiner report: A substantial number of candidates searched for a horizontal tangent or set the gradient equal to one. Those who correctly used the denominator condition generally progressed well, although leaving a logarithmic expression unsimplified made the coordinate calculation harder.

    Question paper p.15 · Mark scheme p.15

15. Question 10 (a)

    3 marks · 57/75

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

    Examiner report: Most candidates knew to construct a direction vector, but sign errors in one component were common. Some used the right ingredients but wrote an expression for the line rather than an equation for its position vector.

    Question paper p.16 · Mark scheme p.15

16. Question 10 (b)

    2 marks · 59/75

    This assesses using a scalar multiple of a displacement vector to find a point's position vector.

    Examiner report: Many candidates found the required displacement but stopped there. The missing step was usually adding that displacement to the position vector of the first point to obtain the position vector of the new point.

    Question paper p.16 · Mark scheme p.15

17. Question 10 (c)

    5 marks · 64/75

    This assesses finding the intersections of a three-dimensional line with a sphere centred at the origin.

    Examiner report: Some candidates calculated unrelated vector lengths without forming a general point on the line. Others obtained the component expression but failed to use the prescribed distance; when a correct quadratic was formed, most candidates completed the remaining work successfully.

    Question paper p.17 · Mark scheme p.16

18. Question 11 (a)

    4 marks · 68/75

    This assesses differentiating a square root of a tangent function by the chain rule and evaluating the derivative at a specified angle.

    Examiner report: Candidates who began with the chain rule usually obtained a usable derivative. Some did not convert it into the requested tangent form, and others substituted the angle but provided no calculation demonstrating the claimed value.

    Question paper p.18 · Mark scheme p.16

19. Question 11 (b)

    4 marks · 72/75

    This assesses converting the condition that the derivative equals one into a polynomial equation and reducing it to the required cubic.

    Examiner report: Many candidates tried to manipulate the cubic stated in the question rather than deriving it from the derivative condition. A more effective route was to square the derivative equation first and then reduce the resulting quartic.

    Question paper p.18 · Mark scheme p.17

20. Question 11 (c)

    3 marks · 75/75

    This assesses numerical solution of the equation by fixed-point iteration with a prescribed formula.

    Examiner report: The starting value was not specified, so several choices can work, but some were taken outside the relevant graph interval. Other difficulties included using degree mode, entering the iteration incorrectly, and rounding a convergent sequence improperly.

    Question paper p.19 · Mark scheme p.17