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A-LevelFurther MathematicsComplex numbers9231/12 · Oct/Nov 2019 · Q911 Marks

(i) Use de Moivre's theorem to show that sec 6θ = sec^6 θ / (32 - 48 sec² θ + 18 sec⁴ θ – sec^6 θ)

  1. (i)Use de Moivre's theorem to show that sec 6θ = sec^6 θ / (32 - 48 sec² θ + 18 sec⁴ θ – sec^6 θ)[6]
  2. (ii)Hence obtain the roots of the equation 3x^6 - 36x^4 + 96x^2 – 64 = 0 in the form sec qπ, where q is rational.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

  1. [M1] Uses binomial theorem.
  2. [A1]
  3. [M1] Uses c² = 1-s².
  4. [A1]
  5. [M1] Divides numerator and denominator by c^6.
  6. [A1] AG
  7. [M1 A1] Relates with equation in part (i).
  8. [M1] Solves cos 6θ = 1/2.
  9. [A1] Gives one correct solution.
  10. [A1] Gives five other solutions. Allow different values of q as long as all six solutions are found.

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About this question

Topic
Complex numbers, A-Level Further Mathematics (syllabus 9231). Structured question, 11 marks.
Source
A-Level Further Mathematics, Oct/Nov 2019 Paper 1 Variant 2, question 9.

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