Complex numbers — AS & A Level Further Mathematics (9231) questions by topic
De Moivre's theorem and root-finding define this topic: finding the roots of z^3 = 27 - 27i in modulus-argument form, using a binomial expansion of (z + 1/z)^7 to express cos^7 theta as a sum of multiple-angle cosines, and showing an expansion for sin 7theta.
Find and show are the two top command words across 17 questions from 2023 to 2025 papers, worth from 5 to 14 marks, where converting cleanly between exponential and multiple-angle cosine or sine forms is the main technical hurdle. With no examiner-report extracts logged for this topic, submitting each root-finding or expansion derivation for instant marking is the way to confirm the argument range and coefficients are exactly right before the final expression is trusted.
May–June 2025, Paper 21
Find the roots of the equation z^3 = 27 - 27i, giving your answers in the form r e^(itheta), where r 0 and -pi < theta <= pi.
Answer this question and get it marked →By considering the binomial expansion of (z - 1/z)^5, where z = cos(theta) + i sin(theta), use de Moivre's theorem to show that cosec^5(theta) = a / (sin(5theta) + b sin(3theta) +…
Answer this question and get it marked →May–June 2025, Paper 22
Find the roots of the equation z^3 = 27 - 27i, giving your answers in the form re^(itheta), where r 0 and -pi < theta <= pi.
Answer this question and get it marked →By considering the binomial expansion of (z - 1/z)^5, where z = cos(theta) + isin(theta), use de Moivre's theorem to show that cosec^5(theta) = a / (sin(5theta) + bsin(3theta) +…
Answer this question and get it marked →May–June 2025, Paper 23
Use de Moivre's theorem to show that sec(5theta) = sec^5(theta) / (5sec^4(theta) - 20sec^2(theta) + 16).
Answer this question and get it marked →May–June 2025, Paper 24
Use de Moivre's theorem to show that sin 7θ = -64 sin^7 θ + 112 sin^5 θ - 56 sin^3 θ + 7 sin θ.
Answer this question and get it marked →May–June 2024, Paper 21
Find the roots of the equation z^3 = -108sqrt(3) + 108i, giving your answers in the form r(cos(theta) + i sin(theta)), where r 0 and 0 < theta < 2pi.
Answer this question and get it marked →May–June 2024, Paper 22
Find the roots of the equation z^3 = -108sqrt(3) + 108i, giving your answers in the form r(cos(theta) + isin(theta)), where r 0 and 0 < theta < 2pi.
Answer this question and get it marked →October–November 2024, Paper 21
By considering the binomial expansion of (z + 1/z)^7, where z = cos theta + i sin theta, use de Moivre's theorem to show that cos^7 theta = a cos 7 theta + b cos 5 theta + c cos 3…
Answer this question and get it marked →October–November 2024, Paper 22
Use de Moivre's theorem to show that cot(6 theta) = (cot^6(theta) - 15 cot^4(theta) + 15 cot^2(theta) - 1) / (6 cot^5(theta) - 20 cot^3(theta) + 6 cot(theta)).
Answer this question and get it marked →October–November 2024, Paper 23
By considering the binomial expansion of (z + 1/z)^7, where z = cos(theta) + i sin(theta), use de Moivre's theorem to show that cos^7(theta) = a cos(7 theta) + b cos(5 theta) + c…
Answer this question and get it marked →May–June 2023, Paper 21
By considering the binomial expansion of (z + z^(-1))^4, where z = cos(theta) + i sin(theta), use de Moivre's theorem to show that cos^4(theta) = (1/8)(cos(4theta) + 4 cos(2theta)…
Answer this question and get it marked →May–June 2023, Paper 22
By considering the binomial expansion of (z + z^(-1))^4, where z = cos(theta) + isin(theta), use de Moivre's theorem to show that cos^4(theta) = (1/8)(cos(4theta) + 4cos(2theta) +…
Answer this question and get it marked →May–June 2023, Paper 23
By considering the binomial expansions of (z + 1/z)^4 and (z - 1/z)^4, where z = cos(theta) + i sin(theta), use de Moivre's theorem to show that cot^4(theta) = (cos 4theta + a cos…
Answer this question and get it marked →October–November 2023, Paper 21
Find the roots of the equation (z + 5i)^3 = 4 + 4 sqrt(3) i, giving your answers in the form r cos(theta) + i(r sin(theta) - 5), where r 0 and 0 < theta < 2pi.
Answer this question and get it marked →October–November 2023, Paper 22
Use de Moivre's theorem to show that cos 5theta = 16 cos^5 theta - 20 cos^3 theta + 5 cos theta.
Answer this question and get it marked →October–November 2023, Paper 23
Find the roots of the equation (z + 5i)^3 = 4 + 4 sqrt(3) i, giving your answers in the form r cos(theta) + i(r sin(theta) - 5), where r 0 and 0 < theta < 2pi.
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