Complex numbers: past paper questions

Where the questions come from

• Question 7May/June 2026, paper 21Structured10 marks

  State the sum of the series $z+z^2+z^3+\dots +z^n$, for $z\mathrm{\ne }1$.

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• Question 8Oct/Nov 2025, paper 21Structured12 marks

  State the sum of the series $z+z^2+\dots +z^n$, for $z\mathrm{\ne }1$.

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• Question 8Oct/Nov 2024, paper 21Structured14 marks

  By considering the binomial expansion of (z + 1/z )⁷, where z = cosθ + i sinθ, use de Moivre's theorem to show that cos⁷θ = a cos 7θ + b cos 5θ + c cos 3θ + d…

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• Question 2Oct/Nov 2023, paper 21Structured5 marks

  Find the roots of the equation $(z+5\mathrm{i}{)}^3=4+4\sqrt{3}\mathrm{i}$, giving your answers in the form $r\cos \theta +\mathrm{i}(r\sin \theta -5)$, where $r>0$ and $0<\theta <2\pi$.

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• Question 8Oct/Nov 2023, paper 21Structured15 marksHas a diagram

  State the sum of the series $1+z+z^2+\dots +z^{n-1}$, for $z\mathrm{\ne }1$.

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• Question 4Oct/Nov 2025, paper 22Structured8 marks

  By considering the binomial expansion of (z + 1/z )⁶, where z = cosθ + i sinθ, use de Moivre's theorem to show that cos⁶θ = a cos 6θ + b cos 4θ + c cos 2θ + d…

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• Question 1May/June 2024, paper 21Structured5 marks

  Find the roots of the equation $z^3=-108\sqrt{3}+108\mathrm{i}$, giving your answers in the form $r(\cos \theta +\mathrm{i}\sin \theta )$, where $r>0$ and $0<\theta <2\pi$.

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• Question 3May/June 2025, paper 21Structured6 marksHas a diagram

  By considering the binomial expansion of (z - 1/z )⁵, where z = cosθ + i sinθ, use de Moivre's theorem to show that cosec⁵θ = a/(sin 5θ + b sin 3θ + c sinθ)…

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• Question 3May/June 2023, paper 21Structured8 marks

  By considering the binomial expansion of $(z+z^{-1}{)}^4$, where $z=\cos \theta +\mathrm{i}\sin \theta$, use de Moivre's theorem to show that ${\cos }^4\theta =\frac{1}{8}(\cos 4\theta +4\cos 2\theta +3)$.

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• Question 3May/June 2025, paper 22Structured6 marksHas a diagram

  By considering the binomial expansion of (z - 1/z )⁵, where z = cosθ + i sinθ, use de Moivre’s theorem to show that cosec⁵θ = a/(sin 5θ + b sin 3θ + c sinθ)…

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