QCE Specialist Maths practice questions

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1. Paper 1 · Multiple choice · 1 mark · Further complex numbers

   In polar form with the principal argument, $(\sqrt{3} + i)^{4}$ is
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2. Paper 1 · Short response · 7 marks · Further complex numbers

   Let $z = a + b i$, where $a$ and $b$ are real and not both zero.
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3. Paper 2 · Multiple choice · 1 mark · Further complex numbers

   The fifth roots of unity are the vertices of a regular pentagon in the complex plane. Given that $\sin\left(\frac{2 \pi}{5}\right) = 0.9511$, the area of the pentagon, correct to two decimal places, is
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4. Paper 2 · Short response · 8 marks · Further complex numbers

   A drawing program makes a spiral of points in the complex plane. It starts at $z_{0} = 10$ and makes each new point by multiplying the previous one by $u = 0.9\,\mathrm{cis}(0.5)$, so $z_{n} = u\,z_{n - 1}$ for $n \ge 1$ and $z_{n} = 10 u^{n}$.
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