Mathematics 9709 · AS & A Level · Complex numbers

Complex numbers — practice question

Let the complex numbers $u$ and $w$ be given by $u = 2e^{\frac{1}{4}\pi i}$ and $w = 3e^{\frac{1}{3}\pi i}$.
(a)[3]

Find $\frac{u^2}{w}$, and write your answer in the form $re^{i\theta}$, where $r > 0$ and $-\pi < \theta \leq \pi$. State the exact values of $r$ and $\theta$.

(b)[1]

State the smallest positive integer $n$ for which both $\operatorname{Im}\, w^n = 0$ and $\operatorname{Re}\, w^n > 0$.

Worked solution & mark scheme

This 4-mark question has a full step-by-step worked solution and mark scheme. One marking point: State or deduce $u^2 = 4e^{\frac{1}{2}i\pi}$

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