The complex number $w$ is given by ${{\left( \frac{-\surd 3+\mathrm{i}}{\surd 2-\mathrm{i}\surd 2} \

Question

The complex number $w$ is given by ${{\left( \frac{-\surd 3+\mathrm{i}}{\surd 2-\mathrm{i}\surd 2} \right)}^{2}}$. Without using a calculator, find $\left| w \right|$ and the exact value of $\arg w$, [4] the set of values of $n$, where $n$ is a positive integer, for which ${{w}^{n}}{{w}^{*}}$ is a real number. [4]

Accepted Answer

(i) $ |w| = 1 $, $ \arg w = -\frac{\pi}{6} $; (ii) $ n = 1 + 6k $ where $ k = 0, 1, 2, \ldots $

2015 IJC P2 Q4

2015 IJC P2 Q4

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