The complex number $z$ is given by $(\surd 3){{\mathrm{e}}^{\mathrm{i}\left( \frac{1}{6}\mathrm{ }\!

Question

The complex number $z$ is given by $(\surd 3){{\mathrm{e}}^{\mathrm{i}\left( \frac{1}{6}\mathrm{ }\!\!\pi\!\!\mathrm{ } \right)}}.$ Given that $w=(1+\mathrm{i})z,$ find $\left| w \right|$ and $\mathrm{arg }w$ in exact form. [2] Without using a calculator, find the smallest positive integer $n$ such that ${{w}^{n}}$ is purely imaginary. State the modulus of ${{w}^{n}}$ when $n$ takes this value. [4]

Accepted Answer

(i) $ |w| = \sqrt{6} $, $ \arg w = \frac{5\pi}{12} $; (ii) $ n = 6 $, $ |w^n| = 216 $

2015 DHS P1 Q4

2015 DHS P1 Q4

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