The complex number $z$ is given by $z=r{{e}^{\mathrm{i} heta }}$, where $r0$ and $0\le heta \le

Question

The complex number $z$ is given by $z=r{{e}^{\mathrm{i}	heta }}$, where $r>0$ and $0\le 	heta \le \frac{1}{2}\pi $. Given that $w=\left( 1-\mathrm{i}\sqrt{3} ight)z$, find $\left| w ight|$ in terms of $r$ and $\arg w$ in terms of $	heta $. [2] Given that $\arg \left( \frac{{{z}^{10}}}{{{w}^{2}}} ight)=\pi $, find $	heta $. [3]

Accepted Answer

(i) $\left| w ight|=2r$, $\arg w=	heta -\frac{\pi }{3}$ (iii) $	heta =\frac{\pi }{24}$, $\frac{7\pi }{24}$ 	ext{ or } alternatively, $	heta =\frac{\pi }{3}$, $\frac{7\pi }{24}$

N2013 P1 Q8

N2013 P1 Q8

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