The complex numbers $u$ and $v$ are such that $\frac{u-2v}{u+2v}\mathrm{i}$ is a real number. Show t

Question

The complex numbers $u$ and $v$ are such that $\frac{u-2v}{u+2v}\mathrm{i}$ is a real number. Show that $\left| u ight|=2\left| v ight|$. [3] It is given that $z=\cos \left( \frac{3\pi }{16} ight)+\cos \left( \frac{7\pi }{16} ight)+\mathrm{i}\left[ \sin \left( \frac{3\pi }{16} ight)+\sin \left( \frac{7\pi }{16} ight) ight]$. Without using a calculator, express $z$ in the form of $r{{e}^{i	heta }}$, where $r>0$ and $-\pi Hence find the smallest positive integer $n$ such that ${{z}^{n}}=k\left( -1-\mathrm{i} ight)$ for some positive real number $k$. [5]

Accepted Answer

$z = 2\cos\dfrac{\pi}{8} \cdot \mathrm{e}^{\mathrm{i}\frac{5\pi}{16}}$; $n = 4$

2018 HCI JC2 MYE P1 Q5

2018 HCI JC2 MYE P1 Q5

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