Show that for any complex number $z=r{{\mathrm{e}}^{\mathrm{i} heta }}$, where $r0$, and $-\pi $\f

Question

Show that for any complex number $z=r{{\mathrm{e}}^{\mathrm{i}	heta }}$, where $r>0$, and $-\pi $\frac{z}{z-r}=\frac{1}{2}-\frac{1}{2}\left( \cot \frac{	heta }{2} ight)\mathrm{i}$. [3] Given that $z=2{{\mathrm{e}}^{\mathrm{i}\left( \frac{\pi }{3} ight)}}$ is a root of the equation ${{z}^{2}}-2z+4=0$. State, in similar form, the other root of the equation. [1] Using parts (i) and (ii), solve the equation $\frac{4{{w}^{2}}}{{{\left( w-1 ight)}^{2}}}-\frac{4w}{w-1}+4=0$. [4]

Accepted Answer

(i) Shown; (ii) $2e^{-i\pi/3}$; (iii) $w = \frac{1}{2} - \frac{\sqrt{3}}{2}i$ \text{ or } $\frac{1}{2} + \frac{\sqrt{3}}{2}i$

2017 SAJC P1 Q7

2017 SAJC P1 Q7

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