Mathematics - Complex Number Question with Solution | TestHub

For equation (1) $z^3=k^3\Rightarrow z=k,k\omega ,k{\omega }^2$ where $\omega ,{\omega }^2$ non real cube roots of unity $\Rightarrow {\mathrm{z}}_1=\mathrm{k},{\mathrm{z}}_2=\mathrm{k}\cdot {\mathrm{e}}^{\mathrm{i}2\pi /3},{\mathrm{z}}_3=\mathrm{k}\cdot {\mathrm{e}}^{\mathrm{i}4\pi /3}$ $z_2$ lies on $y=-\sqrt{3}x$ and $z_3$ lies on $y=\sqrt{3}x$ where $x<0$ and $k>0$ for equation (2) ${\mathrm{z}}^3={\mathrm{ik}}^3$ $\Rightarrow \mathrm{z}={\mathrm{ki}}^{1/3}$

$\Rightarrow {\mathrm{z}}_1=\mathrm{k}\cdot {\mathrm{e}}^{\mathrm{i}\pi /6},{\mathrm{z}}_2=\mathrm{ke}{\mathrm{e}}^{\mathrm{i}5\pi /6}$ and ${\mathrm{z}}_3=\mathrm{k}\cdot {\mathrm{e}}^{\mathrm{i}3\pi /2}$ or $\mathrm{k}\cdot {\mathrm{e}}^{-\mathrm{i}\pi /2}$ ${\mathrm{z}}_3$ is purely imaginary ${\mathrm{z}}_1$ lies on $\mathrm{y}=\frac{1}{\sqrt{3}}\mathrm{x}$ and $z_2$ lies on $y=-\frac{1}{\sqrt{3}}x$ where $y>0$ and $k>0$