If $\overline{\mathrm{z}}={\mathrm{iz}}^2$ then the number of values of $\mathrm{z}$ is

$$\begin{gathered} \overline{\mathrm{z}}={\mathrm{iz}}^2 \\ \mathrm{Taking} \mathrm{modulus} \mathrm{on} \mathrm{both} \mathrm{sides} \\ \begin{array}{r}\left|\mathrm{z}\right|=|\mathrm{z}{|}^2\\ \Rightarrow \left|z\right|=0\text{ or }\left|z\right|=1\end{array} \\ \left|z\right|=0\Rightarrow z=\left(0,0\right) and \\ \begin{array}{r}\overline{\mathrm{z}}={\mathrm{iz}}^2\Rightarrow \mathrm{z}\overline{\mathrm{z}}={\mathrm{iz}}^3\\ \Rightarrow |\mathrm{z}{|}^2={\mathrm{iz}}^3\Rightarrow {\mathrm{iz}}^3=1\\ \Rightarrow {\mathrm{z}}^3=-\mathrm{i}\\ \Rightarrow {\mathrm{z}}^3-{\mathrm{i}}^3=0\\ \Rightarrow (\mathrm{z}-\mathrm{i})\left({\mathrm{z}}^2+\mathrm{iz}-1\right)=0\end{array} \\ \begin{array}{r}\mathrm{z}=\mathrm{i}\text{ and }{\mathrm{z}}^2+\mathrm{iz}-1=0\\ {\mathrm{z}}^2+\mathrm{iz}-1=0\text{ Gives }\\ \mathrm{z}=\frac{-\mathrm{i}\pm \sqrt{-1+4}}{2}\end{array} \\ \mathrm{z}=\frac{-\mathrm{i}\pm \sqrt{3}}{2} \\ \mathrm{z}=\left(0,0\right),\left(0,1\right)\text{ or }\left(\pm \frac{\sqrt{3}}{2},-\frac{1}{2}\right) \end{gathered}$$