# If $z$ is a complex number so that $iz^2-\overline z=0$, then $|z|= ?$

$\begin{array}{1 1}(A) \;0 \;or \;1 \\(B)\;0 \;or\; \large\frac{\sqrt 3}{2} \\(C)\;1\;or \;\large\frac{\sqrt 3}{2} \\(D)\; 0 \;or\;\large\frac{1}{20} \end{array}$

Let $z=x+iy$
$z^2=x^2-y^2+2xyi$
$\overline z=x-iy$
$iz^2-\overline z=0$
$\Rightarrow\:i(x^2-y^2+2xyi)-(x-yi)=0$
$\Rightarrow\:(x^2-y^2+y)i-(2xy+x)=0$
$\Rightarrow\:x^2-y^2+y=0\:and\:2xy+x=0$
$\Rightarrow\:x=0\:or\:y=\large-\frac{1}{2}$
If $x=0,\:then\:-y^2+y=0$
$\Rightarrow\:y=0\:\:or\:\:1$