Question

Write the correct letter from column II against the entry number in column I in your answer book, z $\ne$ 0 is a complex number

Answer 1

Let $z=a+ib$
$Re\left(z\right)=0\Rightarrow a=0$
And
$z=ib\Rightarrow z^2=-b^2\Rightarrow Im\left(z^2\right)=0$
$arg\left(z\right)=\frac{\pi }{4}$
$\Rightarrow z=r(cos\frac{\pi }{4}+isin\frac{\pi }{4})\Rightarrow z^2=r^2({cos}^2\frac{\pi }{4}-{sin}^2\frac{\pi }{4})+2i{r}^{2}cos\frac{\pi }{4}sin\frac{\pi }{4}=i{r}^{2}sin\frac{\pi }{2}=i{r}^{2}Re\left(z^2\right)=0$