IIT-JEE 1992Question 1
$z\neq0$ is a complex number.
Column I: (A) $\operatorname{Re}z=0$; (B) $\operatorname{Arg}z=\dfrac{\pi}{4}$.
Column II: (p) $\operatorname{Re}z^{2}=0$; (q) $\operatorname{Im}z^{2}=0$; (r) $\operatorname{Re}z^{2}=\operatorname{Im}z^{2}$.
Complex Numbers
Solution
$\operatorname{Re}z=0$: $z=iy$, $z^2=-y^2$ real, so $\operatorname{Im}z^2=0$ (q). $\operatorname{Arg}z=\frac\pi4$: $z=re^{i\pi/4}$, $z^2=ir^2$, so $\operatorname{Re}z^2=0$ (p).