Question

If $Z$ is a non zero complex number such that $Re\left(Z\right)=Im\left(Z\right),$ then

• A. $Re\left(Z^2\right)=Im\left(Z^2\right)$
• B. $Im\left(Z^2\right)=0$
• C. $Re\left(Z^2\right)=0$
• D. $Re\left(Z^2\right)+Im\left(Z^2\right)=0$

Answer 1

The correct option is C $Re\left(Z^2\right)=0$

As $Re\left(Z\right)=Im\left(Z\right)$, assuming the complex number be $Z=x+ix$
Now,
$Z^2=x^2(1+i{)}^2\Rightarrow Z^2=x^2(1-1+2i)\Rightarrow Z^2=2x^{2}i\therefore Re(Z^2)=0$