Question

If $z\ne 0$ is a complex number, then

• A. $Re\left(z\right)0⟹Im\left(z^2\right)=0$
• B. $Re\left(z^2\right)=0⟹Im\left(z^2\right)=0$
• C. $Re\left(z\right)=0⟹Re\left(z^2\right)=0$
• D. None of these

Answer 1

The correct option is A $Re\left(z\right)0⟹Im\left(z^2\right)=0$

Let $z=x+iy$......(1) with $x\ne 0,y\ne 0$. [Since $z\ne 0$]

Now $z^2=x^2-y^2+2ixy$.......(2).

i) For $Re\left(z\right)=0$ gives $x=0\Rightarrow 2ixy(=Im(z^2\left)\right)=0$

So (A) is true.

ii) Now $Re\left(z^2\right)=0$ gives $x^2=y^2\nRightarrow 2ixy(=Im(z^2\left)\right)=0$

So (B) is not true.

iii) For $Re\left(z\right)=0$ gives $x=0\nRightarrow x^2-y^2(=Re(z^2\left)\right)=0$

So (C) is not true.