Mathematics - Complex Number Question with Solution | TestHub

Let $z=x+iy$, where $x=\text{Re}(z)$ and $y=\text{Im}(z)$. Then $z^2=(x+iy{)}^2=x^2+2ixy+(iy{)}^2=x^2-y^2+2ixy$. Therefore, $\text{Re}(z^2)=x^2-y^2=(\text{Re}(z){)}^2-(\text{Im}(z){)}^2$. So, Statement-1 is true.

For Statement-2, if $|z|=1$, then $x^2+y^2=1$. From Statement-1, $\text{Re}(z^2)=x^2-y^2$. Substituting $y^2=1-x^2$, we get $\text{Re}(z^2)=x^2-(1-x^2)=2x^2-1=2(\text{Re}(z){)}^2-1$. So, Statement-2 is also true.

Statement-2 is a special case of Statement-1 under the condition $|z|=1$. It does not explain Statement-1, but rather is a consequence of it. Thus, Statement-2 is NOT a correct explanation for Statement-1.