It is known that a particular complex number z is NOT a real number - HSC - SSCE Mathematics Extension 2 - Question 6 - 2022 - Paper 1

The modulus of $z$ is non-negative and is defined as $|z| = ext{sqrt}(x^2 + y^2)$. Thus, ar{z} = |z| would imply $x - iy = ext{sqrt}(x^2 + y^2)$ which cannot be true since $y$ is non-zero (as $z$ is not real). Therefore, this statement is false.

If we write $iz$ as $i(x + iy) = -y + ix$, the real part Re($iz$) is $-y$ and the imaginary part Im($z$) is $y$. For these to be equal, we have $-y = y$, leading to $y = 0$. This contradicts the condition that $z$ is not a real number. Therefore, this statement is false.

The expression simplifies to Arg($z^2$), which is $2 imes ext{Arg}(z)$. It can only equal Arg($z$) if Arg($z$) is zero, which would make $z$ real. Hence, this statement cannot be true either.