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

This statement asserts that the conjugate of $z$ equals the modulus of $z$. Since $|z|$ is always a non-negative real number and the conjugate indicates that the imaginary component must be zero for equality, this option cannot be true for a non-real $z$.

For $z = x + iy$, $iz = i(x + iy) = -y + ix$. Thus, $\text{Re}(iz) = -y$ and $\text{Im}(z) = y$. This leads to the equation $-y = y$, which can hold true when $y = 0$. However, since $z$ is non-real, this can be true under certain conditions, making it a potential candidate.

The argument of a complex number relates to its angle in the complex plane. For a non-real number $z$, $\text{Arg}(z^3) = 3 \cdot \text{Arg}(z)$. This implies that $\text{Arg}(z)$ can equal $\text{Arg}(z^3)$ only if $n \cdot \text{Arg}(z)$ is a multiple of $2\pi$. Hence, this could potentially be true under certain circumstances.