4.7 JEE Advanced Extensions

Method 1 (Algebraic): Let $\omega = \frac{z-1}{z+1}$. We check $\overline{\omega}$. $\overline{\omega} = \frac{\overline{z}-1}{\overline{z}+1}$. Since $|z|=1$, $z\overline{z}=1 \implies \overline{z} = 1/z$. $\overline{\omega} = \frac{1/z - 1}{1/z + 1} = \frac{1-z}{1+z} = -\frac{z-1}{z+1} = -\omega$. Since $\overline{\omega} = -\omega$, the number is purely imaginary.