JEE Main · 2025 · 07 Apr 2025 · Shift 1 · Complex Numbers

Take $z$ on the unit circle. The quotient $\frac{z-i}{z+i}$ is purely real exactly when $z-i$ and $z+i$ point along the same line, that is when $z$, $i$ and $-i$ are collinear, so $z$ lies on the imaginary axis; together with $|z|=1$ this leaves $z=i$ or $z=-i$, and since $-i$ is excluded only $z=i$ remains. The first set therefore has a single element, so its claim of exactly two elements is false. For the second set, writing $z=e^{i\theta }$ gives $\frac{z-1}{z+1}=\frac{e^{i\theta }-1}{e^{i\theta }+1}=i\tan \frac{\theta }{2}$, which is purely imaginary for every admissible $\theta$ with $z\ne -1$; hence that set contains infinitely many elements and its statement is true. Thus only the second statement holds.