JEE Main · 2024 · 01 Feb 2024 · Shift 2 · Complex Numbers

The expression $\left|z+\frac{3}{2}+2i\right|$ represents the distance from $z$ to the point $-\frac{3}{2}-2i$. The distance from the origin to this point is $\left|-\frac{3}{2}-2i\right|=\sqrt{\frac{9}{4}+4}=\frac{5}{2}$. Since $|z|\ge 1$, the set of possible $z$ is the exterior (including boundary) of the unit circle. The minimum distance from this set to $-\frac{3}{2}-2i$ is $\frac{5}{2}-1=\frac{3}{2}$. However, the source solution notes that the minimum value is actually zero, which would occur if $z=-\frac{3}{2}-2i$, but $|z|=\frac{5}{2}>1$, so this is valid. The source answer is marked as Bonus, indicating an issue with the problem statement or options.