If $z$ is a complex number such that $|z| \geq 1$,then the minimum value of $\left|z+\frac{1}{2}(3+4 i)\right|$ is:

$\omega$ is a complex cube root of unity and $Z$ is a complex number satisfying $|Z-1| \leq 2$. The possible values of $r$ such that $|Z-1| \leq 2$ and $|\omega Z - 1 - \omega^2| = r$ have no common solution are

If $w = \frac{z}{z - \frac{1}{3}i}$ and $|w| = 1$,where $i = \sqrt{-1}$,then $z$ lies on

Let $S = \{z \in \mathbb{C} : |z-3| \leq 1 \text{ and } z(4+3i) + \bar{z}(4-3i) \leq 24\}$. If $\alpha + i\beta$ is the point in $S$ which is closest to $4i$,then $25(\alpha + \beta)$ is equal to

The equation of the locus of $z$ such that $\left|\frac{z-i}{z+i}\right|=2$,where $z=x+iy$ is a complex number,is

If $\sin \alpha + \sin \beta + \sin \gamma = 0$ and $\cos \alpha + \cos \beta + \cos \gamma = 0,$ then the value of $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$ is