If $P(x, y)$ denotes $z = x + iy$ in the Argand plane and $\left|\frac{z-1}{z+2i}\right| = 1$,then the locus of $P$ is a/an

If $\cos \alpha + 2 \cos \beta + 3 \cos \gamma = 0$,$\sin \alpha + 2 \sin \beta + 3 \sin \gamma = 0$ and $\alpha + \beta + \gamma = \pi$,then $\sin 3 \alpha + 8 \sin 3 \beta + 27 \sin 3 \gamma$ is equal to

If the center of a regular hexagon is at the origin and one of the vertices on the Argand diagram is $1 + 2i$,then its perimeter is

Let $z = x + iy$ be a point in the Argand plane. If the amplitude of $\left(\frac{z - 3}{z + 2i}\right)$ is $\frac{\pi}{2}$,then the locus of $z$ is

Let $z$ and $w$ be two non-zero complex numbers such that $|z| = |w|$ and $arg(z) + arg(w) = \pi$. Then $z$ is equal to:

If $z_{1}, z_{2}$ are complex numbers such that $\operatorname{Re}(z_{1})=|z_{1}-1|$, $\operatorname{Re}(z_{2})=|z_{2}-1|$ and $\arg(z_{1}-z_{2})=\frac{\pi}{6}$, then $\operatorname{Im}(z_{1}+z_{2})$ is equal to