If $z=x+iy$ satisfies the condition $|z+1|=1$,then $z$ lies on the

The locus of a point on the Argand plane represented by the complex number $z$, when $z$ satisfies the condition $\left|\frac{z-1+i}{z+1-i}\right|=\left|\operatorname{Re}\left(\frac{z-1+i}{z+1-i}\right)\right|$ is

If $|z| = 1$ $(z \neq -1)$ and $z = x + iy$,then $\left( \frac{z - 1}{z + 1} \right)$ is

$\sinh(ix)$ is equal to

If $P$ is the point in the Argand diagram corresponding to the complex number $\sqrt{3}+i$ and if $OPQ$ is an isosceles right-angled triangle,right-angled at $O$,then $Q$ represents the complex number

If $\cos \alpha + \cos \beta + \cos \gamma = 0$ and $\sin \alpha + \sin \beta + \sin \gamma = 0$,then $\sin 2 \alpha + \sin 2 \beta + \sin 2 \gamma = $