If $z = x + iy$ is a complex number and $|1 + iz| = |1 - iz|$, then

The locus of the complex number $z$ such that $\arg \left(\frac{z-2}{z+2}\right)=\frac{\pi}{3}$ is:

$\sinh(ix)$ is equal to

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 ${Z_1} \ne 0$ and ${Z_2}$ are two complex numbers such that $\frac{{{Z_2}}}{{{Z_1}}}$ is a purely imaginary number,then $\left| {\frac{{2{Z_1} + 3{Z_2}}}{{2{Z_1} - 3{Z_2}}}} \right|$ is equal to

If $\sin A+\sin B+\sin C=0$ and $\cos A+\cos B+\cos C=0$,then $\cos (A+B)+\cos (B+C)+\cos (C+A)$ is equal to