$\mathrm{z}=\frac{3+2 \mathrm{i} \sin \theta}{1-2 \mathrm{i} \sin \theta},(\mathrm{i}=\sqrt{-1})$ will be purely imaginary if $\theta=$

If $\mathrm{z}=x+\mathrm{i} y$ is a complex number, then the equation $\left|\frac{z+i}{z-i}\right|=\sqrt{3}$ represents the

The locus of the points represented by $|z+3|-|z-3|=6$, where $z$ is a complex number, is ….

The complex numbers $\sin x+i \cos 2 x$ and $\cos x$ - $\mathrm{i} \sin 2 x,(\mathrm{i}=\sqrt{-1})$ are conjugate to each other for,