If $\frac{2+3i \sin \theta}{1-2i \sin \theta}$ is purely imaginary,then $\cos^2 \theta=$

$z_1$ and $z_2$ are two complex numbers such that $|z_1 + z_2| = 1$ and $|z_1^2 + z_2^2| = 25$. Then the minimum value of $|z_1^3 + z_2^3|$ is

If $\cos \alpha+3 \cos 3 \beta+5 \cos 5 \gamma=0$,$\sin \alpha+3 \sin 3 \beta+5 \sin 5 \gamma=0$ and $\cos 3 \alpha+27 \cos 9 \beta+125 \cos 15 \gamma=\left(\lambda^2-4\right) \cos (\alpha+3 \beta+5 \gamma)$,then $\lambda$ is equal to

The number of complex numbers $z$ satisfying $\overline{z} = i z^2$ is

The maximum value of $|z|$, when the complex number $z$ satisfies the condition $\left|z+\frac{2}{z}\right|=2$ is

The value of the expression $\left( \cos \frac{\pi }{2} + i\sin \frac{\pi }{2} \right) \left( \cos \frac{\pi }{{{2^2}}} + i\sin \frac{\pi }{{{2^2}}} \right) \dots$ to $\infty$ is