If $-i$ and $\alpha$ are the roots of the equation $iz^2 - 2(i+1)z + (2-i) = 0$,$\tan \theta = \frac{-1}{2}$ and $\theta \in 4^{\text{th}}$ quadrant,then $5^3 \cos 6\theta =$

If $Z_1 = \sqrt{3} + i \sqrt{3}$ and $Z_2 = \sqrt{3} + i$,and $\left(\frac{Z_1}{Z_2}\right)^{50} = x + iy$,then the point $(x, y)$ lies in

The sum of the squares of the modulus of all complex numbers $z$ satisfying $\bar{z} = i z^{2} + z^{2} - z$ is equal to

Let $z=x+iy$ and $w=u+iv$ be complex numbers on the unit circle such that $z^2+w^2=1$. Then the number of ordered pairs $(z, w)$ is

If $z$ is a complex number such that $\left|z-\frac{4}{z}\right|=2$,then the greatest value of $|z|$ is

Let $A = \{z : (\frac{z - \bar{z}}{2i})^2 \leqslant 2(\frac{z - \bar{z}}{2i})\}$ where $i = \sqrt{-1}$ and $B = \{z : |z| \leqslant \sqrt{5}\}$. The number of points with integral real and imaginary parts of $z$ lying in $A \cap B$ is -