Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$. In the set of complex numbers, the number of distinct roots of the equation

$\bar{z}-z^2=i\left(\bar{z}+z^2\right)$ is. . . . . .

If $\frac{{2{z_1}}}{{3{z_2}}}$ is a purely imaginary number, then $\left| {\frac{{{z_1} – {z_2}}}{{{z_1} + {z_2}}}} \right|$ =

If $\frac{\pi }{2} < \alpha  < \frac{3}{2}\pi $ , then the modulus and argument of $(1 + cos\, 2\alpha ) + i\, sin\, 2\alpha $ is respectively

The conjugate of the complex number $\frac{{2 + 5i}}{{4 – 3i}}$ is

If $z = x + iy\, (x, y \in R,\, x \neq \, -1/2)$ , the number of values of $z$ satisfying ${\left| z \right|^n}\, = \,{z^2}{\left| z \right|^{n – 2}}\, + \,z{\left| z \right|^{n – 2}}\, + \,1\,.\,\left( {n \in N,n > 1} \right)$ is