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

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

$\frac{(1+i)^{2016}}{(1-i)^{2014}}$ is equal to

If $a = \cos (2\pi /7) + i\sin (2\pi /7)$,then the quadratic equation whose roots are $\alpha = a + a^2 + a^4$ and $\beta = a^3 + a^5 + a^6$ is

For all complex numbers $z$ of the form $1 + i\alpha$,where $\alpha \in R$,if $z^2 = x + iy$,then which of the following relations holds?

Let $\omega = e^{i \pi / 3}$,and $a, b, c, x, y, z$ be non-zero complex numbers such that $a+b+c = x$,$a+b \omega + c \omega^2 = y$,and $a+b \omega^2 + c \omega = z$. Then the value of $\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}$ is