If ${z_1} = 1 + 2i$ and ${z_2} = 3 + 5i$, and then $\operatorname{Re} \left( {\frac{{{{\bar z}_2}{z_1}}}{{{z_2}}}} \right)$ is equal to

For any two complex numbers ${z_1},{z_2}$we have $|{z_1} + {z_2}{|^2} = $ $|{z_1}{|^2} + |{z_2}{|^2}$ then

Let $z_1$ and $z_2$ be two complex number such that $z_1$ $+z_2=5$ and $z_1^3+z_2^3=20+15 i$. Then $\left|z_1^4+z_2^4\right|$ equals-

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

If complex numbers $(x -2y) + i(3x -y)$ and $(2x -y) + i(x -y + 6)$ are conjugates of each other, then $|x + iy|$ is $(x,y \in R)$ 
 

Let ${z_1}$ be a complex number with $|{z_1}| = 1$ and ${z_2}$be any complex number, then $\left| {\frac{{{z_1} - {z_2}}}{{1 - {z_1}{{\bar z}_2}}}} \right| = $