Complex numbers practice · Medium

If ${z_1},{z_2}$ are two complex numbers such that $\left| {\frac{{{z_1} - {z_2}}}{{{z_1} + {z_2}}}} \right| = 1$ and $i\;{z_1} = k\;{z_2}$ where $k \in R,$ then the angle between ${z_1} - {z_2}$ and ${z_1} + {z_2}$ is

If $z$ is a complex number of unit modulus and argument $\theta ,$ then $\arg \left( {\frac{{1 + z}}{{1 + \overline z }}} \right)$ is equal to

Let $w_1$ be the point obtained by the rotation of $z_1=5+4 i$ about the origin through a right angle in the anticlockwise direction, and $w_2$ be the point obtained by the rotation of $z_2=3+5 i$ about the origin through a right angle in the clockwise direction. Then the principal argument $w_1-w_2$ is equal to

The real part of ${\left( {1 - \cos \mathrm{\theta } + 2i\sin \mathrm{\theta }} \right)^{ - 1}}$ is

For two complex numbers ${z_1},\;{z_2}$ the relation $\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|$ holds, if

The solution set of $\left| {{x^2} - 10} \right| \le 6,$ is

If $z^{2}+z+1=0$, where $\mathrm{z}$ is complex number, then the value of $\left(z+\frac{1}{z}\right)^{2}+\left(z^{2}+\frac{1}{z^{2}}\right)^{2}+\left(z^{3}+\frac{1}{z^{3}}\right)^{2}+\ldots \ldots .+\left(z^{6}+\frac{1}{z^{6}}\right)^{2}$ is