The least positive integer $n$ which will reduce ${\left( {\frac{{i – 1}}{{i + 1}}} \right)^n}$ to a real number, is

For two non-zero complex number $z_1$ and $z_2$, if $\operatorname{Re}\left(z_1 z_2\right)=0$ and $\operatorname{Re}\left(z_1+z_2\right)=0$, then which of the following are possible ?

$(A)$ $\operatorname{Im}\left(z_1\right) > 0$ and $\operatorname{Im}\left(z_2\right) > 0$

$(B)$ $\operatorname{Im}\left(z_1\right) < 0$ and $\operatorname{Im}\left(z_2\right) > 0$

$(C)$ $\operatorname{Im}\left(z_1\right) > 0$ and $\operatorname{Im}\left(z_2\right) < 0$

$(D)$ $\operatorname{Im}\left( z _1\right) < 0$ and $\operatorname{Im}\left( z _2\right) < 0$

Choose the correct answer from the options given below :

If $z \ne 0$ is a complex number, then

Solve the equation $x^{2}+3 x+5=0$

Let $z_{1}=2-i, z_{2}=-2+i .$ Find

$ \operatorname{Re}\left(\frac{z_{1} z_{2}}{\bar{z}_{1}}\right)$