If $z=\frac{1}{2}-2 i$, is such that $|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}$ and $\alpha, \beta \in R \quad$, then $\alpha+\beta$ is equal to

The inequality $|z – 4|\, < \,|\,z – 2|$represents the region given by

For any two complex numbers ${z_1}$and${z_2}$ and any real numbers $a$ and $b$; $|(a{z_1} – b{z_2}){|^2} + |(b{z_1} + a{z_2}){|^2} = $

$|{z_1} + {z_2}|\, = \,|{z_1}| + |{z_2}|$ is possible if

$z_1$ and $z_2$ are two complex numbers such that $|z_1 + z_2|$ = $1$ and $\left| {z_1^2 + z_2^2} \right|$ = $25$ , then minimum value of $\left| {z_1^3 + z_2^3} \right|$ is