$z_1$ 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
$24$
$42$
$37$
$33$
The number of solutions of the equation ${z^2} + \bar z = 0$ is
If $z_1, z_2 $ are any two complex numbers, then $|{z_1} + \sqrt {z_1^2 - z_2^2} |$ $ + |{z_1} - \sqrt {z_1^2 - z_2^2} |$ is equal to
Let $A =\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1- i \sin \theta}\right.$ is purely imaginary $\}$. Then the sum of the elements in $A$ is
The product of two complex numbers each of unit modulus is also a complex number, of
Find the complex number z satisfying the equations $\left| {\frac{{z - 12}}{{z - 8i}}} \right| = \frac{5}{3},\left| {\frac{{z - 4}}{{z - 8}}} \right| = 1$