$z_1$ and $z_2$ are two complex numbers such that $|z_1 + z_2| = 1$ and $|z_1^2 + z_2^2| = 25$. Then the minimum value of $|z_1^3 + z_2^3|$ is

If the complex numbers $z_1, z_2$ and the origin form an equilateral triangle,then $z_1^2 + z_2^2 = $

The value of $4x^3 - 4x^2 - 7x + 127$ when $x = \frac{4 + 5i}{2}$ is

Let $z$ be a complex number such that $|z|=1$. If $\frac{2+k^2z}{k+\overline{z}}=kz$,where $k \in R$,then the maximum distance of $k+ik^2$ from the circle $|z-(1+2i)|=1$ is:

If $(x+iy)^{3}=u+iv$,then show that: $\frac{u}{x}+\frac{v}{y}=4(x^{2}-y^{2})$

If $Z_1, Z_2, Z_3$ are three complex numbers with unit modulus such that $|Z_1-Z_2|^2+|Z_1-Z_3|^2=4$,then $Z_1 \overline{Z_2}+\overline{Z_1} Z_2+Z_1 \overline{Z_3}+\overline{Z_1} Z_3=$