If $1+x^2=\sqrt{3} x$,then $\sum_{n=1}^{24}\left(x^n-\frac{1}{x^n}\right)^2$ is equal to

$z_1$ 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

Let $S$ be the set of all complex numbers $z$ satisfying $|z^2+z+1|=1$. Then which of the following statements is/are $TRUE$?
$(A) |z+\frac{1}{2}| \leq \frac{1}{2}$ for all $z \in S$
$(B) |z| \leq 2$ for all $z \in S$
$(C) |z+\frac{1}{2}| \geq \frac{1}{2}$ for all $z \in S$
$(D)$ The set $S$ has exactly four elements

If for $z=\alpha+i \beta$,$|z+2|=z+4(1+i)$,then $\alpha+\beta$ and $\alpha \beta$ are the roots of the equation

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

Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of $(\bar{z})^2+\frac{1}{z^2}$ are integers,then which of the following is/are possible value$(s)$ of $|z|$?