If $e^{it} = \cos t + i \sin t$ and $e^{-it} = \cos t - i \sin t$,then $\cosh(x + iy) - \cosh(x - iy) =$

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 $|z|$?

If $z_1$ and $z_2$ 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

If $a, b, c$ are non-zero real numbers with $c \neq 1$ such that $a^2+b^2+c^2=c$ and if $\alpha=\frac{a+i b}{1-c}$,then $a^2+b^2=$

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

If $z=1-\sqrt{3} i$,then $z^3-3 z^2+3 z=$