The imaginary part of $\cosh(\alpha + i\beta)$ is

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$. In the set of complex numbers,the number of distinct roots of the equation $\bar{z}-z^2=i(\bar{z}+z^2)$ is . . . . . .

In a cubic equation,the coefficient of $x^2$ is $0$ and the remaining coefficients are real. If one root is $\alpha = 3 + 4i$,and the remaining roots are $\beta$ and $\gamma$,then find the value of $\alpha \beta \gamma$.

$\cosh (\alpha + i\beta ) - \cosh (\alpha - i\beta )$ is equal to

$\frac{3 + 2i\sin \theta}{1 - 2i\sin \theta}$ will be purely imaginary,if $\theta = $ [Where $n$ is an integer]

If the equation $x^{2}+bx+45=0$ $(b \in R)$ has conjugate complex roots and they satisfy $|z+1|=2\sqrt{10}$,then