If $z_n = (1 + i \sqrt{2})^n$, $n \in Z$, then $\frac{1}{9} \operatorname{Re}(z_4 \bar{z}_5) = $

Let $z=x+iy$ and $w=u+iv$ be complex numbers on the unit circle such that $z^2+w^2=1$. Then the number of ordered pairs $(z, w)$ is

Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1| = 4|z_2|$. If $z = \frac{3z_1}{2z_2} + \frac{2z_2}{3z_1}$,then:

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

If $a_k = \cos \alpha_k + i \sin \alpha_k$ for $k = 1, 2, 3$ and $a_1, a_2, a_3$ are the roots of the equation $x^3 + bx + c = 0$,then the real part of $b$ is: