If a complex number $z$ is such that $(7+i)(z+\bar{z})-(4+i)(z-\bar{z})+116i=0$,then $z\bar{z}=$

Let $z_1$ and $z_2$ be two complex numbers such that $z_1 + z_2 = 5$ and $z_1^3 + z_2^3 = 20 + 15i$. Then $|z_1^4 + z_2^4|$ equals-

Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg(z_1) = \frac{-\pi}{4}, \arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{2}$,where $\alpha, \beta \in \mathbb{Z}$,then the value of $\alpha^2 + \beta^2$ is:

For all complex numbers $z$ of the form $1 + i\alpha$,where $\alpha \in R$,if $z^2 = x + iy$,then which of the following relations holds?

If $z=x-iy$ and $z^{1/3}=a+ib$,then $\frac{(x/a+y/b)}{a^2+b^2}=$

Let the set of all values of $k \in R$ such that the equation $z(\bar{z} + 2 + i) + k(2 + 3i) = 0, z \in C$, has at least one solution, be the interval $[\alpha, \beta]$. Then $9(\alpha + \beta)$ is equal to: