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:

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:

If $P, Q$ and $R$ are angles of an isosceles triangle and $\angle P = \frac{\pi}{2}$, then the value of $\left(\cos \frac{P}{3} - i \sin \frac{P}{3}\right)^3 + (\cos Q + i \sin Q) (\cos R - i \sin R) + (\cos P - i \sin P) (\cos Q - i \sin Q) (\cos R - i \sin R)$ 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

Let $n$ denote the number of solutions of the equation $z^{2}+3 \bar{z}=0$,where $z$ is a complex number. Then the value of $\sum_{k=0}^{\infty} \frac{1}{n^{k}}$ is equal to:

The $A + iB$ form of $\frac{(\cos x + i\sin x)(\cos y + i\sin y)}{(\cot u + i)(1 + i\tan v)}$ is