If $\cos \alpha + \cos \beta + \cos \gamma = 0$ and $\sin \alpha + \sin \beta + \sin \gamma = 0$,then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = $

If $A = \{z : |\frac{z - 2}{z + 2}| = 3, z \in C\}$ and $z_1, z_2, z_3, z_4 \in A$ are $4$ complex numbers representing points $P, Q, R, S$ respectively on the complex plane such that $z_1 - z_2 = z_4 - z_3$,then the maximum value of the area of quadrilateral $PQRS$ is:

If a complex number $z$ satisfies $|z^2-1|=|z|^2+1$,then $z$ lies on

Let $z_{1}$ and $z_{2}$ be the roots of the equation $z^{2} + az + 12 = 0$. If $z_{1}$,$z_{2}$,and the origin form an equilateral triangle in the complex plane,then the value of $|a|$ is:

The number of elements in the set $\{ z = a + ib \in \mathbb{C} : a, b \in \mathbb{Z} \text{ and } 1 < |z - 3 + 2i| < 4 \}$ is:

For any integer $k$,let $w_k = \cos \left( \frac{k\pi}{11} \right) + i \sin \left( \frac{k\pi}{11} \right)$,where $i = \sqrt{-1}$. The value of the expression $\frac{\sum_{k=1}^8 |w_{2k+1} - w_{2k}|}{\sum_{k=1}^4 |w_{3k-1} - w_{3k-2}|}$ is