$PQ$ and $PR$ are two infinite rays. $QAR$ is an arc. $A$ point lying in the shaded region,excluding the boundary,satisfies:

Let $w = \frac{\sqrt{3} + i}{2}$ and $P = \{w^n : n = 1, 2, 3, \ldots\}$. Further, $H_1 = \{z \in C : \operatorname{Re}(z) > \frac{1}{2}\}$ and $H_2 = \{z \in C : \operatorname{Re}(z) < -\frac{1}{2}\}$, where $C$ is the set of all complex numbers. If $z_1 \in P \cap H_1$, $z_2 \in P \cap H_2$, and $O$ represents the origin, then $\angle z_1 O z_2$ can be:

If at least one value of the complex number $z = x + iy$ satisfies the condition $|z + \sqrt{2}| = a^2 - 3a + 2$ and the inequality $|z + i\sqrt{2}| < a^2$,then

$A$ particle $P$ starts from the point $Z_0 = 1 + 2i$ where $i = \sqrt{-1}$. It moves first horizontally away from the origin by $5$ units and then vertically upwards parallel to the positive $y$-axis by $3$ units to reach a point $Z_1$. From $Z_1$,the particle moves $\sqrt{2}$ units in the direction of vector $\hat{i} + \hat{j}$ and then it moves through an angle $\frac{\pi}{2}$ in the anticlockwise direction on a circle with the centre at the origin to reach point $Z_2$. Then $Z_2 =$

$A$ circle whose radius is $r$ and centre is $z_0$,then the equation of the circle is

If $|z_1|=1, |z_2|=2, |z_3|=3$ and $|9z_1z_2 + 4z_1z_3 + z_2z_3| = 12$,then the value of $|z_1 + z_2 + z_3|$ is equal to :-