The maximum value of $|z|$ where $z$ satisfies the condition $\left| z + \frac{2}{z} \right| = 2$ is

$A$ 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 $Z_2 =$

Geometrically,the set $\{z \in \mathbb{C} : |z - 2 - 2i| \leq 1\}$ represents

Let a complex number be $w = 1 - \sqrt{3} i$. Let another complex number $z$ be such that $|zw| = 1$ and $\arg(z) - \arg(w) = \frac{\pi}{2}$. Then the area of the triangle with vertices at the origin,$z$,and $w$ is equal to ........ .

The polar coordinates of the point,whose Cartesian coordinates are $(-2 \sqrt{3}, 2)$,are

The points representing the complex number $z$ for which $\text{arg}\left(\frac{z-2}{z+2}\right)=\frac{\pi}{3}$ lie on