Let $S = \{z = x + iy : \frac{2z - 3i}{4z + 2i} \text{ is a real number} \}$. Then which of the following is $NOT$ correct?

Let $z$ be a complex number such that $|z + 2| = |z - 2|$ and $\arg\left(\frac{z + 3}{z - i}\right) = \frac{\pi}{4}$. Then $|z|^2$ is equal to:

The locus of $z$ satisfying $\left|\frac{z-i}{z-2i}\right|=2$ is a

$A$ 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 $Z_2 =$

If $P(x, y)$ denotes $z = x + iy$ in the Argand plane and $\left|\frac{z-1}{z+2i}\right| = 1$,then the locus of $P$ is a/an

The region of the complex plane for which $\left| \frac{z - a}{z + \overline{a}} \right| = 1$ where $\text{Re}(a) \neq 0$ is