Let $S = \{z \in \mathbb{C} : 4z^2 + \overline{z} = 0\}$. Then $\sum_{z \in S} |z|^2$ is equal to:

$A$ 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 ${z_1}, {z_2}, {z_3}$ are affixes of the vertices $A, B$ and $C$ respectively of a triangle $ABC$ having centroid at $G$ such that $z = 0$ is the midpoint of $AG$,then:

If the amplitude of $z-2-3i$ is $\frac{\pi}{4}$,then the locus of $z=x+iy$ is:

Let point $P = \alpha + i\beta$,where $\alpha, \beta > 0$,undergo the following three transformations successively on the Argand plane:
$(I)$ Reflection about $\text{amp}(z) = \frac{\pi}{4}$
$(II)$ Transformation through a distance $\beta$ units along the positive direction of the real axis
$(III)$ Rotation through an angle $\frac{\pi}{4}$ about the origin in the counter-clockwise direction
If the final position of the point is given by $Q = -\sqrt{2} + i\sqrt{6}$,then:

The equation $\arg \left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}$ represents a circle with: