Let $z$ be a complex number. Then the angle between vectors $z$ and $-iz$ is

If $\text{Im} \left( \frac{2z + 1}{iz + 1} \right) = -3$,then the locus of $z$ is :-

Let $\omega = z \bar{z} + k_1 z + k_2 i z + \lambda(1 + i)$,where $k_1, k_2 \in R$. Let $\operatorname{Re}(\omega) = 0$ be the circle $C$ of radius $1$ in the first quadrant touching the line $y = 1$ and the $y$-axis. If the curve $\operatorname{Im}(\omega) = 0$ intersects $C$ at $A$ and $B$,then $30(AB)^2$ is equal to $.......$.

If $z$ is a complex number such that $|z+4| \geq 3$,then the smallest value of $|z+3|$ is

If the eight vertices of a regular octagon are given by the complex numbers $z_j = \frac{1}{x_j - 2i}$ for $j = 1, 2, \dots, 8$,where $x_j$ are the roots of $x^8 - 1 = 0$,then the radius of the circumcircle of the octagon is

For any real number $r$, let $A_r = \{e^{i \pi r n} : n \in \mathbb{N}\}$ be a set of complex numbers. Then,