Let $z_{1}$ and $z_{2}$ be two imaginary roots of $z^{2}+pz+q=0$,where $p$ and $q$ are real. The points $z_{1}, z_{2}$ and the origin form an equilateral triangle if

Let $X_{n} = \{z = x + iy : |z|^{2} \leq \frac{1}{n}\}$ for all integers $n \geq 1$. Then,$\bigcap_{n=1}^{\infty} X_{n}$ is

If $z = \frac{3}{2 + \cos \theta + i \sin \theta}$,then the locus of $z$ is :-

Let $w_1$ be the point obtained by the rotation of $z_1=5+4i$ about the origin through a right angle in the anticlockwise direction,and $w_2$ be the point obtained by the rotation of $z_2=3+5i$ about the origin through a right angle in the clockwise direction. Then the principal argument of $w_1-w_2$ is equal to $...........$.

If complex numbers $z_1$ and $z_2$ both satisfy $z + \overline{z} = 2 |z - 1|$ and $\arg(z_1 - z_2) = \frac{\pi}{3},$ then the value of $\text{Im}(z_1 + z_2)$ is,where $\text{Im}(z)$ denotes the imaginary part of $z$.

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: