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 $z=x+iy$ represents a point $P$ in the Argand plane, then the area of the region represented by the inequality $2 < |z-(1+i)| < 3$ is (in $\pi$)

If $z_1$ and $z_2$ are two distinct complex numbers such that $\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2$,then:

If $Z$ is a complex number such that $|Z| \leq 3$ and $-\frac{\pi}{2} \leq \operatorname{amp}(Z) \leq \frac{\pi}{2}$,then the area of the region formed by the locus of $Z$ is

$A$ complex number $z$ is said to be unimodular if $|z| = 1$. Suppose $z_1$ and $z_2$ are complex numbers such that $\frac{z_1 - 2z_2}{2 - z_1 \overline{z_2}}$ is unimodular and $z_2$ is not unimodular. Then the point $z_1$ lies on a:

If $|z+i|-|z-1|=|z|-2=0$ for a complex number $z$, then $z=$