If $z_1, z_2, z_3$ are points in the Argand plane,then $\left| \begin{array}{ccc} z_1 & \overline{z_1} & 1 \\ z_2 & \overline{z_2} & 1 \\ z_3 & \overline{z_3} & 1 \end{array} \right| = $

The largest value of $r$ for which the region represented by the set $\{ \omega \in \mathbb{C} : |\omega - 4 - i| \le r \}$ is contained in the region represented by the set $\{ z \in \mathbb{C} : |z - 1| \le |z + i| \}$ is equal to

If $z=x+iy$ is a complex number satisfying $\left|\frac{z-2i}{z+2i}\right|=2$ and the locus of $z$ is a circle,then its radius is

Let $z$ and $w$ be two non-zero complex numbers such that $|z| = |w|$ and $arg(z) + arg(w) = \pi$. Then $z$ is equal to:

If a complex number $z$ satisfies $|z^2-1|=|z|^2+1$,then $z$ lies on

In the complex plane,let $z_1=\sqrt{3}+i$ and $z_2=\sqrt{3}-i$ be two adjacent vertices of an $n$-sided regular polygon centered at the origin. Then,$n$ equals