In the Argand plane, the distinct roots of $1+z+z^{3}+z^{4}=0$ ($z$ is a complex number) represent vertices of

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 $\log_{\sqrt{3}} \left( \frac{|z|^2 - |z| + 1}{2 + |z|} \right) < 2$,then the locus of $z$ is

In the Argand plane,the vector $z = 4 - 3i$ is turned in the clockwise sense through $180^o$ and stretched three times. The complex number represented by the new vector is

The vertices $B$ and $D$ of a parallelogram are $1 - 2i$ and $4 + 2i$. If the diagonals are at right angles and $AC = 2BD$,the complex number representing $A$ is

$P$ is a point denoting $z$ in the Argand diagram. If $\frac{z-i}{z-1}$ is always purely imaginary,then the locus of $P$ is