Let the curve $z(1+i)+\bar{z}(1-i)=4, z \in \mathbb{C}$,divide the region $|z-3| \leq 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha-\beta|$ equals :

The triangle formed by the complex numbers $z_1$,$z_2$,and $-\omega z_1 - \omega^2 z_2$ on the Argand plane is:

If $C$ is a point on the straight line joining the points $A(-2+i)$ and $B(3-4i)$ in the Argand plane and $\frac{AC}{CB}=\frac{1}{2}$,then the argument of $C$ is

Let $P=\{z \in C:|z+2-3 i| \leq 1\}$ and $Q=\{z \in C: z(1+i)+\bar{z}(1-i) \leq-8\}$. Let in $P \cap Q, |z-3+2 i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If $|z_1|^2+2|z_2|^2=\alpha+\beta \sqrt{2}$,where $\alpha, \beta$ are integers,then $\alpha+\beta$ equals . . . . . . .

Let $z=x+iy$ be a complex number,where $x$ and $y$ are integers and $i=\sqrt{-1}$. Then the area of the rectangle whose vertices are the roots of the equation $z\bar{z}^3+\bar{z}z^3=350$ 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