$\sinh(ix)$ is equal to:

The locus of $z$ given by $\left| \frac{z - 1}{z - i} \right| = 1$ is

Let $z=x+iy$ be a complex number with $x, y \in \mathbb{Z}$. Then,the area (in sq units) of the rectangle whose vertices are the roots of the equation $\bar{z} \cdot z^3+z \cdot \bar{z}^3=350$ is

Let $z_1 = 6 + i$ and $z_2 = 4 - 3i$. Let $z$ be a complex number such that $\arg \left( \frac{z - z_1}{z_2 - z} \right) = \frac{\pi}{2}$,then $z$ satisfies -

Let $A_1, A_2, A_3, \ldots, A_8$ be the vertices of a regular octagon that lie on a circle of radius $2$. Let $P$ be a point on the circle and let $PA_i$ denote the distance between the points $P$ and $A_i$ for $i=1, 2, \ldots, 8$. If $P$ varies over the circle,then the maximum value of the product $PA_1 \cdot PA_2 \cdot \cdots \cdot PA_8$ is:

$A$ rectangle is constructed in the complex plane with its sides parallel to the axes and its centre situated at the origin. If one of the vertices of the rectangle is $a + ib\sqrt{3}$,then the area of the rectangle is