If $(x-iy)^{1/3} = 2-i\sqrt{3}$ and the point $z = (x, y)$ lies on the line $\frac{x}{2} + \frac{y}{\sqrt{3}} = k$,then $k =$

The value of $\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^3$ is

If ${x_n} = \cos \left( \frac{\pi }{3^n} \right) + i\sin \left( \frac{\pi }{3^n} \right)$,then the product ${x_1} \cdot {x_2} \cdot {x_3} \cdots {x_\infty }$ is equal to:

The $A + iB$ form of $\frac{(\cos x + i\sin x)(\cos y + i\sin y)}{(\cot u + i)(1 + i\tan v)}$ is

The values of $x$ for which $\sin x + i \cos 2x$ and $\cos x - i \sin 2x$ are conjugate to each other are

Let $Z_1, Z_2, Z_3$ be three non-zero complex numbers such that $a = |Z_1|, b = |Z_2|, c = |Z_3|$. If the determinant $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = 0$,then: