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:

If $\alpha, \beta$ and $\gamma$ are angles that satisfy the following conditions, find the value of $xyz$.
$1.$ $\tan \alpha + \tan \beta + \tan \gamma = \tan \alpha \tan \beta \tan \gamma$
$2.$ $x = \cos \alpha + i \sin \alpha$
$3.$ $y = \cos \beta + i \sin \beta$
$4.$ $z = \cos \gamma + i \sin \gamma$

The expression $\frac{3+2 i \sin \theta}{1-2 i \sin \theta}$ will be purely imaginary,if $\theta$ is equal to

Let $S$ be the set of all complex numbers $z$ satisfying $|z^2+z+1|=1$. Then which of the following statements is/are $TRUE$?
$(A) |z+\frac{1}{2}| \leq \frac{1}{2}$ for all $z \in S$
$(B) |z| \leq 2$ for all $z \in S$
$(C) |z+\frac{1}{2}| \geq \frac{1}{2}$ for all $z \in S$
$(D)$ The set $S$ has exactly four elements

If $\frac{1-10 i \cos \theta}{1-10 \sqrt{3} i \sin \theta}$ is purely real,then one of the values of $\theta$ is

If $a_k = \cos \alpha_k + i \sin \alpha_k$ for $k = 1, 2, 3$ and $a_1, a_2, a_3$ are the roots of the equation $x^3 + bx + c = 0$,then the real part of $b$ is: