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 $z=1-\sqrt{3} i$,then $z^3-3 z^2+3 z=$

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

$\frac{3 + 2i\sin \theta}{1 - 2i\sin \theta}$ will be purely imaginary,if $\theta = $ [Where $n$ is an integer]

If $z_1, z_2, z_3$ are the roots of the equation $z^3 - z^2(4 + 3i) + z(3 + 8i) - 5i = 0$, then $Re(z_1) + Re(z_2) + Re(z_3)$ is -

For two non-zero complex numbers $z_1$ and $z_2$, if $\operatorname{Re}(z_1 z_2) = 0$ and $\operatorname{Re}(z_1 + z_2) = 0$, then which of the following are possible?
$(A) \operatorname{Im}(z_1) > 0$ and $\operatorname{Im}(z_2) > 0$
$(B) \operatorname{Im}(z_1) < 0$ and $\operatorname{Im}(z_2) > 0$
$(C) \operatorname{Im}(z_1) > 0$ and $\operatorname{Im}(z_2) < 0$
$(D) \operatorname{Im}(z_1) < 0$ and $\operatorname{Im}(z_2) < 0$
Choose the correct answer from the options given below: