In the Argand plane,no value of $\sqrt[3]{1-i \sqrt{3}}$ lies in which quadrant?

If $i = \sqrt{-1}$,then $4 + 5\left(-\frac{1}{2} + \frac{i\sqrt{3}}{2}\right)^{334} + 3\left(-\frac{1}{2} + \frac{i\sqrt{3}}{2}\right)^{365}$ is equal to

$\sum_{k=1}^6 \left[ \sin \left(\frac{2 \pi k}{7}\right) - i \cos \left(\frac{2 \pi k}{7}\right) \right] = $

If $\left(\frac{3}{2}+i \frac{\sqrt{3}}{2}\right)^{50}=3^{25}(x+i y)$ where $x$ and $y$ are real, then the ordered pair $(x, y)$ is

If $1, \omega, \omega^2$ are the cube roots of unity,then
$1(2+\frac{1}{\omega})(2+\frac{1}{\omega^2})+2(3+\frac{1}{\omega})(3+\frac{1}{\omega^2})+3(4+\frac{1}{\omega})(4+\frac{1}{\omega^2})+\ldots 10 \text{ terms} =$

If $\omega$ is a non-real cube root of unity,then the value of $\cos \left[ \left\{ (1-\omega)(1-\omega^2) + (2-\omega)(2-\omega^2) + \dots + (2017-\omega)(2017-\omega^2) \right\} \cdot \frac{\pi}{2017} \right]$ is -