If $x = -2 - \sqrt{3} i$,where $i = \sqrt{-1}$,then the value of $2x^4 + 5x^3 + 7x^2 - x + 41$ is

For a non-zero complex number $z$,let $\arg(z)$ denote the principal argument with $-\pi < \arg(z) \leq \pi$. Then,which of the following statement$(s)$ is (are) $FALSE$?
$(A)$ $\arg(-1-i) = \frac{\pi}{4}$,where $i = \sqrt{-1}$
$(B)$ The function $f: \mathbb{R} \rightarrow (-\pi, \pi]$,defined by $f(t) = \arg(-1+it)$ for all $t \in \mathbb{R}$,is continuous at all points of $\mathbb{R}$,where $i = \sqrt{-1}$
$(C)$ For any two non-zero complex numbers $z_1$ and $z_2$,$\arg\left(\frac{z_1}{z_2}\right) - \arg(z_1) + \arg(z_2)$ is an integer multiple of $2\pi$.
$(D)$ For any three given distinct complex numbers $z_1, z_2$ and $z_3$,the locus of the point $z$ satisfying the condition $\arg\left(\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\right) = \pi$ lies on a straight line.

The set of all values of $\theta$ such that $\frac{1-i \cos \theta}{1+2 i \sin \theta}$ is purely imaginary is

If $\cos A+\cos B+\cos C=0$ and $\sin A+\sin B+\sin C=0$,then $\cos (A-B)=$

If $\cos (u + iv) = \alpha + i\beta ,$ then ${\alpha ^2} + {\beta ^2} + 1$ equals

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