If $e^{i \theta} = \operatorname{cis} \theta$, then find the value of $\sum_{n=0}^{\infty} \frac{\cos (n \theta)}{2^n}$.

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.

If $\sqrt{5}-i \sqrt{15}=r(\cos \theta+i \sin \theta)$ where $-\pi < \theta < \pi$,then find the value of $r^2(\sec \theta+3 \operatorname{cosec}^2 \theta)$.

If $a, b, c$ and $d \in \mathbb{R}$ such that $a^2+b^2=4$ and $c^2+d^2=2$ and if $(a+ib)^2=(c+id)^2(x+iy)$,then $x^2+y^2$ is equal to

The minimum degree of a polynomial equation with rational coefficients having $\sqrt{3}+\sqrt{27}$ and $\sqrt{2}+5i$ as two of its roots is

If $x = -5 + 2 \sqrt{-4}$,then the value of $x^4 + 9x^3 + 35x^2 - x + 4$ is