$(r, \theta)$ denotes $r(\cos \theta + i \sin \theta)$. If $x = (1, \alpha)$,$y = (1, \beta)$,$z = (1, \gamma)$ and $x + y + z = 0$,then $\sum \cos (2\alpha - \beta - \gamma) = $

If $a, b \in \mathbb{R}$ and $i=\sqrt{-1}$,then the number of ordered pairs of real numbers $(a, b)$ satisfying the condition $(a+bi)^3 = a-bi$ is

If $z_1$ and $z_2$ are the roots of the equation $x^2+2x+2=0$,then $\frac{-2^{11}(z_1+1+3i)^{11}}{2^5(z_2+1-3i)^{11}}$ is equal to

The product of the real roots of the equation $4x^4 - 24x^3 + 57x^2 + 18x - 45 = 0$,given that one of the roots is $3 + i\sqrt{6}$,is:

If $z$ and $w$ are two complex numbers such that $|zw| = 1$ and $\arg(z) - \arg(w) = \frac{\pi}{2}$,then

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.