If $3+i$ and $2-\sqrt{3}$ are the roots of the equation $f(x)=a_0+a_1 x+a_2 x^2+\ldots+a_{n} x^{n}$ where $a_0, a_1, \ldots, a_{n} \in \mathbb{Z}$,then the least value of $n$ and the value of $a_0$ are respectively:

For a non-real root $z$ of the equation $z^4+z^2+1=0$,find the value of $\left(z+\frac{1}{z}\right)^3+\left(z^2+\frac{1}{z^2}\right)^2+\left(z^3+\frac{1}{z^3}\right)^3$.

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.

Let the set of all values of $k \in R$ such that the equation $z(\bar{z} + 2 + i) + k(2 + 3i) = 0, z \in C$, has at least one solution, be the interval $[\alpha, \beta]$. Then $9(\alpha + \beta)$ is equal to:

If $z$ is a non-real complex number, then the minimum value of $\frac{\operatorname{Im}(z^5)}{(\operatorname{Im} z)^5}$ is

If $\omega$ is a complex cube root of unity,then $\left(\frac{1-\sqrt{3} i}{2}\right)^{2020}+\left(\frac{1+\sqrt{3} i}{2}\right)^{2026} +\sin \left(\sum_{j=1}^6(j+\omega)(j+\omega^2) \frac{3 \pi}{152}\right)=$