Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^2+|z|=0$,where $z \in \mathbb{C}$. Then $4(\alpha^2+\beta^2)$ is equal to:

If $z=1-\sqrt{3} i$,then $z^3-3 z^2+3 z=$

If $e^{i x}$ is a solution of the equation $z^n+p_1 z^{n-1}+p_2 z^{n-2}+\ldots+p_n=0$,where $p_i$ are real $(i=1, 2, \ldots, n)$,then $p_n \sin nx + p_{n-1} \sin(n-1)x + \ldots + p_1 \sin x + \sin(0) = $ (Note: The constant term in the equation is $p_n$ and the coefficient of $z^0$ is $1$ if we normalize,but here the equation is given as $z^n + p_1 z^{n-1} + \ldots + p_n = 0$. Let us assume the constant term is $p_n$. The expression to evaluate is $p_n \sin nx + p_{n-1} \sin(n-1)x + \ldots + p_1 \sin x + \sin(0)$). Given the standard form,find the value of $p_n \sin nx + p_{n-1} \sin(n-1)x + \ldots + p_1 \sin x$.

Let $n$ denote the number of solutions of the equation $z^{2}+3 \bar{z}=0$,where $z$ is a complex number. Then the value of $\sum_{k=0}^{\infty} \frac{1}{n^{k}}$ is equal to:

If $(x-iy)^{1/3} = a-ib$,then the value of $\frac{x}{a} + \frac{y}{b}$ is

Assertion $(A)$: If $z$ is a complex number such that $|z| \geq 3$,then the least value of $|z + \frac{3}{z}|$ is $1$.
Reason $(R)$: $|z_1 - z_2| \leq |z_1| + |z_2|$,for any two complex numbers $z_1, z_2$.
The correct option among the following is: