Let $a, b \in \mathbb{C}$. Let $\alpha, \beta$ be the roots of the equation $x^2 + ax + b = 0$. If $\beta - \alpha = \sqrt{11}i$ and $\beta^2 - \alpha^2 = 3\sqrt{11}i$, then find the value of $|(\beta^3 - \alpha^3)^2|$.

If $\alpha$ and $\beta$ are the roots of the equation $4x^2 + 3x + 7 = 0$,then $\frac{1}{\alpha} + \frac{1}{\beta} = $

If the ratio of the roots of the equations $x^2 + bx + c = 0$ and $x^2 + qx + r = 0$ are equal,then:

If $p$ and $q$ are the roots of the equation $x^{2}+px+q=0$, then:

Two candidates attempt to solve the equation $x^2 + px + q = 0$. One starts with a wrong value of $p$ and finds the roots to be $2$ and $6$,and the other starts with a wrong value of $q$ and finds the roots to be $2$ and $-9$. The roots of the original equation are

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 5x - 3 = 0$,then what is the equation whose roots are $\frac{1}{2\alpha - 3}$ and $\frac{1}{2\beta - 3}$?