If $\alpha, \beta$ are the roots of $ax^2+bx+c=0$,then $\left(\frac{\alpha}{a\beta+b}\right)^3 - \left(\frac{\beta}{a\alpha+b}\right)^3 = $

If the sum of the squares of the roots of the equation $x^2 - (\sin \alpha - 2)x - (1 + \sin \alpha) = 0$ is least,then $\alpha$ is

The quadratic equation in $t$,such that the $A.M.$ of its roots is $A$ and the $G.M.$ of its roots is $G$,is

Let $A = \left| \begin{matrix} 2 & e^{i \pi} \\ -1 & i^{2012} \end{matrix} \right|$,$C = \left. \frac{d}{dx} \left( \frac{1}{x} \right) \right|_{x=1}$,and $D = \int_{e^2}^{1} \frac{dx}{x}$. If the sum of two roots of the equation $Ax^3 + Bx^2 + Cx - D = 0$ is equal to zero,then $B$ is equal to:

If $\alpha, \beta, \gamma$ are the roots of the equation $4x^3 + 12x^2 - 7x + 165 = 0$ and $\alpha + 5, \beta + 5, \gamma + 5$ are the roots of the equation $ax^3 + bx^2 + cx + d = 0$,then the product of the roots of the second equation is:

The equation $x^5-5x^3+5x^2-1=0$ has three equal roots. If $\alpha$ and $\beta$ are the other two roots of this equation,then $\alpha+\beta+\alpha\beta=$