If $\alpha, \beta, \gamma$ are the roots of $2x^3 - 2x - 1 = 0$,then $(\Sigma \alpha \beta)^2$ is equal to

If $\alpha, \beta, \gamma$ are roots of $x^3-5x+4=0$,then $(\alpha^3+\beta^3+\gamma^3)^2$ is equal to

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha+\beta=1$ and $\alpha \beta=-1$. Let $p_{n}=\alpha^{n}+\beta^{n}$,$p_{n-1}=11$ and $p_{n+1}=29$ for some integer $n \geq 1$. Then,the value of $p_{n}^{2}$ is .... .

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + \frac{a}{2} x + b = 0$ and $(\alpha-\beta)(\alpha-\gamma)$,$(\beta-\alpha)(\beta-\gamma)$,$(\gamma-\alpha)(\gamma-\beta)$ are the roots of the equation $(y+a)^3 + K(y+a)^2 + L = 0$,then $\frac{L}{K} =$

If $\alpha, \beta$ are the roots of the equation $ax^2 + bx + c = 0$,then the equation whose roots are $\alpha + \frac{1}{\beta}$ and $\beta + \frac{1}{\alpha}$ is

If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation $x^{2} + (2 - \lambda)x + (10 - \lambda) = 0$ is minimum,then the magnitude of the difference of the roots of this equation is