Let $\alpha, \beta$ be roots of $x^2+\sqrt{2}x-8=0$. If $U_n = \alpha^n + \beta^n$,then $\frac{U_{10} + \sqrt{2}U_9}{2U_8}$ is equal to ............

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 = $

Let $P_n = \alpha^n + \beta^n, n \in N$. If $P_{10} = 123, P_9 = 76, P_8 = 47$ and $P_1 = 1$,then the quadratic equation having roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is:

If $\alpha, \beta, \gamma$ are the roots of the cubic equation $x^3+p_1 x^2+p_2 x+p_3=0$. Let $S_r=\alpha^r+\beta^r+\gamma^r$. Given $S_1=10, S_2=38$ and $S_3=-1840$,then $p_3=$

If $\tan 15^{\circ}$ and $\tan 30^{\circ}$ are the roots of the equation $x^2+px+q=0$,then $pq=$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-6x^2+11x+6=0$,then $\Sigma \alpha^2 \beta+\Sigma \alpha \beta^2$ is equal to :