For the quadratic equation $2x^2 - 2(p - 2)x - p - 1 = 0$,what should be the value of $p$ such that the sum of the squares of its roots is minimized?

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

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $ax^2 + bx + c = 0$. If $a, b, c$ are in $A.P.$ and $\alpha + \beta = 15$,then $\alpha \beta$ equals:

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3-3x^2+x+5=0$,then $y=\Sigma \alpha^2+\alpha \beta \gamma$ satisfies the equation

If $\alpha, \beta$ are the roots of the equation $x^2 - 6x - 2 = 0$,$\alpha > \beta$ and $a_n = \alpha^n - \beta^n$,$n > 1$,then the value of $\frac{a_{10} - 2a_8}{2a_9}$ is equal to

If $\alpha, \beta, \gamma$ are the real roots of the equation $18x^3-15x^2-4x+4=0$ such that $\alpha=\beta$ and $\alpha>\gamma$,then $\alpha+\beta^2+\gamma^3=$