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

If $\alpha, \beta$ are the roots of the equation $x^2 - 5x + 16 = 0$ and $(\alpha^2 + \beta^2)$ and $\frac{\alpha\beta}{2}$ are the roots of the equation $x^2 + px + q = 0$,then:

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

$p$ is a non-zero real number. If the equation whose roots are the squares of the roots of the equation $x^3 - px^2 + px - 1 = 0$ is identical to the given equation,then $p =$

If the sum of the roots of the quadratic equation $ax^2 + bx + c = 0$ is equal to the sum of the squares of their reciprocals,then $a/c, b/a, c/b$ are in

Let $\alpha, \beta$ with $\alpha > \beta$ be the roots of the equation $x^2 - \sqrt{2}x - \sqrt{3} = 0$. Let $P_n = \alpha^n - \beta^n$ for $n \in \mathbb{N}$. Then $(11\sqrt{3} - 10\sqrt{2})P_{10} + (11\sqrt{2} + 10)P_{11} - 11P_{12}$ is equal to: