If $A$ is the $A.M.$ of the roots of the equation $x^2 - 2ax + b^2 = 0$ and $G$ is the $G.M.$ of the roots of the equation $x^2 - 2bx + a^2 = 0,$ then

The harmonic mean of the roots of the equation $(5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + 8 + 2\sqrt{5} = 0$ is

For the equation $x^4+x^3-4x^2+x-1=0$,the ratio of the sum of the squares of all the roots to the product of the distinct roots is

The roots of the quadratic equation $3x^2 - px + q = 0$ are the $10^{\text{th}}$ and $11^{\text{th}}$ terms of an arithmetic progression with a common difference $d = \frac{3}{2}$. If the sum of the first $11$ terms of this arithmetic progression is $88$,then $q - 2p$ is equal to:

If the roots of the equation $x^2 + px + q = 0$ are $p$ and $q$,then what must be the value of $p$?

If the sum of the roots of the equation $\lambda x^2 + 2x + 3\lambda = 0$ is equal to their product,then $\lambda = $