The condition that one root of the equation $ax^2 + bx + c = 0$ is three times the other 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 $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^{2}+bx+c=0$ and $3b^{2}=16ac$, then:

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+ax^2+bx+c=0$,then $\alpha^{-1}+\beta^{-1}+\gamma^{-1}$ is equal to

If $\operatorname{cosec} \theta$ and $\cot \theta$ are the roots of $cx^2+bx+a=0$ $(bc \neq 0)$,then $b^2(b^2-4ac)=$

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 $\frac{L}{K} =$