If $\alpha, \beta$ and $\gamma$ are roots of the equation $x^3+4x-19=0$,then the value of $\frac{\alpha^3}{19-4\alpha}+\frac{\beta^3}{19-4\beta}+\frac{\gamma^3}{19-4\gamma}$ is equal to

If $\alpha \ne \beta$ but $\alpha^2 = 5\alpha - 3$ and $\beta^2 = 5\beta - 3$,then the equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ is

For what value of $a$ is one root of the quadratic equation $(a^2 - 5a + 3) x^2 + (3a - 1) x + 2 = 0$ twice the other (in $/3$)?

If $a$ and $b$ are roots of $x^2 - px + q = 0$,then $\frac{1}{a} + \frac{1}{b} = $

If one root of the equation $ax^3+bx+c=0$ is twice another root,then

Let $\lambda \neq 0$ be a real number. Let $\alpha, \beta$ be the roots of the equation $14 x^2-31 x+3 \lambda=0$ and $\alpha, \gamma$ be the roots of the equation $35 x^2-53 x+4 \lambda=0$. Then $\frac{3 \alpha}{\beta}$ and $\frac{4 \alpha}{\gamma}$ are the roots of the equation: