If $f(x) = 2x^3 + mx^2 - 13x + n$ and $2, 3$ are the roots of the equation $f(x) = 0$,then the value of $4m + 5n$ is:

For the quadratic equation $ax^2 + bx + c = 0$,if $\alpha$ and $\beta$ are the roots,then $\frac{\alpha}{a\beta + b} + \frac{\beta}{a\alpha + b} = \dots$

Let $f(x) = x^3 - 2x + 2$. If real numbers $a, b, c$ are such that $|f(a)| + |f(b)| + |f(c)| = 0$,then the value of $f^2(a^2 + \frac{2}{a}) + f^2(b^2 + \frac{2}{b}) - f^2(c^2 + \frac{2}{c})$ is equal to:

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $2x^3+3x^2-5x-7=0$,then $\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}=$

If $\alpha$ and $\beta$ are roots of the equation $Ax^2 + Bx + C = 0$,then the value of $\alpha^3 + \beta^3$ is

Let $\alpha$ and $\beta$ be the roots of the equation $p x^2 + q x + r = 0$,where $p \neq 0$. If $p, q, r$ are in $AP$ and $\frac{1}{\alpha} + \frac{1}{\beta} = 4$,then the value of $|\alpha - \beta|$ is