If the roots of the equation $12x^2 + mx + 5 = 0$ are in the ratio $3 : 2$,then $m = ......$

$A$ student notices that the roots of the equation $x^2+bx+a=0$ are each $1$ less than the roots of the equation $x^2+ax+b=0$. Then,$a+b$ is

If $p$ and $q$ are the roots of the equation $x^2 + pq = (p + 1)x$,then $q=$

If $\alpha$ and $\beta$ are the roots of the equation ${x^2} - (1 + {n^2})x + \frac{1}{2}(1 + {n^2} + {n^4}) = 0$,then the value of ${\alpha ^2} + {\beta ^2}$ is

Let $a, b, c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a, c), (2, b)$ and $(a, b)$ be $\left(\frac{10}{3}, \frac{7}{3}\right)$. If $\alpha, \beta$ are the roots of the equation $ax^{2} + bx + 1 = 0$,then the value of $\alpha^{2} + \beta^{2} - \alpha\beta$ is ....... .

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