If the expression $7+6x-3x^2$ attains its extreme value $\beta$ at $x=\alpha$,then the sum of the squares of the roots of the equation $x^2+\alpha x-\beta=0$ is

Let $S = \left\{ \frac{a^2+b^2+c^2}{ab+bc+ca} : a, b, c \in \mathbb{R}, ab+bc+ca \neq 0 \right\}$ where $\mathbb{R}$ is the set of real numbers. Then,$S$ equals

If $x$ is real, then the difference between the greatest and least values of $\frac{x^2-x+1}{x^2+x+1}$ is

The population of cattle in a farm increases such that the difference between the population in year $n+2$ and that in year $n$ is proportional to the population in year $n+1$. If the populations in years $2010, 2011$ and $2013$ were $39, 60$ and $123$ respectively,then the population in $2012$ was:

The sum of all the roots of the equation $|x^2-8x+15|-2x+7=0$ is:

Let $\alpha, \beta (\alpha > \beta)$ be the roots of the quadratic equation $x^{2} - x - 4 = 0$. If $P_{n} = \alpha^{n} - \beta^{n}, n \in N$,then $\frac{P_{15} P_{16} - P_{14} P_{16} - P_{15}^{2} + P_{14} P_{15}}{P_{13} P_{14}}$ is equal to $......$