If $x$ is real,then the maximum and minimum values of $\frac{x^2+14x+9}{x^2+2x+3}$ are respectively

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

The number of ordered pairs $(a, b)$ of positive integers such that $\frac{2a-1}{b}$ and $\frac{2b-1}{a}$ are both integers is

Let $(a-3)x^2+12x+(a+6)>0, \forall x \in R$ and $a \in (\ell, \infty)$. If $\alpha$ is the least positive integral value of $a$,then the roots of $(\alpha-3)x^2+12x+(\ell+2)=0$ are

The curves $y=x^2+9x+20$ and $y=x^2+bx+c$ intersect the $X$-axis at the points $(\alpha_i, 0)$ for $i=1, 2, 3, 4$. If $\alpha_1 < \alpha_2 < \alpha_3 < \alpha_4$ are such that $|\alpha_1-\alpha_3|=|\alpha_2-\alpha_4|=8$,then the sum of all possible values of $b$ and $c$ is:

The equation of lowest degree with rational coefficients having roots $\sqrt{3}+\sqrt{2} i$ and $\sqrt{3}-\sqrt{2}$ is