The sum of all the rational roots of the equation $6x^6-25x^5+31x^4-31x^2+25x-6=0$ is

Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2-12x+[x]+31=0$ and $x^2-5|x+2|-4=0$ respectively,where $[x]$ denotes the greatest integer $\leq x$. Then $m^2+mn+n^2$ is equal to $..............$.

Let $a, b, c, d$ be distinct real numbers such that $a, b$ are roots of $x^2-5cx-6d=0$,and $c, d$ are roots of $x^2-5ax-6b=0$. Then $b+d$ is

Let $\lambda \in R$ and let the equation $E$ be $|x|^2 - 2|x| + |\lambda - 3| = 0$. Then the largest element in the set $S = \{x + \lambda : x \text{ is an integer solution of } E\}$ is $..........$

If $x$ is real,then the value of the expression $\frac{x^2 + 14x + 9}{x^2 + 2x + 3}$ lies between

If $x$ is real,then the value of $\frac{x^2 + 34x - 71}{x^2 + 2x - 7}$ does not lie between