If $x$ is real,then the difference between the greatest and least values of $\frac{x^2-x+1}{x^2+x+1}$ 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 $y = \frac{x^2 + 14x + 9}{x^2 + 2x + 3}$ for all $x \in R$,then the interval of maximum length in which $y$ lies is

If $x$ is real, then the value of $\frac{x^2-3x+4}{x^2+3x+4}$ lies in the interval

For any real value of $x$,if $\frac{11 x^2+12 x+6}{x^2+4 x+2} \notin(a, b]$,then the value of $x$ for which $\frac{11 x^2+12 x+6}{x^2+4 x+2}=b-a+3$ is

If $x = \frac{1}{2} \left( \sqrt{3} + \frac{1}{\sqrt{3}} \right)$, then the value of $\frac{\sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}}$ is equal to