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 $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2 + px + q = 0$ $(p \neq 0)$,then:

The number of different possible values for the sum $x+y+z$,where $x, y, z$ are real numbers such that $x^4+4y^4+16z^4+64=32xyz$ is

If $\left|\frac{x^2+kx+1}{x^2+x+1}\right| < 3$ for all real $x$,then $k$ is in the interval

If $x = \frac{1}{2} \left( \sqrt{7} + \frac{1}{\sqrt{7}} \right)$, then $\frac{\sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}}$ is equal to

$f(x)$ is a quadratic expression such that $f(x)$ is negative when $x \in \left(-\infty, -\frac{5}{3}\right) \cup (3, \infty)$ and positive when $x \in \left(-\frac{5}{3}, 3\right)$. $g(x)$ is another quadratic expression such that $g(x)$ is negative when $x \in \left(3, \frac{9}{2}\right)$ and positive when $x \in \mathbb{R} - \left[3, \frac{9}{2}\right]$. Then,the sign of $f(x)g(x)$ in $[0, 5]$ is