If $x = 2 + \sqrt{3}$ and $xy = 1$,then $\frac{x}{\sqrt{2} + \sqrt{x}} + \frac{y}{\sqrt{2} - \sqrt{y}} = $

If $x$ is real,then the value of $x^2 - 6x + 13$ will not be less than

The number of values of $b$ for which there is an isosceles triangle with sides of lengths $b+5$,$3b-2$,and $6-b$ is

Let $f(x) = x^3 + a x^2 + b x + c$ be a polynomial with integer coefficients. If the roots of $f(x)$ are integers and are in Arithmetic Progression,then '$a$' cannot take the value:

For what values of $a \in \mathbb{Z}$,the quadratic expression $(x+a)(x+1991)+1$ can be factorised as $(x+b)(x+c)$,where $b, c \in \mathbb{Z}$?

Let $E$ denote the set of all integers $a$ such that the point of intersection of the parabola $y = x^2 + 2ax + 2021$ with the $x$-axis has rational coordinates. The largest element of $E$ is