Let $[x]$ denote the greatest integer $\leq x$, where $x \in R$. If the domain of the real valued function $\mathrm{f}(\mathrm{x})=\sqrt{\frac{[\mathrm{x}] \mid-2}{\sqrt{[\mathrm{x}] \mid-3}}}$ is $(-\infty, \mathrm{a}) \cup[\mathrm{b}, \mathrm{c}) \cup[4, \infty), \mathrm{a}\,<\,\mathrm{b}\,<\,\mathrm{c}$, then the value of $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is:

Let $f(x)=x^6-2 x^3+x^3+x^2-x-1$ and $g(x)=x^4-x^3-x^2-1$ be two polynomials. Let $a, b, c$ and $d$ be the roots of $g(x)=0$. Then, the value of $f(a)+f(b)+f(c)+f(d)$ is

Statement $-1$ : The equation $x\, log\, x = 2 - x$ is satisfied by at least one value of $x$ lying between $1$ and $2$

Statement $-2$ : The function $f(x) = x\, log\, x$ is an increasing function in $[1, 2]$ and $g (x) = 2 -x$ is a decreasing function in $[ 1 , 2]$ and the graphs represented by these functions intersect at a point in $[ 1 , 2]$

Period of $f(x) = nx + n - [nx + n]$, $n \in N$

where [ ] denotes the greatest integer function is :

Function $f(x)={\left( {1 + \frac{1}{x}} \right)^x}$ then Range of the function f (x) is

If $y = 3[x] + 1 = 4[x -1] -10$, then $[x + 2y]$ is equal to (where $[.]$ is $G.I.F.$)