$f : R \to R$ is defined as

$f(x) = \left\{ {\begin{array}{*{20}{c}}
{{x^2} + 2mx - 1\,,}&{x \leq 0}\\
{mx - 1\,\,\,\,\,\,\,\,\,\,\,\,\,,}&{x > 0}
\end{array}} \right.$

 If $f (x)$ is one-one then the set of values of $'m'$ is

Let $f : R \rightarrow R$ be a function defined by $f ( x )=$ $\log _{\sqrt{m}}\{\sqrt{2}(\sin x-\cos x)+m-2\}$, for some $m$, such that the range of $f$ is $[0,2]$. Then the value of $m$ is $............$

Let $\mathrm{f}(\mathrm{x})$ be a polynomial of degree $3$ such that $\mathrm{f}(\mathrm{k})=-\frac{2}{\mathrm{k}}$ for $\mathrm{k}=2,3,4,5 .$ Then the value of $52-10 \mathrm{f}(10)$ is equal to :

Set of all values of $x$ satisfying

$\frac{{{x^4} - 4{x^3} + 3{x^2}}}{{({x^2} - 4)({x^2} - 7x + 10)}} \ge 0$

The range of $f(x) = [\cos x + \sin x]$ is (Where $[.]$ is $G.I.F.$)

Prove that the Greatest Integer Function $f: R \rightarrow R ,$ given by $f(x)=[x]$, is neither one-one nor onto, where $[x]$ denotes the greatest integer less than or equal to $x$.