If $\mathrm{R}=\left\{(\mathrm{x}, \mathrm{y}): \mathrm{x}, \mathrm{y} \in \mathrm{Z}, \mathrm{x}^{2}+3 \mathrm{y}^{2} \leq 8\right\}$ is a relation on the set of integers $\mathrm{Z},$ then the domain of $\mathrm{R}^{-1}$ is 

If the function $f\,:\,R - \,\{ 1, - 1\}  \to A$ defined by $f\,(x)\, = \frac{{{x^2}}}{{1 - {x^2}}},$ is surjective, then $A$ is equal to

For $x\,\, \in \,R\,,x\, \ne \,0,$ let ${f_0}(x) = \frac{1}{{1 - x}}$ and ${f_{n + 1}}(x) = {f_0}({f_n}(x)),$ $n\, = 0,1,2,....$  Then the value of ${f_{100}}(3) + {f_1}\left( {\frac{2}{3}} \right) + {f_2}\left( {\frac{3}{2}} \right)$ is equal to

The value of $b$ and $c$ for which the identity $f(x + 1) - f(x) = 8x + 3$ is satisfied, where $f(x) = b{x^2} + cx + d$, are

The function $f\left( x \right) = \left| {\sin \,4x} \right| + \left| {\cos \,2x} \right|$, is a periodic function with period

If the domain of the function $f(x)=\sec ^{-1}\left(\frac{2 x}{5 x+3}\right)$ is $[\alpha, \beta) \cup(\gamma, \delta]$, then $|3 \alpha+10(\beta+\gamma)+21 \delta|$ is equal to $.......$.