If $f(x) = \cos (\log x)$, then $f({x^2})f({y^2}) - \frac{1}{2}\left[ {f\,\left( {\frac{{{x^2}}}{2}} \right) + f\left( {\frac{{{x^2}}}{{{y^2}}}} \right)} \right]$ has the value

The number of one-one function $f :\{ a , b , c , d \} \rightarrow$ $\{0,1,2, \ldots ., 10\}$ such that $2 f(a)-f(b)+3 f(c)+$ $f ( d )=0$ is

Suppose $f$ is a function satisfying $f ( x + y )= f ( x )+ f ( y )$ for all $x , y \in N$ and $f (1)=\frac{1}{5}$. If $\sum \limits_{n=1}^m \frac{f(n)}{n(n+1)(n+2)}=\frac{1}{12}$, then $m$ is equal to $...............$.

Domain of the function $f(x) = {\sin ^{ - 1}}\left( {\frac{{2 - |x|}}{4}} \right) + {\cos ^{ - 1}}\left( {\frac{{2 - |x|}}{4}} \right) + {\tan ^{ - 1}}\left( {\frac{{2 - |x|}}{4}} \right)$ is

If domain of the function $\log _e\left(\frac{6 x^2+5 x+1}{2 x-1}\right)+\cos ^{-1}\left(\frac{2 x^2-3 x+4}{3 x-5}\right)$ is $(\alpha, \beta) \cup(\gamma, \delta]$, then $18\left(\alpha^2+\beta^2+\gamma^2+\delta^2\right)$ is equal to $....$.

Solve $|x\,-\,2| + |x\,-\,1| = x\,-\,3$