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 :

Let $S=\{1,2,3,4,5,6,7\} .$ Then the number of possible functions $f: S \rightarrow S$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for every $m, n \in S$ and $m . n \in S$ is equal to $......$

Let $A = \{ {x_1},\,{x_2},\,............,{x_7}\} $ and $B = \{ {y_1},\,{y_2},\,{y_3}\} $ be two sets containing seven and three distinct elements respectively. Then the total number of functions $f : A \to B$ that are onto, if there exist exactly three elements $x$ in $A$ such that $f(x)\, = y_2$, is equal to

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

where [ ] denotes the greatest integer function is :

If $f(x) = \frac{{{{\cos }^2}x + {{\sin }^4}x}}{{{{\sin }^2}x + {{\cos }^4}x}}$ for $x \in R$, then $f(2002) = $

If $f(x) = \cos (\log x)$, then $f(x)f(y) - \frac{1}{2}[f(x/y) + f(xy)] = $