Range of the function , $f (x) = cot ^{-1}$ $\left( {{{\log }_{4/5}}\,\,(5\,{x^2}\,\, - \,\,8\,x\,\, + \,\,4)\,} \right)$ is :

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

Let $f$ be a function satisfying $f(xy) = \frac{f(x)}{y}$ for all positive real numbers $x$ and $y.$ If $ f(30) = 20,$ then the value of $f(40)$ is-

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be defined as

$f(x+y)+f(x-y)=2 f(x) f(y), f\left(\frac{1}{2}\right)=-1 .$ Then, the value of $\sum_{\mathrm{k}=1}^{20} \frac{1}{\sin (\mathrm{k}) \sin (\mathrm{k}+\mathrm{f}(\mathrm{k}))}$ is equal to:

The number of functions $f$, from the set$A=\left\{x \in N: x^{2}-10 x+9 \leq 0\right\}$ to the set $B=\left\{n^{2}: n \in N\right\}$ such that $f(x) \leq(x-3)^{2}+1$, for every $x \in A$, is.

Let $f(x) = sin\,x,\,\,g(x) = x.$

Statement $1:$ $f(x)\, \le \,g\,(x)$ for $x$ in $(0,\infty )$

Statement $2:$ $f(x)\, \le \,1$ for $(x)$ in $(0,\infty )$ but $g(x)\,\to \infty$ as $x\,\to \infty$