The number of bijective functions $f :\{1,3,5, 7, \ldots \ldots . .99\} \rightarrow\{2,4,6,8, \ldots \ldots, 100\}$, such that $f(3) \geq f(9) \geq f(15) \geq f(21) \geq \ldots \ldots f(99), \quad$ is
${ }^{50} P _{17}$
${ }^{50} P _{33}$
$33 ! \times 17 !$
$\frac{50 !}{2}$
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:
Let $f(x)=\frac{x-1}{x+1}, x \in R-\{0,-1,1)$. If $f^{a+1}(x)=f\left(f^{n}(x)\right)$ for all $n \in N$, then $f^{\prime}(6)+f(7)$ is equal to
If the domain and range of $f(x){ = ^{9 - x}}{C_{x - 1}}$ contains $m$ and $n$ elements respectively, then
The range of $f(x)=4 \sin ^{-1}\left(\frac{x^2}{x^2+1}\right)$ is
Which one of the following best represent the graph of $y = \frac{|x-x^2|}{x^2-x}$ ?