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

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-

The number of points, where the curve $f(x)=e^{8 x}-e^{6 x}-3 e^{4 x}-e^{2 x}+1, x \in R$ cuts $x$-axis, is equal to

If $f$ is an even function defined on the interval $(-5, 5),$ then four real values of $x$ satisfying the equation $f(x) = f\left( {\frac{{x + 1}}{{x + 2}}} \right)$ are

The number of functions $f :\{1,2,3,4\} \rightarrow\{ a \in Z :| a | \leq 8\}$ satisfying $f ( n )+$ $\frac{1}{ n } f ( n +1)=1, \forall n \in\{1,2,3\}$ is

Let $f ^1( x )=\frac{3 x +2}{2 x +3}, x \in R -\left\{\frac{-3}{2}\right\}$ For $n \geq 2$, define $f ^{ n }( x )= f ^1 0 f ^{ n -1}( x )$. If $f ^5( x )=\frac{ ax + b }{ bx + a }, \operatorname{gcd}( a , b )=1$, then $a + b$ is equal to $............$.