Functions

If $f(x)=\log _e\left(6-\left|x^2+x-6\right|\right)$, then domain of $f(x)$ has how many integral values of $x$ ?

The number of positive integral solutions of $x^2+9 \lt (x+3)^2 \lt 8 x+25$, is

If $f(x)=\frac{a^x+a^{-x}}{2}$ and $f(x+y)+f(x-y)$ $=k f(x) f(y)$, then $k$ is equal to

The length of the longest interval, in which $f(x)=3 \sin x-4 \sin ^3 x$ is increasing, is

The domain of the function $f(x)=\frac{\sin ^{-1}(3-x)}{\log (|x|-2)}$ is :

Let: $f: N \rightarrow N$ be defined as $f(n)=\left\{\begin{array}{ll}\frac{n+1}{2}, & \text { if is } n \text { odd } \\ \frac{n}{2}, & \text { if } n \text { is even }\end{array}\right.$ for all $n \in N$. Then $f$ is

The maximum value of $3 \cos \theta+4 \sin \theta$ is

If $f(x)=\cos \left[\pi^2\right] x+\cos \left[-\pi^2\right] x$, then

If $f(x+y)=f(x) \cdot f(y)$ for all $x$ and $y$ and $f(15)=2, f^{\prime}(0)=3$, then $f^{\prime}(5)$ will be

The range of $f(x)=\sec \left(\frac{\pi}{4} \cos ^2 x\right),-\infty \lt x \lt \infty$ is

The range of the function $f(x)=\log _5\left(25-x^2\right)$ is

One mapping (function) is selected at random from all the mappings of the set $A=\{1,2,3, \ldots, n\}$ into itself. The probability that the mapping selected is one-one, is