D.P. GUPTA, Preetima Bajpai and, Sanjeev Kumar Jha Solutions for Chapter: Relations & Functions, Exercise 1: Exercise 1

If $f\left(x\right)=\ln \left(\frac{x^2+e}{x^2+1}\right),$ then range of $f\left(x\right)$ is

The function $y\left(x\right)$ given implicitly by $3\sin y+a\sin x=1, a>0$ exists provided, then

The function $y\left(x\right)$ given implicitly by the equation $3\sin y+5\sin x=1$ is real in

The domain and range of the function $f$ given by $f\left(x\right)=2-\left|x-5\right|$ is

The set of all values of $x$ in $[0,\pi ]$ for which the function $\sqrt{1-2\sin 3x}$ takes real values is

The period of $\tan 3\theta$ is

The range of $f\left(x\right)=\cos \frac{\pi \left[x\right]}{2}$ where $\left[·\right]$ denotes greatest integer function is

If the set $A$ has $p$ elements, $B$ has $q$ elements, then the number of elements in $A\times B$ is