Range of a Function

Q. The range of $f\left(x\right)={\cos }^{-1}x+2{\tan }^{-1}x+3{\mathrm{cosec}}^{-1}x$ is

1. $(-\pi ,2\pi )$
2. $[-\pi ,2\pi ]$
3. $\{-\pi ,2\pi \}$
4. $\varphi$

Q. If $f\left(x\right)={\tan }^{-1}x-{\cot }^{-1}x,$
$g\left(x\right)={\sec }^{-1}x-{\text{cosec}}^{-1}x,$
$h\left(x\right)={\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}x,$
$i\left(x\right)={\sin }^{-1}x-{\cos }^{-1}x,$
$j\left(x\right)=\sqrt{x},$
then

1. Domain of $jof\left(x\right)+jog\left(x\right)$ is $[\sqrt{2},\infty )$
2. Domain of $jof\left(x\right)+joi\left(x\right)$ is $[0,1]$
3. Domain of $joi\left(x\right)+joh\left(x\right)$ is
   $[\frac{1}{\sqrt{2}},1]$
4. Domain of $jog\left(x\right)+joh\left(x\right)$ is $\left\{1\right\}$

Q. The function $f\left(x\right)={\sin }^{-1}(x^2-2x+2)$ is defined at $x=a$ and $f\left(a\right)=b.$ Then

1. $a$ is a positive integer
2. $a$ is a negative integer
3. $b$ is a rational number
4. $b$ is an irrational number

Q. Let $f\left(x\right)=\sin x,g\left(x\right)={\log }_e|x|$. If the ranges of the composite functions $fog$ and $gof$ are $R_1$ and $R_2$ respectively, then

1. $R_1=\{\mu :-1\le \mu <1\}$
2. $R_2=\{\vartheta :-\infty <\vartheta <0\}$
3. $R_1=\{\mu :-1\le \mu \le 1\}$
4. $R_2=\{\vartheta :-\infty <\vartheta \le 0\}$

Q. Let $f:[0,1]\to [0,\infty )$ be a differentiable function with decreasing first derivative in its domain and $f\left(0\right)=0$. If $f^{\prime }\left(x\right)>0$ for all $x\in [0,1],$ then

1. $\underset{0}{\overset{1}{\int }}\frac{dx}{\left(f\right(x){)}^2+1}<\frac{{\tan }^{-1}f\left(1\right)}{f^{\prime }\left(1\right)}$
2. $\underset{0}{\overset{1}{\int }}\frac{dx}{\left(f\right(x){)}^2+1}<\frac{f\left(1\right)}{f^{\prime }\left(1\right)}$
3. $\underset{0}{\overset{1}{\int }}\frac{dx}{\left(f\right(x){)}^2+1}>\frac{{\tan }^{-1}f\left(1\right)}{f^{\prime }\left(1\right)}$
4. $\underset{0}{\overset{1}{\int }}\frac{dx}{\left(f\right(x){)}^2+1}=\frac{f\left(1\right)}{f^{\prime }\left(1\right)}$ for some $x\in [0,1]$

Q. $Iftan(\frac{\pi }{4}+\frac{y}{2})=ta{n}^3(\frac{\pi }{4}+\frac{x}{2}),thensinx\left(\frac{3+si{n}^{2}x}{1+3si{n}^{2}x}\right)equals$

1. sin y
2. sin 2y
3. 0
4. cos y

Q. The range of $f\left(x\right)={\cos }^{-1}x+2{\tan }^{-1}x+3{\mathrm{cosec}}^{-1}x$ is

1. $(-\pi ,2\pi )$
2. $[-\pi ,2\pi ]$
3. $\{-\pi ,2\pi \}$
4. $\varphi$

Q. If the function $f\left(x\right)=\frac{k\sin x+2\cos x}{\sin x+\cos x}$ is strictly increasing for all values of $x$ in its domain, then

1. $k<1$
2. $k>1$
3. $k<2$
4. $k>2$

Q.

The principal value of $si{n}^{-1}tan\left(\frac{-5\pi }{4}\right)$ is

1. π/4
2. - π/4
3. - π/2
4. π/2

Q. The function $f\left(x\right)={\sin }^{-1}(x^2-2x+2)$ is defined at $x=a$ and $f\left(a\right)=b.$ Then

1. $a$ is a positive integer
2. $a$ is a negative integer
3. $b$ is a rational number
4. $b$ is an irrational number

Q. $\sum _{r=1}^n{\sin }^{-1}\left(\frac{\sqrt{r}-\sqrt{r-1}}{\sqrt{r(r+1)}}\right)$ is equal to

1. ${\tan }^{-1}\left(\sqrt{n+1}\right)-\frac{\pi }{4}$
2. ${\tan }^{-1}\left(\sqrt{n}\right)$
3. ${\tan }^{-1}\left(\sqrt{n}\right)-\frac{\pi }{4}$
4. ${\tan }^{-1}\left(\sqrt{n+1}\right)$

Q. Let $f\left(x\right)={\left({\sin }^{-1}x\right)}^2-{\left({\cos }^{-1}x\right)}^2.$ If range of $f$ equals $[\frac{a{\pi }^2}{4},\frac{b{\pi }^2}{4}]$ where $a,b\in Z,$ then the value of $b-a$ is

Q. Let a function $f\left(x\right)=\underset{-\frac{\pi }{2}}{\overset{x}{\int }}(2{\sin }^{2}t+3\cos t)dt$ is defined in $[-\frac{\pi }{2},\frac{\pi }{2}].$ Then which of the following is/are correct

1. $f_{min}=\pi -6$
2. $f_{min}=0$
3. $f_{max}=\pi +6$
4. $f_{max}=6-\pi$

Q. Let $f\left(x\right)=\sqrt{({\sin }^{-1}x-{\cos }^{-1}x)},$ $g\left(x\right)=\sqrt{({\tan }^{-1}x-{\cot }^{-1}x)}$ and $h\left(x\right)=\sqrt{({\sec }^{-1}x-{\text{cosec}}^{-1}x)}$. Then CORRECT statement(s) is(are)

1. Domain of $f\left(x\right)+g\left(x\right)$ is $\left\{1\right\}$
2. Domain of $g\left(x\right)+h\left(x\right)$ is $[\sqrt{2},\infty )$
3. Domain of $h\left(x\right)+f\left(x\right)$ is $[\frac{1}{\sqrt{2}},1]$
4. Domain of $f\left(x\right)+g\left(x\right)+h\left(x\right)$ is $\varphi$

Q. $\frac{x^3\sin \left({\tan }^{-1}{x}^4\right)}{1+x^8}$