Using Monotonicity to Find the Range of a Function

Q. The values of x where the function $f\left(x\right)=\frac{tanxlog(x-2)}{x^2-4x+3}$ is discontinuous are given by:

1. $(-\infty ,2]\cup \left\{3\right\}$
2. $(-\infty ,3)$
3. $(-\infty ,2]\cup \{3,n\pi +\frac{\pi }{2}:n\ge 1\}$
4. R-{0, 1, 3}

Q. Find the value of the expression :
${\tan }^{-1}\left(\tan \frac{3\pi }{4}\right)$

Q. Let $f:R\to \left[0.\frac{\pi }{2}\right)$ be defined by $f\left(x\right)=ta{n}^{-1}(x^2+x+a)$. Then the set of values of a for which f is onto is

1. $[0,\infty )$
   
2. $[\frac{1}{4},\infty )$
   
3. $(-\infty ,\frac{1}{4}]$
   
4. $\left\{\frac{1}{4}\right\}$

Q. Let $f:(1,3)\to R$ be a function defined by $f\left(x\right)=\frac{x\left[x\right]}{x^2+1}$, where $\left[x\right]$ denotes the greatest integer $\le x$. Then the range of $f$ is :

1. $(\frac{2}{5},\frac{1}{2})\cup (\frac{3}{5},\frac{4}{5}]$
2. $(\frac{2}{5},\frac{4}{5}]$
3. $(\frac{2}{5},\frac{3}{5}]\cup (\frac{3}{4},\frac{4}{5})$
4. $(\frac{3}{5},\frac{4}{5})$

Q. If $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x-\frac{1}{2}\sin 2x,$ then the range of $f\left(x\right)$ is

1. $[0,\frac{3}{2}]$
2. $[-\frac{1}{2},\frac{7}{2}]$
3. $[0,\frac{9}{8}]$
4. $[\frac{3}{4},\frac{7}{8}]$

Q. The range of the function $f\left(x\right)=co{s}^2\frac{x}{4}+sin\frac{x}{4},xϵR$ is

1. $[0,\frac{5}{4}]$
2. $[1,\frac{5}{4}]$
3. $(-1,\frac{5}{4})$
4. $[-1,\frac{5}{4}]$

Q.

If $n$ is the natural number.

Then, the range of the function $f\left(n\right)={}^{8-n}{P}_{n-4},4\le n\le 6$, is

1. $\{1,2,3,4\}$

2. $\{1,2,3,4,5,6\}$

3. $\{1,2,3\}$

4. $\{1,2,3,4,5\}$

5. $\varnothing$

Q. Let $p\left(x\right)$ be a polynomial such that $p\left(x\right)–p^{\prime }\left(x\right)=x^n$, where $n$ is a positive integer. Then $p\left(0\right)$ equals

1. $(n-1)!$
2. $n!$
3. $\frac{1}{n!}$
4. $\frac{1}{(n-1)!}$

Q. The function $f\left(x\right)=\sin 2x+\cos 2x\forall x\in [0,\frac{\pi }{2}]$ is strictly decreasing in the interval

1. $(\frac{\pi }{8},\frac{\pi }{2}]$
2. $(\frac{\pi }{8},\frac{\pi }{4}]$
3. $(\frac{\pi }{8},\frac{\pi }{3}]$
4. $(0,\frac{\pi }{4}]$

Q. The domain of the function ${\cos }^{-1}\left(\frac{1}{1-x}\right)$ is equal to

1. $[0,2]$
2. $R-[0,2]$
3. $R-(0,2)$
4. $[-1,1]$

Q. Find the value of ${\cos }^{-1}\left(\cos \frac{13\pi }{6}\right)$

Q. Find the principal value of ${\cot }^{-1}\left(\sqrt{3}\right).$

Q. If the relation $R:A\to B$, where $A=\{1,2,3,4\}$ and $B=\{1,3,5\}$, is defined as $R=\left\{\right(x,y):x<y\text{for}x\in A,y\in B\}$, then which of the following is CORRECT?

1. $(2,3)\in R^{-1}$
2. Domain of $R^{-1}$ is $\{1,3,5\}$
3. Range of $R^{-1}$ is $\{2,3,4\}$
4. Domain of $R^{-1}$ is $\{3,5\}$

Q. Let $S$ be the set of all ordered pairs $(x,y)$ of positive integers satisfying the condition
$x^2–y^2=12345678.\text{Then}$

1. $S\text{is an infinite set}$
2. $S\text{is the empty set}$
3. $S\text{has exactly one element}$
4. $S\text{is finite set and has atleast two elements}$

Q.

Find the principal value of ${\sin }^{-1}(-\frac{1}{2}).$

Q. Find the value of the expression :
${\sin }^{-1}\left(\sin \frac{2\pi }{3}\right).$

Q. If $[.]$ denotes the greatest integer function, then the range of $y=\left[e^x(1+\frac{e}{e^x})\right]$ is

1. $N-\left\{1\right\}$
2. $[2,\infty )$
3. $Z$
4. $R^{+}$

Q. The number of integer(s) in the range of $f\left(x\right)={\sin }^{2}x+{\sin }^2(x+\frac{\pi }{3})+\cos x\cos (x+\frac{\pi }{3})$ is

Q. Solve $\underset{x\to 0}{lim}{\left(\frac{\tan x}{x}\right)}^{1/x}$

Q. Range of $f\left(x\right)=\frac{tan\left(\pi \right[x^2-x\left]\right)}{1+sin\left(cosx\right)}$ is (where [x] denotes the greatest integer function)

1. $(-\infty ,\infty )\sim [0,tan1]$
   
2. $\left\{0\right\}$
3. $(-\infty ,\infty )\sim [tan2,0]$
   
4. $[tan2,tan1]$
   

Q. Let $f\left(x\right)=(1+b^2)x^2+2bx+1$ and let $m\left(b\right)$ be the minimum value of $f\left(x\right).$ As $b$ varies, the range of $m\left(b\right)$ is

Q. Let $f:R\to R$ be defined be $f\left(x\right)=x^4,$ then

1. $f$ is one-one and onto
2. $f$ may be one-one and onto
3. $f$ is one-one but not onto
4. $f$ is neither one-one nor onto

Q. Range of the function $f\left(x\right)=\frac{x^2+x+2}{x^2+x+1};xϵR$ is

1. $(1,\infty )$
2. $(1,\frac{11}{7}]$
3. $(1,\frac{7}{3}]$
4. $[1,\frac{7}{5}]$

Q. The value of the expression :
${\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)$ is equal to

1. $\frac{\pi }{3}$
2. $\frac{5\pi }{6}$
3. $\frac{7\pi }{6}$
4. $\frac{\pi }{6}$

Q. The range of $f\left(x\right)=ta{n}^{-1}(x^2+x+a)$ $\forall xϵ$ R is a subset of $[0,\frac{\pi }{2})$ then the range of a is -

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

Q. The value of the expression:
${\tan }^{-1}\sqrt{3}-{\cot }^{-1}(-\sqrt{3})$ is equal to

1. $0$
2. $-\frac{\pi }{2}$
3. $2\sqrt{3}$
4. $\pi$

Q. If $f\left(x\right)={\tan }^{-1}\left(\log \right(|x|)$, then the set of values of Range $-$ Domain is

1. $\left\{0\right\}$
2. $\varphi$
3. $(-\frac{\pi }{2},\frac{\pi }{2})-\left\{0\right\}$
4. $R-(-\frac{\pi }{2},\frac{\pi }{2})$

Q.

$\underset{x\to 0}{Lt}\frac{cos\left(sinx\right)-cosx}{x^4}$ is equal to

1. $\frac{1}{2}$

2. $\frac{1}{3}$

3. $\frac{2}{3}$

4. $\frac{1}{6}$

Q. Let $f:(-1,1)\to B$, be a function defined by $f\left(x\right)=ta{n}^{-1}\frac{2x}{1-x^2}$, then f is both one - one and onto when B is the interval

1. 
2. 
3. 
4. 

Q. Convert the following complex numbers into polar form.
$i(1+i)$

1. $\sqrt{2}{e}^{i\frac{5\pi }{4}}$
2. $\sqrt{2}{e}^{i\frac{\pi }{4}}$
3. $\sqrt{2}{e}^{i\frac{3\pi }{4}}$
4. $\sqrt{2}{e}^{i3\pi }$