Fractional Part Function

Q. The number of value(s) of $x$ satisfying the equation $|2x-1|=3\left[x\right]+2\left\{x\right\}$ is
($[.]$ and $\{.\}$ represent greatest integer function and fractional part function respectively)

Q. The number of real solution(s) of the equation $2\left[x\right]=x+\left\{x\right\}$ is
($[.]$ denotes the greatest integer function and $\{.\}$ denotes the fractional part function)

1. $1$
2. $2$
3. more than $2$ but finite
4. infinite

Q. The range of the function $y=\frac{x-\left[x\right]}{1-\left[x\right]+x}$, where $[.]$ represents the greatest integer function, is

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

Q. The solution of the equation $(x-2)\left[x\right]=\left\{x\right\}-1$ is ${}^{\prime }{x}^{\prime }$ such that $a\le x<b$ then $|a+b|=$
(where {.} represents fractional part function)

Q. The total number of solutions of $[x{]}^2=x+2(x\},$ where [.] and {.} denote the greatest integer function and the fractional part function, respectively, is equal to

1. 2
2. 3
3. 4
4. 6

Q.

The domain of the function $f\left(x\right)=\frac{\left|x+3\right|}{\left(x+3\right)}$ is

1. $\left\{-3\right\}$

2. $R-\left\{-3\right\}$

3. $R-\left\{3\right\}$

4. $R$

Q. Let $f\left(x\right)=e^{\left\{e^x\text{sgn}x\right\}}$ and $g\left(x\right)=e^{\left[e^x\text{sgn}x\right]},x\in R$ where $\{.\}$ and $[.]$ denote greatest integer function and fractional part function, respectively. Also $h\left(x\right)=\log \left(f\right(x\left)\right)+\log \left(g\right(x\left)\right),$ then for real $x,h\left(x\right)$ is

1. an odd function.
2. an even function.
3. neither odd nor an even function.
4. both odd as well as even function.

Q.

Show that the function $f:R_{\ast }\to R_{\ast }$ defined by $f\left(x\right)=\frac{1}{x}$ is one-one, where $R_{\ast }$ is the set of all non-zero real numbers. Is the result true, if the domain $R_{\ast }$ is replaced by N with co-domain being same as $R_{\ast }$?

Q.

How many real numbers satisfy the relation $\left[x\right]=\frac{3}{2}x$.

__

Q. The domain of the function $f\left(x\right)=\frac{1}{[x{]}^2-7[x]+10}$ is
(where $[.]$ denotes the greatest integer function)

1. $R$
2. $R-[2,3)$
3. $R-[5,6)$
4. $R-[2,3)\cup [5,6)$

Q.

If $A$ and $B$ be subsets of a set $X$. Then

1. $A–B=A\cup B$

2. $A–B=A\cap B$

3. $A–B=A^c\cap B$

4. $A–B=A\cap B^c$

Q. The domain of $f\left(x\right)=\sqrt{{\log }_x\left\{x\right\}}$ is
$\left(\right\{x\}$ represents the fractional part of $x)$

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

Q. The number of integer(s) in the range of $\sqrt{\left[\text{sgn}\right(x){]}^2-\{\text{sgn}\left(x\right){\}}^2}$ is
(Here, $[.],\{.\}$ and $\text{sgn}\left(x\right)$ represent the greatest integer function, fractional part function and signum function respectively)

Q. For a function, $y=f\left(x\right)=\frac{x}{1+|x|},x,y\in R$, which among the following is true :

1. $f\left(x\right)$ is a one-one and an onto function
2. $f\left(x\right)$ is an onto but not a one -one function
3. $f\left(x\right)$ is a one-one but not an onto function
4. $f\left(x\right)$ is neither a one-one nor an onto function

Q. The number of values of $x$ satisfying the equation $(x^2+7x+11{)}^{(x^2-4x-21)}=1$ is

1. $6$
2. $5$
3. $4$
4. $2$

Q. Let $[.]$ and $\{.\}$ represent the greatest integer function and the fractional part function respectively. The number of value(s) of $x$ satisfying the equation $|2x-1|=3\left[x\right]+2\left\{x\right\}$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q.

Decide, among the following sets, which sets are subsets of one and another:

A = {x: x ∈ R and x satisfy x² – 8x + 12 = 0},

B = {2, 4, 6}, C = {2, 4, 6, 8…}, D = {6}.

Q. Consider $f\left(x\right)=\{x+[{\log }_2(2+x)\left]\right\}+\{x+[{\log }_2(2+x^2)\left]\right\}+...+\{x+[{\log }_2(2+x^{10})\left]\right\},$ then correct statement is
(where {} and [] denotes the fractional part function and greatest integer function respectively)

1. $\left[f\right(e\left)\right]=7$
2. $f\left(\pi \right)=20\pi -60$
3. Number of solutions of the equation $f\left(x\right)=x$ is $9$
4. Number of solutions of the equation $f\left(x\right)=x$ is $10$

Q.

Write an expression using all six trigonometric functions such that the value of the expression is $3$.

Q. The number of real solution(s) of the equation $2\left[x\right]=x+\left\{x\right\}$ is
($[.]$ denotes the greatest integer function and $\{.\}$ denotes the fractional part function)

1. $1$
2. $2$
3. more than $2$ but finite
4. infinite

Q. For a suitably chosen real constant $a$, let a function, $f:R-\{-a\}\to R$ be defined by $f\left(x\right)=\frac{a-x}{a+x}$. Further suppose that for any real number $x\ne -a$ and $f\left(x\right)\ne -a,\left(fof\right)\left(x\right)=x$. Then $f(-\frac{1}{2})$ is equal to:

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

Q. If $\left\{x\right\}$ and $\left[x\right]$ represent the fractional and the integral part of $x$ respectively, then $\frac{2019}{2020}\left[x\right]+\frac{x}{2020}+\sum _{r=1}^{2019}\frac{\{x+r\}}{2020}$ is equal to

1. $x$
2. $2021x$
3. $1010x$
4. $\frac{1}{2021}x$

Q. If $\left\{x\right\}$ and $\left[x\right]$ represent the fractional and the integral part of $x$ respectively, then $\frac{2019}{2020}\left[x\right]+\frac{x}{2020}+\sum _{r=1}^{2019}\frac{\{x+r\}}{2020}$ is equal to

1. $x$
2. $2021x$
3. $1010x$
4. $\frac{1}{2021}x$

Q. The solution of the equation $(x-2)\left[x\right]=\left\{x\right\}-1$ is ${}^{\prime }{x}^{\prime }$ such that $a\le x<b$ then $|a+b|=$
(where {.} represents fractional part function)

Q. Let $f:A\to R$ be a function defined by $f\left(x\right)={\log }_{\left\{x\right\}}\left(\frac{x-\left[x\right]}{|x|}\right)$, where $[.]$ and $\{.\}$ represent greatest integer function and fractional part function respectively. If $B$ is the range of $f$, then the number of integer(s) in $R-B$ is

Q. The function $f\left(x\right)=x+\ln \left(\frac{x}{1+x}\right)$ is increasing in

1. $(-1,0)$
2. $(-\infty ,\infty )$
3. only some part of its domain
4. its domain

Q.

If $f$ and $g$ are differentiable functions such that $f\left(2\right)=3,f\left(2\right)=1,f\left(3\right)=7,g\left(2\right)=-5$ and $g\left(2\right)=2$, then find the numbers indicated in problem $\left(\frac{f}{g}\right)\left(2\right)$.

Q. Let $f$ be a function whose domain is all real numbers. If $f\left(x\right)+2f\left(\frac{x+2001}{x-1}\right)=4013-x$ for all $x$ not equal to $1,$ then the value of $f\left(2003\right)$ is

Q. The domain of the function $f\left(x\right)=\frac{4}{\{x-4\}}$ is
(where $\{.\}$ denotes the fractional part of $x$)

1. $R-Z$
2. $R$
3. $Z-\left\{4\right\}$
4. $R-\left\{4\right\}$

Q. Let $f:R\to R$ be defined as $f\left(x\right)=2x–1$ and $g:R–\left\{1\right\}\to R$ be defined as $g\left(x\right)=\frac{x-\frac{1}{2}}{x-1}$. Then the composition function $f\left(g\right(x\left)\right)$ is:

1. both one-one and onto
2. onto but not one-one
3. neither one-one nor onto
4. one-one but not onto