Fractional Part Function

Q. Let $A=\{x\in R:x\text{is not a positive integer}\}$.
Define a function $f:A\to R$ as $f\left(x\right)=\frac{2x}{x-1}$,
then $f$ is :

1. not injective
2. injective but not surjective
3. neither injective nor subjective
4. surjective but not injective

Q.

if non zero numbers $a,b,c$ are in HP then the straight line $\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0$ always passes through a fixed point

1. $\left(1,–\frac{1}{2}\right)$

2. $\left(1,-2\right)$

3. $(-1,-2)$

4. $(-1,2)$

Q. Let the solution of the equation
$5\left\{x\right\}=2\left[x\right]+x$ is $a/4$ then the value of $a$ is
where {.} and [.] represents fractional part function and greatest integer function respectively.

Q. If $x\in R$, ${\left(\log \right\{x\left\}\right)}^2-3\log \left[x\right]+2=0$ and $\frac{1-2x}{3-|x-1|}=1,$ where $\{.\}$ is the fractional part function and $[.]$ is the greatest integer function, then the number of non-integral value(s) of $x$ is

Q. The value of $\left\{2.32\right\}-\left\{3.44\right\}+\{-1.35\}+\{-2.22\}$ is
(where $\{.\}$ represents fractional part function)

Q. Let $N$ be the set of natural numbers and two functions $f$ and $g$ be defined as $f,g:N\to N$ such that
$f\left(n\right)=\{\begin{array}{cc}\frac{n+1}{2}& \text{if}n\text{is odd}\\ \frac{n}{2}& \text{if}n\text{is even}\end{array}$
and $g\left(n\right)=n-(-1{)}^n$. Then $f\circ g$ is :

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

Q. Let $f\left(x\right)$ be a function defined as $f\left(x\right)=\frac{a}{|x|}+b.$ If $f\left(6\right)=3$ and $f(-3)=4,$ then which of the following is/are correct?

1. $f\left(x\right)=\frac{6}{|x|}+2$
2. Domain of $f\left(x\right)$ is $R$
3. Domain of $f\left(x\right)$ is $R-\left\{0\right\}$
4. Range of $f\left(x\right)$ is $(2,\infty )$

Q. $f\left(x\right)=\left\{\frac{|x+5|-2}{x^2+10x+21}\right\},$where {.} denotes fractional part function then which of the following is true?

1. Number of integers in the domain of f(x) is 2.
2. Number of integers in the domain of f(x) is 3.
3. Range of f(x) is $(0,\frac{1}{2}]$
4. Number of integers in the domain of f(x) is infinite.

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

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. $f\left(x\right)=\left\{\frac{|x+5|-2}{x^2+10x+21}\right\},$where {.} denotes fractional part function then which of the following is true?

1. Number of integers in the domain of f(x) is 3.
2. Number of integers in the domain of f(x) is 2.
3. Range of f(x) is $(0,\frac{1}{2}]$
4. Number of integers in the domain of f(x) is infinite.

Q.

Which of the following fractions have the same value as $16÷(-3)$? Check all that apply.

1. $16$ over negative $3$

2. $-16$ over negative $3$

3. $-16$ over $3$

4. $16$ over $3$

5. $-\left(16\mathrm{over}3\right)$

6. $-\left(16\mathrm{over}-3\right)$

Q. Let the solution of the equation $5\left\{x\right\}=2\left[x\right]+x$ be $a/4.$ Then the value of $a$ is where $\{.\}$ and $[.]$ represents fractional part function and greatest integer function respectively.

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

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

Q. The set of values of $x,$ which satisfy $x^2-13x+\left[x\right]+36=0$ is

1. $[5,7)$
2. $(6,7)$
3. $[6,7)$
4. $(5,7)$

Q. The value of $\left\{2.32\right\}-\left\{3.44\right\}+\{-1.35\}+\{-2.22\}$ is
(where $\{.\}$ represents fractional part function)

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