Greatest Integer Function

Q. Let $\left[K\right]$ denotes the greatest integer less than or equal to $K$. If number of positive integral solutions of the equation $\left[\frac{x}{\left[{\pi }^2\right]}\right]=\left[\frac{x}{\left[11\frac{1}{2}\right]}\right]$ is $n$, then the value of $n$ is :

Q. $f:R\to [-1,\infty )$ and $f\left(x\right)=ln\left(\right[|\sin 2x|+|\cos 2x|\left]\right)$ (where [.] denotes greatest integer function) then which of the following is NOT correct?

1. f(x) is periodic but fundamental period is not defined.
2. f(x) is into function.
3. $(R^{-}\cap Rangeoff(x\left)\right)$ is a null set
4. f(x) is invertible in $[0,\frac{\pi }{4}]$

Q.

What is the range of the greatest integer function?

Q. The value of the expression $\left[\frac{10}{27}\right]+\left[\frac{11}{27}\right]+\left[\frac{12}{27}\right]+...+\left[\frac{35}{27}\right]$ is

Q. If $[x+[x+[x+[x+\left[x\right]]]]]=10$, then $x$ lies in

1. $[10,11)$
2. $[2,3)$
3. $Z$
4. $[5,6)$

Q. Which of the following options is/are true for the Greatest integer function [.]?

1. $\left[x\right]\le x<\left[x\right]+1$
2. $x-1<\left[x\right]\le x$
3. $[x+m]=x+\left[m\right],$ if $m\in Z$
4. $[x+m]=\left[x\right]+m,$ if $m\in Z$

Q. $f:R\to [-1,\infty )$ and $f\left(x\right)=ln\left(\right[|\sin 2x|+|\cos 2x|\left]\right)$ (where [.] denotes greatest integer function) then which of the following is NOT correct?

1. $(R^{-}\cap Rangeoff(x\left)\right)$ is a null set
2. f(x) is periodic but fundamental period is not defined.
3. f(x) is invertible in $[0,\frac{\pi }{4}]$
4. f(x) is into function.

Q.

Find the quotient.

$\left(-72\right)÷9$

Q. Which of the following options is/are true for the Greatest integer function [.]?

1. $\left[x\right]\le x<\left[x\right]+1$
2. $x-1<\left[x\right]\le x$
3. $[x+m]=x+\left[m\right],$ if $m\in Z$
4. $[x+m]=\left[x\right]+m,$ if $m\in Z$

Q. If $y=3\left[x\right]+1=4[x-1]-10$, then the value of $[x+2y]$ is

Q. Let $\left[x\right]$ denote the greatest integer function of $x$. If the domain of the function $\frac{1}{[x{]}^2-7[x]+12}$ is $R-[a,b),$ then the value of $a+b$ is

Q. The domain of the function $f\left(x\right)=\frac{7}{\left[|x|\right]-5}$ is
(where $[.]$ denotes the greatest integer function)

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

Q. If $y=2\left[x\right]+30,y=3[x-2]+15$, then $[x+y]$ is equal to
(where [.] denotes greatest integer function)

1. $93$
2. $72$
3. $52$
4. $94$

Q.

Which of the following functions are identical to f(x) =
$f\left(x\right)=\{\begin{array}{cc}x,& 1\le x<2\\ x^2,& 2\le x<3\end{array}$

(1 $\le$ x < 3)

1. $[x{]}^x$

2. $x^{\left[x\right]}$

3. $x^2$

4. $x^3$

Q. Let $\left[x\right]$ denote the greatest integer $\le x$, where $x\in R$. If the domain of the real valued function $f\left(x\right)=\sqrt{\frac{\left|\right[x\left]\right|-2}{|\left[x\right]|-3}}$ is $(-\infty ,a)\cup [b,c)\cup [4,\infty ),$ $a<b<c,$ then the value of $a+b+c$ is

1. $1$
2. $-2$
3. $-3$
4. $8$

Q. If $f\left(x\right)=\sqrt[n]{n{x}^m}$ is an even function, then $m$ is

1. an even integer
2. an odd integer
3. any positive integer
4. any real number

Q. If $f:R\to R$ is defined by $f\left(x\right)=x-\left[x\right]+\frac{1}{4},$ where $\left[x\right]$ is the greaterst integer not exceeding $x$, then the set $\{x\in R:f(x)=\frac{1}{4}\}$ is equal to

1. $Q$, the set of all rational numbers
2. $N$, the set of all natural numbers
3. $W$, the set of all whole numbers
4. $Z$, the set of all integers

Q. The set of values of $x$ for which the function $f\left(x\right)={\log }_{[x+\frac{1}{2}]}|x^2-5x+6|$ is defined is
(where [.] denote the greatest integer function)

1. $[\frac{3}{2},2)$
2. $(2,3)$
3. $(3,\infty )$
4. $[2,\infty )$

Q.

Let $\left[k\right]$ denotes the greatest integer less than or equal to $k.$ Then the number of positive integral solutions of the equation $\left[\frac{x}{\left[{\pi }^2\right]}\right]=\left[\frac{x}{\left[11\frac{1}{2}\right]}\right]$ is

1. $29$
2. $24$
3. $21$
4. $34$

Q. The range of $f\left(x\right)=\frac{x^2-81}{x-9}$ is

1. $R-\left\{18\right\}$
2. $R-\left\{9\right\}$
3. $R-\{\pm 9\}$
4. $R-\{\pm 18\}$

Q.

What is the value of [x] + [-x] , where [x] is the greatest integer function

1. -1

2. 0

3. 1

4. 2

Q. If $f:R\to R$ is defined by $f\left(x\right)=x-\left[x\right]+\frac{1}{4},$ where $\left[x\right]$ is the greaterst integer not exceeding $x$, then the set $\{x\in R:f(x)=\frac{1}{4}\}$ is equal to

1. $Q$, the set of all rational numbers
2. $N$, the set of all natural numbers
3. $W$, the set of all whole numbers
4. $Z$, the set of all integers

Q. The range of the function $f\left(x\right)=\left[\right\{2x+3\left\}\right]$ is
$\left(\right[.]$ represents the greatest integer function and $\left\{x\right\}$ is the fractional part of $x)$

1. $\left\{3\right\}$
2. $\left\{0\right\}$
3. $\{0,1\}$
4. $\left\{1\right\}$

Q. The domain of $f\left(x\right)=\ln \left[x\right]$ is
(where [.] denotes the greatest integer function)

1. $x>0$
2. $x>1$
3. $x\ge 1$
4. $x\ge e$

Q. Let $g\left(x\right)=1+x-\left[x\right]$ and $f\left(x\right)=\{\begin{array}{ccc}-1& x<0& \\ 0,& x=0& \\ 1,& x>0& \end{array}$ then for all $x$, $f\left[g\right(x\left)\right]$ is equal to:

1. $x$
2. 1
3. $f\left(x\right)$
4. $g\left(x\right)$

Q. Let $\left[K\right]$ denotes the greatest integer less than or equal to $K$. If number of positive integral solutions of the equation $\left[\frac{x}{\left[{\pi }^2\right]}\right]=\left[\frac{x}{\left[11\frac{1}{2}\right]}\right]$ is $n$, then the value of $n$ is

Q.

Carry out the following division.

$\left(-4\right)$ by $2$

Q. The domain of $f\left(x\right)$ is $[0,1]$, then the domain of $y=f\left(e^x\right)+f\left(|\right[x\left]|\right)$ is
(where [.] denotes greatest integer function)

1. $(-e,-1)$
2. $[-1,0]$
3. $(-\infty ,2)$
4. $[-1,0]\cap [0,1]$

Q.

Which of the following is true about [x], greatest integer function ?

1. $\left[\right[x\left]\right]=\left[x\right]$

2. [x + n ] = [x] + n , n $\in$ I

3. x - 1 < [x] $\le$ x

4. [-x] = -1 - [x] , when x is not an integer

Q. The domain of the function $f\left(x\right)=3-\frac{1}{[x+3]}$ is
(where [.] denotes the greatest integer function)

1. $R$
2. $R-\{-3\}$
3. $R-[-3,-2)$
4. $\varphi$