Greatest Integer Function

Q. The values of x satisfying the inequality $0\le \frac{2}{3}\left[x\right]\le 1$ is in the interval

1. [1, 2)
2. [0, 2)
3. [1, 3)

Q. Let S be the sum of all x in the interval of [0, 2π] such that 3cotx^2+ cotx +8. = 0, then he value of S/π is a) 3 b)4 c)5 d)6

Q. 70. If cosx+cos7x+cos3x+cos5x=0 then x= (A)n(pi)/4, n=Z (B)n(pi)/2, n=I (C)n(pi)/8;n+8k, n, z=Z (D)n(pi)/3;n=Z

Q. Let g(x)=1 + x - [x] and f(x) =$\{\begin{array}{ccc}-1& x<0& \\ 0,& x=0& \\ 1,& x>0& \end{array}$ then for all x, f[g(x)] is equal to:
(IIT 2001)

1. 1
2. x
3. f(x)
4. g(x)

Q. The domain of the function $f\left(x\right)=\sqrt{x^2-[x{]}^2}$ is
$($ $[.]$ represents the greatest integer function $)$

1. $[0,\infty )$
2. $(-\infty ,0]$
3. $R$
4. None of these

Q. If $[x{]}^2-7[x]+10>0$, then $x$ lies in the interval
(Here, $[.]$ denotes the greatest integer function)

1. $[6,\infty )$
2. $(-\infty ,2)$
3. $(-\infty ,3)$
4. $[5,\infty )$

Q. Let $f:[2,3]\to B$ be a function defined by $f\left(x\right)=\left[{\log }_2\right[x^2+2x-1\left]\right]$, where $[.]$ represents the greatest integer function. If range of $f$ is $B$, then

1. $1\in B$
2. $2\in B$
3. $3\in B$
4. $4\in B$

Q. For $x\in R,\text{let}\left[x\right]$ denote the greatest integer $\le x,$ then the sum of the series $[-\frac{1}{3}]+[-\frac{1}{3}-\frac{1}{100}]+[-\frac{1}{3}-\frac{2}{100}]+....+[-\frac{1}{3}-\frac{99}{100}]$ is :

1. $-135$
2. $-153$
3. $-131$
4. $-133$

Q. The domain of the function $f\left(x\right)=\frac{\sin \left(\log \right(x^2-4x+4\left)\right)}{\left(\right[x]-1)}$, where $[.]$ denotes the greatest integer function, is

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

Q.

Draw the curve $\frac{2^x}{2^{\left[x\right]}}$. Where [.] denotes greatest integer function

1. 

2. 

3. 

4. 

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 I < x < I +1, Find [-x], where I is an integr

1. -I

2. -I+1

3. -I-1

4. I-1

Q. $-8.0,\sqrt{5},\frac{5}{7},-\sqrt{18},\sqrt{32},4.28,\pi ,3,\frac{8}{15},0.075$
How many negative integers are there amongst above mentioned numbers?

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

Q. If the characteristics of ${\log }_{10}0.0000739$ is $a$, then value of $\left[a\right]$ is
(where [.] denotes the greatest integer function)

1. $-4$
2. $-5$
3. $-6$
4. $-7$

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. Solve $-12x>30$, when (i) $x$ is a natural number. (ii) $x$ is an integer

Q.

How many real numbers satisfy the condition 0 $\le$$\frac{2}{3}\left[x\right]\le$1

__

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.

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

1. -1

2. 0

3. 1

4. 2

Q.

If I < x < I +1, Find [-x], where I is an integr

1. -I

2. -I-1

3. -I+1

4. I-1

Q. If $f:[0,2]\to A,f\left(x\right)=\left[x^2\right]-[x{]}^2$ is a real valued function, then minimum elements required in set $A$ is
(where [.] denotes greatest integer function)

1. $(-\infty ,0]$
2. $(-\infty ,1]$
3. $\{0,1,2\}$
4. $N$

Q. The Set $K=\{k:0\le k\le 1,k\in R\},$ where $R$ is the set of all real numbers is

1. an infinite set
2. a finite set
3. a singleton set

Q.

Identify the function from the description given.

1. Its an irrational function

2. Its graph is mirror image of y =$x^{2n}$ about y = x, for some n $\ge$ 1, n $\in$ I and x $\ge$ 0

3. x $\le$ f(x) $\le$ $x^{1/6}$ if x $\in$ [0, 1]

4. Its domain and range are [0, $\infty$)

1. 

2. 

3. 

4. 

Q.

Find the function corresponding to the graph given below, $1\le x\le 3$ ( The function which closely describes the graph )

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

2. $[x{]}^x$

3. $\left[x^3\right]$

4. $\left[x^x\right]$

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. 37.If x=2[ , if x=2(b+sinb) and y=2(1-cosb) then value of dy/dx is

Q. Let $f:(4,6)\to (6,8)$ be defined by $f\left(x\right)=x+\left[\frac{x}{2}\right]$, where $[.]$ represents the greatest integer function. Then the range of $f$ is

1. $[7,8)$
2. $(7,8)$
3. $(6,8)$
4. $(6,7]$