Least Integer Function

Q. If $[x+[2x\left]\right]<3,$ where $[.]$ denotes the greatest integer less than or equal to $x,$ then

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

Q. The function $f:R\to R$ defined by $f\left(x\right)=[x{]}^2+[x+1]-3,$ (where [.] represents the greatest integer function). Then which of the following is/are true?$

1. $f\left(x\right)$ one-one
2. $f\left(x\right)$ many-one
3. $f\left(0.5\right)=-2$
4. $f\left(0.7\right)=0$

Q. If $y={\tan }^{-1}\frac{1}{1+x+x^2}+{\tan }^{-1}\frac{1}{x^2+3x+3}$
$+{\tan }^{-1}\frac{1}{x^2+5x+7}+\cdots +$ upto $n$ terms, then

1. $y^{\prime }\left(0\right)=\frac{-n^2}{1+n^2}$
2. $y^{\prime }\left(0\right)=\frac{n^2}{1+n^2}$
3. $y^{\prime }(-n)=\frac{n^2}{1+n^2}$
4. $y^{\prime }(-n)=\frac{-n^2}{1+n^2}$

Q.

If a, b be positive real numbers such that $a^2$ + $b^2$ = 8, then maximum value of a+b is _____.

1. 8

2. 4

3. 2

4. 1

Q. If $-5<\left[x\right]\le 6,$ then
(where $[.]$ denotes the greatest integer function)

1. $x\in [-4,7)$
2. $x\in (-4,7)$
3. $x\in (-5,7)$
4. $x\in [-4,6]$

Q. Let $f\left(x\right)={\left[x\right]}^2+[x+1]-3$, where $\left[x\right]$ is greatest integer less than or equal to $x$, then

1. $f\left(x\right)$ is a many one and into function
2. $f\left(x\right)=0$ for infinite number of values of $x$
3. None of these
4. $f\left(x\right)=0$ for only two real values

Q. Let $f\left(x\right)+(x+\frac{1}{2})f(1-x)=1$ and $g\left(x\right)=\frac{e^{-x}}{1+\left[\ln x\right]},$ where $[.]$ is the greatest integer function. Then domain of $(f+g)$ is

1. $R-[\frac{1}{e},1)$
2. $(0,\infty )-[-\frac{1}{2},\frac{1}{e})$
3. $(0,\infty )-[\frac{1}{e},1)$
4. $(0,\infty )-[-\frac{1}{2},1)$

Q. Which of the following is/are correct regarding $4\left\{x\right\}=x+\left[x\right]$?
(where {.} represents fractional part function and [.] represents greatest integer function)

1. There exist $3$ solutions.
2. There exist $2$ solutions.
3. The sum of solutions is $\frac{5}{3}$
4. The sum of solutions is $\frac{4}{3}$

Q. Let $g\left(x\right)=\frac{f\left(x\right)}{(x-a)(x-b)(x-c)},$ where $f\left(x\right)$ is a polynomial of degree less than $3.$ If $(a-b)(b-c)(c-a)=k,$ where $k$ is a non-zero constant, then

1. $\int g\left(x\right)dx=\frac{1}{k}\left|\begin{array}{ccc}1& a& f\left(a\right)\ln |x-a|\\ 1& b& f\left(b\right)\ln |x-b|\\ 1& c& f\left(c\right)\ln |x-c|\end{array}\right|+C$
2. $\int g\left(x\right)dx=-\frac{1}{k}\left|\begin{array}{ccc}1& a& f\left(a\right)\ln |x-a|\\ 1& b& f\left(b\right)\ln |x-b|\\ 1& c& f\left(c\right)\ln |x-c|\end{array}\right|+C$
3. $\frac{dg\left(x\right)}{dx}=\frac{1}{k}\left|\begin{array}{ccc}1& a& f\left(a\right)(x-a{)}^{-2}\\ 1& b& f\left(b\right)(x-b{)}^{-2}\\ 1& c& f\left(c\right)(x-c{)}^{-2}\end{array}\right|$
4. $\frac{dg\left(x\right)}{dx}=-\frac{1}{k}\left|\begin{array}{ccc}1& a& f\left(a\right)(x-a{)}^{-2}\\ 1& b& f\left(b\right)(x-b{)}^{-2}\\ 1& c& f\left(c\right)(x-c{)}^{-2}\end{array}\right|$

Q. Draw the graph of the smallest integer function
$f\left(x\right)=\left[x\right]$

Q. We want to find a polynomial f(x) of degree n such that f(1) = $\sqrt{2}$ and f(3) =$\pi$. Which of the following is true?

1. There is exactly one such polynomial and it has degree 1
2. There does not exist such a polynomial
3. There are infinitely many such polynomials for each n $\ge$1
4. There are infinitely many such polynomials for each n $\ge$ 2 but not infinitely many for n =1

Q. Let $x$ be a real number $\left[x\right]$ denotes the greatest integer function, and $\left\{x\right\}$ denotes the fractional part and $\left(x\right)$ denotes the least integer function, then solve the following.$\left[x\right]+|x-2|\le 0$ and $-1\le x\le 3$, Number of elements in set is

Q. Let $f\left(x\right)=\{\begin{array}{cc}{\tan }^{-1}\alpha -5x^2,& 0<x<1\\ -6x,& x\ge 1\end{array}.$
If $f\left(x\right)$ has a maximum at $x=1,$ then

1. $\alpha \in (\tan 1,\infty )$
2. $\alpha \in (\frac{\pi }{4},\infty )$
3. $\alpha \in (-\infty ,-\frac{\pi }{4})$
4. $\alpha \in (-\infty ,-\tan 1)$

Q. If $0<x<1000$ and $\left[\frac{x}{2}\right]+\left[\frac{x}{3}\right]+\left[\frac{x}{5}\right]=\frac{31}{30}x$ where [.] GIF; the number of possible values of x is

1. 34
2. 33
3. 32
4. 35

Q. Match the statements of Column $I$ with values of Column $II$

	Column I		Column II
A.	If $f\left(x\right)=x+1$, when $x<0$, $f\left(x\right)=x^2-1$, for $x\ge 0,$ then $f\left(f\right(x\left)\right)$, for $-1\le x\le 0$ is	1.	$\frac{x-3}{2}$
B.	If $f\left(\frac{2\tan x}{1+{\tan }^{2}x}\right)=$$\frac{(\cos 2x+1)({\sec }^{2}x+2\tan x)}{2}$ then $f\left(x\right)$ is	2.	$x^2+2x$
C.	If $f(x+y+1)=(\sqrt{f\left(x\right)}+\sqrt{f\left(y\right)}{)}^2$ for all $x,y\in R$ and $f\left(0\right)=1,$ then $f\left(x\right)$ is	3.	$1+x$
D.	If $4<x<5$ and $f\left(x\right)=\left[\frac{x}{4}\right]+2x+2$, where $\left[y\right]$ is the greatest integer $\le y,$ then $f^{-1}\left(x\right)$ is	4.	$(x+1{)}^2$

1. $A-2.B-4,C-3,D-1$
2. $A-2.B-3,C-1,D-4$
3. $A-3.B-2,C-4,D-1$
4. $A-1.B-3,C-4,D-2$

Q.

Express the following in words :

$15÷x\ne 2$

Q. The given equation $4x^3-7y^3=2003$ has;

1. one integer solutions
2. two integer solutions
3. no integer solutions
4. three integer solutions