Greatest Integer Function

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.

You know that $\frac{1}{7}=0.\overline{142857}$, can you predict what the decimal expression of $\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7}\mathrm{and}\frac{6}{7}$ without actually doing the long division.

Q. In how many ways can 10 true-false questions be replied

1. 20
2. 512
3. 1024
4. 100

Q.

What is meaning of integral multiples?

Q.

What is the square root of $0.4$?

Q.

If f and g are two functions defined as $f\left(x\right)=x+2,x\le 0;g\left(x\right)=3,x\ge 0,$then the domain of $f+g$ is

1. $\left\{0\right\}$

2. $[0,\infty )$

3. $(-\infty ,\infty )$

4. $(-\infty ,0)$

Q.

Let $\left[t\right]$ denote the greatest integer$\le t$. If for some $\lambda \in R-\left\{0,1\right\}$, $\underset{x\to 0}{lim}\left|\frac{1-x+\left|x\right|}{\lambda -x+\left[x\right]}\right|=L$ then $L$ is equal to.

1. $0$

2. $2$

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

4. $1$

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 $\left[x\right]$ denotes the greatest integer $\le x$, then $\left[\frac{2}{3}\right]+\left[\left(\frac{2}{3}\right)+\left(\frac{1}{99}\right)\right]+\left[\left(\frac{2}{3}\right)+\left(\frac{2}{99}\right)\right]+\dots .+\left[\left(\frac{2}{3}\right)+\left(\frac{98}{99}\right)\right]=?$

1. $99$

2. $98$

3. $66$

4. $65$

5. $33$

Q.

The sum of all five-digit numbers that can be formed using the digits $1,2,3,4,5$ when repetition of digits is not allowed, is

1. $366000$

2. $660000$

3. $360000$

4. $3999960$

Q. If $f:[0,1]\to [-1,3]$ is defined by $f\left(x\right)=x^2+x+1,$ then $f$ is

[1 mark]

1. a many-one function
2. a one-one function
3. an onto function
4. a bijective function

Q.

If $x+y\le 2$ ; $x\ge 0,y\ge 0$ is the point at which maximum value of $3x+2y$ attained will be:

1. $\left(0,2\right)$

2. $\left(0,0\right)$

3. $\left(2,0\right)$

4. $\left(\frac{1}{2},\frac{1}{2}\right)$

Q.

The logarithm of $0.0625$ to the base $2$ is______________?

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.

The number of roots of the quadratic equation $8se{c}^2\theta -6sec\theta +1$ = 0 is

1. Infinite

2. 0

3. 1

4. 2

Q.

Let $\left(\frac{1-ix}{1+ix}\right)=\left(a-ib\right)$ and $a^2+b^2=1$ where $a$ and $b$ are real, then $x$ equal to

Q. The domain of the function $f:R\to R$ defined by $f\left(x\right)=\sqrt{x^2-7x+12}$ is

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

Q. If $x\in [0,10]$, then the number of integral value(s) of $x$ satisfying the equation $x^2+4x+\left[x\right]+6=0$ (where [.] denotes the greatest integer function) is

Q.

The area bounded by the curves $y=x$ and $y=x^3$ is equal to

1. $0$

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

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

4. $1$

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. $94$
4. $52$

Q.

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

1. -1

2. 0

3. 1

4. 2

Q. Prove that cos√x is non periodic.

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. $x\ge e$
4. $x\ge 1$

Q.

The number of ordered pairs $(r,k)$ for which $6·{}^{35}C_r=\left(k^2-3\right){}^{36}C_{r+1}$, where $\mathrm{k}$ is an integer, is

1. $4$

2. $6$

3. $2$

4. $3$

Q. If z be any complex number and if $z^2+az+b=0$ has roots both of which has unit modulus, then

1. 
2. 
3. arg(b) = 2 arg (a)
4. None of these

Q. If $\left[x^2\right]+x-a=0$ has a solution $\left(\right[.]$ represents the greatest integer function$)$, where $a\in N,a\le 20$, then the total number of distinct values of $a$ is

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

Q. Consider the sequence of number $[n+\sqrt{2n}+\frac{1}{2}]$ for $n\ge 1$, where $\left[x\right]$ denotes the greatest integer not exceeding $x$. If the missing integers in the sequence are $n_1<n_2<n_3<\dots \dots$ then find the $n_{12}$.

Q. If $x\in (6,8),$ then the value of $\left[\frac{x}{2}\right]$ is
(where $[.]$ represents the greatest integer function)

Q. The number of integers in the domain of $f\left(x\right)=\sqrt{\frac{\left[x\right]-2}{5-\left[x\right]}},$ where $[.]$ represents the greatest integer function, is

Q. The graphs of the polynomial $x^2-1$ and $\cos x$ intersect

1. at exactly $3$ points
2. at least $4$ but at finitely many points
3. at exactly two points
4. at infinitely many points