Domain

Q. x+1/2 log(x+1/2) (x2+ 2x-3/4x2-4x-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. $-2$
2. $1$
3. $8$
4. $-3$

Q. The domain of the definition of the function $f\left(x\right)=\frac{1}{4-x^2}+{\log }_{10}(x^3-x)$ is :

1. $(-1,0)\cup (1,2)\cup (2,\infty )$
2. $(-1,0)\cup (1,2)\cup (3,\infty )$
3. $(1,2)\cup (2,\infty )$
4. $(-2,-1)\cup (-1,0)\cup (2,\infty )$

Q. The domain of $f\left(x\right)=\sqrt{\frac{1-5^x}{7^{-x}-7}}$ is

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

Q.

The function $f\left(x\right)=\log \left(x+\sqrt{x^2+1}\right)$ is

1. an even function

2. an odd function

3. a periodic function

4. neither an even nor an odd function

Q. The domain of the function $f\left(x\right)=\log |\log x|,$ is

1. $(-\infty ,1)\cup (1,\infty )$
2. $(-\infty ,1)$
3. $(0,1)\cup (1,\infty )$
4. $(1,\infty )$

Q. The domain of the function $f\left(x\right)=\ln \left\{x\right\}$ is
(where {.} represents fractional part function)

1. $R-Z$
2. $R$
3. $R^{+}$
4. $R-W$

Q.

The domain of $f\left(x\right)={\sin }^{-1}\left[{\log }_2\left(\frac{x}{2}\right)\right]$ is

1. $0\le x\le 1$

2. $0\le x\le 4$

3. $1\le x\le 4$

4. $\left[-2,-1\right]\cup \left[1,2\right]$

Q.

How do I find the value of $\tan \left(\frac{\mathrm{\pi }}{12}\right)$.

Q.

The domain of $F\left(x\right)=\frac{{\log }_2\left(x+3\right)}{\left(x^2+3x+2\right)}$ is

1. $R-\left\{-1,-2\right\}$

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

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

4. $\left(-3,\infty \right)-\left\{-1,-2\right\}$

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

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

Q.

If R = {$(x,y):x,yϵZ,x^2+y^2\le 4$} is a relation on Z, then domain of R is

1. None of these

2. {0, 1, 2}

3. {0, -1, -2}

4. {-2, -1, 0, 1, 2}

Q.

The range of the function is $y=3sin\left(\sqrt{\frac{{\mathrm{\pi }}^2}{16}-x^2}\right)$

1. $[0,1]$

2. $\left[0,\sqrt{\frac{3}{2}}\right]$

3. $\left[0,\frac{3}{\sqrt{2}}\right]$

4. $[0,\infty )$

Q.

The domain of definition of the function $f\left(x\right)=\sqrt{x-1}+\sqrt{3-x}$ is

1. $(-\infty ,3)$

2. $(1,3)$

3. $[1,3]$

4. $[1,\infty ]$

Q.

The domain of definition of $f\left(x\right)=\sqrt{4x-x^2}$ is

1. R - (0, 4)

2. R - [0, 4]

3. (0, 4)

4. [0, 4]

Q.

Let R be a relation on N defined by x + 2y = 8. The domain of R is

1. {2, 4, 8}

2. {1, 2, 3, 4}

3. {2, 4, 6}

4. {2, 4, 6, 8}

Q. The domain of $f\left(x\right)=\sqrt{a^2-x^2}$ (a > 0) is

1. (-a, a)
2. [-a, a]
3. [0, a]
4. (-a, 0]

Q. Let $|X|$ denotes the number of elements in a set $X.$ Let $S=1,2,3,4,5,6$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S,$ then the number of ordered pairs $(A,B)$ such that $1\le |B|<|A|,$ equals

Q.

The common difference of the A.P $b_1,b_2,\dots \dots ,b_m$ is $2$ more than the common difference of A.P $a_1,a_2,\dots \dots ,a_n$. If $a_{40}=-159,a_{100}=-399$and $b_{100}=a_{70}$, then $b_1$ is equal to.

1. $-127$

2. $81$

3. $127$

4. $-81$

Q.

$\cos \frac{\pi }{11}+\cos \frac{3\pi }{11}+\cos \frac{5\pi }{11}+\cos \frac{7\pi }{11}+\cos \frac{9\pi }{11}=?$

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

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

3. $1$

4. $-1$

Q. The domain of the function $f\left(x\right)=\sqrt{{\log }_{x^2-1}x}$ is

1. $(1,\infty )$
2. $(\sqrt{2},\infty )$
3. $(0,\sqrt{2})$
4. $(0,\infty )$

Q.

The domain of the function $f\left(x\right)=\sqrt{5|x|-x^2-6}$ is

1. $(-3,-2)\cup (2,3)$

2. $(-3,-2)\cup (2,3)$

3. None of these

4. $[-3,-2]\cup [2,3]$

Q. The number of integral value(s) of $x$ which is/are not in the domain of $f\left(x\right)=\frac{2x}{2x^2+3x-20}$ is

Q.

Show that the greatest integer function f(x) =[x] is continuous at all points except at integer points

Q. The domain of $f\left(x\right)=\sqrt{x-4-2\sqrt{x-5}}-\sqrt{x-4+2\sqrt{x-5}}$ is

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

Q. The set of all real values of ${}^{\prime }{a}^{\prime }$ so that the range of function $y=\frac{x^2+a}{x+1},x\in R-\{-1\}$ is $R$, is

1. $R-\{-1\}$
2. $(-\infty ,-1)$
3. $(-\infty ,-1]$
4. $(1,\infty )$

Q. The domain of the function $f\left(x\right)=\frac{\sqrt{{\log }_{0.4}\left(\frac{x-1}{x+5}\right)}}{x^2-36}$ is

1. $(-5,\infty )-\left\{6\right\}$
2. $(6,\infty )$
3. $(1,\infty )$
4. $(1,\infty )-\left\{6\right\}$

Q. If a, b, c, d are four vectors, then prove that
(a×b).(c×d)+(b×c).(a×d)+(c×a).(b×d)=0

Q.

The range of the function $f\left(x\right)=x^2+2x+2$ is

1. $\left(1,\infty \right)$

2. $(2,\infty )$

3. $(0,\infty )$

4. $[1,\infty )$

Q. The domain of the function $f\left(x\right)=\frac{1}{\sqrt{|[|x|-5]|-5}}$ is
(where [.] denotes the greatest integer function)

1. $(-\infty ,-5]$
2. $(-\infty ,-11]\cup [11,\infty )$
3. $(-\infty ,-5]\cup [5,\infty )$
4. $[11,\infty )$