Method of Intervals

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

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

Q. The outcome of each of $30$ items was observed ; $10$ items gave an outcome $\frac{1}{2}-d$ each, $10$ items gave outcome $\frac{1}{2}$ each and the remaining $10$ items gave outcome $\frac{1}{2}+d$ each. If the variance of this outcome data is $\frac{4}{3}$, then $|d|$ equals to

1. $2$
2. $\frac{\sqrt{5}}{2}$
3. $\frac{2}{3}$
4. $\sqrt{2}$

Q. All real values of $x$ which satisfy $x^2-3x+2>0$ and $x^2-3x-4\le 0$ lie in the interval

1. $[-1,1)\cup (2,4]$
2. $[3,6]$
3. $(-5,0)$
4. $[1,6]$

Q. If $2^x+2^{f\left(x\right)}=2$ , then the domain of the function $f\left(x\right)$ is

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

Q. Let $S$ be the solution set of the inequality $4x+5\le 2x+17$ (where $x$ is a whole number), then $n\left(S\right)$ is equal to

1. $5$
2. $6$
3. $7$
4. $8$

Q. The value of $x$, if ${\log }_4(3x^2+11x)=1$

1. $4$
2. $\frac{1}{3}$
3. $-4$
4. $-\frac{1}{3}$

Q. Solve the irrational inequality:
$\frac{3}{\sqrt{2-x}}-\sqrt{2-x}\le 2$

1. $(-5,3)$
2. $[-7,1]$
3. $(-\infty ,-1)$
4. $(-\infty ,1]$

Q. The number of integral solution(s) of $(x-5{)}^5(x-8{)}^8(x-11{)}^{11}<0$ is

Q. If $x$ satisfies the inequality $(x^2-x-1)(x^2-x-7)<-5,$ then which of the following statements is (are) TRUE?

1. Number of integral values of $x$ satisfying the given inequality is $2$
2. $x^2+1\in (2,5),$ if $x<0$
3. $x^2-1\in (3,8),$ if $x>0$
4. Number of integral values of $x$ satisfying the given inequality is $0$

Q. Solution set of $\frac{2}{x^2-x+1}-\frac{1}{x+1}\ge \frac{2x-1}{x^3+1}$ is

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

Q. The solution set of $\frac{x^2-4}{x^2-16}\le 0$ is

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

Q. If $2^x+2^{f\left(x\right)}=2$ , then the domain of the function $f\left(x\right)$ is

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

Q. The set of the solutions for $\frac{(x+1)(x-3)}{(x+5)}<0$ is

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

Q. A natural number x is chosen at random from the first 100 natural numbers. Then the probability that, $x+\frac{100}{x}>50$ is:

1. $\frac{1}{20}$
2. $\frac{11}{20}$
3. $\frac{1}{3}$
4. $None$

Q. The maximum value of $y=|x-4|-|x-7|$ is

Q. Solution set of $x(2^x-1)(3^x-9)(x-3)<0$ is

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

Q. The solution set of $\frac{3}{x+1}-\frac{2}{x-1}<0$ is

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

Q.

Complete the equation so it has infinitely many solutions. $4x+7=4(x+3)–\_\_$

Q. If $\sqrt{2x-5}<3$, then $x\in$

1. $(\frac{5}{2},\infty )$
2. $[\frac{5}{2},\infty )$
3. $[\frac{5}{2},7)$
4. $(-\infty ,7)$

Q. Number of integral solutions of $\frac{2x-1}{x-7}\le 0$ is

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

Q. The solution set of $x^4-8x^2-9\le 0$ is

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

Q.

If $\left[x\right]$ denotes the greatest integer less than or equal to $x$, then the value of ${\int }_0^2\left[|x-2|+\left[x\right]\right]dx$ is:

1. $2$

2. $3$

3. $1$

4. $4$

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

Q. All the values of $x$ for which $x^2-5x+6$ is non-negative are

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

Q. Solution set of $(x+1)(x-1{)}^2(x-2)\ge 0$ is

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

Q. The set of the solutions for $\frac{(x+1)(x-3)}{(x+5)}<0$ is

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

Q. If $f\left(x\right)=\frac{2x-5}{x+3},$ then

1. Domain of $f\left(x\right)$ is $R$
2. Domain of $f\left(x\right)$ is $R-\{-3\}$
3. Range of $f\left(x\right)$ is $R-\left\{2\right\}$
4. Range of $f\left(x\right)$ is $R$

Q. If ${\left(\frac{x}{2}\right)}^{{\log }_{2}x}=8x,$ then the values of $x$ is/are

1. $\frac{1}{4}$
2. $\frac{1}{2}$
3. $4$
4. $8$

Q. The set of solutions for $x+\frac{1}{x}\ge 2$ is

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

Q. If $\frac{1}{{\log }_4\left(\frac{x+1}{x+2}\right)}>\frac{1}{{\log }_4(x+3)},$ then the sets of values of $x$ that satisfy the expression are

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

Q. Solution of $|x+2|+|2x+6|+|3x-3|=12$ is

1. $\{-1,1\}$
2. $\{-\frac{2}{5},\frac{6}{7}\}$
3. $\{-\frac{5}{2},\frac{7}{6}\}$
4. $\{\frac{5}{2},-\frac{7}{6}\}$