Inequalities Involving Modulus Function

Q. The value(s) of $x$ satisfying the equation $x^9+\frac{9}{8}{x}^6+\frac{27}{64}{x}^3-x+\frac{219}{512}=0$ is/are

1. $\frac{-1+\sqrt{13}}{4}$
2. $-\frac{1}{2}$
3. $\frac{-1-\sqrt{13}}{4}$
4. $\frac{1}{2}$

Q.

If $\text{A}$ and $\text{B}$ are two events such that $\text{A ⊆ B,}$then $\text{P(B/A)}$

1. $0$

2. $1$

3. $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

4. $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

Q. The number of integral values of $k$ for which $\left|\frac{x^2+kx+1}{x^2+x+1}\right|<2$ for all real values of $x,$ is

Q. The solution set of $x^2-7x+12\ge 0$ is

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

Q.

What is the prime factorization of $3780$?

Q. If $\frac{1}{|x^2-2|}\le 2$, then $x$ lies in the interval

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

Q. Solution set of $(x^2-1)(x^3-1)(x^4-1)>0$ is

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

Q. Number of integer values of $x$ satisfying the inequality
$|x-3|+|2x+4|+|x|\le 11$ is

Q. Let $f\left(x\right)$ be a real-valued function such that $\left|f\right(x)+x^2+1|\ge |f\left(x\right)|+|x^2+1|$ and $f\left(x\right)\le 0$ for all real values of $x$. Then the absolute value of $\sum _{r=1}^5(1+f\left(r\right))$ is

Q. The solution set of $(x-1{)}^{99}(x+1{)}^{100}\le 0$ is

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

Q. The value of $k$ for which the equation $|x-2|+|x-6|-|x+1|=k$ has atleast one solution

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

Q. The solution set of $16-x^2\ge 0$ is

1. $(-\infty ,-4]\cup [4,\infty )$
2. $[-4,4]$
3. $(-4,4)$
4. $(-16,16)$

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

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

Q. The solution set of $\left|\frac{3x}{x-3}\right|+|x|=\frac{x^2}{|x-3|}$ is

1. $(3,\infty )$
2. $[3,\infty )$
3. $\left\{0\right\}\cup [3,\infty )$
4. $\left\{0\right\}\cup (3,\infty )$

Q. Which of the following option represent the inequalities $-3\le 1-2x<7$ and $-5<\frac{4-3x}{4}<7$ on the number line?

1. 
2. 
3. 
4. No Solution

Q. If $-7\le x^2+8x+5\le 14$, then the interval(s) in which $x$ lies is/are

1. $(-8,-6)$
2. $(-5,-2)$
3. $(-1,1)$
4. $(1,2)$

Q. If $|x+3|-|x-4|=4$, then $x=$

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

Q. Complete values of $a$ for which the equation $|x-1|+|x-2|+x-a>0$ has two solutions, is

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

Q. The number of prime numbers in the solution set of $|2x-9|+|2x-39|=30$ is equal to

Q. If $|1-\frac{|x|}{1+|x|}|\ge \frac{1}{2}$, then $x\in$

1. $[0,1]$
2. $[-1,0]$
3. $[-1,1]$
4. $R-[-1,1]$

Q. The number of solution(s) of the equation $|x-|6-x||-2x=6$ is

1. $1$
2. $0$
3. $2$
4. infinite

Q. Solve $\frac{|x+3|+x}{x+2}>1$

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

Q.

Find the quotient and remainder and verify your answers:

$345÷10$

Q. The solution set of $\left|\frac{3x}{x-3}\right|+|x|=\frac{x^2}{|x-3|}$ is

1. $(3,\infty )$
2. $[3,\infty )$
3. $\left\{0\right\}\cup (3,\infty )$
4. $\left\{0\right\}\cup [3,\infty )$

Q.

If $AB\times 4=192,$ then $A+B=7.$

State whether the statement given is true (T) or false (F).

1. True
2. False

Q. The solution set of the inequality $||x|-1<1-x,2\forall xϵR$ is equal to

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

Q. The solution set of $x^2+4x+9\ge 0$ is

1. $\varphi \left(\text{empty set}\right)$
2. $R$
3. $[-3,4]$
4. $[0,\infty )$

Q. If $|x|\ge 6,$ then $x$ can be represented on the number line by

1. 
2. 
3. 
4. 

Q. If $|4x-3|=|x+5|$, then $x$ is/are

1. $\frac{8}{3}$
2. $-\frac{2}{5}$
3. $-\frac{8}{3}$
4. $\frac{2}{5}$

Q. If $|x{|}^2-6|x|+9\le 4$, then $x\in$

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