Absolute Value Function

Q. The sum of the solutions of the equation $|\sqrt{x}-2|+\sqrt{x}(\sqrt{x}-4)+2=0,(x>0)$ is equal to:

1. $4$
2. $9$
3. $10$
4. $12$

Q. Number of real solutions of the equation $|x-3{|}^{3x^2-10x+3}=1$ is

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

Q. If $x$ satisfies the inequality $(x^2-x-1)(x^2-x-7)<-5,$ then number of integral value(s) of $x$ satisfying the given inequality is

1. $4$
2. $5$
3. $0$
4. $1$

Q. Number of solutions of the equation $|x-1|+|x-2|+|x-3|=7$ is

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

Q.

Let f (x) be a function such that f (x) = x - [x], where [x] is the greatest integer less than or equal to x. Then the number of solutions of the equation $f\left(x\right)+f\left(\frac{1}{x}\right)=1$ is (are)

1. 0

2. 1

3. 2

4. infinite

Q. If, $2^{|x+1|}-2^x=|2^x-1|+1$ then

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

Q.

Let a relation $r$ in the set $N$ of natural numbers be defined as $(x,y)\iff x^2–4xy+3y^2=0,\forall x,y\in N$. The relation $R$ is

1. Reflexive

2. Symmetric

3. Transitive

4. An equivalence relation

Q. If $f\left(x\right)=|x^2-4x+3|,$ then

1. $f\left(x\right)=x^2-4x+3$ when $x^2-4x+3\ge 0$
2. $f\left(x\right)=x^2-4x+3$ when $x^2-4x+3\le 0$
3. $f\left(x\right)=-(x^2-4x+3)$ when $x^2-4x+3>0$
4. $f\left(x\right)=-(x^2-4x+3)$ when $x^2-4x+3<0$

Q. If $a\le 0$ then the real roots of the equation $x^2-2a|x-a|-3a^2=0$ is/are

1. $a+a\sqrt{2}$
2. $a-a\sqrt{2}$
3. $-a+a\sqrt{6}$
4. $-a-a\sqrt{6}$

Q. The minimum possible value of $|x-1|+|x-2|+\cdots +|x-100|$ is

1. $4900$
2. $2500$
3. $2450$
4. $2550$

Q. If $|x-1|+|x+1|=2$, then $x$ belongs to

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

Q. The real value of $x,$ which satisfies the equation $|x^2-5x+6|+|x^2-8x+12|=0,$ is

Q. [[5(x+1)]^(ln5)]=(2y)^(ln2) and (x+1)^(ln2)=[5^(lny)], then x is , [[ (A) (1)/(5), (B) -(1)/(5)], [ (C) (4)/(5), (D) -(4)/(5)]]

Q. Which among the following has the greatest absolute value?

1. $3$
2. $-\frac{9}{2}$
3. $-\sqrt{36}$
4. $-4$

Q. If $|x-3|+|x-2|=1$, then $x\in$

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

Q. If $\left|\frac{|x|-1}{|x|-2}\right|=4$, then $x\in$

1. $\{\frac{7}{3},\frac{9}{5}\}$
2. $\{-\frac{9}{5},-\frac{7}{3},\frac{7}{3},\frac{9}{5}\}$
3. $\{-\frac{7}{3},\frac{7}{3}\}$
4. $\{-\frac{9}{5},\frac{9}{5}\}$

Q.

Which of the following points lie on the same side of origin with respect to the line

x - 3y + 7 = 0?

1. (9, 3)

2. (3, 9)

3. (3, -9)

4. (-9, -3)

Q. Let $f\left(x\right)=x^{13}+x^{11}+x^9+x^7+x^5+x^3+x+19.$ Then $f\left(x\right)=0$ has

1. $13$ real roots
2. only one positive and only two negative real roots
   
3. not more than one real root
4. has two positive and one negative real root

Q.

A branch of a certain bank in New York City has six ATMs. Let $X$ represent the number of machines in use at a particular time of day. The cdf of $X$ is as follows:
$f\left(x\right)=\left\{\begin{array}{l}\begin{array}{c}0x<1\\ .601\le x<1\\ \begin{array}{c}.191\le x<2\\ .392\le x<3\\ \begin{array}{c}.673\le x<4\\ .924\le x<5\\ .975\le x<6\end{array}\end{array}\end{array}\\ 16\le x\end{array}\right.$
$P(2\le X\le 5)$.

Q. Number of integral values of $x$ satisfying $|x^2-5x+4|=2$ is

Q.

The set $A=\left\{x:\left|2x+3\right|<7\right\}$ is equal to the set

1. $A=\left\{x:0<x+5<7\right\}$

2. $A=\left\{x:-3<x<7\right\}$

3. $A=\left\{x:-7<x<7\right\}$

4. $A=\left\{x:-13<2x<4\right\}$

Q. The difference between the maximum and minimum value of the expression $y=|x-9|-|x+2|$ is

1. $-11$
2. $0$
3. $11$
4. $22$

Q.

If a, b, c are unequal and positive , show that bc/(b+c ) +ac/(c+a) +ab/(a+b) is less than1/2(a+b+c)

Q. Number of solution(s) of equation $|x-1|-|x-2|=10$ is

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

Q. If $f\left(x\right)=|x^2-4x+3|,$ then

1. $f\left(x\right)=x^2-4x+3$ when $x^2-4x+3\ge 0$
2. $f\left(x\right)=x^2-4x+3$ when $x^2-4x+3\le 0$
3. $f\left(x\right)=-(x^2-4x+3)$ when $x^2-4x+3>0$
4. $f\left(x\right)=-(x^2-4x+3)$ when $x^2-4x+3<0$

Q. Form a differential equation representing the given family of curves by eliminating arbitrary constants $a$ and $b$
$y=e^{2x}(a+bx)$

Q. The sum of integral value(s) of $x$ satisfying the equation $\frac{|x+9|}{|x-5|}=8$ is

Q. The function $x^5-5x^4+5x^3-1$ is

1. Maximum at x =0
2. Maximum at and minimum at x=1
3. Minimum at x =1
4. Neither maximum nor minimum at x=0

Q. Set of solutions of the equation $3\left|\right(x-9{)}^2(x-1)|-6|x^2-10x+9|=0$ is

1. $\{1,7,9\}$
2. $\{1,7,9,11\}$
3. $\{1,9,11\}$
4. $\{1,9\}$

Q.

If $|x-7{|}^2-3|x-7|-10=0$, then value(s) of $x$ can be equal to

1. $2$
2. $5$
3. $9$
4. $12$