Inequalities Involving Modulus Function

Q.

The value of $\int {\log }_{}\left(\text{x}\right)\text{dx}$ is

Q.

The greatest integer less than or equal to $(\sqrt{2}+1{)}^6$ is

1. 196

2. 197

3. 198

4. 199

Q.

Solve $\frac{|x-2|-1}{|x-2|-2}\le 0,xϵR$.

Q.

If a, b and c are three positive real numbers, then the minimum value of the expression $\frac{b+c}{a}+\frac{c+a}{b}+\frac{a+b}{c}$ is

1. 1

2. 2

3. 3

4. 6

Q.

Solve $\frac{5}{x-2}>3$ and represent the solution set on the number line.

Or

Solve $|x|<4$ and represent the solution set on the number line.

Q.

If α, β and γ are the roots of $x^3+8=0$, then the equation whose roots are ${\alpha }^2,{\beta }^{2}and{\gamma }^2$ is

1. 

2. 

3. 

4. 

Q.

Solve For $x$:

$lo{g}_{3}x-lo{g}_{3}2=1$

Q. Find x and y, if (x+y, 2)=(3, 2x+y)

Q.

If $z=x+iy,x,y\in R$ and $\left|x\right|+\left|y\right|\le k\left|z\right|$ then $k$ is equal to

1. $1$

2. $\sqrt{2}$

3. $\sqrt{3}$

4. $\sqrt{4}$

Q.

If |x| = - x, then x ≤ 0

1. True

2. False

Q.

Let S denote the set of real values of x for which
$|x^3-x|\le x\left(1\right)and\frac{2}{|x-2|}>\frac{3}{|1-2x|}\left(2\right)$
then S equals

1. $(2,\infty )$
   

2. $(\frac{8}{7},\sqrt{2})$
   

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

4. $(\frac{8}{7},\infty )$

Q. If $1\le |x|<4$, then $x$ belongs to

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

Q.

If $h$and $k$are positive numbers and $h+k=7,$then$\frac{7-k}{h}=?$

Q.

Solve

$|x-1|+|x-2|\ge 4,xϵR$

Q.

If $\alpha ,\beta and\gamma$ are the roots of the equation $x^3+3x+2=0and(\alpha -\beta )(\alpha -\gamma ),(\beta -\gamma )(\beta -\alpha ),(\gamma -\alpha )(\gamma -\beta )$ are the roots of equation $y^3-9y^2-216=0$ . Express y in terms of α.

1. 

2. 

3. 

4. 

Q.

The solution set of the inequality $\frac{(x-1{)}^2(x-2)}{(x^2+1)x}$ > 0 is

1. ( 0, 2 )

2. (-, 0) (2, )

3. ( , )

4. ( , -2 )

Q. Let $f\left(x\right)=\left\{\begin{array}{ll}x+1,& \mathrm{if}x\ge 0\\ x-1,& \mathrm{if}x<0\end{array}.\right.$Prove that $\underset{x\to 0}{\lim }f\left(x\right)$ does not exist.

Q. The number of real roots of the equation $(x^2+2x{)}^2-(x+1{)}^2-55=0$ is
(a) 2
(b) 1
(c) 4
(d) none of these

Q. If the roots of the equation $x^2+bx+ac=0are\alpha ,\beta$ and roots of the equation $x^2+ax+bc=0$ are $\alpha ,\gamma$ then the value of $\alpha ,\beta ,\gamma$ respectively (Previous year JEE question)

1. c, a, b
2. None of these
3. a, b, c
4. b, c, a

Q. If $|(2|x-1|-2)|\ge 1$, then $x\in$

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

Q. Find the condition if the sum of the roots of quadratic equations ax^2-bx-c=0(a is not equal to 0) is equal to the product of its roots.

Q.

Find the common interval among x < 0, x < -2 and x $\le$ $\frac{-5}{2}$

1. 

2. 

3. x ∈ (-∞, -2]

4. x ∈ (0, ∞)

Q.

If $\alpha ,\beta and\gamma$ are the roots of the equation $x^3+3x+2=0$ , Find the equation whose roots are $\alpha -\beta \left)\right(\alpha -\beta ),(\beta -\gamma \left)\right(\beta -\alpha ),(\gamma -\alpha \left)\right(\gamma -\beta$.

1. 

2. - 9 - 216 = 0 \)

3. - 11 - 100 = 0 \)

4. 

Q.

If a>0 and the equation $a{x}^2+bx+c=0$ has two real roots $\alpha$ and $\beta$ such that $\left|\alpha \right|\le 1,\left|\beta \right|\le 1,$ then

1. $a+b+c\ge 0$
   

2. $a-b+c\ge 0$
   

3. $a+\left|b\right|+c\ge 0$
   

4. $a-c\ge 0$

Q. Find $\underset{x\to 1}{\lim }f\left(x\right)$, if $f\left(x\right)=\left\{\begin{array}{ll}{x}^2-1,& x\le 1\\ -x^2-1,& x>1\end{array}\right.$

Q.

If the equations $x^2-x-12=0andk{x}^2+10x+3=0$ may have one common root, then k =

1. 

2. 

3. 

4. 

Q. If C (n, 12) = C (n, 8), then C (22, n) is equal to
(a) 231
(b) 210
(c) 252
(d) 303

Q. three digit numbers are formed such that the sum of the numbers is also a three digit number and in no place addition is going on if the number of such numbers is of the form 36lambda^2.then find the value of lambda

Q. If $2|3x+9|<36$, then the number of integral values of $x$ is

Q. If the distance between -x and -y is 9 units, find the value of $|x-y|$. Where x and y are two points on the number line.
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