Properties of Inequalities

Q.

Find the value of $x$ if $x=\sqrt{6+\sqrt{6+\sqrt{+6+.....}}}$. Where $x$ is a natural number.

Q. If the function $f$ satisfies the relation $f(x+y)+f(x-y)=2f\left(x\right)f\left(y\right)\forall x,y\in R$ and $f\left(0\right)\ne 0$, then

1. $f\left(x\right)$ is an even function
2. $f\left(x\right)$ is an odd function
3. If $f\left(2\right)=a$, then $f(-2)=a$
4. If $f\left(4\right)=b$, then $f(-4)=-b$

Q. The number of solutions of ${\log }_2(x+5)=6-x$ is

Q. Let $f:A\to R^{+}$ be a function defined by $f\left(x\right)={\log }_{\left\{x\right\}}\left(\frac{x-\left[x\right]}{|x|}\right)$, where $[.]$ and $\{.\}$ represent greatest integer function and fractional part function respectively. If $B$ is the range of $f$, then the number of integer(s) in $R^{+}-B$ is

Q. The number of solution(s) of $\left\{x\right\}+\left\{x^2\right\}+\left\{x^3\right\}=3$ is
(where {.} denotes fractional part function)

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

Q. The sum of an infinite terms of a G.P. is $20$ and sum of their squares is $100.$ If $r$ is the common ratio of the G.P., then the value of $10r$ is

Q. The set of real values of $x$ satisfying $||x-1|-1|\le 2,$ is

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

Q. Raj has a boat with a maximum weight capacity of $2700$ kg. He wants to take as many of his friends as possible. If the average weight of each friend considered to be $70$ kg.
Then the maximum number of persons that can travel in the boat if $2$ VIP persons weighing $98$kg and $86$kg have to travel compulsorily is

Q. If $\frac{1}{x-2}>0$, then $x$ lies in

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

Q. The number of terms common to the series $3+7+11+\cdots +2019$ and $1+6+11+\cdots +2021$ is

Q. If $lo{g}_{\frac{1}{\sqrt{2}}}sinx>0,xϵ[0,4\pi ]$ then the number of values of x which are integral multiples of $\frac{\pi }{4}$ is

1. 4
2. 12
3. 3
4. None of these

Q. Raj has a boat with a maximum weight capacity of $2700$ kg. He wants to take as many of his friends as possible. If the average weight of each friend considered to be $70$ kg.
If a new system is added in the boat that increases the weight capacity of boat upto $150$ kg. But requires specific person of weight $90$ kg to use it. Then the maximum number of persons that can travel in the boat is

Q. I.Q. of a person is given by $I=\frac{M}{C}\times 100$, where $M$ is mental age and $C$ is chronological age. If $80\le I\le 140$ for a group of $12$ years old children, then their mental age can be

1. $10$
2. $13$
3. $14$
4. $17$

Q. Let $p,p_1$ be A.M. and G.M. between $a$ and $b$ respectively and $q,q_1$ be the A.M. and G.M. between $b$ and $c$ respectively where $a,b,c>0$. If $a,b,c$ are in A.P., then which of the following is CORRECT?

1. $p^2-q^2=q_1^2-p_1^2$
2. $p^2-q^2=p_1^2-q_1^2$
3. $p^2+q^2=p_1^2-q_1^2$
4. $p^2-q^2=p_1^2+q_1^2$

Q. Raj has a boat with a maximum weight capacity of $2700$ kg. He wants to take as many of his friends as possible. If the average weight of each friend considered to be $70$ kg.
Then the maximum number of persons that can travel in the boat is

Q. Solve $|x-2|\ge 5$

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

Q. Let $S_1,S_2,S_3,\dots ,S_n$ be squares such that for each $n\ge 1$, the length of a side of $S_n$ equals the length of a diagonal of $S_{n+1}$. If the length of a side of $S_1$ is $10$ cm, then for which of the following values of $n$ is the area of $S_n$ less than $1$ sq. cm?

1. $7$
2. $9$
3. $8$
4. $10$

Q. If $-3x>-15$ and $x\in N$, then $x$ is equal to

1. $\{6,7,8,....\}$
2. $\{1,2,3,4\}$
3. $\{1,2,3,4,5\}$
4. $\{....,1,2,3,4\}$

Q. If $x\in [-4,-1]$,
then $\frac{1}{x^2}$ belongs to

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

Q. The interval(s) of $x$ which satisfies the inequality $\frac{1}{x+2}<\frac{1}{3x+7}$ is/are

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

Q. Solve $|x+1|-|1-x|=2$

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

Q. If $A=\{x:x^2-4\le 0,x\in Z\}$ and $B=\{y:y^2-9\ge 0,y\in Z\}$, then $n(A\cap B)$ is equal to

1. $1$
2. $4$
3. $0$
4. $10$

Q. If $x\in [-3,1),$ then $\frac{1}{x^3}$ belongs to

1. $(-\infty ,-\frac{1}{9}]\cup (1,\infty )$
2. $(-\infty ,-\frac{1}{27}]\cup (1,\infty )$
3. $[-\frac{1}{27},1)$
4. ​$(-\infty ,-\frac{1}{27}]\cup [1,\infty )$

Q. If $x\in [-4,-1]$, then $\frac{1}{x^2+4x+7}$ belongs to

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

Q. Which of the following statement(s) is/are always holds true for $k<0$

1. $a>b\Rightarrow \frac{a}{k}<\frac{b}{k},a,b<0$
2. $a>b\Rightarrow a^k>b^k,a,b<0$
3. Maximum value of $a^k+a^{-k}$ is $-2,a<0$
4. $a>b\Rightarrow -ka<-kb,a,b>0$

Q. The maximum value of the expression $\frac{1}{x^2-2x+3}$ is

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

Q.

The possible values of $x^2+4$ lie in the interval

1. (0, 4]

2. $(4,\infty )$

3. (0, 4)

4. $[4,\infty )$

Q. The minimum integral value of the x for which $\frac{x-2}{x+5}>2$ is .

1. -12
2. -11
3. -10
4. -13

Q. The minimum positive value of the expression $(10-5^{-x}-5^x{)}^{-1}$ is

Q. If $I_1=\underset{0}{\overset{1}{\int }}{e}^{-x}{\cos }^{2}xdx$,
$I_2=\underset{0}{\overset{1}{\int }}{e}^{-x^2}{\cos }^{2}xdx$ and
$I_3=\underset{0}{\overset{1}{\int }}{e}^{-x^3}dx$ ; then :

1. $I_3>I_2>I_1$
2. $I_2>I_1>I_3$
3. $I_2>I_3>I_1$
4. $I_3>I_1>I_2$