Squaring an Inequality

Q. Which of the following option(s) is/are correct for exponential function $f\left(x\right)=a^x$ ?

1. Domain of $f\left(x\right)$ is $R$
2. Range of $f\left(x\right)$ is $(0,\infty )$
3. $a\in R-\{0,1\}$
4. $a\in (0,1)\cup (1,\infty )$

Q. The minimum value of the expression $y=|x+3|+|x+1|+|x-5|$ is

Q. Least value of the function $f\left(x\right)=|x-a|+|x-b|+|x-c|+|x-d|$, where $a<b<c<d$ are fixed real numbers & $x$ takes arbitary values, is

1. $b+c-a-d$
2. $a+d-b-c$
3. $a+c-b-d$
4. $c+d-a-b$

Q. If $|x|=1,|y|=2,$ then the least value of $|x-y|$ is

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

Q. The solution of the inequality $\sqrt{x-2}\ge -1$ is

1. $[2,3)$
2. (2, 3)
3. x > 2
4. $x\ge 2$

Q. If $a,b,c$ are the integral values of $x(a<b<c)$ satisfying $\sqrt{-x^2+10x-16}<x-2,$ then which of the following statements is (are) FALSE?

1. The minimum value of $|x-a|+|x-b|+|x-c|$ is $2$
2. The quadratic equation whose roots are $a$ and $b$ is $x^2-15x+56=0$
3. $2a+3b-4c=0$
4. $2b=a+c$

Q. Let $\left[x\right]$ denote the greatest integer less than or equal to $x$. Then, the values of $x\in R$ satisfying the equation $[e^x{]}^2+[e^x+1]-3=0$ lies in the interval:

1. $[0,\frac{1}{e})$
2. $[0,{\log }_{e}2)$
3. $[{\log }_{e}2,{\log }_{e}3)$
4. $[1,e)$

Q.

Solve: $\sqrt{x-2}$ ≥ -1

1. [2, 3)

2. [3, $\infty$)

3. [2, $\infty$)

4. ($-\infty$, 3]

Q. The minimum value of $x^2$, if $x\in \left(-3,5\right]$ is

1. \N
2. 9
3. -9
4. 4

Q. If $x\in [-4,3]$, then $x^2$ lies in

1. $[9,16]$
2. $[0,9]$
3. $[0,16]$
4. $[-16,9]$

Q. $\text{Statement I}$ For every natural number $n\ge 2$
$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\cdots +\frac{1}{\sqrt{n}}>\sqrt{n}$
$\text{Statement II}$ For every natural number $n\ge 2$
$\sqrt{n(n+1)}<n+1$

1. Statement $\text{I}$ is true, Statement $\text{II}$ is true; and Statement $\text{II}$ is correct explanation for Statement $\text{I}.$
2. Statement $\text{I}$ is true, Statement $\text{II}$ is true; and Statement $\text{II}$ is not correct explanation for Statement $\text{I}.$
3. Statement $\text{I}$ is true, Statement $\text{II}$ is false.
4. Statement $\text{I}$ is false, Statement $\text{II}$ is true.

Q. If $\sqrt{2(x+24)}-\sqrt{x-7}\ge \sqrt{x+7},$ then $x\in$

1. $[-25,25]$
2. $(-17,-24)\cup [7,\infty )$
3. $[7,\infty )$
4. $[7,25]$