Squaring an Inequality

Q. Let $C$ be the set of all complex numbers. Let
$S_1=\{z\in C:|z-2|\le 1\}$ and
$S_2=\{z\in C:z(1+i)+\overline{z}(1-i)\ge 4\}$. Then the maximum value of ${|z-\frac{5}{2}|}^2$ for $z\in S_1\cap S_2$ is equal to

1. $\frac{5+2\sqrt{2}}{2}$
2. $\frac{5+2\sqrt{2}}{4}$
3. $\frac{3+2\sqrt{2}}{4}$
4. $\frac{3+2\sqrt{2}}{2}$

Q.

How many two digit positive integers N have the property that the sum of N and number obtained by reversing the order of the digits of N is a perfect square.

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

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

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.

The number of real roots of the equation $x^4+\sqrt{x^4+20}=22$ is

1. $4$

2. $2$

3. $0$

4. $1$

Q. If a, b, c and d are four positive real numbers such that abcd=1 what is the minimum value of (1+a), (1+b), (1+c) and (1+d)

Q.

The real number x when added to its inverse gives the minimum value of the sum at x equal to

A) -2. B) 2. C) 1. D) -1.

Explain in Detail.

Q. If the roots of the equation x square _ ax +b=0 are real and differ by a quantity which is less than c (c>0) then b lies between

Q. Find the minimum value of root asquare+b square, when3a+4b=15

Q. Solve the equation $|z|=z+1+2i.$

Q. If
$\left(1\right)$ ''ASO PAC GLA'' means ''All is well''
$\left(2\right)$ ''LAC MAR ASO'' means ''She sings well''
$\left(3\right)$ ''MOR LAC PAC'' means ''She is good''
$\left(4\right)$ ''GLA SQR MOR'' means ''All was good'',
then which of the following words stands for ''sings''?

1. ASO
2. MOR
3. MAR
4. LAC

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

Q. If $\sqrt{x^2+x-12}<6-x$, then

1. $x\in (-\infty ,-4]\cup [3,\infty )$
2. $x\in [3,\infty )$
3. $x\in (-\infty ,-4]$
4. $x\in (-\infty ,-4]\cup [3,\frac{48}{13})$

Q. If $\sqrt{x^2+x-12}<6-x$, then

1. $x\in (-\infty ,-4]\cup [3,\infty )$
2. $x\in [3,\infty )$
3. $x\in (-\infty ,-4]$
4. $x\in (-\infty ,-4]\cup [3,\frac{48}{13})$

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

Q. Simlify: $\sqrt{\frac{-17}{144}-i}$

1. $\pm \left(\frac{3}{2}-\frac{i}{3}\right)$
2. $\pm \left(\frac{3}{4}-\frac{2i}{3}\right)$
3. $\pm \left(\frac{3}{5}-\frac{5i}{6}\right)$
4. $\pm \left(\frac{2}{3}-\frac{3i}{4}\right)$

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. $[7,25]$

Q. If $a^2$ $\%$ of $b=$ $b^3$ $\%$ of $c$ and $c^4$ $\%$ of $a=b$% of $b$, then the relation between $a$ and $b$ is

1. $10000a^5=b^{12}$
2. $a^9=b^{10}$
3. $a^{12}=b^{21}$
4. $10000a^{14}=b^{27}$

Q. If $\sqrt{x+6}<x-6$, then

1. $x\in (6,\infty )$
2. $x\in (-6,\infty )$
3. $x\in (10,\infty )$
4. $x\in (6,10)$

Q.

The values of $x^2$ for x > -3 is set of

1. All positive real numbers

2. All negative real numbers

3. All non-positive real numbers

4. All non-negative real numbers

Q.

If $(x^2-9)\sqrt{x^2-1}<0$, then what are the possible values of $x$?

1. $x\in (-3,-2]\cup [2,3)$

2. $x\in (-3,-1)\cup (1,3)$

3. $x\in (-3,3)$

4. $x\in (-\infty ,-1]\cup [1,\infty )$

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. $[7,25]$

Q. If,
f(x) = $x^2$ for x < 2
g(x) = $x^2$ for x > -1,
then, Range of f(x) = Range of g(x)

1. False
2. True

Q. Where does $f\left(x\right)=x+\sqrt{1-x};0<x<1$ decrease?

1. $(\frac{3}{4},1)$
2. $(0,1)$
3. $(0,\frac{3}{4})$
4. $(\frac{3}{4},\infty )$

Q. If $\sqrt{x+6}<x-6$, then

1. $x\in (6,\infty )$
2. $x\in (10,\infty )$
3. $x\in (6,10)$
4. $x\in (-6,\infty )$

Q. If $x=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$, $y=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}},$ then the value of $x^2+xy+y^2$ is

1. $5$
2. none of these
3. $98$
4. $99$

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 $(x^2-9)\sqrt{x^2-1}<0$, then what are the possible values of $x$?

1. $x\in (-3,-2]\cup [2,3)$

2. $x\in (-3,-1)\cup (1,3)$

3. $x\in (-3,3)$

4. $x\in (-\infty ,-1]\cup [1,\infty )$

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

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

Q. Find two numbers whose sum is $27$ and product is $182$.