Law of Reciprocal

Q. If $x=\sqrt{7}-\sqrt{5}$ and $y=\sqrt{13}-\sqrt{11}$, then

1. $x>y$
2. $x<y$
3. None of these
4. $x=y$

Q. If [.] denotes the greatest integer function, then the value of natural number $n$ satisfying the equation
$\left[{\log }_{2}1\right]+\left[{\log }_{2}2\right]+\left[{\log }_{2}3\right]+\cdots +\left[{\log }_{2}n\right]=1538$ is

Q. Which of the following is/are true?

1. $\frac{1}{x}\ge 2\Rightarrow x\ge \frac{1}{2}$
2. $\frac{1}{x}>2\Rightarrow x<\frac{1}{2}$
3. $\frac{1}{x}>2\Rightarrow 0<x<\frac{1}{2}$
4. $\frac{1}{x}\ge 2\Rightarrow 0<x\le \frac{1}{2}$

Q. The fundamental period of the function $f\left(x\right)=2cos\frac{1}{3}(x-\pi )$ is

1. $6\pi$
2. $4\pi$
3. $\pi$
4. $2\pi$

Q. The total number of integral solution(s) of $|4x-5|+|6x-12|=|2x-7|$ is/are

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

Q. If $xϵR$ and $m=\frac{x^2}{(x^4-2x^2+4)}$, then m lies in the interval

1. $[0,\frac{1}{4}]$
2. $[0,\frac{1}{3}]$
3. $[0,\frac{1}{2}]$
4. $[0,\frac{1}{5}]$

Q. Let a relation R in the set R of real numbers be defined as (a, b) $ϵ$ R if and only if 1 + ab > 0 for all a, b $ϵ$ R. The
relation R is

1. Reflexive and symmetric
2. symmetric and transitive
3. an equivalence relation
4. None of these

Q. The number of solution(s) of the equation $|x-7|-|x+1|=10$ is

Q.

If $x_1,x_2,x_3,x_4$ are four positive real numbers such that $x_1+\frac{1}{x_2}=4,x_2+\frac{1}{x_3}=1,x_3+\frac{1}{x_4}=4$ and $x_4+\frac{1}{x_1}=1,$ then

1. $x_1=x_3$
2. $x_2=x_4$
3. $x_1{x}_2=1$
4. $x_3{x}_4=1$

Q. Which of the following functions is non – injective?

1. $f\left(x\right)=|x+1|,xϵ[-1,\infty )$
2. $g\left(x\right)=x+\frac{1}{x},xϵ(0,\infty )$
3. $h\left(x\right)=x^2+4x-5,xϵ(1,\infty )$
4. $k\left(x\right)=e^{-x},xϵ[0,\infty )$

Q.

The possible values of the expression $\frac{1}{x^2-2x+5}$

1. (4, $\infty$)

2. [4, $\infty$)

3. $(0,\frac{1}{4}]$

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

Q.

If $x_1,x_2,x_3,x_4$ are four positive real numbers such that $x_1+\frac{1}{x_2}=4,x_2+\frac{1}{x_3}=1,x_3+\frac{1}{x_4}=4$ and $x_4+\frac{1}{x_1}=1,$ then

1. $x_1=x_3$
2. $x_2=x_4$
3. $x_1{x}_2=1$
4. $x_3{x}_4=1$

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. $(-\infty ,-\frac{5}{2})$