Question : If $x=\frac{\sqrt{5}+1}{\sqrt{5}-1}$ and $y=\frac{\sqrt{5}-1}{\sqrt{5}+1}$, then the value of $\frac{x^{2}+xy+y^{2}}{x^{2}-xy+y^{2}}$ is:

Option 1: $\frac{3}{4}$

Option 2: $\frac{4}{3}$

Option 3: $\frac{3}{5}$

Option 4: $\frac{5}{3}$

Question : What is the value of $\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}–\sqrt{5}} \div \frac{\sqrt{14}+\sqrt{10}}{\sqrt{14}–\sqrt{10}}+\frac{\sqrt{10}}{\sqrt{5}}$?

Option 1: $\sqrt{2}+2$

Option 2: $2 \sqrt{2}+2$

Option 3: $\sqrt{2}+1$

Option 4: $2 \sqrt{2}+1$

Question : If $\left(x^2+\frac{1}{x^2}\right)=7$, and $0<x<1$, find the value of $x^2-\frac{1}{x^2}$.

Option 1: $3 \sqrt{5}$

Option 2: $4 \sqrt{5}$

Option 3: $-4\sqrt{3}$

Option 4: $-3\sqrt{5}$

Question : If $x+\left [\frac{1}{(x+7)}\right]=0$, what is the value of $x-\left [\frac{1}{(x+7)}\right]$?

Option 1: $3\sqrt{5}$

Option 2: $3\sqrt{5}-7$

Option 3: $3\sqrt{5}+7$

Option 4: $8$

Question : If $3\sqrt{\frac{1-a}{a}}+9=19-3\sqrt{\frac{a}{1-a}};$ then, what is the value of $a?$

Option 1: $\frac{3}{10}$ and $\frac{7}{10}$

Option 2: $\frac{1}{10}$ and $\frac{9}{10}$

Option 3: $\frac{2}{5}$ and $\frac{3}{5}$

Option 4: $\frac{1}{5}$ and $\frac{4}{5}$