Question : What is the third proportion of $2 \sqrt{3}$ and $6 \sqrt{5}$?

Option 1: $50 \sqrt{6}$

Option 2: $40 \sqrt{3}$

Option 3: $20 \sqrt{6}$

Option 4: $30 \sqrt{3}$

Question : If $x=\frac{4\sqrt{15}}{\sqrt{5}+\sqrt{3}}$, the value of $\frac{x+\sqrt{20}}{x–\sqrt{20}}+\frac{x+\sqrt{12}}{x–\sqrt{12}}$ is:

Option 1: $1$

Option 2: $2$

Option 3: $\sqrt{3}$

Option 4: $\sqrt{5}$

Question : Which of the following is true?

Option 1: $\sqrt 5 + \sqrt 3 > \sqrt 6 + \sqrt 2$

Option 2: $\sqrt 5 + \sqrt 3 < \sqrt 6 + \sqrt 2$

Option 3: $\sqrt 5 + \sqrt 3 = \sqrt 6 + \sqrt 2$

Option 4: $(\sqrt 5 + \sqrt 3 ) (\sqrt 6 + \sqrt 2 )= 1$

Question : If $x=(7+3 \sqrt{5})$, then find the value of $x^2+\frac{1}{x^2}$.

Option 1: $ \frac{580+315 \sqrt{5}}{8}$

Option 2: $\frac{799+328 \sqrt{5}}{8}$

Option 3: $\frac{799+315 \sqrt{5}}{12}$

Option 4: $\frac{799+315 \sqrt{5}}{8}$

Question : Evaluate $\sqrt{20}+\sqrt{12}+\sqrt[3]{729}-\frac{4}{\sqrt{5}-\sqrt{3}}-\sqrt{81}:$

Option 1: $\sqrt{2}$

Option 2: $\sqrt{3}$

Option 3: $0$

Option 4: $2\sqrt{2}$