Question : If $x=5–\sqrt{21}$, the value of $\frac{\sqrt{x}}{\sqrt{32–2x}–\sqrt{21}}$ is:

Option 1: $\frac{1}{\sqrt2}(\sqrt{3}–\sqrt{7})$

Option 2: $\frac{1}{\sqrt2}(\sqrt{7}–\sqrt{3})$

Option 3: $\frac{1}{\sqrt2}(\sqrt{7}+\sqrt{3})$

Option 4: $\frac{1}{\sqrt2}(7–\sqrt{3})$

Question : If $a^2+\frac{1}{a^2}=\frac{7}{3}$, then what is the value of $\left(a^3-\frac{1}{a^3}\right)?$

Option 1: $\frac{5}{3 \sqrt{3}}$

Option 2: $\frac{10}{3 \sqrt{3}}$

Option 3: $\frac{7}{3 \sqrt{3}}$

Option 4: $\frac{8}{3 \sqrt{3}}$

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 : If $\sin \theta = \frac{3}{11}$, then what is the value of $\cot \theta$?

Option 1: $\frac{3 \sqrt{7}}{11}$

Option 2: $\frac{4 \sqrt{7}}{11}$

Option 3: $\frac{4 \sqrt{7}}{3}$

Option 4: $\frac{3 \sqrt{7}}{28}$

Question : If $x=(\sqrt{6}-1)^{\frac{1}{3}}$, then the value of $\left(x-\frac{1}{x}\right)^3+3\left(x-\frac{1}{x}\right)$ is:

Option 1: $\frac{2 \sqrt{6}-6}{5}$

Option 2: $\frac{4 \sqrt{6}-6}{5}$

Option 3: $\frac{4 \sqrt{6}-6}{3}$

Option 4: $\frac{4 \sqrt{3}-6}{5}$