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

Question : If $x=\frac{1}{x-5}(x>0)$, then the value of $x+\frac{1}{x}$ is:

Option 1: $\sqrt{41}$

Option 2: $\sqrt{29}$

Option 3: $\sqrt{23}$

Option 4: $\sqrt{43}$

Question : If $\left(y^2+\frac{1}{y^2}\right)=74$ and $y>1$, then find the value of $\left(y-\frac{1}{y}\right)$.

Option 1: $6 \sqrt{2}$

Option 2: $-2 \sqrt{19}$

Option 3: $2 \sqrt{19}$

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

Question : If $x=\sqrt3+\frac{1}{\sqrt3}$, then the value of $(x-\frac{\sqrt{126}}{\sqrt{42}})(x-\frac{1}{x-\frac{2\sqrt3}{3}})$ is:

Option 1: $\frac{5\sqrt3}{6}$

Option 2: $\frac{2\sqrt3}{3}$

Option 3: $\frac{5}{6}$

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

Question : If $a=\frac{1}{a-\sqrt{6}}$ and $(a>0)$, then the value of $\left(a+\frac{1}{a}\right)$ is:

Option 1: $\sqrt{6}$

Option 2: $\sqrt{10}$

Option 3: $\sqrt{15}$

Option 4: $\sqrt{7}$