Question : What is the value of $\frac{x^2-x-6}{x^2+x-12}÷\frac{x^2+5x+6}{x^2+7x+12}$?

Option 1: $1$

Option 2: $\frac{(x-3)}{(x+3)}$

Option 3: $\frac{(x+4)}{(x-3)}$

Option 4: $\frac{(x-3)}{(x+4)}$

Question : If $x^2+\frac{1}{x^2}=\frac{7}{4}$ for $x>0$, what is the value of $(x^3+\frac{1}{x^3})$?

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

Option 2: $\frac{3\sqrt{15}}{5}$

Option 3: $\frac{3\sqrt{15}}{8}$

Option 4: $\frac{3\sqrt{5}}{8}$

Question : If $x^2-2\sqrt{10}x+1=0$, what is the value of $(x-\frac{1}{x})$?

Option 1: $4$

Option 2: $6$

Option 3: $3$

Option 4: $5$

Question : If $x^2-8x+1=0$, what is the value of $(x^2+\frac{1}{x^2})$?

Option 1: $18$

Option 2: $34$

Option 3: $40$

Option 4: $62$

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

Option 1: 4

Option 2: 5

Option 3: 6

Option 4: 8