Question : If $x^2-\sqrt{7} x+1=0$, then what is the value of $x^5+\frac{1}{x^5} ?$

Option 1: $19 \sqrt{7}$

Option 2: $21 \sqrt{7}$

Option 3: $25 \sqrt{7}$

Option 4: $27 \sqrt{7}$

Question : If $x+\frac{1}{x}=2$. Then, the value of $x^7+\frac{1}{x^5}$ is:

Option 1: $2^{12}$

Option 2: $2$

Option 3: $2^5$

Option 4: $2^7$

Question : Simplify the following expression.
$\frac{x^2-2 x-63}{x^2+14 x+49}$

Option 1: $\frac{x+9}{x+7}$

Option 2: $\frac{x-7}{x+7}$

Option 3: $\frac{x+7}{x-7}$

Option 4: $\frac{x-9}{x+7}$

Question : If $x+\frac{2}{x}=1$, then the value of $\frac{x^2+7x+2}{x^2+13x+2}$ is:

Option 1: $\frac{5}{7}$

Option 2: $\frac{3}{7}$

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

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