Question : If $x^{2}+x=19,$ then what is the value of $(x+5)^{2}+\left [ \frac{1}{(x+5)^{2}} \right ]\;?$

Option 1: 77

Option 2: 79

Option 3: 81

Option 4: 83

Question : If $2 x+\frac{2}{x}=5$, then the value of $\left(x^3+\frac{1}{x^3}+2\right)$ will be:

Option 1: $\frac{81}{11}$

Option 2: $\frac{81}{7}$

Option 3: $\frac{71}{8}$

Option 4: $\frac{81}{8}$

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 $\left(x-\frac{1}{x}\right) =4$, then what is the value of $\left(x^6+\frac{1}{x^6}\right)$?

Option 1: 4689

Option 2: 4786

Option 3: 5832

Option 4: 5778

Question : If $x^{\frac{1}{4}}+x^{\frac{-1}{4}}=2$, then what is the value of $x^{81}+\frac{1}{x^{81}}$?

Option 1: –2

Option 2: 0

Option 3: 1

Option 4: 2

Question : If $x+\frac{1}{x}=3$, then the value of $\frac{3x^{2}-4x+3}{x^{2}-x+1}$ is:

Option 1: $\frac{4}{3}$

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

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

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