Question : If $x+\frac{1}{x}=2$, then the value of $x^{11}+\frac{1}{x^{20}}$ is:
Option 1: 0
Option 2: 1
Option 3: 2
Option 4: 7
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Correct Answer: 2
Solution : Given, $x+\frac{1}{x}=2$ ⇒ $x^2+1=2x$ ⇒ $x^2+1-2x=0$ ⇒ $(x-1)^2=0$ ⇒ $x-1=0$ $\therefore x=1$ Consider, $x^{11}+\frac{1}{x^{20}}$ = $1^{11}+\frac{1}{1^{20}}$ = $1+1$ = $2$ Hence, the correct answer is 2.
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