Question : If $x+\frac{1}{x}=2$, then find the value of $x^{1823}+\frac{1}{x^{1929}}$.
Option 1: 2
Option 2: 1
Option 3: 0
Option 4: –1
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Correct Answer: 2
Solution : According to the question, $x+\frac{1}{x}=2$ ⇒ $\frac{x^2+1}{x} = 2$ ⇒ $x^2+1=2x$ ⇒ $x^2-2x+1=0$ ⇒ $(x-1)^2=0$ ⇒ $x=1$ $\therefore x^{1823}+\frac{1}{x^{1929}} = 1+1 = 2$ Hence, the correct answer is 2.
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