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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