Question : If $x+\frac{1}{x}=2$, then find the value of $x^5+\frac{1}{x^5}$.
Option 1: 1
Option 2: 3
Option 3: 4
Option 4: 2
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Correct Answer: 2
Solution :
$x+\frac{1}{x}=2$
$⇒\frac{x^2+1}{x} = 2$
$⇒x^2+1 = 2x$
$⇒x^2+1-2x = 0$
$⇒(x-1)^2 = 0$
$⇒x-1 = 0$
$⇒x =1$
$\therefore x^5+\frac{1}{x^5} = 1+1 = 2$
Hence, the correct answer is 2.
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