If $x+\frac{1}{x}=2$, then which one of the following is the value of $x^{32}+\frac{1}{x^{32}}$ ?
-1
0
1
2
Answer is Right!
Answer is Wrong!
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UPSC CAPF – 2024
$x(x + \frac{1}{x}) = 2x$
$x^2 + 1 = 2x$
Rearranging the terms gives a quadratic equation:
$x^2 – 2x + 1 = 0$
This equation is a perfect square trinomial, which can be factored as $(x-1)^2 = 0$.
Solving for $x$, we get $x-1 = 0$, which means $x = 1$.
Now, we need to find the value of the expression $x^{32} + \frac{1}{x^{32}}$.
Substitute $x = 1$ into the expression:
$1^{32} + \frac{1}{1^{32}} = 1 + \frac{1}{1} = 1 + 1 = 2$.