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AM,GM,HM Inequality

If x-1x2=1 ...

Question

If ${(x - 1)}^{x^{2}} = 1$ and $x$ is not equal to $1$ , then all the possible values of $x$ are

0

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2

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- 2

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0

and

2

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None of the above

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Solution

The correct option is E $0$ and $2$
Given, ${(x - 1)}^{x^{2}} = 1$
Apply logarithm on both sides, we get
$x^{2} log (x - 1) = 0$
$\Rightarrow x^{2} = 0$ or $log (x - 1) = 0$
$\Rightarrow x = 0$ or $x = 2$

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