\(x\) is a real number and \(x+\frac{1}{x}=-\dfrac{7}{\sqrt{6}}\).

Read the following statements.

\([1]\). \(\sin^{-1} (x+\frac{1}{x})\) is a real number.

\([2]\). \(x^2+\frac{1}{x^2}=\frac{37}{6}\)

\([3]\) The given information can not be true because by the AM-GM inequality, \(x+\frac{1}{x}\) has to be greater than or equal to \(2\). And \(-\dfrac{7}{\sqrt{6}}\) is clearly less than \(2\).

Which of these statements are correct?

This problem is from the set "MCQ Is Not As Easy As 1-2-3". You can see the rest of the problems here.

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