If for real values of x, cos θ = x + 1/x , then(a) θ is an acute angle(b) θ is a right angle(c) θ is an obtuse angle(d) No value of θ is possible

Question

\[ \text{If for real values of } x, \]

\[ \cos\theta=x+\frac1x, \]

\[ \text{then} \]

(a) \(\theta\) is an acute angle
(b) \(\theta\) is a right angle
(c) \(\theta\) is an obtuse angle
(d) No value of \(\theta\) is possible

For real \(x\),

\[ x+\frac1x\ge2 \quad \text{or} \quad x+\frac1x\le-2 \]

But

\[ -1\le\cos\theta\le1 \]

Therefore,

\[ \cos\theta=x+\frac1x \]

is not possible for any real value of \(\theta\).

\[ \boxed{\text{No value of }\theta\text{ is possible}} \]

Correct Option: (d)

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