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If for real values of x,


If for real values of $x, \cos \theta=x+\frac{1}{x}$, 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


Given for real value of $x, \cos \theta=x+\frac{1}{x}$

i. e $\cos \theta=\frac{x^{2}+1}{x}$

$\Rightarrow x \cos \theta=x^{2}+1$

$\Rightarrow x^{2}-x \cos \theta+1=0$

for $x \in \mathbb{R}$ i. e roots should be real

i. e $b^{2}-4 a c \geq 0$

i. e $\cos ^{2} \theta-4(1)(1) \geq 0$

i. e $\cos ^{2} \theta \geq 4$

which is not possible $\quad(\because-1 \leq \cos \theta \leq 1)$

∴ No such value of θ is possible

Hence, the correct answer is option D.

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