Choose the correct answer. If for real values of x, =x+1 x , then… - Vidyadip

MCQ

Choose the correct answer. If for real values of $x, \cos\theta=\text{x}+\frac{1}{\text{x}},$ then:

A
$\theta$ is an acute angle
B
$\theta$ is right angle
C
$\theta$ is an obtuse angle
✓
No value of $\theta$ is possible

✓

Answer

Correct option: D.

No value of $\theta$ is possible

given that, $\cos\theta=\text{x}+\frac{1}{\text{x}}\Rightarrow\cos\theta=\frac{\text{x}^2+1}{\text{x}}$
$\Rightarrow\text{x}^2+1=\text{x}\cos\theta\Rightarrow\text{x}^2-\text{x}\cos\theta+1=0$
$$For real value of $\text{x},\text{b}^2-4\text{a}\text{c}\geq0$
$\Rightarrow(-\cos\theta)^2-4\times1\times1\geq0$
$\Rightarrow\cos^2\theta-4\geq0\Rightarrow\cos^2\theta\geq4$
$\Rightarrow\cos\theta\geq\pm2[-1\leq\cos\theta\leq1]$
so, the value of $\theta$ is not possible.
Hence, the correct options $(d).$

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