Question

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

Answer 1

Given for real value of x, $\cos \theta =x+\frac{1}{x}$
$$\begin{gathered} \mathrm{i}.\mathrm{e}\cos \theta =\frac{x^2+1}{x} \\ \Rightarrow x\cos \theta =x^2+1 \\ \Rightarrow x^2-x\cos \theta +1=0 \\ \mathrm{for}x\in \mathrm{ℝ}\mathrm{i}.\mathrm{e}\mathrm{roots}\mathrm{should}\mathrm{be}\mathrm{real} \\ \mathrm{i}.\mathrm{e}{b}^2-4ac\ge 0 \\ \mathrm{i}.\mathrm{e}{\cos }^2\theta -4\left(1\right)\left(1\right)\ge 0 \\ \mathrm{i}.\mathrm{e}{\cos }^2\theta \ge 4 \end{gathered}$$
which is not possible (∵ –1 ≤ cosθ ≤ 1)
∴ No such value of θ is possible
Hence, the correct answer is option D.