Question

The equation $cos\theta =x+\frac{p}{x}$ for all $xϵR$ has a real solution for $\theta .$ Then

• A. $p=\frac{1}{2}$
• B. $p\le \frac{1}{4}$
• C. $p\ge \frac{1}{4}$
• D. none of these

Answer 1

The correct option is B $p\le \frac{1}{4}$

consider, $cos\theta =x+\frac{p}{x}$

$\Rightarrow x^2-x\cos \theta +p=0$

For real solution,
$D\ge 0$

$\Rightarrow {\cos }^2\theta \ge 4p$

We know ${\cos }^2\theta \le 1$

$\Rightarrow 4p\le 1$

$\Rightarrow p\le \frac{1}{4}$