Question

The set of values of $\lambda$ such that the equation $\cos \theta +\cos 2\theta +\lambda =0$ admits of a solution for $\theta$ is

• A. $[0,+\infty )$
• B. $[0,\frac{9}{8}]$
• C. $[-3,0]$
• D. None of these

Answer 1

The correct option is B $[0,\frac{9}{8}]$

$\cos \theta +\cos 2\theta +\lambda =0$
$\cos \theta +2{\cos }^2\theta -1+\lambda =0$
$2{\cos }^2\theta +\cos \theta +(\lambda -1)=0$
$\cos \theta =\frac{-1\pm \sqrt{9-8\lambda }}{4}$
We know,
$\sqrt{9-8\lambda }\ge 0$
$\lambda \le \frac{9}{8}$
$-1\le \cos \theta \le 1$
$-4\le -1\pm \sqrt{9-8\lambda }\le 4$
$9\ge 9-8\lambda$
$\lambda \ge 0$
so,
$[0,\frac{9}{8}]$