Question

$\cos 2\theta +2\cos \theta$ is always

• A. greater than $-\frac{3}{2}$
• B. less than or equal to $\frac{3}{2}$
• C. greater than or equal to $-\frac{3}{2}$ and less than or equal to $3$
• D. None of these

Answer 1

The correct option is C

greater than or equal to $-\frac{3}{2}$ and less than or equal to $3$

Explanation for the correct option:

Step 1. Let $f\left(\theta \right)=\cos 2\theta +2\cos \theta$

$=2{\cos }^2\theta -1+2\cos \theta$ ; $\left[\mathbf{\because }\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{2}\mathit{\theta }\mathbf{=}\mathbf{2}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{\theta }\mathbf{-}\mathbf{1}\right]$

Step 2. Differentiate $f\left(\theta \right)$ with respect to $\theta$ for minimum value:

$f\text{'}\left(\theta \right)=-4\cos \theta \sin \theta -2\sin \theta$

$=0$

$\Rightarrow$$\cos \theta =-\frac{1}{2}$

$\therefore$minimum of $f\left(\theta \right)=2\times {\left(-\frac{1}{2}\right)}^2-1+2\times \left(-\frac{1}{2}\right)$

$=2\times \frac{1}{4}-1-1$

$=-\frac{3}{2}$

$\Rightarrow$ $f\left(\theta \right)\ge -\frac{3}{2}$

Step 3. when $\theta =0°$, the given expression will be maximum.

That is, the maximum value will be $3$.

Therefore, $\cos 2\theta +2\cos \theta$ is greater than or equal to $-\frac{3}{2}$ and less than or equal to $3$.

Hence, Option ‘C’ is Correct.