Question

The maximum value of $5\cos \theta +3\cos \left(\theta -\frac{\pi }{3}\right)+3$ is

• A. 5
• B. 10
• C. 11
• D. -11

Answer 1

The correct option is B 10

Maximum value of $5\cos \theta +3\cos \theta (\theta -\frac{\pi }{3})+3$

$\Rightarrow 5\cos \theta +3(\cos \theta \cos \frac{\pi }{3}+\sin \theta \sin \frac{\pi }{3})+3$

$\Rightarrow 5\cos \theta +\frac{3}{2}\cos \theta +\frac{3\sqrt{3}}{2}\sin \theta +3$

$\Rightarrow \frac{13}{2}\cos \theta +\frac{3}{2}\sqrt{3}\sin \theta +3$

$\Rightarrow 7\cos (\theta -\beta )+3$ where $\beta ={\cos }^{-1}\left(\frac{13}{14}\right)$

Maximum value of $\cos (\theta -\beta )=1$

$\therefore$ Max value of $7\cos (\theta -\beta )+3=7+3=10$