Question

If the range of 5$\cos \theta +3\cos (\theta +\frac{\pi }{3})+3$ is $[a,b],$ $\forall \theta \in$ R then $a+b$ is equal to

• A. $6$
• B. $2$
• C. $8$
• D. $4$

Answer 1

The correct option is A $6$

$5\cos \theta +3\cos \theta \times \frac{1}{2}-3\sin \theta \times \frac{\sqrt{3}}{2}+3$
$=\frac{13}{2}\cos \theta -3\frac{\sqrt{3}}{2}\sin \theta +3$
$\therefore$ least value of $\frac{13}{2}\cos \theta -3\frac{\sqrt{3}}{2}\sin \theta +3$ is $-4$ and Greatest value of $\frac{13}{2}\cos \theta -3\frac{\sqrt{3}}{2}\sin \theta +3$ is $10$.
$\therefore$ Range of $5\cos \theta -3\cos (\theta +\frac{\pi }{3})+3$ is $[-4,10]$.
$\therefore a=-4,b=10$
$\therefore a+b=6$