Question

The expression $5\cos \theta +3\cos (\theta +\frac{\pi }{3})+3$ lies between-

• A. $-4,10$
• B. $5,12$
• C. $-5,12$
• D. $-4,5$

Answer 1

Answer: (A)

$-4,10$

$5\cos \theta +3(\cos \theta .\cos \frac{\pi }{3}-\sin \theta .\sin \frac{\pi }{3})+3$
$\Rightarrow \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3$
We have $-\sqrt{a^2+b^2}<a\cos \theta +b\sin \theta <\sqrt{a^2+b^2}$
Hence the expression lies in the range $[-7+3,7+3]$
Hence option A is the correct option