Question

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

• A. 5
• B. 11
• C. 10
• D. -1
• E. 2

Answer 1

Answer: (C)

$5\cos \theta +3\cos \left(\theta +\frac{\pi }{3}\right)+3$
$=5\cos \theta +3\left[\cos \theta \cos 60^{\circ }-\sin \theta \sin 60^{\circ }\right]+3$
$=5\cos \theta +3\left[\frac{\cos \theta }{2}-\frac{\sqrt{3}}{2}\sin \theta \right]+3$
$=\frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3$
Let $\frac{13}{2}=a$ and $\frac{3\sqrt{3}}{2}=b$
Then expression becomes,
$a\cos \theta -b\sin \theta +3$
Maximum value of this type of expression is equal to
${\left[a^2+b^2\right]}^{\frac{1}{2}}+3=$ Maximum value
After putting values of $a$ and $b$, we get
$[49{]}^{\frac{1}{2}}+3=$ Max value $10-$ Max value