Question

If $f\left(\theta \right)=5cos\theta +3cos(\theta +\pi /3)+7$ and $a\le f\left(\theta \right)\le b$, then

• A. $a=0$
• B. $a=-7$
• C. $b=14$
• D. $b=7$

Answer 1

Answer: (A), (C)

The correct options are
A $a=0$
C $b=14$

$f\left(\theta \right)=5cos\theta +3cos(\theta +\pi /3)+7$
$=5cos\theta +3(cos\theta .cos(\pi /3)-sin\theta .sin(\pi /3\left)\right)+7$
$=\frac{13}{2}cos\theta -\frac{3\sqrt{3}}{2}sin\theta +7$
Since, $-7\le \frac{13}{2}cos\theta -\frac{3\sqrt{3}}{2}sin\theta \le 7$
$\Rightarrow 0\le f\left(\theta \right)\le 14$
Hence $a=0,b=14$
Hence, option 'C' is correct.