Question

If $a\le 5\cos \theta +3\cos (\theta +\frac{\pi }{3})+5\le b$, then

• A. $a=-2$
• B. $a=2$
• C. $b=12$
• D. $b=7$

Answer 1

Answer: (A), (C)

$a=-2$
$b=12$

$f\left(\theta \right)=5\cos \theta +3\cos (\theta +\frac{\pi }{3})+5$

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

$f\left(\theta \right)=\frac{13\cos \theta -3\sqrt{3}\sin \theta }{2}+5$

as we know,
$\therefore$ range of $a\sin x\pm b\cos x$ is $[-\sqrt{a^2+b^2},\sqrt{a^2+b^2}]$

$\therefore$ So, range of $f\left(\theta \right)ϵ[-\frac{\sqrt{{13}^2+(3\sqrt{3}{)}^2}}{2}+5,\frac{\sqrt{(13{)}^2+(3\sqrt{3}{)}^2}}{2}+5]$

$ϵ[-\frac{14}{2}+5,\frac{14}{2}+5]$

$ϵ[-2,12]=[a,b]$

$a=-2,b=12$