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  4. The maximum value of the expression (1/ sin 2 θ+3 sin θ cos θ+5 cos 2 θ) is

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Q. The maximum value of the expression $\frac{1}{\sin ^{2} \theta+3 \sin \theta \cos \theta+5 \cos ^{2} \theta}$ is _______

JEE AdvancedJEE Advanced 2010

Solution:

$\frac{1}{4 \cos ^{2} \theta+1+\frac{3}{2} \sin 2 \,\theta}$
$\Rightarrow \frac{1}{2[1+\cos 2 \,\theta]+1+\frac{3}{2} \sin 2 \,\theta}$
lies between $\frac{1}{2}$ to $\frac{11}{2}$
$\therefore $ maximum value is $2$
Minimum value of $1+4 \cos ^{2} \theta+3 \sin \theta \cos \theta$
$1+\frac{4(1+\cos 2\, \theta)}{2}+\frac{3}{2} \sin 2 \,\theta $
$=1+2+2 \cos 2 \,\theta+\frac{3}{2} \sin 2\, \theta $
$=3+2 \cos 2 \,\theta+\frac{3}{2} \sin 2\, \theta$
$\therefore =3-\sqrt{4+\frac{9}{4}}=3-\frac{5}{2}=\frac{1}{2}$
So maximum value of $\frac{1}{4 \cos ^{2} \theta+1+\frac{3}{2} \sin 2 \theta}$ is $2$ .