Question

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

• A. $1\frac{1}{2}$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (B)

$2$

Let $A={\sin }^2\theta +3\sin \theta \cos \theta +5{\cos }^2\theta$

$=\frac{3}{2}\sin 2\theta +1+4{\cos }^{2}x$

$=\frac{3}{2}\sin 2\theta +1+2(1+\cos 2\theta )$

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

$=3+\frac{3\sin 2\theta +4\cos 2\theta }{2}$

Hence $\frac{-5}{2}\le \frac{3\sin 2\theta +4\cos 2\theta }{2}\le \frac{5}{2}$

$\therefore 3-\frac{5}{2}\le A\le 3+\frac{5}{2}$

$\Rightarrow \frac{1}{2}\le A\le \frac{11}{2}$

$\Rightarrow \frac{2}{11}\le \frac{1}{A}\le 2$

Hence, the maximum value of the given expression is $2$