Question

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

• A. 2
• B. 3
• C. 4
• D. 5

Answer 1

The correct option is A

2

$f\left(\theta \right)=\frac{1}{si{n}^2\theta +3sin\theta costheta+5co{s}^2\theta }$

Let $g\left(\theta \right)=si{n}^2\theta +3sin\theta cos\theta +5co{s}^2\theta$

= $\frac{1-cos2\theta }{2}+5\left(\frac{1+cos2\theta }{2}\right)+\frac{3}{2}sin2\theta$

= 3 + 2 $co{s}^2\theta +\frac{3}{2}sin2\theta$

$g(\theta {)}_{min}=3-\sqrt{4+\frac{9}{4}}=3+\frac{5}{2}=\frac{1}{2}$

$\Rightarrow$ $f\left(\theta \right)=2$