Question

Let
$A=\{\theta \in R:{(\frac{1}{3}\sin \theta +\frac{2}{3}\cos \theta )}^2=\frac{1}{3}{\sin }^2\left(\theta \right)+\frac{2}{3}{\cos }^2\left(\theta \right)\}.$
Then

• A. $A\cap [0,\pi ]$ has excatly one point
• B. $A\cap [0,\pi ]$ has more than two points
• C. $A\cap [0,\pi ]$ is an empty set
• D. $A\cap [0,\pi ]$ has excatly two point

Answer 1

The correct option is A $A\cap [0,\pi ]$ has excatly one point

${(\frac{1}{3}\sin \theta +\frac{2}{3}\cos \theta )}^2=\frac{1}{3}{\sin }^2\left(\theta \right)+\frac{2}{3}{\cos }^2\left(\theta \right)\Rightarrow \frac{1}{9}{\sin }^2\theta +\frac{4}{9}{\cos }^2\theta +\frac{4}{9}\sin \theta \cos \theta =\frac{1}{3}{\sin }^2\theta +\frac{2}{3}{\cos }^2\theta \Rightarrow \frac{4}{9}\sin \theta \cos \theta =\frac{2}{9}{\sin }^2\theta +\frac{2}{9}{\cos }^2\theta \Rightarrow \sin 2\theta ={\sin }^2\theta +{\cos }^2\theta \Rightarrow \sin 2\theta =1\Rightarrow 2\theta =\frac{\pi }{2},\frac{5\pi }{2}\Rightarrow \theta =\frac{\pi }{4},\frac{5\pi }{4}$

$\Rightarrow \theta =\frac{\pi }{4}\text{in}[0,\pi ]$
Hence, The correct answer is option B.