Q18 / 50 · Manipal (MET) 2022 · Mathematics
For what value of $\theta$ lying between 0 and $\pi$ which satisfy inequality $\sin \theta \cos ^3 \theta\gt\sin ^3 \theta \cos \theta$
- A$\theta \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
- Correct answerB$\theta \in\left(0, \frac{\pi}{4}\right)$
- C$\theta \in\left(0, \frac{\pi}{2}\right)$
- DNone of these
Solution
$\begin{array}{lr}\text { We have, } \sin \theta \cos ^3 \theta\gt\sin ^3 \theta \cos \theta \\ \Rightarrow & \sin \theta \cos ^3 \theta-\sin ^3 \theta \cos \theta\gt0 \\ \Rightarrow & \sin \theta \cos \theta\left(\cos ^2 \theta-\sin ^2 \theta\right)\gt0 \\ \Rightarrow & \sin 2 \theta \cdot \cos 2 \theta\gt0 \\ \Rightarrow & \sin 4 \theta\gt0 \\ \Rightarrow & \quad \theta \in\left(0, \frac{\pi}{4}\right) \quad[\therefore 4 \theta \in(0, \pi)]\end{array}$
Hence, the correct answer is option: 'B'.