The set of values of $\theta$ satisfying the inequation $2 \sin ^2 \theta-5 \sin \theta+2>0$, where $0 < \theta < 2 \pi$, is

• Correct answer
  A

  $\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, 2 \pi\right)$
• B

  $\left[0, \frac{\pi}{6}\right] \cup\left[\frac{5 \pi}{6}, 2 \pi\right]$
• C

  $\left[0, \frac{\pi}{3}\right] \cup\left[\frac{2 \pi}{3}, 2 \pi\right]$
• D

  None of these

Solution

$$ \begin{aligned} & \text { As, } 2 \sin ^2 \theta-5 \sin \theta+2>0 \\ & \Rightarrow(2 \sin \theta-1)(\sin \theta-2)>0 \\ & \quad[\text { where, }(\sin \theta-2) < 0 \text { for all } \theta \in R] \\ & \therefore \quad(2 \sin \theta-1) < 0 \end{aligned} $$
$$ \begin{array}{ll} \Rightarrow & \sin \theta < \frac{1}{2}, \text { shown as } \\ \therefore & \theta \in\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, 2 \pi\right) \end{array} $$

Hence, the correct answer is option: 'A'.