Question

The set of all x in the interval $[0,\pi ]$ for which $2si{n}^{2}x-3sinx+1\ge 0$ is

• A. $[0,\pi /6]\cup \left[\pi /2\right]\cup [5\pi /6,\pi ]$
• B. $[0,\pi /6]$
• C. $[5\pi /6,\pi ]$
• D. $\left[\pi /2\right]\cup [5\pi /6,\pi ]$

Answer 1

The correct option is A $[0,\pi /6]\cup \left[\pi /2\right]\cup [5\pi /6,\pi ]$

$2{\sin }^{2}x-3\sin x+1\ge 0\Rightarrow 2{\sin }^{2}x-2\sin x-\sin x+1\ge 0\Rightarrow (2\sin x-1)(\sin x-1)\ge 0\Rightarrow 2\sin x-1\le 0\sin x\ge 1$
$\Rightarrow \sin x\le \frac{1}{2}$ or $\sin x=1$
$\Rightarrow xϵ[0,\frac{\pi }{6}]\cup \left\{\frac{\pi }{2}\right\}\cup [\frac{5\pi }{6},\pi ]$