Question

Let $A=\{\theta :2{\cos }^2+\theta +\sin \theta \le 2\}$ and $B=\{\theta :\pi /2\le \theta \le 3\pi /2\}.$ Then $A\cap B=\{\theta :\pi /2\le \theta \le 5\pi /6or\pi \le \theta \le 3\pi /4\}$
If true enter 1 else 0

Answer 1

$2{\cos }^2\theta +\sin \theta \le 2$

$2{\sin }^2\theta -\sin \theta \ge 0$

$\sin \theta \in [-1,0]\cup [\frac{1}{2},1]$

$\theta \in [\frac{3\pi }{2},2\pi ]\cup [\frac{\pi }{6},\frac{5\pi }{6}]$

$A\cap B\in [\frac{\pi }{2},\frac{5\pi }{6}]$

The correct answer is $0$