Question

The values of $\alpha$ in the interval $[\frac{\pi }{2},\frac{3\pi }{2}]$, for which the point $(\sin \alpha ,\cos \alpha )$ does not lie outside the parabola $2y^2+x-2=0$ lie in the set

• A. $[\pi ,\frac{3\pi }{2}]$
• B. $[\frac{\pi }{2},\frac{5\pi }{6}]$
• C. $[\frac{\pi }{3},\frac{5\pi }{6}]$
• D. Both A and B

Answer 1

The correct option is D Both A and B

Point $(\sin \alpha ,\cos \alpha )$ not lie outside the parabola
$2y^2+x-2=0$
$\Rightarrow 2{\cos }^2\alpha +\sin \alpha -2\le 0$
$2-2{\sin }^2\alpha +\sin \alpha -2\le 0$
$\sin \alpha \le 0$ or $\sin \alpha \ge \frac{1}{2}$
$\alpha \in [\pi ,\frac{3\pi }{2}]$ or $\alpha \in [\frac{\pi }{2},\frac{5\pi }{6}]$