Practicing Success
If A = $\{\theta : 2 cos^2 \theta + sin \theta ≤ 2\}$ and B = $\{\theta : \frac{\pi}{2} ≤ \theta ≤ \frac{3\pi}{2}$, then A ∩ B is equal to |
$\left\{\theta: \frac{\pi}{2} \leq \theta \leq \frac{5 \pi}{6}\right\}$ $\left\{\theta: \pi \leq \theta \leq \frac{3 \pi}{2}\right\}$ $\left\{\theta: \frac{\pi}{2} \leq \theta \leq \frac{5 \pi}{6}\right.$ or $\left.\leq \theta \leq \frac{3 \pi}{2}\right\}$ None of these |
$\left\{\theta: \frac{\pi}{2} \leq \theta \leq \frac{5 \pi}{6}\right.$ or $\left.\leq \theta \leq \frac{3 \pi}{2}\right\}$ |
Let : $2 cos^2 \theta + sin \theta ≤ 2$ and $\frac{3\pi}{2}$ $\Rightarrow 2-2 \sin ^2 \quad \theta+\sin \theta \leq 2$ $\Rightarrow 2 \sin ^2 \quad \theta-\sin \theta \geq 0 \Rightarrow \sin \theta(2 \sin \theta-1) \geq 0$ Case I. $\sin \theta \geq 0,2 \sin \theta-1 \geq 0$ ∴ $\sin \theta \geq 0, \sin \theta \geq \frac{1}{2} \Rightarrow \sin \theta \geq \frac{1}{2}$ $\Rightarrow \frac{\pi}{2} \leq \theta \leq \frac{5 \pi}{6}$ Case II. $\sin \theta \leq 0,2 \sin \theta-1 \leq 0$ $\sin \theta \leq 0, \sin \theta \leq \frac{1}{2} \Rightarrow \sin \theta \leq 0$ $\Rightarrow \pi \leq \theta \leq \frac{3 \pi}{2}$ ∴ $A \cap B=\left\{\theta: \frac{\pi}{2} \leq \theta \leq \frac{5 \pi}{6} \text { or } \pi \leq \theta \leq \frac{3 \pi}{2}\right\}$ Hence (3) is the correct answer. |