Trigonometrical Equations 3 Question 13
14. The set of all $x$ in the interval $[0, \pi]$ for which $2 \sin ^{2} x-3 \sin x+1 \geq 0$, is…… .
(1987, 2M)
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Answer:
Correct Answer: 14. $x \in 0, \frac{\pi}{6} \cup \frac{\pi}{2} \cup \frac{5 \pi}{6}, \pi$
Solution:
- Given, $2 \sin ^{2} x-3 \sin x+1 \geq 0$
$$ \begin{array}{lr} \Rightarrow & 2 \sin ^{2} x-2 \sin x-\sin x+1 \geq 0 \\ \Rightarrow & (2 \sin x-1)(\sin x-1) \geq 0 \\ \Rightarrow & 2 \sin x-1 \leq 0 \text { or } \sin x \geq 1 \\ \Rightarrow & \sin x \leq \frac{1}{2} \quad \text { or } \quad \sin x=1 \\ \Rightarrow & x \in 0, \frac{\pi}{6} \cup \frac{\pi}{2} \cup \frac{5 \pi}{6}, \pi \end{array} $$
