Question

Set of values of \(x\) lying in \([0,2\pi]\) satisfying the inequality \( |\sin x|>2\sin^2 x \) contains:

• A. \( (0,\frac{\pi}{6}) \cup (\pi,\frac{7\pi}{6}) \)
• B. \( (0,\frac{7\pi}{6}) \)
• C. \( \frac{\pi}{6} \)
• D. None of these

Hint:

Split modulus inequalities into cases — positive and negative parts.

Answer 1

The Correct Option is A

Concept: \[ |\sin x|>2\sin^2 x \]

Step 1: Case 1: \(\sin x \ge 0\) \[ \sin x>2\sin^2 x \Rightarrow \sin x(1 - 2\sin x)>0 \] \[ \Rightarrow 0<\sin x<\frac{1}{2} \] \[ x \in (0,\frac{\pi}{6}) \cup (\pi - \frac{\pi}{6}, \pi) = (0,\frac{\pi}{6}) \cup (\frac{5\pi}{6}, \pi) \]

Step 2: Case 2: \(\sin x<0\) \[ -\sin x>2\sin^2 x \Rightarrow \sin x(2\sin x + 1)<0 \] \[ - \frac{1}{2}<\sin x<0 \] \[ x \in (\pi, \frac{7\pi}{6}) \cup (\frac{11\pi}{6}, 2\pi) \]

Step 3: Combine valid intervals From given options, matching part is: \[ (0,\frac{\pi}{6}) \cup (\pi,\frac{7\pi}{6}) \] \[ \therefore \text{correct answer = (A) \( (0,\frac{\pi}{6}) \cup (\pi,\frac{7\pi}{6}) \)} \]