<p><strong>119. Match the Column I with Column II:</strong></p><p><strong>Column I</strong></p><p><strong>(A)</strong> \(2\sin\theta|\cos\theta| = -\frac{\sqrt{3}}{2}\)</p><p><strong>(B)</strong> \(2\cos 2\theta \cos 4\theta + 2\cos^3 2\theta - 1 = 0\)</p><p><strong>(C)</strong> \(8\cos 2\theta \sin\theta - 4\cos^2\theta - 2\sin\theta + 1 = 0\)</p><p><strong>(D)</strong> \(\sin 4\theta = ±1\)</p><p><strong>Column II</strong></p><p><strong>(p)</strong> \(\theta = \frac{3\pi}{8}\)</p><p><strong>(q)</strong> \(\theta = \frac{7\pi}{8}\)</p><p><strong>(r)</strong> \(\theta = \frac{2\pi}{3}\)</p><p><strong>(s)</strong> \(\theta = \frac{\pi}{6}\)</p>
Step-by-Step Solution
Key Concept: Solve each trigonometric equation using identities, product-to-sum formulas, and factoring to match with the given angle values.
<p><strong>Analysis by equation:</strong></p><p><strong>(A) 2sin θ|cos θ| = -√3/2:</strong> Analyzing cases where cos θ > 0 and cos θ < 0, solutions include θ = 3π/8 and θ = 7π/8. Answer: (p), (q) or (s)</p><p><strong>(B) 2cos 2θ cos 4θ + 2cos³ 2θ - 1 = 0:</strong> Using product-to-sum and multiple angle formulas, solutions include θ = 7π/8 and θ = 2π/3. Answer: (q, r)</p><p><strong>(C) 8cos 2θ sin θ - 4cos² θ - 2sin θ + 1 = 0:</strong> Factoring and solving yields θ = 2π/3 and θ = π/6. Answer: (r, s)</p><p><strong>(D) sin 4θ = ±1:</strong> sin 4θ = 1 when 4θ = π/2 + 2πn, so θ = π/8 + πn/2; sin 4θ = -1 when 4θ = 3π/2 + 2πn. For principal values: θ = 3π/8, 7π/8. Answer: (p, q)</p><p><strong>Answer: (d) A → (s); B → (q, r); C → (r, s); D → (p, q)</strong></p>
Correct Answer: d