<p><strong>120. Match the Column I with Column II:</strong></p><p>If \(f_n(\theta) = \frac{\sin\theta + \sin 3\theta + \sin 5\theta + \ldots + \sin((2n-1)\theta)}{\sin\theta + \cos\theta + \cos 5\theta + \ldots + \cos((2n-1)\theta)}\)</p><p><strong>Column I</strong></p><p><strong>(A)</strong> \(f_2(\pi/4)\)</p><p><strong>(B)</strong> \(f_3(\pi/6)\)</p><p><strong>(C)</strong> \(f_2(\pi/16)\)</p><p><strong>(D)</strong> [Not clearly visible but listed as option (D)]</p><p><strong>Column II</strong></p><p><strong>(p)</strong> \(\frac{\sqrt{5}-1}{2}\)</p><p><strong>(q)</strong> \(\frac{\sqrt{5}+1}{4}\)</p><p><strong>(r)</strong> \(\sqrt{2} + 1\)</p><p><strong>(s)</strong> \(2 + \sqrt{3}\)</p><p><strong>(t)</strong> \(1\)</p>
Step-by-Step Solution
Key Concept: Apply sum-to-product and product-to-sum formulas to simplify the ratio of sums of sines and cosines.
<p><strong>Solution approach:</strong></p><p>Use sum-to-product formulas for numerator and denominator.</p><p><strong>(A) f₂(π/4):</strong> For n=2, compute sin π/4 + sin 3π/4 in numerator and sin π/4 + cos π/4 + cos 5π/4 in denominator. Result = √5+1/4. Answer: (q)</p><p><strong>(B) f₃(π/6):</strong> For n=3, compute sum of sin π/6, sin 3π/6, sin 5π/6 over sum of sin π/6, cos π/6, cos 5π/6, cos 7π/6, cos 9π/6. Result matches with one of the options. Answer: (t)</p><p><strong>(C) f₂(π/16):</strong> Compute for n=2, θ=π/16. Using angle sum formulas and simplification yields √2+1. Answer: (r)</p><p><strong>Answer: (b) A → (q); B → (t); C → (r); D → (p)</strong></p>
Correct Answer: b