JEE Advanced Maths: Sequences, Trigonometry questions with solutions

Q1. Let alpha be a solution of 2[x + 32] = 3[x - 64] (where [x] denotes the greatest integer function), and let beta = product_(r=1)⁹ sin((2r-1)*pi/18). Which of the following can be true?

1. [alpha] = [beta]
2. alpha = 2051 / 8
3. [alpha] * [1/beta] = 1
4. [1/alpha] + [1/beta] = 2⁸

Answer: [1/alpha] + [1/beta] = 2⁸

Step 1: 2[x+32]=3[x-64]. Since 32 and 64 are integers, [x+32]=[x]+32 and [x-64]=[x]-64. Let n=[x]: 2(n+32)=3(n-64) => 2n+64=3n-192 => n=256. So [alpha]=256, alpha in [256,257). Step 2: beta = sin(pi/18)*sin(3pi/18)*...*sin(17pi/18). Using product formula for odd-indexed sines: productₖ₌₀⁸ sin((2k+1)pi/18) = sqrt(2)/2⁹... actually = 1/2⁸ = 1/256 (derived from Chebyshev/product identities). So beta=1/256. [1/beta]=[256]=256. Check options: A: [alpha]=256, [beta]=[1/256]=0. 256≠0. FALSE. B: alpha=2051/8=256.375 in [256,257). POSSIBLE (TRUE). C: [alpha]*[1/beta]=256*256=65536≠1. FALSE. D: [1/alpha]+[1/beta]=[1/256.375]+256=0+256=256=2⁸. TRUE.