<p>The coefficient of \(x^2\) in the expansion of the product \((2-x^2)\cdot((1+2x+3x^2)^6+(1-4x^2)^6)\) is</p>
Step-by-Step Solution
Key Concept: The coefficient of x² comes only from the product of (-x²) with the constant term of the second bracket, and (2) with the x² term of the second bracket. Use binomial theorem to find these terms in each binomial expansion separately.
<p><strong>Step 1:</strong> Identify that we need the coefficient of x² in (2-x²)·[(1+2x+3x²)⁶+(1-4x²)⁶].</p><p><strong>Step 2:</strong> Coefficient of x² = 2·[coefficient of x² in (1+2x+3x²)⁶ + coefficient of x² in (1-4x²)⁶] + (-1)·[constant term in both brackets].</p><p><strong>Step 3:</strong> For (1-4x²)⁶: Contains only even powers. Constant term = 1. Coefficient of x² is C(6,1)·(-4x²) = -24.</p><p><strong>Step 4:</strong> For (1+2x+3x²)⁶: Using multinomial expansion, coefficient of x² comes from (1)⁶⁻ᵏ·(2x)ᵃ·(3x²)ᵇ where a+2b=2. Cases: a=2,b=0 gives C(6,2)·2²=15·4=60; a=0,b=1 gives C(6,1)·3=18. Total = 60+18 = 78.</p><p><strong>Step 5:</strong> Coefficient of x² in second bracket = 78 + (-24) = 54.</p><p><strong>Step 6:</strong> Final answer = 2·54 = 108.</p><p>∴ Answer: C</p>
Correct Answer: C