<p>Given \((2-x^2)\cdot\left((1+2x+3x^2)^6+(1-4x^2)^6\right)\), find the coefficient of \(x^2\).</p>
Step-by-Step Solution
Key Concept: Recognize that the expression contains a product of two parts: (2-x²) and a sum of two sixth powers. Use binomial expansion selectively—only the terms that produce x² when multiplied by (2-x²) contribute to the final coefficient.
<p><strong>Step 1:</strong> Let S = (1+2x+3x²)⁶ + (1-4x²)⁶. We need the coefficient of x² in (2-x²)·S.</p><p><strong>Step 2:</strong> Expanding (2-x²)·S = 2S - x²·S, the x² terms come from:</p><ul><li>2 times the [x²] coefficient of S</li><li>-1 times the [x⁰] coefficient of S (constant term)</li></ul><p><strong>Step 3:</strong> Find [x⁰] in S: Both (1+2x+3x²)⁶ and (1-4x²)⁶ have constant term = 1. So [x⁰]S = 1 + 1 = 2.</p><p><strong>Step 4:</strong> Find [x²] in (1+2x+3x²)⁶ using multinomial theorem: [x²] = C(6,0,1,0)·1⁵·(2x)¹·0 + C(6,0,0,1)·1⁵·(3x²)¹ + C(6,2,0,0)·1⁴·(2x)² = 6·3 + 15·4 = 18 + 60 = 78.</p><p><strong>Step 5:</strong> Find [x²] in (1-4x²)⁶: Using C(6,0)·1⁶ + C(6,1)·1⁵·(-4x²) gives only x⁰ and x² terms. [x²] = C(6,1)·(-4) = -24.</p><p><strong>Step 6:</strong> [x²]S = 78 + (-24) = 54.</p><p><strong>Step 7:</strong> Coefficient of x² in (2-x²)·S = 2(54) - 1(2) = 108 - 2 = 106.</p><p>∴ Answer: A</p>
Correct Answer: A