<p>The coefficient of \(x^2\) in \((1+ax+bx^2)(1-3x)^{15}\) is equal to the coefficient of \(x^3\) in the same expansion. Given this condition (along with another condition), find the values of \(a\) and \(b\).</p>
Step-by-Step Solution
Key Concept: Extract coefficients of x² and x³ from the product by identifying which terms in (1+ax+bx²) multiply with which terms in (1-3x)¹⁵ to produce the required powers, then equate them.
<p><strong>Step 1:</strong> Find the coefficient of x² in (1+ax+bx²)(1-3x)¹⁵</p><p>From (1-3x)¹⁵, using binomial expansion: general term = C(15,r)(-3x)ʳ</p><p>• Coefficient of x⁰: C(15,0) = 1</p><p>• Coefficient of x¹: C(15,1)(-3) = -45</p><p>• Coefficient of x²: C(15,2)(-3)² = 105·9 = 945</p><p><strong>Step 2:</strong> Coefficient of x² in product = 1·945 + a·(-45) + b·1 = 945 - 45a + b</p><p><strong>Step 3:</strong> Find the coefficient of x³ in (1+ax+bx²)(1-3x)¹⁵</p><p>• Coefficient of x³ in (1-3x)¹⁵: C(15,3)(-3)³ = 455·(-27) = -12,285</p><p><strong>Step 4:</strong> Coefficient of x³ in product = 1·(-12,285) + a·945 + b·(-45) = -12,285 + 945a - 45b</p><p><strong>Step 5:</strong> Apply given condition: coefficient of x² = coefficient of x³</p><p>945 - 45a + b = -12,285 + 945a - 45b</p><p>12,230 = 990a - 46b</p><p>6,115 = 495a - 23b ... (Equation 1)</p><p><strong>Note:</strong> A second condition is needed to solve for unique values of a and b. Without it, infinitely many solutions exist satisfying this equation.</p><p>∴ Requires additional constraint (typically a = b or similar, or another coefficient equality condition)</p>
Correct Answer: A