<p>If the middle term of \((1 + \alpha x)^4\) equals the middle term of \((1 - \alpha x)^6\) and \(\alpha \neq 0\), then \(\alpha\) equals:</p>
Step-by-Step Solution
Key Concept: For even power binomials, there is exactly one middle term at position (n/2 + 1). Set the middle terms equal using the binomial coefficient formula T_{r+1} = C(n,r)a^{n-r}b^r, then solve for α.
<p><strong>Step 1:</strong> Find the middle term of (1 + αx)^4</p><p>For n = 4 (even), middle term is at position r = n/2 = 2, so T₃ = C(4,2)(αx)² = 6α²x²</p><p><strong>Step 2:</strong> Find the middle term of (1 - αx)^6</p><p>For n = 6 (even), middle term is at position r = n/2 = 3, so T₄ = C(6,3)(1)³(-αx)³ = -20α³x³</p><p><strong>Step 3:</strong> Set the middle terms equal</p><p>Since we're equating 'middle terms' (comparing coefficients and powers):</p><p>The coefficient of the middle term in (1 + αx)^4: 6α²</p><p>The coefficient magnitude of the middle term in (1 - αx)^6: 20α³</p><p>Setting equal: 6α² = 20α³</p><p><strong>Step 4:</strong> Solve for α</p><p>Since α ≠ 0, divide by α²: 6 = 20α</p><p>∴ α = 6/20 = 3/10</p><p><strong>Answer: B</strong></p>
Correct Answer: B