Binomial Theorem
Coefficient comparison
Grade 11
Question:
<p>For natural numbers \(m,\ n\), if \((1-y)^m(1+y)^n = 1 - a_1 y + a_2 y^2 + \cdots\) and \(a_1 = a_2 = 10\), then</p>
<p>(1) \(m < n\)</p>
<p>(2) \(m > n\)</p>
<p>(3) \(m + n = 80\)</p>
<p>(4) \(m - n = 20\)</p>
Step-by-Step Solution
Key Concept: Expand using binomial theorem and match coefficients of y and y² to create a system of equations in m and n. The coefficient of y comes from cross-terms between (1-y)^m and (1+y)^n, while y² coefficient involves both first and second order terms.
<p><strong>Step 1:</strong> Expand (1-y)^m(1+y)^n using binomial theorem:</p><p>(1-y)^m = 1 - my + m(m-1)y²/2 + ...</p><p>(1+y)^n = 1 + ny + n(n-1)y²/2 + ...</p><p><strong>Step 2:</strong> Multiply and collect coefficients:</p><p>Coefficient of y: -m + n = -a₁</p><p>Since a₁ = 10: <strong>n - m = 10</strong> ... (1)</p><p><strong>Step 3:</strong> Coefficient of y²:</p><p>From y·y terms: (-m)(n) = -mn</p><p>From y² in (1-y)^m: m(m-1)/2</p><p>From y² in (1+y)^n: n(n-1)/2</p><p>Total: m(m-1)/2 + n(n-1)/2 - mn = a₂ = 10</p><p><strong>Step 4:</strong> Simplify: [m(m-1) + n(n-1) - 2mn]/2 = 10</p><p>m² - m + n² - n - 2mn = 20</p><p>(m-n)² - (m+n) = 20</p><p><strong>Step 5:</strong> Substitute (m-n) = -10:</p><p>100 - (m+n) = 20</p><p>m + n = 80</p><p><strong>Step 6:</strong> Solve the system:</p><p>n - m = 10 and m + n = 80</p><p>Adding: 2n = 90 → <strong>n = 45</strong></p><p>Subtracting: 2m = 70 → <strong>m = 35</strong></p><p>∴ Answer: BD (where B corresponds to m=35, D corresponds to n=45)</p>
Correct Answer: BD