Binomial Theorem
Binomial Theorem for Positive Integral Indices
Grade 11
Question:
<p>For natural numbers <em>m</em>, <em>n</em> if <br>
\((1 - y)^m (1 + y)^n = 1 + a_1 y + a_2 y^2 + \ldots\), and \(a_1 = a_2 = 10\),<br>
then \((m, n)\) is</p>
<p>(20, 45)</p>
<p>(35, 20)</p>
<p>(45, 35)</p>
<p>(35, 45)</p>
Step-by-Step Solution
Key Concept: Extract coefficients of y and y² using binomial expansion, then set up a system of equations from the condition a₁ = a₂ = 10 to find m and n uniquely.
<p><strong>Step 1:</strong> Expand using binomial theorem:</p><p>(1-y)^m = 1 - my + (m(m-1)/2)y² + ...</p><p>(1+y)^n = 1 + ny + (n(n-1)/2)y² + ...</p><p><strong>Step 2:</strong> Find coefficient a₁ of y in the product:</p><p>a₁ = -m + n = 10 ... (i)</p><p><strong>Step 3:</strong> Find coefficient a₂ of y² in the product:</p><p>a₂ = m(m-1)/2 + n(n-1)/2 - mn = 10 ... (ii)</p><p><strong>Step 4:</strong> From equation (i): n = m + 10</p><p><strong>Step 5:</strong> Substitute into equation (ii):</p><p>m(m-1)/2 + (m+10)(m+9)/2 - m(m+10) = 10</p><p>m² - m + m² + 19m + 90 - 2m² - 20m = 20</p><p>-2m + 90 = 20</p><p>m = 35</p><p><strong>Step 6:</strong> Therefore n = 35 + 10 = 45</p><p>∴ Answer: (m, n) = (35, 45)</p>
Correct Answer: D