<p>If the coefficients of \(x^7\) and \(x^4\) are equal in magnitude but opposite in sign in the expansion of \(\left(\dfrac{x^2}{a} - \dfrac{b}{x}\right)^{11}\), then</p>
Step-by-Step Solution
Key Concept: In the binomial expansion of (x²/a - b/x)¹¹, the general term is C(11,r)(x²/a)^(11-r)(-b/x)^r. For equal magnitude but opposite sign coefficients of x⁷ and x⁴, we need the same binomial coefficient with opposite signs from the (-1)^r factor, which occurs when the powers of x are symmetric and the sign pattern differs.
<p><strong>Step 1:</strong> Find the general term in the expansion of (x²/a - b/x)¹¹</p><p>T_(r+1) = C(11,r)(x²/a)^(11-r)(-b/x)^r = C(11,r)(-1)^r · (1/a)^(11-r) · b^r · x^(22-2r-r)</p><p>T_(r+1) = C(11,r)(-1)^r · (1/a)^(11-r) · b^r · x^(22-3r)</p><p><strong>Step 2:</strong> Find r values for x⁷ and x⁴ terms</p><p>For x⁷: 22 - 3r = 7 ⟹ r = 5</p><p>For x⁴: 22 - 3r = 4 ⟹ r = 6</p><p><strong>Step 3:</strong> Set up the condition for equal magnitude, opposite sign</p><p>Coefficient of x⁷ = C(11,5)(-1)^5(1/a)^6 b^5 = -C(11,5)(1/a)^6 b^5</p><p>Coefficient of x⁴ = C(11,6)(-1)^6(1/a)^5 b^6 = C(11,6)(1/a)^5 b^6</p><p><strong>Step 4:</strong> Apply the condition: |coeff of x⁷| = |coeff of x⁴| with opposite signs</p><p>C(11,5)(1/a)^6 b^5 = C(11,6)(1/a)^5 b^6</p><p>Since C(11,5) = C(11,6) = 462:</p><p>(1/a)^6 b^5 = (1/a)^5 b^6</p><p>Dividing both sides by (1/a)^5 b^5:</p><p>1/a = b</p><p>∴ <strong>ab = 1</strong> or the relationship establishes the required condition</p>
Correct Answer: A