<p>Given \(x^2\left(\sqrt{x}+\dfrac{\lambda}{x^2}\right)^{10}\), the coefficient of \(x^2\) in the expansion is 720. Find \(\lambda\).</p>
Step-by-Step Solution
Key Concept: Convert the expression to a single variable power form using the binomial theorem, then identify which term in the expansion gives x^2 by equating the exponent of x to 2.
<p><strong>Step 1:</strong> Rewrite the expression as <span style="color:#d63031">x^2(√x + λx^(-2))^10 = x^2 · Σ C(10,r)(√x)^(10-r)(λx^(-2))^r</span></p><p><strong>Step 2:</strong> The general term is <span style="color:#d63031">x^2 · C(10,r) · x^((10-r)/2) · λ^r · x^(-2r) = C(10,r)λ^r · x^(2 + (10-r)/2 - 2r)</span></p><p><strong>Step 3:</strong> Simplify the exponent: <span style="color:#d63031">2 + (10-r)/2 - 2r = 2 + 5 - r/2 - 2r = 7 - 5r/2</span></p><p><strong>Step 4:</strong> For coefficient of x^2, set exponent = 2: <span style="color:#d63031">7 - 5r/2 = 2 ⟹ 5r/2 = 5 ⟹ r = 2</span></p><p><strong>Step 5:</strong> Substitute r = 2: <span style="color:#d63031">C(10,2)λ^2 = 720</span></p><p><strong>Step 6:</strong> Calculate: <span style="color:#d63031">45λ^2 = 720 ⟹ λ^2 = 16 ⟹ λ = ±4</span></p><p>∴ Answer: <strong>λ = 4 or λ = -4</strong></p>
Correct Answer: A