<p>Find the term independent of \(x\) in the expansion of \(\left(\sqrt{\dfrac{x}{3}} + \dfrac{\sqrt{3}}{2x^2}\right)^{10}\).</p>
Step-by-Step Solution
Key Concept: The general term in the binomial expansion is T_{r+1} = C(10,r)(√(x/3))^(10-r)(√3/(2x²))^r. For the term independent of x, the power of x in this term must equal zero: (10-r)/2 - 2r = 0.
<p><strong>Step 1:</strong> Write the general term using binomial expansion:</p><p>T_{r+1} = C(10,r)(√(x/3))^(10-r)(√3/(2x²))^r</p><p><strong>Step 2:</strong> Simplify the general term:</p><p>T_{r+1} = C(10,r) · (1/√3)^(10-r) · x^((10-r)/2) · (√3)^r · (1/2)^r · x^(-2r)</p><p>T_{r+1} = C(10,r) · (1/2)^r · 3^(-((10-r)/2)) · 3^(r/2) · x^((10-r)/2 - 2r)</p><p><strong>Step 3:</strong> For the term independent of x, equate the power of x to zero:</p><p>(10-r)/2 - 2r = 0</p><p>10 - r - 4r = 0</p><p>10 = 5r</p><p>r = 2</p><p><strong>Step 4:</strong> Find T₃ (when r = 2):</p><p>T₃ = C(10,2) · (1/2)² · 3^(-(10-2)/2) · 3^(2/2)</p><p>T₃ = 45 · (1/4) · 3^(-4) · 3^1</p><p>T₃ = 45 · (1/4) · (3/81)</p><p>T₃ = 45 · (1/4) · (1/27)</p><p>T₃ = 45/(108) = 5/12</p><p>∴ Answer: <strong>5/12</strong></p>
Correct Answer: 5