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Find the term independent of x in the expansion of \(\big[\sqrt{\frac{x}{3}}+\frac{\sqrt{3}}{2x^2}\big]^{10}\)

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Let (r + 1)th term be independent of x which is given by

Since the term is independent of x, we have

\(\frac{10-r}{2}\) - 2r = 0 ⇒ r = 2

Hence 3rd term is independent of and its value is given by

T3 =  10C2 \(\frac{3^{-3}}{4} = \frac{10\times9}{2\times1}\times\frac{1}{9\times12} = \frac{5}{12}\)

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