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  • The coefficient of the term independent of x in the expansion of <img alt="\left(\sqrt{\frac{x}{3}}+\frac{3}{2 x^2}\right)^{10}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cleft%28

The coefficient of the term independent of x in the expansion of \left(\sqrt{\frac{x}{3}}+\frac{3}{2 x^2}\right)^{10} is

Option: 1

5/4


Option: 2

7/4


Option: 3

9/4


Option: 4

none of these


Answers (1)

The (r + 1)th term in the expansion of \left(\sqrt{\frac{x}{3}}+\frac{3}{2 x^2}\right)^{10} is given by
\begin{aligned} T_{r+1} & ={ }^{10} C_r\left(\sqrt{\frac{x}{3}}\right)^{10-r}\left(\frac{3}{2 x^2}\right)^r={ }^{10} C_r \frac{x^{5-(r / 2)}}{3^{5-(r / 2)}} \cdot \frac{3^r}{2^r x^{2 r}} \\ & ={ }^{10} C_r \frac{3^{(3 r / 2)-5}}{2^r} x^{5-(5 r / 2)} \end{aligned}

For T_{r+1} to be independent of x, we must have 5 − (5r/2) = 0 or r = 2. Thus, the 3rd term is independent of x and its coefficient is given by
    
{ }^{10} C_2{ }^{\frac{3^{3-5}}{2^2}}=\frac{10 \times 9}{2} \times \frac{3^{-2}}{4}=\frac{5}{4} .

Posted by

Sumit Saini

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