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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

\frac{5}{4}


Option: 2

\frac{7}{4}


Option: 3

\frac{9}{4}


Option: 4

none of these.


Answers (1)

best_answer

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} \mathrm{T}_{r+1} & ={ }^{10} \mathrm{C}_r\left(\sqrt{\frac{\mathrm{x}}{3}}\right)^{10-\mathrm{r}}\left(\frac{3}{2 \mathrm{x}^2}\right)^{\mathrm{r}}={ }^{10} \mathrm{C}_r \frac{\mathrm{x}^{5-(\mathrm{r} / 2)}}{3^{5-(\mathrm{r} / 2)}} \cdot \frac{3^{\mathrm{r}}}{2^{\mathrm{r}} \mathrm{x}^{2 \mathrm{r}}} \\ & ={ }^{10} \mathrm{C}_{\mathrm{r}} \frac{3^{(3 \mathrm{r} / 2)-5}}{2^{\mathrm{r}}} \mathrm{x}^{5-(5 \mathrm{r} / 2)} \end{aligned}


For T_{z+1} to be independent of x, we must have 5-(5 r / 2)=0 or r=2. Thus, the 3 rd term is independent of x and is equal to


{ }^{10} \mathrm{C}_2 \frac{3^{3-5}}{2^2}=\frac{10 \times 9}{2} \times \frac{3^{-2}}{4}=\frac{5}{4}

Posted by

rishi.raj

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