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  4. The term independent of x in the expansion of ( √(x/3)+(3/2x2) )10 is

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

J & K CETJ & K CET 2009Binomial Theorem

Solution:

General term, $ {{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}}{{3}^{r-\frac{10-r}{2}}}{{\left( \frac{1}{2} \right)}^{r}}.{{x}^{\frac{10-r}{2}-2r}} $
$ {{=}^{10}}{{C}_{r}}{{3}^{r-\frac{3r-10}{2}}}{{\left( \frac{1}{2} \right)}^{r}}.{{x}^{\frac{10-5r}{2}}} $
For the term independent of x, put
$ \frac{10-5r}{2}=0 $
$ \Rightarrow $ $ 5r=10 $
$ \Rightarrow $ $ r=2 $
$ \therefore $ The term independent of x,
$ {{T}_{3}}{{=}^{10}}{{C}_{2}}\,{{3}^{\frac{6-10}{2}}}{{\left( \frac{1}{2} \right)}^{2}} $ $ =\frac{10\times 9}{2\times {{3}^{2}}\times 4}=\frac{5}{4} $