Question:
The term independent of $x$ in the expansion of $ {{\left( \sqrt{\frac{x}{3}}+\frac{3}{2{{x}^{2}}} \right)}^{10}} $ is
The term independent of $x$ in the expansion of $ {{\left( \sqrt{\frac{x}{3}}+\frac{3}{2{{x}^{2}}} \right)}^{10}} $ is
Updated On: Jun 23, 2024
- $ \frac{5}{4} $
- $ \frac{7}{4} $
- $ \frac{9}{4} $
- $ 45 $
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The Correct Option is A
Solution and Explanation
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} $
$ {{=}^{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} $
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.