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Find the Term Independent of X in the Expansion of the Expression: (V) ( √ X 3 + 3 2 X 2 ) 10 - Mathematics

Find the term independent of x in the expansion of the expression: 

(v)  \[\left( \frac{\sqrt{x}}{3} + \frac{3}{2 x^2} \right)^{10}\]

 

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Solution

(v) Suppose the (+ 1)th term in the given expression is independent of x.
Now, 

\[\left( \sqrt{\frac{x}{3}} + \frac{3}{2 x^2} \right)^{10} \]
\[ T_{r + 1} =^{10}{}{C}_r \left( \sqrt{\frac{x}{3}} \right)^{10 - r} \left( \frac{3}{2 x^2} \right)^r \]
\[ = ^{10}{}{C}_r . \frac{3^{r - \frac{10 - r}{2}}}{2^r} x^\frac{10 - r}{2} - 2r \]
\[\text{ For this term to be independent of x, we must have} \]
\[\frac{10 - r}{2} - 2r = 0\]
\[ \Rightarrow 10 - 5r = 0\]
\[ \Rightarrow r = 2\]
\[\text{ Hence, the required term is the 3rd term }  . \]
\[\text{ Now, we have } \]
\[^{10}{}{C}_2 \times \frac{3^{2 - \frac{10 - 2}{2}}}{2^2}\]
\[ = \frac{10 \times 9}{2 \times 4 \times 9}\]
\[ = \frac{5}{4}\]

  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Exercise 18.2 | Q 16.05 | Page 39
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