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Find the Term Independent of X in the Expansion of the Expression: (Ii) ( 2 X + 1 3 X 2 ) 9 - Mathematics

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

(ii)  \[\left( 2x + \frac{1}{3 x^2} \right)^9\]

 

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Solution

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

\[\left( 2x + \frac{1}{3 x^2} \right)^9 \]
\[ T_{r + 1} = ^{9}{}{C}_r (2x )^{9 - r} \left( \frac{1}{3 x^2} \right)^r \]
\[ = ^{9}{}{C}_r . \frac{2^{9 - r}}{3^r} x^{9 - r - 2r} \]
\[\text{ For this term to be independent of x, we must have} \]
\[9 - 3r = 0\]
\[ \Rightarrow r = 3\]
\[\text{ Hence, the required term is the 4th term .}  \]
\[\text{ Now, we have } \]
\[ ^{9}{}{C}_3 \frac{2^6}{3^3}\]
\[ = ^{9}{}{C}_3 \times \frac{64}{27}\]

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Exercise 18.2 | Q 16.02 | Page 39
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