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Question
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 (r + 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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