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Question
Find the term independent of x in the expansion of the expression:
(vi) \[\left( x - \frac{1}{x^2} \right)^{3n}\]
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Solution
(vi) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
\[\left( x - \frac{1}{x^2} \right)^{3n} \]
\[ T_{r + 1} = ^{3n}{}{C}_r x^{3n - r} \left( \frac{- 1}{x^2} \right)^r \]
` = ( - 1 )^r "^(3n) C_r x^{3n - r - 2r} `
\[\text{ For this term to be independent of x, we must have} \]
\[3n - 3r = 0\]
\[ \Rightarrow r = n\]
\[\text{ Hence, the required term is the (n + 1)th term .} \]
\[\text{ Now, we have} \]
`( - 1 )^n "^3n C_n`
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