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Question
In the expansion of \[\left( x^2 - \frac{1}{3x} \right)^9\] , the term without x is equal to
Options
\[\frac{28}{81}\]
\[\frac{-28}{243}\]
\[\frac{28}{243}\]
none of these
Solution
\[\frac{28}{243}\]
Suppose the (r + 1)th term in the given expansion is independent of x.
Then , we have:
\[T_{r + 1} = ^{9}{}{C}_r ( x^2 )^{9 - r} \left( \frac{- 1}{3x} \right)^r \]
`= ( - 1 )^r " ^9C_r \frac{1}{3^r} x^{18 - 2r - r}`
\[\text{ For this term to be independent of x, we must have: } \]
\[18 - 3r = 0\]
\[ \Rightarrow r = 6\]
`therefore \text{ Required term } = ( - 1 )^6 " ^9C_6 \frac{1}{3^6} = \frac{9 \times 8 \times 7}{3 \times 2} \times \frac{1}{3^6} = \frac{28}{243}`
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