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Question
Find the term independent of x in the expansion of the expression:
(x) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^6\]
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Solution
(x) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
\[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^6 \]
\[ T_{r + 1} = ^{6}{}{C}_r \left( \frac{3}{2} x^2 \right)^{6 - r} \left( \frac{- 1}{3x} \right)^r \]
`= \left( - 1 \right)^r "^6C_r \times \frac{3^{6 - r - r}}{2^{6 - r}} x^{12 - 2r - r} `
\[\text{ For this term to be independent of x, we must have} \]
\[12 - 3r = 0\]
\[ \Rightarrow r = 4\]
\[\text{ Hence, the required term is the 4th term } . \]
\[^{6}{}{C}_4 \times \frac{3^{6 - 4 - 4}}{2^{6 - 4}}\]
\[ = \frac{6 \times 5}{2 \times 1 \times 4 \times 9}\]
\[ = \frac{5}{12}\]
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