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Question
Find the term independent of x in the expansion of the expression:
(iv) \[\left( 3x - \frac{2}{x^2} \right)^{15}\]
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Solution
(iv) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
\[\left( 3x - \frac{2}{x^2} \right)^{15} \]
\[ T_{r + 1} =^{15}{}{C}_r (3x )^{15 - r} \left( \frac{- 2}{x^2} \right)^r \]
`= ( - 1 )^r "^15 C_r \times 3^{15 - r} \times 2^r x^{15 - r - 2r} `
\[\text{ For this term to be independent of x, we must have} \]
\[15 - 3r = 0\]
\[ \Rightarrow r = 5\]
\[\text{ Hence, the required term is the 6th term .} \]
\[\text{ Now, we have: } \]
`( - 1 )^5 "^15 C_5 . 3^{15 - 5} . 2^5 `
\[ = - 3003 \times 3^{10} \times 2^5\]
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