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Question
In the expansion of \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\] , the term independent of x is
Options
T5
T6
T7
T8
Solution
T6
Suppose the (r + 1)th term in the given expansion is independent of x.
Thus, we have:
\[T_{r + 1} =^{8}{}{C}_r \left( \frac{1}{2} x^{1/3} \right)^{8 - r} ( x^{- 1/5} )^r \]
\[ = ^{8}{}{C}_r \frac{1}{2^{8 - r}} x^\frac{8 - r}{3} - \frac{r}{5} \]
\[\text{ For this term to be independent of x, we must have } \]
\[\frac{8 - r}{3} - \frac{r}{5} = 0\]
\[ \Rightarrow 40 - 5r - 3r = 0\]
\[ \Rightarrow r = 5\]
\[\text{ Hence, the required term is the 6th term, i . e } . T_6\]
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