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Question
If the coefficient of x in \[\left( x^2 + \frac{\lambda}{x} \right)^5\] is 270, then \[\lambda =\]
Options
3
4
5
none of these
Solution
3
\[\text{ The coefficient of x in the given expansion where x occurs at the (r + 1)th term } . \]
\[\text{ We have } \]
\[ ^{5}{}{C}_r ( x^2 )^{5 - r} \left( \frac{\lambda}{x} \right)^r \]
\[ =^{5}{}{C}_r \lambda^r x^{10 - 2r - r} \]
\[\text{ For it to contain x, we must have: } \]
\[10 - 3r = 1\]
\[ \Rightarrow r = 3 \]
\[ \therefore \text{ Coefficient of x in the given expansion: } \]
\[ ^{5}{}{C}_3 \lambda^3 = 10 \lambda^3 \]
\[\text{ Now, we have } \]
\[10 \lambda^3 = 270\]
\[ \Rightarrow \lambda^3 = 27\]
\[ \Rightarrow \lambda = 3\]
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