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प्रश्न
If the coefficient of x in \[\left( x^2 + \frac{\lambda}{x} \right)^5\] is 270, then \[\lambda =\]
विकल्प
3
4
5
none of these
उत्तर
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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\[= ^{5}{}{C}_0 (2x )^5 (3y )^0 +^{5}{}{C}_1 (2x )^4 (3y )^1 + ^{5}{}{C}_2 (2x )^3 (3y )^2 + ^{5}{}{C}_3 (2x )^2 (3y )^3 + ^{5}{}{C}_4 (2x )^1 (3y )^4 +^{5}{}{C}_5 (2x )^0 (3y )^5\]
\[= 32 x^5 + 5 \times 16 x^4 \times 3y + 10 \times 8 x^3 \times 9 y^2 + 10 \times 4 x^2 \times 27 y^3 + 5 \times 2x \times 81 y^4 + 243 y^5 \]
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