If the co-efficient of $x^9$ in ${\left( {\alpha {x^3} + {1 \over {\beta x}}} \right)^{11}}$ and the co-efficient of $x^{-9}$ in ${\left( {\alpha x - {1 \over {\beta {x^3}}}} \right)^{11}}$ are equal, then $(\alpha\beta)^2$ is equal to ___________.
Answer (integer)
1
Solution
Coefficient of $\mathrm{x}^{9}$ in $\left(\alpha x^{3}+\frac{1}{\beta x}\right)={ }^{11} C_{6} \cdot \frac{\alpha^{5}}{\beta^{6}}$
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$\because$ Both are equal
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$\therefore \frac{11}{C_{6}} \cdot \frac{\alpha^{5}}{\beta^{6}}=-\frac{11}{C_{5}} \cdot \frac{\alpha^{6}}{\beta^{5}}$
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$\Rightarrow \frac{1}{\beta}=-\alpha$
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$\Rightarrow \alpha \beta=-1$
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$\Rightarrow(\alpha \beta)^{2}=1$
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Applications of Binomial Theorem
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