Question

If the co-efficient of $x^9$ in $\left(\alpha x^3+\frac{1}{\beta x}\right)^{11}$ and the co-efficient of $x^{-9}$ in $\left(\alpha x-\frac{1}{\beta x^3}\right)^{11}$ are equal, then $(\alpha \beta)^2$ is equal to______

Answer 1

Coefficient of $x ^9$ in $\left(\alpha x^3+\frac{1}{\beta x}\right)={ }^{11} C_6 \cdot \frac{\alpha^5}{\beta^6}$
$\because$ Both are equal
$ \therefore \frac{11}{C_6} \cdot \frac{\alpha^5}{\beta^6}=-\frac{11}{C_5} \cdot \frac{\alpha^6}{\beta^5} $
$ \Rightarrow \frac{1}{\beta}=-\alpha $
$ \Rightarrow \alpha \beta=-1 $
$\Rightarrow(\alpha \beta)^2=1$