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

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
$\therefore \frac{11}{C_6}\cdot \frac{{\alpha }^5}{{\beta }^6}=-\frac{11}{C_5}\cdot \frac{{\alpha }^6}{{\beta }^5}$
$\Rightarrow \frac{1}{\beta }=-\alpha$
$\Rightarrow \alpha \beta =-1$
$\Rightarrow (\alpha \beta {)}^2=1$