Binomial Theorem Question 9
Question 9 - 29 January - Shift 1
If the co-efficient of $x^{9}$ in $(\alpha x^{3}+\frac{1}{\beta x})^{11}$ and the
co-efficient of $x^{-9}$ in $(\alpha x-\frac{1}{\beta x^{3}})^{11}$ are equal, then $(\alpha \beta)^{2}$ is equal to_______
Show Answer
Answer: 1
Solution:
Formula: Important Results Coefficient of $x^{m}$ in the expansion of $(a x^p + \frac{b}{x^q})^n$
Coefficient of $x^{9}$ in $(\alpha x^{3}+\frac{1}{\beta x})={ }^{11} C_6 \cdot \frac{\alpha^{5}}{\beta^{6}}$
$\because$ Both are equal
$\therefore \frac{11}{C_6} \cdot \frac{\alpha^{5}}{\beta^{6}}=-\frac{11}{C_5} \cdot \frac{\alpha^{6}}{\beta^{5}}$
$\Rightarrow \frac{1}{\beta}=-\alpha$
$\Rightarrow \alpha \beta=-1$
$\Rightarrow(\alpha \beta)^{2}=1$
