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_______

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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

$\therefore \frac{11}{C_6} \cdot \frac{\alpha^{5}}{\beta^{6}}=-\frac{11}{C_5} \cdot \frac{\alpha^{6}}{\beta^{5}}$

$\Rightarrow \frac{1}{\beta}=-\alpha$

$\Rightarrow \alpha \beta=-1$

$\Rightarrow(\alpha \beta)^{2}=1$