If the co-efficient of x 9 in ( x 3 + 1 x ) 11 and the co-efficient of x - 9 in ( x - 1 x 3 ) 11 are equal, then 2 is equal to ________ . [2023]

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Solution

Q.
If the co-efficient of $x^{9}$ in ${(α x^{3} + \frac{1}{β x})}^{11}$ and the co-efficient of $x^{- 9}$ in ${(α x - \frac{1}{β x^{3}})}^{11}$ are equal, then ${(α β)}^{2}$ is equal to ________ . [2023]

Ans.
(1)

${(r + 1)}^{th} term in$ ${(α x^{3} + \frac{1}{β x})}^{11} = {}^{11}C_{r} α^{11 - r} \cdot β^{- r} \cdot x^{33 - 4 r}$

$Here, 33 - 4 r = 9 \Rightarrow r = 6$

$∴$ $Coefficient of x^{9} = {}^{11}C_{6} α^{5} β^{- 6}$

${(r + 1)}^{th} term in$ ${(α x - \frac{1}{β x^{3}})}^{11} = {}^{11}C_{r} α^{11 - r} {(- \frac{1}{β})}^{r} \cdot x^{11 - 4 r}$

$Here, 11 - 4 r = - 9 \Rightarrow r = 5$

$∴$ $Coefficient of x^{- 9} = - {}^{11}C_{5} α^{6} β^{- 5}$

$Now, {}^{11}C_{6} \frac{α^{5}}{β^{6}} = - {}^{11}C_{5} \frac{α^{6}}{β^{5}} [∵ Given] \Rightarrow \frac{1}{β} = - α \Rightarrow {(α β)}^{2} = 1$

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