If the coefficient of x 9 in ( x 3 + 1 x ) 11 and the coefficient of x - 9 in ( x - 1 x 3 ) 11 are equal, then 2 is equal to

Question

If the coefficient of $x^{9}$ in ${(α x^{3} + \frac{1}{β x})}^{11}$ and the coefficient of $x^{- 9}$ in ${(α x - \frac{1}{β x^{3}})}^{11}$ are equal, then ${(α β)}^{2}$ is equal to _______.

Smart Achievers · Accepted Answer

(1)

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

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

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

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

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

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

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

$\Rightarrow$ $\frac{1}{β} = - α \Rightarrow {(α β)}^{2} = 1$

Solution

Q.
If the coefficient of $x^{9}$ in ${(α x^{3} + \frac{1}{β x})}^{11}$ and the coefficient of $x^{- 9}$ in ${(α x - \frac{1}{β x^{3}})}^{11}$ are equal, then ${(α β)}^{2}$ is equal to _______.

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Solution

Q. If the coefficient of x9 in (αx3+1βx)11 and the coefficient of x-9 in (αx-1βx3)11 are equal, then (αβ)2 is equal to _______.

Ans. (1) (r+1)th term in (αx3+1βx)11=Cr11α11-r·β-r·x33-4r Here, 33-4r=9⇒r=6 ∴ Coefficient of x9=C611α5β-6 (r+1)th term in (αx-1βx3)11=Cr11α11-r(-1β)r·x11-4r Here, 11-4r=-9⇒r=5 ∴ Coefficient of x-9=-C511α6β-5 Now, C611α5β6=-C511α6β5 [∵ Given] ⇒ 1β=-α⇒(αβ)2=1

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Q.
If the coefficient of $x^{9}$ in ${(α x^{3} + \frac{1}{β x})}^{11}$ and the coefficient of $x^{- 9}$ in ${(α x - \frac{1}{β x^{3}})}^{11}$ are equal, then ${(α β)}^{2}$ is equal to _______.