If the co-efficient of $x^{9}$ in $\left(\alpha x^{3}+\frac{1}{\beta x}\right)^{11}$ and the co-efficient of $\mathrm{x}^{-9}$ in $\left(\alpha \mathrm{x}-\frac{1}{\beta \mathrm{x}^{3}}\right)^{11}$ are equal, then $(\alpha \beta)^{2}$ is equal to ____.

JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Binomial Theorem

If the co-efficient of

x^{9}

(α x^{3} + \frac{1}{β x})^{11}

and the co-efficient of

x^{- 9}

(α x - \frac{1}{β x ^{3}})^{11}

are equal, then

(α β)^{2}

is equal to ____.

Answer: 1

(α x^{3} + \frac{1}{β x})^{11}

the general term carries

x^{4 r - 11}

, so

x^{9}

arises from

r = 5

with coefficient

(5 11) \frac{α ^{5}}{β ^{6}}

. In

(α x - \frac{1}{β x ^{3}})^{11}

the general term carries

x^{4 r - 33}

, so

x^{- 9}

arises from

r = 6

with coefficient

- (6 11) \frac{α ^{6}}{β ^{5}}

. Because

(5 11) = (6 11)

, equality of the two coefficients gives

\frac{α ^{5}}{β ^{6}} = - \frac{α ^{6}}{β ^{5}}

, that is

α β = - 1

, so

(α β)^{2} = 1

JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Integer answer