If the constant term in the binomial expansion of \(\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^{\ell}}\right)^{9}\) is -84 and the Coefficient of \(x^{-3 \ell}\) is \(2^{\alpha} \beta\), where \(\beta<0\) is an odd number, Then \(|\alpha \ell-\beta|\) is equal to

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Binomial Theorem

If the constant term in the binomial expansion of

(\frac{x ^{\frac{5}{2}}}{2} - \frac{4}{x ^{ℓ}})^{9}

is -84 and the Coefficient of

x^{- 3 ℓ}

2^{α} β

, where

β < 0

is an odd number, Then

∣ α ℓ - β ∣

is equal to

Answer: 98

The general term is

T_{r + 1} = (- 1)^{r}^{9} C_{r} 2^{3 r - 9} x^{\frac{45 - 5 r}{2} - r ℓ} .

The constant term needs

\frac{45 - 5 r}{2} = r ℓ

; taking

r = 3

makes

2^{3 r - 9} = 1

and

^{9} C_{3} = 84

, so the constant term is

- 84

and

ℓ = \frac{45 - 15}{6} = 5.

For

x^{- 3 ℓ} = x^{- 15}

\frac{45 - 5 r}{2} - 5 r = - 15

gives

r = 5

, and the coefficient is

(- 1)^{5}^{9} C_{5} 2^{6} = - 126 \cdot 64 = - 63 \cdot 2^{7}

, so

α = 7, β = - 63.

Hence

∣ α ℓ - β ∣ = ∣35 + 63∣ = 98.

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Integer answer