Question

If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<$ is an odd number, then $|\alpha l-\beta|$ is equal to_____

Answer 1

In, (2x25​​−xℓ4​)9
Tr+1​=9Cr​29−r(x5/2)9−r​(xℓ−4​)r
=(−1)r29−r9Cr​​4rx245​−25r​−r
=45−5r−2lr=0
r=5+2145​ ....(1)
Now, according to the question, (−1)r29−r9Cr​​4r=−84
=(−1)r9Cr​23r−9=21×4
Only natural value of r possible if 3r−9=0
r=3 and 9C3​=84
∴1=5 from equation (1)
Now, coefficient of x−31=x245​−25r​−lr at 1=5, gives
r=5
∴9c5​(−1)2445​=2α×β
=−63×27
⇒α=7,β=−63
∴ value of ∣αℓ−β∣=98
So, the correct answer is 98.