Question

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_____

Answer 1

In, $\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^{\ell}}\right)^9$
$ T _{ r +1}={ }^9 C _{ r } \frac{\left( x ^{5 / 2}\right)^{9- r }}{2^{9- r }}\left(\frac{-4}{ x ^{\ell}}\right)^{ r } $
$ =(-1)^{ r } \frac{{ }^9 C _{ r }}{2^{9- r }} 4^{ r } x ^{\frac{45}{2}-\frac{5 r }{2}- r } $
$=45-5 r -2 lr =0 $
$ r =\frac{45}{5+21}$ ....(1)
Now, according to the question, $(-1)^{ r } \frac{{ }^9 C _{ r }}{2^{9- r }} 4^{ r }=-84$
$=(-1)^{ r }{ }^9 C _{ r } 2^{3 r -9}=21 \times 4$
Only natural value of $r$ possible if $3 r-9=0$
$r =3$ and ${ }^9 C _3=84$
$\therefore 1=5$ from equation (1)
Now, coefficient of $x^{-31}=x^{\frac{45}{2}-\frac{5 r}{2}-\operatorname{lr}}$ at $1=5$, gives
$ r =5$
$ \therefore{ }^9 c _5(-1) \frac{4^5}{2^4}=2^\alpha \times \beta$
$ =-63 \times 2^7 $
$\Rightarrow \alpha=7, \beta=-63$
$ \therefore \text { value of }|\alpha \ell-\beta|=98$