If the constant term in the binomial expansion of <img alt="\mathrm{\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^l}\right)^9 is -84}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B

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

Option: 1
98

Option: 2
-

Option: 3
-

Option: 4
-

Answers (1)

$\begin{aligned} & \left(\frac{x^{5 / 2}}{2}-\frac{4}{x^{\ell}}\right)^9 \\ & \mathrm{~T}_{\mathrm{r}+1}={ }^9 \mathrm{C}_{\mathrm{r}}\left(\frac{x^{5 / 2}}{2}\right)^{9-\mathrm{r}}\left(\frac{-4}{x^{\ell}}\right)^r \\ & ={ }^9 C_r\left(\frac{1}{2}\right)^{9-\mathrm{r}}(-4)^{\mathrm{r}} x^{\frac{45-5 \mathrm{r}}{2}-\mathrm{r}} \end{aligned}$
For constant term

$={ }^9 \mathrm{C}_{\mathrm{r}}\left(\frac{1}{2}\right)^{9-\mathrm{r}}(-4)^{\mathrm{r}}=-84$
$={ }^9 \mathrm{C}_{\mathrm{r}}\left(\frac{1}{2}\right)^{9-\mathrm{r}}(-1)^{\mathrm{r}} 2^{2 \mathrm{r}}=-84$
$={ }^9 \mathrm{C}_{\mathrm{r}} 2^{\mathrm{r}-9} 2^{2 \mathrm{r}}(-1)^{\mathrm{r}}=-84$
$={ }^9 \mathrm{C}_{\mathrm{r}} 2^{3 \mathrm{r}-9}(-1)^{\mathrm{r}}=-84$
$\Rightarrow {\mathrm{r}=3}{}$
$\frac{45-5 \mathrm{r}}{2}-\ell \mathrm{r}=0$
$\frac{45-15}{2}-3 \ell=0$
$15=3 \ell$
$\ell=5$