If the constant term in the binomial expansion of ((x(5)/(2))/(2) -...

Given,

The constant term in the binomial expansion of ${\left(\dfrac{x\dfrac{5}{2}}{2} - \dfrac{4}{{x}^{l}}\right)}^{9}$ is $- 84 and the coefficient of${x}^{- 3}$is${2}^{\alpha}\beta$</p><p>Now General Term of the binomial is given by,${T}_{r + 1} = Cr 9 {\left(\dfrac{x\dfrac{5}{2}}{2}\right)}^{9 - r}{\left(\dfrac{- 4}{{x}^{l}}\right)}^{r} = {\left(- 1\right)}^{r}{C}_{r}{x}^{\dfrac{45}{2} - \dfrac{5r}{2} - lr}{2}^{3 - 9}$</p><p>Now for constant term$45 - 5r - 2lr = 0 ⇒ r = \dfrac{45}{5 + 2l}$</p><p>So, coefficient of constant term will be,</p><p>${\left(- 1\right)}^{9}{\times}^{9}{C}_{r}{2}^{3r - 9} = - 84 ⇒ r = 3$</p><p>Now putting the value of$r in r = \dfrac{45}{5 + 2l}$we get,$l = 5$</p><p>Now coefficient of${x}^{- 3l} = {x}^{- 15} ⇒ \dfrac{45}{2} - \dfrac{5r}{2} - lr = - 15 ⇒ r = 5$when$l = 5$<br />Now coefficient of${x}^{- 15} = {\left(- 1\right)}^{59}{C}_{5}{2}^{6} = - 63\left({2}^{7}\right) = \beta\left({2}^{\alpha}\right)$</p><p>Hence, on comparing we get,$\alpha = 7, \beta = - 63 \text{and } l = 5</p><p>So, the value of \left|\alphal - \beta\right| = \left|7 \times 5 - \left(- 63\right)\right| = 98