If the sum of the coefficients of all the positive even powers of x, in the binomial expansion of ${\left(2{\mathrm{x}}^3+\frac{3}{\mathrm{x}}\right)}^{10}\text{, is }{\mathrm{\alpha }}^{10}-83\left({\mathrm{\beta }}^9\right)$, is , then the value of ${\log }_2|\mathrm{\alpha }-\mathrm{\beta }|=$

The general term is${ }^{10}{\mathrm{C}}_{\mathrm{r}}{\left(2{\mathrm{x}}^3\right)}^{10-\mathrm{r}}{\left(\frac{3}{\mathrm{x}}\right)}^{\mathrm{r}}{=}^{10}{\mathrm{C}}_{\mathrm{r}}{2}^{10-\mathrm{r}}{3}^{\mathrm{r}}{\mathrm{x}}^{30-4\mathrm{r}}.$
Taking  as even, we can get $\mathrm{r}\in \{0,1,2,3,4,5,6,7\}\text{ and }\mathrm{r}\notin \{8,9,10\}$
Therefore the sum required, is $5^{10}-\left({ }^{10}{\mathrm{C}}_8{2}^2{3}^8{+}^{10}{\mathrm{C}}_9{2}^1{3}^9{+}^{10}{\mathrm{C}}_{10}{2}^0{3}^{10}\right)=5^{10}-83\left(3^9\right).$
$\text{ So }\mathrm{\alpha }=5;\mathrm{\beta }=3\text{. Then }{\log }_2|\mathrm{\alpha }-\mathrm{\beta }|=1\text{. }$