Let $a$ and $b$ be two nonzero real numbers. If the coefficient of x⁵ in the expansion of ${\left({\mathrm{ax}}^2+\frac{70}{27\mathrm{bx}}\right)}^4$ is equal to the coefficient of x^-5 in the expansion of ${\left(\mathrm{ax}-\frac{1}{{\mathrm{bx}}^2}\right)}^7$ then the value of $2b$ is 
 

$\mathrm{In}{\left({\mathrm{ax}}^2+\frac{70}{27\mathrm{bx}}\right)}^4\text{ General term }{\mathrm{T}}_{\mathrm{r}+1}{=}^4{\mathrm{C}}_{\mathrm{r}}{\left({\mathrm{ax}}^2\right)}^{4-\mathrm{r}}{\left(\frac{70}{27\mathrm{bx}}\right)}^{\mathrm{r}}$
$={ }^4{\mathrm{C}}_{\mathrm{r}}\cdot {\mathrm{a}}^{4-\mathrm{r}}{\left(\frac{70}{27\mathrm{b}}\right)}^{\mathrm{r}}\cdot {\mathrm{x}}^{8-3\mathrm{r}}$
$\mathrm{In}{\left(\mathrm{ax}-\frac{1}{{\mathrm{bx}}^2}\right)}^7$,  General term ${\mathrm{T}}_{\mathrm{S}+1}{=}^7{\mathrm{C}}_{\mathrm{S}}(\mathrm{ax}{)}^{7-\mathrm{S}}{\left(-\frac{1}{{\mathrm{bx}}^2}\right)}^{\mathrm{S}}$
=${ }^7{\mathrm{C}}_{\mathrm{S}}\cdot {\mathrm{a}}^{7-\mathrm{S}}\cdot {\left(\frac{-1}{\mathrm{b}}\right)}^{\mathrm{S}}\cdot {\mathrm{x}}^{7-3\mathrm{S}}$
Given that coefficient of x⁵ in ${\left({\mathrm{ax}}^2+\frac{70}{27\mathrm{bx}}\right)}^4$= coefficient of x^-5 in ${\left(\mathrm{ax}-\frac{1}{{\mathrm{bx}}^2}\right)}^7$
$\begin{array}{l}{\Rightarrow }^4{\mathrm{C}}_1\cdot {\mathrm{a}}^{\mathrm{\gamma }}\left(\frac{70}{276}\right){=}^7{\mathrm{C}}_4\cdot {\mathrm{A}}^{\mathrm{\gamma }}{\left(\frac{-1}{\mathrm{b}}\right)}^4\Rightarrow 4\cdot \frac{{70}^2}{27\mathrm{b}}=35\cdot \frac{1}{{\mathrm{b}}^4}\\ \Rightarrow {\mathrm{b}}^3=\frac{27}{8}\Rightarrow \mathrm{b}=\frac{3}{2}\Rightarrow 2\mathrm{b}=3\end{array}$