The positive value of a so that the coefficients of $x^5$ and $x^{15}$

are equal in the expansion of ${\left(x^2+\frac{a}{x^3}\right)}^{10}$

Suppose $x^5$ occure in $(r+1{)}^{\text{th }}$ term of the expansion of ${\left(x^2+\frac{a}{x^3}\right)}^{10}$

We have,

$\begin{array}{l} T_{r+1}{=}^{10}{C}_r{\left(x^2\right)}^{10-r}{\left(\frac{a}{x^3}\right)}^r{-}^{10}{C}_r{x}^{20-5r}ar\\ \therefore 20-5r=5\Rightarrow r=3\end{array}$

$\therefore$Coefficient of $x^5{=}^{10}{C}_3{a}^3.$

Similarly, Coefficient of $x^{15}{=}^{10}{C}_1{a}^1$

Now,

Coeff. of $x^5=$Coeff. of $x^{15}$

$\begin{array}{l}\Rightarrow {}^{10}{C}_3{a}^3{=}^{10}{C}_{1}a\\ \Rightarrow 120a^3=10a\Rightarrow a^2=\frac{1}{12}\Rightarrow a=\frac{1}{2\sqrt{3}}\end{array}$