The ratio of the coefficient of x

$\ln \left(x^{2}+\frac{2}{x}\right)^{15}$ we have $T_{r+1}={ }^{15} C_{r}\left(x^{2}\right)^{15-r}\left(\frac{2}{x}\right)^{r}$

i.e. General term is $T_{r+1}={ }^{15} C_{r} x^{30-3 r} 2^{r}$

Hence for the term independent of x,

30 – 3r = 0

i.e. r = 10

hence T11 has coefficient 15C10 210       ...(1)

and term with x15 will have 30 – 3r = 15

i.e. 15 = 3r

i.e. r = 5

∴ coefficient will be 15C5 25              ...(2)

∴  ratio of coefficient of x15 to the term independent of x will be

$\frac{e q^{n}(2)}{e q^{n}(1)}=\frac{2^{5} \times{ }^{15} C_{5}}{2^{10} \times{ }^{15} C_{10}}$

$=\frac{1}{2^{5}} \quad\left(\because{ }^{15} C_{10}={ }^{15} C_{5-10}={ }^{15} C_{5}\right)$

i.e. ratio will be 1 : 32

Hence, the correct answer is option B.