A vessel contains a mixture of one mole of oxygen and two moles of nitrogen at 300K. the ratio of the average rotational kinetic energy per ${\mathrm{O}}_2$  molecule to that per ${\mathrm{N}}_2$  molecule is

Kinetic energy per degree of freedom is  $\frac{1}{2}\mathrm{KT}$
As diatomic gas posses two degree of freedom for rotational motion therefore rotational kinetic energy =  $2.\frac{1}{2}\mathrm{KT}=\mathrm{KT}$
In the problem both gases are diatomic and have same temperature. Therefore the average rotational kinetic energy will be equal to one.