But the problem arises on what volume should be used to establish an equation from where to find the composition of the mixture.
You don't need to know this. You simply need to assume that the effused volumes of nitrogen and mixture are the same. (The question doesn't say this, but you can't solve it otherwise.) Then the effusion times give you the relative rates of effusion, from which you can work out an effective molecular weight for the mixture, and hence the composition.
But I wonder if there is an error in the question. They seem to assume (at least, this assumption leads to one of the optional answers) that the effective MW is the arithmetic mean, given by
M = x*32 + (1-x)*44, where x is the mole fraction of oxygen.
However, this does not seem to me to be right. From the additivity of the effusion rates of O
2 and CO
2, I would expect the effective MW to be given by
1/sqrt(M) = x/sqrt(32) + (1-x)/sqrt(44)
This gives a different value for x, which does not correspond to any of the answers. (The composition by the first assumption is very close, but not identical, to the composition I calculate for the effused mixture.)