The approach is correct. However, I'd just make a couple of points.

1. C°_{A} is not 1 mole. What are the units of C?

2. Stoichiometric proportions doesn't mean that the starting concentrations are both 1 M, but simply that they are both the same. Note that for a second-order reaction, unlike a first-order reaction, the half-life is not constant but depends on the initial concentration - which you don't know. However, the ratio of the half-life to the 0.1th life is independent of initial concentration. You might like to try and verify this algebraically. (Calculate k as a function of C° in the first step, then calculate t_{0.1}, and see if C° cancels out.)