Carburated cars are not as efficient as fuel injected cars, however this should not stop a vehicle from passing the smog test. As far as having to drive your Toyota Corolla "a lot" before the smog test, that doesn't have much truth. All vehicles should be properly warmed up before the smog inspection. Smog check procedure requires all smog stations ensure a vehicle is properly warmed up before the smog check.
In this specific case you want to first find out which emission(s) your Toyota Corolla failed for; HC, CO, NOx, all, or just two. From there you can go on to diagnosing the failure. All three of these emissions can be due to an improperly adjusted carburetor, or may having nothing to do with the carburetor. It all depends on the condition of your Toyota Corolla's overall emission systems; the EGR system, EVAP system, the computer system (oxygen sensor, MAP sensor, ECT sensor).
We're afraid you will absolutely need to have a smog check repair station conduct a smog failure diagnosis in order to find out which emission component is at fault - the Carburetor or something else.
Read up on smog check emission failures here:
Research Failed Emissions Test Results
