1) Did I suppose to take back the letter ? --> This sentence is incorrect. You cannot say "Did I supposed to". "Did" is wrong, here. You have to say "Was I supposed to". 2) Was I supposed to take back the letter ? --> This sentence is correct! "Was I supposed to" is correct. Hope this helps ^_^

The first sentence doesn't really make sense. "Did you suppose...?" means "Did you think...?" or "Did you believe...?". "Was I supposed to...?" means "Should I have...?" So, your first sentence would be like saying, "Did I believe to take back the letter?" It would sound very strange. So, why does English have two separate meanings for the word "suppose?" I don't really know--that's just the way our language works!

Personally I would just say the first sentence is grammatically wrong. The second sentence is correct.