"Did you eat?"
Translation:क्या तुमने खाया?
No. In Hindi, transitivity is a property of the verb. So, even though the verb doesn't have any objects in this particular sentence, it is still transitive since it can have objects. The sentence must thus be क्या तुमने खाया? with खाया conjugated in its default masculine singular form in the absence of objects.
(Some regional dialects do use something like क्या तुम खाये? though. )