Can you be efficient without being effective?

No, if something is not at all effective, then it cannot be made efficient. Remember that "effective" is essentially a subjective term. In other words, even when you don't say it, you're implying that a task is effective at something in particular. For example, rubbing rocks together is really effective at making noise, but it takes a special kind of rock to be effective at making sparks as well.

No amount of reduced resources or time will increase your spark-generating efficiency if you're using the wrong rocks, because your task will literally never produce the desired result.