Hi, Ruifeng: I'm going to make the benevolent assumption that it was an accident that you didn't acknowledge my first reply in your follow-up post. However, if you ignore me again, I won't make that assumption. I'll simply ignore you in return.
7. Whenever we had a test, he completed it THE most quickly.
8. Whenever we had a test, he completed it most quickly.
In 7 and 8, I want to use "quickly" to modify "completed". Are they correct and is "the" necessary?
Both sentences are grammatically correct, and in both sentences "quickly" modifies "completed." You should use (7) (i.e., you should use "the") if you want the implied meaning to be "the most quickly (of anyone in the class)." You should use (8) if you want the sentence to be equivalent in meaning to "Whenever we had a test, he completed it very quickly."
ruifeng posted:9. Whenever we had a test , he completed it quickest.
10. Whenever we had a test, he completed it the quickest.
11. Whenever we had a test, he completed it the quickest of anyone in our class.
In these three sentences, I want to say that of all the students, he was the quickest to finish the test. Are they all correct? And do they all express this idea? Is "the" necessary in 9?
For the meaning you want, you need to use "the," whether or not you use the "of"-phrase or leave the idea expressed by the "of"-phrase implied. In other words, both (10) or (11) are correct with the meaning you want. Sentence (9) is not correct with that meaning, even if people succeed in understanding what you are trying to say, as they doubtless would. Omitting this "the" is not the end of the world!
Sentence (9) could, however, be correctly used in a context such as the one I discussed in my first reply (which you ignored), provided "it" does not refer to the test. Let's suppose "he" is your brother, he is not in your class, and "it" refers to his homework. In that case, (9) would mean that, when you (plural) had a test, your brother completed his homework quicker than in any other circumstance (perhaps because you were not around to distract him!). His speed in this circumstance would be being compared with his own speed in other circumstances, not with that of anyone else.