When writing a direct quotation, which word or words are always capitalized?
a. all of the nouns
b. all important words
c. the last word in the quotation
d. the first word in the quotation

A all of the nouns. It could have been d, but if the quote is embedded, capitalizing the first letter would be incorrect grammar. 

The best answer here is B, but only if by "important words" it means "proper nouns."

A is wrong because not ALL nouns will be capitalized in a quotation. Only proper nouns will be capitalized. For instance, you would NOT capitalize a quote in this way:

• The manager proclaimed, "No Visitors to Yosemite National Park will be allowed to bring their Pets or their Cars with them."

In the above sentence, "visitors," "pets," and "cars" are all nouns, but they don't need to be capitalized because they are only common nouns. Only "Yosemite National Park," which is a proper noun (because it refers to a specific, named place) needs to be capitalized.

C is wrong because it simply makes no sense and conforms to no rule of capitalization.

D is wrong because the first word of a quote does not ALWAYS need to be capitalized. If the quote is embedded in a sentence, then its first word does not require capitalization (unless, of course, that word is a proper noun).

But please be away that this question is very misleading, since it's not clear that "important words" means "proper nouns." You might want to point this out to your teacher.