The answer is "My cousin likes to exaggerate when he tells stories." The word "exaggerate" is used to describe something for more than it really is. "The way to school was exaggerate that morning" is wrong because an adjective should be put in place of "exaggerate". "The story was quite exaggerate" is wrong because it incorrectly uses "exaggerate"; for it to be a complete sentence it would have to read "The story was quite <em>exaggerated</em>. "Her exaggerate was unbelievable" is wrong because, well, it just doesn't make sense. It would have to mention some type of noun to describe as unbelieable.
