"Math", "history", and "literature" are all school subjects.  A library can be found in most schools, but it is not an actual subject like the others.

Three of these answers refer to an actual illness.  A headache is just a symptom, and not an actual illness.

"Bench", "couch", and "chair" are all pieces of furniture that you can sit on.  You can sit on a ladder, but it is not designed for that purpose, nor is it a piece of furniture.

"Rain", "sleet", and "snow" all describe types of precipitation.  "Wind" is a weather term, but it does not describe a form of water falling to the earth.

All 4 of these words would be "things that float".  However, "barge", "raft", and "canoe" are all types of floating objects built by people that can also be piloted by people.

The group of similar words here all are synonyms for the general category of “abilities” that a person might have. An aptitude is a natural ability—think of the expression, “It is apt to do that.” This means “it will likely do that naturally.” A “knack” is an acquired skill. “To get the knack” of something is to come to be able to do it. The word “musician” is far too specific in comparison with these other words. Certainly, to be a musician is to have a certain skill and perhaps even to express an aptitude. Still, it is not at all synonymous with the general categories as is the case with the other options.

Note that every one of the related terms pertain to copying or replicating something else. Even the word “mirror” can be used as a verb that basically means “to show a reflection of (something else).” An imitation is something made in the “image” of something else. Imitation vanilla flavoring is an artificially created type of flavoring that is made to match the image of real vanilla flavoring. Do not be tricked by the word “sibling.” While siblings (brothers and sisters) are alike, the word itself does not indicate any kind of reflecting or sameness. The word “twin” could represent such replication. Sibling, however, falls short of such direct correspondence. 

Each of the related words here are types written and printed material that pertain particularly to magazines and newspapers; however, the dissimilar term could be discerned even without thinking of the others in that strict sense. All of them definitely pertain to written forms of communication. The word “leader” might cause some problems, for it is somewhat rarely used in American English. It can mean “leading story,” though the expression is not quite as normal as the other terms. Still, given this meaning, the word “fable” differs from the others in that fables are neither explicitly related to news stories nor necessarily bound to written forms of communication.

All of the related words could be interpreted as pertaining to physical height (or even to some sort of metaphorical “height”). The apex is the highest point of something—whether physically (“the mountain’s apex”) or metaphorically (“the apex of his career”). The zenith likewise pertains to height in a similar way, though its physical use often applies to heavenly bodies. Do not be fooled by “depth” just because it is seemingly opposed to “height.” Depth still pertains to height. “Intensity,” on the other hand does not necessarily pertain to height but more properly describes qualitative changes like “intense light” and “intense attitude.”

All of the related words can be interpreted as pertaining to some sort of building. A domicile is a “house.” (“Domestic” affairs are “affairs of the house”—whether that house be an actual familial household or the “household” of a nation). An edifice is a building—from the Latin for “to build.” Something “edifying” is something that “builds up” morally or intellectually. A lodging is a temporary location of living. “Position” is far too abstract to be used in a manner synonymous to any of these other options.