When someone feigns illness, he or she pretends to be sick. The word "feign" most directly signifies the act of pretending or dissimulating. Often (likely most of the time), this is done so as to lie. The actual act of pretending is the "feigning." The lying may well just be the reason for the pretending. The two are not the same, so do not be tricked by several of the options that try to lead you to say that they are. Thus, the best option is the simple, "pretend."

The word "modulate" contains the same roots that are found in the related word "moderate." The "mod-" prefix has a rich history, though it generally indicates that something has been measured. Thus, a "moderate" temper is one that has just the right amount of anger, excitement, sadness, and so forth.  To "accomodate" is to match the needs of someone else (thus, to meet his or her "measure"). When someone modulates his or her voice, he or she adjusts it to a given situation—thus "measuring" the voice to match the situation.  Thus, the best option for this question is "regulate."

The word "recant" comes from two somewhat common roots. The "re-" prefix can mean again. It can also indicate that something turns around (as in re-verse, for instance). The "-cant" portion of the word is derived from the Latin for to sing. Thus, we have words like "chant" and "cantor." When someone recants something, he or she takes back a remark made in the past. Thus, when we say that someone "recants his allegiance to the king of Spain", that person denies that he has such allegiance, though he did have it once upon a time. Although "renounce" would be an even better synonym, the option "deny" is sufficient for this question.

"Befuddle" means to cause someone to think unclearly. Of the answer choices only "confuse" is related to unclear thoughts. "Enlarge" and "diminish" both refer to changes in size, getting bigger and smaller respectively. "Explaining" makes a concept clearer which makes it an antonym of "befuddle." "Obscure" is the most similar word after "confuse," but it often refers to physical obstruction of a view rather than making an idea unclear.

"Rebut" means to prove a claim is false in the course of an argument.

"Contradict" is closer than "argue" because a "rebuttal" is a rhetorical device often used within an argument rather than the argument itself, "agree" is an antonym of "rebut," "slip" means to fall or trip, and "exclaim" means to say something excitedly.

"Peddle" means to sell or to travel around selling something. To provide further help, "sentence" means determine the punishment for; "portray" means depict, show. 

To "swindle" someone means to cheat him or her out of money. To provide further help, "mollify" means sooth, calm, relieve; "redeem" means make right something done wrong; "plead" means beg, urge. 

To "interrogate" means to question, to investigate for a crime through questioning. To provide further help, "ponder" means think carefully; "replenish" means refuel, restock; "surmise" means deduce, figure out. 

To "ruminate" means to think carefully and deliberately about (something). So the best possible answer choice is "ponder" which has the same meaning. To provide further help, "abscond" means run away with; "meander" means to move in a winding manner, like a river. 

To "infringe" means to actively break the rules or to actively disturb someone. As such the closest answer choice is "violate." To provide further help, "kindle" means the same thing as "ignite," they both mean to start a fire; "fetter" means shackle, tie to something; "embellish" means to adorn with details.