The word "akin" is related to "kin", meaning "relative" or "relation" more generally. A person's "kin" are his or her relatives. To be "akin" to something is to be related to it, thus to be alike to it.  This does not mean that the things are equivalent. Likewise, the options "parental" and "begotten" try to trick you into thinking that "akin" must be directly related to blood relations. Therefore, the best option given here is "analogous", which means alike in some respects but not others. Analogies express this kind of relationship.  For example, "Just as a lightbulb illuminates a house, so too does a brilliant person illuminate the minds of his or her friends." The two senses of "illumination" spoken of here are different, though not totally so. They are analogous, or akin.

The word "abundant" is related to the word "abound," which means to be full or to have a great deal of (something). When someone has, for example, "abundant financial resources," he or she has a lot of money. Because of this meaning, you might think that "wealthy" is correct; however, abundance (and being "abundant") has to deal with having a lot of anything whatsoever. Hence, the best synonym offered here is "plentiful," which is clearly related to the word "plenty" meaning existing in a large quantity.

When we prolong something, we stretch it out. A "prolonged argument" is one that is extended over a long period of time. In one way, you can think of the word like "pro-long." Something that is stretched is made to be long, but this might lead you to pick an answer like "distant" as well. Distant things are, after all, a "long way off." In any case, the essence of "prolonging" something is extending that thing over a larger or longer space—whether it be a longer physical space or perhaps a longer "space of time."

We use many meanings for words like "powerful" and "strong." Here, the best option is "potent," which means powerful as well. It comes from Latin roots that mean able to do something or even, having the power to do something. The words "potential," "possible," and "potency" also all come from the same root. Do not be tempted by many of the other options, which try to get you to think of "strong" in the physical sense. None of these options are actually synonyms for "strong," even though they might deal with things related to physical strength and wellness.

Something that is "lulling" is soothing. A "lull" is a quiet time. For instance, at a restaurant, there can be a "lull" when patrons stop coming—giving the workers a quiet time to recover. Now, a "lulling" voice can put someone to sleep, but this does not mean that something that is "lulling" is itself sleepy. The basic synonym for "lull" that works best is just the simple word "soothing."

"Robust" is an adjective meaning hearty or powerful. For example: "The extremely spicy ingredients caused the sauce to have a ROBUST taste and aroma."

"Fragile" is an antonym to "robust." "Rotund" is an adjective describing an object with a round shape or a corpulent person. "Belligerent" is an adjective used to describe a person who is aggressive or unpleasant.

"Rigorous" is an adjective describing something very difficult, thorough, or demanding. For example: "The RIGOROUS course of study caused many students to either fail or quit."

"Talentless" is an adjective used to describe someone who is without skill at a given task. "Straight" is a concrete adjective using to describe a specific, linear directionality of a movement or the shape or arrangement of an object or objects. 

An "abrasive" is something scratchy and is usually used to help smooth an object or clean something. Sandpaper is abrasive as are the scratchy sides of sponges used to clean dishes. "Smooth" is an antonym to "abrasive." "Clean" means free of dirt. "Sticky" is an adjective describing a gummy or adhesive texture. "Ugly" is an adjective describing an unsightly person or object.

"Surgical" means precise, accurate. If you think about the fact that a "surgeon" needs to be precise and accurate to do their job this makes a lot of sense. 

"Mean" can have many different meanings, but when used as an adjective in a mathematical or statistical sense it means average. To provide further help, "forthright" means honest; "supreme" means best, greatest; "polarizing" means divisive, causing radically different opinions; "abundant" means existing in large quantities.