You should rightly be confused. Either this test was authored by a non-native speaker, or you have transposed the correct and incorrect responses. A.) The first sentence is in active voice, and absent a good reason for the second sentence to be in passive voice, it should be active voice as well. This is actually an elliptical construction: "Even in two-person relationships, a culture develops [between the persons] in time." The alternative: "Even in two-person relationships, a culture is developed [between the persons] in time." The second is less linear and not parallel with the first sentence. Semantically they are identical, so the least verbose is normally preferred.
B.) This is another case of ellipsis. Actually both alternatives are passive. To see this, consider "So dictionaries have [been] developed to mend the gap." To cast the sentence into active voice, use s-v-o word order: Dictionaries have developed an appetite for learning in students. You cannot insert [been] after "have" here; this "appetite" sentence is in active voice. The only way "dictionaries have developed to mend the gap" can be active, is if dictionaries can develop themselves. They cannot. Because of this, the elliptical construction is considered colloquial or off-hand, and the inclusion of "been" is preferred. "Been" marks the phrase as definitively passive, and implies "by scholars" which is the semantic idea that should be conveyed.
Your answer key is wrong on both counts.