A. "developed" is better. "Cultures" can "develop", which is used here as an intransitive verb. If you said "is developed", then it would be correct but less natural. It would beg the question - developed by whom? If, by the two people alone, then "develops" expresses this. So there is no reason to use the passive. I do not know what the person who set the question was thinking. B. "were developed" is best (but is not an option). "have been developed" is perhaps better than "have developed". The context is that there is a problem. A solution to this problem "has been developed" i.e. dictionaries have been invented. It does not matter who (has) developed the dictionaries. If we said in B, the "dictionaries developed", then this means that they existed already but were improved in order to deal with the problem identified. There is no context to indicate that they existed already and so it is not the obvious conclusion. As Andrew said, "mend a gap" is an unnatural collocation. I have never seen it before. I don't like the questions. They are not fair to students. I suspect that they were not checked by a native speaker, which is absolutely crucial for advanced exam questions.

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.

I cannot see a good reason for B. It is possibly worth noting that we do not, typically, "mend," gaps in English. We mend holes (in clothes). We fill gaps. And there are probably other words that could be used. But mend is wrong. So that may mean that the people setting the exam are not all that fluent in English themselves. In A , I do think "is developed," works better.