
Ed M. answered 04/08/16
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The correct answer for the particular sentence you give would seem to be "B) for," though one could argue (rather weakly) that inserting the preposition at of choice C) into the slot in this sentence could, in some contexts, also result in a grammatical English sentence.
The reason that B) is the more likely correct answer is that the two-word verb search for is an idiom in English, specifically a prepositional verb (sometimes included among the verb idioms called phrasal verbs, but I prefer to keep them distinct because included in the latter are other verbs that have particular properties) and it is also transitive meaning it needs a direct object after for. That is, search for, like look for, another two-word verb with principally the same meaning and probably even more commonly used, functions as a unit in which another preposition cannot replace for. That search for is a semantic and grammatical unit can be illustrated by substituting another synonym for search for, namely the single-word verb seek (though seek is considerably less common in Modern English than both search for and look for):
- His brother was searching for King Solomon's mines.
- His brother was seeking King Solomon's mines.
(And of course I trust you recognize that in your sentence, the reduced clause while searching ........ king solomon 's mines [sic] does actually have the same meaning as the above examples, i.e., both his brother and the auxiliary verb are omitted because they can be understood from the context, given that the subject of the main clause is His brother.)
The reason I indicated that His brother has gone missing while searching at King Solomon's mines might be grammatical in the right context is only if at is understood not as an element of a (non-existing) two-verb *search at but rather as the head of the prepositional phrase at King Solomon's mines. That is, on this reading His brother has gone missing while searching at King Solomon's mines uses the single-word verb search as an intransitive verb (though its implied meaning is transitive, i.e., 'search [for something]' with the the direct object not expressed) and the prepositional phrase at King Solomon's mines functions as an adverbial indicating where the searching was taking place (i.e., not the thing being searched for, in other words his brother was searching for something, but it wasn't King Solomon's mines). Again, this interpretation of the sentence seems a lot less likely than the one described above where searching for is an instance of a two-word transitive verb, especially since, as I understand it, no one's yet discovered these supposed mines of this Biblical King of Israel, hence it's not a "real" place that someone could be "at" as far as we know.