<p>For the third question the issue is whether or not “a surprising large number of” is correct.</p>
<p>The intent of “surprising” is to describe “large” in the sense of “under what condition” or “how” large is. That function is meant for an adverb. Surprising is the present participle of surprise, and as such it can function as an adjective. So for example you can say “he demonstrated a surprising trick”.</p>
<p>Here though you need an adverb – and that adverb is surprisingly.</p>
<p>For the second question you ask why is “She … but in searching his face, he …” is incorrect. The issue here is that the subject that the participial phrase “in searching his face” modifies is ambiguous. A participial phrase in the current context functions as an adjective. Who does the searching? It could be “she” or it could be “Hank”. Typically the noun/pronoun closest to the participial phrase is the modified subject. The error is due to the ambiguity.</p>
<p>Choice C " a search of his face …" is grammatically correct. All the other choices are not. Because choice C is passive and awkward it is not very good written English … but since there is nothing better to pick it’s the answer.</p>