The sentences you provided involve prepositions for indicating place and location on test material, which could be printed on a sheet of paper, written on a board or chart, or displayed on a computer screen. The general rule for preposition usage in such situations is “in” for an enclosed space, “on” for a surface, and “at” for a point.
A “blank,” in whatever medium is being used, is an empty surface, so “on” is the correct preposition for Sentence 1: “Write your answer
on the blank.”
A “space” allotted for answers to a particular test item is normally enclosed by the test item before it and the test item after it, so “in” is the correct preposition for Sentence 2: “Write your answer
in the space provided.”
Sentence 3 is, in practice, an exception to the general rule for the use of “in” and “on.” The context here is that the word “blank” consists of a set of spaces for the entry of data, in contrast to the sense of “blank” in Sentence 1 as simply empty surface. For the verb “fill,” native English speakers idiomatically use “in” to form the prepositional phrase “fill in” in this particular instance: “
Fill in the blank.” (“
Fill on the blank” is frowned upon as unidiomatic, and so with “
Fill up the blank.”)
I must qualify that the three particular usages I presented above use the American English Standard; there may be notable variations in the British English Standard. Nonnative English speakers need to be aware that whatever English standard is used, preposition usage is essentially conventional, even quirkish at times, and that many preposition choices actually have no inherent or discernible logic of their own.
For deeper grounding on preposition usage in American English, I suggest the following readings previously posted in the Forum:
Lesson #7 – The Prepositions RevisitedLesson #8 – Specific Rules for Preposition UsageLesson #9 – Getting to Know the Prepositional PhrasesLesson #10 – Dealing with the Prepositional Idioms