The Big Bang was not an explosion. It's an expansion of space that not only is still occurring, but is getting faster.
I agree, that the question was poorly phrased, and leaves room for ambiguity in “true”, “false” category, both in the definition of “explosion” and in the state of our knowledge. This quiz rather reminds me of when Richard Feyman interviewed for a job a Microsoft, or when Neils Bohr almost didn’t receive his doctorate because he refused to say how you could measure the height of a building with a barometer. If one were to select a coordinate system centered on the earth, then the sun does, in fact, go around the earth. Such a coordinate system may not be the most convenient for expression motion in Newtonian Dynamics, but that does not make motions described in such a coordinate system “false”.