The correct option is A (c)
The option with the error is the correct one.
Option C- The error lies here: "flew half mast", the preposition 'at' is missing. It should be: and the national flag flew at half mast. Hence this is the correct option.
Option A: This part "Being a day of national mourning" has no error, where 'being' is a free adjunct that shortens the part which would otherwise be written as: It was a day of national mourning. Hence we do not choose this option. In short, A is incorrect.
Option B: This part, too, has no error. It's in the simple past tense, which agrees to the sentence: 'flew' (in part c) is also in the same tense. Hence B is the incorrect option.
Option D is incorrect because the error lies in C.