Option A: 'Managed' means 'succeed in achieving something despite difficulties that came in the way'. The word relates to the given sentence, as the person was able to start the car despite the difficulties.
Hence option A is correct.
Option B: 'Manoeuvred' means 'carefully guide or manipulate'. Since his car was not working, he could not have guided it.
Option C: 'Coped' means 'to deal with something difficult'.
Option D: 'Worked' means 'to function' or 'bring about an action'.
These words do not complete the sentence very well. The meaning formed is ambiguous and does not refer to the 'starting of the car': 'finally managed to start the car'.
Hence options B, C and D are incorrect.