Statorvane
I'm not suggesting that you have misquoted the question. It is quite possible, indeed quite likely that the examiners have taken a defective question and "improved" it to make an equally defective question. They have done this many times in the past.
If we take the question as stated in your original post
If the value of a balanced V1 is found to be lower than Vmcg then which of the following is correct?
A. Correcting this will increase the ASDR and decrease the 1 eng inop TODR
B. The take off is permitted
C. The one engine out take off distance will become greater than the ASDR
D. Vmcg will be lowered to V1.
Option A is correct is we assume that the aircraft was not field limited and we increased V1 to at least equal Vmcg.
Option B is just plain wrong. You cannot legally take-off with V1 less than Vmcg.
Option C will occur only if we go the wrong way and actually decrease V1, but to do so makes no sense.
Option D could be achieved if we were not field limited , and used reduced thrust.
Overall option A is the best answer.