The question as stated in the JAR CQB was not wuite as it is listed in the OP in this thread.
The CQB question was as follows:
If the value of the balanced V1 is found to be lower than VMCG, which of the following is correct?
a. The ASDR will become greater than the one engine out take-off distance.
b. The take-off is not permitted.
c. The one engine out take-off distance will become greater than the ASDR.
d. The VMCG will be lowered to V1.
The JAR "correct answer" was option b but this has been appealed many times. The problem was not helped by the fact that the CAA examiners did not really understand the arguments.
My comments below were written for the consumption of students, so they are a bit laborious for some of the more expert members who have alreday contributed to this thread.
V1 is the take-off decision speed. If an engine fails before V1 then the take-off must be aborted. If an engine fails after V1 the take-off must be continued. V1 must therefore be selected such that it is possible to complete the take-off within the distance available, or to stop with the distance available from the point at which V1 is reached. If the runway is wet the friction between the tyres and the runway will be reduced. This will reduce the effectiveness of the braking system, thereby increasing the distance required to stop from V1. This in turn will reduce the maximum allowable value of V1.
The distances required to complete the take-off or abort are determined by the mass of the aircraft and the value of V1. V1 is said to be balanced when the two distances TODR and ASDR are equal. An aircraft is said to be field limited when one or both of the distances available are equal to the distances required (TODR = TODA or ASDR = ASDA). Under these circumstances there is no spare distance available.
VMCG is the minimum speed at which it is possible to maintain control following the failure of the critical engine during the take-off run. If the failure occurs before VMCG then the aircraft will go out of control and run off the side of the runway. For this reason V1 must never be less than VMCG. If pre take-off performance calculations reveal that V1 is less than VMCG, then V1 must be increased to equal VMCG.
But this increased V1 will increase the distance required to stop the aircraft if the take-off is aborted. It is not therefore possible to increase V1 if the aircraft is field limited. So if V1 is less than VMCG and the aircraft is already field limited, the take-off is not permitted (option b).
It should however be noted that this question does not specify a field limited condition. Under these circumstances it may be possible to increase V1. This will increase the accelerate stop distance and decrease the take-off distance. This action might result in a situation in which the ASDR is greater than the one engine out take-off distance (option a).
A further possibility might be to carry out a reduced thrust take-off. This would reduce VMCG and might possibly make it less than V1 (option d). But VMCG cannot be reduced arbitrarily simply to make it less than V1.
Overall this is a very badly worded question, which has caused problems for many years.