The correct answer is A. The motor will stop.
The back EMF is an induced voltage that opposes the applied voltage in a DC motor. It is generated by the rotation of the armature coils in the magnetic field of the stator. The back EMF increases with the speed of the motor.
If the back EMF vanishes suddenly, the applied voltage will be unopposed and the current will increase rapidly. This will cause the motor to overheat and stop.
Option B is incorrect because the motor will not continue to run if the back EMF vanishes suddenly.
Option C is incorrect because the armature will not burn if the back EMF vanishes suddenly. The armature may burn if the motor overheats, but this is not caused by the vanishing of the back EMF.
Option D is incorrect because the motor will not run noisy if the back EMF vanishes suddenly. The motor may run noisy if it overheats, but this is not caused by the vanishing of the back EMF.