What determines the nature of path following by the particle?

(a) speed (b) velocity (c) acceleration (d) none of these

Option (c) is correct.
A body moving with certain velocity will move in a straight line unless a force acts on it. So the velocity ( magnitude, direction or both) will change only when a force is applied (F = ma), i.e. when the body is accelerated. So the nature of the path is determined by acceleration.