Among the following, the correct statement is

  I) If the acceleration of an object is constant, then it must move in a straight line.

II) If the acceleration of an object is zero, then its speed remains constant.

            III) If the speed of an object remains constant then its acceleration must be zero.

Let us review the given statements

i) It is true because the body is changing its velocity in a fixed direction with constant magnitude so acceleration will become constant hence it will move in straight line.

ii) It is also true because when acceleration becomes zero then there will not be any change in velocity or direction so body will maintain its speed as constant.

iii) It is not true because when speed is maintained as constant being revolving in a circular path the acceleration of the body will not become zero.

So here I & II are correct.