According to Special Relativity, Mass of an object increases with increase in its velocity. I got a question in my test asking, will mass of a cannon ball fired from a cannon remain constant or increase? I answered increase which was rejected on the basis that these equations are only valid for velocities approaching to that of light. Theoretically mass should increase and even if the increment is of millionth factor, an increase is an increase. Granted that increment will be immeasurable and completely negligible but still it will increase; won't it?
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NO. The mass never increases or decreases with velocity. Mass is a Lorentz scalar, i.e., it remains invariant when you go from one frame to another in which the velocity of the particle can be generically different from the velocity in the original frame. The misleading and confusing concept of relativistic mass was introduced in early days of Special Relativity which tried to define the relativistic mass $M = \dfrac{m}{\sqrt{1-\dfrac{v^2}{c^2}}}$. Here, $m$ is the mass (or the so-called rest mass). But this concept is completely overthrown because of several reasons including the inconsistencies and unnecessary conceptual difficulties it gave rise to. Have a look at this fantastic article by Lev Okun for a detailed discussion.
But yes, for the sake of argument, if we talk in terms of relativistic mass then no matter how small the speed-difference is, as long as it is non-zero, one can't deny the variation in the relativistic mass of the particle.