Why must the mass of an object moving at a relativistic speed increase?
a. in order to compensate for time dilation
b. so that conservation of mass is not violated
c. in order to compensate for length contraction
d. so that conservation of momentum is not violated

The mass is a property that quantifies the resistance to motion offered by an object or, which is the same, its inertia. This means it is a property that defines the amount of energy that an object composes and manifests through its inertia.

In other words:

The energy has inertia .

Therefore, since mass is a measure of the inertia of a body, the mass of an object increases with speed, which simply means that, as an object accelerates, its inertia grows and more and more energy is needed to accelerate it up to an even higher speed.  

Nevertheless, this does not mean that the amount of material that makes up an object increases while accelerating, or anything like that. The only thing that happens is that the faster it moves, the more its energy increases and, therefore, its inertia.

This was inferred by Einstein with his famous equation:

And to reach this equation it was necessary to take into account two important laws:

When two objects collide at different speeds (and therefore different linear momentum) the result of the sum of the momentum of both objects must have the same value before and after.

-The law of conservation of energy:

Energy is neither created nor destroyed, it only transforms. It changes from one form of energy to another.

So, this equation directly relates mass and energy, both being transformable. This means that, according to the theory of relativity, any object with mass can not move at the speed of light, because when that object approaches the speed of light, a stationary observer will "watch" the kinetic energy of the object and its momentum tend to infinity.

However, for very small speeds compared to the speed of light, like those of our normal world, "the mass does not change" is "constant".