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No it doesn't. Relativistic mass = Rest mass + KE. Rest mass is constant. KE is tied up in the velocity of the particle. There will be no increase in the "mass component" of relativistic mass.
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Yes. The full equation is:
E2 = p2c2 + m2c4
where m = rest mass and p = momentum. The shortened version of the equation is just to illustrate the relationship between mass and energy.