What will happen to the kinetic energy if the mass of a moving object is doubled without any change in its speed?

The magnitude of kinetic energy of a body is ‘k’. what is its kinetic energy if its velocity is doubled ?

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Zeldredge's answer is great, answering the question from a mathematical point of view. Since you asked this question in a physics forum, I'll just add to that by answering from a physical point of view, clarifying why the energy is linear in mass and not quadratic:

Suppose you have two objects of equal mass moving parallel to each other with velocity $v$, each of mass $m$, and thus each of energy $\frac{1}{2}mv^2$. Although the two objects don't have anything to do with each other, we can calculate the sum of their energies if we want, and get $mv^2$.

Now attach the masses by a thin thread with negligible mass. This doesn't change the energy of any of the masses, and neither does it change their sum. Regarding now the masses as one single object of mass $2m$, the energy of this object must be $\frac{1}{2}\times(2m)\times v^2 = mv^2$, which is equal to the sum we calculated before.

That is, if an object has twice the mass, it also has twice the energy. If it had four times the energy, that would mean that by attaching two objects by a thin thread, you could increase the total energy, which would mean violating the law of energy conservation — one of the worst offenses you can commit.