Maybe t'internets diminshes the tongue in cheek element of my post, but the physicist within is forcing me to be incredibly pedantic here.
Yes, I do understand the question. Physically, we all know that if you shove on the power (i.e. increase speed) you'll have to push the nose down (reduce the AoA) to avoid climbing. Equally I understand it mathematically and I agree with the answers above; you are keeping the LHS of the lift equation constant whilst changing terms on the RHS, so:
ro * cl(old) * v(old) ^ 2 = ro * cl(new) * v(new) ^ 2
since ro is constant (level flight), you can rearrange to get:
cl(new) = cl(old) x v(old)^2 / v(new)^2
since v(new) = 2*v(old)
cl(new) = cl(old) * (1^2 / 2^2), i.e. cl(new) = 0.25 cl(old)
(pause for standing ovation).
Now back to my point. I argue the question implies we have a CL level in the cockpit - I don't in mine. We don't alter the CL as this set by the aerofoil designer - we change the AoA to that corresponding to appropriate CL. Yes, we all get the meaning of the question, but the authority/question bank asking this should be precise - there is no reason not to be! So after all that, all I am saying is the question should be asking something along the lines of
"An aeroplane maintains straight and level flight while the IAS is doubled. The angle of attack must be altered to that corresponding to a lift coefficient that is: ..."
Semantic? Pendantic? Certainly - but why cut corners and dumb everything down?