Yes we need to look at E_{A} here but I think the answer is still actually D.

What may be confusing is that if you look at the expression for k_{2}/k_{1} using the Arrhenius equation, looks like d(k_{2}/k_{1})/d(E_{A}) is positive for any temperature increase, meaning that the *relative* increase in rate constant, when temperature increases by a certain amount, is greater for higher E_{A}. This would suggest we want largest activation energy (-> reaction 1's reverse direction).

But the wording of the question suggests that the magnitude of increase of rate (and thus of rate constant) is what they want you to identify (find the reaction with the greatest). For this, it can be shown that the non-relative magnitude of increase in rate constant, when temperature increases by a certain amount, decreases as E_{A} increases (for all E_{A} above RT, which can be assumed AFAIK). Therefore we should go for the minimum E_{A} to give maximum magnitude of increase in rate constant with a given temperature. So the answer should indeed be D (reaction 2's reverse direction).

I suppose the question-writer really meant to put "relative increase in rate" but since he didn't it has to be interpreted in the second way IMO.

Edit: note: the sign of d(rate)/dt will be the same as the sign of dk/dT because the concentration and partial pressure and order terms will definitely be positive, as x^y for any positive real x and any real y is always positive. Meanwhile, though not necessary for this problem, r_{2}/r_{1} = k_{2}/k_{1} (so of course d(r_{2}/r_{1})/d(E_{A}) = d(k_{2}/k_{1})/d(E_{A}) is true too, they are the same) since you assume concentration and partial pressure and order terms stay the same for a fair test.