My question is: is "a" or "b" necessarily integer here? The answer as per test is A, I came up with D

This is function question <x,y> is notation for a certain function which give (-1)^x+y.

in quantity A whatever value of a and b you take it will always give you EVEN integer in power which makes quantity A always positive.
And in quantity B whatever value of a and b you take it will always give you ODD integer in power which makes quantity B always negative.

Simple ways is to choose value of a and b as -1,0,1 which gives quantity A is always greater.

I was thinking, can a =1/2 and b=1 ?
SO that will make 2a+4b, then x+y will be odd in either cases.
I think this makes sense as well.

Yeah, that also seems right. I think we are missing something here @Leaderboard pls explain

Yes, the answer is D not A.