assuming the motherboard supports both i3 and i5 and that they are the same socket type LGA 1156 then it should work.
do you have another i5 capable motherboard to test it on, preferbly one actually running an i5 already?
how did it come packed from e-bay? in the manufacters box unopened? or in some bubble wrap?
was the cpu exposed to static during or prior to fitting? exposure to static could have been from yourself or not packed properly in an antistatic bag if it wasn't in the oem box.
did you have any difficulty fitting it into the slot, did you have to force it or did it just slot right in?
yes I have tested
cpu's are very delicate sure we can get away with handling them sometimes but it takes just one incedent of a static charge to kill them, static discharge accounts for a rather large amount of failures in components excluding overheating and overclocking.