The mystery element from category "A" would be copper (Cu), while the mystery element from category "B" could be either chromium (Cr) or titanium (Ti). Henry Mosely discovered the mathematical relationship between an element's atomic number and the frequency of x-rays the element gives off when bombarded with high energy electrons. He found he could plot a straight-line graph between the number of protons in an atom's nucleus and the square root of the frequency of the x-rays given off.
Since the element in group "A" gave off a higher frequency, it would have to be copper, which has the highest atomic number of the group at 29. The two members of group "B" with atomic numbers lower than that would be nickel (28) and iron (26). Both those would give off a lower frequency of x-rays than copper when bombarded by high energy electrons. This combination would represent Mosely's findings accurately.