Why does ionization energy increase from left to right in a period?

a) is true.
From left to right in a period, the increase in nuclear charge leads to stronger attractions between the nucleus and the electrons in the outermost shell. Therefore, increasing energy is needed to remove an electron from the outermost shell, and thus ionization energy increases.

b) is false.
From left to right in a period, the nuclear charge increases.

c) and d) are false.
From left to right in a period, the number of energy levels remains unchanged.

e) is false.
From left to right in a period, the number of electrons increases.