The correct options are
C (a) is anti-aromatic.
D (b) is aromatic.
Compound (a) i.e. cyclobutadiene is cyclic and conjugated. There are two π bonds and zero contributing lone pairs. Two π bonds gives a total of 4 π electrons, i.e. has (4n) π e−. So it is anti-aromatic.
Structure (b)
Cyclobutadienyl dianion:
The cyclobutadiene dianion is cyclic, planar and conjugated. It has a single pi bond, and now two carbons bearing negative charge which can contribute to the pi system, giving us a total of six π electrons. So, it follows Huckel rule and thus the Cyclobutadienyl dianion is aromatic.
Theory :
Special case of cyclobutadiene:
It is Anti aromatic in nature and unstable (more electrons in the antibonding molecular orbitals) so it dimerises at room temperature.
Thus it becomes non-aromatic.
(But whenever cyclobutadiene is mentioned in question we will count it as anti-aromatic only unless specified).