Step 1: Conjugated (direct) hyperbilirubinemia occurs when bilirubin has already been conjugated in the liver but cannot be excreted normally, either due to impaired secretion of conjugated bilirubin into bile or due to impaired bile flow (obstruction).
Step 2: Rotor syndrome is a defect in hepatic storage and excretion of conjugated bilirubin (along with Dubin-Johnson syndrome). It therefore causes a conjugated hyperbilirubinemia. So option (a) is correct.
Step 3: Gilbert syndrome and Crigler-Najjar syndrome are defects in the enzyme UDP-glucuronosyltransferase, which conjugates bilirubin. With less conjugation, bilirubin stays unconjugated, so both cause unconjugated hyperbilirubinemia. Options (c) and (d) are ruled out.
Step 4: Breast milk jaundice also produces an unconjugated hyperbilirubinemia (substances in breast milk inhibit conjugation and increase enterohepatic circulation), so option (b) is wrong. The only conjugated cause listed is Rotor syndrome.