(a) At the disposition hearing, in any case where the court orders the care, custody, and control of the minor to be under the supervision of the probation officer for foster care placement pursuant to subdivision (a) of Section 727, the court shall inquire of the mother and any other appropriate person as to the identity and address of all presumed or alleged fathers. The presence at the hearing of a man claiming to be the father shall not relieve the court of its duty of inquiry. The inquiry may include all of the following:

1. (1) Whether a judgment of paternity already exists.
2. (2) Whether the mother was married or believed she was married at the time of conception of the child or at any time thereafter.
3. (3) Whether the mother was cohabiting with a man at the time of conception or birth of the child.
4. (4) Whether the mother has received support payments or promises of support with respect to the child or in connection with her pregnancy.
5. (5) Whether any man has formally or informally acknowledged or declared his possible paternity of the child.
6. (6) Whether paternity tests have been administered and the results, if any.