Can doctors detect past abortion or a oregnancy that she does not want to tell anybody?

No: There is not a way to know if a woman had an abortion in the past. This is information she may choose to share or choose to keep private.

Doctor detectives: Yes, we can diagnose pregnancy when it is present after examination. However, we are also confidential in everything we do regarding a patient's medical care. Therefore, as a detective we are "great", as a commissioned "investigator" we "fail" as we are bound by medical ethics to not discuss or reveal any "clues" to anyone other than the patient or her designee.