What is the ruling on one who swore falsely by the Qur’an while in a state of ritual impurity (junub), intending thereby to preserve his household? Is expiation (kaffarah) obligatory upon him after his repentance?

Lying under oath is one of the major sins and is considered a ghāmūs oath if it pertains to a past event. There is no expiation for it, only repentance. If the one taking the oath is in a state of major ritual impurity (junub), then touching the Mus'haf (Qur'an) is forbidden for him. Repentance must involve remorse and a firm resolve not to return to lying.