As reported by Oxford Handbook of Commercial Correspondence:
[1.] If the letter begins with Dear Sir, Dear Sirs, Dear Madam, or Dear Sir/Madam,
the COMPLIMENTARY CLOSE should be "Yours faithfully".
[2.] If the letter begins with a personal name, e.g. Dear Mr James, Dear Mrs Robinson, or Dear Ms Jasmin, it should be "Yours sincerely".
Yet why's faithfully used for unnamed recipients, but sincerely for the named?
Namely (pun intended), why do the unnamed expect faith, but the named expect sincerity?