Detective Inspector was a title given to certain police officers in the United Kingdom.
On Christmas Day 1965, landing in London, the First Doctor was brought before a Detective Inspector after his arrest. Steven Taylor, in the guise of a policeman, was able to convince him that the Doctor was harmless. (TV: The Daleks' Master Plan)
A Detective Inspector found the body of Tegan Jovanka's aunt, Vanessa, in 1981 after she became a victim of the Master's Tissue Compression Eliminator. (TV: Logopolis)
Burke was a detective inspector. He worked with Liz Shaw to solve the murders in and near Winterborne School and later applied to join P.R.o.B.e. (HOMEVID: The Devil of Winterborne)