The correct options are
A The name 'Karnataka' has been mentioned in the ancient tamil works of Kavirajamarga and Silappathikaram.
B Kavirajamarga mentions that the state of Karnataka extends from the Cauvery river to the Krishna river.
C The grammatical work, 'Silappadikaram', refers to Karnataka as 'Karunat'.
The name 'Karnataka' has been mentioned in the ancient tamil work of Silappadikaram and also in the grammatical work, Kavirajamarga, written by Nrupatunga. Silappadikaram refers to Karnataka as 'Karunat', which means 'the land on an elevated plateau'. Kavirajamarga mentions that the state extended from the Cauvery to to the Godavari river.
Thus, the first three statements are incorrect.