The correct option is A The, the
An article is 'a word that comes before a noun'. It describes the noun, whether it is general or a particular thing.
'A' and 'an' are indefinite articles that refer to general, non-specific nouns. 'The' is a definite article that refers to a specific noun.
Options A and C: Usually, proper nouns do not require an article. There are exceptions though, when the proper noun is a name of a unique person, or thing in the world, then the definite article 'the' is used.
In the sentence, 'Damodar river' is a unique proper noun, there is only one river called 'Damodar'. In the same way, the title 'Sorrow of Bengal' is a specific title for the river.
Hence in both cases, 'the' will be used. So, option A is correct.
For the same reason, option C is incorrect.
Options B and D: 'A' is not used for proper nouns, (brand names are exceptions). Hence options B and D are incorrect.