The correct option is
C indifferent
The words, 'in spite of' is a conjunction which is used to connect ideas that contrast. The first idea is :'The old woman repeatedly entreats (requests/ pleads)'. When somebody pleads, the natural reaction would be to show sympathy / be moved/ show concern. However, the second idea has to contrast. Hence, the blank can be filled by a word which means unmoved/ unconcerned/ be unsympathetic. The word, 'indifferent' is an adjective which means 'unconcerned/ having no particular interest or sympathy'. Hence, option C is the correct answer.
The word, 'ashamed' is an adjective which means 'embarrassed or guilty because of one's actions/ characteristics'.
The word, 'docile' is an adjective which means 'ready to accept control or instruction/ submissive'.
The word, 'lethargic' is an adjective which means 'sluggish/ slow'.
The words, 'ashamed', 'docile' and 'lethargic' do not contrast with 'showing sympathy/ concern'. Hence, options A, B and D are incorrect.