The correct option is
A listening
The verbs that do not explain the tense of the sentence and do not agree with the subject are called non-finite verbs. Non-finite verbs are usually infinitives, gerunds and participles.
Option B - 'To be listened' can be used in passive voice. As the sentence is in the active voice, we can't use 'to be listened' here. Also, it is grammatically incorrect to use. Hence, option B is incorrect.
Option C - 'Listened' is the past form of the verb. It is a finite verb. Also, as the sentence is in the present tense, we can't use 'listened'. Hence, option C is incorrect.
Option D - 'Listen' is the original form (present tense) of the verb. However, 'listen' is a finite verb. As explained above, non-finite verbs are usually infinitives, gerunds and participles, we can't use 'listen' here. Hence, option D is incorrect.
Option A - 'Listening' is the participle form of the verb and is a non-finite verb here, as it doesn't explain the tense of the sentence. Hence, option A is the correct answer.
'Mother likes listening to the radio while cooking.'