Answer:
3. listlessly
Explanation:
From this excerpt, it is clear that Prince Vasili's character was completely contrary to Anna Pavlovna's. She was impulsive, active, full of energy, joy and laughter. She constantly expressed her vitality and enjoyed in social interactions.
Now, it comes natural to define languid as completely opposite to this. Thus, languidly depicts doing something in a manner that lacks vitality, energy, speed or interest; doing something slowly, lazily, deprived of every effort and motive.
Although all the given answers could, more or less, be regarded as partial synonyms for "languidly", "carefully" and "soothingly" are usually seen as positive qualities, while "infrequently" simply means "rarely", which leaves us with "listlessly" as the best fitting answer.
