The correct answer to the question is option 2, Avadi Session of 1955. During this session, the Indian National Congress officially accepted the "socialistic pattern of society" as part of its official policy.

Let`s analyze the other options to understand why they are not correct:

Option 1, Nagpur Session of 1959: The Nagpur session of 1959 focused on issues of agricultural and rural development, but it did not specifically mention the acceptance of a socialistic pattern of society.

Option 3, Bombay Session of 1953: The Bombay session of 1953 focused on planning for economic development and the adoption of a socialist ideology. However, it did not explicitly mention the acceptance of a socialistic pattern of society.

Option 4, Calcutta Session of 1957: The Calcutta session of 1957 focused on issues of socialist planning and economic policies. However, it did not mention the official acceptance of a socialistic pattern of society.

Therefore, option 2, Avadi Session of 1955, is the correct answer as it was during this session that the Indian National Congress officially accepted the "socialistic pattern of society."