Explanation:

India's National Anthem, 'Jana Gana Mana', composed by Rabindranath Tagore, was first published under the title 'Bharat Vidhata' in the January $1912$ issue of 'Tatwabodhini Patrika', which was the official publication of the Brahmo Samaj. It was first publicly sung on December $27$, $1911$, at the Calcutta Session of the Indian National Congress. Options (1), (2), and (4) provide incorrect dates and publications. The 'Tatwabodhini Patrika' was a significant journal in the intellectual and social reform movements of Bengal. Therefore, (3) January $1912$ Tatwabodhini Patrika is the correct answer.