The correct answer is A. Shri Mata Vaishno Devi Shrine Board.
The Shri Mata Vaishno Devi Shrine Board is a statutory body constituted by the Government of India in 1986 to manage the affairs of the Vaishno Devi Shrine in Katra, Jammu and Kashmir. The Board is headed by a Chairman, who is appointed by the Central Government. The other members of the Board are nominated by the Central Government, the State Government of Jammu and Kashmir, and the Shri Mata Vaishno Devi Shrine Trust.
The Board is responsible for the management of the Shrine, including the construction and maintenance of the Shrine, the provision of facilities for pilgrims, and the collection and management of offerings. The Board also undertakes various welfare activities for the benefit of the local people.
The Board has been successful in managing the affairs of the Shrine and has made it one of the most popular tourist destinations in India. The Shrine is visited by millions of pilgrims every year.
The other options are incorrect.
Option B is incorrect because the Central Government does not directly manage the affairs of the Vaishno Devi Shrine. The Central Government is represented on the Board, but the Board is an autonomous body.
Option C is incorrect because the State Government of Jammu and Kashmir does not directly manage the affairs of the Vaishno Devi Shrine. The State Government is represented on the Board, but the Board is an autonomous body.
Option D is incorrect because the Vaishno Devi Shrine is not managed by both the Central and State Governments. The Shrine is managed by the Shri Mata Vaishno Devi Shrine Board, which is an autonomous body.