The correct answer is: B. 1934
The first Legislature of Jammu and Kashmir was established in 1934. It was a bicameral legislature, consisting of a Legislative Assembly and a Council of State. The Legislative Assembly had 75 members, of whom 60 were elected and 15 were nominated by the Maharaja. The Council of State had 36 members, of whom 24 were elected and 12 were nominated by the Maharaja.
The first elections to the Legislative Assembly were held in 1934. The National Conference won the majority of seats in the Assembly. The Maharaja appointed Sheikh Mohammad Abdullah, the leader of the National Conference, as the Prime Minister of Jammu and Kashmir.
The first Legislature of Jammu and Kashmir was dissolved in 1957. It was replaced by a unicameral legislature, the Jammu and Kashmir Legislative Assembly.
Option A is incorrect. The first Legislature of Jammu and Kashmir was established in 1934, not 1947.
Option C is incorrect. The first Legislature of Jammu and Kashmir was established in 1934, not 1942.
Option D is incorrect. The first Legislature of Jammu and Kashmir was established in 1934, not 1957.