Statement 1 is correct because according to the Indian Constitution, if the Speaker of the Legislative Assembly ceases to be a member of the assembly, he/she automatically loses the position of Speaker. Basically, the Speaker must be an active member of the assembly to hold the position.

Statement 2, however, is incorrect. Even if the Legislative Assembly is dissolved, the Speaker does not vacate his/her office immediately. He/she continues to serve until the new assembly sits for its first meeting and elects the new Speaker.

Therefore, the correct answer is option 1 which claims only the first statement is correct. Options 2, 3, and 4 are incorrect because they either incorrectly validate the second statement or incorrectly invalidate the first statement. Hence, the answer provided is correct.