I am preparing for a competitive exam (GATE) in which questions are asked in Automata about operations among different types of languages. For example,
If $L_1$ is recursive & $L_2$ is recursively enumerable, then is $L_1\cap L_2$ recursively enumerable?
If $L_1$ is regular, $L_2$ is a deterministic CFL and $L_3$ is recursively enumerable, then is $L_1 \cap L_2$ a deterministic CFL? Is $L_3 \cap L_1$ recursive? Is $L_1 \cup L_2$ context free? etc.
Now I can compare such operations between languages of the same type using closure properties, but how do I tackle such questions?