Context free languages are closed under:

Let $${L_D}$$ be the set of all languages accepted by a $$PDA$$ by final state and $${L_E}$$ the set of all languages accepted by empty stack. Which of the following is true?

Consider the grammar with the following productions.
$$S \to a\,\alpha \,\,b\left| {\,\,b\,\alpha } \right.\,c\,\left| {aB} \right.$$
$$S \to \alpha S\,\left| b \right.$$
$$S \to \alpha \,bb\,\left| {ab} \right.$$
$$S\alpha \to bdb\,\left| b \right.$$
the above grammar is

State whether the following statement is TRUE / FALSE.

A minimal $$DFA$$ that is equivalent to an $$NFDA$$ with $$n$$ modes has always 2ⁿ states