CFL Closure Properties prove or disprove for the following languages
1) If $L$ is a CFG then assume $S$ is the start symbol of the grammar rules. Then introduce a new start symbol $S' \rightarrow SS\dots SS$ ($k$ times). So if $w \in L^k$ then $w$ can be written as $u_1u_2\dots u_k$ where $S \Rightarrow^* u_1$, $\dots$, $S \Rightarrow^* u_k$ and so $S' \rightarrow SS\dots S \Rightarrow^* u_1u_2\dots u_k$. Now suppose that $S' \Rightarrow w$. Then since $S' \Rightarrow SS\dots S \Rightarrow^* w$, $w$ clearly in $L^k$. Thus, $L^k$ is CF.

2) $L_1 = \{ 0^i | i \in N\}$, $L_2 = \{ 0^i1^j2^k | i,j,k \in N, i=j \ or  \ j=k\}$, $L_3 = \{ 0^i1^j2^k | i,j,k \in N\}$. Clearly $L_1 \subset L_2 \subset L_3$. But $L_2$ is not CFL, so not DCFL.