GATE Computer Science (CS) 2020 Solved Papers

${\mathit{L}}_1$ is reqular and ${\mathit{L}}_2$ is $\mathit{C}\mathit{F}\mathit{L}$.
${\mathit{L}}_1:\left\{\mathit{\omega }\mathit{x}\mathit{y}\mathit{x}\mid \mathit{\omega },\mathit{x},\mathit{y}\in {\left(0+1\right)}^{*}\right\}$
We can write the regular expression for this,
Let, $\mathit{x}=0$
${\mathit{L}}_{11}=\mathit{\omega }0\mathit{Y}0\Rightarrow {\left(0+1\right)}^{+}0{\left(0+1\right)}^{+}0$
$\mathit{x}=1$
${\mathit{L}}_{12}=\mathit{\omega }1\mathit{Y}1\Rightarrow$ we can generate all the strings and since a regular expression can be written for 4, we can say 4 is regular.
${\mathit{L}}_2$ is CFL:
${\mathit{L}}_2$ generates strings of even length, which can be done by PDA, therefore ${\mathit{L}}_2$ is CFL.