For any two languages $L_1$ and $L_2$ such that $L_1$ is context-free and $L_2$ is recursively enumerable but not recursive, which of the following is/are necessarily true? I. $\bar{L}_1$ (complement of $L_1$) is recursive II. $\bar{L}_2$ (complement of $L_2$) is recursive III. $\bar{L}_1$ is context-free IV. $\bar{L}_1 \cup L_2$ is recursively enumerable

For any two languages L1 and L2 such that L1 is context-free and L2 is recursively enumerable but not recursive, which of the following is/are necessarily true? I. L1 (complement of L1) is recursive II. L2 (complement of L2) is recursive III. L1 is context-free IV. L1 L2 is recursively enumerable

Topic-wise GATE CS PYQs with verified steps