Solution

For $(a)$:

$L_3$ is a recursive language.

Complement of a recursive language is also recursive.

So,

$\overline{L_3}$ is recursive.

Also, $L_4$ is recursively enumerable.

Union of a recursive language and a recursively enumerable language is recursively enumerable.

Therefore,

$\overline{L_3}\cup L_4$ is recursively enumerable.

So,

$(a)\rightarrow(ii)$

For $(b)$:

$L_2$ is context-free language.

Every context-free language is recursive.

So, complement of $L_2$ is also recursive.

Also, $L_3$ is recursive.

Union of two recursive languages is recursive.

Therefore,

$\overline{L_2}\cup L_3$ is recursive.

So,

$(b)\rightarrow(iii)$

For $(c)$:

$L_1$ is regular.

Regular languages are closed under Kleene star.

So,

$L_1^*$ is regular.

Intersection of a regular language and a context-free language is context-free.

Therefore,

$L_1^*\cap L_2$ is context-free.

So,

$(c)\rightarrow(i)$

Hence, correct matching is

$(a)-(ii);\ (b)-(iii);\ (c)-(i)$