Correct option is (4) 1*
Given $L_1 = \phi $, so, $L_1^* = \{ \epsilon \}$. Remember, Kleene's star operator means to repeat the strings any number of times (including 0 times).
$L_2 = \{ 1 \}$, therefore, $L_2^* = \{ \epsilon, 1, 11, 111, ... \}$, i.e $ = 1^*$.
Now, $L_1^* \cup L_2^* . L_1^*$
$= \{ \epsilon \} \cup \{ 1^* \} . \{ \epsilon \}$
$= \{ \epsilon \} \cup \{ 1^* \}$
$= \{ 1^* \}$