Let $L_4$ $\subseteq$ {0,1}$^*$ be the set of all palindromes whose first character is 1. Give a context-free grammar for $L_4$
Not exactly, as your $1B$ and $0B$ productions might cause problems. You will accept 1101 for example. I believe just simplyfing it a bit, like this:

$A → 1B1\ |\ 1\ |\ \epsilon$

$B → 0B0\ |\ 1B1\ |\ 1\ |\ 0\ |\ \epsilon$

is enough (if we say that an empty string is a palindrome).