Solution:The set of Turing machine codes for Turing machines that accept all inputs that are palindromes (along with some other inputs) is undecidable. This is because the set of palindromes is not recursively enumerable, and determining whether a Turing machine accepts all palindromes is equivalent to determining whether it accepts all strings in that set, which is undecidable. So, the statement (A) is incorrect.
(B) The language of codes for Turing machines that, when started with a blank tape, eventually write a 1 some where on the tape is undecidable. This falls under the category of the Halting problem, which is known to be undecidable. So, statement (B) is correct.
(C) The language accepted by a Turing machine M is always recursive. This is not necessarily true. there exist languages that are not recursive, meaning they cannot be decided by a turing machine. so, the statement (C) is incorrect.
(D) Post's correspondence problem (PCP) is known to be undecidable, which means there is no algorithm that exists that can solve all instances of the PCP. So, the statement (D) is correct.