Show that the following language is not recursive:
We say a language $L$ is good if "$0x\in L$ if and only if $1x\in L$". Let $\mathcal{L}$ be the set of all good languages. Then you are trying to determine whether the language recognized by a given TM belongs to $\mathcal{L}$. Since $\mathcal{L}$ is non-trivial, according to Rice's theorem, this is undecidable.

If you want a proof without Rice's theorem, you can take a look at the proof of Rice's theorem and try to write a proof for this specific problem. This is not hard.