NP-complete language as a result of xoring two PTIME languages
This seems like a rather difficult question. Here is one approach.

Every 3CNF on $n$ variables can be encoded as a binary string of length $8n^3$ (how?). Consider the following two languages:
$$
\begin{align*}
L_1 &amp;= \bigcup_{n=1}^\infty \{ xy0^{8n^3} : |x|=n, |y|=8n^3, \text{$x$ is an assignment satisfying the 3CNF $y$} \}, \\
L_2 &amp;= \bigcup_{n=1}^\infty \{ x0^{8n^3}y : |x|=n, |y|=8n^3, \text{$x$ is an assignment satisfying the 3CNF $y$} \}.
\end{align*}
$$
These two languages are clearly in P, and so their XOR is clearly in NP. However,
$$
(L_1 \oplus L_2) \cap \bigcup_{n=1}^\infty \{0^ny^2 : |y|=8n^3\} = \bigcup_{n=1}^\infty \{0^n y^2 : |y|=8n^3, y \text{ is satisfiable}\},
$$
showing that $L_1 \oplus L_2$ is NP-complete.