Does a pushdown automata exists for the following language?
Your languages are still not context-free.

A string homomorphism is a function $f\colon \Sigma \to \Sigma^*$ that replaces each character in a string with a (possibly empty) string over the same alphabet.  For a string $w=w_1\dots w_\ell$, we define $f(w)= f(w_1)\cdots f(w_\ell)$.  Context-free languages are closed under string homomorphisms.  That is, if a language&nbsp;$X$ is context free, so is the language $\{f(w)\mid w\in X\}$ for any string homomorphism&nbsp;$f$.

So, suppose that either of your languages were context-free, and consider the homomorphism defined by $f(a)=a$, $f(b)=b$, $f(c)=c$, $f(d)=\varepsilon$. We have $f(L_1)=\{a^nb^nc^{2n}\mid n\geq 0\}$ and $f(L_2) = \{a^nb^{2n}c^n\mid n\geq 0\}$. Neither of these is context free, which contradicts the assumption that $L_1$ and&nbsp;$L_2$ are context-free.



Also "a mixture" is not a well-defined context-free operation. In general, you should be suspicious of arguments where the main step is just some hand-wave along the lines of "And, yeah, then surely we can just combine them." If it's easy to explain how to "just combine them", then the argument can easily state that; if it's not easy to explain how, then it certainly needs to be explained!