a) Suppose that symbol is 'a'
Got first 'a' push it. Go next state.
Ignore other symbols. Got another 'a'
Now go to final state. If more 'a' comes no problem.
DPDA
b) Even length palindrome.
It totally depends on the type.
abb bba (NPDA)
abb x bba (DPDA) (Do not call it odd length palindrome)
c) Ending with a particular terminal.
It is under regular language. DPDA too.
d) Odd length palindrome.
abb a bba (NPDA only)
How can you know when that middle one will come?
I have not seen ussages of x in these.
Please tell if I went wrong somewhere with proper explanations.