Problem on irreducible palindromic polynomial

Question

Assignment:

Q1. a) Determine the number of polynomials f(x) of degree 6 over  F₂ for which f(1)=1 and f(0)=1.

b) Determine the number of polynomials of degree 6 over  F₂  that are reducible but have no linear factors. How this is possible, please can you explain this?
     
c)  Determine the number of irreducible polynomials of degree 6 over F₂. Explain.
In order to find this, is there a formula o we need to figure out by inspection?

Q2. Show that there is an irreducible palindromic polynomial of degree 6 over F₂.

Provide complete and step by step solution for the question and show calculations and use formulas.