NP-hardness of an extention of 2 sat a 2  sat instance which is unsatisfiable and an integer k are given, decision problem is that: is it possible to delete k variables, also remove clauses contain them, in order to satisfy the 2-sat instance?? and optimization problem is that: find the minimum number of variables to remove to make the 2  sat instance satisfiable. I want to use reduction to show this problem is np-hard but I don't know which np-hard problem should be used. I appreciate if anyone could help me by the reduction or introduce me a reference (I searched but maybe I don't know the name of problem exactly)  

please notice that this problem is different from maximum 2-sat, in this problem each clause can even contain one literal and we reduce 3-sat to show it is np-hard.
You can reduce from independent set. Given a graph, have a variable for each vertex, and for each edge, add the four clauses
$$ x \lor y, x \lor \lnot y, \lnot x \lor y, \lnot x \lor \lnot y, $$
where $x,y$ are the variables corresponding to the vertices connected by the edge. I'll let you complete the proof.