Whew, that's a toughie!
a. True. A system of quadratic-quadratic equations could have exactly one solution if they were to touch at one point, say share the same vertex. Generally, if they cross once, they have to cross again.
b. True. It is possible for the graphs to not touch or cross at all, thus have no solutions.
c. True. The simplest way to explain this is that a quadratic equation forms a parabola and a linear equation forms a line. Two equations have an infinite number of solutions when they produce the same exact graph, and a parabola and a straight line can't be the same.
d. False, but on a technicality. The solution is verified by subsituting the solution into BOTH of the original equations. A solution to one may not be a solution to the other, thus not s solution to the system.