There are a few books of "questions and answers in solid state physics", but unfortunately, the overlap between those books and our syllabus is not very good.
One well known book is called "Problems and Solutions in Solid State Physics" by Martin and Mihaly, but I fear that many of the problems are too high level... and much of the book is not on our syllabus. You may be able to find an illegal pdf of this one the web somewhere. (I think an old edition of it is circulating russian web sites --- but you didn't hear this from me).
A book that I thought was very nice was "Fundamentals of Solid State" by Christman. It is an old book and might be hard to find. I thought it had some nice problems in it, although again you will have to pick and choose which ones are on the syllabus and which ones are not.
Note that the course book has one complete exam with solutions in the back. And I think there is another complete exam with solutions on the course web page (right now the course web page is dead --- sorry about this, I've put a call in to IT support, but no response yet).