MAT 3271: Geometry Name:Second Exam; Due by 5 p.m. on Monday, November 9, 2009.You are expected to work on this exam alone and to refrain from talking about the exam to anyoneexcept the professor until the time and date when it is due. You may use your own notes and anypublished materials that you like.Your signature below attests to a pledge that you have done the exam according to the aboveinstructions. (Please attach this cover page to your solutions.)Signature:1. (a) Prove the following statement for a projective plane: If there exist infinitely many points,then every line is incident with infinitely many points. (You may use the results proved forhomework.)(b) Show that this statement is independent of the axioms of incidence geometry (IA1, IAII- not strengthened, and IAIII). A diagram of your model is sufficient, although you arealso welcome to describe it set theoretically. (Hint: Create a model in which one line hasa countably infinite number of points, P1, P2, P3, . . ., and every other line has exactly twopoints.)(c) Suppose that, in a model of incidence geometry, every line is incident with the same finitenumber of points. Call this number k. Suppose in addition that every point is incident withthe same number of lines. Call this number l. Show that the total number of points is finiteand compute it (in terms of k and l); in addition, show that the total number of lines is finite,and compute it (in terms of k and l).2. If the points and lines of a projective plane are exchanged, keeping the same incidence relation,then the resulting interpretation is also a projective plane, said to be dual to the original one.But if the points and lines of an affine plane are exchanged, keeping the same incidence relation,the resulting interpretation is not even a model of incidence geometry. Which of the three axiomsof incidence geometry are satisfied by the dual interpretation, and which are not? Prove youranswers!3. In the real projective plane, compute the points of interse ction of the lines determined by thefollowing pairs of equations:(a)x + y + z = 0z = 0(b)x + y + z = 0x + y + 2z = 0(c)x + 2y + z = 02x + y + z = 0In the real projective plane, compute an equation for the line through the following pair of points:(d) [1, 0, 1] and [0, 1, 0](You should be able to do all parts of this problem by inspection, without much algebra orcomputation.)Chocolate forYT P O