MATHEMATICS COMPREHENSIVE EXAMINATION JANUARY 2004 Part II Advanced Problems (Three hours) Write your code number on the cover of your envelope and all the materials you submit for this exam. Do not write your name on anything, only your code number. Do any ten of the following problems. Do each problem on a separate sheet of paper. Write the course and problem number and your code number at the top of each sheet. When you are done, put the problems in order and place them in your envelope. Make a list of the problems you submit for part II on the cover of the envelope. Do not include more than ten problems for part II. Be sure your code number is on the cover of your envelope. Complete arguments must be given for each problem. You may use a calculator for arithmetic as convenient. You may use a computer to check your work, but your answers must be independent of the computer (unless instructed otherwise).
Math 22 Geometry () A very small, intelligent bug who knows Euclidean geometry is placed in hyperbolic geometry. Explain why it would not know immediately that it was not in Euclidean geometry, but how it could go about determining that it is not in Euclidean geometry. (2) Given ABC, prove that the angle bisectors of the triangle intersect in a common point. Your proof should be valid in Euclidean, hyperbolic, and spherical geometries. (3) Suppose that ABC and DEF have right angles at B and E,andthatAB = DE and AC = DF. a) Give a proof valid in both Euclidean and hyperbolic geometries that the triangles are congruent. To begin, assume you have an isometry that takes D to A and B to E, which is possible since AB = DE. b) Give a clear example showing that the triangles need not be congruent in spherical geometry. Math 222 Theory of Numbers (4) Using the techniques found in the proof of the Chinese Remainder Theorem, determine a solution to the following system of congruences: x 2(mod 3) 2x (mod 5) x 2(mod 6) x 2(mod 7) (5) If p is prime, prove that for any integer a: p (a + a p (p )!). Hint: Wilson. (6) a) Find the gcd (306, 657) and express it as a linear combination of 306 and 657. b) Determine all solutions in the integers to the Diophantine equation 23x + 360y = 99. c) Solve the linear congruence 6x 5 (mod 2).
Math 224 Elementary Differential Equations (7) Find all solutions of the given differential equation: (a) y 4y +3y = e t (b) dy dt + t y = sin t. t (8) Consider the one parameter family of differential equations y + αy +4y =0. (Note: Unlike the case of an harmonic oscillator, here α can be negative, positive or 0) (a) Rewrite the equation as a system of linear differential equations. (b) Draw the curve in the trace-determinant plane obtained by varying the parameter α. (c) Discuss the different types of behavior that the system exhibits as α varies. Are there any bifurcation values? (9) Find the general solution of the system X = AX for each matrix A below. Sketch the phase ( portrait ) and determine ( the ) behavior of solutions as t. 3 4 2 3 (a) A = (b) A=. 0 3 2 Math 225 Multivariable Calculus (0) A body of mass 2 kg moves in a circular path on a circle of radius 3 meters, making one revolution every 4 seconds. Find the centripetal force. () Find the length of the curve c(t) =(sint, cos t, t) in0 t π. (2) Use Green s theorem to prove that if C is a simple closed curve that bounds a region D, then the area of the region is equal to the value of the line integral xdy. (Note that xdy =0dx + xdy). Use this formula to find the area enclosed D by the ellipse x2 a + y2 2 b = (Hint: The ellipse can be parametrized by x = a cos t, 2 y = b sin t, 0 t 2π.) Math 226 Operations Research (3) Bloomington Breweries produces beer and ale. Beer sells for $ 0 per barrel, and ale sells for $ 5 per barrel. Producing a barrel of beer requires 5 lbs of corn and 2 lbs of hops. Producing a barrel of ale requires 2 lbs of corn and lb of hops. Sixty pounds of corn and 25 lbs of hops are available. Use linear programming and the simplex method to determine how much beer and ale the brewery should produce to maximize the revenue. (4) Player writes an integer between and 20 on a slip of paper. Without showing this slip of paper to player 2, player tells player 2 what he has written. Player
may lie or tell the truth. Player 2 must then guess whether or not player has told the truth. If caught in a lie, player must pay player 2 $ 20; if falsely accused of lying, player collects $ 0 from player 2. If player tells the truth and player 2 guesses player has told the truth, player must pay $ 5 to player 2. If player has lied and player 2 does not guess player has lied, player wins $ 0 from player 2. Give the payoff table for this two-person, zero-sum game, and determine the value of the game and each player s optimal strategy. (5) For your graduation present from college, your parents are offering you your choice of two alternatives. The first alternative is to give you a money gift of $ 38,000. The second is to make an investment in your name. The investment has a 70% chance of increasing to $ 60,000 and a 30% chance of decreasing to $ 20,000. Your utility for receiving M thousand dollars is given by the utility function u(m) = M +6. Which choice should you make to maximize expected utility? Math 227-228 Probability and Statistics (6) A coin is altered so that P (head)=0.6. If we flip the coin over and over again, how likely is it that the consecutive sequence TTT occurs before the consecutive sequence HHH? (7) Recall that the probability density function for the normal distribution is f(x) = e (x µ)2 2σ 2. Suppose that Alex plays a game where he has a 20% chance of 2πσ winning $ 3, a 30% chance of winning $ 4, and a 50% chance of losing $2. If the plays the game 00 times, find the normal pdf approximation to the probability his net profit is exactly $00. (8) Roll a standard die three times. Find the probability that the sum of the first two rolls was 8 given that the sum of all three rolls was 2. (9) If X and Y are independent, f (x) = e x, 0 < x <, and f 2 (y) = ye y, 0 <y<, find the pdf of U = X Y. ( ) 4 (20) Suppose that f(x Π =π) = π x x ( π) 4 x and that f(π) =,0<π<. We want to test H 0 :0< Π 0.5 versush :0.5 < Π <. Find P (H 0 X =3). (2) For the least squares parabola of the form y = β x + β 2 x 2, (there is no β 0 in the model), find ˆβ 2 as a function of Y, Y 2, Y 3,andY 4. X - 0 0 Y Y Y 2 Y 3 Y 4
Math 39 Combinatorics (22) A vendor has four each of red, green, yellow, and blue balloons. A man buys three balloons at random. Assume that any two balloons of the same color are identical; how many possible sets of three balloons are there? (23) Find the number of permutations of the ten digits 0,, 2,..., 9 in which a) An odd digit is in the first position and, 2, 3, 4, or 5 is in the last position b) 5 is not in the first position and 9 is not in the last position. (24) Suppose that a person can climb a ladder taking steps of either one rung at a time or two rungs at a time. Find a recurrence relation for the numbers s n of ways he can climb a ladder that has n rungs. Math 324 Topics in Differential Equations (25) Show that ydx+(2x ye y )dy = 0 is not an exact differential equation. Then, find an integrating factor µ(y) that depends only on y and then solve the resulting exact differential equation. (26) Find a power series solution of Airy s equation y ty =0,y(0) =, y (0) = 0. (27) Consider the PDE 2 u t + u 2 t + u = α2 2 u. Suppose that we use separation of u2 variables to find solutions of the form u(x, t) = X(x)T (t). Find the differential equations that must be satisfied by X(x) andt (t). (Partial answer: X (x) λx(x) =0.) Math 33 Abstract Algebra (28) Give an example of a non-cyclic group all of whose proper subgroups are cyclic. Prove that your answer is correct. (29) (30) a) Define isomorphism from group G to group G. b) Prove that there is no isomorphism from Q, the group of rational numbers under addition, to Q #, the group of non-zero rationals numbers under multiplication. a) State the theorem of Lagrange regarding the order of group G and the order of its subgroup H. b) Prove the theorem of Lagrange. (3) Prove: In a group of order 35, if a 5 = e = b 7,thenab = ba.
Math 332 Abstract Algebra II (32) Let R be a ring and A a subset of R. a) Define what it means to say that A is a prime ideal in R. b) Define integral domain. c) Prove: R/A is an integral domain if and only if A is a prime ideal of R. (33) Prove that x 4 + is irreducible over Q, the rational numbers, but reducible over R (34) Prove that any homomorphism of a field either is an isomorphism or takes each element to the additive identity. Math 333 Real Variables (35) Prove that every Cauchy sequence is bounded. (36) Let p k = 3 ( )k k for each k N. Using epsilons, prove that p k converges to 0. (37) Let f (x) = x 3. Prove that lim x 7 f(x) =2usingtheε δ definition of a limit. (38) Assume that f is Riemann integrable on [, 5]. a) Prove that f is bounded above and below on the interval [, 5]. b) If 4 f (x) 9 for all x [, 5], use the Riemann sum definition of 5 the integral to show that 6 f(x) dx 36. Math 344 Complex Analysis (39) Find the region onto which the function w = z 3 maps the region 0 r, 0 θ π/3. (40) Use the Cauchy-Riemann equations to show that f(z) =e x cos y ie x sin y is not analytic at any point in the complex plane. (4) Let C be the positively oriented boundary of the square whose sides lie along the lines x = ±4, y = ±4. Evaluate each of the following: e z (a) z +2 dz (b) e 2z z 4 dz C C
Numerical Analysis (42) Find the Lagrange form of the polynomial of degree at most 3 that interpolates the data below: x 2 3 4 5 y 2 2. (43) Find values of A, B and C so that the approximation f (x) = is exact for all polynomials of degree 2. Af(x + h)+bf(x) +Cf(x h) h (44) Define a sequence x n by x 0 =,x n+ = for n 0. Verify that x n converges +x n to The Golden Ratio + 5 by showing that this is simply fixed point iteration 2 on an appropriate function, and that the conditions for convergence of fixed point iteration are met. Deduce that = + 5. + 2 + + +