First Round: February 27, 2016 at Regional Testing Centers Second Round: April 9, 2016 at The University of North Alabama COMPREHENSIVE EXAM

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First Round: February 7, 06 at Regional Testing Centers Second Round: April, 06 at The University of North Alabama COMPREHENSIVE EXAM Construction of this test directed by Miranda Bowie and Ashley

1 First Round: February 7, 06 at Regional Testing Centers Second Round: April, 06 at The University of North Alabama COMPREHENSIVE EXAM Construction of this test directed by Miranda Bowie and Ashley Johnson, The University of North Alabama INSTRUCTIONS This test consists of 0 multiple choice questions. The questions have not been arranged in order of difficulty. For each question, choose the best of the five answer choices labeled A, B, C, D and E. The test will be scored as follows: points for each correct answer, point for each question left unanswered and 0 points for each wrong answer. (Thus a perfect paper with all questions answered correctly earns a score of 0, a blank paper earns a score of 0, and a paper with all questions answered incorrectly earns a score of 0.) Random guessing will not, on average, either increase or decrease your score. However, if you can eliminate one or more of the answer choices as wrong, then it is to your advantage to guess among the remaining choices. All variables and constants, except those indicated otherwise, represent real numbers. Diagrams are not necessarily to scale. We use the following geometric notation: If A and B are points, then: AB is the segment between A and B AB is the line containing A and B AB is the ray from A through B AB is the distance between A and B If A is an angle, then m A is the measure of angle A in degrees If A and B are points on a circle, then: AB is the arc between A and B mab is the measure of AB in degrees If AB = CD, then AB and CD are congruent. If l, m are two lines, then l m means l and m are perpendicular. Printing by The University of North Alabama

2 Why Major in Mathematics? What sorts of jobs can I get with a mathematics degree? Examples of occupational opportunities available to math majors: Market Research Analyst Air Traffic Controller Climate Analyst Estimator Research Scientist Computer Programmer Cryptanalyst Professor Pollster Population Ecologist Operations Research Data Mining Mathematician Meteorologist Medical Doctor Lawyer Actuary Statistician Where can I work? What sorts of companies hire mathematicians? Well just to name a few... U.S. Government Agencies such as the National Center for Computing Sciences, the National Institute of Standards and Technology (NIST), the National Security Agency (NSA), and the U.S. Department of Energy. Government labs and research offices such as Air Force Office of Scientific Research, Los Alamos National Laboratory, and Sandia National Laboratory. Engineering research organizations such as AT&T Laboratories - Research, Exxon Research and Engineering, and IBM Research. Computer information and software firms such as Adobe, Google, Mentor Graphics, Microsoft, and Yahoo Research. Electronics and computer manufacturers such as Alcatel-Lucent, Hewlett-Packard, Honeywell, Philips Research, and SGI. Aerospace and transportation equipment manufacturers such as Boeing, Ford, General Motors, and Lockheed Martin. Transportation service providers such as FedEx Corporation and United Parcel Service (UPS). Financial service and investment manangement firms such as Citibank, Morgan Stanley, and Prudential. A Mathematics Major isn t just for those wanting to be Mathematicians! The top scoring major on the Law School Entrance Exam (LSAT) is Mathematics (Source: Journal of Economic Education) Mathematics is also a top scoring major on the Medical School Entrance Exam (MCAT) (Source: American Institute of Physics) Study in the field of mathematics offers an education with an emphasis on careful problem solving, precision of thought and expression, and the mathematical skills needed for work in many other areas. Many important problems in government, private industry, and health and environmental fields require mathematical techniques for their solutions. The study of mathematics provides specific analytical and quantitative tools, as well as general problem-solving skills, for dealing with these problems. The University of North Alabama offers an undergraduate degree in Mathematics and has many great things to offer, including a new Mathematics Fellow program, an active undergraduate research group and a new Dual Degree Engineering program. For more information, go to

3 . Find the slope of the line between the points (, 6) and (, ) Put the following values in order from least to greatest. ( ) ( ) π π ( I. sin II. sec III. cos π ) + 6 IV. tan ( ) 7π 6 I, II, III, IV II, I, III, IV IV, I, III, II II, III, IV, II. In last year s NBA Finals, LeBron James scored a total of 6 points from -pointers and -pointers. If he made a total of 78 shots, how many -pointers did he make? 6 (). Given that the points (, ), (6, ) and (c, 0) are collinear, what is the value of c?...7 (E) none. The equation (x x ) / = 6 has two solutions, a and b. Find a + b. 6. An equilateral triangle and a regular hexagon have equal perimeters. If the area of the regular hexagon is 6, then what is the area of the equilateral triangle? In a trapezoid ABCD with AB parallel to CD, the diagonals intersect at a point E. The area of triangle ABE is and of triangle CDE is 0. Find the area of the trapezoid Find the absolute value of the sum of the two solutions to the equation (x )(x + ) = The College World Series for baseball is decided by a best of three game series; that is, the first team to win two games is the winner. If Texas and Florida are playing in the championship, and for any given game Texas has a 60% chance of beating Florida, what is the probability that the series will need to play the third game? If x + y = and x + y = 6, find x + y Find the value of x so that log (x + ) = (E) None of these

4 . Two tangents to the same circle form a 0 angle. The radius of the circle is 0. Find the length of the smaller of the intercepted arcs. 6π 0π π 0π (E) None of these. On a date night menu, you and your date get to choose one appetizer to share from a list of three, two entrees from a list of five, and one dessert to share from a list of four. Assuming you both could order the same entree, how many different meals are possible? The system of equations listed below has two solutions (a, b) and (c, d), where a < c. Find the number a b + c + d. x + y xy = = Define the function f(x ) = (x ) (x ) +. Find f( ) The area of a square inscribed in a circle is 0 units. Determine the perimeter of the square inscribed in a semicircle of the same radius Find the sum of all solutions to the equation x x + 8 = Assume p q in the figure shown. Find the angle supplementary to angle x. 8 8 (8 ). Find the sum of all solutions to the equation cos(x) = sin(x) for x in [0, π]. π π π π 0. Define two new operations on real numbers as follows a b = a b a b = a + b Calculate the value of (6 ) ( ). 6

5 . Find the value of i i i 6, where i =. i i. Given that it is o clock, exactly how long will it take for the minute hand to catch up with the hour hand? min 6 min 6 min 7 min. If f (x) = x +, is the inverse of the function f(x), find the solution to the equation f(x) =.. A parallelogram s vertices have coordinates of (0, 0), (, ), (, 0) and (, ). What is the area of the parallelogram? 8. Points A, B, C and D lie on a circle with AC a diameter, AB = and BC =, ABD = CBD. What is BD? [ x x + 6. Find the sum of all values of x so that the matrix A = x ] has a determinant of For what value of k are the lines x + y = k and x ky = 7 perpendicular? 6 8. For which of the following values of k does the equation x x = x k x 6 have no solution for x? 7. Find the equation of the line which is the perpendicular bisector of the line segment connecting the points (, ) and (, 6). y = x + 7 y = x + y = x + y = x + 0. Find the exact value of (0!) (!) (0!) + (!)

6 . In the figure shown, the line y = x + intersects the parabola y = x x at points P and Q. What are the coordinates of point Q? (, 0) (, 0) (, ) (, 6). Find the number of distinct solutions to the equation x = x. 0. A box contains four fair coins and six biased coins. Whenever a fair coin is flipped, it lands with heads facing up with a probability of 0.. Whenever a biased coin is flipped it lands with heads facing up with a probability of 0.. A coin is randomly chosen from the box and flipped. What is the probability that it will land with heads facing up? The smallest angle of a rhombus is one-half the measure of the larger angle. The shorter diagonal is 0 m. Find the perimeter of the rhombus. 0 m 60 m 80 m 00 m. Find the sum of all solutions to the equation e x e x = 6. ln 6 ln 6 6. How many solutions are there to the equation cos(x) = for x in [0, π]? 8 (0) 7. Clara is on a bike ride. If she bikes uphill miles at a rate of 8 miles per hour and then bikes downhill miles at a rate of 6 miles per hour, what was her average speed on the trip, rounded to the nearest integer? 0 8. Find the sum of the squares of all values of x so that f(x) = 8 if { x f(x) = + x x < 0 x 7 x 0 0. If f(x) = a + bx, there are two such pairs (a, b) such that f(f()) = and f(f(0)) =. What is the maximum value of a b? Suppose that f(x) = x x and g(x) = x x. Which of the functions below is equivalent to (f g)(x)? x x x xx x xx+ x xx+ (E) (x) x

7 . Quadrilateral ABCD is inscribed in a circle with m C = 8. Find m A. 8 0 (E) Not enough information. Find the number of distinguishable permutations of the letters of the word PARALLEL The difference quotient of a function f(x) is the quotient quotient of x +. h h + h f(x + h) f(x), h 0. Find the difference h x (E) x + h. Find the value of c so that x = is not a vertical asymptote of the graph of y = x x + c x + x.. Which of the following classical constructions is impossible using only a compass and a straight edge? I. Draw the perpendicular bisector of a segment II. Find the midpoint of a segment III. Inscribe a regular hexagon in a circle IV. Trisect an angle IV Only III and IV I and III II Only (E) I and II 6. In the figure shown, the circle is inscribed in an isosceles triangle ABC, with segment AP passing through center O of the circle, with AC = AB = and BP =. Find the radius of the circle. 7. The function p(x) = x + 6x 60 models the daily profits, in hundreds of dollars, for a small company that produces x bicycles per day. How many bicycles should be made each day to maximize profit? Which of the following is equivalent to sin (x) cos (x)? 8 8 cos(x) cos(x) 8 cos(x) (E) 0. Find the area of the region of the plane satisfying x + y and y x. π π 6π 8π 0. Form a 7-digit number, n, at random using the digits through 7 exactly once. What is the probability that n is divisible by? 0

First Round: February 27, 2016 at Regional Testing Centers Second Round: April 9, 2016 at The University of North Alabama GEOMETRY EXAM

First Round: February 27, 2016 at Regional Testing Centers Second Round: April 9, 2016 at The University of North Alabama GEOMETRY EXAM First Round: February 7, 016 at Regional Testing Centers Second Round: April, 016 at The University of North Alabama GEOMETRY EXAM Construction of this test directed by Scott H. Brown and Luke A. Smith,