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First Round : March 26, 2011 Second Round: April 23, 2011 at The University of Alabama ALGEBRA II WITH TRIGONOMETRY EXAM Construction of this test directed by Congxiao Liu and Chunhui Yu, Alabama A&M University INSTRUCTIONS This test consists of 50 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: 5 points for each correct answer, 1 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 250, a blank paper earns a score of 50, 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 of 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: • If A is an angle, then: AB is the segment between A and B m ∠ A is the measure of angle A in degrees ← → AB is the line containing A and B • If A and B are points on a circle, then: −−→ ⌢ AB is the arc between A and B AB is the ray from A through B ⌢ ⌢ AB is the distance between A and B m AB is the measure of AB in degrees Editing by Zhijian Wu, The University of Alabama Printing by The University of Alabama Doing the Math to Find the Good Jobs Mathematicians Land Top Spot in New Ranking of Best and Worst Occupations in the U.S. THE WALL STREET JOURNAL January 6, 2009 The Best The Worst 1. Mathematician 200. Lumberjack 2. Actuary 199. Dairy Farmer 3. Statistician 198. Taxi Driver 4. Biologist 197. Seaman196. EMT 5. Software Engineer 195. Roofer 6. Computer Systems Analyst 194. Garbage Collector 7. Historian 193. Welder 8. Sociologist 192. Roustabout 9. Industrial Designer 191. Ironworker 10. Accountant 190. Construction Worker 11. Economist 189. Mail Carrier 12. Philosopher13. Physicist 188. Sheet Metal Worker 14. Parole Oﬃcer 187. Auto Mechanic 15. Meteorologist 186. Butcher 16. Medical Laboratory Technician 185. Nuclear Decontamination Tech 17. Paralegal Assistant 184. Nurse (LN) 18. Computer Programmer 183. Painter 19. Motion Picture Editor 182. Child Care Worker 20. Astronomer 181. Fireﬁghter What You Can Do With A Mathematics Major Occupational opportunities Actuarial and Insurance Government Accountant Computer & Information Sciences Investment Analyst Financial Planner Researcher Beneﬁts Specialist Mathematician Demographers Computer Programmer Cartographer Data Processor Navigator Meteorologist Applications Programmer Ecologist Health Systems Analyst Biomedical Engineer Bio-mathematician Computer Applications Engineer Operations Analyst Operations Research Control Systems Engineer Control Systems Engineer Systems Engineer Numerical Analyst Teaching BusinessIndustry Statistician Engineering Analyst Financial Analyst Technical Writer Homeland Security Communications Engineer Study in the ﬁeld of mathematics oﬀers an education with an emphasis on careful problem analysis, pre- cision of thought and expression, and the mathematical skills needed for work in many other areas. Many important problems in government, private industry, health and environmental ﬁelds, and the academic world require sophisticated mathematical techniques for their solution. 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 Alabama oﬀers undergraduate and graduates degrees in Mathematics. Please visit www.math.ua.edu and refer to the undergraduate and graduate programs for additional information. ( )2011 1+i 1. If i2 = −1, then what is the value of √ ? 2 1+i 1−i (A) 1 (B) √ (C) √ (D) i (E) −i 2 2 √ 2. The graphs of the two lines 2ax + y = 1 and x − 4 ay = 2 are perpendicular to each other. What is the value of a ? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 √ x−2 3. The limit of as x approaches 4 is ? x−4 1 1 (A) 1 (B) 2 (C) (D) 4 (E) 2 4 4. What is the slope of the line that goes through points (−1, 2) and (2, −1) ? (A) 1 (B) 0 (C) −1 (D) 2 (E) −2 5. For which value of θ listed below is it true that 0.7sin θ < 1 and 0.7tan θ > 1 ? (A) −36◦ (B) 44◦ (C) 124◦ (D) 204◦ (E) 284◦ 4 4 6. Let A be the area of the region enclosed by the graph of |x| 3 + |y| 3 = 1. Then A is in which of the following range ? π π 3π (A) 1 < A < (B) <A<2 (C) 2 < A < π (D) π < A < 3 (E) 3 < A < 2 2 2 1 1 7. Suppose x is a complex number satisfying the equation x + = 1. What is the value of x3 + 3 ? x x (A) −2 (B) −1 (C) 0 (D) 1 (E) 2 8. What is the number of sides of a regular polygon for which the number of diagonals is between 30 and 40 ? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10 9. If a is a nonzero integer and b is a positive number such that ab2 = log10 b. What is the median of { 1} the set 0, 1, a, b, ? b 1 (A) (B) b (C) a (D) 1 (E) 0 b 10. Given the function f (x) = 2x4 − 3x3 + 4x2 − 5x + 6, what is the sum of A, B, C, D, and E if f (x) = A(x − 1)4 + B(x − 1)3 + C(x − 1)2 + D(x − 1) + E ? (A) 16 (B) 17 (C) 18 (D) 19 (E) 20 11. If a polynomial function with real coeﬃcients has 3 distinct x-intercepts, what is the maximal degree of the polynomial? (A) 3 (B) 4 (C) 5 (D) 6 (E) None of these 12. Let S = i2n−2 + i2n+2 , where n is an integer and i2 = −1. The total number of possible distinct values of S is (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 13. Given the sequence {an }, where a1 = 1 and an+1 = an + n for n ≥ 1. Find a11 . (A) 50 (B) 55 (C) 60 (D) 65 (E) 66 14. If the age of a person 16 years ago was 5 times the current age of his son, and two years ago the sum of his age and his son’s age was 30, what is the age of his son now ? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 15. Which of the following is equal to sec5 x cot x − sec3 x cot x ? cos x cos x sin x sin x (A) (B) (C) (D) (E) None of these sin2 x sin4 x cos2 x cos4 x 16. Express the rational number s = sin2 (10◦ ) + sin2 (20◦ ) + sin2 (30◦ ) + · · · + sin2 (80◦ ) in lowest terms . (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 17. Given x > y, and z ̸= 0, the inequality which is always correct is x y (A) x + z > y + z (B) xz > yz (C) > (D) x2 z > y 2 z (E) None of these z z x y 18. If xy = 2 and x2 + y 2 = 5, then + is equal to y x √ 5 5 2 (A) 0 (B) (C) 1 (D) (E) 2 2 5 ( π) 19. What is the exact value of sin x − · tan x · csc x ? 2 (A) 2 (B) 1 (C) 0 (D) −1 (E) −2 20. The largest solution of the equation 2 log10 x = log10 (3x − 20) + 1 is 20 23 (A) 10 (B) (C) (D) 20 (E) 100 3 3 1 21. Let f (x) = . Deﬁne f1 (x) = f (x) and fn+1 (x) = f (fn (x)) for n ≥ 1. What is f2011 (x) ? 1−x 1 x−1 x x (A) x (B) (C) (D) (E) 1−x x x−1 1−x 22. If z is a complex number satisfying z 2 − z + 1 = 0, then z 24 is equal to (A) 1 (B) −1 (C) i (D) −i (E) None of these 23. For a certain integer n, 5n + 16 and 8n + 29 have a common factor larger than one. That common factor is (A) 11 (B) 13 (C) 17 (D) 19 (E) 23 24. Function f satisﬁes f (x) + 2f (5 − x) = x for all real numbers x. The value of f (1) is 7 3 5 2 (A) (B) (C) (D) (E) None of these 3 7 2 5 √ 1 − cos x 3 25. How many solutions does the trigonometric equation = have in the interval [0, 2π] ? sin x 3 (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 26. How many 5 digit numbers that are divisible by 3 can be formed using 0,1,2,3,4,5 if no repetitions allowed ? (A) 216 (B) 120 (C) 240 (D) 126 (E) 96 27. Points A and B are in the ﬁrst quadrant and O is the origin. If the slope of OA is 1, the slope of OB √ is 7, and the length of OA is equal to the length of OB, then the slope of AB is √ √ 1 √ 2 (A) − 7 (B) 1 − 7 (C) − (D) 2 − 7 (E) − 7 7 28. Given that logb (a3 ) = 2, the value of loga (b3 ) is 2 2 9 3 (A) 3 (B) (C) (D) (E) 9 3 2 2 29. If greeting cards cost $20 for a box of 12 pieces, $10 for a packet of 3 pieces, or $4 each, then the greatest number of cards that can be purchased by $118 is (A) 60 (B) 61 (C) 65 (D) 66 (E) 67 30. Consider the region deﬁned by the inequality |x| + 2|y| ≤ 2. What is the area of this region ? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 1 31. If sin x − cos x = , then what is the value of sin3 x − cos3 x ? 2 1 3 9 5 11 (A) (B) (C) (D) (E) 2 4 16 8 16 9 9 9 9 A 32. Let A be the positive integer satisfying the following equation + + 5 + · · · + 19 = 19 . 10 103 10 10 10 How many 9’s appear in the decimal representation of A ? (A) 7 (B) 8 (C) 9 (D) 10 (E) 11 33. A ﬁnite sequence a0 , a1 , · · · , an of integers is called a curious sequence if it has the property that for every k = 0, 1, 2, · · · , n, the number of times k appears in the sequence is ak . For example, a0 = 1, a1 = 2, a2 = 1, a3 = 0 ∑ forms a curious sequence. Let a0 , a1 , · · · , a2011 be a curious sequence. 2011 What is the value of the sum k=0 ak ? (A) 4022 (B) 2012 (C) 2011 (D) 22011 (E) None of these 34. A used car dealer sold two cars and received $1120 for each car. One of these transactions amounted to a 40% proﬁt for the dealer, whereas the other amounted to a 20% loss. What is the dealer’s net proﬁt on the two transactions ? (A) $40 (B) $72 (C) $112 (D) $168 (E) $224 ( )9 2 x 35. Determine the coeﬃcient of x−3 in the expansion of + . x 4 (A) 80 (B) 81 (C) 82 (D) 83 (E) 84 36. A box contains nine balls, labeled 1, 2, 3, · · · , 9. Suppose four balls are drawn simultaneously. What is the probability that the sum of the labels on the balls is odd ? 1 3 5 7 10 (A) (B) (C) (D) (E) 2 4 9 13 21 37. How many positive integer solutions are there to 1! + 2! + 3! + 4! + · · · + x! ≤ x3 ? (A) 1 (B) 2 (C) 4 (D) 5 (E) None of these 38. Find the number of distinguishable permutations of the letters in the word: MATHEMATICS ? ( )( ) 11! 11 8 (A) 11! (B) 3 (C) (D) 118 (E) None of these (2!) 2 2 ax − a−x 1 39. Let g(x) = , where a > 0 and a ̸= 1. If g(p) = , then g(4p) is equal to ax + a−x 3 15 7 13 6 11 (A) (B) (C) (D) (E) 17 8 15 7 13 40. Legally married in Alabama, my neighbor has reached a square age. The product of the digits of his age is his wife’s age. The age of their daughter is the sum of the digits of the father’s age, and the age of their son is the sum of the digits of the mother’s age. How old is the son ? (A) 9 (B) 10 (C) 11 (D) 12 (E) 13 41. Three fair 6-sided dice, each labeled 1, 2, · · · , 6 are tossed. One is colored red, one is colored blue and one is colored yellow. What is the probability that the numbers at the top of the dice satisfy the inequality: red<blue<yellow ? 1 5 7 4 5 (A) (B) (C) (D) (E) 6 12 25 45 54 42. Let f (x) be a function such that, for every real number x, f (−x) + 3f (x) = sin x. What is the value (π) of f ? 2 1 1 (A) −1 (B) − (C) 0 (D) (E) 1 2 2 43. Using pennies, nickels, dimes, and quarters, what is the largest possible total value of coins you can have without having exact change for a dollar ? (A) 99 cents (B) 101 cents (C) 105 cents (D) 109 cents (E) None of these 44. How many polynomials are there of the form x3 − 9x2 + cx + d such that c and d are real numbers and the three roots of the polynomial are distinct positive integers ? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 45. If it is true that a < a5 < a4 , then which of the following must be also true ? (A) a < 1 (B) −1 < a < 0 (C) 0 < a < 1 (D) a > 1 (E) None of these 46. What is the maximal number of pieces you can form with 10 straight cuts across the face of a pizza ? (A) 90 (B) 100 (C) 55 (D) 56 (E) 96 √ 5 47. Which of the following is equal to 729 ? √ √ √ √ √ (A) 2 5 3 (B) 2 5 9 (C) 353 (D) 3 5 9 (E) 9 5 3 48. Suppose a, b < 1. If ax = by and ay = bx , then (A) a=b (B) a > b (C) a < b (D) a ̸= b (E) None of these 49. Let 1 Z= . 1 a+ 1 b+ 1 c+ 1 d+ 1 e+ f If a, b, c, d, e and f are all either 1 or 2, what is the minimal value of Z ? 7 15 8 17 9 (A) (B) (C) (D) (E) 20 41 21 43 22 17 50. The average of three numbers is 5. The average of their reciprocals is . Their product is 96. What 72 is the median of the three numbers ? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7