Soal Kompetisi Matematika : Alabama - Aljabar

					                                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

				
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