algebra ans 11

7/29/2019 Algebra Ans 11

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First Round : March 26, 2011Second 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 ascore of 250, a blank paper earns a score of 50, and a paper with all questions answered incorrectly earns ascore of 0.)

Random guessing will not, on average, either increase or decrease your score. However, if you can eliminateone of more of the answer choices as wrong, then it is to your advantage to guess among the remainingchoices.

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 BAB is the line containing A and BAB is the ray from A through BAB is the distance between A and B

If A is an angle, then:mA 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

m

AB is the measure of

AB in degrees

Editing by Zhijian Wu, The University of AlabamaPrinting by The University of Alabama


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The Worst

200. Lumberjack199. Dairy Farmer198. Taxi Driver197. Seaman196. EMT195. Roofer194. Garbage Collector193. Welder

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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. Manyimportant problems in government, private industry, health and environmental fields, and the academicworld require sophisticated mathematical techniques for their solution. The study of mathematics providesspecific analytical and quantitative tools, as well as general problem-solving skills, for dealing with theseproblems. The University of Alabama offers undergraduate and graduates degrees in Mathematics. Pleasevisit www.math.ua.edu and refer to the undergraduate and graduate programs for additional information.


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1. If i2

= 1, then what is the value of1 + i2 2011

?

(A) 1 (B)1 + i

2(C)

1 i2

(D) i (E) i

2. The graphs of the two lines 2ax + y = 1 and x 4ay = 2 are perpendicular to each other. What isthe value of a ?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

3. The limit of

x 2

x 4

as x approaches 4 is ?

(A) 1 (B) 2 (C)1

2(D) 4 (E)

1

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 listed below is it true that 0.7sin < 1 and 0.7tan > 1 ?

(A) 36 (B) 44 (C) 124 (D) 204 (E) 284

6. Let A be the area of the region enclosed by the graph of |x| 43 + |y| 43 = 1. Then A is in which of thefollowing range ?

(A) 1 < A y + z (B) xz > yz (C)x

z>

y

z(D) x2z > y2z (E) None of these

18. If xy = 2 and x2 + y2 = 5, thenx

y+

y

xis equal to

(A) 0 (B)5

2(C) 1 (D)

5

2(E)

2

5

19. What is the exact value of sin(

x

2) tan x csc x ?

(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

(A) 10 (B)20

3(C)

23

3(D) 20 (E) 100

21. Let f(x) =1

1 x . Define f1(x) = f(x) and fn+1(x) = f(fn(x)) for n 1. What is f2011(x) ?

(A

)x

(B

)

1

1 x (C

)

x

1

x (D

)

x

x 1 (E

)

x

1 x


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22. If z is a complex number satisfying z2

z + 1 = 0, then z24 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 commonfactor 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

(A)7

3(B)

3

7(C)

5

2(D)

2

5(E) None of these

25. How many solutions does the trigonometric equation1 cos x

sin x=

33

have in the interval [0, 2] ?

(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 repetitionsallowed?

(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 ofOA is 1, the slope of OBis

7, and the length of OA is equal to the length of OB, then the slope of AB is

(A) 7 (B) 1 7 (C) 1

7 (D) 2 7 (E) 2

7

28. Given that logb(a3) = 2, the value of loga(b

3) is

(A) 3 (B)2

9(C)

2

3(D)

9

2(E)

3

2

29. If greeting cards cost $20 for a box of 12 pieces, $10 for a packet of 3 pieces, or $4 each, then thegreatest 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

31. If sin x cos x = 12

, then what is the value of sin3 x cos3 x ?

(A)1

2(B)

3

4(C)

9

16(D)

5

8(E)

11

16

32. Let A be the positive integer satisfying the following equation9

10+

9

103+

9

105+ + 9

1019=

A

1019.

How many 9s appear in the decimal representation of A ?

(A) 7 (B) 8 (C) 9 (D) 10 (E) 11


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33. A finite sequence a0, a1,

, an of integers is called a curious sequence if it has the property thatfor 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.What is the value of the sum

2011k=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 amountedto a 40% profit for the dealer, whereas the other amounted to a 20% loss. What is the dealers netprofit on the two transactions ?

(A) $40 (B) $72 (C) $112 (D) $168 (E) $224

35. Determine the coefficient of x3 in the expansion of

2

x+

x

4

9

.

(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. Whatis the probability that the sum of the labels on the balls is odd ?

(A)1

2(B)

3

4(C)

5

9(D)

7

13(E)

10

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 ?

(A) 11! (B)11!

(2!)3(C)

11

2

8

2

(D) 118 (E) None of these

39. Let g(x) =ax axax + ax

, where a > 0 and a = 1. Ifg(p) =1

3, then g(4p) is equal to

(A)15

17

(B)7

8

(C)13

15

(D)6

7

(E)11

13

40. Legally married in Alabama, my neighbor has reached a square age. The product of the digits of hisage is his wifes age. The age of their daughter is the sum of the digits of the fathers age, and theage of their son is the sum of the digits of the mothers 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 blueand one is colored yellow. What is the probability that the numbers at the top of the dice satisfy theinequality: red


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42. Let f(x) be a function such that, for every real number x, f(

x) + 3f(x) = sin x. What is the value

of f(

2

)?

(A) 1 (B) 12

(C) 0 (D)1

2(E) 1

43. Using pennies, nickels, dimes, and quarters, what is the largest possible total value of coins you canhave 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 numbersand 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

47. Which of the following is equal to 5

729?

(A) 2 5

3 (B) 2 5

9 (C) 3 5

3 (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

Z =1

a +1

b +1

c +1

d + 1

e +1

f

.

If a,b,c,d,e and f are all either 1 or 2, what is the minimal value of Z?

(A)7

20(B)

15

41(C)

8

21(D)

17

43(E)

9

22

50. The average of three numbers is 5. The average of their reciprocals is17

72. Their product is 96. What

is the median of the three numbers ?

(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

algebra ans 11

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