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ACT考试模拟练习题1.

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  • Question #1: The population of the city of Boston was 520,000 in 2005. Assume the population grows at a constant rate of 7% per year. Which of the following is an expression for the population of Boston in the year 2010?

    (a) 520,000(1.07)4 (b) 520,000(1.07)6 (c) 520,000(1.07)5 (d) 520,000(1.17)1/5 (e) 520,000(1.17)5

    • Solution: After one year the population of Boston is 520,000(1 + 7/100) = 520,000(1.07) After two years the population of Boston is 520,000(1.07)(1.07) = 520,000(1.07)2 After n years the population of Boston is 520,000(1.07)n In conclusion, after 5 years the population of Boston is 520,000(1.07)5

  • Question #2: What is the real value of x in the equation log345 - log35 = log7x?

    (a) 49 (b) 16 (c) 5 (d) 7 (e) 81

    • Answer: log345 - log35 = log7x log3(45/5) = log7x log39 = log7x log332 = log7x 2 = log7x x = 72 x = 49

  • Question #3: What is the length of the segment BD in the figure below, if AD is 5 inches? Figure not drawn to scale

    (a) 6 (b) 7 (c) 8 (d) 10 (e) 12

    • Answer: If angle BDC is 120o, then angle BDA is 180o - 120o = 60o. Triangle ABD is right and cos(angleBDA) = AD/BD cos(60o) = AD/BD. 1/2 = AD/BD. BD = 2�AD = 10 inches.

  • Question #4: Which of the following graphs could be the correct representation of 2 functions f(x) and g(x) that satisfy the equation g(x) = - f(-x)?

    (a) (b) (c) (d)

    • Answer: (b) is the correct solution. On the graph from solution (b), if we consider a random negative value for x, g(x) is negative. f(-x) will be positive and -f(-x) will be negative, same as g(x). This is not the case with solutions(a),(c) or (d).

  • Question #5: The price of wheat was $5/bushel on January 2007 and $9/bushel on January 2008. By what percentage did the price change from January 2007 to January 2008?

    (a) 80%

    (b) 90%

    (c) 40%

    (d) -40%

    (e) 50%

    • The price change is [(9 - 5)/5]�100 = (4/5)�100 = 80%.

  • Question #6: If 5y + 3x = 4, what is the value of 15y + 9x?

    (a) 11 (b) 12 (c) 13 (d) 14 (e) 15

    • Answer: We realize that 15y + 9x = 3(5y + 3x). 3(5y + 3x) = 3�4 = 12.

  • Question #7: The number of real solutions of the equation sin(x) = cos(x) + 2 is:

    (a) 1 (b) 2 (c) 3 (d) The equation does not have any real solutions (e) The equation has an infinity of real solutions

    • Answer: The range of the function sin(x) is [-1, 1] while the range of the cos(x) + 2 is [1, 3].

      The 2 functions never intersect each other because while they both can reach the value 1, they don’t do it at the same x.

      Sin(x) is 1 for multiples of �/2 while Cos(x) + 2 is 1 for multiples of �. (d) is the correct answer. �


  • Question #8: What is the value of the sum of angles a and b in the figure below? (a) 120o (b) 200o (c) 220o (d) 240o (e) 260o

    • Answer: Angle BAC is 60o. The sum of the angles of a triangle is 180o. angle(BAC) + angle(CBA) + angle(ACB) = 180o. angle(CBA) + angle(ACB) = 180o - 60o. angle(CBA) + angle(ACB) = 120o.

      a + b = 180o - angle(CBA) + 180o - angle(ACB). a + b = 360o - 120o. a + b = 240o. �


  • Question #9: If a,b and c are consecutive integers, a < b < c, and a + b + c = 96, what is the value of b?

    (a) 30 (b) 31 (c) 32 (d) 33 (e) 34

    • Answer: If a, b and c are consecutive numbers, b = a + 1 and c = a + 2. Then, a + a + 1 + a + 2 = 96, so 3�a + 3 = 96. a = (96 - 3)/3 = 31. b = a + 1 = 32.


  • Question #10: Suppose f(x) = x3 + 1 and g(x) = |x - 5|, what is f(g(1))?

    (a) 23 (b) 24 (c) 65 (d) 55 (e) 81

    • Answer: g(1) = |1 - 5|. g(1) = 4. f(g(1)) = 43 + 1. f(g(1)) = 65.

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