• 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

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• 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

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• 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 120^o, then angle BDA is 180^o - 120^o = 60^o. Triangle ABD is right and cos(angle_BDA) = AD/BD cos(60^o) = 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) 120^o (b) 200^o (c) 220^o (d) 240^o (e) 260^o

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

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

• 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.

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• 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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