SAT数学Problem Solving类型练习二

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　　Question #1: f(x) = 1/(x + 1) and g(x) = x + 1. What are the values of x for which f(x) = g(x)?

　　(a) x1 = 0 and x2 = 2

　　(b) x1 = 0 and x2 = 1

　　(c) x1 = 2 and x2 = -2

　　(d) x1 = 0 and x2 = -2

　　(e) x1 = 1 and x2 = -2

　　Answer: f(x) = g(x) means x + 1 = 1/(x + 1);

　　(x + 1) · (x + 1) = 1;

　　x2 + 2 · x = 0

　　x · (x + 2) = 0 so x1 = 0 and x2 = -2.

　　(d) is the correct answer.

　　Question #2: A website had 2,000 unique visitors in 5 days. At the same rate, how many days are needed to achieve 12,000 unique visitors?

　　(a) 15 days

　　(b) 30 days

　　(c) 40 days

　　(d) 120 days

　　(e) 20 days

　　Answer: 2000 visitors in 5 days means that the website has 2000/5 = 400 visitors/day.

　　12,000 visitors / (400visitors / day) = 30 days.

　　Question #3: If a is an integer chosen randomly from the set {3, 4, 5, 9} and b is an integer chosen randomly from the set {3, 8, 12, 15}, what is the probability that a/b is an integer?

　　(a) .25

　　(b) .2

　　(c) .125

　　(d) .375

　　(e) .4

　　Answer: We have 4 possible integers for a and 4 for b, so the number of possible combinations for (a / b) is 16

　　a/b is an integer only for 2 combinations:

　　1. a = 3 and b = 3

　　2. a = 9 and b = 3

　　The probability that a/b is an integer is 2/16 = 1/8 = .125 and (c) is the correct answer.

　　Question #4: Two of the sides of a triangle are 7 and 6. What of the following could be the perimeter of the triangle?

　　(a) 14

　　(b) 26

　　(c) 18

　　(d) 9

　　(e) 12

　　Answer: The angle between the two sides should be higher than 0 degrees and the third side should be higher than 7 - 6 = 1. The perimeter should be higher than 7 +6 +1 = 14

　　The angle between the two sides should be lower than 180 degrees and the third side should be lower than 7 + 6 = 13. The perimeter should be lower than 7 + 6 + 13 = 26

　　(c) is the only value between 14 and 26.

　　Question #5: What is the line in the coordinate plane that is perpendicular to y = m · x + n and passes through the origin?

　　(a) y = (-1/m) · x

　　(b) y = (-1/m) · x - 1/n

　　(c) y = (-m) · x

　　(d) y = (-m) · x - n

　　(e) y = (-2m) · x

　　Answer: The line y = a · x + b, passes through the origin (0,0), so 0 = a · 0 + b, b = 0

　　(b) and (d) answers excluded, a simple test can be done to find the right answer, e.g. y = 2 · x will be perpendicular to y = -(1/2) · x. The right answer is (a).

　　Question #6: Given the table below, which of the following answers describes the relation between x and y?

　　xy

　　01

　　25

　　417

　　526

　　(a) y = x + 5

　　(b) y = x2 + 1

　　(c) y = 2 · x + 1

　　(d) y = 3 · x - 1

　　(e) y = 4 · x + 1

　　Answer: From the pair x = 0 and y = 1, we realize that only answers (b), (c) and (e) could be correct answers.

　　Examining the 3 choices, we find that the correct answer is (b)

　　Question #7: The median m divides the ABC triangle in 2 triangles, ABM and ACM. What is the ratio between the areas of the 2 triangles, AreaABM/AreaACM ?

　　(a) 1

　　(b) 1/2

　　(c) 2

　　(d) It cannot be determined from the information given

　　(e) 3

　　Answer: The areas of the 2 triangles will be equal because they are both (1/2) · h ·(BC/2), where h is the altitude. The correct answer is (a).

　　Question #8: Out of 25 mining companies, 14 extract copper, 16 extract silver and 1 extracts neither copper nor silver. How many companies extract both silver and copper?

　　(a) 1

　　(b) 6

　　(c) 2

　　(d) 15

　　(e) 3

　　Answer: If x is the number of companies that extract both metals,

　　1. the number of companies that extract copper only is 14 - x;

　　2. the number of companies that extract silver only is 16 - x;

　　There are 4 types of companies:

　　-that extract copper only,

　　-silver only,

　　-both copper and silver

　　-neither copper nor silver.

　　Their total should be 25:

　　25= 14 - x + 16 - x + x + 1

　　x = 31 - 25 = 6

　　Question #9: The isosceles triangle ABC from the figure below has the BAC angle = 400. What is the angle ADE if DE is half of BC?

　　(a) 70o

　　(b) 35o

　　(c) 40o

　　(d) It cannot be determined from the information given

　　(e) 75o

　　Answer: There is an infinite number of segments DE that will have half the length of BC. Fig.1 and Fig.2 show 2 possibilities. The correct answer is (d).

　　Question #10: For any a≠1, (a4 - 1) / (a - 1) =

　　(a) a3 + a2 + a

　　(b) a3 + a2 + a + 1

　　(c) a2 + a + 1

　　(d) 1

　　(e) a + 1

　　Answer: A first inspection of the choices eliminates the c, d and e answers since the answer should be a polynomial with a degree of 4 - 1 = 3.

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