SAT数学Problem Solving练习题二.

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?

(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, Area_ABM/Area_ACM ?

(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) 70^o

(b) 35^o

(c) 40^o

(d) It cannot be determined from the information given

(e) 75^o

• 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. The correct answer is b.