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?
x y 0 1 2 5 4 17 5 26
(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?#p#分页标题#e#
(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.
The correct answer is b.
