Question #1: In the x,y plane, which of the following statements are true?
I. Line y + x = 5 is perpendicular to line y – x = 5.
II. Lines y + x = 5 and y – x = 5 intersect each other on the y axis.
III. Lines y + x = 5 and y – x = 5 intersect each other on the x axis.
(a) I and III are both true.
(b) I is the only true statement.
(c) II is the only true statement.
(d) I and II are both true.
Answer: y + x = 5 can be written as y = -x + 5. The slope of this equation is m1 = -1.
y – x = 5 can be written as y = x + 5. The slope of this equation is m2 = 1.
m2 = -1/m1 so the 2 lines are perpendicular.
We also need to find where the 2 lines intersect. If we add the 2 equations, 2·y = 10, y = 5.
From the first equation, x = 5 – y = 5 – 5 = 0. In conclusion the lines intersect at (0, 5) and this point is on the y axis.
In conclusion I and II statements are correct.
Question #2: If a is an integer chosen randomly from the set {3, 5, 6, 9} and b is an integer chosen randomly from the set {2, 3, 4}, what is the probability that a/b is an integer?
(a) .125
(b) .250
(c) .333
(d) .5
(e) .55
Answer: We have 4 possible integers for a and 3 for b, so the number of possible combinations for a/b is 4 · 3 = 12.
a/b is an integer only for 4 combinations:
1. a = 3 and b = 3
2. a = 6 and b = 2
3. a = 6 and b = 3
4. a = 9 and b = 3
The probability that a/b is an integer is 4/12 = 1/3 = .333.
Question #3: What is the value of integer a, if x = 2 is a solution of the equation √(a + x) = 2·x?
(a) a = 10
(b) a = 12
(c) a = 14
(d) a = 16
(e) a = 18
Answer: If we square the equation we get a + x = 4·x2
By replacing x with 2, a + 2 = 4·22, so a + 2 = 16.
In conclusion, a = 14.
Question #4: What is the value of (3x + 1 – 3x) / (3x – 3x – 1)?
(a) 6
(b) 3x
(c) 3x + 1
(d) 3x – 1
(e) 3
Answer: The numerator of the fraction is: 3x + 1 – 3x = 3x·(3 – 1) = 2 · 3x
The denominator of the fraction is: 3x – 3x – 1 = 3x – 1·(3 – 1) = 2 · 3x – 1
We can write the fraction as (2 · 3x) / (2 · 3x – 1) = 3x / 3x – 1 = 3
Question #5: Two diameters of a circle create an angle AOB of 45o between them. What is the length of arc AB if the radius of the circle is 10/¶?
(a) 5/2
(b) 3/2
(c) 2
(d) 4
(e) 6
Answer: The circumference of the circle is 2·¶·r = 2·¶·10/¶ = 20.
The ratio between the length of arc AB and the circumference of the circle is equal between the ratio between the 45o angle and 360o.
In conclusion, AB = 20 · 45o/360o = 20/8 = 5/2.
Question #6: A bus travels from town A to town B for 2 hours at a speed of 60 miles/hour. The bus stops in town B for 2 hours and then travels from town B to town C for 1 hour, at a speed of 50 miles/hour. What is the average speed of the bus?
(a) 30miles/hour
(b) 31miles/hour
(c) 32miles/hour
(d) 34miles/hour
(e) 40miles/hour
Answer: The distance the bus travels from A to B is 60 miles/hour · 2 hours = 120 miles.
Then, the bus travels from B to C: 50 miles/hour · 1 hour = 50 miles.
The total distance traveled is 120 + 50 = 170 miles and the total time is 2 hours + 2 hours stop + 1 hour = 5 hours.
In conclusion the average speed was 170 miles / 5 hours = 34 miles/hour.
Question #7: If a·b + b·c + c·a = 0, what is (a + b)2 + (b + c)2 + (c + a)2?
(a) a2 + b2 + c2
(b) 2·(a2 + b2 + c2)
(c) (a2 + b2 + c2)/2
(d) a2 + a + b2 + b + c2 +c
(e) (a + b + c)/2
Answer: (a + b)2 + (b + c)2 + (c + a)2 = 2·(a2 + b2 + c2) + 2·(a·b + b·c + c·a)
Since a·b + b·c + c·a = 0, the correct result is 2·(a2 + b2 + c2).
Question #8:
| Column A | Column B |
| x2 + 1 | x + 1 |
(a) The quantity in Column A is greater then the quantity in Column B.
(b) The quantity in Column B is greater then the quantity in Column A.
(c) The two quantities are equal.
(d) The relationship cannot be determined from the information given.
Answer: We need to compare x2 + 1 with x + 1. This results in a comparison between x2 and x.
For some x, x2 will be greater than x, e.g. for x = 2. For others, e.g. x = 1/2, x2 will be lower so the relationship cannot be determined from the information given.
Question #9:
2·m – n = 4
m + 2·n = 12
| Column A | Column B |
| (m + n)2 | 61 |
(a) The quantity in Column A is greater then the quantity in Column B.
(b) The quantity in Column B is greater then the quantity in Column A.
(c) The two quantities are equal.
(d) The relationship cannot be determined from the information given.
Answer: From the first equation, n = 2·m – 4. Then, the first equation will be m + 2·(2·m – 4) = 12
m + 4·m – 8 = 12 so 5·m = 20 and m = 4
From the first equation, n = 2·m – 4 = 2·4 – 4 = 4
Column A expression will be (m + n)2 = (4 + 4)2 = (8)2 = 64
The quantity in Column A is greater than the quantity in Column B.
Question #7: If a and b are positive integers and a·b = 200, which of the following can be the sum a + b?
(a) 40
(b) 46
(c) 33
(d) 55
(e) 50
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Answer: 200 = 2·2·2·5·5.
If a and b are positive integers, the 2 numbers and their sum can be:
2 + 100 = 102
4 + 50 = 54
5 + 40 = 45
8 + 25 = 33
10 + 20 = 30
(c) is the correct answer.
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