Question #1: What is the length of the transverse diameter of the following ellipse?
x2/49 + y2/19 = 1
(a) 7
(b) 14
(c) 49
(d) 2
(e) 19
In our case a = 7, so the transverse diameter of the ellipse is 14.
Question #2: Mr Jones plans to drive 660 miles at an average speed of 60 miles/hour. How many miles per hour faster needs he to average, while driving, to reduce his total time by 1 hour?
(a) 5
(b) 6
(c) 7
(d) 8
(e) 9
At 60 miles/hour, the trip would take 660/60 = 11 hours. A 10 hours trip would require a speed of 660/10 = 66 miles/hour.
The answer to the question is 66miles/hour - 60 miles/hour = 6 miles/hour.
Question #3: If AC and BD are the diagonals of the ABCD rectangle, what is the ratio between the area of triangle EDC and the area of triangle EBC?
(a) .5
(b) .8
(c) 1
(d) 1.2
(e) 1.5
(1/2) ·h·ED
The area of triangle EBC is (1/2) ·h·EB
Since ED = EB, the areas of the 2 triangles are equal and their ratio is 1.
Question #4: If a is an integer chosen randomly from {3, 4, 5, 9} and b is an integer chosen randomly from {3, 8, 12, 15}, what is the probability that a/b is an integer?
(a) .125
(b) .25
(c) .1
(d) 1
(e) .5
The probability is 2/16 = .125.
Question #5:
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.
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 #6: The edges of a cube are each 4 inches long. What is the surface area, in square inches, of this cube?
(a) 66
(b) 60
(c) 76
(d) 96
(e) 65
The total surface area of the cube is (6 faces)x(16 square inches) = 96 square inches.
Question #7: In triangle ABC, MN is parallel with BC and AM/AB = 2/3. What is the ratio between the area of triangle AMN and the area of triangle ABC?
(a) 2/3
(b) 2/9
(c) 4/5
(d) 4/9
(e) 5/9
The area of triangle ABC = (BC · H)/2 where H is the altitude of ABC.
The area of triangle AMN = (MN · h)/2 where h is the altitude of AMN.
AreaAMN / AreaABC = (2/3) · (2/3) = 4/9.
Question #8: In the standard (x, y) plane, the line described by the equation y = mx + n intersects the y axis at a higher point than the line y = ax + b. Which of the following must be true?
(a) n > b
(b) m > a
(c) n + m > b + a
(d) n < b
(e) m < a
The line described by the equation y = ax + b intersects the y axis when x = 0, at y = b.
n > b is the correct answer.
Question #9: If a, b and c are the sides of any triangle, which of the following inequalities is not true?
(a) a·b > 0
(b) a + b > c
(c) a + c/2 >b
(d) b + c > a
(e) (a + b)·(b + c) > a·c
The second and the fourth answers will also be true, since the sum of 2 sides of a triangle is always higher than the third side.
The fifth answers is also true because it is just a multiplication of the second and fourth inequalities.
Answer three should be the one that is not true, and we can verify this result with an example: an isosceles triangle with a = 3, b = 3, c= 10 will not satisfy the inequality.
Question #10: For any x such that 0 < x < ¶/2, the expression (1 - sin2x)/cos(x) + (1 - cos2x)/sin(x) is equivalent to:
(a) sin(x)
(b) cos(x)
(c) sin(x) - cos(x)
(d) sin(x) + cos(x)
(e) 2sin(x)
(1 - sin2x)/cos(x) + (1 - cos2x)/sin(x) = cos2x/cos(x) +
sin2x/sin(x) = sin(x) + cos(x).
