• 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

  • Solution: For any x²/a² + y²/b² = 1 ellipse, the transverse diameter is 2a. In our case a = 7, so the transverse diameter of the ellipse is 14.

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• 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

  • Answer: 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.

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• 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

  • Answer: If h is the altitude of triangle EDC from point C to ED, the area of triangle EDC is: (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

  • There are 4�4 = 16 combinations possible if an integer is chosen randomly from {3, 4, 5, 9} and another integer chosen randomly from {3, 8, 12, 15}. Out of these combinations, only (a = 3, b = 3) and (a = 9, b = 3) result in integers a/b. 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.

  • Answer: From the first equation, n = 2�m - 4. Then, the second 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 #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

  • Answer: Each face of the cube has an area of 4�4 = 16 square inches 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

  • Answer: Triangles ABC and AMN are similar so the lengths of their corresponding sides are proportional. This means MN/BC = 2/3 and the ratio between their altitudes is also 2/3. 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. Area_AMN / Area_ABC = (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

  • Answer: The line described by the equation y = mx + n intersects the y axis when x = 0, at y = n. 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

  • Answer: The first answer is true, since the product of 2 positive reals will be positive. 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.

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• 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)

  • Answer: (1 - sin²x)/cos(x) + (1 - cos²x)/sin(x) = cos²x/cos(x) + sin²x/sin(x) = sin(x) + cos(x).

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