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  1. Image Upload 1
    The figure above shows a solid formed by joining the bases of two square pyramids to opposite faces of a cube. If each edge of the solid has length  inches, what is the total surface area, in square inches, of this solid
    (A) 100 + 25√3
    (B) 100 + 50√3
    (C) 150 + 25√3
    (D) 200
    (E) 150 + 50√3
    Choice (B) is correct. The surface of the solid is made up of 4 square faces of the cube plus the triangular (lateral) faces of the pyramids. Each face of the cube has an area of,(5)(5)=25 and so the 4 faces of the cube have a total surface area of  100 square inches. Each lateral face of the pyramids is an equilateral triangle, which can be divided into two 30°,60°,90° triangles by a perpendicular dropped from any of the triangle’s vertices. Each 30°,60°,90° triangle will have a base of 2.5, which is half of the side length 5, since the hypotenuse of the triangle is 5 and the triangle is 30°,60°,90°; the second leg of each triangle is 2.5√3. Since the area of a triangle is 1/2bh, the area of each triangle is Image Upload 2 . Each lateral face of each of the pyramids is made up of two congruent triangles, thus the area of each lateral face of the pyramids is Image Upload 3. There are a total of 8 equilateral triangular faces on the solid, so the total area of triangular faces is Image Upload 4. Therefore, the total surface area of the solid is Image Upload 5.
  2. The four points Image Upload 6Image Upload 7,Image Upload 8 , and Image Upload 9 lie in the xy -coordinate plane. If  Image Upload 10 and Image Upload 11 are opposite sides of parallelogram Image Upload 12 and Image Upload 13, what is one possible value of Image Upload 14 ?
    The correct answer is any number greater than 3 and less than 5. Since Image Upload 15 and Image Upload 16 are opposite sides of a parallelogram, they must be parallel and equal in length. Since Image Upload 17 and Image Upload 18 are parallel, they are of equal slope, and so Image Upload 19, which simplifies to Image Upload 20. Since Image Upload 21 and Image Upload 22 are equal in length, Image Upload 23. This simplifies to Image Upload 24. The conditions Image Upload 25 and Image Upload 26 together are the same as the single condition Image Upload 27. Therefore, Image Upload 28, or Image Upload 29. Any fraction or decimal greater than 3 and less than 5 may be gridded as the correct answer.
  3. If Image Upload 30, which of the following could be true?

    I. Image Upload 31
    II. Image Upload 32
    III. Image Upload 33

    (A) I only
    (B) II only
    (C) I and III only
    (D) II and III only
    (E) I, II, and III
    Choice (E) is correct. To determine whether I, II, or III could be true, it is only necessary to find single examples of x and y that satisfy Image Upload 34 while also satisfying the statement in question. An example of I being true is Image Upload 35 and Image Upload 36. An example of II being true is Image Upload 37 and Image Upload 38. An example of III being true is Image Upload 39 and Image Upload 40.
  4. If x and y are two different numbers selected from the integers from 500 to 1000, inclusive, what is the greatest possible value for Image Upload 41 ?
    The correct answer is 1999. To achieve the greatest possible value for the expression, the numerator must be maximized, and the denominator should be the smallest positive number possible. Since x cannot equal y, and x and y are both integers, the denominator should be 1. Thus (taking x to be the larger number),Image Upload 42 . In order to maximize the numerator, both x and y must be maximized. Since 1000 is the greatest value for either x or y, it follows that x must be 1000 and y must be 1000 - 1 = 999. Therefore, the greatest possible value of Image Upload 43 is Image Upload 44.
  5. Two functions f and g are defined by Image Upload 45 and Image Upload 46, where a, b, and c are constants. If Image Upload 47 and Image Upload 48 , what is the value of a?
    The correct answer is 3. Since Image Upload 49 and Image Upload 50, it follows that Image Upload 51. Since Image Upload 52 and Image Upload 53, it follows that Image Upload 54. Subtracting the corresponding sides of Image Upload 55 from Image Upload 56 gives Image Upload 57. Hence Image Upload 58, and so the value of a is 3.
  6. Image Upload 59

    The line graph above shows the number of students in the computer lab during seven periods on one school day. Each of the students in the computer lab during lunch was also in the lab during two of the other periods. What is the maximum number of different students who could have been in the computer lab on that day?

    (A) 156
    (B) 148
    (C) 132
    (D) 124
    (E) 116
    Choice (D) is correct. From the graph, the sum of the number of students who were in the computer lab periods throughout the day is Image Upload 60. Each of the 8 students in the computer lab during lunch was also in the computer lab during 2 other periods. These students were counted 3 times each (for a total of Image Upload 61). In determining the maximum number of different students who could have been in the classroom that day, each student can only be counted once. Therefore, students who were in the computer lab during lunch should be counted as 8 students instead of 24, and the sum 140 is at least Image Upload 62 more than the number of different students. Since Image Upload 63, it follows that 124 is the maximum number of different students who could have been in the computer lab that day.
  7. Image Upload 64

    The figure above represents four offices that will be assigned randomly to four employees, one employee per office. If Karen and Tina are two of the four employees, what is the probability that each will be assigned an office indicated with an X ? 

    (A) 1/16 
    (B) 1/12 
    (C) 1/6 
    (D) 1/4 
    (E) 1/2
    Choice (C) is correct because two of the four employees will have a chance to get the x box, thus the probability is 2/4 or 1/2 then that must be multiplied by the second probability which is now 1/3. The answer is 1/6
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