Math Power ENTRANCE EXAM...2

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1 Math Power October 26, 2007 : site: Visit for more product info. Direct your questions and comments to DL@MathEnglish.com. Sample Only ENTRANCE EXAM...2 1

2 Entrance Exam 5. What is the value of x in the equation 5x - 7 = y, 1. What is the least common multiple of 18 if y = 8? and 24? (A) -1 (A) 6 (B) 1 (B) 54 (C) 2 (C) 72 (D) 3 (D) 96 (E) 70 (E) What is the difference between 90% of 9 and 9% of 90? (A) 0 (B) 2 (C) 7.29 (D) 10 (E) Let R = and S = The greatest common factor of R and S is (A) 12 (B) 24 (C) 56 (D) 168 (E) At 9 A.M. it was 12 degrees below zero. By noon the temperature had dropped 7 degrees. Over the next two hours, the temperature rose 5 degrees. What was the temperature at 2 P.M.? (A) 0 (B) 10 below zero (C) 14 below zero (D) 24 below zero (E) none of these 7. For how many positive integer values of N 18 will the expression be an integer? N + 2 (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 4. Consider the following statements X: 7 > Y: 5 < 22 Z: (.1) 2 >.1 (A) X Y, and Z are all true (B) Z is true (C) X, Y, and Z are all false (D) Y is false (E) X is false 2

3 8. All of the students in Mr. Jacobs s class 12. What is the smallest prime number that is are taking either French or algebra or both 2 more than a positive multiple of 7? subjects. Exactly 16 students are taking (A) 9 French, exactly 18 students are taking (B) 19 algebra, and exactly 6 are taking both (C) 23 subjects. What is the total number of (D) 37 students in Mr. Jacobs s class? (E) 41 (A) 22 (B) 28 (C) 31 (D) 34 (E) If xˆ = x, what is the value of 3 3ˆ? 9. In the sequence of numbers 1, 2, 2, 3, 3, 3, [assume the pattern of one 1, two 2s, three 3s, four 4s, and so on continues], the number in the 40 th position will be (A) 8 (B) 9 (C) 10 (D) 11 (E) If P = , Q = and R = (A) P < Q < R (B) P < R < Q (C) Q < R < P (D) R < Q < P (E) P = Q 11. What is the value of a in the equation 3a - 6 = b, if b = 18? (A) 17 (B) 37 (C) 65 (D) 101 (E) 122 (A) 8 (B) 2 (C) 1 (D) 3 (E) If x + y = 7 and x 2 + y 2 = 25, what does 2xy equal? (A) 14 (B) 24 (C) 28 (D) 36 (E) Find the smallest positive integer that is a multiple of both 21 and 77. (A) 7 (B) 33 (C) 98 (D) 231 (E)

4 16. If x = 199, then the value of x 18. If 2 blobs = 3 glops and 3 blobs = 2 is chunks, then 1 chunk = (A) 520 (A) 1 glop (B) 510 (B) B glop (C) 210 (C) G glops (D) (D) (E) 11 9 glop 9 (E) 4 glops 17. Which of the following shows an isosceles triangle? (A) 19. The areas of ΔABC and ΔXYZ are each o B Y W (B) (C) 30 o 120 o A X C The area of the shaded region is 14. Then the area of ΔXWC is (A) 12 (B) 8 (C) 6 (D) 3 (E) 2 Z 120o 25 o (D) (E) 58 o 62 o 20. If n is positive and n 2 = 51, between which two numbers does n lie? (A) 5 and 6 (B) 6 and 7 (C) 7 and 8 (D) 8 and 9 (E) 9 and o 30 o 21. Let x = 3y and y = 3z. If y = 30, then the value of (x + y + z) is (A) 390 (B) 130 (C) 90 (D) 43 (E) 41 4

5 2 ( x+ 1) 22. For what value of x is 3 = 1? 25. Point B is the center of the circle. (A) 0 A (B) 1 4" (C) 1 (D) 2 B C 5 (E) Sammy pitched eight games and won five of them. To the nearest percent, what percent of games did Sammy lose? (A) 50% (B) 30% (C) 40% (D) 63% (E) 38% 24. What is the median of the set of numbers {4, 16, 12, 10, 6, 8, 12, 12}? (A) 9 (B) 10 (C) 11 (D) 12 (E) 13 AB = 4 inches and ABC is a right angle. The area of the shaded region is closest to: (A) 20 sq in (B) 25 sq in (C) 30 sq in (D) 33 sq in (E) 40 sq in 26. What fraction, reduced to its lowest terms, is halfway between K and 0.9? (A) 1 (B) B 17 (C) 20 6 (D) 7 (E) If x is an even number, how many values of x satisfy both of the following conditions: x 22 and x > 10? (A) 6 (B) 7 (C) 5 (D) 8 (E) 9 5

6 28. When N is divided by 15 the result is Use the following table. with a remainder of 9. What is the Name Sex Age Height Weight remainder when N is divided by 6? Jack M 20 6 ft. 0 in. 181 lbs Penelope F 9 4 ft. 8 in. 90 lbs (A) 0 Marge F 20 5 ft. 2 in. 110 lbs (B) 1 Peter M 20 5 ft. 11 in. 220 lbs (C) 2 Joe M 4 4 ft. 0 in. 45 lbs (D) 3 Julie F 36 6 ft. 1 in. 220 lbs (E) 4 Sam M 15 5 ft. 3 in. 110 lbs 29. A square of side 4 is flanked by a semicircle, as shown. 4 A P Which person is 5 times as old as Joe and weighs twice as much as Sam? (A) Peter (B) Jack (C) Penelope (D) Marge (E) Julie Point P is the center of the arc at the right. Then the shaded area is equal to (A) π + 2 (B) 1π + 2 (C) 2π + 2 (D) 2π - 4 (E) 2π What is the greatest common factor of 105 and 126? (A) 3 (B) 7 (C) 9 (D) 21 (E) What is the greatest integer less than 6 71? (A) 9 (B) 10 (C) 11 (D) 12 (E) 13 Question set [34-35] The two pie charts show the ice cream flavor preferences of the students at two high schools. Montague High Capulet High 31. ( ) B 2 (A) 5 (B) 1 3 (C) 5 21 (D) (E) 20 20% Strawberry 10% Coffee 30% Chocolate 40% Vanilla 19% Strawberry 36% Chocolate 25% Coffee 20% Vanilla 6

7 34. The ratio of Montague students who 38. Point B is on line segment AD, and point prefer vanilla to Montague students who E is on line segment BC. prefer chocolate is C (A) 10:9 70 (B) 4:3 E x (C) 3:10 (D) 3:4 (E) 2:5 50 o 20 o 35. If the number of Montague students who prefer vanilla is M and the number of Capulet students who prefer vanilla is C, then (A) M > 2C (B) M = 2C (C) M = C (D) M < C (E) the relationship cannot be determined from the given information A B D The value of x is (A) 20 (B) 40 (C) 50 (D) 60 (E) If 25% of x is 120, what is the value of x? (A) 30 (B) 120 (C) 145 (D) 240 (E) The square root of 6.25 is (A) (B) 0.25 (C) 2.5 (D) 25 (E) The ratio of 2.2 hours to 10 minutes is (A) (B) 5 1 (C) 22 5 (D) 66 (E) The points A, B, and C lie on a line (not necessarily in that order). If AB = 3 cm and BC = 8 cm, then which of the following must be true? (A) AC = 11 cm (B) AC = 5 cm (C) B lies between A and C. (D) A lies between B and C. (E) C does not lie between A and B (1-2) 3 + (-2) 3 = (A) -12 (B) -4 (C) 0 (D) 4 (E) 12 7

8 42. If 100 grams is half the mass of one object 46. A sheet of cardboard measuring 12 inches and twice the mass of another, the mass by 54 inches is to be cut into squares with of the heavier object is how many grams equal sides. What is the smallest number more than the mass of the lighter object? of squares if they are all to be equal, (A) 200 without any waste? (B) 150 (A) 9 squares (C) 100 (B) 16 squares (D) 75 (C) 18 squares (E) 50 (D) 36 squares (E) 72 squares What is the least integer greater than 4? (A) 5 (B) 6 (C) 7 (D) 8 (E) If 6(7x - 14) = 3(4 + 8x), then x = (A) -4 (B) -3 (C) -2 (D) 3 (C) If x = -5 and y = 4, then the value of x 2 y - y 2 x is (A) 180 (B) 18 (C) 20 (D) 60 (E) Express the product of and in scientific notation. (A) (B) (C) (D) (E) (A) 4 (B) 6 (C) 24 (D) 48 (E) One-third the product of two numbers is 24. One-half the product of these same two numbers is (A) 72 (B) 54 (C) 48 (D) 36 (E) 12 8

9 50. Here are two different ways to add: 54. After Richard gave 10% of his money to - First add the numbers, then round it to charity, he had $180 left. How much the nearest 10. money did he have originally? - First round the numbers to the nearest (A) $216 10, then add. (B) $200 In which of the following expressions will (C) $195 the result be the same with either method? (D) $172 (A) (E) $144 (B) (C) (D) (E) Circles A and B touch but do not overlap, as shown. 51. The equation 2(3x + 8) = 3(2x + 4) is satisfied by (A) no value of x (B) only negative values of x (C) only x = 0 (D) only positive values of x (E) all values of x 52. 3( ) (A) 1.00 (B) 0.75 (C) 0.50 (D) 0.25 (E) What is the greatest prime number less than 40? (A) 31 (B) 37 (C) 38 (D) 39 (E) 41 A B The area of circle A is 9π and the area of circle B is 16π. What is the area of the smallest circle that can contain the entire shaded figure? (A) 25π (B) 36π (C) 49π (D) 64π (E) 196π 56. What is the value of x(y - 2) + xz, if x = 2, y = 5, and z = 7? (A) 12 (B) 20 (C) 22 (D) 28 (E) 32 9

10 57. If x and y are integers, in which equation must x be negative? (A) xy = -1 (B) xy 2 = -1 (C) x 2 y = -1 (D) x 2 y 2 = -1 (E) x 2 y 3 = x is not even number when x = (A) 330 (B) 436 (C) 752 (D) 861 (E) Zim buys a calculator that is marked 30% off. If he pays $35. What was the original price? (A) $24.50 (B) $45.50 (C) $47.00 (D) $50.00 (E) $ When 11 1 is expressed as a decimal, what digit appears in the 16 th place to the right of the decimal point? (A) 0 (B) 1 (C) 2 (D) 5 (E) Box A contains exactly four cards, numbered 3, 5, 6, and 9. Box B contains exactly three cards, numbered 1, 4, and 7. One card is selected at random from Box A, and one card is selected at random from Box B. What is the probability that the sum of the numbers on the two selected cards is even? (A) 4 (B) 2 (C) 12 5 (D) 1 (E) P is a point that is not on line m. How many lines can be drawn through P that form a 30 angle with line m? (A) 0 (B) 1 (C) 2 (D) 3 (E) The number varies. 63. (a - 1) - (a - 2) + (a - 3)= (A) -a + 4 (B) a + 4 (C) a - 6 (D) a - 2 (E) 3a If 20 lies halfway between 5 and x, then x = (A) 1 3 (B) 5 (C) 7 3 (D) K (E) H 10

11 65. The following are coordinates of points in 12 rem 6 the xy-plane. Which of these points is n nearest the origin? 69. In the division problem, what is the value (A) (-1, -1) of n? (B) (1, 1) (A) 0 (C) (1, -1) (B) 6 (D) (0, -1) (C) 10 (E) (0, 1) (D) 12 (E) N, M, and T are integers. N + M is an odd number. M + T is an odd number. Which of the following must be true? (A) N T is even. (B) N T is odd. (C) N + T is even. (D) N + T is odd. (E) N - T is odd. 70. The weight of a barrel when empty is 3 of the weight of the water it contains when full. If, when full, the barrel and water weigh a total of 20 pounds, what is the weight, in pounds, of the empty barrel? (A) 4 (B) 5 (C) 15 (D) 16 (E) n is an even integer and 10 < n < 19. What is the average of all possible values of n? (A) 14 (B) 14.5 (C) 15 (D) 15.5 (E) The total value of some coins is $1.20. The coins contain nickels, dimes, and quarters only (at least one of each kind). What is the fewest number of coins possible? (A) 5 (B) 6 (C) 7 (D) 8 (E) If M and N are positive integers, then the statement two-ninths of M is greater than three-eighths of N is equivalent to (A) M N > 9 (B) MN > 72 (C) N - M > 1 (D) 16M > 27N (E) 27M > 16N 72. The union of {1, 3, 5} and {3, 4, 6} intersects with {1, 2, 3, 4, 5} = (A) {1, 3, 5} (B) {1, 2, 3, 4, 5} (C) {3, 4, 5} (D) {1, 3, 4, 5} (E) an empty set 11

12 73. Each of the integers from -3 to 5 inclusive 76. Two numbers are in the ratio of 2:3. If the is placed in the diagram, with one number larger number is 45, the smaller number is going into each box. If the sum of the (A) 30 numbers in each row is the same, what is (B) 671 that sum? (C) 15 (D) 221 (E) 12 (A) 0 (B) 3 (C) 5 (D) 6 (E) If the average of 5 consecutive odd numbers is 11, then the largest number is (A) 17 (B) 15 (C) 13 (D) 11 (E) Which of the following is NOT less than.25? (A) (B) (C) (D) 80 (E) A room is 20 feet long, 15 feet wide, and 12 feet high. If it costs 20 per square foot to paint walls, 30 per square foot to paint a ceiling, and 10 per square foot to paint a floor, how much does it cost to paint just the four walls of the room? (A) $48 (B) $84 (C) $96 (D) $168 (E) $ A grocer sells eggs only in boxes of 8. If he starts with 175 eggs and fills as many boxes as he can with eggs, how many eggs will he have left over? (A) 1 (B) 3 (C) 4 (D) 5 (E) If 7 4 of a quantity is 24, then M of the same quantity is (A) 10 (B) 14 (C) 28 (D) 35 (E) 30 12

13 80. Which of the following represents the 83. A is 3 times C. B is 8 less than C. For phrase 5 less than 8 times n? what value of C does A = B? (A) 5 < 8n (A) -4 (B) 5-8n (B) 0 (C) 8n + 5 (C) 1 (D) 8n - 5 (D) 2 5 (E) n - 3 (E) In this figure, AB = AD = DB and ABC is a straight line. What is the measure of DBC? D 84. If 4z - 3 = -19, then z = (A) -16 (B) -51 (C) -4 (D) 0 (E) 4 (A) 60 (B) 90 (C) 100 (D) 120 (E) 150 A B C 82. From a box containing 5 black marbles, 8 pink marbles, 6 white marbles, and 5 yellow marbles, Ingrid removed 4 marbles, one of which was black. If she removes one more from the box at random, what is the probability of getting a black marble? (A) 5 4 (B) 21 (C) 4 5 (D) 21 (E) An experiment requires that 5 2 grams of nitrate be added to a soil bed each day. If 40 grams of nitrate are available, how long may the experiment last? (A) 18 days (B) 40 days (C) 100 days (D) 200 days (E) 400 days 86. The sides of a triangle all have integer lengths. Two sides have lengths 7 and 10. If the largest and smallest possible perimeters for the triangle are L and S, the value of L + S (A) is 20 (B) is 51 (C) is 52 (D) is 54 (E) Cannot be determined from the in formation given. 13

14 87. If x = 2 and y = 3, what is the value of 90. Susie and Dennis are training for a y x x? marathon. On Monday. They both run 3.2 y miles. On Tuesday, Susie runs 54 miles 8 (A) 27 and Dennis runs 3.6 miles. On 4 (B) 9 Wednesday, Susie runs 4.8 miles and 2 (C) B Dennis runs 2 5 miles. During those 3 (D) U days, how many more miles does Susie (E) 1 run than Dennis? (A) 4.8 (B) 4.0 (C) 3.2 (D) 3.0 (E) At 7:00 A.M., the temperature was 12 below zero. Then the temperature rose 3 per hour for 9 hours. What was the temperature at 2:00 P.M.? (A) 21 (B) 9 (C) 6 (D) 3 (E) The angle between the hands of a clock at 10:00 is (A) 15 (B) 30 (C) 45 (D) 60 (E) Today is Saturday. What day of the week will it be 65 days from today? (A) Sunday (B) Monday (C) Tuesday (D) Wednesday (E) Thursday 92. Which fraction has the greatest value? (A) H (B) M (C) U (D) N (E) James buys one halibut steak and two salmon steaks for $30. Dave buys two halibut steaks and four salmon steaks for $60. What is the cost of a halibut steak? (A) $5.00 (B) $8.00 (C) $10.00 (D) It cannot be determined from the information given. 14

15 94. In rectangle ABCD, point E is on side 98. What is the least integer greater than AB. 200? D C (A) 20 (B) 18 4x o (C) 16 (D) 15 (E) 14 A 5x o 6x o E B What is the measure of DEC? (A) 10 (B) 48 (C) 60 (D) 70 (E) (A) 1 (B) 2 (C) 8 (D) 16 (E) 24 = 96. Three consecutive multiples of 4 have a sum of 60. What is the greatest of these numbers? (A) 8 (B) 12 (C) 16 (D) 20 (E) There were 20 socks in a drawer. Of these, 8 were blue, 10 were black, and 2 were white. Someone then removed 4 blue socks. If one sock is now drawn at random, what is the probability that it is not white? (A) / 9 (B) 10 (C) 7 (D) Q (E) none of these 100. Mr. Diaz gets $6 an hour for doing 8 hours work, and 11 times that for overtime. How much does he earn if he works 12 hours? (A) $72 (B) $60 (C) $80 (D) $96 (E) $ Let n be an integer from 9 to 38. For how many values of n will 5 n be a prime? (A) 3 (B) 4 (C) 6 (D) 8 (E) 29 15

16 Answer Key Entrance Exam 1. C 2. A 3. D 4. E 5. D 6. C 7. D 8. B 9. B 10. B 11. C 12. C 13. C 14. B 15. D 16. C 17. B 18. E 19. C 20. C 21. B 22. B 23. E 24. C 25. E 26. C 27. A 28. D N = = = 16R3 29. A Let BP divide the figure into two halves. Since the area of the whole area is (2 2 π) = π Half of it is 8 + π. Now that the area of ΔABP = 1(2)(6) = 6 The area of the shaded region is 8 + π = 6 = 2 + π 30. C 31. E 32. A 33. D 16 2 B A 34. B 40:30 = 4:3 35. E 36. C 37. B 38. B 39. E 40. E 41. C 6 P

17 42. B 43. C Blair & IB Entrance Exam Math Prep Issue E 44. B 6(7x - 14) = 3(4 + 8x) x - 2 = 1 + 2x x = A 46. C The greatest common factor of 12 and 54 is 6 (inches) = = = 18 (squares) 47. A = = C 72 8 = 72 8 = = 6 4 = D 50. D 51. A 52. D 53. B 54. B 55. C 56. B 57. B 58. D 59. D 60. E 61. E 62. C 63. D 64. A 66. C 67. C 12, 14, 16, 18 The average is 1( ) = C 69. C 70. A 71. D 2 9 M > IN Multiply both sides by 72: 16M > 27N 72. B 73. B 74. B 75. C 76. A 77. D 78. E 79. D 80. D 81. D 82. C = 20 (all) 5-1 = 4 (black) 4 = A 84. C 85. C

18 86. D 94. D The third side < = 17. The longest 4x + 5x = 90 possible length is 16. x = 10 The third side > 10-7 = 3. The shortest 5x + 6x = 11x = 110 possible length is 4. CED = = 70 L = = 33. S = = D L + S = E 87. D 97. B 88. B 98. D 89. B 99. D 90. B 100. E 91. D 92. C 93. D 18