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- 1. ACT MATH PREPARATION QUESTIONS … DR……
- 2. Athanasios Drougas http://mathhmagic.blogspot.gr/ 1 ACT MATH PREPARATION QUESTIONS 1.The rectangular prism below has a length of 8,a width of 3, and a height of 4.That means its volume is equal to: Α.69 B.96 C.86 D.106 E.68 2. Andrew and Bob are sharing a pack of gum. There are 9 sticks of gum in the pack,and Andrew takes 3 of them. Since Andrew has 3 of 9 sticks in the pack of gum he has ….of gum. Α.9/3 B.6/9 C.0.3 D.1/3 E.3.333 3. if a=9,b=5,and c=-6 what does (a+b-c)(b+c) equal ? Α.-20 B.-8 C.19 D.20 E.20 4. The sum of two numbers is S. Suppose 3 is added to each number and then each of the resulting number is doubled. What is the sum of the final two numbers? A.2S+3 B.3S+2 C.3S+6 D.2S+6 E.2S+12 5. A bolt making machine makes 60 bolts per minute. A second bolt making machine makes 80 bolts per minute. The second machine starts making bolts 2 minutes after the first machine starts . Both machines stops making bolts 8 minutes after the first machine starts. Together, the 2 machines made how many bolts? Α.480 B.600 C.680 D.720 E.960 6.The first term is -1 in the geometric sequence -1,3,-9,27,….What is the seventh term of geometric sequence Α.-243 B.-30 C.81 D.-729 E.189 8 3 4
- 3. Athanasios Drougas http://mathhmagic.blogspot.gr/ 2 7.Bob ,Alice ,and John shared a pizza .Bob ate 1/3 of the pizza , Alice ate ½ of the pizza and John ate the rest. What is the ratio of John’s share to Alice ‘ s share to Bob ‘s share; Α.6:2:3 Β.3:6:2 C.2:3:6 D.2:6:3 E.3:2:6 8.For triangle FGH ,shown below, which of the following is an expression of x in terms of y? Α.y+3 Β. 2 3y + C. 2 3y − D. 2 9y − E. 2 9y + 9.The inequality 5(x+2) < 6(x-5) is equivalent for which of the following inequalities Α. 30x > Β. 40x > C. 20x > D. 20x > − E. 30x > − 10.A bag contains 10 red marbles,5 yellow marbles ,and 12 blue marbles .How many marbles must be added to the 27 marbles already in the bag so that the probability of randomly drawing a yellow marble is 14 25 ? Α. 22 Β. 23 C. 11 D. 10 E. 45 11.The coordinates of the endpoints of CD ,in the standard (x,y) coordinate plane are (-2,4) and (-6,18) .What is the y-coordinate of the midpoint of CD . Α.10 Β. 11 C. -4 D. -3 E. 9 H GF y cm x cm 3 cm
- 4. Athanasios Drougas http://mathhmagic.blogspot.gr/ 3 12.Peter’s bicycle has a wheel circumference of 96 cm (or 0.96 m). It is a three-speed bicycle with a low, a middle and a high gear. The gear ratios of Peter’s bicycle are: Low 3:1 Middle 6:5 High 1:2 How many pedal turns would Peter take to travel 960 m in middle gear? A gear ratio of 3:1 means 3 complete pedal turns yields 1 complete wheel turn. Α.1200 pedal turns B.1300 pedal turns C. 300 pedal turns D. 9600 pedal turns E. 1000 pedal turns 13.The coordinates of the endpoints of CD ,in the standard (x,y) coordinate plane are (2,14) and (6,-18) .What is the x-coordinate of the midpoint of CD . Α.16 Β.-4 C.4 D.-2 E. -16 14.What is the surface area ,in square inches,of an 10-inch cube? Α.400 Β. 600 C. 4000 D. 6000 E. 800 15. if a=2017,b=2016 what does 3(a-2)+2(a+3)-5(a-2b) equal ? Α.20170 B.20160 C.0 D.20017 E.20016 16.If 2w+z is equal to 50% of 2w ,what is the value of w z ? Α. 1 3 Β. 1 9 C. 1 4 D. 1 5 E. 1 2
- 5. Athanasios Drougas http://mathhmagic.blogspot.gr/ 4 17.What is the surface area ,in square inches, of an 10-inch cube Α.400 Β. 600 C. 4000 D. 6000 E. 800 18.The rectangular prism below has a length of 8,a width of 3, and a height of 4.That means its surface area is equal to: Α.156 B.96 C.120 D.256 E.154 19.Space station Mir remained in orbit for 15 years and circled Earth some 86500 times during its time in space.The longest stay of one cosmonaut in the Mir was around 680 days. Approximately how many times did this cosmonaut fly around Earth? A. 110 B. 1100 C. 11000 D. 110000 E. 1100000 20.The table below shows the temperatures, in degrees Fahrenheit ,in a city in over a one week period .If m represents the median temperature, f represents the temperature that occurs more often, an a represents the average (arithmetic mean) of the seven temperatures, which of the following is the correct order of m,f and a? A. a m f< < B. a f m< < C. m a f< < D. m f a< < E. a m f= < 21.A number is increased by 45% and the resulting number is then decreased by 40%.The final number what percent is of the original number ? Α.85% B. 87% C. 89% D. 92% E.94% 8 3 4 Mon Tue Wed Thu Fri Sat Sun 66 78 75 69 78 77 70
- 6. Athanasios Drougas http://mathhmagic.blogspot.gr/ 5 22.Bob buys three sandwiches for two dollars each. He also buys a few bottles of water .Each bottle of water costs seventy-five cents. Bob pays with fifteen dollars and gets six dollars and seventy-five cents in change. How many bottles of water did he buy? Α.3 B.4 C.2 D.5 E.6 23.We write all integers from 1 to 100.How many times appears 3? Α.9 B.10 C.11 D.12 E.3 24.A bag contains 10 red marbles,5 yellow marbles ,and 12 blue marbles.How many marbles must be added to the 27 marbles already in the bag so that the probability of randomly drawing a yellow marble is 14 25 ? Α. 22 Β. 23 C. 11 D. 10 E. 45 25.The equations below are linear equations of a system ,a,b,c are positive integers ax=c+1 -by ax+bc=c Which of the following describes the graph of at least 1 such system of equations in the standard(x,y) coordinate plane? I. two parallel lines II.two intersecting lines III. a single line Α.I only B.II only C.III only D.I or II only E.I,II or III 26.Which of the following mathematical expressions is equivalent to the verbal expression ”A number x squared is 16 more than the product of 6 and x”? Α.x2 =16+6x B.2x=16+6x C.2x=16x+6x D. x2 =16+x6 E. x2 =16-6x
- 7. Athanasios Drougas http://mathhmagic.blogspot.gr/ 6 27.The determinant of a matrix a b c d equals ad-bc.What must be the value of x for the matrix 1 5 10 x x − to have a determinant of 55? Α. 15 B. 12 C. 10 D. 14 E. 11 28. For 3 consecutive integers ,which of the following statements is false ? Α.Half the sum of the smaller and the larger integer is equal to the middle integer. B. The difference between the third number and the second is equal to the difference between the second number and the first. The average for 3 numbers is the number in the middle. C. At least one of the three integers is odd D. At least one of the three integers is even. E. At least one of the three integers is prime. 29. A function f(x) is defined as f(x)=-256x-3 .What is f(-4)? Α.3 B.-2 C.2 D.-4 E. 4 30.What is the sum of all roots of (2x+3)(x-4)+(2x+3)(x-6)=0? A.7/2 B.4 C.5 D.7 E.13 31.A function f(x) is defined as 10 5 1 ( ) 1 x f x x − = − . 1x ≠ .What is 5 ( 13)f ? Α.14 B.12 C.11 D. 10 E. 13 32. If 2 63x = ,the which of the following must be true? Α. 4 5x< < B. 3 4x< < C. 5 6x< < D. 2 3x< < E. 6 x<
- 8. Athanasios Drougas http://mathhmagic.blogspot.gr/ 7 33.What is the least common multiple of 30,70,130? Α. 140 B. 26000 C. 21000 D. 910 E. 273000 34.A building with 100 flours has 3 elevators. The elevator A stops only every 3th floor , the elevator B stops only every 5th floor ,and the elevator C stops only every 7th floor . Which floor of the building is possible for the three elevators (A,B,C) to meet? Α. 15th floor B. 35th floor C. 21th floor D. 91th floor E. There is no such a floor 35. A rectangle has an area of 40 square feet and a perimeter of 26 feet .What is the shortest of the side lengths ,in feet, of the rectangle? Α.8 B.5 C.2 D.4 E.3 36. The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest .The median of the three numbers is 5.What is their sum? A.5 B.20 C.25 D.30 E.36 37. A rectangle has an area of 12 square feet and a perimeter of 14 feet .What is the length ,in feet, of the diagonal ? Α.5 B.6 C.3 D.4 E.7 38.The expression (4x-7y)(3x+7y) is equivalent to: F. 8x2 -49y2 G. 16x2 +49y2 H. 49y2 -16x2 J. 16x2 -49y2 K. 16x2 -14y2
- 9. Athanasios Drougas http://mathhmagic.blogspot.gr/ 8 39. Discount tickets to a Baseball tournament sell for $ 3.00 each .Bob spent $40.00 on discount tickets , $30.00 less than if he had bought the tickets at the regular price. What was the regular ticket price? Α. $ 5.00 B. $ 4.50 C. $ 5.50 D. $ 10.00 E. $ 23.33 40. In triangle ABC , the sum of the measures of A∠ and B∠ is 57o .What is the measure of C∠ ? Α. 57o B. 67o C. 77o D. 133o E. 123o 41. If A is a two- digit positive integer ,then ( 10) ( 11) ( 12) .... ( 97) ( 98) ( 99) ?A A A x x x− × − × − × × − × − × − = Α. 100 B. -100 C. 10 D. 0 E. cannot be determined from the given information 42.For what value of b would the following system of equations have an infinite number of solutions? -3x+5y=10 9x-15y=2b Α. 10 B. -10 C. 3 D. 15 E. -15 43.A bag contains 10 red marbles,5 yellow marbles ,and 12 blue marbles.How many marbles must be added to the 27 marbles already in the bag so that the probability of randomly drawing a yellow marble is 14 25 ? Α. 22 Β. 23 C. 11 D. 10 E. 45
- 10. Athanasios Drougas http://mathhmagic.blogspot.gr/ 9 44.The determinant of a matrix a b c d equals ad-bc.What must be the value of x for the matrix 14x x x to have a determinant of -49? Α. -7 B. -6 C. 7 D. 49 13 − E. 49 13 − 45.The solution set of which the following equations is the set of all real numbers that are 6 units from -4? Α. 4 1 6x − = B. 6 4x − = C. 6 4x + = D. 4 6x + = E. 4 6x − = 46.The rectangular prism below has a length of 8,a width of 3, and a height of 4.That means,its surface area not visible from your point of view is equal to: Α.120 B.60 C.72 D36 E.196 47.A function f is defined as follows : For x>0, 4 3 2 ( ) 23 23f x x x x= + − − For x <0, 4 3 2 ( ) 23 23f x x x x= − + + − What is the value of f(-1) Α. 46 B. -46 C. 0 D. -1 E. -2 8 3 4
- 11. Athanasios Drougas http://mathhmagic.blogspot.gr/ 10 48.The number line graph below is the graph of which of the following inequalities? Α. 2 x− ≤ and 5 x≤ B. 2 x− ≤ and 5 x≥ C. 2 x− ≤ or 5 x≤ D. 2 x− ≥ or 5 x≤ E. 2 x− ≥ or 5 x≥ 49. If two sides of a triangle have lengths 7 and 8 , the perimeter of the triangle could be which of the following? A.11 B.15 C.24 D.31 E.30 50. The solution set of which the following equations is the set of all real numbers that are less than 3 units from 7? Α. 3 7x − > B. 3 7x + > C. 3 7x + < D. 3 7x − < E. 7 3x − < 51.What is the sum of the first 5 terms of the arithmetic sequence in which the 7th term is 10 and the 8th term is 12? Α.14 B. 16 C. 20 D. 18 E. 22 -2 5
- 12. Athanasios Drougas http://mathhmagic.blogspot.gr/ 11 52.Triangles ABC∆ and PQR∆ are shown below. The given side lengths are in centimeters. The area of ABC∆ is 40 square centimeters. What is the area of PQR∆ ? Α.20 B.10 C.40 D.30 E. 60 53. In a poll 33 people were in favor pf voting John Doe ,15 people were against him, and two persons had no opinion. What percent of those polled were in favor of voting John Doe? Α. 33 100 B. 66 100 C. 30 100 D. 12 100 E. 70 100 54.Which of the following mathematical expressions is equivalent to the verbal expression ”A number x plus his squared product with 25 is 60 less than the product of 343 and twice x”? Α.x+625x2 =686x-60 B. x+625x2 =686x+60 C. x-25x2 =343x+60 D. x+25x2 =343x-60 E. x+25x2 =686x-60 1200 o R P Q y x B A C x y 60o
- 13. Athanasios Drougas http://mathhmagic.blogspot.gr/ 12 55.For the population that grows at a constant rate of r% per year ,the formula ( ) 1 100 t r P t Po= + models the population t years after an initial population of Po people is counted. The population of city of San Antonio was 1,151,979 in 2010.Assume the population grows at a constant rate of 5% per year .According to this formula ,which of the following is an expression for the population of San Antonio in the year 2016? Α. 1,151,979 (6)6 B. 1,151,979 (1.5)6 C. 1,151,979 (1.05)6 D. (1,151,979 x 1.5)6 E. (1,151,979 x 1.05)6 56.Triangle ABC∆ is shown in the figure below .The measure of A∠ is 40o ,AB=18cm and AC=12 cm. Which of the following is the length , in centimeters ,of BC ? (Note: For a triangle with sides of length a,b and c opposite angles A∠ , B∠ , C∠ , respectively the law of sines states sin sin sinA B C a b c ∠ ∠ ∠ = = and the law of cosines states 2 2 2 2 cosc a b ab C= + − ∠ ) Α. 0 18cos70 B. 0 12sin 70 C. 2 2 12 18+ D. 2 2 18 12+ E. 2 2 18 12 2(12)(18)sin 70o + − 57.In the equation 2 2 0x mx n+ + = , m,n are integers .The only possible value for x is -2.What is the value of m? Α. 2 B. -2 C. 4 D. -4 E. 1 A C B 12 cm 18 cm 70o
- 14. Athanasios Drougas http://mathhmagic.blogspot.gr/ 13 58.In the equation 2 2 0x mx+ + = , 0m ≠ .If a,b are the roots of the equation .The expression 2 2 3a b abm m + + is equal to: Α.6 B.4 C.-2 D.2 E. 3 59. The length of a rectangle is 5 times the length of a smaller rectangle .The two rectangles have the same width .The area of the smaller rectangle is A square units .The area of the larger rectangle is kA square units .Which of the following is the value of k? Α. 1 5 B. 1 25 C. 5 D.25 E.1 60.The dimensions of the right triangle shown below are given in feet( k positive integer). What is sinθ? Α. 5 13 B. 12 13 C. 5 12 D. 12 5 E. 13 12 θ 5κ 13κ 12κ
- 15. Athanasios Drougas http://mathhmagic.blogspot.gr/ 14 61.The hair color of 190 people were determined for a study as shown in the figure below. If one person from the study is randomly selected, what is the probability that this person has either red hair or blond hair. Α. 89 190 B. 12 190 C. 112 190 D. 25 190 E. 77 190 62. Given that a=bc5 ,a=2017,c=10,what is b? Α. 2.017 B. 0.2017 C. 0.02017 D. 20.17 E. 4 201.7 10x − 63. If f(x)=(5x-6)2 then f(f(1))=? Α.-1 B.1 C. 0 D. 10 E. 100 55 33 77 20 10 30 40 50 60 70 80 black brow n red blond Hair color Number of people
- 16. Athanasios Drougas http://mathhmagic.blogspot.gr/ 15 64. If f(x)=(5x-6)2 then f(f(1))=? Α.-1 B.1 C. 0 D. 10 E. 100 65.Bob the scientist studying motion of his parrot Jim ,so Bob observed Jim flying at a constant rate along a straight line .The table below gives the distance ,d feet, Jim the parrot was from a reference point at 2-second intervals from t=0 second to t=6 seconds . Which of the following equations represents this relationship between d and t? Α. 4t+29 B. t+25 C. 4t+25 D. t-25 E. 5t+24 66.In scientific notation 2350,000,000-450,000,000=? Α.1.9 x 10-9 B. 1.9 x 107 C. 1.9 x 108 D. 1.9 x 1015 E. 1.9 x 109 67. If f(x)=(6x-16)3 then f(1)=? Α.10 B.100 C. 1 D. 1000 E. 10000 68.In scientific notation 6250,000,000/250=? Α.2.5 x 10-7 B. 2.5 x 107 C. 2.5 x 108 D. 2.5 x 10-9 E. 2.5 x 10-8 69.We write all integers from 1 to 100.How many integers of them have at least a 5? Α.9 B.10 C.11 D.12 E.8 t d 0 25 2 33 4 41 6 49
- 17. Athanasios Drougas http://mathhmagic.blogspot.gr/ 16 70.In triangle ABC , the measure of A∠ is 90o and the measure of B∠ is twice the measure of C∠ .What is the measure of C∠ ? Α. 60o B. 30o C. 45o D. 120o E. 90o 71.In which of the following are 2 6 3 , , 3 7 2 , 6 11 and arranged in ascending order ? Α. 2 6 6 3 3 7 11 2 < < < B. 2 6 6 3 3 11 7 2 < < < C. 3 6 6 2 2 11 7 3 < < < D. 3 2 6 6 2 3 11 7 < < < E. 6 6 3 2 11 7 2 3 < < < 72. 3 2 2 3 3 2 6a ab a b⋅ ⋅ is equivalent to: Α. 6 5 11a b B. 5 6 12a b C. 6 5 12a b D. 5 5 11a b E. 6 6 11a b 73. If 3( 4) 4 11 12x x x− + = − , the x=? Α. 1 B. 3 C. every real number D. 0 E. 2 74. If 20 2a t= − then 5 2 ? 2 a t+ = Α. 100 B. 1000 C. 10000 D. 500 E. 5000 75.Which of the following numbers is prime? Α. 28713 B. 25726 C.15675 D.78250 E. 2244
- 18. Athanasios Drougas http://mathhmagic.blogspot.gr/ 17 76.For all pairs of positive integers a,b, which of the following numbers is even? Α. ab B. 2 2 a a b b+ + + C. 2 2 2a ab b+ + D. 2 2 a b ab+ E. a b− 77.What is the y-coordinate of the point in the standard (x,y) coordinate plane at which the three lines y-3=0,y=25x+25, 50 0 150 y x + − = ? Α. 4 B. 5 C. 50 D. 0 E. 3 78.If x and y are positive numbers such that the greatest common factor of 4 x y and 2 3 x y is 147,which of the following is a possible value for x? Α. 147 B. 21 C. 7 D. 3 E. 9 79.The sum of two numbers is 12,and the product of the same two numbers is 4.The sum of the reciprocals of the two numbers is equal to: Α. 4 2 B. 23 C. 4 D. 3 E. 2 (Reciprocal of number x( 0x ≠ ) is the 1 x . ) 80.If 2 19a b ab a∇ = + + for all nonnegative numbers ,what is the value of 6 10∇ ? Α. 25 B. 35 C. 45 D. 615 E. 635 81.A cat chases a mouse , who has a 60–m head start. For every 7 m the mouse runs ,the cat runs 9 m. How far must the cat run to catch the mouse? Α. 72 B. 73 C. 74 D. 84 E. 124
- 19. Athanasios Drougas http://mathhmagic.blogspot.gr/ 18 82.If 7 hats cost a dollars and b hats would cost how many dollars? Α. 7a b B. 7ab C. 7 ab D. 7 a b E. 7 ab 83.For x such that 2 x π π < < , the expression 2 2 1 sin 1 cos cos sin x x x x − − + is equivalent to: Α. 0 B. 1 C. 2 D. 1 tan x E. tan x 84.For x such that 0 2 x π < < , the expression 2 2 1 sin 1 cos cos sin x x x x − − − is equivalent to: Α. 0 B. 1 C. 2 D. 1 tan x E. tan x 85.For x such that 0 2 x π < < , the expression ( ) ( ) 2 2 sin cos sin cos 2 x x x x− + + is equivalent to: Α. 2 B. 0 C. 1 D. cos x E. cos x−
- 20. Athanasios Drougas http://mathhmagic.blogspot.gr/ 19 86.Consider the functions ( )f x x= and ( ) 2g x x b= + .In the standard (x,y) coordinate plane , ( ( ))y f g x= passes through (3,5).What is the value of b? Α.19 B. 20 C. 21 D. 22 E. -22 87.What is the real value of x in the equation 2 2 3log 96 log 3 log x− = Α.32 B.27 C.9 D.125 E.243 88.What is the real value of x in the equation 2 2 15 2 log 5 2log 3 log log 45 x + = Α.5 B.45 C.3 D.15 E.9 89.What is the average of the following numbers 1+w, 2-2w, 3+4w,16, 3+3w , 7-6w Α.8+2w B.4+2w C.4 D.8 E. 32 90.What are the real solutions to the equation 2 4 5 0?x x+ − = Α. 5± B. 1± C.1 and 5 D. -1 and -5 E. 5± and 1± 91.What are the real solutions to the equation 2 9 6 1 0?x x− + = Α. 3± B. 1 3 ± C.1 and 1 3 D. -1 and 1 3 − E. 3± and 1 3 ±
- 21. Athanasios Drougas http://mathhmagic.blogspot.gr/ 20 92.The parallel sides of the isosceles trapezoid shown below are 10 feet long and 16 feet long, respectively. What is the distance, in feet , between these 2 sides? Α.3 B.4 C.5 D.10 E.16 93.Normally, a penguin couple produces two eggs every year. Usually the chick from the larger of the two eggs is the only one that survives. With rockhopper penguins, the first egg weighs approximately 78 g and the second egg weighs approximately 110 g. By approximately how many percent is the second egg heavier than the first egg? A. 29% B. 32% C. 41% D. 71% D. 61% 94.The parallel sides of the isosceles trapezoid shown below are 10 feet long and 16 feet long, respectively. What is the distance, in feet, between these 2 sides? Α.13 B.14 C.12 D.20 E.16 95.The parallel sides of the isosceles trapezoid shown below are 10 feet long and 16 feet long, respectively. What is the area of triangle ABE, in square feet? Α.240 B.120 C.260 D.160 E.100 5 ft 5 ft 16 ft 10 ft 13 ft 13 ft 20 ft 10 ft 13 ft 13 ft 20 ft 10 ftA B C D E
- 22. Athanasios Drougas http://mathhmagic.blogspot.gr/ 21 96.For all x in the domain of the function 3 2 1 4 x x x − − , this function is equivalent to: Α. 2 1 2x x− B. 3 2 1 2x x− C. 2 2 1 4 1 4 1x x − − − D. 2 1 1 2x x + E. 2 1 2x x+ 97.The number line graph below is the graph of which of the following inequalities? Α. 21 x≤ and 10 x≤ B. 21 x≥ and 10 x≤ C. 21 x≥ or 10 x≤ D. 21 x≥ or 10 x≤ E. 21 x≥ or 10 x≥ 98. For the pentagram ABCD shown below , ?A B C D F∠ + ∠ + ∠ + ∠ + ∠ = Α.360o B.180o C.270o D.540o E.120o A B C D E 10 21
- 23. Athanasios Drougas http://mathhmagic.blogspot.gr/ 22 99. 3 3 3 3 321 2 13 13 7 3 a a b a b⋅ ⋅ is equivalent to: Α. 27 9 2a b B. 27 9 3a b C. 9 6 2a b D. 9 6 3a b E. 9 9 2a b 100.In the diagram below, the congruent circles are tangent to the larger square and each other as shown and their centers are vertices of the smaller square. The area of the smaller square is 4.The area of the larger square is equal to: Α.8 B.16 C.8π D.16π E.4π 101.Jack rode his bike from home to the river, which is 4 km away. It took him 9 minutes. He rode home using a shorter route of 3 km. This only took him 6 minutes. Jack's average speed, in km/h, for the trip to the river and back is equal to: A. 60 km/h B. 50 km/h C. 40 km/h D. 30 km/h E. 46.6 km/h 102. Given the six numbers 1916, 2817,3718,4619,5520,6421 The percent of their average to their sum is equal to: Α.20% B.25% C.30% D.35% E.15% 103. If a=1-i then (a(1+i)-1)20 =? Α.0 B.1 C. -1 D.i E.-i
- 24. Athanasios Drougas http://mathhmagic.blogspot.gr/ 23 B C E A 104. If 6 4x + < which of the following statements are true? I. 2x < − II. 2x > − III. 10x > − Α.I only B.II only C.I and II only D.II and III only E. I and III only 105. If 23 3x − < − which of the following statements are true? I. 20x < II. 23x > III. 26x > − Α.I only B.II only C.I and II only D.II and III only E. none the three statements is true 106.In the figure below, EC=10 centimeters AB=5 centimeters and the inscribed circle has a radii of 2 centimeters then AC is equal to: Α.12 B.13 C.14 D.10 E.9 107.Which of the following is true for all consecutive integers m and n such m <n? Α. m is odd B. n is odd C. n-m is even D. n2 -m2 is odd E. n2 +m2 is even 108.Two numbers are reciprocals if their product is equal to 1.If x and y are reciprocals and x>1 ,the y must be: Α. less than -1 B. between 0 and -1 C. between 0 and 1 D. equal to 0 E. greater than 1
- 25. Athanasios Drougas http://mathhmagic.blogspot.gr/ 24 109.Which of the following is true for all consecutive integers m and n such m <n? Α. m is odd B. n is odd C. mn is even D. mn+1 is even E. n2 +1 is odd 110.Mount Fuji is only open to the public for climbing from 1 July to 27 August each year. About 200,000 people climb Mount Fuji during this time. On average, about how many people climb Mount Fuji each day? A. 340 B. 710 C. 3,400 D. 7,100 E. 7,400 111.The Gotemba walking trail up Mount Fuji is about 9 km long. Walkers need to return from the 18km walk by 8pm. Toshi estimates that he can walk up the mountain at 1.5km/h on average, and down at twice that speed. These speeds take into account meal breaks and rest times. Using Toshi's estimated speeds, what is the latest time he can begin his walk so that he can return by 8pm? A. 8:30 am B. 9:30 am C. 9:00 am D.10:00 am E.11:00 am 112.Toshi wore a pedometer to count his steps on his walk along the Gotemba trail. His pedometer showed that he walked 22,500 steps on the way up. Estimate Toshi's average step length for his walk up the 9km Gotemba trail. Give your answer in centimeters. A. 30 cm B. 40 cm C. 25 cm D. 35 cm E. 45 cm 113.If 3 3 3 log log log b a b a b a = + ⊙ for all nonnegative numbers ,what is the value of 1 2017⊙ ? Α. 1 B. 0 C. 2017 D.2018 E. 3
- 26. Athanasios Drougas http://mathhmagic.blogspot.gr/ 25 114.A revolving door includes three wings which rotate within a circular-shaped space. The inside diameter of this space is 2 meters (200 centimeters). The three door wings divide the space into three equal sectors. The plan below shows the door wings in three different positions viewed from the top. The door makes 4 complete rotations in a minute. There is room for a maximum of two people in each of the three door sectors. What is the maximum number of people that can enter the building through the door in 30 minutes? A. 60 B. 180 C. 300 D. 240 E. 720 115.What is the slope of the line through (-2,4) and (-3,9) in the standard(x,y) coordinate plane? A.5 B.-5 C.6 D. 7 5 E. 7 5 − 116.The figure below consists of congruent squares and has total area 100 square feet. What is the outside perimeter of the figure? A.80 B.40 C.50 D.20 E.60
- 27. Athanasios Drougas http://mathhmagic.blogspot.gr/ 26 117.According to the measurements given in the figure below, which of the following expressions give the distance, in miles from the dog to the cat. A.30cos52o B.30tan52o C.30sin52o D. 0 30 sin52 E. 0 30 cos52 118.The sum of the real numbers x and y is 36.Their difference is 5.What is the value of xy? A.3 B.6 C.9 D.36 E.41 119.Each side of the smaller square in the figure below is x inches long, and each side of the larger square is y inches longer than a side of the smaller square. The area of the larger square is how many square inches greater than the area of the smaller square? A. y2 B. xy C. 4y(x + y)2 D. 2xy + y2 E. 2xy+ x2 x x y y 30 miles 52o cat dog Dog house
- 28. Athanasios Drougas http://mathhmagic.blogspot.gr/ 27 120.On the number line above ,there are 9 equal intervals between 0 and 1.What is the value of x? A. 1 9 B. 1 81 C. 1 27 D. 1 3 E. 1 121.A cube with edges 1 2 inch long is shown below. What is the length, in inches, of a diagonal that runs from one corner of the cube to the opposite corner? A. 1 4 B. 3 2 C. 3 4 D. 2 2 E. 3 2 1 2 inch 1 2 inch 1 2 inch 0 1 x
- 29. Athanasios Drougas http://mathhmagic.blogspot.gr/ 28 122.In the 8–sided figure below, adjacent sizes meet at right angles and the lengths given are in meters.What is the perimeter of the figure, in meters? A.90 B.120 C.180 D.200 E. 400 123. When 1 1 1 5 3 a a+ = ,what is the value of k? A. 1 15 B. 8 15 C. 15 8 D. 8 E. 15 124.Hexagon ABCDFG shown below was drown on a grid with unit squares.Each vertex is at intersection of 2 grid lines .What is the area of hexagon? A. 62 B. 63 C.64 D.65 E.66 40 38 12
- 30. Athanasios Drougas http://mathhmagic.blogspot.gr/ 29 125.In the figure above, what is the median of the slopes of OA,OB ,OC ,OD and OE ? Α. 1 4 B. 1 2 C.1 D. 2 E. 2 4 126. The hair color of 180 people were determined for a study as shown in the figure below. If one person from the study is randomly selected, what is the probability that this person has either red or blond or brow hair. Α. 77 180 B. 80 180 C. 103 180 D. 23 180 E. 134 180 0 B C D EA 2 1 2 3 4 5 1 6 7 8 3 4 5 6 7 x y 77 20 10 30 40 50 60 70 80 black brow n red blond Hair color Number of people
- 31. Athanasios Drougas http://mathhmagic.blogspot.gr/ 30 127.The median of a set of 11 consecutive integers 23.What is the greatest of these 11 integers? Α.26 B.27 C.28 D.29 E.30 128.Let the function f defined by f(x)=x-1.If 3f(k)=3,what is the value of f(5k)? Α.2 B.3 C.4 D.5 E.6 129.In the figure above, what is the median of the slopes of GA,GB ,GC ,GD and GE ? Α.5 B.4 C.3 D.2 E.1 130.Two identical jars A and B are filled with equal numbers of marbles .The marbles are colored black or white. The ratio of red to white marbles is 7:1 in Jar A and 9:1 in Jar B. If there are 90 white marbles all together, how many black marbles are in Jar B? F. 180 G. 300 H. 360 J. 400 K. 420 G 2 1 2 3 4 5 1 6 7 8 3 4 5 6 7 x y D C B A E 0
- 32. Athanasios Drougas http://mathhmagic.blogspot.gr/ 31 131.A cube is illustrated in the diagram below. In this cube what is the measure of ABC∠ ? F. 90o G. 60o H. 30o J. 120o K. 45o 132.If A is the set of all odd integers ,B is the set of positive integers and C is the set of all integers more than 8 which of the following integer will be in all three sets? F. -10 G. 10 H. 7 J. 12 K. 9 133.If 7 19w+ = then ?w = F. 9 G. 90 H. 100 J. 81 K. -90 134.Given that the number 1948a8 is divisible by 18,what are the possible values for a? F.3 G.4 H.5 J.6 K.7 135.The range of values for a in the triangle shown below is equal to: Α. 11a < B. 6 11a< < C.5 11a< < D. 5 12a< < E. 6 12a< < B A C 3 8 α
- 33. Athanasios Drougas http://mathhmagic.blogspot.gr/ 32 136. 17 17 17 17 2 2 2 2 ?+ + + = F. 18 2 G. 19 2 H. 20 2 J. 21 2 K. 1028 137. OEC is a quarter circle. ABCO is a rectangle with sides AB = 4 and OA = 3. What is the length of the arc EBC ? A. 5π B. 10π C. 2.5π D. 16 E. 14 138. The smallest positive integer value for x which 6 1x − yields an positive integer is equal to: A.0 B.1 C. 2 D.4 E. 6 139. In the figure below the ratio a:b is 1:3.What is the diagonal in terms of b? A. 12b B. 10 3 b C. 10b D. 2 3 b E. 3 b O A E D C B α b
- 34. Athanasios Drougas http://mathhmagic.blogspot.gr/ 33 140. In the figure below ,what is the ratio of ABC∠ to DBC∠ ? A.2:7 B.2:6 C.7:2 D.6:7 E. 3:7 141. In the figure below ,what is value of x? A.110o B.100o C.30o D.40o E.50o 142.If 3 3 ( ) 5 5f x x x= − − ,what is the value of 3 3 ( 1) 5 5f x x− = − − A.10 B.0 C.-10 D.25 E.-25 143.If 2 2 5x y+ = and ( ) 2 9y x− = what is the value of xy ? A.1 B.2 C.-2 D.3 E.4 3α+20ο 2α-40ο Α Β C D 70o x0 A C B
- 35. Athanasios Drougas http://mathhmagic.blogspot.gr/ 34 144.If 2 1 25a a + = what is the value of 2 2 1 a a + ? A. 5 B.27 C. 23 D. 7 E. 3 145.If 1 ( ) 1 f x x x = + what is 1 f x ? A. 1 B. ( )f x C. ( )f x− D. 1 ( )f x E. 1 ( )f x − 146.What is the range of the function 2 ( ) 9f x x= − A. All real numbers B. All real numbers greater than or equal to 0. C. All real numbers greater than or equal to 3. D.All real numbers greater than or equal to 3 or less than or equal to -3.. E. All real numbers between -3 and 3 ,inclusive. 147.What is the domain of the function 2 ( ) 16f x x= − − A. All real numbers B. The empty set. C. All real numbers greater than or equal to 4. D. All real numbers greater than or equal to 4 or less than or equal to -4. E. All real numbers between -4 and 4 ,inclusive. 148.What is the domain of the function 4 3 ( 4) ( ) ( 4) x f x x − = − ? A. All real numbers B. { : 4 4}x x− < < C. { : 4}x x ≠ D. { : 4}x x ≠ − E. { : 4 4}x x or x< − >
- 36. Athanasios Drougas http://mathhmagic.blogspot.gr/ 35 149.The figure below shows the graph of an equation y=αx2 +bx+c .Which of the answer choices CANNOT be true? A. a b> B. a c< C. a b= D. b c< E. a c> 150. Which of the following function has a range of ( ) 2f x ≥ A. ( ) 2f x x= + B. ( ) 2f x x= − C. ( ) 2 2f x x= − + D. ( ) 2 2f x x= − − E. ( ) 2 2f x x= + − 151. Of the 133 employees of a factory ,9 are women. What is the ratio of workers who are women to those who are men? A. 1:11 B. 1:12 C. 1:13 D. 1:14 E. 1:15 y x
- 37. Athanasios Drougas http://mathhmagic.blogspot.gr/ 36 152.In the figure above ,each of the four angles in the parallelogram is as shown.What is the value of c? A. 40o B. 60o C. 120o D. 80o E. 50o Solution 153.For the real numbers x and y ,when is the equation 0x y x y+ + − = true? A. only x=0 B. only y=0 C. always D. x=0 or y=0 E. x=0 and y=0 0 2 100a − α Α Β C D α-20ο b c
- 38. Athanasios Drougas http://mathhmagic.blogspot.gr/ 37 154.As shown below, rectangle ABCD is divided into 1 large square (labeled K) x inches on a side , 10 rectangles (Labeled R) each x inches by y inches and 6 small squares(labeled M ) each side 2y inches .What is the total area, of ABCD? A.x2 +10xy+4y2 B. x2 +10xy+6y2 C. 2x+10xy+6y2 D. x2 +10xy+24y2 E. 2x+10xy+6y2 155.What are the Quadrants of the standard (x,y) coordinate plane below that contain points on the graph of the equation 6x+3y=3 A.I and III only B. I and II and III only C. II and III only D. II and III and IV only E. I and II and IV only 156.For 2 1i = − ,( ) 2 1 ?i− = A. 0 B. 2 C. 2i D. 2i− E. i− K E E E E E E B CD A M M M M M M E E E E III III IV Quadrants of the standard (x,y) coordinate plane
- 39. Athanasios Drougas http://mathhmagic.blogspot.gr/ 38 157. The graph of y=-2x2 +3 passes through (2, ) 5 a in the standard (x,y) coordinate plane . What is the value of a? A.5 B.-5 C.25 D.-25 E.-20 158. If 22x =44x-3 ,what is the value of x? A.2 B.1 C.3 D.4 E.-2 159. What is the average (arithmetic mean) of the numbers 3 2 , 2 3 ,and 5 6 A. 1 6 B.1 C. 8 3 D. 5 6 E.2 160. If (x-α)(x-b)=x2 -7x-13,what is the value of a b ab + ? A. 8 13 B. 13 7 C. 7 13 D. 1 E. it cannot be determined from the given information
- 40. Athanasios Drougas http://mathhmagic.blogspot.gr/ 39 161. If (x-α)(x-b)=x2 -7x-13,what is the value of a b ab + ? A. 8 13 B. 13 7 C. 7 13 D. 1 E. it cannot be determined from the given information 162.Ann and Bob pick two digits numbers .If you multiply Ann ‘s number with 4 and subtract Bob’s number the result is 100.If you multiply Bob ‘s number with 6 then add Ann‘s number the sum is 200.What is the sum of their two numbers ? A.50 B.100 C.150 D.200 E.250 163.In the figure below ,Square ABCD is inscribed with circle P. If the area of the circle is 12π,what is the area of the square? A. 2 3 B.24 C.48 D. 24 3 E. 16 3 164.In the figure below the rectangle ABCD consists of two congruent squares(EBCF,AEFD) .If diagonal BD= 5 .The diagonal BF is equal to: A. 3 B. 2 C.4 D. 1 E. 5 2 D C BA .P F E D C BA
- 41. Athanasios Drougas http://mathhmagic.blogspot.gr/ 40 165.What is the fraction of the square is shaded? A. 1 3 B. 2 5 C. 5 12 D. 3 7 E. 1 2 166.Figures A1,A2 ,A3 are squares.The perimeter of A3 is 8 and the perimeter of A2 is 20.The perimeter of A1 is : A.25 B.28 C.32 D.24 E.26 167.If X is 50 percent of 4Y, then Y what percent of X? A.50% B.75% C.100% D.200% E.250% 168.For a linear function f, f(0)=2 and f(2)=-4 .If k=f(6) what is the value of k? A.-13 B.-14 C.-15 D.-16 E.-17 A1 A2 A3
- 42. Athanasios Drougas http://mathhmagic.blogspot.gr/ 41 169.If 10% of 10% of a certain number is 2, this certain number is: A.20 B.100 C.120 D.200 E.300 170.Bob can eat 20 sandwiches every 3 days. How many sandwiches can Bob eat in 12 days ? A.160 B.120 C.80 D.60 E. 40 170.The average of eleven 11’s is : A.121 B.11 C.111 D.1 E.131 171.If f(2x)= 4x+1,what is the value of α for which 11 2 a f = − ? A.0 B.-1 C.-2 D.-3 E.-4 178.If log 1000 log a b = the numerical value of log log a b b is: A.900 B.899 C.999 D.1999 E.1000
- 43. Athanasios Drougas http://mathhmagic.blogspot.gr/ 42 179.If ( ) 2 1x yi i+ = + ,where x,y are real , what is the value of x+y A.x=1 and y=2 B. x=1 and y=-2 C. x=0 and y=-2 D. x=0 and y=2 E. x=1 and y=0 180.If a and b , are the roots of 2 2 5 4 0x x− + = , what is the value of 2 2log loga b+ ? A.0 B.1 C.2 D.3 E.4 180.For positive integers a,b,and c let a b c be defined by 2 2 a b abc b c c = − − .What is the value of 2 3 4 ? A.0 B.1 C. 2 D. 3 E.4 181. For positive integers a,b, let a#b be defined by 2 a ##b (2 )(2 )a b a b b= − + + .What is the value of4##2017? A.0 B. 4 C.8 D.12 E.16
- 44. Athanasios Drougas http://mathhmagic.blogspot.gr/ 43 182.If 7 10 0.20 is expressed as a fraction in lowest terms ,what number will be in the denominator? A.1 B.2 C.4 D.5 E.10 183.If 180 270o o θ< < and tanθ=0.75 what is the value of sinθ+cosθ? A.3 B. -3 C. 1.4 D.-1.4 E.0.71 184.What is the value of 12 x− if 15x = ? A.-3 B. 27 C. 3 D.-27 E.-9 185. Which of the following is equal to 15 3 ? A. 5 3 B. 3 5 C. 9 5 D. 3 3 E. 5 186. For the integers a and b such that ab=10, which of the following is NOT possible value of x? A. 5− B. 4− C. 2− D. 5 E. 4 187. How many pairs of prime numbers are there whose sum is 825? A. 0 B. 1 C. 2 D. 3 E. 4

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