1- Choice B is correct
The correct answer is 180^\circ The sum of all angles in a quadrilateral is 360 degrees. Let x be the smallest angle in the quadrilateral. Then the angles are: x, 2 \ x, 3 \ x, 6 \ x x \ + \ 2 \ x \ + \ 3 \ x \ + \ 6 \ x=360→12 \ x=360→x=30 The angles in the quadrilateral are: 30^\circ, 60^\circ, 90^\circ, and 180^\circ
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2- Choice B is correct
The correct answer is $1,458 Use simple interest formula: I=prt (I = interest, p = principal, r = rate, t = time) I=(18000) \ (0.027) \ (3)=1458
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3- Choice A is correct
The correct answer is \frac{6 \ x \ + \ 3}{2 \ x^2 \ + \ 2 \ x} (\frac{f}{g})(x) = \frac{f(x)}{g(x)}=\frac{6 \ x \ + \ 3}{2 \ x^2 \ + \ 2\ x}
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4- Choice C is correct
The correct answer is \frac{6 \ \sqrt{π}}{π} Formula for the area of a circle is: A =π \ r^2 Using 36 for the area of the circle we have: 36=π \ r^2 Let’s solve for the radius (r). \frac{36}{π}=r^2→r=\sqrt{\frac{36}{π}}=\frac{6}{\sqrt{π}}=\frac{6}{\sqrt{π}} \ × \ \frac{\sqrt{π}}{\sqrt{π}}=\frac{6\ \sqrt{π}}{π}
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5- Choice A is correct
The correct answer is 12,000 Number of visiting fans: \frac{4 \ × \ 30000}{10}=12,000
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6- Choice C is correct
The correct answer is y=8 \ x \ − \ 22 The equation of a line is: y=m \ x \ + \ b, where m is the slope and is the y-intercept. First find the slope: m=\frac{y_{2} \ - \ y_{1}}{x_{2} \ - \ x_{1}}=\frac{18 \ -\ (- \ 6)}{5 \ - \ 2}=\frac{24}{3}=8 Then, we have: y=8 \ x \ + \ b Choose one point and plug in the values of x and y in the equation to solve for b. Let’s choose the point (2, - \ 6) y=8 \ x \ + \ b→- \ 6=8 \ (2) \ + \ b→- \ 6=16 \ + \ b→b=- \ 22 The equation of the line is: y=8 \ x \ - \ 22
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7- Choice D is correct
The correct answer is 120 degree The angle x and 35 are complementary angles. Therefore: x \ + \ 60=180 180^\circ \ - \ 60^\circ=120^\circ
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8- Choice B is correct
The correct answer is \frac{\sqrt{20}}{4} sin A=\frac{6}{4}⇒ Since sin θ=\frac{opposite}{hypotenuse}, we have the following right triangle. Then: c=\sqrt{6^2 \ - \ 4^2 }=\sqrt{36 \ - \ 16}=\sqrt{20} cos =\frac{\sqrt{20}}{4}
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9- Choice E is correct
The correct answer is 90 Length of the rectangle is: \frac{3}{2} \ × \ 18=27 perimeter of rectangle is: 2 \ × \ (27 \ + \ 18)=90
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10- Choice B is correct
The correct answer is 40 First, find the number. Let x be the number. Write the equation and solve for x. 120\% of a number is 60, then: 1.2 \ × \ x=60 ⇒ x=60 \ ÷ \ 1.2=50 80\% of 50 is: 0.9 \ × \ 50=40
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11- Choice A is correct
The correct answer is 96,000 Three times of 32,000 is 128,000. One sixth of them cancelled their tickets. One sixth of 128,000 equals 32,000 \ (\frac{1}{4} \ × \ 128000=32000). 96,000 \ (128000 \ – \ 32000=96000) fans are attending this week
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12- Choice D is correct
The correct answer is 512 cm^3 If the length of the box is 32, then the width of the box is one Fourth of it, 8, and the height of the box is 2 (one Fourth of the width). The volume of the box is: V = (length) × (wdth) × (height) =(32) \ × \ (8) \ × \ (2)=512 cm^3
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13- Choice D is correct
The correct answer is \frac{5}{13} tan=\frac{opposite}{adjacet}, and tanx=\frac{5}{12}, therefore, the opposite side of the angle x is 5 and the adjacent side is 12. Let’s draw the triangle. Using Pythagorean theorem, we have: a^2 \ + \ b^2=c^2→5^2 \ + \ 12^2=c^2→25 \ + \ 144=c^2→c=13 sin x=\frac{opposite}{hypotenuse}=\frac{5}{13}
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14- Choice C is correct
The correct answer is 8.6 \ × \ 10^5 860000=8.6 \ × \ 10^5
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15- Choice A is correct
The correct answer is 1 Plug in the value of each option in the inequality. A. 1 \ \ \ (1 \ - \ 3)^2 \ + \ 2 \ > \ 4 \ (1) \ - \ 5→6 \ > - \ 1 Bingo! B. 6 \ \ \ (6 \ - \ 3)^2 \ + \ 2 \ > \ 4 \ (6) \ - \ 5→11 \ > \ 19 No! C. 8 \ \ \ (8 \ - \ 3)^2 \ + \ 2 \ > \ 4 \ (8) \ - \ 5→27 \ > \ 27 No! D. 3 \ \ \ (3 \ - \ 3)^2 \ + \ 2 \ > \ 4 \ (3) \ - \ 5→2 \ > \ 7 No! E. 4 \ \ \ (4 \ - \ 3)^2 \ + \ 3 \ > \ 4 \ (4) \ - \ 5→4 \ > \ 11 No!
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16- Choice C is correct
The correct answer is x \ ≥ \ 3 \ ∪ \ x \ ≤ − \ 11 x \ + \ 4 \ ≥ \ 7→x \ ≥ \ 7 \ - \ 4→x \ ≥ \ 3 Or x \ + \ 4 \ ≤ \ - \ 7→x \ ≤ \ - \ 7 \ - \ 4→x \ ≤ \ - \ 11 Then, solution is: x \ ≥ \ 3 \ ∪ \ x \ ≤ \ − \ 11
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17- Choice B is correct
The correct answer is 5 \ \sqrt{5} Based on triangle similarity theorem: \frac{a}{a \ + \ b}=\frac{c}{4}→c=\frac{4 \ a}{a \ + \ b}=\frac{4 \ \sqrt{5}}{4 \ \sqrt{5}}=1→ area of shaded region is: (\frac{c \ + \ 4}{2}) \ (b)=\frac{5}{2} \ × \ 2 \ \sqrt{5}=5 \ \sqrt{5}
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18- Choice C is correct
The correct answer is 82\% the population is increased by 30\% and 40\%. 15\% increase changes the population to 130\% of original population. For the second increase, multiply the result by 140\%. (1.30) \ × \ (1.40)=1.82=182\% 82 percent of the population is increased after two years.
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19- Choice C is correct
The correct answer is 1 \ - \ \sqrt{5} x_{1,2} = \frac{- \ b \pm \sqrt{b^2 \ - \ 4 \ a \ c}}{2 \ a } a \ x^2 \ + \ b \ x \ + \ c=0 2 \ x^2 \ - \ 4 \ x \ – \ 8=0 ⇒ then: a=1, \ b=- \ 2 and c= – \ 4 x =\frac{ 2 \ + \ \sqrt{- \ 2 ^2 \ - \ (4) .(1) .(- \ 4)} }{2 .1}=1 \ - \ \sqrt{5 } x =\frac{2 \ - \ \sqrt{- \ 2^2 \ - \ (4) .(1) .(- \ 4)} }{2 .1}= 1 \ + \ \sqrt{5 }
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20- Choice B is correct
The correct answer is x^{\frac{12}{9}} (x^2)^{\frac{6}{9}} = x^{2 \ × \ \frac{6}{9}} = x^{ \frac{12}{6}} =
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21- Choice E is correct
The correct answer is $3 Let x be the cost of one-kilogram orange, then: 4\ x \ + \ (3 \ × \ 6)=30→ 4 \ x \ + \ 18= 30→ 4 \ x=30 \ - \ 18→ 4 \ x=12→x=\frac{12}{4}=$3
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22- Choice D is correct
The correct answer is 40 Let x be the length of AB, then: 30=\frac{x \ × \ 2}{2}→x=15 The length of AC =\sqrt{15^2 \ + \ 8^2}=\sqrt{289}=17 The perimeter of \triangleABC =15 \ + \ 8 \ + \ 17=40
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23- Choice B is correct
The correct answer is \frac{2 \ x}{5} Simplify the expression. \sqrt{\frac{x^2}{5} \ - \ \frac{x^2}{25}}=\sqrt{\frac{5 \ x^2}{25} \ - \ \frac{x^2}{25}}=\sqrt{\frac{4 \ x^2}{25}}=\sqrt{\frac{4}{25} \ x^2}= \sqrt{\frac{4}{25}} \ × \ \sqrt{x^2}=\frac{2}{5} \ × \ x=\frac{2 \ x}{5}
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24- Choice D is correct
The correct answer is \frac{y}{25} Solve for x. 5 \ \sqrt{ x}=\sqrt{y} Square both sides of the equation: ( 5\ \sqrt{ x})^2=(\sqrt{y})^2 25 \ x=y x=\frac{y}{25}
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25- Choice A is correct
The correct answer is \frac{2}{9} The probability of choosing a Hearts is \frac{16}{72} = \frac{2}{9}
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26- Choice C is correct
The correct answer is 57.81 average = \frac{sum \ of \ terms }{number \ of \ terms} The sum of the weight of all girls is: 23 \ × \ 50=1150 kg The sum of the weight of all boys is: 25 \ × \ 65=1625 kg The sum of the weight of all students is: 1150 \ + \ 1625=2775 kg average = \frac{2775}{48}=57.81
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27- Choice D is correct
The correct answer is 135 \ x \ + \ 18,000 \ ≤ \ 35,000 Let x be the number of shoes the team can purchase. Therefore, the team can purchase 120 \ x. The team had $35,000 and spent $18000. Now the team can spend on new shoes $17000 at most. Now, write the inequality: 135 \ x \ + \ 18,000 \ ≤ \ 35,000
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28- Choice C is correct
The correct answer is - \ 9 Plug in the value of x and, x=4 and y= 2 4 \ (x \ + \ 3 \ y) \ - \ (3 \ + \ x)^2= 4 \ (4 \ + \ 3 \ ( 2)) \ - \ (3 \ + \ 4)^2= 4 \ (4 \ + \ 6) \ - \ (7)^2 = 40 \ - \ 49=- \ 9
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29- Choice A is correct
The correct answer is 25 Let x be the smallest number. Then, these are the numbers: x, x \ + \ 1, x \ + \ 2, x \ + \ 3, x \ + \ 4 average = \frac{sum \ of \ terms}{number \ of \ terms} ⇒ 27= \frac{x \ + \ (x \ + \ 1) \ + \ (x \ + \ 2) \ + \ (x \ + \ 3) \ + \ (x \ + \ 4)}{5}⇒ 27=\frac{5 \ x \ + \ 10}{5} ⇒ 135 = 5 \ x \ + \ 10 ⇒ 125 = 5 \ x ⇒ x=25
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30- Choice E is correct
The correct answer is 16 \ x^6 y=(- \ 4 \ x^3)^2=(- \ 4)^2 \ (x^3)^2=16 \ x^6
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31- Choice B is correct
The correct answer is 8 cm Formula for the Surface area of a cylinder is: SA =2 \ π \ r^2 \ + \ 2 \ π \ h→192 \ π=2 \ π \ r^2 \ + \ 2 \ π \ r \ (4)→r^2 \ + \ 4 \ r \ - \ 96=0 Factorize and solve for r. (r \ + \ 12) \ (r \ - \ 8)=0→r=8 or = - \ 12 (unacceptable)
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32- Choice D is correct
The correct answer is 27\% The question is this: 616.85 is what percent of 845? Use percent formula: part = \frac{percent}{100} \ × whole 616.85 = \frac{percent}{100} \ × \ 845 ⇒ 616.85=\frac{percent \ × \ 845}{100} ⇒ 61685= percent × \ 845 ⇒ percent = \frac{61685}{845} =73 616.85 is 73\% of 845. Therefore, the discount is: 100\% \ – \ 73\%=27\%
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33- Choice E is correct
The correct answer is 30 The area of rectangle is: 6 \ × \ 3=18 cm^2 The area of circle is: π \ r^2=π \ × \ (\frac{8}{2})^2=3 \ × \ 16=48 cm^2 Difference of areas is: 48 \ - \ 18=30
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34- Choice B is correct
The correct answer is 130 miles Use the information provided in the question to draw the shape. Use Pythagorean Theorem: a^2 \ + \ b^2=c^2 50^2 \ + \ 120^2=c^2⇒ 2500 \ + \ 14400 = c^2⇒ 16900 = c^2⇒ c = 130
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35- Choice C is correct
The correct answer is − \ \frac{1}{3} The equation of a line in slope intercept form is: y=m \ x \ + \ b Solve for y. 9 \ x \ - \ 3 \ y=24 ⇒ - \ 3 \ y=24 \ - \ 9 \ x ⇒ y=(24 \ - \ 9 \ x) \ ÷ \ (- \ 3) ⇒ y=3\ x \ - \ 8 The slope is 3. The slope of the line perpendicular to this line is: m_{1} \ × \ m_{2} = - \ 1 ⇒ 3 \ × \ m_{2} = - \ 1 ⇒ m_{2} = - \ \frac{1}{3}
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36- Choice B is correct
The correct answer is \frac{ x^3}{54} \ - \ 5 f(g(x))=4 \ × \ (\frac{x}{6})^3 \ - \ 5=\frac{4 \ x^3}{216} \ - \ 5 = \frac{x^3 }{54} \ - \ 5
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37- Choice E is correct
The correct answer is 125 The ratio of boy to girls is 1:5. Therefore, there are 1 boys out of 6 students. To find the answer, first divide the total number of students by 6, then multiply the result by 1. 750 \ ÷ \ 6=125 ⇒ 125 \ × \ 1=125
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38- Choice C is correct
The correct answer is \frac{1}{14} Write the ratio of 7\ a to 4 \ b. \frac{7 \ a}{4 \ b}=\frac{1}{8} Use cross multiplication and then simplify. 7 \ a \ × \ 8=4 \ b \ × \ 1→56 \ a=4 \ b→a=\frac{4 \ b}{56}=\frac{b}{14} Now, find the ratio of a to b. \frac{a}{b}=\frac{\frac{b}{14}}{b}→\frac{b}{14} \ ÷ \ b=\frac{b}{14} \ × \ \frac{1}{b}=\frac{b}{14 \ b}=\frac{1}{14}
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39- Choice A is correct
The correct answer is \frac{\sqrt{3}}{2} The relationship among all sides of right triangle 30^\circ \ - \ 60^\circ \ - \ 90^\circ is provided in the following triangle: Sine of 60^\circ equals to: \frac{opposite}{hypotenuse}=\frac{x\sqrt{3}}{2 \ x}=\frac{\sqrt{3}}{2}
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40- Choice D is correct
The correct answer is 31 Plug in the value of x in the equation and solve for y. 4 \ y=\frac{4 \ x^2}{2} \ - \ 4→ 4 \ y = \frac{ 4 \ (8)^2}{2} \ - \ 4→ 4 \ y= \frac{4 \ (64)}{2} \ - \ 4→ 4 \ y= 128 \ - \ 4=124 4 \ y = 124→y=31
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