39 Created by Master Student Quantitative Reasoning Practice Test #3 1 / 40 If 3(x + 5) - (x + 2) = 2x- 3x + 4, the n x = -3 -1 0 1 3 2 / 40 If x2 - 4x -12 = 0, what is the sum of the two possible values of x? -8 -6 -2 2 4 This answer can be found by finding both roots and adding them together. 3 / 40 If f(x) = 2 ln x + 3 and g(x) = ex, then f(g(3)) = 9 11 43.17 47.13 180.77 (A) Since g(3) = e3, f (g(3)) = 2 ln e3 + 3. ln e3 = 3. So f(g(3)) = 6 + 3 = 9. [1.4] 4 / 40 The set of points (x, y, z) such that x = 5 is a point a line a plane a circle a cube (C) Since the y- and z-coordinates can have any values, the equation x = 5 is a plane where all points have an x-coordinate of 5. [2.2] 5 / 40 Rodney is starting a small business selling pumpkins. If he spends $200 on supplies and sells his pumpkins for $4 each, which of the following functions correctly shows the amount of money Rodney has gained or lost when he has sold x pumpkins? f(x) = 800x f(x) = 200x + 4 f(x) = 200x - 4 f(x) = 4x + 200 f(x) = 4x - 200 E Plug In x = 1,000. Then Rodney has earned 4 Ã— 1,000 = $4,000 and spent $200, so he has made a total of $3,800. That's (E). 6 / 40 If i = , then (5 - 3i)(4 + 2i) = 14 - 2i 16 24 26 - 2i 28 For this question, you need to remember how FOIL works, and that i2 can be replaced by -1. 7 / 40 If f(x) = 4x - 5 and g(x) = 3x, then f(g(2)) = 3 9 27 31 none of the above (D)g(2) = 32 = 9. f(g(2)) = f(9) = 31. 8 / 40 Jean canpaint a house in 10 hours, and Dan can paint the same house in 12 hours.If Jean begins the job and doesof it and the n Dan takes over and finishes the job, what is the total time it takes the m to paint the house? 10 hours, 40 minutes 11 hours 11 hours, 20 minutes 11 hours, 40 minutes 12 hours If Jean can do the whole job in 10 hours, then it takes her to do of the job. If Dan can do the whole job in 12 hours, then it takes him hours to do the other of the job. Thus, together, it takes them hours, or 11 hours and 20 minutes, to paint the house. 9 / 40 At what points does the circle given by the equation (y - 3)2 + (x - 2)2 = 16 intersect the y-axis? (0, -5.66) and (0, 5.66) (0, -0.46) and (0, 6.46) (0, -1.00) and (0, 7.00) (-0.65, 0) and (4.65, 0) (-2.00, 0) and (6.00, 0) B At all points on the y-axis of the coordinate plane, x = 0, so eliminate (D) and (E) right away. Then PITA. In this case, plug each point (x, y) from the answer choices into the equation, to see if it makes the equation true. If the first point in the answer choice works, then try the other point. If they both work, you've found the right answer. If either point fails, cross off that answer choice. Only the two points in (B) fit the given equation. Notice that (D) gives you the x-intercepts. 10 / 40 If for all real numbers x, a function f(x) is defined by f(x) = , then f(15) - f(14) = -2 0 1 2 4 The statement f(x) = can be read, "f of x equals 2 when x does not equal 13, and f of x equals 4 when x equals 13." Since neither of the values you're given equals 13, the function will always come out to 2. f(15) - f(14) = 2 - 2 = 0. The correct answer is (B). 11 / 40 If vector and vector = (3,-2), find the value of 5.4 6 7 7.2 52 (D) 12 / 40 If and g(x) = x2 + 1, then f(g(2)) = 2.24 3 3.61 6 6.16 (C) Enter the function f into Y1 and the function g into Y2. Evaluate Y1(Y2(2)) to get the correct answer choice C. An alternative solution is to evaluate g(2) = 5 and f(5) = , and either use your calculator to evaluate or observe that 3 < < 4, indicating 3.61 as the only feasible answer choice. [1.1] 13 / 40 The point (5, -10) is at a distance of 26 from point Q, and the point (2, -10) is at a distance of 25 from Q. Which of the following could be the coordinates of Q ? (-5, 14) (-3, 18) (-1, 19) (0, 21) (2, 16) A The simple, grinding way to do this one is to use the distance formula on the answer choices. Any answer choice that produces a distance other than 25 or 26 can be discarded immediately. Only (A) produces distances of 25 and 26 from the two points given. 14 / 40 If a+ b = 10 and both a and b are positive integers,what is the largest possible value for a? 6 7 8 9 10 For a to be as large as possible, make b as small as possible. If b = 1, then a = 9. 15 / 40 Which of the following ordered pairs is the solution to the equations 2y +x = 5 and -2y + x = 9? (-7, -1) (-1, 7) (7, -1) (-7, 1) (1, 7) If you add the equations as presented, the y's drop out: 16 / 40 ||-5|- 16 + |-1|| = -12 -10 10 12 22 17 / 40 At a certain software company, the cost, C, of developing and producing a computer software program is related to the number of copies produced, x, by the equation C = 30,000 + 2x. the company's total revenues, R, are related to the number of copies produced, x, by the equation R = 6x - 10,000. How many copies must the company produce so that the total revenue is equal to the cost? 5,000 6,000 7,500 9,000 10,000 Put much more simply, what the question is asking is: At what point does cost equal revenue? In algebraic terms, for what value of x does C equal R? 18 / 40 a dance competition, each of six couples must compete against the other five couples in a dance-off three times before the winning couple can be declared. How many such dance-offs will occur? 12 33 45 60 63 The first couple must compete against 5 couples. The second couple has already faced the first couple, so they have to compete against only four new couples. The third couple has to compete against three new couples, and so on. This means that to make sure everyone competes in a dance-off with everyone else, there have to be 5 + 4 + 3 + 2 + 1 = 15 dance-offs. Multiply 15 by the 3 times this must happen, and you get 19 / 40 The solution set of 3x + 4y < 0 lies in which quadrants? I only I and II I, II, and III II, III, and IV I, II, III, and IV (D) Use the standard window to graph , by moving the cursor all the way left (past Y =) and keying Enter until a "lower triangle" is observed. The shaded portion of the graph will lie in all but Quadrant I. An alternative solution is to graph the related equation and test points to determine which side of the line contains solutions to the inequality. This will indicate the quadrant that the graph does not enter. [1.2] 20 / 40 What is the solution set for the equation |2x - 3| = 13? {-8} {-5} {-5, -8} {-5, 8} {5, -8} To solve an equation with absolute value, consider the two possibilities. 21 / 40 If there are known to be 4 broken transistors in a box of 12, and 3 transistors are drawn at random, what is the probability that none of the 3 is broken? 0.25 0.255 0.375 0.556 0.75 (B) Since there are 4 broken transistors, there must be 8 good ones. P (first pick is good) = . Of the remaining 11 transistors, 7 are good, and so P(second pick is good) = . Finally, P(third pick is good) = . Therefore, P(all three are good) = . Alternative Solution: Note that there are ways to select 3 good transistors. There are ways to select any 3 transistors. P(3 good ones) = 22 / 40 If the vertices of a triangle are (u,0), (v,8), and (0,0), then the area of the triangle is 4|u | 2|v | |uv | 2|uv | |uv | (A) Sketch a graph of the three vertices. The base is |u | and the altitude is 8. Therefore, the area is 4 |u |. [2.1] 23 / 40 M & M plain candies come in six colors: brown, green, orange, red, tan, and yellow. Assume there are at least 3 of each color. If you pick three candies from a bag, how many color possibilities are there? 18 20 120 216 729 (D) There are 6 choices of color for each of the three candies selected. Therefore, there are 6 Ã— 6 Ã— 6 = 216 color possibilities altogether. 24 / 40 What is the equation of a line with a y-intercept of 3 and an x-intercept of -5 ? y = 0.6x + 3 y = 1.7x - 3 y = 3x + 5 y = 3x - 5 y = -5x + 3 25 / 40 If fn+1 = fn-1 + 2 fn for n = 2, 3, 4, . . . , and f1 = 1 and f2 = 1, then f5 = 7 11 17 21 41 (C) Let n = 2, f3 = 3; then let n = 3, f4 = 7; and finally let n = 4, f5 = 17. [3.4] 26 / 40 f(x) = 4x2 + 4x + 4, which of the following is equal to f(-3.5) ? f(-14) f(-7) f(-0.5) f(0.5) f(2.5) E If you have a graphing calculator, press the Y= key and enter the function. If you check the values of the TABLE, you can find that f(-3.5) and f(2.5) both equal 39. If you don't have a graphing calculator, you can PITA. It may take a while, but you'll get it. 27 / 40 At the end of a meeting all participants shook hands with each other. Twenty-eight handshakes were exchanged. How many people were at the meeting? 7 8 14 28 56 (B) 28 / 40 All of the elements of list M and list N are arranged in exactly 20 pairs, such that every element from list M is paired with a distinct element from list N. If in each such pair, the element from list M is larger than the element from list N, then which of the following statements must be true? The median of the elements in M is greater than the median of the elements in N. Any element of M is greater than any element of N. The mode of the elements in M is greater than the mode of the elements in N. The range of the elements in M is greater than the range of the elements in N. The standard deviation of the elements in M is greater than the standard deviation of the elements in N. All the fancy language in this question basically boils down to this: List M and list N each contains 20 elements; each element in list M is larger than the corresponding element in list N. 29 / 40 If = 0.625, then is equal to which of the following? 1.6 2.67 2.7 3.33 4.25 A The fraction is the reciprocal of ; it's just flipped over. To find the numerical value of , just flip the numerical value of , which is 0.625. Your calculator will tell you that = 1.6. 30 / 40 z = logx (yx), then xz = xx yx xyx y2x x2y B If logx(yx) = z, then z is the exponent that turns x into yx. If you think about it that way, then it's clear that x raised to the power of z would be yx. If that doesn't make sense to you, then review Chapter 3. If that doesn't work for you, then it's possible (but a little tricky) to Plug In. Do it this way. Plug 10 in for x so that you're working with a common logarithm, the kind your calculator can compute. Plug In 2 for y and you get: log10(210) = z. This can be written simply as log 1,024 = z. Your calculator can then compute the value of z. z = 3.0103. You can then compute the value of xz. You get 1,024. The only answer choice that equals 1,024 is (B). 31 / 40 If 7a + 2b = 11 and a - 2b = 5, then what is the value of a ? -2 -1.5 -0.5 1.4 2 E When you're given two equations extremely similar in form, you're probably looking at classic ETS-style simultaneous equations. The best way to solve these? Rack 'em, stack 'em, add or subtract 'em! (Isn't that satisfying?) In this case, adding the two equations cancels out the b term, leaving you with the equation 8a = 16, so a = 2. 32 / 40 A wire is stretched from the top of a two-foot-tall anchor to the top of a 50-foot-tall antenna. If the wire is straight and has a slope of , then what is the length of the wire in feet? 89.18 120 123.26 129.24 134.63 It's helpful to draw this one. The wire has a slope of , meaning that it rises 2 feet for every 5 feet it runs. Since its total rise is 48, or (2 Ã— 24), its total run must be 120, or (5 Ã— 24). Don't pick answer choice (B), though. The question asks for the length of the wire, not the distance between the anchor and antenna. The wire's length is the hypotenuse of a right triangle with legs 48 and 120. The Pythagorean theorem will tell you that its length is 129.24. 33 / 40 If the cube root of the square root of a number is 2, what is the number? 16 64 128 256 1,024 B Translate the words into math: = 2. Now peel away. First cube both sides: = 8. Now square both sides: x = 64. PITA works well, too. 34 / 40 xand Y are independent events. If the probability event xwill occur is 0.2 and the probability event Y will occur is 0.9,what is the probability both x and Y will occur? 18% 20% 50% 72% 90% When two events are independent, the probability they will both occur is the product of each of their probabilities. 35 / 40 Write the product of (2 + 3i )(4 - 5i ) in standard form. -7 - 23i -7 + 2i 23 - 7i 23 + 2i 23 - 2i (D) If you enter imaginary numbers into the calculator, it will do imaginary arithmetic without changing mode. The imaginary unit is 2nd decimal point. Enter the product, and read the solution 23 + 2i . 36 / 40 Two identical rectangular solids, each of dimensions 3 Ã— 4 Ã— 5, are joined face to face to form a single rectangular solid with a length of 8. What is the length of the longest line segment that can be drawn within this new solid? 8.6 9.9 10.95 11.4 12.25 It's a good idea to sketch the situation that's described in this question: 37 / 40 Sarah is scheduling the first four periods of her school day. She needs to fill those periods with calculus, art, literature, and physics, and each of the se courses is offered during each of the first four periods. How many dIf ferent schedules can Sarah choose from? 1 4 12 24 120 Sarah can fill her first-period slot with any of the 4 subjects, leaving 3 for the second, 2 for the third, and 1 for the fourth: 38 / 40 A sphere of radius 5 has the same volume as a cube with an edge of approximately what length? 5 5.5 6.24 8.06 9.27 A sphere of radius 5 has volume If that's also the volume of a cube, then an edge of that cube is the cube root of that. Use your calculator: 39 / 40 A boat sights the top of a 40-foot lighthouse at an angle of elevation of 25degrees. How far away is the boat from the lighthouse ( the horizontal distance), to the nearest tenth of a foot? 16.9 feet 18.7 feet 44.1 feet 85.8 feet 94.6 feet Draw a picture of the problem situation: 40 / 40 A rectangular solid has three faces with areas of 28, 20, and 35 square centimeters. What is the volume of this solid? 83 cubic cm 140 cubic cm 166 cubic cm 196 cubic cm 19,600 cubic cm To solve this problem, sketch an "unfolded" drawing of this solid: Your score is The average score is 33% LinkedIn Facebook Twitter VKontakte 0% Restart quiz Previous Quiz Next Quiz