584 Created by Master Student Quantitative Reasoning Practice Test #1 1 / 40 When 4x2 + 6x + L is divided by x + 1, the remainder is 2. Which of the following is the value of L ? 4 6 10 12 15 Don't bother doing long division with the polynomial. Just Plug In x = 10. The problem now reads, "When 460 + L is divided by 11, the remainder is 2." Well, 460 divided by 11 gives you 41.818. 11 Ã— 41 = 451, and 11 Ã— 42 = 462. To get a remainder of 2, you would divide 464 by 11. So 460 + L = 464, and L = 4. 2 / 40 If five coins are flipped and all the different ways they could fall are listed, how many elements of this list will contain more than three heads? 5 6 10 16 32 (B) "More than 3" implies 4 or 5, and so the number of elements is . [3.1] 3 / 40 If r - s > r + s, then which of the following must be true? r > s s < 0 r < 0 r < s s > 0 Algebraic manipulation is the easiest way to solve this one. Start by adding s to each side, producing the inequality r > r + 2s. Then subtract r from each side to get 0 > 2s. If 2s < 0, then s < 0 (just divide both sides by 2). This can also be solved by Plugging In, but since only certain values will make the original inequality true, it can take some time to Plug In enough different values to eliminate all the wrong answers. Algebraic manipulation is faster here. 4 / 40 Rob rented a minivan at A's Auto for $60.00 per day with tax included, plus $0.25per mile. If he rented it for 3 days and was charged $206.00, how many miles did he drive the minivan? 6.5 26 104 584 824 Since Rob rented the minivan for 3 days, he will pay 3($60) = $180.00 plus 0.25 times the number of miles he drove it. 5 / 40 If each side of an equilateral triangle is 8, what is the approximate measure of the altitude? 1.73 2 3.46 4 6.93 Use the Pythagorean theorem: 6 / 40 If ln (xy) < 0, which of the following must be true? xy < 0 xy < 1 xy > 1 xy > 0 none of the above (B) Since ln stands for loge, and e > 1, xy < 1. 7 / 40 If a certain car can travel 240 miles on 12 gallons of gasoline, the n at the same rate, how many gallons of gasoline are needed to travel 300miles? 10 15 20 25 30 Set up a proportion: 8 / 40 A man piles 150 toothpicks in layers so that each layer has one less toothpick than the layer below. If the top layer has three toothpicks, how many layers are there? 15 17 20 148 11,322 (A) This is an arithmetic series with t1 = 3, d = 1, and S = 150. . n = 15. [3.4] 9 / 40 Points A, B, C, and D are arranged on a line in that order. If AC = 13, BD = 14, and AD = 21, then BC = 12 9 8 6 3 D This one's much clearer if you draw it. You can think of segments AC and BD as overlapping segments, where BC is the amount of the overlap. The lengths of AC and BD add up to 27, but it's only a distance of 21 from A to D. The difference, a distance of 6, is the overlap. That's the length of BC. 10 / 40 Last week, police ticketed 13 men traveling 18 miles per hour over the speed limit and 8 women traveling 14 miles per hour over the speed limit. What was the mean speed over the limit of all 21 drivers? 16 miles per hour 16.5 miles per hour 17 miles per hour none of these cannot be determined (B) There are 13 eighteens and 8 fourteens, so the total over the speed limit is 346. Divide this by the 21 people to get 16.5. 11 / 40 Six congruent circles are arranged so that each circle is externally tangent to at least two other circles. The centers of these six circles are then connected to form a polygon. If each circle has a radius of 2, then what is the perimeter of this polygon? 6 12 24 36 48 C The best way to attack this question is just to try it. You'll find that no matter how you arrange the circles and connect them, the resulting polygon (which will be a hexagon) always has a perimeter of 24. That's because the polygon is always made up of two radii from each circle, for a total of 12 radii-each with a length of 2. 12 / 40 The mean ofa set of 5 numbers is 90. If one of the numbers is removed, the mean of the remaining numbers is 92. What number was removed? 82 84 87 90 92 Since the mean of a set of 5 numbers is 90, the sum of those numbers is 5 Ã— 90 = 450. When one number is removed, the mean of the set of the 4 remaining numbers is 92. The sum of those numbers is 4 Ã— 92 = 368. Then 450 - 368 = 82, so 82 must have been the number that was removed. 13 / 40 If 2 tons of snow fall in 1 minute, how many tons of snow fall in 3 hours? 6 36 360 3,600 4,000 Set up a proportion. 14 / 40 If tan x = 3, the numerical value of is 0.32 0.97 1.03 1.78 3.16 (C) Since tan x = 3, x could be in the first or third quadrants. Since, however, is only defined when csc x 0, we need only consider x in the first quadrant. Thus, we can enter to get the correct answer choice C. [1.3] 15 / 40 If 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). 16 / 40 The pendulum on a clock swings through an angle of 1 radian, and the tip sweeps out an arc of 12 inches. How long is the pendulum? 3.8 inches 6 inches 7.6 inches 12 inches 35 inches (D)s = r. 12 = r. [1.3] 17 / 40 Which of the following inequalities is equivalent to -2(x + 5) <-4? x > -3 x < -3 x > 3 x < 3 x > 7 Simplify: 18 / 40 What is the radius of a sphere, with center at the origin, that passes through point (2,3,4)? 3 3.31 3.32 5.38 5.39 (E) Since the radius of a sphere is the distance between the center, (0,0,0), and a point on the surface, (2,3,4), use the distance formula in three dimensions to get 19 / 40 How many 3-person committees can be selected from a fraternity with 25 members? 15,625 13,800 2,300 75 8 (C) This is the number of ways 3 objects can be chosen from 25, or = 25nCr3 = 2,300. 20 / 40 Which of the following functions transforms y = f(x) by moving it 5 units to the right? y = f(x + 5) y = f(x - 5) y = f(x) + 5 y = f(x) - 5 y = 5f(x) (B) Horizontal translation (right) is accomplished by subtracting the amount of the translation (5) from x before the function is applied. 21 / 40 What is the distance between the points with coordinates (-3,4,1) and (2,7,-4)? 5.24 7.68 11.45 13 19.26 (B) Use the distance formula for three-dimensional space: 22 / 40 The formula A = Pe0.04t gives the amount A that a savings account will be worth if an initial investment P is compounded continuously at an annual rate of 4 percent for t years. Under these conditions, how many years will it take an initial investment of $10,000 to be worth approximately $25,000? 1.9 2.5 9.9 22.9 25.2 (D) Substitute the given values into the formula to get 25000 = 10000e0.04t. To solve for t, first divide both sides by 10000. Then take logarithms (either ln or log) of both sides to get ln2.5 = 0.04t. Therefore 23 / 40 A taxicab company wanted to determine the fuel cost of its fleet. A sample of 30 vehicles was selected, and the fuel cost for the last month was tabulated for each vehicle. Later it was discovered that the highest amount was mistakenly recorded with an extra zero, so it was 10 times the actual amount. When the correction was made, this was still the highest amount. Which of the following must have remained the same after the correction was made? mean median mode range standard deviation (B) Since the median is the middle value, it does not change if the number (quantity) of values above and below it remain the same. [4.1] 24 / 40 After 8:00 p.m., a ride in a taxi costs $2.50 plus $0.30 for every fifth of a mile traveled. If a passenger travels b miles, then what is the cost of the trip, in dollars, in terms of b ? 2.5 + 0.3b 2.5 + 1.5b 2.8b 30 + 250b 250 + 30b You've got variables in the question and the answer choices, so Plug In. Suppose that you're traveling 5 miles (b = 5). That means that your fare will include an original $2.50 plus five $0.30 charges ($1.50) for each of the 5 miles traveled (which is $7.50), for a total of $9.00. To find the correct answer, plug b = 5 into the answer choices and see which one gives you a value of 9. Only (B) does the trick. 25 / 40 If x= 6 when y = 5 and x varies directly as y, what is the value of x when y = 2? 2.4 7.5 15 60 1.6 If x varies directly as y, 26 / 40 In a cube with edge length 2, the distance from the center of one face to a vertex of the opposite face is 2 2.24 2.45 2.65 2.83 Sketch a diagram: 27 / 40 If m varies directly as n and = 5, then what is the value of m when n = 2.2 ? 0.44 2.27 4.1 8.2 11 E Direct variation between two quantities means that they always have the same quotient. In this case, it means that must always equal 5. To find the value of m when n = 2.2, set up the equation = 5, and solve for m. You'll find that m = 11. 28 / 40 If a cube and a sphere intersect at exactly eight points, then which of the following must be true? The sphere is inscribed in the cube. The cube is inscribed in the sphere. The diameter of the sphere is equal in length to an edge of the cube. The sphere and the cube have equal volumes. The sphere and the cube have equal surface areas. B This one isn't really vulnerable to shortcuts or techniques. You pretty much have to visualize the situation described in each answer choice and find the one that produces exactly eight points of intersection. When a cube is inscribed in a sphere, each of the cube's 8 corners touches the inside of the sphere. 29 / 40 If sin x = -0.6427, what is csc x? -1.64 -1.56 0.64 1.56 1.7 (B). 30 / 40 If f(x) = x2 - 6x + 11, then what is the minimum value of f(x) ? -8 -7 3.2 6 11 This is the equation of a parabola which opens upward. The minimum value will be the y-value of the vertex, which you can find using the vertex formula. The x-coordinate of the vertex is given by x = which gives you x = 6 in this case. Plug this value back into the equation to get the y-coordinate of the vertex, (6)2 - 6(6) + 11 = -7. The function's minimum value is -7. 31 / 40 Sally is interested in buying a car. If she has a choice of 3 colors (red, green,or blue), 2 body types (two or four doors), and 3 engine types (four,six, or eight cylinders), how many dIf ferent models can she choose from? 6 8 12 16 18 To get the total number of possibilities, multiply: 3 colors, 2 body types, and 3 engine types means 3 Ã— 2 Ã— 3 = 18 different models. 32 / 40 A rectangular room has walls facing due north, south, east, and west. On the southern wall, a tack is located 85 inches from the floor and 38 inches from the western wall, and a nail is located 48 inches from the floor and 54 inches from the western wall. What is the distance in inches between the tack and the nail? 21 26.4 32.6 37 40.3 This is a Pythagorean theorem question disguised by a complicated physical description. Making a sketch of the southern wall can help clear this up. 33 / 40 A basketball team has 5 centers, 9 guards, and 13 forwards. Of these, 1 center, 2 guards, and 2 forwards start a game. How many possible starting teams can a coach put on the floor? 56,160 14,040 585 197 27 (B) There are ways of choosing the one center, ways of choosing the two guards, and ways of choosing the two forwards. Therefore, there are 5 Ã— 36 Ã— 78 = 14,040 possible starting teams. 34 / 40 Given the set of data 1, 1, 2, 2, 2, 3, 3, x, y, where x and y represent two different integers. If the mode is 2, which of the following statements must be true? If x = 1 or 3, then y must = 2. Both x and y must be > 3. Either x or y must = 2. It does not matter what values x and y have. Either x or y must = 3, and the other must = 1. (A) The number of 2s must exceed the number of other values. Some of the choices can be eliminated. B: one integer could be 2. C: x or y could be 2, but not necessarily. D and E: if x = 3 and y = 1, there will be no mode. Therefore, Choice A is the answer. [4.1] 35 / 40 Find the number of radians in cot-1(-5.2418). -10.8 -5.3 -1.38 -0.19 none of these (E) The range of inverse cotangent functions consists of only positive numbers. 36 / 40 The average of your first three test grades is 78. What grade must you get on your fourth and final test to make your average 80? 80 82 84 86 88 (D) Since the average of your first three test grades is 78, each test grade could have been a 78. If x represents your final test grade, the average of the four test grades is , which is to be equal to 80. Therefore, . 37 / 40 Rob and Sherry together weigh 300 pounds. Sherry and Heather together weigh 240 pounds. If all three people together weigh 410 pounds, then what is Sherry's weight in pounds? 110 115 120 130 145 All three people weigh 410 pounds, but Rob and Sherry weigh 300 pounds; therefore, Heather must weigh 410 - 300 = 110 pounds. Because Sherry and Heather weigh 240 pounds, Sherry must weigh 240 - 110 = 130 pounds. If you don't want to handle the question this way, you can simply PITA. That means starting with (C) and using the answer choices as Sherry's weight. For example, if Sherry's weight is 120 pounds, that makes Rob's weight 180 pounds and Heather's weight 120 pounds. Those three weights don't add up to 410 pounds, so move on and try the next answer choice. Answer choice (D) works. 38 / 40 Log7 5 = 1.2 1.1 0.9 0.8 -0.7 (D). 39 / 40 A cylinder whose base radius is 3 is inscribed in a sphere of radius 5. What is the difference between the volume of the sphere and the volume of the cylinder? 88 297 354 448 1345 (B) Height of cylinder is 8. 40 / 40 The area of square ABCD is three-fourths the area of parallelogramEFGH. the area of parallelogram EFGH is one-third the area of trapezoid IJKL. If square ABCD has an area of 125square feet, what is the area of trapezoid IJKL, in square feet? 75 225 350 500 625 Area of the square (Area of the parallelogram). 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