25 Created by Master Student Quantitative Reasoning Practice Test #5 1 / 40 Which of the following has the greatest value? 1.73999 2799 3500 4400 250100 D No ordinary calculator can work with exponents this big, and there's no way to spot the biggest values here by looking at them; you've got to get tricky. The important fact about this question is that it's not necessary to find the exact value of any expression merely to compare them. The best way to compare these expressions is to get them into similar forms. To start with, rearrange as many answer choices as possible so that they have exponents of 100. (C) can be expressed as (35)100, or 243100; (D) can be expressed as (44)100, or 256100; and (E) is already there-250100. Suddenly it's easy to see that (D) is the biggest of the three, and eliminate (C) and (E). Next, take a look at (A). The exponent 999 is approximately 1,000. The expression is therefore worth a little less than (1.7310)100, or (240.14)100. That's definitely smaller than (D), so you can eliminate (A) as well. Finally, take a look at (B). The expression 2799 is almost equal to 4400. How can you tell? Well, 4400 can be written as (22)400, or 2800. That makes it clear that (D) is bigger than (A). Answer choice (D) reigns supreme. 2 / 40 The length, width, and height of a rectangular solid are 5 cm, 3 cm, and 7 cm, respectively. What is the length of the longest segment whose endpoints are vertices of the rectangular solid? 5.8 cm 7.6 cm 8.6 cm 9.1 cm 15 cm (D) The length of the longest segment is 3 / 40 If x = i - 1, then x2 + 2x + 2 = 2i + 4 4 + 2i 0 i -2 You're dealing with imaginary numbers here, so your calculator won't be much use. To solve this problem, just plug (i - 1) into the expression in place of x and do the math. 4 / 40 If the perimeter of a square is 60, what is the area of the square? 15 20 80 150 225 E All the sides of a square are the same. So each side must be 15. Since the area of a square is (side)2, the area must be 225. 5 / 40 If 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. 6 / 40 Points and C are collinear. If B is the midpoint of line segment AC, approximately what are the (x, y) coordinates of point C? (3.71, 1.13) (3.71, 5.73) (7.41, -7.46) (10.59, -7.46) (10.59, 5.73) If is the midpoint of and C(x, y), then 6 is the average of and x, and is the average of 4 and y : 7 / 40 If f(x) = 5 - 2x and g(x) = x2 + 7, then f(g(2)) = -17 -8 8 17 24 A Work from the inside out. g(2) = 22 + 7 = 11. Now put in 11 for x: f(11) = 5 - 2(11) = -17. 8 / 40 For what positive value of a does a - equal -4 ? 0.56 1 1.12 2.06 4.12 PITA. Starting with (C), Plug In the values from the answer choices for a in the original expression. If the value makes the equation true, you've got the right answer. If not, then pick another answer choice and try again. 9 / 40 If the mean of the set of data 1, 2, 3, 1, 2, 5, x is what is the value of x ? -10.7 2.5 5.6 7.4 8.9 (E) Mean Therefore, x = 8.9. [4.1]. 10 / 40 What is the value of x: logx125 = 3? 5 41.7 122 128 375 Rewrite the equation to be x3 = 125. Take the cube root of both sides to get x = 5. 11 / 40 If f(x) = x2 - 4x + 1,f(x) crosses the x-axis closest to which of the following points? (-0.33, 0) (0.27, 0) (1.73, 0) (3.27, 0) (4.33, 0) The graph crosses the x-axis at the point where f(x) = 0. So 12 / 40 If f(x) = x - 7 and g(x) = , what is the domain of gof? x 0 x -7 x 0 x 7 all real numbers (D) (g of)(x) = g(f(x)) = . Therefore, x 7. [1.1] 13 / 40 If Pis a point on the line y = 2x in the first quadrant, and the distance from the origin to point P is 5, what are the approximate coordinates of point P? (2.24, 4.47) (3.00, 6.00) (4.00, 8.00) (4.47, 2.24) (4.72, 2.36) Sketch a diagram: 14 / 40 If 2y2 + x - 4 = 0 and then x = 1 2 3 4 5 The second equation expresses y2 in terms of x, so you can substitute that expression for y2 in the first equation and solve for x: 15 / 40 A fair cube is one that is labeled with the numbers 1, 2, 3, 4, 5, and 6, such that there is an equal probability of rolling each of those numbers. If Jade rolls two fair cubes at the same time, then what is the probability that the product of the two numbers she rolls will be greater than 18 ? 0.222 0.278 0.5 0.6 0.778 A To find the probability, first figure out the total number of possibilities, and then figure out how many meet the condition you want. Since there are 6 possible rolls on a fair cube, the total number of possibilities for two rolls is 6 Ã— 6 = 36. Now you need to figure out all the ways to get a product greater than 18. If you roll a 1, 2, or 3 on the first cube, you're out of luck, since the most you could roll would be 3 Ã— 6 = 18, but you want more than 18. The rolls that will work are 4 Ã— 5, 4 Ã— 6, 5 Ã— 4, 5 Ã— 5, 5 Ã— 6, 6 Ã— 4, 6 Ã— 5, and 6 Ã— 6. That's 8 rolls out of 36, which is a probability of =0.222. 16 / 40 What is the median of the frequency distribution shown below? 2 3 4 5 Cannot be determined (B) There are 49 data values altogether, so the median is the 25th largest. Adding the frequencies up to 25 puts the 25th number at 3. 17 / 40 The inverse of f = {(1,2),(2,3),(3,4),(4,1),(5,2)} would be a function if the domain of f is limited to {1,3,5} {1,2,3,4} {1,5} {1,2,4,5} {1,2,3,4,5} (B) The inverse is {(2,1),(3,2),(4,3),(1,4),(2,5)}, which is not a function because of (2,1) and (2,5). Therefore, the domain of the original function must lose either 1 or 5. 18 / 40 A plumbing company charges $42.50 plus x dollars per hour to make a house call. Which expression below represents the cost of a 90-minute house call? 90(42.50 + x) 1.5(42.50 + x) 1.5(42.50x) 42.50 + 90x 42.50 + 1.5x The company charges a flat fee of $42.50, plus x dollars times the number of hours. The problem asks for the cost of a 90-minute call, which is 1.5 hours. The correct expression is therefore 42.50 + 1.5x. 19 / 40 If g(x) = |5x2 - x3|, then g(6) = -54 -36 36 216 396 This simple function question just requires you to plug 6 into g(x). You can start by eliminating (A) and (B), because the entire function is contained within an absolute value sign, so it can't produce negative values. To find the exact value of g(6), Plug In the number. You get |5(6)2 - (6)3|, or |-36|, which equals 36. 20 / 40 If f(x) = x2 - 5x and f(n) = -4, then which of the following could be the value of n ? -5 -4 -1 1 5 PITA. Plug each answer choice into the function for x, to see which one spits out -4. (D) works, because f (1) = (1)2 - 5(1) = 1 - 5 = -4. 21 / 40 If 4x + 2 = 48, then 4x = 3 6.4 6.9 12 24 This question tests your understanding of exponent rules. You're told that 4x + 2 = 48, but how do you solve for 4x- Remember that when you multiply exponential terms that have a common base, you add the exponents. In the same way, you can express addition in an exponent as multiplication. 22 / 40 P(x) = ax4 + x3 - bx2 - 4x + c. If P(x) increases without bound as x increases without bound, then, as x decreases without bound, P(x) increases without bound decreases without bound approaches zero from above the x-axis approaches zero from below the x-axis cannot be determined (A) Since the degree of the polynomial is an even number, both ends of the graph go off in the same direction. Since P(x) increases without bound as x increases, P(x) also increases without bound as x decreases. 23 / 40 If f(x) = 2 for all real numbers x, then f(x + 2) = 2 4 x The value cannot be determined. (B) Regardless of what is substituted for x, f(x) still equals 2 Alternative Solution: f(x + 2) causes the graph of f(x) to be shifted 2 units to the left. Since f(x) = 2 for all x, f(x + 2) will also equal 2 for all x. [1.1] 24 / 40 In an engineering test, a rocket sled is propelled into a target. If the sled's distance d in meters from the target is given by the formula d = -1.5t2 + 120, where t is the number of seconds after rocket ignition, then how many seconds have passed since rocket ignition when the sled is 10 meters from the target? 2.58 8.56 8.94 9.31 11.26 PITA. Plug each answer choice into the equation for t, to see which one makes d = 10. (B) works. 25 / 40 There are 50 people in a room. Twenty-eight are male, and 32 are under the age of 30. Twelve are males under the age of 30. How many women over the age of 30 are in the group? 2 3 4 5 6 (A) A Venn diagram will help you solve this problem. 26 / 40 If f(x) = 2x2 + 2, then what is the value of f(x + 4)? 2x2 + 4 2x2 + 6 2x2 + x + 6 2x2 + 16x + 32 2x2 + 16x + 34 E Since there are variables in the answer choices, you should Plug In. Try x = 3. We are trying to find f(3 + 4) = f(7) = 2(7)2 + 2, which is 100, our target number. Now plug 3 in for x in the answer choices, to see which answer choice hits the target. Only (E) works. 27 / 40 Solve forx to the nearest hundredth: -0.76 -0.69 -0.67 0.69 0.76 First, raise both sides of the equation to the third power to eliminate the cube root on the left-hand side. This results in or Cross multiply to get 27(2x + 3) = 40, or 54x + 81 = 40. Subtract 81 from both sides and divide both sides by 54 to get 28 / 40 Jackie uses30 percent of her monthly earnings for rent and 50 percent of the remaining amount for food and transportation. If she spends $525 for food and transportation, how much does she pay for rent? $400 $450 $500 $550 $600 After spending 30% on rent, Jackie has 70% of her earnings left. Half of that 70% is 35%, which goes for food and transportation. So the ratio of rent to food and transportation is 30:35, or 6:7. Now you can set up a proportion and solve for the rent: 29 / 40 As , find the limit of the product . 1.9 2 2.1 2.2 2.3 (C) The infinite product can be approximated by using your calculator and a "large" value of n. The exponents form a geometric sequence with a first term of and constant ratio of . Enter prod to approximate the product of the first 50 terms as 2.08. Evaluating the product of 75 terms yields the same approximation to 9 decimals, so choose C. 30 / 40 Which of the following ordered pairs is the solution to the equations 2y - 4x = 4 and y + x + 1 = 0? (1, 1) (1, 0) (0, -1) (0, 1) (-1, 0) First isolate one of the variables: 31 / 40 Which of the following is a zero of f(x) = x2 + 6x - 12 ? -15.16 -7.58 0.67 3.16 7.58 PITA. Plug each answer choice in for x to see which one makes f(x) = 0. (B) works. Alternately, you could graph y = x2 + 6x - 12 on your calculator and see where it crosses the x-axis. 32 / 40 On a number line, the coordinate of point A is 0, and the coordinate of pointB is 6. If point P is located on the number line so that the distance from P to A is twice the distance fromP to B, which of the following could be the coordinate of point P? 4 or 12 12 only 2 only 4 only 2 or 4 Sketch a diagram: 33 / 40 Dick and Suzanne drove for 13 hours at 55 miles per hour. If the y had traveled ata rate of 65 miles per hour, how much time, in hours, would the y have saved? 0.5 1 2 6 11 Distance = rate Ã— time. 34 / 40 A U.S. dollar equals 0.716 European euros, and a Japanese yen equals 0.00776 European euros. How many U.S. dollars equal a Japanese yen? 0.0056 0.011 0.71 94.2 179.98 (B) 1 yen equals 0.0076 euros, and 1 euro equals dollars. Therefore, 1 yen equals 0.0076 Ã— 1.40 = 0.011 dollars. [algebra] 35 / 40 A 3-digitcode is made up of three dIf ferent digits from the set {2, 3, 4, 5, 6}.If 4 is always the first digit in the code, how many 3-digit codes can be formed using each digit only once? 3 5 12 13 20 The first digit can only be the number 4. Therefore, there is only one option for this digit. There are 4 digits left to choose from for the second digit and 3 digits to choose for the third. Multiply to get 12, which is the number of 3-digit codes that can be formed using each digit only once. 36 / 40 Which of the following statements is logically equivalent to: "If he studies, he will pass the course." He passed the course; therefore, he studied. He did not study; therefore, he will not pass the course. He did not pass the course; therefore he did not study. He will pass the course only if he studies. None of the above. (C) Relative to the statement, answer choice A is the converse, B is the inverse, C is the contrapositive, and D is another form of the inverse. Of these, the contrapositive is the logical equivalent of the original statement. [logic] 37 / 40 Coach Hathaway is arranging 9 players in a batting order that will include all9 of his players. How many dIf ferent arrangements are possible? 18 72 81 181,440 362,880 This is the number of permutations of 9 things taken 9 at a time. 38 / 40 A line, m, is parallel to a plane, X, and is 6 inches from X. The set of points that are 6 inches from m and 1 inch from X form a line parallel to m two lines parallel to m four lines parallel to m one point the empty set (B) Points 6 in. from m form a cylinder, with m as axis, which is tangent to plane X. Points 1 in. from X are two planes parallel to X, one above and one below X. The cylinder intersects only one of the planes in two lines parallel to m. [2.2] 39 / 40 The region in the first quadrant bounded by the line 3x + 2y = 7 and the coordinate axes is rotated about the x-axis. What is the volume of the resulting solid? 8 units3 20 units3 30 units3 90 units3 120 units3 (C) The line 3x + 2y = 7 has x -intercept and y -intercept . The part of this line that lies in the first quadrant forms a triangle with the coordinate axes. Rotating this triangle about the x -axis produces a cone with radius and height . The volume of this cone is . 40 / 40 The distance between two points in space, P (x,-1,-1) and Q (3,-3,1), is 3. Find the possible values of x . 1 or 2 2 or 3 -2 or -3 2 or 4 -2 or -4 (D) The square of the distance between P and Q is 9, so Your score is The average score is 24% LinkedIn Facebook Twitter VKontakte 0% Restart quiz Previous Quiz Next Quiz