26 Created by Master Student Quantitative Reasoning Practice Test #4 1 / 40 A recent survey reported that 60 percent of the students at a high school are girls and 65 percent of girls at this high school play a sport. If a student at this high school were selected at random, what is the probability that the student is a girl who plays a sport? 0.1 0.21 0.32 0.39 0.42 (D) The probability that a student is a girl at this high school is 0.6. The probability that a student who plays a sport is a girl is 0.65. Therefore, the probability that a girl student plays a sport is (0.6)(0.65) = 0.39. [4.2] 2 / 40 If f(x) = x2 - 10 and g(x)= 4x + 3, what is f(g(2))? -24 -21 12 27 111 In a composition of functions, first evaluate the innermost function, in this case g(2): 4 Ã— 2 + 3 = 8 + 3 = 11. Now use this result in the outermost function f: f(11) = 112 - 10 = 121 - 10 = 111. 3 / 40 In the function g(x) = A[sin (Bx + C)] + D, constants are represented by A, B, C, and D. If g(x) is to be altered in such a way that both its period and amplitude are increased, which of the following constants must be increased? A only B only C only A and B only C and D only This is a little tricky; you've got to pay attention to that underlined word, "increased." The values in the equation g(x) = A[sin (Bx + C)] + D that determine amplitude and period are A and B, respectively. Quantities C and D do not have to change to alter the amplitude or period, so (C) and (E) are out. The trick is that while you increase the amplitude by increasing A, you increase the period by decreasing B. If the amplitude and period of the curve both increase, that means that A increases and B decreases. 4 / 40 When 3x3 + 2x2 + 2x + kis divided by x + 2, the remainder is 4. What is the value ofk? 4 24 34 54 x + 4 Divide this polynomial using long division: 5 / 40 If a1 = 3 and an = n + an-1, the sum of the first five terms is 17 30 42 45 68 (D)a2 = 5, a3 = 8, a4 = 12, a5 = 17. Therefore, S5 = 45 6 / 40 In Titheland, the first 1,000 florins of any inheritance are untaxed. After the first 1,000 florins, inheritances are taxed at a rate of 65%. How large must an inheritance be, to the nearest florin, in order to amount to 2,500 florins after the inheritance tax? 7,143 5,286 4,475 3,475 3,308 This is a great PITA question. When you receive an inheritance in Titheland, you get the first 1,000 florins free and clear. After that, you get only 35% of the remaining amount; the government keeps the other 65%. To find the right answer, take the numbers from the answer choices and see which one would give you 2,500 florins after taxes. 7 / 40 If f(x) =|x| + 10, for which of the following values of x does f(x) = f(-x) ? -10 only -10 and 10 only All real x All real x except 10 All real x except -10 and 10 You'll want to use Process of Elimination here to knock out some answer choices. The best numbers to plug into the function are the ones in the answer choices: 10 and -10. Start by finding out whether the statement f(x) = f(-x) is true when x = 10. f(10) = |10| + 10 = 20. f(-10) = |-10| +10 = 10 + 10 = 20. Since f(10) and f(-10) are equal, f(x) = f(-x) is true when x = 10, and every answer choice that does not include 10 can be eliminated. That gets rid of (A), (D), and (E). To choose between the remaining answers (B) and (C), Plug In a number included in (C) but not in (B). A simple number like zero works best. f(0) = |0| + 10 = 10. f(-0) = |-0| + 10 = 0 + 10 = 10. You can see that f(0) = f(-0), so zero must be part of the correct answer. (B) does not include zero, so you can eliminate it. (C) is correct. 8 / 40 e 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. 9 / 40 If logy 2 = 8, then y = 0.25 1.04 1.09 2.83 3 Use the definition of a logarithm to rewrite the equation logy 2 = 8 in exponential form, y8 = 2. To find the value of y, take the eighth root of both sides. This can be done on your calculator by raising 2 to the power of one-eighth, or 0.125. You'll find that y = 1.09. The correct answer is (C). You can also PITA, starting with (C), which works. 10 / 40 In how many ways can a committee of four be selected from nine men so as to always include a particular man? 48 56 70 84 126 (B) Since one particular man must be on the committee, the problem becomes: "Form a committee of 3 from 8 men." Calculate 8nCr3 = 56. [3.1] 11 / 40 Mary spends40% of her monthly earnings on rent and 10% of the remaining amount on entertainment. If she spends $800 for rent, how much does she use for entertainment? $200 $800 $2,000 $120 $180 Mary spends 40% of her earnings on rent. 12 / 40 If f(x, y) = for all x, y, f(a, b) = 15, f(b, c) = 20, and f(a, c) = 10, which of the following could be the product of a, b, and c ? 18.26 54.77 284.6 1,800.00 3,000.00 The statements f(a, b) = 15, f(b, c) = 20, and f(a, c) = 10 can be rewritten as equations by inserting the definition of the function f(x, y). 13 / 40 at is the value of |6 - 3i| ? -3 3 3 9 15 This is a complex number, because it has both real and imaginary components. To find its absolute value, visualize the term 6 - 3i on the complex plane, where one axis represents real values and the other represents imaginary values. 14 / 40 If a1 = 5 and find a3. 2.623 2.635 2.673 2.799 3.323 (D) Press 5 ENTER into your graphing calculator. Then enter and press ENTER twice more to get a3. 15 / 40 If a - x = 12, b - y = 7, c - z = 15, and a + b + c = 50, then x + y + z = 16 18 34 66 84 This is a fancy simultaneous-equations question. It's impossible to solve for the value of x, y, or z, but you can find the value of their sum. It's done by adding the equations, like 16 / 40 At 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 45. 17 / 40 The area bound by the relation |x| + |y| = 2 is 8 1 2 4 There is no finite area. (A) Subtract |x| from both sides of the equation. Since |y| cannot be negative, graph the piecewise function 18 / 40 If a geometric sequence begins with the terms , 1, ? ? ? , what is the sum of the first 10 terms? 9841 6561 3280 33 6 (A) Calculate the common ratio as . The first term is so the nth term is . Use the sum and sequence features of your calculator to evaluate the sum of the first 10 terms in the generated sequence: 19 / 40 Which statement is true for the set of numbers {6, 7, 11, 6, 8}? mean > mode median > mean mode > median mean = median median = mode 20 / 40 Take 6 less than a number n. If you raise this result to the 5th power, it is equal to 32. What is the value of n? 2 2.07 4 only 4 or 8 8 only Translate the words into algebra. Six less than a number n is n - 6. The equation is thus (n - 6)5 = 32. Take the fifth root of each side to get n - 6 = 2. Add 6 to both sides of the equation, and n = 8. 21 / 40 The length of the segment joining the points with coordinates (-2,4) and (3,-5) is 2.8 3.7 10 10.3 none of these (D) Length = 22 / 40 |x - y| + |y - x| x - y y - x 2|x - y| 2|x + y| This is a great question for Plugging In. Plug In a couple of simple values, like x = 3 and y = 5, and solve. |3 - 5| + |5 - 3| = |- 2| + |2| = 2 + 2 = 4 The correct answer must also equal 4. Using x = 3 and y = 5, you'll find that only (D) gives you the correct value. 23 / 40 If pointA(3,5) is located on a circle in the coordinate plane, and the center of the circle is the origin, which of the following points must lie outside this circle? (1.0, 6.0) (1.5, 5.5) (2.5, 4.5) (4.0, 4.0) (5.0, 3.0) is a radius of the circle, and 24 / 40 If the point (5, 2) is reflected across the x-axis, then what are the coordinates of the resulting point? (5, 0) (0, 2) (5, -2) (-5, 2) (2, 5) Draw a rough sketch: 25 / 40 If f(x) = and g(x) = + 1, then f(g(2.3)) = 0.1 1.2 1.3 1.8 2.3 This is a compound-function question, in which you'll have to apply two functions in combination. The most important rule to remember is that you have to work from the inside out. Start by finding g(2.3). That means putting the number 2.3 in place of the x in the definition of g(x), g(2.3) = + 1 = (1.52) + 1 = 0.76 + 1 = 1.76. Once you know that g(2.3) = 1.76, you know that f(g(2.3)) is equal to f(1.76), which is easily solved. f(1.76) = = 1.21. The correct answer is (B). 26 / 40 If 2x + y = 2y - x, the n y = 3x 4x 2x + 2 3x + 1 4x + 1 Isolate y: 27 / 40 Suppose Find the magnitude of 2 3 4 5 6 (D) Add the components to get The magnitude is 28 / 40 The number of employees who work at Company A increased by 18% during the year2003. During 2004, the number of employees increased by 26%. What was the approximate percent of increase, to the nearest percent, in employment over the two-year period? 31% 44% 49% 53% 65% Let P = the number of employees at the beginning of 2003. By the end of 2003, the number of employees increases by 18% of P or 0.18P, so the number becomes P + 0.18P = 1.18P. 29 / 40 If a varies directly as b2, and a = 14 when b = 2, then what is the value of a when b = 5 ? 3.6 14 35 70 87.5 Set up a proportion: = . Simplify: 4a = 350, and a = 87.5. You can also approximate. When b goes up, a goes up. So you can eliminate (A) and (B). 30 / 40 If 25percent of a certain number is 36, what would 40 percent of the same number be? 13.6 22.5 42.6 52.8 57.6 If 25% of a number is 36, then that number is 36 times 4, or 144. Forty percent of that is (0.40)(144) = 57.6. 31 / 40 Students in a certain research program are either engineers or doctoral candidates; some students graduate each year. In a certain year, no doctoral candidates graduate. Which of the following statements must be true? The program then contains more engineers than doctoral candidates. Doctoral candidates are poorer students than engineers. More doctoral candidates will graduate in following years. Every student graduating in that year is an engineer. All engineers in the program graduate in that year. D With few exceptions, logic questions on the Math Level 1 test you on the contrapositive. The idea is that, given a statement like "If A, then B," the only thing you automatically know is the contrapositive: "If not B, then not A." Here, you're told that no doctoral candidates graduate. In Cookie Monster language, that would read "If doctoral student, then no graduate." The contrapositive would be "If graduate, then not doctoral student." That's almost exactly what (D) says (since there are only engineers and doctoral candidates at this school, not being a doctoral candidate is the same as being an engineer). The other answer choices all talk about subjects you don't know anything about, like the number of students or the quality of their scholarly skills. You can eliminate those answer choices. 32 / 40 Joan and Grant are shopping at the deli for lunchmeat. If Joan buys 3 pounds of bologna for $2.80 per pound, and Grant buys 2 pounds of pastrami for $1.80 per pound, then what is the average (arithmetic mean) price per pound of all the lunchmeat they buy? $2.30 $2.40 $3.60 $4.60 $8.40 B Use three average pies: 33 / 40 If 3x3 - 9x2 + Kx - 12 is divisible by x - 3, then K = -40 -3 3 4 22 (D) Substitute 3 for x set equal to zero and solve for K. 34 / 40 If = 6.25, then x could equal which of the following? 0.11 2.42 9 10.24 13.76 A PITA. Plug In values from the answer choices, and see which one makes the equation true. Only (A) does it. When solving this question algebraically, the trick is to isolate x one step at a time. The correct order is below. 35 / 40 For what value(s) of x, 0 < x < , is sin x < cos x? x < 0.79 x < 0.52 0.52 < x < 0.79 x > 0.52 x > 0.79 (A) Graph y = sin x and y = cos in radian mode using the Xmin = 0 and Xmax = . Observe that the first graph is beneath the second on [0,0.79]. 36 / 40 If the second term in an arithmetic sequence is 4, and the tenth term is 15, what is the first term in the sequence? 1.18 1.27 1.38 2.63 2.75 An arithmetic series is one that increases by a constant added amount. From 4 to 15 is a total increase of 11, which happens from the second term in the arithmetic series to the tenth, taking 8 steps. This means that each step is worth one-eighth of 11, , or 1.375. That's the constant amount added to each term in the series to get the next term. To find the first term in the series, just take one step backward from the second term, that is, subtract 1.375 from 4. You get 2.625. 37 / 40 If g(x) = 3x + 2 and g(f(x)) = x, then f(2) = 1 2 6 8 (A) Since f(2) implies that x = 2, g(f(2)) = 2. Therefore, g(f(2)) = 3(f(2)) + 2 = 2. Therefore, f(2) = 0. 38 / 40 If x < |x| and x2 + 2x - 3 = 0, then 2x + 4 = -2 2 6 8 10 Solving x2 + 2x - 3 = 0 is just a matter of factoring. 39 / 40 Of the 850 stores in Noel-Bentley County, 250 have alarm systems and 450 have guard dogs. If 350 stores have neither alarm systems nor guard dogs, then how many stores have both alarm systems and guard dogs? 100 150 200 500 700 Use the Group Problem Formula, Total = Group 1 + Group 2 + Neither - Both. Here, you've got 850 = 250 + 450 + 350 - Both. So Both = 200. Pick (C). You need to memorize this formula! 40 / 40 A rubber ball is dropped from a height of 10 meters. If the ball always rebounds the distance it has fallen, how far, in meters, will the ball have traveled at the moment it hits the ground for the fourth time? 4.1 5.12 29.52 43.92 49.04 Sketch a diagram. First the ball drops 10 meters and rebounds of that, or 8 meters: Your score is The average score is 27% LinkedIn Facebook Twitter VKontakte 0% Restart quiz Previous Quiz Next Quiz