110 Created by Master Student Quantitative Reasoning Practice Test #2 1 / 40 (x +y + 3)(x + y + 3) = (x +y)2 + 9 x2+y2 + 9 x2 +y2 + 6xy + 9 (x +y)2 + 6(x + y) + 9 (x +y)2 + 9(x + y) + 9 It helps to put parentheses around x + y. Then use FOIL. 2 / 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. 3 / 40 In ABC, if sin A = and sin B = , sin C = 0.14 0.54 0.56 3.15 2.51 (C) Use your calculator to evaluate 4 / 40 If f(x) = ex and g(x) = sin x, then the value of (fg)( ) is -0.01 -0.8 0.34 1.8 2.7 (E) 5 / 40 Which of the following is a vertex of 16x2 - y2 - 32x - 6y - 57 = 0? (1, -1) (1, 3) (1, 5) (1, -3) (-1, 3) (D) Complete the square in both x and y to get the standard equation: 6 / 40 How many distinct 3-digit numbers contain only nonzero digits? 909 899 789 729 504 D In a 3-digit number containing no zeros, there are nine possibilities for the first digit (1-9); nine possibilities for the second digit (1-9); and nine possibilities for the third digit (1-9). That makes a total of 9 Ã— 9 Ã— 9 possible 3-digit numbers, or 729. 7 / 40 Circle O is centered at (-3, 1) and has a radius of 4. Circle P is centered at (4, -4) and has a radius of n. If circle O is externally tangent to circle P, then what is the value of n ? 4 4.37 4.6 5.28 6.25 Drawing this one is helpful. You'll find that the radii of the two circles have to add up to the distance between the circles' centers. You can find that distance using the distance formula, d = You'll find that the two centers are separated by a distance of 8.6023. Since one circle has a radius of 4, the other must have a radius of 4.6023. 8 / 40 Harry had grades of 70, 80, 85, and 80 on his quizzes. If all quizzes have the same weight, what grade must he get on his next quiz so that his average will be 80? 85 90 95 100 more than 100 (A) Average . Therefore, x = 85. [4.1] 9 / 40 The positive zero of y = 3x2 - 4x - 5 is, to the nearest tenth, equal to 0.8 0.7 + 1.1i 0.7 2.1 2.2 (D) Use the quadratic formula program on your graphing calculator to get both zeros and choose the positive one. [1.2] 10 / 40 If a circle has a circumference of 16 inches, the area of a sector with a central angle of 4.7 radians is 10 12 15 25 48 (C) 11 / 40 If f(x) = - 1, for all x > 0, then f-1(x) (x + 1)2 x2 + 2 x2 + 1 (x - 1)2 (x + 2)2 The best way to solve inverse-function questions on the Math Level 2 is to Plug In numbers. Be sure to pick a number that will work out neatly. For example, for the function f(x) = - 1, a good value for x would be 4. Then, f(4) = - 1 = 2 - 1 = 1. You can see that f(x) turns 4 into 1. So f-1(1) = 4. To find the correct answer, plug 1 into all of the answer choices. The correct answer will produce a value of 4. Only (A) comes out to 4, since (1 + 1)2 = 4, and so (A) is the correct answer. 12 / 40 The parabola with the equationhas how many points with (x, y) coordinates that are both positive integers? 3 4 7 8 Infinitely many The question is asking, in other words, for how many positive integer values of x does y turn out to be a positive integer? 13 / 40 Which of the following most closely approximates (5.5e4)^2 ? 3.0e5 3.0e6 3.0e7 3.0e8 3.0e9 Type this into your calculator, being careful to use parentheses. You should get either 3025000000 or something like 3.025e9. To get 3025000000 into scientific notation, you need to move the decimal point 9 places to the left, which means your power of 10 will be 9; pick (E). 14 / 40 If a remainder of 4 is obtained when x3 +2x2 - x - k is dividedby x - 2, what is the value of k? 4 6 10 12 14 15 / 40 A salad bar has 7 ingredients, excluding the dressing. How many different salads are possible where two salads are different if they don't include identical ingredients? 5,040 823,543 7 14 128 (C) You can either include or exclude each of the seven ingredients in your salad, which means there are 2 choices for each ingredient. According to the multiplication rule, there are 27 = 128 ways of making these yes-no choices. 16 / 40 If x mod y is the remainder when x is divided by y, then (61 mod 7) - (5 mod 5) = 2 3 4 5 6 Don't panic because you've never seen a term like "x mod y" before. This isn't something you slept through in math class. It's just one of those terms ETS throws at you sometimes. Some will be little-known math terms, and others will be made up. Either way, it doesn't matter whether you've seen it before, because ETS defines it for you. To find the value of any "x mod y," just take the number in the x position and divide it by the number in the y position. The remainder is the value of "x mod y" for those numbers. The value of 61 mod 7 is 5, because the remainder when 61 is divided by 7 is 5. The value of 5 mod 5 is 0, because the remainder when 5 is divided by 5 is zero. The expression (61 mod 7) - (5 mod 5) can be rewritten as 5 - 0, which equals 5. The correct answer is (D). 17 / 40 The plane whose equation is 5x + 6y + 10z = 30 forms a pyramid in the first octant with the coordinate planes. Its volume is 15 21 30 36 45 (A) The plane cuts the x-axis at 6, the y-axis at 5, and the z-axis at 3. The base is a right triangle with area 18 / 40 What value(s) must be excluded from the domain of ? -2 0 2 2 and -2 no value (C) Since division by zero is forbidden, x cannot equal 2. 19 / 40 If f(x) = x2 - 3x, then f(-3) = 3.3 6 9.9 18 E Plug -3 into the function: (-32) - 3(-3) = 18. 20 / 40 If log2m = x and log2n = y, then mn = 2x+y 2xy 4xy 4x+y cannot be determined (A) Add the two equations: log2 m + log2 n = x + y, which becomes log2 mn = x+y (basic property of logs). 2x + y = mn. [1.4] 21 / 40 If the roots of the equation 2x2 + k = 8x are real, which of the following expresses all the possible values fork? k Less Than or Equal to 8 k Greater Than or Equal to 8 k Greater Than or Equal to 4 k Less Than or Equal to -8 k Greater Than or Equal to -8 Re-express the equation in standard form: 22 / 40 Sheila leaves her house and starts driving due south for 30 miles, the n drives due west for 60 miles, and finally drives due north for 10 miles to reach her office. Which of the following is the approximate straight-line distance, in miles, from her house to her office? 63 67 71 75 80 Sketch a diagram: 23 / 40 If (x + y)2 = (x -y)2, which of the following must be true? x = y x = -y x = 0 y = 0 x = 0 or y =0 Expand both sides and simplify: 24 / 40 A code consists of two letters of the alphabet followed by 5 digits. How many such codes are possible? 7 10 128 20,000 67,600,000 (E) The multiplication rule applies. There are (26)(26)(10)5 = 67,600,000 possible codes. 25 / 40 What is the area of a triangle with vertices (1,1), (3,1), and (5,7)? 6 7 9 10 12 Sketch the diagram: 26 / 40 All of the following can be formed by the intersection of a cube and a plane EXCEPT a triangle a point a rectangle a line segment a circle This is a visual perception question, and there's no hard and fast technique to follow to solve it. The best plan is to experiment with sketches in your test booklet and use common sense. Remember that this is an EXCEPT question, so you're looking for a shape that can't be made. Any time you find a way to make one of the shapes in the answer choices, that choice can be eliminated. The intersection of a cube and a plane can be a triangle: 27 / 40 If x - 3 = 3(1 - x), then what is the value of x ? 0.33 0.67 1.5 1.67 2.25 C PITA, starting with (C). Use your calculator to see which value makes the equation true. The first one you try, (C), is correct. To solve the problem algebraically, isolate x one step at a time. First, multiply through by 3 on the right: x - 3 = 3 - 3x. Next, add 3x to each side, and then add 3 to each side, to get 4x = 6. Divide each side by 4 to get x = , or 1.5. 28 / 40 Which of the following lines is perpendicular to the line 3x - 2y = 16 ? 3x - 2y = 25 3x + 2y = 16 2x - 3y = 7 6x + 9y = 16 6x - 9y = 32 D Move the equation around so that it's in y = mx + b formula: y = x - 8. So an equation perpendicular would have a slope of -. The only one is (D). 29 / 40 If f(x) = i, where i is an integer such that i x < i + 1, the range of f(x) is the set of all real numbers the set of all positive integers the set of all integers the set of all negative integers the set of all nonnegative real numbers (C) Since f(x) = an integer by definition, the answer is Choice C. 30 / 40 The menu of a certain restaurant lists 10 items in column A and 20 items in column B. A family plans to share 5 items from column A and 5 items from column B. If none of the items are found in both columns, then how many different combinations of items could the family choose? 25 200 3,425 3,907,008 5.63 Ã— 1010 This is a combinations question, since rearranging the order of the dishes doesn't change the dinner. Since the dinner's being ordered in two parts (the part from column A and the part from column B), you should calculate the number of combinations in two parts. Start with the five dishes from column A. The number of permutations for five items selected from a group of ten is given by 10 Ã— 9 Ã— 8 Ã— 7 Ã— 6 = 30,240. To find the number of combinations, divide that number by 5! = 252. There are 252 possible combinations of five dishes from column A. You're also selecting five dishes from column B, which has twenty selections. The number of combinations for column B will therefore be given by = 15,504 So there are 15,504 combinations of 5 dishes from column B. To find the total number of possible combinations, multiply these figures together, 15,504 Ã— 252 = 3,907,008. The correct answer is (D). 31 / 40 If f(g(x)) = 4x2 - 8x and f(x) = x2 - 4, then g(x) = 4 - x x 2x - 2 4x x2 (C) To get from f(x) to f(g(x)), x2 must become 4x2. Therefore, the answer must contain 2x since (2x)2 = 4x2. 32 / 40 In a group of 30 students, 20 take French, 15 take Spanish, and 5 take neither language. How many students take both French and Spanish? 5 10 15 20 (C) From the Venn diagram below you get the following equations: a + b + c + d = 30 (1) b + c = 20 (2) 33 / 40 Within a given area code, how many seven-digit numbers are available, given the restrictions that the first three digits cannot be 911 or 411 and the first digit cannot be a 1 or a 0? 604,800 2,893,401 6,380,000 7,980,000 8,000,000 First, determine how many numbers are available with just the restriction on the first digit. For the first digit, there are 8 choices, and then for the remaining 6 digits, there are 10 choices. So the pool of available numbers is 8 Ã— 10 Ã— 10 Ã— 10 Ã— 10 Ã— 10 Ã— 10, or 8,000,000. If phone numbers cannot start with the digit pattern 911, then this eliminates 1 Ã— 1 Ã— 1 Ã— 10 Ã— 10 Ã— 10 Ã— 10 numbers, or 10,000. There are another 10,000 numbers starting with 411. So the actual number of available phone numbers is 8,000,000 - 20,000 = 7,980,000. 34 / 40 If the 20th term of an arithmetic sequence is 20 and the 50th term is 100, what is the first term of the sequence? -33.33 -30.67 1 2 2.67 An arithmetic sequence is one that increases by adding a constant amount again and again. The most important information to have when you're working with an arithmetic sequence is the size of the interval between any two consecutive terms in the sequence. Since the 20th term in the sequence is 20 and the 50th term is 100, you know that 30 steps in the sequence produce an increase of 80. That makes each step worth or . To find the first term in the sequence, you can use the formula for the nth term in an arithmetic sequence, an = a1 + (n - 1)d, where n is the number of the term and d is the interval between any two consecutive terms. In this case, you know that a20 = 20, and that d = , or . You can then fill those values into the formula, so 35 / 40 If f(x) = logbx and f(2) = 0.231, the value of b is 0.3 1.3 13.2 20.1 32.5 (D)f(2) = logb 2 = 0.231. Therefore, b0.231 = 2, and so b = 21/0.231 , which (using your calculator) is approximately 20.1. [1.4] 36 / 40 For f(x) = 3x2 + 4, g(x) = 2, and h = {(1,1), (2,1), (3,2)}, f is the only function h is the only function f and g are the only functions g and h are the only functions f, g, and h are all functions (E) For each value of x there is only one value for y in each case. Therefore, f, g, and h are all functions. 37 / 40 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. 38 / 40 Which of the following expresses the range of values of y = g(x), if g(x) = ? {y: y Not Equal 0} {y: y Not Equal 1.25} {y: y Not Equal -4.00} {y: y > 0} {y: y Less Than or Equal to -1 or y Greater Than or Equal to 1} A good way to tackle this one is by trying to disprove each of the answer choices. If you start with (A), you're lucky. There's no way to divide 5 by another quantity and get zero; it's the right answer. Even if you weren't sure, the other answer choices are pretty easy to disprove. Just set equal to a quantity prohibited by each answer choice, and solve for x. If there's a real value of x that solves the equation, then the value is in the range after all, and the answer choice is incorrect. Another method is to graph the function on your calculator and see what y-values seem impossible. 39 / 40 What is the length of the major axis of the ellipse whose equation is 10x2 + 20y2 = 200? 3.16 4.47 6.32 8.94 14.14 (D) Divide both sides of the equation by 200 to write the equation in standard form . The length of the major axis is [2.1] 40 / 40 What is the sum of the infinite geometric series ? 18 36 45 60 There is no sum. (A) The formula for the sum of a geometric series is , where a1 is the first term and r is the common ratio. In this problem a1 = 6 and Your score is The average score is 22% LinkedIn Facebook Twitter VKontakte 0% Restart quiz Previous Quiz Next Quiz