CLEP College Mathematics Study Questions
study.com Which of the following is the inverse of the below argument? If I am in Kansas, then I'm in the United States. A. If I'm not in the United States, then I'm not in Kansas. B. If I'm in the United States, then I'm in Kansas. C. If I'm not in Kansas, then I'm not in the United States. D. If I am in Kansas, then I'm in the United States.
C. - The inverse is the negative form of the conditional statement.
Suppose a trapezoid has base lengths 8 and 14. What does its height need to be in order for the trapezoid to have area 30? A. 3 B. 2.7273 C. 3.4505 D. 2 E. 3.9082
correct answer: B. 2.7272
#3 diagnostic Which set does not represent a function? A. {(0, 0), (1, 1), (2, 2), (3, 3)} B. {(-1, 6), (3, 4), (5, 6), (8, -4)} C. {(-3, -1), (-1, -3), (1, -1), (3, -3)} D. {(-2, -8), (4, -16), (5, 1), (-2, -20)}
(D) In a function, each input has exactly one output. For the set {(-2, -8), (4, -16), (5, 1),(-2, -20)}, the input value of -2 has two different outputs and is, therefore, not a function.
REA Practice test two #54 The number n is known to have exactly two composite factors, not counting itself. Which one of the following could be the value of n? A. 30 B. 20 C. 15 D. 8
(B) The factors of 20 are 1, 2, 4, 5, 10, and 20. Not counting the number itself, the two composite factors are 4 and 10. Answer choice (A) is wrong because 30 has composite factors of 6, 10, and 15. Answer choice (C) is wrong because 15 has no other composite factors besides itself. Answer choice (D) is wrong because 8 has only one other composite factor besides itself, namely 4.
#6 Practice Test 1 If f(x) = 2x + 4 and g(x) = x² - 2, then (f∘g)(x), where (f∘g)(x) is a composition of functions, is: A. 2x² - 8 B. 2x² + 8 C. 2x² D. 2x³ + 4x² - 4x - 8
(C) (f∘g) (x) = f(x) ∘ g(x) = f(g(x)) This is the definition of the composition of functions. For the functions given in this problem we have 2(g(x)) + 4 = 2(x²-2) + 4 = 2x²-4+4 = 2x²
#12 Practice Test 1 There are six Knights of the Round Table. Given that Sir Lancelot must sit in a specific chair and that Sir Gawain must be directly on either side of him, in how many ways may the knights be seated? A. 24 B. 120 C. 48 D. 25
(C) Since Sir Lancelot must sit in an assigned chair, and Sir Gawain on either side of him, there are 4! or 24 ways of seating the other four. For each of these arrangements, Sir Gawain can be in either of two seats, so the total number of ways of seating the knights is 24 x 2 or 48.
Modern States #62 Let P(E) represent the probability of event E. Let A and B be events with P(A) = 0.3 , P(B) = 0.7, and P(A∩B) = 0.2 . What is the value of P(A∪B) ? A. 1.0 B. 0.9 C. 0.8 D. 0.7
Correct Answer: C. 0.8
REA Book probability and statistics drill #10 At the MNP company, each person gets an identification number. The number contains five digits, chosen from the digits 1 through 8, inclusive. The left-most digit must be 2. If repetition of any digit is allowed, how many different identification numbers are possible? A. 32,768 B. 8192 C. 6720 D. 4096
Correct Answer: D. Since only the number 2 is allowed for the left-most digit, there is only one choice for this digit. For each of the other four digits, there are no restrictions on digits 1 through 8. This means that there are eight choices for each of these four digits. Thus, there are (1)(8)(8)(8)(8) = 4096 different identification numbers.
REA Practice test two #17 Which one of the following is the conjugate for the complex number -5 + 8i? A. 5 - 8i B. -5 - 8i C. 5 + 8i D. -8 + 5i
(B) For any complex number in the form a + bi, the conjugate is given by the expression a - bi. In this example, a = -5 and b = 8.
REA Practice test two #1 What is the domain of the function defined by f(x) = √(-x+1) + 5 A. {x | x ≥ 0} B. {x | x ≤ 1} C. {x | 0 ≤ x ≤ 1} D. {x | x ≥ -1}
(B) The only restriction for the domain is that -x + 1 must be greater than or equal to zero. -x + 1 ≥ 0 ⇒ 1 ≥ x, which means x ≤ 1
Modern States #19 The width of a rectangular garden is x feet. If 300 feet of fencing is needed to enclose the garden, which of the following represents the length of the garden, in feet? A. 300 - x B. 300 - 2x C. 150 - x D. 150 - 2x
Answer: C. 150 -x
modern states Which of the following ordered pairs is NOT in the Cartesian product A×B? (A) (a,b) (B) (b,a) (C) (b,b) (D) (b,c)
B. (b,a) Note that in this case A × B ≠ B × A, i.e., the Cartesian product is not commutative.
REA Book logic drill #9 Given three statements, P, Q, and R, suppose it is known that R is true. Which one of the following must be true? A. ( P ∧ Q ) → R B. R → ( P ∨ Q ) C. ( P → Q ) ∧ R D. ( R → Q) ∨ P
Correct Answer: A A conditional statement "X → Y" is true in all instances except when X is true and Y is false. Let X be represented by P ∧ Q and let Y be represented by R. Since R is known to be true and we do not know the truth value of "P ∧ Q," we have either "True → True" or "False → True." In either case "( P ∧ Q ) → R" must be true.
REA Book logic drill #5 Given any two statements P and Q, where Q is a false statement, which one of the following must be false? A. Not P and Q B. Not (P and Q) C. P or not Q D. P implies Q
Correct Answer: A. Given that Q is a false statement, the statement "Not P and Q" must be false. Any compound statement with the conjunction operator (which is "and") is false unless both component parts are true.
Study.com practice test #42 Rebecca wants to borrow $500 to buy her favorite video game player and a couple of games. She asks a friend to lend her this amount, offering to pay him back $548 in 9 months. What is the annual interest rate for this short term loan? A. 0.2 B. 0.128 C. 0.165 D. 0.0825
Correct Answer: B. 0.128 Using the simple interest formula, I=Prt, we substitute $500 for P, 9/12 for t (time expressed in years) and $48 for I (the interest paid, calculated by $548-$500). Solve the equation for r. r + 12.80%
REA Practice test two #24 What is the value of | -(-3) - (15) | + | -6² + 6 |?
The correct answer is 42. |-(-3) - 15 | = | 3 - 15 | = | -12 | = 12 and | -6² + 6 | = | -36 + 6 | = | -30 | = 30. Then 12 + 30 + 42
#58 practice test 1 What is the domain of f(x) = √(7-x) ? A. All numbers less than or equal to 7 B. All numbers greater than or equal to 7 C. All numbers less than or equal to -7 D. All numbers greater than or equal to -7
(A) The domain is defined as all numbers for which 7 - x is at least zero. Solving 7 - x ≥ 0, we get x ≤ 7.
Modern states #6 A company manufactures electronic components that each must weigh from 29.5 grams to 30.5 grams, inclusive. Which of the following inequalities describes all acceptable weights x, in grams, for each component? A. | 30 - x | ≤ 0.5 B. | 30 - x | > 0.5 C. 30 - x ≤ 0.5 D. 30 - x > 0.5
Answer: A. | 30 - x | ≤ 0.5 29.5 ≤ x ≤ 30.5 -0.5 ≤ x - 30 ≤ 0.5 -0.5 ≤ 30 -x ≤ 0.5 | 30 - x | ≤ 0.5
Modern states #24 Let A be a nonempty set and let B and C be any two subsets of A. Which of the following statements must be true? A. B ∪ C = A B. B ∩ C = { } , the empty set C. B ⊂ C ⊂ A D. B ∪ C ⊂ A
Answer: D. B ∪ C ⊂ A
college board practice test #9 3x - 4 ≥ 0 Which of the following subsets of real numbers best describes the solution set of the inequality above? A. [ 0 , 4/3 ) B. [ 4/3 , ∞ ) C. ( -∞ , ∞ ) D. ( -∞ , -4 ] ∪ [ 3 , ∞ )
Correct Answer: B. [ 4/3 , ∞ ) Given: 3x - 4 ≥ 0 ⇒ 3x ≥ 4 ⇒ x ≥ 4/3 Therefore subsets of the real numbers best describes the solution set of the given inequality is [ 4/3 , ∞ )
modern states "If it snows, then school is closed." Which of the following is logically equivalent to the statement above? (A) If it snows, then school is not closed. (B) If school is closed, then it snows. (C) If it does not snow, then school is not closed. (D) If school is not closed, then it does not snow.
D. If school is not closed, then it does not snow. (contrapositive)
#16 practice test 1 If f(x) = {(2, 5), (6, 9), (11, 2), (x, 4)} is a function, what is the smallest value x may NOT have?
The answer is 2. If f(x) is a function, then each member of the domain is mapped into one and only one member of the range. Hence, x may not be 2, 6, or 11; these numbers are already first components of ordered pairs in f(x).
REA Practice test two #48 Let J represent any non-empty set. Consider the following group of combinations of sets: J ∪ ∅, J ∩ ∅, J - J' , J ∩ J' , J - ∅ , ∅ - J How many of these six combinations are equivalent to ∅?
The correct answer is 3, namely J ∩ ∅, J ∩ J' , and ∅ - J . Note that J ∪ ∅ is equivalent to J, J - J' is equivalent to J, and J - ∅ is equivalent to J .
REA Practice test two #40 Using an ordinary deck of 52 playing cards, what is the probability of drawing three black cards in a row, without replacement? A. 1/8 B. 2/17 C. 3/25 D. 4/33
correct answer: (B) Pr (black card on first draw) = 26/52 Pr (black card on second draw) = 25/51 Pr (black card on third draw) = 24/50 Pr (drawing three straight black cards = (26/52) x (25/51) x (25/50) = (2/17)
REA algebra drill #7 If x + 2 is a factor 3x³ - x + k, what is the value of k? A. 22 B. 16 C. -16 D. -22
correct answer: A Based on the factor theorem, if x + 2 is a factor of 3x³ - x + k = 0 . This means that (3)(-2)³ - (-2) + k = 0 . This equation simplifies to (3)(-8) + 2 + k = ) . Thus, k = 24 - 2 = 22 .
#5 Practice Test 1 Not counting the empty set, how many proper subsets are there for R = {2, 3, 4}? A. 5 B. 6 C. 7 D. 8
(B) Given a set of n elements, the number of proper subsets is given by the expression 2ⁿ - 1. The set of proper subsets does not include the set itself. However the expression 2ⁿ - 1 does include the empty set. Thus, the answer is 2ⁿ - 2 = 2³ - 2 = 6.
REA Practice test two #26 How many composite numbers are there between 30 and 40 that also end in an odd digit? A. 2 B. 3 C. 4 D. 5
(B) The three composite numbers between 30 and 40 that end in an odd digit are 33, 35, and 39. Note that the numbers 31 and 37 are prime numbers.
study.com Jenny has a bowl of M&M's that has 6 brown, 3 green, 4 red, and 12 yellow M&M's. She selects a yellow M&M and does not replace it. What is the probability that her second selection will be a brown M&M? A. 6/24 B. 1/4 C. 6/25 D. 3/5 E. 1/5
Correct answer: B. We are looking for one event, the possibility of selecting a brown M&M. If a yellow M&M is removed, the number of M&Ms remaining is 24 (out of which 6 are brown) P (selecting a brown M&M) = 6/24 = 1/4
REA Book logic drill #1 If P and Q represent statements, which one of the following is equivalent to "Not P and not Q"? A. Not P or not Q B. Not P or Q C. Not (P or Q) D. Not (P and Q)
Correct answer: C. One of De Morgan's Laws for sentences is ~(X ∨ Y) ↔ ~X ∧ ~Y. Substituting P for X and Q for Y, we have ~(P ∨ Q) ↔ ~P ∧ ~Q. This statement is read as follows: "Not (P or Q)" is equivalent to "Not P and not Q."
REA Practice test two #29 If Pr(A) = 0.2 (the probability of event A is 1/5), Pr(B) = 0.6, and Pr(A ∩ B) = 0.1, what is Pr(A ∪ B)?
The answer is 0.7. The probability of the union of two events equals the sum of the probabilities of each event separately minus the probability of the intersection. Pr (A ∪ B ) = Pr (A) + Pr (B) - Pr (A ∩ B) = 0.2 + 0.6 - 0.1 = 0.7
study.com practice test #50 If Brian borrows $150,000 at 5% annual interest on a 30-year loan, what is his monthly payment? A. 631.3 B. 627.6 C. 625.6 D. 625
correct answer: B. 627.6 A ={ i x P x (1 + i) ^n } / (1 + i)^n -1 0.0041667 x $150,000 x (1 + 0.0041667)^360 = $625 x 4.4677977 = $2,792.37357 (1 + 0.0041667)^359 = 4.4492590 $2,792.37357 / 4.4492590 = $627.60
Which of the following pairs of lines are perpendicular? A. 2x = y + 1, -2x = y - 2 B. 7y = x - 2, 7x = 3 - y C. 2y = x, y = -x + 1 D. y = 0, y = x + 1 E. y = 3x + 3, y = -3x - 3
correct answer: B. 7y = x - 2, 7x = 3 - y
study.com practice test #48 What is a graphical representation of the distribution of data? A. Range B. Histogram C. Pie Chart D. Frequency Table E. Cumulative Frequency Table
correct answer: B. Histogram A histogram is a graphical representation of the distribution of data.
Modern states #83 Which of the following functions exhibits odd symmetry? A. f(x) = x³ + 1 B. f(x) = x⁵ + 3x³ C. f(x) = x⁷ - x² D. f(x) = x³ + x -2 E. f(x) = x⁵ + 2x⁴
correct answer: B. f(x) = x⁵ + 3x³
#54 practice test 1 Which one of the following is true for any function? A. A horizontal line may only intersect the graph of the function once. B. The inverse must also be a function. C. A vertical line may only intersect the graph of the function once. D. The function must be defined for all real numbers.
(C) Since each x value in the domain can correspond only to one y value in the range, any vertical line can intersect the graph at most once. Answer choice (A) is wrong because the same y value may correspond to two different x values. An example to show why answer choice (B) is wrong would be f (x) = {(1, 2), (2, 3), (4, 3)}. f (x) is a function consisting of three points. Its inverse would be {(2, 1), (3, 2), (3, 4)}, which is not a function. An example to show why answer choice (D) is wrong would be g (x) = {(1, 5), (3, 7)}. f (x) is only defined for x = 1 or x = 3.
#28 diagnostic For which of the following does the reflection of f(x) about the y-axis equal f(x)? A. f (x) = (x+2)² B. f (x) = x C. f (x) = |x| + 5 D. f (x) = √(x²-16)
(C) The graph of is the absolute value "v" translated up to 5 units. This graph is symmetric about the y-axis and, therefore, the reflection of f(x) about the y-axis will be identical to the graph of f(x) .
Assume that x is the standard deviation of the set of the nonzero numbers {a, b, c, d, e} For each of the following sets, indicate which sets must have a standard deviation equal to x. Note: Make sure you select all of the correct options—there may be more than one! A. {a+2, b+2, c+2, d+2, e+2} B. {a-2, b-2, c-2, d-2, e-2} C. {2a, 2b, 2c, 2d, 2e} D. {a/2, b/2, c/2, d/2, e/2}
Answer A. {a+2, b+2, c+2, d+2, e+2} B. {a-2, b-2, c-2, d-2, e-2}
Modern States #4 Which of the following statements is NOT true for all real numbers a and b ? A. (a+b)² - (a-b)² = 4ab B. (a-b)(a+b) = a² - b² C. (a+b)³ = a³ + 3a²b + 3ab² + b³ D. a³ - b³ = (a + b²)(a² - b)
Answer: D. a³ - b³ = (a + b²)(a² - b)
study.com practice test #44 Which of the following connectors is the logical conjunction? A. Both B. Neither C. But D. Or E. And
correct answer: E. And In logic, mathematics and linguistics, And is the truth-functional operator of logical conjunction
#52 practice test 1 The function f (x) is defined as follows: f(x) = 4x-1 , if x ≤ -5 = 5x + 1 , if x > -5 What is the value of f (-7) + f (10)?
The correct answer is 22. f (-7) = (4)(-7) - 1 = -29 and f (10) = (5)(10) + 1 = 51. Then -29 + 51 = 22.
#59 practice test 1 The number 0.8̅ is equivalent to what reduced fraction?
The correct answer is 8/9 Let N = 0.8̅ Multiply both sides of the equation by 10 to get 10N = 8.8̅ Subtract N = 0.8̅ from 10N = 8.8̅ to get 9N = 8 then, N = 8/9
Free Practice test #4 The mayor is going to visit 4 libraries on Monday from a list of 10 possible libraries. The mayor asked her assistant to list every possible permutation of 4 libraries from the list. Which of the following correctly shows the number of different possible permutations? A. 4¹⁰ = 1,048,576 permutations B. 10⁴ = 10,000 permutations C. 10! / (10-4)! = 5,040 permutations D. 10! / 4! (10 - 4)! = 210 permutations
We are told that this is a permutation question and that the order in which the libraries are chosen is important. Method 1 (if we don't know the permutation formula): There are 10 possible choices for the first library. Once we have chosen the first library, there are 9 possible choices remaining for the second library. Once we have chosen the second library, there are 8 possible choices remaining for the fourth library. We simply multiply 10*9*8*7 and get 5,040 permutations. Method 2 (if we know the permutation formula): The formula for the number of ways to arrange r items from a pool of n is nPr = n! / (n-r)! . Substituting 10 for n for and 4 for r , we get 10P4 = 10! / (10 - 4)! , which is what we see in choice (C). (Note: n! means to multiply every integer from n down to 1.) Either way, let's hope that the mayor's assistant didn't waste time writing out every single possible list.
#60 practice test 1 In a room of 20 people, if each person shakes hands with every other person, how many different handshakes are possible? A. 40 B. 190 C. 380 D. 400
(B) Each handshake involves two people, so the number of handshakes for 20 people is given by the expression 20C2 = (20)(19) ÷ 2 = 190.
#31 Practice test 1 How many four-digit numbers are there such that the first digit is odd, the second is even, and there is no repetition of digits? A. 1,200 B. 1,625 C. 200 D. 1,400
(D) There are five possibilities for the first (1, 3, 5, 7, 9), five for the second (0, 2, 4, 6, 8), eight possibilities for the third (10 - 2), and seven for the fourth (8 - 1) = 7. We subtracted 2 possibilities on the third and 3 on the fourth because there is no repetition, as shown below: 5 x 5 x (10 - 2)(8 - 1) = 5 x 5 x 8 x 7 = 1,400
#8 Practice Test 1 Given that i = √-1 , what is the simplified expression for 3i³ - 4i² + 5i? A. -2i - 4 B. -2i + 4 C. 2i - 4 D. 2i + 4
(D) i² = -1 and i³ = i² (i) = (-1) (i) = -i. So, 3i³ - 4i² + 5i = (3)(-i) - (4)(-1) + 5i = 5i - 3i + 4 = 2i + 4.
modern states #64 If f(x) = x² + x - 6 and g(x) = (x-2)/(x+3), then f(g(2)) = A. -56/9 B. -6 C. 0 D. f(g(2)) is undefined
Correct Answer: B. -6
#11 diagnostic What is the simplified expression for (3i+ 2)(3i - 2)? A. -13 B. 6i C. 9i - 4 D. 9i² - 4i - 4
Correct answer: (A) By expanding the two binomials, the product of (3i+2)(3i-2) = 9i² - 6i + 6i -4 combine like terms. = 9i² -4 replace i² with -1. = 9(-1) - 4 simplify. = -13
study.com The probability of selecting a green ball from a bag is 1 out of 8, assuming there are 8 balls in the bag. If you select 3 balls, replacing them each time, what is the probability of at least one of them being green? A. 0.33 B. 0.67 C. 0.875 D. 0.125 E. 0.25
Correct answer: A. P(selecting a green ball) = 1/8 P(not selecting a green ball) = 7/8 Since 3 selections are to be made = 7/8 x 7/8 x 7/8 = 0.67 P(selecting 3 balls and at least one of them being green) = 1-0.67 = 0.33
study.com The local bowling team plays in a 7-team league where each team plays other teams 4 times in a season. Using the combination formula, how many different games will be played in a season? A. 84 B. 35 C. 420 D. 320 E. 105
correct answer: A. There are a total of 7 teams in the league. Each game requires 2 teams. There are, therefore, 7C2 ways of playing each other. This must happen four times since every team must play every other team four times. So the answer is 4 times 7C2 = 4 x (7 x 6 x 5!)/(2 x 1 x 5!) = 4 x (7 x 6 / 2) = 84.
#36 Practice test 1 Which one of the following is a valid argument? A. All rainy days are cloudy.Yesterday was not cloudy.Yesterday was not rainy. B. All trees have brown leaves.This plant has brown leaves.This plant is a tree. C. Some wolves are vicious.This animal is vicious.This animal is a wolf. D. Some people have stocks and bonds.Charles has stocks.Charles has bonds.
(A) An argument is valid if given that the premises are true, then the conclusion must be true. Only answer choice (A) would satisfy this definition. Answer choice (B) is wrong because other objects besides trees may have brown leaves. Answer choice (C) is wrong because animals other than wolves may be vicious. Answer choice (D) is wrong because people may own only stocks, only bonds, both stocks and bonds, or neither stocks nor bonds.
REA Practice test two #55 In how many different ways can all the letters of the word LEASES be arranged without repeating the same sequence of letters? A. 180 B. 360 C. 540 D. 720
(A) The number of different arrangements of these letters is given by the expression(6!)/(2!)(2!) = 720 / 4 = 180 . . Recall that the n! = (n)(n - 1)(n - 2)(...)(1), where n is any whole number.
#24 diagnostic Given two statements, R and S, suppose it is known that S is false. Which of the following MUST be true? A. R ∧ S B. ~ (R ∧ S) C. R ∨ ( S ∧ ~S ) D. (R ∧ S) ∨ ( R ∧ ~S )
(B) Since S is known to be false, the negation of R AND S is true whether R is true or false. Answer choice (A) is never true since S is false. Answer choice (C) and (D) are true only if R is true.
#1 diagnostic Simon has a total of 7 textbooks and a backpack that is capable of holding 3 textbooks at a time. How many combinations of three textbooks are possible in Simon's backpack? A. 21 B. 35 C. 210 D. 343
(B) The combination of 7 things taken 3 at a time is
#7 diagnostic Which one of the following is the converse of the statement shown below? If an animal does not eat meat, then the animal is not a carnivore. A. If an animal eats meat, then the animal is a carnivore. B. If an animal is a carnivore, then the animal eats meat. C. If an animal is not a carnivore, then the animal does not eat meat. D. If an animal is a carnivore, then the animal does not eat meat.
(C) Given a statement in the form "If P, then Q," the converse is in the form "If Q, then P." In this example, P is represented by "an animal does not eat meat" and Q is represented by "the animal is not a carnivore."
#22 diagnostic Which of the following choices represents mutually exclusive events? A. The probability of pulling a red marble out of a sack followed by a green marble without replacement. B. The probability of drawing a red card from a standard deck after drawing five black cards. C. The probability of a flipped coin landing heads up and then tails up. D. The probability of drawing a card that is a red 2.
(C) Since the outcome of a tossed coin landing on heads does not affect the odds for the next time that the coin is tossed, two coin toss outcomes are mutually exclusive events.
REA Practice test two #44 Which one of the following is the negation for the statement "Some women enjoy shopping"? A. Some women do not enjoy shopping. B. At least one woman enjoys shopping. C. No women enjoy shopping. D. All women enjoy shopping.
(C) The statement "Some women enjoy shopping" is equivalent to the statement "At least one woman enjoys shopping." The negation of this statement would mean that it is not true that at least one woman enjoys shopping, which becomes the statement "No women enjoy shopping."
#44 practice test 1 Suppose that point K is reflected across the line y = x. If the original coordinates of K are (-9, 3), what will be K's new coordinates? A. (-9, -3) B. (-3, 9) C. (3, -9) D. (9, -3)
(C) When any point (x, y) is reflected across the line y = x, the coordinates are simply reversed so that the new coordinates become (y, x). Thus, (-9, 3) becomes (3, -9).
#28 Practice test 1 One of the following is equivalent to the statement "If Joan sings, then I will play my guitar"? A. Joan sings and I will play my guitar. B. Joan does not sing and I will not play my guitar. C. Joan sings or I will not play my guitar. D. Joan does not sing or I will play my guitar.
(D) Given a conditional statement in the form "If P then Q," an equivalent statement is in the form "Not P or Q." In this example, P is the statement "Joan sings" and Q is the statement "I will play my guitar."
#29 diagnostic Which of the following sets of numbers combine to form the set of all real numbers? A. The set of odd numbers and even numbers B. The set of prime numbers and composite numbers C. The set of positive numbers and negative numbers D. The set of rational numbers and irrational numbers
(D) The set of all real numbers is composed of the set of rational numbers and the set of irrational numbers. All of the other answer choices contain subsets of the rational numbers only.
#4 diagnostic Let: L = {x|x is an even even integer} , let M = {x|x is a multiple of 6} , and let P = {x|x is an odd negative} Which one of the following statements is NOT true? A. L ∩ M = M∩L B. L ∩ P = ∅ C. M ∩ M ′ = ∅ D. L − M = M − L
(D) The set of even integers minus the set of multiples of 6 is not equal to the set of multiples of 6 minus the set of even integers. For example: 8 is in the set of even integers minus the set of multiples of 6, while it is not in the set of multiples of 6 minus the set of even integers.
#12 diagnostic How many different passwords can be created by first using 4 letters followed by 2 digits including zero, if a letter or number cannot be repeated? A. 48 B. 26!10! C. 26⁴10² D. (26!10!) / (22!8!)
(D) There are 26 different letters in the alphabet; hence, there are 26 choices for the first letter of the password. Since no letter can repeat, there are only 25 choices for the second, and 24 for the third, and 23 for the fourth. Similarly, there are ten digits from 0 to 9. There are 10 choices for the first, and 9 choices for the second. The total number of passwords is found by calculating .
REA Practice test two #4 Which one of the following is equivalent to the statement"If roses are red, then the sky is blue"? A. Roses are red and the sky is blue. B. Roses are not red and the sky is blue. C. Roses are red or the sky is blue. D. Roses are not red or the sky is blue.
(D) When a statement is in the form "If P, then Q," the equivalent statement is in the form "Not P or Q." Let P represent the statement "Roses are red" and let Q represent the statement "The sky is blue." Then "Not P" is represented by "Roses are not red."
#1 Practice test 1 Which one of the following is equivalent to the negation of the statement "Cats are friendly and Bob has a hamster"? A. If cats are friendly, then Bob does not have a hamster. B. If Bob has a hamster, then cats are friendly. C. If cats are not friendly, then Bob has a hamster. D. If Bob does not have a hamster, then cats are not friendly.
A) When a statement is in the form "If P, then Q," the equivalent statement is in the form "Not P or Q." The negation of "Not P or Q" is the statement "P and not Q." Let P represent the statement "Cats are friendly." Let "not Q" represent the statement "Bob has a hamster." Then the given statement in the stem of this question is written in the form "P and not Q," so the negation will be in the form "If P, then Q." Note that Q represents the statement "Bob does not have a hamster."
study.com What kind of logic is presented below? Your dog loves rock and roll music because my dog loves rock and roll music. A. Hasty generalization B. Circular reasoning C. False cause D. Limited choice E. Valid logic
A. - a hasty generalization is when someone makes assumptions without sufficient evidence, such as assuming all dogs love rock and roll because one person's dog does.
Modern States #60 Assume that a and b are positive integers. For each statement below, check if the statement is true (and not false). Note: Make sure you select all of the correct options—there may be more than one! statement: A. If a is divisible by b, then a² is divisible by b² B. If a² is divisible by b, then a is divisible by b C. If a is divisible by b2, then a is divisible by b
Answer: A. If a is divisible by b, then a² is divisible by b² C. If a is divisible by b2, then a is divisible by b
Modern states #15 The faces of a fair cube are numbered 1 through 6; the probability of rolling any number from 1 through 6 is equally likely. If the cube is rolled twice, what is the probability that an even number will appear on the top face in the first roll or that the number 1 will appear on the top face in the second roll? A. 1/12 B. 7/12 C. 2/3 D. 3/4
Answer: B. 7/12 There is just a single word messing you up. The problem says "or", but you are solving as if it said "and". You are getting the probability that they BOTH happen. But you don't need them both to happen -- only one needs to occur to be counted as a success. Find it as follows: P(even on first roll) + P(1 on second roll) - P(even on first AND 1 on second) 3/6 + 1/6 - (3/6)*(1/6) = 4/6 - 3/36 = 4/6 - 1/12 = 8/12 - 1/12 = 7/12
Modern states #78 Suppose the weight of mice is normally distributed with mean 10 and standard deviation 2. Which of the following is the most accurate approximation to the probability a mouse has weight less than 8? A. 0.16 B. 0.20 C. 0.68 D. 0.34 E. 0.32
Correct Answer: A. 0.16 The 2 is the value of one standard deviation. So, there is a 0.68 probability that a mouse has a weight between 8 and 12 grams, and thus there is a 0.32 probability that the weight of a mouse is outside that interval, i.e. below 8 or above 12. By symmetry, this means that there is a 0.16 probability it is below 8.
REA Book logic drill #2 Let R and S represent statements. Consider the following: I. If R then S II. Not R and S III. If S then R Which of the above statements is (are) equivalent to the statement "R is a necessary condition for S"? A. Only I B. I and II C. II and III D. Only III
Correct Answer: D. If R is a necessary condition for S, then by definition S implies R. Also, the statement that "S implies R" is equivalent to the statement "If S then R," which represents item III. Neither of items I or II is equivalent to the given statement.
Rea Sets Drill #5 Which one of the following is an example of disjoint sets? A. {0,1,2,3} and {3,2,1,0} B. {0,2,4,6} and {2,4,6,8} C. {0,3,6,9} and {9,16, 25, 36} D. {0,4,8,12} and {6, 10, 14, 18}
Correct Answer: D. Disjoint sets are those that do not contain any common elements, such as {0, 4, 8, 12} and {6, 10, 14, 18}
college board practice test #27 In a group of students at a four-year college, 60 percent of the students are 25 years old or older and 25 percent of the students have a grade point average of 3.0 or better. The age of a student is independent of their grade point average. What is the probability that a student randomly selected from this group will be 25 years old and will have a grade point average of 3.0 or better? A. 0.85 B. 0.42 C. 0.35 D. 0.15
Correct Answer: D. 0.15 keyword is AND 0.60 x 0.25 = 0.15
#27 diagnostic There are 2 green marbles, 4 yellow marbles, and 7 red marbles in a bag. Three marbles will be randomly selected from the bag without replacement. What is the probability that all three marbles will be red or yellow? A. 2/13 B. 11/13 C. 15/26 D. 11³/13³
Correct answer: (C) The probability that the first marble is red or yellow = ( # of red and yellow marbles) / (total # of marbles) = (4 + 7) / (13) = 11/13 . The probability of the second marble changes since now there is one less red or yellow marble and one less total marbles in the bag. Therefore, the probability that the second marble is red or yellow = 10/12 . Likewise, the probability that the third marble drawn is red or yellow = 9/11 . The total probability that all three marbles drawn are red or yellow = (11/13) x (10/12) x (9/11) = 990/1716 = 15/26
Rea Sets Drill #7 If P ⊆ Q, which one of the following conclusions must be true? A. P is either equal to Q or P is the proper subset of Q B. P is a proper subset of Q C. Q is a proper subset of P D. P is either equal to Q or P is the empty set
Correct answer: A P ⊆ Q means that each element of set P is also an element of set Q. This implies that either (a) P and Q are identical or (b) Q contains all the elements of P, plus at least one element not found in P. By definition, if part (b) applies, then P is a proper subset of Q.
REA algebra drill #2 The solution set of the inequality 5 - 7x ≥ -9 is: A. X = { x | x ≤ 2 } B. X = { x | x ≥ 2 } C. X = { x | x ≤ 4/7 } D. X = { x | x ≥ 4/7 }
Correct answer: A subtract 5 from each side of 5 - 7x ≥ -9, so that 5 - 7x - 5 ≥ -9 - 5, which simplifies to -7x ≥ -14. Next, divide each side by -7. Since we are dividing by a negative number, we must switch the order of the inequality. Then (-7x/-7) ≤ (-14/-7), which becomes x ≤ 2 . This is equivalent to x = { x | x ≤ 2 }
REA real number system drill #9 n² is a proper fraction in reduced form. Which one of the following must be true? A. n is a proper fraction B. n is an improper fraction C. n is a positive number D. n is a negative number
Correct answer: A. Proper fraction = a fraction that is less than one, with the numerator less than the denominator. Improper fraction = a fraction in which the numerator is greater than the denominator, such as 5/4. If n² is a proper fraction in reduced form, then its numerator and denominator must contain no common factors. This implies that its square root, namely n, also contains no common factors. So n must be proper. Note that numerator and denominator of n² must be a perfect square. As an example, let n² = 81/100 . Then n = √(81 / 100 ) = 9 / 10, which is a proper fraction.
REA Book logic drill #3 What is the inverse of the statement "If it is snowing, then people stay indoors"? A. If people stay indoors, then it is snowing B. If it is not snowing, then people do not stay indoors C. If people do not stay indoors, then it is not snowing D. If it is not snowing, then people stay indoors
Correct answer: B. The inverse of "If P then Q" is "If not P then not Q." Let P represent the statement "It is snowing" and let Q represent the statement "People stay indoors." By substitution, for the statement "If it is snowing, then people stay indoors," the inverse statement is "If it is not snowing, then people do not stay indoors."
study.com practice practice test #34 What is the relative frequency of wins if a football team's record is 3 wins and 12 losses? A. 0.25 B. 0.8 C. 0.2 D. 0.75
Correct answer: C. 0.2 Relative frequency is the number of times a specific event occurs divided by the total number of events that occur.
college board practice test #43 If the diameter of the Sun is about 100 times the diameter of the Earth, which of the following is the best approximation for the ratio of the volume of the Sun to the volume of the Earth? A. 10² B. 10⁴ C. 10⁶ D. 10⁸
Correct answer: C. 10⁶ Area of a sphere = (4/3) (π) (r³) Diameter of Earth = 10 , Radius of Earth = 5 Diameter of Sun = 1000, Radius of Sun = 500 Area of Earth = 4/3 * π * 125 Area of Sun = 4/3 * π * 125,000,000
REA real number system drill #10 Which one of the following has no solution for x? A. | x | > 0 B. | x | = 0 C. | x | = -x D. | x | < 0
Correct answer: D. The absolute value of any number must be non negative. This means that | x | must be greater than or equal to zero. Thus, | x | < 0 has no solution.
REA Book logic drill #4 What is the negation for the statement "Image is important or personality matters"? A. Image is important and personality does not matter B. Image is not important or personality does not matter C. Image is important or personality does not matter D. Image is not important and personality does not matter
Correct answer: D. The negation of ( P ∨ Q ) is ~( P ∨ Q ). By one of De Morgan's Laws, the statement ~( P ∨ Q ) is equivalent to ~P ∧ ~Q. Let P represent the statement "Image is important" and let Q represent the statement "Personality matters." Then the negation for "Image is important or personality matters" is "Not (image is important or personality matters)." This latter statement is equivalent to "Image is not important and personality does not matter."
#30 practice test 1 A floor that measures 10 feet by 20 feet is to be tiled with square tiles that are 36 square inches in area. How many tiles are needed to cover the entire floor?
The answer is 800. The floor that measures 10 ft. by 20 ft. has an area of 10 x 20 = 200 sq. ft. The tiles with 36 sq. in. of area must measure 6 in. by 6 in. or sq. ft. of area. Because it would take four tiles to cover 1 sq. ft., 4 x (200 sq. ft.) = 800 tiles would be needed to cover the entire floor.
#11 Practice Test 1 The sum of three different prime numbers is 22. One of these numbers is 2. What is the largest possible value of either of the other numbers?
The correct answer is 17. The sum of the other two numbers must be 20. The possible pairs of integers for which both numbers are prime are 7 and 13 or 3 and 17. Thus, 17 is the largest possible value. Note that 1 and 19 cannot be considered since 1 is not a prime.
Modern States #34 A painting was purchased for $22 million and sold one year later for $26.4 million. The profit on the most recent sale was what percent of the purchase price? A. 25% B. 20% C. 22% D. 30%
answer: B. 20%
Modern States #21 m = 4 and n = 2 For each of the following expressions, check if the value will be a rational number. Note: Make sure you select all of the correct options—there may be more than one! A. ∛(m-n) B. √(m³) C. ∛(m/n) D. √(m+n²)
answer: B. √(m³)
study.com On a 5 question, multiple-choice test, what is the probability that you will get at least one problem correct while guessing? Each question has 5 choices. A. 0.672 B. 0.2 C. 0.328 D. 0.8 E. 0.25
correct answer: A. P(selecting correct answer) = 1/5 P(not selecting correct answer) = 4/5 Since there are 5 questions in the test = (4/5)^5 = 0.328 P(at least one problem correct) = 1-0.328 = 0.672
study.com The name Joe is very common at a school and 1 out of every 10 students go by that name. If there are 15 students in one class, what is the probability that at least one of them is named Joe? A. 0.79 B. 0.1 C. 0.21 D. 0.15 E. 0.68
correct answer: A. P(selecting the name Joe) = 1/10 P(not selecting the name Joe) = 9/10 Since 15 selections are to be made = (0.9)^15 = 0.206 P(at least one of 15 students is named Joe) = 1-0.206 = 0.794
Modern states #72 Suppose a colony of mushrooms triples in size every 10 days. If there are 10 mushrooms to start, how many days until there are 1000 mushrooms? A. 41.9181 days B. 56.4091 days C. 37.9010 days D. 29.8974 days E. 49.3955 days
correct answer: A. 41.9181 days A quantity is said to grow exponentially if it increases by a common factor. A general equation for exponential growth is given by Q=abᵗ where a is the initial value, Q is the quantity at time t and b is the growth factor. Some examples of exponential growth are populations and compounding interest in investments. This is an exponential growth problem, since the mushroom population triples every 10 days. We need to write an equation that will multiply our mushroom population by 3 every ten days. An exponential equation can be written in the form Q = abᵗ . In this problem, our initial value is a = 10 and if t = 10 we have Q = 30. We can use this information to solve for the growth factor b. Q = abᵗ 30 = 10b¹⁰ 3 = b¹⁰ ¹⁰√3 = b Now that we have the growth factor, we can write the general model for the mushroom colony growth as Q = 10(¹⁰√3)ᵗ . To find how many days until there are 1000 mushrooms, substitute Q=1000. Q = 10(¹⁰√3)ᵗ 1000 = 10(¹⁰√3)ᵗ 100 = (¹⁰√3)ᵗ To solve for a variable that is in the exponent, we need to take the logarithm of both sides of the equation. The type of logarithm does not matter - we will use the common logarithm here. 100 = (¹⁰√3)ᵗ log (100) = log(((¹⁰√3)ᵗ) log(100) = t log (¹⁰√3) ( log (100) / log(¹⁰√3) ) = t t = 41.92 Therefore, It will take about 42 days for the mushroom colony to reach 1000 mushrooms. Approach2: Power function approach For every 10 days, the count becomes triple. after how many multiple of 10 (n) days, the count becomes 1000. given by the function: So, 1000= 3n X 10 3n=100 apply logarithm on both sides, we get n = log 100/ log3 n = 4.1918065485787692085931350440428 So, in 10n days, the count becomes 1000. implies, 41.9181 days is the answer.
REA probability drill #5 From four textbooks of college math and three textbooks of college physics, a researcher will choose two math textbooks and one physics text book. How many different groupings of three textbooks are there? A. 12 B. 18 C. 24 D. 36
correct answer: B The number of ways to choose a group of two out of four math textbooks is given by the formula ₄C₂ = 4! / 2! (4-2)! = 24 / (2)(2) = 24 / 4 = 6 . In addition, there are only 3 ways to select one out of three physics textbooks. Thus, the number of different groupings for choosing the two math books and one physics book is (6)(3) = 18
REA functions drill #7 The function g(x) is known to have a range of all real numbers except zero? Which one of the following expressions could represent g(x)? A. 4x² - 4 B. 4 / x C. 4 / x² D. -√(x² - 4)
correct answer: B. Domain - the set of all the values for x for which the function is defined Range - the set of corresponding values of y First determine if there are any values that would make the function undefined (i.e., division by 0). The range is represented by the g(x) values. For g(x) = 4 / x, g(x) can assume any value (including negative numbers) except zero. Also notice that x ≠ 0. A graph of g(x) = 4 / x would also confirm that the range is all numbers except zero.
study.com practice test #29 What is the simplified form of the following exponential term?(5x² / 2x⁴)³? A. 25x⁻⁶ B. (125/8)x⁻⁶ C. 125x⁻⁶ D. (25/4)x
correct answer: B. (125/8)x⁻⁶ The power to a power property of exponents helps to simplify this expression. 5^3 = 125, x^2 raised to 3rd power is x^6. 2^3=8, x^4 raised to third power is x^12. After simplifying, all the x's in the numerator cancel out and leave x^6 in denominator and 125/8 as the constant.
REA geometry drill #2 Parallelogram RSTU is similar to parallelogram WXYZ. if ∠RST = 60° , ∠XYZ = ? A. 60° B. 90° C. 120° D. Not enough information is given
correct answer: C Corresponding angles of similar geometric figures must be congruent. This means that ∠XYZ in parallelogram WXYZ must be congruent to ∠STU in parallelogram RSTU . We note that ∠RST and ∠STU are consecutive angles in RSTU and the sum of and the sum of any two consecutive angles in a parallelogram is 180° . Since ∠RST = 60° , ∠STU = 180° - 60° = 120° . Thus, ∠XYZ = 120° .
REA functions drill #6 If f(x) is a linear function such that f(1) = 5 and f(4) = -7, then what is the value of (-1)? A. 11 B. 12 C. 13 D. 14
correct answer: C The points on the graph of this linear function are (1,5) and (4, -7). The function can be written as y = mx + b, where m is the slope and b is the y-intercept. By substitution of the two given points, we get 5 = (m)(1) + b and -7 = (m)(4) + b . Now subtracting the second of these equations from the first equation leads to 5 - (-7) = (m)(1) + b - (m)(4) - b, which becomes 12 = -3m. So m = -4. Returning to the equation 5 = (m)(1) + b we can substitute -4 for m so that 5 = (-4)(1) + b . Then b = 9 . The equation of this linear function is y = -4x + 9 . Let's replace y with f(x) . Finally, f(-1) = (-4) (-1) + 9 = 4 + 9 = 13
REA functions drill #8 The point (5, -8) is reflected across the line y = x. What is the new location? A. (-5, 8) B. (-5, -8) C. (-8, 5) D. (-8, -5)
correct answer: C. If the point (x ,y ) is reflected across the line y = x, its new location is given by ( y , x ) . Let x = 5 and y = -8 . Then for the point ( 5, -8) , its new location after being reflected across the line y = x is found by simply interchanging the coordinates. The correct answer is (-8, 5)
modern states #88 Which of the following lines is perpendicular to the line y = 1/2x + 4 A. y = 2x -2 B. y = -x + 4 C. y = -2x + 3 D. y = -1/2x - 2 E. y = 4x - 1/2
correct answer: C. y = -2x + 3
REA functions drill #5 Each point of the function f(x) = x² + 10 is moved 3 units to the left and 2 units down to create a new function g(x). What would be the y-coordinate of a point on the graph of g(x) whose x-coordinate is -1 on the graph of f(x)? A. 6 B. 7 C. 8 D. 9
correct answer: D Substitute -1 for x in the function f(x) to get (-1)² + 10 = 1 + 10 = 11 . The corresponding point on the graph f(x) is (-1, 11) . In order to find the corresponding point for g(x), this point will be moved 3 units to the left and 2 units down. Thus, the point (-1 , 11) will become the point ( -4, 9) on the graph of g(x). So, y-coordinate becomes 9.
Modern States #69 Roll two dice. What is the probability of getting a five or higher on the first roll and getting a total of 7 on the two dice. A. 0 B. 1/6 C. 1 D. 1/12 E. 1/18
correct answer: D. 1/18
Modern states #82 Which of the following does not have a well-defined inverse function, defined on its range? A. f(x) = 3x + 1 B. f(x) = e³ˣ C. f(x) = log₁₀(x) D. f(x) = x⁴ + 1 E. f(x) = -x³ + 4
correct answer: D. f(x) = x⁴ + 1 a) f(x)=3x+1f(x)=3x+1 is a linear function which is one-to-one and onto. Also its graph passes the horizontal line test in that no horizontal line cross the curve at two or more points. Hence the function DOES have an inverse. (b) f(x)=e3xf(x)=e3x is an exponential function which is one-to-one and onto. Also its graph passes the horizontal line test in that no horizontal line cross the curve at two or more points. Hence the function DOES have an inverse. (c) f(x)=log10xf(x)=log10x is a common logarithm function which is one-to-one and onto. Also its graph passes the horizontal line test in that no horizontal line cross the curve at two or more points. Hence the function DOES have an inverse. (d) f(x)=x4+1f(x)=x4+1 is a quartic polynomial function which is NOT one-to-one or onto. Also its graph fails the horizontal line test in that there exists a horizontal line that crosses the curve at two points. Hence the function DOES NOT have an inverse. (e) f(x)=−x3+4f(x)=−x3+4 is a cubic function which is one-to-one and onto. Also its graph passes the horizontal line test in that no horizontal line cross the curve at two or more points. Hence the function DOES have an inverse.
