Quantitative Solve the problem and indicate the best of the answer choices given. Numbers: All numbers used are real numbers. Figures: A figure accompanying a problem solving question is intended to provide information useful in solving the problem. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.
1. If 3＜x＜100, for how many values of x is the square of a prime number?
A.Two
B.Three
C.Four
D.Five
E.Nine
A B C D E
B
[解析] Arithmetic Properties of numbers If is the square of a prime number, then possible values of are 22, 32, 52, 72,.... Therefore, possible values of x are 3×22=12, 3×32=27, 3×52=75, 3×72=147,...Since only three of these values, namely 12, 27, and 75, are between 3 and 100, there are three values of x such that is the square of a prime number. The correct answer is B.
2. A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?
A.4
B.5
C.6
D.7
E.8
A B C D E
B
[解析] Arithmetic Elementary combinatorics None of the essential aspects of the problem is affected if the letters are restricted to be the first n letters of the alphabet, for various positive integers n. With the 3 letters a, b, and c, there are 6 codes: a, b, c, ab, ac, and bc. With the 4 letters a, b, c, and d, there are 10 codes: a, b, c, d, ab, ac, ad, bc, bd, and cd. Clearly, more than 12 codes are possible with 5 or more letters, so the least number of letters that can be used is 5. The correct answer is B.
3.
The graph of which of the following equations is a straight line that is parallel to line e in the figure above?
A.3y-2x=0
B.3y+2x=0
C.3y+2x=6
D.2y-3x=6
E.2y+3x=-6
A B C D E
A
[解析] Algebra Coordinate geometry From the graph, line e contains points (-3,0) and (0,2), so the slope of line e is . Any line parallel to line e has slope . Rewrite each of the equations given in the answer choices in slope-intercept form y=mx+b, where m is the slope and b is the y-intercept, to find the equation whose graph is a line with slope . For answer choice A, 3y-2x=0 so 3y=2x and . The graph of this equation is a line with slope The correct answer is A.
4. An object thrown directly upward is at a height of h feet after t seconds, where h=-16(t-3)2+150. At what height, in feet, is the object 2 seconds after it reaches its maximum height?
A.6
B.86
C.134
D.150
E.166
A B C D E
B
[解析] Algebra Applied problems Since (t-3)2 is positive when t≠3 and zero when t=3, it follows that the minimum value of (t-3)2 occurs when t=3. Therefore, the maximum value of -16(t-3)2, and also the maximum value of -16(t-3)2+150, occurs when t=3. Hence, the height 2 seconds after the maximum height is the value of b when t=5, or -16(5-3)2+150=86. The correct answer is B.
5. Which of the following is equivalent to the pair of inequalities x+6＞10 and x-3＜5?
A.2≤x＜16
B.2≤x＜4
C.2＜x≤8
D.4＜x≤8
E.4≤x＜16
A B C D E
D
[解析] Algebra Inequalities Solve the inequalities separately and combine the results. x+6＞10 x＞4 x-3≤5 x≤8 Since x＞4, then 4＜x. Combining 4＜x and x≤8 gives 4＜x≤8. The correct answer is D.
6. David has d books, which is 3 times as many as Jeff and as many as Paula. How many books do the three of them have altogether, in terms of d? A. B. C. D. E.
A B C D E
C
[解析] Algebra Applied problems; Simultaneous equations Let J be the number of books that Jeff has, and let P be the number of books Paula has. Then, the given information about David's books can be expressed as d=3J and . Solving these two equations for J and P gives and 2d=P. Thus, The correct answer is C.
7. There are 8 teams in a certain league and each team plays each of the other teams exactly once. If each game is played by 2 teams, what is the total number of games played?
A.15
B.16
C.28
D.56
E.64
A B C D E
C
[解析] Arithmetic Operations on rational numbers Since no team needs to play itself, each team needs to play 7 other teams. In addition, each game needs to be counted only once, rather than once for each team that plays that game. Since two teams play each game, games are needed. The correct answer is C.
8. At his regular hourly rate, Don had estimated the labor cost of a repair job as $336 and he was paid that amount. However, the job took 4 hours longer than he had estimated and, consequently, he earned $2 per hour less than his regular hourly rate. What was the time Don had estimated for the job, in hours?
A.28
B.24
C.16
D.14
E.12
A B C D E
B
[解析] Algebra Second-degree equations Let r be Doffs regular hourly rate and t be the number of hours he estimated the repair job to take. Then rt=336 is Doffs estimated labor cost. Since Don was paid $336 for doing t+4 hours of work at an hourly rate of r-2, it also follows that (r-2)(t+4)=336. Then Alternatively, from the third line above, So, t-24=0, which means t=24, or t+28=0, which means t=-28. Since an estimated time cannot be negative, t=24. The correct answer is B.
9. , and p and q are positive integers, which of the following must be greater than 1? A. B. C. D. E.
A B C D E
E
[解析] Arithmetic Properties of numbers Since p and q are positive integers, A. Since then q＞p. Taking the square root of both sides of the inequality gives so here the denominator will still be larger than the numerator. CANNOT be greater than 1. B. Squaring the denominator increases the denominator, which decreases the value of the fraction. CANNOT be greater than 1. C. Multiplying the denominator by 2 increases the denominator, which decreases the value of the fraction. CANNOT be greater than 1. D. Since , then q＞p. When p2＜q, this expression will be greater than 1, but p2 need not be less than q. For example, if p=2 and q=100, and However, if p=3 and q=4, then and NEED NOT be greater than 1. E. Again, since , then q＞p. Thus, the reciprocal, , always has a value greater than 1 because the numerator will always be a larger positive integer than the denominator. MUST be greater than 1. The correct answer is E.
10. To mail a package, the rate is x cents for the first pound and y cents for each additional pound, where x＞y. Two packages weighing 3 pounds and 5 pounds, respectively, can be mailed separately or combined as one package. Which method is cheaper, and how much money is saved?
A.Combined, with a savings of x-y cents
B.Combined, with a savings of y-x cents
C.Combined, with a savings of x cents
D.Separately, with a savings of x-y cents
E.Separately, with a savings of y cents
A B C D E
A
[解析] Algebra Applied problems Shipping the two packages separately would cost 1x+2y for the 3-pound package and 1x+4y for the 5-pound package. Shipping them together (as a single 8-pound package) would cost 1x+7y. By calculating the sum of the costs for shipping the two packages separately minus the cost for shipping the one combined package, it is possible to determine the difference in cost, as shown. Since x＞y, this value is positive, which means it costs more to ship two packages separately. Thus it is cheaper to mail one combined package at a cost savings of x-y cents. The correct answer is A.
11. If money is invested at r percent interest, compounded annually, the amount of the investment will double in approximately years. If Pat's parents invested $5,000 in a long-term bond that pays 8 percent interest, compounded annually, what will be the approximate total amount of the investment 18 years later, when Pat is ready for college?
A.$20,000
B.$15,000
C.$12,000
D.$10,000
E.$9,000
A B C D E
A
[解析] Algebra Applied problems Since the investment will double in years, the value of the investment over 18 years can be approximated by doubling its initial value twice. Therefore, the approximate value will be ($5,000 )(2)(2)=$20,000. The correct answer is A.
12. On a recent trip, Cindy drove her car 290 miles, rounded to the nearest 10 miles, and used 12 gallons of gasoline, rounded to the nearest gallon. The actual number of miles per gallon that Cindy's car got on this trip must have been between A. B. C. D. E.
A B C D E
D
[解析] Arithmetic Estimation The lowest number of miles per gallon can be calculated using the lowest possible miles and the highest amount of gasoline. Also, the highest number of miles per gallon can be calculated using the highest possible miles and the lowest amount of gasoline. Since the miles are rounded to the nearest 10 miles, the number of miles is between 285 and 295. Since the gallons are rounded to the nearest gallon, the number of gallons is between 11.5 and 12.5. Therefore, the lowest number of miles per gallon is and the highest number of miles per gallon is The correct answer is D.
13.
Which of the following inequalities is an algebraic expression for the shaded part of the number line above?
A.|x|≤3
B.|x|≤5
C.|x-2|≤3
D.|x-1|≤4
E.|x+1|≤4
A B C D E
E
[解析] Algebra Inequalities The number line above shows -5≤x≤3. To turn this into absolute value notation, as all the choices are written, the numbers need to be opposite signs of the same value. Since the distance between -5 and 3 is 8 (3-(-5)=8), that distance needs to be split in half with -4 to one side and 4 to the other. Each of these two values is 1 more than the values in the inequality above, so adding 1 to all terms in the inequality gives -4≤x+1≤4, which is the same as |x+1|≤4. The correct answer is E.
14. In a small snack shop, the average (arithmetic mean) revenue was $400 per day over a 10-day period. During this period, if the average daily revenue was $360 for the first 6 days, what was the average daily revenue for the last 4 days?
A.$420
B.$440
C.S450
D.$460
E.$480
A B C D E
D
[解析] Arithmetic; Algebra Statistics, Applied problems Let x be the average daily revenue for the last 4 days. Using the formula , the information regarding the average revenues for the 10-day and 6-day periods can be expressed as follows and solved for x: The correct answer is D.
15. If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be
A.2
B.5
C.6
D.7
E.14
A B C D E
E
[解析] Arithmetic Properties of numbers To find the smallest positive integer y such that 3,150y is the square of an integer, first find the prime factorization of 3,150 by a method similar to the following: 3,150=10×315 =(2×5)×(3×105) =2×5×3×(5×21) =2×5×3×5×(3×7) =2×32×52×7 To be a perfect square, 3,150y must have an even number of each of its prime factors. At a minimum, y must have one factor of 2 and one factor of 7 so that 3,150y has two factors of each of the primes 2, 3, 5, and 7. The smallest positive integer value of y is then (2)(7)=14. The correct answer is E.
16. If [x] is the greatest integer less than or equal to x, what is the value of [-1.6]+[3.4]+[2.7]?
A.3
B.4
C.5
D.6
E.7
A B C D E
A
[解析] Arithmetic Profit and loss The greatest integer that is less than or equal to -1.6 is -2. It cannot be -1 because -1 is greater than -1.6. The greatest integer that is less than or equal to 3.4 is 3. It cannot be 4 because 4 is greater than 3.4. The greatest integer that is less than or equal to 2.7 is 2. It cannot be 3 because 3 is greater than 2.7. Therefore, [-1.6]+[3.4]+[2.7]=-2+3+2=3. The correct answer is A.
17. In the first week of the year, Nancy saved $1. In each of the next 51 weeks, she saved $1 more than she had saved in the previous week. What was the total amount that Nancy saved during the 52 weeks?
A.$1,326
B.$1,352
C.$1,378
D.$2,652
E.$2,756
A B C D E
C
[解析] Arithmetic Operations on rational numbers In dollars, the total amount saved is the sum of 1, (1+1), (1+1+1), and so on, up to and including the amount saved in the 52nd week, which was $52. Therefore, the total amount saved in dollars was 1+2+3+...+50+51+52. This sum can be easily evaluated by grouping the terms as (1+52)+(2+51)+(3+50)+...+(26+27), which results in the number 53 added to itself 26 times. Therefore, the sum is (26)(53)=1,378. Alternatively, the formula for the sum of the first n positive integers is Therefore, the sum of the first 52 positive integers is The correct answer is C.
18. In a certain sequence, the term xn is given by the for all n≥2. If x0=3 and x1=2, what is the value of x3?
A.2.5
B.3.125
C.4
D.5
E.6.75
A B C D E
C
[解析] Algebra Simplifying algebraic expressions Given the formula with x0=3 and x1=2, then The correct answer is C.
19. During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine's average speed for the entire trip? A. B. C. D. E.
A B C D E
E
[解析] Algebra Applied problems Assume for simplicity that the total distance of Francine's trip is 100 miles. Then the table below gives all of the pertinent information.
The correct answer is E.
20. If n=(33)43+(43)33, what is the units digit of n?
A.0
B.2
C.4
D.6
E.8
A B C D E
A
[解析] Arithmetic Properties of numbers If the units digit of an integer n is 3, then the units digits of n1, n2, n3, n4, n5, n6, n7, and n8 are, respectively, 3, 9, 7, 1, 3, 9, 7, and 1. Thus, the units digit of the powers of n form the sequence in which the digits 3, 9, 7, and 1 repeat indefinitely in that order. Since 43=(10)(4)+3, the 43rd number in the sequence is 7, and therefore the units digit of(33)43 is 7. Since 33=(8)(4)+1, the 33rd number of this sequence is 3, and therefore, the units digit of (43)33 is 3. Thus, the units digit of (33)43+(43)33 is the units digit of 7+3, which is 0. The correct answer is A.
21. Team A and Team B are competing against each other in a game of tug-of-war. Team A, consisting of 3 males and 3 females, decides to line up male, female, male, female, male, female. The lineup that Team A chooses will be one of how many different possible lineups?
A.9
B.12
C.15
D.36
E.720
A B C D E
D
[解析] Arithmetic Elementary combinatorics Any of the 3 males can be first in the line, and any of the 3 females can be second. Either of the 2 remaining males can be next, followed by either of the 2 remaining females. The last 2 places in the line are filled with the only male left followed by the only female left. By the multiplication principle, there are 3×3×2×2×1×1=36 different lineups possible. The correct answer is D.
22. A border of uniform width is placed around a rectangular photograph that measures 8 inches by 10 inches. If the area of the border is 144 square inches, what is the width of the border, in inches?
A.3
B.4
C.6
D.8
E.9
A B C D E
A
[解析] Algebra Second-degree equations
Let x be the width, in inches, of the border. The photograph with the border has dimensions (10+2x) inches and (8+2x) inches with an area of (10+2x)(8+2x)=(80+36x+4x2) square inches. The photograph without the border has dimensions 10 inches and 8 inches with an area of (10)(8)=80 square inches. The area of the border is then the difference between the areas of the photograph with and without the border or (80+36x+4x2)-80=36x+4x2 square inches. It is given that the area of the border is 144 square inches so, So, x-3=0, which means x=3, or x+12=0, which means x=-12. Thus, after discarding x=-12 since the width of the border must be positive, x=3. The correct answer is A.
23. is expressed as a terminating decimal, how many nonzero digits will d have?
A.One
B.Two
C.Three
D.Seven
E.Ten
A B C D E
B
[解析] Arithmetic Operations on rational numbers It will be helpful to use the fact that a factor that is an integer power of 10 has no effect on the number of nonzero digits a terminating decimal has. The correct answer is B.
24. For any positive integer n, the sum of the first n positive integers equals . What is the sum of all the even integers between 99 and 301?
A.10,100
B.20,200
C.22,650
D.40,200
E.45,150
A B C D E
B
[解析] Algebra Simplifying expressions; Arithmetic Computation with integers The given formula translates into The sum of the even integers between 99 and 301 is the sum of the even integers from 100 through 300, or the sum of the 50th even integer through the 150th even integer. To get this sum, find the sum of the first 150 even integers and subtract the sum of the first 49 even integers. In symbols, The correct answer is B.
25. How many prime numbers between 1 and 100 are factors of 7,150?
A.One
B.Two
C.Three
D.Four
E.Five
A B C D E
D
[解析] Arithmetic Rate To find the number of prime numbers between 1 and 100 that are factors of 7,150, find the prime factorization of 7,150 using a method similar to the following: Thus, 7,150 has four prime factors: 2, 5, 11, and 13. The correct answer is D.
26. A sequence of numbers a1, a2, a3...is defined as follows: a1=3, a2=5, and every term in the sequence after a2 is the product of all terms in the sequence preceding it, e.g., a3=(a1)(a2) and a4=(a1)(a2)(a3). If an=t and n＞2, what is the value of an+2 in terms of t?
A.4t
B.t2
C.t3
D.t4
E.t8
A B C D E
D
[解析] Algebra Sequences It is given that an=(a1)(a2)...(an-1) and an=t. Therefore, an+1=(a1)(a2)...(an-1)(an)=(an)(an)=t2 and an+2=(a1)(a2)...(an)(an+1)=(an+1)(an+1)=(t2)(t2)=t4. The correct answer is D.
27. Last year the price per share of Stock X increased by k percent and the earnings per share of Stock X increased by m percent, where k is greater than m. By what percent did the ratio of price per share to earnings per share increase, in terms of k and m? A. B. (k-m)% C. D. E.
A B C D E
D
[解析] Algebra Percents If P and E are the price and earnings per share before the increase, then P and E are the price and earnings per share1+100 after the increase. Therefore, the percent increase in the ratio of price per share to earnings per share can be expressed as follows: The correct answer is D.
28. Of the 300 subjects who participated in an experiment using virtual-reality therapy to reduce their fear of heights, 40 percent experienced sweaty palms, 30 percent experienced vomiting, and 75 percent experienced dizziness. If all of the subjects experienced at least one of these effects and 35 percent of the subjects experienced exactly two of these effects, how many of the subjects experienced only one of these effects?
A.105
B.125
C.130
D.180
E.195
A B C D E
D
[解析] Arithmetic Applied problems Let a be the number who experienced only one of the effects, b be the number who experienced exactly two of the effects, and c be the number who experienced all three of the effects. Then a+b+c=300, since each of the 300 participants experienced at least one of the effects. From the given information, b=105 (35% of 300), which gives a+105+c=300, or a+c=195 (Eq.1). Also, if the number who experienced sweaty palms (40% of 300, or 120) is added to the number who experienced vomiting (30% of 300, or 90), and this sum is added to the number who experienced dizziness (75% of 300, or 225), then each participant who experienced only one of the effects is counted exactly once, each participant who experienced exactly two of the effects is counted exactly twice, and each participant who experienced all three of the effects is counted exactly 3 times. Therefore, a+2b+3c=120+90+225=435. Using b=105, it follows that a+2(105)+3c=435, or a+3c=225 (Eq.2). Then solving the system defined by Eq.1 and Eq.2, -2a=-360, or a=180 The correct answer is D.
29. then m-2 is equal to A. -9 B. -3 C. D. E. 9
A B C D E
D
[解析] Arithmetic Negative exponents Using rules of exponents, m2=m-1 2=(m-1)2, and since The correct answer is D.
30. A photography dealer ordered 60 Model X cameras to be sold for $250 each, which represents a 20 percent markup over the dealer's initial cost for each camera. Of the cameras ordered, 6 were never sold and were returned to the manufacturer for a refund of 50 percent of the dealer's initial cost. What was the dealer's approximate profit or loss as a percent of the dealer's initial cost for the 60 cameras?
A.7% loss
B.13% loss
C.7% profit
D.13% profit
E.15% profit
A B C D E
D
[解析] Arithmetic Percents Given that $250 is 20% greater than a camera's initial cost, it follows that the initial cost for each camera was . Therefore, the initial cost for the 60 cameras was 60 . The total revenue is the sum of the amount obtained from selling 60-6=54 cameras for $250 each and the refund for each of 6 cameras, or . The total profit, as a percent of the total initial cost, is Finally, (0.13×100)%=13%, which represents a profit since it is positive. The correct answer is D.
31. Seven pieces of rope have an average (arithmetic mean) length of 68 centimeters and a median length of 84 centimeters. If the length of the longest piece of rope is 14 centimeters more than 4 times the length of the shortest piece of rope, what is the maximum possible length, in centimeters, of the longest piece of rope?
A.82
B.118
C.120
D.134
E.152
A B C D E
D
[解析] Algebra Statistics Let a, b, c, d, e, f and g be the lengths, in centimeters, of the pieces of rope, listed from least to greatest. From the given information it follows that d=84 and g=4a+14. Therefore, listed from least to greatest, the lengths are a, b, c, 84, e, f, and 4a+14. The maximum value of 4a+14 will occur when the maximum value of a is used, and this will be the case only if the shortest 3 pieces all have the same length. Therefore, listed from least to greatest, the lengths are a, a, a, 84, e, f and 4a+14. The maximum value for 4a+14 will occur when e and fare as small as possible. Since e and fare to the right of the median, they must be at least 84 and so 84 is the least possible value for each of e and f. Therefore, listed from least to greatest, the lengths are a, a, a, 84, 84, 84, and 4a+14. Since the average length is 68, it follows that or a=30. Hence, the maximum length of the longest piece is (4a+14)=[4(30)+14]=134 centimeters. The correct answer is D.
32. What is the difference between the sixth and the fifth terms of the sequence 2, 4, 7...whose nth term is n+2n-1?
A.2
B.3
C.6
D.16
E.17
A B C D E
E
[解析] Algebra Simplifying algebraic expressions According to the given formula, the sixth term of the sequence is 6+26-1=6+25 and the fifth term is 5+25-1=5+24. Then, The correct answer is E.
33. From the consecutive integers -10 to 10, inclusive, 20 integers are randomly chosen with repetitions allowed. What is the least possible value of the product of the 20 integers?
A.(-10)20
B.(-10)10
C.0
D.-(10)19
E.-(10)20
A B C D E
E
[解析] Arithmetic Properties of numbers If -10 is chosen an odd number of times and 10 is chosen the remaining number of times (for example, choose -10 once and choose 10 nineteen times, or choose -10 three times and choose 10 seventeen times), then the product of the 20 chosen numbers will be -(10)20. Note that-(10)20 is less than -(10)19, the only other negative value among the answer choices. The correct answer is E.
34. The letters D, G, I, I, and T can be used to form 5-letter strings such as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter?
A.12
B.18
C.24
D.36
E.48
A B C D E
D
[解析] Arithmetic Elementary combinatorics There are 6 ways to select the locations of the 2 occurrences of the letter Ⅰ, and this number can be determined by listing all such ways as shown below, where the symbol * is used in place of the letters D, G, and T: Ⅰ*Ⅰ**, Ⅰ**Ⅰ*, Ⅰ***Ⅰ, *Ⅰ*Ⅰ*, *Ⅰ**Ⅰ, **Ⅰ*Ⅰ Alternatively, the number of ways to select the locations of the 2 occurrences of the letter Ⅰ can be determined by using , which is the number of ways to select 2 of the 5 locations minus the 4 ways in which the 2 selected locations are adjacent. For each of these 6 ways to select the locations of the 2 occurrences of the letter Ⅰ, there are 6 ways to select the locations of the letters D, G, and T, which can be determined by using 3!=6 or by listing all such ways: DGT, DTG, GDT, GTD, TDG, TGD It follows that the number of ways to select the locations of the 5 letters to form 5-letter strings is (6)(6)=36. The correct answer is D.
35.
A.10-8
B.3(10-8)
C.3(10-4)
D.2(10-4)
E.10-4
A B C D E
D
[解析] Arithmetic Operations on rational numbers Calculations with lengthy decimals can be avoided by writing 0.99999999 as 1-10-8, 0.99999991 as 1-9(10-8), 1.0001 as 1+10-4, and 1.0003 as 1+3(10-4). Doing this gives The correct answer is D.
36. Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for $1.25 each, and the store sold no other newspapers that day. If r percent of the store's revenue from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of Newspaper A, which of the following expresses r in terms of p? A. B. C. D. E.
A B C D E
D
[解析] Algebra Simultaneous equations Let N be the total number of newspapers that the store sold. Then, the number of copies of Newspaper A the store sold was p% of and the revenue from those copies of Newspaper A, in dollars, was . The number of copies of Newspaper B the store sold was (100-p)% of and the revenue from those copies of Newspaper B, in dollars, was The store's total revenue from newspaper sales in dollars, was and the fraction of that revenue from the sale of Newspaper A was Since r percent of the store's newspaper sales revenue was from Newspaper A, and so The correct answer is D.
37. For the past n days, the average (arithmetic mean) daily production at a company was 50 units. If today's production of 90 units raises the average to 55 units per day, what is the value of n?
A.30
B.18
C.10
D.9
E.7
A B C D E
E
[解析] Arithmetic; Algebra Statistics; Applied problems; Simultaneous equations Let x be the total production of the past n days. Using the formula average the information in the problem can be expressed in the following two equations Solving the first equation for x gives x=5On. Then substituting 50n for x in the second equation gives the following that can be solved for n: The correct answer is E.
38.
In the coordinate system above, which of the following is the equation of line e?
A.2x-3y=6
B.2x+3y=6
C.3x+2y=6
D.2x-3y=-6
E.3x-2y=-6
A B C D E
B
[解析] Geometry Simple coordinate geometry The line is shown going through the points (0,2) and (3,0). The slope of the line can be found with the formula slope for two points (x1,y1) and (x2,y2). Thus, the slope of this line equals . Using the formula for a line of y=mx+b, where m is the slope and b is the y-intercept (in this case, 2), an equation for this line is . Since this equation must be compared to the available answer choices, the following further steps should be taken: This problem can also be solved as follows. From the graph, when x=0, y is positive; when y=0, x is positive. This eliminates all but B and C. Of these, B is the only line containing (0,2). Still another way is to use (0,2) to eliminate A, C, arid E, and then use (3,0) to eliminate D. The correct answer is B.
39. If a two-digit positive integer has its digits reversed, the resulting integer differs from the original by 27. By how much do the two digits differ?
A.3
B.4
C.5
D.6
E.7
A B C D E
A
[解析] Algebra Applied problems Let the one two-digit integer be represented by 10t+s, where s and t are digits, and let the other integer with the reversed digits be represented by 10s+t. The information that the difference between the integers is 27 can be expressed in the following equation, which can be solved for the answer. Thus, it is seen that the two digits s and t differ by 3. The correct answer is A.
40. In an electric circuit, two resistors with resistances x and y are connected in parallel. In this case, if r is the combined resistance of these two resistors, then the reciprocal of r is equal to the sum of the reciprocals of x and y. What is r in terms of x and y? A. xy B. x+y C. D. E.
A B C D E
D
[解析] Algebra Applied problems Note that two numbers are reciprocals of each other if and only if their product is 1. Thus the reciprocals of r, x, and y are , and , respectively. So, according to the problem, . To solve this equation for r, begin by creating a common denominator on the right side by multiplying the first fraction by and the second fraction by The correct answer is D.
41. Xavier, Yvonne, and Zelda each try independently to solve a problem. If their individual probabilities for success are respectively, what is the probability that Xavier and Yvonne, but not Zelda, will solve the problem? A. B. C. D. E.
A B C D E
E
[解析] Arithmetic Probability Since the individuals' probabilities are independent, they can be multiplied to figure out the combined probability, The probability of Xavier's success is given as , and the probability of Yvonne's success is given as Since the probability of Zelda's success is given as , then the probability of her NOT solving the problem is Thus, the combined probability is The correct answer is E.
42. then x could be
A.0
B.-1
C.-2
D.-3
E.-4
A B C D E
C
[解析] Algebra Second-degree equations Solve the equation for x. Begin by multiplying all the terms by x(x+1)(x+4) to eliminate the denominators. This problem can also be solved as follows. Rewrite the left side as, then set equal to the right side to get . Next, cross multiply: (1)(x+4)=x(x+1)(1). Therefore, x+4=x2+x, or x2=4, so x=±2. The correct answer is C.
43. A. B. C. D. E.
A B C D E
B
[解析] Arithmetic Operations on rational numbers It is clear from the answer choices that all three factors need to be written with a common denominator, and they thus become The correct answer is B.
44.
The figure shown above consists of a shaded 9-sided polygon and 9 unshaded isosceles triangles. For each isosceles triangle, the longest side is a side of the shaded polygon and the two sides of equal length are extensions of the two adjacent sides of the shaded polygon. What is the value of a?
A.100
B.105
C.110
D.115
E.120
A B C D E
A
[解析] Geometry Polygons
Let x°represent the measure of each base angle of the triangle with vertex angle labeled a°. Each base angle of this triangle and one base angle of a triangle with which it shares a vertex are vertical angles and have the same measure. THUS, the base angles of these triangles also have measure x°. This pattern continues for the base angles of each pair of triangles that share a vertex, so each base angle of each of the 9 triangles has measure x°, as shown above. Also, the vertex angle of each of the 9 triangles has measure a°=180°-2x°. Each interior angle of the shaded polygon has measure (180-x)° since each forms a straight angle with an angle that has measure x°, and the sum of the measures is (9)(180-x)°. But the sum of the interior angles of a polygon with n sides is (n-2)(180°), so the sum of the interior angles of a 9-sided polygon is (7)(180°)=1,260°. Therefore, (9)(180-x)°=1,260° and x=40. Finally, a=180-2x=180-2(40)=100. The correct answer is A.
45. List T consists of 30 positive decimals, none of which is an integer, and the sum of the 30 decimals is S. The estimated sum of the 30 decimals, E, is defined as follows. Each decimal in T whose tenths digit is even is rounded up to the nearest integer, and each decimal in T whose tenths digit is odd is rounded down to the nearest integer; E is the sum of the resulting integers. If of the decimals in T have a tenths digit that is even, which of the following is a possible value of E-S? Ⅰ. -16 Ⅱ. 6 Ⅲ. 10
A.Ⅰ only
B.Ⅰ and Ⅱ only
C.Ⅰ and Ⅲ only
D.Ⅱ and Ⅲ only
E.Ⅰ, Ⅱ, and Ⅲ
A B C D E
B
[解析] Arithmetic Operations on rational numbers; Since of the 30 decimals in T have an even tenths digit, it follows that decimals in T have an even tenths digit. Let TE represent the list of these 10 decimals, let SE represent the sum of all 10 decimals in TE, and let EE represent the estimated sum of all 10 decimals in TE after rounding, The remaining 20 decimals in T have an odd tenths digit. Let To represent the list of these 20 remaining decimals, let So represent the sum of all 20 decimals in To, and let EO represent the estimated sum of all 20 decimals in To after rounding. Note that E=EE+EO and S=SE+SO and hence E-S=(EE+EO)-(SE+SO)=(EE-SE)(EO-SO). The least values of EE-SE occur at the extreme where each decimal in TE has tenths digit 8. Here, the difference between the rounded integer and the original decimal is greater than 0.1. (For example, the difference between the integer 15 and 14.899 that has been rounded to 15 is 0.101.) Hence, EE-SE＞10(0.1)=1. The greatest values of EE-SE occur at the other extreme, where each decimal in TE has tenths digit 0. Here, the difference between the rounded integer and the original decimal is less than 1. (For example, the difference between the integer 15 and 14.001 that has been rounded to 15 is 0.999.) Hence, EE-SE＜10(1)=10. Thus, 1＜EE-SE＜10. Similarly, the least values of EO-SO occur at the extreme where each decimal in TO has tenths digit 9. Here, the difference between the rounded integer and the original decimal is greater than -1. (For example, the difference between the integer 14 and 14.999 that has been rounded to 14 is -0.999.) Hence EO-SO＞20(-1)=-20. The greatest values of EO-SO occur at the other extreme where each decimal in TO has tenths digit 1. Here, the difference between the rounded integer and the original decimal is less than or equal to -0.1. (For example, the difference between the integer 14 and 14.1 that has been rounded to 14 is -0.1.) Hence, EO-SO≤20(-0.1)=-2. Thus, -20＜EO-SO≤-2. Adding the inequalities 1＜EE-SE＜10 and -20＜ EO-SO≤-2 gives-19＜(EE-SE)+(EO-SO)＜8. Therefore, -19＜(EE+EO)-(SE+SO)＜8 and -19＜E-8＜8. Thus, of the values -16, 6, and 10 for E-S, only -16 and 6 are possible. Note that if T contains 10 repetitions of the decimal 1.8 and 20 repetitions of the decimal 1.9, S=10(1.8)+20(1.9)=18+38=56, E=10(2)+20(1)=40, and E-S=40-56=-16. Also, if T contains 10 repetitions of the decimal 1.2 and 20 repetitions of the decimal 1.1, S=10(1.2)+20(1.1)=12+22=34, E=10(2)+20(1)=40, and E-S=40-34=6. The correct answer is B.
46. , then x has how many possible values?
A.None
B.One
C.Two
D.A finite number greater than two
E.An infinite number
A B C D E
C
[解析] Algebra Second-degree equations Solve the equation to determine how many values are possible for x. The correct answer is C.
47. Seed mixture X is 40 percent ryegrass and 60 percent bluegrass by weight; seed mixture Y is 25 percent ryegrass and 75 percent fescue. If a mixture of X and Y contains 30 percent ryegrass, what percent of the weight of the mixture is X? A. 10% B. C. 40% D. 50% E.
A B C D E
B
[解析] Algebra Applied problems Let X be the amount of seed mixture X in the final mixture, and let Y be the amount of seed mixture Y in the final mixture. The final mixture of X and Y needs to contain 30 percent ryegrass seed, so any other kinds of grass seed are irrelevant to the solution to this problem. The information about the ryegrass percentages for X, Y, and the final mixture can be expressed in the following equation and solved for X. Using this, the percent of the weight of the combined mixture (X+Y) that is X is The correct answer is B.
48. How many of the integers that satisfy the inequality are less than 5?
A.1
B.2
C.3
D.4
E.5
A B C D E
D
[解析] Algebra Inequalities
Pictorially, the number line above shows the algebraic signs of the expressions (x+3), (x+2), and (x-2). For example, x+3 is 0 when x=-3, x+3 is negative when x＜-3, and x+3 is positive when x＞-3. The expression will be positive in the intervals of the number line where the number of minus signs is even. Therefore is positive for values of x such that -3＜x＜-2 and for values of x such that x＞2. The only integer values of x in these intervals that are also less than 5 are 3 and 4. Also, will be zero if and only if (x+2)(x+3)=0, which has two integer solutions less than 5, namely, x=-2 and x=-3. Therefore, there are four integers less than 5 that satisfy and those integers are -3, -2, 3, and 4. Alternatively, will be zero if and only if (x+2)(x+3)=0, which has two integer solutions less than 5, namely, x=-2 and x=-3. Also, will be positive if (x+2)(x+3) and x-2 are both positive or both negative, and for no other values of x. On the one hand, (x+2)(x+3) will be positive when x+2 and x+3 are both positive, which will be the case when x＞-2 and x＞-3 and thus when x＞-2. On the other hand, (x+2)(x+3) will be positive when x+2 and x+3 are both negative, which will be the case when x＜-2 and x＜-3 and thus when x＜-3. So, (x+2)(x+3) will be positive when x＜-3 or x＞-2. (This result can also be deduced from the fact that the graph of y=(x+2)(x+3) is a parabola with x-intercepts (-2,0) and (-3,0) that opens upward.) Since x-2 will be positive when x＞2, it follows that (x+2)(x+3) and x-2 are both positive when x＞2, which includes exactly two integer values less than 5, namely, x=3 and x=4. There are no integer values of x such that (x+2)(x+3) and x-2 are both negative, since (x+2)(x+3) is negative if and only if x lies between -3 and -2 and there are no integers between -3 and -2. Therefore, there are exactly 4 integer values of x less than 5 such that Two of the values, x=-2 and x=-3, arise from solutions to , and two of the values, x=3 and x=4, arise from solutions to The correct answer is D.
49. Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30 percent have a swimming pool. If 5 of the houses have all three of these amenities and 5 have none of them, how many of the houses have exactly two of these amenities?
A.10
B.45
C.50
D.55
E.65
A B C D E
D
[解析] Arithmetic Sets
Since 60% of the 150 houses have air-conditioning, b+c+d+5=0.6(150)=90, so b+c+d=85 (ⅰ). Similarly, since 50% have a sunporch, a+b+f+5=0.5(150)=75, so a+b+f=70 (ii). Likewise, since 30% have a swimming pool, d+e+f+5=0.3(150)=45, so d+e+f=40 (ⅲ). Adding equations (ⅰ), (ⅱ), and (ⅲ) gives (b+c+d)+(a+b+f)+(d+e+f)=195, or a+2b+c+2d+e+2f=195 (ⅳ). But a+b+c+d+e+f+5+5=150, or a+b+c+d+e+f=140 (ⅴ). Subtracting equation (ⅴ) from equation (ⅳ) gives b+d+f=55, so 55 houses have exactly two of the amenities. The correct answer is D.
50. The value of is how many times the value of 2(-17)? A. B. C. 3 D. 4 E. 5
A B C D E
C
[解析] Arithmetic Negative exponents If the value of is x times the value of 2-17, then The correct answer is C.
the square of a prime number?
is the square of a prime number, then possible values of 
are 22, 32, 52, 72,....
is the square of a prime number.
. Any line parallel to line e has slope 
. Rewrite each of the equations given in the answer choices in slope-intercept form y=mx+b, where m is the slope and b is the y-intercept, to find the equation whose graph is a line with slope 
. For answer choice A, 3y-2x=0 so 3y=2x and 
. The graph of this equation is a line with slope 
as many as Paula. How many books do the three of them have altogether, in terms of d?
. Solving these two equations for J and P gives 
and 2d=P. Thus, 
games are needed.
, and p and q are positive integers, which of the following must be greater than 1?
then q＞p. Taking the square root of both sides of the inequality gives 
so here the denominator will still be larger than the numerator. CANNOT be greater than 1.
, then q＞p. When p2＜q, this expression will be greater than 1, but p2 need not be less than q. For example, if p=2 and q=100, 
and 
and 
NEED NOT be greater than 1.
, then q＞p. Thus, the reciprocal, 
, always has a value greater than 1 because the numerator will always be a larger positive integer than the denominator. MUST be greater than 1.
years. If Pat's parents invested $5,000 in a long-term bond that pays 8 percent interest, compounded annually, what will be the approximate total amount of the investment 18 years later, when Pat is ready for college?
years, the value of the investment over 18 years can be approximated by doubling its initial value twice. Therefore, the approximate value will be ($5,000 )(2)(2)=$20,000.
and the highest number of miles per gallon is 
, the information regarding the average revenues for the 10-day and 6-day periods can be expressed as follows and solved for x:
Therefore, the sum of the first 52 positive integers is 
for all n≥2. If x0=3 and x1=2, what is the value of x3?
with x0=3 and x1=2, then
is expressed as a terminating decimal, how many nonzero digits will d have?
. What is the sum of all the even integers between 99 and 301?
The sum of the even integers between 99 and 301 is the sum of the even integers from 100 through 300, or the sum of the 50th even integer through the 150th even integer. To get this sum, find the sum of the first 150 even integers and subtract the sum of the first 49 even integers. In symbols,
P and 
E are the price and earnings per share1+100 after the increase. Therefore, the percent increase in the ratio of price per share to earnings per share can be expressed as follows:
then m-2 is equal to
. Therefore, the initial cost for the 60 cameras was 60 
. The total revenue is the sum of the amount obtained from selling 60-6=54 cameras for $250 each and the
refund for each of 6 cameras, or 
. The total profit, as a percent of the total initial cost, is
or a=30. Hence, the maximum length of the longest piece is (4a+14)=[4(30)+14]=134 centimeters.
, which is the number of ways to select 2 of the 5 locations minus the 4 ways in which the 2 selected locations are adjacent.
and the revenue from those copies of Newspaper A, in dollars, was 
. The number of copies of Newspaper B the store sold was (100-p)% of 
and the revenue from those copies of Newspaper B, in dollars, was 
The store's total revenue from newspaper sales in dollars, was 
and the fraction of that revenue from the sale of Newspaper A was
and so 
the information in the problem can be expressed in the following two equations
for two points (x1,y1) and (x2,y2). Thus, the slope of this line equals 
. Using the formula for a line of y=mx+b, where m is the slope and b is the y-intercept (in this case, 2), an equation for this line is 
. Since this equation must be compared to the available answer choices, the following further steps should be taken:
, and 
, respectively. So, according to the problem, 
. To solve this equation for r, begin by creating a common denominator on the right side by multiplying the first fraction by 
and the second fraction by 
respectively, what is the probability that Xavier and Yvonne, but not Zelda, will solve the problem?
, and the probability of Yvonne's success is given as 
Since the probability of Zelda's success is given as 
, then the probability of her NOT solving the problem is 
then x could be
then set equal to the right side to get 
. Next, cross multiply:
of the decimals in T have a tenths digit that is even, which of the following is a possible value of E-S?
of the 30 decimals in T have an even tenths digit, it follows that 
decimals in T have an even tenths digit. Let TE represent the list of these 10 decimals, let SE represent the sum of all 10 decimals in TE, and let EE represent the estimated sum of all 10 decimals in TE after rounding, The remaining 20 decimals in T have an odd tenths digit. Let To represent the list of these 20 remaining decimals, let So represent the sum of all 20 decimals in To, and let EO represent the estimated sum of all 20 decimals in To after rounding. Note that E=EE+EO and S=SE+SO and hence E-S=(EE+EO)-(SE+SO)=(EE-SE)(EO-SO).
, then x has how many possible values?
are less than 5?
will be positive in the intervals of the number line where the number of minus signs is even. Therefore 
is positive for values of x such that -3＜x＜-2 and for values of x such that x＞2. The only integer values of x in these intervals that are also less than 5 are 3 and 4. Also, 
will be zero if and only if (x+2)(x+3)=0, which has two integer solutions less than 5, namely, x=-2 and x=-3.
and those integers are -3, -2, 3, and 4.
will be zero if and only if (x+2)(x+3)=0, which has two integer solutions less than 5, namely, x=-2 and x=-3. Also, 
will be positive if (x+2)(x+3) and x-2 are both positive or both negative, and for no other values of x. On the one hand, (x+2)(x+3) will be positive when x+2 and x+3 are both positive, which will be the case when x＞-2 and x＞-3 and thus when x＞-2. On the other hand, (x+2)(x+3) will be positive when x+2 and x+3 are both negative, which will be the case when x＜-2 and x＜-3 and thus when x＜-3. So, (x+2)(x+3) will be positive when x＜-3 or x＞-2. (This result can also be deduced from the fact that the graph of y=(x+2)(x+3) is a parabola with x-intercepts (-2,0) and (-3,0) that opens upward.) Since x-2 will be positive when x＞2, it follows that (x+2)(x+3) and x-2 are both positive when x＞2, which includes exactly two integer values less than 5, namely, x=3 and x=4. There are no integer values of x such that (x+2)(x+3) and x-2 are both negative, since (x+2)(x+3) is negative if and only if x lies between -3 and -2 and there are no integers between -3 and -2. Therefore, there are exactly 4 integer values of x less than 5 such that 
, and two of the values, x=3 and x=4, arise from solutions to 
is how many times the value of 2(-17)?
is x times the value of 2-17, then
深色：已答题 浅色：未答题