2018年GMAT年度回顾——数学|沃邦年终总结


来源:   时间:2019-01-31 13:29:32

根据官方指南,GMAT 数学中可能出现的考点可归为四大类:算术、代数、几何、应用题。具体涉及的概念如下:
 
算术:
整数性质、分数、小数、实数、比值及比例、百分比、次方及次方根、描述性统计、集合、排列组合、离散概率;
代数:代数式化简、方程、解一元一次方程、解二元一次方程组、因式分解、解二次方程、指数运算、不等式、绝对值、函数;
几何:直线、相交线及角度、垂线、平行线、凸多边形、三角形、四边形、圆、长方体及圆柱体、坐标几何;
应用题:速率问题、工作效率问题、混合物问题、利率问题、打折问题、利润问题、集合问题、几何问题、单位换算问题、图表数据理解。

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对于广大中国考生而言,GMAT 数学通常都不是一项高要求的挑战,然而这并不意味着考试中所有题目都易如反掌:首先,GMAT 数学部分不允许参试者使用计算器,这一方面对手算的可靠度提出了要求,另一方面也迫使考生在某些原本可依赖计算器的题目中必须动脑筋巧算;其次,数论、集合容斥原理、排列组合、概率、统计等几个概念可称得上是相对的难点,尤其对于数学基础本不雄厚的文科生而言,更可能带来不小困难;最后,GMAT 数学特有的题型——数据充分性解题思路比较贴别,也值得额外留意。下面就看一看上述几个难点的例题:

1. 数论

GMAT 数学中会出现一些涉及数论的题目,尽管其中仅仅要求运用最为基本的数论定理,但往往它们也足以对考生造成不小的困扰。
 
例1
The positive two-digit integers x and y have the same digits, but in reverse order. Which of the following must be a factor of x+y?
A. 6
B. 9
C. 10
D. 11
E. 14
答案为 D。
 
例2
If x and y are positive integers such that y is a multiple of 5 and 3x+2y=200, then x must be a multiple of which of the following?
A. 3
B. 6
C. 7
D. 8
E. 10
答案为 E。
 
例3
If k is a positive integer, what is the remainder when (k+2)(k^3-k) is divided by 6?
A. 0
B. 1
C. 2
D. 3
E. 4
答案为 A。
 
例4
If n=20!+17, then n is divisible by which of the following?
I. 15
II. 17
III. 19
A. None
B. I only
C. II only
D. I and II
E. II and III
答案为 C。
 
例5
If n is an integer greater than 6, which of the following must be divisible by 3?
A. n(n+1)(n-4)
B. n(n+2)(n-1)
C. n(n+3)(n-5)
D. n(n+4)(n-2)
E. n(n+5)(n-6)
答案为 A。
 
例6
If n is a positive integer and the product of all the integers from 1 to n, inclusive, is divisible by 990, what is the least possible value of n?
A. 8
B. 9
C. 10
D. 11
E. 12
答案为 D。
 
例7
When positive integer x is divided by positive integer y, the remainder is 9. If x/y=96.12, what is the value of y?
A. 96
B. 75
C. 48
D. 25
E. 12
答案为 B。
 
例8
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18
答案为 C。
 
例9
If y is the smallest positive integer such that 3150 multiplied by y is the square of an integer, then y must be
A. 2
B. 5
C. 6
D. 7
E. 14
答案为 E。
 
例10
If n=33^43+43^33, what is the units digit of n?
A. 0
B. 2
C. 4
D. 6
E. 8
答案为 A。
 
例11
If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonzero digits will d have?
A. One
B. Two
C. Three
D. Seven
E. Ten
答案为 B。
 
例12
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
答案为 A。
 
例13
If n is a positive integer, for which of the following values of k i 25*10^n+k*10^2n divisible by 9?
A. 9
B. 16
C. 23
D. 35
E. 47
答案为 E。
 
例14
What is the remainder when 3^24 is divisible by 5?
A. 0
B. 1
C. 2
D. 3
E. 4
答案为 B。
 
例15
A positive integer n is a perfect number provided that the sum of all the positive factors of n, including 1 and n, is equal to 2n. If k is a perfect number, what is the sum of the reciprocals of all the positive factors of k?
A. 1/4
B. 56/27
C. 2
D. 3
E. 4
答案为 C。
 
例16
For every even positive integer m, f(m) represents the product of all even integers from 2 to m, inclusive. For example, f(12)=2*4*6*8*10*12. What is the greatest prime factor of f(24)?
A. 23
B. 19
C. 17
D. 13
E. 11
答案为 E。
 
例17
The function f is defined for each positive three-digit integer n by f(n)=(2^x)(3^y)(5^z), where x, y, and z are the hundreds, tens, and units digits of n, respectively. If m and v are three-digit positive integers such that f(m)=9f(v), then m-v=
A. 8
B. 9
C. 18
D. 20
E. 80
 
例18
If 10^50-74 is written as an integer in base 10 notation, what is the sum of the digits in that integer?
A. 424
B. 433
C. 440
D. 449
E. 467
 
例19
If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is
A. 6
B. 12
C. 24
D. 36
E. 48
  

2. 集合的容斥原理

GMAT 数学中涉及集合的难题主要在于容斥原理的应用。
 
例20
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
答案为 D。
  

3. 排列组合

GMAT 数学中也会出现排列组合问题,而它们常常就会给考生带来相当大的挑战。
 
例21
Clarissa will create her summer reading list by randomly choosing 4 books from the 10 book approved for summer reading. She will list the books in the order in which they are chosen. How many different lists are possible?
A. 6
B. 40
C. 210
D. 5040
E. 151200
答案为 D。
 
例22
A three-digit code for certain locks uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 according to the following constraints. The first digit cannot be 0 or 1, the second digit must be 0 or 1, and the second and third digits cannot both be 0 in the same code. How many different codes are possible?
A. 144
B. 152
C. 160
D. 168
E. 176
答案为 B。
 
例23
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
答案为 D。
 
例24
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 occurences of the letter I are separated by at least one other letter?
A. 12
B. 18
C. 24
D. 36
E. 48
答案为D。
 
例25
A certain university will select 1 of 7 candidates eligible to fill a position in the mathematics department and 2 of 10 candidates eligible to fill 2 identical positions in the computer science department. If none of the candidates is eligible for a position in both departments, how many different sets of 3 candidates are there to fill the 3 positions?
A. 42
B. 70
C. 140
D. 165
E. 315
 
例26
How many subsets of the set {w, x, y, z} contain w?
A. Four
B. Five
C. Seven
D. Eight
E. Sixteen
 
例27
There are 5 cars to the displayed in 5 parking spaces, with all the cars facing the same direction. Of the 5 cars, 3 are red, 1 is blue, and 1 is yellow. If the cars are identical except for color, how many different display arrangements of the 5 cars are possible?
A. 20
B. 25
C. 40
D. 60
E. 125
 
例28
There are 10 books on a shelf, of which 4 are paperbacks and 6 are hardbacks. How many possible selections of the 5 books from the shelf contain at least one paperback and at least one hardback?
A. 75
B. 120
C. 210
D. 246
E. 252
  

4. 概率与统计

GMAT 数学涉及的概率及统计问题对于没接触过统计课的考生来说,或会造成一点困难。
 
例29
In a box of 12 pens, a total of 3 are defective. If a customer buys 2 pens selected at random from the box, what is the probability that neither pen will be defective?
A. 1/6
B. 2/9
C. 6/11
D. 9/16
E. 3/4
答案为 C。
 
例30
For each student in a certain class, a teacher adjusted the student’s test score using the formula y=0.8x+20, where x is the student’s original test score and y is the student’s adjusted test score. If the standard deviation of the original test scores of the students in the class was 20, what was the standard deviation of the adjusted test scores of the students in the class?
A. 12
B. 16
C. 28
D. 36
E. 40
答案为 B。
 
例31
The probability that event M will not occur is 0.8 and the probability that event R will not occur is 0.6. If events M and R cannot both occur, which of the following is the probability that either event M or event R will occur?
A. 1/5
B. 2/5
C. 3/5
D. 4/5
E. 12/25
答案为 C。
 
例32
A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 percent of the distribution lies within one standard deviation d of the mean, what percent of the distribution is less than m+d?
A. 16%
B. 32%
C. 48%
D. 84%
E. 92%
答案为 D。
 
例33
A couple decides to have 4 children. If they succeed in having 4 children and each child is equally likely to be a boy or a girl, what is the probability that they will have exactly 2 girls and 2 boys?
A. 3/8
B. 1/4
C. 3/16
D. 1/8
E. 1/16
  

5. 数据充分性

在 GMAT 数学考试中,有一类比较独特的题型:数据充分性。这类题目一方面测试数学知识,另一方面更着重考查学生的逻辑思维。面对这类问题,许多考生会对相应的解题思路比较陌生,也更容易犯错。
 
例34
If x and y are positive integers, what is the remainder when 10^x+y is divided by 3?
(1) x=5
(2) y=2
答案为 B。
 
例35
If x is a positive integer, is x^0.5 an integer?
(1) (4x)^0.5 is an integer
(2) (3x)^0.5 is not an integer
答案为 A。
 
例36
If a and b are positive integers, what is the value of the product ab?
(1) The least common multiple of a and b is 48.
(2) The greatest common factor of a and b is 4.
答案为 C。
 
例37
Last year in a group of 30 businesses, 21 reported a net profit and 15 had investments in foreign markets. How many of the businesses did not report a net profit nor invest in foreign markets last year?
(1) Last year 12 of the 30 businesses reported a net profit and had investments in foreign markets.
(2) Last year 24 of the 30 businesses reported a net profit or invested in foreign markets, or both.
答案为 D。
 
例38
If a, b, and c are consecutive integers and 0<a<b<c, is the product abc a multiple of 8?
(1) The product ac is even.
(2) The product bc is a multiple of 4.
答案为 A。
 
例39
Of a group of 50 households, how many have at least one cat or at least one dog, but not both?
(1) The number of households that have at least one cat and at least one dog is 4.
(2) The number of households that have no cats and no dogs is 14.
答案为 C。
 
例40
If x and y are positive integers, is xy even?
(1) x^2+y^2-1 is divisible by 4.
(2) x+y is odd.
答案为 D。
 
例41
Jill has applied for a job with each of the two different companies. What is the probability that she will get the job offer from both companies?
(1) The probability that she will get a job offer from neither company is 0.3.
(2) The probability that she will get a job offer from exactly one of the two companies is 0.5.
答案为 C。
 
例42
A scientist recorded the number of eggs in each of 10 birds’ nests. What was the standard deviation of the numbers of eggs in the 10 nests?
(1) The average (arithmetic mean) number of eggs for the 10 nests was 4.
(2) Each of the 10 nests contained the same number of eggs.
答案为 B。
 
例43
The range of the numbers in set S is x, and the range of the numbers in set T is y. If all of the numbers in set T are also in set S, is x greater than y?
(1) Set S consists of 7 numbers.
(2) Set T consists of 6 numbers.
答案为 E。
 
例44
A certain list consists of 3 different numbers. Does the median of the 3 numbers equal the mean of the 3 numbers?
(1) The range of the 3 numbers is equal to twice the difference between the greatest numbers and the median.
(2) The sum of the 3 numbers is equal to 3 times one of the numbers.
答案为 D。
 
例45
If x, y, and z are three-digit positive integers and if x=y+z, is the hundreds digit of x equal to the sum of the hundreds digits of y and z?
(1) The tens digit of x is equal to the sum of the tens digits of y and z.
(2) The units digit of x is equal to the sum of the units digits of y and z.
答案为 A。
 
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