GMAT Handout Math

新东方在线 [www.koolearn.com]网络课堂电子教材系列 GMAT 数学

GMAT数学电子讲义

主讲：王燚

欢迎使用新东方在线电子教材

GMAT数学备考关键词

一、知识点：准确掌握二、词汇、表达法：读懂题目三、熟练：平均两分钟一道题

考试相关问题

一、时间与题量二、题型三、机经与换题库四、其它

If a and b are positive integers such that a – b and a/b are both even

integers, which of the following must be an odd integer?

（A） a/2 （B） b/2 （C）(a+b)/2

（D） (a+2)/2 （E） (b+2)/2If M is the least common multiple of 90, 196, and 300, which of the

following is NOT a factor of M?

(A) 600 (B)700 (C) 900 (D) 2,100 (E) 4,900

复习注意事项

*战略上重视

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*初等数学的思维 *解法力求稳妥清晰*把握好 DS 题型*熟练重于技巧

推荐复习步骤

*知识点查缺补漏 *背熟词汇 *复习课上所学*OG，及其它相关资料*机经：www.chasedream.com

第一章算术

1. integer (whole number): 整数

* positive integer：正整数，从 1 开始，不包括 0。

2. odd & even number 奇数与偶数

* 凡整数均具有奇偶性，如－1 是奇数，0 是偶数。* 奇+奇=偶，奇+偶=奇… 若干个整数相乘，除非都是奇数，其乘积才会是奇数…

例： If a and b are positive integers such that a – b and are both even integers, which

of the following must be an odd integer?

(A） (B） (C) (D) (E)

3. prime number & composite number 质数与合数* A prime number is a positive integer that has exactly two different positive

divisors,1 and itself. * A composite number is a positive integer greater than 1 that has more than two

divisors. * The numbers 1 is neither prime nor composite, 2 is the only even prime number.

4. factor(divisor) & prime factor 因子和质因子

* 一个数能被哪些数整除，这些数就叫它的因子（因数、约数）。* 因子里的质数叫质因子（数）。

例 1： If n=4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n?

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(A) 2 (B) 3 (C) 4 (D) 6 (E) 8

例 2 ： If the integer n has exactly three positive divisors, including 1 and n, how many positive divisors does n2 have?

(A) 4 (B) 5 (C) 6 (D) 8 (E) 9

例 3：1225 有几个因子？例：What is the greatest prime factor of 2100 - 296? (A) 2 (B) 3 (C) 5 (D) 7 (E) 11

例： A positive integer n is said to be “prime-saturated” if the product of all the different positive prime factors of n is less than the square root of n. What is the greatest two-digit

prime-saturated integer?

(A) 99 (B) 98 (C) 97 (D) 96 (E) 95

5. the greatest common divisor (GCD)& the least

common multiple(LCM) 最大公约数和最小公倍数

例：If M is the least common multiple of 90, 196, and 300, which of the following is NOT a factor of M?

(A) 600 (B)700 (C) 900 (D) 2,100 (E) 4,900

例：What is the lowest positive integer that is divisible by each of the integers 1 through 7,inclusive?

(A) 420 (B) 840 (C) 1,260 (D) 2,520 (E) 5,040

6. decimals & fractions 小数和分数

* 相关词汇： reaccuring decimal ; terminating decimal ; numerator ; denominator ; improper fracion ; mixed number *整数位与分位：后面加 s 的是整数位（小数点前面的某位），加 th 或 ths 的是分位（小数点后面的某位），如 tens 是十位数，而 tenth 是十分位*What is the fractional part of ….这样的表达法意为“谁的几分之几”*小数和分数的互相转换：例 1： 0．373737…=? (将其转换成一个分数)

例 2：Which of the following fractions has a decimal equivalent that is a terminating decimal? (A) 10/189 (B) 15/196 (C) 16/225 (D) 25/144 (E) 39/128

7. consecutive numbers 连续数

例 1：In an increasing sequence of 10 consecutive integers, the sum of the first 5 integers is 560. What is the sum of the last 5 integers in the sequence?

(A) 585 (B) 580 (C )575 (D)570 (E) 565

例 2：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)

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8. divisibility & remainder 整除及余数问题

* 一个数是否能够被 5 整除，只要看它的最后一位（是 0 或 5）。* 一个数是否能够被 4 整除，只要看它的后两位（是否是 4 的倍数）。* 一个数是否能够被 8 整除，只要看它的后三位（是否是 8 的倍数）。* 一个数能否被 3 整除，取决于各位之和能否被 3 整除。* 一个数能否被 9 整除，也取决于各位之和能否被 9 整除。* 0 能被所有数整除。* 余数包括 0，如 24 除以 6，商为 4 余数为 0。例：1912 257 的个位数字是几？

例：If s and t are positive integers such that ,which of the following could be the

remainder when s is divided by t?

(A) 2 (B) 4 (C) 8 (D) 20 (E) 45

9. 数字问题

例：1001 位数字组成的数，任意相邻的两位数字组成的数能被 17 或 23 整除，这个1001 位的数字以 6 开头，则它的最后六位是（）

10. 算术部分的几种常用方法

*参数法例：两个两位数个位与十位恰好颠倒，问下面哪个不能是两数之和？A.181，B.121，C.77，D.132，E.154

解法：设两数分别为 ab 和 ba，则(ab)+(ba)=(10a+b)+(10b+a)=11(a+b)，即和必为 11 的倍数，答案为 A。

*代数法*试错法例：

□△

× △□The product of the two-digit numbers above is the three-digit number □◇□, where

□,△,and◇ are three different nonzero digits. If □×△＜10, what is the two-digit number

□△?

(A) 11 (B) 12 (C) 13 (D) 21 (E) 31

第二章代数

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1． Quadratic equations: 一元二次方程ax2+bx+c=0

但一般更常用的是因式分解法：x2-2x-3=0 (x-3)(x+1)=0 x1=3, x2=-1

2. Simultaneous linear equations: 多元一次方程组

* 基本方法：消元法。例 1：3x+y=5 (1) 2x+y=4 (2)

(1)－(2), 消去 y, 得 x=1,y=2

* 注意：并不是任何二元一次方程组都有唯一解。例 2: 3x+y=5 (1) 6x+2y=10 (2)

上述方程有无穷多组解。因此，方程的数量须等于未知数的数量，此时多元一次方程有唯一的一组解。

3. Simultaneous quadratic equations: 二元二次方程组

一般只考如下形式：a1x+b1y=c1 (1)

a2x2+b2x+a3y2+b3y=c2 (2)

即其中一个方程为一次。这种形式等价于一元二次方程，把（1）代入（2）即可。

4. Inequalities: 不等式

*不等式部分不会像中国高考那样考推导、证明，注意两边乘以负数变号等最基本原则即可。

5. Arithmetic sequence: 等差数列an=a1+(n-1)d

sn=(a1+an)n/2

n=(an-a1)/d +1

6．Geometric sequence: 等比数列an=a1qn-1

当∣q∣<1 时，

例：

例： 0．373737…=? (将其转换成一个分数)

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7．Sets: 集合

例 1：全班 50 个人，选音乐课的有 20 人，选体育课的有 18 人，两课都选的有 5 人，问两课都没选的几人？例 2： A marketing firm determined that, of 200 households surveyed, 80 used neither Brand A nor Brand B soap, 60 used only Brand A soap, and for every household that used both

brands of soap, 3 used only Brand B sop. How many of the 200 households surveyed used

both brands of soap?

(A) 15 (B)20 (C)30 (D)40 (D)45

例 3：五个人排队，甲不能在首位，乙不能在末位，有几种不同的排法？

第三章几何

1． Lines & planes 直线与平面

* 两直线平行并为第三条直线所截后，相应角的关系。* 直线与平面的关系。例：If n distinct planes intersect in a line, and another line L intersects one of these planes in a single point, what is the least number of these n planes that L could intersect?

(A) n (B) n－1 (C) n－2 (D) n/2 (E)(n－1)/2

2. Triangles 三角形

* 勾股定理：a2+b2=c2

* 构成三角形的条件：两边之和大于第三边。* 三角形内部边和角的关系：大边对大角。

3. Quadrilaterals 四边形

* parallelogram(平行四边形) : 面积=a×h; 周长=2（a+b）

* rectangle(矩形) : 面积=a×h; 周长=2（a+b）

* square(正方形) : 面积=a2 ; 周长=4a

* trapezoid(梯形) : 面积=（a+b）×h/2

4. Circles 圆

* 面积=πR2

* 周长=2πR

5. Polygons 多边形

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* 多边形内角和：（n-2）180º

6. Rectangular Solids 长方体

* 体积=a×b×c

* 表面积=2（a×b+b×c+c×a）

7. Cubes 正方体

* 体积=a3

* 表面积=6a2

8. Cylinders 圆柱

* 体积=πR2h* 表面积=2πR2+2πR×h例：一个圆锥内接于一个半球，圆锥的底面与半球的底面重合，则圆锥的高与半球的半径的比是多少？

9. Coordinate Geometry 解析几何

* 直线的标准方程： y=kx+b ；即斜截式，其中 k 为斜率 slope，b 为 y轴截距 y-intercept* 斜率的计算：K= (Y2-Y1)/( X2-X1) * 两点或一点加斜率确定一条直线。* 两直线垂直，其斜率的乘积为－1。

第四章统计1. arithmetic mean (average) 算术平均值

E=

2. median 中位数* The median is the middle value of a list when the numbers are in order.* 先排序，后取中。

3. mode 众数* The mode of a list of numbers is the nmuber that occurs most frequently in the list.* A list of numbers may have more than one mode.

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4. expectation 期望

* 期望就是算术平均值。

5. deviation 偏差

di=ai-E

6. variance 方差

7. standard deviation 标准差

例：Ⅰ.72,73,74,75,76.74,74,74,74,74Ⅱ

.62,74,74,74,89Ⅲ

The data sets , ,and above are orderedⅠ Ⅱ Ⅲ from greatest standard deviation to least standard

deviation in which of the following ?

(A) , , Ⅰ Ⅱ Ⅲ (B) , ,Ⅰ Ⅲ Ⅱ (C) , Ⅱ Ⅲ, Ⅰ (D) Ⅲ, ,Ⅰ Ⅱ (E) ,Ⅲ Ⅱ, Ⅰ

8. range 范围

* 最大数减去最小数所得的差就是该组数据的范围。例 1：150, 200, 250, nWhich of the following could be the median of the 4 integers listed above?

Ⅰ. 175 Ⅱ. 215 Ⅲ. 235

(A) Ⅰonly (B) Ⅱonly (C) Ⅰand Ⅱonly (D) Ⅱand Ⅲ only (E) Ⅰ,Ⅱ,and Ⅲ

例 2：The least and greatest numbers in a list of 7 real numbers are 2 and 20,respectively. The median of the list is 6,and the number 3 occurs most often in the list. Which of the following

could be the average of the numbers in the list?

Ⅰ. 7 Ⅱ. 8.5 Ⅲ. 10.5

(A)Ⅰonly (B) ⅠandⅡonly (C) Ⅰand Ⅲ only (D) Ⅱand Ⅲ only (E) Ⅰ,Ⅱ,and Ⅲ

第五章数据充分性题

*每道 DS 题的选项都是固定的：

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A Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is

sufficient.

D Each statement ALONE is sufficient.

E Statement (1) and (2) TOGETHER are not sufficient.(additional data are needed).

* DS 题的本质是一种判断型的选择题，并非判断正误，而是判断根据条件给的信息能否回答主题干里提出的问题。

*要注意的几大问题：<1> 唯一性例：x=?

（1）x=2

（2）x2=4

<2> 否定性例：x>0?

（1）x2>0

（2）x3<0

<3> 不矛盾性例：A，B 两车在长直道路上相对行驶，现距离为 500英里，问多长时间后相遇？(1) 其中一辆速度为 200英里每小时。(2) 其中一辆速度为 300英里每小时。

<4> 独立性例：x>0?

（1）√x=5

（2）x3<0

<1> If n is an integer, is n+1 odd?

(1) n+2 is an even integer.

(2) n-1 is an odd integer.

<2> In △PQR,if PQ=x, QR=x+2, PR=y, which of the three angles of △PQR has the greatest

degree measure?

(1) y=x+3

(2) x=2

<3> Tom, Jane, and Sue each purchased a new house . The average (arithmetic mean )price of

the three houses was $120,000.What was the median price of the three houses?

(1) The price of Tom’s house was $110,000.

(2) The price of Jane’s house was $120,000.

<4> 3.2□△6, □=?

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(1) 3.2□△6 四舍五入到十分位后是 3.2。(2) 3.2□△6 四舍五入到百分位后是 3.24。

<5>If °represents one of the operations +, -,and ×,is k°(l +m)=(k°l)+(k°m)for all numbers k, l ,

and m?

(1) k°1 is not equal to 1°k for some numbers k.

(2) °represents subtraction.

<6>On Jane's credit card account, the average daily balance for a 30-day billing cycle is the

average of the daily balances at the end of each of 30 days. At the beginning of a certain 30-

day billing cycle, Jane's credit card account had a balance of $600. Jane made a payment of

$300 on the account during the billing cycle. If no other amounts were added to or subtracted

from the account during the billing cycle, what was the average daily balance on Jane’s

account for the billing cycle?

(1) Jane’s payment was credited on the 21st day of the billing cycle.

(2) The average daily balance through the 25th day of the billing cycle was $540.

第六章排列组合与概率1. Permutation & combination: 排列与组合

*

从 m 个元素中挑出 n 个并进行排列（需要考虑 n 个元素的内部顺序）的所有情况

的数量。

*

从 m 个元素中挑出哪 n 个元素（不考虑 n 个元素的内部顺序）的所有情况的数量。

*

2．Probability: 概率

* 概率的古典定义：P(A)=A 所包含的基本事件数/基本事件总数。例：掷一个骰子，掷出的是个奇数的概率是多少？

练习题：<1> 一只袋中装有五只乒乓球，其中三只白色，两只红色。现从袋中取球两次，每次一只，取出后不再放回。求：①两只球都是白色的概率；②两只球颜色不同的概率；③至少有一只白球的概率。

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<2> 从 5 位男同学和 4 位女同学中选出 4 位参加一个座谈会，要求与会成员中既有男同学又有女同学，有几种不同的选法？

<3> 电话号码由四个数字组成，每个数字可以是 0,1,2,…,9 中的任一个数，求电话号码是由完全不同的数字组成的概率。

<4> 晚会上有 5 个不同的唱歌节目和 3 个不同的舞蹈节目，问：分别按以下要求各可排出几种不同的节目单？* 3 个舞蹈节目排在一起；

* 3 个舞蹈节目彼此隔开；

* 3 个舞蹈节目先后顺序一定。<5> 6张同排连号的电影票，分给 3名男生和 3名女生，如欲男女相间而坐，则不同的分法数为多少？

<6> 用 0,2,4,6,9 这五个数字可以组成数字不重复的五位偶数共有多少个？

<7> 从 6双不同的手套中任取 4 只，求其中恰有一双配对的概率。

<8> 3封不同的信，有 4 个信箱可供投递，共有多少种投信的方法？

<9> 3 个打字员为 4家公司服务，现在每家公司各有 1份文件需要录入，问每个打字员都收到文件的概率？

<10> A certain roller coaster has 3 cars, and a passenger is equally likely to ride in any 1 of the 3

cars each time that passenger rides the roller coaster. If a certain passenger is to ride the roller

coaster 3 times, what is the probability that the passenger will ride in each of the 3 cars?

（A）0 （B）1/9（C）2/9 （D）1/3（E）1

<11>A gardener is going to plant 2 red rosebushes and 2 white rosebushes. If the gardener is to

select each of the bushes at random, one at a time, and plant them in a row, what is the probability

that the 2 rosebushes in the middle of the row will be the red rosebushes?

（A）1/12 （B）1/6 （C）1/5 （D）1/3 （E）1/2

<12> If a committee of 3 people is to be selected from among 5 married couples so that the

committee does not include two people who are married to each other, how many such committees

are possible?

（A）20 （B）40 （C）50 （D）80 （E）120

<13> There are 5 cars to be 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

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<14> How many different 6-letter sequences are there that consist of 1 A, 2 B’s, and 3 C’s ?

(A)6 (B) 60 (C) 120 (D) 360 (E) 720

<15> A photographer will arrange 6 people of 6 different heights for photograph by placing them

in two rows of three so that each person in the first row is standing in front of someone in the

second row. The heights of the people within each row must increase from left to right, and each

person in the second row must be taller than the person standing in front of him or her. How

many such arrangements of the 6 people are possible?

(A)5 (B) 6 (C) 9 (D) 24 (E)36

<16>Pat will walk from Intersection X to Intersection Y along a route that is confined to the

square grid of four streets and three avenues shown in the map above.How many routes from X to

Y can Pat take that have the minimum possible length?

(A) 6 (B) 8 (C) 10 (D) 14 (E) 16

<17>In the integer 3589 the digits are all different and increase from left to right. How many

integers between 4000 and 5000 have digits that are all different and that increase from left to

right?

数学词汇

1．数学符号等于： = equal to, the same as, is

不等： > more than < less than

≥ no less than

≤ no more than

加： + add, plus, more than; sum

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减： - minus, subtract; less than; difference

乘： × multiply, times; product

除： ÷ divide; quotient

绝对值：∣…∣ absolute value

平方： X2 square

立方： X3 cube

开平方： square root

开立方： cube root

平行： ∥ parallel to

垂直： ⊥ perpendicular to

2．数字前缀1: uni,mono, 2: bi,di 3: tri,ter 4: tetra,quad 5: penta,quint 6: hex,sex 7: sept,hapta

8: oct 9: enn 10:dec,deka

3. 方程和函数 equation 方程 solution, root, zero 解 variable 变量 constant 常量（数） term 项 coefficient 系数

4. 数列和集合arithmetic progression 等差数列geometric progression 等比数列set 集合subset 子集sequence 序列term 序列中的项inclusive 包含序列的首末项exclusive 不包含序列的首末项

5. 排列组合与概率permutation 排列combination 组合probability，possibility 概率

6. 数论common division 公约数common factor 公因子composite number 合数(质数和 1 以外的自然数)

consecutive integer 连续整数digit 数字divide 除以divisor 因子,除数

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（evenly）divisible by 可整除的even number 偶数factor 因子integer 整数irrational 无理数least common multiple 最小公倍数multiple 倍数,公倍数natural number 自然数negative number 负数nonzero 非零odd number 奇数positive number 正数prime factor 质因子prime number 质数quotient 商rational 有理数real number 实数remainder 余数whole number 整数units'digit 个位数tens'digit 十位数hundreds'digit 百位数two-digit number 两位数

7. 单利复利和价格

compound interest 复利 cost 成本 discount 折扣 down payment 预付款,现付款 interest rate 利率 list price 标价 margin 利润 mark up 涨价 mark down 降价 markup 毛利 profit 利润 simple interest 单利

8. 其它代数 addition 加 arithmetic mean 算术平均数 average 平均数 base 底数 closest approximation 近似

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decimal 小数 base 10 notation; decimal notation 十进制 decimal point 小数点 decreased 下降后的 decrease … to… 从…下降到… decrease by … 下降了… define 定义 denominator 分母 denote 表示,代表 distinct 不同的 expression 表达式 fraction 分数 improper fraction 假分数 increased 增加后的 increase… to … 从…增加到… increase by… 增加了… in terms of 用…表达 least possible 最小值 maximum 最大值 minimum 最小值 multiply 乘 multiplier 乘数 numerator 分子 per capita 人均 power 指数 proportional to 正比于 proper faction 真分数 ratio 比率 reciprocal 倒数 reduced 降低后的 rounded to the nearest tenth 四舍五入到十分位 Successive; consecutive; in a row 连续的 tenth 十分位 tenths'digit 十分位 tie 平局 times 几倍 two digits 两个数字 twice as many A as B A 是 B 的两倍 3/2 as many A as B A 是 B 的 3/2 倍 A is 20% more than B A比 B 多 20%,(A-B)/B=20%

9. 几何 abscissa 横坐标 acute angle 锐角

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altitude 高 arc 弧 area 面积 angle bisector 角平分线 bisect 平分 center 中心 chord 弦 circle 圆 circumference 圆周长 circumscribe 外接,外切 clockwise 顺时针 concentric circle 同心圆 cone 圆锥 congruent 全等的 coordinate 坐标 counterclockwise 逆时针 cube 正方体 cylinder 圆柱 decagon 十边形 degree 角度 diameter 直径 diagonal 对角线 dimension 大小,维度 distance 距离 due north 正北方 equilateral triangle 等边三角形 face 面 height 高 hexagon 六边形 hypotenuse 斜边 isosceles triangle 等腰三角形 inscribe 内接,内切 intersect 相交 length 长度 median of a triangle 三角形的中线 mid point 中点 number lines 数轴 obtuse angle 钝角 octagon 八边形 ordinate 纵坐标 overlap 交叠 parallelogram 平行四边形 pentagon 五边形 perimeter 周长

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parallel lines 平行线 perpendicular lines 垂直线 plane 平面 polygon 多边形 quadrant 象限 quadrilateral 四边形 radius 半径 radian 弧度(弧长/半径)

regular polygon 正多边形 rectangular solid 长方体 rectangle 长方形 right angle 直角 right triangle 直角三角形 square 正方形 sphere 球 side 边 surface area 表面积 straight angle 平角 segment 线段 tangent 切线 triangle 三角形 vertex（vertices）angle 顶角

作业<1> If a and b are positive integers such that a – b and are both even integers, which

of the following must be an odd integer?

(A) (B） (C) (D) (E)

<2> If y≠3, and 3x/y is a prime integer greater than 2, which of the following must be true?

Ⅰ. x=y

Ⅱ. y=1

Ⅲ. x and y are prime integers

(A) None (B) Ⅰonly (C) Ⅱonly

(D)Ⅲ only (E) Ⅰand Ⅲ

<3> What is the greatest prime factor of 2100 - 296?

(A) 2 (B) 3 (C) 5 (D) 7 (E) 11

<4> A positive integer n is said to be “prime-saturated” if the product of all the different

positive prime factors of n is less than the square root of n. What is the greatest two-digit

prime-saturated integer?

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(A) 99 (B) 98 (C) 97 (D) 96 (E) 95

<5> If 1050 – 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<6> If P and Q are different points in a plane, the set of all points in this plane that are closer to P

than to Q is

(A) the region of the plane on one side of a line

(A) the interior of a square

(B) a wedge-shaped region of the plane

(C) the region of the plane bounded by a parabola

(D) the interior of a circle

<7> A researcher computed the mean, the median, and the standard deviation for a set of

performance scores. If 5 were to be added to each score, which of these three statistics would

change?

(A)Ⅰonly (B) Ⅱonly (C) Ⅰand Ⅱonly (D) Ⅱand Ⅲ only (E) Ⅰ,Ⅱ,and Ⅲ

<8>

Test Sonya's score Mean score Standard deviation

Biology physics 90 84 2

biology 96 86 5

history 89 81 4

chemistry 88 85 3

math 80 88 4

The chart above shows data for five tests that Sonya took. On which of the five tests did she score

highest relative to the rest of the test takers?

(A) math (B) biology (C) physics (D) history (E) chemistry

<9> Some people in New York express 2/8 as 8th Feb and others express 2/8 as 2nd Aug. This can

be confusing as when we see 2/8, we don’t know whether it is 8th Feb or 2nd Aug. However, it is

easy to understand 9/22 or 22/9 as 22nd Sept, because there are only 12 months in a year. How

many dates in a year can cause this confusion?

（A）48 （B) 53 （C）120 （D）132 （E）144<10> Beginning in January of last year, Carl made deposits of $120 into his account on the 15 th of each

month for several consecutive months and then made withdrawals of $50 from the account on the 15 th of

each of the remaining months of 1ast year. There were no other transactions in the account last year. If the

closing balance of Carl's account for May of last year was $2600, what was the range of the monthly closing

balances of Carl's account last year?

(1) Last year the closing balance of Carl's account for April was less than $2625.

(2) Last year the closing balance of Carl's account for June was less than $2675.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

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E. Statements (1) and (2) TOGETHER are NOT sufficient.

<11> At the beginning of a five-day trading week, the price of a certain stock was $10 per share. During the

week, four of the five closing prices of the stock exceeded $10. Did the average closing price of the stock

during the week exceed its price at the beginning of the week?

(1) The stock's closing price on Tuesday was the same as its closing price on Thursday.

(2) The sum of the stock's highest and lowest closing prices during the week was 20.






<12> If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is the

value of r?

(1) 2 is not a factor of n.

(2) 3 is not a factor of n.






<13> 三人独立地去破译一个密码，他们能译出的概率分别为 1/5,1/3,1/4, 求此密码被译出的概率。<14> 某市共有 10000辆自行车，其牌照号码从 00001到 10000，求偶然遇到的一辆自行车，其牌照号码中有数字 8 的概率。<15> 用 0,2,4,6,9 这五个数字可以组成数字不重复的五位偶数共有多少个？<16>有 5 个队伍参加了足球联赛，任意两队之间都要进行主客场各一场比赛，问总共将有多少场比赛？<17> 甲，乙，丙，丁，戊五人并排站成一排，如果乙必须站在甲的右边（甲乙可以不相邻），那么不同的排法共有多少种？<18> 4 本不同的书分给 2 人，每人 2 本，不同的分法有多少种？<19> 两把 keys,放到有 5 个 keys 的 keychain 中，相邻的概率为多少(分直线和环形)？<20> 6张同排连号的电影票，分给 3名男生和 3名女生，如欲男女相间而坐，则不同的分法数为多少？<21> 有 4对人，从中取 3 个人，不能从任意一对中取 2 个，问有多少种取法？<22> 从 6双不同的手套中任取 4 只，求其中恰有一双配对的概率。<23> 3封不同的信，有 4 个信箱可供投递，共有多少种投信的方法？<24> 3 个打字员为 4家公司服务，现在每家公司各有 1份文件需要录入，问每个打字员都收到文件的概率？<25> A发生的概率是 0.6，B发生的概率是 0.5，问 A,B 都不发生的最大概率？<26> 袋中有 a 只白球，b 只红球，依次将球一只只摸出，不放回。求第 k 次摸出的是个白球的概率（1≤k≤a+b）。<27> 掷一枚均匀硬币 2n 次，求出现正面 k 次的概率。<28> 有 4 组人，每组一男一女，从每组各取一人，问取出两男两女的概率？

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<29> A certain roller coaster has 3 cars, and a passenger is equally likely to ride in any 1 of the 3

cars each time that passenger rides the roller coaster. If a certain passenger is to ride the roller

coaster 3 times, what is the probability that the passenger will ride in each of the 3 cars?

（A）0 （B）1/9（C）2/9 （D）1/3（E）1<30>A gardener is going to plant 2 red rosebushes and 2 white rosebushes. If the gardener is to

select each of the bushes at random, one at a time, and plant them in a row, what is the probability

that the 2 rosebushes in the middle of the row will be the red rosebushes?

（A）1/12 （B）1/6 （C）1/5 （D）1/3 （E）1/2<31> If a committee of 3 people is to be selected from among 5 married couples so that the

committee does not include two people who are married to each other, how many such committees

are possible?

（A）20 （B）40 （C）50 （D）80 （E）120<32> How many seven-digit numbers contain the digit '7' at least once?

<33> In how many ways can seven students A, B, C, D, E, F and G line up in one row if students

B and C are always next to each other?

<34> There are 5 cars to be 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

<35> How many different 6-letter sequences are there that consist of 1 A, 2 B’s, and 3 C’s ?

(A)6 (B) 60 (C) 120 (D) 360 (E) 720

<36> In how many distinguishable ways can the seven letters in the word MINIMUM be

arranged, if all the letters are used each time?

(A) 7 (B) 42 (C) 420 (D) 840 (E)5040

<37> A photographer will arrange 6 people of 6 different heights for photograph by placing them

in two rows of three so that each person in the first row is standing in front of someone in the

second row. The heights of the people within each row must increase from left to right, and each

person in the second row must be taller than the person standing in front of him or her. How

many such arrangements of the 6 people are possible?

(A)5 (B) 6 (C) 9 (D) 24 (E)36

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