08新版GMAT数学-讲义
【小站教育】GMAT数学知识点专题一 - 算术和数论
专题一算术和数论1. Integer (whole number): 整数①Positive integer: 正整数，从1 开始，不包括0。
②奇数：不能被2 整除的整数（可正可负），通式：2n+1。
如-1，1。
③偶数：能被2 整除的整数（可正可负），零是偶数。
通式：2n。
如-4，-2，0，2，4。
2. Odd & even number: 奇数与偶数①偶数＝偶数＋偶数或奇数＋奇数，偶数＝偶数×偶数或奇数×偶数②奇数＝奇数＋偶数③奇数个奇数相加减，结果为奇数④偶数个奇数相加减，结果为偶数⑤任意个偶数相加减，结果为偶数⑥若n 个整数相乘结果为奇数，则这n 个整数为奇数⑦若n 个连续的整数相加等于零，则n 为奇数。
⑧若n 个连续的奇数相加等于零，则n 为偶数。
⑨若n 个连续的偶数相加等于零，则n 为奇数。
⑩两个质数之和为奇数，其中必有一个是2。
例：若a2+b2=c2，其中a, b, c 均为整数，下面那个不可能是a+b+c 的值？A.2B.1C.-2D.4E.6【解析】因为a2+b2=c2，如果a,b,c 中有奇数存在，则必然为2 个。
所以a+b+c 必为偶数，正确答案选择B。
3. Prime number & Composite number: 质数与合数①质数又称素数。
指在一个大于1 的自然数中，除了1 和此整数自身外，不能被其他自然数整除的数。
②合数指指自然数中除了能被1 和本身整除外，还能被其他的数整除的数③2 是最小的质数、4 是最小的合数、1 既不是质数也不是合数。
4. Factor (divisor) & Prime factor: 因子和质因子a) 一个数能被那些书整除，这些书就叫它的因子（因数、约数）。
b) 因子里的质数叫做质因子（数）。
*小技巧：⑴分解质因数：讲一个整数拆分成全部由质数表示，如36 = 22 + 32⑵A b（A 为质数）有(b+1)个因子：A0, A1 ……A b⑶如果X=a m·b n（分解质因数后），X 有(m+1)(n+1)个因子。
gmat数学次方-概述说明以及解释
gmat数学次方-概述说明以及解释1.引言1.1 概述概述部分的内容如下：概述在GMAT数学考试中，次方是一个非常重要的数学概念。
次方通常被用于表示一个数的幂次运算，它在数学中有着广泛的应用。
掌握了次方的定义、运算规则以及应用，有助于我们在GMAT数学考试中更好地理解和解决与次方相关的问题。
次方的概念可以简单地解释为一个数（被称为底数）经过若干次乘法运算，而得到的结果（被称为幂）。
例如，我们可以将2的3次方表示为2^3，其结果是8。
在这个例子中，数字2是底数，数字3是指数（也可以称为幂）。
遵循定义，我们可以总结出次方运算的一些规则，这些规则有助于我们在解决问题时简化计算过程。
例如，当两个数的指数相加或相减时，我们可以使用指数的乘法与除法规则来简化计算。
在GMAT数学考试中，次方运算的应用非常广泛。
在代数、几何和概率等各个领域，我们都可以看到次方的身影。
在解决问题时，理解和应用次方运算的规则和性质是非常重要的。
通过掌握次方的相关知识点，我们能够更加灵活地运用次方来解决各种数学题目。
因此，在备考GMAT数学考试时，对于次方的理解和掌握是不可或缺的。
在本文中，我们将详细介绍次方的定义、运算规则以及应用。
同时，我们将总结GMAT数学中次方的相关知识点，并提供一些建议，帮助大家更好地备考GMAT数学考试。
总之，次方是GMAT数学考试中的一个关键概念。
掌握次方的定义、运算规则和应用，对于解决与次方相关的问题至关重要。
希望通过本文的讲解，读者们能够对次方有更深入的理解，并能在GMAT数学考试中得到更好的成绩。
1.2文章结构文章结构:本文将分为三个主要部分进行讨论。
首先，我们将在引言部分概述GMAT数学中次方的重要性和应用。
接下来，正文部分将详细介绍次方的定义、次方运算的规则以及次方运算在GMAT数学中的应用。
最后，在结论部分我们将总结GMAT数学中次方的相关知识点，并提供备考建议。
引言部分介绍了该文章的目的和概述。
张广GMAT数学讲义
目录第一章Data Sufficiency题型-------------------------------------------2 第二章数论---------------------------------------------------------------4第三章代数--------------------------------------------------------------17第四章几何--------------------------------------------------------------24第五章文字应用题-------------------------------------------------------32第六章综合练习题-------------------------------------------------------44第七章GMAT数学常见术语---------------------------------------------72第一章 Data Sufficiency 题型【DS 题形式】Information required (Introduction or Background)QuestionTwo statements labeled (1) and (2)Option:(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.【答题步骤】1. 分析问句类型数值计算——特殊疑问句 (Special Question).判断是非——一般疑问句 (General Question).2. 罗列各种充分与不充分条件特殊疑问句——答案：唯一确定实数解.充分：能得到诸如0, –27, 4.567, 2311, , π等唯一解的Statement .不充分：能得到诸如x = x <1等两个或更多解，及其它一切无法计算出解的Statement .例题1：What is the value of x ?(1) 3x = 15(2) 5x < 30例题2：Tom and Jack are in a line to purchase tickets. How many people are in the line?(1) There are 20 people behind Tom and 20 people in front of Jack.(2) There are 5 people between Tom and Jack.一般疑问句——答案：明确回答”YES”或者”NO”.充分：完全符合或者完全不符合Question提出的内容，即能理直气壮回答”YES”或者”NO”，不留任何余地的Statement.不充分：不完全符合Question提出的内容，即只能心虚回答”Yes, but…”或者”No, but…”，及其它一切无法判断结果的Statement.例题3：Is x equal to 1 ?(1)x2 = 1(2)x2 = 4例题4：体会下列两个Question的区别.There are eight balls in the pocket.Question 1: Are all the balls in the pocket red?充分：”YES”:”NO”:Question 2: Are there any red balls in the pocket?充分：”YES”:”NO”:Statement 1: Three balls are removed; whose colors are brown, green, and red, respectively. Statement 2: Three balls are removed; whose colors are brown, green, and yellow, respectively. Statement 3: Three balls are removed; whose colors are red, red, and red, respectively.3.按照Problem Solving常规题型继续思考牢记：当分析Statement (1) 时，不要预测Statement (2)；当分析Statement (2) 时，确信忘记Statement (1)；第二章数论【奇数与偶数(Odd and Even Numbers) 】1．奇数个奇数相加减，其结果必为奇数。
GMAT备考资料
GMAT核心资料篇所谓核心资料，也可以理解为准备GMAT考试的最高优先级的资料。
原则上来说，非真题的备考资料基本不推荐。
Official Guide for GMAT Review（官方指南，以下简称OG）OG即official guide for GMAT Review，由GMAC出版的官方指南，目前最新版本为第12版，论坛上简称OG12。
依次类推以前的版本有OG11，OG10。
下表列出目前可以见到的三个OG版本的始发年份和构成。
版本始发年份构成OG 10 1999 综合，共一本OG 11 2005 综合+语文分册+数学分册，共三本OG 12 2008 综合+语文分册+数学分册，共三本OG是最基本同时也是最官方的备考资料，其中所有的题目均是过去实战中的原题，给出的题目解析也充分体现了GMAC的出题角度和考察内容。
在备考GMAT时，我唯一强烈建议大家购买纸质档的两本书：OG12综合与OG12语文分册。
在OG中最有用的部分是语法部分的解释，认真地用心体会，解析里面的每一句话都不应该错过。
“OG每看一遍都会有新的收获！”——这是很多前辈都赞成的备考真理。
据观察，OG本身所体现的考点和标准也不断有所变化。
例如曾经OG10中允许使用的部分语法点在OG12中视为绝对错误的用法。
所以推荐大家使用最新版的OG。
切记，OG的解释是最官方的说法，如果遇到与其他非真题资料相矛盾的地方，请选择相信OG；在语法部分，细心研读的同学会发现阅读和逻辑的文章中出现语法并不允许的用法，也请不要钻牛角尖，相信语法部分的解释。
因为阅读和逻辑的文章原稿并不是GMAC本身所写，也不是商业类文章，难免出现与语法相矛盾的地方。
PREP破解GMAT官方免费提供了GMAT模考软件GMATPrep，其界面与真实考试完全相同，但是我们只能做两套模拟题。
有前辈发现GMATPrep中事实上内置了一个很大的题库，Prep破解即前辈从GMATPrep中还原出来的完整题库，全部属于真题，且有标准答案，是分项突破时很好的练习材料。
2008年2月23日GMAT数学真题总结
假设I为绝对值符号 Key: -20+2-4=-22 2. a= 2^4, b=2^a, 2^x=a^b, what is value of x? a=2^4=16; b=2^a=2^16; 2^x=16^(2^16)=2^(2^18) Key:à x=2^18=1024*256 3. * , the answer is -1. The last quesion in Math. Key: ((x+y)/(x-y))*((y-x)/(y+x))=-1 4. X, Y are positive integers. Is Y odd? a. XY=even b. X+Y=even “Is” type question a: Y/N (yes or no, means Y can be either odd or even) not sufficient b: Y/N not sufficient a+b: (从问题倒推) If Y odd, then (a) x is even and then (b) x is odd->can not be true!! If Y is even, then (a) x is even or odd and then (b) x is even->can be true. Thus Y MUST be even. Therefore, to answer “Is Y odd”, NO. So a+b together sufficient. Key C: a, b alone not sufficient, but together are sufficient. 5. How many mutiples of 3 between -100 and 100. I choose 67, as I consider 0. 1.倍数的定义是：⼀个数能够被另⼀数整除，这个数就是另⼀数的倍数。
GMATGWD数学部分考试复习资料
Math Section------------------------------------------------------------------------------------------------------------ Q1:If x2 = 4 and y2 =7, then x6–y4 =A.33B.25C.15D.-3E.-33Answer:------------------------------------------------------------------------------------------------------------ Q2:There are 11 women and 9 men in a certain club. If the club is to select a committee of 2 women and 2 men, how many different such committees are possible?A. 120B. 720C.1,060D.1,520E.1,980Answer:------------------------------------------------------------------------------------------------------------ Q3:If p, s, and t are positive prime numbers, what is the value of p3s3t3 ?(1) p3st = 728(2) t = 13A. 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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q4:If x and y are integers, what is the value of 2x6y- 4 ?(1) x2y = 16(2) xy = 4A. 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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q5:A department manager distributed a number of pens, pencils, and pads among the staff in the department, with each staff member receiving x pens, y pencils, and z pads. How many staff members were in the department ?(1) The numbers of pens, pencils, and pads that each staff member received were inthe ratio 2 : 3 : 4, respectively.(2) The manager distributed a total of 18 pens, 27 pencils, and 36 pads. 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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q6:The function f is defined for each positive three-digit integer n by f (n ) = 2x 3y 5z , 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 ) = 9 f (v ), then m - v =A. 8B. 9C. 18D. 20E. 80Answer:------------------------------------------------------------------------------------------------------------ Q7:If N = 31 + 231 + 331, then N is betweenA. 0 and 91B. 91 and 31C. 31 and 98D. 98 and 34E. 34and 2Answer:------------------------------------------------------------------------------------------------------------ Q8:A cash register in a certain clothing store is the same distance from two dressing rooms in the store. If the distance between the two dressing rooms is 16 feet, which of the following could be the distance between the cash register and either dressing room?I. 6 feetII. 12 feetIII. 24 feetA. I onlyB. II onlyC. III onlyD. I and IIE. II and IIIAnswer:------------------------------------------------------------------------------------------------------------ Q9:Is x < 0 ?(1) x3 < x2(2) x3 < x4A. 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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q10:A rectangular floor that measures 8 meters by 10 meters is to be covered with carpet squares that each measure 2 meters by 2 meters. If the carpet squares cost $12 apiece, what is the total cost for the number of carpet squares needed to cover the floor?A.$200B.$240C.$480D.$960E.$1,920Answer:------------------------------------------------------------------------------------------------------------ Q11:In the sequence 1, 2, 4, 8, 16, 32, …, each t erm after the first is twice the previous term. What is the sum of the 16th, 17th, and 18th terms in the sequence ?A.218B.3(217)C.7(216)D.3(216)E.7(215)Answer:------------------------------------------------------------------------------------------------------------ Q12:Some computers at a certain company are Brand X and the rest are Brand Y. If the ratio of the number of Brand Y computers to the number of Brand X computers at the company is 5 to 6, how many of the computers are Brand Y ?(1) There are 80 more Brand X computers than Brand Y computers at the company. (2) There is a total of 880 computers at the company.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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q13:Each of 20 parents chose one of five days from Monday through Friday to attend parent-teacher conferences. If more parents chose Monday than Tuesday, did at least one of the parents choose Friday?(1) None of the five days was chosen by more than 5 parents. (2) More parents chose Monday than Wednesday.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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q14:A certain pilot flew 400 miles to City K at an average speed of 350 miles per hour with the wind and made the trip back at an average speed of 250 miles per hour against the wind. Which of the following is closest to the pilot’s average speed, in miles per hour, for the round-trip?A. 280B. 290C. 300D. 310E. 320Answer:------------------------------------------------------------------------------------------------------------ Q15:If R = QP, is R ≤ P ?(1) P > 50 (2) 0 < Q ≤ 20A. 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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q16:If 2, x, y, and z are different positive integers whose average (arithmetic mean) is 10, what is the greatest possible value of z ?A.10B.24C.34D.36E.40Answer:QX:If the average (arithmetic mean) of positive integers x, y, and z is 10, what is the greatest possible value of z ?A.8B.10C.20D.28E.30Answer:------------------------------------------------------------------------------------------------------------ Q17:Which of the following is closest to 10180– 1030 ?A.10210B.10180C.10150D.1090E.106Answer:------------------------------------------------------------------------------------------------------------ Q18: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%Answer:------------------------------------------------------------------------------------------------------------ Q19:If x, y, and z are consecutive even positive integers, which of the following could be equal to x + y + z ?A.141B.200C.318D.391E.412Answer:------------------------------------------------------------------------------------------------------------ Q20: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 inforeign markets.(2) Last year 24 of the 30 businesses reported a net profit or invested in foreignmarkets, or both.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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q21:y––––●xO–P–––In the figure shown, the circle has center O and radius 50, and point P has coordinates (50,0). If point Q (not shown) is on the circle, what is the length of line segment PQ ?(1) The x-coordinate of point Q is – 30.(2) The y-coordinate of point Q is – 40.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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q22:If –2x > 3y, is x negative?(1) y > 0(2) 2x + 5y– 20 = 0A. 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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q23:The toll for crossing a certain bridge is $0.75 each crossing. Drivers who frequently use the bridge may instead purchase a sticker each month for $13.00 and then pay only $0.30 each crossing during that month. If a particular driver will cross the bridge twice on each of x days next month and will not cross the bridge on any other day, what is the least value of x for which this driver can save money by using the sticker?A.14B.15C.16D.28E.29Answer:------------------------------------------------------------------------------------------------------------ Q24:The positive numbers w, x, y, and z are such that x is 20 percent greater than y, y is 20 percent greater than z, and w is 20 percent less than x. What percent greater than z is w ?A.15.2%B.16.0%C.20.0%D.23.2%E.24.8%Answer:------------------------------------------------------------------------------------------------------------ Q25:Machine M processes a certain chemical product at a constant rate. Does machine M process the product at a rate that is greater than 25 grams per second? (1 kilogram =1,000 grams)(1) Machine M processes the product at a rate that is greater than 92 kilograms perhour.(2) Machine M processes the product at a rate that is less than 95 kilograms per hour.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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q26:Is –7 < y < 10 ?(1) y > -6 (2) y < 0A. 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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q27:If k is an integer and (0.0025)(0.025)(0.00025) × 10k is an integer, what is the least possible value of k ?A. -12B. -6C. 0D. 6E. 12Answer:------------------------------------------------------------------------------------------------------------ Q28:Jacob purchased 100 boxes of oranges at $8.00 per box. He sold 8 of the boxes at $4.00 per box to Company A , and he sold the rest of the boxes at x dollars per box to Company B . If Jacob’s profit from the purchase and sale of the 100 boxes of oranges was $336.00, at what price per box did he sell the boxes to Company B ?A. $11.36B. $11.60C. $12.00D. $12.35E. $12.90Answer:------------------------------------------------------------------------------------------------------------ Q29:A sum of money was divided between John and Anne such that the ratio of John’s shareto Ann e’s share was 5 to 3. If John’s share exceeded 95of the sum of money by $50,what was Anne’s share?A. $180B. $270C. $340D. $450Answer:------------------------------------------------------------------------------------------------------------ Q30:From May 1 to May 30 in the same year, the balance in a checking account increased. What was the balance in the checking account on May 30 ?(1) If, during this period of time, the increase in the balance in the checking accounthad been 12 percent, then the balance in the account on May 30 would have been $504.(2) During this period of time, the increase in the balance in the checking accountwas 8 percent.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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q31:In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle?A.4B. 3C. 2Answer:------------------------------------------------------------------------------------------------------------ Q32:What is the probability that event E or event F or both will occur?(1) The probability that event E will occur is 0.6.(2) The probability that event F will occur is 0.4.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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:QX:If the probability is 0.54 that Stock A will increase in value during the next month and the probability is 0.68 that Stock B will increase in value during the next month, what is the greatest possible value for the probability that neither of these two events will occur?0.22A.0.32B.0.37D. 0.63Answer:------------------------------------------------------------------------------------------------------------ Q33:If Diana’s stamp collection, 54 of the stamps are Canadian, and 73of the Canadianstamps were issued before 1940. If 192 stamps in Diana’s collection are Canadian stamps that were issued in 1940 or later, how many stamps in her collection are not Canadian?A. 84B. 88C. 96D. 104E. 112Answer:------------------------------------------------------------------------------------------------------------ Q34:If a rectangular room measures 10 meters by 6 meters by 4 meters, what is the volume of the room in cubic centimeters? (1 meter = 100 centimeters)A. 24,000B. 240,000C. 2,400,000D. 24,000,000E. 240,000,000Answer:------------------------------------------------------------------------------------------------------------ Q35:For a certain car repair, the total charge consisted of a charge for parts, a charge for labor, and a 6 percent sales tax on both the charge for parts and the charge for labor. If thecharge for parts, excluding sales tax, was $50.00, what was the total charge for the repair?(1) The sales tax on the charge for labor was $9.60. (2) The total sales tax was $12.60.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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q36:If x and y are integers, is y an even integer?(1) 2y – x = x 2 – y 2 (2) x is an odd integer.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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q37:For any number x , ⎣⎦x denotes the greatest integer less than or equal to x . What is thevalue of ⎥⎦⎥⎢⎣⎢-410?A. 0B. –1C. –2D. –3E. –4Answer:------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------ Answers:CEAAE, DCECB, EDABE, C(D )BDCD, ADBAA, CECBC, CE(B )AED, ADF. MATH---------------------------------------------------------------------------------------------------------------------- Q1:If Henry were to add 5 gallons of water to a tank that is already 3/4 full of water, the tank would be 7/8 full. How many gallons of water would the tank hold if it were full?A. 25B. 40C. 64D. 80E. 96Answer:---------------------------------------------------------------------------------------------------------------------- Q2:The function f is defined for each positive three-digit integer n by f(n) = 2x 3y 5z , 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. 8B. 9C. 18D. 20E. 80Answer:----------------------------------------------------------------------------------------------------------------------Q3:At a certain food stand, the price of each apple is ¢ 40 and the price of each orange is ¢60. Mary selects a total of 10 apples and oranges from the food stand, and the average (arithmetic mean) price of the 10 pieces of fruit is ¢ 56. How many oranges must Mary put back so that the average price of the pieces of fruit that she keeps is ¢ 52?A. 1B. 2C. 3D. 4E. 5Answer:----------------------------------------------------------------------------------------------------------------------Q4:Professor Vasquez gave a quiz to two classes. Was the range of scores for the first class equal to the range of scores for the second class?(1) In each class, the number of students taking the quiz was 26, and the lowestscore in each class was 70.(2) In each class, the average (arithmetic mean) score on the quiz was 85.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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:----------------------------------------------------------------------------------------------------------------------Q5:If x and y are positive integers, what is the value of x?(1) 3x5y = 1,125(2) y = 3A. 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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:----------------------------------------------------------------------------------------------------------------------Q6:If ∣d - 9∣ = 2d, then d =A. -9B. -3C. 1D. 3E. 9Answer:----------------------------------------------------------------------------------------------------------------------Q7:There are 11 women and 9 men in a certain club. If the club is to select a committee of 2 women and 2 men, how many different such committees are possible?A. 120B. 720C. 1,060D. 1,520E. 1,980Answer:----------------------------------------------------------------------------------------------------------------------Q8:In the figure shown, two identical squares are inscribed in the rectangle. If the perimeter of the rectangle is 18√2, then what is the perimeter of each square?A. 8√2B. 12C. 12√2D. 16E. 18Answer:----------------------------------------------------------------------------------------------------------------------Q9:Is the measure of one of the interior angles of quadrilateral ABCD equal to 60 degrees?(1) T wo of the interior angles of ABCD are right angles.(2) The degree measure of angle ABC is twice the degree measure of angleBCD.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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:----------------------------------------------------------------------------------------------------------------------Q10:$10,000 is deposited in a certain account that pays r percent annual interest compounded annually, the amount D(t), in dollars, that the deposit will grow to in t years is given by D(t) = 10,000 {1+(r/100)}t . What amount will the deposit grow to in 3 years?(1) D (t) = 11,000 (2) r =10A. 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. Statements (1) and (2) TOGETHER are NOT sufficient. Answer:---------------------------------------------------------------------------------------------------------------------- Q11:If a and b are integers, is b even?(1) 3a + 4b is even. (2) 3a + 5b is even.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. Statements (1) and (2) TOGETHER are NOT sufficient. Answer:---------------------------------------------------------------------------------------------------------------------- Q12:Working alone at its constant rate, machine K took 3 hours to produce 41of the unitsproduced last Friday. Then machine M started working and the two machines, working simultaneously at their respective constant rates, took 6 hours to produce the rest of the units produced last Friday. How many hours would it have taken machine M , working alone at its constant rate, to produce all of the units produced last Friday?A. 8B. 12C. 16D. 24E. 30Answer:------------------------------------------------------------------------------------------------------------ Q13:If x + y is an integer, is y an integer?(1) x–y is an integer.(2) x + 2y is an integer.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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:------------------------------------------------------------------------------------------------------------ Q14:A certain computer program generates a sequence of numbers a1, a2, … , an such that a1 = a2 = 1 and a k = a k-1 + 2a k-2 for all integers k such that 3 ≤ k ≤ n. If n > 6, then a7 = ?A. 32B. 43C. 64D. 100E. 128Answer:----------------------------------------------------------------------------------------------------------------------Q15:At a certain bookstore, the regular price of each book is 20 percent less than its list price. If during a sale the price of each book at the store was 15 percent less than its regular price, then the price of a book during the sale was what percent less than its list price?A. 30%B. 32%C. 35%D. 38%E. 40%Answer:----------------------------------------------------------------------------------------------------------------------Q16:What is the value of the ratio of x to y2?(1) T he ratio of x2 to y is 5/3.(2) The ratio of x to 1 is 5/3.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. Statements (1) and (2) TOGETHER are NOT sufficient.Answer:。
张广GMAT数学讲义
张广GMAT数学讲义目录第一章Data Sufficiency题型-------------------------------------------2 第二章数论---------------------------------------------------------------4第三章代数--------------------------------------------------------------17第四章几何--------------------------------------------------------------24第五章文字应用题-------------------------------------------------------32第六章综合练习题-------------------------------------------------------44第七章GMAT数学常见术语---------------------------------------------72第一章 Data Sufficiency 题型【DS 题形式】Information required (Introduction or Background) Question Two statements labeled (1) and (2) Option:(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.【答题步骤】1. 分析问句类型数值计算——特殊疑问句 (Special Question). 判断是非——一般疑问句 (General Question).2. 罗列各种充分与不充分条件特殊疑问句——答案：唯一确定实数解.充分：能得到诸如0, –27, 4.567,2311,π等唯一解的Statement .不充分：能得到诸如x = x <1等两个或更多解，及其它一切无法计算出解的Statement .例题1：What is the value of x ?(1) 3x = 15 (2) 5x < 30例题2：T om and Jack are in a line to purchase tickets. How many people are in the line? (1) There are 20 people behind Tom and 20 people in front of Jack. (2) There are 5 people between Tom and Jack.一般疑问句——答案：明确回答”YES”或者”NO”.充分：完全符合或者完全不符合Question提出的内容，即能理直气壮回答”YES”或者”NO”，不留任何余地的Statement.不充分：不完全符合Question提出的内容，即只能心虚回答”Yes, but…”或者”No, but…”，及其它一切无法判断结果的Statement.例题3：Is x equal to 1 ?(1)x2 = 1(2)x2 = 4例题4：体会下列两个Question的区别.There are eight balls in the pocket.Question 1: Are all the balls in the pocket red?充分：”YES”:”NO”:Question 2: Are there any red balls in the pocket?充分：”YES”:”NO”:Statement 1: Three balls are removed; whose colors are brown, green, and red, respectively. Statement 2: Three balls are removed; whose colors are brown, green, and yellow, respectively. Statement 3: Three balls are removed; whose colors are red, red, and red, respectively.3.按照Problem Solving常规题型继续思考牢记：当分析Statement (1) 时，不要预测Statement (2)；当分析Statement (2) 时，确信忘记Statement (1)；第二章数论【奇数与偶数(Odd and Even Numbers) 】1．奇数个奇数相加减，其结果必为奇数。
GMAT讲义
Argument 部分一.Argument的写作特点三个重大逻辑错误。
1. 典型的外推类逻辑错误：过去不能推出现在。
2. 攻击它条件不充分：并不是只由新东方导致了中国留学事业的发展。
3. 错误类比：中国不代表越南。
False analogy1.调查类错误。
样本选择不随机，不具备代表性，样本数量少。
而且在西安市做的，但是要陕西省。
由个体推整体。
还有本科生、研究生。
2.因果关系错误：不是由于结婚才导致他取得科技进步。
时序因果类错误。
3.由个体推整体。
广告练习：喝了哇哈哈，吃饭就是香。
条件不充分要想皮肤好，早晚用大宝。
条件不充分。
牙好胃口就好：样本小，条件不充分。
二.Argument整体结构第一段：开头段。
主要是归纳论点，说明论点有问题，存在逻辑漏洞，准备发起进攻。
第二段和第三段甚至第四段：分类别去攻击各个逻辑错误。
第五段：结尾段。
作者的结论似乎是合理的，但是通过论证，不是这样的。
因此作者在做出决定之前，应该还要考虑其他情况。
三.如何写开头段？例.重复结论+扼要重述论据+转折（总论）In this memo the vice president of Nature's Way CNW), a chain of stores selling health food and health-related products, recommends opening a store in Plamesville. To support this recommendation the vice president cites the following facts about Plainesville: (1) sales of exercise shoes and clothing are at all-time highs; (2) the local health club is more popular than ever; and (3) the city's schoolchildren are required to participate in a fitness program. Close scrutiny of each of these facts, however, reveals that none of them lend credible support to the recommendation.四.开头段的写法Merely based on unfounded assumption and dubious (suspicious) evidence, the statement draws a conclusion that_____. To substantiate (support) the conclusion, the arguer points out evidence that_____. In addition, he indicates that_____. Furthermore, he cites the result of a recent survey in support of this recommendation. At first glance, the author’s argument appears to be somewhat convincing, but further reflection reveals that it omits some substantial concerns that should be addressed to substantiate the argument. In my point of view, this argument suffers from N logical flaws.1. 先在开头段指出原文的conclusion2. 指出作者引用哪些evidence or assumption3. 指出存在几个flawIn this argument, the author concludes that, to support his conclusion, he points out that ____. In addition, he infers that____. Furthermore, the arguer sites as a typical example in support of his recommendation. However, this alone / only this do not constitute a logical argument in favor of its conclusion, and fails to provide convincing support, making this argument sound and invulnerable.五.如何写正文段逻辑错误的攻击顺序1. 按照逻辑错误出现的顺序，分段顺序攻击。
GMAT数学讲义.ppt
• A. 12/11
B. 4/3
C.11/8
• D. 3/2
E.2
有关数学的运算
熟悉： sqrt(2)=1.414 sqrt(3)=1.732 sqrt(5)=2.236
Round off Round to
有关数学的运算
熟悉： sqrt(2)=1.414 sqrt(3)=1.732 sqrt(5)=2.236
有关数学的运算
Add,plus Subtract Difference Multiply, times Product divide
有关数学的运算
Add,plus
•加
Subtract
•减
Difference
•差
Multiply,times
•乘
Product
•积
divide
•除
Divisible Divided evenly Divident Divisor Quotient Remainder
有关数学的运算
Divisible Divided evenly Divident Divisor ent Remainder
有关数学的运算
• 可被整除的 • 被整除 • 被除数，红利 • 因子，除数 •商 • 余数
• Which of the following could be the remainder when four consecutive positive integers are each divided by 3?
• 集合 • 子集 • 真子集 • 解集 • 补集 • 交集
• 最近演了3部电影。调查了100个人，其中20人看 了所有3部，10人什么都没看。看了每部电影的 人数分别为：47，48，55。问：只看了2部的人 数？
[江苏省苏州市2008年高考数学二轮复习讲义]第38讲：化归与转化
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第38讲：化归与转化 张家港高级中学 施曙光一、高考要求等价转化是把未知解的问题转化到在已有知识范围内可解的问题的一种重要的思想方法．通过不断的转化，把不熟悉、不规范、复杂的问题转化为熟悉、规范、简单的问题．历年高考，等价转化思想无处不见，我们要不断培养和训练自觉的转化意识，将有利于强化解决数学问题中的应变能力，提高思维能力和技能、技巧．二、两点解读重点：注意转化的等价性与非等价性的不同要求，实施等价转化时确保其等价性，保证逻辑上的正确．难点：①转化的等价性与非等价性；②常作为高考压轴题． 三、课前训练1． f (x )是R 上的奇函数，f (x ＋2)＝f (x )，当0≤x ≤1时，f (x )＝x ，则f (7.5)等于 （ ）(A ) 0.5 (B ) －0.5 (C ) 1.5 (D )－1.52．等差数列{a n }和{b n }的前n 项和分别用S n 和T n 表示，若534+=n nT S n n ，则88a b 的值为( )(A )34(B )1 (C )5 (D )653． 函数f (x )=x 3–3bx +3b 在（0，1）内有极小值，则b 的取值范围是 ．4．设椭圆y a22＋x b 22＝1 （a >b >0）的半焦距为c ，直线L 过(0，a )和(b ，0)，已知原点到L 的距离等于2217c ，则椭圆的离心率为_____． 四、典型例题例1已知两条直线l 1:y =x ，l 2:ax –y =0，其中a ∈R ，当这两条直线的夹角在(0，2π)内变动时，a 的取值范围是 ( )(A )（0，1） (B )（33，3）(C )（33，1）∪（1，3） (D )（1，3）例2 已知三棱锥S -ABC 的三条侧棱两两垂直，SA ＝5，SB ＝4，SC ＝3，D 为AB 的中点，E 为AC 的中点，则四棱锥S -BCED 的体积为 ( )(A )152(B ) 10 (C ) 252 (D ) 352例3一条路上共有9个路灯，为了节约用电，拟关闭其中3个，要求两端的路灯不能关闭，任意两个相邻的路灯不能同时关闭，那么关闭路灯的方法总数为 ．例4在连接长方体各顶点的直线中，成异面直线的共有 对?例5 对任意函数f (x )， x ∈D ，可按图示构造一个数列发生器，其工作原理如下： ①输入数据x 0∈D ，经数列发生器输出x 1=f (x 0)； ②若x 1∉D ，则数列发生器结束工作；若x 1∈D ，则将x 1反馈回输入端，再输出x 2=f (x 1)，并依此规律继续下去．现定义124)(+-=x x x f (Ⅰ)若输入x 0=6549，则由数列发生器产生数列{x n }，请写出{x n }的所有项；(Ⅱ)若要数列发生器产生一个无穷的常数列，试求输入的初始数据x 0的值；(Ⅲ)若输入x 0时，产生的无穷数列{x n }，满足对任意正整数n 均有x n ＜x n +1；求x 0的取值范围．例6设椭圆C 1的方程为12222=+b y a x (a ＞b ＞0)，曲线C 2的方程为y =x1，且曲线C 1与C 2在第一象限内只有一个公共点P ．(Ⅰ)试用a 表示点P 的坐标；(Ⅱ)设A 、B 是椭圆C 1的两个焦点，当a 变化时，求△ABP 的面积函数S (a )的值域； (Ⅲ)记min{y 1，y 2，…，y n }为y 1，y 2，…，y n 中最小的一个．设g (a )是以椭圆C 1的半焦距为边长的正方形的面积，试求函数f (a )=min{g (a )，S (a )}的表达式．第38讲 化归与转化 过关练习1．设f (x )＝3x －2，则f -1[f (x )]等于 ( ) （A ）x +89（B ） 9x －8 （C ）x （D ）132x -2．函数f (x ) = | lg x | ，若0<a <b 时有f （A ）>f （B ），则下列各式中成立的是 ( ) （A ） ab ≤1 （B ） ab <1 （C ） ab >1 （D ） a >1且b >13．正方形ABCD 与正方形ABEF 成90°的二面角，则AC 与BF 所成的角为 ( ) （A ）45° （B ） 60° （C ）30° （D ） 90°4．(a ＋b ＋c )10展开式的项数是 ( ) （A ）11 （B ） 66 （C ） 132 （D ）3105．某房间有4个人，那么至少有2人生日是同一个月的概率是 ．（列式表示即可）6．直线y = a 与函数y = x 3–3x 的图象有相异三个交点，则a 的取值范围为 ．7．已知平面向量a = (3，–1)，b = (23,21)． （Ⅰ） 证明a ⊥b ；（Ⅱ）若存在不同时为零的实数k 和t ，使x = a + (t 2 – 3) b ， y = – ka + tb ，且x ⊥y ，试求函数关系式k = f (t )；（Ⅲ）据（Ⅱ）的结论，讨论关于t 的方程f (t ) – k = 0的解的情况．8．设A 、B 是双曲线x 2–22y =1上的两点，点N （1，2）是线段AB 的中点． (Ⅰ)求直线AB 的方程；(Ⅱ)如果线段AB 的垂直平分线与双曲线相交于C 、D 两点，那么A 、B 、C 、D 四点是否共圆？为什么？第38讲化归与转化参考答案课前训练部分1．B2．D3．0<b <14．12典型例题部分例1. 分析直线l 2的变化特征，化数为形，已知两直线不重合，因此问题应该有两个范围即得解答案：C例2.由S ∆ADE ＝14S ∆ABC 和三棱椎的等体积转化容易求，选A ．例3.解析：9个灯中关闭3个等价于在6个开启的路灯中，选3个间隔（不包括两端外边的装置）插入关闭的过程故有C 35=10种．例4 [分析] 过长方体各顶点的连线共有＝28条；这些连线中有多少对是异面直线?如果采用分类计算的方法则十分繁杂，且易重复或易遗漏．但考虑到每对异面直线对应着长方体的四个不共面的顶点，即对应着一个三棱锥，而每个三棱锥中有3对异面的棱，故可把命题等价转化为以长方体的顶点为顶点的三棱锥有多少个，从而知原题答案是对．例5 解：（1）∵f (x )的定义域D =（–∞,–1)∪(–1,+∞)∴数列{x n }只有三项，1,51,1911321-===x x x ； （2）∵x x x x f =+-=124)(,即x 2–3x +2=0 ∴x =1或x =2，即x 0=1或2时，n n n n x x x x =+-=+1241故当x 0=1时，x n =1，当x 0=2时，x n =2（n ∈N *）； （3）解不等式124+-<x x x ，得x ＜–1或1＜x ＜2， 要使x 1＜x 2，则x 1＜–1或1＜x 1＜2， 对于函数164124)(+-=+-=x x x x f ， 若x 1＜–1，则x 2=f (x 1)＞4，x 3=f (x 2)＜x 2；若1＜x 1＜2时，x 2=f (x 1)＞x 1且1＜x 2＜2；依次类推可得数列{x n }的所有项均满足x n +1＞x n （n ∈N *) 综上所述，x 1∈(1,2)． 由x 1=f (x 0),得x 0∈(1,2).例6解：（1）将y =x1代入椭圆方程，得112222=+x b a x化简，得b 2x 4–a 2b 2x 2+a 2=0，由条件，有Δ=a 4b 4–4a 2b 2=0，得ab =2， 解得x =2a 或x =–2a （舍去），故P 的坐标为(a a 2,2). (2)∵在△ABP 中，｜AB ｜=222b a -，高为a2, ∴)41(22221)(422aa b a a S -=⋅-⋅=∵a ＞b ＞0,b =a 2，∴a ＞a 2,即a ＞2,得0＜44a＜1， 于是0＜S （a ）＜2，故△ABP 的面积函数S (a )的值域为(0,2)． (3)g (a )=c 2=a 2–b 2=a 2–24a ， 解不等式g (a )≥S (a )，即a 2–24a ≥)41(24a -， 整理，得a 8–10a 4+24≥0，即(a 4–4)(a 4–6)≥0，解得a ≤2（舍去）或a ≥46.故f (a )=min{g (a ), S (a )}⎪⎪⎩⎪⎪⎨⎧<-≤<-=)6()41(262(444422a a a a a过关练习1． C 2． B 3． B 4．B 5．441212A 1- 6．–2<a <27．(1)证明：∵a ·b =23)1(213⋅-+⨯=0，∴a ⊥b ； (2)解：∵x ⊥y ,∴x ·y =0即［a +（t 2–3)b ］·(–k a +t b )=0，整理后得 –k a 2+［t –k (t 2–3)］a ·b +t (t 2–3)·b 2=0 ∵a ·b =0,a 2=4,b 2=1∴上式化为–4k +t (t 2–3)=0，∴k =41t (t 2–3). (3)解：讨论方程41t (t 2–3)–k =0的解的情况，可以看作曲线f (t )=41t (t 2–3)与直线y =k 的交点个数于是f ′(t )=43(t 2–1)=43(t +1)(t –1). 令f ′(t )=0,解得t t(–∞,–1)–1 (–1,1) 1 (1,+∞) f ′(t ) + 0 – 0 + f (t )↗极大值↘极小值↗当t =–1时，f (t )有极大值，f (t )极大值=2； 当t =1时，f (t )有极小值，f (t )极小值=–21.而f (t )=41(t 2–3)t =0时，得t =–3,0,3.所以f (t )的图象大致如右： 于是当k >21或k <–21时，直线y =k 与曲线y =f (t )仅有一个交点，则方程有一解；当k =21或k =–21时，直线与曲线有两个交点，则方程有两解；当k =0，直线与曲线有三个交点，但k 、t 不同时为零，故此时也有两解；当–21<k <0或0<k <21时，直线与曲线有三个交点，则方程有三个解 8．(1)设AB ∶y =k (x –1)+2代入x 2–22y =1. 整理得（2–k 2）x 2–2k (2–k )x –(2–k )2–2=0 ① 设A (x 1,y 1)、B （x 2,y 2),x 1,x 2为方程①的两根 所以2–k 2≠0且x 1+x 2=22)2(2k k k --.又N 为AB 中点，有21（x 1+x 2）=1.∴k (2–k )=2–k 2,解得k =1.故AB ∶y =x +1. (2)解出A （–1，0）、B （3，4）得CD 的方程为y =3–x .与双曲线方程联立.消y 有x 2+6x –11=0 ②记C (x 3,y 3)、D (x 4,y 4)及CD 中点M (x 0,y 0)由韦达定理可得x 0=–3,y 0=6.∵|CD |=104)()(243243=-+-y y x x ∴|MC |=|MD |=21|CD |=210. 又|MA |=|MB |=102)()(210210=-+-y y x x .即A 、B 、C 、D 四点到点M 的距离相等，所以A 、B 、C 、D 四点共圆.。
GMAT数学考试中余数精讲
Introduction (赶时间的童鞋可以略过。
只是一些概念帮助童鞋们回忆余数~~~) DefinitionIf x and y are positive integers, there exist unique integers q and r, called the quotient and remainder, respectively, such that y= divisor * quotient + remainder = xq + r; and 0<=r < x.For example, when 15 is divided by 6, the quotient is 2 and the remainder is 3 since 15 = 6*2+3.Notice that 0<= r < x means that remainder is a non-negative integer and always less than divisor.This formula can also be written as y/x = q + r/x.PropertiesWhen y is divided by x the remainder is0 if y is a multiple of x.For example, 12 divided by 3 yields the remainder of 0 since 12 is a multiple of 3 and 12 = 3*4+0.When a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer.For example, 7 divided by 11 has the quotient 0 and the remainder 7 since 7=11*0+7The possible remainders when positive integer y is divided by positive integer x can range from 0 to x-1.For example, possible remainders when positive integer y is divided by 5 can range from 0 (when y is a multiple of 5) to 4 (when y is one less than a multiple of 5).If a number is divided by 10, its remainder is the last digit of that number. If it is divided by 100 then the remainder is the last two digits and so on.For example, 123 divided by 10 has the remainder 3 and 123 divided by 100 has the remainder of 23.1. Collection of MethodsMethod 1：小数法（妹纸自己取的名字，包括后面的方法也都是妹纸取的~欢迎讨论）A way that the GMAT will test remainders is what you would typically just divide back into the problem to determine the decimals:25/4 = 6 remainder 1Divide that 1 back by 4 to get 0.25, so the answer is 6.25.Any number with a remainder could be expressed as a decimal.The remainder provides the data after the decimal point, and the quotient gives you the number to the left of the decimal point.Consider this problem (which appears courtesy of GMAC):Example： 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) 12Sol：Going back to the concept of the remainder, the remainder of 9 is what will give us that 0.12 after the decimal place. The answer to the division problem x/y is either:96 remainder 9Or96.12Therefore, when the remainder of 9 is divided back over y, we get 0.12. Mathematically, this means that:9/y = 0.120.12y = 912y = 900y = 900/12y = 300/4y = 75The correct answer is B.方法二：重建法Given that an integer "n" when divided by an integer "a" gives "r" as reminder then the integer"n" can be written asn = ak + rwhere k is a constant integer.Example 1： What is the remainder when B is divided by 6 if B is a positive integer?(1) When B is divided by 18, the remainder is 3(2) When B is divided by 12, the remainder is 9Sol：STAT1 : When B is divided by 18, the remainder is 3So, we can write B asB = 18k + 3Now, to check the reminder when B is divided by 6, we essentially need to check the reminder when 18k + 3 is divided by 618k goes with 6 so the reminder will 3So, it is sufficientSTAT2 : When B is divided by 12, the remainder is 9So, we can write B asB = 12k + 9Now, to check the reminder when B is divided by 6, we essentially need to check the reminder when 12k + 9 is divided by 612k goes with 6 so the remainder will be the same as the reminder for 9 divided by 6 which is 3So, reminder is 3So, it is sufficient.Answer will be DPractice:What is the remainder when positive integer t is divided by 5?(1) When t is divided by 4, the remainder is 1(2) When t is divided by 3, the remainder is 1这题请大家自己试一试哦。
名师解析2008MBA各科——数学篇
名师解析2008MBA各科——数学篇名师郑家俊解析2008年MBA联考大纲之数学部分郑家俊：大家好！很高兴近距离和大家交流。
我从99年开始从事MBA联考数学的备考工作，今年是数学大纲变化最大的一次，微积分、线性代数不再做考试要求。
数学分值仍然是75分；概率部分只考概率初步，相当于高中的内容，在样题中只占5分的分值。
这样，初等数学占70分，就是说MBA联考数学部分基本上只考初等数学了。
初等数学相比07年大纲，增加的内容有：实数的概念、性质、运算及应用；整式、分式及其运算；排列组合；常见平面图形(三角形、四边形、圆)；平面直角坐标及直线与圆的方程。
减少的部分为：绝对值，比和比例，算术平均值和几何平均值。
还有一些是大纲里没有，但是还是会考的。
比如说大纲里没有比和比例，但是照考不误，还有二项式定理，绝对值，考实数不可能不涉及运算应用。
现在大纲说是考一元一次方程，一元二次方程的解法和应用。
考应用的话，那出题几乎就不再受约束了。
现在初数除了复数之外几乎没有什幺不考的了。
对于初数，深度，范围有所增加。
另外，样题里有一道平均值的题，在大纲里是不作要求的；还有一道推理题，这道题不属于任何考试范围。
就以上变化，对广大备考考生来说不一定是利好消息，考试范围缩小了，初数内容增加了，感觉备考就相对轻松些。
其实不是这样，大纲的改变，让我们重新回到了高考的时代，我们不能轻视，应该加以重视，因为高考不容易得高分，失分点太多，原因是初数涵盖了初中、高中六年的知识，面多，量大、范围广，机巧性强等特点，从历届MBA考试成绩看，初数得分率极底，出错率最高。
大纲的变化我就说这幺多。
问：就数学大纲的变化，郑老师给大家提个备考建议，谢谢！郑家俊：因08年大纲增加的数学部分几何占相当大的比例，从往年数学考试失分情况来看初数所。
《gmat数学讲义》PPT课件
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《GMAT数学讲义》课件
2
中位数
了解中位数的计算方法及其在数据分析中的意义。
3
样本均值与总体均值
区分样本均值和总体均值的计算方法。
概率与统计
概率基本概念 离散型与连续型随机变量 概率分布及其特点 统计推断及假设检验
几何
平面几何
学习基本的平面几何概念和 性质，如平行线、垂直线、 三角形、四边形等。
重点强调
掌握考试重点，提高解题速度和准确性。
技巧分享
发现解题的有效技巧和策略，让您事半功倍。
基本算术和代数
1 四则运算
2 代数方程
学习加法、减法、乘法和除法的基本规则 及运用。
解读和解决各类代数方程，包括线性和二 次方程。
3 因式分解
4 函数与图像
掌握多项式的因式分解法，简化算式和求 解方程。
理解函数的定义，绘制函数图像及其特征。
分数，小数和百分比
分数运算
学习分数的四则运算及应用。
小数运算
掌握小数的四则运算及应用。
百分比
了解百分比的计算方法及实际 应用。
整数
1
正整数与负整数
了解整数的正负概念及其在数学运算中的应用。
2
整数运算
掌握整数的加法、减法、乘法和除法规则。
3
整数的应用
理解整数在实际问题中的应用，如温度、海拔等。
平均数和中位数
立体几何
理解球体、立方体和圆柱体 等立体图形的计算和性质。
几何推理
通过几何推理解决几何问题， 包括证明三角形相似和全等 的方法。
《GMAT数学讲义》PPT课 件
准备参加GMAT考试？这个课件将为您提供全面的数学知识以及解题技巧。 从基础算术到高级几何，准备解开GMAT数学难题的秘密吧！
shu08
a11 a12 a13 0.3 0.4 0.1 A = a21 a22 a23 ＝ 0.5 0.2 0.6 a a a33 0.1 0.3 0.1 31 32
d1 20 d 2 = 10 d 30 3
Mathematics for Economists
投入产出模型
• 投入产出矩阵：
a11 a12 L a1n a21 a22 L a2n A= M a a L a nn n1 n 2
aij表示生产1单位的产品j需要的第i种投入的数量
China Agri. Univ.
d1 10 d2 = 5 d 6 3
China Agri. Univ.
College of Economics and Mangement
求解
Mathematics for Economists
投入产出模型例子1
• 投入产出矩阵：
< 0 ，n为奇数 An > 0 ，n为偶数
China Agri. Univ.
College of Economics and Mangement
Mathematics for Economists
• 练习：判断二次型的正定与负定：
q = 5u 2 + 2v 2 + 3uv
2 2 q = u12 + 6u2 + 3u3 − 2u1u2 − 4u2u3 2 2 q = 2u12 + 3u2 − u3 + 6u1u2 − 8u1u3 − 2u2u3
C −1 = C '
Q AC = CD ∴ C ' AC = D A = CDC '
GMATppt课件
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• 例：The following appeared in a memorandum from the owner of Carlo’s Clothing to the staff:
• “Since Disc Depot, the music store on the next block, began a new radio advertising campaign last year, its business has grown dramatically, as evidenced by the large increase in foot traffic into the store. While the Disc Depot’s owners have apparently become wealthy enough to retire, profits at Carlo’s Clothing have remained stagnant for the past three years. In order to boost our sales and profits, we should therefore switch from newspaper advertising to frequent radio advertisements like those for Disc Depot.”
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图表解读题（Graphics Interpretation）
• 图表解读题考查对图形数据的解析。每道 题都有填空设问和下拉菜单;考生需做出正 确选择。
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多源推理（Multi-Source Reasoning）
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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
parallel lines 平行线
perpendicular lines 垂直线
（A）0.004
（B）0.02775
（C）2.775
（D）3.6036
（）36.036
例2：蓝皮书P148第98题
Of the following,which if closest to ?
(A)7.5
(B)15
(C)75
(D)150
(E)750
例3：蓝皮书P144第76题
(C) 120
(D) 40
(E) 20
1.一元二次方程
一般常用因式分解法：
十字交叉法 例1：蓝皮书P140第50题
If x is a number such that and , what is the value of ?
(A)-2
(B)-1
(C)0
is between
(A)3 and 4
(B)4 and 5
(C)5 and 6
(D)6 and 7
(E)7 and 8
2.二元一次方程组
消去其中一个元素即可
例1：
（1）
（2）
（1）—（2），消去y，得x=1,y=2
蓝皮书P147第92题 If and ,what is the value of ?
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 抵押贷款 mort
interest rate 利率
list price 标价
margin 利润
mark up 涨价
mark down 降价
digit 位
dividend 被除数
divide 除以
divisor 除数
divisible by 可整除的
evenly divisible 可整除的
even number 偶数
factor 因子
integer 整数
irrational 无理数
(A)5
(B)4
(C)3
(D)2
(E)0
例4：蓝皮书P155第148题
If and are positive integers and , what is the value of
(A)3
(B)5
?6
(D)8
(E)9
界定范围法
例1：蓝皮书P143第69题
closest approximation 近似
decimal 小树
decimal notation 十进制
decimal point 小数点
decreased 下降后的
decrease???to??? 从???下降至???
decerease by???下降了
define 定义
denominator 分母
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 外切，外接
clockwise 顺时针
concentric circle 同心圆
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
GMAT数学讲义
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数学词汇
1. 数学符号
等于： ≡ equal to, the same as, is
不等于： ＞ more than
＜ less than
≥ no more than
right triangle 直角三角形
square 正方形
sphere 球
side 边
surface area 表面积
straight angle 平面
segment 线段
tangent 切线
triangle 三角形
vertex（vertices） angle 顶角
vertical angle 对顶角
Ⅰ. =
Ⅱ. =1
Ⅲ. and are prime integers
(A) None
(B) Ⅰ only
(C) Ⅱ only
(D) Ⅲ only
(E) Ⅰand Ⅲ
例3：蓝皮书P136第24题
If 2x=3y=10,then 12xy=?
(A) 1,200
(B) 200
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
2-digit number 两位数
markup毛利
profit 利润
purchasing price 购买价
retail value 零售价
sale price 销售价
simple interest 单利
8.其他代数
addition 加
arithmetic mean 算术平均数
average 平均数
base 底数
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
(D)1
(E)2
例2：蓝皮书P173第264题
(E) none of the above
例3：蓝皮书P210第499题
In a certain game, a large container is filled with red, yellow, green, and blue beads worth, respectively, 7, 5, 3, and 2 points each .A number of beads are then removed from the container. If the product of the point values of the removed beads is 147,000, how many red beads were removed?
least common multiple 最小公倍数
multiple 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
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
4.数列和集合
arithmetic progression 等差数列
geometric progression 等比数列
set 集合
term 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
绝对值： |…| absolute value
平方： square X2
立方： cube
开平方： square root
开立方： cube root
平行： // parallel to
垂直： ⊥ perpendicular to
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 质数
proportional to 正比于