GMAT数学讲义 难题集合.pdf

Data Sufficiency 题型解题分析题型解题分析 DS 题形式 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 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 分析什么是充分与不充分 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 23 11 5 等唯一解的 Statement 不充分 能得到诸如x 7 x 1等两个或更多解 及其它一切无法计算 出解的 Statement 例例 1 What is the value of x 1 3x 15 2 5x 30 例例 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 例例 3 Is x equal to 1 1 x2 1 2 x2 4 1 2 例例 4 体会下列两个 Question 的区别 There are eight balls in the pocket 已知袋中有 8 个球 Question 1 Are all the balls in the pocket red 袋中所有的球都是红色的吗 充分 YES NO Question 2 Are there any red balls in the pocket 袋中有红色的的球吗 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 Question 1 Question 2 Statement 1 Statement 2 Statement 3 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 GMAT 数学重点解析数学重点解析 第一章第一章 基本数论基本数论 I 奇数与偶数奇数与偶数 Odd and Even Numbers 1 奇数 奇数 偶数 奇数 奇数 奇数 偶数 偶数 偶数 奇数 偶数 偶数 奇数 偶数 奇数 偶数 偶数 偶数 2 多个整数之和为奇数 其中包含奇数个奇数 多个整数之和为偶数 其中包含偶数个奇数 多个整数之积为奇数 全部都是奇数 多个整数之积为偶数 其中包含至少一个偶数 任何一个大于 2 的偶数都能够表示成为两个质数的和 哥德巴赫猜想 例例1 1 If x and y are integers and xy2 is a positive odd integer which of the following must be true xy is positive xy is odd x y is even 例例1 2 Is x an even integer 1 x is the square of an integer 2 x is the cube of an integer II 因数与质因数因数与质因数 Factors and Prime Factors 分解质因数 将一个大的合数写成质数相乘的形式 是极其重要的基本数学技能 方法 1 将大合数写成显而易见的若干小合数相乘的形式 例如 4500 45 100 2 反复用 2 除这个小合数 直至不能被 2 整除 再反复用 3 除 以此类推 直至被完全分解 写成完全由质数相乘的形式 例例1 3 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 例例1 4 How many factors does 360 have A 24 3 B 36 C 48 D 120 E 360 III 最大公约数与最小公倍数最大公约数与最小公倍数 Greatest Common Divisors and Least Common Multiples 两个数的最大公约数与最小公倍数的求解方法 1 将两个数分别各自分解质因数 2 每一个质数 取较小的指数 含 0 相乘得到最大公约数 每一个质数 取较大的指数 相乘得到最小公倍数 例例1 5 The greatest common divisor of a and b is 21 and the least common multiple of a and b is 126 where a and b are positive integers what is the sum of a and b A 105 B 147 C 150 D 105 or 147 E 105 or 150 例例1 6 Is the integer n a multiple of 140 c59 1 n is a multiple of 10 2 n is a multiple of 14 IV 余数余数 Remainders 牢记 ykxb 表示 y 除以 x 商是 k 余数是 b 例例1 7 If x and y are integers is xy 1 divisible by 3 c52 1 When x is divided by 3 the remainder is 1 2 When y is divided by 9 the remainder is 8 例例1 8 If n is a positive integer what is the remainder when 38n 3 2 is divided by 5 A 0 B 1 C 2 D 3 E 4 V 连续数连续数 Consecutive Numbers 2 个连续整数之积至少是 2 的倍数 3 个连续整数之积至少是 6 的倍数 4 4 个连续整数之积至少是 24 的倍数 例例1 9 The product of two consecutive positive integers CANNOT be A a prime number B divisible by 11 C a multiple of 13 D an even number less than 10 E a number having 4 as its units digit 例例1 10 If x is the product of three consecutive positive integers which of the following must be true I x is an integer multiple of 3 II x is an integer multiple of 4 III x is an integer multiple of 6 A I only B II only C I and II only D I and III only E I II and III VI 小数 分数与科学计数法小数 分数与科学计数法 Decimals Fractions and Scientific Notation 1 识别各位数字名称 7654 321 其中 7 thousands 6 hundreds 5 tens 4 units or ones decimal point 3 tenths 2 hundredths 1 thousandths 例例1 11 3 2 6 If and each represent single digits in the decimal above what digit does represent 1 When the decimal is rounded to the nearest tenth 3 2 is the result 2 When the decimal is rounded to the nearest hundredth 3 24 is the result c54 例例1 12 Any decimal that has only a finite number of nonzero digits is a terminating decimal For example 24 0 82 and 5 096 are three terminating decimals If r and s are positive integers and the ratio r s is expressed as a decimal is r s a terminating decimal 5 1 90 r 100 2 s 4 c59 2 除了无限不循环小数之外 其它任何小数都能转化成分数的形式 例例1 13 Of the following which best approximates 0 1667 0 8333 0 3333 0 2222 0 6667 0 1250 A 2 00 B 2 40 C 2 43 D 2 50 E 3 43 3 科学计数法表示 0 0000486 4 86 10 5 8 245 000 8 245 106 例例1 14 If k is an integer and 0 0010101 10k is greater than 1 000 what is the least possible value of k og62 A 2 B 3 C 4 D 5 E 6 例例1 15 If 1050 74 is written as an integer in base decimal notation what is the sum of the digits in that integer A 424 B 433 C 440 D 449 E 467 VI 比率与比例比率与比例 Ratios and Proportions the ratio of A to B 表示为 A B 或者 A B There is twice as much A as B 表示为 A 2B 或者 A B 2 例例1 16 Among registered voters in a certain district the ratio of men to women is 3 5 Of the district currently includes 24 000 registered voters how many additional men must register to make the ratio 4 5 A 2 000 B 3 000 6 C 4 000 D 5 000 E 6 000 例例1 17 A certain fraction is equivalent to 2 5 If the numerator of the fraction is increased by 4 and the denominator is doubled the new fraction is equivalent to 1 3 What is the sum of the numerator and denominator of the original fraction A 21 B 26 C 28 D 35 E 49 例例1 18 A merchant purchased a jacket for 60 and then determined a selling price that equaled the purchase price of the jacket plus a markup that was 25 percent of the selling price During a sale the merchant discounted the selling price by 20 percent and sold the jacket What was the merchant s gross profit on this sale A 0 B 3 C 4 D 12 E 15 例例1 19 In a certain formula p is directly proportional to s and inversely proportional to r If p 1 when r 0 5 and s 2 what is the value of p in terms of r and s A s r B r s4 C s r4 D r s E r s 4 7 第二章第二章 代代 数数 I 指数运算指数运算 Rules of Exponents 1 am an am n am an am n am n amn 2 am bm a b m am bm a b m 例例2 1 234 2 832 96 A 3 B 6 C 9 D 12 E 18 例例2 2 Of the following values of n the value of n 1 5 will be greatest for n A 3 B 2 C 0 D 2 E 3 II 解方程解方程 Equations 1 一元二次方程 0 axbxc 2 标准根的公式为 x1 2 bbac a 2 4 2 但真正实用的是分解因式法 十字相乘法 应熟练掌握分解质因数 试图将 a 和 c 分 解成 a a1 a2 c c1 c2 以成功凑出 b a1 c2 c1 a2 的形式 原方程转化为 a1x c1 a2x c2 0 根x1 c a 1 1 x2 c a 2 2 例例2 3 A certain theater has 100 balcony seats For every 2 increase in the price of a balcony seat above 10 5 fewer seats will be sold If all the balcony seats are sold when the price of each seat is 10 which of the following could be the price of a balcony seat if the revenue from the sale of balcony seats is 1 360 A 12 8 B 14 C 16 D 17 E 18 2 二元一次方程组 将一个方程中x与y 的关系代入另一个方程 先后解出两个未知数的数值 通常单纯的计算题 计算量并不大 关键要对等价的方程有高度的敏感 例如 方程组中 两个方程最终化简形式其实一模一样 根本没有特定的解 xy yx 3 226 例例2 4 If the sum of two positive integers is 24 and the difference of their squares is 48 what is the product of the two integers A 108 B 119 C 128 D 135 E 143 例例2 5 If xy 6 what is the value of xy x y 1 x y 5 2 xy2 18 例例2 6 What is Steve s annual salary and Maria s annual salary 1 The combined total of the annual salaries of Steve and Maria is 80 000 2 If Steve were to receive a 10 percent increase in annual salary and Maria an 10 percent increase their combined annual salaries would be 88 000 III 不等式不等式 Inequalities 1 对已有的不等式两边取倒数或负数 不等号要改变方向 例如 73 73 11 73 11 73 2 对x 4 x 3 x x x 2 x3 x4 等函数的性质有一定的认识 在几个函数的比较大小中 对的取值范围要有清醒的分段意识 x x 1 x 10 x 01x 1 3 绝对值 9 x 恒非负 xa 表示数轴上x 到a 的距离 例例2 7 If x 0 9 which of the following could be the value of x A 0 81 B 0 9 C 20 9 D E 0 9 0 99 10 01 例例2 8 If x 0 is x 1 1 x 2 1 2 x x 1 例例2 9 Which of the following inequalities has a solution set that when graphed in the number line is a single line segment of finite length A x 4 16 B x 3 27 C x 2 16 D x 25 E x 2346 IV 函数函数 Functions 题目会定义新的算符和运算法则 模仿运算即可 例例2 10 If the operation is defined for all integers a and b by a b a b ab which of the following statements must be true for all integers a b and c a b b a a 0 a a b c a b c V 数列数列 Sequences 1 等差数列 Arithmetic Sequence 第 n 项 n aand 1 1 前 n 项之和 n n n aan nd Sna 1 1 1 22 10 2 等比数列 Geometric Sequence 第 n 项 n n aa q 1 1 前 n 项之和 n n aq S q 1 1 1 q 1 当 q 1 时 n a S q 1 1 例例2 11 If the sum of 7 consecutive integers is 434 then the greatest of the 7 integers is A 62 B 65 C 67 D 69 E 71 例例2 12 In the sequence 1 2 4 8 16 32 each term after the first is twice the previous term What is the sum of the 16th 17th and 18th terms in the sequence A 218 B 3 217 C 7 216 D 3 216 E 7 215 例例2 13 In a certain sequence the first term is 1 and each successive term is 1 more than the reciprocal of the term that immediately precedes it What is the fifth term of the sequence A 3 5 B 5 8 C 8 5 D 5 3 E 9 2 11 第三章第三章 几几 何何 I 三角形与四边形三角形与四边形 Triangles and Quadrilaterals 1 三角形的某些性质 三角形两边之和大于第三边 两边之差小于第三边 三角形中 较大角的对边也较大 2 勾股定理 直角边a 2 直角边b 2 斜边c 2 要对以下形状的直角三角形要特别熟悉 3 三角形 面积 1 2 底 高 矩形 Rectangles 面积 长 宽 周长 2 长 宽 正方形 Squares 面积 边长 2 周长 4 边长 例例3 1 In the figure above square CDEF has area 4 What is the area of ABF A 2 2 B 2 3 C 4 D 3 3 E 6 例例3 2 If each side of ACD above has length 3 and if AB has length 1 what is the area of region BCDE A 9 4 B 7 3 4 C 9 3 4 D 7 3 2 E 63 1 2 3 60 5 2 4 1 45 30 A B C F D E 45 60 3 1 C B A D 12 E A D 例例3 3 In the rectangle above AB 6 AD 8 If point P is selected from segment BC at random what is the probability that the length of segment AP is less than 3 5 A 3 8 B 19 8 C 3 5 B P C D 3 4 E 3 5 8 例例3 4 A ladder 25 feet long is leaning against a wall that is perpendicular to level ground The bottom of the ladder is 7 feet from the base of the wall If the top of the ladder slips down 4 feet how many feet will the bottom of the ladder slip A 4 B 5 C 8 D 9 E 15 II 直线与角直线与角 Lines and Angles 同位角相等 两直线平行 内错角相等 同旁内角互补 P 例例3 5 In the figure above QRS is a straight line and line TR bisects PRS Is it true that lines TR and PQ are parallel T 1 PQ PR 2 QR PR R Q S III 圆圆 Circles 半径为 r 的圆 面积 r 2 周长 r 2 角度为 x 的圆弧 弧长 x r 360 2 同一段圆弧所对圆心角是圆周角的两倍 例例3 6 In the circle above PQ is parallel to diameter OR and OR has length 18 What is the length of minor arc PQ 13 R P 35 Q A 2 B 9 4 C 2 7 D 2 9 E 3 O 例例3 7 In the figure above A is the point of tangency for two circles and also the center of the third circle If the radii of three circles are 1 what is the external perimeter of the figure A A 7 3 B 10 3 C 4 D 3 14 E 6 例例3 8 In the figure above a circle is inscribed in a square with side b and a square with side a is inscribed in the circle What is the area of the square with side b b a 1 a 4 2 The radius of the circle is 22 IV 立体几何立体几何 Solid Geometry 长方体 Rectangular Solids 体积 长 宽 高 正方体 Cubes 体积 边长 3 圆柱 Cylinders 体积 底面半径 2 高 例例3 9 What is the volume of a certain rectangular solid 1 Two adjacent faces of the solid have areas 15 and 24 respectively 2 Each of two opposite faces of the solid has area 40 例例3 10 The inside dimensions of a rectangular wooden box are 6 inches 8 inches by 10 inches A cylindrical canister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces Of all such canisters that could be used what is the radius in inches of the one that has maximum volume A 3 B 4 C 5 D 6 E 8 14 V 平面直角坐标几何平面直角坐标几何 Plane Rectangular Coordinate Geometry 平面直角坐标上两点间距离为 xxyy 22 1212 通过两点直线方程为 yyyy xxxx 12 12 1 1 化简为斜截式 ykxb 其中 k 为斜率 Slope b为y轴截距 Intercept 若两直线垂直 其斜率乘积为 1 例例3 11 In the rectangular coordinate system above the line yx is perpendicular bisector of segment AB not shown and the x axis is the perpendicular bisector of segment BC not shown If the coordinates of point A are 2 3 what are the coordinates of point C y A 3 2 B 3 2 C 2 3 D 3 2 E 2 3 例例3 12 In the rectangular coordinate system shown above does the line k not shown intersect quadrant 1 The x intercept of k is negative 2 The slope of k is positive 例例3 13 In the xy plane does the point 4 12 lie on line k 1 The point 1 7 lies on line k 2 The point 2 2 lies on line k 例例3 14 In the rectangular coordinate system above both of two tangent circles are tangent to the x axis If the radii of the two circles are 4 and 6 respectively what is the slope of the line on which two centers lie A 1 B 2 6 1 C 3 2 1 3 D 1 E 5 1 2 O yx A x y O x y O x 15 16 例例3 15 An isosceles triangle lies on the rectangular coordinate plane the coordinates of point A are 0 0 and the coordinates of point B are 3 1 point C could lie at one of 6 positions such that 1 3 1 3 3 1 1 3 1 3 3 1 How many lengths of side BC are possible A 2 B 3 C 4 D 5 E 6 第四章第四章 文字应用题文字应用题 I 集合集合 Sets 1 文氏图题 准确画出文氏图 认真分析题目中每一个数字在文氏图中的归属 例例4 1 All trainees in a certain aviator training program must take both a written test and a flight test If 70 percent of the trainees passed the written test and 80 percent of the trainees passed the flight test what percent of the trainees passed both tests 1 10 percent of the trainees did not pass either test 2 20 percent of the trainees passed only the flight test 例例4 2 In a marketing survey for products some people were asked which of the products if any they use Of the people surveyed a total of 400 use A a total of 400 use B a total of 450 use C a total of 200 use A and B simultaneously a total of 175 use B and C simultaneously a total of 200 use C and A simultaneously a total of 75 use A B and C simultaneously and a total of 200 use none of the products How many people were surveyed A 950 B 975 C 1 000 D 1 025 E 1 050 例例4 3 How many integers between 1 and 100 inclusive can be divided by none of 2 3 and 5 A 24 B 26 C 28 D 30 E 32 例例4 4 In a certain class 10 students can play the piano 14 students can play the violin 11 students can play the flute If 3 students can play exactly three instruments 20 students can play exactly one instrument how many students can play exactly two instruments A 3 B 6 C 9 D 12 E 18 17 2 双标准分类题 一个集合的元素按照两种标准各自进行分类 将集合分成四个部分 作表仔细分析 例例4 5 A shipment of banners contains banners of two different shapes triangular and square and two different colors red and green In a particular shipment 26 of the banners are square and 35 of the banners are red If 60 of the red banners in the shipment are square what is the ratio of red triangular banners to green triangular banners A 7 50 B 3 13 C 7 30 D 13 37 E 35 26 例例4 6 One fifth of the light switches produced by a certain factory are defective Four fifths of the defective switches are rejected and 1 20 of the nondefective switches are rejected by mistake If all the switches not rejected are sold what percent of the switches sold by the factory are defective A 4 B 5 C 6 25 D 11 E 16 II 排列 组合与概率 排列 组合与概率 Permutation Combination and Probability 1 组合 Combination n m m C n mn nm mm CC n 排列 Permutation n n Pn nnn mmn m PCP mn mm PCm 11 2 概率 Probability 1 等可能事件的概率 P Am n 如 7 个白球 5 个红球中 取出一个白球的概率 2 互斥事件的概率 ABP ABP AP B 如 一个骰子掷出奇数的概率或偶数的概率 18 3 相互独立事件同时发生的概率 P ABP A P B 如 一个骰子连续掷出两次奇数的概率 4 n次独立重复试验发生k次的概率 kkn k n C pp1 如 一股骰子连续掷9次 只有2次是奇数的概率 例例4 7 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 例例4 8 In how many distinguishable ways can the 7 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 例例4 9 一只袋中装有5只乒乓球 其中3只白色 2只红色 现从袋中取球2次 每次1 只 取出后不再放回 试求 2只球都是白色的概率 2只球颜色不同的概率 至少有1只白球的概率 例例4 10 Six cards numbered from 1 to 6 are placed in an empty bowl First one card is drawn and then put back into the bowl then a second card is drawn If the cards are drawn at