數(shù)學(xué)獨(dú)家12126全程班

1、GRE Quantitative Reasoning趙立儀2017.1.261. (a)What is the prime factorization of 372?2. (b)What are the positive divisors of 372?3. 2. (a)What are the prime divisors of 100?4. (b)What are the prime divisors of144?5. Which of the integers 2,9,19,29,30,37,45,49,51,83,90 and 91 are prime numbers?6. What

2、is the prime factorization of 585?7. The integer a is even and the integer b is odd. 8. For each of the following integers, 9. indicate whether the integer is even or odd.10. 6. When the positive integer n is divided by 3, 11. the remainder is 2 and when n is divided by 5, 12. the remainder is 1. Wh

3、at is the least possible value of n?n=3a+2n=5b+13a+2=5b+1-a=(5b-1)/31. 372=22*3*311 2 3 4 6 12 31 62 93 124 186 3722. 2&52&33. 2 19 29 37 834. 32*5*135. EOEEOO6. 111. 690,70If represents a single digit in the integer above, which of the following CANNOT be a factor of this integer?A. 2 B. 3

4、C. 4 D. 5 E. 72. n is a positive integer with exactly two different divisors greater than1, how many positive factors does n2 have?A. 4 B. 5 C. 6 D. 8 E. 93. 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. 4,9004. If

5、 n is an integer and 99n2200, then n could have at most how many values?A. Two B. Four C. Six D. Eight E. Tenn=4 1 2 4 n2=16=24 1 2 4 8 16 n=9 1 3 9n=25 1 5 25n=a2(a為質(zhì)數(shù))，n2=a4，正因子個數(shù)為4+1=590=2*32*5196=22*72300=22*3*52 LCM=22*32*52*7210，11，12，13，14C B A E C E A AC D CD D B C E5. n and k are positive e

6、ven integers.Quantity A Quantity BThe remainder when k2+n is divided by 2 The remainder when kn is divided by 2Quantity A is greater. B. Quantity B is greater.C. The two quantities are equal D. The relationship cannot be determined from the information given.6. The “reflection” of a positive integer

7、 is obtained by reversing its digits. For example, 321 is the reflection of 123. The difference between a five-digit integer and its reflection must be divisible by which of the following?A. 2 B. 4 C. 5 D. 6 E. 97. If n and m are positive integers and m is a factor of 26, what is the greatest possib

8、le number of integers that can be equal to both 3n and 26/m?A.Zero B. One C. Three D. Four E. Six xyzmn-nmzyx=10000 x y+100z+10m+n -(x+10y+100z m 0n) =9999x+990y m-9999n8. Let S be the set of all positive integers n such n2 is a multiple of both 24 and 108. Which of the following integers are diviso

9、rs of every integer n in S?Indicate all such integers.A 12B 24C 36D 729. When 20 is divided by the positive integer k, the remainder is k-2, which of the following is a possible value of k?A. 8B. 9C. 10D. 11E. 1220=ka+k-2=k(a+1)-222=k(a+1)1&22; 2&1124=23*3108=22*33LCM=23*3324*34-22*32set:n,

10、2n, 3n, 4n.10. If 1050-74 is written as an integer in base decimal notation, what is the sum of the digits in that integer?A. 424B. 433C. 440D. 449E. 46711. When 20 is divided by the positive integer k, the remainder is k-2, which of the following is a possible value of k?A. 8B. 9C. 10D. 11E. 12100.

11、00（1后面有50個0，一共51位）1050-74后退了一位變成了50位99.9926, 除了2和6這兩位，剩余48位，均為948*9+2+6=44012. Of the following values of n, the value of will be greatest for n= A. -3B. -2C. 0D. 2E. 313. If is expressed as a terminating decimal, how many zeros will t have between the decimal point and the first nonzero digit to th

12、e right of the decimal point?A. ThreeB. FourC. FiveD. SixE. Nine1()5n93125t (-5)-1)-n=(-5)n23*53=103=10001/1000=0.0010.001/26=0.001/6414. Of the following, which is the closest approximation to ?A. 1B. 5C. 10D. 20E. 10015. If the sum of two positive integers is 24 and the difference of their squares

13、 is 48, what is the product of the two integers?A. 108B. 119C. 128D. 135E. 1432(97.942)(0.261)(0.51)a+b=24a2-b2=48a2-b2=(a+b)(a-b)=48a-b=2a=13, b=11ab=143第一節(jié) Arithmetic-核心考點(diǎn)2. 注意在比較類里面變化的基數(shù)是什么？ 第一類： 自我變化類| 公式： （|新-舊|）/ 舊 例子： 從2到3增加了百分之多少？ 從3到2減少了百分之多少？ 第二類： 兩個不同的數(shù)之間的比較| 公式： （|先-后|）/ 后 例子： 1比2少百分之多少？

14、 2比1多百分之多少？) 1(11bbbababa%100*323%100*223%100*112%100*212%100*8 .1816 .1408 .181At Megalomania Industries, factory workers were paid $20 per hour in 1990 and $10 per hour in 2000. The CEO of Megalomania Industries was paid $5 million per year in 1990 and $50 million per year in 2000. The percent inc

15、rease in the pay of Megalomanias CEO from 1990 to 2000 was what percent greater than the percent decrease in the hourly pay of Megalomanias factory workers over the same period? (A) 850%(B) 900%(C) 950%(D) 1700%(E) 1900% 1990 2000workers 20 10CEO 5m 50m%1700%100*%50%50%900%50%100*201020:ker%900%100*

16、5550:sWorCEO第一節(jié) Arithmetic算數(shù)- 綜合注意事項(xiàng)（OG+總結(jié)經(jīng)驗(yàn)?。?. If a persons salary increased from $200 per week to $234 , what was the percent increase in the persons salary?2. If an athletes weight decreased from 160 pounds to 152 pounds, what was the percent decrease in the athletes weight?3. A particular stock

17、 is valued at $40 per share. If the value increase by 20 percent and then decreases by 25 percent, what will be the value of the stock per share after the decrease?4. If the ratio of the number of men to the number of women on a committee of 20 members is 3 to 2, how many members of the committee ar

18、e women?52*20%)251%)(201 (40%100*160152160%100*200200234Algebra(代數(shù))- 01. Operations with Algebraic Expressions（代數(shù)運(yùn)算） 1. 專業(yè)詞匯VariableLike termsConstantCoefficientIdentityLinear equationQuadratic equation變量同類項(xiàng)常數(shù)系數(shù)恒等式線性方程二次方程(xy)2=x22xy+y2x2-y2=(x+y)(x-y) 2. 知識點(diǎn)講解.2xy2+3xy+6z-21xz-xy+0.5 is called an a

19、lgebraic expression 代數(shù)表達(dá)式2xy2, 3xy, 6z, -21xz, -xy, 0.5稱為terms（代數(shù)）項(xiàng)3xy和-xy稱為like terms or similar terms（同類項(xiàng)）0.5稱為constant term 常數(shù)項(xiàng)coefficient系數(shù)：2xy2這一項(xiàng)的系數(shù)是2， -21xz的系數(shù)是-212xy2+3xy+6z-21xz-xy+0.5 =1稱為equation等式，等式的solution解就是滿足等式的值。如x=2是x2-2x=0的一個解Algebra(代數(shù))- 02. Rules of Exponents（指數(shù)運(yùn)算法則） 1. 專業(yè)詞匯Bas

20、e Exponent基數(shù)指數(shù)指數(shù)運(yùn)算規(guī)則 x-a=1/xa (xa)*(xb)=xa+b (x)a)b=xab xa/xb=xa-b x0=1 (xy)a=xayan is an integer Quantity A Quantity B (-1)n(-1)n+2 1Quantity A is greater Quantity B is greaterThe two quantities are equalThe relationship cannot be determined from the information givenn+n+2=2n+2:偶數(shù)（-1）的偶次方=1Algebra(

21、代數(shù))- 03. Solving Linear Equations and Inequalities（解線性方程和線性不等式） 1. 專業(yè)詞匯Satisfy the equationEquivalent equationsSubstitutionElimination滿足方程等價方程式代入消去 2. 知識點(diǎn)講解. 注意不等式符號的可逆性： 左邊同時乘以或除以一個負(fù)數(shù) 開方的對象是不是介于0和1之間 x+y=a (一正二定三相等)2x+y=3x+3y=6x0,y0 x+y=a:x和y的和是一個定值當(dāng)x=y時，x和y的乘積最大22020)(2yxxyyxxyxyyxyxInequalities:

22、不等式不等式部分不會像中國高考那樣考推導(dǎo)、證明，注意兩邊乘以負(fù)數(shù)變號等最基本原則即可。對已有的不等式兩邊取倒數(shù)或者負(fù)數(shù)，不等式可能會改變方向，例如：-73 (-1/7)(-1/3)絕對值（absolute value）|x|恒非負(fù)，|x-a|表示數(shù)軸上x到a的距離1. 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. x416 B. x327 C. x216 D. 2

23、 |x|5 E. 23x+46 2. x1 Quantity A Quantity B |x-1| |-x-1|3232552225522242xxorxxxorxorxxxyx 2y-x和0比較大小Algebra(代數(shù))- 04 Solving Quadratic Equations（解二次方程） 1. 專業(yè)詞匯Quadratic formula二次方程 2. 知識點(diǎn)講解.第二節(jié) Algebra(代數(shù))- 05 Function （方程） 1. 專業(yè)詞匯 Domain定義域 2. 知識點(diǎn)講解.y=f(x)稱為一個函數(shù)Domain定義域?yàn)楹瘮?shù)有定義的所有x值Range值域?yàn)楹瘮?shù)所有可能的取值如

24、y=f(x)=x2-1,那么定義域就是所有實(shí)數(shù)，值域?yàn)閥-1注意文字轉(zhuǎn)換為數(shù)學(xué)方程時候的過程 principal interest interest rate simple interest compound interest semiannually quarterly 本金 利息 利率 單利 復(fù)利 半年 一季度 1000 10% 單利存10年1000 10% 復(fù)利存10年 1000 10% 季度復(fù)利存1年1000 *10%*101000*(1+10%)101000*(1+2.5%)101. Ellen has received the following scores on 3 exams

25、:82,74, and 90. What score will Ellen need to receive on the next exam so that the average (arithmetic mean)score for the 4 exams will be 85?2. In a driving competition, Jeff and Dennis drove the same course at average speeds of 51 miles per hour and 54 miles per hour, respectively. If it took Jeff

26、40 minutes to drive the course, how long did it take Dennis?3. If an amount P is to be invested at an annual interest rate of 3.5 percent, compounded annually, what should be the value of P so that the value of the investment is 1,000 at the end of 3 years?4. A college student expects to earn at lea

27、st $1,000 in interest on an initial investment of $20,000. If the money is invested for one year at interest compounded quarterly, what is the least annual interest rate that would achieve the goal?P(1+3.5)3=100020000(1+x/4)420000 1. 942. 37.83. 9024. 4.8%Algebra(代數(shù))- 07 Coordinate Geometry（幾何坐標(biāo)） 1.

28、 專業(yè)詞匯Xy-coordinate systemOriginQuadrantsReflection of P about x-axisSlope (rise over run)InterceptParallelPerpendicularparabola平面直角坐標(biāo)系原點(diǎn)象限P點(diǎn)關(guān)于x軸的對稱點(diǎn)斜率截距平行垂直拋物線 2. 知識點(diǎn)講解.計算直線的斜率 （rise over run）平面直角坐標(biāo)上兩點(diǎn)間距離為 通過兩點(diǎn)直線方程為： 化簡為斜截式：其中k為斜率（slope），b為y軸截距（intercept）平行，垂直的斜率條件若兩直線垂直，其斜率乘積為-1 術(shù)語了解： P1 is the ref

29、lection of P about x-axis = P1 and P are symmetric about the x-axis221221)()(yyxxPQy=kx+bQ(x1,y1) Q(x2,y2)帶入112121xxyyxxyygraphs of functionsy=kx+b 的圖像是一條直線，斜率為k, y軸的截距為b（截距可正可負(fù)可為0）y=ax2+bx+c , 其中a0的圖像是一個拋物線parabola如y=x2-2x-3的頂點(diǎn)vertex是（1, -4），x=1是對稱軸line of symmetry，開口向上opens upward(x-a)2+(y-b)2=r2的

30、圖像是一個圓circle，圓心center為（a, b), radius半徑為ry=f(x)和y=g(x)的圖像交點(diǎn)就是f(x)=g(x)的解上加下減&左加右減y=x2-2x-3=x2-2x+1-1-3=(x-1)2-4第二節(jié) Algebra(代數(shù))- 例題綜合練習(xí)1. Find an algebraic expression to represent each of the following.2. (a) The y is subtracted from 5, and the result is multiplied by 37.3. (b) Three times x is sq

31、uared, and the result is divided by 7.4. (c) The product of (x+4) and y is added to 18.5. A group can charter a particular aircraft at a fixed total cost. If 36 people charter the aircraft rather than 40 people, then the cost per person is greater by $12.6. (a) What is the fixed total cost to charte

32、r the aircraft?7. (b) What is the cost per person if 40 people charter the aircraft?8. An antiques dealer bought c antique chairs for a total of x dollars. The dealer sold each chair for y dollars.9. (a) Write an algebraic expression for the profit, P, earned from buying and selling the chairs.10. (

33、b) Write an algebraic expression for the profit per chair.1. (5-y)*37(3x)2/7(x+4)y+182. 4320;1083. P=cy-x;y-(x/c)4. In the xy-plane, find the following.5. (a)Slope and y-intercept of the line with equation 2y+x=6 6. (b)Equation of the line passing through the point (3,2) with y-intercept 17. (c)The

34、y-intercept of a line with slope 3 that passes through the point (-2,1)8. (d) The x-intercepts of the graphs in (a),(b), and (c)9. For the parabola y=x2-4x-12 in the xy-plane, find the following.10. (a)The x-intercepts11. (b)The y-intercept12. (c)Coordinates of the vertex13.For a given two-digit pos

35、itive integer, the tens digit is 5 more than the14. units digit. The sum of the digits is 11. Find the integer.15.7. If the ratio of 2x to 5y is 3 to 4, what is the ratio of x to y?4. -0.5,3; y=x/3+1; 7; 6,-3,-7/35. -2,6; -12 (2,-16)6. 837. 15:88. n is a positive integer.Quantity A Quantity BThe remainder when 34n is divided by 10 1A. Quantity A is greater.B. Quantity B is greater.C. The two quantities are equalD. The relationship cannot be determined from the information given.9. (4,6) is t