《毕出21套》解析:数学第2套

毕出 ·

数学第 2 套DS

1.The cardinality of a finite set is the number of elements in the set. What is the cardinality of set A?

(1) 2 is the cardinality of exactly 6 subsets of set A.

(2) Set A has a total of 16 subsets, including the empty set and set A itself.


(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.


解:

题目的意思为:一个有限集的cardinality 就是这个集合元素的个数,问集合A的元素个数。

(1)集合A中恰好有6个子集的元素个数为2

(2)集合A共有16个子集,包括空集和它本身

首先需要知道,一个有限集合有2n个子集,其中n为该有限集合中的元素个数,非空子集有2n - 1 个。

条件1:集合A中元素取2个有6中不同的取法,Cn2 = 6, n = 4, 即,集合A有4个元素。

条件2:2n = 16, n = 4, 即,集合A有4个元素。

本题答案为D。


2. X,81,73,71,98,73,64

What is the value of X in the above list of 7 numbers?

(1) The average (arithmetic mean) of these 7 numbers is 80.

(2) The range of these 7 numbers is 36.


(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.


解:

条件1,若知道这7个数的平均数,则X值必然可求。

条件2,纵然不算x的range为98-64 = 34,比36小,但我们不知道x是整个数列的极小值还是极大值,所以无法确定x的值。答案为A。


3.If 1 < d <2, is the tenths digit of the decimal representation of d equal to 9?

(1)d+0.01<2

(2)d+0.05>2


(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.


解:

通过条件1可得1d>1.95,此时d的十分位必然是9,答案为B。


4.If b is the product of three consecutive positive integers c, c+ 1, and c+2, is b a multiple of 24?

(1)b is a multiple of 8.

(2)c is odd.


(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.


解:

3个连续的正整数的乘积一定是3的倍数,因为每3个连续的正整数中必有一个是3的倍数。因此,根据条件1,若b是8的倍数,则b同时是3和8的倍数,自然也是24的倍数。条件2说c是奇数,则只有c+1是偶数,无法保证其中有8,所以单独2不行,答案为A。


5.If x and z are integers, is x+z2odd?

(1)x is odd and z is even.

(2)x-z is odd.


(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.


解:

条件1说x是奇数,z是偶数,则x+z2 为奇数,条件1是充分的;条件2说x-z为奇数,则x和z必然一奇一偶。无论谁是奇谁是偶,x+z2均为奇数。答案为D。


6.If x an integer?

(1) x2is an integer.

(2) x/2 is not 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.


解:

本题有一些陷阱。根据条件1,请注意,题目中的x可以是根号下的数字。例如,如果x是根号2,则x的平方是整数,但x显然不是整数,不充分。条件2,如果x =3,则x/2就不是整数,但x是整数(当然,x本身不是整数且除2不是整数的数太多了,此处就不列举了)。条件1+条件2:x =3和x=根号2均同时满足两个条件,即,x依然不一定是整数。答案为E。


7.If m and n are positive integers, what is the value of 3/m + n/4?

(1) mn=12

(2) 3/m is in lowest terms and n/4 is in lowest terms.


(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.


解:

本题首先要将题目看懂。in lowest terms是最简分式的意思。首先将题干通分,则有:

3/m + n/4 = (12+mn)/4m。根据条件1可得24/(4m),因为无法确定m的值,所以单独1是不充分的。单独根据条件2显然更不充分。把两个条件相加,如果3/m和4/n都是最简式,则m不能是3的倍数,n不能是2的倍数。m*n = 12,则m = 4,n= 3。显然可以求得题干中算式的答案,答案为C。


8.If x, y, and d are integers and d is odd are both x and y divisible by d?

(1) x+y is divisible by d.

(2) x-y is divisible by d.


(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.


解:

显然单独两个条件均不能推出x和y是否都能单独被d整除。先将条件1+条件2,则有:

因为x-y, x+y都能被d整除,根据高斯同余的加法定理,则(x+y)+(x-y)和(x+y)-(x-y)均可以被d整除,(x+y)+(x-y)=2x且(x+y)-(x-y)=2y,d是奇数,因此,x, y都能被d整除,答案为C。


9.If K is a positive integer less than 10 and N=4321+K, what is the value of K?

(1) N is divisible by 3.

(2) N is divisible by 7.


(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.


解:

条件1,能被3整除的数,要求各个数位相加可以被三整除。4+3+2+1 = 10。此时,K = 2, 5, 8均满足N能被3整除,不充分。

条件2,能被7整除的数要求比较高,即,一个整数的个位数字去掉,并用剩下的数减去这个个位数字的2倍,得到的数如果是7的倍数,那么这个数便可以被7整除。432-(1+K)*2如果是7的倍数,即可满足条件。显然,420,427均是7的倍数。仅当K = 5时,N = 420(427时K不是整数)。

答案为B。


10.If m and n are positive integers, is n even?

(1) m(m+2)+1=mn

(2) m(m+n) is odd.


(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.


解:

条件1:m(m+2)+1=mn,可以推出mn=(m+1)*(m+1),即,n=(m+1)*(m+1)/m

因为m和m+1互质,n又是整数,两个互质的数相除是个整数,被除数只能是1,所以这个条件的解只有一个:m=1,n=4,n必然为偶数。

条件2: m(m+n)。若m是偶数,则乘积必然为偶数。若m是奇数,则m+n必为奇数才可以,此时n必为偶数。

答案为D。


11.If @ denotes a mathematical operation, does x@y=y@x for all x and y?

(1) For all x and y, x@y=2(x2+y2).

(2) For all y, 0@y=2y2.


(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.


解:

本题实际上是把@当成了一个新函数。只要交换条件中的x和y,且和原式相等,即是成立的。

条件1显然是正确的。

条件2中只有y,x并不知道,所以无法确定是否x@y=y@x。

答案为A。


12.The 9 participants in a race were divided into 3 teams with 3 runners on each team. A team was awarded 6-n points if one of its runners finished in nth place, where 1≤n≤5. If all of the runners finished the race and if there were no ties was each team awarded at least 1 point?

(1) No team was awarded more than a total of 6 points.

(2) No pair of teammates finished in consecutive places among the top five places.


(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.


解:

条件1,一共9个人,得分应分别为:5,4,3,2,1,0,0,0,0。如果没有任意一组超过6分,则第一组最多是5,1,0;第二组最多是4,2,0,第三组必为1,0,0。因此,条件1必然是充分的。

条件2说的是,没有任意一对队友是在前五名相邻的。条件1所提到的情况是满足的,也可以是第一组5,3,1;第二组4,2,0;第三组0,0,0。因此,条件2是不充分的。

答案为A。


13. If x and y are integers, is xy+1 divisible by 3?

(1)When x is divided by 3, the remainder is 1.

(2)When y is divided by 9, the remainder is 8.


(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.


解:

因为xy+1能否被3整除,不但取决于x的值,而且还取决于y的值,所以(1)和(2)单独都很明显不充分。

根据(1)+(2)可得:

x=3m+1,y=9n+8,其中m,n是整数。

把这两个式子代入xy+1可得:

xy+1=(3m+1)(9n+8)+1

=27mn+24m+9n+84-1

=3(9mn4-8m+3n+3)

因此xy+1能被3整除,所以(1)+(2)充分。


14.If x is a positive integer, how many positive integers less than x are divisors of x?

(1)x^2 is divisible by exactly 4 positive integers less than x^2.

(2)2x is divisible by exactly 3 positive integers less than 2x.


(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.


解:

将一个数分解为质因子相乘的形式,将相同的质因子写成幂指数的形式,因子的个数=每个因子的幂指数+1的乘积,包含1及其本身。

如:30=(2*3*5)的因子的个数=2*2*2=8个;20=(22*5)的因子的个数=3*2=6个

条件1:设x有两个质因子,分别为a, b, x=am*bn, x2=a2m*b2n, (2m+1)*(2n+1)=5, m, n都是整数,解得m=2, n=0,x是由两个相同的质因子构成。条件1充分。

条件2:设x=am, 当a=2时,2x=2(m+1),m+1+1=4,m=2, x=22,有3个因子; 当a不等于2时,2m=2*am, 2*(m+1)=4, m=1, x=a1, 有两个因子。条件2不充分。


15.Sprinklers are being installed to water a lawn. Each sprinkler waters in a circle. Can the lawn be watered completely by 4 installed sprinklers?

(1) The lawn is rectangular and its area is 32 square yards.

(2) Each sprinkler can completely water a circular area of lawn with a maximum radius of 2 yards.


(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.


解:

条件1,我们不知道喷头喷水范围的半径多大,所以不知道是否能覆盖整个的长方形区域。

条件2,只知道半径,不知道lawn的大小。

两个条件相加,按照最大半径算,可以覆盖整个lawn。但若不是最大半径,则不一定覆盖。因此依然不充分。答案为E。


16.In quadrilateral ABCD, is angle BCD a right angle?

(1) Angle ABC is a right angle.

(2) Angle ADC is a right angle.


(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.


解:

单独确定四边形的任意一个内角,均不能确定BCD的值。

条件1+条件2相对较难判断。但其实不规则四边形是可以满足这个条件的,即,就算两个条件均满足,BCD也可以不是直角。方法是:先画abc,然后bad大于90度画一个,最后找一个点线连上dc,可以构造该四边形。

答案为E。


17.

In rectangular region PQRS above, T is a point on side PS. If PS =4, what is the area of region PQRS?

(1)ΔQTR is equilateral.

(2) Segments PT and TS have equal length.


(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.


解:

通过条件1,我们可以做一条辅助线,则有:


从点T引出一条对QR的垂线。因为三角形QTR为等边三角形,则角QTM为30度(基于等边三角形的三线合一)。QM = 1/2QR。由此可以推出线段TM和QR的关系。因为QR = PS = 4,所以该长方形面积可求。

通过条件2,我们只知道T是中点,无法推得长方形的长和宽的关系,不充分。

答案为A。


18.The top surface area of a square table top was changed so that one of the dimensions was reduced by 1 inch and the other dimension was increased by 2 inches. What was the surface area before these changes were made?

(1) After the changes were made, the surface area was 70 square inches.

(2) There was a 25 percent increase in one of the dimensions.


(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.


解:

桌子是正方形的。题干的意思是一对边减少1英尺,另外一对边就增加2英尺。

条件1,告诉我们改变后的面积,又因为我们知道了两对边长的改变,自然可以求得原边长,进而可以求面积。

条件2,告诉了我们增长2英尺等于增长了25%,相当于直接告诉了我们边长,可以求得面积。

答案为D。


19.Is the perimeter of a certain rectangular garden greater than 50 meters?

(1) The two shorter sides of the garden are each 15 meters long.

(2) The length of the gardens 5 meters greater than the width of the garden.


(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.


解:

条件1: 两个短边都是15m 因此短边之和 2b=30。因此,两个长边的和 2a>30,周长=2 (a + b)>60>50。条件1是充分的。

条件2: 长方形的长比长方形的宽多了5m,无法判断周长的情况,不充分。

答案为A。


20.

In the figure above, points A, B, C, and D are collinear and AB, BC, and CD are semicircles with diameters d1cm, d2cm, and d3cm, respectively. What is the sum of the lengths of AB, BC and CD in centimeters?

(1) d1: d2: d3 is 3:2:1 

(2) The length of AD is 48 cm.


(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.


解:

首先注意,如果题目问AB的长度,则表示问的是AB的线段长度。若问弧长,则会写为arc AB的长度。

条件1:只知道比值,无法得知准确数字,不充分。

条件2:AD的长度等于AB+BC+CD。充分。

答案为B。


21.In the rectangular coordinate system, line k passes through the point(n,-1). Is the slope of line k greater than zero?

(1) Line k passes through the origin.

(2) Line k passes through the point(1, n+2).


(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.


解:

单独通过条件1和条件2显然都不能确定斜率。

条件1+条件2:

如果k通过原点,则直线方程为y = kx。已知通过两点,必然可以求出n和k的值,自然可以断定是否k大于0。

答案为C。


22.What is the length of the hypotenuse of △ABC?

(1) The lengths of the three sides of △ABC are consecutive even integers .

(2) The hypotenuse of △ABC is 4 units longer than the shorter leg.


(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.


解:

本题主要的难度在于hypotenuse的词义。这个词特指直角三角形的斜边。

条件1:直角三角形的三边长度是连续的偶整数,只有一组可能,即,6,8,10。(要满足勾股定理),充分。

条件2:只知道斜边比最短边大4。根据勾股定理可以列出一个方程两个未知数,无法得解。不充分。

答案为A。


23.In the standard (x, y) coordinate plane, what is the slope of the line containing the distinct points P and Q?

(1) Both P and Q lie on the graph of |x|+|y| =1

(2) Both P and Q lie on the graph of |x+y|=1


(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.


解:

条件1:图像是个菱形,p和q两个点可以在任何线上,斜率不固定。

条件2:图形是两条平行的直线,p和q只要在不同的直线上斜率就不固定。

条件1+条件2: p和q只是条件二的基础上加了区间,斜率还是不固定。

答案为E。


24.Alan and Sue have each been saving one dollar a day and will continue to do so for the next month. If Sue began saving several days before Alan, in how many days from today will Alan have saved one-half as much as Sue?

(1) As of today, Alan has saved 7 dollars and Sue has saved 27 dollars.

(2) Three days from today, Alan will have saved one-third as much as Sue.


(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.


解:

条件1给出了现在Alan和Sue已存的钱数。题干中给出了两个人每天都向账户里存1块钱。因此,很容易确定多少天后Alan会存到Sue的一半。

条件2只给出了比值,没有给出具体数字,无法推出准确的时间。

答案为A。


25.What is the value of x/yz?

(1) x=y/2 and z=2x/5

(2) x/z=5/2 and 1/y=1/10


(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.


解:

条件1:x = y/2;z = y/5,带入题干无法约掉y,不能确定数值。

条件2:y = 10;x/z可以确定,则带入即可直接确定x/yz的值,答案为B。


26.Of the numbers q, r, s, and t, which is greatest?


(1)The average(arithmetic mean) of q and r is s.

(2)The sum of q and r is t.


(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.


解:

条件1,我们无从得知t的值,无法判断谁是最大的。

条件2,我们无从得知s的值,无法判断谁是最大的。

条件1+条件2:通过条件1可以知道q和r有一个数比s小,一个比s大。通过条件2可以知道,若q和r都是正数,则t是最大的。但如果q和r是负数,则t是最小的,最大未知。

因此,答案为E。


27. A box of light bulbs contains exactly 3 light bulbs that are defective. What is the probability that a sample of light bulbs picked at random from this box will contain at least 1 defective light bulb?

(1) The light bulbs in the sample will be picked 1 at a time without replacement.

(2) The sample will contain exactly 20 light bulbs.


(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.

解:

本题其实非常简单。一定要记住的一点是,只要题干中没说的信息,不要自己假设。譬如本题,题干中没有说过那个box里的灯泡数总量,所以无论两个条件的哪一个或者是两个条件加和,均无法得出概率。答案为E。


28. Is xy<6?


(1)x<3 and y<2

(2)1/2 < x < 2/3 and y2<64


(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.

解:

条件1略有迷惑性。如果不考虑负数,则本条件很充分。但题干中没说过x和y均为正数。所以,条件1不充分(例如-9和-10相乘就远大于6)。

条件2可得-8

答案为B。



29. What is the median of the data set S that consists of the integers 17, 29, 10, 26, 15, and x?

(1) The average (arithmetic mean) of S is 17.

(2) The range of S is 24.


(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.

解:

条件1:通过平均数可以求得x的值,自然可以确定整个集合的中位数。充分。

条件2:极差为24。x可能是最大的数34,也可能是最小的数5,不充分。

答案为A。





30. If x≠1,is y equal to x+1?

(1)(y−2)/(x−1)=1

(2)y2=(x+1)2


(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.

解:

条件1:等式两边同乘以x -1,则有(y−2) = (x−1),即,y = x + 1,充分。

条件2:两边同时开方,可得y = x +1 或 y = -x – 1,不充分。

答案为A。


31. Three dice, each with faces numbered 1 through 6, were tossed onto a game board. If one of the dice turned up 4, what was the sum of the numbers that turned up on all three dice?

(1)The sum of two of the numbers that turned up was 10.

(2)The sum of two of the numbers that turned up was 11.


(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.

解:

本题主要的难度在于读题。掷三颗骰子,第一颗是4,问三颗骰子的数字加一起是多少。

条件1:其中的两颗加一起是10。这有很多种可能,如果前两颗加一起是10,最后一颗可以是1-6的任意一个,所以不充分。

条件2:其中的两颗加一起是11。加一起是11的一定是后两颗。如果是前两颗,则第二颗为7,不符合骰子的特点。因此总数为15,充分。

答案为B。


32. Is x2-y2 a positive number?

(1) x-y is a positive number.

(2) x+y is a positive number.


(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.

解:

题干中的多项式可以变为(x + y)(x - y)。

条件1:无从得知x+y的值,不充分。

条件2:无从得知x -y的值,不充分。

条件1+条件2:题干中两项均为正,乘积必为正,答案为C。


33. An investment has been growing at a fixed annual rate of 20% since it was first made;no portion of the investment has been withdrawn, and all interest has been reinvested. How much is the investment now worth?

(1) The value of the investment has increased by 44% since it was first made.

(2) If one year ago $600 had been withdrawn, today the investment would be worth 12% less than it is actually now worth.


(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.

解:

条件1:只给出百分比,没有任何一个实际值是不可能得到现在价值多少的。

条件2: 假设现在是x,则一年以前是x/1.2,1年以前拿掉了600,则有(x/1.2-600)*1.2=(1-0.12)x,可以求出x,充分。

答案为B。


34. If x+y+z>0, is z>1?

(1) z>x+y+1

(2) x+y+1<0


(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.

解:

条件1:z>x+y+1且x+y+z>0,得出x+y+x+y>-1,即,x+y>-1/2. 由z>x+y+1可知,z要大于1则必须x+y>0,无法得知,不充分。

条件2:由x+y+1<0得出,一个小于-1的数+z>0,因此z必须大于1,充分。

答案为B。


35. In the first 2 hours after Meadow's self-service laundry opens, m large washing machines and n small washing machines are in continual use. Including the time for filling and emptying the washing machines, each load of laundry takes 30 minutes in a large washing machine and 20 minutes in a small washing machine. What is the total number of loads of laundry done at Meadow's self-service laundry during this 2-hour period?

(1) n=3m

(2) 2m+3n=55


(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.

解:

题干的意思为,2小时用m个大洗衣机和n个小洗衣机工作。大洗衣机30分钟处理1 load,小洗衣机20分钟处理1 load,问2小时内一共处理了多少load。

显然,依题意可以直接列方程,则有:

总量 = 4m + 6n

显然,条件2可以,因为直接等式两侧同乘以2,即可得知4m + 6n = 110,答案为B。


36. Let S be a set of outcomes and let A and B be events with outcomes in S. Let ~B denote the set of all outcomes in S that are not in B and let P(A) denote the probability that event A occurs. What is the value of P(A)?

(1) P(A∪B)=0.7

(2) P(A∪~B)=0.9


(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.

解:

条件1:P(A∪B) = P(A) + P(B) - P(AB) = 0.7,显然单独无法判断P(A),不充分。

条件2:P(A∪~B) = 1 - P(B) + P(AB) = 0.9,显然单独无法判断P(A),不充分。

条件1+条件2:两个等式相加,即可得出P(A) = 0.6,充分。

因此,答案为C。


37. How many people in Town X read neither the Word newspaper nor the Globe newspaper?

(1) Of the 2500 people in Town X, 1000 read no newspaper.

(2) Of the people in Town X, 700 read the globe only and 600 read the World only.


(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.

解:

条件1:我们知道1000人不读报纸,但我们不知道是否这个城镇只有2种报纸,所以不充分。

条件2并没有告诉我们一共有多少人,有没有人什么也不读等等,所以单独不充分。

条件1+条件2:依然不充分,还是不知道有没有其它的报纸。

答案为E。


38. What is the sum of 3 consecutive integers?


(1) The sum of the 3 integers is less than the greatest of the 3 integers.

(2) Of the 3 integers, the ratio of the least to the greatest is 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.

解:

条件1:三个数的加和小于三个数中的最大的数。这只能表明这三个数不都是正数,但不能确定它们的加和。

条件2:三个整数中,最小和最大的比值是3。之后-3,-2,-1这三个连续的整数是符合条件的,充分。

因此,答案为B。


39. What is the number of integers that are common to both set S and set T?

(1) The number of integers in S is 7, and the number of integers in T is 6.

(2) U is the set of integers that are in S only or in T only or in both, and the number of integers in U is 10.


(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.

解:

条件1:只知道S和T各自的元素数量,完全无法知道交集的情况。

条件2:知道U=(S+T+交集)= 10,还是不知道交集的元素数量。

条件1+条件2:可以知道S,T,以及交集的数量,充分。

答案为C。


40. A country's per capita national debt is its national debt divided by its population. Is the per capita national debt of Country G within $5 of $500?

(1) Country G's national debt to the nearest $1,000,000,000 is $43,000,000,000

(2) Country G's population to the nearest 1, 000,000 is 86,000,000


(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.

解:

问题问的是,人均债务是否能在495-505之间。显然单独条件1和条件2都不充分。

条件1+条件2:

债务的范围是(42500~43400),人口的范围是(85.5~86.4),债务/人口的范围是(42500/86.4~43400/85.5)=(491.89~507.60),依然不充分。

答案为E。





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