Math Section

1When 20 is divided by the positive integer k, the remainder is k – 2. Which of the following is a possible value of k ?

2In the figure above, is quadrilateral PQRS a parallelogram?


(1) The area of ∆PQS is equal to the area of ∆QRS.

(2) QR = RS

3If n is an integer and xn – xn = 0, what is the value of x ?

(1)x is an integer.

(2)n ≠ 0

4If xy = 165, where x and y are positive integers and x > y, what is the least possible value of x – y ?

5According to a survey, 93 percent of teenagers have used a computer to play games, 89 percent have used a computer to write reports, and 5 percent have not used a computer for either of these purposes. What percent of the teenagers in the survey have used a computer both to play games and to write reports?

6The value of –3 + √(9-k) is positive for which of the following values of k ?

I. -1

II. 0

III. 1

7There were 2 apples and 5 bananas in a basket. After additional apples and bananas were placed in the basket, the ratio of the number of apples to the number of bananas was 1/2 . How many apples were added?

(1) The number of apples added was 2/3 the number of bananas added.

(2) A total of 5 apples and bananas were added.

8S is a set of points in the plane. How many distinct triangles can be drawn that have three of the points in S as vertices?

(1) The number of distinct points in S is 5.

(2) No three of the points in S are collinear.

9A total of 100 customers purchased books at a certain bookstore last week. If these customers purchased a total of 200 books, how many of the customers purchased only 1 book each?

(1) None of the customers purchased more than 3 books.

(2) 20 of the customers purchased only 2 books each.

10In the figure, AB = AE = 8, BC = CD = 13, and DE = 2. What is the area of region ABCDE ?


11A certain business produced x rakes each month form November through February and shipped x/2 rakes at the beginning of each month from March through October. The business paid no storage costs for the rakes from November through February, but it paid storage costs of $0.10 per rake each month from March through October for the rakes that had not been shipped. In terms of x, what was the total storage cost, in dollars, that the business paid for the rakes for the 12 months form November through October?

12If the integer k is a multiple of 3, which of the following is also a multiple of 3 ?

13For any positive integer x, the 2-height of x is defined to be the greatest nonnegative integer n such that 2n is a factor of x. If k and m are positive integers, is the 2-height of k greater than the 2-height of m ?

(1) k > m

(2) is an even integer.

14If 4x < x < x3 < x2, which of the following is a possible value of x ?

15What was the range of the selling prices of the 30 wallets sold by a certain store yesterday?

(1) 1/3 of the wallets had a selling price of $24 each.

(2) The lowest selling price of the wallets was 1/3 the highest selling price of the wallets.

16On a certain nonstop trip, Marta averaged x miles per hour for 2 hours and y miles per hour for the remaining 3 hours. What was her average speed, in miles per hour, for the entire trip?

(1) 2x + 3y = 280

(2) y = x + 10

17In a box of 12 pens, a total of 3 are defective. If a customer buys 2 pens selected at random from the box, what is the probability that neither pen will be defective?

18A scientist recorded the number of eggs in each of 10 birds’ nests. What was the standard deviation of the numbers of eggs in the 10 nests?

(1) The average (arithmetic mean) number of eggs for the 10 nests was 4.

(2) Each of the 10 nests contained the same number of eggs.

19If p, r, s, and t are nonzero integers, is p/r = s/t ?

(1) s = 3p and t = 3r.

(2) 3p = 2r and 3s = 2t.

20If a1 = 1 and an + 1 = 1 + 1/an for all n ≥ 1, what is the value of a5 ?

21A toy store’s gross profit on a computer game was 10 percent of the cost of the game. If the store increased the selling price of the game from $44 to $46 and the cost of the game remained the same, then the store’s gross profit on the game after the price increase was what percent of the cost of the game?

22Four extra-large sandwiches of exactly the same size were ordered for m students, where m > 4. Three of the sandwiches were evenly divided among the students. Since 4 students did not want any of the fourth sandwich, it was evenly divided among the remaining students. If Carol ate one piece from each of the four sandwiches, the amount of sandwich that she ate would be what fraction of a whole extra-large sandwich?

23When the wind speed is 9 miles per hour, the wind-chill factor w is given by w = -17.366 + 1.19t, where t is the temperature in degrees Fahrenheit. If at noon yesterday the wind speed was 9 miles per hour, was the wind-chill factor greater than 0 ?

(1) The temperature at noon yesterday was greater than 10 degrees Fahrenheit.

(2) The temperature at noon yesterday was less than 20 degrees Fahrenheit.

24Which of the following is closest in value to 99 – 92 ?

(1) The temperature at noon yesterday was greater than 10 degrees Fahrenheit.

(2) The temperature at noon yesterday was less than 20 degrees Fahrenheit.

25An office supply store received a shipment of boxes of pencils. How many pencils were in the shipment altogether? (1 gross = 12 dozen)

(1) The number of boxes of pencils in the shipment was 6 gross.

(2) There were 3 dozen pencils in each box of pencils in the shipment.

26Of the following values of n, the value of (-1/5)-n will be greatest for n =

27For all numbers x, the function f is defined by f(x) = 3x + 1 and the function g is defined by g(x) = (x-1)/3. If c is a positive number, what is the value of g(c)?

(1) f(c) = 13

(2) f(1) = c

28A certain company expects quarterly earnings of $0.80 per share of stock, half of which will be distributed as dividends to shareholders while the rest will be used for research and development. If earnings are greater than expected, shareholders will receive an additional $0.04 per share for each additional $0.10 of per share earnings. If quarterly earnings are $1.10 per share, what will be the dividend paid to a person who owns 200 shares of the company’s stock?

29For which real numbers x is the expression 2/{(x-2)(x+1)} not defined?

30What is the value of (2x – 5)3 ?

(1) (2x)3 = 216

(2) (2x + 3)2 = 81

31Phobus, one of the satellites of Mars, revolves about Mars at a distance of approximately 5,800 miles from the center of Mars and approximately 3,700 miles from the surface of Mars. If it is assumed that Mars is spherical and the orbit of Phobus is circular, what is the approximate circumference, in miles, of Mars at its equator?

32In the decimal representation of x, where 0 < x < 1, is the tenths digit of x nonzero?

(1) 16x is an integer.

(2) 8x is an integer.

33A small business invests $9,900 in equipment to produce a product. Each unit of the product costs $0.65 to produce and is sold for $1.20. How many units of the product must be sold before the revenue received equals the total expense of production, including the initial investment in equipment?

34A merchant discounted the sale price of a coat and the sale price of a sweater. Which of the two articles of clothing was discounted by the greater dollar amount?

(1) The percent discount on the coat was 2 percentage points greater than the percent discount on the sweater.

(2) Before the discounts, the sale price of the coat was $10 less than the sale price of the sweater.

35If g(x) = x + 4 for all x and f(x) = (2x2-5)/x for all nonzero x, then f(x) = g(x) for which of the following values of x ?

36Is x > 5 ?

(1) x2 > 25

(2) 2x + 7 > 0

37Pumping alone at their respective constant rates, one inlet pipe fills an empty tank to 1/2 of capacity in 3 hours and a second inlet pipe fills the same empty tank to 2/3 of capacity in 6 hours. How many hours will it take both pipes, pumping simultaneously at their respective constant rates, to fill the empty tank to capacity?

Verbal Section

1More than fifty years after the Second World War, a number of African American soldiers were awarded―some of them posthumously―with the Congressional Medal of Honor, which was the nation’s highest military award, and which was long overdue in recognition of their outstanding bravery.

2The discovery that Earth’s inner core rotates independently of and more quickly than Earth’s outer layers is responsible for advancing studies of the flow of heat from the inner through the outer planet and of the formation and periodic reversal in direction of Earth’s magnetic field.

Q3 to Q6:

In a 1984 book, Claire C.

Robertson argued that,

before colonialism, age was

linea more important indicator

5of status and authority than

gender in Ghana and in

Africa generally. British

colonialism imposed

European-style male-

10dominant notions upon

more egalitarian local

situations to the detriment

of women generally, and

gender became a defining

15characteristic that weak-

ened women’s power and


Subsequent research in

Kenya convinced Robertson

20that she had overgeneralized

about Africa. Before colo-

nialism, gender was more

salient in central Kenya than

it was in Ghana, although age

25was still crucial in determin-

ing authority. In contrast with

Ghana, where women had

traded for hundreds of years

and achieved legal majority

30(not unrelated phenomena),

the evidence regarding

central Kenya indicated that

women were legal minors

and were sometimes treated

35as male property, as were

European women at that

time. Factors like strong

patrilinearity and patrilocality,

as well as women’s inferior

40land rights and lesser

involvement in trade, made

women more dependent on

men than was generally the

case in Ghana. However,

45since age apparently

remained the overriding

principle of social organiza-

tion in central Kenya, some

senior women had much

50authority. Thus, Robertson

revised her hypothesis

somewhat, arguing that

in determining authority in

precolonial Africa age was a

55primary principle that super-

seded gender to varying

degrees depending on the


3The primary purpose of the passage is to

4The passage indicates that Robertson’s research in Kenya caused her to change her mind regarding which of the following?

5The passage suggests that after conducting the research mentioned in line 18, but not before, Robertson would have agreed with which of the following about women’s status and authority in Ghana?

6The author of the passage mentions the status of age as a principle of social organization in precolonial central Kenya in lines 24-26 most likely in order to

7Until a few centuries ago, any large bones discovered in the fields or caves of Europe, now known to be large prehistoric animals, were usually assumed to be the remains of giants and were often displayed as curiosities in castles, palaces, town halls, churches, and monasteries.

8Which of the following, if true, provides evidence that most logically completes the argument below?

According to a widely held economic hypothesis, imposing strict environmental regulations reduces economic growth. This hypothesis is undermined by the fact that the states with the strictest environmental regulations also have the highest economic growth. This fact does not show that environmental regulations promote growth, however, since ______.

9Manatees, aquatic mammals inhabiting Florida’s rivers and coastal waters, swim close to the surface and are frequently killed in collisions with boats. To address the problem, boat traffic in manatee-populated waters is being required to maintain very low speeds. Unfortunately, manatees are unable to hear low-pitched sounds and a boat’s sound lowers in pitch as the boat slows. Therefore, this approach may in fact make things worse rather than better.

Which of the following, if true, casts most doubt on the conclusion?

10The principal feature of the redesigned checks is a series of printed instructions that the company hopes will help merchants confirm a check’s authenticity, which includes reminders to watch the endorsement, compare signatures, and view the watermark while holding the check to the light.

Q11 to Q13:

Findings from several studies on

corporate mergers and acquisitions

during the 1970’s and 1980’s raise

linequestions about why firms initiate and

5consummate such transactions. One

study showed, for example, that acquir-

ing firms were on average unable to

maintain acquired firms’ pre-merger

levels of profitability. A second study

10concluded that post-acquisition gains

to most acquiring firms were not ade-

quate to cover the premiums paid

to obtain acquired firms. A third

demonstrated that, following the

15announcement of a prospective

merger, the stock of the prospective

acquiring firm tends to increase in

value much less than does that of

the firm for which it bids. Yet merg-

20ers and acquisitions remain common,

and bidders continue to assert that

their objectives are economic ones.

Acquisitions may well have the desir-

able effect of channeling a nation’s

25resources efficiently from less to

more efficient sectors of its economy,

but the individual acquisitions execu-

tives arranging these deals must see

them as advancing either their own or

30their companies’ private economic

interests. It seems that factors hav-

ing little to do with corporate economic

interests explain acquisitions. These

factors may include the incentive

35compensation of executives, lack

of monitoring by boards of directors,

and managerial error in estimating the

value of firms targeted for acquisition.

Alternatively, the acquisition acts of

40bidders may derive from modeling:

a manager does what other man-

agers do.

11According to the passage, during the 1970’s and 1980’s bidding firms differed from the firms for which they bid in that bidding firms

12It can inferred from the passage that the author would be most likely to agree with which of the following statements about corporate acquisitions?

13The author of the passage implies that which of the following is a possible partial explanation for acquisition behavior during the 1970’s and 1980’s?

14In 1850 Lucretia Mott published her Discourse on Women, arguing in a treatise for women to have equal political and legal rights and for changes in the married women’s property laws.

15In little more than a decade, Argentina has become the world’s leading exporter of honey, with nearly 90,000 tons a year sold to foreign markets, almost half of which going to the United States.

16Political Advertisement:

Mayor Delmont’s critics complain about the jobs that were lost in the city under Delmont’s leadership. Yet the fact is that not only were more jobs created than were eliminated, but each year since Delmont took office the average pay for the new jobs created has been higher than that year’s average pay for jobs citywide. So it stands to reason that throughout Delmont’s tenure the average paycheck in this city has been getting steadily bigger.

Which of the following, if true, most seriously weakens the argument in the advertisement?

Compare with an old edition of this question listed below. There are many tiny changes in the main body of the question and its answer choices. Code No. Pam

Political Advertisement:

Mayor Delmont’s critics complain about the jobs that were lost in the city under Delmont’s leadership. Yet the fact is that not only were more jobs created than were eliminated, but the average pay for these new jobs has been higher than the average pay for jobs citywide every year since Delmont took office. So there can be no question that throughout Delmont’s tenure the average paycheck in this city has been getting steadily bigger.

Which of the following, if true, most strengthens the argument in the advertisement?

17The total market value of real estate in Altonville has steadily declined over the past four years. This decline has meant that the overall figure on which the city’s property tax is based—the assessed value of that real estate—has also declined. The percentage of assessed value that was paid as property taxes to the city, however, did not change from year to year during this period.

The information above most strongly supports which of the following?

Compare with an old edition of this question listed below. There are many tiny changes in the main body of the question and its answer choices.
Code No. Ttm

The total market value of real estate in Altonville has steadily declined over the past four years. This decline has meant that the overall figure on which the city’s property tax is based—the assessed value of that real estate—has also declined. Moreover, the percentage of assessed value that was paid as property taxes to the city did not change during this period.

The information above most strongly supports which of the following?

18In the early part of the twentieth century, many vacationers found that driving automobiles and sleeping in tents allowed them to enjoy nature close at hand and tour at their own pace, with none of the restrictions of passenger trains and railroad timetables or with the formalities, expenses, and impersonality of hotels.

19Critical-thinking instruction is predicted on two assumptions: that there are clearly identifiable thinking skills that students can be taught to recognize and apply appropriately, and if recognized and applied, students will become more effective thinkers.

Q20 to Q22:

Acting on the recommen-

dation of a British government

committee investigating the

linehigh incidence in white lead

5factories of illness among

employees, most of whom

were women, the Home Sec-

retary proposed in 1895 that

Parliament enact legislation

10that would prohibit women from

holding most jobs in white lead

factories. Although the

Women’s Industrial Defence

Committee (WIDC), formed

15in 1892 in response to earlier

legislative attempts to restrict

women’s labor, did not dis-

count the white lead trade’s

potential health dangers, it

20opposed the proposal, view-

ing it as yet another instance

of limiting women’s work

opportunities. Also opposing

the proposal was the Society

25for Promoting the Employment

of Women (SPEW), which

attempted to challenge it by

investigating the causes of ill-

ness in white lead factories.

30SPEW contended, and WIDC

concurred, that controllable

conditions in such factories

were responsible for the devel-

opment of lead poisoning.

35SPEW provided convincing

evidence that lead poisoning

could be avoided if workers

were careful and clean and

if already extant workplace

40safety regulations were

stringently enforced. How-

ever, the Women’s Trade

Union League (WTUL), which

had ceased in the late 1880’s

45to oppose restrictions on

women’s labor, supported the

eventually enacted proposal,

in part because safety regu-

lations were generally not

50being enforced in white lead

factories, where there were no

unions (and little prospect of

any) to pressure employers to

comply with safety regulations.

20The passage is primarily concerned with

21The passage suggests that WIDC differed from WTUL in which of the following ways?

22Which of the following, if true, would most clearly support the contention attributed to SPEW in lines 30-34 (“SPEW contended … lead poisoning”) ?

23Because they can reproduce several times each year and produce large numbers of offspring, many insect species can generally adapt quickly to changes in their environment.

24According to recent studies comparing the nutritional value of meat from wild animals and meat from domesticated animals, wild animals have less total fat than do livestock fed on grain and more of a kind of fat they think is good for cardiac health.

25In 1983 Argonia’s currency, the argon, underwent a reduction in value relative to the world’s strongest currencies. This reduction resulted in a significant increase in Argonia’s exports over 1982 levels. In 1987 a similar reduction in the value of the argon led to another increase in Argonia’s exports. Faced with the need to increase exports yet again, Argonia’s finance minister has proposed another reduction in the value of the argon.

Which of the following, if true, most strongly supports the prediction that the finance minister’s plan will not result in a significant increase in Argonia’s exports next year?

26In his experiments with gravity, Isaac Newton showed how the motion of each planet in the solar system results from the combined gravitational pull of the Sun and of all the other planets, each contributing according to their mass and distance from the others.

27Though sucking zinc lozenges has been promoted as a treatment for the common cold, research has revealed no consistent effect. Recently, however, a zinc gel applied nasally has been shown to greatly reduce the duration of colds. Since the gel contains zinc in the same form and concentration as the lozenges, the greater effectiveness of the gel must be due to the fact that cold viruses tend to concentrate in the nose, not the mouth.

In order to evaluate the argument, it would be most helpful to determine which of the following?

28Astronomer: Observations of the Shoemaker-Levi comet on its collision course with Jupiter showed that the comet broke into fragments before entering Jupiter’s atmosphere in 1994, but they did not show how big those fragments were. Nevertheless, some indication of their size can be inferred from spectrographic analyses of Jupiter’s outer atmosphere. After the fragments’ entry, these analyses revealed unprecedented traces of sulfur. The fragments themselves almost certainly contained no sulfur, but astronomers believe that the cloud layer below Jupiter’s outer atmosphere does contain sulfur. Since sulfur would have seeped into the outer atmosphere if comet fragments had penetrated this cloud layer, it is likely that some of the fragments were at least large enough to have passed through Jupiter’s outer atmosphere without being burned up.

In the astronomer’s argument, the two portions in boldface play which of the following roles?

29Most of Portugal’s 250,000 university students boycotted classes in a one-day strike to protest a law that requires them to contribute $330 a year toward the cost of higher education, previously paying $7 per year.

30Biologists working in Spain say that their discovery of teeming life in a highly acidic river may not only broaden the search for life, or for evidence of past life, no other planets but also show that a number of forms of microscopic life can adapt to conditions that scientists have long thought hostile to all but the hardiest bacteria.

Q31 to Q34:

Recently biologists have been

interested in a tide-associated

periodic behavior displayed by

linethe diatom Hantzschia virgata, a

5microscopic golden-brown alga that

inhabits that portion of a shoreline

washed by tides (the intertidal zone).

Diatoms of this species, sometimes

called “commuter” diatoms, remain

10burrowed in the sand during high

tide, and emerge on the sand sur-

face during the daytime low tide.

Just before the sand is inundated by

the rising tide, the diatoms burrow

15again. Some scientists hypothesize

that commuter diatoms know that it

is low tide because they sense an

environmental change, such as an

alteration in temperature or a change

20in pressure caused by tidal move-

ment. However, when diatoms are

observed under constant conditions

in a laboratory, they still display

periodic behavior, continuing to bur-

25row on schedule for several weeks.

This indicates that commuter diatoms,

rather than relying on environmental

cues to keep time, possess an inter-

nal pacemaker or biological clock

30that enables them to anticipate peri-

odic changes in the environment.

A commuter diatom has an unusu-

ally accurate biological clock, a

consequence of the unrelenting

35environmental pressures to which

it is subjected; any diatoms that do

not burrow before the tide arrives

are washed away.

This is not to suggest that the

40period of this biological clock is

immutably fixed. Biologists have

concluded that even though a

diatom does not rely on the envi-

ronment to keep time, environmental

45factors—including changes in the

tide’s hydrostatic pressure, salin-

ity, mechanical agitation, and

temperature—can alter the period

of its biological clock according to

50changes in the tidal cycle. In short,

31The passage suggests which of the following about the accuracy of the commuter diatom’s biological clock?

32E. The accuracy of the commuter diatom’s biological clock tends to fluctuate when the diatom is observed under variable laboratory conditions.

33The author of the passage compares the relationship between an organism’s biological clock and its environment to the relation between a wristwatch and its owner most probably in order to

34The primary purpose of the passage is to

35In Patton City, days are categorized as having heavy rainfall (more than two inches), moderate rainfall (more than one inch, but no more than two inches), light rainfall (at least a trace, but no more than one inch), or no rainfall. In 1990, there were fewer days with light rainfall than in 1910 and fewer with moderate rainfall, yet total rainfall for the year was 20 percent higher in 1990 than in 1910.

If the statements above are true, then it is also possible that in Patton City

36Section 301 of the 1988 Omnibus Trade and Competitiveness Act enables the United States Trade Representative to single out a country as an unfair trader, begin trade negotiations with that country, and, if the negotiations do not conclude by the United States government’s being satisfied, to impose sanctions.

37Rabbits were introduced to Numa Island in the nineteenth century. Overgrazing by the enormous population of rabbits now menaces the island’s agriculture. The government proposes to reduce the population by using a virus that has caused devastating epidemics in rabbit populations elsewhere. There is, however, a chance that the virus will infect the bilby, an endangered native marsupial. The government’s plan, therefore, may serve the interests of agriculture but will clearly increase the threat to native wildlife.

Which of the following, if true, most seriously weakens the argument?

38In the past, every ten-percentage-point increase in cigarette prices in the country of Coponia has decreased per capita sales of cigarettes by four percent. Coponia is about to raise taxes on cigarettes by 9 cents per pack. The average price of cigarettes in Coponia is and has been for more than a year 90 cents per pack. So the tax hike stands an excellent chance of reducing per capita sales of cigarettes by four percent.

Which of the following is an assumption on which the argument depends?

39Scientists have recently discovered that the ultrathin, layered construction of a butterfly’s wings, the same as the one making some butterflies shimmer via the phenomenon of iridescence, are enabling the insect to control how much heat energy is absorbed by its wings and how much is reflected away.

40Mice that have been given morphine are very likely to develop blood poisoning because bacteria that normally reside in the intestine typically respond to morphine by migrating into the bloodstream. However, when mice are given both morphine and the new drug naltrexone, blood poisoning is much less frequent, although it does still occur. These results provide support for researchers’ prediction that naltrexone will turn out to be toxic to certain types of bacteria.

Which of the following, if discovered to be true, would most seriously weaken the support for the researchers’ prediction?

41Gas hydrates, chemical compounds of water and natural gas, are increasingly being studied for their potential to be huge reservoirs of energy, possibly causing sea floor instability, and significant contributors to global warming.