gre math

Graduate Record Examinations®

This book is provided FREE withtest registration by the

Graduate Record Examinations Board.

MATHEMATICSTEST

(RESCALED)PRACTICE

BOOK

This practice bookcontains

n one actual full-lengthGRE Mathematics Test(Rescaled)

n test-taking strategies

Become familiar with

n test structure and content

n test instructions andanswering procedures

Compare your practicetest results with theperformance of thosewho took the test at aGRE administration.

Visit GRE Online at www.gre.org

Copyright © 2001 by Educational Testing Service. All rights reserved.EDUCATIONAL TESTING SERVICE, ETS, the ETS logos, GRADUATE RECORD EXAMINATIONS,

and GRE are registered trademarks of Educational Testing Service.

Note to Test Takers:

Keep this practice book until you receive your score report.

This book contains important information about content specifications and scoring.

This practice book and the interpretive information included in it are appropriatefor individuals taking the Mathematics Test (Rescaled) after October 1, 2001. TheGRE Mathematics Test will be rescaled effective October 2001 and renamed “Math-ematics Test (Rescaled)”; the content of the test will not change. Although therange of scores for the Mathematics Test (Rescaled) will continue to extend from200 to 990, scores earned on the test after October 2001 should not be comparedto scores earned earlier.

The primary reason for rescaling the Mathematics Test is to make the test scoresmore useful for admissions committees and other score users by spreading out high-ability examinees on the score scale. The scores of the Mathematics Test populationhave increased substantially since the test was first scaled in 1952, andan increasingly large percentage of individuals each year have been earning 990, thehighest possible score on the scale. Based on recent interpretive data (individualswho were tested between October 1996 and October 1999), 18 percent of theexaminees received 990 on the test (in other words, the percentage of test takersscoring below 990 was 82). The scale of the new Mathematics Test (Rescaled) willenable admissions committees and other score users to distinguish among high-scoring examinees, something the former scale was no longer able to do.

3MATHEMATICS TEST (RESCALED)PRACTICE BOOK

Purpose of theGRE Subject TestsThe GRE Subject Tests are designed to help graduateschool admission committees and fellowship sponsorsassess the qualifications of applicants in specific fieldsof study. The tests also provide you with an assessmentof your own qualifications.

Scores on the tests are intended to indicate knowl-edge of the subject matter emphasized in many under-graduate programs as preparation for graduate study.Because past achievement is usually a good indicator offuture performance, the scores are helpful in predictingsuccess in graduate study. Because the tests are stan-dardized, the test scores permit comparison of studentsfrom different institutions with different undergraduateprograms. For some Subject Tests, subscores are pro-vided in addition to the total score; these subscoresindicate the strengths and weaknesses of your prepara-tion, and they may help you plan future studies.

The GRE Board recommends that scores on theSubject Tests be considered in conjunction with otherrelevant information about applicants. Because numer-ous factors influence success in graduate school,reliance on a single measure to predict success is notadvisable. Other indicators of competence typicallyinclude undergraduate transcripts showing coursestaken and grades earned, letters of recommendation,the GRE Writing Assessment score, and GRE GeneralTest scores. For information about the appropriate useof GRE scores, write to GRE Program, EducationalTesting Service, Mail Stop 57-L, Princeton, NJ 08541,or visit our Web site at www.gre.org/codelst.html.

Development of theSubject TestsEach new edition of a Subject Test is developed by acommittee of examiners composed of professors in thesubject who are on undergraduate and graduate facul-ties in different types of institutions and in differentregions of the United States and Canada. In selectingmembers for each committee, the GRE Program seeksthe advice of the appropriate professional associationsin the subject.

The content and scope of each test are specifiedand reviewed periodically by the committee of exam-iners. Test questions are written by the committee andby other faculty who are also subject-matter specialistsand by subject-matter specialists at ETS. All questionsproposed for the test are reviewed by the committeeand revised as necessary. The accepted questions areassembled into a test in accordance with the contentspecifications developed by the committee to ensureadequate coverage of the various aspects of the fieldand, at the same time, to prevent overemphasis on anysingle topic. The entire test is then reviewed andapproved by the committee.

Table of ContentsPurpose of the GRE Subject Tests ........................ 3

Development of the Subject Tests ....................... 3

Content of the Mathematics Test (Rescaled) ...... 4

Preparing for a Subject Test ................................. 5

Test-Taking Strategies .......................................... 5

What Your Scores Mean ...................................... 6

Practice Mathematics Test (Rescaled) ................. 9

Scoring Your Subject Test .................................. 55

Evaluating Your Performance ............................. 58

Answer Sheet ..................................................... 59

4 MATHEMATICS TEST (RESCALED)PRACTICE BOOK

Subject-matter and measurement specialists on theETS staff assist the committee, providing informationand advice about methods of test construction andhelping to prepare the questions and assemble the test.In addition, each test question is reviewed to eliminatelanguage, symbols, or content considered potentiallyoffensive, inappropriate for major subgroups of the test-taking population, or likely to perpetuate any negativeattitude that may be conveyed to these subgroups. Thetest as a whole is also reviewed to ensure that the testquestions, where applicable, include an appropriatebalance of people in different groups and different roles.

Because of the diversity of undergraduate curricula,it is not possible for a single test to cover all thematerial you may have studied. The examiners, there-fore, select questions that test the basic knowledge andskills most important for successful graduate study inthe particular field. The committee keeps the testup-to-date by regularly developing new editions andrevising existing editions. In this way, the test contentchanges steadily but gradually, much like most cur-ricula. In addition, curriculum surveys are conductedperiodically to ensure that the content of a testreflects what is currently being taught in the under-graduate curriculum.

After a new edition of a Subject Test is first admin-istered, examinees’ responses to each test question areanalyzed in a variety of ways to determine whethereach question functioned as expected. These analysesmay reveal that a question is ambiguous, requiresknowledge beyond the scope of the test, or is inappro-priate for the total group or a particular subgroup ofexaminees taking the test. Answers to such questionsare not used in computing scores.

Following this analysis, the new test edition isequated to an existing test edition. In the equatingprocess, statistical methods are used to assess thedifficulty of the new test. Then scores are adjusted sothat examinees who took a difficult edition of the testare not penalized, and examinees who took an easieredition of the test do not have an advantage. Varia-tions in the number of questions in the differenteditions of the test are also taken into account inthis process.

Scores on the Subject Tests are reported as three-digit scaled scores with the third digit always zero.The maximum possible range for all Subject Test totalscores is from 200 to 990. The actual range of scores fora particular Subject Test, however, may be smaller. Themaximum possible range of Subject Test subscores is20 to 99; however, the actual range of subscores forany test or test edition may be smaller. Subject Testscore interpretive information is provided in Interpret-ing Your GRE Scores, which you will receive withyour GRE score report, and on the GRE Web site atwww.gre.org/codelst.html.

Content of the Mathematics Test (Rescaled)The test consists of 66 multiple-choice questions,drawn from courses commonly offered at the under-graduate level. Although the Mathematics Test hasbeen rescaled, the content of the test has not changed.Approximately 50 percent of the questions involvecalculus and its applications —subject matter that canbe assumed to be common to the backgrounds ofalmost all mathematics majors. About 25 percent ofthe questions in the test are in elementary algebra,linear algebra, abstract algebra, and number theory.The remaining questions deal with other areas ofmathematics currently studied by undergraduates inmany institutions.

The following content descriptions may assiststudents in preparing for the test. The percentagesgiven are estimates; actual percentages will varysomewhat from one edition of the test to another.

Calculus — 50%

Material learned in the usual sequence of elementarycalculus courses — differential and integral calculusof one and of several variables — including calculus-based applications and connections with coordinategeometry, trigonometry, differential equations, andother branches of mathematics

Algebra — 25%

Elementary algebra: basic algebraic techniques andmanipulations acquired in high school and usedthroughout mathematics


Linear algebra: matrix algebra, systems of linearequations, vector spaces, linear transformations, char-acteristic polynomials, eigenvalues and eigenvectors

Abstract algebra and number theory: elementarytopics from group theory, the theory of rings andmodules, field theory, and number theory

Additional Topics — 25%

Introductory real analysis: sequences and series ofnumbers and functions, continuity, differentiabilityand integrability, elementary topology of R and Rn

Discrete mathematics: logic, set theory, combina-torics, graph theory, and algorithms

Other topics: general topology, geometry, complexvariables, probability and statistics, and numericalanalysis

The above descriptions of topics covered in the testshould not be considered exhaustive; it is necessary tounderstand many other related concepts. Prospectivetest takers should be aware that questions requiring nomore than a good precalculus background may be quitechallenging; some of these questions turn out to beamong the most difficult questions on the test. Ingeneral, the questions are intended not only to testrecall of information, but also to assess the test taker’sunderstanding of fundamental concepts and the abilityto apply these concepts in various situations.

Preparing for a Subject TestGRE Subject Test questions are designed to measureskills and knowledge gained over a long period of time.Although you might increase your scores to some extentthrough preparation a few weeks or months before youtake the test, last minute cramming is unlikely to be offurther help. The following information may be helpful.

n A general review of your college courses isprobably the best preparation for the test. How-ever, the test covers a broad range of subjectmatter, and no one is expected to be familiarwith the content of every question.

n Use this practice book to become familiar withthe types of questions in the GRE MathematicsTest (Rescaled), paying special attention to thedirections. If you thoroughly understand thedirections before you take the test, you will

have more time during the test to focus onthe questions themselves.

Test-Taking StrategiesThe questions in the practice test in this book illus-trate the types of multiple-choice questions in the test.When you take the test, you will mark your answers ona separate machine-scorable answer sheet. Total testingtime is two hours and fifty minutes; there are noseparately timed sections. Following are some generaltest-taking strategies you may want to consider.

n Read the test directions carefully, and work asrapidly as you can without being careless. Foreach question, choose the best answer from theavailable options.

n All questions are of equal value; do not wastetime pondering individual questions you findextremely difficult or unfamiliar.

n You may want to work through the test quiterapidly, first answering only the questions aboutwhich you feel confident, then going back andanswering questions that require more thought,and concluding with the most difficult questionsif there is time.

n If you decide to change an answer, make sureyou completely erase it and fill in the ovalcorresponding to your desired answer.

n Questions for which you mark no answer or morethan one answer are not counted in scoring.

n As a correction for haphazard guessing, one-fourth of the number of questions you answerincorrectly is subtracted from the number ofquestions you answer correctly. It is improbablethat mere guessing will improve your scoresignificantly; it may even lower your score.If, however, you are not certain of the correctanswer but have some knowledge of the questionand are able to eliminate one or more of theanswer choices, your chance of getting the rightanswer is improved, and it may be to your advan-tage to answer the question.


n Record all answers on your answer sheet.Answers recorded in your test book will notbe counted.

n Do not wait until the last five minutes of atesting session to record answers on youranswer sheet.

What Your Scores MeanYour raw score — that is, the number of questions youanswered correctly minus one-fourth of the numberyou answered incorrectly — is converted to the scaledscore that is reported. This conversion ensures that ascaled score reported for any edition of a Subject Testis comparable to the same scaled score earned on anyother edition of the same test. Thus, equal scaledscores on a particular Subject Test indicate essentiallyequal levels of performance regardless of the testedition taken. Test scores should be compared onlywith other scores on the same Subject Test. (Forexample, a 680 on the Computer Science Test is notequivalent to a 680 on the Mathematics Test.)

Note that the Mathematics Test has been rescaledeffective October 2001 and renamed “MathematicsTest (Rescaled).” Scores earned on the MathematicsTest (Rescaled) after October 2001 should not becompared to Mathematics Test scores earned beforeOctober 2001. Additional information about therescaled test is available on the GRE Web site and willalso be included in the score interpretive leaflet thatwill accompany score reports.

Before taking the test, you may find it useful toknow approximately what raw scores would berequired to obtain a certain scaled score. Severalfactors influence the conversion of your raw scoreto your scaled score, such as the difficulty of the testedition and the number of test questions included inthe computation of your raw score. Based on recenteditions of the Mathematics Test, the following tablegives the range of raw scores associated with selectedscaled scores for three different test editions that havebeen rescaled. (Note that when the number of scoredquestions for a given test is greater than the range ofpossible scaled scores, it is likely that two or more rawscores will convert to the same scaled score.) The

three test editions in the table that follows wereselected to reflect varying degrees of difficulty. Examin-ees should note that future test editions may be some-what more or less difficult than the test editionsillustrated in the table.

Range of Raw Scores* Neededto Earn Selected Scaled Scores onThree Mathematics Test (Rescaled)

Editions That Differ in Difficulty

Raw Scores

Scaled Score Form A Form B Form C

900 66 62 60 – 61

800 55 51 49

700 43 40 38

600 30 29 28

Number of Questions Used to Compute Raw Score

66 66 65

*Raw Score = Number of correct answers minus one-fourth thenumber of incorrect answers, rounded to the nearest integer.

For a particular test edition, there are many ways toearn the same raw score. For example, on the editionlisted above as “Form A,” a raw score of 30 would earna scaled score of 600. Below are a few of the possibleways in which a scaled score of 600 could be earned onthat edition.

Examples of Ways to Earna Scaled Score of 600 on theEdition Labeled As “Form A”

Number ofQuestions

Questions Questions Questions Used toRaw Answered Answered Not ComputeScore Correctly Incorrectly Answered Raw Score

30 30 0 36 66

30 33 14 19 66

30 37 27 2 66


PR A C T I C E TE S T

To become familiar with how the administration will be conducted at the test center, first remove theanswer sheet (pages 59 and 60). Then go to the back cover of the test book (page 54) and follow theinstructions for completing the identification areas of the answer sheet. When you are ready to begin thetest, note the time and begin marking your answers on the answer sheet.

10

MATHEMATICS TEST (RESCALED)

Time—170 minutes

66 Questions

Directions: Each of the questions or incomplete statements below is followed by five suggested answers orcompletions. In each case, select the one that is the best of the choices offered and then mark the correspondingspace on the answer sheet.

Computation and scratchwork may be done in this examination book.

Note: In this examination:

(1) All logarithms with an unspecified base are natural logarithms (that is, with base e).(2) The set of all real numbers x such that a ≤ x ≤ b is denoted by [a, b].(3) The symbols �, �, �, and � denote the sets of integers, rational numbers, real numbers, and

complex numbers, respectively.

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SCRATCHWORK

I NOTE: To ensure prompt processing of test results, it is important that you fill in the blanks exactly as directed.

SUBJECT TEST

A. Print and signyour full namein this box:

Copy this code in box 6 onyour answer sheet. Then fillin the corresponding ovalsexactly as shown.

GRADUATE RECORD EXAMINATIONS SUBJECT TEST

B. The Subject Tests are intended to measure your achievement in a specialized field of study. Most of the questions are concernedwith subject matter that is probably familiar to you, but some of the questions may refer to areas that you have not studied.

Your score will be determined by subtracting one-fourth the number of incorrect answers from the number of correct answers.Questions for which you mark no answer or more than one answer are not counted in scoring. If you have some knowledge of aquestion and are able to rule out one or more of the answer choices as incorrect, your chances of selecting the correct answer areimproved, and answering such questions will likely improve your score. It is unlikely that pure guessing will raise your score; itmay lower your score.

You are advised to use your time effectively and to work as rapidly as you can without losing accuracy. Do not spend too muchtime on questions that are too difficult for you. Go on to the other questions and come back to the difficult ones later if you can.

YOU MUST INDICATE ALL YOUR ANSWERS ON THE SEPARATE ANSWER SHEET. No credit will be given for anythingwritten in this examination book, but you may write in the book as much as you wish to work out your answers. After you havedecided on your response to a question, fill in the corresponding oval on the answer sheet. BE SURE THAT EACH MARK ISDARK AND COMPLETELY FILLS THE OVAL. Mark only one answer to each question. No credit will be given for multipleanswers. Erase all stray marks. If you change an answer, be sure that all previous marks are erased completely. Incomplete erasuresmay be read as intended answers. Do not be concerned that the answer sheet provides spaces for more answers than there arequestions in the test.

PRINT:(LAST) (FIRST) (MIDDLE)

SIGN:

Example:

What city is the capital of France?(A) Rome(B) Paris(C) London(D) Cairo(E) Oslo

CORRECT ANSWERPROPERLY MARKED

IMPROPER MARKS

DO NOT OPEN YOUR TEST BOOK UNTIL YOU ARE TOLD TO DO SO.

Copy the Test Name and Form Code in box 7 on your answer sheet.

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Scoring Your Subject TestThe GRE Mathematics Test (Rescaled) scores typicallyrange from 440 to 810. The range for different editionsof a given test may vary because different editions arenot of precisely the same difficulty. The differences inranges among different editions of a given test, however,usually are small. This should be taken into account,especially when comparing two very high scores. Ingeneral, differences between scores at the 99th percen-tile should be ignored. The score conversion table onpage 57 shows the score range for this edition of thetest only.

The worksheet on page 56 lists the correct answersto the questions. Columns are provided for you to markwhether you chose the correct (C) answer or an incor-rect (I) answer to each question. Draw a line across anyquestion you omitted, because it is not counted in thescoring. At the bottom of the page, enter the totalnumber correct and the total number incorrect. Dividethe total incorrect by 4 and subtract the resultingnumber from the total correct. This is the adjustmentmade for guessing. Then round the result to the nearestwhole number. This will give you your raw total score.Use the total score conversion table to find the scaledtotal score that corresponds to your raw total score.

Example: Suppose you chose the correct answersto 48 questions and incorrect answers to 15. Dividing 15by 4 yields 3.75. Subtracting 3.75 from 48 equals 44.25,which is rounded to 44. The raw score of 44 correspondsto a scaled score of 710 on the new GRE Mathematicsscale, effective October 2001. The scaled scores in thisbook, and those received by individuals taking the GREMathematics Test (Rescaled) after October 2001, areNOT comparable to Mathematics Test scores reportedprior to October 2001.


Worksheet for the GRE Mathematics Test (Rescaled), Form GR9768Answer Key and Percentages* of Examinees

Answering Each Question Correctly

QUESTION TOTALNumber Answer P + C I

1 C 732 D 863 C 884 B 875 E 83

6 C 877 B 818 E 769 A 77

10 B 55

11 A 8012 C 7513 A 7214 C 6315 B 61

16 D 8317 E 8018 E 6319 B 5520 D 52

21 D 7122 B 7223 C 7224 C 5025 D 70

26 B 6727 E 6528 C 6229 B 6330 D 55

31 C 6532 D 5933 A 7234 A 5235 B 50

QUESTION TOTALNumber Answer P + C I

36 D 3837 B 3238 B 5439 B 5340 E 54

41 E 4942 D 6043 B 6444 D 3945 A 47

46 C 5047 E 3248 A 3649 E 4150 D 34

51 D 3552 D 5253 D 3554 A 3155 D 28

56 E 1957 C 2058 C 3459 A 2860 E 23

61 E 2562 A 3663 D 1764 C 3565 A 44

66 E 58

Correct (C)

Incorrect (I)

Total Score:

C – I/4 = ____________

Scaled Score (SS) = ____________

* The P+ column indicates the percentage of Mathematics Test examinees that answered each questioncorrectly; it is based on a sample of December 1997 examinees selected to represent all MathematicsTest examinees tested between October 1, 1997, and September 30, 2000.


TOTAL SCORE

Raw Score Scaled Score % Raw Score Scaled Score %

66 890 9965 880 9864 870 97

62-63 860 9761 850 9560 840 9459 830 93

57-58 820 9256 810 9055 800 88

54 790 8752-53 780 85

51 770 8350 760 8249 750 80

47-48 740 7846 730 7645 720 7444 710 7243 700 70

41-42 690 6840 680 6639 670 6338 660 61

36-37 650 5835 640 5634 630 5333 620 51

31-32 610 4830 600 46

29 590 4328 580 41

26-27 570 3725 560 3524 550 3323 540 31

21-22 530 2920 520 2719 510 2418 500 22

17 490 2015-16 480 17

14 470 1513 460 1312 450 11

10-11 440 99 430 88 420 67 410 5

5-6 400 3

4 390 23 380 12 370 1

0-1 360 1

Score Conversions* and Percents Below** forGRE Mathematics Test (Rescaled), Form GR9768

*Score conversions presented in this table are based on the rescaledMathematics Test.

**Percentage scoring below the scaled score is based on the performance of6,753 examinees who took the Mathematics Test between October␣ 1, 1997,and September 30, 2000.


Evaluating Your PerformanceNow that you have scored your test, you may wish tocompare your performance with the performance ofothers who took this test. Both the worksheet on page56 and the table on page 57 use performance data fromGRE Mathematics Test (Rescaled) examinees.

The data in the worksheet on page 56 are based onthe performance of a sample of the examinees who tookthis test in December 2000. This sample was selected torepresent the total population of GRE Mathematics Test(Rescaled) examinees tested between October 1997 andSeptember 2000. The numbers in the column labeled“P+” on the worksheet indicate the percentages ofexaminees in this sample who answered each questioncorrectly. You may use these numbers as a guide forevaluating your performance on each test question.

The table on page 57 contains, for each scaled score,the percentage of examinees tested between October1997 and September 2000 who received lower scores.Interpretive data based on the scores earned by examin-ees tested in this three-year period will be used byadmissions officers in the 2001 – 02 testing year. Thesepercentages appear in the score conversion table in a

column to the right of the scaled scores, when their rawscores were placed on the new scale. For example, in thepercentage column opposite the scaled score of 620 isthe number 51. This means that 51 percent of the GREMathematics Test (Rescaled) examinees tested betweenOctober 1997 and September 2000 scored lower than620 when their raw scores are placed on the new scale.To compare yourself with this population, look at thepercentage next to the scaled score you earned on thepractice test.

It is important to realize that the conditions underwhich you tested yourself were not exactly the same asthose you will encounter at a test center. It is impossibleto predict how different test-taking conditions will affecttest performance, and this is only one factor that mayaccount for differences between your practice test scoresand your actual test scores. By comparing your perform-ance on this practice test with the performance of otherGRE Mathematics Test (Rescaled) examinees, however,you will be able to determine your strengths and weak-nesses and can then plan a program of study to prepareyourself for taking the GRE Mathematics Test(Rescaled) under standard conditions.

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