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G R A D U A T E R E C O R D E X A M I N A T I O N S ®

Mathematics Test Practice Book This practice book contains � one actual, full-length GRE® Mathematics Test

� test-taking strategies

Become familiar with � test structure and content

� test instructions and answering procedures

Compare your practice test results with the performance of those who

took the test at a GRE administration.

This book is provided FREE with test registration by the Graduate Record Examinations Board.

www.ets.org/gre

Copyright © 2008 by Educational Testing Service. All rights reserved. ETS, the ETS logos, LISTENING. LEARNING. LEADING., GRADUATE RECORD EXAMINATIONS, and GRE are registered trademarks of Educational Testing Service (ETS) in the United States of America

and other countries throughout the world.

®

Note to Test Takers: Keep this practice book until you receive your score report. This book contains important information about scoring.

3MATHEMATICS TEST PRACTICE BOOK

Purpose of the GRE Subject Tests The GRE Subject Tests are designed to help graduate school admission committees and fellowship sponsors assess the qualifi cations of applicants in specifi c fi elds of study. The tests also provide you with an assessment of your own qualifi cations.

Scores on the tests are intended to indicate knowledge of the subject matter emphasized in many undergraduate programs as preparation for graduate study. Because past achievement is usually a good indicator of future performance, the scores are helpful in predicting success in graduate study. Because the tests are standardized, the test scores permit comparison of students from different institutions with different undergraduate programs. For some Subject Tests, subscores are provided in addition to the total score; these subscores indicate the strengths and weaknesses of your preparation, and they may help you plan future studies.

The GRE Board recommends that scores on the Subject Tests be considered in conjunction with other relevant information about applicants. Because numer- ous factors infl uence success in graduate school, reliance on a single measure to predict success is not advisable. Other indicators of competence typically include undergraduate transcripts showing courses taken and grades earned, letters of recommendation, and GRE General Test scores. For information about the appropriate use of GRE scores, see the GRE Guide to the Use of Scores at ets.org/gre/stupubs.

Development of the Subject Tests Each new edition of a Subject Test is developed by a committee of examiners composed of professors in the subject who are on undergraduate and graduate faculties in different types of institutions and in different regions of the United States and Canada. In selecting members for each committee, the GRE Program seeks the advice of the appropriate professional associations in the subject.

The content and scope of each test are specifi ed and reviewed periodically by the committee of exam iners. Test questions are written by committee members and by other university faculty members who are subject-matter specialists. All questions proposed for the test are reviewed and revised by the committee and subject-matter specialists at ETS. The tests are assembled in accordance with the content specifi cations developed by the committee to ensure adequate coverage of the various aspects of the fi eld and, at the same time, to prevent overemphasis on any single topic. The entire test is then reviewed and approved by the committee.

Table of Contents Purpose of the GRE Subject Tests ........................ 3

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

Content of the Mathematics Test ........................ 4

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

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

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

Practice Mathematics Test .................................. 9

Scoring Your Subject Test .................................. 65

Evaluating Your Performance ............................. 68

Answer Sheet...................................................... 69

4 MATHEMATICS TEST PRACTICE BOOK

Subject-matter and measurement specialists on the ETS staff assist the committee, providing information and advice about methods of test construction and helping to prepare the questions and assemble the test. In addition, each test question is reviewed to eliminate language, symbols, or content considered potentially offensive, inappropriate for major subgroups of the test- taking population, or likely to perpetuate any negative attitude that may be conveyed to these subgroups.

Because of the diversity of undergraduate curricula, it is not possible for a single test to cover all the material you may have studied. The examiners, therefore, select questions that test the basic knowledge and skills most important for successful graduate study in the particular fi eld. The committee keeps the test up-to- date by regularly developing new editions and revising existing editions. In this way, the test content remains current. In addition, curriculum surveys are conducted periodically to ensure that the content of a test refl ects what is currently being taught in the undergraduate curriculum.

After a new edition of a Subject Test is fi rst administered, examinees’ responses to each test question are analyzed in a variety of ways to determine whether each question functioned as expected. These analyses may reveal that a question is ambiguous, requires knowledge beyond the scope of the test, or is inappropriate for the total group or a particular subgroup of examinees taking the test. Such questions are not used in computing scores.

Following this analysis, the new test edition is equated to an existing test edition. In the equating process, statistical methods are used to assess the diffi culty of the new test. Then scores are adjusted so that examinees who took a more diffi cult edition of the test are not penalized, and examinees who took an easier edition of the test do not have an advantage. Variations in the number of questions in the different editions of the test are also taken into account in this 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 total scores is from 200 to 990. The actual range of scores for a particular Subject Test, however, may be smaller. For Subject Tests that report subscores, the maximum possible range is 20 to 99; however, the actual range of

subscores for any test or test edition may be smaller. Subject Test score interpretive information is provided in Interpreting Your GRE Scores, which you will receive with your GRE score report. This publication is also available at ets.org/gre/stupubs.

Content of the Mathematics Test The test consists of approximately 66 multiple-choice questions drawn from courses commonly offered at the undergraduate level. Approximately 50 percent of the questions involve calculus and its applications— subject matter that can be assumed to be common to the backgrounds of almost all mathematics majors. About 25 percent of the questions in the test are in elementary algebra, linear algebra, abstract algebra, and number theory. The remaining questions deal with other areas of mathematics currently studied by undergraduates in many institutions.

The following content descriptions may assist students in preparing for the test. The percents given are estimates; actual percents will vary somewhat from one edition of the test to another.

Calculus—50%

� Material learned in the usual sequence of elementary calculus courses—differential and integral calculus of one and of several variables—includes calculus-based applications and connections with coordinate geometry, trigonometry, differential equations, and other branches of mathematics

Algebra—25%

� Elementary algebra: basic algebraic techniques and manipulations acquired in high school and used throughout mathematics

� Linear algebra: matrix algebra, systems of linear equations, vector spaces, linear transformations, characteristic polynomials, and eigenvalues and eigenvectors

� Abstract algebra and number theory: elementary topics from group theory, theory of rings and modules, fi eld theory, and number theory

5MATHEMATICS TEST PRACTICE BOOK

Additional Topics—25%

� Introductory real analysis: sequences and series of numbers and functions, continuity, differentiability and integrability, and elementary topology of � and �n

� Discrete mathematics: logic, set theory, combinatorics, graph theory, and algorithms

� Other topics: general topology, geometry, complex variables, probability and statistics, and numerical analysis

The above descriptions of topics covered in the test should not be considered exhaustive; it is necessary to understand many other related concepts. Prospective test takers should be aware that questions requiring no more than a good precalculus background may be quite challenging; such questions can be among the most diffi cult questions on the test. In general, the questions are intended not only to test recall of information but also to assess test takers’ understanding of fundamental concepts and the abili