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GRE - Mathematics test

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This practice book and the interpretive information included in it are appropriate for individuals taking the Mathematics Test (Rescaled) after October 1, 2001. The GRE Mathematics Test will be rescaled effective October 2001 and renamed “Math- ematics Test (Rescaled)”; the content of the test will not change. Although the range of scores for the Mathematics Test (Rescaled) will continue to extend from 200 to 990, scores earned on the test after October 2001 should not be compared to scores earned earlier....

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  1. This book is provided FREE with test registration by the Graduate Record Examinations Board. Graduate Record Examinations® This practice book contains one actual full-length GRE Mathematics Test MATHEMATICS TEST (Rescaled) test-taking strategies Become familiar with test structure and content (RESCALED) test instructions and answering procedures PRACTICE Compare your practice test results with the performance of those who took the test at a BOOK GRE administration. Visit GRE Online at
  2. This practice book and the interpretive information included in it are appropriate for individuals taking the Mathematics Test (Rescaled) after October 1, 2001. The GRE Mathematics Test will be rescaled effective October 2001 and renamed “Math- ematics Test (Rescaled)”; the content of the test will not change. Although the range of scores for the Mathematics Test (Rescaled) will continue to extend from 200 to 990, scores earned on the test after October 2001 should not be compared to scores earned earlier. The primary reason for rescaling the Mathematics Test is to make the test scores more useful for admissions committees and other score users by spreading out high- ability examinees on the score scale. The scores of the Mathematics Test population have increased substantially since the test was first scaled in 1952, and an increasingly large percentage of individuals each year have been earning 990, the highest possible score on the scale. Based on recent interpretive data (individuals who were tested between October 1996 and October 1999), 18 percent of the examinees received 990 on the test (in other words, the percentage of test takers scoring below 990 was 82). The scale of the new Mathematics Test (Rescaled) will enable admissions committees and other score users to distinguish among high- scoring examinees, something the former scale was no longer able to do. Note to Test Takers: Keep this practice book until you receive your score report. This book contains important information about content specifications and scoring. 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.
  3. The GRE Board recommends that scores on the Table of Contents Subject Tests be considered in conjunction with other relevant information about applicants. Because numer- Purpose of the GRE Subject Tests ........................ 3 ous factors influence success in graduate school, Development of the Subject Tests ....................... 3 reliance on a single measure to predict success is not advisable. Other indicators of competence typically Content of the Mathematics Test (Rescaled) ...... 4 include undergraduate transcripts showing courses Preparing for a Subject Test ................................. 5 taken and grades earned, letters of recommendation, the GRE Writing Assessment score, and GRE General Test-Taking Strategies .......................................... 5 Test scores. For information about the appropriate use What Your Scores Mean ...................................... 6 of GRE scores, write to GRE Program, Educational Practice Mathematics Test (Rescaled) ................. 9 Testing Service, Mail Stop 57-L, Princeton, NJ 08541, or visit our Web site at Scoring Your Subject Test .................................. 55 Evaluating Your Performance ............................. 58 Development of the Answer Sheet ..................................................... 59 Subject Tests Each new edition of a Subject Test is developed by a committee of examiners composed of professors in the Purpose of the subject who are on undergraduate and graduate facul- GRE Subject Tests ties in different types of institutions and in different regions of the United States and Canada. In selecting The GRE Subject Tests are designed to help graduate members for each committee, the GRE Program seeks school admission committees and fellowship sponsors the advice of the appropriate professional associations assess the qualifications of applicants in specific fields in the subject. of study. The tests also provide you with an assessment The content and scope of each test are specified of your own qualifications. and reviewed periodically by the committee of exam- Scores on the tests are intended to indicate knowl- iners. Test questions are written by the committee and edge of the subject matter emphasized in many under- by other faculty who are also subject-matter specialists graduate programs as preparation for graduate study. and by subject-matter specialists at ETS. All questions Because past achievement is usually a good indicator of proposed for the test are reviewed by the committee future performance, the scores are helpful in predicting and revised as necessary. The accepted questions are success in graduate study. Because the tests are stan- assembled into a test in accordance with the content dardized, the test scores permit comparison of students specifications developed by the committee to ensure from different institutions with different undergraduate adequate coverage of the various aspects of the field programs. For some Subject Tests, subscores are pro- and, at the same time, to prevent overemphasis on any vided in addition to the total score; these subscores single topic. The entire test is then reviewed and indicate the strengths and weaknesses of your prepara- approved by the committee. tion, and they may help you plan future studies. MATHEMATICS TEST (RESCALED) 3 PRACTICE BOOK
  4. Subject-matter and measurement specialists on the Scores on the Subject Tests are reported as three- ETS staff assist the committee, providing information digit scaled scores with the third digit always zero. and advice about methods of test construction and The maximum possible range for all Subject Test total helping to prepare the questions and assemble the test. scores is from 200 to 990. The actual range of scores for In addition, each test question is reviewed to eliminate a particular Subject Test, however, may be smaller. The language, symbols, or content considered potentially maximum possible range of Subject Test subscores is offensive, inappropriate for major subgroups of the test- 20 to 99; however, the actual range of subscores for taking population, or likely to perpetuate any negative any test or test edition may be smaller. Subject Test attitude that may be conveyed to these subgroups. The score interpretive information is provided in Interpret- test as a whole is also reviewed to ensure that the test ing Your GRE Scores, which you will receive with questions, where applicable, include an appropriate your GRE score report, and on the GRE Web site at balance 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 the Content of the material you may have studied. The examiners, there- fore, select questions that test the basic knowledge and Mathematics Test (Rescaled) skills most important for successful graduate study in The test consists of 66 multiple-choice questions, the particular field. The committee keeps the test drawn from courses commonly offered at the under- up-to-date by regularly developing new editions and graduate level. Although the Mathematics Test has revising existing editions. In this way, the test content been rescaled, the content of the test has not changed. changes steadily but gradually, much like most cur- Approximately 50 percent of the questions involve ricula. In addition, curriculum surveys are conducted calculus and its applications —subject matter that can periodically to ensure that the content of a test be assumed to be common to the backgrounds of reflects what is currently being taught in the under- almost all mathematics majors. About 25 percent of graduate curriculum. the questions in the test are in elementary algebra, After a new edition of a Subject Test is first admin- linear algebra, abstract algebra, and number theory. istered, examinees’ responses to each test question are The remaining questions deal with other areas of analyzed in a variety of ways to determine whether mathematics currently studied by undergraduates in each question functioned as expected. These analyses many institutions. may reveal that a question is ambiguous, requires The following content descriptions may assist knowledge beyond the scope of the test, or is inappro- students in preparing for the test. The percentages priate for the total group or a particular subgroup of given are estimates; actual percentages will vary examinees taking the test. Answers to such questions somewhat from one edition of the test to another. are not used in computing scores. Following this analysis, the new test edition is Calculus — 50% equated to an existing test edition. In the equating Material learned in the usual sequence of elementary process, statistical methods are used to assess the calculus courses — differential and integral calculus difficulty of the new test. Then scores are adjusted so of one and of several variables — including calculus- that examinees who took a difficult edition of the test based applications and connections with coordinate are not penalized, and examinees who took an easier geometry, trigonometry, differential equations, and edition of the test do not have an advantage. Varia- other branches of mathematics tions in the number of questions in the different Algebra — 25% editions of the test are also taken into account in this process. Elementary algebra: basic algebraic techniques and manipulations acquired in high school and used throughout mathematics 4 MATHEMATICS TEST (RESCALED) PRACTICE BOOK
  5. Linear algebra: matrix algebra, systems of linear have more time during the test to focus on equations, vector spaces, linear transformations, char- the questions themselves. acteristic polynomials, eigenvalues and eigenvectors Abstract algebra and number theory: elementary topics from group theory, the theory of rings and Test-Taking Strategies modules, field theory, and number theory The questions in the practice test in this book illus- Additional Topics — 25% trate the types of multiple-choice questions in the test. Introductory real analysis: sequences and series of When you take the test, you will mark your answers on numbers and functions, continuity, differentiability a separate machine-scorable answer sheet. Total testing and integrability, elementary topology of and n time is two hours and fifty minutes; there are no Discrete mathematics: logic, set theory, combina- separately timed sections. Following are some general torics, graph theory, and algorithms test-taking strategies you may want to consider. Other topics: general topology, geometry, complex Read the test directions carefully, and work as variables, probability and statistics, and numerical rapidly as you can without being careless. For analysis each question, choose the best answer from the The above descriptions of topics covered in the test available options. should not be considered exhaustive; it is necessary to All questions are of equal value; do not waste understand many other related concepts. Prospective time pondering individual questions you find test takers should be aware that questions requiring no extremely difficult or unfamiliar. more than a good precalculus background may be quite challenging; some of these questions turn out to be You may want to work through the test quite among the most difficult questions on the test. In rapidly, first answering only the questions about general, the questions are intended not only to test which you feel confident, then going back and recall of information, but also to assess the test taker’s answering questions that require more thought, understanding of fundamental concepts and the ability and concluding with the most difficult questions to apply these concepts in various situations. if there is time. If you decide to change an answer, make sure Preparing for a Subject Test you completely erase it and fill in the oval corresponding to your desired answer. GRE Subject Test questions are designed to measure Questions for which you mark no answer or more skills and knowledge gained over a long period of time. than one answer are not counted in scoring. Although you might increase your scores to some extent through preparation a few weeks or months before you As a correction for haphazard guessing, one- take the test, last minute cramming is unlikely to be of fourth of the number of questions you answer further help. The following information may be helpful. incorrectly is subtracted from the number of questions you answer correctly. It is improbable A general review of your college courses is that mere guessing will improve your score probably the best preparation for the test. How- significantly; it may even lower your score. ever, the test covers a broad range of subject If, however, you are not certain of the correct matter, and no one is expected to be familiar answer but have some knowledge of the question with the content of every question. and are able to eliminate one or more of the Use this practice book to become familiar with answer choices, your chance of getting the right the types of questions in the GRE Mathematics answer is improved, and it may be to your advan- Test (Rescaled), paying special attention to the tage to answer the question. directions. If you thoroughly understand the directions before you take the test, you will MATHEMATICS TEST (RESCALED) 5 PRACTICE BOOK
  6. Record all answers on your answer sheet. three test editions in the table that follows were Answers recorded in your test book will not selected to reflect varying degrees of difficulty. Examin- be counted. ees should note that future test editions may be some- Do not wait until the last five minutes of a what more or less difficult than the test editions testing session to record answers on your illustrated in the table. answer sheet. Range of Raw Scores* Needed What Your Scores Mean to Earn Selected Scaled Scores on Three Mathematics Test (Rescaled) Your raw score — that is, the number of questions you Editions That Differ in Difficulty answered correctly minus one-fourth of the number Raw Scores you answered incorrectly — is converted to the scaled score that is reported. This conversion ensures that a Scaled Score Form A Form B Form C scaled score reported for any edition of a Subject Test 900 66 62 60 – 61 is comparable to the same scaled score earned on any 800 55 51 49 other edition of the same test. Thus, equal scaled 700 43 40 38 scores on a particular Subject Test indicate essentially 600 30 29 28 equal levels of performance regardless of the test Number of Questions Used to Compute Raw Score edition taken. Test scores should be compared only with other scores on the same Subject Test. (For 66 66 65 example, a 680 on the Computer Science Test is not *Raw Score = Number of correct answers minus one-fourth the equivalent to a 680 on the Mathematics Test.) number of incorrect answers, rounded to the nearest integer. Note that the Mathematics Test has been rescaled effective October 2001 and renamed “Mathematics For a particular test edition, there are many ways to Test (Rescaled).” Scores earned on the Mathematics earn the same raw score. For example, on the edition Test (Rescaled) after October 2001 should not be listed above as “Form A,” a raw score of 30 would earn compared to Mathematics Test scores earned before a scaled score of 600. Below are a few of the possible October 2001. Additional information about the ways in which a scaled score of 600 could be earned on rescaled test is available on the GRE Web site and will that edition. also be included in the score interpretive leaflet that will accompany score reports. Examples of Ways to Earn Before taking the test, you may find it useful to a Scaled Score of 600 on the know approximately what raw scores would be Edition Labeled As “Form A” required to obtain a certain scaled score. Several Number of factors influence the conversion of your raw score Questions to your scaled score, such as the difficulty of the test Questions Questions Questions Used to edition and the number of test questions included in Raw Answered Answered Not Compute the computation of your raw score. Based on recent Score Correctly Incorrectly Answered Raw Score editions of the Mathematics Test, the following table 30 30 0 36 66 gives the range of raw scores associated with selected 30 33 14 19 66 scaled scores for three different test editions that have been rescaled. (Note that when the number of scored 30 37 27 2 66 questions for a given test is greater than the range of possible scaled scores, it is likely that two or more raw scores will convert to the same scaled score.) The 6 MATHEMATICS TEST (RESCALED) PRACTICE BOOK
  7. PRACTICE TEST To become familiar with how the administration will be conducted at the test center, first remove the answer sheet (pages 59 and 60). Then go to the back cover of the test book (page 54) and follow the instructions for completing the identification areas of the answer sheet. When you are ready to begin the test, note the time and begin marking your answers on the answer sheet. MATHEMATICS TEST (RESCALED) 7 PRACTICE BOOK
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  9. MATHEMATICS TEST (RESCALED) Time—170 minutes 66 Questions Directions: Each of the questions or incomplete statements below is followed by five suggested answers or completions. In each case, select the one that is the best of the choices offered and then mark the corresponding space 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. 10
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