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 ®
This practice book contains
one actual, full-length GRE® Mathematics Test
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.
Note to Test Takers: Keep this practice book until you receive your score report.
This book contains important information about scoring.
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.
The GRE Board recommends that scores on the
Subject Tests be considered in conjunction with other
Table of Contents
relevant information about applicants. Because numer-
Purpose of the GRE Subject Tests ........................ 3
ous factors inﬂ uence success in graduate school,
reliance on a single measure to predict success is not
Development of the Subject Tests ........................ 3
advisable. Other indicators of competence typically
Content of the Mathematics Test ........................ 4
include undergraduate transcripts showing courses
Preparing for a Subject Test .................................. 5
taken and grades earned, letters of recommendation,
and GRE General Test scores. For information about
Test-Taking Strategies .......................................... 5
the appropriate use of GRE scores, see the GRE Guide
What Your Scores Mean ....................................... 6
to the Use of Scores at ets.org/gre/stupubs.
Practice Mathematics Test .................................. 9
Scoring Your Subject Test .................................. 65
Development of the
Evaluating Your Performance ............................. 68
Answer Sheet...................................................... 69
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
Purpose of the
faculties in different types of institutions and in
different regions of the United States and Canada.
GRE Subject Tests
In selecting members for each committee, the
GRE Program seeks the advice of the appropriate
The GRE Subject Tests are designed to help graduate
professional associations in the subject.
school admission committees and fellowship sponsors
The content and scope of each test are speciﬁ ed
assess the qualiﬁ cations of applicants in speciﬁ c ﬁ elds
and reviewed periodically by the committee of
of study. The tests also provide you with an assessment
exam iners. Test questions are written by committee
of your own qualiﬁ cations.
members and by other university faculty members
Scores on the tests are intended to indicate
who are subject-matter specialists. All questions
knowledge of the subject matter emphasized in many
proposed for the test are reviewed and revised by the
undergraduate programs as preparation for graduate
committee and subject-matter specialists at ETS. The
study. Because past achievement is usually a good
tests are assembled in accordance with the content
indicator of future performance, the scores are helpful
speciﬁ cations developed by the committee to ensure
in predicting success in graduate study. Because the tests
adequate coverage of the various aspects of the ﬁ eld
are standardized, the test scores permit comparison
and, at the same time, to prevent overemphasis on
of students from different institutions with different
any single topic. The entire test is then reviewed and
undergraduate programs. For some Subject Tests,
approved by the committee.
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
Subject-matter and measurement specialists on the
subscores for any test or test edition may be smaller.
ETS staff assist the committee, providing information
Subject Test score interpretive information is provided
and advice about methods of test construction and
in Interpreting Your GRE Scores, which you will receive
helping to prepare the questions and assemble the test.
with your GRE score report. This publication is also
In addition, each test question is reviewed to eliminate
available at ets.org/gre/stupubs.
language, symbols, or content considered potentially
offensive, inappropriate for major subgroups of the test-
Content of the
taking population, or likely to perpetuate any negative
attitude that may be conveyed to these subgroups.
Because of the diversity of undergraduate curricula,
The test consists of approximately 66 multiple-choice
it is not possible for a single test to cover all the material
questions drawn from courses commonly offered at
you may have studied. The examiners, therefore, select
the undergraduate level. Approximately 50 percent of
questions that test the basic knowledge and skills
the questions involve calculus and its applications—
most important for successful graduate study in the
subject matter that can be assumed to be common to
particular ﬁ eld. The committee keeps the test up-to-
the backgrounds of almost all mathematics majors.
date by regularly developing new editions and revising
About 25 percent of the questions in the test are in
existing editions. In this way, the test content remains
elementary algebra, linear algebra, abstract algebra,
current. In addition, curriculum surveys are conducted
and number theory. The remaining questions deal
periodically to ensure that the content of a test reﬂ ects
with other areas of mathematics currently studied by
what is currently being taught in the undergraduate
undergraduates in many institutions.
The following content descriptions may assist
After a new edition of a Subject Test is ﬁ rst
students in preparing for the test. The percents given
administered, examinees’ responses to each test
are estimates; actual percents will vary somewhat from
question are analyzed in a variety of ways to determine
one edition of the test to another.
whether each question functioned as expected. These
analyses may reveal that a question is ambiguous,
requires knowledge beyond the scope of the test, or
Material learned in the usual sequence of
is inappropriate for the total group or a particular
elementary calculus courses—differential
subgroup of examinees taking the test. Such questions
and integral calculus of one and of several
are not used in computing scores.
variables—includes calculus-based applications
Following this analysis, the new test edition is
and connections with coordinate geometry,
equated to an existing test edition. In the equating
trigonometry, differential equations, and other
process, statistical methods are used to assess the
branches of mathematics
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
Elementary algebra: basic algebraic techniques
an easier edition of the test do not have an advantage.
and manipulations acquired in high school and
Variations in the number of questions in the different
used throughout mathematics
editions of the test are also taken into account in this
Linear algebra: matrix algebra, systems of linear
equations, vector spaces, linear transformations,
Scores on the Subject Tests are reported as three-
characteristic polynomials, and eigenvalues and
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
Abstract algebra and number theory: elementary
for a particular Subject Test, however, may be smaller.
topics from group theory, theory of rings and
For Subject Tests that report subscores, the maximum
modules, ﬁ eld theory, and number theory
possible range is 20 to 99; however, the actual range of
Introductory real analysis: sequences and
The questions in the practice test in this book
series of numbers and functions, continuity,
illustrate the types of multiple-choice questions in the
differentiability and integrability, and elementary
test. When you take the actual test, you will mark your
topology of and n
answers on a separate machine-scorable answer sheet.
Discrete mathematics: logic, set theory,
Total testing time is two hours and ﬁ fty minutes; there
combinatorics, graph theory, and algorithms
are no separately timed sections. Following are some
Other topics: general topology, geometry,
general test-taking strategies you may want to consider.
complex variables, probability and statistics, and
Read the test directions carefully, and work as
rapidly as you can without being careless. For
The above descriptions of topics covered in the test
each question, choose the best answer from the
should not be considered exhaustive; it is necessary to
understand many other related concepts. Prospective
All questions are of equal value; do not waste
test takers should be aware that questions requiring no
time pondering individual questions you ﬁ nd
more than a good precalculus background may be quite
extremely difﬁ cult or unfamiliar.
challenging; such questions can be among the most
You may want to work through the test quite
difﬁ cult questions on the test. In general, the questions
rapidly, ﬁ rst answering only the questions about
are intended not only to test recall of information but
which you feel conﬁ dent, then going back and
also to assess test takers’ understanding of fundamental
answering questions that require more thought,
concepts and the ability to apply those concepts in
and concluding with the most difﬁ cult questions
if there is time.
Preparing for a Subject Test
If you decide to change an answer, make sure
you completely erase it and ﬁ ll in the oval
GRE Subject Test questions are designed to measure
corresponding to your desired answer.
skills and knowledge gained over a long period of time.
Questions for which you mark no answer or more
Although you might increase your scores to some extent
than one answer are not counted in scoring.
through preparation a few weeks or months before you
Your score will be determined by subtracting
take the test, last minute cramming is unlikely to be of
one-fourth the number of incorrect answers from
further help. The following information may be helpful.
the number of correct answers. If you have some
A general review of your college courses is
knowledge of a question and are able to rule out
probably the best preparation for the test.
one or more of the answer choices as incorrect,
However, the test covers a broad range of subject
your chances of selecting the correct answer are
matter, and no one is expected to be familiar
improved, and answering such questions will
with the content of every question.
likely improve your score. It is unlikely that pure
Use this practice book to become familiar with
guessing will raise your score; it may lower your
the types of questions in the GRE Mathematics
Test, taking note of the directions. If you
Record all answers on your answer sheet.
understand the directions before you take the
Answers recorded in your test book will not
test, you will have more time during the test to
focus on the questions themselves.
Do not wait until the last ﬁ ve minutes of a testing
session to record answers on your answer sheet.
Range of Raw Scores* Needed
What Your Scores Mean
to Earn Selected Scaled Score on
Your raw score — that is, the number of questions you
Three Mathematics Test
answered correctly minus one-fourth of the number
Editions that Differ in Difﬁ culty
you answered incorrectly — is converted to the scaled
score that is reported. This conversion ensures that
a scaled score reported for any edition of a Subject
Test is comparable to the same scaled score earned
on any other edition of the same test. Thus, equal
scaled scores on a particular Subject Test indicate
essentially equal levels of performance regardless of
the test edition taken. Test scores should be compared
Number of Questions Used to Compute Raw Score
only with other scores on the same Subject Test. (For
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.
Before taking the test, you may ﬁ nd it useful
to know approximately what raw scores would be
For a particular test edition, there are many ways to
required to obtain a certain scaled score. Several
earn the same raw score. For example, on the edition
factors inﬂ uence the conversion of your raw score
listed above as “Form A,” a raw score of 28 would earn
to your scaled score, such as the difﬁ culty of the test
a scaled score of 600. Below are a few of the possible
edition and the number of test questions included in
ways in which a scaled score of 600 could be earned on
the computation of your raw score. Based on recent
editions of the Mathematics Test, the following table
gives the range of raw scores associated with selected
scaled scores for three different test editions. (Note
Examples of Ways to Earn
that when the number of scored questions for a given
a Scaled Score of 600 on the
test is greater than the number of actual scaled score
Edition Labeled as “Form A”
points, it is likely that two or more raw scores will
convert to the same scaled score.) The three test
editions in the table that follows were selected to
reﬂ ect varying degrees of difﬁ culty. Examinees should
note that future test editions may be somewhat more
or less difﬁ cult than the test editions illustrated in the
P R A C T I C E T E S T
To become familiar with how the administration will be conducted at the test center, ﬁ rst remove the
answer sheet (pages 69 and 70). Then go to the back cover of the test book (page 64) and follow the
instructions for completing the identiﬁ cation 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.
GRADUATE RECORD EXAMINATIONS®
Do not break the seal
until you are told to do so.
The contents of this test are conﬁ dential.
Disclosure or reproduction of any portion
of it is prohibited.
THIS TEST BOOK MUST NOT BE TAKEN FROM THE ROOM.
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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 scratch work may be done in this examination book.
Note: In this examination:
logarithms with an unspecified base are natural logarithms, that is, with base e.
(2) The set of all real numbers x such that a
b is denoted by a, b .
, and denote the sets of integers, rational numbers, real numbers,
and complex numbers, respectively.
1. In the xy-plane, the curve with parametric equations x
cos t and y
sin t, 0
p, has length
3 (B) p (C) 3p (D) (E)
2. Which of the following is an equation of the line tangent to the graph of
e at x
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