GRE ® Mathematics Test Practice BookThis practice book contains Mathematics Test ◾ test-taking strategies Become familiar with ◾ test structure and content ◾ test instructions and answe
Trang 1GRE ® Mathematics Test Practice Book
This practice book contains
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.
www.ets.org/gre
Trang 2Table of Contents
Overview 3
Test Content 3
Preparing for the Test 3
Test-Taking Strategies 4
What Your Scores Mean 4
Taking the Practice Test 4
Scoring the Practice Test 5
Evaluating Your Performance 5
Practice Test 7
Worksheet for Scoring the Practice Test 63
Score Conversion Table 64
Answer Sheet 65
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Trang 3GRE Mathematics Test Practice Book 3 Page
approximately 66 multiple-choice questions
drawn from courses commonly offered at the
undergraduate level Testing time is 2 hours and
50 minutes; there are no separately-timed sections
This publication provides a comprehensive
overview of the GRE Mathematics Test to help you
get ready for test day It is designed to help you:
To learn more about the GRE Subject Tests, visit
www.ets.org/gre
Test Content
Approximately 50 percent of the questions involve
calculus and its applications — subject matter that
is 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 — including
calculus-based applications and connections
with coordinate geometry, trigonometry,
differential equations, and other branches
• 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, ield theory, and number theory
Additional Topics (25%)
• Introductory real analysis: sequences and series of numbers and functions, continuity, differentiability and integrability, and
• 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 dificult questions on the test In general, the questions are intended not only to test recall of information, but also to assess the understanding of fundamental concepts and the ability to apply those concepts in various situations.
Preparing for the Test
GRE Subject Test questions are designed to measure skills and knowledge gained over a long period of time Although you might increase your scores to some extent through preparation a few weeks or months before you take the test, last minute cramming is unlikely to be of further help
differentiability and integrabilitytopology of and
differentiability and integrability
and n
Trang 4Page GRE Mathematics Test Practice Book 4
• A general review of your college courses is
probably the best preparation for the test
However, the test covers a broad range of
subject matter, and no one is expected to be
familiar with the content of every question
• Use the practice test to become familiar
with the types of questions in the GRE
Mathematics Test, taking note of the
directions If you understand the directions
before you take the test, you will have
more time during the test to focus on the
questions themselves
Test-Taking Strategies
The questions in the practice test in this book
illustrate the types of multiple-choice questions in
the test When you take the actual test, you will
mark your answers on a separate machine-scorable
answer sheet
Following are some general test-taking strategies
you may want to consider
• Read the test directions carefully, and work
as rapidly as you can without being careless
For each question, choose the best answer
from the available options
• All questions are of equal value; do not
spend time pondering individual questions
you ind extremely dificult or unfamiliar
• You may want to work through the test quite
rapidly, irst answering only the questions
about which you feel conident, then going
back and answering questions that require
more thought, and concluding with the most
dificult questions if there is time.
• If you decide to change an answer, make sure
you completely erase it and ill in the oval
corresponding to your desired answer
• Questions for which you mark no answer or
more than one answer are not counted in
scoring
• Your score will be determined by subtracting
one-fourth the number of incorrect answers
unlikely that pure guessing will raise your score; it may lower your score However,
if you have some knowledge of a question and are able to rule out one or more of the answer choices as incorrect, your chances of selecting the correct answer are improved, and answering such questions will likely improve your score
• Record all answers on your answer sheet
Answers recorded in your test book will not
be counted
• Do not wait until the last ive minutes of
a testing session to record answers on your answer sheet
What Your Scores Mean
Your raw score — that is, the number of questions you answered correctly minus one-fourth of the number 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 GRE Mathematics Test is comparable to the same scaled score earned on any other edition
of the test Thus, equal scaled scores indicate essentially equal levels of performance regardless of the test edition taken
GRE Mathematics Test scores are reported on a
200 to 990 score scale in ten-point increments
Test scores should be compared only with other scores on the GRE Mathematics Test For example,
a 680 on the GRE Mathematics Test is not equivalent to a 680 on the GRE Physics Test.
Taking the Practice Test
The practice test begins on page 7 The total time that you should allow for the practice test is
2 hours and 50 minutes An answer sheet is provided for you to mark your answers to the test questions
It is best to take the practice test under timed conditions Find a quiet place to take the test and make sure you have a minimum of 2 hours and
50 minutes available
Trang 5GRE Mathematics Test Practice Book 5 Page
(pages 65 and 66) Then go to page 62 and follow
the instructions for completing the identiication
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 Stop working on
the test when 2 hours and 50 minutes have elapsed.
Scoring the Practice Test
The worksheet on page 63 lists the correct answers
to the questions Columns are provided for you to
mark whether you chose the correct (C) answer or
an incorrect (I) answer to each question Draw a
line across any question you omitted, because it is
not counted in the scoring
At the bottom of the page, enter the total
number correct and the total number incorrect
Divide the total incorrect by 4 and subtract the
resulting number from the total correct Then
round the result to the nearest whole number
This will give you your raw score Use the score
conversion table on page 64 to ind the scaled
score that corresponds to your raw score
Example: Suppose you chose the correct answers
to 46 questions and incorrect answers to 9 Dividing
9 by 4 yields 2.25 Subtracting 2.25 from 46 equals
43.75, which is rounded to 44 The raw score of
44 corresponds to a scaled score of 770
Evaluating Your Performance
Now that you have scored your test, you may wish
to compare your performance with the performance
of others who took this test
The data in the worksheet on page 63 are based
on the performance of a sample of the test takers
who took the GRE Mathematics Test in October
2012 This sample was selected to represent
the total population of GRE Mathematics Test
examinees tested between July 1, 2011, and
June 30, 2014
The numbers in the column labeled “P+” on the
worksheet indicate the percentages of examinees in
this sample who answered each question correctly
Interpretive data based on the scores earned
by test takers in a recent three-year period are
available in Table 2, Subject Tests Total Score Interpretive Data Used on Score Reports, at
www.ets.org/gre/subject/scores/understand
The table shows selected scaled scores and the corresponding percentage of test takers who received lower scores Note that these interpretive data are updated annually and appear on GRE score reports To compare yourself with this population, look at the percentage next to the scaled score you earned on the practice test If the scaled score you earned does not appear in the table, look at the scaled scores (and corresponding percentages) above and below the score you earned to assess your performance.
It is important to realize that the conditions under which you tested yourself were not exactly the same as those you will encounter at a test center It is impossible to predict how different test-taking conditions will affect test performance, and this is only one factor that may account for differences between your practice test scores and your actual test scores By comparing your performance on the practice test with the performance of other individuals who took GRE Mathematics Test, however, you will be able to determine your strengths and weaknesses and can then plan a program of study to prepare yourself for taking the GRE Mathematics Test under standard conditions
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MATHEMATICS TEST
Trang 8MATHEMATICS TEST 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 best and then completely fill in the corresponding space on the answer sheet
Computation and scratch work may be done in this test book
In this test:
(1) All logarithms with an unspecified base are natural logarithms, that is, with base e
(2) The symbols , , , and denote the sets of integers, rational numbers, real numbers,
and complex numbers, respectively
2 0
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Trang 104 Let V and W be 4-dimensional subspaces of a 7-dimensional vector space X Which of the following
CANNOT be the dimension of the subspace V «W?
Trang 127 The figure above shows the graph of the derivative f¢ of a function f, where f is continuous on the interval
[ ]0, 4 and differentiable on the interval ( )0, 4 Which of the following gives the correct ordering of the values
8 Which of the following is NOT a group?
(A) The integers under addition
(B) The nonzero integers under multiplication
(C) The nonzero real numbers under multiplication
(D) The complex numbers under addition
(E) The nonzero complex numbers under multiplication
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Trang 1410 In the xy-plane, the set of points whose coordinates satisfy the equation above is
(A) a line (B) a circle (C) an ellipse (D) a parabola (E) one branch of a hyperbola
Trang 1611 The region bounded by the curves y = x and y = x2 in the first quadrant of the xy-plane is rotated about the
y-axis The volume of the resulting solid of revolution is
12 For which integers n such that 3 £ £n 11 is there only one group of order n (up to isomorphism) ?
(A) For no such integer n
(B) For 3, 5, 7, and 11 only
(C) For 3, 5, 7, 9, and 11 only
(D) For 4, 6, 8, and 10 only
(E) For all such integers n
13 If f is a continuously differentiable real-valued function defined on the open interval (-1, 4) such that f( )3 = 5and f¢( )x ≥ -1 for all x, what is the greatest possible value of f( )0 ?
Trang 1814 Suppose g is a continuous real-valued function such that 3 5+96 = Úx ( )
c
x g t dt for each x Πwhere c is ,
a constant What is the value of c ?
16 Suppose A, B, and C are statements such that C is true if exactly one of A and B is true If C is false,
which of the following statements must be true?
(A) If A is true, then B is false
(B) If A is false, then B is false
(C) If A is false, then B is true
(D) Both A and B are true
(E) Both A and B are false
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Trang 20z z
Æ
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Trang 2220 Let g be the function defined by ( ) = 2x+ 1
(A) 2e (B) 4e (C) 2 2e+1
e
4 cos 1 sin cos ?
22 What is the volume of the solid in xyz-space bounded by the surfaces y = x2, y = -2 x2, z = 0,
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Trang 2423 Let ( 10,+ , ) be the ring of integers modulo 10, and let S be the subset of 10 represented by {0, 2, 4, 6, 8 }
Which of the following statements is FALSE?
(A) (S,+ is closed under addition modulo 10 , )
(B) (S,+, is closed under multiplication modulo 10 )
(C) (S,+, has an identity under addition modulo 10 )
(D) (S,+, has no identity under multiplication modulo 10 )
(E) (S,+, is commutative under addition modulo 10 )
24 Consider the system of linear equations
(A) The system is consistent
(B) The system has infinitely many solutions
(C) The sum of any two solutions is a solution
(D) (-5, 1, 1, 0) is a solution
(E) Every solution is a scalar multiple of (-5, 1, 1, 0 )
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Trang 2828 Let f be a one-to-one (injective), positive-valued function defined on Assume that f is differentiable at
1
=
x and that in the xy-plane the line y- =4 3(x -1) is tangent to the graph of f at x =1. Let g be the
function defined by g x( ) = x for x ≥ 0 Which of the following is FALSE?
(A) ¢ f ( )1 = 3
43
Trang 32(C) The last row of A2 is (0 0 0 0 25 )
(D) A can be transformed into the 5¥ identity matrix by a sequence of elementary row operations 5
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