Mathematics Test Practice Book

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GRE  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.

www.ets.org/gre

Table 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................................................................................................................................6 Worksheet for Scoring the Practice Test..................................................................................62 Score Conversion Table ...........................................................................................................63 Answer Sheet............................................................................................................................64

Test takers with disabilities or health-related needs who need test preparation materials in an alternate format should contact the ETS Office of Disability Services at [email protected] For additional information, visit www.ets.org/gre/disabilities.

Copyright © 2017 by Educational Testing Service. All rights reserved. ETS, the ETS logo, MEASURING THE POWER OF LEARNING, GRADUATE RECORD EXAMINATIONS, and GRE are registered trademarks of Educational Testing Service (ETS) in the United States and other countries.

Overview The GRE ® Mathematics Test consists of 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: • Understand what is being tested • Gain familiarity with the question types • Review test-taking strategies • Understand scoring • Practice taking the test To learn more about the GRE Subject Tests, visit www.ets.org/gre.

Test Content Approximately 50 percent of the Mathematics Test 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 percentages given are estimates; actual percentages will vary somewhat from one edition of the test to another. I. 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 of mathematics

GRE ® Mathematics Test Practice Book

II. 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, field theory, and number theory III. Additional TopicsTopics—25% (25%) Additional Additional Topics—25%

Introductory real analysis:real sequences  Introductory analysis: sequences Introductory real analysis:and sequencesand and series of numbers functions, continuity, series ofofnumbers and functions, seriesand numbers and functions,continuity, continuity, differentiability and integrability, and differentiability and and differentiability andintegrability, integrability, andeleme eleme and elementary topology topology ofofand topologyof andnn

Discrete mathematics: logic, set theory,  Discrete logic, Discretemathematics: mathematics: logic,set settheory, theory, combinatorics, graph theory,graph and theory, algorithms combinatorics, combinatorics, graph theory,and andalgorithms algorithm

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

The ofoftopics Theabove above descriptions topics coveredin inthe th The above descriptions ofdescriptions topics covered in covered should not be considered exhaustive; it is necessa should not be considered exhaustive; it is necess the test should not be considered exhaustive; it many other related concepts. understand many other related concepts.Prospect Prospec is necessary understand to understand many other related test takers should be aware that questions requirin test takers should be aware that questions requirin concepts. Prospective test takers should be aware more than precalculus background more thanaagood good precalculus backgroundmay maybe be that questions requiring no more than a good challenging; such questions can be among the mo challenging; such questions can be among the mo precalculus background may be quite challenging; diffi cult questions test. In diffican cultbe questions onthe the test. Ingeneral, general,the theques que such questions among on the most difficult are intended not only to test recall of information are intended not only to test recall of information questions on the test. In general, the questions are to assess takers’ understanding also toto assess test takers’ understanding fundam intended notalso only testtest recall of information, butofoffundame concepts and the ability to apply those concepts in concepts and the ability to apply those concepts i also to assess the understanding of fundamental various situations. situations. concepts andvarious the ability to apply those concepts in various situations.

Preparing Preparing for for aa Subject Subject Test Tes Preparing for the GRE Test questions GRESubject Subject TestTest questionsare aredesigned designedto tomeasu meas

skills skillsand andknowledge knowledgegained gainedover overaalong longperiod periodofof GRE Subject Test questions are designed to scoresto Although Althoughyou youmight mightincrease increaseyour yourscores tosome somee measure skills and knowledge gained over a long through throughpreparation preparationaafew fewweeks weeksor ormonths monthsbefore befor period of time. Although youminute might cramming increase your take the isisunlikely take thetest, test,last last minute cramming unlikelyto to scores to some extent through preparation a few further furtherhelp. help.The Thefollowing followinginformation informationmay maybe behe h weeks or months before you take the test, last  AA review college general review your college coursesisis minute cramming isgeneral unlikely to beofof ofyour further help.courses probably probablythe thebest bestpreparation preparationfor forthe thetest. test. However, However,the thetest testcovers coversaabroad broadrange rangeofofsub su matter, matter,and andno noone oneisisexpected expectedto tobe befamiliar familia with withthe thecontent contentofofevery everyquestion. question. 3 Page  Use Usethis thispractice practicebook bookto tobecome becomefamiliar familiarwi w the thetypes typesofofquestions questionsin inthe theGRE GREMathemat Mathema

The following information may be helpful. • 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. • Become familiar with the types of questions in the GRE Mathematics Test, paying special attention to the directions. If you thoroughly 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 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. The 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 waste time pondering individual questions you find extremely difficult or unfamiliar. • You may want to work through the test quickly, first answering only the questions about which you feel confident, then going back and answering questions that require more thought, and concluding with the most difficult questions if there is time. • If you decide to change an answer, make sure you completely erase it and fill in the oval corresponding to your desired answer. • Your score will be determined by the number of questions you answer correctly. Questions you answer incorrectly or for which you mark no answer or more than one answer are counted as incorrect. Nothing is subtracted from a score if you answer a question incorrectly. Therefore, to maximize your

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score it is better for you to guess at an answer than not to respond at all. • Record all answers on your answer sheet. Answers recorded in your test book will not be counted. • Do not wait until the last few minutes of a testing session to record answers on your answer sheet.

What Your Scores Mean The number of questions you answered correctly on the whole test (total correct score) is converted to the total reported scaled score. 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 on a particular test indicate essentially equal levels of performance regardless of the test edition taken. GRE Mathematics Test total 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 Mathematics Test. For example, a 680 on the Mathematics Test is not equivalent to a 680 on the Physics Test.

Taking the Practice Test The practice test begins on page 6. The total time that you should allow for this 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 this 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. To simulate how the administration will be conducted at the test center, print the answer sheet (pages 64 and 65). Then go to the back cover of the test book (page 61) 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. Stop working on the test when 2 hours and 50 minutes have elapsed.

GRE ® Mathematics Test Practice Book

Scoring the Practice Test The worksheet on page 62 lists the correct answers to the questions on the practice test. The “Correct Response” columns are provided for you to mark those questions for which you chose the correct answer. Mark each question that you answered correctly. Then, add up your correct answers and enter your total number of correct answers in the space labeled “Total Correct” at the bottom of the page. Next, use the “Total Score” conversion table on page 63 to find the corresponding scaled score. For example, suppose you chose the correct answers to 50 questions on the test. The “Total Correct” entry in the conversion table of 50 shows that your total scaled score is 790.

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 this 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. Note that question 53 was slightly modified for this practice test.

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 62 are based on the performance of a sample of the test takers who took the GRE Mathematics Test in the United States. The numbers in the column labeled “P+” on the worksheet indicate the percentages of examinees in this sample who answered each question correctly. You may use these numbers as a guide for evaluating your performance on each test question. Interpretive data based on the scores earned by a recent cohort of test takers are available on the GRE website at www.ets.org/gre/subject/scores/ understand. The interpretive data shows, for selected scaled score, the percentage of test takers who received lower scores. To compare yourself with this population, look at the percentage next to the scaled score you earned on the practice test. Note that these interpretive data are updated annually and reported on GRE score reports. 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

GRE ® Mathematics Test Practice Book

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FORM GR1768

68 GRADUATE RECORD EXAMINATIONS®

MATHEMATICS TEST

Copyright © 2012, 2007 by Educational Testing Service. All rights reserved.

GRE, GRADUATE RECORD EXAMINATIONS, ETS, EDUCATIONAL TESTING

SERVICE and the ETS logos are registered trademarks of Educational Testing Service.

6

MATHEMATICS 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 C denote the sets of integers, rational numbers, real numbers, and complex numbers, respectively.

lim

1.

xÆ0

(A)

9 2

cos (3 x ) - 1 x2 (B)

=

3 2

(C) -

2 3

(D) -

3 2

(E) -

9 2

2. What is the area of an equilateral triangle whose inscribed circle has radius 2 ? (A) 12

3.

e -2

Úe (A) 1

-3

(B) 16

(C) 12 3

(D) 16 3

(E) 4 (3 + 2 2 )

1 dx = x log x (B)

2 3

(C)

3 2

(D) log

( 23 )

(E) log

( 23 )

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8

4. 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 ? (A) 0

(B) 1

(C) 2

(D) 3

(E) 4



5. Sofia and Tess will each randomly choose one of the 10 integers from 1 to 10. What is the probability that neither integer chosen will be the square of the other? (A) 0.64

(B) 0.72

(C) 0.81

(D) 0.90

(E) 0.95

6. Which of the following shows the numbers 21 2, 31 3, and 61 6 in increasing order? (A) 21 2 < 31 3 < 61 6

(B) 61 6 < 31 3 < 2 1 2

(C) 61 6 < 21 2 < 3 1 3

(D) 31 3 < 2 1 2 < 6 1 6

(E) 31 3 < 6 1 6 < 2 1 2

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7. 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 f (0 ) , f (2 ) , and f (4 ) ? (A) f (0 ) < f (2 ) < f (4 )

f (2 ) (B) f (0 ) < f (4 ) = (C) f (0 ) < f (4 ) < f (2 ) (D) f= (4)

f (2 ) < f (0 )

(E) f (4 ) < f (0 ) < f (2 )

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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9. Let g be a continuous real-valued function defined on � with the following properties.

g¢ (0 ) = 0 g¢¢ ( -1 ) > 0 g ¢¢ ( x ) < 0 if 0 < x < 2.

Which of the following could be part of the graph of g ?

(A)

(B)

(D)

(E)

( x + 3 ) 2 + ( y - 2 )2 =

(C)

( x - 3)2 + y 2

10. 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

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11. The region bounded by the curves y = x and y = x 2 in the first quadrant of the xy-plane is rotated about the y-axis. The volume of the resulting solid of revolution is

(A)

p 12

(B)

p 6

(C)

p 3

(D)

2p 3

(E)

3p

2

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) = 5 and f ¢ ( x ) ≥ -1 for all x, what is the greatest possible value of f (0 ) ? (A) 3

(B) 4

(C) 5

(D) 8

(E) 11

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16

x

Úc

14. Suppose g is a continuous real-valued function such that 3 x 5 + 96 =g (t ) dt for each x Œ �, where c is a constant. What is the value of c ? (A) -96

(B) -2

(C) 4

(D) 15

(E) 32

15. Let S, T, and U be nonempty sets, and let f : S Æ T and g : T Æ U be functions such that the function g D f : S Æ U is one-to-one (injective). Which of the following must be true? (A) f is one-to-one. (B) f is onto. (C) g is one-to-one. (D) g is onto. (E) g D f is onto.

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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17. Which of the following equations has the greatest number of real solutions? 3 10 - x (A) x=

(B) x 2 + 5 x - 7 = x + 8

(C) 7 x + 5 = 1 - 3 x

(D) e x = x (E) sec x = e - x

2

18. Let f be the function defined by f ( x ) = (A)

1 1- x

(B)

x 1- x

(C)



n

 xn n =1

1 1+ x

for all x such that -1 < x < 1. Then f ¢ ( x ) = (D)

x 1+ x

(E) 0

19. If z is a complex variable and z denotes the complex conjugate of z, what is lim

zÆ0

(A) 0

(B) 1

(C) i

(D) •

( z )2 z2

?

(E) The limit does not exist.

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20

g ( g ( x )) - g ( e ) = x xÆ0

20. Let g be the function defined by g ( x ) = e2 x +1 for all real x. Then lim (A) 2e

(B) 4e2

21. What is the value of (A) 0

(B)

(C) e2 e +1

Ú- p 4 (cos t +

2

p 4

(C)

(D) 2e2 e +1

(E) 4e2 e + 2

)

1 + t 2 sin 3 t cos3 t dt ? 2 -1

(D)

2 2

(E)

2 -1 2

22. What is the volume of the solid in xyz-space bounded by the surfaces y = x 2 , y= 2 - x 2 , z = 0, and z= y + 3 ? (A)

8 3

(B)

16 3

(C)

32 3

(D)

104 105

(E)

208 105

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23. 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

w + 3x + 2y + 2z w + 4x + y 3w + 5x + 10 y + 14z 2w + 5x + 5y + 6z

= 0 = 0 = 0 = 0

with solutions of the form (w, x, y, z ) , where w , x, y, and z are real. Which of the following statements is FALSE?

(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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25. The graph of the derivative h¢ is shown above, where h is a real-valued function. Which of the following open intervals contains a value c for which the point (c, h (c )) is an inflection point of h ? (A)

( -2, -1)

(B)

( - 1, 0 )

(C) (0, 1)

(D) (1, 2 )

(E) (2, 3)

3x ∫ 5 (mod 11) 2y ∫ 7 (mod 11) 26. If x and y are integers that satisfy the congruences above, then x + y is congruent modulo 11 to which of the following? (A) 1

27.

(B) 3

(C) 5

(D) 7

(C) 32

(D) 32i

(E) 9

(1 + i )10 = (A) 1

(B) i

(E) 32 (i + 1)

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28. Let f be a one-to-one (injective), positive-valued function defined on �. Assume that f is differentiable at x = 1 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?

3

(A) f ¢ (1) = (B)

( f -1 )¢ (4) = 13

(C)

( fg )¢ (1) = 5

(D) ( g D f )¢ (1) = (E)

1 2

( g D f )(1) = 2

29. A tree is a connected graph with no cycles. How many nonisomorphic trees with 5 vertices exist? (A) 1

(B) 2

(C) 3

(D) 4

(E) 5

30. For what positive value of c does the equation log x = cx 4 have exactly one real solution for x ? (A)

1 4e

(B)

1 4e 4

(C)

e4 4

(D)

4 14

e

(E) 4e1 4

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28

Ê 3 5 3ˆ 31. Of the numbers 2, 3, and 5, which are eigenvalues of the matrix Á 1 7 3˜ ? Á ˜ Ë1 2 8¯ (A) None

32.

(B) 2 and 3 only

(D) 3 and 5 only

(E) 2, 3, and 5

x4 t2

d dx

Úx

x6

(e

(A) e

(C) 2 and 5 only

3

e dt =

x8 - x 6

)

-1

3 x8

(B) 4x e

33. What is the 19th derivative of

(C)

1

2 1 - ex

2

ex (D) 2 - 1 x

6

(

(E) x 2 e x 4xe x

8

- x6

-3

)

x -1 ? ex

(A) (18 - x ) e - x (B) (19 - x ) e - x (C) (20 - x ) e - x

(D)

( x - 19) e - x

(E)

( x - 20 ) e - x

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Ê1 Á0 Á A = Á0 Á0 ÁË 0

2 2 0 0 0

3 3 3 0 0

4 4 4 4 0

5ˆ 5˜ ˜ 5˜ 5˜ ˜ 5¯

34. Which of the following statements about the real matrix shown above is FALSE? (A) A is invertible. (B) If x Œ�5 and Ax = x, then x = 0. (C) The last row of A2 is (0 0 0 0 25) . (D) A can be transformed into the 5 ¥ 5 identity matrix by a sequence of elementary row operations. (E) det ( A) = 120

35. In xyz-space, what are the coordinates of the point on the plane 2 x + y + 3z = 3 that is closest to the origin? (A) (0, 0, 1)

(B)

( 37 , 143 , 149 )

(C)

(157 , 158 , 151 )

(D)

( 65 , 13 , 13 )

( 13 )

(E) 1, 1,

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36. Suppose S is a nonempty subset of �. Which of the following is necessarily true? (A) For each s, t Œ S, there exists a continuous function f mapping [0, 1] into S with f (0 ) = s and f (1) = t. (B) For each u œ S, there exists an open subset U of � such that u Œ U and U « S =∆. (C)

{v Œ S : there exists an open subset V of � with v Œ V Õ S} is an open subset of �.

(D)

{w œ S : there exists an open subset W of � with w Œ W and W « S =∆} is a closed subset of �.

(E) S is the intersection of all closed subsets of � that contain S.

37. Let V be a finite-dimensional real vector space and let P be a linear transformation of V such that P 2 = P. Which of the following must be true? I. P is invertible. II. P is diagonalizable. III. P is either the identity transformation or the zero transformation. (A) None

(B) I only

(C) II only

(D) III only

(E) II and III

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38. The maximum number of acute angles in a convex 10-gon in the Euclidean plane is (A) 1

(B) 2

(C) 3

(D) 4

(E) 5

39. Consider the following algorithm, which takes an input integer n > 2 and prints one or more integers. input(n)

set i = 1

while i < n

begin

replace i by i + 1

set k = n

while k ≥ i

begin

if i = k then print(i)

replace k by k - 1

end

end

If the input integer is 88, what integers will be printed? (A) Only the integer 2 (B) Only the integer 88 (C) Only the divisors of 88 that are greater than 1 (D) The integers from 2 to 88 in increasing order (E) The integers from 88 to 2 in decreasing order

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40. Let S be the set of all functions f : � Æ �. Consider the two binary operations + and D on S defined as pointwise addition and composition of functions, as follows.

( f + g )( x ) = f ( x ) + g ( x ) ( f D g ) ( x ) = f ( g ( x )) Which of the following statements are true? I. D is commutative. II. + and D satisfy the left distributive law f D ( g + = h) III. + and D satisfy the right distributive law ( g + h ) D f = (A) None

(B) II only

(C) III only

( f D g) + ( f D h ). ( g D f ) + (h D f ) .

(D) II and III only

(E) I, II, and III

3 and x - y + z = 5 in �3 . An equation 41. Let A be the line that is the intersection of the planes x + y + z = of the plane that contains (0, 0, 0 ) and is perpendicular to A is

0

(A) x - z =

(B) x + y + z = 0

0 (C) x - y - z = (D) x + z = 0

0 (E) x + y - z =

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42. Let : + be the set of positive integers and let d be the metric on : + defined by

d (m, n ) =

{

0 if m = n 1 if m π n

(

)

for all m, n Π: +. Which of the following statements are true about the metric space : +, d ? I. If n Π: +, then {n} is an open subset of : +. II. Every subset of : + is closed. III. Every real-valued function defined on : + is continuous. (A) None

(B) I only

(C) III only

(D) I and II only

(E) I, II, and III

43. A curve in the xy-plane is given parametrically by

= x t 2 + 2t = y 3t 4 + 4t 3 for all t > 0. The value of (A) 4

(B) 24

d2 y dx 2

at the point (8, 80 ) is

(C) 32

(D) 96

(E) 192

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y¢ + xy = x y (0 ) = -1 44. If y is a real-valued function defined on the real line and satisfying the initial value problem above, then lim y ( x ) = xÆ -•

(A) 0

(B) 1

(C) -1

(D) •

(E) -•

45. How many positive numbers x satisfy the equation cos (97 x ) = x ? (A) 1

(B) 15

(C) 31

(D) 49

(E) 96

46. A ladder 9 meters in length is leaning against a vertical wall on level ground. As the bottom end of the ladder is moved away from the wall at a constant rate of 2 meters per second, the top end slides downward along the wall. How fast, in meters per second, will the top end of the ladder be sliding downward at the moment the top end is 3 meters above the ground? (A) 12 2

(B) 6 2

(C) 4 2

(D)

1 2 2

(E)

2 3

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47. The function f : � Æ � is defined as follows.

ÏÔ 3 x 2 f (x) = Ì 2 ÓÔ -5 x

if x Œ � if x œ �

Which of the following is true? (A) f is discontinuous at all x Œ�. (B) f is continuous only at x = 0 and differentiable only at x = 0. (C) f is continuous only at x = 0 and nondifferentiable at all x Œ�. (D) f is continuous at all x Œ� and nondifferentiable at all x Œ�. (E) f is continuous at all x œ� and nondifferentiable at all x Œ�.

48. Let g be the function defined by g ( x, y= , z ) 3x 2 y + z for all real x, y, and z. Which of the following is the best approximation of the directional derivative of g at the point (0, 0, p ) in the direction of the vector i + 2 j + 3k ? (Note: i, j, and k are the standard basis vectors in �3 . ) (A) 0.2

(B) 0.8

(C) 1.4

(D) 2.0

(E) 2.6

49. What is the largest order of an element in the group of permutations of 5 objects? (A) 5

(B) 6

(C) 12

(D) 15

(E) 120

GO ON TO THE NEXT PAGE.

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44

50. Let R be a ring and let U and V be (two-sided) ideals of R. Which of the following must also be ideals of R ? I. U + V =

{u + v : u ŒU and v ŒV }

II. U � V ={uv : u ŒU and v ŒV } III. U « V (A) II only

(B) III only

(C) I and II only

(D) I and III only

(E) I, II, and III

Ê 1 -1 2 -3ˆ 51. Which of the following is an orthonormal basis for the column space of the real matrix Á -1 1 -3 2˜ ? Á ˜ Ë 2 -2 5 -5¯ Ï Ê 1ˆ Ô (A) Ì Á 0˜ , ÔÓ ÁË 0˜¯

Ê 0ˆ Á 1˜ Á ˜ Ë 0¯

Ï Ê 1ˆ Ô (B) Ì Á 0˜ , Á ˜ ÓÔ Ë 0¯

Ê 0ˆ Á 1˜ , Á ˜ Ë 0¯

Ê 0ˆ Á 0˜ Á ˜ Ë 1¯

Ï 1 ÔÔ Ê ˆ (C) Ì Á 0˜ , Ô ÁË 0˜¯ ÔÓ

Ê 25 ˆ Á 1˜ Á 5˜ Á0˜ Ë ¯

¸ ÔÔ ˝ Ô Ô˛

Ï Ê 1ˆ ÔÁ ˜ (D) Ì -1 , Á ˜ ÓÔ Ë 2¯ ÏÊ ÔÔ Á (E) Ì Á ÔÁ ÓÔ Ë

¸ Ô ˝ Ô˛

Ê 2ˆ Á -3˜ Á ˜ Ë 5¯

1 ˆ 6 ˜ 1 , 6˜ 2 ˜ 6¯

¸ Ô ˝ Ô˛

¸ Ô ˝ Ô˛

Ê 12 ˆ Á 1˜ Á 2˜ Á ˜ Ë0¯

¸ ÔÔ ˝ Ô ˛Ô

GO ON TO THE NEXT PAGE.

45

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46

52. A university’s mathematics department has 10 professors and will offer 20 different courses next semester. Each professor will be assigned to teach exactly 2 of the courses, and each course will have exactly one professor assigned to teach it. If any professor can be assigned to teach any course, how many different complete assignments of the 10 professors to the 20 courses are possible? (A)

20! 210

(B)

10! 29

(C) 1020 - 210

(D) 1020 - 100

53. Let f and g be continuous functions of a real variable such that g( x ) =

(E)

20!10! 210

x

Ú0 f ( y)( y - x ) dy for all x. If g is three

times continuously differentiable, what is the greatest integer n for which f must be n times continuously differentiable? (A) 1

(B) 2

(D) 4

(C) 3

(E) 5

54. If a real number x is chosen at random in the interval [0, 3] and a real number y is chosen at random in the interval [0, 4], what is the probability that x < y ? (A)

1 2

(B)

7 12

(C)

5 8

(D)

2 3

(E)

3 4

GO ON TO THE NEXT PAGE.

47

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48

55. If a and b are positive numbers, what is the value of

(A) 0

(B) 1

(C) a - b

(D)



e ax - ebx dx ? 1 + e ax 1 + ebx

Ú0 (

)(

(a - b ) log 2

)

(E)

a-b log 2 ab

56. Which of the following statements are true? I. There exists a constant C such that log x £ C x for all x ≥ 1. n

II. There exists a constant C such that

 k 2 £ Cn2 for all integers n ≥ 1. k =1

III. There exists a constant C such that sin x - x £ C x 3 for all real x. (A) None

(B) I only

(C) III only

(D) I and III only

(E) I, II, and III

GO ON TO THE NEXT PAGE.

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50

( 1n ). Which of the following

57. For each positive integer n, let xn be a real number in the open interval 0, statements must be true? I. lim xn = 0 nƕ



II. If f is a continuous real-valued function defined on (0, 1) , then { f ( xn )}n =1 is a Cauchy sequence. III. If g is a uniformly continuous real-valued function defined on (0, 1) , then lim g ( xn ) exists. nƕ

(A) I only

(B) I and II only

(C) I and III only

(D) II and III only

(E) I, II, and III

58. A circular helix in xyz-space has the following parametric equations, where q Œ�.

x (q ) = 5 cos q y (q ) = 5 sin q z (q ) = q Let L (q ) be the arc length of the helix from the point P (q ) = ( x (q ) , y (q ) , z (q )) to the point (5, 0, 0 ) , and let

D (q ) be the distance between P (q ) and the origin. If L (q0 ) = 26, then D (q0 ) =

(A) 6

(B)

51

(C)

52

(D) 14 3

(E) 15 3

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52

59. Let A be a real 3 ¥ 3 matrix. Which of the following conditions does NOT imply that A is invertible? (A) - A is invertible.

( )

(B) There exists a positive integer k such that det Ak π 0.

0, where I is the 3 ¥ 3 identity matrix. (C) There exists a positive integer k such that ( I - A) = k

(D) The set of all vectors of the form Av, where v Œ �3 , is �3 . (E) There exist 3 linearly independent vectors v1 , v 2 , v3 Œ �3 such that Av i π 0 for each i.

60. A real-valued function f defined on � has the following property. For every positive number e, there exists a positive number d such that f ( x ) - f (1) ≥ e whenever x - 1 ≥ d.

This property is equivalent to which of the following statements about f ?

(A) f is continuous at x = 1. (B) f is discontinuous at x = 1. (C) f is unbounded. (D) lim f ( x ) = • x Æ•

(E)



Ú0

f ( x ) dx = •

GO ON TO THE NEXT PAGE.

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54

61. A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed? (A) 2

(B) 2 - e -2

(C) 2 + e -2

(D) 2 - e -4

(E) 2 + e -4

62. Let S be the subset of �2 consisting of all points ( x, y ) in the unit square [0, 1] ¥ [0, 1] for which x or y, or both, are irrational. With respect to the standard topology on �2, S is (A) closed (B) open (C) connected (D) totally disconnected (E) compact

GO ON TO THE NEXT PAGE.

55

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56

63. For any nonempty sets A and B of real numbers, let A � B be the set defined by

A � B ={ xy : x Œ A and y Œ B}. If A and B are nonempty bounded sets of real numbers and if sup ( A) > sup ( B ) , then sup ( A � B ) = (A) sup ( A) sup ( B ) (B) sup ( A) inf ( B ) (C) max {sup ( A) sup ( B ) , inf ( A) inf ( B )} (D) max {sup ( A) sup ( B ) , sup ( A) inf ( B )} (E) max {sup ( A) sup ( B ) , inf ( A) sup ( B ) , inf ( A) inf ( B )}

64. What is the value of the flux of the vector field F, defined on �3 by F ( x, y, z ) = xi + yj + zk, through the surface z = 1 - x 2 - y 2 oriented with upward-pointing normal vector field? (Note: i, j, and k are the standard basis vectors in �3 . ) (A) 0

(B)

2p 3

(C) p

(D)

4p 3

(E) 2 p

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57

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58

65. Let g be a differentiable function of two real variables, and let f be the function of a complex variable z defined by

= f ( z ) e x sin y + ig ( x, y ) , where x and y are the real and imaginary parts of z, respectively. If f is an analytic function on the complex plane, then g (3, 2 ) - g (1, 2 ) = (A) e2 (B) e2 (sin 3 - sin 1) (C) e2 (cos 3 - cos 1) (D) e - e3 sin 2

(E)

(e - e 3 ) cos 2

66. Let �17 be the ring of integers modulo 17, and let �17 ¥ be the group of units of �17 under multiplication. Which of the following are generators of �17 ¥ ? I. 5

II. 8

III. 16

(A) None

(B) I only

(C) II only

(D) III only

(E) I, II, and III

STOP

If you finish before time is called, you may check your work on this test.

59

SCRATCH WORK

60

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 sign your full name in this box:

PRINT: ___________________________________________________________________ (LAST) (FIRST) (MIDDLE) SIGN: ____________________________________________________________________ 6. TITLE CODE

Copy this code in box 6 on your answer sheet. Then fill in the corresponding ovals exactly as shown.

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

6 8 1 7 9 0

0

0

0

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Mathematics TEST NAME ___________________________________ GR1768 FORM CODE ____________________________________

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 concerned with 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 the number of questions you answer correctly. Questions you answer incorrectly or for which you mark no answer or more than one answer are counted as incorrect. Nothing is subtracted from a score if you answer a question incorrectly. Therefore, to maximize your score, it is better for you to guess at an answer than not to respond at all. You are advised to use your time effectively and to work as rapidly as you can without losing accuracy. Do not spend too much time 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 anything written in this examination book, but you may write in the book as much as you wish to work out your answers. After you have decided on your response to a question, fill in the corresponding oval on the answer sheet. BE SURE THAT EACH MARK IS DARK AND COMPLETELY FILLS THE OVAL. Mark only one answer to each question. No credit will be given for multiple answers. Erase all stray marks. If you change an answer, be sure that all previous marks are erased completely. Incomplete erasures may be read as intended answers. Do not be concerned that the answer sheet provides spaces for more answers than there are questions in the test. Example:

Sample Answer

What city is the capital of France? (A) (B) (C) (D) (E)

Rome Paris London Cairo Oslo

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

CORRECT ANSWER PROPERLY MARKED IMPROPER MARKS

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

®

Educational Testing Service Princeton, New Jersey 08541

61

Worksheet for the GRE Mathematics Test, Form GR1768 Answer Key and Percentages* of Test Takers Answering Each Question Correctly

QUESTION

P+

Number Answer 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

E C D A E C C B A A B B D B A B B A E E B C D E A D D D C A C E C B B

88 62 80 80 77 58 73 80 82 87 72 78 86 80 58 75 49 75 53 78 44 45 80 59 68 77 85 65 45 41 58 72 72 68 65

CORRECT RESPONSE

QUESTION

P+

Number Answer 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

C C C D C A E A B C C B B B D E A A C E D C B E D E C E E E B

CORRECT RESPONSE

49 31 37 63 64 56 43 27 44 35 67 52 43 30 24 43 53 22 53 33 38 36 38 27 25 36 32 41 24 53 27

Total Correct:__________________ Total Scaled:__________________

* The numbers in the P+ column indicate the percentages of test takers in the United States who answer each question correctly.

62 Page

GRE ® Mathematics Test Practice Book

Score Conversions for the GRE Mathematics Test, Form GR1768

Total Score Total Correct

Scaled Score

Total Correct

Scaled Score

63-66 61-62

910 900

59-60 58 57 56 55

890 880 870 860 850

33 32 31 30 29

590 580 570 560 550

54 53 52 51

840 830 820 810

28 27 26 25

540 520 510 500

50 49 48 47

790 780 770 760

24 22-23 21 20 19

490 480 470 460 450

46 45 44 43

740 730 720 710

18 17 16 15 14

440 430 420 410 400

42 41 40 39 38

690 680 670 660 650

12-13 11 9-10 8 7

390 380 370 360 350

37 36 35 34

630 620 610 600

6 4-5 3 2 1 0

340 330 320 310 300 280

GRE ® Mathematics Test Practice Book

63 Page

Q

O

G H I J K L M N

G H I J K L M N

G G G G G G G G G G G G G G G

H H H H H H H H H H H H H H H

I I I I I I I I I I I I I I I

J J J J J J J J J J J J J J J

K K K K K K K K K K K K K K K

L L L L L L L L L L L L L L L

M M M M M M M M M M M M M M M

N N N N N N N N N N N N N N N

SIGNATURE:





City

Center Number



Room Number

State or Province

ZIP or Postal Code

State or Province

P.O. Box or Street Address

Country

City



Country

CENTER:

(Print)



Last Name (Family or Surname) First Name (Given) M.I.

MAILING ADDRESS:

(Print)

2. YOUR NAME:



Z

Z

Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z



Y

Y

Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y



X

X

X X X X X X X X X X X X X X X



W

W

W W W W W W W W W W W W W W W



V

V

V V V V V V V V V V V V V V V



U

U

U U U U U U U U U U U U U U U



T

T

T T T T T T T T T T T T T T T



S

S

S S S S S S S S S S S S S S S



R

R

R R R R R R R R R R R R R R R



Q

Q

Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q

P

F

F

F F F F F F F F F F F F F F F

P

E

E

E E E E E E E E E E E E E E E

P P P P P P P P P P P P P P P

D

D

D D D D D D D D D D D D D D D

O

C

C

C C C C C C C C C C C C C C C

O

B

B

O O O O O O O O O O O O O O O

A

A

B B B B B B B B B B B B B B B

First Name Middle Initial Initial

A A A A A A A A A A A A A A A

Last Name only (Family Name or Surname) - First 15 Letters

Omit spaces, apostrophes, Jr., II., etc. Use only a pencil with soft, black lead (No. 2 or HB) to complete this answer sheet. Be sure to fill in completely the space that corresponds to your answer choice. Completely erase any errors or stray marks.

- SUBJECT TEST

2

April

6 7 8 9

Sept. Oct. Nov. Dec.

5

Aug.

4

3

2

1

0

July

June

3

1

May

0

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Day

Feb.

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9

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4

3

2

1

0

9

8

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5

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2

1

0

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3. DATE OF BIRTH

22 A B C D E 23 A B C D E

17 A B C D E 18 A B C D E 19 A B C D E 20 A B C D E 21 A B C D E

1 A B C D E 2 A B C D E 3 A B C D E 4 A B C D E 5 A B C D E 6 A B C D E 7 A B C D E 8 A B C D E 9 A B C D E 10 A B C D E 11 A B C D E 12 A B C D E 13 A B C D E 14 A B C D E 15 A B C D E 16 A B C D E

9 9 9 9

8 8 8 8

7 7 7 7

6 6 6 6

5 5 5 5

4 4 4 4

3 3 3 3

2 2 2 2

1 1 1 1

0 0 0 0

(U.S.A. only)

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

A B C D E

4. LAST FOUR DIGITS OF SOCIAL SECURITY NUMBER

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 107 A 108 A 109 A 110 A 111 A 112 A 113 A 114 A 115 A

84 A B C D E 85 A B C D E 86 A B C D E 87 A B C D E 88 A B C D E 89 A B C D E 90 A B C D E 91 A B C D E 92 A B C D E

9 9 9 9 9 9 9

8 8 8 8 8 8 8

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6 6 6 6 6 6 6

5 5 5 5 5 5 5

4 4 4 4 4 4 4

3 3 3 3 3 3 3

2 2 2 2 2 2 2

1 1 1 1 1 1 1

0 0 0 0 0 0 0

9 9 9 9 9

8 8 8 8 8

7 7 7 7 7

6 6 6 6 6

5 5 5 5 5

4 4 4 4 4

3 3 3 3 3

2 2 2 2 2

1 1 1 1 1

0 0 0 0 0

(on back cover of your test book)

6. TITLE CODE

1 2 3 4

SIDE 1

SHADED AREA FOR ETS USE ONLY

8. TEST BOOK SERIAL NUMBER (number in upper right corner of front cover of your test book)

your test book)

FORM CODE (on back cover of

7. TEST NAME (on back cover of your test book)

Item responses continued on reverse side.

B C D E

B C D E

B C D E

B C D E

B C D E

B C D E

B C D E

B C D E

B C D E

93 A B C D E 94 A B C D E 95 A B C D E 96 A B C D E 97 A B C D E 98 A B C D E 99 A B C D E 100 A B C D E 101 A B C D E 102 A B C D E 103 A B C D E 104 A B C D E 105 A B C D E 106 A B C D E

70 A B C D E 71 A B C D E 72 A B C D E 73 A B C D E 74 A B C D E 75 A B C D E 76 A B C D E 77 A B C D E 78 A B C D E 79 A B C D E 80 A B C D E 81 A B C D E 82 A B C D E 83 A B C D E

(from your admission ticket)

5. REGISTRATION NUMBER

47 A B C D E 48 A B C D E 49 A B C D E 50 A B C D E 51 A B C D E 52 A B C D E 53 A B C D E 54 A B C D E 55 A B C D E 56 A B C D E 57 A B C D E 58 A B C D E 59 A B C D E 60 A B C D E 61 A B C D E 62 A B C D E 63 A B C D E 64 A B C D E 65 A B C D E 66 A B C D E 67 A B C D E 68 A B C D E 69 A B C D E

YOU MAY FIND MORE RESPONSE SPACES THAN YOU NEED. IF SO, PLEASE LEAVE THEM BLANK.

BE SURE EACH MARK IS DARK AND COMPLETELY FILLS THE INTENDED SPACE AS ILLUSTRATED HERE: .

DO NOT USE INK

GRADUATE RECORD EXAMINATIONS ® -

763338

Enter your last name, first name initial (given name), and

1. NAME middle initial if you have one. Q3442/1-2/2 89176-02954 • TF212E70 Copyright ® 2012 by Educational Testing Service, Princeton, NJ 08541 All rights reserved. Printed in U.S.A. ®

64 Page

GRE ® Mathematics Test Practice Book

SIDE 2

CERTIFICATION STATEMENT Please write the following statement below, DO NOT PRINT. “I certify that I am the person whose name appears on this answer sheet. I also agree not to disclose the contents of the test I am taking today to anyone.” Sign and date where indicated.

SUBJECT TEST

COMPLETE THE CERTIFICATION STATEMENT, THEN TURN ANSWER SHEET OVER TO SIDE 1. SIGNATURE:

DATE:

C

D

E

148

A

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180

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C

D

145

A

B

C

D

146 147

A

B

C

A

B

C

A

B

C

D

E

A

B

C

D

E

TW

164 165 166 167 168

B B

173 174 175 176

141 142 143 144

TR

159 160 161 162 163

A A

TFS

FOR ETS USE ONLY

TCS

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

E

237 238 239 240

A

B

C

D

E

E

241

A

B

C

D

E

D

E

242

A

B

C

D

E

D

E

A

B

C

D

E

A

B

C

D

E

1R

1W

1FS

1CS

2R

2W

2FS

2CS

3R

3W

3FS

3CS

4R

4W

4FS

4CS

5R

5W

5FS

5CS

6R

6W

6FS

6CS

GRE ® Mathematics Test Practice Book

B. Sign your full name here:

B

A

Year

A. Fill in both circles here . . .

A

117 118 119 120 121 122 123 124 125 126

Day

To cancel your scores from this test administration, you must:

116

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IF YOU DO NOT WANT THIS ANSWER SHEET TO BE SCORED

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