ets gremath

Practicing to take the Mathematics Test

Two Actual Full-length GRE Mathematics Tests

Plus Additional Practice Questions

Strategies and Tips from the Test Maker

Test Prep From The Test Maker

/v 'T'C Educatiottal ~,.. Testing Service

The only source for authentic GRE test questions from beginning to end! POWERPREP Software: Test Preparation for the GRE General Test, Version 2.0 Contains TWO actual computer-adaptive tests, hundreds of questions from past tests, a comprehensive practice section for the new GRE Writing AssessmeQt, and study techniques to give you the most complete GRE test preparation program on the market. Hardware requirements: IBM-compatible computer, 486 or beHer processor, Windows 3.1/95/98/NT, 8 MB RAM, 20 MB hard disk space, CD-ROM drive, VGA monitor, mouse

GRE: Practicing to Take the General Test - 9th Edition Contains SEVEN actual GRE General Tests (different from those in POWERPREP) including one test complete with explanations, test-taking strategies, and score conversion tables. Also includes the content from the Math Review Booklet.

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Practicing to take the Mathematics Test

3rd edition Two Actual Full-length GRE Mathematics Tests

Plus Additional Practice Questions Strategies and Tips from the Test Maker

T A B L E O F C O N T E N T S

B A C K G R O U N D F O R T H E T E S T

H o w t o U s e T h i s P u b l i c a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

A d d i t i o n a l I n f o r m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

P u r p o s e o f t h e G R E S u b j e c t T e s t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

D e v e l o p m e n t o f t h e G R E M a t h e m a t i c s T e s t . . . . . . . . . . . . . . . . . . . . . . . . . . 6

C o n t e n t o f t h e G R E M a t h e m a t i c s T e s t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

S a m p l e Q u e s t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

T A K I N G T H E T E S T

P r e p a r i n g f o r t h e G R E M a t h e m a t i c s T e s t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3

T e s t - t a k i n g S t r a t e g i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3

H o w t o S c o r e Y o u r T e s t ( G R 9 3 6 7 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4

E v a l u a t i n g Y o u r P e r f o r m a n c e ( G R 9 3 6 7 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7

P r a c t i c e G R E M a t h e m a t i c s T e s t , F o r m G R 9 3 6 7 . . . . . . . . . . . . . . . . . . . . . . 2 9

H o w t o S c o r e Y o u r T e s t ( G R 9 7 6 7 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1

E v a l u a t i n g Y o u r P e r f o r m a n c e ( G R 9 7 6 7 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4

P r a c t i c e G R E M a t h e m a t i c s T e s t , F o r m G R 9 7 6 7 . . . . . . . . . . . . . . . . . . . . . . 8 5

A n s w e r S h e e t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 1

3

4

BACKGROUND FOR~THE TEST HOW TO USE THIS PUBLICATION

This practice book has been published on behalf of the Graduate Record Exami-

nations Board to help potential graduate students prepare to take the GRE

Mathematics Test. The book contains two actual GRE Mathematics Tests and

includes information about the purpose of the GRE Subject Tests, along with a section of sample questions, a detailed description of the content specifications

for the GRE Mathematics Test, and a description of the procedures for develop-

ing the test. All test questions that were scored have been included in the

practice test. The sample questions included in this practice book are organized by content

category and represent the types of questions included in the test. The purpose of

these questions is to provide some indication of the range of topics covered in the

test as well as to provide some additional questions for practice purposes. These

questions do not represent either the length of the actual test or the proportion of

actual test questions within each of the content categories. Before you take a full-length test, you may want to answer the sample ques-

tions. A suggested time limit is provided to give you a rough idea of how much

time you would have to complete the sample questions if you were answering

them on an actual timed test. After answering the sample questions, evaluate your

performance within content categories to determine whether you would benefit

by reviewing certain courses. This practice book contains two complete test books, including the general

instructions printed on the back covers of the tests. When you take the test at the

test center, you will be given time to read these instructions. They show you how

to mark your answer sheet properly and give you advice about guessing.

Try to take these practice tests under conditions that simulate those in an

actual test administration. Use the answer sheets provided on pages 131 to 136

and mark your answers with a No.2 (soft-lead) pencil as you will do at the test center. Give yourself 2 hours and 50 minutes in a quiet place and work through

the tests without interruption, focusing your attention on the questions with the

same concentration you would use in taking a test to earn a score. Since you will

not be permitted to use them at the test center, do not use keyboards, dictionaries

or other books, compasses, pamphlets, protractors, highlighter pens, rulers, slide

rules, calculators (including watch calculators), stereos or radios with head-phones, watch alarms (including those with flashing lights or alarm sounds), or paper of any kind.

After you complete each practice test, use the work sheets and conversion

tables on pages 25-26 and 82-83 to score your tests. The work sheets also show

the estimated percent of a sample of GRE Mathematics Test examinees who

answered each question correctly. This will enable you to compare your perfor-

mance on the questions with theirs. Evaluating your performance on the actual

test questions, as well as the sample questions, should help you determine

whether you would benefit by further reviewing certain courses before taking the

test at the test center. We believe that if you use this practice book as we have suggested, you will

be able to approach the testing experience with increased confidence.

A D D I T I O N A L I N F O R M A T I O N

I f y o u h a v e a n y q u e s t i o n s a b o u t a n y o f t h e i n f o r m a t i o n i n t h i s b o o k , p l e a s e s e n d

a n e m a i l m e s s a g e t o t h e G R E P r o g r a m a t g r e - i n f o @ e t s . o r g , o r w r i t e t o :

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

E d u c a t i o n a l T e s t i n g S e r v i c e

P . O . B o x 6 0 0 0

P r i n c e t o n , N J 0 8 5 4 1 - 6 0 0 0

P U R P O S E O F T H E G R E S U B J E C T T E S T S

T h e G R E S u b j e c t T e s t s a r e d e s i g n e d t o h e l p g r a d u a t e s c h o o l a d m i s s i o n c o m m i t -

t e e s a n d f e l l o w s h i p s p o n s o r s a s s e s s t h e q u a l i f i c a t i o n s o f a p p l i c a n t s i n t h e i r

s u b j e c t f i e l d s . T h e t e s t s a l s o p r o v i d e s t u d e n t s w i t h a n a s s e s s m e n t o f t h e i r o w n

q u a l i f i c a t i o n s .

S c o r e s o n t h e t e s t s a r e i n t e n d e d t o i n d i c a t e k n o w l e d g e o f t h e s u b j e c t m a t t e r

e m p h a s i z e d i n m a n y u n d e r g r a d u a t e p r o g r a m s a s p r e p a r a t i o n f o r g r a d u a t e s t u d y .

S i n c e p a s t a c h i e v e m e n t i s u s u a l l y a g o o d i n d i c a t o r o f f u t u r e p e r f o r m a n c e , t h e

s c o r e s a r e h e l p f u l i n p r e d i c t i n g s u c c e s s i n g r a d u a t e s t u d y . B e c a u s e t h e t e s t s a r e

s t a n d a r d i z e d , t h e t e s t s c o r e s p e r m i t c o m p a r i s o n o f s t u d e n t s f r o m d i f f e r e n t i n s t i t u -

t i o n s w i t h d i f f e r e n t u n d e r g r a d u a t e p r o g r a m s .

T h e 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 B o a r d r e c o m m e n d s t h a t s c o r e s o n t h e

S u b j e c t T e s t s b e c o n s i d e r e d i n c o n j u n c t i o n w i t h o t h e r r e l e v a n t i n f o r m a t i o n a b o u t

a p p l i c a n t s . B e c a u s e n u m e r o u s f a c t o r s i n f l u e n c e s u c c e s s i n g r a d u a t e s c h o o l ,

r e l i a n c e o n a s i n g l e m e a s u r e t o p r e d i c t s u c c e s s i s n o t a d v i s a b l e . O t h e r i n d i c a t o r s

o f c o m p e t e n c e t y p i c a l l y i n c l u d e u n d e r g r a d u a t e t r a n s c r i p t s s h o w i n g c o u r s e s t a k e n

a n d g r a d e s e a r n e d , l e t t e r s o f r e c o m m e n d a t i o n , a n d G R E G e n e r a l T e s t s c o r e s .

F o r i n f o r m a t i o n a b o u t t h e a p p r o p r i a t e u s e o f G R E s c o r e s , v i s i t t h e G R E

W e b s i t e a t w w w . g r e . o r g o r w r i t e t o t h e G R E P r o g r a m , E d u c a t i o n a l T e s t i n g

S e r v i c e , M a i l s t o p 0 6 - L , P r i n c e t o n , N J 0 8 5 4 1 .

5

6

DEVELOPMENT OF THE GRE MATHEMATICS TEST

Each new edition of the GRE Mathematics Test is developed by a committee of examiners composed of professors in the subject who are on undergraduate and graduate faculties in different types of institutions and in different regions of the United States. In selecting members for the committee of examiners, the GRE Program seeks the advice of the Mathematical Association of America.

The content and scope of each test are specified and reviewed periodically by the committee of examiners. Test questions are written by the committee and by other faculty who are also subject-matter specialists and by subject-matter specialists at ETS. All questions proposed for the test are reviewed by the com-mittee and revised as necessary. The accepted questions are assembled into a test in accordance with the content specifications developed by the committee to ensure adequate coverage of the various aspects of the field and at the same time to prevent overemphasis on any single topic. The entire test is then reviewed and approved by the committee.

Subject-matter and measurement specialists on the ETS staff assist the committee, providing information and advice about methods of test construction and helping to prepare the questions and assemble the test. In addition, individual test questions and the test as a whole are reviewed to eliminate language, sym-bols, or content considered to be potentially offensive, inappropriate for major subgroups of the test-taking population, or serving to perpetuate any negative attitude that may be conveyed to these subgroups. The test as a whole is also reviewed to make sure that the test questions, where applicable, include an appropriate balance of people in different groups and different roles.

Because of the diversity of undergraduate curricula in mathematics, it is not possible for a single test to cover all the material an examinee may have studied. The examiners, therefore, select questions that test the basic know ledge and understanding most important for successful graduate study in the particular field. The committee keeps the test up-to-date by regularly developing new editions and revising existing editions. In this way, the test content changes steadily but gradually, much like most curricula. In addition, curriculum surveys are conducted periodically to ensure that the content of a test reflects what is currently being taught in the undergraduate curriculum.

After a new edition of a GRE Mathematics Test is first administered, examin-ees' responses to each test question are analyzed in a variety of ways to determine whether each question functioned as expected. These analyses may reveal that a question is ambiguous, requires knowledge beyond the scope of the test, or is inappropriate for the total group or a particular subgroup of examinees taking the test. Answers to such questions are not used in computing scores.

,,

C O N T E N T O F T H E

G R E M A T H E M A T I C S T E S T

T h e t e s t c o n s i s t s o f 6 6 m u l t i p l e - c h o i c e q u e s t i o n s , d r a w n f r o m c o u r s e s c o m m o n l y

o f f e r e d a t t h e u n d e r g r a d u a t e l e v e l . A p p r o x i m a t e l y 5 0 p e r c e n t o f t h e q u e s t i o n s

i n v o l v e c a l c u l u s a n d i t s a p p l i c a t i o n s - s u b j e c t m a t t e r t h a t c a n b e a s s u m e d t o b e

c o m m o n t o t h e b a c k g r o u n d s o f a l m o s t a l l m a t h e m a t i c s m a j o r s . A b o u t 2 5 p e r c e n t

o f t h e q u e s t i o n s i n t h e t e s t a r e i n e l e m e n t a r y a l g e b r a , l i n e a r a l g e b r a , a b s t r a c t

a l g e b r a , a n d n u m b e r t h e o r y . T h e r e m a i n i n g q u e s t i o n s d e a l w i t h o t h e r a r e a s o f

m a t h e m a t i c s c u r r e n t l y s t u d i e d b y u n d e r g r a d u a t e s i n m a n y i n s t i t u t i o n s .

T h e f o l l o w i n g c o n t e n t d e s c r i p t i o n s m a y a s s i s t s t u d e n t s i n p r e p a r i n g f o r t h e

t e s t . T h e p e r c e n t a g e s g i v e n a r e e s t i m a t e s ; a c t u a l p e r c e n t a g e s w i l l v a r y s l i g h t l y

f r o m o n e e d i t i o n o f t h e t e s t t o a n o t h e r .

C a l c u l u s ( 5 0 o / o )

T h e u s u a l m a t e r i a l o f t w o y e a r s o f c a l c u l u s , i n c l u d i n g t r i g o n o m e t r y , c o o r d i n a t e

g e o m e t r y , i n t r o d u c t o r y d i f f e r e n t i a l e q u a t i o n s , a n d a p p l i c a t i o n s b a s e d o n t h e

c a l c u l u s .

A l g e b r a ( 2 5 / o )

E l e m e n t a r y a l g e b r a : t h e k i n d o f a l g e b r a t a u g h t i n p r e c a l c u l u s c o u r s e s .

L i n e a r a l g e b r a : m a t r i c e s , l i n e a r t r a n s f o r m a t i o n s , c h a r a c t e r i s t i c p o l y n o m i a l s ,

e i g e n v e c t o r s , a n d o t h e r s t a n d a r d m a t e r i a l

A b s t r a c t a l g e b r a a n d n u m b e r t h e o r y : t o p i c s f r o m t h e e l e m e n t a r y t h e o r y o f

g r o u p s , r i n g s , a n d f i e l d s ; e l e m e n t a r y t o p i c s f r o m n u m b e r t h e o r y

A d d i t i o n a l T o p i c s ( 2 5 o / o )

I n t r o d u c t o r y r e a l v a r i a b l e t h e o r y : t h e e l e m e n t a r y t o p o l o g y o f i l l a n d i l l n ;

p r o p e r t i e s o f c o n t i n u o u s f u n c t i o n s : d i f f e r e n t i a b i l i t y a n d i n t e g r a b i l i t y

O t h e r t o p i c s : g e n e r a l t o p o l o g y , c o m p l e x v a r i a b l e s , p r o b a b i l i t y a n d s t a t i s t i c s ,

s e t t h e o r y a n d l o g i c , c o m b i n a t o r i c s a n d d i s c r e t e m a t h e m a t i c s , a l g o r i t h m s a n d

n u m e r i c a l a n a l y s i s

T h e r e m a y a l s o b e q u e s t i o n s t h a t a s k t h e t e s t t a k e r t o m a t c h " r e a l - l i f e "

s i t u a t i o n s t o a p p r o p r i a t e m a t h e m a t i c a l m o d e l s .

T h e a b o v e d e s c r i p t i o n s o f t o p i c s c o v e r e d i n t h e t e s t s h o u l d n o t b e c o n s i d e r e d

e x h a u s t i v e ; i t i s n e c e s s a r y t o u n d e r s t a n d m a n y o t h e r r e l a t e d c o n c e p t s . K n o w l e d g e

o f t h e m a t e r i a l i n c l u d e d i n t h e d e s c r i p t i o n s i s a n e c e s s a r y , b u t n o t s u f f i c i e n t ,

c o n d i t i o n f o r a n s w e r i n g t h e q u e s t i o n s o n t h e t e s t . P r o s p e c t i v e t e s t t a k e r s s h o u l d

b e a w a r e t h a t q u e s t i o n s r e q u i r i n g n o m o r e t h a n a g o o d p r e c a l c u l u s b a c k g r o u n d

m a y b e q u i t e c h a l l e n g i n g ; s o m e o f t h e s e q u e s t i o n s t u r n o u t t o b e a m o n g t h e m o s t

d i f f i c u l t q u e s t i o n s o n t h e t e s t . I n g e n e r a l , t h e q u e s t i o n s a r e i n t e n d e d n o t o n l y t o

t e s t r e c a l l o f i n f o r m a t i o n , b u t a l s o t o a s s e s s t h e t e s t t a k e r ' s u n d e r s t a n d i n g o f

f u n d a m e n t a l c o n c e p t s a n d t h e a b i l i t y t o a p p l y t h e s e c o n c e p t s i n v a r i o u s s i t u a t i o n s .

7

SCRATCHWORK ~

8

SAMPLE QUESTIONS The sample questions included in this practice book represent the types of questions included in the test. The purpose of the sample questions is to provide some indication of the range of topics covered in the test as well as to provide some additional questions for practice purposes. These questions do not represent either the length of the actual test or the proportion of actual test questions within each of the content categories. A time limit of 140 minutes is suggested to give you a rough idea of how much time you would have to complete the sample questions if you were answering them on an actual timed test. Correct answers to the sample questions are listed on page 22.

Directions: Each of the questions or incomplete statements is followed by five suggested answers or completions. Select the one that is best in each case.

CALCULUS

I. If S is a plane in Euclidean 3-space containing (0, 0, 0), (2, 0, 0), and (0, 0, I), then S is the

(A) xy-plane (8) xz-plane (C) yz-plane (D) plane y - z = 0 (E) plane x + 2y - 2z = 0

2. J~J; xy dy dx (A) 0 (C) ! 3

4. All functions f defined on the xy-plane such that of = 2x + y and of = X + 2y ax oy

are given by f(x, y) =

(D) 1 (E) 3

(A) x 2 + xy + y 2 + C (D) x 2 + 2xy + y 2 + C

(B) x 2 - xy + y 2 + C (E) x 2 - 2xy + y 2 + C

(C) x 2 - xy - y 2 + C

9

y

5. Which of the following could be the graph of the derivative of the function whose graph is shown in the figure above?

(A) y (8) y (C) y

(D) y (E) y

y

( 1,3)

(2, 2)

(-1,1)

6. Which of the following integrals represents the area of the shaded portion of the rectangle shown in the figure

above?

(A) f 1 (x + 2 - lxl) dx

(D) f, lxl dx

oc n 7. L n+t =

n=J

(B) log 2

10

(B) f 1 (lxl + X + 2) dx

(E) f, 2 dx

(C) 1 (D) e

(C) J~ 1 (x + 2) dx

(E) +oo

8. If sin- 1 x = ~, then the acute angle value of cos- 1 x ts

(A) 5n 6

(A) n

(B) ~ 3

(B) en

~ (C) ~ - 62 7[ (D) 1 - 6

10. Which of the following is true of the behavior of f( x) = :! : + 8 as x -+ 2 ? X -4

(A) The limit is 0. (B) The limit is 1. (C) The limit is 4. (D) The graph of the function has a vertical asymptote at 2. (E) The function has unequal, finite left-hand and right-hand limits.

(E) 0

(E) en - 1

11. Suppose that an arrow is shot from a point p and lands at a point q such that at one and only one point in its -flight is the arrow parallel to the line of sight between p and q. Of the following, which is the best mathematical

model for the phenomenon described above?

(A) A function f differentiable on [a, b] such that there is one and only one point c in [a. b] with b

Ja f'(x) dx = c(b - a)

(B) A function f whose second derivative is at all points negative such that there is one and only one point c in [a, b] with f'(c) = f(b~ =~(a)

(C) A function f whose first derivative is at all points positive such that there is one and only one point c b

in [a, b] with Jaf(x) dx = f(c) (b - a)

(D) A function f continuous on [a, b] such that there is one and only one point c in [a, b] with b

Jaf(x) dx = f(c) (b - a)

(E) A function f continuous on [a, b] and f(a) < d < f(b) such that there is one and only one point c in [a, b] with f(c) = d

12. If c > 0 and f(x) = ex - ex for all real numbers x, then the minimum value off is

(A) f(c)

13. For all x > 0, if f(log x) = Jx, then f(x) = X 1

(A) e 2 (B) logJx (C) efi

(D) f(log c)

(D) jiV

(E) nonexistent

(E) log x 2

11

rl (rsiny 1 ) 14. J, J, !1=---:-2 dx dy = 0 0 v I - X

(A) ! (B) l 3 2 (C) ~ 4 (D)

15. For what triples of real numbers (a, b, c) with a :f. 0 is the function

{X, if X ~ I

defined by f(x) = 2 . ax + bx + c, If x > I

differentiable at all real x ?

(A) {(a, - 2a, a): a is a nonzero real number}

(B) {(a, - 2a, c): a, c are real numbers and a :f. 0}

(C) {(a, b, c): a, b, c are real numbers, a :f. 0, and a + b + c = 1}

(D) {0, 0, 0 )} (E) {(a, I - 2a, 0): a is a nonzero real number}

Questions 16-18 are based on the following information.

(E) ~ 3

Let f be a function such that the graph off is a semicircle S with endpoints (a, 0) and (b, 0) where a < b.

(A) f(b) - f(a) f(b) - f(a)

(B) b - a TC (C) (b - a) 4

17. The graph of y = 3 f(x) IS a

(A) translation of S (D) subset of a parabola

(B) semicircle with radius three times that of S (E) subset of a hyperbola

b 18. The improper integral Ja f(x)f'(x)dx is

(A) necessarily zero (B) possibly zero but not necessarily (C) necessarily nonexistent (D) possibly nonexistent but not necessarily (E) none of the above

--1t -t 19 lim e . - e . =

x-+n sm x

(A) - oo

12

(C) 0 (D) e n

(E) (b -a)2 ~

(C) subset of an ellipse

(E) 1

y

20. The shaded region in the figure above indicates the graph of which of the following?

(A) x2 < y and y < ! X

1 (D) x 2 > y or y > -X

1 (B) x2 < y or y < -X (E) x 2 < y and xy < 1

'f 21. The shortest dis~a!!~ from the curve xy = 8 to the origin is (A) 4 (B) 8 (C) 16 (D) 2~

22. If f(x) = ( l;l' for x # 0 then (f(x) dx is 0, for x = 0,

(A) -2 (B) 0 (C) 2 (E) none of the above

1 (C) x 2 > y and y > -X

(E) 4~

(D) not defined

23. Let y = f(x) be a solution of the differential equation x dy + (y - xex) dx = 0 such that y = 0 when x = 1. What is the value of /(2) ?

(A) j_ 2e 2 (C)~ 2 (D) 2e

13

24. Of the following, which best represents a portion of the graph of y = _!_ + x - ! near ( 1, 1) ? ex ~ e

(A) y

(D) y 1

) (B) y

I

-0~------+_,... X

(E) y I

-+------+-+-X -t------+- X 0 0

25. In xyz-space, the degree measure of the angle between the rays

Z =X~ 0, y = 0 and

z = y ~ 0, x = 0 is

(B) 30

(C) y

~+------+- X 0

(E) 90

26. Suppose f is a real function such that f'(x0) exists. Which of the following is the value of . f(xo + h) - f(x0 - h) ?

hm h . h-t>O

(A) 0 (B) 2f'(x0) (C)/'( -xo)

oo n 27. The radius of convergence of the series L ~! xn is

n=O

(A) 0 (C)

28. In the xy-plane, the graph of xlog Y = ylog x is

(A) empty (B) a single point (D) a closed curve (E) the open first quadrant

14

(D) - f'(xo) (E) - 2f'(x0)

(D) e (E) +oo

(C) a ray in the open first quadrant

29. Let x 1 = I and Xn+ 1 = J3 + 2xn for all positive integers n. If it is assumed that {xn} converges, then lim Xn =

n-.oo

(A) -I (B) 0 (C)~ (D) e (E) 3

30. Let f(x, y) = x 3 + y 3 + 3xy for all real x and y. Then there exist distinct points P and Q such that f has a (A) local maximum at P and at Q (B) saddle point at P and at Q (C) local maximum at P and a saddle point at Q (D) local minimum at P and a saddle point at Q (E) local minimum at P and at Q

31. The polynomial p(x) = I + 1 (x - I) - k (x - I)2 is used to approximate y'tm. Which of the following most closely approximates the error y'tm - p ( 1.0 I) ?

(A) (I~) x 10-6 (D) -(i) x 10-10

(B) (Js) x w- (C) (i) x 10-10 (E) -U6) x w-

y

(0, I) ....,__,....._____,(1, I)

s

0 .....___ ... :--"--~X (I, 0)

32. If B is the boundary of S as indicated in the figure above, then S (3 ydx + 4xdy) = B

(A) 0 (B) I (C) 3 (D) 4 (E) 7

33. Let f be a continuous, strictly decreasing, real-valued function such that So+ 00

f(x) dx is finite and /(0) = I. +OO

In terms of f- 1 (the inverse function of f), So f(x) dx is

(A) less than S1+

00 f- 1 (y) dy

(D) equal to S~ f- 1 (y) dy (B) greater than S~ f- 1 (y) dy

+00

(E) equal to So f- 1 (y) dy

(C) equal to S1+

00

/-1 (y) dy

15

34. Which of the following indicates the graphs of two functions that satisfy the differential equation

( dy)2 + 2ydy + y2 = 0? dx dx

(A) y (B) y (C) y

(D) y (E) y

ALGEBRA

35. If a, b, and c are real numbers, which of the following are necessarily true? I 1 I. If a < b and ab # 0, then a > b .

II. If a < b, then ac -b.

(A) I only (B) I and III only (C) III and IV only (D) II, HI, and IV only (E) I, II, III, and IV

36. In order to send an undetected message to an agent in the field, each letter in the message is replaced by the

number of its position in the alphabet and that number is entered in a matrix M. Thus, for example, "DEAD"

16

becomes the matrix M = ( 1 ~). In order to further avoid detection, each message with four letters is sent to the agent encoded as M C, where C = ( i -:) . If the agent receives the matrix G: :::: ~). then the message is (A) RUSH (B) COME (C) ROME (D) CALL (E) not uniquely determined by the information given

37. Iff is a linear transformation from the plane to the real numbers and if /(I, 1) = 1 and /( -1, 0) = 2, then /(3, 5) =

(A) -6 (B) -5 (C) 0 (D) 8 (E) 9

38. Let be the binary operation on the rational numbers given by a b = a + b + 2ab. Which of the following are true?

I. is commutative. II. There is a rational number that is a -identity.

III. Every rational number has a -inverse.

(A) I only (B) II only (C) I and II only (D) I and III only

39. A group G in which (ab )2 = a2b2 for all a, b in G is necessarily

(A) finite (8) cyclic (C) of order two (D) abelian (E) none of the above

(E) I, II, and III

40. Suppose that /(I + x) = f(x) for all real x. If f is a polynomial and /( 5) = II, then tCi) is (A) -11 (8) 0 (C) 11 (D) 3i (E) not uniquely determined by the information given

41. Let x and y be positive integers such that 3x + 1y is divisible by 11. Which of the following must also be divisible by II ?

(A) 4x + 6y (8) X + y + 5 (C) 9x + 4y (D) 4x - 9y (E) X + y - I

42. The dimension of the subspace spanned by the real vectors

m. m. m. m. rn m is (A) 2 (B) 3 (C) 4 (D) 5 (E) 6

43. The rank of the matrix

(~~ 2 3 4 I~) 7 8 9 12 13 14 15 is 16 17 18 19 20 21 22 23 24 25

(A) I (B) 21 (C) 3 (D) 4 (E) 5

17

44. If M is the matrix (~ 6 ~) , then M 100 is 1 0 0

(0 1 0) (A) 0 0 1 1 0 0 (0 0 1) (B) 1 0 0 0 1 0

(E) none of the above

( 1 0 0) (C) 0 1 0 0 0 1 ( 0 0 0) (D) 0 0 0 0 0 0

45. If a polynomial f(x) over the real numbers has the complex numbers 2 + i and 1 - i as roots, then f(x) could be

(A) x 4 + 6x 3 + 10 (D) x 3 + 5x 2 + 4x +

(B) x 4 + 7x 2 + 10 (E) x 4 - 6x 3 + 15x2 - 18x + 10

(C) x 3 - x 2 + 4x + I

46. Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x)-+xp(x) and S:p(x)-+p'(x) = Jxp(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?

(A) ST = 0 (B) ST = T (C) ST = TS (D) ST - TS is the identity map of V onto itself. (E) ST + TS is the identity map of V onto itself.

47. If the finite group G contains a subgroup of order seven but no element (other than the identity) is its own inverse, then the order of G could be

(A) 27 (B) 28 (C) 35 (D) 37 (E) 42

48. Which of the following is the larger of the eigenvalues (characteristic values) of the matrix ( ~ ~) ? (A) 4 (B) 5 (C) 6 (D) 10 (E) 12

49. Let V be the vector space, under the usual operations, of real polynomials that are of degree at most 3. Let W be the subspace of all polynomials p(x) in V such that p(O) = p(l) = p( -1) = 0. Then dim V + dim W is (A) 4 (B) 5 (C) 6 (D) 7 (E) 8

50. The map x --+ axa 2 of a group G into itself is a homomorphism if and only if

(A) G is abelian (B) G = {e} (C) a = e (D) a2 =a (E) a3 = e

18

51. Let I :1: A :1: -I, where I is the identity matrix and A is a real 2 x 2 matrix. If A = A - 1, then the trace of A is

(A) 2 (B) 1 (C) 0 (D) -1

52. Which of the following subsets are subrings of the ring of real numbers?

I. {a + bj2: a and b are rational}

II. { 3':., : n is an integer and m is a non-negative integer}

III. {a + b.j5: a and b are real numbers and a2 + h2 ~ 1}

(E) -2

(A) I only (B) I and II only (C) I and III only (D) II and III only (E) I, II, and III

ADDITIONAL TOPICS

53. k digits are to be chosen at random (with repetitions allowed) from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. What is the probability that 0 will not be chosen?

I (B) 10 (C) k - 1 k

54. S(n) is a statement about positive integers n such that whenever S(k) is true, S(k + 1) must also be true. Furthermore, there exists some positive integer n0 such that S(n0) is not true. Of the following, which is the strongest conclusion that can be drawn?

(A) S(n0 + 1) is not true. (B) S(n0 - I) is not true. (C) S(n) is not true for any n ~ n0 (D) S(n) is not true for any n ~ n0 (E) S(n) is not true for any n.

55. Let f and g be functions defined on the positive integers and related in the following way:

and {

1, if n = 1 f(n) = 2/(n - 1), if n :1: 1

Gg(n + 1 ), if n #: 3 g(n) = . (n), tf n = 3. The value of g(1) is

(A) 6 (B) 9 (C) 12 (D) 36 (E) not uniquely determined by the information given

19

56. If k is a real number and

f(x) ={sin ~ for x =I= 0 k for x = 0

and if the graph of f is not a connected subset of the plane, then the value of k (A) could be - 1 (B) must be 0 (C) must be 1 (D) could be less than 1 and greater than - 1 (E) must be less than - 1 or greater than 1

57. What is wrong with the following argument?

Let R be the real numbers. (1) "For all x, y R, f(x) + f(y) = f(xy)."

is equivalent to (2) "For all x, y R, f( -x) + f(y) = /(( -x)y)."

which is equivalent to (3) "For all x, y R, f( -x) + f(y) = f(( -x)y) = f(x(- y)) = f(x) + f(- y)."

From this for y = 0, we make the conclusion (4) "For all x R, f( -x) = f(x)."

Since the steps are reversible, any function with property (4) has property (1). Therefore, for all x, y R, cos x + cos y = cos(xy ).

(A) (2) does not imply (1). (D) (4) does not imply (3).

(B) (3) does not imply (2). (E) ( 4) is not true for f = cos.

(C) (3) does not imply (4).

58. Suppose that the space S contains exactly eight points. If 1J is a collection of 250 distinct subsets of S, which of the following statements must be true?

20

(A) S is an element of }J.

(B) (IG = S Gc}J

(C) (IG is a nonempty proper subset of S. Gc}J

(D) }J has a member that contains exactly one element.

(E) The empty set is an element of 1J.

59. In a game two players take turns tossing a fair coin; the winner is the first one to toss a head. The probability that the player who makes the first toss wins the game is

(D)~

60. Acceptable input for a certain pocket calculator is a finite sequence of characters each of which is either a digit or a sign. The first character must be a digit, the last chara

ANSWER KEYFORSAMPLE QUESTIONS f'

Calculus 1. B 18. A 2. B 19. B 3. A 20. E 4. A 21. A 5. c 22. B 6. A 23. c 7. E 24. D 8. B 25. D 9. B 26. B

10. D 27. E 11. B 28. E 12. D 29. E 13. A 30. c 14. B 31. A 15. A 32. B 16. E 33. D 17. c 34. A

Algebra Additional Topics 35. c 53. E 36. c 54. c 37. E 55. D 38. c 56. E 39. D 57. D 40. c 58. D 41. D 59. D 42. B 60. c 43. B 61. B 44. A 62. c 45. E 63. c 46. D 47. c 48. c 49. B 50. E 51. c 52. B

22

L

TAKING THE TEST PREPARING FOR THE GRE MATHEMATICS TEST

ORE Mathematics 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. The following information will help guide you if you decide to spend time preparing for the test.

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 official ORE publications, published by ETS, to familiarize yourself with questions used on the ORE Mathematics Test.

Become familiar with the types of questions used in the 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

When you take the test, you will be marking your answers on a separate machine-scorable answer sheet. Total testing time is two hours and 50 minutes; there are no separately timed sections. 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, you should 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 quite rapidly, 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.

Questions for which you mark no answer or more than one answer are not counted in scoring.

As a correction for haphazard guessing, one-fourth of the number of ques-tions you answer incorrectly are subtracted from the number of questions you answer correctly. It is improbable that mere guessing will improve your score

i

significantly; it may even lower your score. If, however, you are not certain of the correct answer but have some knowledge of the question and are able 23

24

to eliminate one or more of the answer choices, your chance of getting the right answer is improved, and it may be to your advantage to answer such a question.

Record all answers on your answer sheet. Answers recorded in your test book will not be counted.

Do not wait until the last five minutes of a testing session to record answers on your answer sheet.

HOW TO SCORE YOUR TEST (GR9367) Total Subject Test scores are reported as three-digit scaled scores with the third digit always zero. The maximum possible range for all Subject Test total scores is from 200 to 990. The actual range of scores for a particular Subject Test, however, may be smaller. The possible range for GRE Mathematics Test scores is from 400 to 990. The range for different editions of a given test may vary because different editions are not of precisely the same difficulty. The differences in ranges among different editions of a given test, however, usually are small. This should be taken into account, especially when comparing two very high scores. In general, differences between scores at the 99th percentile should be ignored. The score conversion table provided shows the score range for this edition of the test only.

The work sheet on page 25 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. This is the adjust-ment made for guessing. Then round the result to the nearest whole number. This will give you your raw total score. Use the total score conversion table to find the scaled total score that corresponds to your raw total score.

Example: Suppose you chose the correct answers to 48 questions and incor-rect answers to 15. Dividing 15 by 4 yields 3.75. Subtracting 3.75 from 48 equals 44.25, which is rounded to 44. The raw score of 44 corresponds to a scaled score of870.

W O R K S H E E T f o r t h e G R E M a t h e m a t i c s T e s t , F o r m G R 9 3 6 7

A n s w e r K e y a n d P e r c e n t a g e s * o f E x a m i n e e s A n s w e r i n g E a c h Q u e s t i o n C o r r e c d y

Q U E S T I O N

T O T A L Q U E S T I O N

T O T A L

N u m b e r A n s w e r

P +

c I

N u m b e r A n s w e r

P +

c I

1 D

9 6

3 6

D 7 0

2

c 9 5

3 7 B

6 6

3 A

8 4

3 8

D 2 3

4

c

7 6 3 9 D

6 9

5 A 7 0

4 0 A

4 0

6

B

8 6

4 1 B

7 5

7 D 8 0

4 2 D 3 1

8

B 7 6

4 3 E

1 7

9

E 7 4

4 4 D 3 1

1 0 A 9 4 4 5 c 5 3

1 1

c 8 2

4 6 E 8 3

1 2 E 6 9

4 7

c

1 8

1 3 A 6 5

4 8 D 5 8

1 4 D 6 2

4 9 D 2 8

1 5 D

5 6

5 0

A 3 8

1 6 A 6 3

5 1 B 3 7

1 7 E 6 9

5 2 D

7 2

1 8 D 4 8 5 3

A 4 9

1 9 A

5 7 5 4 c

5 7

2 0

B 5 2 5 5

E 2 8

2 1 D 7 9 5 6

B

1 7

2 2 E

7 3

5 7

c

7 2

2 3 c

6 1 5 8 c 6 5

2 4 A

5 1 5 9 A

4 9

2 5

D 3 3 6 0

D 2 2

2 6

c

6 9

6 1

c

5 2

2 7 A 5 9

6 2 B 1 0

2 8 B

4 5 6 3 D

4 9

2 9 E 6 8

6 4

c

5 0

3 0

E 5 6

6 5

A

6 0

3 1 c

6 4

6 6

B 5 3

3 2 B 3 6

3 3 c 8 8

3 4 E 6 0

3 5

B 4 5

C o r r e c t ( C ) _ _ _ _ _

I n c o r r e c t ( I ) _ _ _ _ _

T o t a l S c o r e :

c - 1 / 4 = - - - - -

S c a l e d S c o r e ( S S ) = _ _ _ _ _

* T h e P + c o l u m n i n d i c a t e s t h e p e r c e n t o f G R E M a t h e m a t i c s T e s t e x a m i n e e s w h o a n s w e r e d e a c h

q u e s t i o n c o r r e c t l y ; i t i s b a s e d o n a s a m p l e o f F e b r u a r y 1 9 9 3 e x a m i n e e s s e l e c t e d t o r e p r e s e n t

a l l G R E M a t h e m a t i c s T e s t e x a m i n e e s t e s t e d b e t w e e n O c t o b e r 1 , 1 9 9 6 a n d S e p t e m b e r 3 0 , 1 9 9 9 .

2 5

26

Score Conversions and Percents Below* for GRE Mathematics Test, Form GR93'67

TOTAL SCORE Raw Score Scaled Score % Raw Score Scaled Score

55-66 990 82 26 690 54 980 81 25 680 53 970 80 24 670 52 960 79 23 660 51 950 77 22 650 50 930 74 21 640 49 920 72 20 630 48 910 71 19 620 47 900 69 18 600

46 890 67 17 590 45 880 66 16 580 44 .870 64 15 570 43 860 62 14 560 42 850 61 13 550 41 840 59 12 540 40 830 57 11 530 39 820 55 10 520 38 810 54 9 510 37 800 52 8 500

36 790 51 7 490 35 780 49 6 480 34 770 47 5 470 33 760 45 4 460 32 750 43 3 450 31 740 41 2 440 30 730 39 1 430 29 720 38 0 420 28 710 37 27 700 35

o/o

33 31 30 28 26 24 23 22 18

17 15 13 12 11 10 8 7 6 5

4 2 2 1 1 0 0 0

*Percent scoring below the scaled score is based on the performance of 7,092 examinees who took the GRE Mathematics Test between October 1, 1996 and September 30, 1999. This percent below information was used for score reports during the 2000-01 testing year.

-

E V A L U A T I N G Y O U R P E R F O R M A N C E ( G R 9 3 6 7 )

N o w t h a t y o u h a v e s c o r e d y o u r t e s t , y o u m a y w i s h t o c o m p a r e y o u r p e r f o r m a n c e

w i t h t h e p e r f o r m a n c e o f o t h e r s w h o t o o k t h i s t e s t . A w o r k s h e e t a n d t a b l e a r e

p r o v i d e d , b o t h u s i n g p e r f o r m a n c e d a t a f r o m G R E M a t h e m a t i c s T e s t e x a m i n e e s .

T h e w o r k s h e e t ( o n p a g e 2 5 ) i s b a s e d o n t h e p e r f o r m a n c e o f a s a m p l e o f t h e

e x a m i n e e s w h o t o o k t h i s p a r t i c u l a r t e s t i n F e b r u a r y 1 9 9 3 . T h i s s a m p l e w a s

s e l e c t e d t o r e p r e s e n t t h e t o t a l p o p u l a t i o n o f G R E M a t h e m a t i c s T e s t e x a n 1 i n e e s

t e s t e d b e t w e e n O c t o b e r 1 9 9 6 a n d S e p t e m b e r 1 9 9 9 . O n t h e w o r k s h e e t y o u u s e d

t o d e t e r m i n e y o u r s c o r e i s a c o l u m n l a b e l e d " P + . " T h e n u m b e r s i n t h i s c o l u m n

i n d i c a t e t h e p e r c e n t o f t h e e x a m i n e e s i n t h i s s a m p l e w h o a n s w e r e d e a c h q u e s t i o n

c o r r e c t l y . Y o u m a y u s e t h e s e n u m b e r s a s a g u i d e f o r e v a l u a t i n g y o u r p e r f o r m a n c e

o n e a c h t e s t q u e s t i o n .

T h e t a b l e o n p a g e 2 6 c o n t a i n s , f o r e a c h s c a l e d s c o r e , t h e p e r c e n t a g e o f

e x a m i n e e s t e s t e d b e t w e e n O c t o b e r 1 9 9 6 a n d S e p t e m b e r 1 9 9 9 w h o r e c e i v e d l o w e r

s c o r e s . I n t e r p r e t i v e d a t a b a s e d o n t h e s c o r e s e a r n e d b y e x a m i n e e s t e s t e d i n t h i s

t h r e e - y e a r p e r i o d w e r e u s e d b y a d m i s s i o n s o f f i c e r s i n 2 0 0 0 - 2 0 0 1 . T h e s e p e r c e n t -

a g e s a p p e a r i n t h e s c o r e c o n v e r s i o n t a b l e i n a c o l u m n t o t h e r i g h t o f t h e s c a l e d

s c o r e s . F o r e x a m p l e , i n t h e p e r c e n t c o l u m n o p p o s i t e t h e s c a l e d s c o r e o f 8 7 0 i s t h e

n u m b e r 6 4 . T h i s m e a n s t h a t 6 4 p e r c e n t o f t h e G R E M a t h e m a t i c s T e s t

e x a m i n e e s t e s t e d b e t w e e n O c t o b e r 1 9 9 6 a n d S e p t e m b e r 1 9 9 9 s c o r e d l o w e r t h a n

8 7 0 . T o c o m p a r e y o u r s e l f w i t h t h i s p o p u l a t i o n , l o o k a t t h e p e r c e n t n e x t t o t h e

s c a l e d s c o r e y o u e a r n e d o n t h e p r a c t i c e t e s t . T h i s n u m b e r i s a r e a s o n a b l e i n d i c a -

t i o n o f y o u r r a n k a m o n g G R E M a t h e m a t i c s T e s t e x a m i n e e s i f y o u f o l l o w e d t h e

t e s t - t a k i n g s u g g e s t i o n s i n t h i s p r a c t i c e b o o k .

I t i s i m p o r t a n t t o r e a l i z e t h a t t h e c o n d i t i o n s u n d e r w h i c h y o u t e s t e d y o u r s e l f

w e r e n o t e x a c t l y t h e s a m e a s t h o s e y o u w i l l e n c o u n t e r a t a t e s t c e n t e r . I t i s

i m p o s s i b l e t o p r e d i c t h o w d i f f e r e n t t e s t - t a k i n g c o n d i t i o n s w i l l a f f e c t t e s t p e r f o r -

m a n c e , a n d t h i s i s o n l y o n e f a c t o r t h a t m a y a c c o u n t f o r d i f f e r e n c e s b e t w e e n y o u r

p r a c t i c e t e s t s c o r e s a n d y o u r a c t u a l t e s t s c o r e s . B y c o m p a r i n g y o u r p e r f o r m a n c e

o n t h i s p r a c t i c e t e s t w i t h t h e p e r f o r m a n c e o f o t h e r G R E M a t h e m a t i c s T e s t

e x a m i n e e s , h o w e v e r , y o u w i l l b e a b l e t o d e t e r m i n e y o u r s t r e n g t h s a n d w e a k -

n e s s e s a n d c a n t h e n p l a n a p r o g r a m o f s t u d y t o p r e p a r e y o u r s e l f f o r t a k i n g t h e

G R E M a t h e m a t i c s T e s t u n d e r s t a n d a r d c o n d i t i o n s .

2 7

FORMGR9367

67

THE GRADUATE RECORD EXAMINATIONS

MATHEMATICS TEST

Do not break the seal until you are told to do so.

The contents of this test are confidential. Disclosure or reproduction of any portion of it is prohibited.

THIS TEST BOOIS MUST NOT BE TAKEN FROM THE ROOM. GRE. GRADUATE RECORD EXAMINATIONS, ETS, EDUCATIONAL TESTING SERVICE and the ETS logo

are registered trademarks of Educational Testing Service. Copyright 1990, 1991, 1993 by Educational Testing Service. All rights reserved. 29

Princeton, N.J. 08541

MATHEMATICS TEST Time- 170 minutes

66 Questions

Directions: Each of the questions or incomplete statements below is followed by five suggested answers or comple-tions. 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 are to the base e unless otherwise specified. (2) The set of all x such that a ~ x ~ b is denoted by [a, b ].

1. If f(g(x)) = 5 and f(x) = x + 3 for all real x, then g(x) =

(A) X - 3

2. lim tan x = X-+O COS X

(A) - oo

[log4 2x 3. e dx =

O

(A) J1 ' 2

(B) 3 - x

(B) -1

(B) 8

5 (C) X + 3

(C) 0

(D) 2

(D) 1

(D) log2

16 _ 1

4. Let A - B denote {x A: x B}. If (A - B) U B = A, which of the following must be true?

(A) B is empty. (B) A s B (C) B s A (D) (B -A)UA =B (E) None of the above

30

GO ON TO THE NEXT PAGE.

(E) 8

(E) +oo

1 (E) log 4 - 2

-

1 SCRATCHWORK

31

5. If f(x) = lxl + 3x2 for all real x, then f' ( -1) is

(A) -7 (B) -5 (C) 5 (D) 7 (E) nonexistent

b+t 6. For what value of b is the value of Jb (x 2 + x) dx a minimum?

(A) 0 (B) -1 (C) -2 (D) -3 (E) -4

7. In how many of the eight standard octants of xyz-space does the graph of z = ex+y appear?

(A) One (B) Two (C) Three (D) Four (E) Eight

8. Suppose that the function f is defined on an interval by the formula f(x) = Jtan2x - I . Iff is continuous, which of the following intervals could be its domain?

32

(A)

(B)

(C)

(3;' n) (~~) (~. 3;)

(D) ( -~.o) (E) ( _3;' -~)


-

33

SCRATCHWORK

9. J.l 2 x 2 dx = o -x

1 (A) -2 (B) ~ 3 (C) log 2 - e 2 (D) _lo~ 2

10. If /"(x) = f'(x) for all real x, and if /(0) = 0 and /'(0) = -I, then f(x) =

(A) I -ex (B) ex - I (C) e-x- I

3

II. If cp (x, y, z) = x 2 + 2xy + xz2, which of the following partial derivatives are identically ~ro? a2c;

I. oy2

a2c; II. oxoy

a2c; III. ozoy

(A) III only (B) I and II only (C) I and III only (D) II and III only (E) I, II, and III

34


(E) log 2 2

..

35

SCRATCHWORK

12. lim sin 2x x-o (1 + x)log(l + x)

(A) -2 1 (B) -2 (C) 0 (E) 2

13. lim r ~dx = n-+oo I X

(A) 0 (B) I (C) e (D) 7t (E) +oo

14. At a 15 percent annual inflation rate, the value of the dollar would decrease by approximately one-half every

36

5 years. At this inflation rate, in approximately how many years would the dollar be worth 1 .~,000 of its present value?

(A) 25 (B) 50 (C) 75 (D) 100 (E) 125


37

SCRATCHWORK

15. Let f (x) = Jx -1 1 2 dt for all real x. An equation of the line tangent to the graph of f at the point (2, f (2)) is I + t 1 1 (A) y - 1 = 5 (x - 2) (B) y - Arctan 2 = 5 (x - 2) (C) y - 1 = (Arctan 2) (x - 2)

7t 1 7t (D) y -Arctan 2 + 4 = 5(x - 2) (E) y - 2 =(Arctan 2)(x - 2)

16. Let f(x) = eg(x)h(x) and h' (x) = -g' (x)h(x) for all real x. Which of the following must be true?

(A) f is a constant function. (B) f is a linear nonconstant function. (C) g is a constant function. (D) g is a linear nonconstant function. (E) None of the above

17. 1 -sin\ Arccos 1;) =

(A) )1- ~OS~ (B) J 1 - ;os~ (C) 7t 1 +cos 24 2


38

(D)~ 6

-

39

SCRATCHWORK

00

18. If f(x) = L ( -l)n x 2n for all x e (0, 1), then f'(x) = n=O

(A) sin x (B) cos X I (C) I + x2

19. Which of the following is the general solution of the differential equation d3y d2y dy --3- + 3-- y = O? dt 3 dt 2 dt .

(A) c1e1 + c2te 1 + c3t2e1 (B) c1e-t + c2te-t + c3t 2e-t

t -1 ~ (C) c1e - c2e + c3te (D) c1e 1 + c2e21 + c3e31 (E) c1e21 + c2te - 21

40


-

2x (E) (I - 2x)2

41

SCRATCHWORK

20. Which of the following double integrals represents the volume of the solid bounded above by the graph of z = 6 - x 2 - 2y2 and bounded below by the graph of z = -2 + x 2 + 2y2 ?

x-2 J~-J2 (A) 4J (8 - 2x2 - 4y2) dy dx x-o y-o

Jx~2 (B) x -2

(C) 4JY~ __ 0J2 Jx= J4=2Y2 dx dy

x=-J4=2Yl

(E) 2J~-J2 Jx-J4=2Yl (8 - 2x2 - 4y2) dx dy y=O X=O

21. Let a be a number in the interval [0, 1] and let I be a function defined on [0, 1] by

42

/(x) = {a2 if 0 ~ .x ~ a, ax otherwtse. I

Then the value of a for which J0 I (x) dx = 1 is


(D) 1 ,(E) nonexistent

43

SCRATCHWORK

'2:2':) If b and c are elements in a group G, and if b5 = c3 = e, where e is the unit element of G, then the inverse of b 2cb4c2 must be

23. Let f be a real-valued function continuous on the closed interval (0, 1] and differentiable on the open interval (0, 1) with /(0) = 1 and /(1) = 0. Which of the following must be true?

I. There exists x (0, 1) such that f (x) = x. II. There exists x (0, 1) such that f' (x) = -1.

III. f(x) > 0 for all x [0, 1).

(A) I only (8) II only (C) I and II only (D) II and III only (E) I, II, and III

24. If A and B are events in a probability space such that 0 < P(A) = P(B) = P(A ()B) < 1, which of the following CANNOT be true?

(A) A and B are independent. (D) A('tB = AUB

44

(B) A is a proper subset of B. (E) P(A )P(B) < P(A ()B)


(C) A :F B

45

SCRATCHWORK

25. Let f be a real-valued function with domain [0, 1]. If there is some K > 0 such that /(x) - f(y) ~ K I x - y 1 for all x and y in [0, I], which of the following must be true?

(A) f is discontinuous at each point of (0, I). (B) f is not continuous on (0, I), but is discontinuous at only countably many points of (0, I). (C) f is continuous on (0, I), but is differentiable at only countably many points of (0, I). (D) f is continuous on (0, I), but may not be differentiable on (0, 1 ). (E) f is differentiable on (0, 1 ).

26. Let i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1). The vectors v1 and v2 are orthogonal if v1 =I + j - k and v2 =

(A) i + J- k (B) i- j + k (C) i + k (D) J- k (E) i + j

27. If the curve in the yz-plane with equation z = f(y) is rotated around they-axis, an equation of the resulting surface of revolution is

(A) x2 + z2 = [/(y)]2 (B) x 2 + z2 = /(y) (C) x 2 + z 2 = 1/(y) I (D) y 2 + z2 = 1/(y) I (E) y 2 + z2 = [/(x)jl

46


47

SCRATCHWORK

48

28. Let A and B be subs paces of a vector space V. Which of the following must be subspaces of V? I. A + B = {a + b: a e A and b E B}

II. AU B III. A() B IV. {x e V: x A }

(A) I and II only (8) I and III only (C) III and IV only (D) I, II, and III only (E) I, II, III, and IV

29. lim I (! - 2~) = n--+oo k=l

(A) 0 (B) 1 (C) 2

30 Iff(x X ) - ~ xx then of = 1 , ' n - i..J 1 J ' ox l~i

49

SCRATCHWORK

{J I x 2 for 0 s; x s; I 31. If f (x) = x _- 1 for I < x s; 2, 2

then J0

f(x) dx is

(A) ~ 2

(B) .j2 2

(C) ! + ~ 2 4 (D) ! + ~ 2 2 (E) undefined

32. Let R denote the field of real numbers, Q the field of rational numbers, and Z the ring of integers. Which of the following subsets F; of R, I s; i s; 4, are subfields of R?

F 1 = {ajb: a, b e Z and b is odd} F2 = {a + b.jf: a, b e Z} F3 ={a+ h.jf: a,b e Q} F4 = {a + b.:/2: a, b e Q}

(A) No F; is a subfield of R. (B) F3 only (C) F2 and F3 only (D) F 1, F2, and F3 only (E) F 1, F2, F3, and F4

50


51

SCRATCHWORK

33. If n apples, no two of the same weight, are lined up at random on a tab~e, what is the probability that they are lined up in order of increasing weight from left to right? ~

(A)! (B) ! (C) l_ 1 (E) (~ )n 2 n n! (D) 2n

34. _!!_ r2 e-(2 dt = dx o (A) e-x2 (B) 2e-x2 (C) 2e-x4 (D) x2e-x2 (E) 2xe-x4


52

53

SCRATCHWORK

35. Let I be a real-valued function defined on the set of integers and satisfying I (x) = !1 (x - 1) + !l

55

SCRATCHWORK

(C) 3 (D)~

39. For a real number x, log( 1 + sin 2nx) is not a real number if and only if x is

(A) an integer

(8) nonpositive 2n -I (C) equal to 2 for some integer n

4n -I (D) equal to 4 for some integer n

(E) any real number

40. If x, y, and z are selected independently and at random from the interval [0, I], then the probability that x ~ yz is

(A) ~ 4

56

(B) ~ 3


(E) J.! 6

57

SCRATCHWORK

41. If A = ( ~ _ 0 then the set of all vectors X for which AX = X ii (A) {(~)I a = 0 and b is arbitrary} (B) {(~)I a is arbitrary and b = o} (C) {(~)I a = -b and b is arbitrary} (D) {(~)} (E) the empty set

42. What is the greatest value of b for which any real-valued function f that satisfies the following properties must also satisfy /(1) < 5?

(i) f is infinitely differentiable on the real numbers; (ii) f(O) = I, /'(0) = I, and /"(0) = 2; and

(iii) 1/"'(x) I < b for all x in [0, 1]. (A) I (8) 2 (C) 6


58

(D) 12 (E) 24

59

SCRATCHWORK

43. Let n be an integer greater than 1. Which of the following conditions guarantee that the equation f' n-1

xn = L a;xi has at least one root in the interval (0, 1)? i = 0

n-1 I. a0 > 0 and L a; < 1

i = 0

n-1 II. a0 > 0 and L a; > 1

i = 0

n-1 III. a0 < 0 and L a; > 1

(A) None (B) I only (C) II only (D) III only (E) I and III

i = 0

44. If x is a real number and P is a polynomial function, then lim P(x + Jh) + P(~ - 3h) - 2P(x) = h-o h (A) 0 (B) 6P'(x) (C) 3P"(x) (D) 9P"(x)


60

(E) oo

61

SCRATCHWORK

45. Consider the system of equations

ax 2 + by3 = c dx2 + ey3 =!

where a, b, c, d, e, and f are real constants and ae '#; bd. The maximum possible number of real solutions (x, y) of the system is

(A) none

(A) (I, ~ 0, - k) (B) (1, -1, 0, 1) (C) (7, 6, 10, 1)

(D) (7, 11, 12, 6)

(E) (7, 11, 6, 1)

62

(B) one (C) two (D) three (E) five


63

SCRATCHWORK

47. Let C be the ellipse with center (0, 0), major axis of length 2a, and minor axis of length 2b. The value f.c of

65

SCRATCHWORK

49. Let X be a random variable with probability density function

f (x) = { ~ (I - x 2) if - I ~ x ~ I, 0 otherwise.

What is the standard deviation of X ?

(A) 0 I (B) --5 j3o

(C) 15

50. The set of all points (x, y, z) in Euclidean 3-space such that

X y z 1 0 0

=0 0 I 0 0 0 I

is

(A) a plane containing the points (I, 0, 0), (0, I; 0), and (0, 0, I) (B) a sphere with center at the origin and radius I (C) a surface containing the point (I, I, I) (D) a vector space with basis {(1, 0, 0), (0, I, 0), (0, 0, I)} (E) none of the above


66

(D) _1_ .j5 (E) I

67

SCRATCHWORK

51. An automorphism ~ of a field F is a one-to-one mapping of F onto itself such that ~(a + b) = ~(a) + ~(b) and ~(ab) = ~(a )~(b) for all a, b e F. If F is the field of rational numbers', then the number of distinct automorphisms of F is

(A) 0 (B) 1 (C) 2 (D) 4 (E) infinite

52. Let T be the transformation of the xy-plane that reflects each vector through the x-axis and then doubles the vector's length.

If A is the 2 X 2 matrix such that r([; ]) = A r; 1 for each vector [; J ' then A (A) [~ ~]

[f I ] (B) -fi

1 --2

fi -fi 2 -2-

(C) j2 fi 2 -2-

[ 0 -2] (E) -2 0 GO ON TO THE NEXT PAGE.

68

'

69

SCRATCHWORK

53. Let r > 0 and let C be the circle lz I = r in the complex plane. If P is a polynomial function, then Jc P(z) dz =

(A) 0 (B) nr2 (C) 2ni (D) 2nP(O)i (E) P(r)

54. If 1 and g are real-valued differentiable functions and iff' (x) ;;::: g' (x) for all x in the closed interval [0, 1 ], which of the following must be true?

(A) I (0) ;;::: g(O) (B) 1(1) ;;::: g(l) (C) l(l) -g(1) ;;:>::1(0) -g(O) (D) I- g has no maximum on [0, 1]. (E) [ is a nondecreasing function on [0, 1]. g

55. Let p and q be distinct primes. There is a proper subgroup J of the additive group of integers which contains

exactly three elements of the set {p, p + q, pq, pq, qP}. Which three elements are in J?

(A) pq, pq, qP (B) P + q,pq,pq (C) p,p + q,pq (D) p, pq, qP (E) p, pq, pq

70


71

SCRATCHWORK

56. For a subset S of a topological space X, let cl(S) denote the closure of S in X, and let S I = {x: x E cl(S - {x }) } denote the derived set of S. If A and B fcare subsets of X, which of the following statements are true?

I. (A u B)' = A I u Bl II.

73

SCRATCHWORK

58. Which of the following is an eigenvalue of the matrix

I - ;) -2

over the complex numbers?

(A) 0 (B) (C) J6 (D) i (E) 1 + i

59. Two subgroups H and K of a group G have orders 12 and 30, respectively. Which of the following could NOT be the order of the subgroup of G generated by H and K?

(A) 30 (B) 60 (C) 120 (D) 360 (E) Countable infinity

60. Let A and B be subsets of a set M and let S0 = {A, B}. For i ~ 0, define S; + 1 inductively to be the

collection of subsets X of M that are of the form C U D, C n D, or M - C (the complement of C in M), 00

where C, DeS;. Let S = U S;. What is the largest possible number of elements of S?

(A) 4 (B) 8 (C) 15 (D) 16 (E) S may be infinite.

74

i= 0


75

SCRATCHWORK

61. A city has square city blocks formed by a grid of north-south and east-west streets. One automobile route from City Hall to the main firehouse is to go exactly 5 blocks east and 7 blocks niorth. How many different routes from City Hall to the main firehouse traverse exactly 12 city blocks?

(A) 5 7

7! (B) 5! 12!

(C) 7!5!

(D) 2'2

(E) 7!5!

62. Let R be the set of real numbers with the topology generated by the basis {[a, b): a < b, where a, b R }. If X is the subset [0, 1] of R, which of the following must be true?

I. X is compact. II. X is Hausdorff.

III. X is connected.

(A) I only (B) II only (C) III only (D) I and II (E) II and III

76


77

SCRATCHWORK

63. Let R be the circular region of the xy-plane with center at the origin and radius 2.

J J 2 2 " Then R e-dx dy = (A) 4n (B) ne - 4 (C) 4ne - 4 (D) n(l - e - 4) (E) 4n(e - e - 4)

64. Let V be the real vector space of real-valued functions defined on the real numbers and having derivatives of all orders. If D is the mapping from V into V that maps every function in V to its derivative, what are all the eigenvectors of D ?

(A) All nonzero functions in V

(B) All nonzero constant functions in V

(C) All nonzero functions of the form keA.x, where k and A. are real numbers k

(D) All nonzero functions of the form L c;xi, where k > 0 and the c;'s are real numbers i = 0

(E) There are no eigenvectors of D.


78

79

SCRATCHWORK

65. Iff is a function defined by a complex power series expansion in z - a which converges for I z - a I < 1 and diverges for I z - a l > 1, which of the following must be true? ~ (A) f (z) is analytic in the open unit disk with center at a. (B) The power series for f (z + a) converges for I z + a I < 1. (C) f'(a) = 0 (D) J c f (z )dz = 0 for any circle C in the plane. (E) f(z) has a pole of order 1 at z =a.

66. Let n be any positive integer and 1 ~ x1 < x2 < . . . < Xn + 1 ~ 2n, where each x; is an integer. Which of the following must be true?

I. There is an x; that is the square of an integer. II. There is an i such that x;+ 1 = x; + 1.

III. There is an x; that is prime.

(A) I only (B) II only (C) I and II (D) I and III (E) II and III

IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST.

80

HOW TO SCORE YOUR TEST (GR9767) Total Subject Test scores are reported as three-digit scaled scores with the third digit always zero. The maximum possible range for all Subject Test total scores is from 200 to 990. The actual range of scores for a particular subject test, however, may be smaller. The possible range for GRE Mathematics Test scores is from 400 to 990. The range for different editions of a given test may vary because different editions are not of precisely the same difficulty. The differences in ranges among different editions of a given test, however, usually are small. This should be taken into account, especially when comparing two very high scores. In general, differences between scores at the 99th percentile should be ignored. The score conversion table provided shows the score range for this edition of the test only.

The work sheet on page 82 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. This is the adjust-ment made for guessing. Then round the result to the nearest whole number. This will give you your raw total score. Use the total score conversion table to find the scaled total score that corresponds to your raw total score.

Example: Suppose you chose the correct answers to 48 questions and incorrect answers to 15. Dividing 15 by 4 yields 3.75. Subtracting 3.75 from 48 equals 44.25, which is rounded to 44. The raw score of 44 corresponds to a scaled score of 930.

81

82

WORK SHEET for the GRE Mathematics Test, Form GR9767 Answer Key and Percentages* of Examinees Answering Each Question Correctly

QUESTION TOTAL QUESTION TOTAL Number Answer P+ c I Number Answer P+ c I

1 c 76 36 D 37 2 D 86 37 B 32 3 c 87 38 B 51 4 B 85 39 B 50 5 E 81 40 E 49

6 c 88 41 E 43 7 B 80 42 D 59 8 E 72 43 B 60 9 A 79 44 D 37

10 B 55 45 A 51

11 A 82 46 c 45 12 c 73 47 E 32 13 A 70 48 A 35 14 c 65 49 E 42 15 B 61 50 D 37

16 D 83 51 D 32 17 E 80 52 D 51 18 E 63 53 D 30 19 B 59 54 A 38 20 D 48 55 D 34

21 D 70 56 E 17 22 B 73 57 c 21 23 c 71 58 c 39 24 c 48 59 A 29 25 D 73 60 E 23

26 B 69 61 E 25 27 E 61 62 A 32 28 c 63 63 D 21 29 B 58 64 c 35 30 D 56 65 A 39

31 c 62 66 E 53 32 D 56 33 A 70 34 A 54 35 B 48

Correct (C) ____ _ Incorrect (I) ____ _ Total Score: c- 1/4 = -----Scaled Score (SS) = ____ _

*The P+ column indicates the percent of GRE Mathematics Test examinees who answered each question correctly; it is based on a sample of December 1997 examinees selected to represent all GRE Mathematics Test examinees tested between October 1, 1996 and September 30, 1999.

S c o r e C o n v e r s i o n s a n d P e r c e n t s B e l o w *

f o r G R E M a t h e m a t i c s T e s t , F o r m G R 9 7 6 7

T O T A L S C O R E

R a w S c o r e S c a l e d S c o r e

% R a w S c o r e S c a l e d S c o r e

%

5 0 - 6 6 9 9 0 8 2

2 3 6 9 0 3 3

4 9 9 8 0 8 1 2 2 6 8 0

3 1

4 8 9 7 0

8 0

2 1

6 7 0 3 0

4 7

9 6 0

7 9 2 0 6 6 0 2 8

4 6 9 5 0

7 7 1 9 6 5 0 2 6

4 5 9 4 0 7 5 1 8 6 4 0

2 4

4 4

9 3 0

7 4 1 7 6 3 0

2 3

4 3

9 2 0

7 2 1 6

6 1 0 2 0

4 2

9 1 0 7 1 1 5 6 0 0

1 8

4 1 8 9 0 6 7

1 4

5 9 0

1 7

4 0

8 8 0 6 6

1 3 5 8 0

1 5

3 9

8 7 0 6 4 1 2

5 7 0 1 3

3 8 8 6 0

6 2 1 1 5 6 0

1 2

3 7 8 5 0 6 1

1 0

5 5 0

1 1

3 6 8 4 0 5 9 9

5 4 0

1 0

3 5 8 3 0 5 7 8 5 3 0

8

3 4 8 2 0 5 5

7 5 1 0 6

3 3 8 0 0 5 2 6 5 0 0

5

3 2

7 9 0 5 1 5 4 9 0

4

3 1 7 8 0 4 9

4 4 8 0 2

3 0 7 7 0 4 7 3

4 7 0 2

2 9 7 6 0 4 5 2 4 6 0

1

2 8

7 5 0 4 3 1 4 5 0

1

. z z

7 4 0 4 1 0

4 4 0

0

' 2 6

7 3 0 3 9

2 5 7 2 0 3 8

2 4 7 0 0 3 5

* P e r c e n t s c o r i n g b e l o w t h e s c a l e d s c o r e i s b a s e d o n t h e p e r f o r m a n c e o f

7 , 0 9 2 e x a m i n e e s w h o t o o k t h e G R E M a t h e m a t i c s T e s t b e t w e e n O c t o b e r 1 ,

1 9 9 6 a n d S e p t e m b e r 3 0 , 1 9 9 9 . T h i s p e r c e n t b e l o w i n f o r m a t i o n w a s u s e d f o r

s c o r e r e p o r t s d u r i n g t h e 2 0 0 0 - 0 1 t e s t i n g y e a r .

8 3

84

EVALUATING YOURPERFORMANCE(GR9767) t'

Now that you have scored your test, you may wish to compare your performance with the performance of others who took this test. A worksheet and table are provided, both using performance data from GRE Mathematics Test examinees.

The worksheet (on page 82) is based on the performance of a sample of the examinees who took this particular test in December 1997. This sample was selected to represent the total population of GRE Mathematics Test examinees tested between October 1996 and September 1999. On the worksheet you used to determine your score is a column labeled "P+." The numbers in this column indicate the percent of the examinees in this sample who answered each question correctly. You may use these numbers as a guide for evaluating your perfonnance on each test question.

The table (on page 83) contains, for each scaled score, the percentage of examinees tested between October 1996 and September 1999 who received lower scores. Interpretive data based on the scores earned by examinees tested in this three-year period were used by admissions officers in 2000-2001. These percent-ages appear in the score conversion table in a column to the right of the scaled scores. For example, in the percent column opposite the scaled score of 870 is the number 64. This means that 64 percent of the GRE Mathematics Test examinees tested between October 1996 and September 1999 scored lower than 870. To compare yourself with this population, look at the percent next to the scaled score you earned on the practice test. This number is a reasonable indica-tion of your rank among GRE Mathematics Test examinees if you followed the test-taking suggestions in this practice book.

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 perfor-mance, 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 GRE Mathematics Test examinees, however, you will be able to determine your strengths and weak-nesses and can then plan a program of study to prepare yourself for taking the GRE Mathematics Test under standard conditions.

FORM GR9767

67

THE GRADUATE RECORD EXAMINATIONS

MATHEMATICS TEST

Do not break the seal until you are told to do so.

The contents of this test are confidential. Disclosure or reproduction of any portion of it is prohibited.

THIS TEST BOOK MUST NOT BE TAKEN FROM THE ROOM . ..

ORE, GRADUATE RECORD EXAMINATIONS, ETS, EDUCATIONAL TESTING SERVICE and the ETS logo are registered trademarks of Educational Testing Service.

Copyright 1997 by Educational Testing Service. All rights reserved. Princeton, N.J. 08541

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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 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 are to the base e unless otherwise specified. (2) The set of all x such that a S x S b is denoted by [a, b ].

1. If F(x) = f' Jog t dt for all positive x, then F'(x) = e

(A) x

(B)! X

(C) log x

(D) X log X

(E) X log X- 1

2. If F(l) = 2 and F(n) = F(n - 1) + ~ for all integers n > 1, then F(lOl) = (A) 49 (B) 50 (C) 51 (D) 52 (E) 53


86

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SCRATCHWORK

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3. If (~ -:) is invertible under matrix multiplication, then its in~rse is (A) (~ -:) (B) a2 ~ b2 (~ -:) (C) a2 ~ b2 (_~ :) (D) ( a b)

-b a

1 (-b a) (E) az- b2 a b y

b b 4. If b > 0 and if I x dx = I x2 dx, then the area of the shaded region in the figure above is

0 0

1 (A) 12

(B) 1 6

(C) 1 4

(D) 1 3

(E) 1 2


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SCRATCHWORK

10

5

y

y= f'(x)

5. If the figure above is the graph of y = f'(x), which of the following could be the graph of y = f(x)?

(A)

(B)

-10

(C)

90

y

-10

y

10

-5

-10

y

10

(D) y

(E) y

10

5 10 X

-10


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SCRATCHWORK

6. Consider the following sequence of instructions.

1. Set k = 999, i = 1, and p = 0. 2. If k > i, then go to step 3; otherwise go to step 5. 3. Replace i with 2i and replace p with p + 1. 4. Go to step 2. 5. Print p.

If these instructions are followed, what number will be printed at step 5 ?

(A) 1 (B) 2 (C) 10 (D) 512 (E) 999

7. Which of the following indicates the graph of {(sin t, cos t ): - i ~ t ~ 0} (A) (D) y

1

X X -1 1 -1 0 1

-1 -1

(B) y (E)

1

X X -1 0 1

-1 (C) y

1

X -1 1


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in the xy-plane?

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SCRATCHWORK

8. Jl X 2 dx = o 1 +X

(A) 1 (B)~ 4 (C) tan -I -V2

2 (D) log 2 (E) log -V2

9. If S is a nonempty finite set with k elements, then the number of one-to-one functions from S onto S is

(A) k! (B) k 2 (C) kk (D) 2k (E) 2k+1

10. Let g be the function defined on the set of all real numbers by

( ) _ { 1 if x is rational, g x - ex if x is irrational.

Then the set of numbers at which g is continuous is

(A) the empty set (B) {0} (C) { 1} (D) the set of rational numbers (E) the set of irrational numbers

x + y + lx- y I 11. For all real numbers x and y, the expression 2 is equal to

(A) the maximum of x and y (B) the minimum of x and y (C) lx + y I (D) the average of I x I and I y I (E) the average of lx + y I and x- y


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SCRATCHWORK

12. Let B be a nonempty bounded set of real numbers and let b be t~e least upper bound of B. If b is not a member of B, which of the following is necessarily true? f'

(A) B is closed. (B) B is not open. (C) b is a limit point of B. (D) No sequence in B converges to b. (E) There is an open interval containing b but containing no point of B.

13. A drawer contains 2 blue, 4 red, and 2 yellow socks. If 2 socks are to be randomly selected from the drawer, what is the probability that they will be the same color?

(A)~ 7

(B)~ 5 (C)~

7

(D)! 2

(E) ~ 5

14. Let R be the set of real numbers and let f and g be functions from R into R. The negation of the statement

"For each s in R, there exists an r in R such that if f(r) > 0, then g(s) > 0." is which of the following?

(A) For each s in R, there does not exist an r in R such that if f(r) > 0, then g(s) > 0. (B) For each s in R, there exists an r in R such that f(r) > 0 and g(s) ~ 0. (C) There exists an s in R such that for each r in R, f(r) > 0 and g (s) ~ 0. (D) There exists an s in R and there exists an r in R such that f(r) ~ 0 and g(s) ~ 0. (E) For each r in R, there exists an s in R such that f(r) ~ 0 and g(s) ~ 0.

15. If g is a function defined on the open interval (a, b) such that a < g ( x) < x for all x e (a, b), then g is

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(A) an unbounded function (B) a nonconstant function (C) a nonnegative function (D) a strictly increasing function (E) a polynomial function of degree 1


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SCRATCHWORK

16. For what value (or values) of m is the vector ( 1, 2, m, 5) a linear combination of the vectors (0, 1, 1, 1 ), (0, 0, 0, 1), and (1, 1, 2, 0) ? fc

(A) For no value of m (B) -1 only (C) 1 only (D) 3 only (E) For infinitely many values of m

17. For a function f, the finite differences ll.f(x) and ll.2/(x) are defined by ll.f(x) = f(x + 1) - f(x) and ll.2f(x) = ll.f(x + 1) - ll.f(x). What is the value of /(4), given the following partially completed finite difference table?

X f(x) ll.f(x) ll.2/(x) 1 -1 4 2 -2 6 3 4

(A) -5 (B) -1 (C) 1 (D) 3 (E) 5

18. In the figure above, the annulus with center C has inner radius r and outer radius 1. As r increases, the

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circle with center 0 contracts and remains tangent to the inner circle. If A (r) is the area of the annulus

and a(r) is the area of the circular region with center 0, then lim A(r) = r-+1- a(r)

(A) 0 (B)~

1t

(C) 1 (D)~

2 (E) oo


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SCRATCHWORK

19. Which of the following are multiplication tables for groups with four elements? J.c

I. a b c d II. a b c d III. a b c d a a b c d a a b c d a a b c d b b c d a b b a d c b b a d c c c d a b c c d a a c c d c d d d a b c d d c a b d d c d c

(A) None (B) I only (C) I and II only (D) II and III only (E) I, II, and lll

20. Which of the following statements are true for every function f, defined on the set of all real numbers, such that lim /( x) is a real number L and /(0) = 0 ?

x~o X

I. f is differentiable at 0. IT. L = 0

III. lim f(x) = 0 x~o

(A) None (B) I only (C) III only (D) I and III only (E) I, II, and III

21. What is the area of the region bounded by the coordinate axes and the line tangent to the graph of 1 2 1 y = -gx + 2x + 1 at the point (0, 1) ?

1 (A) 16

(B) 1 8 (C) 1 4 (D) 1

(E) 2


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SCRATCHWORK

22. Let Z be the group of all integers under the operation of addition. Which of the following subsets of Z is NOT a subgroup of Z ? fc

(A) {0} (B) {n e Z: n ~ 0} (C) { n e Z: n is an even integer} (D) {n e Z: n is divisible by both 6 and 9} (E) Z

23. In the Euclidean plane, point A is on a circle centered at point 0, and 0 is on a circle centered at A. The circles intersect at points B and C. What is the measure of angle BAC ?

(A) 60 (B) 90 (C) 120 (D) 135 (E) 150

24. Which of the following sets of vectors is a basis for the subspace of Euclidean 4-space consisting of all vectors that are orthogonal to both (0, 1, 1, 1) and (1, 1, 1, 0) ?

(A) { (0, -1, 1, 0)} (B) {(1, 0, 0, 0), (0, 0, 0, 1)} (C) {(-2, 1, 1, -2), (0, 1, -1, 0)} (D) {(1, -1, 0, 1), (-1, 1, 0, -1), (0, 1, -1, 0)} (E) {(0, 0, 0, 0), (-1, 1, 0, -1), (0, 1, -1, 0)}

25. Let f be the function defined by f(x, y) = 5x- 4y on the region in the .xy-plane satisfying the inequali-ties x ~ 2, y ~ 0, x + y ~ 1, and y - x ~ 0. The maximum value of f on this region is

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(A) 1 (B) 2 (C) 5 (D) 10 (E) 15


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SCRATCHWORK

26. Let f be the function defined by

f(x) = {-x~ + 4x - 2 ~f x < 1, -X + 2 If X~ 1.

Which of the following statements about f is true?

(A) f has an absolute maximum at x = 0. (B) f has an absolute maximum at x = 1. (C) f has an absolute maximum at x = 2. (D) f has no absolute maximum. (E) f has local maxima at both x = 0 and x = 2.

27. Let f be a function such that /( x) = f( 1 - x) for all real numbers x. If f is differentiable everywhere, then /'(0) =

(A) /(0) (B) /(1) (C) -/(0) (D) /'(1) (E) -/'(1)

28. If V1 and V2 are 6-dimensional subspaces of a 1 0-dimensional vector space V, what is the smallest possible dimension that VI (") v2 can have? (A) 0 (B) 1 (C) 2 (D) 4 (E) 6

29. Assume that p is a polynomial function on the set of real numbers. If p (0) = p (2) = 3 and 2

p'(O) = p'(2) = -1, then J xp"(x) dx =

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(A) -3 (B) -2 (C) -1 (D) 1 (E) 2

0

GO ON 10 THE NEXT PAGE.

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SCRATCHWORK

30. Suppose B is a basis for a real vector space V of dimension greatFr than 1. Which of the following statements could be true?

(A) The zero vector of V is an element of B. (B) B has a proper subset that spans V. (C) B is a proper subset of a linearly independent subset of V. (D) There is a basis for V that is disjoint from B. (E) One of the vectors in B is a linear combination of the other vectors in B.

31. Which of the following CANNOT be a root of a polynomial in x of the form 9x5 + ax3 + b, where

a and b are integers?

(A) -9

(B) -5

(C).! 4 (D).!_

3

(E) 9

32. When 20 children in a classroom line up for lunch, Pat insists on being somewhere ahead of Lynn. If Pat's demand is to be satisfied, in how many ways can the children line up?

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(A) 20!

(B) 19!

(C) 18! (D) 20!

2 (E) 20 19


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SCRATCHWORK

33. How many integers from 1 to 1,000 are divisible by 30 but not by 1~?

(A) 29 (B) 31 (C) 32 (D) 33 (E) 38

34. Suppose f is a differentiable function for which lim f(x) and lim f'(x) both exist and are finite. Which of the following must be true? x-+oo x-+oo

(A) lim f'(x) = 0 X-+oo

(B) lim f"(x) = 0 X-+oo

(C) lim f(x) = lim /'(x) X-+oo X-+oo

(D) f is a constant function.

(E) f' is a constant function.

35. In .xyz-space, an equation of the plane tangent to the surface z = e-x sin y at the point where x = 0 and

1t. y = 2 IS

(A) X+ y = 1 (B) X+ z = 1 (C) X- z = 1 (D) y + z = 1 (E) y - z = 1

36. For each real number x, let J.l(x) be the mean of the numbers 4, 9, 7, 5, and x; and let 11(x) be the median of these five numbers. For how many values of x is J.l(x) = 11(x)?

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(A) None (B) One (C) Two (D) Three (E) Infinitely many


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SCRATCHWORK

37.

(A) e (B) 2e (C) (e + l)(e - 1) (D) e 2 (E) oo

38. Which of the following integrals on the interval [ 0, i J has the greatest value? 7t

110

(A) f sin t dt 0

7t

(B) f cos tdt 0

7t

(C) f cos2t dt 0

7t

(D) r cos 2t dt 0

7t

(E) f sin t cos t dt 0

GO ON TO 1HE NEXT PAGE.

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SCRATCHWORK

39. Consider the function f defined by f(x) = e-x on the interval [0, JO]. Let n > 1 and let x0 , x1, , xn be numbers such that 0 = x0 < x1 < x2 < < Xn-l < xn = 10. Which of the following is greatest?

n

(A) L f(xj)(xj- Xj_1) j=l n

(B) L f(xj_ 1}(xj- Xj_1} j=l

n (X+ X ) (C) ~ f I 2 J-1 (xj - Xj-1) J=l

10

(D) J f(x) dx 0

(E) 0

40. A fair coin is to be tossed 8 times. What is the probability that more of the tosses will result in heads will result in tails?

(A) 1 4

(B) 1 3 87 (C) 256

(D) 23 64 93 (E) 256

41. The function f( x, y) = xy - x 3 - y 3 has a relative maximum at the point

112

(A) (0, 0)

(B) (1, 1) (C) (-1, -1)

(D) (1, 3)

(E) (~. ~) GO ON TO THE NEXT PAGE.

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SCRATCHWORK

42. Consider the points A = (-1, 2), B = (6, 4), and C = (1, -20) in the plane. For how many different

points D in the plane are A , B, C, and D the vertices of a parallelogram?

(A) None (B) One (C) Two (D) Three (E) Four

43. ff A is a 3 x 3 mabix such iliat A(D = G) and A(D = (!). ilien ilie prod~ AG) ~ (A) G) (B) CD (C) ( -i) (D) (!D (E) not uniquely determined by the information given

44. Let f denote the function defined for all x > 0 by f(x) = (-vi)x. Which of the following statements

FALSE?

114

(A) lim f(x) = 1 x~o+

(B) lim f(x) = oo x~oo

(C) f(x) = Xx12 for all X > 0.

(D) The derivative f'(x) is positive for all x > 0.

(E) The derivative f'(x) is increasing for all x > 0.


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SCRATCHWORK

45. An experimental car is found to have a fuel efficiency E(v), in miles per gallon of fuel, where v is the speed of the car, in miles per hour. For a certain 4-hour trip, if v = ~(t) is the speed of the car t hours after the trip started, which of the following integrals represents the number of gallons of fuel that the car used on the trip?

f4 v(t) (A) o E(v(t)) dt (B) f4E(v(t)) dt

0 v(t) 4 f tv(t) (C) E(v(t)) dt 0

(D) f4tE(v(t)) dt 0 v(t)

(E) fv(t)E(v(t)) dt 0

. (cos t -sin t) 46. For 0 < t < 1t, the matnx . t t has distinct complex eigenvalues A 1 and A2 For what value sm cos of t, 0 < t < 1t, is A 1 + A 2 = 1 ?

(A) 1t 6

(B) 1t 4 (C) 1t 3 (D) 1t 2 (E) 21t

3


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