# GRE Mathematics Test Practice Book

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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 ®

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.

This book is provided FREE with test registration by the Graduate Record Examinations Board.

www.ets.org/gre

This book contains important information about scoring.

Copyright © 2008 by Educational Testing Service. All rights reserved.

ETS, the ETS logos, LISTENING. LEARNING. LEADING., GRADUATE RECORD EXAMINATIONS,

®

and GRE are registered trademarks of Educational Testing Service (ETS) in the United States of America

and other countries throughout the world.

The GRE Board recommends that scores on the

Subject Tests be considered in conjunction with other

relevant information about applicants. Because numer-

Purpose of the GRE Subject Tests ........................ 3

ous factors inﬂ uence success in graduate school,

reliance on a single measure to predict success is not

Development of the Subject Tests ........................ 3

advisable. Other indicators of competence typically

Content of the Mathematics Test ........................ 4

include undergraduate transcripts showing courses

Preparing for a Subject Test .................................. 5

taken and grades earned, letters of recommendation,

and GRE General Test scores. For information about

Test-Taking Strategies .......................................... 5

the appropriate use of GRE scores, see the

What Your Scores Mean ....................................... 6

Practice Mathematics Test .................................. 9

Scoring Your Subject Test .................................. 65

Evaluating Your Performance ............................. 68

Answer Sheet...................................................... 69

Each new edition of a Subject Test is developed by

a committee of examiners composed of professors in

the subject who are on undergraduate and graduate

faculties in different types of institutions and in

different regions of the United States and Canada.

In selecting members for each committee, the

GRE Program seeks the advice of the appropriate

The GRE Subject Tests are designed to help graduate

professional associations in the subject.

school admission committees and fellowship sponsors

The content and scope of each test are speciﬁ ed

assess the qualiﬁ cations of applicants in speciﬁ c ﬁ elds

and reviewed periodically by the committee of

of study. The tests also provide you with an assessment

exam iners. Test questions are written by committee

of your own qualiﬁ cations.

members and by other university faculty members

Scores on the tests are intended to indicate

who are subject-matter specialists. All questions

knowledge of the subject matter emphasized in many

proposed for the test are reviewed and revised by the

undergraduate programs as preparation for graduate

committee and subject-matter specialists at ETS. The

study. Because past achievement is usually a good

tests are assembled in accordance with the content

indicator of future performance, the scores are helpful

speciﬁ cations developed by the committee to ensure

in predicting success in graduate study. Because the tests

adequate coverage of the various aspects of the ﬁ eld

are standardized, the test scores permit comparison

and, at the same time, to prevent overemphasis on

of students from different institutions with different

any single topic. The entire test is then reviewed and

undergraduate programs. For some Subject Tests,

approved by the committee.

subscores are provided in addition to the total score;

these subscores indicate the strengths and weaknesses

of your preparation, and they may help you plan future

studies.

PRACTICE BOOK

Subject-matter and measurement specialists on the

subscores for any test or test edition may be smaller.

ETS staff assist the committee, providing information

Subject Test score interpretive information is provided

and advice about methods of test construction and

in

helping to prepare the questions and assemble the test.

with your GRE score report. This publication is also

In addition, each test question is reviewed to eliminate

available at

language, symbols, or content considered potentially

offensive, inappropriate for major subgroups of the test-

taking population, or likely to perpetuate any negative

attitude that may be conveyed to these subgroups.

Because of the diversity of undergraduate curricula,

The test consists of approximately 66 multiple-choice

it is not possible for a single test to cover all the material

questions drawn from courses commonly offered at

you may have studied. The examiners, therefore, select

the undergraduate level. Approximately 50 percent of

questions that test the basic knowledge and skills

the questions involve calculus and its applications—

most important for successful graduate study in the

subject matter that can be assumed to be common to

particular ﬁ eld. The committee keeps the test up-to-

the backgrounds of almost all mathematics majors.

date by regularly developing new editions and revising

About 25 percent of the questions in the test are in

existing editions. In this way, the test content remains

elementary algebra, linear algebra, abstract algebra,

current. In addition, curriculum surveys are conducted

and number theory. The remaining questions deal

periodically to ensure that the content of a test reﬂ ects

with other areas of mathematics currently studied by

what is currently being taught in the undergraduate

undergraduates in many institutions.

curriculum.

The following content descriptions may assist

After a new edition of a Subject Test is ﬁ rst

students in preparing for the test. The percents given

administered, examinees’ responses to each test

are estimates; actual percents will vary somewhat from

question are analyzed in a variety of ways to determine

one edition of the test to another.

whether each question functioned as expected. These

analyses may reveal that a question is ambiguous,

requires knowledge beyond the scope of the test, or

Material learned in the usual sequence of

is inappropriate for the total group or a particular

elementary calculus courses—differential

subgroup of examinees taking the test. Such questions

and integral calculus of one and of several

are not used in computing scores.

variables—includes calculus-based applications

Following this analysis, the new test edition is

and connections with coordinate geometry,

equated to an existing test edition. In the equating

trigonometry, differential equations, and other

process, statistical methods are used to assess the

branches of mathematics

difﬁ culty of the new test. Then scores are adjusted so

that examinees who took a more difﬁ cult edition of

the test are not penalized, and examinees who took

Elementary algebra: basic algebraic techniques

an easier edition of the test do not have an advantage.

and manipulations acquired in high school and

Variations in the number of questions in the different

used throughout mathematics

editions of the test are also taken into account in this

Linear algebra: matrix algebra, systems of linear

process.

equations, vector spaces, linear transformations,

Scores on the Subject Tests are reported as three-

characteristic polynomials, and eigenvalues and

digit scaled scores with the third digit always zero.

eigenvectors

The maximum possible range for all Subject Test total

scores is from 200 to 990. The actual range of scores

Abstract algebra and number theory: elementary

for a particular Subject Test, however, may be smaller.

topics from group theory, theory of rings and

For Subject Tests that report subscores, the maximum

modules, ﬁ eld theory, and number theory

possible range is 20 to 99; however, the actual range of

PRACTICE BOOK

Introductory real analysis: sequences and

The questions in the practice test in this book

series of numbers and functions, continuity,

illustrate the types of multiple-choice questions in the

differentiability and integrability, and elementary

test. When you take the actual test, you will mark your

topology of and

answers on a separate machine-scorable answer sheet.

Discrete mathematics: logic, set theory,

Total testing time is two hours and ﬁ fty minutes; there

combinatorics, graph theory, and algorithms

are no separately timed sections. Following are some

Other topics: general topology, geometry,

general test-taking strategies you may want to consider.

complex variables, probability and statistics, and

Read the test directions carefully, and work as

numerical analysis

rapidly as you can without being careless. For

The above descriptions of topics covered in the test

each question, choose the best answer from the

available options.

understand many other related concepts. Prospective

All questions are of equal value; do not waste

test takers should be aware that questions requiring no

time pondering individual questions you ﬁ nd

more than a good precalculus background may be quite

extremely difﬁ cult or unfamiliar.

challenging; such questions can be among the most

You may want to work through the test quite

difﬁ cult questions on the test. In general, the questions

rapidly, ﬁ rst answering only the questions about

are intended not only to test recall of information but

which you feel conﬁ dent, then going back and

also to assess test takers’ understanding of fundamental

answering questions that require more thought,

concepts and the ability to apply those concepts in

and concluding with the most difﬁ cult questions

various situations.

if there is time.

If you decide to change an answer, make sure

you completely erase it and ﬁ ll in the oval

GRE Subject Test questions are designed to measure

corresponding to your desired answer.

skills and knowledge gained over a long period of time.

Questions for which you mark no answer or more

Although you might increase your scores to some extent

than one answer are not counted in scoring.

through preparation a few weeks or months before you

Your score will be determined by subtracting

take the test, last minute cramming is unlikely to be of

one-fourth the number of incorrect answers from

further help. The following information may be helpful.

the number of correct answers. If you have some

A general review of your college courses is

knowledge of a question and are able to rule out

probably the best preparation for the test.

one or more of the answer choices as incorrect,

However, the test covers a broad range of subject

your chances of selecting the correct answer are

matter, and no one is expected to be familiar

improved, and answering such questions will

with the content of every question.

likely improve your score. It is unlikely that pure

Use this practice book to become familiar with

guessing will raise your score; it may lower your

the types of questions in the GRE Mathematics

score.

Test, taking note of the directions. If you

Record all answers on your answer sheet.

understand the directions before you take the

Answers recorded in your test book will not

test, you will have more time during the test to

be counted.

focus on the questions themselves.

Do not wait until the last ﬁ ve minutes of a testing

session to record answers on your answer sheet.

PRACTICE BOOK

Your raw score — that is, the number of questions you

answered correctly minus one-fourth of the number

you answered incorrectly — is converted to the scaled

score that is reported. This conversion ensures that

Raw Scores

a scaled score reported for any edition of a Subject

Scaled Score

Form A

Form B

Form C

Test is comparable to the same scaled score earned

800

49

47

45

on any other edition of the same test. Thus, equal

700

39

36

35

scaled scores on a particular Subject Test indicate

600

28

25

25

essentially equal levels of performance regardless of

500

18

14

16

the test edition taken. Test scores should be compared

Number of Questions Used to Compute Raw Score

only with other scores on the same Subject Test. (For

66

66

66

example, a 680 on the Computer Science Test is not

*Raw Score = Number of correct answers minus one-fourth the

equivalent to a 680 on the Mathematics Test.)

number of incorrect answers, rounded to the nearest integer.

Before taking the test, you may ﬁ nd it useful

to know approximately what raw scores would be

For a particular test edition, there are many ways to

required to obtain a certain scaled score. Several

earn the same raw score. For example, on the edition

factors inﬂ uence the conversion of your raw score

listed above as “Form A,” a raw score of 28 would earn

to your scaled score, such as the difﬁ culty of the test

a scaled score of 600. Below are a few of the possible

edition and the number of test questions included in

ways in which a scaled score of 600 could be earned on

the computation of your raw score. Based on recent

the edition:

editions of the Mathematics Test, the following table

gives the range of raw scores associated with selected

scaled scores for three different test editions. (Note

that when the number of scored questions for a given

test is greater than the number of actual scaled score

points, it is likely that two or more raw scores will

Number of

convert to the same scaled score.) The three test

Questions

editions in the table that follows were selected to

Questions

Questions

Questions

Used to

reﬂ ect varying degrees of difﬁ culty. Examinees should

Raw

Answered

Answered

Not

Compute

note that future test editions may be somewhat more

Score

Correctly

Incorrectly

Answered

Raw Score

or less difﬁ cult than the test editions illustrated in the

28

28

0

38

66

table.

28

32

15

19

66

28

36

30

0

66

PRACTICE BOOK

To become familiar with how the administration will be conducted at the test center, ﬁ rst remove the

answer sheet (pages 69 and 70). Then go to the back cover of the test book (page 64) and follow the

instructions for completing the identiﬁ cation areas of the answer sheet. When you are ready to begin the

test, note the time and begin marking your answers on the answer sheet.

PRACTICE BOOK

FORM GR0568

68

GRADUATE RECORD EXAMINATIONS®

MATHEMATICS TEST

Copyright © 1999, 2000, 2003, 2005 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.

completions. In each case, select the one that is the best of the choices offered and then mark the corresponding

space on the answer sheet.

Computation and scratch work may be done in this examination book.

(1)

All

logarithms with an unspecified base are natural logarithms, that is, with base

(2) The set of all real numbers

(3)

The

symbols

,

,

, and denote the sets of integers, rational numbers, real numbers,

and complex numbers, respectively.

1. In the

cos

sin

3

(A)

3 (B)

2

2

2. Which of the following is an equation of the line tangent to the graph of

0 ?

(A)

(B)

(C)

2

(D)

2

(E)

2

1

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.

This book is provided FREE with test registration by the Graduate Record Examinations Board.

www.ets.org/gre

**Note to Test Takers:**Keep this practice book until you receive your score report.This book contains important information about scoring.

Copyright © 2008 by Educational Testing Service. All rights reserved.

ETS, the ETS logos, LISTENING. LEARNING. LEADING., GRADUATE RECORD EXAMINATIONS,

®

and GRE are registered trademarks of Educational Testing Service (ETS) in the United States of America

and other countries throughout the world.

The GRE Board recommends that scores on the

Subject Tests be considered in conjunction with other

**Table of Contents**relevant information about applicants. Because numer-

Purpose of the GRE Subject Tests ........................ 3

ous factors inﬂ uence success in graduate school,

reliance on a single measure to predict success is not

Development of the Subject Tests ........................ 3

advisable. Other indicators of competence typically

Content of the Mathematics Test ........................ 4

include undergraduate transcripts showing courses

Preparing for a Subject Test .................................. 5

taken and grades earned, letters of recommendation,

and GRE General Test scores. For information about

Test-Taking Strategies .......................................... 5

the appropriate use of GRE scores, see the

*GRE Guide*What Your Scores Mean ....................................... 6

*to the Use of Scores*at**ets.org/gre/stupubs**.Practice Mathematics Test .................................. 9

Scoring Your Subject Test .................................. 65

**Development of the**Evaluating Your Performance ............................. 68

**Subject Tests**Answer Sheet...................................................... 69

Each new edition of a Subject Test is developed by

a committee of examiners composed of professors in

the subject who are on undergraduate and graduate

**Purpose of the**faculties in different types of institutions and in

different regions of the United States and Canada.

**GRE Subject Tests**In selecting members for each committee, the

GRE Program seeks the advice of the appropriate

The GRE Subject Tests are designed to help graduate

professional associations in the subject.

school admission committees and fellowship sponsors

The content and scope of each test are speciﬁ ed

assess the qualiﬁ cations of applicants in speciﬁ c ﬁ elds

and reviewed periodically by the committee of

of study. The tests also provide you with an assessment

exam iners. Test questions are written by committee

of your own qualiﬁ cations.

members and by other university faculty members

Scores on the tests are intended to indicate

who are subject-matter specialists. All questions

knowledge of the subject matter emphasized in many

proposed for the test are reviewed and revised by the

undergraduate programs as preparation for graduate

committee and subject-matter specialists at ETS. The

study. Because past achievement is usually a good

tests are assembled in accordance with the content

indicator of future performance, the scores are helpful

speciﬁ cations developed by the committee to ensure

in predicting success in graduate study. Because the tests

adequate coverage of the various aspects of the ﬁ eld

are standardized, the test scores permit comparison

and, at the same time, to prevent overemphasis on

of students from different institutions with different

any single topic. The entire test is then reviewed and

undergraduate programs. For some Subject Tests,

approved by the committee.

subscores are provided in addition to the total score;

these subscores indicate the strengths and weaknesses

of your preparation, and they may help you plan future

studies.

*3***MATHEMATICS TEST**PRACTICE BOOK

Subject-matter and measurement specialists on the

subscores for any test or test edition may be smaller.

ETS staff assist the committee, providing information

Subject Test score interpretive information is provided

and advice about methods of test construction and

in

*Interpreting Your GRE Scores,*which you will receivehelping to prepare the questions and assemble the test.

with your GRE score report. This publication is also

In addition, each test question is reviewed to eliminate

available at

**ets.org/gre/stupubs.**language, symbols, or content considered potentially

offensive, inappropriate for major subgroups of the test-

**Content of the**taking population, or likely to perpetuate any negative

attitude that may be conveyed to these subgroups.

**Mathematics Test**Because of the diversity of undergraduate curricula,

The test consists of approximately 66 multiple-choice

it is not possible for a single test to cover all the material

questions drawn from courses commonly offered at

you may have studied. The examiners, therefore, select

the undergraduate level. Approximately 50 percent of

questions that test the basic knowledge and skills

the questions involve calculus and its applications—

most important for successful graduate study in the

subject matter that can be assumed to be common to

particular ﬁ eld. The committee keeps the test up-to-

the backgrounds of almost all mathematics majors.

date by regularly developing new editions and revising

About 25 percent of the questions in the test are in

existing editions. In this way, the test content remains

elementary algebra, linear algebra, abstract algebra,

current. In addition, curriculum surveys are conducted

and number theory. The remaining questions deal

periodically to ensure that the content of a test reﬂ ects

with other areas of mathematics currently studied by

what is currently being taught in the undergraduate

undergraduates in many institutions.

curriculum.

The following content descriptions may assist

After a new edition of a Subject Test is ﬁ rst

students in preparing for the test. The percents given

administered, examinees’ responses to each test

are estimates; actual percents will vary somewhat from

question are analyzed in a variety of ways to determine

one edition of the test to another.

whether each question functioned as expected. These

analyses may reveal that a question is ambiguous,

**Calculus—50%**requires knowledge beyond the scope of the test, or

Material learned in the usual sequence of

is inappropriate for the total group or a particular

elementary calculus courses—differential

subgroup of examinees taking the test. Such questions

and integral calculus of one and of several

are not used in computing scores.

variables—includes calculus-based applications

Following this analysis, the new test edition is

and connections with coordinate geometry,

equated to an existing test edition. In the equating

trigonometry, differential equations, and other

process, statistical methods are used to assess the

branches of mathematics

difﬁ culty of the new test. Then scores are adjusted so

that examinees who took a more difﬁ cult edition of

**Algebra—25%**the test are not penalized, and examinees who took

Elementary algebra: basic algebraic techniques

an easier edition of the test do not have an advantage.

and manipulations acquired in high school and

Variations in the number of questions in the different

used throughout mathematics

editions of the test are also taken into account in this

Linear algebra: matrix algebra, systems of linear

process.

equations, vector spaces, linear transformations,

Scores on the Subject Tests are reported as three-

characteristic polynomials, and eigenvalues and

digit scaled scores with the third digit always zero.

eigenvectors

The maximum possible range for all Subject Test total

scores is from 200 to 990. The actual range of scores

Abstract algebra and number theory: elementary

for a particular Subject Test, however, may be smaller.

topics from group theory, theory of rings and

For Subject Tests that report subscores, the maximum

modules, ﬁ eld theory, and number theory

possible range is 20 to 99; however, the actual range of

*4***MATHEMATICS TEST**PRACTICE BOOK

**Additional Topics—25%****Test-Taking Strategies**Introductory real analysis: sequences and

The questions in the practice test in this book

series of numbers and functions, continuity,

illustrate the types of multiple-choice questions in the

differentiability and integrability, and elementary

test. When you take the actual test, you will mark your

topology of and

*n*answers on a separate machine-scorable answer sheet.

Discrete mathematics: logic, set theory,

Total testing time is two hours and ﬁ fty minutes; there

combinatorics, graph theory, and algorithms

are no separately timed sections. Following are some

Other topics: general topology, geometry,

general test-taking strategies you may want to consider.

complex variables, probability and statistics, and

Read the test directions carefully, and work as

numerical analysis

rapidly as you can without being careless. For

The above descriptions of topics covered in the test

each question, choose the best answer from the

**should not be considered exhaustive**; it is necessary toavailable options.

understand many other related concepts. Prospective

All questions are of equal value; do not waste

test takers should be aware that questions requiring no

time pondering individual questions you ﬁ nd

more than a good precalculus background may be quite

extremely difﬁ cult or unfamiliar.

challenging; such questions can be among the most

You may want to work through the test quite

difﬁ cult questions on the test. In general, the questions

rapidly, ﬁ rst answering only the questions about

are intended not only to test recall of information but

which you feel conﬁ dent, then going back and

also to assess test takers’ understanding of fundamental

answering questions that require more thought,

concepts and the ability to apply those concepts in

and concluding with the most difﬁ cult questions

various situations.

if there is time.

**Preparing for a Subject Test**If you decide to change an answer, make sure

you completely erase it and ﬁ ll in the oval

GRE Subject Test questions are designed to measure

corresponding to your desired answer.

skills and knowledge gained over a long period of time.

Questions for which you mark no answer or more

Although you might increase your scores to some extent

than one answer are not counted in scoring.

through preparation a few weeks or months before you

Your score will be determined by subtracting

take the test, last minute cramming is unlikely to be of

one-fourth the number of incorrect answers from

further help. The following information may be helpful.

the number of correct answers. If you have some

A general review of your college courses is

knowledge of a question and are able to rule out

probably the best preparation for the test.

one or more of the answer choices as incorrect,

However, the test covers a broad range of subject

your chances of selecting the correct answer are

matter, and no one is expected to be familiar

improved, and answering such questions will

with the content of every question.

likely improve your score. It is unlikely that pure

Use this practice book to become familiar with

guessing will raise your score; it may lower your

the types of questions in the GRE Mathematics

score.

Test, taking note of the directions. If you

Record all answers on your answer sheet.

understand the directions before you take the

Answers recorded in your test book will not

test, you will have more time during the test to

be counted.

focus on the questions themselves.

Do not wait until the last ﬁ ve minutes of a testing

session to record answers on your answer sheet.

*5***MATHEMATICS TEST**PRACTICE BOOK

**Range of Raw Scores* Needed****What Your Scores Mean****to Earn Selected Scaled Score on**Your raw score — that is, the number of questions you

**Three Mathematics Test**answered correctly minus one-fourth of the number

**Editions that Differ in Difﬁ culty**you answered incorrectly — is converted to the scaled

score that is reported. This conversion ensures that

Raw Scores

a scaled score reported for any edition of a Subject

Scaled Score

Form A

Form B

Form C

Test is comparable to the same scaled score earned

800

49

47

45

on any other edition of the same test. Thus, equal

700

39

36

35

scaled scores on a particular Subject Test indicate

600

28

25

25

essentially equal levels of performance regardless of

500

18

14

16

the test edition taken. Test scores should be compared

Number of Questions Used to Compute Raw Score

only with other scores on the same Subject Test. (For

66

66

66

example, a 680 on the Computer Science Test is not

*Raw Score = Number of correct answers minus one-fourth the

equivalent to a 680 on the Mathematics Test.)

number of incorrect answers, rounded to the nearest integer.

Before taking the test, you may ﬁ nd it useful

to know approximately what raw scores would be

For a particular test edition, there are many ways to

required to obtain a certain scaled score. Several

earn the same raw score. For example, on the edition

factors inﬂ uence the conversion of your raw score

listed above as “Form A,” a raw score of 28 would earn

to your scaled score, such as the difﬁ culty of the test

a scaled score of 600. Below are a few of the possible

edition and the number of test questions included in

ways in which a scaled score of 600 could be earned on

the computation of your raw score. Based on recent

the edition:

editions of the Mathematics Test, the following table

gives the range of raw scores associated with selected

scaled scores for three different test editions. (Note

**Examples of Ways to Earn**that when the number of scored questions for a given

**a Scaled Score of 600 on the**test is greater than the number of actual scaled score

**Edition Labeled as “Form A”**points, it is likely that two or more raw scores will

Number of

convert to the same scaled score.) The three test

Questions

editions in the table that follows were selected to

Questions

Questions

Questions

Used to

reﬂ ect varying degrees of difﬁ culty. Examinees should

Raw

Answered

Answered

Not

Compute

note that future test editions may be somewhat more

Score

Correctly

Incorrectly

Answered

Raw Score

or less difﬁ cult than the test editions illustrated in the

28

28

0

38

66

table.

28

32

15

19

66

28

36

30

0

66

*6***MATHEMATICS TEST**PRACTICE BOOK

**P R A C T I C E T E S T**To become familiar with how the administration will be conducted at the test center, ﬁ rst remove the

answer sheet (pages 69 and 70). Then go to the back cover of the test book (page 64) and follow the

instructions for completing the identiﬁ cation areas of the answer sheet. When you are ready to begin the

test, note the time and begin marking your answers on the answer sheet.

*7***MATHEMATICS TEST**PRACTICE BOOK

FORM GR0568

68

GRADUATE RECORD EXAMINATIONS®

MATHEMATICS TEST

*Do not break the seal**until you are told to do so.**The contents of this test are conﬁ dential.**Disclosure or reproduction of any portion**of it is prohibited.***THIS TEST BOOK MUST NOT BE TAKEN FROM THE ROOM.**Copyright © 1999, 2000, 2003, 2005 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.

*9***MATHEMATICS TEST****Time—170 minutes****66 Questions****Directions:**Each of the questions or incomplete statements below is followed by five suggested answers orcompletions. In each case, select the one that is the best of the choices offered and then mark the corresponding

space on the answer sheet.

Computation and scratch work may be done in this examination book.

**Note:**In this examination:(1)

All

logarithms with an unspecified base are natural logarithms, that is, with base

*e*.(2) The set of all real numbers

*x*such that*a**x**b*is denoted by*a*,*b*.(3)

The

symbols

,

,

, and denote the sets of integers, rational numbers, real numbers,

and complex numbers, respectively.

1. In the

*xy*-plane, the curve with parametric equations*x*cos

*t*and*y*sin

*t*, 0*t**p*, has length3

*p*(A)

3 (B)

*p*(C) 3*p*(D) (E)2

2

2. Which of the following is an equation of the line tangent to the graph of

*x**y**x**e*at*x*0 ?

(A)

*y**x*(B)

*y**x*1(C)

*y**x*2

(D)

*y*2

*x*(E)

*y*2

*x*1

**Unauthorized copying or reuse of**

any part of this page is illegal.any part of this page is illegal.

**G****O****O****N O****T****E****H****T****G****A****P****T****X****E****N****.****E***10*