master sat ii math

8/18/2019 Master SAT II Math

1/404


2/404

ARCO

MASTER THESAT* SUBJECT TEST:

MATH LEVELS

1 AND 2

4th Edition

Mark N. Weinfeld

Lalit A. AhujaDavid Alan Miller


3/404

An ARCO Book

ARCO is a registered trademark of Thomson Learning, Inc., and is used herein under licenseby Thomson Peterson’s.

About Thomson Peterson’s

Thomson Peterson’s (www.petersons.com) is a leading provider of education information and advice, with books and

online resources focusing on education search, test preparation, and financial aid. Its Web site offers searchable

databases and interactive tools for contacting educational institutions, online practice tests and instruction, and planning

tools for securing financial aid. Thomson Peterson’s serves 110 million education consumers annually.

For more information, contact Thomson Peterson’s, 2000 Lenox Drive, Lawrenceville, NJ 08648;

800-338-3282; or find us on the World Wide Web at www.petersons.com/about.

© 2006 Thomson Peterson’s, a part of The Thomson Corporation

Thomson Learning™ is a trademark used herein under license.

Previous editions published as SAT II Success Math 1C and IIC © 2000, 2001, 2004

Special thanks to Joan Marie Rosebush

Editor: Wallie Walker Hammond; Production Editor: Alysha Bullock; Manufacturing

Manager: Raymond Golaszewski; Composition Manager: Gary Rozmierski

ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be

reproduced or used in any form or by any means—graphic, electronic, or mechanical,

including photocopying, recording, taping, Web distribution, or information storage and

retrieval systems—without the prior written permission of the publisher.

For permission to use material from this text or product, submit a request online at

www.thomsonrights.com

Any additional questions about permissions can be submitted by e-mail to

[email protected]

ISBN 13: 978-0-7689-2304-9

ISBN 10: 0-7689-2304-2

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1 08 07 06

Fourth Edition


4/404

Contents

Before You Begin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

How This Book Is Organized . . . . . . . . . . . . . . . . . . . . . . . . . . viiComprehensive Answer Explanations . . . . . . . . . . . . . . . . . . . viiiSpecial Study Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

You’re Well on Your Way to Success . . . . . . . . . . . . . . . . . . . ixTop 10 Ways to Raise Your Score . . . . . . . . . . . . . . . . . . . . . . x

PART I: SAT SUBJECT TEST BASICS

1 All About the SAT Subject Test: Math Levels 1and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Get to Know the Exam Format. . . . . . . . . . . . . . . . . . . . . . . . . 3 When to Use Calculators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4How the Test Is Scored . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Some Test-wise Strategies for SAT Subject Test Success . . . . 4Educated Guessing Will Boost Your Score! . . . . . . . . . . . . . . . 6

Getting Ready: The Night Before and the Day of the Test . . 6Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

PART II: DIAGNOSING STRENGTHS AND WEAKNESSES

2 Practice Test 1: Diagnostic Level 1. . . . . . . . . . . . . . . . . 11

Preparing to Take the Diagnostic Test. . . . . . . . . . . . . . . . . . . 11Practice Test 1: Diagnostic Level 1 . . . . . . . . . . . . . . . . . . . . . . 11Reference Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Answer Key and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Practice Test 2: Diagnostic Level 2. . . . . . . . . . . . . . . . . 25

Preparing to Take the Diagnostic Test. . . . . . . . . . . . . . . . . . . 25Practice Test 2: Diagnostic Level 2 . . . . . . . . . . . . . . . . . . . . . . 25Reference Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Answer Key and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . 35

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii


5/404

PART III: SAT SUBJECT TEST: MATH REVIEW

4 Multiple-Choice Math Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Solving Multiple-choice Math Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Know When to Use Your Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Learn the Most Important Multiple-choice Math Tips . . . . . . . . . . . . . . . . . . 45Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 Numbers and Operations Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Properties of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Exercise: Properties of Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Whole Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Operations with Whole Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Exercise: Whole Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Operations with Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Exercise: Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Operations with Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Decimals and Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Exercise: Decimals and Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Operations with Percents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Exercise: Percents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Exercise: Percent Word Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Systems of Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Exercise: Systems of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Signed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Exercise: Signed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Algebra and Functions Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Powers, Exponents, and Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Exercise: Exponents, Powers, and Roots . . . . . . . . . . . . . . . . . . . . . . . . . 117

Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Exercise: Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Exercise: Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

Contentsiv

www.petersons.com


6/404

Exercise: Word Problems Involving One Unknown . . . . . . . . . . . . . . . . 138 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Exercise: Equation-Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147


Exercise: Linear Inequalities and Equations. . . . . . . . . . . . . . . . . . . . . . . 156 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Exercise: Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7 Geometry and Measurement Review . . . . . . . . . . . . . . . . . . . . . . . . . 177

Plane Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Three-dimensional Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Exercise: Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Exercise: Coordinate Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219Exercise: Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225Trigonometric Functions of the General Angle. . . . . . . . . . . . . . . . . . . . . . . . 228

Exercise: Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235Exercise: Trigonometric Identity and Equations . . . . . . . . . . . . . . . . . . . 241

Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242Solving Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Exercise: Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251Exercise: Graphs of the Trigonometric Functions . . . . . . . . . . . . . . . . . 255

Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

8 Data Analysis, Statistics, Probability, and Advanced

Math Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Exercise: Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264Exercise: Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

Contents v

www.petersons.com


7/404

Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270Exercise: Permutations and Combinations. . . . . . . . . . . . . . . . . . . . . . . . 273

Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Exercise: Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

Exponents and Logarithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281Exercise: Exponents and Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Exercise: Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Systems of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296Exercise: Systems of Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299


Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301Exercise: Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

Exercise: Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

PART IV: FOUR PRACTICE TESTS

Practice Test 3: Level 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

317Reference Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

Answer Key and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

Practice Test 4: Level 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

Reference Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 Answer Key and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351





. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

Contentsvi

www.petersons.com


8/404

Before You Begin

HOW THIS BOOK IS ORGANIZED

Almost a qua rter of a million students ta ke SAT Subject Tests every y ear. In

t h e p a st , t h e se t e st s w e re k n ow n a s t h e Co lleg e B oa rd Ach ievem e n t Te st s .

These tests are important for several reasons. Because many of the colleges

requir e SAT Subject Tests, these a re importa nt exams for you. The purpose of

these tests is t o measure a nd demonstra te y our knowledge an d/or skills insp ecif ic su b je ct s a n d t o t e st you r a b il it y t o a p ply t h a t k n ow le d ge t o e a ch

particular examination. The better your score is, the better your application

w ill look t o th e colleges of your choice.

I f yo u ’re re a d in g t h is b o o k , i t ’s l ik e ly t h a t yo u a re p re p a rin g f o r t h e S A T

Subject Test Ma thema tics—either L evel 1 or Level 2. We ha ve tried t o make

this a workable book. In other words, the book is set up so that regardless of

the level exam y ou’re ta king, you w ill be able to find t he ma terial n ecessary to

study and to take those tests that are most applicable to your level.

As a f u rt h er e n h a n ce m en t t o yo ur a b il it y t o p rep a re f or t h is e x a m , w e h a ve

prepared in-depth mathematics review material and highlighted those areas

tha t a re required primarily for Level 2, so tha t t hose studying for the Level 1

test can focus on only those areas that are appropriate.

Divided int o sections, the book begins w ith t w o diagnostic exams. There is one

each for Level 1 and Level 2. Take t hese exam s (and all of t he tests) under

simulated exam conditions, if you can. Find a quiet place in which to work, set

up a clock, a nd t ake th e test w ithout stopping. When you ar e finished, ta ke a

b r ea k a n d t h en g o b a ck a n d ch eck y ou r a n s w er s . Al w a y s r er ea d t h os e

qu e st ion s you g ot w ro n g , s in ce so m et im es you r e rrors com e f rom m e relymisreading the question. Again, double-check your answers, and if they’re still

not clear, read the appropriate section in the review material.

Once you’ve completed your diagnostic test(s), it ’s t ime to move on to the

review section. Study the m at erial car efully, but feel free to skim t he portion

of the review section that is easy for you.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii


9/404

Then, ta ke the a ctual pra ctice tests. These simulated exams ar e designed t o give you a broad

spectrum of question types th at ar e similar to those you will f ind on t he a ctual S AT Subject

Tests: Ma thema tics. We suggest tha t, rega rdless of the level exam y ou are plann ing to ta ke, it

would be extremely helpful to tak e all of th e tests i n the book

. I f you ar e ta king Level 2, takingthe lower-level test will give you that much more practice for the exam. And if you are taking

Level 1, it would be helpful to test your skills and stretch your thinking to give you a stronger

grounding for the L evel 1 exam.

As you complete each exam, take some t ime to review your answers. We think you’ll f ind a

m a rk e d im provem e n t f rom t a k in g t h e d ia g n ost ic t e st s t o com p le t in g a l l of t h e f u ll-len g t h

practice tests. Always take the t ime to check the review section for clarificat ion, and if you

still don’t un derstan d t he ma terial, go to your teacher for help.

COMPREHENSIVE ANSWER EXPLANATIONSA t t h e e n d o f e a ch p ra ct ice se ssio n , re a d a l l t h e a n sw e rs a n d e x p la n a t io n s, e ve n f o r t h e

questions that you answered correctly. There are comprehensive explanations for every one of

the book’s 1,000+ questions! By reading t he an swer explana tions, you can learn from your

mistakes.

Ou r o b je ct ive is t o h e lp yo u d ra m a t ica l ly ra ise yo u r sco re s so t h a t yo u ca n m a x im iz e t h e

likelihood of getting into the college of your choice. And if you use this book properly, we can

help you reach that goal.

SPECIAL STUDY FEATURES

ARCO M aster th e SAT Su bject Test: Mat h L evels 1 and 2 is designed to be as user-friendly as

it is complete. To this end, it includes severa l feat ures t o ma ke your prepara tion much more

efficient.

Overview

Ea ch chapter begins w ith a bulleted overview list ing the t opics t o be covered in the chapter.

This will allow you to quickly target the areas in which you are most interested.

Summing It Up

Each chapter ends with a point-by-point summary that captures the most important points

conta ined in t he chapter. They a re a convenient wa y to review key points. As y ou work your

way through the book, keep your eye on the margins to find bonus information and advice.

Informat ion can be found in the following forms:

viii Before You Begin

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

www.petersons.com


10/404

Note

Notes highlight critical informa tion about t he SAT Subject Test forma t—for example, tha t t he

a n sw e rs in t h e t e st a lw a ys g o fro m sm a ller t o la rg er.

Tip

Tips d ra w you r a t t e n t ion t o va lu a b le con ce pt s, a d vice a n d sh ort cu t s f o r t a ck lin g M a t h :

Levels 1 and 2. By reading the t ips you will learn how to approach different question types,

use process-of-eliminat ion techniqu es, pa ce yourself, a nd guess m ost effectively.

Alert!

Wherever you need to be car eful of a common pit fall or test-ta ker tr ap, y ou’ll f ind a n Alert! .

This information reveals a nd eliminat es the m isperceptions a nd w rong turn s so ma ny people

ta ke on the exam. B y ta king full adva nta ge of all features presented in ARCO M aster t he SAT

Subject Test: M ath L evels 1 an d 2 you will become much more comfortable with the SAT and

considerably more confident about getting a good score.

YOU’RE WELL ON YOUR WAY TO SUCCESS

Re m e m b e r t h a t k n o w le d g e is p o w e r. B y u sin g ARCO M aster the SAT Subject Test: M ath

L evels 1 and 2 you w ill be stu dying the most comprehensive SAT Subject Tests prepara tion

guide a va ilable a nd y ou will become extremely know ledgeable about the S AT. We look forw a rd

to helping you raise your scores and improve your college prospects.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

Before You Begin ix

www.petersons.com


11/404

Top 10 Waysto Raise Your ScoreWhen it comes t o ta king t he SAT, some test -ta king skills w ill do you more good t ha n

others. There are concepts you can learn, techniques you can follow, and tricks you

can use t hat will help you to do your very best . H ere are our picks for t he top 10 wa ys

to ra ise your score:

1. Regardless of which plan you will follow, get started by reading

Chapter 1 to familiarize yourself with the test format.

2. Take the diagnostic practice tests.

3. Compute your category percentages to assess your relative strengths

and areas for improvement.

4. If you have time, read the book from cover to cover. S t a rt a t t he

b e g in n in g o r st a rt w it h t h e k in d o f qu e st io n o r t h e t o p ic t h a t yo u f in d m o st

difficult.

5. Complete the exercises in each chapter you read and assess your

performance against your diagnostic scores.

6. When you are one third of the way through your preparation, take a

practice test. Compare your scores w ith y our original r esults. Make sure you

are applying new test-taking strategies.

7. Revisit problematic chapters and chapter summaries. Th e n re a d

a d d it ion a l ch a p t e rs, d o e x ercises, a n d com p a re you r p ercen t a g e s w it h you r

original category percenta ges.

8. After you have reviewed all the chapters or all of your weaknesses,

take another practice test.

9. During the last phase of your review, go back over the practice tests.

10. The night before the SAT: RE LAX. You’ll be prepared.

10


12/404

Petersons.com/publishing

Check out our Web site at www.petersons.com/publishing to see if there is any new information

regarding the tests and any revisions or corrections to the content of this book. We’ve made sure the

information in this book is accurate and up-to-date; however, the test format or content may have

changed since the time of publication.


13/404


14/404


15/404


16/404

All About the SATSubject Test:

Math Levels 1 and 2

OVERVIEW

• Get to know the exam format

• When to use calculators

• How the test is scored

• Some test-wise strategies for SAT subject test success

• Educated guessing will boost your score!

• Getting ready: the night before and the day of the test

• Summing it up

GET TO KNOW THE EXAM FORMAT

Ea ch of the S AT Subject Tests in Ma thema tics (Level 1 an d Level 2) is simila r

in format. They both contain 50 multiple-choice questions. In addit ion, the

current versions of the tests includes a background q uestionna ire on the first

page. The College Board uses this information for stat ist ical purposes; your

answers will not affect your scores in any way.

The Level 1 examinat ion tests ma terial tha t covers t he following topics:

• Algeb ra (30%)

• G eometr y (plan e Euclidean, t hree-dimensional, coordina te) (38%)

• B a sic Trigonomet ry (8%)

• Algebra ic funct ions (12%)

• E le m en t a ry st a t ist ics (p rob a b ili t y, cou n t in g p ro blem s, d a t a in t erp ret a -

tion, mea n, m edian , a nd mode) (6%)

• M iscella n e ou s t o pics (log ic, n u m b er t h e ory, a ri t h m e t ic, g e om e t ric

seq uen ces) (6%)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c

h a p

t er 1

3


17/404

The Level 2 examinat ion tests ma terial tha t covers th e following topics:

• Algeb ra (18%)

• G eometr y (coordina te a nd th ree-dimensional) (20%)

• Tr igon omet ry (20%)

• Fu nct ions (24%)

• St at istics (probability, permuta tions, combinat ions) (6%)

• Miscella neous topics (logic a nd proof, num ber th eory, sequences, limit s) (12%)

You ca n q uickly see where your focus sh ould be for ea ch level, depending upon the percenta ge

of questions that are on the tests.

WHEN TO USE CALCULATORS

You ma y use a lmost a ny scientific or gra phic calculator, and it is est imat ed tha t y ou will f ind

it useful, a nd often necessar y, for a bout 40 percent of th e questions on the Level 1 exam and

for about 60 percent on the Level 2 exam. You should, therefore, be very familiar with the

operation of the calculat or you plan to bring to th e exam .

HOW THE TEST IS SCORED

While it ’s not imperative that you completely understand how the test is scored, since the

p ro ce ss sh o u ld n ’ t d e t e r yo u f ro m t ryin g t o d o yo u r b e st , yo u a re p ro b a b ly a w a re t h a t t h e

scores are reported on the 200–800 range.

Ea ch question a nsw ered correctly receives one point . You lose a fraction of a point for each

incorrect answer. However, you do not lose points if you don’t answer a question. (If you skip

an y a nsw ers, make sure that the next question you answ er is f illed in on your answ er page in

the correct space.) Thus, it ma kes sense to guess at those questions t hat you don’t know, a nd

of course, as w ith most m ultiple-choice questions, you should use t he process of elimina tion to

increase the odds of guessing correctly. The more choices you eliminate, the better your odds

ar e for choosing the correct an swer.

SOME TEST-WISE STRATEGIES FOR SAT SUBJECT TEST SUCCESS

What makes some people better test-takers than others? The secret isn’t just knowing the

su b je ct ; i t ’s k n ow in g sp ecif ic t e st -t a k in g st ra t e gies t h a t ca n a d d u p t o e xt ra p oin t s . Th is

means psyching out the t est , knowing how t he test-ma kers think a nd w hat they’re looking for,

and using this knowledge to your advantage. Smart test-takers know how to use pacing and

guessing t o add points to their score.

4 PART I: SAT Subject Test Basics

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

www.petersons.com


18/404

Pace Yourself

You know there a re 50 questions in one of the mat h sections a nd t hey need to be answ ered in

60 minutes. That means tha t y ou ha ve 1 minute and 10 seconds to answ er each question. B ut

sma rt t est-ta kers know t hat ’s not the best w ay to use their t ime. If you use less tha n a minuteto answer the easier questions, you’ll have extra t ime to help you answer the more difficult

ones. Tha t ’s wh y lear ning t o pace yourself is so importa nt .

Question Sets Usually Go from Easiest to Most Difficult—You Should, Too

SAT questions follow the pattern of easiest to hardest . Work your way through the earlier,

e a sier qu e st ion s a s qu ick ly a s you ca n . Th a t w a y you ’l l h a ve m ore t im e f o r t h e la t e r, m o re

difficult ones.

B ut t wo w ords of caut ion. Wha t ma y be easy t o the test-wr iter ma y not be to you. Don’t pa nic

if question 3 seems h ar d. Try to w ork through it , but don’t spend t oo much t ime on it if i t ’s at o pic su ch a s f a ct orin g t h a t h a s ju st n e ve r b e en e a sy f or you t o u n d e rst a n d . S e con d , w ork

quickly but carefully. Don’t work so fast that you make a silly mistake and lose a point that

you should ha ve aced.

You Can Set Your Own Speed Limit

All right , how will you know what your speed limit is? Use the practice tests to check your

timing and see how it affects your answers. I f you’ve answered most of the questions in the

time limit , but also have a lot of incorrect answers, better slow down. On the other hand, if

you are very accurate in your answers but aren’t answering every question in a section, you

can probably pick up th e pace a bit .

It’s Smart to Keep Moving

It’s hard to let go, but sometimes you have to. Don’t spend too much time on any one question

before you’ve tried a ll the questions in a section. There ma y be questions later on in t he test

that you can answer easily, and you don’t want to lose points just because you didn’t get to

them.

You Don’t Have to Read the Directions

Wha t? Yes, you read it correctly th e first time—you don’t ha ve to read th e directions. L ook, bythe t ime you actua lly sit down to t ake the S AT Subject Test , you’ve read this book, y ou’ve

taken all the practice tests you could find, and you’ve read enough SAT directions to fill a

l ib ra ry. S o w h e n t h e e x a m clo ck st a rt s t ick in g , d o n ’ t w a st e t im e re re a d in g d ire ct io n s yo u

alrea dy know. Instea d, go directly to Question 1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

Chapter 1: All About the SAT Subject Test: Math Levels 1 and 2 5

www.petersons.com


19/404

You’re Going to Need a Watch

If you’re going to pace yourself , you need to keep track of the t ime—and what if there is no

clock in your room or if th e only clock is out of your line of vision? Tha t’s w hy it’s a good idea

to bring a wa tch to the test . Aw ord of wa rning: don’t use a w at ch alarm or your w at ch will endup on the proctor’s desk.

EDUCATED GUESSING WILL BOOST YOUR SCORE!

T h e f ra ct io n a l d e d u ct io n f o r w ro n g a n sw e rs m a k e s ra n d o m g u e ssin g a w a sh — st a t ist ica l ly

speaking, you’re unlikely to change your score. This means that if you come to a question that

you have a bsolutely no idea how t o answ er, you’re probably better off skipping it and moving

on , ra t h e r t h a n ju st ch oosin g a n a n sw e r a t ra n d om .

Although random guessing won’t help you, anything better than random guessing will . Onmost questions, you should be able to guess better t han ra ndomly by using common sense and

the process-of-eliminat ion techniques tha t ar e developed throughout this book. Even if y ou

aren’t certain which answer is correct , you might be certain that one or more of the answer

choices is definitely wrong. If you can knock out one choice out of five, you have a 25 percent

chance of guessing correctly. If you can knock out two choices, the odds go up to 331

3 percent .

If y ou can kn ock out t hree, you ha ve a 50/50 cha nce of guessing t he right an sw er. With odds

like this, it ma kes sense to guess, especially wh en you realize tha t a single correct guess can

ra ise your scaled score a s much a s 10 points.

Obviously, the better you get at eliminating implausible choices, the more points you will pickup from educated guessing. But no matter what , if you have t ime to read through a question

and eliminate at least one choice, it is always to your benefit to guess.

Don’t worry about the fact that you will probably get the question wrong, because you don’t

need to guess correctly on too many questions to gain points. As long as you are guessing

better t han ra ndomly, you w ill do considerably better.

GETTING READY: THE NIGHT BEFORE AND THE DAY OF THE TEST

If you follow the guidelines in this book, you will be extremely well prepared for the SAT

Subject Test . You will know the format inside and out; you will know how t o approach every

type of question; you will ha ve worked har d to strengthen your weak a reas; a nd you will have

taken multiple practice tests under simulated testing conditions. The last 24 hours before the

SAT Subject Test is not t he t ime to cram—it’s a ctually the t ime to r elax.

Tha t sa id, there a re plenty of steps you can ta ke over the fina l 24 hours t o get ready. For one

thing, don’t do anything too stressful. On the night before the big day, find a diversion to keep

yourself from obsessing a bout the SATS ubject Test. Ma ybe sta y home and w a tch some of your


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

ALERT!Don’t spin your

wheels by

spending too

much time on

any one question.

Give it some

thought, take

your best shot,

and move along.

www.petersons.com


20/404

f a vo ri t e t e le visio n sh o w s. Or g o o u t t o a n e a rly m o vie . Or t a lk f o r h o u rs a n d h o u rs o n t h e

phone about another subject . Do whatever is best for you. J ust make sure you get plenty of

sleep.

You should also lay out the following items before you go to bed:

• Registration ticket: Un le ss yo u a re t a k in g t h e t e st a s a “st a n d b y” t e st e r , yo u sh o u ld

ha ve received one of t hese in the ma il.

• Identification: A driver’s license is prefera ble, but an ything wit h a picture w ill do.

• Pencils: Make sure you bring at least three number-2 pencils; those are the only pencils

t h a t t h e m a ch in es ca n re a d .

• Calculator: B oth tests r equire the use of a calculator. You should bring a calculator w ith

which you are most familiar, but don’t bring a calculator that makes noise, requires an

electrical outlet, or is part of a cell phone.

• Wristwatch: Your classroom should have an operational clock, but if it doesn’t, you want

to come prepared. Again, don’t w ear a wa tch tha t beeps, unless you can t urn off the ala rm

function. You w on’t be allowed to w ear a noise-ma king w at ch during t he exam.

Make sure you allow enough t ime to a rrive at the test sit e at least 15 minutes before the sta rt

t ime. You don’t wa nt to ra ise your level of a nxiety by ha ving to rush to get there.

I n t h e m or n in g , t a k e a s h ow e r t o w a k e u p a n d t h en e a t a s en s ib le b r ea k f a s t . I f y o u a r e a

person who usua lly eats breakfast , you should probably ea t y our customary meal. I f y ou don’t

u su a lly e a t b re a k f a st , d o n ’ t g o rg e yo u rse lf o n t e st d a y, b e ca u se i t w il l b e a sh o ck t o yo u r

system. Eat something light (like a granola bar and a piece of fruit) and pack a snack.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

Chapter 1: All About the SAT Subject Test: Math Levels 1 and 2 7

www.petersons.com


21/404

SUMMING IT UP

The following information is the foundation on which you’re going to build as you prepare for

the SAT Subject Test: M at h L evels 1 an d 2.

• Lear ning the SAT quest ion ty pes is the best wa y to prepare for the SAT. Knowing the test

format and question types will relieve test anxiety because you’ll know exactly what to

expect on test day.

• You may use a calculator on t he SAT, but wh ile it ma y be helpful for some problems, it

isn’t absolutely n ecessary for an y of th em. So use your calculator, but don’t rely on it too

much.

• Questions are usually arra nged from easiest to most difficult .

• Every S AT question is worth 1 point , wh ether it is an easy question or a difficult one. So

nail the ea sier questions—and quickly a ccumulat e points.

• Fill in the an sw er bubbles cleanly a nd completely. Otherw ise you won’t get credit for your

a n sw e rs.

• Ra ndom guessing will have no effect on your score, but educa ted guessing can boost your

score. So if you’ve read t hrough a question and can eliminate a t least one an swer, guess!

• P ace yourself and move through the test relat ively quickly. But you can get a very good

score even if you don’t answer every question.

• Relax th e day before the SAT, but a lso be sure you’re prepared.

• Assemble the supplies you will need for t he test .

• P ick out w hat you’ll wear a nd remember to layer your clothes.

• B e sure your calculator has fresh batt eries.

• On the morning of the test , eat breakfast , pack your snack, and leave for the test site in

plenty of t ime t o get t here 15 minutes before the st ar t t ime.

Remember that because you are working through this book, you’re giving yourself the best

prepara tion a va ilable for succeeding on th e SAT Subject Test: Ma th Levels 1 and 2. Let your

prepar at ion give you th e confidence you need to be calm and focused dur ing th e test .


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

www.petersons.com


22/404

PART II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .DIAGNOSING STRENGTHSAND WEAKNESSES

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 2 Practice Test 1:

Diagnostic Level 1

CHAPTER 3 Practice Test 2:

Diagnostic Level 2


23/404


24/404

Practice Test 1:

Diagnostic Level 1

B efore you begin preparing for t he SAT, it’s importa nt to know your st rengt hs

an d t he a reas wh ere you need improvement. Taking the D iagnostic Test in

this chapter and then working out your scores will help you determine how

you should a pportion y our study t ime.

PREPARING TO TAKE THE DIAGNOSTIC TESTIf possible, ta ke the test in one sit t ing. Doing this w ill give you an idea of how

it feels to t ake t he test in one hour. You will a lso get a sense of how long y ou

can spend on each q uestion, so you can begin t o work out a pa cing schedule for

yourself.

First , assemble all the things you will need to take the test . These include:

• Lined paper for the essay, at least t wo sheets

• Number 2 pencils, at least three

• A calculator with fresh bat teries

• A timer

• The answ er sheet , page 15

S e t t h e t im e r f o r o n e h ou r. S t ick t o t h e t im e , so yo u a re sim u la t in g t h e re a l

t e st . A t t h is p o in t , i t ’s a s im p o rt a n t t o k n o w h o w m a n y qu e st io n s yo u ca n

answer in the t ime allotted as it is to answer questions correctly.

PRACTICE TEST 1: DIAGNOSTIC LEVEL 1

Wh il e y ou h a v e t a k en m a n y s t a n da r d iz ed t es t s a n d k now t o b la ck en

completely the ovals on th e answ er sheets an d t o era se completely a ny errors,

th e instru ctions for t he SAT Subject Test: Ma thema tics Level 1 exam differs in

t h re e im p o rt a n t w a ys f ro m t h e d ire ct io n s f o r o t h e r st a n d a rd iz e d t e st s yo u

ha ve ta ken. You need to indicat e on th e answ er key which test you are t aking.

The instructions on th e answ er sheet of the actua l test will tell you to fill out

the top portion of the answer sheet exactly as shown.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p

r a ct i c et e s t 1

11


25/404

1. P rin t M ATHE M ATI CS L E V E L 1 o n t h e l in e t o t h e r igh t u n d er t h e w o rd s Subject

Test (print).

2. I n t h e sh a d e d box la b e le d Test Code fill in four ovals:

—Fill in oval 3 in t he row labeled V.

—Fill in oval 2 in t he row labeled W.

—Fill in oval 5 in the r ow la beled X.

—Fill in oval A in the row labeled Y.

—Leave the ovals in row Q blank.

Test Code

V ÞO1 ÞO2 Þ ÞO4 ÞO5 ÞO6 ÞO7 ÞO8 ÞO9

W ÞO1 Þ ÞO3 ÞO4 ÞO5 ÞO6 ÞO7 ÞO8 ÞO9

X ÞO1 ÞO2 ÞO3 ÞO4 Þ Y Þ ÞOB ÞOC ÞOD ÞOE

Q ÞO1 ÞO2 ÞO3 ÞO4 ÞO5 ÞO6 ÞO7 ÞO8 ÞO9

Subject Test (print)

MATHEMATICS LEVEL 1

D u rin g t h e a ct u a l t e st , w h e n e ve ryo n e h a s co m p le t e d f i l l in g in t h is p o rt io n o f t h e a n sw e r

s h e e t , t h e s u p e r v i s o r w i l l t e l l y o u t o t u r n t h e p a g e a n d b e g i n t h e M a t h e m a t i c s L e v e l 1

examination. The answer sheet has 100 numbered ovals on the sheet , but there are only 50

multiple-choice questions on the test , so be sure to use only ovals 1 to 50 to record your

a n sw e rs.

REFERENCE INFORMATION

The follow ing informa tion is for your r eference in ans w ering some of the qu estions on this test .

Volume of a right circular cone with ra dius r and height h :

V 51

3pr 2h

L a t e ra l a re a o f a r ig h t circu la r co n e w it h circu m f e re n ce o f t h e b a se c a n d s l a n t

height l :

S 51

2cl

S u rf a ce a re a o f a sp h e re w it h ra d iu s r :

S 5 4pr 2

Volume of a pyramid with base area B and height h :

V 51

3B h

Volume of a sphere w ith r adius r :

V 543pr

3

12 PART II: Diagnosing Strengths and Weaknesses

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

www.petersons.com


26/404

Notes

You will need a calculator to a nsw er some (but not a ll) of the q uestions. You must

decide whether or not t o use a calculator for each question. You must use a t least a

s ci en t if ic ca l cu la t or ; y ou a r e per m it t ed t o u se g r a ph in g or pr og r a m ma b lecalculators.

Degree measure is the only a ngle measure used on this test . B e sure your calculator

is set in degree mode.

The figures tha t accompany questions on t he test ar e designed t o give you informa-

tion that is useful in solving problems. They are drawn as accurately as possible

E X CE PT w h e n st a t e d t h a t a f ig u re is n o t d ra w n t o sca le . Un le ss o t h e rw ise in d i-

cated, a ll f igures lie in plan es.

The doma in of an y funct ion f is a ssumed to be the set of all real numbers x for which

f (x ) is a real number, unless otherwise specified.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

Chapter 2: Practice Test 1: Diagnostic Level 1 13

www.petersons.com


27/404


28/404

ANSWER SHEET PRACTICE TEST 1: DIAGNOSTIC LEVEL 1

1. OA OB OC OD OE

2. OA OB OC OD OE

3. OA OB OC OD OE

4. OA OB OC OD OE

5. OA OB OC OD OE

6. OA OB OC OD OE

7. OA OB OC OD OE

8. OA OB OC OD OE

9. OA OB OC OD OE

10. OA OB OC OD OE

11. OA OB OC OD OE

12. OA OB OC OD OE

13. OA OB OC OD OE

14. OA OB OC OD OE

15. OA OB OC OD OE

16. OA OB OC OD OE

17. OA OB OC OD OE

18. OA OB OC OD OE

19. OA OB OC OD OE

20. OA OB OC OD OE

21. OA OB OC OD OE

22. OA OB OC OD OE

23. OA OB OC OD OE

24. OA OB OC OD OE

25. OA OB OC OD OE

26. OA OB OC OD OE

27. OA OB OC OD OE

28. OA OB OC OD OE

29. OA OB OC OD OE

30. OA OB OC OD OE

31. OA OB OC OD OE

32. OA OB OC OD OE

33. OA OB OC OD OE

34. OA OB OC OD OE

35. OA OB OC OD OE

36. OA OB OC OD OE

37. OA OB OC OD OE

38. OA OB OC OD OE

39. OA OB OC OD OE

40. OA OB OC OD OE

41. OA OB OC OD OE

42. OA OB OC OD OE

43. OA OB OC OD OE

44. OA OB OC OD OE

45. OA OB OC OD OE

46. OA OB OC OD OE

47. OA OB OC OD OE

48. OA OB OC OD OE

49. OA OB OC OD OE

50. OA OB OC OD OE

------------

-------------

--------------------------------------

--

✁ an s w

er s h e e

t

15


29/404


30/404

MATHEMATICS LEVEL 1

Directions: F or e a ch of t h e f ol low i n g p rob lem s , i de nt i fy t h e B E S T a n s w e r of t h e

choices given. If the exact numerical value is not one of the choices, select the answ ertha t is closest t o this va lue. Then fill in th e corresponding oval on t he a nsw er sheet .

1. I f x 5 2, then (x 2 1)(2x 2 3) 5

(A) 1(B) 3(C) 21(D) 0(E) 23

2. In t he figure below, the side of theinside square is x ; the side of the

outside square is x 1 2. How muchgreater is the area of the outside

squ a re t h a n t h e a re a o f t h e in sid e

squ a re ?

(A) 4(B) 4x (C) 4x 1 4(D) 4x 1 1(E) x 1 4

3. Given the figure below, what is thea re a in t e rm s of a a n d b of the large

squ a re ?

c

(A) a 2 1 b 2

(B) (a 1 b )2

(C) (a 2 b )2

(D) c 2

(E) (a 2 2b )2 1 b 2

4. What is the sine of 30 degrees?

(A)1

2

(B)3

2

(C) =2

2

(D) =3

2

(E) 2

di a g

n o s t i ct e

s t

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Chapter 2: Practice Test 1: Diagnostic Level 1 17

www.petersons.com


31/404

5. A gumball ma chine conta ins 40 bluegum balls, 20 red gumba lls, 15 green

gumballs, a nd 25 purple gumballs.

What is the probability that a person

gets a red gumball?(A) .10(B) .20(C) .25(D) .15(E) .30

6. Find x if x

41

~x 1 1!

3 5

3

2.

(A) 17

2

(B) 15

7

(C) 2

(D) 1

(E) 22

7. At what point(s) does f(x) 5x 2 1 3x 1 2 intersect the x -axis?

(A) (2, 0) a nd (1, 0)(B) (0, 2) a nd (0, 1)(C) (0, 21) and (0, 22)(D) (21, 0) and (22, 0)

(E) (21, 0) and (2, 0)

8. Sequential a rra ngements of blocks areformed according to a pattern. Each ar-

ra ngement aft er the first one is gener-

a t e d b y a d d in g a ro w o f squ a re s t o t h e

b ot t o m of t h e p reviou s a rra n g em e n t ,a s sh ow n in t h e f ig ure t o t h e r ig h t . I f

this pa tt ern continues, w hich of the fol-

lowing gives t he number of squa res in

t he n t h a rra n g e m e n t ?

(A) 2n 2

(B) 2(2n 2 1)

(C) n ~n 1 1!

2

(D) n (n 1 1)

(E) n 2 1 1

9. I f t h e d ist a n ce b et w e en t h e t w opoints P (a , 3) and Q (4, 6) is 5, t hen

find a .

(A) 0(B) 24(C) 8

(D) 24 a n d 0(E) 0 a n d 8

10. Line m h a s a n e ga t ive slop e a n d anegative y -intercept. Line n is

parallel to line m a n d h a s a p osit ive

y -int ercept. The x -intercept of n

m u st b e

(A) n e ga t ive a n d g re a t e r t h a n t h ex -intercept of m

(B) n e ga t ive a n d le ss t h a n t h ex -intercept of m

(C) zero(D) p osi t ive a n d g re a t e r t h a n t h ex -intercept of m

(E) p osi t ive a n d less t h a n t h ex -intercept of m


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

www.petersons.com


32/404

11. Given the list of numbers {1, 6, 3, 9,16, 11, 2, 9, 5, 7, 12, 13, 8}, w ha t is

the median?

(A) 7

(B) 8(C) 9(D) 11(E) 6

12. I f f(x) 5 x 1 1 a n d g(x) 5 x 1 1, thenfind f(g(x) ).

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

13. The equation x 2 1 9 5 2y 2 i s a nexample of which of the following

curves?

(A) hyperbola(B) circle(C) ellipse(D) parabola(E) line

14. I f x 1 1 , 0 then ?x ? is

(A) 0(B) x 1 1

(C) 2x (D) 2x 2 1(E) x

15. The an gle of the sun a bove the horizonis 27.5 degrees. Find th e approxima te

length of the shadow of a person who

is 4.75 feet ta ll.

(A) 4.75(B) 2.47(C) 4.65(D) 9.12

(E) 4.86

16. Define the x -coordina te of themidpoint between (21, 1) and (4, 1)a s m a n d d e f in e t h e y -coordinate of

the midpoint between (4, 1) and (4, 3)

a s n

. Wh a t a re t h e x

- a n d y

-coordi-nates of (m , n )?

(A) S52, 32D(B) S52, 2D(C) S32, 32D(D) S32, 2D(E) (4, 1)

17. What is the slope of the line that isperpendicular to y 2 2x 5 1?

(A) 2

(B) 22

(C) 1

2

(D) 21

2

(E) 1

18. Trees are t o be plant ed inside acircular tree orchard so that there

ar e 5 trees per squa re meter. The

circumference of the tree orchard is

30 meters. I f trees are available only

in a llotments of packages of 6, how

m a n y a l lot m e n t s w ill t h e ca re t a k e r

need to purchase?

(A) 59(B) 117(C) 58

(D) 60(E) 118

di a g

n o s t i ct e

s t

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .


www.petersons.com


33/404

19. P lace the following list of numberswit h t he given labels in order of

g re a t e st t o le a st .

F 5 (1010)10

G 5 10101010

H 510100

1010

I 5 10

(A) F, G, H , I (B) G, F, H , I (C) F, H , G, I (D) H , F, G, I (E) G, H , F, I

20. The statement “y va ries inverselyw it h t h e squ a re of x ” m e a n s

(A) y 5 k x

(B) y 5k

x

(C) y 5 k

(D) y 5 k

x 2

(E) y 5 k x 2

21. Suppose one takes a tetra hedronA B C D w i t h b a s e A CD a s sh o w n

below and combines it with another

tetra hedron by connecting their

corresponding ba ses. How m an y

dist inct edges does the r esult ing

object ha ve?

C

B

DA

(A) 9(B) 12(C) 10(D) 11(E) 8

22. The area of the para llelogra m A B C D below is

(A) 20(B) 10

(C) 5=3(D) 10=3

(E) 1823. I f f (x ) 5 3x 2 1 a n d i f f 21 is t h e

inverse function of f , w h a t i s f 21(5)?

(A) 18

(B) 4

3

(C) 5

3

(D) 2

(E) 0

24. I f (x 2 1)x 2

5 1 a n d x is n o t e qu a l t o1, then all the possible values of

x a re

(A) 0(B) 2(C) 22(D) 0 a n d 2(E) None of the a bove

25. Wha t is t he sum of the first 40 evenpositive integers?

(A) 1,600(B) 1,560(C) 820(D) 1,640(E) 400


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

www.petersons.com


34/404

ANSWER KEY AND EXPLANATIONS

1. A

2. C3. A

4. A

5. B

6. C

7. D8. C

9. E

10. D

11. B

12. C13. A

14. C

15. D

16. D

17. D18. D

19. C

20. D

21. A

22. A23. D

24. D

25. D

1. The correct answer is (A). I f x 5 2 then (x 2 1)(2x 2 3) 5 (2 2 1)(2(2) 2 3) 5 1(1) 5 1.

2. The correct answer is (C). The area of the larger square is (x 1 2)2, a n d t h e a re a o ft h e sm a ller squ a re is x 2. Hence, the area of the larger square is (x 1 2)2 2 x 2 5 4x 1 4g rea t e r t h a n t h e a re a of t h e sm a ller squ a re .

3. The correct answer is (A). Use t h e P yt h a g o rea n t h e ore m o n o n e of t h e t r ia n g les t oge t c 2 5 a 2 1 b 2.

4.The correct answer is (A). U se a 30-60-90 special tr ian gle to g et s in(30) 5

1

2.

5. The correct answer is (B). The probability that a person gets a red gumball is just

the number of red gumballs divided by the total number of gumballs. That is, 20

100.

6. T he correct answer is (C). I f x

41

~x 1 1!

3 5

3

2 then add the two algebraic fractions by

first gett ing a common denominat or of 12 and then a dding the t wo fra ctions. The result

o f t h i s i s 7x 1 4

12 5

3

2. N ow m u lt i pl y b ot h s id es of t h e e q u a t i on b y 12 t o cl ea r t h e

denominators. The result of this is 7x 1 4 5 18, and now solving for x , w e g et t h e

desired result .

7. The correct answer is (D). S et f (x ) 5 0 a n d s ol ve b y f a ct or i ng a n d u s in g t h ezero-fact or property. Tha t is, f (x ) 5 (x 1 2)(x 1 1) 5 0 then x 1 2 5 0 a n d x 1 1 5 0,w h ich sa ys t h e x -intercepts are (22, 0) an d (21, 0).

8. The correct answer is (C). Since we ar e adding t he numbers from 1 up to the number

t h a t n equals, we use the formula n ~n 1 1!

2 .

9. The correct answer is (E). The distan ce between the t wo points P and Q is defined to

b e 5. Th a t is , =~a 2 4!2 1 ~3 2 6!2 5 5. To f in d t h e a n sw e r, w e n e ed t o so lve t h isequat ion for a . S o, a f t er s q u a r i ng b ot h s id es a n d s im pl if y in g , w e g et t h e eq u a t i on

(a 2 4)2 5 16. So, a 5 0 a n d 8.

10. The correct answer is (D). If l ine m h a s a n e ga t ive slop e a n d a n e ga t ive y -intercept,i t h a s a n e ga t ive x -intercept. Since line n is par allel to line m with a posit ive y -intercept

w e k n ow t h a t l ine n will have a posit ive x -intercept. So, the answer follows.

11. The correct answer is (B). G iven t he list , f irst wr ite the list in increasing order fromt h e le a st n u m b e r t o t h e g re a t e st n u m b e r a n d t h e n f in d t h e n u m b e r t h a t h a s a n e qu a l

n u m ber of n u m be rs on e a ch s id e. Th u s, w e f in d t h a t 8 i s t h e n u m b er t h a t h a s 6

n u m b ers le ss t h a n i t a n d 6 g rea t e r t h a n i t .

an s w

er s

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .


www.petersons.com


35/404

12. The correct answer is (C). E v a l u a t e f a t g (x ) t o g e t f (g (x )) 5 g (x ) 1 1 , a n d n o w substitute g (x ) to get the answer.

13. The correct answer is (A). J ust r ewrite th e equation to 2y 2 2 x 2 5 9 to see that it is

a hyperbola. The equa tion 2y

2

2 x 2

5 9 ca n b e w ri t t e n a s

y 2

1 2

x 2

2 5

9

2, or equivalently,y

2

9

2

2x

2

9 5 1, which represents the general formula of a h yberbola.

14. The correct answer is (C). I f x 1 1 , 0 t h e n x , 21. S o , t h e a b so lu t e va lu e o f anegative number is defined to be the sa me number but posit ive. So, ?x ? 5 2x .

15. The correct answer is (D). G iven the informa tion, w e can dra w the following t riangle:

T h e a n sw e r ca n b e f o u n d b y lo o k in g a t t h e t a n g e n t o f 27.5 a n d so lvin g f o r x . The

equation becomes tan 27.5° 54.75

x , so x 5

4.75

tan27.5°

16. The correct answer is (D). Ca lcu la t e t h e a ve ra g e o f t h e x coordinates of 21 a n d 4(define as m ) a n d a lso ca lcula t e t h e a ve ra g e o f t h e y coordina tes 1 and 3 (define a s n ).

So, (m , n ) is given by choice (D).

17. The correct answer is (D). Write in s lope-intercept form to get the slope of th e givenl i n e t o b e 2 5 m 1. T h u s, t h e l in e t h a t is p e rp e n d icu la r t o t h e g ive n l in e h a s slo p e

21

m 15 2

1

2.

18. The correct answer is (D). Circumference is given by the formula C 5 2pr . So, using

the given information, we can calculate tha t r 5 30

~2p!. Now plug r into A 5 pr 2 to get th e

ar ea in meters squar ed. We know t ha t th ere ar e 5 trees per squar e meter a nd there are

6 trees in a package, so multiply the last result by 5 and divide by 6. That is, 5A

6 '

5

6~71.62! ' 59.68. Therefore, we would need to buy 60 packages.

19. The correct answer is (C). By exponent rules, F 5 10100, G 5 1020, H 5 1090, a n dI 5 10. So, the order of greatest to least is choice (C).

20. The correct answer is (D). Value of y va rie s in ve rse ly w it h t h e squ a re o f x , w h ich

m e a n s y 5 k

x 2.

21. The correct answer is (A). One tetra hedron ha s 6 edges, an d if w e place tw o togethera lo n g t h e b a ses of t h e t w o t e t ra h e d ron s, t h e y sh a re 3 com m on e d ge s f or a t o t a l o f

12 2 3 5 9 edges.

22. The correct answer is (A). The area of the parallelogram is just 4(5) 5 20.


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

www.petersons.com


36/404

23. The correct answer is (D). f 21(5) implies that f (x ) 5 5, wh ich sa ys 3x 2 1 5 5 or x 5 2.

24. The correct answer is (D). For (x 2 1)x 2

5 1, either x 2 1 5 61 or x 2 5 0 because theonly va lues th at can be ra ised to a n on-zero power to yield 1 a re 11 a n d 21. x 2 1 5 1⇒ x 5 2; x 2 1 5 21 ⇒ x 5 0; x 2 5 0 ⇒ x 5 0; hence x 5 2 or x 5 0.

25. The correct answer is (D). 2 1 4 1 6 1 .. . 1 80 5 2[1 1 2 1 3 1 .. . 1 40]. The sum

of t he first n positive integers is n ·~n 11!

2 , so t h e t o t a l is 2F40·41

2 G 5 1,640.

an s w

er s

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .


www.petersons.com


37/404


38/404

Practice Test 2:

Diagnostic Level 2

B efore you begin preparing for t he SAT, it’s importa nt to know your st rengt hs

an d t he a reas wh ere you need improvement. Taking the D iagnostic Test in

this chapter and then working out your scores will help you determine how

you should a pprotion y our study t ime.

PREPARING TO TAKE THE DIAGNOSTIC TESTIf possible, ta ke the test in one sit t ing. Doing this w ill give you an idea of how

it feels to t ake t he test in one hour. You will a lso get a sense of how long y ou

can spend on each q uestion, so you can begin t o work out a pa cing schedule for

yourself.

First , assemble all the things you will need to take the test . These include:

• Lined paper for the essay, at least t wo sheets

• Number 2 pencils, at least three

• A calculator with fresh bat teries

• A timer

• The answ er sheet , page 29

S e t t h e t im e r f o r o n e h ou r. S t ick t o t h e t im e , so yo u a re sim u la t in g t h e re a l

t e st . A t t h is p o in t , i t ’s a s im p o rt a n t t o k n o w h o w m a n y qu e st io n s yo u ca n

answer in the t ime allotted as it is to answer questions correctly.

PRACTICE TEST 2: DIAGNOSTIC LEVEL 2

Wh il e y ou h a v e t a k en m a n y s t a n da r d iz ed t es t s a n d k now t o b la ck en

completely the ovals on th e answ er sheets an d t o era se completely a ny errors,

th e instru ctions for t he SAT Subject Test: Ma thema tics Level 1 exam differs in

t h re e im p o rt a n t w a ys f ro m t h e d ire ct io n s f o r o t h e r st a n d a rd iz e d t e st s yo u

ha ve ta ken. You need to indicat e on th e answ er key which test you are t aking.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p

r a ct i c et e s t 2

25


39/404

The instructions on t he a nswer sheet of the a ctual t est w ill tell you t o fill out t he top portion

of the answer sheet exactly as shown.

1. P rin t M ATHE M ATI CS L E V E L 2 o n t h e l in e t o t h e r igh t u n d er t h e w o rd s Subject

Test (print).

2. I n t h e sh a d e d box la b e le d Test Code fill in four ovals:

—Fill in oval 5 in t he row labeled V.

—Fill in oval 3 in t he row labeled W.

—Fill in oval 5 in the r ow la beled X.

—Fill in oval E in the row labeled Y.

—Leave the ovals in row Q blank.

Test Code

V ÞO1 ÞO2 ÞO3 ÞO4 Þ ÞO6 ÞO7 ÞO8 ÞO9

W ÞO1 ÞO2 Þ ÞO4 ÞO5 ÞO6 ÞO7 ÞO8 ÞO9

X ÞO1 ÞO2 ÞO3 ÞO4 Þ Y ÞOA ÞOB ÞOC ÞOD Þ

Q ÞO1 ÞO2 ÞO3 ÞO4 ÞO5 ÞO6 ÞO7 ÞO8 ÞO9

Subject Test (print)

MATHEMATICS LEVEL 2

Dur ing the actu a l test, wh en everyone has completed filling in this port ion of th e answ er sheet,

the supervisor w ill tell you to turn the page a nd begin the Ma thema tics Level 2 exam inat ion.

The answer sheet has 100 numbered ovals on the sheet, but there are only 50 multiple-choice

questions on t he test , so be sure to use only ovals 1 t o 50 to record your a nsw ers.

REFERENCE INFORMATION

The follow ing informa tion is for your r eference in ans w ering some of the qu estions on this test .

Volume of a right circular cone with ra dius r and height h :

V 51

3pr 2h

L a t e ra l a re a o f a r ig h t circu la r co n e w it h circu m f e re n ce o f t h e b a se c a n d s l a n t

height l :

S 51

2cl

S u rf a ce a re a o f a sp h e re w it h ra d iu s r :

S 5 4pr 2

Volume of a pyramid with base area B and height h :

V 51

3B h

Volume of a sphe

master sat ii math

Documents