BARRON’S SAT* SUBJECT TEST MATH LEVEL 4TH EDITION Ira K Wolf, Ph.D President, PowerPrep, Inc Former High School Math Teacher, College Professor of Mathematics, and University Director of Teacher Preparation * SAT is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product About the Author Dr Ira Wolf has had a long career in math education In addition to teaching math at the high school level for several years, he was a professor of mathematics at Brooklyn College and the Director of the Mathematics Teacher Preparation program at SUNY Stony Brook Dr Wolf has been helping students prepare for the PSAT, SAT, and SAT Subject Tests in Math for 35 years He is the founder and president of PowerPrep, a test preparation company on Long Island that currently works with more than 1,000 high school students each year © Copyright 2012, 2010, 2008 by Barron’s Educational Series, Inc Previous edition © Copyright 2005 under the title How to Prepare for the SAT II: Math Level 1C All rights reserved No part of this work may be reproduced or distributed in any form or by any means without the written permission of the copyright owner All inquiries should be addressed to: Barron’s Educational Series, Inc 250 Wireless Boulevard Hauppauge, New York 11788 www.barronseduc.com eISBN: 978-1-4380-8376-6 Second eBook publication: August, 2012 Contents What You Need to Know About SAT Subject Tests What You Need to Know About the SAT Subject Test in Math Level TEST-TAKING STRATEGIES Important Tactics An Important Symbol Used in This Book Calculator Tips The Inside Scoop for Solving Problems TOPICS IN ARITHMETIC Basic Arithmetic The Number Line Absolute Value Addition, Subtraction, Multiplication, Division Integers Exponents and Roots Squares and Square Roots Logarithms PEMDAS Exercises Answers Explained Fractions, Decimals, and Percents Fractions Arithmetic Operations with Fractions Arithmetic Operations with Mixed Numbers Complex Fractions Percents Percent Increase and Decrease Exercises Answers Explained Ratios and Proportions Ratios Proportions Exercises Answers Explained ALGEBRA Polynomials Polynomials Algebraic Fractions Exercises Answers Explained Equations and Inequalities First-Degree Equations and Inequalities Absolute Value, Radical, and Fractional Equations and Inequalities Quadratic Equations Exponential Equations Systems of Linear Equations The Addition Method The Substitution Method The Graphing Method Solving Linear-Quadratic Systems Exercises Answers Explained Word Problems Rate Problems Age Problems Percent Problems Integer Problems A Few Miscellaneous Problems Exercises Answers Explained PLANE GEOMETRY Lines and Angles Angles Perpendicular and Parallel Lines Exercises Answers Explained Triangles Sides and Angles of a Triangle Right Triangles Special Right Triangles Perimeter and Area Similar Triangles Exercises Answers Explained 10 Quadrilaterals and Other Polygons The Angles of a Polygon Special Quadrilaterals Perimeter and Area of Quadrilaterals Exercises Answers Explained 11 Circles Circumference and Area Tangents to a Circle Exercises Answers Explained SOLID AND COORDINATE GEOMETRY 12 Solid Geometry Rectangular Solids Cylinders Prisms Cones Pyramids Spheres Exercises Answers Explained 13 Coordinate Geometry Distance Between Two Points The Midpoint of a Segment Slope Equations of Lines Circles and Parabolas Exercises Answers Explained TRIGONOMETRY 14 Basic Trigonometry Sine, Cosine, and Tangent What You Don’t Need to Know Exercises Answers Explained FUNCTIONS 15 Functions and Graphs Relations Functions Combining Functions Composition of Functions Inverse Functions Exercises Answers Explained STATISTICS, COUNTING, AND PROBABILITY 16 Basic Concepts of Statistics, Counting, and Probability Statistics Counting Probability Exercises Answers Explained MISCELLANEOUS TOPICS 17 Imaginary and Complex Numbers Imaginary Numbers Complex Numbers Exercises Answers Explained 18 Sequences Repeating Sequences Arithmetic Sequences Geometric Sequences Exercises Answers Explained 19 Logic Statements Negations Conditional Statements Exercises Answers Explained MODEL TESTS Guidelines and Scoring 20 Model Test 21 Model Test 22 Model Test What You Need to Know About SAT Subject Tests *The importance of the College Board’s Score Choice policy* • What Are SAT Subject Tests? • How Many SAT Subject Tests Should You Take? • How Are SAT Subject Tests Scored? • How Do You Register for an SAT Subject Test? This e-Book contains hyperlinks that will help you navigate through content, bring you to helpful resources, and allow you to click between exam questions + answers *Please Note: This e-Book may appear differently depending on which device you are using Please adjust accordingly Since you are reading this book, it is likely that you have already decided to take the SAT Subject Test in Math Level 1; at the very least, you are seriously considering taking it Therefore, you probably know something about the College Board and the tests it administers to high school students: PSAT, SAT, and SAT Subject Tests In this short introductory chapter, you will learn the basic facts you need to know about the Subject Tests In the next chapter, you will learn everything you need to know about the Math Level test in particular In 2009, the College Board instituted a Score Choice policy for all SAT Subject Tests, as well as for the SAT What this means is that at any point in your high school career you can take (or even retake) any Subject Tests you want, receive your scores, and then choose whether or not the colleges to which you eventually apply will ever see those scores In fact, you don’t have to make that choice until your senior year when you are actually sending in your college applications Suppose, for example, that you take the Biology test one year and the Chemistry test the following year If you earn very good scores on both exams, then, of course, you can send the colleges both scores; if, however, your Chemistry score is much better than your Biology score, you can send the colleges only your 36 (E) The domain of f is the set of all real numbers for which f (x) is defined The only potential problem is that the denominator could be However, for any real number x, x + is positive The denominator of f (x) is never equal to 0, and the domain of f is the set of all real numbers **If you graph on a graphing calculator, you can see that all four of the answer choices (–4, –3, 3, and 4) are in the domain 37 (A) **Use TACTIC Pick any number for x and use your calculator to evaluate the given expression and each of the answer choices 38 (D) When 21, 22, and 23 are divided by 4, the remainders are 1, 2, and 3, respectively Therefore, by KEY FACT P2, i 21 = i, i 22 = i = –1, and i 23 = i = –i So i 21 is not a real number (I is false); i 21 + i 22 = i + (–1), which is also not a real number (II is false); i 21 + i 22 + i 23 = i + –1 + (–i) = –1, which is a real number (III is true) Only III is true 39 (D) By KEY FACT H11, the area of is times as great, its area is length of side AB is RST is Since the area of the shaded region and the total area of the square is Finally, the 40 (D) If s is the side of the cube, then by KEY FACT H7, the length of EG, a diagonal of the square base, is s So in right AEG, by the Pythagorean theorem, we have: So by KEY FACT K1, the volume of the cube is: 41 (E) If r + si is a root of any quadratic equation with real coefficients, then r – si is also a root (I is true) Since (r + si ) + (r – si) = 2r, the sum of the roots is 2r However, by KEY FACT E3, the sum of the roots of a quadratic equation is (II is true) Since (r + si )(r – si ) = r – s 2i = r – (s 2)(–1) = r + s 2, the product of the roots is r + s However, by KEY FACT E3, the product of the roots of a quadratic equation is (III is true) I, II, and III are true 42 (A) Use TACTIC 4: draw and label a diagram Then AOD is a 30-60-90 right triangle whose shorter leg is r, the radius of the circle, and whose longer leg, AD, is But by KEY FACT H8, AD = 43 (E) y = f (g (x)) = f (2x – 5) = 7(2x – 5) + = 14x – 35 + = 14x – 32 y = g(f (x)) = g(7x + 3) = 2(7x + 3) – = 14x + – = 14x + By KEY FACT L8, the slope of each line is equal to 14 So by KEY FACT L7, the lines are parallel and, therefore, not perpendicular (I is true and II is false) y = f (g(x)) – g(f (x)) = (14x – 32) – (14x + 1) = –33 By KEY FACT L8, y = –33 is the equation of a horizontal line (III is true) Only I and III are true 44 (B) By KEY FACT I9, the formula for the area of a parallelogram is A = bh Use side base and draw in height Then by KEY FACT M1, sin Finally: as the 45 (D) By KEY FACT R4, a conditional statement ( p q) is equivalent to its contrapositive (~p ~q) Choice D is the contrapositive of the given statement Note that whether or not the original statement is true is irrelevant In this case it isn’t and, therefore, neither is its contrapositive 46 (E) By KEY FACT N6: To find h–1(x), use KEY FACT N8 Write y = 28x + 23, switch x and y, and solve for y : 47 (B) Since each side of square BCDE is 2, hypotenuse KEY FACT H7, of isosceles right ABC is 2, and by Since is a radius of the circle whose center is A and that passes through B and C, the area of the circle is m∠A = 90° sector ABC is of the circle So its area is The area of square ABCD is 22 = and the area of of the original figure is + = 5, and the area of the shaded region is So the total area **Use TACTICS and 8: trust the diagram and eliminate absurd choices Since the area of square BCDE is 4, the area of the shaded region must be slightly less than Immediately eliminate choice A, which is greater than 4, and choice C, which is negative Choices B, D, and E are all positive numbers less than 4, but choices D and E are much too small The answer must be B 48 (E) If x pounds of peanuts are added to the existing mixture, the result will be a mixture whose total weight will be (70 + x) pounds, of which (20 + x) pounds will be peanuts Then, by expressing 60% as we have **Use TACTIC 2: backsolve If 42 pounds of peanuts are added, peanuts will be of the mixture Since that is not enough, eliminate choice C and choices A and B, which are even smaller Test D or E 49 (B) By KEY FACTS K8 and K6, the volume of a sphere is circular cone is and the volume of a right (Remember that both of these formulas are given to you on the first page of the test.) Then: **Use TACTIC Plug in a number for the radius, say r = Then the volume of the sphere is and the volume of the cone is 50 (D) Write out enough terms of the sequence until you see a pattern: 4, 5, 9, 14, 23, 37, 60, 97, 157, The sequence of odds and evens repeats indefinitely in groups of three: , , Since each group contains one even and two odds, the first 333 groups of contain 999 terms—333 of which are even and 666 of which are odd The 1,000th term is the first term of the next group and is even So in all, there are 334 even terms and 666 odd terms ...BARRON’S SAT* SUBJECT TEST MATH LEVEL 4TH EDITION Ira K Wolf, Ph. D President, PowerPrep, Inc Former High School Math Teacher, College Professor of Mathematics, and University Director of Teacher... * SAT is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product About the Author Dr Ira Wolf has had a long career in math. .. appear differently depending on which device you are using Please adjust accordingly Since you are reading this book, it is likely that you have already decided to take the SAT Subject Test in Math