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Welcome to the Math Level 2 Subject Test This chapter will help you get familiar with this book and learn how to use it most effectively. We’ll also talk about when to take the test and how to determine whether to take Math Level 1 or 2. (So, if you’re flipping through this book in the bookstore, this chapter’s for you) New for this edition, we at The Princeton Review have created individual books for the Math Level 1 and the Math Level 2 Subject Tests. Though you’ll notice overlap if you’re working through both books, you’ll also find that creating separate books has let us go further indepth with many topics. In addition, chapters 310 now end with comprehensive drills to test your knowledge of the material in each chapter. We’re excited to bring you this updated, improved book. So let’s get started
Editorial Rob Franek, Senior VP, Publisher Casey Cornelius, VP Content Development Mary Beth Garrick, Director of Production Selena Coppock, Managing Editor Calvin Cato, Editor Colleen Day, Editor Aaron Riccio, Editor Meave Shelton, Editor Orion McBean, Editorial Assistant Random House Publishing Team Tom Russell, Publisher Alison Stoltzfus, Publishing Manager Melinda Ackell, Associate Managing Editor Ellen Reed, Production Manager Kristin Lindner, Production Supervisor Andrea Lau, Designer The Princeton Review 24 Prime Parkway, Suite 201 Natick, MA 01760 E-mail: editorialsupport@review.com Copyright © 2014 by TPR Education IP Holdings, LLC All rights reserved Published in the United States by Penguin Random House LLC, 1745 Broadway, New York, NY 10019, and in Canada by Random House of Canada, a division of Penguin Random House Ltd., Toronto Cover art © Seamartini/Alamy eBook ISBN: 978-0-8041-2561-1 Trade Paperback ISBN: 978-0-8041-2560-4 SAT is a registered trademark of the College Board, which does not sponsor or endorse this product The Princeton Review is not affiliated with Princeton University Editor: Calvin S Cato Production Editor: Harmony Quiroz Production Artist: Deborah A Silvestrini v3.1 Acknowledgments Thanks to Tom Watts, Morgan Chase and Alexandra Schaffer for their work on previous iterations of this title Special thanks to Jonathan Chiu, Aaron Lindh, and their content development team for their hard work in the creation of the new SAT Math Subject Test book Special thanks to Adam Robinson, who conceived of and perfected the Joe Bloggs approach to standardized tests, and many other techniques in the book Contents Cover Title Page Copyright Acknowledgments …So Much More Online! Part I: Orientation Introduction Strategy Part II: Content Review Algebra Algebra on the Math Level Subject Test Solving Equations Factoring and Distributing Plugging In Plugging In The Answers Inequalities Working with Ranges Direct and Indirect Variation Work and Travel Questions Simultaneous Equations FOIL Method Factoring Quadratics The Quadratic Formula Graphing Calculator to the Rescue! Comprehensive Algebra Drill Summary Fundamentals Definitions Doing Arithmetic Fractions, Decimals, and Percentages Averages Irrational Numbers Exponents Roots Special Exponents More Important Exponent Stuff Comprehensive Fundamentals Drill Summary Plane and Solid Geometry Definitions Plane Geometry Formulas Prisms Rectangular Solids Cubes Cylinders Cones Spheres Pyramids Tricks of the Trade Comprehensive Plane and Solid Geometry Drill Summary Coordinate Geometry Definitions The Coordinate Plane The Equation of a Line Linear Inequalities General Equations Triaxial Coordinates: Thinking in 3D Comprehensive Coordinate Geometry Drill Summary Trigonometry Definitions The Basic Functions Trigonometric Identities Graphing Trigonometric Functions Trigonometry in Non-Right Triangles Polar Coordinates Comprehensive Trigonometry Drill Summary Functions Definitions Weird Symbols as Functions Functions Using Standard Notation Compound Functions Inverse Functions Domain and Range Functions Within Intervals: Domain Meets Range Graphing Functions Range and Domain in Graphs Roots of Functions in Graphs Symmetry in Functions Degrees of Functions Comprehensive Functions Drill Summary Statistics and Sets Definitions Working with Statistics Probability Permutations, Combinations, and Factorials Group Questions Comprehensive Statistics and Sets Drill Summary 10 Miscellaneous Logarithms Visual Perception Arithmetic and Geometric Sequences Limits Vectors Logic Imaginary Numbers Polynomial Division Matrices Comprehensive Miscellaneous Drill Summary Part III: Drills: Answers and Explanations Chapter 3: Algebra Drill Explanations Chapter 4: Fundamentals Drill Explanations Chapter 5: Plane and Solid Geometry Drill Explanations Chapter 6: Coordinate Geometry Drill Explanations Chapter 7: Trigonometry Drill Explanations Chapter 8: Functions Drill Explanations Chapter 9: Statistics and Sets Drill Explanations Chapter 10: Miscellaneous Drill Explanations Part IV: Practice Tests 11 Practice Test 12 Practice Test 1: Answers and Explanations Practice Test Answer Key Practice Test Explanations How to Score Practice Test 13 Practice Test 14 Practice Test 2: Answers and Explanations Practice Test Answer Key Practice Test Explanations How to Score Practice Test About the Authors Double click on the image to enlarge Part I Orientation Introduction Strategy 10 smallest value in list M You don’t know anything about the modes of the two lists, so cross off (C) For (D), suppose that each element in M is exactly greater than the corresponding element in N The ranges would be identical, so you can cross off (D) In that same situation, the standard deviations of the two lists would be identical, so you can cross off (E) 43 D This problem looks scary, but it’s really just a matter of Plugging In Each answer choice says that the first term is 3, and then each one gives a different definition of the sequence In the given sequence a0 = 3, a1 = 5, a2 = 8.333, and a3 = 13.889 This obviously isn’t an arithmetic sequence, because the difference between the terms is not consistent So Plug In values of n to see if the terms you are given fit the equations in each answer choice If you plug n = into answer choice (A), you get a0 + = a0 + Well, = + 2, so this seems to work Now plug n = into answer choice (A) This gives a1 + = a1 + But 8.333 ≠ + 2, so (A) doesn’t define the sequence In (B), Plug In n = again, which gives a0 + = 2a0 – This works, because = 2(3) – But when you try n = 1, you get a1 + = 2a1 – 1, which is wrong, because 8.333 ≠ 2(5) – Cross off (B) Repeating this process for answer choice (C), Plug In n = 1, which fails Choice (D) works for all the terms given It turns out that this is a geometric sequence with a constant factor of 44 D The statement 0≤n≤ tells you that you’re working in the first quadrant of the unit circle where both sine and cosine are never negative The unit also tells you that you’ll be working with angles in radians, not degrees Make sure your calculator is in the correct mode The question tells you that the cosine of the cosine of n is 0.8 To find n, just take the inverse cosine of 0.8, and then take the inverse cosine of the result You should get 0.8717— that’s n If you get an error, your calculator is probably in degree mode Finally, take the tangent of 0.8717 You should get 1.1895 45 C This cylinder has a radius of n (because n is half the diameter) and a height of Just plug these values into the formula for the surface area of a cylinder, SA = 2πr2 + 2πrh You get 2πn2 + πn2, or 3πn2 794 46 A PITA Plug In each answer choice for θ to see which one makes the equation true Make sure your calculator is in radian mode Only (A) works 47 A A good way to tackle this one is by trying to disprove each of the answer choices If you start with (A), you’re lucky There’s no way to divide by another quantity and get zero; it’s the right answer Even if you weren’t sure, the other answer choices are pretty easy to disprove Just set equal to a quantity prohibited by each answer choice, and solve for x If there’s a real value of x that solves the equation, then the value is in the range after all, and the answer choice is incorrect Another method is to graph the function on your calculator and see what y-values seem impossible 48 C cscθ , so rewrite the given equation as That means sin θ = 3t Now, Plug In an easy number for θ, such as 30° So you have sin 30° = 3t Therefore, t = 0.167 The question asks for cos 30°, which is 0.866, the target number Now, Plug In 0.167 for t in the answer choices to see which one becomes 0.866 Choice (C) is the only one that comes close 49 C There’s no sophisticated math here It’s just an annoying function question with a lot of steps As usual with I, II, III questions, tackle the statements one at a time and remember the Process of Elimination Statement I must be true —if x = 0, then the function comes out to which equals no matter what y is (since y can’t be zero) Answer choice (B) can be eliminated since it doesn’t contain Statement I Statement II must be true; since the value is being plugged into the function in the x and y positions, the function will always equal , or Answer choices (A) and (D) can be eliminated because they don’t include Statement II Finally, Statement III is not necessarily true; f(x,y) = = and f(y,x) = If x and y have different values, then these expressions will not be equal Answer choice (E) can thus be eliminated, because Statement III is false That leaves only (C) 50 B Start by Plugging In Because there are no numerical values given for m and b, you can plug in numbers as long as they satisfy the equation m = –b3 Make m = –8 and b = The line’s equation is therefore y = –8x + At this 795 point, it helps to draw the triangle: As shown, the equation gives you an x-intercept of 0.25 and a yintercept of If you rotate this triangle around the x-axis, you will create a cone with a radius of and a height of 0.25 Using the formula for volume of a cone (V = of the cone is πr2h), you find that the volume Therefore, the answer is choice (B) 796 HOW TO SCORE PRACTICE TEST When you take the real exam, the proctors will collect your test booklet and bubble sheet and send your bubble sheet to a processing center where a computer looks at the pattern of filled-in ovals on your bubble sheet and gives you a score We couldn’t include even a small computer with this book, so we are providing this more primitive way of scoring your exam Determining Your Score STEP Using the answer key, determine how many questions you got right and how many you got wrong on the test Remember: Questions that you not answer don’t count as either right answers or wrong answers STEP List the number of right answers here (A) STEP List the number of wrong answers here Now divide that number by (Use a calculator if you’re feeling particularly lazy.) (B) ÷ = (C) STEP Subtract the number of wrong answers divided by from the number of correct answers Round this score to the nearest whole number This is your raw score (A) − (C) = STEP To determine your real score, take the number from Step and look it up in the left column of the Score Conversion Table on the next page; the corresponding score on the right is your score on the exam 797 PRACTICE TEST SCORE CONVERSION TABLE 798 799 800 801 802 803 About the Authors Jonathan Spaihts was born in 1970 He is a graduate of Princeton University, and by pure coincidence works for The Princeton Review as a teacher, researcher, and writer In that capacity, he has helped to develop Princeton Review courses for the SAT Math Subject Tests and a number of other standardized tests He may also be seen in thrilling full-motion video on The Princeton Review’s Inside the SAT and Inside the GRE CD-ROMs When not working for The Princeton Review, Jonathan creates various arcane writings of his own Aaron Lindh has been with The Princeton Review since 2007 Based in Toronto and teaching both in person and worldwide through TPR’s LiveOnline division, Aaron teaches and tutors myriad subjects, including the SAT, ACT, and SAT Subject Tests He is also a Master Trainer for the SAT and LSAT Aaron’s work with The Princeton Review has garnered rave reviews, with students praising his patience and expertise He was Canada’s Tutor of the Year for 2012 and 2013 Furthermore, Aaron is an experienced member of The Princeton Review’s Research and Development division, creating content for the GMAT, GRE, Redesigned SAT, and College Admission Counseling programs in addition to this title Aaron holds a Bachelor of Music in Composition degree from the University of Wisconsin and a Bachelor of Civil Law and Bachelor of Laws degrees from McGill University When not teaching, Aaron enjoys hiking, reading, and painting miniatures 804 805 806 Click here to download a PDF of the Answer Sheets 807 808 ... you The Math Level 2, on the other hand, is not for just anyone—it’s a much harder test The great majority of students who take a Math Subject Test choose to take the Math Level Taking the Math. .. graduate school The Math Level Subject Test has 50 multiple-choice questions and is one hour long The test is scored from 20 0 to 800 points The Math Level Subject Test covers a range of mathematical... can take both tests 13 WHEN SHOULD I TAKE THE MATH LEVEL SUBJECT TEST? The right time to take the Math Level Subject Test varies from person to person Many students take the test at the end of a