About the Authors Richard Ku has been teaching secondary mathematics, including Algebra and 2, Geometry, Precalculus, AP Calculus, and AP Statistics, for almost 30 years He has coached math teams for 15 years and has also read AP Calculus exams for years and began reading AP Statistics exams in 2007 Howard P Dodge spent 40 years teaching math in independent schools before retiring Acknowledgments I would like to dedicate this book to my wonderful wife, Doreen I would also like to thank Barron’s editor Pat Hunter for guiding me through the preparation of this new edition R.K © Copyright 2012, 2010, 2008 by Barron’s Educational Series, Inc Previous edition © Copyright 2003, 1998 under the title How to Prepare for the SAT II: Math Level IIC Prior editions © Copyright 1994 under the title How to Prepare for the SAT II: Mathematics Level IIC and © Copyright 1991, 1987, 1984, 1979 under the title How to Prepare for the College Board Achievement Test—Math Level II by Barron’s Educational Series, Inc 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 e-ISBN: 978-1-4380-8377-3 e-Book revision: August, 2012 Contents Introduction PART DIAGNOSTIC TEST Diagnostic Test Answer Key Answers Explained Self-Evaluation Chart for Diagnostic Test PART REVIEW OF MAJOR TOPICS Functions 1.1 Overview Definitions Exercises Combining Functions Exercises Inverses Exercises Odd and Even Functions Exercises Answers and Explanations 1.2 Polynomial Functions Linear Functions Exercises Quadratic Functions Exercises Higher-Degree Polynomial Functions Exercises Inequalities Exercises Answers and Explanations 1.3 Trigonometric Functions and Their Inverses Definitions Exercises Arcs and Angles Exercises Special Angles Exercises Graphs Exercises Identities, Equations, and Inequalities Exercises Inverse Trig Functions Exercises Triangles Exercises Answers and Explanations 1.4 Exponential and Logarithmic Functions Exercises Answers and Explanations 1.5 Rational Functions and Limits Exercises Answers and Explanations 1.6 Miscellaneous Functions Parametric Equations Exercises Piecewise Functions Exercises Answers and Explanations Geometry and Measurement 2.1 Coordinate Geometry Transformations and Symmetry Exercises Conic Sections Exercises Polar Coordinates Exercises Answers and Explanations 2.2 Three-Dimensional Geometry Surface Area and Volume Exercises Coordinates in Three Dimensions Exercises Answers and Explanations Numbers and Operations 3.1 Counting Venn Diagrams Exercise Multiplication Rule Exercises Factorial, Permutations, Combinations Exercises Answers and Explanations 3.2 Complex Numbers Imaginary Numbers Exercise Complex Number Arithmetic Exercises Graphing Complex Numbers Exercises Answers and Explanations 3.3 Matrices Addition, Subtraction, and Scalar Multiplication Exercises Matrix Multiplication Exercises Determinants and Inverses of Square Matrices Exercises Solving Systems of Equations Exercises Answers and Explanations 3.4 Sequences and Series Recursive Sequences Arithmetic Sequences Geometric Sequences Series Exercises for Sequences and Series Answers and Explanations 3.5 Vectors Exercises Answers and Explanations Data Analysis, Statistics, and Probability 4.1 Data Analysis and Statistics Measures and Regression Exercises Answers and Explanations 4.2 Probability Independent Events Mutually Exclusive Events Exercises Answers and Explanations PART MODEL TESTS Model Test Answer Key Answers Explained Self-Evaluation Chart Model Test Answer Key Answers Explained Self-Evaluation Chart Model Test Answer Key Answers Explained Self-Evaluation Chart Model Test Answer Key Answers Explained Self-Evaluation Chart Model Test Answer Key Answers Explained Self-Evaluation Chart Model Test Answer Key Answers Explained Self-Evaluation Chart Summary of Formulas 47 * (B) Temperature increases by 3.4°F for each additional chirp Therefore, additional chirps indicate an increase of 5(3.4) = 17.0°F [1.2] 48 * (C) Complete the square on the ellipse formula, and put the equation in standard form: x2 – 4x + + 4(y2 + 2y – 1) = 28 + + This leads to the length of the major axis: = 12 Therefore, the radius of the circle is 6, and the area = 36π 113 [2.1] 49 (C) Since the velocity of a 45-mile-per-hour wind is times that of a 15-mile-per-hour wind and the force on the sail is proportional to the square of the wind velocity, the force on the sail of a 45-mile-per-hour wind is times that of a 15-mile-per-hour wind: · 45 = 405 [algebra] 50 * (B) Total horizontal distance traveled = (4)(8) = 32 Total vertical distance traveled = (5)(6) = 30 If a coordinate system is superimposed on the diagram with A at (0,0), then B is at (32,30) Use the program on your calculator to find the distance between two points to compute the correct answer choice [2.1] Self-Evaluation Chart for Model Test Evaluate Your Performance Model Test Rating Excellent Very good Above average Average Below average Number Right 41–50 33–40 25–32 15–24 Below 15 Calculating Your Score Raw score R = number right – (number wrong), rounded = Approximate scaled score S = 800 – 10(44 – R) = If R 44, S = 800 Summary of Formulas CHAPTER 1: FUNCTIONS 1.2 Polynomial Functions Linear Functions General form of the equation: Ax + By + C = Slope-intercept form: y = mx + b, where m represents the slope and b the y-intercept Point-slope form: y – y1 = m(x – x1), where m represents the slope and (x1,y1) are the coordinates of some point on the line Slope: , where (x1,y1) and (x2,y2) are the coordinates of two points Parallel lines have equal slopes Perpendicular lines have slopes that are negative reciprocals If m1 and m2 are the slopes of two perpendicular lines, m1 · m2 = –1 Distance between two points with coordinates (x1,y1) and ( x2 , y2 ) = Coordinates of the midpoint between two points = Distance between a point with coordinates (x1,y1) and a line Ax + By + C = = If is the angle between two lines, tan lines , where m1 and m2 are the slopes of the two Quadratic Functions General quadratic equation: ax2 + bx + c = General quadratic formula: General quadratic function: y = ax2 + bx + c Coordinates of vertex: Axis of symmetry equation: Sum of zeros (roots) = Product of zeros (roots) = Nature of zeros (roots): If b2 – 4ac < 0, two complex numbers If b2 – 4ac = 0, two equal real numbers If b2 – 4ac > 0, two unequal real numbers 1.3 Trigonometric Functions and Their Inverses Length of arc in circle of radius r and central angle is given by Area of sector of circle of radius r and central angle is given by Trigonometric Reduction Formulas In any ABC: Law of sines: Law of cosines: Area = 1.4 Exponential and Logarithmic Functions Exponents Logarithms LogbN = x if and only if bx = N 1.6 Miscellaneous Functions Absolute Value If x 0, then |x| = x If x < 0, then |x| = –x Greatest Integer Function [x] = i, where i is an integer and i x