SAT Math Level I-II. Problem solving & SAT Math.
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We believe that the present work should help considerably the students to their maths examinations, either for the mathematical competitions as well as in SAT math Level 2 exams.
In case you are going to take the SAT math Level 2 examinations, then:
• 1. Read some pages from the first part of this book. You will learn techniques on How you can help your mind generate more ideas in problem solving.
Then go to the other part.
• 2. Here, there is a selection of problems for SAT Mathematics examinations, from all chapters.
The solutions are presented in a very easy way, using simple and effective techniques.
• 3. We apply, in a very simple and instructive way, knowledge from Cognitive Psychology (see: How the mind works, 1980) which is the easiest and fastest way to learn and remember. In few days, or in two weeks, you can go through this point, and then you will gain
much self – confidence in solving problems.
• 4. If you already have any textbook, use it parallel to this part. You will very soon be able to achieve a high score in the examinations. The more you practise, the greater the speed of the brain increases. Mathematics will be very easy for you, it will be a
pleasure . . .

1 Introduction 1
1.1 Technique A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 More applications of Technique A . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Categories & a longer discussion on short notes . . . . . . . . . . . . . . . . 9
1.4 Applications – Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Technique B – How to get out of loops . . . . . . . . . . . . . . . . . . . . . 17
1.6 A few applications to problems for mathematical competitions . . . . . . . . 20
2 Topic 1 – Arithmetic-Algebra & Geometry 35
2.1 Problems on percentages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Ratio & Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2.1 Examples & Techniques for problems on directly, or, inversely propor-
tional values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 Inverse, odd-even & Combining functions . . . . . . . . . . . . . . . . . . . . 48
2.3.1 Odd – Even Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.2 Combining functions & inverse functions . . . . . . . . . . . . . . . . 53
2.4 Absolute value & Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.5 Geometry & Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.5.1 A summary of geometric formulas . . . . . . . . . . . . . . . . . . . . 71
2.5.2 Geometry Examples & Problems . . . . . . . . . . . . . . . . . . . . 75
3 Topic 2 – Functions 91
3.1 Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.1.1 General form of the equation . . . . . . . . . . . . . . . . . . . . . . . 92
3.1.2 Slope formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.1.3 Parallel lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.1.4 Perpendicular lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.1.5 Midpoint & Distance of points . . . . . . . . . . . . . . . . . . . . . . 93
3.1.6 Distance of a point from a straight line . . . . . . . . . . . . . . . . . 93
3.1.7 Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.2 Examples of Linear functions & graphs . . . . . . . . . . . . . . . . . . . . . 94
3.2.1 Multiple Choice Problems . . . . . . . . . . . . . . . . . . . . . . . . 98
3.3 Quadratic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.3.1 Problems on quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.4 Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.5 Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.5.1 The remainder & factor theorems . . . . . . . . . . . . . . . . . . . . 123
3.5.2 Problems on the remainder theorem . . . . . . . . . . . . . . . . . . . 124
3.6 Rational Functions & Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.6.1 Domains of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.6.2 How to find limits – 2 mnemonic rules . . . . . . . . . . . . . . . . . . 142
3.7 Exponential & Logarithmic function . . . . . . . . . . . . . . . . . . . . . . 150
3.7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.7.2 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.7.3 Problem Solving now . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.8 Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4 Topic 3 169
4.1 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.1.1 Basic formulae for trigonometry . . . . . . . . . . . . . . . . . . . . . 169
4.1.2 Examples & problems for trigonometry . . . . . . . . . . . . . . . . . 172
4.2 Analytic & Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . 186
4.2.1 Basic equations – Analytic Geometry . . . . . . . . . . . . . . . . . . 186
4.2.2 Problems on 2 dimensional coordinate plane . . . . . . . . . . . . . . 188
4.2.3 Polar Coordinates in plane geometry . . . . . . . . . . . . . . . . . . 201
4.2.4 3 – dimensional geometry . . . . . . . . . . . . . . . . . . . . . . . . . 203
5 Topic 4 – Sequences – Complex numbers – Vectors – Matrices 207
5.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.1.2 Arithmetic sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.1.3 Geometric sequences or Progressions . . . . . . . . . . . . . . . . . . 213
5.1.4 Multiple Choice Problems . . . . . . . . . . . . . . . . . . . . . . . . 215
5.2 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
5.2.1 Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
5.3 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
5.3.1 Basic theory on vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 228
5.3.2 Examples & problems on vectors . . . . . . . . . . . . . . . . . . . . 228
5.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
5.4.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
6 Topic 5 – Data Analysis – Combining & Probabilities 253
6.1 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
6.2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
6.2.1 Mutually exclusive events . . . . . . . . . . . . . . . . . . . . . . . . 258
6.3 Some practice now . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
6.4 Counting rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
6.5 Sample Space & Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 274
7 Practice Tests 285
7.1 Practice Test A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
7.1.1 Solutions of Practice Test A . . . . . . . . . . . . . . . . . . . . . . . 292
7.2 Practice Test B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
7.2.1 Solutions of Practice Test B . . . . . . . . . . . . . . . . . . . . . . . 312
Appendices 327
Contents
A To the Editors 329
A.1 Comments regarding Kalomitsines’ ideas . . . . . . . . . . . . . . . . . . . . 329