My A Level Maths Notes

622 Pages·2013·4.35 MB·English

My A Level Maths Notes
Core Maths C1−C4
Kathy
Updated 07-Apr-2013
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My A Level Maths Notes
‘My A Level Core Maths Notes’
These are my private notes, and it is up to you, dear reader, to ensure the facts are correct. Although I have done
my best to proof read the notes, please use sensibly - check your facts.
The notes can be downloaded from www.brookhaven.plus.com/maths
If you see any problems please send corrections to: [e-mail]
My thanks to Fritz K for his comments and corrections.
These notes have been produced entirely on an RISC OS Iyonix computer, using Martin Würthner’s TechWriter
for the typesetting and equations. Illustrations have been created in Martin Würthner’s Artworks vector drawing
package.
See www.mw-software.com for further information.
Please check the website for the latest version.
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Contents
Preface xv
Introduction xv
Required Knowledge 17
Algebra 17
Studying for A Level 17
Meaning of symbols 17
Sets of Numbers 17
Calculators in Exams 18
Exam Tips 18
Module C1 19
Core 1 Basic Info 19
C1 Contents 19
Disclaimer 19
C1 Assumed Basic Knowledge 20
C1 Brief Syllabus 21
1 • C1 • Indices 23
1.1 The Power Rules - OK 23
1.2 Examples 24
2 • C1 • Surds 29
2.1 Intro to Surds 30
2.2 Handling Surds — Basic Rules 30
2.3 Factorising Surds 30
2.4 Simplifying Surds 30
2.5 Multiplying Surd Expressions 31
2.6 Surds in Exponent Form 31
2.7 Rationalising Denominators (Division of Surds) 32
2.8 Geometrical Applications 33
2.9 Topical Tip 34
2.10 The Difference of Two Squares 34
2.11 Heinous Howlers 34
3 • C1 • Algebraic Fractions 35
3.1 Handling Algebra Questions 35
3.2 Simplifying Algebraic Fractions 35
3.3 Adding & Subtracting Algebraic Fractions 36
3.4 Multiplying & Dividing Algebraic Fractions 37
3.5 Further Examples 38
4 • C1 • Straight Line Graphs 39
4.1 Plotting Horizontal & Vertical Lines 39
4.2 Plotting Diagonal Lines 40
4.3 The Equation of a Straight Line 41
4.4 Plotting Any Straight Line on a Graph 42
4.5 Properties of a Straight Line 43
4.6 Decoding the Straight Line Equation 47
4.7 Plotting a Straight Line Directly from the Standard Form 48
4.8 Parallel Lines 48
4.9 Straight Line Summary 49
4.10 Topical Tips 50
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5 • C1 • Geometry of a Straight Line 51
5.1 General Equations of a Straight Line 51
5.2 Distance Between Two Points on a Line 52
5.3 Mid Point of a Line Segment 52
5.4 Gradient of a Straight Line 53
5.5 Parallel Lines 54
5.6 Perpendicular Lines 55
5.7 Finding the Equation of a Line 55
5.8 Heinous Howlers 58
6 • C1 • The Quadratic Function 59
6.1 Intro to Polynomials 59
6.2 The Quadratic Function 59
6.3 Quadratic Types 60
6.4 Quadratic Syllabus Requirements 60
7 • C1 • Factorising Quadratics 61
7.1 Methods for Factorising 61
7.2 Zero Factor Property 61
7.3 Expressions with a Common Factor 61
7.4 Expressions of the form (u + v)2 = k 62
7.5 Difference of Two Squares 62
7.6 Perfect Squares 63
7.7 Finding Possible Factors 63
7.8 Quadratic Factorisation, type x2 + bx + c 64
7.9 Factorising Quadratic of Type: ax2 + bx + c 65
8 • C1 • Completing the Square 75
8.1 General Form of a Quadratic 75
8.2 A Perfect Square 75
8.3 Deriving the Square or Vertex Format 76
8.4 Completing the Square 76
8.5 Completing the Square in Use 78
8.6 Solving Quadratics 78
8.7 Solving Inequalities 78
8.8 Graphing − Finding the Turning Point (Max / Min Value) 79
8.9 A Geometric View of Completing the Square 81
8.10 Topic Digest 82
9 • C1 • The Quadratic Formula 83
9.1 Deriving the Quadratic Formula by Completing the Square 83
9.2 Examples of the Quadratic Formulae 84
9.3 Finding the Vertex 86
9.4 Heinous Howlers 86
9.5 Topical Tips 86
10 • C1 • The Discriminant 87
10.1 Assessing the Roots of a Quadratic 87
10.2 Discriminant = 0 88
10.3 Topical Tips 88
10.4 Examples 88
10.5 Complex & Imaginary Numbers (Extension) 91
10.6 Topic Digest 92
11 • C1 • Sketching Quadratics 93
11.1 Basic Sketching Rules for any Polynomial Function 93
11.2 General Shape & Orientation of a Quadratic 93
11.3 Roots of a Quadratic 93
11.4 Crossing the y-axis 94
11.5 Turning Points (Max or Min Value) 94
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Contents
11.6 Sketching Examples 95
11.7 Topical Tips 96
12 • C1 • Further Quadratics 97
12.1 Reducing Other Equations to a Quadratic 97
12.2 Reducing to Simpler Quadratics: Examples 97
12.3 Pairing Common Factors 102
13 • C1 • Simultaneous Equations 103
13.1 Solving Simultaneous Equations 103
13.2 Simultaneous Equations: Worked Examples 103
14 • C1 • Inequalities 105
14.1 Intro 105
14.2 Rules of Inequalities 105
14.3 Linear Inequalities 105
14.4 Quadratic Inequalities 106
14.5 Inequality Examples 108
14.6 Heinous Howlers 110
14.7 Topical Tips 110
15 • C1 • Standard Graphs I 111
15.1 Standard Graphs 111
15.2 Asymptotes Intro 111
15.3 Power Functions 111
15.4 Roots and Reciprocal Curves 117
15.5 Exponential and Log Function Curves 118
15.6 Other Curves 119
15.7 Finding Asymptotes 120
15.8 Worked Examples 126
16 • C1 • Graph Transformations 127
16.1 Transformations of Graphs 127
16.2 Vector Notation 127
16.3 Translations Parallel to the y-axis 128
16.4 Translations Parallel to the x-axis 128
16.5 One Way Stretches Parallel to the y-axis 130
16.6 One Way Stretches Parallel to the x-axis 131
16.7 Reflections in both the x-axis & y-axis 133
16.8 Translating Quadratic Functions 133
16.9 Translating a Circle Function 133
16.10 Transformations Summary 134
16.11 Recommended Order of Transformations 134
16.12 Example Transformations 135
16.13 Topical Tips 136
17 • C1 • Circle Geometry 137
17.1 Equation of a Circle 137
17.2 Equation of a Circle Examples 138
17.3 Properties of a Circle 139
17.4 Intersection of a Line and a Circle 140
17.5 Completing the Square to find the Centre of the Circle 142
17.6 Tangent to a Circle 143
17.7 Tangent to a Circle from Exterior Point 145
17.8 Points On or Off a Circle 147
17.9 Worked Examples 149
17.10 Circle Digest 150
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18 • C1 • Calculus 101 151
18.1 Calculus Intro 151
18.2 Historical Background 151
18.3 What’s it all about then? 151
18.4 A Note on OCR/AQA Syllabus Differences 152
19 • C1 • Differentiation I 153
19.1 Average Gradient of a Function 153
19.2 Limits 154
19.3 Differentiation from First Principles 155
19.4 Deriving the Gradient Function 156
19.5 Derivative of a Constant 157
19.6 Notation for the Gradient Function 157
19.7 Differentiating Multiple Terms 157
19.8 Differentiation: Worked Examples 158
19.9 Rates of Change 159
19.10 Second Order Differentials 160
19.11 Increasing & Decreasing Functions 161
20 • C1 • Practical Differentiation I 163
20.1 Tangent & Normals 163
20.2 Stationary Points 166
20.3 Maximum & Minimum Turning Points 167
20.4 Points of Inflection & Stationary Points (Not in Syllabus) 170
20.5 Classifying Types of Stationary Points 170
20.6 Max & Min Problems (Optimisation) 171
20.7 Differentiation Digest 176
Module C2 177
Core 2 Basic Info 177
C2 Contents 177
C2 Assumed Basic Knowledge 177
C2 Brief Syllabus 179
21 • C2 • Algebraic Division 181
21.1 Algebraic Division Intro 181
21.2 Long Division by ax + b 181
21.3 Comparing Coefficients 182
22 • C2 • Remainder & Factor Theorem 183
22.1 Remainder Theorem 183
22.2 Factor Theorem 184
22.3 Topic Digest 186
23 • C2 • Sine & Cosine Rules 187
23.1 Introduction 187
23.2 Labelling Conventions & Properties 187
23.3 Sine Rule 188
23.4 The Ambiguous Case (SSA) 190
23.5 Area of a Triangle 190
23.6 Cosine Rule 191
23.7 Bearings 194
23.8 Topic Digest 195
23.9 Digest in Diagrams 196
23.10 Heinous Howlers 196
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24 • C2 • Radians, Arcs, & Sectors 197
24.1 Definition of Radian 197
24.2 Common Angles 197
24.3 Length of an Arc 198
24.4 Area of Sector 198
24.5 Area of Segment 198
24.6 Length of a Chord 198
24.7 Worked Examples 199
24.8 Common Trig Values in Radians 201
24.9 Topical Tips 201
24.10 Topic Digest 201
25 • C2 • Logarithms 203
25.1 Basics Logs 203
25.2 Uses for Logs 204
25.3 Common Logs 204
25.4 Natural Logs 204
25.5 Log Rules - OK 205
25.6 Log Rules Revision 206
25.7 Change of Base 206
25.8 Worked Examples in Logs 207
25.9 Inverse Log Operations 209
25.10 Further Worked Examples in Logs 210
25.11 Use of Logs in Practice 212
25.12 Log Rules Digest 213
25.13 Heinous Howlers 214
26 • C2 • Exponential Functions 215
26.1 General Exponential Functions 215
26.2 The Exponential Function: e 215
26.3 Exponential Graphs 216
26.4 Translating the Exponential Function 217
26.5 The Log Function Graphs 218
26.6 Exponentials and Logs 219
26.7 Exponential and Log Worked Examples 219
27 • C2 • Sequences & Series 221
27.1 What is a Sequence? 221
27.2 Recurrence Relationship 221
27.3 Algebraic Definition 222
27.4 Sequence Behaviour 222
27.5 Worked Example 224
27.6 Series 224
27.7 Sigma Notation ∑ 225
27.8 Sequences & Series: Worked Examples 227
27.9 Finding a likely rule 228
27.10 Some Familiar Sequences 229
27.11 Sequences in Patterns 230
28 • C2 • Arithmetic Progression (AP) 231
28.1 Intro to Arithmetic Progression 231
28.2 n-th Term of an Arithmetic Progression 232
28.3 The Sum of All Terms of an Arithmetic Progression 232
28.4 Sum to Infinity of an Arithmetic Progression 233
28.5 Sum to Infinity Proof 233
28.6 Arithmetic Progression: Worked Examples 233
28.7 Heinous Howlers 236
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29 • C2 • Geometric Progression (GP) 237
29.1 The n-th Term of a Geometric Progression 237
29.2 The Sum of a Geometric Progression 238
29.3 Divergent Geometric Progressions 239
29.4 Convergent Geometric Progressions 240
29.5 Oscillating Geometric Progressions 240
29.6 Sum to Infinity of a Geometric Progression 241
29.7 Geometric Progressions: Worked Examples 241
29.8 Heinous Howlers 245
29.9 AP & GP Topic Digest 246
30 • C2 • Binomial Theorem 247
30.1 Binomials and their Powers 247
30.2 Pascal’s Triangle 247
30.3 Factorials & Combinations 249
30.4 Binomial Theorem 250
30.5 Binomial Theorem: Special Case 250
30.6 Binomial Coefficients 251
30.7 Finding a Given Term in a Binomial 252
30.8 Binomial Theorem: Worked Examples 253
30.9 Heinous Howlers 255
30.10 Some Common Expansions 255
30.11 Binomial Theorem Topic Digest 256
31 • C2 • Trig Ratios for all Angles 257
31.1 Trig Ratios for all Angles Intro 257
31.2 Standard Angles and their Exact Trig Ratios 257
31.3 The Unit Circle 258
31.4 Acute Related Angles 259
31.5 The Principal Value 260
31.6 The Unit Circle and Trig Curves 261
31.7 General Solutions to Trig Equations 262
31.8 Complementary and Negative Angles 263
31.9 Angles of 0°, 90°, 180° & 270° 263
31.10 Trig Ratios Worked Examples 264
31.11 Trig Ratios for all Angles Digest 266
32 • C2 • Graphs of Trig Functions 267
32.1 Graphs of Trig Ratios 267
32.2 Transformation of Trig Graphs 268
32.3 Transformation Summary 270
33 • C2 • Trig Identities 271
32.1 Basic Trig Ratios 271
32.2 Identity tan x ≡ sin x / cos x 271
32.3 Identity sin2x + cos2x ≡ 1 271
32.4 Solving Trig Problems with Identities 272
32.5 Trig Identity Digest 274
34 • C2 • Trapezium Rule 275
34.1 Estimating Areas Under Curves 275
34.2 Area of a Trapezium 275
34.3 Trapezium Rule 275
34.4 Trapezium Rule Errors 276
34.5 Trapezium Rule: Worked Examples 277
34.6 Topical Tips 278
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35 • C2 • Integration I 279
35.1 Intro: Reversing Differentiation 279
35.2 Integrating a Constant 279
35.3 Integrating Multiple Terms 280
35.4 Finding the Constant of Integration 280
35.5 The Definite Integral − Integration with Limits 281
35.6 Area Under a Curve 282
35.7 Compound Areas 286
35.8 More Worked Examples 288
35.9 Topical Tips 288
Module C3 289
Core 3 Basic Info 289
C3 Contents 289
C3 Assumed Basic Knowledge 290
C3 Brief Syllabus 291
36 • C3 • Functions 293
36.1 Function Intro 293
36.2 Mapping Relationships, Domain & Ranges 294
36.3 Vertical Line Test for a Function 296
36.4 Compound or Composite Functions 298
36.5 Inverse Functions 300
36.6 Horizontal Line Test for an Inverse Function 303
36.7 Graphing Inverse Functions 304
36.8 Odd, Even & Periodic Functions 305
36.9 Functions: Worked Examples 306
36.10 Heinous Howlers 308
37 • C3 • Modulus Function & Inequalities 309
37.1 The Modulus Function 309
37.2 Graphing y = f (x) 310
37.3 Graphing y = f (|x|) 311
37.4 Inequalities and the Modulus Function 312
37.5 Algebraic Properties 312
37.6 Solving Equations Involving the Modulus Function 312
37.7 Solving Modulus Equations by Critical Values 313
37.8 Squares & Square Roots Involving the Modulus Function 314
37.9 Solving Modulus Equations by Graphing 316
37.10 Solving Modulus Equations by Geometric Methods 317
37.11 Heinous Howlers 318
37.12 Modulus Function Digest 318
38 • C3 • Exponential & Log Functions 319
38.1 Exponential Functions 319
38.2 THE Exponential Function: e 320
38.3 Natural Logs: ln x 321
38.4 Graphs of ex and ln x 322
38.5 Graph Transformations of The Exponential Function 323
38.6 Solving Exponential Functions 324
38.7 Exponential Growth & Decay 324
38.8 Differentiation of ex and ln x 326
38.9 Integration of ex and ln x 326
38.10 Heinous Howler 326
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39 • C3 • Numerical Solutions to Equations 327
39.1 Intro to Numerical Methods 327
39.2 Locating Roots Graphically 328
39.3 Change of Sign in f(x) 328
39.4 Locating Roots Methodically 329
39.5 Limitations of the Change of Sign Methods 332
39.6 Iteration to find Approximate Roots 333
39.7 Staircase & Cobweb Diagrams 335
39.8 Limitations of the Iterative Methods 337
39.9 Choosing Convergent Iterations 337
39.10 Numerical Solutions Worked Examples 338
39.11 Numerical Solutions Digest 342
40 • C3 • Estimating Areas Under a Curve 343
40.1 Estimating Areas Intro 343
40.2 Trapezium Rule − a Reminder 343
40.3 Mid-ordinate Rule 344
40.4 Simpson’s Rule 346
40.5 Relationship Between Definite Integrals and Limit of the Sum 349
41 • C3 • Trig: Functions & Identities 351
41.1 Degrees or Radians 351
41.2 Reciprocal Trig Functions 351
41.3 Reciprocal Trig Functions Graphs 352
41.4 Reciprocal Trig Functions Worked Examples 353
41.5 Pythagorean Identities 354
41.6 Compound Angle (Addition) Formulae 355
41.7 Double Angle Formulae 358
41.8 Triple Angle Formulae 362
41.9 Half Angle Formulae 363
41.10 Factor Formulae 364
41.11 Topical Tips on Proving Identities 366
41.12 Trig Identity Digest 367
42 • C3 • Trig: Inverse Functions 369
42.1 Inverse Trig Functions Intro 369
42.2 Inverse Sine Function 370
42.3 Inverse Cosine Function 371
42.4 Inverse Tangent Function 372
42.5 Inverse Trig Function Summary Graphs 373
43 • C3 • Trig: Harmonic Form 375
43.1 Form of a cos x + b sin x 375
43.2 Proving the Identity 376
43.3 Geometric View of the Harmonic Form 377
43.4 Choosing the Correct Form 377
43.5 Worked Examples 378
43.6 Harmonic Form Digest 382
44 • C3 • Relation between dy/dx and dx/dy 383
44.1 Relation between dy/dx and dx/dy 383
44.2 Finding the Differential of x = g(y) 384
44.3 Finding the Differential of an Inverse Function 385
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