Advanced Level Pure Mathematics Notes Pdf

Unformatted text preview: My A Level Maths Notes Core Maths Class Notes for C1−C4 Kathy Updated 10-May-2015 i My A Level Maths Notes 'My A Level Core Maths Class Notes' These are my private class 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 If you see any problems, please send corrections to: [email protected] 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 for further information. Please check the website for the latest version. Disclaimer These are my class notes for C1 to C4 which my Dad has transcribed on to the computer for me, although he has gone a bit OTT with them! My cousin has been studying the AQA syllabus, so some of the chapters have been marked to show the differences. Although a lot of my hand written mistakes have been corrected - there may be a few deliberate errors still in the script. If you find any, then please let us know so that we can correct them. Kathy, Feb 2013 Module Navigation ii Module C1 Module C2 Module C3 Module C4 21 187 327 495 69 • Annex • Catalogue of Basic Graphs 70 • Annex • Facts, Figures & Formulæ 71 • Annex • Trig Rules & Identities 72 • Annex • Logs & Exponentials 73 • Annex • Calculus Techniques 74 • Annex • Standard Calculus Results 75 • Annex • Integration Flow Chart 76 • Annex • Set Theory Symbols 651 663 671 671 681 683 685 686 ALevelNotesv8Fbb 18-May-2015 Contents Preface xvi Introduction Required Knowledge 2 • C1 • 3 • C1 • 4 • C1 • 19 Algebra Requirements 19 Studying for A Level 19 Meaning of Some Common Symbols 19 Sets of Numbers 19 Calculators in Exams 20 Exam Tips 20 Module C1 1 • C1 • xvi 21 Core 1 Basic Info 21 C1 Contents 21 C1 Assumed Basic Knowledge 22 C1 Brief Syllabus 23 Indices & Power Rules 25 1.1 25 The Power Rules - OK Surds 31 2.1 Intro to Surds 31 2.2 Handling Surds — Basic Rules 32 2.3 Factorising Surds 32 2.4 Simplifying Surds 32 2.5 Multiplying Surd Expressions 33 2.6 Surds in Exponent Form 33 2.7 Rationalising Denominators (Division of Surds) 34 2.8 Geometrical Applications 35 2.9 The Difference of Two Squares 36 2.10 Topical Tips 36 2.11 Heinous Howlers 36 Algebraic Fractions 37 3.1 Handling Algebra Questions 37 3.2 Simplifying Algebraic Fractions 37 3.3 Adding & Subtracting Algebraic Fractions 38 3.4 Multiplying & Dividing Algebraic Fractions 39 Straight Line Graphs 41 4.1 Plotting Horizontal & Vertical Lines 41 4.2 Plotting Diagonal Lines 42 4.3 The Equation of a Straight Line 43 4.4 Plotting Any Straight Line on a Graph 44 4.5 Properties of a Straight Line 45 4.6 Decoding the Straight Line Equation 49 4.7 Plotting a Straight Line Directly from the Standard Form 50 4.8 Parallel Lines 50 4.9 Straight Line Summary 51 4.10 Topical Tips 52 iii My A Level Maths Notes 5 • C1 • 6 • C1 • 7 • C1 • Geometry of a Straight Line 53 5.1 General Equations of a Straight Line 53 5.2 Distance Between Two Points on a Line 54 5.3 Mid Point of a Line Segment 54 5.4 Gradient of a Straight Line 55 5.5 Parallel Lines 56 5.6 Perpendicular Lines 57 5.7 Finding the Equation of a Line 57 5.8 Heinous Howlers 60 The Quadratic Function 61 6.1 Intro to Polynomials 61 6.2 The Quadratic Function 61 6.3 Quadratic Types 62 6.4 Quadratic Syllabus Requirements 62 Factorising Quadratics 63 7.1 Methods for Factorising 63 7.2 Zero Factor Property 63 7.3 Expressions with a Common Factor 63 7.4 Expressions of the form (u + v)2 = k 64 7.5 Difference of Two Squares 64 7.6 Perfect Squares 65 7.7 Finding Possible Factors 7.8 8 • C1 • 9 • C1 • Quadratic Factorisation, type 65 x2 + bx + c 66 ax2 67 7.9 Factorising Quadratic of Type: + bx + c 7.10 Pairing or Grouping Common Factors Completing the Square 79 8.1 General Form of a Quadratic 79 8.2 A Perfect Square 79 8.3 Deriving the Square or Vertex Format 80 8.4 Completing the Square 80 8.5 Completing the Square in Use 82 8.6 Solving Quadratics 82 8.7 Solving Inequalities 82 8.8 Graphing − Finding the Turning Point (Max / Min Value) 83 8.9 A Geometric View of Completing the Square 83 8.10 Topic Digest 86 The Quadratic Formula 87 9.1 Deriving the Quadratic Formula by Completing the Square 87 9.2 Examples of the Quadratic Formulae 88 9.3 Finding the Vertex 90 9.4 Heinous Howlers 90 9.5 Topical Tips 90 10 • C1 • The Discriminant Test iv 77 91 10.1 Assessing the Roots of a Quadratic 91 10.2 'Quadratic' is a Perfect Square: Discriminant = 0 92 10.3 'Discriminant' is a Perfect Square 92 10.5 Complex & Imaginary Numbers (Extension) 95 10.6 Topical Tips 96 ALevelNotesv8Fbb 18-May-2015 Contents 10.7 Topic Digest 96 11 • C1 • Sketching Quadratics 97 11.1 Basic Sketching Rules for any Polynomial Function 97 11.2 General Shape & Orientation of a Quadratic 97 11.3 Roots of a Quadratic 97 11.4 Crossing the y-axis, (the y-intercept) 98 11.5 Turning Points (Max or Min Value) 98 11.6 Sketching Examples 99 11.7 Topical Tips 12 • C1 • Further Quadratics 100 101 12.1 Reducing Other Equations to a Quadratic 101 12.2 Reducing to Simpler Quadratics: Examples 101 13 • C1 • Simultaneous Equations 105 13.1 Solving Simultaneous Equations 105 13.2 Simultaneous Equations: Worked Examples 105 14 • C1 • Inequalities 107 14.1 Intro 107 14.2 Rules of Inequalities 107 14.3 Linear Inequalities 107 14.4 Quadratic Inequalities 108 14.5 Solving Inequalities by Sketching 108 14.6 Critical Values Table 109 14.7 Inequality Examples 110 14.8 Special Case of Inequality 112 14.9 Heinous Howlers 112 14.10 Topical Tips 15 • C1 • Standard Graphs I 112 113 15.1 Standard Graphs 113 15.2 Asymptotes Intro 113 15.3 Power Functions 113 15.4 Roots and Reciprocal Curves 119 15.5 Exponential and Log Function Curves 120 15.6 Other Curves 121 15.7 Finding Asymptotes 122 16 • C1 • Graph Transformations 129 16.1 Transformations of Graphs 129 16.2 Vector Notation 129 16.3 Translations Parallel to the y-axis 130 16.4 Translations Parallel to the x-axis 130 16.5 One Way Stretches Parallel to the y-axis 132 16.6 One Way Stretches Parallel to the x-axis 133 16.7 Reflections in both the x-axis & y-axis 135 16.8 Translating Quadratic Functions 135 16.9 Translating a Circle Function 135 16.10 Recommended Order of Transformations 136 16.11 Example Transformations 137 16.12 Topical Tips 140 16.13 Transformation Summaries 140 v My A Level Maths Notes 17 • C1 • Circle Geometry 17.1 Equation of a Circle 141 17.2 Equation of a Circle Examples 142 17.3 Properties of a Circle 143 17.4 Intersection of a Line and a Circle 144 17.5 Completing the Square to find the Centre of the Circle 146 17.6 Tangent to a Circle 147 17.7 Tangent to a Circle from Exterior Point 149 17.8 Points On or Off a Circle 151 17.10 Circle Digest 18 • C1 • Calculus 101 154 155 18.1 Calculus Intro 155 18.2 Historical Background 155 18.3 What's it all about then? 155 18.4 A Note on OCR/AQA Syllabus Differences 156 19 • C1 • Differentiation I 157 19.1 Average Gradient of a Function 157 19.2 Limits 158 19.3 Differentiation from First Principles 159 19.4 Deriving the Gradient Function 160 19.5 Derivative of a Constant 161 19.6 Notation for the Gradient Function 161 19.7 Differentiating Multiple Terms 161 19.8 Differentiation: Worked Examples 162 19.9 Rates of Change 163 19.10 Second Order Differentials 164 19.11 Increasing & Decreasing Functions 165 20 • C1 • Practical Differentiation I 20.1 Tangent & Normals 167 167 20.2 Stationary Points 170 20.3 Properties of Max & Min Turning Points 171 20.4 Properties of Inflection Points 171 20.5 Testing & Classifying Types of Stationary Points 172 20.6 Max & Min Problems 175 20.7 Optimisation Problems 177 20.8 Differentiation Digest 182 21 • C1 • Crib Sheet 183 21.1 Algebra, Indices, Surds 183 21.2 Handling Surds — Basic Rules 184 21.3 Straight Line Geometry 184 21.4 Quadratic Functions 185 21.5 Simultaneous Equations 185 21.6 Inequalities 185 21.7 Standard Graphs 185 21.8 Transformations Summary 186 21.9 Circle Geometry 186 21.10 Differential Calculus vi 141 ALevelNotesv8Fbb 186 18-May-2015 Contents Module C2 187 Core 2 Basic Info 187 C2 Contents 187 C2 Assumed Basic Knowledge 188 C2 Brief Syllabus 189 22 • C2 • Algebraic Division 191 22.1 Algebraic Division Intro 191 22.2 Long Division by ax + b 191 22.3 Comparing Coefficients 192 23 • C2 • Remainder & Factor Theorem 193 23.1 Remainder Theorem 193 23.2 Factor Theorem 194 23.3 Topic Digest 196 24 • C2 • Sine & Cosine Rules 197 24.1 Introduction 197 24.2 Labelling Conventions & Properties 197 24.3 Sine & Cosine Revisited 197 24.4 Sine Rule 198 24.5 The Ambiguous Case (SSA) 200 24.6 Cosine Rule 201 24.7 Bearings 204 24.8 Area of a Triangle 205 24.9 Cosine & Sine Rules in Diagrams 207 24.10 Heinous Howlers 207 24.11 Digest 208 25 • C2 • Radians, Arcs, & Sectors 209 25.1 Definition of Radian 209 25.2 Common Angles 209 25.3 Length of an Arc 210 25.4 Area of Sector 210 25.5 Area of Segment 210 25.6 Length of a Chord 210 25.7 Radians, Arcs, & Sectors: Worked Examples 211 25.7 Radians, Arcs, & Sectors: Worked Examples 213 25.8 Topical Tips 216 25.9 Common Trig Values in Radians 216 25.10 Radians, Arcs, & Sectors Digest 216 26 • C2 • Logarithms 217 26.1 Basics Logs 217 26.2 Uses for Logs 218 26.3 Common Logs 218 26.4 Natural Logs 218 26.5 Log Rules - OK 219 26.6 Log Rules − More Derivations 220 26.7 Change of Base 220 26.8 Solving Log & Exponential Problems 221 26.9 Worked Examples in Logs of the form 222 26.10 Worked Examples in Logs of the form 223 vii My A Level Maths Notes 26.11 Inverse Log Operations 225 26.12 Further Worked Examples in Logs 227 26.13 Use of Logs in Practice 231 26.14 Heinous Howlers 231 26.15 Log Rules Digest 232 27 • C2 • Exponential Functions 27.1 General Exponential Functions 233 27.2 The Exponential Function: e 233 27.3 Exponential Graphs 234 27.4 Translating the Exponential Function 235 27.5 The Log Function Graphs 236 27.6 Exponentials and Logs 237 27.7 Exponential and Log Worked Examples 237 28 • C2 • Sequences & Series 239 28.1 What is a Sequence? 239 28.2 Recurrence Relationship 239 28.3 Algebraic Definition 240 28.4 Sequence Behaviour 240 28.5 Worked Example 242 28.6 Sequence & Series 242 28.7 Sigma Notation S 243 28.8 Sigma Notation: Worked Examples 245 28.9 Finding a likely rule 246 28.10 Some Familiar Sequences 247 28.11 Sequences in Patterns 248 29 • C2 • Arithmetic Progression (AP) 249 29.1 Intro to Arithmetic Progressions or Sequences 249 29.2 The n-th Term of an Arithmetic Progression 250 29.3 The Sum to n Terms of an Arithmetic Series 251 29.4 Sum to n Terms of an Arithmetic Series: Proof 252 29.5 No Sum to Infinity for an Arithmetic Series 252 29.6 Arithmetic Series: Worked Examples 252 30 • C2 • Geometric Progression (GP) 257 30.1 Intro to Geometric Progressions or Sequences 257 30.2 The n-th Term of a Geometric Progression 257 30.3 The Sum of a Geometric Series 258 30.4 Divergent Geometric Progressions 259 30.5 Convergent Geometric Series 260 30.6 Oscillating Geometric Progressions 260 30.7 Sum to Infinity of a Geometric Series 261 30.8 Geometric Progressions: Worked Examples 261 30.9 Heinous Howlers for Geometric & Arithmetic Progressions 267 30.10 AP & GP Topic Digest 31 • C2 • Binomial Theorem viii 233 268 269 31.1 Binomials and their Powers 269 31.2 Pascal's Triangle 269 31.3 Factorials & Combinations 271 31.4 Binomial Coefficients 272 ALevelNotesv8Fbb 18-May-2015 Contents 31.5 Binomial Theorem 274 31.6 Properties of the Binomial Theorem 275 31.7 Binomial Theorem: Special Case 275 31.8 Finding a Given Term in a Binomial 276 31.9 Binomial Theorem: Worked Examples 277 31.10 Alternative Method of Expanding a Binomial 281 31.11 Heinous Howlers 283 31.12 Topical Tips 283 31.13 Some Common Expansions in C2 283 31.14 Binomial Theorem Topic Digest 284 32 • C2 • Trig Ratios for all Angles 285 32.1 Trig Ratios for all Angles Intro 285 32.2 Standard Angles and their Exact Trig Ratios 285 32.3 The Unit Circle 286 32.4 Related Acute Angles 287 32.5 The Principal & Secondary Value 288 32.6 The Unit Circle and Trig Curves 289 32.7 General Solutions to Trig Equations 290 32.8 Complementary and Negative Angles 294 32.9 Coordinates for Angles 0°, 90°, 180° & 270° 294 32.10 Solving Trig Problems − Worked Examples 295 32.11 Trig Ratios for all Angles Digest 300 33 • C2 • Graphs of Trig Functions 301 33.1 Graphs of Trig Ratios 301 33.2 Transformation of Trig Graphs 302 33.3 Graphs of Squared Trig Functions 303 33.4 Worked Examples 305 33.5 Summary of Trig Functions Transformations 306 34 • C2 • Trig Identities 307 34.1 Trig Identities Intro 307 34.2 Recall the Basic Trig Ratios 307 34.3 Deriving the Identity tan x ≡ sin x / cos x 308 34.4 Deriving the Identity sin2x + cos2x ≡ 1 308 34.5 Solving Trig Problems of the form: p sin x ± q cos x = k 308 34.6 Solving Trig Problems of the form: p sinm x ± q cosn x = k 309 34.7 Proving Other Identities 311 34.8 Trig Identity Digest 312 35 • C2 • Trapezium Rule 313 35.1 Estimating Areas Under Curves 313 35.2 Area of a Trapezium 313 35.3 Trapezium Rule 313 35.4 Trapezium Rule Errors 314 35.5 Trapezium Rule: Worked Examples 315 35.6 Topical Tips 316 ix My A Level Maths Notes 36 • C2 • Integration I 36.1 Intro: Reversing Differentiation 317 36.2 Integrating a Constant 317 36.3 Integrating Multiple Terms 318 36.4 Finding the Constant of Integration 318 36.5 The Definite Integral − Integration with Limits 319 36.6 Area Under a Curve 320 36.7 Compound Areas 324 36.8 More Worked Examples 326 36.9 Topical Tips 326 Module C3 327 Core 3 Basic Info & C3 Contents 327 C3 Assumed Basic Knowledge 328 C3 Brief Syllabus 329 37 • C3 • Functions 331 37.1 Function Intro 331 37.2 Domains & Ranges 332 37.3 Mapping Relationships between the Domain & Range 334 37.4 Vertical Line Test for a Function 336 37.5 Inverse Functions 338 37.6 Procedure for finding the Inverse of a Function 339 37.7 Horizontal Line Test for an Inverse Function 341 37.8 Derivative Test for an Inverse Function 341 37.9 Graphing Inverse Functions 342 37.10 Compound or Composite Functions 343 37.11 The Domain of a Composite Function 346 37.12 Simple Decomposition of a Composite Function 352 37.13 Odd, Even & Periodic Functions 353 37.14 Worked Examples in Functions 354 37.15 Heinous Howlers 356 37.16 Function Notation 357 37.17 Functions Digest 358 38 • C3 • Modulus Function & Inequalities x 317 359 38.1 The Modulus Function 359 38.2 Relationship with Absolute Values and Square Roots 360 38.3 Graphing y = f (x) 360 38.4 Graphing y = f (|x|) 362 38.5 Inequalities and the Modulus Function 363 38.6 Algebraic Properties 364 38.7 Solving Equations Involving the Modulus Function 364 38.8 Solving Modulus Equations Algebraically 365 38.9 Squares & Square Roots Involving the Modulus Function 367 38.10 Solving Modulus Equations by Graphing 370 38.11 Solving Modulus Equations by Critical Values 371 38.12 Gradients not Defined 372 38.13 Heinous Howlers 372 38.14 Modulus Function Digest 372 ALevelNotesv8Fbb 18-May-2015 Contents 39 • C3 • Exponential & Log Functions 373 39.1 Exponential Functions 373 39.2 THE Exponential Function: e 374 39.3 Natural Logs: ln x 375 ex 39.4 Relationship between and ln x, and their Graphs 376 39.5 Graph Transformations of THE Exponential Function 377 39.6 Solving Exponential Functions 378 39.7 Exponential Growth & Decay 380 39.8 Differentiation of ex and ln x 384 39.9 Integration of ex and ln x 384 39.10 Heinous Howler 384 40 • C3 • Numerical Solutions to Equations 385 40.1 Intro to Numerical Methods 385 40.2 Locating Roots Graphically 386 40.3 Change of Sign in f(x) 386 40.4 Locating Roots Methodically 387 40.5 Limitations of the Change of Sign Methods 390 40.6 Iteration to find Approximate Roots 391 40.7 Staircase & Cobweb Diagrams 393 40.8 Limitations of the Iterative Methods 395 40.9 Choosing Convergent Iterations 395 40.10 Numerical Solutions Worked Examples 396 40.11 Numerical Solutions Digest 400 41 • C3 • Estimating Areas Under a Curve 401 41.1 Estimating Areas Intro 401 41.2 Trapezium Rule − a Reminder 401 41.3 Mid-ordinate Rule 402 41.4 Simpson's Rule 404 41.5 Relationship Between Definite Integrals and Limit of the Sum 407 41.6 Estimating Areas Digest 408 42 • C3 • Trig: Functions & Identities 409 42.1 Degrees or Radians 409 42.2 Reciprocal Trig Functions 409 42.3 Reciprocal Trig Functions Graphs 410 42.4 Reciprocal Trig Functions Worked Examples 411 42.5 Trig Function Summary 411 42.6 Pythagorean Identities 412 42.7 Compound Angle (Addition) Formulae 414 42.8 Exact Values of Trig Functions 417 42.9 Double Angle Formulae 418 42.10 Half Angle Formulae 422 42.11 Triple Angle Formulae 423 42.12 Factor Formulae 424 42.13 Guidelines for Proving Trig Identities 426 42.14 Proving Trig Identities Worked Examples 427 42.15 Trig Identity Digest 429 xi My A Level Maths Notes 43 • C3 • Trig: Inverse Functions 431 43.1 Inverse Trig Functions Intro 431 43.2 Inverse Sine Function 432 43.3 Inverse Cosine Function 433 43.4 Inverse Tangent Function 434 43.5 Inverse Trig Function Summary Graphs 435 44 • C3 • Trig: Harmonic Form 437 44.1 Function of the Form: f(x) = a cos x + b sin x 437 44.2 Proving the Identity 438 44.3 Geometric View of the Harmonic Form 439 44.4 Choosing the Correct Form 439 44.5 Worked Examples 440 44.6 Harmonic Form Digest 444 45 • C3 • Relation between dy/dx and dx/dy 445 45.1 Relation between dy/dx and dx/dy 445 45.2 Finding the Differential of x = g(y) 446 45.3 Finding the Differential of an Inverse Function 447 46 • C3 • Differentiation: The Chain Rule 449 46.1 Composite Functions Revised 449 46.2 Intro to the Chain Rule 449 46.3 Applying the Chain Rule 450 46.4 Using the Chain Rule Directly 452 46.5 Chain Rule Applied to Linear Functions 452 46.6 Connecting More than One Variable 452 46.7 Related Rates of Change 453 46.8 Deriving the Chain rule 455 46.9 Differentiating Trig with the Chain Rule 455 46.10 Chain Rule Digest 47 • C3 • Differentiation: Product Rule 456 457 47.1 Differentiation: Product Rule 457 47.2 Deriving the Product Rule 457 47.3 Product Rule: Worked Examples 458 47.4 Topical Tips 460 48 • C3 • Differentiation: Quotient Rule 461 48.1 Differentiation: Quotient Rule 461 48.2 Quotient Rule Derivation 461 48.3 Quotient Rule: Worked Examples 462 48.4 Topical Tips 460 49 • C3 • Differentiation: Exponential Functions 465 49.1 Differentiation of ex 50 • C3 • Differentiation: Log Functions 50.1 Differentiation of ln x 51 • C3 • Differentiation: Rates of Change 467 467 469 51.1 Connected Rates of Change 469 51.2 Rate of Change Problems 469 52 • C3 • Integration: Exponential Functions xii 465 ex 52.1 Integrating 52.2 Integrating 1/x ALevelNotesv8Fbb 475 475 475 18-May-2015 Contents 52.3 Integrating other Reciprocal Functions 476 53 • C3 • Integration: By Inspection 53.1 53.2 53.3 477 Integration by Inspection 477 Integration of (ax+b)n by Inspection 477 Integration of (ax+b)−n by Inspection 478 54 • C3 • Integration: Linear Substitutions 479 54.1 Integration by Substitution Intro 479 54.2 Integration of (ax+b)n by Substitution 479 54.3 Integration Worked Examples 481 54.4 Derivation of Substitution Method 486 55 • C3 • Integration: Volume of Revolution 487 55.1 Intro to the Solid of Revolution 487 55.2 Volume of Revolution about the x-axis 487 55.3 Volume of Revolution about the y-axis 488 55.4 Volume of Revolution Worked Examples 489 55.5 Volume of Revolution Digest 492 56 • C3 • Your Notes 493 Module C4 495 Core 4 Basic Info & C4 Contents 495 C4 Brief Syllabus 496 C4 Assumed Basic Knowledge 497 57 • C4 • Differentiating Trig Functions 499 57.1 Defining other Trig Functions 499 57.2 Worked Trig Examples 501 57.3 Differentiation of Log Functions with Trig 507 58 • C4 • Integrating Trig Functions 58.1 509 Intro 509 sec2 58.2 Integrals of: sin x, cos x and x 58.3 Using Reverse Differentiation: 509 58.4 Integrals of tan x and cot x 511 58.5 Recog...
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Advanced Level Pure Mathematics Notes Pdf

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