11 – 19 PROGRESSION
Edexcel AS and A level Mathematics
Pure Mathematics
Year 1/AS
Series Editor: Harry Smith
Authors: Greg Attwood, Jack Barraclough, Ian Bettison, Alistair Macpherson,
Bronwen Moran, Su Nicholson, Diane Oliver, Joe Petran, Keith Pledger, Harry Smith,
Geoff Staley, Robert Ward-Penny, Dave Wilkins
,11 – 19 PROGRESSION
Edexcel AS and A level Mathematics
Pure Mathematics
Year 1/AS
Series Editor: Harry Smith
Authors: Greg Attwood, Jack Barraclough, Ian Bettison, Alistair Macpherson,
Bronwen Moran, Su Nicholson, Diane Oliver, Joe Petran, Keith Pledger, Harry Smith,
Geoff Staley, Robert Ward-Penny, Dave Wilkins
, Contents
Contents
Overarching themes iv
4.6 Stretching graphs 75
Extra online content vi
4.7 Transforming functions 79
Mixed exercise 4 82
1 Algebraic expressions 1
1.1 Index laws 2
Review exercise 1 85
1.2 Expanding brackets 4
1.3 Factorising 6
5 Straight line graphs 89
1.4 Negative and fractional indices 9
5.1 y = mx + c 90
1.5 Surds 12
5.2 Equations of straight lines 93
1.6 Rationalising denominators 13
5.3 Parallel and perpendicular lines 97
Mixed exercise 1 15
5.4 Length and area 100
5.5 Modelling with straight lines 103
2 Quadratics 18
Mixed exercise 5 108
2.1 Solving quadratic equations 19
2.2 Completing the square 22
6 Circles 113
2.3 Functions 25
6.1 Midpoints and perpendicular
2.4 Quadratic graphs 27 bisectors 114
2.5 The discriminant 30 6.2 Equation of a circle 117
2.6 Modelling with quadratics 32 6.3 Intersections of straight lines
Mixed exercise 2 35 and circles 121
6.4 Use tangent and chord properties 123
3 Equations and inequalities 38 6.5 Circles and triangles 128
3.1 Linear simultaneous equations 39 Mixed exercise 6 132
3.2 Quadratic simultaneous equations 41
3.3 Simultaneous equations on graphs 42 7 Algebraic methods 137
3.4 Linear inequalities 46 7.1 Algebraic fractions 138
3.5 Quadratic inequalities 48 7.2 Dividing polynomials 139
3.6 Inequalities on graphs 51 7.3 The factor theorem 143
3.7 Regions 53 7.4 Mathematical proof 146
Mixed exercise 3 56 7.5 Methods of proof 150
Mixed exercise 7 154
4 Graphs and transformations 59
4.1 Cubic graphs 60 8 The binomial expansion 158
4.2 Quartic graphs 64 8.1 Pascal’s triangle 159
4.3 Reciprocal graphs 66 8.2 Factorial notation 161
4.4 Points of intersection 68 8.3 The binomial expansion 163
4.5 Translating graphs 71 8.4 Solving binomial problems 165
ii
, Contents
8.5 Binomial estimation 167 12.5 Differentiating functions with two
Mixed exercise 8 169 or more terms 266
12.6 Gradients, tangents and normal 268
9 Trigonometric ratios 173 12.7 Increasing and decreasing functions 270
9.1 The cosine rule 174 12.8 Second order derivatives 271
9.2 The sine rule 179 12.9 Stationary points 273
9.3 Areas of triangles 185 12.10 Sketching gradient functions 277
9.4 Solving triangle problems 187 12.11 Modelling with differentiation 279
9.5 Graphs of sine, cosine and tangent 192 Mixed exercise 12 282
9.6 Transforming trigonometric graphs 194
Mixed exercise 9 198 13 Integration 287
13.1 Integrating x n
288
10 Trigonometric identities and 13.2 Indefinite integrals 290
equations 202 13.3 Finding functions 293
10.1 Angles in all four quadrants 203 13.4 Definite integrals 295
10.2 Exact values of trigonometrical ratios 208 13.5 Areas under curves 297
10.3 Trigonometric identities 209 13.6 Areas under the x-axis 300
10.4 Simple trigonometric equations 213 13.7 Areas between curves and lines 302
10.5 Harder trigonometric equations 217 Mixed exercise 13 306
10.6 Equations and identities 219
Mixed exercise 10 222 14 Exponentials and logarithms 311
14.1 Exponential functions 312
Review exercise 2 226 14.2 y = e x
314
14.3 Exponential modelling 317
11 Vectors 230 14.4 Logarithms 319
11.1 Vectors 231 14.5 Laws of logarithms 321
11.2 Representing vectors 235 14.6 Solving equations using logarithms 324
11.3 Magnitude and direction 239 14.7 Working with natural logarithms 326
11.4 Position vectors 242 14.8 Logarithms and non-linear data 328
11.5 Solving geometric problems 244 Mixed exercise 14 334
11.6 Modelling with vectors 248
Mixed exercise 11 251 Review exercise 3 338
12 Differentiation 255 Practice exam paper 342
12.1 Gradients of curves 256
12.2 Finding the derivative 259 Answers 345
12.3 Differentiating x n
262
12.4 Differentiating quadratics 264 Index 399
iii
Edexcel AS and A level Mathematics
Pure Mathematics
Year 1/AS
Series Editor: Harry Smith
Authors: Greg Attwood, Jack Barraclough, Ian Bettison, Alistair Macpherson,
Bronwen Moran, Su Nicholson, Diane Oliver, Joe Petran, Keith Pledger, Harry Smith,
Geoff Staley, Robert Ward-Penny, Dave Wilkins
,11 – 19 PROGRESSION
Edexcel AS and A level Mathematics
Pure Mathematics
Year 1/AS
Series Editor: Harry Smith
Authors: Greg Attwood, Jack Barraclough, Ian Bettison, Alistair Macpherson,
Bronwen Moran, Su Nicholson, Diane Oliver, Joe Petran, Keith Pledger, Harry Smith,
Geoff Staley, Robert Ward-Penny, Dave Wilkins
, Contents
Contents
Overarching themes iv
4.6 Stretching graphs 75
Extra online content vi
4.7 Transforming functions 79
Mixed exercise 4 82
1 Algebraic expressions 1
1.1 Index laws 2
Review exercise 1 85
1.2 Expanding brackets 4
1.3 Factorising 6
5 Straight line graphs 89
1.4 Negative and fractional indices 9
5.1 y = mx + c 90
1.5 Surds 12
5.2 Equations of straight lines 93
1.6 Rationalising denominators 13
5.3 Parallel and perpendicular lines 97
Mixed exercise 1 15
5.4 Length and area 100
5.5 Modelling with straight lines 103
2 Quadratics 18
Mixed exercise 5 108
2.1 Solving quadratic equations 19
2.2 Completing the square 22
6 Circles 113
2.3 Functions 25
6.1 Midpoints and perpendicular
2.4 Quadratic graphs 27 bisectors 114
2.5 The discriminant 30 6.2 Equation of a circle 117
2.6 Modelling with quadratics 32 6.3 Intersections of straight lines
Mixed exercise 2 35 and circles 121
6.4 Use tangent and chord properties 123
3 Equations and inequalities 38 6.5 Circles and triangles 128
3.1 Linear simultaneous equations 39 Mixed exercise 6 132
3.2 Quadratic simultaneous equations 41
3.3 Simultaneous equations on graphs 42 7 Algebraic methods 137
3.4 Linear inequalities 46 7.1 Algebraic fractions 138
3.5 Quadratic inequalities 48 7.2 Dividing polynomials 139
3.6 Inequalities on graphs 51 7.3 The factor theorem 143
3.7 Regions 53 7.4 Mathematical proof 146
Mixed exercise 3 56 7.5 Methods of proof 150
Mixed exercise 7 154
4 Graphs and transformations 59
4.1 Cubic graphs 60 8 The binomial expansion 158
4.2 Quartic graphs 64 8.1 Pascal’s triangle 159
4.3 Reciprocal graphs 66 8.2 Factorial notation 161
4.4 Points of intersection 68 8.3 The binomial expansion 163
4.5 Translating graphs 71 8.4 Solving binomial problems 165
ii
, Contents
8.5 Binomial estimation 167 12.5 Differentiating functions with two
Mixed exercise 8 169 or more terms 266
12.6 Gradients, tangents and normal 268
9 Trigonometric ratios 173 12.7 Increasing and decreasing functions 270
9.1 The cosine rule 174 12.8 Second order derivatives 271
9.2 The sine rule 179 12.9 Stationary points 273
9.3 Areas of triangles 185 12.10 Sketching gradient functions 277
9.4 Solving triangle problems 187 12.11 Modelling with differentiation 279
9.5 Graphs of sine, cosine and tangent 192 Mixed exercise 12 282
9.6 Transforming trigonometric graphs 194
Mixed exercise 9 198 13 Integration 287
13.1 Integrating x n
288
10 Trigonometric identities and 13.2 Indefinite integrals 290
equations 202 13.3 Finding functions 293
10.1 Angles in all four quadrants 203 13.4 Definite integrals 295
10.2 Exact values of trigonometrical ratios 208 13.5 Areas under curves 297
10.3 Trigonometric identities 209 13.6 Areas under the x-axis 300
10.4 Simple trigonometric equations 213 13.7 Areas between curves and lines 302
10.5 Harder trigonometric equations 217 Mixed exercise 13 306
10.6 Equations and identities 219
Mixed exercise 10 222 14 Exponentials and logarithms 311
14.1 Exponential functions 312
Review exercise 2 226 14.2 y = e x
314
14.3 Exponential modelling 317
11 Vectors 230 14.4 Logarithms 319
11.1 Vectors 231 14.5 Laws of logarithms 321
11.2 Representing vectors 235 14.6 Solving equations using logarithms 324
11.3 Magnitude and direction 239 14.7 Working with natural logarithms 326
11.4 Position vectors 242 14.8 Logarithms and non-linear data 328
11.5 Solving geometric problems 244 Mixed exercise 14 334
11.6 Modelling with vectors 248
Mixed exercise 11 251 Review exercise 3 338
12 Differentiation 255 Practice exam paper 342
12.1 Gradients of curves 256
12.2 Finding the derivative 259 Answers 345
12.3 Differentiating x n
262
12.4 Differentiating quadratics 264 Index 399
iii
