A Level Fur ther Mathematics for AQA

Contents
Contents
Introduction.............................................................. iv
How to use this book................................................v
1 Complex numbers
1: Definition and basic arithmetic of i......................1
2: Division and complex conjugates.......................6
3: Geometric representation....................................9
4: Locus in the complex plane...............................20
5: Operations in modulus–argument form...........24
2 Roots of polynomials
1: Factorising polynomials......................................31
2: Complex solutions to polynomial
equations.............................................................34
3: Roots and coefficients........................................38
4: Finding an equation with given roots...............43
5: Transforming equations......................................48
3 Graphs and inequalities
1: Cubic and quartic inequalities...........................55
2: Functions of the form y =
ax + b
cx + d .......................58
2
3: Functions of the form y =
ax + bx + c
2
...............62
dx + ex + f
4: Oblique asymptotes...........................................65
5: Reciprocal transformations of functions...........66
6: Modulus transformation.....................................71
4 Hyperbolic functions
1: Defining hyperbolic functions and
hyperbolic identities...........................................80
2: Hyperbolic identities..........................................85
3: Solving harder hyperbolic equations................87
5 Series
1: Sigma notation....................................................92
2: Using standard formulae....................................94
3: Method of differences........................................99
4: Maclaurin series.................................................103
Focus on … Proof 1.............................................111
Focus on … Problem solving 1...........................113
Focus on … Modelling 1.....................................115
Cross-topic review exercise 1.............................117
6 Matrices
1: Addition, subtraction and
scalar multiplication..........................................120
2: Matrix multiplication.........................................126
3: Determinants and inverses of
2×
2 matrices..................................................... 131
4: Linear simultaneous equations........................140
7 Matrix transformations
1: Matrices as linear transformations................... 147
2: Further transformations in 2D..........................155
3: Invariant points and invariant lines..................162
4: Transformations in 3D....................................... 167
8 Further vectors
1: Vector equation of a line..................................177
2: Cartesian equation of the line.........................184
3: Intersections of lines.........................................190
4: Angles and the scalar product.........................193
5: The vector product...........................................202
9 Polar coordinates
1: Curves in polar coordinates.............................212
2: Some features of polar curves.........................215
3: Changing between polar and Cartesian
coordinates........................................................219
10 Further calculus
1: Volumes of revolution.......................................225
2: Average value of a function.............................230
Focus on … Proof 2............................................ 236
Focus on … Problem solving 2.......................... 238
Draft sample
Focus on … Modelling 2.....................................241
Cross topic review exercise 2............................244
Practice paper 1 ...................................................248
Formulae................................................................250
Answers to exercises ............................................253
Glossary .................................................................296
Index ..................................................................... 297
Acknowledgements............................................. 000
© Cambridge University Press 2017
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