Via Afrika
Mathematics
Grade 12 Study Guide
M. Malan, L.J. Schalekamp, E.C. Brown, L. Bruce,
G. du Toit, C.R. Smith, L.M. Botsane, J. Bouman, M. Pillay
, M. Malan
St udy G u id e
Via Afrika Mathematics
Grade 12
ISBN: 978-1-41546-335-2
, Exponents and Surds
Contents
Introduction ......................................................................................................................... 3
Chapter 1 Number patterns, sequences and series ........................................ 4
OVERVIEW ............................................................................................................................... 4
Unit 1 Arithmetic sequences and series
Unit 2 Geometric sequences and series
Unit 3 The sum to n terms(Sn): Sigma notation
Unit 4 Convergence and sum to infinity
Mixed exercises .................................................................................................................... 8
Chapter 2 Functions ..................................................................................... 10
OVERVIEW ............................................................................................................................... 10
Unit 1 The definitions of a function
Unit 2 The inverse of a function
Unit 3 The inverse of y = ax + q
Unit 4 The inverse of the quadratic function y = ax2
Mixed exercises .................................................................................................................... 21
Chapter 3 Logarithms ................................................................................... 24
OVERVIEW ............................................................................................................................... 24
Unit 1 The definition of a logarithm
Unit 2 Solving exponential equations using logarithms
Unit 3 The graph of y = logbx where b > 1 and 0 < b < 1
Mixed exercises .................................................................................................................... 28
Chapter 4 Finance, growth and decay ............................................................ 29
OVERVIEW ............................................................................................................................... 29
Unit 1 Future value annuities
Unit 2 Present value annuities
Unit 3 Calculating the period
Unit 4 Analysing investments and loans
Mixed exercises .................................................................................................................... 35
Chapter 5 Compound angles ......................................................................... 37
OVERVIEW ............................................................................................................................... 37
Unit 1 Deriving a formula for cos(𝛼 − 𝛽)
Unit 2 Formula for cos(𝛼 + 𝛽) and 𝑠𝑖𝑖(𝛼 ± 𝛽)
Unit 3 Double angles
Unit 4 Identities
Unit 5 Equations
Unit 6 Trigonometric graphs and compound angles
Mixed exercises .................................................................................................................... 46
Chapter 6 Solving problems in three dimensions ........................................... 48
OVERVIEW ............................................................................................................................... 48
Unit 1 Problems in three dimensions
Unit 2 Compound angle formulae in three dimensions
Mixed exercises .................................................................................................................... 51
© Via Afrika ›› Mathematics Grade 12 1
, Exponents and Surds
Chapter 7 Polynomials .................................................................................. 53
OVERVIEW ............................................................................................................................... 53
Unit 1 The Remainder Theorem
Unit 2 The Factor Theorem
Mixed exercises .................................................................................................................... 57
Chapter 8 Differential calculus ...................................................................... 58
OVERVIEW ............................................................................................................................... 58
Unit 1 Limits
Unit 2 The gradient of a graph at a point
Unit 3 The derivative of a function
Unit 4 The equation of a tangent to a graph
Unit 5 The graph of a cubic function
Unit 6 The second derivative (concavity)
Unit 7 Applications of differential calculus
Mixed exercises .................................................................................................................... 70
Chapter 9 Analytical geometry ...................................................................... 72
OVERVIEW ............................................................................................................................... 72
Unit 1 Equation of a circle with centre at the origin
Unit 2 Equation of a circle centred off the origin
Unit 3 The equation of the tangent to the circle
Mixed exercises .................................................................................................................... 77
Chapter 10 Euclidean geometry ..................................................................... 81
OVERVIEW ............................................................................................................................... 81
Unit 1 Proportionality in triangles
Unit 2 Similarity in triangles
Unit 2 Theorem of Pythagoras
Mixed exercises .................................................................................................................... 94
Chapter 11 Statistics: regression and correlation ........................................... 97
OVERVIEW ............................................................................................................................... 97
Unit 1 Symmetrical and skewed data
Unit 2 Scatter plots and correlation
Mixed exercises .................................................................................................................... 106
Chapter 12 Probability .................................................................................. 108
OVERVIEW ............................................................................................................................... 108
Unit 1 Solving probability problems
Unit 2 The counting principle
Unit 3 The counting principle and probability
Mixed exercises .................................................................................................................... 112
ANSWERS TO MIXED EXERCISES ...................................................................................................... 113
EXEMPLAR PAPER 1 ..................................................................................................................... 151
EXEMPLAR PAPER 2 .................................................................................................................... 168
© Via Afrika ›› Mathematics Grade 12 2
Mathematics
Grade 12 Study Guide
M. Malan, L.J. Schalekamp, E.C. Brown, L. Bruce,
G. du Toit, C.R. Smith, L.M. Botsane, J. Bouman, M. Pillay
, M. Malan
St udy G u id e
Via Afrika Mathematics
Grade 12
ISBN: 978-1-41546-335-2
, Exponents and Surds
Contents
Introduction ......................................................................................................................... 3
Chapter 1 Number patterns, sequences and series ........................................ 4
OVERVIEW ............................................................................................................................... 4
Unit 1 Arithmetic sequences and series
Unit 2 Geometric sequences and series
Unit 3 The sum to n terms(Sn): Sigma notation
Unit 4 Convergence and sum to infinity
Mixed exercises .................................................................................................................... 8
Chapter 2 Functions ..................................................................................... 10
OVERVIEW ............................................................................................................................... 10
Unit 1 The definitions of a function
Unit 2 The inverse of a function
Unit 3 The inverse of y = ax + q
Unit 4 The inverse of the quadratic function y = ax2
Mixed exercises .................................................................................................................... 21
Chapter 3 Logarithms ................................................................................... 24
OVERVIEW ............................................................................................................................... 24
Unit 1 The definition of a logarithm
Unit 2 Solving exponential equations using logarithms
Unit 3 The graph of y = logbx where b > 1 and 0 < b < 1
Mixed exercises .................................................................................................................... 28
Chapter 4 Finance, growth and decay ............................................................ 29
OVERVIEW ............................................................................................................................... 29
Unit 1 Future value annuities
Unit 2 Present value annuities
Unit 3 Calculating the period
Unit 4 Analysing investments and loans
Mixed exercises .................................................................................................................... 35
Chapter 5 Compound angles ......................................................................... 37
OVERVIEW ............................................................................................................................... 37
Unit 1 Deriving a formula for cos(𝛼 − 𝛽)
Unit 2 Formula for cos(𝛼 + 𝛽) and 𝑠𝑖𝑖(𝛼 ± 𝛽)
Unit 3 Double angles
Unit 4 Identities
Unit 5 Equations
Unit 6 Trigonometric graphs and compound angles
Mixed exercises .................................................................................................................... 46
Chapter 6 Solving problems in three dimensions ........................................... 48
OVERVIEW ............................................................................................................................... 48
Unit 1 Problems in three dimensions
Unit 2 Compound angle formulae in three dimensions
Mixed exercises .................................................................................................................... 51
© Via Afrika ›› Mathematics Grade 12 1
, Exponents and Surds
Chapter 7 Polynomials .................................................................................. 53
OVERVIEW ............................................................................................................................... 53
Unit 1 The Remainder Theorem
Unit 2 The Factor Theorem
Mixed exercises .................................................................................................................... 57
Chapter 8 Differential calculus ...................................................................... 58
OVERVIEW ............................................................................................................................... 58
Unit 1 Limits
Unit 2 The gradient of a graph at a point
Unit 3 The derivative of a function
Unit 4 The equation of a tangent to a graph
Unit 5 The graph of a cubic function
Unit 6 The second derivative (concavity)
Unit 7 Applications of differential calculus
Mixed exercises .................................................................................................................... 70
Chapter 9 Analytical geometry ...................................................................... 72
OVERVIEW ............................................................................................................................... 72
Unit 1 Equation of a circle with centre at the origin
Unit 2 Equation of a circle centred off the origin
Unit 3 The equation of the tangent to the circle
Mixed exercises .................................................................................................................... 77
Chapter 10 Euclidean geometry ..................................................................... 81
OVERVIEW ............................................................................................................................... 81
Unit 1 Proportionality in triangles
Unit 2 Similarity in triangles
Unit 2 Theorem of Pythagoras
Mixed exercises .................................................................................................................... 94
Chapter 11 Statistics: regression and correlation ........................................... 97
OVERVIEW ............................................................................................................................... 97
Unit 1 Symmetrical and skewed data
Unit 2 Scatter plots and correlation
Mixed exercises .................................................................................................................... 106
Chapter 12 Probability .................................................................................. 108
OVERVIEW ............................................................................................................................... 108
Unit 1 Solving probability problems
Unit 2 The counting principle
Unit 3 The counting principle and probability
Mixed exercises .................................................................................................................... 112
ANSWERS TO MIXED EXERCISES ...................................................................................................... 113
EXEMPLAR PAPER 1 ..................................................................................................................... 151
EXEMPLAR PAPER 2 .................................................................................................................... 168
© Via Afrika ›› Mathematics Grade 12 2
