COMPLETE grade 12 maths paper 1 summary

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CONTENTS PAPER 1 Page

ALGEBRA GR 11 RECAP
Exponent Expressions 3
Exponential Equations 6
Quadratic Inequalities 8
Root Equations 10
Quadratic Theory 12
The Parabola and the Nature of the Roots 17

NUMBER PATTERNS : SEQUENCES AND SERIES 18
Second Difference Patterns 20
The General Term of a sequence 26
Arithmetic Sequences : Basic concepts , finding terms 27
Simultaneous Equations and Arithmetic Sequences 29
The Sum of terms of Arithmetic Sequences 31
The Geometric Sequence : Basic concepts , finding terms 34
Simultaneous Equations and the Geometric Sequence 35
Sum of terms of the Geometric Series 38
Sigma Notation 40
Using the Sn formula to find the value of a term 45
Strange Sequences 46
Infinite Series and the sum to Infinity 47

FUNCTIONS - GRAPHS 50
Drawing the Straight Line graph 50
Finding the equation of the Straight Line graph 53
The Parabola 54
The Parabola in the form 𝑦 = 𝑎(𝑥 − 𝑝)2 + 𝑞 Sketching given this form 55
Sketching the Parabola in the form 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 57
Finding the equation of the Parabola when given the x intercepts 60
Finding the equation of the Parabola when giving the Turning Point 61
Graph Interpretation and mixed questions 62
Finding the Maximum and Minimum length between two graphs 66
The Hyperbola 68
The hyperbola with no shifts 68
The hyperbola with vertical shifts 69
The hyperbola with horizontal shifts 70
The hyperbola with both shifts 71
Finding the equation of the hyperbola when given the diagram 73
The Exponential graph - basic graph with no shifts 75
The exponential graph with vertical shift 77
The exponential graph with horizontal shift 78

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Finding the equation of the exponential graph given the diagram 80
Mixed Graph Applications 82
Log : What is a Log? 86
Changing from Log to Index form 89
Changing from Exponential form to Log form 90
Exponential Graphs and its Inverse : the Log Graph 91
Drawing the Log Graph 95
Summary of Log and Exponential Graphs 96
Definition of a Function 98
The Parabola and its Inverse Graph 99
Using Logs in Finance to find “n” 102

FINANCIAL MATHS 104
Recap of Gr 11 Finance 104
Different Compounding periods 105
Nominal and Effective rates 107
Depreciation : Straight Line and Reducing Balance Depreciation 108
Timelines 109
Gr 12 Finance 114
Using Logs to calculate ‘n’ 114
The Future Value of Annuities 116
Sinking Funds 123
Loans and Loan Repayment : the Present Value 126
The Balance of a Loan pg 131 Deferred Payments pg 135 ; Final payment pg 138 131

CALCULUS 141
Finding The gradient at a point on a curve 142
The Derivative and what you can do with the derivative 144
Derivative with problems 148
Finding the Average Gradient between two points on a curve 151
Finding the equation of a Tangent to a curve 153
Finding the Derivative by First Principles 155
Solving Cubic Equations : Division by Inspection 160
Curve Sketching : Finding the Turning Point of Cubic Curves 163
Points of Inflection 167
Graph Interpretation 171
Important Calculus questions 172
Maximum and Minimum applications 174
Rates of Change and Calculus of Motion 129
Cubic Functions and the graph of their Derivatives 185

PROBABILITY
Some basic terminology 189
Venn Diagrams 191
Tree Diagrams 197
Contingency Tables 203
Applying Probability Rules 209

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ALGEBRA

EXPONENTS

NB : WITH ALL EXPONENT WORK , YOU MUST FIRST REDUCE THE BASE TO THE
LOWEST PRIME NUMBER!!

Method : 1. Break down to lowest base.
2. Apply exponent law 3 (a n ) m  a nm
an nm
3. Apply exponent law 1 [ a .a  a n m nm
] or law 2 [ m  a ]
a
(a) 3 x.9 x 1 (b) 16 x 1.8 2 x.4 x
 3 x.(32 ) x1  (2 4 ) x1 .(23 ) 2 x .(2 2 ) x
 3 x.32 x  2  2 4 x 4.2 6 x.2 2 x
 33 x  2  212x4


3 x.9 x 1 6 n.9 n1.2 n1
(c) (d)
27 x 1 12 n1
3 x.(3 2 ) x 1 (2.3) n .(3 2 ) n 1 .2 n 1
 
(33 ) x 1 (3.2 2 ) n 1
3 x.3 2 x 2 2 n.3n.32 n2.2 n1
 3 x 3 
3 3n1.2 2 n 2
33 x  2 2 2 n 1.33n 2
 
33 x 3 3 n 1.2 2 n  2
 33 x2(3 x3)  2 1.32 n3
3 2 n 3
3 
2

4 n  2.9 n 1
(e)
72 n.21 n

(2 2 ) n  2 .(3 2 ) n 1

(32.2 3 ) n .21n

2 2 n 4 32 n2

3 2 n 2 3n.21n

2 2 n  4.3 2 n 2

3 2 n.2 2 n 1
8

9
3

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NOW WE EXTEND THE LAWS OF EXPONENTS TO INCLUDE RATIONAL EXPONENTS
( FRACTION EXPONENTS)
c x

The “root” definition : a
b b
c a
OR m  a mx
a

2 2
Eg : 8  (2 )  2 2  4
3 3 3

1 1
 
 25  2  52  2 5 1 2 2 4
Eg:     4    
 16  2  2 2 51 5



(a) NB 3
(0,125) 2 Always change decimals to FRACTIONS!
2
1
( )3
8
2
1
( 3) 3
2
1
 2
2

4

(b) NB 3
27 x 6  81x 4  4 256 x 8
1 1 1
 (3 x )  (3 x )  (2 x )
3 6 3 4 4 2 8 8 4


 3x 2  3 2 x 2  2 2 x 2
 3x 2  9 x 2  4 x 2
= 8x 2

729 9
(c) 81x 4  3  512 x18
x 6
9
 9x 2 -  2x 2
x 2
= 9x 2  9x 2  2x 2

 2x 2

343
(d) 5
1024m15  3
m 9
7
 4m 3 
m 3
 4m 3  7m 3
 3m 3


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