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THE PASSWORD TO PASS GRADE 12 MATHEMATICS

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THIS BOOK INCLUDES ALL OF THE NCAPS CHAPTERS AND TOPICS FOR GRADE 12. AS AN ELECTRICAL ENGINEERING GRADUATE, BY THE GRACE OF GOD OF PROPHET PHILIPPE KACOU I WAS GIVEN THE OPPORTUNITY TO TEACH MATHEMATICS TO GRADE 12 STUDENTS IN A FINISHING SCHOOL IN 2018, 2019, AND 2020. IT WAS THROUGH MY INV...

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  • November 13, 2023
  • 186
  • 2020/2021
  • Book review
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  • 12th Grade
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Quadratic sequence ................................ 32
Table of Contents Arithmetic sum ....................................... 33
MATHEMATICS RULES................................... 4
Geometric sum ....................................... 36
Addition and subtraction rules ................... 4
Sum to infinity......................................... 37
Multiplication rules.................................... 4
Convergent ............................................. 38
Division rules ............................................. 4
Sigma notation ........................................ 38
Brackets rules ............................................ 4
Exam type questions examples................ 42
Exponent’s rules ........................................ 4
Number pattern exercises ....................... 43
Common factor rule .................................. 5
FUNCTIONS ................................................ 47
Fraction rule .............................................. 5
Straight line............................................. 47
ALGEBRA ....................................................... 6
Hyperbola ............................................... 48
Linear equation ......................................... 6
Parabola.................................................. 52
Quadratic equation.................................... 6
Equation of the tangent .......................... 54
Quadratic formula ..................................... 8
Exponential graph ................................... 56
Inequalities ................................................ 8
The inverse functions .............................. 57
Completing the square .............................. 9
Exam type questions examples................ 66
Simultaneously equations ........................ 10
Functions exercises ................................. 68
Exponents ............................................... 11
CALCULUS ................................................... 73
Nature of the roots .................................. 13
Derivative ............................................... 73
Exam type questions examples ................ 14
First principle .......................................... 75
Algebra exercises ..................................... 15
Cubic function ......................................... 76
FINANCIAL MATHS ...................................... 17
Application of calculus ............................ 84
Simple growth ......................................... 17
The rate of change and calculus motion .. 86
Compound growth................................... 17
Exam type questions examples................ 87
Nominal and effective rates ..................... 19
Calculus exercises ................................... 88
Depreciation ............................................ 20
PROBABILITY .............................................. 93
Future value annuities ............................. 20
The complementary rule ......................... 93
Sinking fund............................................. 21
Mutually exclusive events ....................... 93
Present value annuities ........................... 22
Independent events ................................ 94
Exam type questions examples ................ 24
Venn diagram.......................................... 94
Financial maths exercises ........................ 25
The fundamental counting principle ........ 97
NUMBER PATTERN...................................... 28
Permutations .......................................... 98
Arithmetic sequence................................ 28
Exam type questions examples.............. 102
Geometric sequence ................................ 30
Probability exercises ............................. 102



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,TRIGONOMETRY ....................................... 105 Collinear points ..................................... 139
Trig ratios .............................................. 105 Parallel lines ......................................... 140
Reductions ............................................ 106 Perpendicular lines................................ 140
Special angles ........................................ 107 Equation of the line ............................... 141
Identities rules....................................... 108 The angle of inclination ......................... 141
Compound angles identities .................. 111 Exam type questions examples.............. 143
Double angle identities .......................... 112 Equation of the cycle ............................. 146
Exam type questions examples .............. 113 Equation of the tangent ........................ 147
Aid diagram problems ........................... 115 Exam type questions examples.............. 149
The general solution .............................. 119 Analytical geometry exercises ............... 151
Sine, cosine and area rule ...................... 122 EUCLIDEAN GEOMETRY ............................ 155
Area rule ............................................... 122 Theorems.............................................. 155
Sine rule ................................................ 123 Similarity ............................................... 169
Cosine rule ............................................ 123 Proportionality theorem ....................... 171
Exam type questions examples .............. 124 Euclidean geometry exercises ............... 174
Trigonometric functions ........................ 127 STATISTICS................................................ 178
Exam type questions examples .............. 131 Five number summary .......................... 178
Trigonometry exercises ......................... 132 Box and whisker diagram ...................... 178
Trigonometry functions exercises .......... 135 Ogive (S shape) ..................................... 180
Area, sine and cosine rule exercises ....... 136 Scatter plot ........................................... 183
ANALYTICAL GEOMETRY ........................... 138 Statistics exercises ................................ 185
Distance or length ................................. 138
Midpoint ............................................... 138
Gradient or slope ................................... 139




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,I PRESENT THIS GRADE 12 NCAPS MATHEMATICS
TEXTBOOK TO YOU
THIS BOOK INCLUDES ALL OF THE NCAPS CHAPTERS AND TOPICS FOR GRADE 12. AS AN ELECTRICAL
ENGINEERING GRADUATE, BY THE GRACE OF GOD OF PROPHET PHILIPPE KACOU I WAS GIVEN THE
OPPORTUNITY TO TEACH MATHEMATICS TO GRADE 12 STUDENTS IN A FINISHING SCHOOL IN 2018,
2019, AND 2020. IT WAS THROUGH MY INVOLVEMENT IN THE BASIC EDUCATION ENVIRONMENT
THAT I DISCOVERED THAT MOST LEARNERS STRUGGLE IN MATHEMATICS NOT BECAUSE OF THEIR
LOW INTELLIGENCE QUOTIENT, BUT BECAUSE OF HOW MATHEMATICS HAS BEEN PRESENTED TO
THEM, AND THAT THE STRUGGLING IS FURTHER EXACERBATED BY A LACK OF ADEQUATE
INFORMATION AND RESOURCES, PARTICULARLY FOR THOSE WHO LIVE IN RURAL AND
UNDERDEVELOPED AREAS.

THE PURPOSE OF THIS BOOK IS TO SIMPLIFY THE LANGUAGE AND TERMS OF REFERENCE IN
MATHEMATICAL KNOWLEDGE. THE BOOK IS INTENDED TO HELP STUDENTS AND INSTRUCTORS
BETTER UNDERSTAND AND EXCEL IN MATHEMATICS BY ALLOWING THEM TO READ AND INTERPRET
MATHEMATICAL WRITING AND LANGUAGE. THE BOOK WILL TEACH STUDENTS AND EDUCATORS HOW
TO TEACH AND EXPLAIN A TOPIC IN A VARIETY OF WAYS (ESPECIALLY FOR THE TOPICS LIKE
PROBABILITY, FUNCTIONS, TRIGONOMETRY AND EUCLIDEAN GEOMETRY).

THE BENEFIT OF THE BOOK IS THAT ANYONE WHO USES THIS BOOK WILL BE ACQUAINTED WITH THE
EXAM QUESTIONS BECAUSE I DESIGNED IT USING PREVIOUS QUESTION PAPERS, PARTICULARLY
WHEN IT COMES TO EXAMPLES AND EXERCISES. THIS BOOK WILL EFFECTIVELY WORK FOR A TEACHER,
USED AS A STUDY GUIDE AND MASTER GUIDE.



THANK YOU FOR TAKING THIS BOOK AS YOUR MATHEMATICS ABSOLUTE




3

, MATHEMATICS RULES Division rules
These following rules you need to know them These four following rules are the first rules
by heart because they are the backbone of that you must know when you are dividing
mathematics your terms.
 Addition and subtraction Rules  + ÷ += +
 Multiplication Rules  + ÷ −= −
 Division Rules  − ÷ −= +
 Exponents Rules  − ÷ += −
 Fraction Rule
 Common factor Rule EXAMPLES
−2𝑥 −4𝑥
 Brackets Rules = −𝑥 ; =2
2 −2𝑥
4𝑦 −4𝑥𝑦 1
Addition and subtraction rules = −4 ; = 𝑦
−𝑦 −8𝑥 2
Adding or subtracting like terms 𝑥 𝑥 −8𝑡
EXAMPLES =2 ; 4 = −2𝑡
2
𝑥 + 𝑥 = 2𝑥 ; 4𝑥 − 2𝑥 = 2𝑥
−8𝑥 + 9𝑥 = 𝑥 ; 2𝑥 − 7𝑥 = −5𝑥 Brackets rules
3𝑡 − 11𝑡 = −8𝑡 ; 4𝑦 − 2𝑦 = 2𝑦 Rule1
7 3 𝐴(𝐵 + 𝐶) = 𝐴𝐵 + 𝐴𝐶 𝑜𝑟 (𝐷 + 𝐶)𝐵 = 𝐵𝐷 + 𝐵𝐶
−7𝑦 − 4𝑦 = −11𝑦 ; − 2 𝑦 + 5𝑦 = 2 𝑦
Adding or subtracting unlike terms
EXAMPLES
EXAMPLES
a. 2(𝑥 − 8) = (2)(𝑥) + (2)(−8)
𝑥+𝑝 = 𝑥+𝑝 ; 2𝑦 + 2 = 2𝑦 + 2
= 2𝑥 − 16
𝑦+4 =𝑦+4 ; 2𝑥 − 4 = 2𝑥 − 4
𝑡−𝑦 = 𝑡−𝑦 ; 𝑝 + 9𝑡 = 𝑝 + 9𝑡
𝑏. (𝑥 − 2)𝑥 = (𝑥)(𝑥) − (2)(𝑥)
𝑥 + 2𝑦 = 𝑥 + 2𝑦 ; 12𝑝 + 2𝑡 = 12𝑝 + 2𝑡
= 𝑥 2 − 2𝑥
Multiplication rules Rule2
(𝐴 + 𝐵)(𝐶 + 𝐷) = 𝐴𝐶 + 𝐴𝐷 + 𝐵𝐶 + 𝐵𝐷
Multiplying like terms
These four following rules are the first rules
EXAMPLES
that you must know when you are multiplying a. (𝑥 + 4)(𝑥 − 3)
your terms. (𝑥)(𝑥) − (3)(𝑥) + (4)(𝑥) − (3)(4 )
𝑥 2 − 3𝑥 + 4𝑥 − 12
 + × += + 𝑥 2 + 𝑥 − 12
 + × −= −
 − × −= + b. (8𝑥 − 2)(−𝑥 + 9)
 − × += − (8𝑥)(−𝑥) + (9)(8𝑥) − (𝑥)(−2) + (9)(−2 )
−8𝑥 2 + 72𝑥 + 2𝑥 − 18
EXAMPLES −8𝑥 2 + 74𝑥 − 18
2×2=4 ; −3 × 9 = −27
𝑥 × 𝑥 = 𝑥2 ; 𝑥 × −𝑥 = −𝑥 2 Exponent’s rules
2𝑥 × 2𝑥 = 4𝑥 2 ; −3𝑥 × −9𝑥 = 27𝑥 2 Rule1
5𝑦 × 𝑦 = 5𝑦 2 ; −4𝑦 × 3𝑦 = −12𝑦 2 (𝑥𝑚 )𝑛 = 𝑥𝑚×𝑛
= 𝑥 𝑚𝑛
Multiplying unlike terms
EXAMPLES EXAMPLES
−4 × 𝑥 = −4𝑥 ; −𝑥 × 𝑦 = −𝑥𝑦
(𝑥2)𝑛 = 𝑥2×𝑛 (𝑥−2)−𝑛 = 𝑥−2×−𝑛
−3 × 2𝑦 = −6𝑦 ; −2𝑥 × 3𝑦 = −6𝑥𝑦
−6 × 𝑡 = −6𝑡 ; 𝑝 × 𝑦 = 𝑝𝑦 = 𝑥2𝑛 = 𝑥2𝑛
−9𝑝 × 2 = −18𝑝 ; 𝑥 × 4𝑦 = 4𝑥𝑦




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