2016
REA’S NOTES
GRADE 12 FINAL EXAM
3 HOURS
150 MARKS
Table of Contents
Algebra and Equations (and inequalities) ------------------------------------------------------------------------------- 1
Patterns and Sequences---------------------------------------------------------------------------------------------------- 12
Finance, growth and decay ------------------------------------------------------------------------------------------------ 15
Functions and Graphs ------------------------------------------------------------------------------------------------------ 20
Differential Calculus -------------------------------------------------------------------------------------------------------- 31
Probability -------------------------------------------------------------------------------------------------------------------- 37
Mathematical Modelling --------------------------------------------------------------------------------------------------- 41
NB & not on formula sheet
[MATHEMATICS
PAPER I]
STUDY NOTES
, Algebra and Equations (and inequalities)
Grade 9 Revision
The Real Number System
Name Description Example
natural numbers counting numbers from 1
whole numbers natural numbers including 0 0
integers positive and negative whole numbers
any number that can be written as (b
rational numbers terminating and recurring decimals ℚ
0)
non-terminating and non-recurring
irrational numbers numbers that are not irrational ℚ'
decimals; ; ;
real numbers all of the above all of the above
non-real / imaginary
anything that is undefined '
numbers
Interval Notation
end value is included [min max]
end value not included (min max)
[min; max] OR [min; max) OR (min; max) OR
separated by ‘;’
(min; max]
if goes on forever to positive infinity “ ” or
[min; ) OR (min; )
negative infinity “ ”
( ; max] OR ( ; max)
the bracket would always be ROUND
Set Builder Notation
what variable we will be using
what the restriction(s) on the variable is/are
which set of numbers we can use
such that: is less than 5 but greater than or equal to 2; x is an element of real
numbers
Summary
Name Example Sets used
inequality real
interval notation real
number line all sets
list natural, whole, integer
set builder notation all sets
Factorisation
Type 1: HCF (Highest Common Factor)
reverse of distribution
Type 2: Common Bracket
Instead of common factor, there is a common bracket
Change of sign rule:
Rea’s Notes | Mathematics Paper I | Algebra and Equations (and inequalities) 1
, Type 3: DOTS (Difference of Two Squares)
Only common fractions can be cancelled to simplify a fraction
You cannot cancel if there is more than one term in the numerator or denominator
Type 4: Trinomials/FOIL
Cross method; NB:
sign of middle term sign of last term sign of lower value factor sign of higher value factor
+ + + +
- + - -
+ - - +
- - + -
Rea’s Notes | Mathematics Paper I | Algebra and Equations (and inequalities) 2
REA’S NOTES
GRADE 12 FINAL EXAM
3 HOURS
150 MARKS
Table of Contents
Algebra and Equations (and inequalities) ------------------------------------------------------------------------------- 1
Patterns and Sequences---------------------------------------------------------------------------------------------------- 12
Finance, growth and decay ------------------------------------------------------------------------------------------------ 15
Functions and Graphs ------------------------------------------------------------------------------------------------------ 20
Differential Calculus -------------------------------------------------------------------------------------------------------- 31
Probability -------------------------------------------------------------------------------------------------------------------- 37
Mathematical Modelling --------------------------------------------------------------------------------------------------- 41
NB & not on formula sheet
[MATHEMATICS
PAPER I]
STUDY NOTES
, Algebra and Equations (and inequalities)
Grade 9 Revision
The Real Number System
Name Description Example
natural numbers counting numbers from 1
whole numbers natural numbers including 0 0
integers positive and negative whole numbers
any number that can be written as (b
rational numbers terminating and recurring decimals ℚ
0)
non-terminating and non-recurring
irrational numbers numbers that are not irrational ℚ'
decimals; ; ;
real numbers all of the above all of the above
non-real / imaginary
anything that is undefined '
numbers
Interval Notation
end value is included [min max]
end value not included (min max)
[min; max] OR [min; max) OR (min; max) OR
separated by ‘;’
(min; max]
if goes on forever to positive infinity “ ” or
[min; ) OR (min; )
negative infinity “ ”
( ; max] OR ( ; max)
the bracket would always be ROUND
Set Builder Notation
what variable we will be using
what the restriction(s) on the variable is/are
which set of numbers we can use
such that: is less than 5 but greater than or equal to 2; x is an element of real
numbers
Summary
Name Example Sets used
inequality real
interval notation real
number line all sets
list natural, whole, integer
set builder notation all sets
Factorisation
Type 1: HCF (Highest Common Factor)
reverse of distribution
Type 2: Common Bracket
Instead of common factor, there is a common bracket
Change of sign rule:
Rea’s Notes | Mathematics Paper I | Algebra and Equations (and inequalities) 1
, Type 3: DOTS (Difference of Two Squares)
Only common fractions can be cancelled to simplify a fraction
You cannot cancel if there is more than one term in the numerator or denominator
Type 4: Trinomials/FOIL
Cross method; NB:
sign of middle term sign of last term sign of lower value factor sign of higher value factor
+ + + +
- + - -
+ - - +
- - + -
Rea’s Notes | Mathematics Paper I | Algebra and Equations (and inequalities) 2
