CURRICULUM GRADE 10 -12 DIRECTORATE
NCS (CAPS) SUPPORT
JUST IN TIME LEARNER REVISION
DOCUMENT
MATHEMATICS
GRADE 11
2025
,Mathematics KZN-GRADE Gr 11 Revision 2025
This document has been compiled by the FET Mathematics Subject Advisors together with Top Teachers.
TABLE OF CONTENTS
PAGE
TOPICS
NUMBERS
1. ALGEBRA 3-9
2. NUMBER PATTERNS 10 - 14
3. FUNCTIONS 15 - 27
4. FINANCE GROWTH AND DECAY 28 - 31
5. PROBABILITY 32 - 36
6. STATISTICS 37 - 43
7. ANALYTICAL GEOMETRY 44 - 55
8. TRIGONOMETRY 56 - 67
9. EUCLIDEAN GEOMETRY 68 - 80
10. MEASUREMENT 81 - 83
2
,Mathematics KZN-GRADE Gr 11 Revision 2025
TOPIC 1. ALGEBRA
GUIDELINES, SUMMARY NOTES, & STRATEGIES
Rational exponents (fractional exponents): expressions with exponents that are rational numbers
➢ Laws of exponents Quadratic Equations Quadratic exponents
• a n a m = a m+ n Highest power of unknown is twice the It is an equation with three terms,
power of a middle term (usually 2) one term has the highest power
am
• n
= a m−n Always has 2 solutions (known as roots) that is multiple of 2, the middle
a term has one- half the exponent of
Standard form is ax 2 + bx + c = 0
• (a ) m n
=a mn
Step 1: write in standard form
the term with a highest power
x − 2 x.0,5 − 3 = 0
• (a b) n
= a b n n
Step2: factorise.
m
let x 0,5 = k
a Step3: equate each factor to 0
m
a
• = m k 2 − 2k − 3 = 0
b b Step4: solve
• a0 = 1 BEWARE: NEVER DIVIDE BY AN (k − 3)(k + 1) = 0
UKNOWN
p
x = x p ; x 0; q 0
q q
•
• NOTE: when adding or subtracting powers with numerical bases, factorise.
• when a m = a n m = n and a m = b m a = b
p p q q
r
• when x = t x q r q p
=t p
provided p is not an even number
➢ Adding, subtracting, dividing and multiplying surds
n
a n b = n ab ➢ Solving surds equation
•
n
a n a Steps:
n
=
• b b • isolate the surd
pn a qn a = p qn a • square both sides
•
• simplify and solve the equation
➢ Factorisation: • test solutions to an original equation and reject if
• common factor applicable
• trinomial using “k” method
• difference of two squares • substitute repeated expression with “k”.
• grouping • solve for k
• sum and difference of two cubes. • substitute back
• solve the original uunknown
➢ completing the square
steps
b
• “take out” the coefficient of x 2 for the first 2 terms a x 2 + x + c
a
3
, Mathematics KZN-GRADE Gr 11 Revision 2025
2
1
2 2
Add and subtract (the coefficient of x ) a x 2 +
b b b
• x+ 2 − 2 + c
2 2a 4a 4a
2
b b2
• Factorise the perfect square trinomial and multiply a x + − 2 +c
2a 4a
➢ Quadratic equation
from completing the square 2
b b2 c
b b2
2 x+ = 2−
a x + − 2 +c =0 2a 4a a
2a 4a
2
b b2 c
x + − 2 = − − b b 2 − 4ac
2a 4a a If ax 2 + bx + c = 0 x = where a 0
2a
➢ Inequalities(notes) • Where y 0, above x axis x -1 or x 2
• Where y 0 below the x axis - 1 x 2
• NOTE:
When it is impossible to find a critical value,
answer the question using the graphs e.g
Y=0 3 x (x − 3) 0 .in this expression we want the
-1 2
values of x where the exponential function
is above the x-axis while the linear
function is below or an opposite.
Simultaneous equations
When solving a pair of simultaneous linear equations, we are, in fact, finding a common point – the point
of intersection of the two functions
Elimination method Substitution method
• Make the coefficients of one of the • Use the simplest of the two given equations to express one
variables the same in both of the variables in terms of the other.
equations, Eliminate the variable by • Substitute into the second equation. By doing this we
adding equation (1) and equation (2) reduce the number of equations and the number of
together. variables by one.
• Simplify and solve for x • We now have one equation with one unknown variable
• Substitute x back into either original which can be solved.
equation and solve for y • Use the solution to substitute back into the first equation
to find the value of the other unknown variable.
Nature of roots
− b b 2 − 4ac
ax + bx + c = 0 x =
2
2a
4
NCS (CAPS) SUPPORT
JUST IN TIME LEARNER REVISION
DOCUMENT
MATHEMATICS
GRADE 11
2025
,Mathematics KZN-GRADE Gr 11 Revision 2025
This document has been compiled by the FET Mathematics Subject Advisors together with Top Teachers.
TABLE OF CONTENTS
PAGE
TOPICS
NUMBERS
1. ALGEBRA 3-9
2. NUMBER PATTERNS 10 - 14
3. FUNCTIONS 15 - 27
4. FINANCE GROWTH AND DECAY 28 - 31
5. PROBABILITY 32 - 36
6. STATISTICS 37 - 43
7. ANALYTICAL GEOMETRY 44 - 55
8. TRIGONOMETRY 56 - 67
9. EUCLIDEAN GEOMETRY 68 - 80
10. MEASUREMENT 81 - 83
2
,Mathematics KZN-GRADE Gr 11 Revision 2025
TOPIC 1. ALGEBRA
GUIDELINES, SUMMARY NOTES, & STRATEGIES
Rational exponents (fractional exponents): expressions with exponents that are rational numbers
➢ Laws of exponents Quadratic Equations Quadratic exponents
• a n a m = a m+ n Highest power of unknown is twice the It is an equation with three terms,
power of a middle term (usually 2) one term has the highest power
am
• n
= a m−n Always has 2 solutions (known as roots) that is multiple of 2, the middle
a term has one- half the exponent of
Standard form is ax 2 + bx + c = 0
• (a ) m n
=a mn
Step 1: write in standard form
the term with a highest power
x − 2 x.0,5 − 3 = 0
• (a b) n
= a b n n
Step2: factorise.
m
let x 0,5 = k
a Step3: equate each factor to 0
m
a
• = m k 2 − 2k − 3 = 0
b b Step4: solve
• a0 = 1 BEWARE: NEVER DIVIDE BY AN (k − 3)(k + 1) = 0
UKNOWN
p
x = x p ; x 0; q 0
q q
•
• NOTE: when adding or subtracting powers with numerical bases, factorise.
• when a m = a n m = n and a m = b m a = b
p p q q
r
• when x = t x q r q p
=t p
provided p is not an even number
➢ Adding, subtracting, dividing and multiplying surds
n
a n b = n ab ➢ Solving surds equation
•
n
a n a Steps:
n
=
• b b • isolate the surd
pn a qn a = p qn a • square both sides
•
• simplify and solve the equation
➢ Factorisation: • test solutions to an original equation and reject if
• common factor applicable
• trinomial using “k” method
• difference of two squares • substitute repeated expression with “k”.
• grouping • solve for k
• sum and difference of two cubes. • substitute back
• solve the original uunknown
➢ completing the square
steps
b
• “take out” the coefficient of x 2 for the first 2 terms a x 2 + x + c
a
3
, Mathematics KZN-GRADE Gr 11 Revision 2025
2
1
2 2
Add and subtract (the coefficient of x ) a x 2 +
b b b
• x+ 2 − 2 + c
2 2a 4a 4a
2
b b2
• Factorise the perfect square trinomial and multiply a x + − 2 +c
2a 4a
➢ Quadratic equation
from completing the square 2
b b2 c
b b2
2 x+ = 2−
a x + − 2 +c =0 2a 4a a
2a 4a
2
b b2 c
x + − 2 = − − b b 2 − 4ac
2a 4a a If ax 2 + bx + c = 0 x = where a 0
2a
➢ Inequalities(notes) • Where y 0, above x axis x -1 or x 2
• Where y 0 below the x axis - 1 x 2
• NOTE:
When it is impossible to find a critical value,
answer the question using the graphs e.g
Y=0 3 x (x − 3) 0 .in this expression we want the
-1 2
values of x where the exponential function
is above the x-axis while the linear
function is below or an opposite.
Simultaneous equations
When solving a pair of simultaneous linear equations, we are, in fact, finding a common point – the point
of intersection of the two functions
Elimination method Substitution method
• Make the coefficients of one of the • Use the simplest of the two given equations to express one
variables the same in both of the variables in terms of the other.
equations, Eliminate the variable by • Substitute into the second equation. By doing this we
adding equation (1) and equation (2) reduce the number of equations and the number of
together. variables by one.
• Simplify and solve for x • We now have one equation with one unknown variable
• Substitute x back into either original which can be solved.
equation and solve for y • Use the solution to substitute back into the first equation
to find the value of the other unknown variable.
Nature of roots
− b b 2 − 4ac
ax + bx + c = 0 x =
2
2a
4
