TABLE OF CONTENTS
2
CHAPTER 1
3
CHAPTER 2
3
CHAPTER 3
Functions
Quadratic Functions
Equations, Inequalities and Graphs
4 Indices & Surds
CHAPTER 4
4 Factors of Polynomials
CHAPTER 5
4 Simultaneous Equations
CHAPTER 6
4 Logarithmic & Exponential Functions
CHAPTER 7
4 Straight Line Graphs
CHAPTER 8
5
CHAPTER 9
5
CHAPTER 10
Circular Measure
Trigonometry
6
CHAPTER 11
6
CHAPTER 12
6
CHAPTER 13
7
CHAPTER 14
Permutations & Combinations
Series
Vectors in 2 Dimensions
Differentiation & Integration
CIE IGCSE ADDITIONAL MATHEMATICS//0606
1. FUNCTIONS
• One-to-one functions: each 𝑥 value maps to one distinct
𝑦 value (check using vertical line test)
e.g. 𝑓(𝑥) = 3𝑥 − 1
• Many-to-one functions: there are some 𝑓(𝑥) values
which are generated by more than one 𝑥 value
e.g. 𝑓(𝑥) = 𝑥 2 − 2𝑥 + 3
Domain = 𝑥 values
Range = 𝑦 values
• Notation: 𝑓(𝑥) can also be written as 𝑓: 𝑥 ↦
• To find range:
o Complete the square
𝑥 2 − 2𝑥 + 3 ⇒ (𝑥 − 1)2 + 2
o Work out min/max point
Minimum point = (1,2)
∴ all 𝑦 values are greater than or equal to 2. 𝑓(𝑥) ≥ 2
One-to-many functions do not exist
• Domain of 𝑔(𝑥) = Range of 𝑔−1 (𝑥)
• Solving functions:
o 𝑓(2):
substitute 𝑥 = 2 and solve for 𝑓(𝑥)
o 𝑓𝑔(𝑥):
substitute 𝑥 = 𝑔(𝑥)
−1 (𝑥):
o𝑓
let 𝑦 = 𝑓(𝑥) and make 𝑥 the subject
• Composite Functions:
o 𝑓(𝑔(𝑥))𝑜𝑟 𝑓 ∙ 𝑔(𝑥)
o Substitute all instances of x in f(x) with g(x)
o Simplify
o If it is 𝑓 2 (𝑥), 𝑜𝑟 𝑓(𝑓(𝑥)), then for every x in f(x)
substitute f(x)’s contents
• Inverse Functions
o 𝑂𝑛𝑙𝑦 1 𝑡𝑜 1 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 ℎ𝑎𝑣𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑠
o If f(x) is a function, equate f(x) to y
o Replace all occurrences of x in f(x) with y
o Try to make x the subject of the function again
o That is the 𝑓 −1 (𝑥)
• Transformation of graphs:
o 𝑓(−𝑥): reflection in the 𝑦-axis
o −𝑓(𝑥): reflection in the 𝑥-axis
o 𝑓(𝑥) + 𝑎: translation of 𝑎 units parallel to 𝑦-axis
o 𝑓(𝑥 + 𝑎): translation of – 𝑎 units parallel to 𝑥-axis
2. QUADRATIC FUNCTIONS
• To sketch 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 ; 𝑎 ≠ 0
o Determine the shape
▪ 𝑎 > 0 – u-shaped ∴ minimum point
▪ 𝑎 < 0 – n-shaped ∴ maximum point
o Use the turning point
Express 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 as 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 by
completing the square
𝑛 2
𝑛 2
𝑥 2 + 𝑛𝑥 ⟺ (𝑥 + ) − ( )
2
2
𝑎(𝑥 + 𝑛)2 + 𝑘
Where the vertex is (−𝑛, 𝑘)
• Find the 𝒚-intercept:
o Substitute 𝑥 as 0 to get 𝑦 intercept
• Find the 𝒙-intercept:
o Factorize or use formula
• Type of root by calculating discriminant 𝑏 2 − 4𝑎𝑐
o If 𝑏 2 − 4𝑎𝑐 = 0, real and equal roots
o If 𝑏 2 − 4𝑎𝑐 > 0, real and distinct roots
o If 𝑏 2 − 4𝑎𝑐 < 0, no real roots
• Intersections of a line and a curve: if the equations of
the line and curve leads to a quadratic equation then:
o If 𝑏 2 − 4𝑎𝑐 = 0, line is tangent to the curve
o If 𝑏 2 − 4𝑎𝑐 > 0, line meets curve in two points
o If 𝑏 2 − 4𝑎𝑐 < 0, line does not meet curve
• Quadratic inequality:
o (𝑥 − 𝑑)(𝑥 − 𝛽) < 0 ⟹ 𝑑 < 𝑥 < 𝛽
o (𝑥 − 𝑑)(𝑥 − 𝛽) > 0 ⟹ 𝑥 < 𝑑 or 𝑥 > 𝛽
3. EQUATIONS, INEQUALITIES AND GRAPHS
• Transformation of graphs:
o 𝑓(−𝑥): reflection in the 𝑦-axis
o −𝑓(𝑥): reflection in the 𝑥-axis
o 𝑓(𝑥) + 𝑎: translation of 𝑎 units parallel to 𝑦-axis
o 𝑓(𝑥 + 𝑎): translation of – 𝑎 units parallel to 𝑥-axis
1
o 𝑓(𝑎𝑥): stretch, scale factor 𝑎 parallel to 𝑥-axis
o 𝑎𝑓(𝑥): stretch, scale factor 𝑎 parallel to 𝑦-axis
• Modulus function:
o Denoted by |𝑓(𝑥)|
o Modulus of a number is its absolute value
o Never goes below 𝑥-axis
o Makes negative graph into positive by reflecting
negative part into 𝑥-axis
• Solving modulus function:
o Sketch graphs and find points of intersection
o Square the equation and solve quadratic
• Relationship of a function and its inverse:
o The graph of the inverse of a function is the reflection
of a graph of the function in 𝑦=𝑥
PAGE 3 OF 8
CIE IGCSE ADDITIONAL MATHEMATICS//0606
• The points of intersection of two graphs are given by the
solution of their simultaneous equations
4. INDICES & SURDS
4.1 Indices
7. LOGARITHMIC & EXPONENTIAL FUNCTIONS
• Definitions:
o for 𝑎 > 0 and positive integers 𝑝 and 𝑞
𝑎0 = 1
1
𝑝
𝑎𝑝 = √𝑎
𝑎−𝑝
𝑝
• Definition
o for 𝑎 > 0 and 𝑎 ≠ 1
𝑦 = 𝑎 𝑥 ⇔ 𝑥 = log 𝑎 𝑦
• For log 𝑎 𝑦 to be defined
𝑦 > 0 and 𝑎 > 0, 𝑎 ≠ 1
• When the logarithms are defined
log 𝑎 1 = 0
log 𝑎 𝑏 + log 𝑎 𝑐 ≡ log 𝑎 𝑏𝑐
1
= 𝑝
𝑎
𝑞
𝑝
𝑎 𝑞 = ( √𝑎)
• Rules:
o for 𝑎 > 0, 𝑏 > 0 and rational numbers 𝑚 and 𝑛
𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛
𝑎𝑛 × 𝑏 𝑛 = (𝑎𝑏)𝑛
𝑎𝑚
𝑎𝑛
𝑎 𝑛
𝑚−𝑛
=
𝑎
=
(
)
𝑎𝑛
𝑏𝑛
𝑏
𝑚
𝑛
𝑚𝑛
(𝑎 ) = 𝑎
log 𝑎 𝑎 = 1
log 𝑎 𝑏 − log 𝑎 𝑐 ≡ log 𝑎
log 𝑏
log 𝑎 𝑏 ≡ log 𝑎
4.2 Surds
Definition
An irrational root is a surd, not all roots are surds
Rationalizing the Denominator
When the denominator is a surd, we can simplify by
multiplying both the numerator and the denominator by
the rationalization factor to rationalize
log 𝑎 𝑏 𝑛 ≡ 𝑛 log 𝑎 𝑏
• When solving logarithmic equations, check solution with
original equation and discard any solutions that causes
logarithm to be undefined
• Solution of 𝑎 𝑥 = 𝑏 where 𝑎 ≠ −1, 0, 1
• If 𝑏 can be easily written as 𝑎𝑛 , then
𝑎 𝑥 = 𝑎𝑛 ⇒ 𝑥 = 𝑛
• Otherwise take logarithms on both sides,
log 𝑏
i.e. log 𝑎 𝑥 = log 𝑏 and so 𝑥 = log 𝑎
• log ⇒ log10
• ln ⇒ log 𝑒
• Change of base rule:
5. FACTORS OF POLYNOMIALS
• To find unknowns in a given identity
o Substitute suitable values of 𝑥
OR
o Equalize the given coefficients of like powers of x
Factor Theorem:
• If (𝑥 − 𝑡) is a factor of the function 𝑝(𝑥) then 𝑝(𝑡) = 0
Remainder Theorem:
• If a function 𝑓(𝑥) is divided by (𝑥 − 𝑡) then:
𝑅𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 = 𝑓(𝑡)
• The formula for remainder theorem:
𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 = 𝐷𝑖𝑣𝑖𝑠𝑜𝑟 × 𝑄𝑢𝑜𝑡𝑖𝑒𝑛𝑡 + 𝑅𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟
𝑙𝑜𝑔𝑏 (𝑥)
𝑙𝑜𝑔𝑎 (𝑥)
Logarithmic & Exponential Graphs
𝑙𝑜𝑔𝑎 (𝑥) =
6. SIMULTANEOUS EQUATIONS
• Simultaneous linear equations can be solved either by
substitution or elimination
• Simultaneous linear and non-linear equations are
generally solved by substitution as follows:
o Step 1: obtain an equation in one unknown & solve it
o Step 2: substitute the results from step 1 into the
linear equation to find the other unknown
𝑏
𝑐
8. STRAIGHT LINE GRAPHS
• Equation of a straight line:
𝑦 = 𝑚𝑥 + 𝑐
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
PAGE 4 OF 8
CIE IGCSE ADDITIONAL MATHEMATICS//0606
• Gradient:
• Area of a sector:
𝑦2 − 𝑦1
𝑚=
𝑥2 − 𝑥1
• Length of a line segment:
1
𝐴 = 𝑟2𝜃
2
10. TRIGONOMETRY
Length = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
• Trigonometric ratio of special angles:
• Midpoint of a line segment:
𝑥1 + 𝑥2 𝑦1 + 𝑦2
(
,
)
2
2
• Point on line segment with ratio m:n
𝑛𝑥1 + 𝑚𝑥2 𝑛𝑦1 + 𝑚𝑦2
(
,
)
𝑚+𝑛
𝑚+𝑛
• Parallelogram:
o ABCD is a parallelogram ⟺ diagonals AC and BD have
a common midpoint
o Special parallelograms = rhombuses, squares,
rectangles
• Special gradients:
o Parallel lines: 𝑚1 = 𝑚2
o Perpendicular lines: 𝑚1 𝑚2 = −1
• Perpendicular bisector: line passes through midpoint
• To work out point of intersection of two lines/curves,
solve equations simultaneously
• Find Tangent: Once the gradient is obtained, substitute
the point into the slope-intercept form to get c and the
equation.
• Find normal: Obtain the gradient by taking the negative
reciprocal (see perpendicular gradients). Once the
gradient is obtained, substitute the point (original point)
into the slope-intercept form to get c and the equation.
• Find Area, using two methods
• Straight Line graphs: find variables when an equation
that does not involve x and y but rather other forms of x
and y example: (𝑥 3 ) or ln (𝑦) . This is represented as a
straight line.
Mostly in the form 𝑦 = 𝑎𝑥 𝑛 or 𝑦 = 𝐴𝑏 𝑛 , that
must be converted to the form y = mx+c.
9. CIRCULAR MEASURE
• Radian measure:
𝜋 = 180°
Degree to Rad =×
• Arc length:
2𝜋 = 360°
𝜋
180
Rad to Degree =×
180
𝜋
SINE CURVE
TANGENT CURVE
COSINE CURVE
CAST DIAGRAM
• Trigonometric ratios:
1
1
1
sec 𝜃 =
cosec 𝜃 =
cot 𝜃 =
cos 𝜃
sin 𝜃
tan 𝜃
• Trigonometric identities:
sin 𝜃
sin2 𝜃 + cos 2 𝜃 = 1
tan 𝜃 =
cos 𝜃
cot 2 𝜃 + 1 = cosec 2 𝜃
tan2 𝜃 + 1 = sec 2 𝜃
𝑠 = 𝑟𝜃
PAGE 5 OF 8
CIE IGCSE ADDITIONAL MATHEMATICS//0606
• Sketching trigonometric graphs:
11. PERMUTATIONS & COMBINATIONS
• Basic counting principle: to find the number of ways of
performing several tasks in succession, multiply the
number of ways in which each task can be performed:
e.g. 5 × 4 × 3 × 2
• Factorial: 𝑛! = 𝑛 × (𝑛 − 1) × (𝑛 − 2) … × 3 × 2 × 1
o NOTE: 0! = 1
• Permutations:
o The number of ordered arrangements of r objects
taken from n unlike objects is:
𝑛!
𝑛
𝑃𝑟 =
(𝑛 − 𝑟)!
o Order matters
• Combinations:
o The number of ways of selecting 𝑟 objects from 𝑛
unlike objects is:
𝑛!
𝑛
𝐶𝑟 =
𝑟! (𝑛 − 𝑟)!
• Order does not matter
12. SERIES
12.1 Binomial Expansion
• Where n is the number of the term, a (𝑈1 ) is the first
term and d is the common difference
• Formula for the sum of the first n terms between
𝑢𝑠𝑡𝑎𝑟𝑡 to 𝑢𝑒𝑛𝑑
𝑛
𝑆𝑛 = (𝑢𝑠𝑡𝑎𝑟𝑡 + 𝑢𝑒𝑛𝑑 )
2
• Example:
Sequence: 1,2,3,4,5,6
Sum: 21
Geometric Progression
• A sequence made by multiplying by the same value each
time.
• A common ration r is multiplied or divided (n-1) times
• General form: 𝑈𝑛 = 𝑎𝑟 𝑛−1
• Where n is the number of the term, a is the first term
and r is the common ratio
Example:
Sequence: 2, 4, 8, 16, 32
Sum: 62
• Formula for the sum of the first n numbers of a
geometric series
1 − 𝑟𝑛
𝑆𝑛 = 𝑎1 ×
1−𝑟
Sum to infinity
• Where the common ratio satisfies the condition:
−1 < 𝑟 < 1, it is an infinite geometric progression
(convergent progression)
1
𝑆∞ = 𝑎1 ×
1−𝑟
13. VECTORS IN 2 DIMENSIONS
• The binomial theorem allows expansion of any
expression in the form (𝑎 + 𝑏)𝑛
(𝑥 + 𝑦)𝑛 = 𝑛𝐶0 𝑥 𝑛 + 𝑛𝐶1 𝑥 𝑛−1 𝑦 + 𝑛𝐶2 𝑥 𝑛−2 𝑦 2 + ⋯ + 𝑛𝐶𝑛 𝑦 𝑛
• e.g. Expand (2𝑥 − 1)4
(2𝑥 − 1)4 = 4𝐶0 (2𝑥)4 + 4𝐶1 (2𝑥)3 (−1)
+ 4𝐶2 (2𝑥)2 (−1)2 + 4𝐶3 (2𝑥) (−1)3 + 4𝐶4 (−1)4
= 1(2𝑥)4 + 4(2𝑥)3 (−1) +6(2𝑥)2 (−1)2 +
4(2𝑥) (−1)3 + 1(−1)4
= 16𝑥 4 − 32𝑥 3 + 24𝑥 2 − 8𝑥 + 1
• The powers of 𝑥 are in descending order
12.2 Sequences & Series
Arithmetic Progression
• A sequence made by adding the same value each time.
• A common difference d is added or subtracted (n-1)
times
• General form: 𝑈𝑛 = 𝑎 + (𝑛 − 1)𝑑
⃗⃗⃗⃗⃗
• Position vector: position of point relative to origin, 𝑂𝑃
• Forms of vector:
𝑎
⃗⃗⃗⃗⃗
( )
𝑝
𝑎𝐢 − 𝑏𝐣
𝐴𝐵
𝑏
• Parallel vectors: same direction but different magnitude
⃗⃗⃗⃗⃗ = 𝑂𝐵
⃗⃗⃗⃗⃗ − 𝑂𝐴
⃗⃗⃗⃗⃗
• Generally, 𝐴𝐵
• Magnitude = √i2 + j2
• Unit vectors: vectors of magnitude 1
⃗⃗⃗⃗⃗
o Examples: consider vector 𝐴𝐵
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗ | = √13
𝐴𝐵 = 2𝐢 + 3𝐣
|𝐴𝐵
1
(2𝒊 + 3𝒋)
∴ 𝑈𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 =
√13
• Collinear vectors: vectors that lie on the same line
• Velocity Vector:
𝑎
( )
𝑏
PAGE 6 OF 8
CIE IGCSE ADDITIONAL MATHEMATICS//0606
• Getting velocity from speed: Find k to get velocity based
on speed
𝑎
𝑘 × |( )| = 𝑠𝑝𝑒𝑒𝑑
𝑏
• Point of intersection:
𝑎
𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑥
Object 1 = (
)
+
𝑡
(
)
𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑦
𝑏
𝑐
𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑥
Object 2 = (
)
+
𝑡
(
)
𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑦
𝑑
Object 1 = Object 2 at time t. If both x and y are not same
at intersection time then they will never meet.
14. DIFFERENTIATION & INTEGRATION
14.1 Differentiation
FUNCTION
𝑦 = 𝑥𝑛
1ST DERIVATIVE
𝑑𝑦
= 𝑛𝑥 𝑛−1
𝑑𝑥
2ND DERIVATIVE
𝑑2 𝑦
= 𝑛(𝑛 − 1)𝑥 𝑛−2
𝑑𝑥 2
INCREASING FUNCTION
DECREASING FUNCTION
𝑑𝑦
𝑑𝑦
>0
<0
𝑑𝑥
𝑑𝑥
• Stationary point: equate first derivative to zero
𝑑𝑦
=0
𝑑𝑥
• 2nd Derivative: finds nature of the stationary point
o If
o If
𝑑2 𝑦
𝑑𝑥 2
𝑑2 𝑦
𝑑𝑥 2
> 0 → minimum stationary point
< 0 →maximum stationary point
• Chain rule:
• Product rule:
• Quotient rule:
𝑑𝑦 𝑑𝑦 𝑑𝑢
=
×
𝑑𝑥 𝑑𝑢 𝑑𝑥
𝑑𝑦
𝑑𝑣
𝑑𝑢
=𝑢
+𝑣
𝑑𝑥
𝑑𝑥
𝑑𝑥
𝑑𝑢
𝑑𝑣
𝑑𝑦 𝑣 𝑑𝑥 − 𝑢 𝑑𝑥
=
𝑑𝑥
𝑣2
Special Differentials
𝑑𝑦
(sin 𝑎𝑥) = 𝑎 cos 𝑎𝑥
𝑑𝑥
𝑑𝑦
(cos 𝑎𝑥) = −𝑎 sin 𝑎𝑥
𝑑𝑥
𝑑𝑦
(tan 𝑎𝑥 ) = 𝑎 sec 2 𝑎𝑥
𝑑𝑥
𝑑𝑦
( 𝑒 𝑎𝑥+𝑏 ) = 𝑎𝑒 𝑎𝑥+𝑏
𝑑𝑥
𝑑𝑦
1
(ln 𝑥) =
𝑑𝑥
𝑥
𝑑𝑦
𝑓 ′ (𝑥)
(ln(𝑓(𝑥)) =
𝑑𝑥
𝑓(𝑥)
• Related rates of change:
o If 𝑥 and 𝑦 are related by the equation 𝑦 = 𝑓(𝑥), then
𝑑𝑥
𝑑𝑦
the rates of change 𝑑𝑡 and 𝑑𝑡 are related by:
𝑑𝑦 𝑑𝑦 𝑑𝑥
=
×
𝑑𝑡 𝑑𝑥 𝑑𝑡
• Small changes:
o If 𝑦 = 𝑓(𝑥) and small change 𝛿𝑥 in 𝑥 causes a small
change 𝛿𝑦 in 𝑦, then
𝑑𝑦
𝛿𝑦 ≈ ( )
× 𝛿𝑥
𝑑𝑥 𝑥=𝑘
14.2 Integration
𝑥 𝑛+1
+𝑐
(𝑛 + 1)
(𝑎𝑥 + 𝑏)𝑛+1
∫(𝑎𝑥 + 𝑏)𝑛 =
+𝑐
𝑎(𝑛 + 1)
• Definite integral: substitute coordinates/values & find 𝑐
• Indefinite integral: has c (constant of integration)
• Integrating by parts:
𝑑𝑣
𝑑𝑢
∫𝑢
𝑑𝑥 = 𝑢𝑣 − ∫ 𝑣
𝑑𝑥
𝑑𝑥
𝑑𝑥
o What to make 𝑢: LATE
Logs
Algebra
Trig
𝒆
∫ 𝑎𝑥 𝑛 = 𝑎
• To find area under the graph (curve and 𝒙-axis):
o Integrate curve
o Substitute boundaries of 𝑥
o Subtract one from another (ignore c)
𝑏
∫ 𝑦 𝑑𝑥
𝑎
• To find area between curve and 𝒚-axis:
o Make 𝑥 subject of the formula
o Follow above method using 𝑦-values instead of 𝑥values
Special Integrals
1
∫ sin(𝑎𝑥 + 𝑏) = − cos(𝑎𝑥 + 𝑏) + 𝑐
𝑎
1
∫ cos(𝑎𝑥 + 𝑏) = sin(𝑎𝑥 + 𝑏) + 𝑐
𝑎
1
∫ sec 2 (𝑎𝑥 + 𝑏) = tan(𝑎𝑥 + 𝑏) + 𝑐
𝑎
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CIE IGCSE ADDITIONAL MATHEMATICS//0606
∫
1
1
= ln|𝑎𝑥 + 𝑏| + 𝑐
𝑎𝑥 + 𝑏 𝑎
1
∫ 𝑒 𝑎𝑥+𝑏 = 𝑒 𝑎𝑥+𝑏 + 𝑐
𝑎
14.3 Kinematics
• Particle at instantaneous rest, 𝑣 = 0
• Maximum displacement from origin, 𝑣 = 0
• Maximum velocity, 𝑎 = 0
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