
Co-ordinate Geometry
The angle sum of a quadrilateral is
360
Straight line
Tests for special quadrilaterals:


Gradient form: 𝑦 = 𝑚𝑥 + 𝑐
General form: 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0

Distance: 𝑑 =
√(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2

Midpoint: ( 1 2 2 , 1 2 2 )
Perpendicular distance of a point
from a line.

𝑥 +𝑥
𝑑=|



𝑦 +𝑦
𝑎𝑥1 + 𝑏𝑦1 + 𝑐
√𝑎2 + 𝑏 2
|
Relationship between gradient and
angle 𝑚 = 𝑡𝑎𝑛 ∝
Angle between two lines
𝑚2 − 𝑚1
𝑡𝑎𝑛𝜑 =
1 + 𝑚1 𝑚2
𝑥=
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
Geometrical Properties
Parallelograms:
 Two opposite sides equal and
parallel or
 Opposite sides are equal or
 Opposite angles are equal or
 Diagonals bisect each other
Rhombus:
 All sides equal or
 Diagonals bisect each other at right
angles
Rectangle:
 All angles are right angles or
 Parallelogram with equal diagonals
Square:
 All sides equal and one angle right
or
 All angles right and two adjacent
sides equal.
Tests for congruent triangles




SSS
SAS
AAS
RHS
Tests for similar shapes













Complementary angles add to 90
Supplementary angle add to 180
Vertically opposite angles are equal
Angles at a point add to 360
Angle sum of a triangle is 180
The exterior angle of a triangle is
equal to the sum of the opposite
interior angles
An isosceles triangle has equal base
angles
Equilateral triangles have all angles
60
Alternate angles on parallel lines are
equal
Corresponding angles on parallel
lines are equal
Co-interior angles between parallel
lines are supplementary
The angle sum of a polygon is
(n-2)x180
The sum of the exterior angles of
any polygon is equal to 360
All angles are the same, therefore the
overall shape is the same.
All equivalent sides on each shape are
in the same proportion to each other.
Applications of Differentiation
𝑑𝑦

First derivative 𝑑𝑥
- Stationary point when equals 0
- Curve increasing>0
- Curve decreasing<0
-Max turning point if second
derivative negative
-Minimum turning point if second
derivative positive

Second derivative 2
𝑑𝑥
- Point of inflexion when equals 0
-Concave up when >0
-Concave down when <0
Horizontal point of inflexion if both
first and second derivatives equal
zero.

𝑑2 𝑦
1
Logarithmic Functions
Integration
𝑛
∫ 𝑥 𝑑𝑥 = 𝑥

𝑛+1
𝑑
1
(𝑙𝑛𝑥) =
𝑑𝑥
𝑥
𝑑
𝑓 ′ (𝑥)
(𝑙𝑛𝑓(𝑥)) =
𝑑𝑥
𝑓(𝑥)
+𝑐
Area between curve and axis
1
∫ 𝑑𝑥 = ln 𝑥 + 𝑙𝑛𝐴
𝑥
𝑓 ′ (𝑥)
∫
𝑑𝑥 = ln 𝑓(𝑥) + 𝑙𝑛𝐴
𝑓(𝑥)
𝑏
𝐴 = ∫ 𝑦𝑑𝑥
𝑎

Volume of revolution
𝑏
𝑉 = 𝜋 ∫ 𝑦 2 𝑑𝑥
Log laws
𝑎

Area between two curves
A =  top curve -  bottom curve

Volume between two curves
A =   (top curve)2 – (bottom curve)2
Approximating integrals
𝑙𝑛𝑒 2 = 2𝑙𝑛𝑒 = 2
𝑙𝑜𝑔𝑘𝑥 = 𝑙𝑜𝑔𝑘 + 𝑙𝑜𝑔𝑥
𝑘
𝑙𝑜𝑔 = 𝑙𝑜𝑔𝑘 − 𝑙𝑜𝑔𝑥
𝑥
𝑙𝑜𝑔𝑐 𝑏
𝑙𝑜𝑔𝑎 𝑏 =
𝑙𝑜𝑔𝑐 𝑎
2.5
Simpson’s Rule
𝒉
𝒂+𝒃
𝑨 = {𝒇(𝒂) + 𝟒 × 𝒇(
+ 𝒇(𝒃))}
𝟑
𝟐
2
1.5
1
Trapezium Rule
𝒉
𝑨 = (𝒚𝟎 + 𝒚𝒏 + 𝟐(𝒚𝟏 + 𝒚𝟐 … . +𝒚𝒏−𝟏 ))
𝟐
Logarithmic and Exponential
Functions
0.5
0
0
2
4
6
8
10
Trigonometric Functions
Exponential functions
𝑑
(𝑒 𝑥 )
𝑑𝑥
𝐴𝑟𝑐 𝐿𝑒𝑛𝑔𝑡ℎ = 𝑟𝜃
= 𝑒𝑥
1
𝐴𝑟𝑒𝑎 𝑜𝑓𝑎 𝑠𝑒𝑐𝑡𝑜𝑟 = 𝑟 2 𝜃
2
y = Sin x
Period = 2
Amplitude = 1
𝑑 𝑎𝑥
(𝑒 ) = 𝑎𝑒 𝑎𝑥
𝑑𝑥
𝑑 𝑓(𝑥)
(𝑒
) = 𝑓 ′ (𝑥)𝑒 𝑓(𝑥)
𝑑𝑥
8
1.5
7
6
1
5
0.5
4
0
3
-0.5
0
2
1
2
3
4
5
6
7
-1
1
-1.5
0
-4
-3
-2
-1
0
1
2
3
2

y = Cos x
Period = 2
Amplitude = 1
Decay y = Ae-k
2.5
2
1.5
1.5
1
1
0.5
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
0
0
1
2
3
4
5
6
7
-0.5

Exponential Growth
If the rate of change is proportional
to P, ie dP/dt = kP
Then P = Poekt
 Exponential Decay
If dP/dt = -kP
Then P = Poe-kt
Where Po is the initial value of P
k is the constant of proportionality
P is the amount of quantity present at
time t
-1
-1.5
y = tan x
Period = 
𝑠𝑖𝑛𝑥
lim
=1
𝑥→0 𝑥
Series and Applications
Arithmetic Series
𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 = 𝑎 + (𝑛 − 1)𝑑
𝑛
𝑆𝑛 = (𝑓𝑖𝑟𝑠𝑡 + 𝑙𝑎𝑠𝑡)
2
𝑛
𝑆𝑛 = (2𝑎 + (𝑛 − 1)𝑑)
2
Kinematics
Displacement = x
𝑑𝑥
Velocity 𝑣 = 𝑥̇ =
𝑑𝑡
Acceleration 𝑎 = 𝑣
𝑑𝑣
𝑑𝑥
= 𝑥̈ =
𝑑2 𝑥
𝑑𝑡 2
Geometric Series
𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 = 𝑎𝑟 𝑛−1
𝑎(𝑟 𝑛 − 1)
𝑆𝑛 =
(𝑟 − 1)
𝑎
|𝑟| < 1
𝑠∞ =
1−𝑟
𝑥 = ∫ 𝑣𝑑𝑡
𝑣 = ∫ 𝑎𝑑𝑡
Exponential Growth and Decay
 If e = a, then  = logea

Growth y = aek
Compound Interest
A=P
Superannuation
If $P is invested at the beginning of
each year in a superannuation fund
earning interest at r% pa, the investment
after n years will amount to T
A1 = P
A2 = P
3
Parabolas
And so on, so that investment = A1 +
A2…
=P
+P
…
(-b)2 = 4a(y-c)
where (b,c) is the vertex
a is the focal length
General Solutions of Trig Equations
forms a geometric series with
X = nπ+ (-1)nsin-1(k)
a=P
n = number of years
X = 2nπ ± cos-1(k)
and r =
X = nπ + tan-1(k)
Angle between two lines of slopes m1
and m2
Time payments
A person borrows $P at r% per term,
where the interest is compounded per
term on the amount owing. If they pay
off the loan in equal term instalments
over n terms, their equal term instalment
is M, where
M=
Deriving the equation:
An = P (rate)n – M (1 + rate + rate2…)
𝑡𝑎𝑛∅ = |
Polynomials
𝛼 2 − 𝛽 2 = (𝛼 + 𝛽)2 − 4𝛼𝛽
𝛼 2 + 𝛽 2 = (𝛼 + 𝛽)2 − 2𝛼𝛽
𝛼 2 + 𝛽 2 + 𝛾 2 = (𝛼 + 𝛽 + 𝛾)2 − 2(𝛼𝛽
+ 𝛽𝛾 + 𝛼𝛾)
𝛼 3 + 𝛽 3 = (𝛼 + 𝛽)3 − 3𝛼𝛽(𝛼 + 𝛽)
For 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
After fully paid An = 0
Rearrange to find M, using (1 + rate +
rate2…) as a geometric series.
𝑏
𝛼+𝛽 =−
𝑎
𝑐
𝛼𝛽 =
𝑎
Probability
Probability of an event occurring =
𝑚1 − 𝑚2
|
1 + 𝑚1 𝑚2
For 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑 = 0
𝑏
𝑎
𝑐
𝛼𝛽 + 𝛽𝛾 + 𝛼𝛾 =
𝑎
𝑑
𝛼𝛽𝛾 = −
𝑎
𝛼+𝛽+𝛾 =−
The probability of two independent
events A and B occurring is given by:
Complex Numbers
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵)
(𝐶𝑜𝑠𝜃 + 𝑖𝑆𝑖𝑛𝜃)𝑛 = 𝐶𝑜𝑠𝑛𝜃 + 𝑖𝑆𝑖𝑛𝑛𝜃
Sum and Difference of Two Cubes
𝑥 3 + 𝑦 3 = (𝑥 + 𝑦)(𝑥 2 − 𝑥𝑦 + 𝑦 2 )
𝑥 3 − 𝑦 3 = (𝑥 − 𝑦)(𝑥 2 + 𝑥𝑦 + 𝑦 2 )
1
𝑧𝑘+1 = 𝑟 𝑛 (𝐶𝑜𝑠
𝜃+2𝑘𝜋
𝑛
+ 𝑖𝑆𝑖𝑛
𝜃+2𝑘𝜋
)
𝑛
Ellipse
4
(𝑥−ℎ)2
(𝑦−𝑘)2
Equation 𝑎2 + 𝑏2 = 1
Foci S and S’ (±𝑎𝑒, 0)
𝑎
Directrices
𝑥 = ±𝑒
Where 𝑏 2 = 𝑎2 (1 − 𝑒 2 )
Parametrics (𝑎𝐶𝑜𝑠𝜃, 𝑏𝑆𝑖𝑛𝜃)
𝑥𝐶𝑜𝑠𝜃
𝑦𝑆𝑖𝑛𝜃
Eqn of Tangent 𝑎 + 𝑏 = 1
Eqn of Normal
𝑎𝑥𝑆𝑒𝑐𝜃 − 𝑏𝑦𝐶𝑜𝑠𝑒𝑐𝜃 = 𝑎2 − 𝑏 2
Hyperbola
𝑥2
𝑦2
Equation
− 2=1
𝑎2
𝑏
Foci S and S’ ±(𝑎𝑒, 0)
𝑎
Directrices
𝑥 = ±𝑒
Where
𝑏 2 = 𝑎2 (𝑒 2 − 1)
Parametrics (𝑎𝑆𝑒𝑐𝜃, 𝑏𝑇𝑎𝑛𝜃)
𝑥𝑆𝑒𝑐𝜃
𝑦𝑇𝑎𝑛𝜃
Eqn of Tangent 𝑎 − 𝑏 = 1
𝑥 = 𝐴𝐶𝑜𝑠(𝑛𝑡 + 𝛼)
𝑥̈ = −𝑛2 𝑥
𝑣 = −𝑛𝐴𝑆𝑖𝑛(𝑛𝑡+∝)
𝑣 2 = 𝑛2 (𝐴2 − 𝑥 2 )
2𝜋
𝑇=
𝑛
Circular Motion
𝑣 = 𝑟𝜔
𝑎 = 𝑟𝑤 2
𝑚𝑣 2
𝐹=
= 𝑚𝑟𝑤 2
𝑟
Complex Numbers
𝑧 𝑛 = 𝑟 𝑛 (𝐶𝑜𝑠𝑛𝜃 + 𝑖𝑆𝑖𝑛𝑛𝜃)
Rectangular Hyperbola
Equation 𝑥𝑦 = 𝑐 2
Eccentricity = √2
𝑐
Parametrics 𝑥 = 𝑐𝑡, 𝑦 =
𝑡
Integration
𝑥
If 𝑡 = 𝑇𝑎𝑛 2 then
2𝑡
𝑆𝑖𝑛 𝑥 = 1+𝑡 2
𝐶𝑜𝑠 𝑥 =
and
2𝑑𝑡
1−𝑡 2
1+𝑡 2
𝑑𝑥 = 1+𝑡 2
Volumes
𝑏
About x axis 𝑉 = 𝜋 ∫𝑎 𝑦 2 𝑑𝑥
𝑏
About y axis 𝑉 = 𝜋 ∫𝑎 𝑥 2 𝑑𝑦
Without a uniform cross section
𝑏
𝑉 = ∫𝑎 𝐴(𝑥)𝑑𝑥
Cylindrical Shell
𝑏
𝑉 = 2𝜋 ∫𝑎 𝑥𝑦𝑑𝑥
Simple Harmonic Motion
5