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
Co-ordinate Geometry
The angle sum of a quadrilateral is
360
Straight line
Tests for special quadrilaterals:
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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
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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
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

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SSS
SAS
AAS
RHS
Tests for similar shapes
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
…
(x-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=
𝑡𝑎𝑛∅ = |
𝑚1 − 𝑚2
|
1 + 𝑚1 𝑚2
Polynomials
𝛼 2 − 𝛽 2 = (𝛼 + 𝛽)2 − 4𝛼𝛽
𝛼 2 + 𝛽 2 = (𝛼 + 𝛽)2 − 2𝛼𝛽
Deriving the equation:
An = P (rate)n – M (1 + rate + rate2…)
After fully paid An = 0
Rearrange to find M, using (1 + rate +
rate2…) as a geometric series.
Probability
Probability of an event occurring =
The probability of two independent
events A and B occurring is given by:
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵)
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 )
4