Algebra
Index Laws
Standard index laws
Standard index laws
(a m) n = a mn
am × an = am + n
a1 = a
a n = a × a × a × ... × a
n times
m
am ÷ an = a n
a
= am - n
a0 = 1
(ab) n = a nb n
for a ≠ 0
a n an
= n
b
b
( )
for b ≠ 0
Negative Indices
a-1 =
1
a
a - n = 1n
a
1
a-n
= an
a -1
b
=
b
a
( )
for a ≠ 0
for a ≠ 0
a -n
=
b
( )
for a ≠ 0
=
for a ≠ 0, b ≠ 0
b n
a
( )
bn
an
for a ≠ 0, b ≠ 0
Fractional Indices
1
a2 = a
a
a
a
1
n
m
2
m
n
-
1
2
for a ≥ 0
a
= a
for a ≥ 0, n � 2
a n
= am
for a ≥ 0
a
-
for a ≥ 0, n � 2
a
-n
n
n
= a m
Formula Sheet
1
m
2
m
=
=
for a � 0
1
for a � 0, n � 2
1
for a � 0
1
for a � 0, n � 2
a
n
=
=
1
a
am
n
am
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Algebra
Operations
Arithmetic rules
Commutative rule
Associative rule
Distributive rule
a+b=b+a
(a + b) + c = a + (b + c)
a(b + c) = ab + ac
a×b=b×a
(a × b) × c = a × (b × c)
a(b - c) = ab - ac
Expanding
Basic expansions
Binomial expansions
(a + b) (c + d) = ac + ad + bc + bd
a(b + c) = ab + ac
a(b - c) = ab - ac
(a + b)2 = a2 + 2ab + b2
a(b + c) + d (e + f ) = ab + ac + de + df
(a - b)2 = a2 - 2ab + b2
(a + b) (a - b) = a2 - b2
Factorising
Common factors
Grouping in pairs
ab + ac = a(b + c)
ac + ad + bc + bd = a(c + d) + b(c + d)
ab - ac = a(b - c)
= (a + b) (c + d)
Perfect squares
Difference of two squares
a2 + 2ab + b2 = (a + b)2
a2 - b2 = (a + b) (a - b)
a2 - 2ab + b2 = (a - b)2
Formula Sheet
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Measurement
Perimeter and Area 1 (of 2)
Square
Rectangle
s
Perimeter = 4s
Area = s 2
s
where s = side length
Triangle
l
Perimeter = 2l + 2w
Area = lw
w
where l = length
w = width
Heron’s formula for area of a triangle
a
b
h
c
b
Area =
1
2
bh
Area =
s(s - a) (s - b) (s - c)
where b = base
h = perpendicular height from base
where s = 1
Parallelogram
Trapezium
2
(a + b + c)
and a, b, c are the side lengths of triangle
a
h
h
b
b
1
2
Area = bh
Area =
where b = base
h = perpendicular height from base
where a and b are the lengths of the parallel sides
h = perpendicular height between the parallel sides
Formula Sheet
h(a + b )
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Measurement
Perimeter and Area 2 (of 2)
Rhombus
1
2
Area =
Kite
xy
Area =
x
where x and y are the
lengths of the diagonals
1
2
xy
x
where x and y are the
lengths of the diagonals
y
y
Annulus
Circle
r
Circumference = πd
or
Circumference = 2 πr
R
d
r
Area = πr 2
where r = radius
d = diameter
Area = π (R 2 - r 2)
where R = radius of larger circle
r = radius of smaller circle
Sector
Pythagoras’ Theorem
r
θ
Arc length =
Area =
where
r
θ
θ
360°
θ
360°
× 2πr
× πr
2
= radius
= angle at the centre (in degrees)
Formula Sheet
c
a
c =a +b
2
2
2
b
where c is the hypotenuse of a right-angled triangle
and a, b are the shorter sides.
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Measurement
Surface Area and Volume
Cube
Rectangular Prism
Surface area = 6s 2
Surface area = 2hl + 2hw + 2lw
Volume = s 3
where s = side length
Volume = lwh
where
s
h
w
l
= height
= width
= length
h
w
l
Prism
Pyramid
Volume = Ah
Volume =
where A = cross-sectional area
h = perpendicular height
where A = area of the base h = perpendicular height
1
3
Ah
h
A
A
h
Cylinder
Cone
Sphere
Surface area = 2πr 2 + 2πrh
Surface area = πr 2 + πrs
Surface area = 4πr 2
Volume = πr 2h
Volume =
where r = radius
h = perpendicular height
1
3
πr 2h
Volume =
where r = radius
h = perpendicular height
s = slant height
4
3
πr 3
where r = radius
r
h
s
h
r
r
Formula Sheet
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Measurement
Unit Conversions (Metric)
Length
Area
Mass
1 cm = 10 mm
1 cm2 = 100 mm2
1 g = 1000 mg
1 m = 100 cm
1 m2 = 10 000 cm2
1 m = 1000 mm
1 ha = 10 000 m2
1 kg = 1000 g
1 t = 1000 kg
1 km2 = 1 000 000 m2
1 km = 1000 m
1 km2 = 100 ha
Capacity
1 L = 1000 mL
Volume
Volume and capacity
1 cm3 = 1000 mm3
1 mL = 1 cm3
1 m3 = 1 000 000 cm3
1 kL = 1000 L
1 m3 = 1000 L
1 m3 = 1 kL
1 ML = 1000 kL
1 ML = 1000 m3
Time
Digital information and file size
1 minute = 60 seconds
1 byte = 8 bits
1 kilobyte = 2 10 bytes = 1024 bytes
1 hour = 60 minutes
1 megabyte = 2 20 bytes = 1024 kilobytes
1 day = 24 hours
1 year = 365 days (in a non-leap year)*
1 gigabyte = 2 30 bytes = 1024 megabytes
1 year = 366 days (in a leap year)
1 terabyte = 2 40 bytes = 1024 gigabytes
*Some mathematical calculations use:
1
1 year = 365 4 days
Formula Sheet
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Surds & Logarithms
Surds
Logarithms
For a ≥ 0, b ≥ 0:
Definitions
For b � 0, b ≠ 1 and x � 0:
2
( a) = a
If log b x = y then x = b y
a2 = a
log b (bx) = x
blog b x = x
ab = a × b
a
=
b
a b
(since x log b b = x × 1)
(b ≠ 0)
Change of base law
For a � 0, b � 0, x � 0 and a ≠ 1, b ≠ 1:
log x
log a x = log b a
b
For a ≥ 0, n > 2:
a=a
n
a=
1
2
1
Log laws
For b � 0, b ≠ 1, x � 0 and y � 0:
an
am = a
m
log b b = 1
(since b1 = b)
log b 1 = 0
(since b0 = 1)
2
log b (x a) = a log b x
n
am = a
m
n
log b (xy) = log b x + log b y
log b x = log b x - log b y
(y)
Formula Sheet
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The Cartesian Plane
Midpoint of an interval
Distance between two points
The midpoint, M, of an interval with endpoints
( x1, y1 ) and ( x2, y2 ) has coordinates:
The distance, d units, between two points
( x1, y1 ) and ( x2, y2 ) is given by:
M=
(
x1 + x2 y1 + y2
,
2
2
)
d=
(x2 - x1 )2 + ( y2 - y1 )2
Gradient of a straight line
Gradient-intercept form of a straight line
The gradient, m, of a straight line joining two
points ( x1 , y1 ) and ( x2 , y2 ) is given by:
The equation of a straight line with gradient m
and y-intercept c is given by:
rise
m = run
or
y2 - y1
m = x -x
2
1
y = mx + c
Point-gradient formula for the equation of a straight line
The equation of a line with gradient m and passing through ( x1 , y1 ) is given by:
y - y1 = m(x - x1)
Two-point formula for the equation of a straight line
The equation of a straight line joining two points ( x1 , y1 ) and ( x2 , y2 ) is given by:
y - y1
y2 - y1
x - x1 = x2 - x1
Parallel lines
Perpendicular lines
If two lines with gradients m1 and m2
are parallel then:
If two lines with gradients m1 and m2
are perpendicular then:
m1 = m2
Formula Sheet
m1 × m2 = -1
or
m2 = - m1
1
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Trigonometry 1
Right-angled triangles
In a right-angled triangle:
opposite
sin θ =
hypotenuse
cos θ =
adjacent
hypotenuse
hypotenuse
opposite
θ
opposite
tan θ =
adjacent
adjacent
Exact ratios
30°
1
sin 30° =
2
sin 45° =
1
2
3
cos 30° =
2
cos 45° =
1
2
tan 30° =
1
3
tan 45° = 1
2
3
3
sin 60° =
2
60°
1
1
cos 60° =
2
tan 60° =
45°
2
1
45°
3
1
Angles of elevation and depression
line of sight
angle of depression
angle of elevation
line of sight
Formula Sheet
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Trigonometry 2
All triangles
For any triangle with vertices A, B and C and opposite sides a, b and c respectively:
A
b
C
c
a
B
Sine rule
a
sin A
sin A
a
=
=
b
sin B
sin B
b
Cosine rule
=
=
c
Area of triangle
Area =
c = a + b - 2ab cosC
2
sin C
sin C
2
cos C =
c
2
1
2
absinC
a2 + b2 - c2
2ab
Complementary angles
Supplementary angles
For complementary angles θ and (90° - θ):
For supplementary angles θ and (180° - θ):
sin θ = cos (90° - θ)
sin (180° - θ) = sin θ
cos θ = sin (90° - θ)
cos (180° - θ) = - cos θ
tan θ = cot (90° - θ)
tan (180° - θ) = - tan θ
Relationships
tan θ =
sin θ
cos θ
Formula Sheet
cosec θ =
1
sin θ
sec θ =
1
cos θ
cot θ =
1
tan θ
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