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MESSAGE FROM AUTHOR
The formulae listed represents a comprehensive list of the formulae that are encountered in CSEC
Mathematics. However, students should be aware that memorizing different formulae in this
format may not be the best option to increasing your skill in mathematics.
This book is best used as a tool for REVISION coupled with doing actual practice questions and
learning the theory behind different sections of the syllabus. If you need additional help in a topic
search the “Kerwin Springer” and the topic on YouTube.
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NUMBER THEORY AND COMPUTATION
NUMBER THEORY
ℕ ⊂ 𝕎 ⊂ ℤ ⊂ ℚ ⊂ ℝ;
DEFINITIONS
ℕ = {1, 2, 3, . . .}
natural numbers
𝕎 = {0, 1, 2, 3, . . .}
whole numbers
ℤ = {. . . −2, −1, 0, 1, 2, . . .}
integers
𝑝
ℚ = { ∶ 𝑝 and 𝑞 are integers, 𝑞 ≠ 𝜊}
𝑞
rational numbers
OPERATIONS - BODMAS
Brackets Of Division Multiplication Addition Subtraction
INDICES/EXPONENTS
𝑎𝑚 𝑎 𝑛 = 𝑎𝑚+𝑛
𝑎𝑚
= 𝑎𝑚−𝑛
𝑎𝑛
(𝑎𝑚 )𝑛 = 𝑎𝑚𝑛
(𝑎𝑏)𝑛 = 𝑎𝑛 𝑏𝑛
𝑎 𝑛 𝑎𝑛
( ) = 𝑛
𝑏
𝑏
𝑎 −𝑛 =
𝑎
𝑎𝑛
1
𝑎 𝑛 = 𝑛√𝑎
𝑎0 = 1
𝑎1 = 𝑎
(These formulae are best memorized and understood by doing questions)
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CONSUMER ARITHMETIC
PROFIT, LOSS, PERCENTAGE PROFIT, PERCENTAGE LOSS
Discount = Marked Price − Selling Price
PROFIT
• Profits occur when the Selling Price is more than the Cost Price
• Profit = Selling Price − Cost Price
when Selling Price > Cost Price
LOSS
•
•
Losses occur when the Selling Price is less than the Cost Price
Loss = Cost Price − Selling Price
when Selling Price < Cost Price
PERCENTAGE PROFIT AND LOSS
Percentage Profit =
Percentage Loss =
Profit
× 100
Cost Price
Loss
× 100
Cost Price
SIMPLE INTEREST
Simple Interest =
𝑃×𝑅×𝑇
100
Where,
P – Principal
R- Rate as a percentage
T- Time in years
FINDING AMOUNT
Amount = Simple Interest + Principal
FINDING OTHER QUANTITIES
𝑆𝐼 × 100
𝑅×𝑇
𝑆𝐼 × 100
Rate =
𝑃×𝑇
𝑆𝐼 × 100
Time =
𝑃×𝑅
Principal =
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COMPOUND INTEREST
𝑅 𝑛
Amount = 𝑃 (1 +
)
100
Where,
P – Principal
R – Rate as a percentage
n – Number of years
𝑅
Amount = 𝑃 (1 −
100
)
𝑛
(Use a negative sign in cases of depreciation)
MEASUREMENT
PLANE SHAPES (AREA, PERIMETER )
PERIMETER
Perimeter = Sum of all sides
AREA OF TRIANGLE (3 METHODS)
1
•
Area of triangle = 𝑏ℎ
•
Area of triangle = 2 𝑎𝑏𝑠𝑖𝑛𝐶
•
Area of triangle = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐 ) 𝑤ℎ𝑒𝑟𝑒 𝑠 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑒𝑚𝑖 − 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟:
2
1
s =
𝑎+𝑏+𝑐
2
FORMULAE OF COMMON PLANE SHAPES
Area of parallelogram = 𝑏ℎ
Area of square = 𝑠 × 𝑠 𝑜𝑟 𝑠 2
Area of rectangle = 𝑙 × 𝑤
1
Area of trapezium = 2 (𝑎 + 𝑏)ℎ
(half the sum of the parallel sides × the height)
Area of circle = 𝜋𝑟 2
Circumference of circle = 2𝜋𝑟 or 𝜋𝑑
𝜃
Area of sector = 360 × 𝜋𝑟 2
𝜃
Length of arc = 360 × 2𝜋𝑟
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(a fraction of a circle)
(a fraction of the circumference)
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SOLIDS AND PRISMS
VOLUMES
Volume of a Prism = 𝐶𝑟𝑜𝑠𝑠 𝑆𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐴𝑟𝑒𝑎 × 𝑙𝑒𝑛𝑔𝑡ℎ
Volume of Cuboid = 𝑙𝑤ℎ
(length by width by height)
Volume of Cylinder = 𝜋𝑟 2 ℎ
4
Volume of Sphere = 𝜋𝑟 3
3
1
Volume of Cone = 𝜋𝑟 2 ℎ
3
SURFACE AREA
Surface Area of Cuboid = 2𝑙ℎ + 2ℎ𝑤 + 2𝑙𝑤
Surface Area of Cylinder = 2𝜋𝑟ℎ + 2𝜋𝑟 2 or 2𝜋𝑟(ℎ + 𝑟)
Surface Area of Sphere = 4𝜋𝑟 2
Surface Area of Cone = 𝜋𝑟 2 + 𝜋𝑟𝑠
(s is length of slope)
COORDINATE GEOMETRY
EQUATION OF LINE
Equation of line if you know the slope and gradient
𝑦 = 𝑚𝑥 + 𝑐
Equation of line with point and gradient
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
DISTANCE MIDPOINT AND GRADIENT
Distance
𝑑 = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2
Mid-Point Formula
(𝑥, 𝑦) = (
𝑥1 + 𝑥2 𝑦1 + 𝑦2
,
)
2
2
Gradient Formula
𝑚=
𝑦2 − 𝑦1
𝑥2 − 𝑥1
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GRADIENT RULES FOR LINES
Parallel lines have equal gradients
𝑚1 = 𝑚2
Perpendicular lines gradients have the following relationship:
1
𝑚2 = − 𝑚
1
(negative reciprocal)
REGULAR POLYGONS
Sum of Interior Angle = 180(𝑛 − 2)
One interior angle =
180(𝑛−2)
𝑛
Sum of Exterior Angles = 360
One interior angle =
360
𝑛
SETS
Main sets formula:
𝑛(𝐴ᴜ𝐵) = 𝑛(𝐴) + (𝑛(𝐵) − 𝑛(𝐴 ∩ 𝐵)
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TRIGONOMETRY
BASIC
Area of a triangle (See section on Measurement)
PYTHAGORAS THEOREM (FOR RIGHT ANGLE TRIANGLES ONLY)
𝑐 2 = 𝑎2 + 𝑏2
(where c is the hypotenuse and a and b represents the other two sides)
TRIGONOMETRIC RATIOS (FOR RIGHT ANGLE TRIANGLES ONLY)
ℎ𝑦𝑝 − ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑠𝑖𝑛𝜃 =
𝑜𝑝𝑝
ℎ𝑦𝑝
𝑐𝑜𝑠𝜃 =
𝑎𝑑𝑗
ℎ𝑦𝑝
𝑡𝑎𝑛𝜃 =
𝑜𝑝𝑝
𝑎𝑑𝑗
𝑎𝑑𝑗 − 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑜𝑝𝑝 − 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ADVANCED
COSINE RULE
𝑎2 = 𝑏2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
SINE RULE
𝑎
𝑏
𝑐
=
=
sin 𝐴 sin 𝐵 sin 𝐶
OR
sin 𝐴 sin 𝐵 sin 𝐶
=
=
𝑎
𝑏
𝑐
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CIRCLE THEOREM
There are 9 Theorems tested in CSEC Mathematics
A line from the center of the circle to a chord
on the circle bisects the chord into two equal
lengths.
The line bisecting a chord into two equal parts
from the center of the circle meets the chord
at 90 ˚.
Look out for the creation of Isosceles triangles
when this appears in questions.
The angle in a semi-circle is 90˚
Angles from a common chord in the same
segment are equal.
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The angles from the same chord at the center
is twice the angle at the circumference, in the
same segment.
Opposite angles in a cyclic quadrilateral are
supplementary. (add up to 180 ˚)
The length of the two tangents from a point to
a circle are equal
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The angle between the tangent and the chord
at the point of contact is equal to the angle
drawn from the chord in the alternate
segment
The angle between the tangent and the radius
is 90 ˚
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RELATIONS, FUNCTIONS AND GRAPHS
QUADRATICS
Difference of two squares
𝑥 2 − 𝑦 2 = (𝑥 − 𝑦)(𝑥 + 𝑦)
General form of quadratic equation:
𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Quadratic Formula
𝑥=
−𝑏 ± √𝑏2 − 4𝑎𝑐
2𝑎
MATRICES
Identity Matrix
1
(
0
0
)
1
Multiplication
(𝑎
𝑐
𝑎𝑒 + 𝑏𝑔 𝑎𝑓 + 𝑏ℎ
𝑏) 𝑒 𝑓
(
)=(
)
𝑐𝑒 + 𝑑𝑔 𝑐𝑓 + 𝑑ℎ
𝑑 𝑔 ℎ
DETERMINANT, ADJOINT AND INVERSE
𝐴=(
𝑎
𝑐
𝑏)
𝑑
Determinant
|𝐴| = 𝑎𝑑 = 𝑏𝑐
Adjoint
𝑑
𝐴𝑑𝑗(𝐴) = (
−𝑐
−𝑏)
𝑎
Inverse
The inverse is one over the determinant multiplied by the adjoint
𝐴−1 =
1
(𝑑
𝑎𝑑 − 𝑏𝑐 −𝑐
−𝑏)
𝑎
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TRANSFORMATION MATRICES
REFLECTION OR FLIP
𝑥 axis
1
(
0
0
)
−1
𝑦 axis
−1 0
(
)
0 1
The line 𝑦 = 𝑥
0
(
1
1
)
0
The line 𝑦 = −𝑥
0 −1
(
)
−1 0
TRANSLATION
Also called slide
𝑥
Use the vector (𝑦)
Where 𝑥 represents the movement in the horizontal and 𝑦 represents the vertical movement
COMMON ROTATIONS
Rotation 90 degrees clockwise
0
(
1
−1
)
0
Rotation 180 degrees
−1 0
(
)
0 −1
Rotation 270 degrees clockwise
(
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0 1
)
−1 0
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SYMBOLS USED ON CSEC MATHEMATICS EXAM PAPERS.
All units used are SI Units.
You can also use any additional symbol or nomenclature in your answer provided that it is
consistent and understandable in the given context.
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FORMULAE SHEET
Below are the formulae included at the front of your exam paper.
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