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Number Systems

Number Theory

Number Set Hierarchy

\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

Natural Numbers

Counting numbers

\mathbb{N} = \{1,\ 2,\ 3,\ \ldots\}

Whole Numbers

Natural numbers + zero

\mathbb{W} = \{0,\ 1,\ 2,\ 3,\ \ldots\}

Integers

Positive and negative whole numbers

\mathbb{Z} = \{\ldots,\ -2,\ -1,\ 0,\ 1,\ 2,\ \ldots\}

Rational Numbers

Can be expressed as a fraction

\mathbb{Q} = \left\{ \frac{p}{q} \mid p, q \in \mathbb{Z},\ q \neq 0 \right\}

Operations

BODMAS

Order of operations (Order = powers & roots)

Brackets, Order, Division, Multiplication, Addition, Subtraction

Indices / Exponents

Multiplying

a^m \times a^n = a^{m+n}

Dividing

\frac{a^m}{a^n} = a^{m-n}

Power of a power

(a^m)^n = a^{mn}

Power of a product

(ab)^n = a^n b^n

Power of a fraction

\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

Negative index

a^{-n} = \frac{1}{a^n}

Fractional index

a^{\frac{1}{n}} = \sqrt[n]{a}

Zero index

Any non-zero number to the power 0 is 1

a^0 = 1

Index of one

a^1 = a

Algebra

Expanding

Distributive Law

a(b + c) = ab + ac

Double Brackets

FOIL method

(a + b)(c + d) = ac + ad + bc + bd

Factorisation

Common Factor (HCF)

Take out the common factor

ax + bx = x(a + b)

Difference of Two Squares

a^2 - b^2 = (a - b)(a + b)

Perfect Square (+)

a^2 + 2ab + b^2 = (a + b)^2

Perfect Square (-)

a^2 - 2ab + b^2 = (a - b)^2

Grouping

ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)

Simultaneous Equations

Elimination Method

Make coefficients equal, then add or subtract. Same sign → subtract. Different sign → add.

Quadratics

General Quadratic

ax^2 + bx + c = 0

Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Completing the Square

Vertex at (-h, k). Axis of symmetry: x = -h

y = a(x + h)^2 + k

Finding h

From ax² + bx + c

h = \frac{b}{2a}

Finding k

k is the min/max value of the function

k = c - \frac{b^2}{4a}

Finding k (alternative)

Equivalent formula for k

k = \frac{4ac - b^2}{4a}

Discriminant

D > 0: two roots. D = 0: one root. D < 0: no real roots.

D = b^2 - 4ac

Axis of Symmetry

Or x = -h from completed square form

x = -\frac{b}{2a}

Variation

Direct Variation

k is the constant of variation

y \propto x \implies y = kx

Inverse Variation

As x increases, y decreases

y \propto \frac{1}{x} \implies y = \frac{k}{x}

Properties of Operations

Commutative (Addition)

Order does not matter

a + b = b + a

Commutative (Multiplication)

Order does not matter

a \times b = b \times a

Associative (Addition)

Grouping does not matter

(a + b) + c = a + (b + c)

Associative (Multiplication)

Grouping does not matter

(a \times b) \times c = a \times (b \times c)

Additive Identity

0 is the identity for addition

a + 0 = a

Multiplicative Identity

1 is the identity for multiplication

a \times 1 = a

Additive Inverse

a + (-a) = 0

Multiplicative Inverse

a \times \frac{1}{a} = 1, \quad a \neq 0

Scientific Notation

Standard Form

Large: n positive. Small: n negative.

a \times 10^n, \quad 1 \leq a < 10

Large Number Example

Move decimal left → positive power

759000 = 7.59 \times 10^5

Small Number Example

Move decimal right → negative power

0.00759 = 7.59 \times 10^{-3}

Relations, Functions & Graphs

Quadratics

Difference of Two Squares

x^2 - y^2 = (x-y)(x+y)

General Quadratic

ax^2 + bx + c = 0

Quadratic Formula

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Discriminant

D > 0: two real roots, D = 0: one root, D < 0: no real roots

D = b^2 - 4ac

Functions

Function Notation

f(x) means "the value of f at x"

f(x) = ax^2 + bx + c

Inverse Function

Swap x and y, then solve for y. f(f⁻¹(x)) = x

f^{-1}(x)

Composite Function

Apply g first, then f. Note: fg ≠ gf in general.

fg(x) = f[g(x)]

Quadratic Graphs

Completed Square Form

Turning point at (−h, k)

y = a(x+h)^2 + k

Axis of Symmetry

Vertical line through the turning point

x = -h \quad\text{or}\quad x = -\dfrac{b}{2a}

Consumer Arithmetic

Profit, Loss & Discount

Discount

\text{Discount} = \text{Marked Price} - \text{Selling Price}

Profit

When Selling Price > Cost Price

\text{Profit} = \text{Selling Price} - \text{Cost Price}

Loss

When Selling Price < Cost Price

\text{Loss} = \text{Cost Price} - \text{Selling Price}

Percentage Profit

\text{\% Profit} = \dfrac{\text{Profit}}{\text{Cost Price}} \times 100

Percentage Loss

\text{\% Loss} = \dfrac{\text{Loss}}{\text{Cost Price}} \times 100

Simple Interest

P = Principal, R = Rate (%), T = Time (years)

SI = \dfrac{P \times R \times T}{100}

Amount (SI)

A = P + SI

Find Principal

P = \dfrac{SI \times 100}{R \times T}

Find Rate

R = \dfrac{SI \times 100}{P \times T}

Find Time

T = \dfrac{SI \times 100}{P \times R}

Compound Interest

P = Principal, R = Rate (%), n = years

A = P\left(1 + \dfrac{R}{100}\right)^n

Depreciation

Use negative sign for depreciation

A = P\left(1 - \dfrac{R}{100}\right)^n

Taxes & Charges

Sales Tax

Total = Price + Tax

\text{Tax} = \dfrac{\text{Rate}}{100} \times \text{Price}

Hire Purchase

Extra paid = HP Price − Cash Price

\text{HP Price} = \text{Deposit} + (\text{Monthly} \times \text{Months})

Currency Conversion

Or Local = Foreign ÷ Rate

\text{Foreign} = \text{Local} \times \text{Exchange Rate}

Markup

Same as profit. Often expressed as % of cost price.

\text{Markup} = \text{Selling Price} - \text{Cost Price}

Measurement

Plane Shapes

Perimeter

\text{Perimeter} = \text{Sum of all sides}

Area of Triangle

Area (base x height)

b = base, h = perpendicular height

A = \dfrac{1}{2}bh

Area (two sides + angle)

When you know two sides and the included angle

A = \dfrac{1}{2}ab\sin C

Heron's Formula

where s = (a+b+c)/2 (semi-perimeter)

A = \sqrt{s(s-a)(s-b)(s-c)}

Common Plane Shapes

Area of Parallelogram

A = bh

Area of Square

A = s^2

Area of Rectangle

A = l \times w

Area of Trapezium

Half the sum of parallel sides times height

A = \dfrac{1}{2}(a+b)h

Area of Circle

A = \pi r^2

Circumference

C = 2\pi r = \pi d

Area of Sector

Fraction of the circle area

A = \dfrac{\theta}{360} \times \pi r^2

Arc Length

Fraction of the circumference

l = \dfrac{\theta}{360} \times 2\pi r

Solids & Prisms

Volume of Prism

Cross-sectional area times length

V = A_\text{cross} \times l

Volume of Cuboid

length times width times height

V = lwh

Volume of Cylinder

V = \pi r^2 h

Volume of Sphere

V = \dfrac{4}{3}\pi r^3

Volume of Cone

V = \dfrac{1}{3}\pi r^2 h

Surface Area

SA of Cuboid

SA = 2lh + 2hw + 2lw

SA of Cylinder

SA = 2\pi rh + 2\pi r^2 = 2\pi r(h+r)

SA of Sphere

SA = 4\pi r^2

SA of Cone

s = slant height

SA = \pi r^2 + \pi rs

Speed, Distance & Time

Speed

Speed = Distance ÷ Time

S = \dfrac{D}{T}

Distance

D = S \times T

Time

T = \dfrac{D}{S}

Average Speed

NOT the average of two speeds

\text{Avg Speed} = \dfrac{\text{Total Distance}}{\text{Total Time}}

D-T Graph: Gradient = Speed

Steeper line = faster speed. Flat line = stationary. Downward = returning.

\text{Gradient of D-T graph} = \text{Speed}

S-T Graph: Gradient = Acceleration

Steeper = accelerating faster. Flat = constant speed. Downward = decelerating.

\text{Gradient of S-T graph} = \text{Acceleration}

S-T Graph: Area = Distance

Use area of trapezium/triangle/rectangle to calculate

\text{Area under S-T graph} = \text{Distance}

D-T Graph Shapes

Line going UP = moving away. FLAT line = stationary. Line going DOWN = returning.

S-T Graph Shapes

Line going UP = accelerating. FLAT line = constant speed. Line going DOWN = decelerating.

Geometry

Basic Angle Facts

Angles on a Straight Line

Supplementary angles

\text{Sum} = 180°

Angles at a Point

Full rotation

\text{Sum} = 360°

Vertically Opposite Angles

Formed by two intersecting lines

\text{Are equal}

Complementary Angles

Two angles that add to 90°

a + b = 90°

Supplementary Angles

Two angles that add to 180°

a + b = 180°

Triangles

Angle Sum of Triangle

Interior angles of any triangle

a + b + c = 180°

Exterior Angle Theorem

Exterior angle = sum of two interior opposite angles

d = a + b

Isosceles Triangle

\text{Two equal sides} \implies \text{two equal base angles}

Equilateral Triangle

\text{All sides equal, all angles} = 60°

Parallel Lines

Alternate Angles

Z-shape between parallel lines. The angles inside the Z are equal.

\text{Are equal (Z-angles)}

Corresponding Angles

F-shape between parallel lines. The angles at matching positions are equal.

\text{Are equal (F-angles)}

Co-interior Angles

C-shape (or U-shape) between parallel lines. The angles add to 180°.

a + b = 180°\text{ (C-angles)}

Transformations

Translation

Shape, size, orientation preserved. No fixed points.

\text{Describe using column vector } \begin{pmatrix} x \\ y \end{pmatrix}

Reflection

Shape, size preserved. Orientation reversed. Mirror line is perpendicular bisector of object-image pairs.

\text{Describe: mirror line (equation)}

Rotation

Shape, size preserved. Orientation preserved.

\text{Describe: centre, angle, direction}

Finding Centre of Rotation

Join object to image points, find perpendicular bisectors — they intersect at the centre

\text{Perp. bisectors of } AA^{\prime} \text{ and } BB^{\prime} \text{ meet at centre}

Enlargement

k > 1: bigger. 0 < k < 1: smaller. k < 0: inverted.

\text{Describe: centre, scale factor } k

Scale Factor

k = \dfrac{\text{image length}}{\text{object length}}

Finding Centre of Enlargement

Draw lines through corresponding object-image points — they intersect at the centre

\text{Lines through } A \to A^{\prime} \text{ and } B \to B^{\prime} \text{ meet at centre}

Area Scale Factor

Area of image = k² × area of object

\text{Area factor} = k^2

Similar Figures

Similar Triangles

Same shape, different size. All corresponding angles are equal and sides are in the same ratio.

\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}

Similar Figures Definition

Corresponding angles equal. Corresponding sides in proportion (same ratio).

\text{Same shape, different size}

Congruence

Congruent Figures Definition

All corresponding sides and angles are equal. One fits exactly on top of the other.

\text{Same shape AND same size}

Congruence Tests

SSS: 3 sides equal. SAS: 2 sides + included angle. ASA: 2 angles + included side. RHS: right angle + hypotenuse + side.

\text{SSS, SAS, ASA, RHS}

Coordinate Geometry

Equation of Line

Slope-Intercept Form

m = gradient, c = y-intercept

y = mx + c

Point-Slope Form

When you know gradient and a point

y - y_1 = m(x - x_1)

Distance, Midpoint & Gradient

Distance Formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Midpoint Formula

M = \left(\dfrac{x_1 + x_2}{2},\; \dfrac{y_1 + y_2}{2}\right)

Gradient Formula

m = \dfrac{y_2 - y_1}{x_2 - x_1}

Gradient Rules

Parallel Lines

Parallel lines have equal gradients

m_1 = m_2

Perpendicular Lines

Product of gradients = -1

m_2 = -\dfrac{1}{m_1}

Regular Polygons

Sum of Interior Angles

n = number of sides

S = 180(n-2)

One Interior Angle

\text{Interior} = \dfrac{180(n-2)}{n}

Sum of Exterior Angles

Always 360 degrees for any convex polygon

\text{Sum} = 360^\circ

One Exterior Angle

\text{Exterior} = \dfrac{360}{n}

Sets

Sets Formula

n(A \cup B) = n(A) + n(B) - n(A \cap B)

Trigonometry

Basic

Pythagoras Theorem

c = hypotenuse; right-angle triangles only

c^2 = a^2 + b^2

Trigonometric Ratios

Sine

\sin\theta = \dfrac{\text{opp}}{\text{hyp}}

Cosine

\cos\theta = \dfrac{\text{adj}}{\text{hyp}}

Tangent

\tan\theta = \dfrac{\text{opp}}{\text{adj}}

Advanced

Cosine Rule

For any triangle

a^2 = b^2 + c^2 - 2bc\cos A

Sine Rule

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

Sine Rule (alt)

Use this form when finding an angle

\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}

Circle Theorems

Angle in Semi-circle

The angle in a semi-circle is 90 degrees

Angles in Same Segment

Angles from a common chord in the same segment are equal

Centre vs Circumference

The angle at the centre is twice the angle at the circumference from the same chord

Cyclic Quadrilateral

Opposite angles in a cyclic quadrilateral are supplementary (add to 180 degrees)

Perpendicular from Centre

A line from the centre perpendicular to a chord bisects the chord. Conversely, a line from the centre to the midpoint of a chord meets it at 90°.

Equal Tangents

The two tangents from an external point to a circle are equal in length

Alternate Segment

The angle between the tangent and chord equals the angle in the alternate segment

Tangent-Radius

The angle between the tangent and the radius is 90 degrees

Statistics & Probability

Central Tendency

Mean (ungrouped)

Sum of all values ÷ number of values

\bar{x} = \dfrac{\sum x}{n}

Mean (frequency table)

Sum of (frequency × value) ÷ total frequency

\bar{x} = \dfrac{\sum fx}{\sum f}

Estimated Mean (grouped)

xₘ = class midpoint (midpoint of each class interval)

\bar{x} \approx \dfrac{\sum f \cdot x_m}{\sum f}

Median Position

Middle value when data is ordered. Median is also Q₂ (the second quartile).

\text{Position} = \dfrac{n+1}{2}

Mode

Can have more than one mode, or no mode

\text{Most frequent value}

Measures of Spread

Range

\text{Range} = \text{highest} - \text{lowest}

Interquartile Range

Spread of the middle 50% of data

IQR = Q_3 - Q_1

Semi-Interquartile Range

SIQR = \dfrac{Q_3 - Q_1}{2}

Standard Deviation

Measures how spread out values are from the mean. Higher σ = more spread out (less consistent). Lower σ = closer to mean (more consistent). No calculation required at CSEC.

\sigma \text{ (sigma)}

Q₁ Position

Lower quartile — 25% of data below

Q_1 = \dfrac{n+1}{4}\text{th value}

Q₃ Position

Upper quartile — 75% of data below

Q_3 = \dfrac{3(n+1)}{4}\text{th value}

Probability

Probability of Event

0 ≤ P(A) ≤ 1

P(A) = \dfrac{\text{favourable outcomes}}{\text{total outcomes}}

Complement

Probability of event NOT happening

P(A^{\prime}) = 1 - P(A)

P(A or B) — Mutually Exclusive

Events cannot happen at the same time

P(A \cup B) = P(A) + P(B)

P(A and B) — Independent

Events do not affect each other

P(A \cap B) = P(A) \times P(B)

Grouped Data

Class Midpoint

Used for estimating mean of grouped data

x_m = \dfrac{\text{lower} + \text{upper}}{2}

Class Width

\text{Width} = \text{upper boundary} - \text{lower boundary}

Vectors & Matrices

Matrices

Identity Matrix

I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

Matrix Multiplication

\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{pmatrix}

Determinant, Adjoint & Inverse

Determinant

For A = (a b; c d)

|A| = ad - bc

Adjoint

Swap a with d, negate b and c

\text{Adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Inverse Matrix

Only exists if det is not 0

A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Transformation Matrices

Reflection in x-axis

\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

Reflection in y-axis

\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}

Reflection in y = x

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

Reflection in y = -x

\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}

Rotation 90 degrees clockwise

\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}

Rotation 180 degrees

\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}

Rotation 270 degrees clockwise

\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}

Translation

x = horizontal, y = vertical movement

\text{Use vector: } \begin{pmatrix} x \\ y \end{pmatrix}

Vectors

Triangle Law

\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}

Column Vector

\overrightarrow{AB} = \begin{pmatrix} x \\ y \end{pmatrix}

Magnitude

|\overrightarrow{AB}| = \sqrt{x^2 + y^2}

Unit Vector

Magnitude = 1

\hat{u} = \dfrac{\overrightarrow{AB}}{|\overrightarrow{AB}|}

Parallel Vectors

Have a common scalar factor

\mathbf{a} = k\mathbf{b}

Collinear Vectors

Parallel + share a common point means same line

\overrightarrow{AB} = k\overrightarrow{AC}

Symbols & Formula Sheet

This section shows the symbols and formulae typically provided on the formula sheet in the CSEC Mathematics examination.

Mathematical Symbols

Set of Natural Numbers {1, 2, 3, ...}

\mathbb{N}

Set of Whole Numbers {0, 1, 2, 3, ...}

\mathbb{W}

Set of Integers {..., -2, -1, 0, 1, 2, ...}

\mathbb{Z}

Set of Rational Numbers

\mathbb{Q}

Set of Real Numbers

\mathbb{R}

Is an element of

\in

Is not an element of

\notin

Is a subset of

\subset

Union

\cup

Intersection

\cap

Complement of set A

A'

Empty set

\emptyset

Pi (approximately 3.14159)

\pi

Square root of x

\sqrt{x}

Absolute value of x

|x|

Approximately equal to

\approx

Not equal to

\neq

Less than or equal to

\leq

Greater than or equal to

\geq

Therefore

\therefore

Angle

\angle

Triangle

\triangle

Parallel to

\parallel

Perpendicular to

\perp

Similar to

\sim

Congruent to

\cong

Vector a

\vec{a}

Unit vector in direction of a

\hat{a}

Implies

\implies

If and only if

\iff

Key Formulas Quick Reference

Triangle

A = \frac{1}{2}bh

Parallelogram

A = bh

Trapezium

A = \frac{1}{2}(a+b)h

Circle

A = \pi r^2

Circumference

C = 2\pi r

Cylinder (V)

V = \pi r^2 h

Cone (V)

V = \frac{1}{3}\pi r^2 h

Sphere (V)

V = \frac{4}{3}\pi r^3

Pythagoras

c^2 = a^2 + b^2

Sine Rule

\frac{a}{\sin A} = \frac{b}{\sin B}

Cosine Rule

a^2 = b^2 + c^2 - 2bc\cos A

Quadratic

x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

Simple Interest

SI = \frac{PRT}{100}

Compound Interest

A = P(1+\frac{R}{100})^n

Gradient

m = \frac{y_2-y_1}{x_2-x_1}

Distance

d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Midpoint

M = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})