All topics with interactive examples, calculators, and study tips for effective exam preparation
Number Theory & Computation
Types of Numbers
Natural: {1, 2, 3, ...}
Whole: {0, 1, 2, 3, ...}
Integers: {..., -2, -1, 0, 1, 2, ...}
Rational: Can be expressed as fractions (e.g., ½, 0.75, -3)
Irrational: Cannot be expressed as fractions (e.g., √2, π)
Key Operations
Factors: Numbers that divide exactly (factors of 12: 1, 2, 3, 4, 6, 12)
Multiples: Results of multiplying by integers (multiples of 5: 5, 10, 15, ...)
HCF: Largest factor shared (HCF of 12 & 18 is 6)
LCM: Smallest shared multiple (LCM of 4 & 6 is 12)
Prime Numbers: Only divisible by 1 and itself (2, 3, 5, 7, 11, ...)
Example: Find LCM of 8 and 12
Multiples of 8: 8, 16, 24, 32, 40...
Multiples of 12: 12, 24, 36, 48...
LCM = 24
Interactive LCM & HCF Calculator
Study Guide: Number Theory
Remember: Prime numbers are the building blocks of all numbers. Every whole number greater than 1 is either prime or can be made by multiplying primes together (prime factorization).
Practice: To find LCM, list multiples until you find the first common one. For HCF, list factors and find the greatest common one.
Common Mistake: Confusing factors (numbers that divide exactly) with multiples (results of multiplication).
Consumer Arithmetic
Percentages
Percentage increase: Original × (1 + %/100)
Percentage decrease: Original × (1 - %/100)
Example: 15% increase on $200
200 × 1.15 = $230
Interest Formulas
Simple Interest
I = P × R × T ÷ 100
Where: P = principal, R = rate (%), T = time (years)
Compound Interest
A = P(1 + r/100)^n
Where: A = final amount, r = rate (%), n = periods
Example: $1000 at 5% compound interest for 2 years
A = 1000(1 + 0.05)² = $1102.50
Interest Calculator
Ratio Calculator
Example: Share $60 in ratio 2:3
:
Sets
Set Notation
Set builder: {x | x > 5}
Roster: {1, 2, 3, 4}
Universal set: U
Complement: A' or Aᶜ
Empty set: ∅ or {}
Set Operations
Union (A ∪ B): Elements in A OR B
Intersection (A ∩ B): Elements in A AND B
Example: Set Operations
U = {1,2,3,4,5,6,7}
A = {1,2,3,4}
B = {3,4,5,6}
A ∪ B = {1,2,3,4,5,6}
A ∩ B = {3,4}
Set Operations Simulator
Enter elements for two sets (comma separated):
Measurement
Perimeter & Area
Rectangle: P = 2(l + w), A = l × w
Triangle: A = ½ × base × height
Circle: C = 2πr, A = πr²
Volume & Surface Area
Prism: V = base area × height
Cylinder: V = πr²h, SA = 2πr(h + r)
Sphere: V = ⁴⁄₃πr³, SA = 4πr²
Example: Speed Calculation
Distance: 150 km, Time: 2 hours
Speed = Distance ÷ Time = 150 ÷ 2 = 75 km/h
Geometry Calculator
Statistics
Data Collection & Representation
Primary Data: Collected directly by researcher
Secondary Data: Collected by someone else
Frequency Tables: Organize data with counts
Graph Types:
Bar charts (categorical data)
Histograms (continuous data)
Pie charts (proportions)
Line graphs (trends over time)
Central Tendency
Mean: Sum of values ÷ number of values
Median: Middle value when arranged in order
Mode: Most frequent value
Example: Data: 4, 7, 5, 9, 5, 6
• Mean = (4+7+5+9+5+6) ÷ 6 = 6
• Arranged: 4,5,5,6,7,9 → Median = (5+6)÷2 = 5.5
• Mode = 5 (most frequent)
Statistics Calculator
Enter numbers separated by commas:
Probability
P(event) = favorable outcomes ÷ total outcomes
Examples:
• Die roll: P(odd number) = 3/6 = ½
• Coin toss: P(heads) = ½
• Deck of cards: P(ace) = 4/52 = 1/13
Probability Simulator
Algebra
Algebraic Expressions
Terms: Parts separated by + or -
Coefficients: Numbers multiplying variables
Like Terms: Same variables and exponents
Simplify: Combine like terms
3x² + 2x - 5 + x² - 4x + 7
= (3x² + x²) + (2x - 4x) + (-5 + 7)
= 4x² - 2x + 2
Factorization
Common Factor: 2x + 4 = 2(x + 2)
Difference of Squares: a² - b² = (a+b)(a-b)
Quadratic Trinomials: x² + bx + c
Example: x² + 5x + 6
Find factors of 6 that add to 5: 2 and 3
x² + 5x + 6 = (x + 2)(x + 3)
Equation Solver
Enter an equation to solve (e.g., 3x + 5 = 17):
Indices/Laws of Exponents
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
a⁰ = 1 (a ≠ 0)
a⁻ⁿ = 1/aⁿ
Algebraic Fractions
Example: (2x/3) + (x/4)
Find LCD: 12
= (8x/12) + (3x/12)
= 11x/12
Algebraic Fractions Calculator
Relations, Functions & Graphs
Cartesian Plane
x-axis: Horizontal axis
y-axis: Vertical axis
Origin: (0,0)
Quadrants: I, II, III, IV
Linear Functions
y = mx + c
Where: m = slope, c = y-intercept
Example: y = 2x + 1
• Slope (m) = 2
• y-intercept = 1
• When x=0, y=1
• When x=1, y=3
Graph Plotter
Enter a function (e.g., 2*x+1 or x^2-4):
Domain & Range
Domain: All possible x-values
Range: All possible y-values
Example: y = √(x-2)
Domain: x ≥ 2 (can't take square root of negative)
Range: y ≥ 0 (square root is never negative)
Quadratic Functions
y = ax² + bx + c
• Parabola shape
• Vertex at x = -b/(2a)
• Axis of symmetry through vertex
Geometry & Trigonometry
Plane Geometry
Angles on straight line: 180°
Angles at a point: 360°
Vertically opposite angles: equal
Triangle sum: 180°
Pythagoras Theorem
a² + b² = c²
For right-angled triangles only
Example: Triangle with sides 3 and 4
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5
Triangle Calculator
Trigonometric Ratios (SOH-CAH-TOA)
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent
Example: Right triangle, angle = 30°, opposite = 5
sin 30° = 0.5 = 5/hypotenuse
Hypotenuse = 5/0.5 = 10
Sine & Cosine Rules
Sine Rule:
a/sinA = b/sinB = c/sinC
Cosine Rule:
a² = b² + c² - 2bc·cosA
For any triangle ABC
Circle Theorems
Angle at center = 2 × angle at circumference
Angles in same segment are equal
Angle in semicircle = 90°
Opposite angles of cyclic quadrilateral sum to 180°
Vectors & Matrices
Vectors
Notation: **a** or ā or column vector
Magnitude: |**v**| = √(x² + y²)
Addition: (2,3) + (1,4) = (3,7)
Scalar Multiplication: k×(x,y) = (kx, ky)
Vector Addition:
**a** = (2,3), **b** = (1,4)
**a** + **b** = (2+1, 3+4) = (3,7)
Matrices
Dimensions: rows × columns
Addition: Same dimensions only
Multiplication: (m×n) × (n×p) = (m×p)
Matrix Calculator
Transformations
Translation: T = [a, b] moves (x,y) to (x+a, y+b)
Rotation 90° clockwise: (x,y) → (y,-x)
Rotation 180°: (x,y) → (-x,-y)
Reflection in x-axis: (x,y) → (x,-y)
Determinant & Inverse
Determinant (2×2):
For A = [[a, b], [c, d]]
det(A) = ad - bc
Example: [[2,3],[1,4]]
det = 2×4 - 3×1 = 8 - 3 = 5
Inverse (2×2):
A⁻¹ = (1/det) × [[d, -b], [-c, a]]
Study Tips & Exam Strategies
Practice regularly with past papers - This helps you understand the exam format and identify your weak areas.
Show all working steps clearly - Even if your final answer is wrong, you may get partial credit for correct steps.
Check units in measurement problems - Always convert to the same units before calculating.
Verify answers make sense - Does your answer seem reasonable? For example, a person's age shouldn't be 250 years.
Memorize key formulas - Create flashcards for important formulas like the quadratic formula, area formulas, etc.
Draw diagrams for geometry problems - Visualizing the problem often makes it easier to solve.
Use calculator efficiently - Know how to use your calculator's memory functions and how to check your work.
Remember: Understanding concepts is more important than memorization! Focus on why formulas work, not just what they are.