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Other note source Complex Numbers Summary This guide covers the essential theory and formulas for Complex Numbers (Pure Mathematics 3), including imaginary numbers, Argand diagrams, polar forms, and Loci. 1. Imaginary Numbers The set of complex numbers combines real numbers and imaginary numbers. We define the imaginary unit i as: i = √ -1 i 2 = -1 Calculating with roots: √ -a = i√ a (for a > 0) Example: √ -9 = √ (9 × -1) = 3i 2. Cartesian Form A complex number z is written as: z = x + iy x : Real part (Re z) y : Imaginary part (Im z) Complex Conjugate The conjugate of z = x + iy is denoted by z* (or z̄ ). It reflects the number across the Real axis. If z = x + iy , then z* = x - iy Property: The product of a complex number and its conjugate is always real: zz...

Other note source 📐 STRAIGHT LINES & QUADRATIC EQUATIONS Cambridge IGCSE Mathematics - Quick Reference Guide PART 1: STRAIGHT LINES 📊 Standard Form y = mx + c m = gradient (slope/steepness) c = y-intercept (crosses y-axis) x, y = coordinates Example: y = 3x + 2 • Gradient = 3 (rises 3 for every 1 across) • Y-intercept = 2 (crosses at (0,2)) Gradient Formula m = (y₂ - y₁) / (x₂ - x₁) Type Value Direction Positive m > 0 ↗ Up right Negative m ↘ Down right Zero m = 0 → Horizontal Undefined - ↑ Vertical Example: Points (2, 4) and (6, 12) m = (12 - 4)/(6 - 2) = 8/4 = 2 Special Lines Horizontal: y = k (e.g. y = 3) Vertical: x = k (e.g. x = -2) Parallel Lines Same gradient: m₁ = m₂ Lines y = 2x + 1 and y = 2x - 5 are parallel (both m = 2) Perpendicular Lines m₁ × m₂ = -1 or m₂ = -1/m₁ Line 1 Line 2 (⊥) m = 2 m = -½ m = 3 m = -⅓ m = -4 m = ¼ Intercepts Y-intercept: Set x = 0, solv...

              Other note source      📚 DIFFERENTIAL EQUATIONS Cambridge AS & A Level Mathematics (Chapter 10) Sections 10.1 - 10.2: Separating Variables & Forming Equations 🎯 What is a Differential Equation? Definition: An equation containing derivatives such as dy/dx or d²y/dx² is called a differential equation . KEY CONCEPT: The solution of a differential equation is a function , not just a number! Types of Solutions Solution Type Description Example General Solution Contains arbitrary constant(s) y = x³ + C Particular Solution Specific solution using initial conditions y = x³ + 5 First-Order Differential Equation: Contains only dy/dx (first derivative) Example: dy/dx = 3x² Differential Equation General Solution Particular Solution 🔧 Section 10.1: Separation of Variables The Basic Method STEP-BY-STEP PROCEDURE: Separate the variables: Rearrange to get f(y)...

Other note source 📚 SEQUENCES, SURDS & SETS Cambridge IGCSE Mathematics 📊 9.1 SEQUENCES Definition & Notation Sequence: A list of numbers following a specific rule. Each number = term . T₁, T₂, T₃ = 1st, 2nd, 3rd terms Tₙ = nth term (general term) Types of Sequences Type Rule Example Formula Arithmetic Common diff (d) 2,6,10,14 Tₙ=dn+c Geometric Common ratio (r) 3,6,12,24 Tₙ=ar^(n-1) Quadratic 2nd diff constant 2,5,10,17 ...

Other note source 📊 Introduction to Probability - Complete Theory & Formulas What is Probability? Probability measures how likely an event is to happen. It is always a number between 0 and 1, where 0 = impossible and 1 = certain. 🔑 Key Terms (Essential Vocabulary) Experiment: An action with an uncertain outcome (e.g., tossing a coin, rolling a die) Outcome: One possible result of an experiment (e.g., "Head" or "4") Event: A set of outcomes (e.g., "getting an even number") Sample Space: The complete list of all possible outcomes, written as {1,2,3,4,5,6} Favourable Outcomes: Outcomes that match the event you want Frequency: How many times an outcome appears in trials Probability Scale (0 to 1) 0 0.5 1 Impossible Even Chance Certain 📚 Section 8.1: Basic Probability (Experimental) Experimental Probability...

📐 INTEGRATION - Theory & Formulas O Level Mathematics | Complete Guide 1. Definition Integration = Reverse of Differentiation DIFFERENTIATION ↓ f(x) = x³ → f'(x) = 3x² ↑ INTEGRATION ⚠️ Important: Integration is many-to-one process. Different functions have same derivative! x² + 5 → 2x | x² + 10 → 2x | x² - 3 → 2x 2. Three Basic Rules Rule 1: Power Rule ⭐ ∫ x n dx = x n+1 /(n+1) + c Steps: (1) Add 1 to power (2) Divide by new power (3) Add + c Ex: ∫ x³ dx = x⁴/4 + c | ∫ x⁷ dx = x⁸/8 + c | ∫ 10x⁴ dx = 2x⁵ + c Rule 2: Constant Multiplier ∫ k·f(x) dx = k ∫ f(x) dx Rule 3: Sum/Difference ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx Ex: ∫ (x³ - x + 3) dx = x⁴/4 - x²/2 + 3x + c 3. Constant ...

Other note source 📐 UNDERSTANDING MEASUREMENT Cambridge IGCSE Mathematics - Grade 10 Complete Study Guide - Chapter 13 📏 13.1 UNDERSTANDING UNITS What Are Measurement Units? Measurement is how we describe the size, length, weight, or volume of objects. In everyday life, you measure things constantly - the distance to school, the amount of water in a bottle, or how heavy your backpack is. To communicate these measurements clearly, we use standardized units that everyone understands. The Metric System The metric system is the most widely used measurement system in the world. It's based on powers of 10, which makes conversions much easier than other systems. The metric system has three main categories: length , mass (weight) , and volume (capacity) . 🔑 Golden Rule of Conversion: To convert to a SMALLER unit → MULTIPLY To convert to a LARGER unit → DIVIDE 📏 LENGTH MEASUREMENTS Length measures how long or far something is. The basic unit i...