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Complex Numbers PM3

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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...
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Straight lines and quadratic equations

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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...
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Differential equations

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              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)...
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Polymers IGCSE

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Quiz 1: Fullscreen Mode Quiz 2: Fullscreen Mode 🧪 POLYMERS - Complete Notes Cambridge IGCSE Chemistry | Chapter 19 1️⃣ What are Polymers? POLYMER = A very large molecule made from many small repeating units called monomers MONOMER = A small molecule that joins to other monomers to form a polymer POLYMERISATION = The process of joining monomers to form a polymer M + M + M Polymerisation M M M ... Monomers Polymer 📊 Types of Polymers Natural Polymers Synthetic Polymers ✓ Proteins (meat, eggs) ✓ Poly(ethene) (plastic bags) ✓ Starch (rice, bread) ✓ Poly(propene) (ropes) ✓ Natural rubber ✓ PVC (pipes) ✓ Silk ✓ Nylon (clothing) ✓ Cellulose ✓ PET (bottles) 2️⃣ Addition Polymerisation Addition polymerisation occurs...
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Chemistry Experimental Techniques and Chemical Analysis

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Quiz 1: Fullscreen Mode Other note source Quiz 2: Fullscreen Mode Chapter 20: Experimental Techniques & Chemical Analysis This guide covers the essential theory, apparatus, and chemical tests required for IGCSE Chemistry analysis. 20.1 Experimental Design (Measurement) 1. Units of Measurement Time: S.I. Unit is the Second (s). Measured with stopwatches (accuracy ± 0.01s). Temperature: Kelvin (K) or Celsius (°C). Mass: Kilogram (kg) or gram (g). Measured with electronic balances. Volume: Cubic meter (m 3 ), dm 3 , or cm 3 . Temp (K) = Temp (°C) + 273 1 kg = 1000 g 1 dm 3 = 1000 cm 3 2. Measuring Liquid Volume Apparatus Accuracy Use ...
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