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Other note source 📊 SAMPLING - Theory & Formulas Cambridge AS & A Level Mathematics 📖 Part 1: Introduction to Sampling Key Definitions Population: Complete set of ALL items of interest Sample: Part of the population (size = n) Representative Sample: Accurately reflects population characteristics Biased Sample: Does NOT properly represent population Random Sample: ALL possible samples of size n have equal probability of selection 💡 Why Use Samples? Reason Example 💰 Cost-Effective Test 50 products vs 10,000 ⏰ Time-Saving Survey 100 people vs millions 🔨 Destructive Testing Crash testing helmets 🌍 Impossible to Survey All All fish in the ocean 🎲 Random Sampling Methods Using Random Number Tables: Number population: 000 to 499 (for 500 items) Pick starting point in table Read digits matching your numbering Ignore numbers outside range Ignore repeats Using Excel: =RAN...

Quiz: Fullscreen Mode AN INTRODUCTION TO ORGANIC CHEMISTRY Complete Study Notes for Cambridge IGCSE Chemistry SECTION 16.1: Understanding Organic Chemistry Basics What Are Organic Compounds? Organic compounds are special chemicals that contain the element carbon . You can find them everywhere in your daily life! The food you eat (like bread, rice, and meat), your hair, plastic bottles, soap, and even medicines are all made from organic compounds. There are two types of organic compounds: Natural organic compounds - These come from nature, like the proteins in your hair or the sugar in fruits Synthetic organic compounds - These are man-made in factories, like plastics, detergents, and modern medicines Important Note: Not ALL compounds with carbon are organic! Carbon dioxide (CO₂) and calcium carbonate (CaCO₃) contain carbon but are NOT organic compounds. Wha...

Other note source PDF Source 📐 Perimeter, Area and Volume Complete Guide for Grade 7 - Cambridge Curriculum 📏 Section 1: Perimeter and Area in Two Dimensions 1.1 Understanding Perimeter Perimeter is the total distance around the outside of a shape. To find the perimeter, add up the lengths of all sides. For any polygon: Perimeter = Sum of all sides Circle - Circumference The perimeter of a circle is called circumference . r d C = πd or C = 2πr where π ≈ 3.14 or 22/7 Example 1: Circle Circumference Find the circumference of a circle with radius 7 cm. Solution: C = 2πr C = 2 × 3.14 × 7 C = 43.96 cm 1.2 Understanding Area Area is the total space contained...

Other note source NUMERICAL SOLUTIONS OF EQUATIONS Concise Theory & Formula Guide Cambridge AS & A Level Mathematics 1. INTRODUCTION Numerical methods find approximate solutions to equations that cannot be solved algebraically. Examples include x³ + x - 4 = 0 , eˣ = 2x + 1 , and sin(x) = x - 1 . Historical Fact: Quintic equations (degree 5 and higher) generally have no algebraic solutions, making numerical methods essential. 2. LOCATING ROOTS (Section 6.1) Root Definition α is a root of f(x) = 0 if f(α) = 0 Method 1: Graphical Approach Rearrange equation as g(x) = h(x) , sketch both graphs, and find intersection points. Each intersection represents a root. Method 2: Change of Sign Method Change of Sign Principle: If f(x) is continuous and f(a) · f(b) If f(a) 0 → Root exists between a and b Example: Change of Sign Problem: Show f(x) = x⁵ + x - 1 = 0 has a root between 0 ...

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Algebra Guide: Equations, Factors & Formulae 📐 Algebra Essentials: Equations, Factors & Formulae Welcome! This guide covers fundamental algebra concepts: solving equations (finding unknowns), factorization (breaking expressions into factors), and rearranging formulae (changing the subject). These skills apply to real-world problems from cooking times to physics calculations. 1️⃣ Solving Equations Linear Equations A linear equation has variables with power ≤ 1 (e.g., 3x + 1 = 13 ). Golden Rule: Whatever you do to one side, do to the other side to keep it balanced. Method 1: Function Machine x → [×3] → 3x → [+1] → 3x+1 = 13 ↓ Work backwards ↓ 4 ← [÷3] ← 12 ← [-1] ← 13 Result: x = 4 Method 2: Algebraic Steps Solve: 3x + 1 = 13 Step 1: 3x + 1 - 1 = 13 - 1 → 3x = 12 Step 2: 3x ÷ 3 = 12 ÷ 3 → x = 4 Type 1: Variable on ...

Other note source 📐 INTEGRATION - Theory & Formulas Cambridge AS & A Level Mathematics - Pure Mathematics 2 1. Introduction to Integration Integration is the reverse process of differentiation . Symbol: ∫ f(x) dx means "integrate f(x) with respect to x" ⚠️ Always add + c for indefinite integrals (constant of integration) Example: • Differentiate x² → get 2x • Integrate 2x → get x² + c 2. Integration of Exponential Functions Basic Rules ∫ e x dx = e x + c ∫ e (ax+b) dx = (1/a) e (ax+b) + c Example 1: ∫ e (3x) dx a = 3, so answer = (1/3)e (3x) + c Example 2: ∫ 6e (3x) dx = 6 × (1/3)e (3x) + c = 2e (3x) + c Example 3: Evaluate ∫₀² e (3x) dx Step 1: Integrate: [(1/3)e (3x) ]₀² Step 2: Upper limit: (1/3)e⁶ Step 3: Lower limit: (1/3)e⁰ = 1/3 Answer: (1/3)(e⁶ - 1) 3. Integration of 1/(ax+b) Basic Rules ∫ (1/x) dx = ln|x| + c ∫ 1/(ax+b) dx = (1/a) ln|ax+b| + c ⚠️ Always use |x| (absolute value) b...