Golden Education

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      📚 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)...

📚 Further Calculus - Complete Guide Pure Mathematics 3| Cambridge AS & A Level 8.1 Derivative of tan⁻¹(x) Understanding tan⁻¹(x): The inverse tangent function (arctan) reverses the tangent operation. If tan(θ) = x, then θ = tan⁻¹(x) Derivation Let y = tan⁻¹(x) Then: tan(y) = x Differentiate both sides: sec²(y) × dy/dx = 1 Therefore: dy/dx = 1/sec²(y) Using identity: sec²(y) = 1 + tan²(y) = 1 + x² Result: dy/dx = 1/(1 + x²) KEY FORMULA: d/dx[tan⁻¹(x)] = 1/(1 + x²) With chain rule: d/dx[tan⁻¹(f(x))] = f'(x)/(1 + [f(x)]²) Example 1: Differentiate tan⁻¹(3x) Solution: Let f(x) = 3x, so f'(x) = 3 d/dx[tan⁻¹(3x)] = 3/(1 + 9x²) Example 2: Differentiate tan⁻¹(√x) Solution: f(x) = x^(1/2), f'(x) = 1/(2√x) d/dx[tan⁻¹(√x)] = [1/(2√x)] / [1 + x] = 1/[2√x(1 + x)] 8.2 Integration of 1/(x² + a²) The Reverse Process: Since d/dx[tan⁻¹(x)] = 1/(1 + x²), we can integrate backwards! KEY FORMULAS: ∫ 1...

📐 VECTORS: Complete Study Guide Cambridge AS & A Level Mathematics - Pure Mathematics 3 1. INTRODUCTION TO VECTORS What is a Vector? A vector is a quantity that has two essential properties: 📏 Magnitude (Size/Length) 🧭 Direction Scalars vs Vectors Scalars (Number only) Vectors (Number + Direction) Temperature (25°C) Velocity (60 km/h North) Mass (5 kg) Force (10 N Downward) Distance (10m) Displacement (10m East) 2. DISPLACEMENT VECTORS (Section 9.1) 2.1 Writing Vectors Method 1: Column Vector ⎛ x ⎞ v = ⎜ y ⎟ ⎝ z ⎠ Where x is movement along the x-axis, y al...

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...

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 ...

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...

Differentiation - Cambridge AS & A Level Mathematics 📐 DIFFERENTIATION Cambridge AS & A Level Mathematics - Pure Mathematics 2 📚 Apa yang Akan Anda Pelajari: Mendiferensiasikan produk dan hasil bagi Menggunakan turunan dari e x , ln(x), sin(x), cos(x), tan(x) Mencari turunan dari fungsi implisit dan parametrik Menerapkan diferensiasi untuk menyelesaikan masalah nyata 4.1 ATURAN PRODUK (PRODUCT RULE) Apa itu Aturan Produk? Ketika kita perlu mendiferensiasikan fungsi yang dikalikan bersama , kita menggunakan Aturan Produk. Contoh: y = (x + 1)⁴ × (3x - 2)³ 📌 RUMUS ATURAN PRODUK Jika y = u × v, maka: dy/dx = u(dv/dx) + v(du/dx) 💡 Dalam Kata-kata: Fungsi pertama × turunan kedua + Fungsi kedua × turunan pertama y = u × v | _________|_________ | | | | u(dv/dx) v(du/dx) | | |_______TAMBAH______| ...