Physical Properties of Metals Pure metals allow their ions to slide over each other, a characteristic shown in the provided table and figure. Alloys like brass and steel are created by mixing metallic elements with others to enhance strength and prevent deformation. The differing sizes of ions in alloys inhibit sliding, improving shape retention compared to pure metals. Reactivity Series Highly reactive metals like potassium and sodium react vigorously with cold water, while others react more slowly. Metals above hydrogen in the series can react with dilute acids to produce salts and hydrogen. The reactivity of metals influences their ability to displace other metals from compounds. Ionic Equations and Displacement Reactions Magnesium can displace copper due to its higher reactivity, forming positive ions more readily. Zinc powder heated with copper(II) oxide demonstrates a similar displacement reaction. No reaction occurs when zinc oxide is heated with copper,

  Mindmap: Market Failures and Socially Undesirable Outcomes Central Theme: Market Failures Externalities Positive Externalities Benefits to others not involved in the transaction Examples: Education, healthcare Solutions: Subsidies, government provision Negative Externalities Costs imposed on others not involved in the transaction Examples: Pollution, noise Solutions: Taxes, regulations Public Goods Characteristics: Non-excludable, non-rivalrous Examples: Street lighting, national defense Issue: Free-rider problem Solution: Government provision Asymmetric Information Definition: Imbalance of information between parties Consequences: Market inefficiencies, adverse selection Solutions: Regulation, transparency Equity vs. Efficiency Equity: Fair distribution of resources

Surface Area, Volume, and Capacity: Comprehensive Guide 1. Surface Area Surface area is the total area of all faces or surfaces of a three-dimensional object. Common Shapes and Their Surface Area Formulas: Cube Surface Area = 6s², where s = length of side Rectangular Prism Surface Area = 2(lw + lh + wh), where: l = length w = width h = height Sphere Surface Area = 4πr², where r = radius Cylinder Surface Area = 2πr² + 2πrh, where: r = radius of base h = height 2. Volume Volume is the amount of three-dimensional space enclosed by a closed surface. Cube Volume = s³, where s = length of side Rectangular Prism Volume = l × w × h Sphere Volume = (4/3)πr³ Cylinder Volume = πr²h 3. Capacity Capacity refers to the maximum amount that something can contain, usually measured in units of volume. Common Conversions: 1 m³ = 1000 liters 1 liter = 1000 ml 1 cm³ = 1 ml 4. Practical Applica

Comprehensive Guide to Trigonometry: A/AS Level Trigonometry is a fundamental branch of mathematics that deals with the relationships between the sides and angles of triangles. This comprehensive guide covers key concepts, formulas, and applications of trigonometry at the A/AS level. 1. Basic Trigonometric Ratios The three primary trigonometric ratios are sine, cosine, and tangent. These are defined in terms of the sides of a right-angled triangle: x (adjacent) y (opposite) r (hypotenuse) θ x = r cos θ y = r sin θ r² = x² + y² tan θ = y/x For an angle θ in a right-angled triangle: $$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}$$ $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}$$ $$\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x}$$ 2. Important Trigonometric Identities Several key identities are essential in trigono

Circle Theorems 1. Alternate Segment Theorem The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. θ θ 2. Angle at the Centre Theorem The angle at the center is twice the angle at the circumference when both angles subtend the same arc. 2θ θ 3. Angles in the Same Segment Theorem Angles in the same segment are equal. θ θ 4. Angles in a Semicircle The angle in a semicircle is 90 degrees (a right angle). 90° 5. Chord of a Circle The perpendicular from the center of a circle to a chord bisects the chord (splits the chord into two equal parts). x x 6. Cyclic Quadrilateral The opposite angles in a cyclic quadrilateral are supplementary (they add up to 180°). a b a + b = 180° 7. Tangent of a Circle

  The Impact of Government Subsidies on Electric Motorcycle Adoption in Indonesia The Impact of Government Subsidies on Electric Motorcycle Adoption in Indonesia Introduction The Indonesian government has recently announced a significant subsidy initiative aimed at promoting the adoption of electric motorcycles. With a budget of $455 million (Rp 7 trillion), this subsidy is designed to make electric motorcycles more affordable and accessible to consumers. The initiative is part of a broader strategy to enhance sustainable transportation and reduce carbon emissions across the nation. This paper explores the implications of such subsidies on market dynamics, using economic models to illustrate the changes in supply and demand. Discussion of Table and Graph Table Analysis Market Outcome Description Equilibrium Quantity Increases from Q* to Q sb

Permutations and Combinations: A Comprehensive Summary 1. Permutations Permutations are arrangements of objects where order matters. Formula: For n distinct objects, the number of permutations is: P(n) = n! For r objects selected from n distinct objects: P(n,r) = n! / (n-r)! Example: Arranging the letters in "REQUIREMENT": 11! / (2!3!) = 3,326,400 (There are 11 letters, with 2 Rs and 3 Es repeated) 2. Combinations Combinations are selections of objects where order doesn't matter. Formula: Selecting r objects from n distinct objects: C(n,r) = n! / (r!(n-r)!) Example: Selecting 4 toys from 9 toys: C(9,4) = 9! / (4!5!) = 126 3. Conditional Probability in Selections When selecting objects with conditions, break down the problem into cases: Example: Selecting 6 people from 8 men and 4 women, with at least twice as many men as women: Case 1: 4 men, 2 women: C(8,4) * C(4,2) Case 2: 5 men, 1 woman: C(8,5) * C(4,1) Case 3: 6 men, 0 women: C(8,6) Total

Permutations and Combinations Permutations and combinations are fundamental concepts in combinatorics used to count the number of ways to arrange or select items. The choice between permutations and combinations depends on whether the order of the items matters. Permutations Permutations are used when the order of items matters. The formula for permutations of n items taken r at a time is: $$P(n, r) = \frac{n!}{(n-r)!}$$ Where n! (n factorial) is the product of all positive integers up to n . Combinations Combinations are used when the order of items does not matter. The formula for combinations of n items taken r at a time is: $$C(n, r) = \frac{n!}{r!(n-r)!}$$ Example Applications (1) Selecting Teams or Committees Problem: How many ways can a team be chosen from a group? Solution: Use combinations when the order does not matter. Formula: $$C(n, r)$$ where n is the total number of people and r is the number of people to choose. (2) Arra

Coordinate Geometry Coordinate Geometry 1. Midpoint and Length of a Line Segment The midpoint \(M\) of a line segment joining points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is given by: \( M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \) The length of the line segment \(PQ\) is: \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) 2. Gradient of a Line The gradient (slope) of the line joining the points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is: \( \frac{y_2 - y_1}{x_2 - x_1} \) 3. Parallel and Perpendicular Lines For parallel lines, the gradients \(m_1\) and \(m_2\) are equal: \( m_1 = m_2 \) For perpendicular lines, the product of their gradients is: \( m_1 \times m_2 = -1 \) 4. Equation of a Straight Line The equation of a straight line with gradient \(m\) passing through point \((x_1, y_1)\) is: \( y - y_1

Permutations and Combinations Permutations and Combinations Factorial: The factorial of a number \( n \) is the product of all positive integers less than or equal to \( n \). It is denoted as \( n! \). \( n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1 \) \( 0! = 1 \) Permutations: A permutation is a way of selecting and arranging objects in a particular order. The keyword is arranged . The number of permutations of all \( n \) objects is given by: \( ^nP_n = n! \) The number of permutations of \( r \) objects from \( n \) distinct objects is: \( ^nP_r = \frac{n!}{(n-r)!} \) For permutations where there are \( p, q, r, \ldots \) of each type: \( \frac{n!}{p! \times q! \times r! \times \ldots} \) Example: How many ways can you arrange the letters in the word "BOOK"? The word "BOOK" has 4 letters where 'O' is repeated. The number of permutations is: \( \frac{4!}{2!} = \frac{24}{2} = 12 \) Combinat

Approximations and Error   Rounding Numbers Approximate answers are often necessary, especially for constantly changing values. The symbol ≈ or ≐ is used to indicate approximation.   Rules for Rounding Off Round down if the digit after the one being rounded is less than 5 (0, 1, 2, 3, or 4) Round up if the digit is 5 or more (5, 6, 7, 8, 9)   Rounding to Significant Figures The first significant figure is the leftmost non-zero digit. To round to a specific number of significant figures: Count off the specified number of significant figures Look at the next digit Round down if less than 5, up if 5 or more Delete remaining figures, replacing with zeros if necessary   Measurement Errors Measurements often involve reading scales, which can lead to approximations. A measurement is typically accurate to ±½ of the smallest division on the scale.   Types of Errors Error: The difference between the measured and actual value Absolute Error: The magnitude of the error, calculat