This document contains a table of contents for a mathematics reference book. It lists 58 topics covered in the book, ranging from natural numbers to symmetry, along with the page number for each topic. The document also includes sections on important points and formulas for various mathematical concepts such as algebraic expressions, quadratic equations, trigonometry, geometry, and statistics. The sections provide definitions, properties, and formulas for key concepts in mathematics.
The document is a mathematics lecture on integers. It discusses the four integer operations of addition, subtraction, multiplication, and division. It provides examples of how to perform each operation on integers and the rules for determining if the result is positive or negative. Addition and subtraction are explained using rules about combining positive and negative integers. Multiplication and division are covered together, as their rules are the same - the result is positive if the signs are the same and negative if the signs are different.
The document provides examples of factorizing algebraic expressions. It begins with expanding simple expressions like 3g + 4 and finding the common factors of terms like 12 and 20. The next section provides more examples of factorizing expressions like x2 + 2x by finding the highest common factor. It emphasizes factorizing completely, like writing 3abc(ab4 + 9) instead of 9abc(ab4 + 9). The document concludes with an exercise for students to practice factorizing expressions and an extension with more challenging examples. It provides the opportunity for a 10 minute practice session to reinforce the steps of factorizing algebraic expressions.
The document provides information about various chemistry concepts related to air and water:
- It describes chemical tests to identify water and the purification of water supplies through filtration and chlorination.
- The composition of clean air is described as 78% nitrogen, 21% oxygen and small quantities of other gases. Common air pollutants like carbon monoxide and their sources are stated.
- Fractional distillation is outlined as the process used to separate oxygen and nitrogen from liquid air based on their different boiling points.
- Rusting is described as a reaction between iron, air and water that can be prevented by methods like painting and galvanizing to exclude oxygen.
Simultaneous equations are two or more equations with the same unknown variables. There are two main methods to solve simultaneous equations:
1) Using graphs - make a table of values, plot the equations on a graph, and find where the graphs intersect.
2) Using algebra - organize the equations, make coefficients equal, eliminate a variable, solve the resulting equation, substitute values back into the original equations, and check the answer. The algebraic method follows the steps of NO MESS: Organize, Make equal, Eliminate, Solve, Substitute.
This document provides revision notes for the Edexcel IGCSE Chemistry exam. It covers topics like atomic structure, bonding, organic chemistry, calculations, the periodic table, acids and bases. For each topic, it provides definitions, explanations and examples. It also explains techniques for separating mixtures and compounds, including filtration, distillation, chromatography and dissolving. In addition, it discusses the structure of atoms and ions, including electrons, protons and neutrons. It describes the three main types of bonding - ionic, covalent and metallic - and provides examples of how bonding occurs between metals and non-metals to form ions.
This document provides a summary of topics related to algebra, functions, and calculus including: linear and quadratic expressions, simultaneous equations, completing the square, trigonometric ratios, differentiation, tangents, normals, and finding stationary points through higher derivatives. It outlines key steps and methods for solving various types of problems within these topics.
The document discusses rules for simplifying expressions involving indices (exponents). It defines indices as powers and explains that the plural of index is indices. It then presents four rules:
1) Multiplication of Indices: an × am = an+m
2) Division of Indices: an ÷ am = an-m
3) For negative indices: a-m = 1/am
4) For Powers of Indices: (am)n = amn
The document applies these rules to simplify various expressions involving integer indices. It also extends the rules to expressions involving fractional indices obtained from roots.
The document is a mathematics lecture on integers. It discusses the four integer operations of addition, subtraction, multiplication, and division. It provides examples of how to perform each operation on integers and the rules for determining if the result is positive or negative. Addition and subtraction are explained using rules about combining positive and negative integers. Multiplication and division are covered together, as their rules are the same - the result is positive if the signs are the same and negative if the signs are different.
The document provides examples of factorizing algebraic expressions. It begins with expanding simple expressions like 3g + 4 and finding the common factors of terms like 12 and 20. The next section provides more examples of factorizing expressions like x2 + 2x by finding the highest common factor. It emphasizes factorizing completely, like writing 3abc(ab4 + 9) instead of 9abc(ab4 + 9). The document concludes with an exercise for students to practice factorizing expressions and an extension with more challenging examples. It provides the opportunity for a 10 minute practice session to reinforce the steps of factorizing algebraic expressions.
The document provides information about various chemistry concepts related to air and water:
- It describes chemical tests to identify water and the purification of water supplies through filtration and chlorination.
- The composition of clean air is described as 78% nitrogen, 21% oxygen and small quantities of other gases. Common air pollutants like carbon monoxide and their sources are stated.
- Fractional distillation is outlined as the process used to separate oxygen and nitrogen from liquid air based on their different boiling points.
- Rusting is described as a reaction between iron, air and water that can be prevented by methods like painting and galvanizing to exclude oxygen.
Simultaneous equations are two or more equations with the same unknown variables. There are two main methods to solve simultaneous equations:
1) Using graphs - make a table of values, plot the equations on a graph, and find where the graphs intersect.
2) Using algebra - organize the equations, make coefficients equal, eliminate a variable, solve the resulting equation, substitute values back into the original equations, and check the answer. The algebraic method follows the steps of NO MESS: Organize, Make equal, Eliminate, Solve, Substitute.
This document provides revision notes for the Edexcel IGCSE Chemistry exam. It covers topics like atomic structure, bonding, organic chemistry, calculations, the periodic table, acids and bases. For each topic, it provides definitions, explanations and examples. It also explains techniques for separating mixtures and compounds, including filtration, distillation, chromatography and dissolving. In addition, it discusses the structure of atoms and ions, including electrons, protons and neutrons. It describes the three main types of bonding - ionic, covalent and metallic - and provides examples of how bonding occurs between metals and non-metals to form ions.
This document provides a summary of topics related to algebra, functions, and calculus including: linear and quadratic expressions, simultaneous equations, completing the square, trigonometric ratios, differentiation, tangents, normals, and finding stationary points through higher derivatives. It outlines key steps and methods for solving various types of problems within these topics.
The document discusses rules for simplifying expressions involving indices (exponents). It defines indices as powers and explains that the plural of index is indices. It then presents four rules:
1) Multiplication of Indices: an × am = an+m
2) Division of Indices: an ÷ am = an-m
3) For negative indices: a-m = 1/am
4) For Powers of Indices: (am)n = amn
The document applies these rules to simplify various expressions involving integer indices. It also extends the rules to expressions involving fractional indices obtained from roots.
The document provides an explanation of the Pythagorean theorem using examples of right triangles found in baseball diamonds and ladders. It begins by defining a right triangle and its components - the hypotenuse and two legs. It then states the Pythagorean theorem formula that the sum of the squares of the two legs equals the square of the hypotenuse. Several word problems are worked through step-by-step using the theorem to calculate missing side lengths of right triangles.
Solving Systems of Equations using Substitution
Step 1) Solve one equation for one variable.
Step 2) Substitute the expression from Step 1 into the other equation.
Step 3) Solve the resulting equation to find the value of the variable. Step 4) Plug this value back into either original equation to find the value of the other variable. Step 5) Check that the solution satisfies both original equations.
This document discusses linear equations in one variable. It defines a linear equation as an equation for a straight line in the form of ax + b, where a and b are real numbers and x is the variable. There is only one unknown value in a linear equation. The document provides examples of linear equations and explains how to solve them using addition, subtraction, multiplication, division and mixed operations. It also contains practice problems for the reader to try.
Atoms are the basic building blocks of matter and are made up of protons, neutrons, and electrons. Elements are pure substances made of only one type of atom, such as sodium (Na) or oxygen (O). Compounds are formed when two or more elements chemically combine, resulting in a substance with totally new properties compared to the original elements. The periodic table provides information about each element's atomic number, chemical symbol, and atomic mass.
- The document discusses proportional relationships between distance and time when speed is constant. It provides examples of using ratios, proportions, and linear equations to solve problems about distance and time given average or constant speed.
- Key concepts covered include defining average speed, identifying when constant speed can be assumed to use proportions, and writing linear equations relating distance and time under conditions of constant speed.
- Students are encouraged to practice basic ratio and proportion skills in preparation for working on proportional relationship word problems over two class periods.
1) Atoms are the smallest units that make up matter and consist of protons, neutrons and electrons. Electrons orbit the nucleus in shells.
2) There are two main types of bonds between atoms - ionic bonds which involve transfer of electrons between metals and non-metals, and covalent bonds which involve sharing of electrons between non-metals.
3) Isotopes are atoms of the same element that have different numbers of neutrons, giving them different masses but the same chemical properties. Radioactive isotopes are unstable and decay over time.
The document discusses moles, molar mass, and empirical and molecular formulas.
It defines key terms like mole, Avogadro's number, and molar mass. A mole represents 6.02x1023 particles of a substance. Molar mass is the mass in grams of one mole of a substance.
Examples are provided for calculating moles from mass and vice versa using molar mass. Empirical formulas give the lowest whole number ratio of elements in a compound, while molecular formulas specify the actual number of each atom in a molecule or formula unit.
This document provides steps for solving equations with variables on both sides:
1. Expand any brackets first.
2. Identify the smaller term with the variable.
3. Apply the opposite operation (+ or -) to that term on both sides.
4. Simplify and solve the resulting equation normally using techniques like onion skins or backtracking.
Worked examples demonstrate subtracting and adding the smaller variable term to move it to one side.
This presentation compares the development of procedural fluency and conceptual understanding and argues for a systematic approach of teaching one before the other.
This document discusses slope, y-intercept, and how to find and graph linear equations. It defines slope as the ratio that describes a line's tilt and explains how to calculate slope using rise over run between two points on a line. It also discusses how to find the y-intercept, and then use slope and y-intercept to write the equation of a line in y=mx+b form. Examples are provided for finding slope from tables of values and graphing linear equations on a coordinate plane.
The document discusses various chemical topics including:
- Physical and chemical changes when combining hydrogen and oxygen to form water, and red and white clay to form pink clay.
- The meaning of exothermic and endothermic reactions, with burning/combustion identified as an exothermic reaction.
- Factors that affect the rate of a chemical reaction including concentration, temperature, and surface area.
- Enzymes being described as biological catalysts that help chemical reactions occur in the body.
- The law of conservation of mass as it applies to balancing chemical equations so atoms are equal on both sides.
This document discusses working with surds, which are expressions involving square roots. It provides rules for multiplying, dividing, adding, subtracting, and simplifying surds. Some key rules covered are combining like terms under a square root, rationalizing denominators by multiplying the numerator and denominator by the conjugate of the denominator, and squaring both sides of an equation to clear surds before solving. Examples are provided to demonstrate applying these rules to simplify expressions and solve equations involving surds.
After this presentation students will be able to define
Identify Base, Exponents/Indices, value
Laws of Exponents/Indices
Product law
Quotient law
Power law
This document provides an overview of algebraic expressions and identities. It defines terms, factors, coefficients, monomials, binomials, polynomials, like and unlike terms. It explains how to perform addition, subtraction, multiplication, and division of algebraic expressions. It also defines what an identity is and how to apply identities.
Atoms are made up of protons, neutrons and electrons. Atoms become ions when they gain or lose electrons. The atomic number is the number of protons and electrons, while the mass number includes protons and neutrons. Isotopes are forms of the same element with different numbers of neutrons. Alkali metals are reactive and found in Group 1. They react vigorously with water. Halogens are in Group 7 and become more reactive going up the group. They react with metals like iron wool. Ionic compounds have high melting points due to strong attractions between oppositely charged ions. Covalent substances share electrons in bonds of varying strengths.
This document discusses linear equations in two variables. It begins by presenting the general form of a linear equation as ax + by + c = 0, where a, b, and c are real numbers. It then explains that a linear equation can have infinitely many solutions (x,y value pairs) that satisfy the equation, and these solutions lie on a straight line. The document provides an example of a single linear equation and shows its graph on the Cartesian plane. It also discusses systems of two linear equations, explaining that their solutions occur where the lines intersect. The document covers various algebraic methods for solving systems of linear equations, including elimination by substitution or equating coefficients, and solving by cross multiplication. It provides examples to illustrate these solution
This document introduces basic geometry concepts including points, lines, line segments, rays, and planes. It defines each concept and provides examples of their notation and relationships. Points have no dimension, lines extend in one dimension, and planes extend in two dimensions. Examples demonstrate identifying collinear points and opposite rays, as well as the intersection of lines and planes. The summary concludes by stating the document defines fundamental geometric objects and their properties.
This document provides important points and formulas for mathematics. It includes:
1) Definitions and properties of numbers like natural numbers, integers, rational numbers, and irrational numbers.
2) Formulas for topics like percentages, simple interest, compound interest, speed, distance, time, and areas of shapes.
3) Explanations of concepts in algebra like quadratic equations, expansion and factorization of expressions, and ordering.
4) Formulas for area and perimeter of shapes, Pythagoras' theorem, surface area and volume of solids like cylinders and cones.
The document serves as a quick reference guide for key mathematical definitions, properties, formulas, and concepts across various topics
This document contains notes and formulas for SPM Mathematics for Forms 1-4. It covers topics such as solid geometry, circle theorems, polygons, factorisation, expansion of algebraic expressions, indices, algebraic fractions, linear equations, simultaneous linear equations, algebraic formulas, linear inequalities, statistics, quadratic expressions and equations, sets, mathematical reasoning, the straight line, trigonometry, angle of elevation and depression, lines and planes. Formulas and properties are provided for calculating areas and volumes of solids, solving different types of equations, and relationships in geometry, trigonometry and statistics. Examples are included to demonstrate solving problems and using the various formulas and concepts.
The document provides an explanation of the Pythagorean theorem using examples of right triangles found in baseball diamonds and ladders. It begins by defining a right triangle and its components - the hypotenuse and two legs. It then states the Pythagorean theorem formula that the sum of the squares of the two legs equals the square of the hypotenuse. Several word problems are worked through step-by-step using the theorem to calculate missing side lengths of right triangles.
Solving Systems of Equations using Substitution
Step 1) Solve one equation for one variable.
Step 2) Substitute the expression from Step 1 into the other equation.
Step 3) Solve the resulting equation to find the value of the variable. Step 4) Plug this value back into either original equation to find the value of the other variable. Step 5) Check that the solution satisfies both original equations.
This document discusses linear equations in one variable. It defines a linear equation as an equation for a straight line in the form of ax + b, where a and b are real numbers and x is the variable. There is only one unknown value in a linear equation. The document provides examples of linear equations and explains how to solve them using addition, subtraction, multiplication, division and mixed operations. It also contains practice problems for the reader to try.
Atoms are the basic building blocks of matter and are made up of protons, neutrons, and electrons. Elements are pure substances made of only one type of atom, such as sodium (Na) or oxygen (O). Compounds are formed when two or more elements chemically combine, resulting in a substance with totally new properties compared to the original elements. The periodic table provides information about each element's atomic number, chemical symbol, and atomic mass.
- The document discusses proportional relationships between distance and time when speed is constant. It provides examples of using ratios, proportions, and linear equations to solve problems about distance and time given average or constant speed.
- Key concepts covered include defining average speed, identifying when constant speed can be assumed to use proportions, and writing linear equations relating distance and time under conditions of constant speed.
- Students are encouraged to practice basic ratio and proportion skills in preparation for working on proportional relationship word problems over two class periods.
1) Atoms are the smallest units that make up matter and consist of protons, neutrons and electrons. Electrons orbit the nucleus in shells.
2) There are two main types of bonds between atoms - ionic bonds which involve transfer of electrons between metals and non-metals, and covalent bonds which involve sharing of electrons between non-metals.
3) Isotopes are atoms of the same element that have different numbers of neutrons, giving them different masses but the same chemical properties. Radioactive isotopes are unstable and decay over time.
The document discusses moles, molar mass, and empirical and molecular formulas.
It defines key terms like mole, Avogadro's number, and molar mass. A mole represents 6.02x1023 particles of a substance. Molar mass is the mass in grams of one mole of a substance.
Examples are provided for calculating moles from mass and vice versa using molar mass. Empirical formulas give the lowest whole number ratio of elements in a compound, while molecular formulas specify the actual number of each atom in a molecule or formula unit.
This document provides steps for solving equations with variables on both sides:
1. Expand any brackets first.
2. Identify the smaller term with the variable.
3. Apply the opposite operation (+ or -) to that term on both sides.
4. Simplify and solve the resulting equation normally using techniques like onion skins or backtracking.
Worked examples demonstrate subtracting and adding the smaller variable term to move it to one side.
This presentation compares the development of procedural fluency and conceptual understanding and argues for a systematic approach of teaching one before the other.
This document discusses slope, y-intercept, and how to find and graph linear equations. It defines slope as the ratio that describes a line's tilt and explains how to calculate slope using rise over run between two points on a line. It also discusses how to find the y-intercept, and then use slope and y-intercept to write the equation of a line in y=mx+b form. Examples are provided for finding slope from tables of values and graphing linear equations on a coordinate plane.
The document discusses various chemical topics including:
- Physical and chemical changes when combining hydrogen and oxygen to form water, and red and white clay to form pink clay.
- The meaning of exothermic and endothermic reactions, with burning/combustion identified as an exothermic reaction.
- Factors that affect the rate of a chemical reaction including concentration, temperature, and surface area.
- Enzymes being described as biological catalysts that help chemical reactions occur in the body.
- The law of conservation of mass as it applies to balancing chemical equations so atoms are equal on both sides.
This document discusses working with surds, which are expressions involving square roots. It provides rules for multiplying, dividing, adding, subtracting, and simplifying surds. Some key rules covered are combining like terms under a square root, rationalizing denominators by multiplying the numerator and denominator by the conjugate of the denominator, and squaring both sides of an equation to clear surds before solving. Examples are provided to demonstrate applying these rules to simplify expressions and solve equations involving surds.
After this presentation students will be able to define
Identify Base, Exponents/Indices, value
Laws of Exponents/Indices
Product law
Quotient law
Power law
This document provides an overview of algebraic expressions and identities. It defines terms, factors, coefficients, monomials, binomials, polynomials, like and unlike terms. It explains how to perform addition, subtraction, multiplication, and division of algebraic expressions. It also defines what an identity is and how to apply identities.
Atoms are made up of protons, neutrons and electrons. Atoms become ions when they gain or lose electrons. The atomic number is the number of protons and electrons, while the mass number includes protons and neutrons. Isotopes are forms of the same element with different numbers of neutrons. Alkali metals are reactive and found in Group 1. They react vigorously with water. Halogens are in Group 7 and become more reactive going up the group. They react with metals like iron wool. Ionic compounds have high melting points due to strong attractions between oppositely charged ions. Covalent substances share electrons in bonds of varying strengths.
This document discusses linear equations in two variables. It begins by presenting the general form of a linear equation as ax + by + c = 0, where a, b, and c are real numbers. It then explains that a linear equation can have infinitely many solutions (x,y value pairs) that satisfy the equation, and these solutions lie on a straight line. The document provides an example of a single linear equation and shows its graph on the Cartesian plane. It also discusses systems of two linear equations, explaining that their solutions occur where the lines intersect. The document covers various algebraic methods for solving systems of linear equations, including elimination by substitution or equating coefficients, and solving by cross multiplication. It provides examples to illustrate these solution
This document introduces basic geometry concepts including points, lines, line segments, rays, and planes. It defines each concept and provides examples of their notation and relationships. Points have no dimension, lines extend in one dimension, and planes extend in two dimensions. Examples demonstrate identifying collinear points and opposite rays, as well as the intersection of lines and planes. The summary concludes by stating the document defines fundamental geometric objects and their properties.
This document provides important points and formulas for mathematics. It includes:
1) Definitions and properties of numbers like natural numbers, integers, rational numbers, and irrational numbers.
2) Formulas for topics like percentages, simple interest, compound interest, speed, distance, time, and areas of shapes.
3) Explanations of concepts in algebra like quadratic equations, expansion and factorization of expressions, and ordering.
4) Formulas for area and perimeter of shapes, Pythagoras' theorem, surface area and volume of solids like cylinders and cones.
The document serves as a quick reference guide for key mathematical definitions, properties, formulas, and concepts across various topics
This document contains notes and formulas for SPM Mathematics for Forms 1-4. It covers topics such as solid geometry, circle theorems, polygons, factorisation, expansion of algebraic expressions, indices, algebraic fractions, linear equations, simultaneous linear equations, algebraic formulas, linear inequalities, statistics, quadratic expressions and equations, sets, mathematical reasoning, the straight line, trigonometry, angle of elevation and depression, lines and planes. Formulas and properties are provided for calculating areas and volumes of solids, solving different types of equations, and relationships in geometry, trigonometry and statistics. Examples are included to demonstrate solving problems and using the various formulas and concepts.
This document contains notes and formulae on solid geometry, circle theorems, polygons, factorisation, expansion of algebraic expressions, algebraic formulae, linear inequalities, statistics, significant figures and standard form, quadratic expressions and equations, sets, mathematical reasoning, straight lines, and trigonometry. The key concepts covered include formulas for calculating the volume and surface area of various 3D shapes, properties of angles in circles and polygons, factorising and expanding algebraic expressions, solving linear and quadratic equations, set notation and Venn diagrams, types of logical arguments, equations of straight lines, and defining the basic trigonometric ratios.
Definitions biology-igcse-biodeluna2011
Compilation of definitions taken from the 2011 Biology IGCSE Syllabus.
IES Pedro de Luna.
www.Biodeluna.wordpress.com
The document provides study material for mathematics for class 10 students of Kendriya Vidyalaya Sangathan. It was prepared by the Patna regional office in accordance with instructions from KVS headquarters. The study material aims to help students understand concepts well and meet quality expectations. It was prepared under the guidance of the Deputy Commissioner with contributions from teachers of KV No. 2 Gaya.
The document provides information on key economic concepts including opportunity cost, factors of production, productivity, and different types of market systems. It defines opportunity cost as the next best alternative forgone when making a choice. The four factors of production are land, labor, capital, and entrepreneurship. Productivity is defined as a measure of efficiency and is increased by using fewer resources to produce the same output or using the same resources to produce more output. The summary discusses the different types of market systems including free markets, mixed economies, and planned economies along with their advantages and disadvantages.
1) The document is a formula sheet for quantitative ability topics for CAT and management entrance tests provided by the website snapwiz.co.in.
2) It includes formulas and properties for arithmetic, percentages, fractions, logarithms, progressions, roots of quadratic equations, counting principles, probability, geometry, triangles, polygons, and circles.
3) Visitors to the website can access free mock CAT tests and other resources after reviewing this formula sheet.
This document contains formulas for calculating the areas, volumes, and surface areas of various 2D and 3D shapes. It includes formulas for calculating the area of triangles, parallelograms, trapezoids, circles, rhombi/kites, and regular polygons. For 3D shapes it includes formulas for calculating the volume, surface area, and lateral area of rectangular prisms, other prisms, cylinders, pyramids, and cones. It also contains the Pythagorean theorem and formulas for calculating trigonometric ratios, circumferences, and the altitude of a triangle.
This document provides information about plane (2D) and 3D figures, including their definitions, examples, and formulas to calculate their areas and volumes. Plane figures are flat shapes that can be made of straight lines, curved lines, or both. 3D figures have height, depth and width and do not lie entirely in a plane. Examples of plane figures include squares, rectangles, trapezoids and circles, while 3D figures include cubes, cuboids, cylinders, cones, spheres and hemispheres. Formulas are given to calculate the areas of common plane shapes like squares, rectangles, triangles and circles, as well as the volumes and surface areas of various 3D solids.
The document provides definitions for mathematical terms that students in 5th/6th class primary school and junior cycle secondary school may encounter. It includes over 50 terms defined with diagrams and examples. The glossary is designed to inform students, parents, and teachers about the vocabulary and meanings of key mathematical terms as students transition between primary and post-primary education in Ireland.
There are so many mathematical symbols that are important for students. To make it easier for you we’ve given here the mathematical symbols table with definitions and examples
This document defines various math terms across several categories:
1) It begins by defining types of numbers such as natural numbers, integers, decimals, and irrational numbers. It also defines basic math operations like addition, subtraction, multiplication, and division.
2) It then defines geometry terms like points, lines, angles, polygons, triangles, circles, and 3D shapes. It also covers perimeter, area, volume, and surface area.
3) Finally, it defines coordinate geometry terms like the coordinate plane, axes, ordered pairs, intercepts, slope, domain, and range. It discusses parallel and perpendicular lines on a graph.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document provides an overview of topics covered in a mathematics guide for civil service examinations. It includes: 1) classification of numbers like rational, irrational, integers; 2) operations on numbers such as addition, subtraction, multiplication, division of signed numbers; 3) concepts like prime factorization, LCM, GCF, ratios, proportions, percentages; 4) geometry, algebra, sets, probability, series; and 5) measurement conversions between metric units. Worked examples are provided to illustrate key rules and procedures.
ICSE class X maths booklet with model paper 2015APEX INSTITUTE
APEX INSTITUTE has been established with sincere and positive resolve to do something rewarding for ENGG. / PRE-MEDICAL aspirants. For this the APEX INSTITUTE has been instituted to provide a relentlessly motivating and competitive atmosphere.
As It is very important to discover the basic weaknesses and problems of students not succeeding in IIT-JEE / PRE-MEDICAL exams. In fact, as question patterns are changing, now you need to have a different approach for these exams. As far as ENGG. / PRE-MEDICAL preparations is concerned, students have been wasting time and energy, studying Physics, Chemistry and Maths at different places. At APEX INSTITUTE, the scope of the subject has been deliberately made all- inclusive to free them of this burden. APEX INSTITUTE offers you complete preparation for IIT-JEE/PRE-MED. exams under one roof.
This PowerPoint was created to help out graduating seniors who are taking the TAKS Mathematics Exit-Level test. It includes formulas, rules & things that they need to remember to pass the test.
The document provides information on various number system concepts in Vedic maths including:
1. Methods for multiplying numbers with 11, 9, 99, and 999 using place value concepts.
2. Methods for multiplying two-digit and three-digit numbers using the "criss-cross" method.
3. Shortcuts for finding squares and square roots of numbers.
4. Divisibility rules and their applications.
5. Concepts like remainder theorem, power cycles, and unit digit patterns that are useful for solving problems involving remainders and exponents.
6. Information on factors, multiples, and their properties like total number of factors and sum of factors.
This document provides an overview of key concepts in number theory that will be covered in a lecture, including natural numbers, integers, rational and irrational numbers, and their basic properties. It also discusses divisibility, greatest common divisors, least common multiples, prime numbers, and modular arithmetic. The concepts of congruences, the Euclidean algorithm, and different representations of integers in numerical bases are introduced.
The document discusses divisibility rules and tests for determining if a number is divisible by numbers from 2 to 13. It provides examples and explanations of divisibility rules for each number, along with related concepts like remainders, prime factorization, and properties of factorials. Practice problems are also included at the end related to divisibility, remainders, factors, and factorials.
1) Fractions represent parts of a whole. They are written as a/b where a is the numerator and b is the denominator. Fractions can be reduced by dividing the numerator and denominator by common factors.
2) To multiply fractions, multiply the numerators and multiply the denominators. To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
3) Mixed numbers represent an integer plus a fraction, such as 5 3/4. Improper fractions have a numerator larger than or equal to the denominator, such as 7/2, and can be converted to mixed numbers.
The document discusses various topics related to numbers including:
1) Perfect numbers which are numbers whose factors sum to the number.
2) Classification of numbers as natural, whole, integers, rational, and irrational.
3) Rules for divisibility including by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.
4) Formulas for finding cubes of two-digit numbers and number of zeros in expressions.
The document provides information on various topics in mathematics including:
- Number systems and notation for numbers written in words, Roman numerals, and different number bases.
- Interest calculation formulas for compound and continuous interest.
- Basic algebra concepts such as properties of addition/multiplication and the definition of conic sections.
- Geometric shapes including the definition of polygons, formulas for calculating polygon properties, and special polygon names.
This document provides information on mathematical concepts and formulas relevant to economics, including:
- Exponential functions such as y=ex and their graphs showing exponential growth and decay
- Quadratic functions of the form y=ax2+bx+c and total cost functions
- Differentiation rules for common functions like exponentials, logarithms, and the product, quotient and chain rules
- Integration basics and formulas for integrating common functions
- Concepts like inverse functions, the mean, variance and standard deviation in statistics
- Information is also provided on fractions, ratios, percentages, and algebraic rules involving exponents, logarithms and sigma notation.
2021 noveno guia en casa matematicas periodo 3Ximena Zuluaga
The document provides information about mathematics courses and teachers for 9th grade students at Institución Educativa Inem "Jorge Isaacs" including:
- A list of the different mathematics and geometry courses/teachers for each of the 16 9th grade groups.
- Criteria for evaluating student activities and assignments including developing work individually, justifying work, submitting photo evidence of work, and meeting deadlines.
- Learning objectives, standards, and competencies related to real numbers, algebraic expressions, linear equations and systems of linear equations.
- An introduction to the third study guide which will cover linear equations of lines and quadratic functions including definitions, examples, and how to calculate elements of parabolas
This document provides information about percentages, fractions, ratios, averages, and using a calculator. It includes steps for converting fractions to percentages and vice versa. It also covers finding percentages of amounts, increasing and decreasing percentages, and calculating percentage profit/loss. Other topics covered include factors, multiples, primes, prime factorization, indices, fractions, rounding, estimation, ratios, BODMAS order of operations, and Venn diagrams. The document provides examples and explanations for solving problems involving these various mathematical concepts.
This document is an eBook containing math formulas and concepts for the CAT, XAT, and other MBA entrance exams. It includes summaries of topics like number systems, arithmetic, algebra, geometry, averages, percentages, interest, profit and loss, ratios, and more. The eBook is intended to help students revise key concepts in the days leading up to their exam. It was compiled by Ravi Handa, who runs a website providing online CAT coaching courses.
The document provides information about the format and techniques for answering Additional Mathematics SPM Paper 1 exam questions in Malaysia. It discusses the contents and structure of the exam, including the breakdown of question types between knowledge/understanding and application skills. It also offers tips for candidates on how to effectively answer questions, including showing workings, following instructions, and presenting neat and precise solutions.
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2. Content
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Topic / sub topic
Natural numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Terminating Decimals
Recurring Decimals
Significant figures
Decimal Places
Standard Form
Conversion Factors
Time
Percentages
Simple Interest
Compound Interest
Speed, Distance and Time
Quadratic Equations
Expansion of algebraic expressions
Factorization of algebraic expressions
Ordering
Variation
PYTHAGORAS’ THEOREM
Area and Perimeter
Surface Area and Volume
Angles on a straight line
Vertically opposite angles
Different types of triangles
Parallel Lines
Types of angles
Angle properties of triangle
Congruent Triangles
Similar Triangles
Areas of Similar Triangles
Polygons
Similar Solids
CIRCLE
Chord of a circle
Tangents to a Circle
Laws of Indices
Solving Inequalities
TRIGONOMETRY
Bearing
Cartesian co-ordinates
Distance – Time Graphs
Speed – Time Graphs
Velocity
Acceleration
SETS
Page
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Loci and construction
Vectors
Column Vectors
Parallel Vectors
Modulus of a Vector
MATRICES
The Inverse of a Matrix
Transformations
Transformation by Matrices
STATISTICS
Probability
Symmetry
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3. NUMBER
Natural Numbers: Numbers which are used for
counting purpose are called natural numbers.
Ex: 1, 2, 3, 4, …………….100, ……………….
Whole Numbers: Natural numbers including 0 are
called Whole Numbers.
Ex: 0, 1, 2, 3, 4, ……………………..
Integers: Positive natural numbers, negative natural
numbers along with 0 are called integers.
Ex.: …………………, -4, -3, -2, -1, 0, 1, 2, 3, 4, ……………
Rational Numbers: Numbers which are in the form
𝑝
of 𝑞 (q ≠ 0) where p and q are positive or negative
whole numbers are called rational numbers.
1 3 −5 49
Ex: 2 , 4 , 7 , −56 …………………..
Irrational Numbers: Numbers like 2 , 𝜋 cannot
be expressed as rational numbers. Such types of
numbers are called as irrational numbers.
Ex:
5 , 17 , ………….
Terminating Decimals
These are decimal numbers which stop after a
certain number of decimal places.
For example,7/8 = 0.875, is a terminating decimal
because it stops (terminates) after 3 decimal places.
Recurring Decimals
These are decimal numbers which keep repeating a
digit or group of digits; for example
137/259,=0.528 957 528 957 528 957 ...., is a
recurring decimal. The six digits 528957 repeat in
this order. Recurring decimals are written with dots
over the first and last digit of the repeating digits,
e.g. 0.528 957
The order of operations follows the BODMAS
rule:
Brackets
Powers Of
Divide
Multiply
Add
Subtract
Even numbers: numbers which are divisible
by 2, eg, 2, 4, 6, 8, …
Odd numbers: numbers which are not
divisible by 2, eg; 1, 3, 5, 7 …
Real numbers are made up of all possible
rational and irrational numbers.
An integer is a whole number.
A prime number is divisible only by itself and
by one (1). 1 is not a prime number. It has
only two factors. 1 and the number itself.
The exact value of rational number can be
written down as the ratio of two whole
numbers.
The exact value of an irrational number
cannot be written down.
A square number is the result of multiplying
a number by itself.
Ex: 12, 22, 32, ……………. i.e. 1, 4, 9, ……………..
A cube number is the result of multiplying a
number by itself three times.
Ex: 13, 23, 33, …………………. i.e. 1, 8, 27,………
The factors of a number are the numbers
which divide exactly into two.
eg. Factors of 36
1, 2, 3, 4, 6, 9, 12, 18
Multiples of a number are the numbers in its
times table.
eg. Multiples of 6 are 6, 12, 18, 24, 30, …
Significant figures;
Example;
8064 = 8000 (correct to 1 significant figures)
8064 = 8100 (correct to 2 significant figures)
8064 = 8060 (correct to 3 significant figures)
0.00508 =0.005 (correct to 1 significant figures)
0.00508 = 0.0051 (correct to 2 significant figures)
2.00508 = 2.01 (correct to 3 significant figures)
Decimal Places
Example
0.0647 = 0.1 (correct to 1 decimal places)
0.0647 = 0.06 (correct to 2 decimal places)
0.0647 = 0.065 (correct to 3 decimal places)
2.0647 = 2.065 (correct to 3 decimal places)
Standard Form:
The number a x 10n is in standard form when
1≤ a < 10 and n is a positive or negative integer.
Eg: 2400 = 2.4 x 103
0.0035 = 3.5 x 10-3
____________________________________________________________________________________
Mathematics - important points and formulas 2009
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Page 1 of 21
4. Conversion Factors:
Length:
1 km = 1000 m
1 m = 100 cm
1 cm = 10 mm
𝑘𝑚 means kilometer
𝑚 means meter
𝑐𝑚 means centimeter
𝑚𝑚 means millimeter
Mass:
1 kg = 1000 gm
1 gm = 1000 mgm
1 tonne = 1000 kg
where kg means kilogram
gm means gram
mgm means milligram
Volume:
1 litre
= 1000 cm3
1 m3
= 1000 litres
1 kilo litre = 1000 litre
1 dozen = 12
Time:
1 hour = 60 minutes = 3600
seconds
1 minute = 60 seconds.
1 day = 24 hours
1 year = 12 months
= 52 weeks
= 365.25 days.
1 week = 7 days
1 leap year = 366 days
1 light year = 9.46 × 1012 km.
Percentages:
Percent means per hundred.
To express one quantity as a percentage of another, first write the first quantity as a fraction of
the second and then multiply by 100.
Profit = S.P. – C.P.
Loss = C.P. – S.P.
𝑆𝑃−𝐶𝑃
Profit percentage = 𝐶𝑃 × 100
Loss percentage =
where CP = Cost price
𝐶𝑃−𝑆𝑃
𝐶𝑃
× 100
and SP = Selling price
Simple Interest:
To find the interest:
𝑖=
𝑃𝑅𝑇
100
Compound Interest:
r
A = 𝑝 1 + 100
where
P = money invested or borrowed
R = rate of interest per annum
T = Period of time (in years)
To find the amount:
𝐴 = 𝑃 + 𝐼
where A = amount
n
Where,
𝑨 stands for the amount of money accruing after 𝑛
year.
𝑷 stands for the principal
𝑹 stands for the rate per cent per annum
𝒏 stands for the number of years for which the
money is invested.
____________________________________________________________________________________
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5. Speed, Distance and Time:
Distance = speed x time
Speed =
Time =
Units of speed: km/hr, m/sec
Units of distance: km, m
Units of time:
hr, sec
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑡𝑖𝑚𝑒
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
5
= m / sec
18
18
m / sec ×
= km / hr
5
Average speed =
D
km / hr ×
𝑆𝑝𝑒𝑒𝑑
𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑡𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒
S
T
ALGEBRA
Quadratic Equations:
An equation in which the highest power of the variable is 2 is called quadratic equation. Thus
ax2 + bx + c = 0 where a, b, c are constants and a ≠ 0 is a general equation.
Solving quadratic equations:
We can solve quadratic equation by method of,
a) Factorization
b) Using the quadratic formula
c) Completing the square
(a) Solution by factors:
Consider the equation c × d = 0, where c and d are numbers. The product c × d can only be zero if either c
or d (or both) is equal to zero.
i.e. c = 0 or d = 0 or c = d = 0.
(b)Solution by formula:
The solutions of the quadratic equation
ax2 + bx + c = 0 are given by the formula:
𝑥 =
−𝑏± 𝑏 2 −4𝑎𝑐
2𝑎
(c) Completing the square
Make the coefficient of x2 , i.e. a = 1
Bring the constant term, i.e. c to the right side of equation.
𝑏
Divide coefficient of x, i.e. by 2 and add the square i.e. ( 2)2 to both sides of the equation.
Factorize and simplify answer
Expansion of algebraic expressions
𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
a2 + b2 = (a + b)2 – 2ab
a2 – b2
= (a + b)(a – b)
Ordering:
= is equal to
≠ is not equal to
> is greater than
Factorization of algebraic expressions
𝑎2 + 2𝑎𝑏 + 𝑏 2 = (𝑎 + 𝑏)2
𝑎2 − 2𝑎𝑏 + 𝑏 2 = (𝑎 − 𝑏)2
𝑎2 − 𝑏 2 = 𝑎 + 𝑏 (𝑎 − 𝑏)
≥ is greater than or equal to
< is less than
≤ is less than or equal to
____________________________________________________________________________________
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6. Variation:
Direct Variation:
y is proportional to x
Inverse Variation:
y is inversely proportional to x
1
y
x
k
y=
x
y x
y = kx
MENSURATION
PYTHAGORAS’ THEOREM
For all the right angled triangles “ the square on the hypotenuse is equal to the
sum of the squares on the other two sides”
𝑐 2 = 𝑎2 + 𝑏 2
𝒄=
𝑎2 + 𝑏 2
𝒃=
Area and Perimeter:
Figure
Rectangle
𝑐 2 − 𝑎2
𝒂=
Diagram
𝑐 2 − 𝑏2
Area
b
Perimeter
Area = l × b
perimeter = 2( 𝑙 + 𝑏 )
Area = side × side
= a×a
perimeter = 4 × side
=4×a
l
a
Square
a
a
a
Parallelogram
Area = b × h
perimeter = 2(a + b )
Area = ab sin 𝜃
where a, b are sides and 𝜃 is
the included angle
Triangle
Area =
1
2
× 𝑏𝑎𝑠𝑒 × 𝑒𝑖𝑔𝑡
1
Area = 2 𝑎𝑏 𝑠𝑖𝑛 𝐶
= 𝑠 𝑠− 𝑎
where s =
perimeter = a + b + c
𝑠 − 𝑏 (𝑠 − 𝑐)
𝑎+𝑏+𝑐
2
____________________________________________________________________________________
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7. Trapezium
1
Area = 2 𝑎 + 𝑏
perimeter = Sum of all
sides
Circle
r
Area = r2
Semicircle
r
Sector
𝑟
𝜃
Surface Area and Volume:
Figure
Diagram
Cylinder
Area =
1 2
r
2
2
Area = r 360
1
perimeter = d + d
2
length of an arc = 2 r
Surface Area
curved surface area = 2πrh
total surface area = 2πr(h + r)
Cone
circumference = 2 r
curved surface area = 𝜋𝑟𝑙
where l = (r 2 h 2 )
total surface area = 𝜋𝑟(𝑙 + 𝑟)
Sphere
Surface area = 4πr2
Pyramid
Base area + area of the shapes in
the sides
360
Volume
2
Volume = r h
Volume =
1 2
πr h
3
Volume =
4 3
πr
3
1
× base area ×
3
perpendicular height
Volume =
Cuboid
Surface area = 2(𝑙𝑏 + 𝑏 + 𝑙)
Cube
Hemisphere
Volume = 𝑙 × 𝑏 ×
Surface area = 6𝑙 2
Volume = 𝑙 3
Curved surface area = = 2 r2
Volume =
2 3
r
3
____________________________________________________________________________________
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8. GEOMETRY
(a) Angles on a straight line
The angles on a straight line add up to 180o.
Parallel Lines:
When lines never meet, no matter how far they are
extended, they are said to be parallel.
x + y + z =180o
Vertically opposite angles are equal.
a = c; b = d; p = s and q =r
(b) Angle at a point
Corresponding angles are equal.
𝑎 = 𝑞; 𝑏 = 𝑝; 𝑐 = 𝑟 and 𝑑 = 𝑠
Alternate angles are equal.
c= q and d = p.
Sum of the angles of a triangle is 180o.
Sum of the angles of a quadrilateral is 360o.
The angles at a point add up to 360o.
a + b + c + d = 360o
(c) Vertically opposite angles
If two straight line intersect, then
𝑎= 𝑐
𝑏 = 𝑑 (Vert,opp.∠𝑠)
Types of angles
Given an angle , if
𝜃 < 90° , then 𝜃 is an acute angle
90° < 𝜃 < 180° , then 𝜃 is an obtuse angle
180° < 𝜃 < 360° , then 𝜃 is an reflex angle
Triangles
Different types of triangles:
1. An isosceles triangle has 2 sides and 2 angles the same.
AB = AC
ABC = BCA
2. An equilateral triangle has 3 sides and 3 angles the same.
AB = BC = CA and ABC = BCA = CAB
3. A triangle in which one angle is a right angle is called the right angled triangle.
ABC = 90o
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9. Angle properties of triangle:
The sum of the angles of a triangle is equal to 180o.
In every triangle, the greatest angle is opposite to the longest side. The smallest angle is
opposite to the shortest side.
Exterior angle is equal to the sum of the opposite interior angles.
x=a+b
Congruent Triangles:
Two triangles are said to be congruent if they are equal in every aspect.
a = x
b = y
c = z
AB = XY
BC = YZ
AC = XZ
Similar Triangles:
If two triangles are similar then they have a pair of corresponding equal angles and the three ratios of
corresponding sides are equal.
AB
BC
AC
=
=
XY
YZ
XZ
a = x; b = y and c = z
If you can show that one of the following conditions is true for two triangles, then the two triangles are
similar.
i)
The angles of one triangle are equal to the corresponding angles of the other triangle.
∆ ABC is similar to ∆ XYZ because a= x; b = y and c = z
ii) The ratio of corresponding sides is equal.
F
R
D
P
If
PQ PR
QR
=
=
DE DF
EF
then ∆ PQR is similar to ∆ DEF
Q
E
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10. iii)
The ratios of the corresponding sides are equal and the angles between them are equal.
R
Z
Y
X
P
Q
∆ PQR is similar to ∆ XYZ (if, for eg: P = X and
PQ PR
=
)
XY XZ
Areas of Similar Triangles:
The ratio of the areas of similar triangles is equal to the ratio of the square on corresponding sides.
C
R
A
B
P
Q
AC 2
area of ABC AB 2
BC 2
=
=
=
area of PQR PQ 2 QR 2
PR 2
Polygons:
i)
The exterior angles of a polygon add up to 360o.
ii)
The sum of the interior angles of a polygon is (𝑛 – 2) × 180o where 𝑛 is the number of sides
of the polygon.
iii)
A regular polygon has equal sides and equal angles.
iv)
v)
360
If the polygon is regular and has 𝑛 sides, then each exterior angle = 𝑛
3 sides = triangle
6 sides = hexagon
9 sides = nonagon
4 sides = quadrilateral
7 sides = heptagon
10 sides = decagon
5 sides = pentagon
8 sides = octagon
Similar Solids:
If two objects are similar and the ratio of corresponding sides is k, then
the ratio of their areas is 𝑘2.
the ratio of their volumes is 𝑘3.
Length
𝑙1
𝑟
= 1
𝑙2
𝑟2
=
1
2
Area
2
2
A1
r
h
= 12 = 12
A2
r2
h2
Volume
3
3
V1
r
h
= 13 = 13
V2
r2
h2
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11. CIRCLE
The angle subtended by an arc at the centre is twice the angle subtended at the circumference
The angle in a semi-circle is a right angle. [or if a triangle is inscribed in a semi-circle the angle
opposite the diameter is a right angle]. ∠𝐴𝑃𝐵 = 90°
Angles subtended by an arc in the same segment of a circle are equal.
Opposite angles of a cyclic quadrilateral add up to 180o (supplementary). The corners touch the
circle. A+C = 180o, B+D 180o
The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.(𝑏 = 𝑝)
Chord of a circle:
A line joining two points on a circle is called a chord.
The area of a circle cut off by a chord is called a segment.
AXB is the minor arc and AYB is the major arc.
a) The line from the centre of a circle to the mid-point of
a chord bisects the chord at right angles.
b) The line from the centre of a circle to the mid-point of a
chord bisects the angle subtended by the chord at the centre of the circle.
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12. Tangents to a Circle:
The angle between a tangent and the radius drawn to
the point of contact is 90o.
ABO = 900
From any point outside a circle just two tangents to the circle may be drawn and they are of
equal length.
TA = TB
Alternate Segment Theorem
The angle between a tangent and a chord through the point
of contact is equal to the angle subtended by the chord
in the alternate segment.
QAB = ACB (p = q)
TC2 = AC × BC
P
T
INDICES:
am × an = am + n
am ÷ an = am – n
(am)n = amn
a0 = 1
a-n =
=
a
m
n
am
bm
m
= am / n
1
an
m
a
b
m
m
(a × b) = a × b
a× b
=
a
b
=
a
2
ab
a
b
= a
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13. Solving Inequalities:
When we multiply or divide by a negative number the inequality is reversed.
Eg: 4 > -2
By multiplying by -2 [4(-2) < (-2)(-2) ]
-8 < +4
TRIGONOMETRY
Let ABC be a right angled triangle, where B = 90o
Sin 𝜃 =
Cos 𝜃 =
Tan 𝜃 =
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑆𝑖𝑑𝑒
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
=
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑆𝑖𝑑𝑒
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑆𝑖𝑑𝑒
=
=
𝑂
𝐻
𝐴
SOH CAH TOA
𝐻
𝑂
𝐴
Sine Rule:
a
b
c
=
=
sin A sin B sin C
Cosine Rule:
To find the length of a side:
a2 = b2 + c2 - 2bc cosA
b2 = a2 + c2 – 2ac cos B
c2 = a2 + b2 – 2ab cos C
To find an angle when all the three sides are
given:
b2 c2 a2
cos A =
2bc
2
a c2 b2
cos B =
2ac
2
a b2 c2
cos C =
2ab
Bearing
The bearing of a point B from another point A is;
(a) an angle measured from the north at A.
(b) In a clockwise direction.
(c) Written as three-figure number (i.e. from 000 ° to 360°)
Eg: The bearing of B from A is 050° .
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14. Cartesian co-ordinates
Gradient and equation of a straight line
The gradient of the straight line joining any two given points A( x1 , y1 ) and B( x2 , y2 ) is;
y y1
m 2
x2 x1
The gradient/intercept form of the equation of a straight line is y mx c , where m gradient and
c intercept on y – axis.
The midpoint of the line joining two points A( x1 , y1 ) and B( x2 , y2 ) is; 𝑀 =
The distance between two points A( x1 , y1 ) and B( x2 , y2 ) is; 𝐴𝐵 =
Parallel lines have the same gradient.
In a graph, gradient =
Vertical height
or
Horizontal height
𝑥 1 +𝑥 2
𝑥2 − 𝑥1
2
2
,
𝑦 1 +𝑦 2
2
+ 𝑦2 − 𝑦1
2
𝑦
𝑥
Distance – Time Graphs:
From O to A : Uniform speed
From B to C : uniform speed
From A to B : Stationery (speed = 0)
The gradient of the graph of a distance-time graph gives the speed of the moving body.
Speed – Time Graphs:
From O to A : Uniform speed
From A to B : Constant speed (acceleration = 0)
From B to C : Uniform deceleration / retardation
The area under a speed –time graph represents the distance
travelled.
The gradient of the graph is the acceleration. If the acceleration is
negative, it is called deceleration or retardation. (The moving body is slowing down.)
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15. Velocity:
Velocity is the rate of change of distance with respect to the time.
Acceleration:
Acceleration is the rate of change of velocity with respect to time.
SETS:
Notations
𝜉 = universal set
∪ (union) = all the elements
∩ (intersection) = common elements
Ø or { } = empty set
∈ = belongs to
∉ = does not belongs to
⊆ = Subset
Subset ⊆
𝐵 ⊆ 𝐴 means every elements of set B is also an
element of set A.
𝐴′ = compliment of A (i.e. the elements of
𝜉 - the elements of A)
n(A) = the number of elements in A.
De Morgan’s Laws: (𝐴 ∪ 𝐵)′ = (𝐴′ ∩ 𝐵 ′ )
(𝐴 ∩ 𝐵)′ = (𝐴′ ∪ 𝐵 ′ )
Proper subset ⊂
B ⊂ A means every element of B is an element
of set A but B≠A.
or
Disjoint sets
Disjoint set do not have any element in
common. If A and B are disjoint sets, then
𝐴∩ 𝐵=∅
Union ∪
𝐴 ∪ 𝐵 is the set of elements in either A , B or
both A and B.
Intersection ∩
𝐴 ∩ 𝐵 is the set of elements which are in A
and also in B
Complement
The complement of A, written as 𝐴′ refers to
the elements in 𝜀 but not in A.
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16. Loci and construction
The locus of a point is a set of points satisfying a given set of conditions.
(a) Locus of points at a distance x from a given point, O.
Locus: The circumference of a circle centre O, radius x.
(b) Locus of a points at a distance x from a straight line AB
Locus: A pair of parallel lines to the given line AB.
(c) Locus of points equidistance between 2 points.
Locus: Perpendicular bisector of the two points.
(d) Locus of points equidistant from two given lines AB and AC
Locus: Angle bisector of ∠𝐵𝐴𝐶
Vectors:
A vector quantity has both magnitude and direction.
a
Vectors a and b represented by the line segments can be added
using the parallelogram rule or the nose- to- tail method.
b
A scalar quantity has a magnitude but no direction. Ordinary numbers are scalars.
The negative sign reverses the direction of the vector.
The result of a – b is a + -b
i.e. subtracting b is equivalent to adding the negative of b.
Addition and subtraction of vectors
𝑂𝐴 + 𝐴𝐶 = 𝑂𝐶 (Triangular law of addition)
𝑂𝐵 + 𝑂𝐴 = 𝑂𝐶 ( parallelogram law of addition)
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17. Column Vectors:
The top number is the horizontal component and the bottom number is the vertical component
x
y
Parallel Vectors:
Vectors are parallel if they have the same direction. Both components of one vector must be
in the same ratio to the corresponding components of the parallel vector.
a
a
In general the vector k is parallel to
b
b
Modulus of a Vector:
The modulus of a vector a, is written as a and represents the length (or magnitude) of the vector.
m
In general, if x = , x =
n
(m 2 n 2 )
MATRICES:
Addition and Subtraction:
Matrices of the same order are added (or subtracted) by adding (or subtracting) the corresponding
elements in each matrix.
a b p q a p b q
c d + r s = c r d s
a b p q a p b q
c d - r s = c r d s
Multiplication by a Number:
Each element of a matrix is multiplied by the multiplying number.
a b ka kb
k×
c d = kc kd
Multiplication by another Matrix:
Matrices may be multiplied only if they are compatible. The number of columns in the left-hand matrix
must equal the number of rows in the right-hand matrix.
a b p q ap br aq bs
c d × r s = cp dr cq ds
In matrices A2 means A × A. [you must multiply the matrices together]
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18. The Inverse of a Matrix:
1
a b
then A-1 =
If A =
c d
(ad bc)
d b
c a
AA-1 = A-1A = I where I is the identity matrix.
The number (ad – bc ) is called the determinant of the matrix and is written as A
If A = 0, then the matrix has no inverse.
Multiplying by the inverse of a matrix gives the same result as dividing by the matrix.
e.g.
if AB = C
A-1AB = A-1C
B = A-1C
x
r
xr
If C = and D = then C + D =
y
s
y s
Transformations:
a) Reflection:
When describing a reflection, the position of the mirror line is essential.
b) Rotation:
To describe a rotation, the centre of rotation, the angle of rotation and the direction of rotation
are required.
A clockwise rotation is negative and an anticlockwise rotation is positive.
>> (angle) (Direction)rotation about (centre)
c) Translation:
When describing a translation it is necessary to give the translation vector
x
y
+ x represents movement to the right
- x represents movement to the left
+ y represents movement to the top
- y represents movement to the bottom.
>> Translation by the column vector -----
d) Enlargement:
To describe an enlargement, state;
i.
The scale factor, K
ii.
The centre of enlargement (the invariant point)
Scale factor =
length of the image
length of the object
>> Enlargement by the scale factor --- centre ------
If K > 0, both the object and the image lie on the same side of the centre of enlargement.
If K < 0, the object and the image lie on opposite side of the centre of enlargement.
If the scale factor lies between 0 and 1, then the resulting image is smaller than the object.
[although the image is smaller than the object, the transformation is still known as an
enlargement]
Area of image = K2 area of object
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19. Repeated Transformations:
XT(P) means ‘perform transformation T on P and then perform X on the image.’
XX(P) may be written X2(P).
Inverse Transformations:
The inverse of a transformation is the transformation which takes the image back to the object.
x
x
If translation T has a vector , then the translation which ahs the opposite effect has vector .
y
y
This is written as T-1.
If rotation R denotes 90o clockwise rotation about (0, 0), then R-1 denotes 90o anticlockwise rotation
about (0, 0).
For all reflections, the inverse is the same reflection.
Base vectors
1
0
The base vectors are considered as I = and J =
0
1
The columns of a matrix give us the images of I and J after the transformation.
Shear:
Shear factor =
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎 𝑝𝑜𝑖𝑛𝑡 𝑚𝑜𝑣𝑒𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑡𝑒 𝑠𝑒𝑎𝑟
𝑃𝑒𝑎𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡𝑒 𝑝𝑜𝑖𝑛𝑡 𝑓𝑟𝑜𝑚 𝑡𝑒 𝑓𝑖𝑥𝑒𝑑 𝑙𝑖𝑛𝑒
=
𝑎
𝑏
[The shear factor will be the same calculated from any point on the object with the exception of those
on the invariant line]
Area of image = Area of object
Stretch:
To describe a stretch, state;
i.
the stretch factor, p
ii.
the invariant line,
iii.
the direction of the stretch
(always perpendicular to the invariant line)
Scale factor =
𝑃𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶 ′ 𝑓𝑟𝑜𝑚 𝐴𝐵
𝑃𝑒𝑎𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶 𝑓𝑟𝑜𝑚 𝐴𝐵
Where, P is the stretch factor
Area of image = 𝑝 × Area of object
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20. Transformation by Matrices
Reflection
Matrix
1 0
0 −1
−1 0
0 1
0 1
1 0
0 −1
−1 0
Transformation
Reflection in the x-axis
Reflection in the y-axis
Reflection in the line y = x
Reflection in the line y = - x
Rotation
Matrix
Angle
Direction
centre
0 −1
1 0
90°
anticlockwise
(0, 0)
90°
clockwise
(0, 0)
180°
Clockwise/ anticlockwise
(0, 0)
0
−1
−1
0
1
0
0
−1
Enlargement
𝑘
0
0
where 𝑘= scale factor and centre of enlargement = (0, 0)
𝑘
Stretch
Matrix
Stretch factor
Invariant line
Direction
𝑘
0
0
1
𝑘
y-axis
Parallel to x - axis
1
0
0
𝑘
𝑘
x - axis
Parallel to y - axis
Shear factor
Invariant line
Direction
𝑘
x-axis
Parallel to x - axis
𝑘
y - axis
Parallel to y - axis
Shear
Matrix
1 𝑘
0 1
1 0
𝑘 1
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21. STATISTICS
Bar Graph:
A bar chart makes numerical information easy to see by showing it in a pictorial form.
The width of the bar has no significance. The length of each bar represents the quantity.
Pie Diagram:
The information is displayed using sectors of a circle.
Histograms:
A histogram displays the frequency of either continuous or grouped discrete data in the form of bars.
The bars are joined together.
The bars can be of varying width.
The frequency of the data is represented by the area of the bar and not the height.
[When class intervals are different it is the area of the bar which represents the frequency not the
height]. Instead of frequency being plotted on the vertical axis, frequency density is plotted.
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
Frequency density =
𝑐𝑙𝑎𝑠𝑠 𝑤𝑖𝑑𝑡
Mean:
The mean of a series of numbers is obtained by adding the numbers and dividing the result by the
number of numbers.
Mean =
fx
f
where ∑ fx means ‘the sum of the products’
i.e. ∑ (number × frequency)
and ∑f means ‘ the sum of the frequencies’.
Median:
The median of a series of numbers is obtained by arranging the numbers in ascending order and then
choosing the number in the ‘middle’. If there are two ‘middle’ numbers the median is the average
(mean) of these two numbers.
Mode:
The mode of a series of numbers is simply the number which occurs most often.
Frequency tables:
A frequency table shows a number x such as a score or a mark, against the frequency f or number of
times that x occurs.
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22. Cumulative frequency:
Cumulative frequency is the total frequency up to a given point.
Cumulative frequency Curve:
A cumulative frequency curve shows the median at the 50 th percentile of the cumulative frequency.
The value at the 25th percentile is known as the lower quartile and that at the 75 th percentile as the
upper quartile.
A measure of the spread or dispersion of the data is given by the inter-quartile range where
inter-quartile range = upper quartile – lower quartile.
Probability:
Probability is the study of chance, or the likelihood of an event happening.
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒
𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
Probability of an event =
If the probability = 0 it implies the event is impossible
If the probability = 1 it implies the event is certain to happen.
All probabilities lie between 0 and 1.
Probabilities are written using fractions or decimals.
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑞𝑢𝑎𝑙𝑙𝑦 𝑙𝑖𝑘𝑒𝑙𝑦 𝑜𝑢𝑡𝑐𝑜𝑚 𝑒
Exclusive events:
Two events are exclusive if they cannot occur at the same time.
The OR Rule:
For exclusive events A and B
p(A or B) = p(A) + p(B)
Independent events:
Two events are independent if the occurrence of one even is unaffected by the occurrence of the other.
The AND Rule:
p(A and B) = p(A) × p(B)
where p(A) = probability of A occurring
p(B) = probability of B occurring
Tree diagrams:
A tree diagram is a diagram used to represent probabilities when two or more events are combined.
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