This document provides a summary of basic algebra concepts for entrepreneurs and social entrepreneurship participants. It includes example algebra problems with step-by-step solutions on topics like equations, factors, exponents, and scientific notation. It also provides links to download additional educational resources on algebra, statistics, reasoning, and other relevant subjects.
This document provides an introduction to rounding methods and tricks for squaring numbers. It explains how to square numbers by rounding them to the nearest multiple of 10 or 100 and then using the properties of exponents. Several examples are shown of squaring two-digit numbers like 45, 73, 58, 105, and 428. The key steps are to round the number to the nearest decade, express the square as the rounded number multiplied by itself plus the difference squared. This technique breaks down the square into easier to calculate components. In the examples, the rounded square is calculated first before adding the difference squared at the end.
This material is for PGPSE / CSE students of AFTERSCHOOOL. PGPSE / CSE are free online programme - open for all - free for all - to promote entrepreneurship and social entrepreneurship
This document discusses factorizing quadratic expressions. It provides examples of expanding and factorizing expressions of the form (x + a)(x + b). A pattern is observed where the constant term is the product of a and b, and the coefficient of x is the sum of a and b. Students are asked to factorize additional quadratic expressions using this pattern.
This document contains solutions to 100 equations of the first degree. The equations involve variables like x and y, and involve operations like addition, subtraction, multiplication and division. Each equation is presented along with its corresponding solution (e.g. x=7).
The document discusses complex roots of the characteristic equation arising from assuming exponential solutions to a differential equation. It shows that complex roots lead to complex-valued solutions, but linear combinations of solutions can give real-valued solutions in the form of sine and cosine functions. Several examples are worked out to find the general solution of differential equations and determine the time for the solution to drop below a given value based on its graph.
Algebra is a branch of mathematics that studies structure, relations, and quantities. The quadratic formula provides a method for solving quadratic equations of the form ax^2 + bx + c = 0 by using the coefficients a, b, and c. There are three main methods for solving quadratic equations: factoring, completing the square, and using the quadratic formula.
This document contains solutions to equations of 1st and 2nd degree. For equations of 1st degree (problems 1-80), it provides the step-by-step work to solve each equation for the variable. For equations of 2nd degree (problems 110-162), it states the solutions for each quadratic equation in factored form.
This document provides an introduction to rounding methods and tricks for squaring numbers. It explains how to square numbers by rounding them to the nearest multiple of 10 or 100 and then using the properties of exponents. Several examples are shown of squaring two-digit numbers like 45, 73, 58, 105, and 428. The key steps are to round the number to the nearest decade, express the square as the rounded number multiplied by itself plus the difference squared. This technique breaks down the square into easier to calculate components. In the examples, the rounded square is calculated first before adding the difference squared at the end.
This material is for PGPSE / CSE students of AFTERSCHOOOL. PGPSE / CSE are free online programme - open for all - free for all - to promote entrepreneurship and social entrepreneurship
This document discusses factorizing quadratic expressions. It provides examples of expanding and factorizing expressions of the form (x + a)(x + b). A pattern is observed where the constant term is the product of a and b, and the coefficient of x is the sum of a and b. Students are asked to factorize additional quadratic expressions using this pattern.
This document contains solutions to 100 equations of the first degree. The equations involve variables like x and y, and involve operations like addition, subtraction, multiplication and division. Each equation is presented along with its corresponding solution (e.g. x=7).
The document discusses complex roots of the characteristic equation arising from assuming exponential solutions to a differential equation. It shows that complex roots lead to complex-valued solutions, but linear combinations of solutions can give real-valued solutions in the form of sine and cosine functions. Several examples are worked out to find the general solution of differential equations and determine the time for the solution to drop below a given value based on its graph.
Algebra is a branch of mathematics that studies structure, relations, and quantities. The quadratic formula provides a method for solving quadratic equations of the form ax^2 + bx + c = 0 by using the coefficients a, b, and c. There are three main methods for solving quadratic equations: factoring, completing the square, and using the quadratic formula.
This document contains solutions to equations of 1st and 2nd degree. For equations of 1st degree (problems 1-80), it provides the step-by-step work to solve each equation for the variable. For equations of 2nd degree (problems 110-162), it states the solutions for each quadratic equation in factored form.
College algebra real mathematics real people 7th edition larson solutions manualJohnstonTBL
This document contains information about the College Algebra Real Mathematics Real People 7th Edition Larson textbook including:
- A link to download the solutions manual and test bank for the textbook
- An overview of the content covered in Chapter 2 on solving equations and inequalities, including linear equations, identities, conditionals, and more.
- 51 example problems from Chapter 2 with step-by-step solutions.
This document provides examples and explanations of operations involving polynomials and rational expressions. It covers factoring polynomials, evaluating polynomial expressions, adding, subtracting, multiplying and dividing rational expressions, and simplifying complex fractions and expressions with radicals. Step-by-step solutions are shown for problems such as factoring expressions, evaluating polynomials for given values, combining like rational expressions, rationalizing denominators, and more. The document demonstrates various techniques for working with polynomials and rational expressions.
The document provides examples of factoring polynomials completely. It begins by outlining guidelines for factoring polynomials completely, such as factoring out the greatest common monomial factor, looking for differences of squares or perfect square trinomials, and factoring trinomials and polynomials with four terms by grouping. It then works through examples of factoring various polynomials completely by applying these guidelines. These include factoring trinomials, solving a polynomial equation, and solving a multi-step problem involving factoring a polynomial to find dimensions.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and finding the nearest integer of a difference. For algebra, it solves systems of equations and determines values based on given equations. For geometry, it calculates areas and volumes. For probability, it finds probabilities of arrangements and outcomes of dice rolls and ball draws.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and nearest integers. For algebra, it solves systems of equations and determines values based on relationships between variables. For geometry, it calculates areas and volumes. For probability, it determines probabilities of events occurring based on arrangements and selections from sets.
The document provides examples of solving mathematical expressions and equations using order of operations (BODMAS) and other simplification rules. It includes 25 word problems with step-by-step solutions showing the calculations and reasoning. The problems cover topics like percentages, ratios, proportions, time/work problems and involve setting up and solving equations. The document aims to help students practice simplifying complex expressions and solving different types of mathematical word problems.
Factors of po lynomials + solving equationsShaun Wilson
This document discusses factorizing polynomials of degree 3 or higher using the factor theorem or "The Big L" method. It provides examples of factorizing polynomials and using the factors to find the roots or solutions of polynomial equations. The examples show setting a polynomial equal to 0, finding a factor using the factor theorem, fully factorizing the polynomial, and then setting each factor equal to 0 to obtain the roots. The document emphasizes that the factor theorem can be used to determine if an expression is a factor if the remainder is 0 upon dividing the polynomial by the expression.
1) The document discusses synthetic division, which is a method for dividing polynomials without using variables.
2) It provides an example of using synthetic division to determine if 1 is a root of the polynomial 4x - 3x + x + 5.
3) Another example uses synthetic division to find the quotient and remainder of (4x - 7x - 11x + 5) divided by (4x - 5).
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.
This document introduces integration as the reverse process of differentiation. It establishes the rule that if dy/dx = axn, then the integral is y = axn+1/(n+1) + c, where c is the constant of integration. Several examples are worked out applying this rule. The document also introduces notation for integration, including the integral sign ∫ and discusses some cases where the power is not a whole number. It emphasizes that the power in the integral must be positive.
This document provides the answers to a mathematics exam for 10th grade students. It includes multiple choice questions with explanations and word problems with step-by-step solutions. The topics covered include algebra, logarithms, quadratic equations, and inequalities.
The document discusses the proper order of mathematical operations known as PEMDAS. It covers absolute values, addition and subtraction of signed numbers, multiplication and division of signed numbers, and the order of operations using PEMDAS. Examples are provided to illustrate how to use PEMDAS to simplify expressions involving multiple operations. Exercises with answers are included to help readers practice applying these concepts.
The document introduces key concepts in algebra including variables, constants, types of numbers (counting, integers, rational, irrational, real), graphs, averages, and positive and negative numbers. It provides examples and guidelines for understanding these concepts. Variables represent quantities that can vary, while constants represent fixed values. Different number sets are explained and visualized on a number line. Averages are calculated by adding values and dividing by the total count. Positive numbers are greater than zero, while negative numbers are less than zero.
The document contains a series of algebra problems involving simplifying and factorizing expressions. The problems cover topics like combining like terms, distributing terms, factorizing quadratics and binomials, and multiplying polynomials.
1. The document provides 20 number series questions with multiple choice answers. It tests the ability to recognize patterns in numeric sequences and determine the next number in the series.
2. For each question, the number series is presented and the test taker must determine if the numbers are increasing or decreasing, and by what amount, to arrive at the next number.
3. An answer key is provided with explanations for the pattern in each series and the correct answer. The questions cover a variety of common numeric patterns like addition, subtraction, multiplication, and division series.
The document provides examples of converting between different units of measurement for capacity, weight, temperature, and more. It includes conversions between cups, quarts, gallons, ounces, pounds, grams, kilograms, Celsius and Fahrenheit. Step-by-step calculations are shown for converting between units like converting between pounds and ounces, kilograms and pounds, and between Celsius and Fahrenheit temperatures. Conversion factors are also provided for different units.
Length of each side = 4 units
To find the area of a square, multiply the length of one side by itself.
Area of square = Length x Length
= 4 x 4
= 16 square units
Therefore, the area of the square is 16 square units.
Mathematics Quiz on Area and Perimeter for Class 7LAL CHAND GOYAL
Power point presentation made by Arnav Goyal, class-7 student, Amity International School, Sector-43,Gurgaon, for the benefit of the students who want to learn area and perimeter for competitive exams.
This document provides information about converting between metric and customary units of measurement for length, volume, weight, and time. It includes examples of conversions between units like centimeters to inches, meters to feet, kilograms to pounds, minutes to hours, and provides conversion factors. Interactive questions and answers are provided to help the reader practice unit conversions.
This free teaching resource is from Innovative Teaching Resources. You can access hundreds of their excellent resources here. https://www.teacherspayteachers.com/Store/Innovative-Teaching-Ideas
Lesson plans and teaching
This document provides an overview of linear algebra concepts including vectors, matrices, and matrix decompositions. It begins with definitions of vectors as ordered tuples of numbers that represent quantities with magnitude and direction. Vectors are elements of vector spaces, which are sets that satisfy properties like closure under addition and scalar multiplication. The document then discusses linear independence, bases, norms, inner products, orthonormal bases, and linear operators. It concludes by stating that these concepts will be applied to image compression.
College algebra real mathematics real people 7th edition larson solutions manualJohnstonTBL
This document contains information about the College Algebra Real Mathematics Real People 7th Edition Larson textbook including:
- A link to download the solutions manual and test bank for the textbook
- An overview of the content covered in Chapter 2 on solving equations and inequalities, including linear equations, identities, conditionals, and more.
- 51 example problems from Chapter 2 with step-by-step solutions.
This document provides examples and explanations of operations involving polynomials and rational expressions. It covers factoring polynomials, evaluating polynomial expressions, adding, subtracting, multiplying and dividing rational expressions, and simplifying complex fractions and expressions with radicals. Step-by-step solutions are shown for problems such as factoring expressions, evaluating polynomials for given values, combining like rational expressions, rationalizing denominators, and more. The document demonstrates various techniques for working with polynomials and rational expressions.
The document provides examples of factoring polynomials completely. It begins by outlining guidelines for factoring polynomials completely, such as factoring out the greatest common monomial factor, looking for differences of squares or perfect square trinomials, and factoring trinomials and polynomials with four terms by grouping. It then works through examples of factoring various polynomials completely by applying these guidelines. These include factoring trinomials, solving a polynomial equation, and solving a multi-step problem involving factoring a polynomial to find dimensions.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and finding the nearest integer of a difference. For algebra, it solves systems of equations and determines values based on given equations. For geometry, it calculates areas and volumes. For probability, it finds probabilities of arrangements and outcomes of dice rolls and ball draws.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and nearest integers. For algebra, it solves systems of equations and determines values based on relationships between variables. For geometry, it calculates areas and volumes. For probability, it determines probabilities of events occurring based on arrangements and selections from sets.
The document provides examples of solving mathematical expressions and equations using order of operations (BODMAS) and other simplification rules. It includes 25 word problems with step-by-step solutions showing the calculations and reasoning. The problems cover topics like percentages, ratios, proportions, time/work problems and involve setting up and solving equations. The document aims to help students practice simplifying complex expressions and solving different types of mathematical word problems.
Factors of po lynomials + solving equationsShaun Wilson
This document discusses factorizing polynomials of degree 3 or higher using the factor theorem or "The Big L" method. It provides examples of factorizing polynomials and using the factors to find the roots or solutions of polynomial equations. The examples show setting a polynomial equal to 0, finding a factor using the factor theorem, fully factorizing the polynomial, and then setting each factor equal to 0 to obtain the roots. The document emphasizes that the factor theorem can be used to determine if an expression is a factor if the remainder is 0 upon dividing the polynomial by the expression.
1) The document discusses synthetic division, which is a method for dividing polynomials without using variables.
2) It provides an example of using synthetic division to determine if 1 is a root of the polynomial 4x - 3x + x + 5.
3) Another example uses synthetic division to find the quotient and remainder of (4x - 7x - 11x + 5) divided by (4x - 5).
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.
This document introduces integration as the reverse process of differentiation. It establishes the rule that if dy/dx = axn, then the integral is y = axn+1/(n+1) + c, where c is the constant of integration. Several examples are worked out applying this rule. The document also introduces notation for integration, including the integral sign ∫ and discusses some cases where the power is not a whole number. It emphasizes that the power in the integral must be positive.
This document provides the answers to a mathematics exam for 10th grade students. It includes multiple choice questions with explanations and word problems with step-by-step solutions. The topics covered include algebra, logarithms, quadratic equations, and inequalities.
The document discusses the proper order of mathematical operations known as PEMDAS. It covers absolute values, addition and subtraction of signed numbers, multiplication and division of signed numbers, and the order of operations using PEMDAS. Examples are provided to illustrate how to use PEMDAS to simplify expressions involving multiple operations. Exercises with answers are included to help readers practice applying these concepts.
The document introduces key concepts in algebra including variables, constants, types of numbers (counting, integers, rational, irrational, real), graphs, averages, and positive and negative numbers. It provides examples and guidelines for understanding these concepts. Variables represent quantities that can vary, while constants represent fixed values. Different number sets are explained and visualized on a number line. Averages are calculated by adding values and dividing by the total count. Positive numbers are greater than zero, while negative numbers are less than zero.
The document contains a series of algebra problems involving simplifying and factorizing expressions. The problems cover topics like combining like terms, distributing terms, factorizing quadratics and binomials, and multiplying polynomials.
1. The document provides 20 number series questions with multiple choice answers. It tests the ability to recognize patterns in numeric sequences and determine the next number in the series.
2. For each question, the number series is presented and the test taker must determine if the numbers are increasing or decreasing, and by what amount, to arrive at the next number.
3. An answer key is provided with explanations for the pattern in each series and the correct answer. The questions cover a variety of common numeric patterns like addition, subtraction, multiplication, and division series.
The document provides examples of converting between different units of measurement for capacity, weight, temperature, and more. It includes conversions between cups, quarts, gallons, ounces, pounds, grams, kilograms, Celsius and Fahrenheit. Step-by-step calculations are shown for converting between units like converting between pounds and ounces, kilograms and pounds, and between Celsius and Fahrenheit temperatures. Conversion factors are also provided for different units.
Length of each side = 4 units
To find the area of a square, multiply the length of one side by itself.
Area of square = Length x Length
= 4 x 4
= 16 square units
Therefore, the area of the square is 16 square units.
Mathematics Quiz on Area and Perimeter for Class 7LAL CHAND GOYAL
Power point presentation made by Arnav Goyal, class-7 student, Amity International School, Sector-43,Gurgaon, for the benefit of the students who want to learn area and perimeter for competitive exams.
This document provides information about converting between metric and customary units of measurement for length, volume, weight, and time. It includes examples of conversions between units like centimeters to inches, meters to feet, kilograms to pounds, minutes to hours, and provides conversion factors. Interactive questions and answers are provided to help the reader practice unit conversions.
This free teaching resource is from Innovative Teaching Resources. You can access hundreds of their excellent resources here. https://www.teacherspayteachers.com/Store/Innovative-Teaching-Ideas
Lesson plans and teaching
This document provides an overview of linear algebra concepts including vectors, matrices, and matrix decompositions. It begins with definitions of vectors as ordered tuples of numbers that represent quantities with magnitude and direction. Vectors are elements of vector spaces, which are sets that satisfy properties like closure under addition and scalar multiplication. The document then discusses linear independence, bases, norms, inner products, orthonormal bases, and linear operators. It concludes by stating that these concepts will be applied to image compression.
This document discusses units and measurements conversions. It explains that prefixes like centi and milli indicate fractions of base units and conversions can be done using ratios of equivalent units. For example, 1 inch is equal to 2.54 centimeters. It also notes that algebraic manipulation of units is similar to variables, where units cancel out in operations like division. Rounding may be needed in conversions to avoid exact answers. Practice problems are provided to convert between units like centimeters and inches using equivalent ratios.
The document discusses units and measurement. It begins by providing examples of the longest bridges in Malaysia and worldwide, and the tallest building in Malaysia and worldwide. It then introduces the International System of Units (SI) and its seven base units: kilogram, meter, second, ampere, kelvin, mole, and candela. Several derived units are also described such as area, volume, velocity, and acceleration. Prefixes used with SI units are defined, ranging from deca to yocto. Methods for converting between units are demonstrated, such as kilometers to meters and feet. Examples of solving unit conversion problems are provided.
This presentation teaches how to perform basic fraction operations:
- Adding fractions requires having a common denominator
- Subtracting and multiplying fractions also require a common denominator and use similar processes as addition and multiplication of whole numbers
- Dividing fractions involves flipping the second fraction and turning division into multiplication
- To get a common denominator when adding or subtracting, multiply the denominators and adjust the numerators proportionately
The document discusses various units of measurement for length, volume, mass, and temperature in both the metric and imperial systems. It provides examples to convert between units and explains how to measure quantities using tools like rulers, graduated cylinders, balances, and thermometers. Key metric units include meters, centimeters, millimeters, liters, milliliters, grams, and degrees Celsius.
Introductory Physics - Physical Quantities, Units and MeasurementSutharsan Isles
This document provides an introduction to physical quantities, units, and measurement in physics. It begins with definitions of key terminology like physical property, scalar and vector quantities, and standard form notation. It then discusses the International System of Units (SI) including the seven base units, common prefixes, and how to convert between multiples and submultiples of units. The document also covers derived SI units and examples of converting between derived units. It emphasizes the importance of understanding whether a quantity is scalar or vector.
This document provides a summary of concepts related to surds and indices in business mathematics. It includes example problems and solutions involving operations with surds, indices, and fractions with surds and indices. The document was created by Dr. T.K. Jain for students of CSE and PGPSE programs. It encourages students to prepare well for exams and provides links to download additional educational resources.
This material is for PGPSE / CSE students of AFTERSCHOOOL. PGPSE / CSE are free online programme - open for all - free for all - to promote entrepreneurship and social entrepreneurship PGPSE is for those who want to transform the world. It is different from MBA, BBA, CFA, CA,CS,ICWA and other traditional programmes. It is based on self certification and based on self learning and guidance by mentors. It is for those who want to be entrepreneurs and social changers. Let us work together. Our basic idea is that KNOWLEDGE IS FREE & AND SHARE IT WITH THE WORLD
The document discusses square roots, decimals, and number systems. It provides download links for educational materials on topics like permutations, combinations, differentiation, integration, and unitary methods. It encourages working together to promote education and entrepreneurship for all.
This material is for PGPSE / CSE students of AFTERSCHOOOL. PGPSE / CSE are free online programme - open for all - free for all - to promote entrepreneurship and social entrepreneurship PGPSE is for those who want to transform the world. It is different from MBA, BBA, CFA, CA,CS,ICWA and other traditional programmes. It is based on self certification and based on self learning and guidance by mentors. It is for those who want to be entrepreneurs and social changers. Let us work together. Our basic idea is that KNOWLEDGE IS FREE & AND SHARE IT WITH THE WORLD
The document discusses questions related to decimals, fractions, and quantitative aptitude tests. It provides examples of questions on topics like square roots, scientific notation, and permutations and combinations. It also provides links to download presentations on related topics like profit and loss, ratios, and business mathematics.
This document provides an outline of topics in algebra including: indices, expanding single and double brackets, substitution, solving equations, solving equations from angle problems, finding the nth term of sequences, simultaneous equations, inequalities, factorizing using common factors, quadratics, grouping and the difference of two squares. It also includes examples and explanations for each topic.
This document provides practice questions and tips in business mathematics. It contains multiple choice questions related to topics like ratios, percentages, profit and loss, time and work, averages, simple and compound interest, discounts, and permutations and combinations. The questions are intended to help students prepare for competitive exams in subjects like commerce and management.
This document provides practice questions and tips in business mathematics. It contains multiple choice questions related to topics like ratios, percentages, profit and loss, time and work, averages, simple and compound interest, discounts, and permutations and combinations. The questions are intended to help students prepare for competitive exams in subjects like commerce and management.
This document provides a review of various algebra topics including: indices, expanding single and double brackets, substitution, solving equations, solving equations from angle problems, finding the nth term of sequences, simultaneous equations, inequalities, factorizing using common factors, quadratics, grouping and the difference of two squares. It also includes examples to practice each topic.
This document provides an overview of topics covered in intermediate algebra revision including: collecting like terms, multiplying terms, indices, expanding single and double brackets, substitution, solving equations, finding nth terms of sequences, simultaneous equations, inequalities, factorizing common factors and quadratics, solving quadratic equations, rearranging formulas, and graphing curves and lines. The document contains examples and practice problems for each topic.
The document provides 12 examples of solving problems involving surds and indices. It covers laws of indices, definitions and laws of surds, and examples of simplifying expressions with surds and indices, evaluating expressions, and determining relative sizes of surds. The examples progress from basic operations to more complex multi-step problems.
This document provides information about polynomial operations including:
1) Defining polynomials as algebraic expressions involving integer powers of a variable and real number coefficients.
2) Examples of adding, subtracting, and multiplying polynomials by using vertical or FOIL methods.
3) Important formulas for polynomial operations such as (a + b)(a - b) = a2 - b2 and (a + b)2 = a2 + 2ab + b2.
4) Worked examples of applying these formulas and methods to polynomials involving single and multiple variables.
Here are the steps to solve this equation by factorising:
1) Factorise the left hand side: 3(x - 1)(x + 2)
2) Set each factor equal to 0:
x - 1 = 0
x + 2 = 0
3) Solve for x:
x = 1
x = -2
Therefore, the solutions are x = 1 or x = -2.
1. The document shows methods for calculating the area of rectangles by splitting them into smaller rectangles.
2. It demonstrates that the area of the original rectangle equals the sum of the areas of the smaller rectangles.
3. Algebraic formulas are developed to represent splitting rectangles and multiplying sums and differences.
1. The document provides various shortcuts and methods for solving problems involving numbers and operations like multiplication, division, finding sums, squares, cubes etc.
2. Shortcuts are given for multiplying multi-digit numbers, finding sums of series where digits are repeated, evaluating expressions with decimals where one number is repeated, and determining properties of squares, cubes, primes and other numbers.
3. Various methods are outlined for testing divisibility by different numbers, finding highest common factors and lowest common multiples, algebraic identities and other quantitative reasoning concepts.
The document provides examples and step-by-step workings for evaluating algebraic expressions by substituting number values for variables. It demonstrates evaluating expressions including powers, integers, fractions, and evaluating expressions to find volumes. The examples show applying order of operations and substitution to simplify expressions down to a single numerical value.
in
I. The rule for solving expressions (BODMAS) is explained, with brackets having the highest priority, followed by division, multiplication, addition and subtraction.
II. The modulus of a real number a is defined as |a|=a if a>0 and |a|=-a if a<0.
III. When an expression contains a vinaculum (bar), the expression under the bar is simplified first before applying BODMAS.
The document discusses evaluating expressions using the proper order of operations. It provides the mnemonic device PEMDAS (Parentheses, Exponents, Multiply, Divide, Add, Subtract) to remember the correct sequence. Several examples are worked through step-by-step to demonstrate applying the order of operations. The objective is for students to be able to evaluate expressions like 7 + 4 * 3 as 19, not 33, by following PEMDAS.
Factoring polynomials involves finding common factors that can be divided out of terms, similar to factoring numbers but with variables; this is done by looking for a single variable or number that is a common factor of all terms that can be pulled out in front of parentheses. The document provides examples of different types of factoring polynomials including using the greatest common factor, difference of squares, grouping, and perfect squares and cubes.
This document provides an overview of quadratic equations and inequalities. It defines quadratic equations as equations of the form ax2 + bx + c = 0, where a, b, and c are real number constants and a ≠ 0. Examples of quadratic equations are provided. Methods for solving quadratic equations are discussed, including factoring, completing the square, and the quadratic formula. Properties of inequalities are outlined. The chapter also covers solving polynomial and rational inequalities, as well as equations and inequalities involving absolute value. Practice problems are included at the end.
Examination reforms are essential to transform the education system according to the document. The current examination system focuses only on rote memorization but needs to evaluate creativity and problem-solving. The document outlines steps to reform examinations including setting goals based on program and course objectives, evaluating whether objectives are achieved through direct and indirect methods, using continuous evaluations, and adopting open book exams and multiple evaluation methods.
1. BASIC ALGEBRA FOR ENTREPRENEURS by : DR. T.K. JAIN AFTERSCHO ☺ OL centre for social entrepreneurship sivakamu veterinary hospital road bikaner 334001 rajasthan, india FOR – PGPSE / CSE PARTICIPANTS [email_address] mobile : 91+9414430763
2. My words..... My purpose here is to give a few questions on fundamentals of Algebra . I welcome your suggestions. I also request you to help me in spreading social entrepreneurship across the globe – for which I need support of you people – not of any VIP. With your help, I can spread the ideas – for which we stand....
3. If (x^3 – 1/x^3)= 14, what is (x-1/x) ? Options : 5,4,3,2 let us assume 1/x = y x^3 -y^3 = 14 (x-y)^3 = x^3 -3x^2y+3xy^2-y^3 (x-y)^3 =14 -3xy(x-y) (x-y)^3 +3xy(x-y)=14 (x-Y)^3 +3(x-y) – 14=0 (we can cancel xy, as y is 1/x) at this stage put the options, when you put x-y = 2, you are able to solve it so answer is 2
4. What must be added to 1/x to make it equal to x ? X – 1/x = (x^2 -1) / x answer
9. X^3 -3x^2+4x+k contains (x+6) as factor, what is the value of k ? X+6 = 0, so x = -6 Take x=-6 -216 -108 -24 +k = 0 k = 348 answer
10. (x+2) and (x-1) are factors of (x^3+10x^2+mx+n), what is the value of m and n ? Options : 5,-3 17,-8, 23,-19, 7,-18 solution : take x = 1 1+10+mx+n or m+n = -11 take x = -2 -8+40-2m+n or -2m+n = -32 or 3n=-54 or n=-18 so last option is correct
11. If 5^x /125 = 1, what is X ? We know that 5^3 = 125, 125/125=1, so x=3 answer
12. If 3^x – 3^(x-1) = 18, what is x^x Options : 3,8,27,216 try with 27, x should be 3 let us put the value of x in the equation we find : 27 – 9 = 18 so X = 3 and x^x = 27 answer
13. If 2^x – 2^(x-1) = 16, then what is the value of x^4 ? Options : 16, 81, 256, 625 let us try with option 625, here x = 5 so put the value of X in the equation, we get : 32 – 16 = 16 so x= 5, and x^4 = 625 answer
14. If (a+b+c) = 0, then a^3+b^3+c^3 is equal to what ? Options : 0, 3abc, abc, (ab+bc+ac) answer : = 3abc formula : (a+b+c)=0 then a^3+b^3+c^3 = 3abc
15. a+b = 5, and ab = 6, what is (a^3+b^3) ? (a-b)^2 = (a+b)^2 – 4ab ab =6, a+b = 5 (a-b)^2=25-24 = 1 (a+b)^3 = a^3 +b^3 +3a^2b+3ab^2 or : (a+b)^3 = a^3 +b^3 +3ab(a+b) (5)^3 = a^3+b^3 +3*6(5) 125 – 90 = a^3+b^3 a^3+b^3 = 35 ans
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17. What will you get from (.4)^(1/2) ? Options : 1. .2 2. .-2 3. .02 4 none of these answer : square root of .4 will be close to .6, so answer is none of these
18. What will you get from (.8)^(1/3) ? a. .2 b .02 c .04 d none of these answer : answer will be more than .9, so answer is none of these
19. What will you get from the following : (.9)^(1/2) a. .3 b. .03 c. between .8 and .9 d. none of these answer : none of these
20. Due to clerical mistake , + is written as *, / is written as - , * is written as +, - is written as =. what will you get from this : 2 of 2 =4 * 6- 3 (2+4-2) Solution : first remove clerical mistake : 2 of 2-4 + 6 / 3 (2*4/2) first solve bracket : 2 of 2-4+ 6 /3 * 4 solve of 4 -4+2 * 4 now solve multiply 4 – 4 + 8 = 8 answer
21. What should we add or subtract or multiply or divide from 13333 to make it a perfect square ? Find minimum digit If we add 125, we get : 13458, which is square of 116 if we deduct 108, we get 13225 so we should deduct 108
23. Which of these is highest : 2/5, 15/18 , 223/226, 1202/1205 ? Here difference between denominator and numerator is same in each case, so the answer will be 1202 / 1205 answer
24. What is the .the sum of first 20 terms of series is 1/5*6 +1/6*7+1/7*8----- We can write it as : 1/30, 1/42, 1/56 we can write it as (1/5-1/6)+(1/6 – 1/7)+ (1/7-1/8) . . . . here you can see that in first bracket we have -1/6 and in 2 nd bracket we have + 1/6 and so on.. when we open bracket, we get : 1/5+1/25 we get 6/25 answer
25. The least among the following :- a. 0.2 b.1/0.2 c. 0.3^2 d. 0.22 a. 0.2 b.1/0.2 = 5 c. 0.3^2 =.09 d. 0.22 so the answer = C
26. If 1/3.718 =0.2689,then find value of 1/0.0003718 ? =2689 just shift the decimals as per the question
34. What will you get from : .3879999999.......? Here 9 is repeating and 387 is non-repeating deduct 387 from the number and divide by 9000 we use as many 9 as there are repeating digits we use as many 0 as we have non-repeatig digits. =(3879-387)/(9000) =3492 / 9000 answer
35. What will you get from .358585858.....? Here 58 are being repeated we will deduct 3 (non repeating) from 358 and divide it by 990 (as many 9 as there are repeating digits and as many 0 as there are non repeating digits) = (358 – 3) / (990) =355/990 =71/198 answer
36. Put these in ascending order : 2/3, 25/39, 18/-19, 55/70, .04555..? Ascending means from small to big the smallest number is 18/-19 (as it is in negative). Then we have .04555..... now compare 2/3 and 25/39, in order to compare these, let us multiply the numerators of these by denominators of the other numbers, we get : 78 and 75, the bigger number is 78 so 2/3 is bigger. Now compare it with 55/70. We get : 140 and 165, 165 is bigger so final order : -18/19, .45555.., 25/39, 2/3, 55/70 answer
37. What will you get from (91.9432)^2 – (41.9432)^2 ? (a^2 – b^2) = (a-b) * ( a+b) = (91.9432 – 41.9432) * ( 91.9432+41.9432) (50) * (133.8864) =6694.32 answer
39. What is the remainder when (2x^3 +5x^2 -4x -6) is divided by (2x -1) ? 2x-1 = 0 so x = ½ or .5 put this value in the equation .25+1.25-2-6 =1.5 – 8 = -6.5 answer
40. Which of these is / are factor of (x^2 +4) ? (x+2) , (x-2) , (x^2 -2), (none of these ) answer = none of these
41. Which of these is a factor of (X^3 -5x^2+8x-4) ? (x-1), (x-2)^2, (x+2), (x+1) , (none) answer = (x-1)
42. What is the remainder when (x^11 +1) is divided by (x+1) ? Answer = zero
43. (x+1/x) = 3, what is the value of (x^6 +1/x^6) ? Options : 927, 414, 364, 322 let us assume y = 1/x find its cube : x^3+3x^2y+3xy^2+y^3 = 27 =x^3+y^3 + 9 or x^3+y^3 = 18 now square it : x^6+2x^3y^3+y^6 = 324 (we can cancel x^3 and y^3) x^6+y^6 = 324-2 = 322 answer
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