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The document contains two examples of maximum and minimum problems involving differentiation. Example 1 asks the reader to find the minimum volume of a cone given that a sphere must fit inside it. It is found that the minimum volume occurs when the radius of the cone is 28.577 cm. Example 2 involves finding the maximum volume of a cylinder inscribed in a sphere. The maximum volume is calculated to be 104,000 cm3, occurring when the height of the cylinder is 28.5 cm. The document provides guidance on solving maximum and minimum problems using differentiation, illustrated through these two examples involving geometric shapes.

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Sec 3 A Maths Notes Indices

1. The document discusses solving exponential equations with one, two, or three terms using properties of exponents such as changing bases to the same term and equating powers.
2. Examples are provided for solving two-term exponential equations by making the bases equal and equations with three terms by substituting variables, changing bases to the same term, and equating powers.
3. Solving exponential equations as products using properties such as treating exponents as multipliers is also demonstrated through examples.

Sec 3 E Maths Notes Coordinate Geometry

This document provides examples and explanations of using the distance formula and equations of lines in coordinate geometry. It defines the distance formula and shows how to calculate the distance between two points with given coordinates. It also demonstrates how to determine the gradient and y-intercept of a line given its equation, find the equation of a line given the gradient and a point or two points, and find values related to lines parallel or intersecting given lines.

Sec 3 A Maths Notes Indices

1) The document provides examples of solving exponential equations with various methods depending on whether the equation has two terms, three or more terms, or involves indices as products or quotients.
2) Key steps include splitting equations, letting one term equal a variable, raising both sides to the same power, and changing all terms to have the same base before equating exponents.
3) Examples range from simple equations like 82=x to more complex ones involving subtraction, addition, and multiplication of terms with different bases and exponents like (23)3=x+2-x.

Complex Numbers 1 - Math Academy - JC H2 maths A levels

The document provides lessons on complex numbers. It defines a complex number as being of the form z = x + iy, where x and y are real numbers. It discusses operations like addition, subtraction, multiplication and division of complex numbers. It also defines the complex conjugate and gives some examples of performing operations on complex numbers.

Form 5 Additional Maths Note

This document contains notes on additional mathematics including topics on progression, linear laws, integration, and vectors. Some key points:
- It discusses arithmetic and geometric progressions, defining the terms and formulas for finding terms and sums. Examples are worked through finding terms, sums, and differences between sums.
- Linear laws are explained including lines of best fit, converting between linear and non-linear forms using logarithms, and working through examples of finding equations from graphs.
- Integration techniques are outlined including formulas for integrals of powers, areas under and between curves, volumes of revolution, and the basic rules of integration. Worked examples find areas and volumes.
- Vectors are introduced including addition using the triangle

Chapter 2(limits)

This document provides information on calculating limits using limit laws and discusses one-sided limits and limits at infinity. It includes theorems on limit laws and examples of applying the laws to calculate limits. There are also 36 practice problems with answers provided to find specific limits algebraically or using limit laws for rational functions, functions with noninteger or negative powers, and limits approaching positive or negative infinity.

2 3 Bzca5e

The document discusses linear functions and slopes. It provides examples of finding the slope of a line between two points, writing the equation of a line in point-slope and slope-intercept form, graphing linear equations, finding the x- and y-intercepts of a line, and applications of linear functions including using a graphing calculator.

Nota math-spm

This document provides fully worked solutions to exam questions from Form 4 mathematics chapters on standard form, quadratic expressions and equations, sets, mathematical reasoning, the straight line, and statistics. The solutions include:
1) Detailed working to obtain the answers for multiple choice and structured questions.
2) Explanations of mathematical concepts and reasoning such as determining gradients, interpreting graphs, and identifying argument forms.
3) Step-by-step derivations to find equations of lines from given points and gradients.

Sec 3 A Maths Notes Indices

1. The document discusses solving exponential equations with one, two, or three terms using properties of exponents such as changing bases to the same term and equating powers.
2. Examples are provided for solving two-term exponential equations by making the bases equal and equations with three terms by substituting variables, changing bases to the same term, and equating powers.
3. Solving exponential equations as products using properties such as treating exponents as multipliers is also demonstrated through examples.

Sec 3 E Maths Notes Coordinate Geometry

This document provides examples and explanations of using the distance formula and equations of lines in coordinate geometry. It defines the distance formula and shows how to calculate the distance between two points with given coordinates. It also demonstrates how to determine the gradient and y-intercept of a line given its equation, find the equation of a line given the gradient and a point or two points, and find values related to lines parallel or intersecting given lines.

Sec 3 A Maths Notes Indices

1) The document provides examples of solving exponential equations with various methods depending on whether the equation has two terms, three or more terms, or involves indices as products or quotients.
2) Key steps include splitting equations, letting one term equal a variable, raising both sides to the same power, and changing all terms to have the same base before equating exponents.
3) Examples range from simple equations like 82=x to more complex ones involving subtraction, addition, and multiplication of terms with different bases and exponents like (23)3=x+2-x.

Complex Numbers 1 - Math Academy - JC H2 maths A levels

The document provides lessons on complex numbers. It defines a complex number as being of the form z = x + iy, where x and y are real numbers. It discusses operations like addition, subtraction, multiplication and division of complex numbers. It also defines the complex conjugate and gives some examples of performing operations on complex numbers.

Form 5 Additional Maths Note

This document contains notes on additional mathematics including topics on progression, linear laws, integration, and vectors. Some key points:
- It discusses arithmetic and geometric progressions, defining the terms and formulas for finding terms and sums. Examples are worked through finding terms, sums, and differences between sums.
- Linear laws are explained including lines of best fit, converting between linear and non-linear forms using logarithms, and working through examples of finding equations from graphs.
- Integration techniques are outlined including formulas for integrals of powers, areas under and between curves, volumes of revolution, and the basic rules of integration. Worked examples find areas and volumes.
- Vectors are introduced including addition using the triangle

Chapter 2(limits)

This document provides information on calculating limits using limit laws and discusses one-sided limits and limits at infinity. It includes theorems on limit laws and examples of applying the laws to calculate limits. There are also 36 practice problems with answers provided to find specific limits algebraically or using limit laws for rational functions, functions with noninteger or negative powers, and limits approaching positive or negative infinity.

2 3 Bzca5e

The document discusses linear functions and slopes. It provides examples of finding the slope of a line between two points, writing the equation of a line in point-slope and slope-intercept form, graphing linear equations, finding the x- and y-intercepts of a line, and applications of linear functions including using a graphing calculator.

Nota math-spm

This document provides fully worked solutions to exam questions from Form 4 mathematics chapters on standard form, quadratic expressions and equations, sets, mathematical reasoning, the straight line, and statistics. The solutions include:
1) Detailed working to obtain the answers for multiple choice and structured questions.
2) Explanations of mathematical concepts and reasoning such as determining gradients, interpreting graphs, and identifying argument forms.
3) Step-by-step derivations to find equations of lines from given points and gradients.

Integration SPM

The document contains examples of indefinite integrals of various functions:
1) Finding the antiderivatives of polynomials like 4x3 + 3x - 2.
2) Finding an antiderivative involving a differential equation like dy/dx = 4x3 - 4x.
3) Evaluating integrals involving rational functions like ∫(3 - 2/x2 + 6x3)dx.
4) Finding antiderivatives of expressions involving radicals like ∫((x2+3)2/x2)dx.
5) Solving differential equations and evaluating integrals using substitution.

Chapter 9- Differentiation Add Maths Form 4 SPM

This document provides an explanation of differentiation and examples of calculating limits and derivatives using the first principle definition of the derivative. It begins by defining the limit of a function and providing examples of evaluating limits. It then introduces the concept of the derivative as the slope of the tangent line to a curve and explains how to calculate derivatives using small changes in x and y. The document provides examples of finding derivatives using this first principle definition. It also discusses rules for deriving composite functions and products of polynomials. Exercises are provided throughout for students to practice differentiation.

Chapter 3 quadratc functions

The document provides information about quadratic functions including:
- The general form of a quadratic function is f(x) = ax2 + bx + c.
- A quadratic function has a minimum or maximum point which can be used to find the axis of symmetry.
- The relationship between the discriminant (b2 - 4ac) and the position of the graph is explained. If it is greater than 0, the graph cuts the x-axis at two points. If it is equal to 0, the graph touches the x-axis at one point. If it is less than 0, the graph does not cut or touch the x-axis.
- Quadratic inequalities can be solved by sketching

Polynomials Test Answers

This document contains a polynomials test with multiple questions:
1) Find the products of several polynomials, including (x-1)(x+3) and (x^2 - x - 1)(x + 7).
2) For a triangle with base x + 1 and height 3x, express the area in terms of x using the area formula A = 1/2bh.
3) Divide the polynomial (x^2 + 10x + 21) by (x + 3).

Notes and-formulae-mathematics

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.

Rumus matematik-tambahan

This document contains mathematical formulas and concepts from algebra, calculus, geometry, trigonometry, and statistics. Some key points include:
- Formulas for addition, subtraction, multiplication, and division of algebraic expressions.
- Rules for differentiation, integration, and limits in calculus.
- Formulas for triangle properties like area, distance between points, and midpoint.
- Trigonometric identities for sine, cosine, and tangent functions of single and summed angles.
- Statistical concepts like mean, standard deviation, binomial distribution, and normal distribution.

Algebra formulas

This document provides an algebra cheat sheet that summarizes many common algebraic properties, formulas, and concepts. It covers topics such as arithmetic operations, properties of inequalities and absolute value, exponent properties, factoring formulas, solving equations, graphing functions, and common algebraic errors. The cheat sheet is a concise 3-page reference for the basics of algebra.

KSSM Form 4 Additional Mathematics Notes (Chapter 1-5)

Project Leader: Lim Jun Hao
Associate Editor: Lai Zhi Jun
Associate Project Editor: Siaw Jia Qi
Senior Managing Editor: Lim Jun Hao
Production Coordinator: Ooi Ming Yang
Cover Design: Ooi Hian Gee
Cover image: Pixabay
A collaboration work by Faculty of Science, University of Malaya Applied
Mathematics undergraduate students.
Copyright © 2020 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. The work is published in electronic form only. For
information on obtaining permission for use of material in this work, please email us
at ooi_hiangee@yahoo.com.

Maths Revision Notes - IGCSE

This document provides notes on various mathematics topics for the IGCSE including: decimals and standard form, accuracy and error, powers and roots, ratio and proportion, and trigonometry. It includes examples and practice problems for each topic. The notes are intended to help with revision for IGCSE mathematics question papers and assessments.

Form 4 add maths note

This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
1. A function f maps objects to images. To find the inverse function f-1, change f(x) to y and solve for x in terms of y.
2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solve

Additional Mathematics Revision

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.

Matematika Kalkulus ( Limit )

1. The document contains examples of evaluating limits as the variable approaches certain values.
2. Several limits were found to be indeterminate forms that require further algebraic manipulation to find the limit.
3. Key observations were made about the behavior of functions as the variable approaches values like noticing a function approaches a certain value as the variable nears another value.

Skills In Add Maths

This document provides examples of solving equations, expanding and factorizing expressions, solving simultaneous equations, working with indices and logarithms. It includes over 100 problems across these topics for students to practice. The problems range in complexity from basic single-step equations to multi-part logarithmic expressions and systems of simultaneous equations.

Chapter 5 indices & logarithms

This document provides a summary of Chapter 5 on Indices and Logarithms from an Additional Mathematics textbook. It includes examples and explanations of:
1. Laws of indices such as addition, subtraction, multiplication and division of indices.
2. Converting expressions between index form and logarithmic form using common logarithms and other bases.
3. Applying the laws of logarithms including addition, subtraction, and change of base.
4. Solving equations involving indices and logarithms through appropriate applications of index laws and logarithmic properties.

Modul bimbingan add maths

The document is a mathematics textbook for Additional Mathematics Form 4. It covers topics on functions, simultaneous equations, quadratic equations, and quadratic functions. It contains examples and practice questions for students to work through with answers. The questions range from simple calculations to solving equations and inequalities involving quadratic expressions.

Online math tutors (1)

The document contains a 15 question multiple choice test on polynomials. It covers topics like finding the number of zeros of a polynomial from its graph, identifying polynomials based on their zeros, finding the sum and product of the zeros of a quadratic polynomial, and determining the remaining zeros if one zero is given. It also includes word problems involving dividing polynomials and finding zeros. The test is meant to assess students' conceptual understanding of polynomials and their ability to solve subjective questions similar to those in board exams.

Add maths module form 4 & 5

1. The document provides an overview of important topics covered in Form 4 and Form 5 mathematics. These include functions, quadratic equations, trigonometry, statistics, calculus, and coordinate geometry.
2. Examples of how to solve different types of problems are given for each topic, such as finding the sum and product of roots for quadratic equations or using rules of logarithms to simplify logarithmic expressions.
3. Strategies for solving problems involving concepts like differentiation, integration, progressions, and linear laws are outlined. Methods for finding volumes or areas under curves are also summarized briefly.

2016 10 mathematics_sample_paper_sa2_03_ans_z657f

This document contains the solutions to a sample paper for Class 10 Mathematics. Some of the key questions solved include:
- Finding the discriminant of a quadratic equation.
- Calculating the perimeter of a figure involving an arc of a circle.
- Using the volume formula to find the number of lead shots that will fill a solid cube.
- Calculating the rainfall given the volume of a cylindrical vessel.
- Solving simultaneous equations to find the coordinates of a point.
- Calculating probabilities related to drawing cards from a standard deck.
The document provides detailed step-by-step workings for 22 other mathematical problems covering a range of topics.

2016 10 mathematics_sample_paper_sa2_01_ans_z9gf4

This document contains the solutions to a mathematics sample paper for Class 10. It includes solutions to 26 multiple choice questions on topics like geometry, trigonometry, and algebra. It also provides step-by-step workings for 4 multi-part questions involving concepts such as ratios, proportions, constructions, and coordinate geometry. Overall, the document analyzes solutions to 30 questions from a CBSE Class 10 math exam.

Exams in college algebra

This document contains a 25 question practice exam for college algebra. It covers topics such as solving linear, quadratic, and rational equations; simplifying expressions; factoring polynomials; graphing lines and parabolas; and solving systems of equations. The exam is broken down by standard to indicate which questions relate to different skills like evaluating expressions, adding and multiplying polynomials, solving various types of equations, and modeling real world problems.

Mathpracticeforfinalexam

This document provides a math practice for a final exam, containing questions on topics like:
1) Dividing expressions and evaluating algebraic expressions
2) Writing expressions in exponential form and working with fractions
3) Identifying terms, properties, and solving equations
4) Word problems involving rates, percentages, areas, perimeters, and patterns
5) Choosing correct formulas and expressions to solve missing number problems

Answer Key Practice For 4th Period Exam

Here are the solutions to the exponent problems:
27) 32 = 9 (because 3 x 3 = 9)
28) (y + 3)2 = 25, when y = 2
(2 + 3)2 = 25
52 = 25
25 = 25
29) x2 + 8 = 12, when x = 2
22 + 8 = 12
4 + 8 = 12
12 = 12
30) x3 = 64, when x = 4
43 = 64
64 = 64
So in summary:
27) 9
28) y = 2
29) x = 2
30) x = 4

Integration SPM

The document contains examples of indefinite integrals of various functions:
1) Finding the antiderivatives of polynomials like 4x3 + 3x - 2.
2) Finding an antiderivative involving a differential equation like dy/dx = 4x3 - 4x.
3) Evaluating integrals involving rational functions like ∫(3 - 2/x2 + 6x3)dx.
4) Finding antiderivatives of expressions involving radicals like ∫((x2+3)2/x2)dx.
5) Solving differential equations and evaluating integrals using substitution.

Chapter 9- Differentiation Add Maths Form 4 SPM

This document provides an explanation of differentiation and examples of calculating limits and derivatives using the first principle definition of the derivative. It begins by defining the limit of a function and providing examples of evaluating limits. It then introduces the concept of the derivative as the slope of the tangent line to a curve and explains how to calculate derivatives using small changes in x and y. The document provides examples of finding derivatives using this first principle definition. It also discusses rules for deriving composite functions and products of polynomials. Exercises are provided throughout for students to practice differentiation.

Chapter 3 quadratc functions

The document provides information about quadratic functions including:
- The general form of a quadratic function is f(x) = ax2 + bx + c.
- A quadratic function has a minimum or maximum point which can be used to find the axis of symmetry.
- The relationship between the discriminant (b2 - 4ac) and the position of the graph is explained. If it is greater than 0, the graph cuts the x-axis at two points. If it is equal to 0, the graph touches the x-axis at one point. If it is less than 0, the graph does not cut or touch the x-axis.
- Quadratic inequalities can be solved by sketching

Polynomials Test Answers

This document contains a polynomials test with multiple questions:
1) Find the products of several polynomials, including (x-1)(x+3) and (x^2 - x - 1)(x + 7).
2) For a triangle with base x + 1 and height 3x, express the area in terms of x using the area formula A = 1/2bh.
3) Divide the polynomial (x^2 + 10x + 21) by (x + 3).

Notes and-formulae-mathematics

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.

Rumus matematik-tambahan

This document contains mathematical formulas and concepts from algebra, calculus, geometry, trigonometry, and statistics. Some key points include:
- Formulas for addition, subtraction, multiplication, and division of algebraic expressions.
- Rules for differentiation, integration, and limits in calculus.
- Formulas for triangle properties like area, distance between points, and midpoint.
- Trigonometric identities for sine, cosine, and tangent functions of single and summed angles.
- Statistical concepts like mean, standard deviation, binomial distribution, and normal distribution.

Algebra formulas

This document provides an algebra cheat sheet that summarizes many common algebraic properties, formulas, and concepts. It covers topics such as arithmetic operations, properties of inequalities and absolute value, exponent properties, factoring formulas, solving equations, graphing functions, and common algebraic errors. The cheat sheet is a concise 3-page reference for the basics of algebra.

KSSM Form 4 Additional Mathematics Notes (Chapter 1-5)

Project Leader: Lim Jun Hao
Associate Editor: Lai Zhi Jun
Associate Project Editor: Siaw Jia Qi
Senior Managing Editor: Lim Jun Hao
Production Coordinator: Ooi Ming Yang
Cover Design: Ooi Hian Gee
Cover image: Pixabay
A collaboration work by Faculty of Science, University of Malaya Applied
Mathematics undergraduate students.
Copyright © 2020 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. The work is published in electronic form only. For
information on obtaining permission for use of material in this work, please email us
at ooi_hiangee@yahoo.com.

Maths Revision Notes - IGCSE

This document provides notes on various mathematics topics for the IGCSE including: decimals and standard form, accuracy and error, powers and roots, ratio and proportion, and trigonometry. It includes examples and practice problems for each topic. The notes are intended to help with revision for IGCSE mathematics question papers and assessments.

Form 4 add maths note

This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
1. A function f maps objects to images. To find the inverse function f-1, change f(x) to y and solve for x in terms of y.
2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solve

Additional Mathematics Revision

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.

Matematika Kalkulus ( Limit )

1. The document contains examples of evaluating limits as the variable approaches certain values.
2. Several limits were found to be indeterminate forms that require further algebraic manipulation to find the limit.
3. Key observations were made about the behavior of functions as the variable approaches values like noticing a function approaches a certain value as the variable nears another value.

Skills In Add Maths

This document provides examples of solving equations, expanding and factorizing expressions, solving simultaneous equations, working with indices and logarithms. It includes over 100 problems across these topics for students to practice. The problems range in complexity from basic single-step equations to multi-part logarithmic expressions and systems of simultaneous equations.

Chapter 5 indices & logarithms

This document provides a summary of Chapter 5 on Indices and Logarithms from an Additional Mathematics textbook. It includes examples and explanations of:
1. Laws of indices such as addition, subtraction, multiplication and division of indices.
2. Converting expressions between index form and logarithmic form using common logarithms and other bases.
3. Applying the laws of logarithms including addition, subtraction, and change of base.
4. Solving equations involving indices and logarithms through appropriate applications of index laws and logarithmic properties.

Modul bimbingan add maths

The document is a mathematics textbook for Additional Mathematics Form 4. It covers topics on functions, simultaneous equations, quadratic equations, and quadratic functions. It contains examples and practice questions for students to work through with answers. The questions range from simple calculations to solving equations and inequalities involving quadratic expressions.

Online math tutors (1)

The document contains a 15 question multiple choice test on polynomials. It covers topics like finding the number of zeros of a polynomial from its graph, identifying polynomials based on their zeros, finding the sum and product of the zeros of a quadratic polynomial, and determining the remaining zeros if one zero is given. It also includes word problems involving dividing polynomials and finding zeros. The test is meant to assess students' conceptual understanding of polynomials and their ability to solve subjective questions similar to those in board exams.

Add maths module form 4 & 5

1. The document provides an overview of important topics covered in Form 4 and Form 5 mathematics. These include functions, quadratic equations, trigonometry, statistics, calculus, and coordinate geometry.
2. Examples of how to solve different types of problems are given for each topic, such as finding the sum and product of roots for quadratic equations or using rules of logarithms to simplify logarithmic expressions.
3. Strategies for solving problems involving concepts like differentiation, integration, progressions, and linear laws are outlined. Methods for finding volumes or areas under curves are also summarized briefly.

Integration SPM

Integration SPM

Chapter 9- Differentiation Add Maths Form 4 SPM

Chapter 9- Differentiation Add Maths Form 4 SPM

Chapter 3 quadratc functions

Chapter 3 quadratc functions

Polynomials Test Answers

Polynomials Test Answers

Notes and-formulae-mathematics

Notes and-formulae-mathematics

Rumus matematik-tambahan

Rumus matematik-tambahan

Algebra formulas

Algebra formulas

KSSM Form 4 Additional Mathematics Notes (Chapter 1-5)

KSSM Form 4 Additional Mathematics Notes (Chapter 1-5)

Maths Revision Notes - IGCSE

Maths Revision Notes - IGCSE

Form 4 add maths note

Form 4 add maths note

Additional Mathematics Revision

Additional Mathematics Revision

Matematika Kalkulus ( Limit )

Matematika Kalkulus ( Limit )

Skills In Add Maths

Skills In Add Maths

Chapter 5 indices & logarithms

Chapter 5 indices & logarithms

Modul bimbingan add maths

Modul bimbingan add maths

Online math tutors (1)

Online math tutors (1)

Add maths module form 4 & 5

Add maths module form 4 & 5

2016 10 mathematics_sample_paper_sa2_03_ans_z657f

This document contains the solutions to a sample paper for Class 10 Mathematics. Some of the key questions solved include:
- Finding the discriminant of a quadratic equation.
- Calculating the perimeter of a figure involving an arc of a circle.
- Using the volume formula to find the number of lead shots that will fill a solid cube.
- Calculating the rainfall given the volume of a cylindrical vessel.
- Solving simultaneous equations to find the coordinates of a point.
- Calculating probabilities related to drawing cards from a standard deck.
The document provides detailed step-by-step workings for 22 other mathematical problems covering a range of topics.

2016 10 mathematics_sample_paper_sa2_01_ans_z9gf4

This document contains the solutions to a mathematics sample paper for Class 10. It includes solutions to 26 multiple choice questions on topics like geometry, trigonometry, and algebra. It also provides step-by-step workings for 4 multi-part questions involving concepts such as ratios, proportions, constructions, and coordinate geometry. Overall, the document analyzes solutions to 30 questions from a CBSE Class 10 math exam.

Exams in college algebra

This document contains a 25 question practice exam for college algebra. It covers topics such as solving linear, quadratic, and rational equations; simplifying expressions; factoring polynomials; graphing lines and parabolas; and solving systems of equations. The exam is broken down by standard to indicate which questions relate to different skills like evaluating expressions, adding and multiplying polynomials, solving various types of equations, and modeling real world problems.

Mathpracticeforfinalexam

This document provides a math practice for a final exam, containing questions on topics like:
1) Dividing expressions and evaluating algebraic expressions
2) Writing expressions in exponential form and working with fractions
3) Identifying terms, properties, and solving equations
4) Word problems involving rates, percentages, areas, perimeters, and patterns
5) Choosing correct formulas and expressions to solve missing number problems

Answer Key Practice For 4th Period Exam

Here are the solutions to the exponent problems:
27) 32 = 9 (because 3 x 3 = 9)
28) (y + 3)2 = 25, when y = 2
(2 + 3)2 = 25
52 = 25
25 = 25
29) x2 + 8 = 12, when x = 2
22 + 8 = 12
4 + 8 = 12
12 = 12
30) x3 = 64, when x = 4
43 = 64
64 = 64
So in summary:
27) 9
28) y = 2
29) x = 2
30) x = 4

Practice For 4th Period Exam

Here are the solutions to the exponent problems:
27) 32 = 9 (because 3 x 3 = 9)
28) (y + 3)2 = 25, when y = 2
(2 + 3)2 = (5)2 = 25
29) x2 + 8 = 12, when x = 2
(2)2 + 8 = 4 + 8 = 12
30) x3 = 64, when x = 4
43 = 64
So in summary:
27) 9
28) y = 2
29) x = 2
30) x = 4

2period review andanswers

The document contains a math review with examples of different techniques for counting combinations and permutations, as well as examples involving polynomials, solid geometry, and volume. It provides the steps to solve problems involving multiplication to find combinations, factoring polynomials, finding volumes of 3D shapes, and determining angle measures using parallel lines and a transversal.

2period review andanswers

The document contains a math review with examples of different techniques for counting combinations and permutations, as well as examples involving polynomials, solid geometry, and volume. It provides the steps to solve problems involving multiplication to find combinations, factoring polynomials, finding volumes of geometric shapes, and determining angle measures using parallel lines and a transversal.

2period review and answers

The document contains a math review with examples of different techniques for counting combinations and permutations, as well as examples involving polynomials, solid geometry, and other math concepts. It provides the steps and work for 14 different math problems, showing how to use techniques like multiplication, combinations, and permutations to calculate things like the number of combinations for choosing food items or essays. It also includes examples of multiplying polynomials, finding volumes of solids, and solving geometry problems involving parallel lines and transversals.

2period ReviewANDANSWERS

The document provides answers to math review questions covering topics like counting techniques, grouping and analyzing data, multiplying polynomials, solid geometry, and volume calculations. It includes step-by-step workings for problems involving multiplication, combinations, permutations, creating tables and graphs from data, multiplying binomial expressions, finding volumes of geometric solids, and solving a word problem about maintaining the same volume with changes in dimensions.

Cbse sample-papers-class-10-maths-sa-ii-solved-1

The document provides information about a math exam including:
- It is divided into 4 sections with various question types and marks.
- Section A has 8 multiple choice 1-mark questions.
- Section B has 6 2-mark questions.
- Section C has 10 3-mark questions.
- Section D has 10 4-mark questions.
- Calculators are not permitted and additional time is given to read the paper.

Answer Key Period Exam

Here are the solutions to the exponent problems:
27) 32 = 9 (because 3 x 3 = 9)
28) (y + 3)2 = 25, when y = 2
(2 + 3)2 = (5)2 = 25
29) x2 + 8 = 12, when x = 2
(2)2 + 8 = 4 + 8 = 12
30) x3 = 64, when x = 4
43 = 64
So in summary:
27) 9
28) y = 2
29) x = 2
30) x = 4

Tcs 2014 saved in 97-2003 format

This document contains multiple questions related to programming, geometry, probability, and other topics. It provides the questions, solutions, and explanations for the following:
1) A question about the number of lines of code that can be written by a certain number of programmers in a given time period.
2) A geometry question about the minimum number of "1-sets" (points separated from others by a line) for 5 points in a plane.
3) A question about the number of times a particular number appears when labeling items with a numeric system of a certain base.
4) A question regarding the angle between the hands of a clock on the planet Oz, which has different timekeeping conventions than

C 14-met-mng-aei-102-engg maths-1

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Answers for 4th period exam (review)

Here are the steps to solve this probability problem:
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2) Identify the favorable outcomes (outcomes we want): The outcomes that result in a number greater than 2 are: 3, 4, 5, 6. There are 4 of these.
3) Calculate the probability:
Probability = Favorable Outcomes / Total Possible Outcomes
= 4 / 6
= 4/6
So the probability of rolling a number greater than 2 on a standard 6-sided die is 4/6.

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This document contains 5 questions regarding a mathematics exam. It covers topics like algebra, geometry, calculus, differential equations, and matrices. Some key details:
- The exam has 5 questions worth a total of 80 marks.
- Question 1 has 8 short answer parts worth 16 marks total.
- Questions 2-4 have 4 medium length parts each worth 16 marks total.
- Question 5 has 2 long answer parts worth 16 marks total.
- The questions cover topics such as finding GCDs, eigenvalues, limits, differential equations, and geometry concepts.Additional mathematics

The document discusses solving quadratic equations by factorizing. It provides examples of factorizing quadratic expressions and equations to find their roots. In one example, the quadratic equation x^2 - 2x + 2 = 4 is factored into (x - 2)(x + 2) = 0, showing it has only one real root of x = 2. Another example factors a quadratic expression f(x) = x^2 - x - 1 to find its two roots of 1 and -1. The document demonstrates how to factorize quadratic expressions and equations in order to solve for their real roots.

xy2.5 3.0 3.5-1.0 6 7 81.0 0 1 23.0 -6 -5 .docx

x
y
2.5 3.0 3.5
-1.0 6 7 8
1.0 0 1 2
3.0 -6 -5 -4
MATH 223
FINAL EXAM REVIEW PACKET ANSWERS
(Fall 2012)
1. (a) increasing (b) decreasing
2. (a) 2 2( 3) 25y z− + = This is a cylinder parallel to the x-axis with radius 5.
(b) 3x = , 3x = − . These are vertical planes parallel to the yz-plane.
(c) 2 2 2z x y= + . This is a cone (one opening up and one opening down) centered on the z-axis.
3. There are many possible answers.
(a) 0x = produces the curve 23y z= − .
(b) 1y = produces the curves 23 cosz x= − and 23 cosz x= − − .
(c)
2
x
π
= produces the curves 3z = and 3z = − .
4. (a) (b) (i) 1 (ii) Increase (iii) Decrease
5. (a) Paraboloids centered on the x-axis, opening up in the positive x direction. 2 2x y z c= + +
(b) Spheres centered at the origin with radius 1 ln c− for 0 c e< ≤ . 2 2 2 1 lnx y z c+ + = −
6. (a) 6 am 11:30 am
(b) Temperature as a function of time at a depth of 20 cm.
(c) Temperature as a function of depth at noon.
7. ( , ) 2 3 2z f x y x y= = − −
8. (a) II, III, IV, VI (b) I (c) I, III, VI (d) VI (e) I, V
9. (a)
12
4 12
5
z x y= − + (b) There are many possible answers.
12
4
5
i j k+ −
(c)
3 569
2
10. (a) iii, vii (b) iv (c) viii (d) ii (e) v, vi (f) i, ix
11. There are many possible answers.
(a) ( )5 4 3
26
i j k− +
or ( )5 4 3
26
i j k− − +
(b) 2 3i j− +
(c)
4
cos
442
θ = , 1.38θ ≈ radians (d) ( )4 4 3
26
i j k− +
(e) 4 11 17i j k− − −
12. (a)
3
5
a = − (b)
1
3
a = (c) 2( 1) ( 2) 3( 3) 0x y z− − + + − = (d)
1 2 , 2 , 3 3x t y t z t= + = − − = +
13. 6 39i
or 6 39i−
14. (a)
( )
2
23 2 2
3 2
3 1
z x y x
x x y x y
∂
= −
∂ + + +
(b)
( )4
10 4 3
5
H
H T
f
H
+ +
=
−
(c)
2
2 2
1 1z
x y y x
∂
= − −
∂ ∂
15. (a) 2 2 24 ( 1) 3 ( 2) 2z e x e y e= − + − + (b) 4( 3) 8( 3) 6( 6) 0x y z− + − + − =
16. (a)
2sin(2 ) cos(2 )
5 5
v v
ds dv d
α α
α= +
(b) The distance s decreases if the angle α increases and the initial speed v remains constant.
(c) 0.0886α∆ ≈ − . The angle decreases by about 0.089 radians.
17. (a) The water is getting shallower.
4
( 1, 2)
17
uh − = −
(b) There are many possible answers. 3i j+
(c) 72 ft/min
18. (a)
( )
2 2 2
22 2 22
2 2
1 1 11
yz xyz z yz
grad i j k
x x xx
= − + + + + + +
(b) ( ) ( ) ( )( )2 2 2 2curl x y z i y z j xz k i zj yk+ + − + + = + −
(c) ( ) ( ) ( )( )2 3 3cos sec 2 cos sin sec tan 3z zdiv x i x y j e k x x x y y e+ + = − + +
(d)
37
3
(e) ( , , ) sin zg x y z xy e c= + +
19. ( , ) 4 3vG a b = −
20. (a) positive (b) negative (c) negative (d) negative (e) positive (f) zero
21. (a)
(.

2016 10 mathematics_sample_paper_sa2_03_ans_z657f

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2016 10 mathematics_sample_paper_sa2_01_ans_z9gf4

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Mathpracticeforfinalexam

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Answer Key Practice For 4th Period Exam

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2period review andanswers

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Howard, anton calculo i- um novo horizonte - exercicio resolvidos v1

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F.Y.B.Sc(2013 pattern) Old Question Papers:Dr.Kshirsagar

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Additional mathematics

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xy2.5 3.0 3.5-1.0 6 7 81.0 0 1 23.0 -6 -5 .docx

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This document provides an introduction to partial fractions. It defines key terms like polynomials, rational functions, and proper and improper fractions. It then outlines the three main cases for splitting a fraction into partial fractions: (1) a linear factor (ax+b), (2) a repeated linear factor, and (3) a quadratic factor (ax^2+bx+c). For each case, it provides an example of how to write the fraction as a sum of partial fractions. It concludes by emphasizing two important checks: (1) the fraction must be proper, and (2) the denominator must be completely factorized before attempting to write it as partial fractions.

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This document provides examples of recurrence relations and their solutions. It begins by defining convergence of sequences and limits. It then provides examples of recurrence relations, solving them using algebraic and graphical methods. One example finds the 6th term of a sequence defined by a recurrence relation to be 2.3009. Another example solves a recurrence relation algebraically to express the general term un in terms of n. The document emphasizes using graphical methods like sketching graphs to prove properties of sequences defined by recurrence relations.

Functions 1 - Math Academy - JC H2 maths A levels

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www.mathacademy.sg
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- 1. 4047 4AM Differentiation (9) Math Academy® ©All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. Math Academy® 1 “If you make a mistake and do not correct it, this is called a mistake.” ― Confucius Application of Differentiation – Maximum and Minimum Problems [A] With Magic Number The diagram shows a box in the shape of a cuboid with a square cross-section of side cm. The volume of the box is 3500 cm . Four pieces of tape are fastened round the box as shown. The pieces of tape are parallel to the edges of the box. (i) Given that the total length of the four pieces of tape is cm, show that . [3] (ii) Given that can vary, find the stationary value of and determine the nature of this stationary value. [5] Solution: (i) Use the magic number 3500 to form an equation (ii) Let the length of the cuboid be cm. Total length (shown) Ans: (ii) L = 210, minimum x 3 L 2 7000 14 x xL += x L y 2 3500x y = 2 3500 y x = L 14 2x y= + 2 3500 14 2x x æ ö = + ç ÷ è ø 2 7000 14x x = + 3D 0 dL dx = Example 1: Observe and make the subject. y
- 2. 4047 4AM Differentiation (9) Math Academy® ©All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. Math Academy® 2 [B] Without Magic Number (may need to draw extra lines to form ) Example 2: The diagram shows a right circular cone with radius cm and height cm, where and can vary. A sphere with radius 9 cm is to be placed inside the cone, such that it will touch the top surface of the cone. (i) Show that the volume of the cone, cm , can be expressed as . [3] (ii) Hence, find the minimum volume of the cone. [5] Solution: Ans: (ii) , min Vol = 6110 cm (3sf) r h r h V 3 2 27 18 h V h p = - 36h = 3 2D Join from centre of circle to tangent Similar triangles / Toa Cah Soh / Pythagoras Theorem 0 dV dh = cm 9 cm cm Right Angled Triangle
- 3. 4047 4AM Differentiation (9) Math Academy® ©All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. Math Academy® 3 Our greatest weakness lies in giving up. The most certain way to succeed is always to try just one more time. Thomas A. Edison Worksheet 7: Maximum and Minimum Problems Qns compiled from exam papers 1 A piece of wire, 48 cm in length, is used to form the framework of a regular prism as shown in the diagram below. The cross-section of the prism is a right-angled triangle with two perpendicular sides of length cm and cm. The length of the prism is cm. (a) Express in terms of . [2] (b) Show that the volume of the prism is given by . [2] (c) Find the value of for which has a stationary value. [3] (d) Find this value of and determine whether it is a maximum or a minimum. [3] Solution: x 1 1 3 x y y x 216 (6 ) 9 V x x= - x V V x cm 1 x cm y cm Magic no. (a) By pythagoras theorem Length = 48 cm (b) Volume 2 2 21 1 3 x x w æ ö + =ç ÷ è ø 2 225 9 x w= 5 3 x w = 1 5 2 2 1 2 3 48 3 3 x x x y æ ö æ ö + + + =ç ÷ ç ÷ è ø è ø 8 3 48x y+ = 8 16 3 y x= - 1 1 ( ) 1 ( ) 2 3 x x y æ ö = ç ÷ è ø 22 8 16 3 3 x x æ ö = -ç ÷ è ø 216 (6 ) 9 x x= - (c) or (rejected) (d) cm When , Volume is a maximum at 2 332 16 3 9 V x x= - 264 16 3 3 dV x x dx = - 64 16 0 3 3 x x æ ö - =ç ÷ è ø 0x = 4x = 216 8 (4) (6 4) 56 9 9 V = - = 3 2 2 64 32 3 3 d V x dx = - 4x = 2 2 64 0 3 d V dx = - < 4x =
- 4. 4047 4AM Differentiation (9) Math Academy® ©All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. Math Academy® 4 3 The diagram shows a cylinder of height cm and base radius cm inscribed in a sphere of radius 35 cm. (i) Show that the height of the cylinder, cm is given by . [2] (ii) Given that can vary, find the maximum volume of the cylinder. [4] Ans: (ii) 104 000 cm Solution: h r h 2 2 1225h r= - r 3 positive 0 negative Slope r 28.5 28.577 28.6 dT dx (i) By Pythagoras’ Theorem, (shown) (ii) (rejected) Volume = 104 000 cm (3 sf) Volume is a maximum when . 2 2 2 35 2 h r æ ö + =ç ÷ è ø 2 2 1225 4 h r= - 2 2 4(1225 )h r= - 2 2 1225h r= - 2 2 (2 1225 )V r rp= - ( ) 1 2 2 21225r rp= - ( ) ( ) 1 1 2 2 22 2 1 2 1225 ( 2 ) 1225 (4 ) 2 dV r r r r r dr p p -æ ö = - - + -ç ÷ è ø ( ) ( ) 1 1 3 2 22 22 1225 1225 (4 )r r r rp p - = - - + - ( ) ( ) 1 2 3 221225 2 (1225 )(4 )r r r rp p - = - - + - ( ) ( ) 1 2 2 222 1225 2450 2r r r rp - = - - + - ( ) ( ) 1 2 222 1225 3 2450r r rp - = - - + 1 2 2 2 2 ( 3 2450) 0 (1225 ) r r r p - + = - 0r = 2 3 2450 0r- + = 28.577r = 3 28.577r = Keyword Max/min qn®
- 5. 4047 4AM Differentiation (9) Math Academy® ©All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. Math Academy® 5 5 The diagram shows a cuboid of height units inside a right pyramid of height 8 units and with square base of side 4 units. The base of the cuboid sits on the square base of the pyramid. The points and are corners of the cuboid and lie on the edges and , respectively, of the pyramid The pyramids and are similar. (i) Find an expression for in terms of and hence show that the volume of the cuboid is given by units . [4] (ii) Given that can vary, find the value of for which is a maximum. [4] Solutions: h OPQRS PQRS A, B, C D OP, OQ, OR OS OPQRS. OPQRS OABCD AD h V V = h3 4 − 4h2 +16h 3 h h V Similar Triangles (i) By similar triangles, (ii) Volume of cuboid = 8 4 8 AD h- = 8 2 h AD - = 2 8 2 h h -æ ö = ç ÷ è ø 2 (64 16 ) 4 h h h- + = 3 2 4 16 4 h h h= - + (ii) or (rejected) When , Volume is a maximum when . 2 3 8 16 0 4 dV h h dh = - + = ( 8)(3 8) 0h h- - = 8h = 8 3 h = 2 2 3 8 2 d V h dh = - 8 3 h = 2 2 4 0 d V dh = - < 8 3 h =
- 6. 4047 4AM Differentiation (9) Math Academy® ©All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. Math Academy® 6 7 A lifeguard at a beach resort is stationed at point along the coastline, as shown in the diagram below. When he detects a swimmer who needs help at a point , he would run along the coastline over a distance of m to a point , and then swim in a straight line, , towards the swimmer. The lifeguard runs at a speed of 4 m/s and swims at a speed of 2 m/s. A swimmer in distress is detected at a position that is 40 m away from the coastline, and the foot of the perpendicular from the swimmer to the coastline is at a distance of 60 m away from the lifeguard. (i) Show that the time taken by the lifeguard to swim from to is seconds. [2] (ii) Find, in terms of , the total time taken by the lifeguard to reach the swimmer. [1] (iii) Obtained an expression for . [2] (iv) Find the value of such that the lifeguard would be able to reach the swimmer in the shortest possible time. [4] Ans: (ii) (iii) 36.9 G S x H HS H S 2 1600 (60 ) 2 x+ - x T dT dx x 2 60 1 42 1600 (60 ) dT x dx x - = + + - Keyword Max/min qn®
- 7. 4047 4AM Differentiation (9) Math Academy® ©All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. Math Academy® 7 9 At 8 a.m, Ship is 100 km due North of Ship . Ship is sailing due South at 20 km/h. Ship is sailing at a bearing of 120 at 10 km/h. (i) Show that the distance between the two ships after hours is given by . [3] (ii) At what time is Ship closest to Ship . [3] Solution: (i) Assume that the distance moved by Ship is less than 100 km. Let’s draw the diagram out. By Cosine Rule, (shown) (ii) For Ship closest to Ship , Ship is closest to Ship after 5 hours. A B A B ° t 2 10 3 30 100AB t t= - + A B A 2 2 2 (100 20 ) (10 ) 2(100 20 )(10 )cos120AB t t t t= - + - - ° 2 2 2 2 10000 4000 400 100 (2000 400 )( 0.5)AB t t t t t= - + + - - - 2 2 300 3000 10000AB t t= - + 2 2 100(3 30 100)AB t t= - + 2 10 3 30 100AB t t= - + A B 0 dAB dt = 1 2 2 1 10 (3 30 100) (6 30) 0 2 t t t -æ ö - + - =ç ÷ è ø 6 30 0t - = 5t = A B Keyword Max/min qn® 100 20t- 10t