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This document provides 10 problems related to differentiating functions. The problems cover differentiating various functions with respect to variables like x and y, finding maximum and minimum values, rates of change, and determining equations of tangents and normals. Solutions are not provided. The document is from an Additional Mathematics module for Form 4 students and focuses on differentiation techniques.
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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.
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This document contains sample problems related to quadratic equations and quadratic functions for Form 4 Additional Mathematics. It is divided into three sections - the first two sections contain sample problems testing concepts related to solving quadratic equations and inequalities. The third section contains sample problems related to identifying properties of quadratic functions such as finding the minimum or maximum value, range of a quadratic function, expressing a quadratic in standard form and sketching its graph.
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This document provides practice problems for additional mathematics Form 4 students in Terengganu, Malaysia. It covers topics on quadratic equations and quadratic functions, with multiple choice and short answer questions. The problems are divided into three sections: quadratic equations, quadratic functions for paper 1, and quadratic functions for paper 2. The document is copyrighted material from the Terengganu State Education Department.
MODULE 4- Quadratic Expression and Equations
(1) The document is a math worksheet containing 20 quadratic equations to solve.
(2) It provides the steps to solve each equation, factorizing the expressions and setting each factor equal to zero to find the roots.
(3) The answers section lists the factored forms and solutions for each of the 20 equations.
Spm add math 2009 paper 1extra222
This document contains a 2010 Additional Mathematics exam paper from the Sijil Pelajaran Malaysia (SPM). It consists of 25 multiple choice and short answer questions covering topics like:
- Relations and functions
- Quadratic equations
- Geometric and arithmetic progressions
- Trigonometry
- Probability and statistics
The questions require students to apply concepts like domain and range, inverse functions, maximum/minimum values, and normal distributions to solve problems involving graphs, equations, and word problems.
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
modul 5 add maths 07
This document is a module on differentiation for Additional Mathematics Form 4 students in Terengganu, Malaysia. It contains 20 practice problems on various topics related to differentiation, including finding derivatives of functions, finding maximum/minimum values, related rates, and finding equations of tangents and normals. The problems are presented without solutions for students to practice solving. The module is published by the Terengganu State Education Department.
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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.
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This document contains a mathematics exam with questions on statistics and circular measure. The statistics section includes questions about mean, median, variance, standard deviation, and data distributions. The circular measure section focuses on converting between degrees and radians, calculating arc lengths and sector areas, and solving geometry problems involving circles. The exam provides the framework for a high school level additional mathematics assessment covering important topics in probability and trigonometry.
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This document contains sample problems related to quadratic equations and quadratic functions for Form 4 Additional Mathematics. It is divided into three sections - the first two sections contain sample problems testing concepts related to solving quadratic equations and inequalities. The third section contains sample problems related to identifying properties of quadratic functions such as finding the minimum or maximum value, range of a quadratic function, expressing a quadratic in standard form and sketching its graph.
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This document provides practice problems for additional mathematics Form 4 students in Terengganu, Malaysia. It covers topics on quadratic equations and quadratic functions, with multiple choice and short answer questions. The problems are divided into three sections: quadratic equations, quadratic functions for paper 1, and quadratic functions for paper 2. The document is copyrighted material from the Terengganu State Education Department.
MODULE 4- Quadratic Expression and Equations
(1) The document is a math worksheet containing 20 quadratic equations to solve.
(2) It provides the steps to solve each equation, factorizing the expressions and setting each factor equal to zero to find the roots.
(3) The answers section lists the factored forms and solutions for each of the 20 equations.
Spm add math 2009 paper 1extra222
This document contains a 2010 Additional Mathematics exam paper from the Sijil Pelajaran Malaysia (SPM). It consists of 25 multiple choice and short answer questions covering topics like:
- Relations and functions
- Quadratic equations
- Geometric and arithmetic progressions
- Trigonometry
- Probability and statistics
The questions require students to apply concepts like domain and range, inverse functions, maximum/minimum values, and normal distributions to solve problems involving graphs, equations, and word problems.
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
modul 5 add maths 07
This document is a module on differentiation for Additional Mathematics Form 4 students in Terengganu, Malaysia. It contains 20 practice problems on various topics related to differentiation, including finding derivatives of functions, finding maximum/minimum values, related rates, and finding equations of tangents and normals. The problems are presented without solutions for students to practice solving. The module is published by the Terengganu State Education Department.
modul 1 add maths 07
This document provides a module on functions and simultaneous equations for Additional Mathematics Form 4 students in Terengganu, Malaysia. It contains 15 problems on functions and 12 problems on simultaneous equations to help students prepare for their SPM examinations. The module is published by the Terengganu Education Department and involves several teachers from technical and science schools in the state.
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This document provides 15 multi-part math word problems involving indices, logarithms, and coordinate geometry. The problems cover topics such as simplifying expressions with indices, solving logarithmic and exponential equations, finding equations of lines and loci, determining properties of geometric figures defined by coordinate points, and calculating areas. Students must use their understanding of indices, logarithms, coordinate geometry, and geometric relationships to solve the problems.
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.
Skills In Add Maths
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35182797 additional-mathematics-form-4-and-5-notes
1) The function f(x) = 2x^2 + 8x + 6 can be written as f(x) = 2(x+2)^2 - 2. The maximum point is (-2, -2) and the equation of the tangent at this point is y = -2.
2) The function f(x) = -(x-4)^2 + h has a maximum point at (k, 9) so k = 4 and h = 9.
3) The function y = (x+m)^2 + n has an axis of symmetry at x = -m. Given the axis is x = 1, m = -1 and the minimum point is (1
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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.
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1. The document is a math test for Additional Mathematics Form 4 consisting of 18 questions. It provides instructions to candidates to answer all questions clearly in the spaces provided and show their working. Diagrams are not drawn to scale unless stated.
2. The questions cover topics on solving simultaneous equations, functions, relations, composite functions, inverse functions and sketching graphs. Candidates are required to find values, images, objects, domains, ranges and relations in function notation.
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This document provides a summary of various math formulae for Form 4 students in Malaysia, including:
1. Functions, quadratic equations, and quadratic functions
2. Simultaneous equations, indices and logarithms, and coordinate geometry
3. Statistics, circular measures, and differentiation
It lists common formulae for topics like the quadratic formula, completing the square, differentiation rules, and measures of central tendency and dispersion. The document is intended as a study guide for students to review essential formulae.
Soalan Jawapan Omk 2007
There are 5 possible pairs: (0,0), (0,9), (9,0), (1,8), (8,1)
The zero can be in any of the 5 positions.
The other two digits in the pair can be arranged in 2 ways.
Hence total numbers = 5 × 2 × 1000 = 10,000
Jawapan: 10,000
SOALAN A4
BM Diberi ABCD satu segiempat sama sisi dengan sisi AB = 4 cm. E dan F
dua titik di atas AB dan BC masing-masing dengan AE = 2 cm dan BF = 3
cm
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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
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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
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This document contains information about the format and topics covered in papers 1 and 2 of an exam. Paper 1 has 25 questions to be answered in 2 hours, with 10 questions of low difficulty, 6 of moderate difficulty, and 1 of high difficulty. Paper 2 has 3 sections, with the first section containing 6 questions to answer, the second 5 questions where the test taker must choose 4, and the third 4 questions where they must choose 2. The total time for Paper 2 is 2.5 hours.
The document then lists topics that will be covered in the exam, grouped under the categories of Algebra, Geometry, Calculus, Trigonometry, Statistics, Science and Technology. Specific topics include functions, quadratic equations
02[anal add math cd]
1. This document discusses solving quadratic equations by factorizing, using the quadratic formula, and finding the roots given information about the sum and product of the roots.
2. Methods covered include factorizing quadratic expressions, setting each factor equal to zero to find roots, using the quadratic formula, and deriving the quadratic equation when given the sum and product of the roots.
3. Examples are provided to demonstrate each method, such as factorizing (x-1)(x+2)=1 to find the roots x=1 and x=-2, or using the quadratic formula to solve equations like x2-3x-10=0.
Add Maths Module
The composite function is gf(x) = 2x - 2. We are given f(x) = 2 - x. To find g, let f(x) = u in gf(x). Then u = 2 - x. Substitute u = 2 - x in gf(x) = 2u - 2. This gives g(x) = 2 - 2x. Therefore, fg(x) = f(2 - 2x) = (2 - (2 - 2x)) = 2x - 2.
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This document provides a marking scheme for an Additional Mathematics paper 2 trial examination from 2010. It consists of 7 questions, each with multiple parts. For each question, it lists the number of marks awarded for various steps in the solutions, such as setting up the correct formula, performing calculations accurately, obtaining the right solution, plotting points correctly, and using appropriate mathematical reasoning. The highest number of marks for a single question is 8 marks. The marking scheme evaluates multiple aspects of students' work and reasoning for 7 multi-step mathematics problems.
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This document provides a summary of Chapter 5 on Indices and Logarithms from an Additional Mathematics textbook. It includes examples and explanations of:
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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.
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This document provides an overview of solving simultaneous equations between linear and quadratic equations with two unknowns. It includes 4 examples of solving simultaneous equations involving a linear equation equal to a quadratic equation. The examples show the steps of substituting one equation into the other and solving. The document also includes a chapter review with 6 practice problems involving simultaneous equations.
Beams Latest 24 Mac 2010 Edited Version
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This document provides a module on functions and simultaneous equations for Additional Mathematics Form 4 students in Terengganu, Malaysia. It contains 15 problems on functions and 12 problems on simultaneous equations to help students prepare for their SPM examinations. The module is published by the Terengganu Education Department and involves several teachers from technical and science schools in the state.
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This document provides 15 multi-part math word problems involving indices, logarithms, and coordinate geometry. The problems cover topics such as simplifying expressions with indices, solving logarithmic and exponential equations, finding equations of lines and loci, determining properties of geometric figures defined by coordinate points, and calculating areas. Students must use their understanding of indices, logarithms, coordinate geometry, and geometric relationships to solve the problems.
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.
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.
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1) The function f(x) = 2x^2 + 8x + 6 can be written as f(x) = 2(x+2)^2 - 2. The maximum point is (-2, -2) and the equation of the tangent at this point is y = -2.
2) The function f(x) = -(x-4)^2 + h has a maximum point at (k, 9) so k = 4 and h = 9.
3) The function y = (x+m)^2 + n has an axis of symmetry at x = -m. Given the axis is x = 1, m = -1 and the minimum point is (1
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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.
Test 1 f4 add maths
1. The document is a math test for Additional Mathematics Form 4 consisting of 18 questions. It provides instructions to candidates to answer all questions clearly in the spaces provided and show their working. Diagrams are not drawn to scale unless stated.
2. The questions cover topics on solving simultaneous equations, functions, relations, composite functions, inverse functions and sketching graphs. Candidates are required to find values, images, objects, domains, ranges and relations in function notation.
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This document provides guidance on solving additional mathematics problems and preparing for the additional mathematics paper 1 and paper 2 exams. It discusses exam formats, common mistakes students make, key strategies for achieving high marks, and examples of solved exam questions. The document emphasizes understanding the problem, planning a strategy, checking answers, and showing working clearly. It also provides tips on time management and the types of questions that may appear.
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This document provides a summary of various math formulae for Form 4 students in Malaysia, including:
1. Functions, quadratic equations, and quadratic functions
2. Simultaneous equations, indices and logarithms, and coordinate geometry
3. Statistics, circular measures, and differentiation
It lists common formulae for topics like the quadratic formula, completing the square, differentiation rules, and measures of central tendency and dispersion. The document is intended as a study guide for students to review essential formulae.
Soalan Jawapan Omk 2007
There are 5 possible pairs: (0,0), (0,9), (9,0), (1,8), (8,1)
The zero can be in any of the 5 positions.
The other two digits in the pair can be arranged in 2 ways.
Hence total numbers = 5 × 2 × 1000 = 10,000
Jawapan: 10,000
SOALAN A4
BM Diberi ABCD satu segiempat sama sisi dengan sisi AB = 4 cm. E dan F
dua titik di atas AB dan BC masing-masing dengan AE = 2 cm dan BF = 3
cm
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
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
Add maths complete f4 & f5 Notes
This document contains information about the format and topics covered in papers 1 and 2 of an exam. Paper 1 has 25 questions to be answered in 2 hours, with 10 questions of low difficulty, 6 of moderate difficulty, and 1 of high difficulty. Paper 2 has 3 sections, with the first section containing 6 questions to answer, the second 5 questions where the test taker must choose 4, and the third 4 questions where they must choose 2. The total time for Paper 2 is 2.5 hours.
The document then lists topics that will be covered in the exam, grouped under the categories of Algebra, Geometry, Calculus, Trigonometry, Statistics, Science and Technology. Specific topics include functions, quadratic equations
02[anal add math cd]
1. This document discusses solving quadratic equations by factorizing, using the quadratic formula, and finding the roots given information about the sum and product of the roots.
2. Methods covered include factorizing quadratic expressions, setting each factor equal to zero to find roots, using the quadratic formula, and deriving the quadratic equation when given the sum and product of the roots.
3. Examples are provided to demonstrate each method, such as factorizing (x-1)(x+2)=1 to find the roots x=1 and x=-2, or using the quadratic formula to solve equations like x2-3x-10=0.
Add Maths Module
The composite function is gf(x) = 2x - 2. We are given f(x) = 2 - x. To find g, let f(x) = u in gf(x). Then u = 2 - x. Substitute u = 2 - x in gf(x) = 2u - 2. This gives g(x) = 2 - 2x. Therefore, fg(x) = f(2 - 2x) = (2 - (2 - 2x)) = 2x - 2.
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This document provides a marking scheme for an Additional Mathematics paper 2 trial examination from 2010. It consists of 7 questions, each with multiple parts. For each question, it lists the number of marks awarded for various steps in the solutions, such as setting up the correct formula, performing calculations accurately, obtaining the right solution, plotting points correctly, and using appropriate mathematical reasoning. The highest number of marks for a single question is 8 marks. The marking scheme evaluates multiple aspects of students' work and reasoning for 7 multi-step mathematics problems.
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.
Chapter 4 simultaneous equations
This document provides an overview of solving simultaneous equations between linear and quadratic equations with two unknowns. It includes 4 examples of solving simultaneous equations involving a linear equation equal to a quadratic equation. The examples show the steps of substituting one equation into the other and solving. The document also includes a chapter review with 6 practice problems involving simultaneous equations.
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KSSM Form 4 Additional Mathematics Notes (Chapter 1-5)
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35182797 additional-mathematics-form-4-and-5-notes
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MA 243 Calculus III Fall 2015 Dr. E. JacobsAssignmentsTh.docx
MA 243 Calculus III Fall 2015 Dr. E. Jacobs
Assignments
These assignments are keyed to Edition 7E of James Stewart’s “Calculus” (Early Transcendentals)
Assignment 1. Spheres and Other Surfaces
Read 12.1 - 12.2 and 12.6
You should be able to do the following problems:
Section 12.1/Problems 11 - 18, 20 - 22 Section 12.6/Problems 1 - 48
Hand in the following problems:
1. The following equation describes a sphere. Find the radius and the coordinates of the center.
x2 + y2 + z2 = 2(x + y + z) + 1
2. A particular sphere with center (−3, 2, 2) is tangent to both the xy-plane and the xz-plane.
It intersects the xy-plane at the point (−3, 2, 0). Find the equation of this sphere.
3. Suppose (0, 0, 0) and (0, 0, −4) are the endpoints of the diameter of a sphere. Find the
equation of this sphere.
4. Find the equation of the sphere centered around (0, 0, 4) if the sphere passes through the
origin.
5. Describe the graph of the given equation in geometric terms, using plain, clear language:
z =
√
1 − x2 − y2
Sketch each of the following surfaces
6. z = 2 − 2
√
x2 + y2
7. z = 1 − y2
8. z = 4 − x − y
9. z = 4 − x2 − y2
10. x2 + z2 = 16
Assignment 2. Dot and Cross Products
Read 12.3 and 12.4
You should be able to do the following problems:
Section 12.3/Problems 1 - 28 Section 12.4/Problems 1 - 32
Hand in the following problems:
1. Let u⃗ =
⟨
0, 1
2
,
√
3
2
⟩
and v⃗ =
⟨√
2,
√
3
2
, 1
2
⟩
a) Find the dot product b) Find the cross product
2. Let u⃗ = j⃗ + k⃗ and v⃗ = i⃗ +
√
2 j⃗.
a) Calculate the length of the projection of v⃗ in the u⃗ direction.
b) Calculate the cosine of the angle between u⃗ and v⃗
3. Consider the parallelogram with the following vertices:
(0, 0, 0) (0, 1, 1) (1, 0, 2) (1, 1, 3)
a) Find a vector perpendicular to this parallelogram.
b) Use vector methods to find the area of this parallelogram.
4. Use the dot product to find the cosine of the angle between the diagonal of a cube and one of
its edges.
5. Let L be the line that passes through the points (0, −
√
3 , −1) and (0,
√
3 , 1). Let θ be the
angle between L and the vector u⃗ = 1√
2
⟨0, 1, 1⟩. Calculate θ (to the nearest degree).
Assignment 3. Lines and Planes
Read 12.5
You should be able to do the following problems:
Section 12.5/Problems 1 - 58
Hand in the following problems:
1a. Find the equation of the line that passes through (0, 0, 1) and (1, 0, 2).
b. Find the equation of the plane that passes through (1, 0, 0) and is perpendicular to the line in
part (a).
2. The following equation describes a straight line:
r⃗(t) = ⟨1, 1, 0⟩ + t⟨0, 2, 1⟩
a. Find the angle between the given line and the vector u⃗ = ⟨1, −1, 2⟩.
b. Find the equation of the plane that passes through the point (0, 0, 4) and is perpendicular to
the given line.
3. The following two lines intersect at the point (1, 4, 4)
r⃗ = ⟨1, 4, 4⟩ + t⟨0, 1, 0⟩ r⃗ = ⟨1, 4, 4⟩ + t⟨3, 5, 4⟩
a. Find the angle between the two lines.
b. Find the equation of the plane that contains every point o ...
Class xii assignment chapters (1 to 9)
1. Find the equation of tangent to the curve x = sin 3t, y = cos 2t at t = π/4.
2. Solve the differential equation: cos2x dy/dx + y = tan x.
3. Using integration, find the area of the region bounded by the parabola y2 = 4x and the circle 4x2 + 4y2 = 9.
4. Evaluate several integrals using properties of integrals, derivatives, and determinants.
2.QuadraticEquations.doc
The document contains 14 problems involving solving quadratic equations. The problems cover expressing quadratic equations in standard form, finding roots of equations given information about the roots, determining values of constants that satisfy equations, and identifying possible values of parameters that satisfy conditions about the roots. The document provides space for writing the answer to each problem.
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.
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.
Trial kedah 2014 spm add math k2
This document is a past year exam paper for Additional Mathematics. It consists of 3 sections - Section A with 6 multiple choice questions worth a total of 40 marks, Section B with 4 structured questions worth a total of 40 marks, and Section C with 2 essay questions worth a total of 20 marks. The document provides relevant formulae, instructions for candidates, diagrams, questions, and spaces for working. It tests students' knowledge and skills in algebra, calculus, geometry, trigonometry and statistics.
F.Y.B.Sc(2013 pattern) Old Question Papers:Dr.Kshirsagar
F.Y.B.Sc(2013 pattern) Old Question Papers:Dr.KshirsagarDr.Ravindra Kshirsagar P.G Department of Zoology Modern College Ganeshkhind Pune
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.Similar to add maths module 5 (20)
MA 243 Calculus III Fall 2015 Dr. E. JacobsAssignmentsTh.docx
MA 243 Calculus III Fall 2015 Dr. E. JacobsAssignmentsTh.docx
F.Y.B.Sc(2013 pattern) Old Question Papers:Dr.Kshirsagar
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- 1. MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4 2007mozac 1 ADDITIONAL MATHEMATICS FORM 4 MODULE 5 DIFFERENTIATIONS
- 2. ADDITIONAL MATHEMATICS FORM 4 2007mozac 2 9 DIFFERENTIATIONS PAPER 1 1 Given y = 4(1 – 2x)3 , find dy dx . Answer : ………………………………… 2 Differentiate 3x2 (2x – 5)4 with respect to x. Answer : ………………………………… 3 Given that 2 1 (3 5) ( ) x h x , evaluate h’’(1). Answer : …………………………………
- 3. ADDITIONAL MATHEMATICS FORM 4 2007mozac 3 4 Differentiate the following expressions with respect to x. (a) (1 + 5x2 )3 (b) 2 43 4 x x Answer : (a) ………………………………… (b) ………………………………… 5 Given a curve with an equation y = (2x + 1)5 , find the gradient of the curve at the point x = 1. Answer : ………………………………… 6 Given y = (3x – 1)5 , solve the equation 2 2 12 0 d y dy dx dx Answer : …………………………………
- 4. ADDITIONAL MATHEMATICS FORM 4 2007mozac 4 7 Find the equation of the normal to the curve 53 2 xy at the point (1, 2). Answer : ………………………………… 8 Given that the curve qxpxy 2 has the gradient of 5 at the point (1, 2), find the values of p and q. Answer : p = ……………………………… q = ……………………………… 9 Given (2, t) is the turning point of the curve 142 xkxy . Find the values of k and t. Answer : k = ……………………………… t = ……………………………… 10 Given 22 yxz and xy 21 , find the minimum value of z. Answer : …………………………………
- 5. ADDITIONAL MATHEMATICS FORM 4 2007mozac 5 11 Given 12 tx and 54 ty . Find (a) dx dy in terms of t , where t is a variable, (b) dx dy in terms of y. Answer : (a) …………………………… (b) …………………………… 12 Given that y = 14x(5 – x), calculate (a) the value of x when y is a maximum, (b) the maximum value of y. Answer : (a) ………………………………… (b) ………………………………… 13 Given that y = x2 + 5x, use differentiation to find the small change in y when x increases from 3 to 301. Answer : …………………………………
- 6. ADDITIONAL MATHEMATICS FORM 4 2007mozac 6 14 Two variables, x and y, are related by the equation y = 3x + 2 x . Given that y increases at a constant rate of 4 units per second, find the rate of change of x when x = 2. Answer : ………………………………… 15 The volume of water, V cm3 , in a container is given by 31 8 3 V h h , where h cm is the height of the water in the container. Water is poured into the container at the rate of 10 cm3 s1 . Find the rate of change of the height of water, in cm s1 , at the instant when its height is 2 cm. Answer : ……………………………
- 7. ADDITIONAL MATHEMATICS FORM 4 2007mozac 7 PAPER 2 16 (a) Given that graph of function 2 3 )( x q pxxf , has gradient function 2 3 192 ( ) 6f x x x where p and q are constants, find (i) the values of p and q , (ii) x-coordinate of the turning point of the graph of the function. (b) Given 3 29 ( 1) 2 p t t . Find dt dp , and hence find the values of t where 9. dp dt 17 The gradient of the curve 4 k y x x at the point (2, 7) is 1 2 4 , find (a) value of k, (b) the equation of the normal at the point (2, 7), (c) small change in y when x decreases from 2 to 197. 18 The diagram above shows a piece of square zinc with 8 m sides. Four squares with 2x m sides are cut out from its four vertices.The zinc sheet is then folded to form an open square box. (a) Show that the volume, V m3 , is V = 128x – 128x2 + 32x3 . (b) Calculate the value of x when V is maximum. (c) Hence, find the maximum value of V. 8 m 8 m 2x m 2x m2x m 2x m 2x m 2x m 2x m 2x m
- 8. ADDITIONAL MATHEMATICS FORM 4 2007mozac 8 19 (a) Given that 12p q , where 0p and 0.q Find the maximum value of .2 qp (b) The above diagram shows a conical container of diameter 8 cm and height 6 cm. Water is poured into the container at a constant rate of 3 cm3 s1 . Calculate the rate of change of the height of the water level at the instant when the height of the water level is 2 cm. [Use = 3142 ; Volume of a cone = hr2 3 1 ] 20 (a) The above diagram shows a closed rectangular box of width x cm and height h cm. The length is two times its width and the volume of the box is 72 cm3 . (i) Show that the total surface area of the box, A cm2 is x xA 216 4 2 , (ii) Hence, find the minimum value of A. (b) The straight line 4y + x = k is the normal to the curve y = (2x – 3)2 – 5 at point E. Find (i) the coordinates of point E and the value of k, (ii) the equation of tangent at point E. 6 cm 8 cm h cm x cm 2x cm