DIPLOMA IN ENGINEERING I YEAR


 Nora French
 7 years ago
 Views:
Transcription
1 GOVERNMENT OF TAMILNADU DIRECTORATE OF TECHNICAL EDUCATION DIPLOMA IN ENGINEERING I YEAR SEMESTER SYSTEM L  SCHEME 00 I SEMESTER ENGINEERING MATHEMATICS  I CURRICULUM DEVELOPMENT CENTER
2 STATE BOARD OF TECHNICAL EDUCATION & TRAINING, TAMILNADU DIPLOMA IN ENGINEERING  SYLLABUS LSCHEME (Implements from the Academic Year 00 on wards) Course Name : All Branches of Diploma in Engineering and Technology and Special Programmes except DMOP, HMCT and Film & TV Subject Code : 00 Semester : I Semester Subject Title : ENGINEERING MATHEMATICS  I TRAINING AND SCHEME OF EXAMINATION: TOPICS AND ALLOCATION OF HOURS: No. of Weeks per Semester: 6 Weeks Subject Instructions Examination Hours / Hours / Duration Week Semester Marks Engineering Mathematics  I Internal Board Total 5 Hrs. 80 Hrs. Assessment Examination Hrs Sl.No. Topic Time (Hrs.) Determinants and Matrices 4. Binomial Theorem and Partial Fraction 4 3. Straight lines 4 4. Trigonometry I 4 5. Trigonometry II 4 Tutorial 0 Total 80 Rationale: This subject being a branch of Logic is classified as one of the basic sciences and intends to teach students, basic facts, concepts and principles of mathematics as a tool to analyse Engineering problems. Mathematics lay down foundation for understanding core technology subjects. Objectives: This subject helps the students to develop logical thinking which is useful in Comprehending the principles of all other subjects. Analytical and systematic approach towards any problem is developed through learning of this subject. Mathematics being a versatile subject can be used at every stage of human life. CDC DOTE ENGINEERING MATHEMATICSI Page of
3 LEARNING STRUCTURE: Application To understand the techniques and methods for solving Engg. Problems such as simultaneous equation involved in vibrations, ckts. Laws of friction, projections, Lami s thorem, stress strain curves etc. Work done moment of force about a point and line. Procedure To explain use of properties of determinants to solve problems, use of cramer s rule, matrix inversion Binomial Theorem for positive integral index and for rational index partial fraction of proper and improper fractions To find distance of a point from a fixed line, distance between parallel lines, derivation of angle between two lines and pair of lines, parallel line and perpendicular line condition for two lines and pair of lines. Using various techniques proving standard results which will be used in solving engineering problems. Using various technique proving standard results which will be used in solving engineering problems. Concepts Cramer s rule, Applications of properties to solve problems, Cramer s rule, Algebra of matrices. Inverse of a matrix Binomial Expansion for positive and negative integral index. Cases of finding partial fractions. Angle between lines combined equation of two lines through origin and not through origin. Angle between pair of lines. Trigonometrical ratios of allied angles compound angles and multiple angles. Inverse Trigonometrical ratios. Facts Definition of determinants properties of determinants definition of matrix. Definition of permutation and combination. Definition of polynomial fractions. Combined equation of two lines. Definition of Trigonometrical ratios, allied angles, compound angles and multiple angles. Definition of inverse Trigonometric functions. CDC DOTE ENGINEERING MATHEMATICSI Page 3 of
4 DETAILED SYLLABUS CONTENTS UNIT NAME OF TOPICS Hours Mark I II III DETERMINANTS. Definition and expansion of determinants of order and 3. Properties of determinants. Cramer s rule to solve simultaneous equations in and unknowns Simple Problems. Problems involving properties of Determinants 4 7 MATRICES.3 Definition of matrix. Types of matrices. Algebra of matrices such as equality, addition, subtraction, scalar multiplication and multiplication of matrices. Transpose a matrix, Adjoint matrix and Inverse matrix. Simple Problems 5 8 BINOMIAL THEOREM. Definition of factorial notation, definition of permutation and combinations with formula. Binomial theorem for positive integral index 5 8 (statement only), finding of general and middle terms. Simple Problems. Problems finding coefficient of x n, independent terms. Simple Problems. Binomial Theorem for rational index, expansions only upto for negative integers. Simple Expansions. PARTIAL FRACTIONS.3 Definition of Polynomial fraction, proper and improper fractions and definition of partial fractions. To resolve proper fraction into partial fraction with denominator containing 5 7 non repeated linear factors, repeated linear factors and irreducible non repeated quadratic factors. Simple Problems. STRAIGHT LINES 3. Length of perpendicular distance from a point to the line and perpendicular distance between parallel lines. Simple Problems 4 7 Angle between two straight lines and condition for parallel and perpendicular lines. Simple Problems PAIR OF STRAIGHT LINES THROUGH ORIGIN 3. Pair of lines passing through the origin ax +hxy+by =0 expressed in the form (ym x)(ym x)=0. Derivation of tan = ± parallel and perpendicular lines. Simple Problems h ab, condition of a + b 5 8 CDC DOTE ENGINEERING MATHEMATICSI Page 4 of
5 UNIT NAME OF TOPICS Hours Mark III IV V PAIR OF STRAIGHT LINES NOT THROUGH ORIGIN 3.3 Condition for General equation of the second degree ax +hxy+by +gx+fy+c=0 to represent pair of lines. a h g h b f = 0 (statement only) g f c Angle between them, condition for parallel and perpendicular lines. Simple Problems 5 7 TRIGONOMENTRY 4. Trigonometrical ratios of allied angles. Expansion of sin (A ± B) and 5 8 cos (A ± B). Problems using above expansion. 4. Expansion of tan (A ± B) and problems using this expansion Trigonometrical ratios of multiple angles (A only) and sub multiple angles. Simple Problems. 5 7 TRIGNOMENTRY 5. Trigonometrical ratios of multiple angels (3A only). Simple Problems Sum and product formulae. Simple Problems Definition of inverse trigonometric ratios relation between inverse trigonometric ratios. Simple Problems 5 7 Text Book: Mathematics for Higher Secondary I year and II year (Tamil nadu Text Book Corporation) Reference Book: Engineering Mathematics  Dr.M.K.Venkatraman, National Publishing Co, Chennai Engineering Mathematics Dr.P.Kandasamy & Others, S.Chand & Co Ltd, New Delhi CDC DOTE ENGINEERING MATHEMATICSI Page 5 of
6 MODEL QUESTION PAPER ENGINEERING MATHEMATICS I Time : 3Hrs PART A Max Marks:75 I. Answer Any 5 Questions 5x=5. Solve : x 3 x x = 0. Find the value of a b c b c a c =0 a b If A =, B Find AB Find the adjoint matrix of 5. Find the value of 0C 7 6. Find the general term of 3x y 8 7. Expand 3 x upto three terms when x 8. Split x into partial fraction without finding the constant x x 9. Find the value of m if the lines x+my=4 and x+5y6=0 are perpendicular. 0. Find the combined the equation of the lines x+5y=0 and x+3y=0.. Show that the pair of lines x 8y 6y 0 are parallel. Write down the condition for the equation ax hxy by gx fy c 0 to represent a pair of straight lines. CDC DOTE ENGINEERING MATHEMATICSI Page 6 of
7 3. Show that sin (330) x sin 40 = Find the value of cos50 cos40 sin 50 sin If tana= tan B = 3 Find tan (A+B) 6. Find the value of sin75 cos75 7 If sin=/3,find the value of sin3 8. Find the value of 4cos 3 03cos0 9. Show that SinAsinB = Cot(A+B) CosAcosB 0. Show that tan  x = tan  x x PART B (Answer Any two subdivision in each question ) 5x=60 All Questions carry Equal Marks a. Solve by using Cramer s Rule : x+y+z=3, xy+z = and 3x+yz=3 b. Short that x x =x (x+3) x c. Find the inverse of a 3 Find the middle terms in the expansion of x 3 x b. Find the term independent of x in the expansion of (x +/x) c. Resolve x 3 in to partial fraction x( x 5)( x 6) CDC DOTE ENGINEERING MATHEMATICSI Page 7 of
8 3. a Find the angle between the lines 7x+y= and x5y=0 b. Find the separate equation of the pair of straight lines 9x +xy+4y =0. Also prove that the lines are parallel. c. Show that the equations represented by x 7xy 3y 5x 5y 0 is a pair of straight lines 4. a If sin A = 0 and sin B prove that A+B= 5 4 b. If A+B =45 prove that (+tana) (+tanb)= and hence deduce the value of tan c. sin A sin A Prove that tan A cos A cos A 5. a Prove that sin0 sin 40 sin 60 sin80 = 3 6 b. Prove that (cosαcosβ) + (sinαsinβ) =4 sin c. Show that tan  xy +xy =tan  x+tan  y ******** αβ CDC DOTE ENGINEERING MATHEMATICSI Page 8 of
9 MODEL QUESTION PAPER ENGINEERING MATHEMATICS I Time : 3Hrs PART A Max Marks:75 I. Answer Any 5 Questions 5x=5. Find x if x 0 0 x  =0 x x 3x. Prove that 4x 5x 6x 0 7x 8x 9x If A = Find A 4. Find the inverse of 0 5. Find the th term of 3x x 0 6. How many middle terms are in the expansion of (5xy) 9 7. Write the first three terms in the expansion of (x)  8. Without finding the constants split 3 x 3 in to partial fraction. ( x )( x ) 9. Find the perpendicular distances from the point (,) to the straight line 3x+y+=0. 0. Write down the condition for the pair of lines given by ax +hxy+by =0 to be parallel.. Find a if the lines represented by 3x +4xy+ay =0 are perpendicular. State the expression for angle between pair of line given by ax +hxy+by +gx+fy+c=0 3 Find the value sin5 o without using tables or calculator. 4. Find the value of cos40 o Cos 0 o + sin 40 o sin0 o CDC DOTE ENGINEERING MATHEMATICSI Page 9 of
10 5. Simplify tan tan 3 tan tan 3 6. Prove that sin A = tana cos A 7. Find the value of 3sin0 o 4 sin 3 0 o 8. If Cos A = 5 3 find the value of cos3a. 9. Show that Cos0 +Cos00+Cos40=0. x 0. Show that sin  = tan  x. x PART B (Answer any two subdivision in each Question) 5x=60 All Questions carry Equal Marks.a) Solve the equations 4x+y+z=6, xyz=6 and x+y+z=3, using Cramer s rule x a b c b) Prove that a x b c =x (x+a+b+c) a b x c c) If A = show that A 5A 4I 0. a Find the middle terms in the expansion of (x+/x) 3 b. Find the term independent of x in the expansion of 7x 4 c. Resolve in to a partial function. ( x )( x ) x x 3. a Derive the expression for angle between two lines y=m x+c and y=m x+c b. If the slope of one of the straight lines ax hxy by 0 is twice that of the other. Show that 8h =9ab. c. Find the value of ' ' so that the equation 3x +4xy+8y 8xy+λ=0 represents a pair of straight lines. CDC DOTE ENGINEERING MATHEMATICSI Page 0 of
11 3 4. a If sin A =, cos B =, find the values of sin (AB) and cos (AB) 5 3 b. If A+B=45 o Prove that (cota) (cotb)=. Also find the value of cot c. Show that +sin cos = tan/ +sin +cos 3 3 cos A cos3a sin A sin 3A 5. a Prove that 3 cos A sin A b. If a=sina+sinb, B=cosA+cosB, Show that tan AB = c. Show that tan  3xx 3 = 3tan  x 3x a +b ******** 4(a +b ) a +b CDC DOTE ENGINEERING MATHEMATICSI Page of
Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationx(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3
CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract 
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationNational 5 Mathematics Course Assessment Specification (C747 75)
National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for
More informationAlgebra I Credit Recovery
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,
More informationDRAFT. Further mathematics. GCE AS and A level subject content
Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed
More informationPRECALCULUS GRADE 12
PRECALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationExtra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationMath Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices.
Math Placement Test Study Guide General Characteristics of the Test 1. All items are to be completed by all students. The items are roughly ordered from elementary to advanced. The expectation is that
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationUnit 4: 15 Progression : Definition of A.P. & G.P., To find Tn & Sn, Simple practical commercial problems.
Solapur University, Solapur. Semester Pattern Syllsbus B.Com Part I Business Mathematics (w. e. f. June 2010) Semester I Periods Unit 1: 15 Determinant & Matrices : Definition of second & Third order Determinant,
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationis identically equal to x 2 +3x +2
Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3
More informationDEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x
Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationKEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 Curriculum. September 2007
KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 447) Surveys and
More informationPrerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303.
Course Syllabus Math 1314 College Algebra Revision Date: 82115 Catalog Description: Indepth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationMathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 PreAlgebra 4 Hours
MAT 051 PreAlgebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT
More informationPrecalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES
Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 01 Sets There are no statemandated Precalculus 02 Operations
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationNotes and questions to aid Alevel Mathematics revision
Notes and questions to aid Alevel Mathematics revision Robert Bowles University College London October 4, 5 Introduction Introduction There are some students who find the first year s study at UCL and
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More information3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes
Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same
More informationMATH. ALGEBRA I HONORS 9 th Grade 12003200 ALGEBRA I HONORS
* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSAMAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationFOREWORD. Executive Secretary
FOREWORD The Botswana Examinations Council is pleased to authorise the publication of the revised assessment procedures for the Junior Certificate Examination programme. According to the Revised National
More informationPCHS ALGEBRA PLACEMENT TEST
MATHEMATICS Students must pass all math courses with a C or better to advance to the next math level. Only classes passed with a C or better will count towards meeting college entrance requirements. If
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationMathematics PreTest Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}
Mathematics PreTest Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {1, 1} III. {1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationAMSCO S Ann Xavier Gantert
AMSCO S Integrated ALGEBRA 1 Ann Xavier Gantert AMSCO SCHOOL PUBLICATIONS, INC. 315 HUDSON STREET, NEW YORK, N.Y. 10013 Dedication This book is dedicated to Edward Keenan who left a profound influence
More informationHere are some examples of combining elements and the operations used:
MATRIX OPERATIONS Summary of article: What is an operation? Addition of two matrices. Multiplication of a Matrix by a scalar. Subtraction of two matrices: two ways to do it. Combinations of Addition, Subtraction,
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More information5.3 SOLVING TRIGONOMETRIC EQUATIONS. Copyright Cengage Learning. All rights reserved.
5.3 SOLVING TRIGONOMETRIC EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use standard algebraic techniques to solve trigonometric equations. Solve trigonometric equations
More informationMath Placement Test Sample Problems PREALGEBRA
Math Placement Test Sample Problems The Math Placement Test is an untimed, multiplechoice, computerbased test. The test is composed of four sections: prealgebra, algebra, college algebra, and trigonometry.
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationPartial Fractions Examples
Partial Fractions Examples Partial fractions is the name given to a technique of integration that may be used to integrate any ratio of polynomials. A ratio of polynomials is called a rational function.
More informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More information1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).
.7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational
More information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationThe Mathematics Diagnostic Test
The Mathematics iagnostic Test Mock Test and Further Information 010 In welcome week, students will be asked to sit a short test in order to determine the appropriate lecture course, tutorial group, whether
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationAnalysis of Stresses and Strains
Chapter 7 Analysis of Stresses and Strains 7.1 Introduction axial load = P / A torsional load in circular shaft = T / I p bending moment and shear force in beam = M y / I = V Q / I b in this chapter, we
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More informationContent. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11
Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationNumber Sense and Operations
Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents
More informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationMath Common Core Sampler Test
High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests
More informationHow To Understand And Solve Algebraic Equations
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More informationMATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!
MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Prealgebra Algebra Precalculus Calculus Statistics
More informationOxford Cambridge and RSA Examinations
Oxford Cambridge and RSA Examinations OCR FREE STANDING MATHEMATICS QUALIFICATION (ADVANCED): ADDITIONAL MATHEMATICS 6993 Key Features replaces and (MEI); developed jointly by OCR and MEI; designed for
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 20072008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 20072008 Pre s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationAlgebra 1. Curriculum Map
Algebra 1 Curriculum Map Table of Contents Unit 1: Expressions and Unit 2: Linear Unit 3: Representing Linear Unit 4: Linear Inequalities Unit 5: Systems of Linear Unit 6: Polynomials Unit 7: Factoring
More informationMathematics Georgia Performance Standards
Mathematics Georgia Performance Standards K12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by
More informationMathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework
Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010  A.1 The student will represent verbal
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationMath 1050 Khan Academy Extra Credit Algebra Assignment
Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In
More informationHIBBING COMMUNITY COLLEGE COURSE OUTLINE
HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE:  Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement Primary
Shape, Space, and Measurement Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two and threedimensional shapes by demonstrating an understanding of:
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationGRE Prep: Precalculus
GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach
More informationCM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation
CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of
More informationMatrices 2. Solving Square Systems of Linear Equations; Inverse Matrices
Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices Solving square systems of linear equations; inverse matrices. Linear algebra is essentially about solving systems of linear equations,
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationAlgebra 2 PreAP. Name Period
Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationAlgebra Practice Problems for Precalculus and Calculus
Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials
More information