Contents. (Term II) Latest Syllabus
|
|
- Solomon Jessie Dalton
- 6 years ago
- Views:
Transcription
1 Contents (Term II) Preface Latest Syllabus (v) (vii). Quadratic Equations Arithmetic Progressions Coordinate Geometry Some Applications of Trigonometry Circles Constructions Areas Related to Circles Surface Areas and Volumes Probability Value Based Questions Practice Papers (I III) xv
2 Each Chapter Contains: TIPS AND TRICKS FORMATIVE ASSESSMENT SUMMATIVE ASSESSMENT CBSE and Other Important Questions Objective Type Questions Higher Order Thinking Skills (HOTS) Questions NCERT Textbook Exercises
3 Quadratic Equations CHAPTER Tips and Tricks Standard form of a quadratic equation: The standard form of a quadratic equation in variable x is ax + bx + c, a 0 Roots of a quadratic equation: A real number α is said to be a root of the quadratic equation ax + bx + c = 0 if aα + bα + c = 0. Note: The roots of the quadratic equation ax + bx + c = 0 are the same as the zeroes of the quadratic polynomial ax + bx + c. Solving a quadratic equation (i) by factorisation: If we can factorise the quadratic polynomial ax + bx + c into two real linear factors, then the roots of the quadratic equation ax + bx + c = 0 can be found by equating each factor to zero. (ii) by completing the square: By adding and subtracting a suitable constant, as required, we club the x and x terms in the quadratic equation so that they become a complete square and then solve for x. (iii) by quadratic formula: The roots of the quadratic equation ax + bx + c = 0 are given by x = b± b ac, provided b ac 0 a Note: The expression b ac is called the discriminant of the quadratic equation. Nature of roots: Value of b ac Nature of roots (i) > 0 two distinct real roots (ii) = 0 two equal real roots (iii) < 0 no real roots Sol. 1. Solve for x: We have, ILLUSTRATIVE EXAMPLES CBSE Exam. 5 3 = x x + 3, x 0, 3 3 x = 5 x x = 5 x x + 3 ( 3x) (x + 3) = 5x 8x + 1 6x 9x = 5x 6x + 6x 1 = 0 x + x = 0 x + x x = 0 x(x + ) 1(x + ) = 0 (x + ) (x 1) = 0 x + = 0 or x 1 = 0 x = or x = 1 x =, 1.. Solve for x: x + 1 x x + = 3; x 1, 1 x + Sol. We have, x + 1 x x + 1 x + = 3 A-
4 QUADRATIC EQUATIONS 3 ( x + 1)( x + ) + ( x )( x 1) = 3 ( x 1)( x + ) x + x + x + + x x x + = 3 x + x x x + = 3 x + x x + = 3(x + x ) x + = 3x + 3x 6 x + 3x 10 = 0 x + 5x x 10 = 0 x(x + 5) (x + 5) = 0 (x + 5) (x ) = 0 x + 5 = 0 or x = 0 x = 5 or x = x = 5,. 3. Two water taps together can fill a tank in 9 3 hours. The 8 ( x 10) + x x( x 10) = 8 75 x 10 = 8 x 10x 75 ( x 5 ) x 10x = 8 75 x 5 = x 10x 75 (x 10x) = 75(x 5) x 0x = 75x 375 x 115x = 0 x 100x 15x = 0 x (x 5) 15(x 5) = 0 (x 5) (x 15) = 0 x 5 = 0 or x 15 = 0 x = 5 or x = 15 Sol. tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank. Let the tap of smaller diameter take x hours to fill the tank alone. Then, the tap of larger diameter will take (x 10) hours to fill the tank. Tank filled by the tap of smaller diameter in 1 hour = 1 x Tank filled by the tap of larger diameter in 1 hour 1 = x 10 Tank filled by both the tanks together in 1 hour = x x 10 Tank filled by both the tanks together in hours = F HG F HG x x 10 I KJ I KJ = x x 10 According to the question, 75 F 1 1 I + = 1 8 HG x x 10KJ = 8 x x x = 5, 15 x = 15 is inadmissible as then x 10 is negative which is not possible. [Time cannot be ve] x = 5 x 10 = 15 Hence, tap of smaller and larger diameter take respectively 5 hours and 15 hours to fill the tank alone.. Find value of p such that the quadratic equation (p 1)x (p 1)x + = 0 has equal roots. Sol. The given quadratic equation is (p 1)x (p 1)x + = 0...(1) Comparing it with Ax + Bx + C = 0, we get A = p 1 B = (p 1) C = For equal roots, Discriminant = 0 B AC = 0 B = AC { (p 1)} = (p 1) () (p 1) = (p 1) () (p 1) = (p 1)
5 CCE MATHEMATICS X Sol. (p 1) (p 1) = 0 (p 1) {(p 1) } = 0 (p 1) (p 1) = 0 p 1 = 0 or p 1 = 0 p = 1 or p = 1 p = 1, 1 p = 1 is inadmissible as then from the given equation = 0 which is impossible. p = 1 5. Solve: 1 1 x + x 7 The given equation is 1 1 x + x 7 ( x 7) ( x + ) ( x + )( x 7) 11 = ( x + )( x 7) = ; x, 7 = = (x + ) (x 7) = 30 x + x 7x 8 = 30 x 3x + = 0 x x x + = 0 x (x 1) (x 1) = 0 (x 1) (x ) = 0 x 1 = 0 or x = 0 x = 1 or x = x = 1, 6. The length of a rectangular plot is greater than thrice its breadth by m. The area of the plot is 10 sq. m. Find the length and breadth of the plot. Sol. Let the breadth of the plot be x m. Then, length of the plot = (3x + ) m Area of the plot = length breadth = (3x + )x m According to the question, (3x + )x = 10 3x + x = 10 3x + x 10 = 0 3x + 0x 18x 10 = 0 x(3x + 0) 6(3x + 0) = 0 (3x + 0) (x 6) = 0 3x + 0 = 0 or x 6 = 0 3x = 0 or x = 6 x = 0 3 or x = 6 x = 0 3, 6 x = 0 is inadmissible as breadth cannot be ve. 3 x = 6 3x + = 3(6) + = 0 Hence, the length and breadth of the plot are 0 m and 6 m, respectively. Formative Assessment ORAL QUESTIONS (Conversation Type) 1. What is the standard form of a quadratic equation in variable x?. What is the expression for the discriminant of the quadratic equation ax + bx + c = 0? 3. What is the maximum number of roots of a quadratic equation?. In a quadratic equation ax + bx + c = 0, if a = 0, then how will you call this equation? 5. What is the condition that a root of the quadratic equation ax + bx + c = 0 is 1? TRUE OR FALSE 1. If the coefficient of x and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots.. If the coefficient of x and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots. 3. Every quadratic equation has at most two roots.. Every quadratic equation has at least two roots. 5. Every quadratic equation has exactly one root.
6 QUADRATIC EQUATIONS 5 Assignments Name:... Class:... Section:... Roll No.:... Grade:... Teacher s sign.:... CLASS ASSIGNMENT 1 1. Find the roots of the equation x 5 = 0.. Find the discriminant of the quadratic equation x + x 5 = Is 3 a root of the equation x 3 = 0?. What must be added and subtracted to solve the quadratic equation x 3x + = 0 by the method of completing the square? 5. If 1 is a root of the quadratic equation x + x + k = 0, then find the value of k. 6. The sum of a number and its reciprocal is Form a quadratic equation for this situation. 7. Does the equation ( x 3) = 0 have two equal roots? 8. Is x 3 x = (x + 1) 3 a quadratic equation? 9. The sum of the squares of two consecutive natural numbers is 5. Form a quadratic equation for this situation. 10. What is the sum of the roots of the quadratic equation x 3 x + 1 = 0?
7 6 CCE MATHEMATICS X Name:... Class:... Section:... Roll No.:... Grade:... Teacher s sign.:... CLASS ASSIGNMENT (From CBSE Examination Paper) 1. If r = 3 is a root of the quadratic equation kr kr 3 = 0, find the value of k.. Find the nature of the roots of the equation 3x x + 3 =0. 3. If 1 is a root of the equation x + kx + 5 = 0, then find the value of k.. Prove that x = 3 is a root of the equation x 5x 3 = If one root of the equation x 10x + p = 0 is, then find the value of p. 6. Find the roots of the equation x + x p(p + 1) = 0, where p is a constant. 7. Find the roots of the equation x 3x m (m + 3) = 0, where m is a constant. 8. For what value of k, the quadratic equation 9x + 8kx + 16 = 0 has equal roots? 9. For the quadratic equation x x + 1 = 0, find the value of x + 1 x. 10. Find the nature of roots of the quadratic equation 3 1 x x + = 0.
8 QUADRATIC EQUATIONS 7 Name:... Class:... Section:... Roll No.:... Grade:... Teacher s sign.:... HOME ASSIGNMENT 1 1. The area of a triangle is 5 cm. The altitude to the base is cm less than its corresponding base. Form a quadratic equation for this situation. 6. Is x + x + 1 = ( x) + 1 a quadratic equation? 7. Find the discriminant of the quadratic equation 5x 3x + 1 = 0.. One diagonal of a rhombus is cm less than the other. The area of the rhombus is 36 cm. Form a quadratic equation for this situation. 8. What are the roots of the equation x x = 0? 3. Can you say that 0.1 is a root of the equation x 0.01 =0?. What must be added and subtracted to solve the qua- 3 dratic equation 9x + x = 0 by the method of completing the square? 5. If 1 is a root of the quadratic equation x + x + k = 0, then find the value of k. 9. What is the product of the roots of the quadratic equation x 5x + 1 = 0? 10. For the quadratic equation ax + bx + c = 0, what is the expression for the value of x using quadratic formula?
9 8 CCE MATHEMATICS X Name:... Class:... Section:... Roll No.:... Grade:... Teacher s sign.:... HOME ASSIGNMENT (From NCERT Exemplary Problems) 1. Is 0. a root of the equation x 0. = 0?. Does (x 1) + (x + 1) = 0 have a real root? 6. Find the roots of 6x x = 0 by the factorisation of the corresponding quadratic polynomial. 7. The square of a natural number diminished by 8 is equal to thrice of 8 more than the given number. Form a quadratic equation for this situation. 3. Which constant should be added and subtracted to solve the quadratic equation x 3 x 5 = 0 by the method of completing the square? 8. A natural number, when increased by 1, equals 160 times its reciprocal. Form a quadratic equation for this situation.. How many real roots does the equation (x + 1) x = 0 have? F H 9. Is x = (5 x) x 3 I K a quadratic equation? 5. Find the roots of the quadratic equation x 5 x = 0 by using the quadratic formula. 10. Find the value of k for which the quadratic equation x kx + k = 0 has equal roots.
QUADRATIC EQUATIONS CHAPTER 4. (A) Main Concepts and Results
CHAPTER QUADRATIC EQUATIONS (A) Main Concepts and Results Quadratic equation : A quadratic equation in the variable x is of the form ax + bx + c = 0, where a, b, c are real numbers and a 0. Roots of a
More informationCONTENTS. iii v viii FOREWORD PREFACE STUDENTS EVALUATION IN MATHEMATICS AT SECONDARY STAGE
CONTENTS FOREWORD PREFACE STUDENTS EVALUATION IN MATHEMATICS AT SECONDARY STAGE iii v viii CHAPTER Real Numbers CHAPTER Polynomials 8 CHAPTER 3 Pair of Linear Equations in Two Variables 6 CHAPTER 4 Quadratic
More informationExercise 4.1. Question 1: Check whether the following are quadratic equations: Answer:
Question 1: Exercise 4.1 Check whether the following are quadratic equations: It is of the form. Hence, the given equation is a quadratic equation. It is of the form. Hence, the given equation is a quadratic
More informationQUADRATIC EQUATIONS. 4.1 Introduction
70 MATHEMATICS QUADRATIC EQUATIONS 4 4. Introduction In Chapter, you have studied different types of polynomials. One type was the quadratic polynomial of the form ax + bx + c, a 0. When we equate this
More information4. QUADRATIC EQUATIONS
. QUADRATIC EQUATIONS Important Terms, Definitions and Results The roots of a quadratic equation ax + bx + c = 0 An equation of the form ax + bx + c = 0, where a, b, c are real numbers and a 0, is called
More informationSection 7.1 Quadratic Equations
Section 7.1 Quadratic Equations INTRODUCTION In Chapter 2 you learned about solving linear equations. In each of those, the highest power of any variable was 1. We will now take a look at solving quadratic
More informationPlan for Beginning of Year 2: Summer assignment (summative) Cumulative Test Topics 1-4 (IB questions only/no retakes) IA!!
Summer Assignment 018 The IB Math SL class covers six different mathematical topics (Algebra, Functions, Trigonometry, Vectors, Probability and Statistics, and Calculus). In an effort to best prepare you
More informationUNIT I : NUMBER SYSTEMS
CLASS X First Term Marks : 80 UNITS MARKS I. NUMBER SYSTEMS 10 II. ALGEBRA 20 III. GEOMETRY 15 IV TRIGONOMETRY 20 V STATISTICS 15 TOTAL 80 UNIT I : NUMBER SYSTEMS 1. REAL NUMBERS (15) Periods Euclid's
More informationOSWAAL BOOKS MARCH 2019 EXAM LEARNING MADE SIMPLE. Published by : FOR CBSE
OSWAAL BOOKS LEARNING MADE SIMPLE FOR CBSE MARCH 2019 EXAM SOLVED PAPER 2018 MATHEMATICS CLASS 10 Published by : OSWAAL BOOKS 1/11, Sahitya Kunj, M.G. Road, Agra - 282002, UP (India) Ph.: 0562 2857671,
More informationMATHEMATICS (IX-X) (Code No. 041)
MATHEMATICS (IX-X) (Code No. 041) The Syllabus in the subject of Mathematics has undergone changes from time to time in accordance with growth of the subject and emerging needs of the society. The present
More informationPLC Papers Created For:
PLC Papers Created For: Year 11 Topic Practice Paper: Factorising Quadratics Factorising difficult quadratic expressions 1 Grade 7 Objective: Factorise a quadratic expression of the form ax 2 + bx + c
More informationDownload PDF Syllabus of Class 10th CBSE Mathematics Academic year
Download PDF Syllabus of Class 10th CBSE Mathematics Academic year 2018-2019 Download PDF Syllabus of Class 11th CBSE Mathematics Academic year 2018-2019 The Syllabus in the subject of Mathematics has
More informationWritten as per the syllabus prescribed by the Central Board of Secondary Education. CBSE CLASS X MATHEMATICS
Written as per the syllabus prescribed by the Central Board of Secondary Education. CBSE CLASS X MATHEMATICS TERM - II Salient Features Etensive coverage of the syllabus for Term - II in an effortless
More informationCCE RR. ( / English Version ) ( / New Syllabus ) ( / Regular Repeater )
CCE RR 1 81-E : 81-E Code No. : 81-E CCE RR Subject : MATHEMATICS ( / English Version ) ( / New Syllabus ) ( / Regular Repeater ) General Instructions : i) The Question-cum-Answer Booklet consists of objective
More informationDinwiddie County Subject: Trigonometry Scope and Sequence
Dinwiddie County Subject: Trigonometry Scope and Sequence GRADE: High School 9 WKS Topics Equations (linear, quadratic, and absolute value) and Radicals (simplest radical form, including rationalizing
More informationPLC Papers Created For:
PLC Papers Created For: Quadratics intervention Deduce quadratic roots algebraically 1 Grade 6 Objective: Deduce roots algebraically. Question 1. Factorise and solve the equation x 2 8x + 15 = 0 Question
More informationTERMWISE SYLLABUS SESSION CLASS-X SUBJECT : MATHEMATICS Course Structure
TERMWISE SYLLABUS SESSION--19 CLASS-X SUBJECT : MATHEMATICS Course Structure Units Unit Name Marks I NUMBER SYSTEMS 06 II ALGEBRA 20 III COORDINATE GEOMETRY 06 IV GEOMETRY 15 V TRIGONOMETRY 12 VI MENSURATION
More informationMathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) May 2010
Link to past paper on OCR website: http://www.mei.org.uk/files/papers/c110ju_ergh.pdf These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or
More informationClass IX Chapter 2 Polynomials Maths
NCRTSOLUTIONS.BLOGSPOT.COM Class IX Chapter 2 Polynomials Maths Exercise 2.1 Question 1: Which of the following expressions are polynomials in one variable and which are No. It can be observed that the
More informationJUST THE MATHS UNIT NUMBER 1.6. ALGEBRA 6 (Formulae and algebraic equations) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.6 ALGEBRA 6 (Formulae and algebraic equations) by A.J.Hobson 1.6.1 Transposition of formulae 1.6. of linear equations 1.6.3 of quadratic equations 1.6. Exercises 1.6.5 Answers
More informationKENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD-32. SECTION A Questions 1 to 6 carry 1 mark each.
KENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD-32 SAMPLE PAPER TEST 03 (2018-19) (ANSWERS) SUBJECT: MATHEMATICS MAX. MARKS : 80 CLASS : X DURATION : 3 HRS General Instruction: (i) All questions are compulsory.
More informationANSWERS. CLASS: VIII TERM - 1 SUBJECT: Mathematics. Exercise: 1(A) Exercise: 1(B)
ANSWERS CLASS: VIII TERM - 1 SUBJECT: Mathematics TOPIC: 1. Rational Numbers Exercise: 1(A) 1. Fill in the blanks: (i) -21/24 (ii) -4/7 < -4/11 (iii)16/19 (iv)11/13 and -11/13 (v) 0 2. Answer True or False:
More informationQUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE
6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write them in standard form. You will
More information3.3 Real Zeros of Polynomial Functions
71_00.qxp 12/27/06 1:25 PM Page 276 276 Chapter Polynomial and Rational Functions. Real Zeros of Polynomial Functions Long Division of Polynomials Consider the graph of f x 6x 19x 2 16x 4. Notice in Figure.2
More informationGovernment of Karnataka MATHEMATICS KTBS TENTH STANDARD PART-II
Government of Karnataka MATHEMATICS TENTH STANDARD PART-II KARNATAKA TEXT BOOK SOCIETY (R) No.4, 100 Feet Ring Road Banashankari 3rd Stage, Bengaluru - 560 085 CONTENTS Unit No PART - II Unit Name Page
More informationLesson 3.5 Exercises, pages
Lesson 3.5 Exercises, pages 232 238 A 4. Calculate the value of the discriminant for each quadratic equation. a) 5x 2-9x + 4 = 0 b) 3x 2 + 7x - 2 = 0 In b 2 4ac, substitute: In b 2 4ac, substitute: a 5,
More informationMathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) June 2010
Link to past paper on OCR website: www.ocr.org.uk The above link takes you to OCR s website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS (MEI), VIEW ALL DOCUMENTS,
More informationMath 1 Unit 1 EOC Review
Math 1 Unit 1 EOC Review Name: Solving Equations (including Literal Equations) - Get the variable to show what it equals to satisfy the equation or inequality - Steps (each step only where necessary):
More informationPreCalc 11 Final Review Pack v1 Answer Section
PreCalc Final Review Pack v Answer Section MULTIPLE CHOICE. ANS: A PTS: DIF: Difficult REF:. Arithmetic Sequences LOC:.RF9. ANS: C PTS: DIF: Difficult REF:. Arithmetic Sequences LOC:.RF9. ANS: C PTS: DIF:
More informationRoots and Coefficients Polynomials Preliminary Maths Extension 1
Preliminary Maths Extension Question If, and are the roots of x 5x x 0, find the following. (d) (e) Question If p, q and r are the roots of x x x 4 0, evaluate the following. pq r pq qr rp p q q r r p
More informationAlgebra of. polynomials2
polynomials2 Algebra of Polynomial functions are used to model many physical situations. One such example, the effect of gravity on falling objects, was investigated experimentally by Galileo Galilei (1564
More informationDOWNLOAD OR READ : GET SOLUTIONS FROM QUADRATIC EQUATION PDF EBOOK EPUB MOBI
DOWNLOAD OR READ : GET SOLUTIONS FROM QUADRATIC EQUATION PDF EBOOK EPUB MOBI Page 1 Page 2 get solutions from quadratic equation get solutions from quadratic pdf get solutions from quadratic equation Free
More information27 th ARYABHATTA INTER-SCHOOL MATHEMATICS COMPETITION 2010 CLASS = VIII
7 th ARYABHATTA INTER-SCHOOL MATHEMATICS COMPETITION 00 CLASS = VIII Time Allowed: Hours Max. Marks: 00 Roll No. of the Participant: GENERAL INSTRUCTIONS :. Participant should not write his/her name on
More informationA Study Guide for. Students PREPARING FOR GRADE. Nova Scotia Examinations in Mathematics
A Study Guide for Students PREPARING FOR 12 GRADE Nova Scotia Examinations in Mathematics A Study Guide for Students PREPARING FOR 12 GRADE Nova Scotia Examinations in Mathematics For more information,
More informationQuantile Textbook Report
Quantile Textbook Report Algebra 2 Author Charles, Randall I., et al StateEdition West Virginia Grade Algebra 2 1 Expressions, Equations, and Inequalities 1.1 Patterns and Expressions 930Q 1.2 Properties
More informationQuestion Bank Quadratic Equations
Question Bank Quadratic Equations 1. Solve the following equations for x : (i) 8x + 15 6x (ii) x (x + 5) 5 Solution. (i) Given 8x + 15 6x 8x 6x + 15 0 [Putting in the form as ax + bx + c 0] 8x 0x 6x +
More informationMathematics: Year 12 Transition Work
Mathematics: Year 12 Transition Work There are eight sections for you to study. Each section covers a different skill set. You will work online and on paper. 1. Manipulating directed numbers and substitution
More informationAlgebraic. techniques1
techniques Algebraic An electrician, a bank worker, a plumber and so on all have tools of their trade. Without these tools, and a good working knowledge of how to use them, it would be impossible for them
More informationSOLUTION OF QUADRATIC EQUATIONS LESSON PLAN. A3 Topic Overview ALGEBRA
ALGEBRA A Topic Overview A SOLUTION OF QUADRATIC EQUATIONS This topic describes three methods of solving Quadratic equations. assumes you understand and have practised using the algebraic methods described
More informationAPPENDIX : PARTIAL FRACTIONS
APPENDIX : PARTIAL FRACTIONS Appendix : Partial Fractions Given the expression x 2 and asked to find its integral, x + you can use work from Section. to give x 2 =ln( x 2) ln( x + )+c x + = ln k x 2 x+
More informationIn a quadratic expression the highest power term is a square. E.g. x x 2 2x 5x 2 + x - 3
A. Quadratic expressions B. The difference of two squares In a quadratic expression the highest power term is a square. E.g. x + 3x x 5x + x - 3 If a quadratic expression has no x term and both terms are
More informationCM2104: Computational Mathematics General Maths: 2. Algebra - Factorisation
CM204: Computational Mathematics General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of simplifying algebraic expressions.
More informationCHAPTER 2 POLYNOMIALS KEY POINTS
CHAPTER POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, and 3 are called linear, quadratic and cubic polynomials respectively.. A quadratic polynomial in x with real coefficient is of the form a x
More informationPROGRAM OF WORK( ) SUBJECT: MATHEMATICS
0 TH STANDARD ENGLISH MEDIUM PROGRAM OF WORK(08-9) (NEW SYLLABUS) SUBJECT: MATHEMATICS PART- LESSON NO. UNIT NAME ALLOTED PERIODS ARITHMETIC PROGRESSION 0 TRIANGLES 8 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
More information3.0 INTRODUCTION 3.1 OBJECTIVES 3.2 SOLUTION OF QUADRATIC EQUATIONS. Structure
UNIT 3 EQUATIONS Equations Structure 3.0 Introduction 3.1 Objectives 3.2 Solution of Quadratic Equations 3.3 Quadratic Formula 3.4 Cubic and Bioquadratic Equations 3.5 Answers to Check Your Progress 3.6
More informationCOURSE STRUCTURE CLASS -X
COURSE STRUCTURE CLASS -X Units Unit Name Marks I NUMBER SYSTEMS 06 II ALGEBRA 20 III COORDINATE GEOMETRY 06 IV GEOMETRY 15 V TRIGONOMETRY 12 VI MENSURATION 10 VII STATISTICS & PROBABILTY 11 Total 80 UNIT
More informationA101 ASSESSMENT Quadratics, Discriminant, Inequalities 1
Do the questions as a test circle questions you cannot answer Red (1) Solve a) 7x = x 2-30 b) 4x 2-29x + 7 = 0 (2) Solve the equation x 2 6x 2 = 0, giving your answers in simplified surd form [3] (3) a)
More informationJakarta International School 8 th Grade AG1 Summative Assessment
Jakarta International School 8 th Grade AG1 Summative Assessment Unit 6: Quadratic Functions Name: Date: Grade: Standard Advanced Highly Advanced Unit 6 Learning Goals NP Green Blue Black Radicals and
More informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:
More informationModesto Junior College Course Outline of Record MATH 122
Modesto Junior College Course Outline of Record MATH 122 I. OVERVIEW The following information will appear in the 2009-2010 catalog MATH 122 Pre-Calculus 2 5 Units Together with MATH 121, a two-semester
More informationMATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Summative Assessment -II. Revision CLASS X Prepared by
MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Summative Assessment -II Revision CLASS X 06 7 Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.), B. Ed. Kendriya Vidyalaya GaCHiBOWli
More informationCHAPTER - 2 EQUATIONS. Copyright -The Institute of Chartered Accountants of India
CHAPTER - EQUATIONS EQUATIONS LEARNING OBJECTIVES After studying this chapter, you will be able to: u Understand the concept of equations and its various degrees linear, simultaneous, quadratic and cubic
More informationRoots of quadratic equations
CHAPTER Roots of quadratic equations Learning objectives After studying this chapter, you should: know the relationships between the sum and product of the roots of a quadratic equation and the coefficients
More informationHigher Portfolio Quadratics and Polynomials
Higher Portfolio Quadratics and Polynomials Higher 5. Quadratics and Polynomials Section A - Revision Section This section will help you revise previous learning which is required in this topic R1 I have
More informationLecture 27. Quadratic Formula
Lecture 7 Quadratic Formula Goal: to solve any quadratic equation Quadratic formula 3 Plan Completing the square 5 Completing the square 6 Proving quadratic formula 7 Proving quadratic formula 8 Proving
More informationQuadratics. SPTA Mathematics Higher Notes
H Quadratics SPTA Mathematics Higher Notes Quadratics are expressions with degree 2 and are of the form ax 2 + bx + c, where a 0. The Graph of a Quadratic is called a Parabola, and there are 2 types as
More informationComplex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i
Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i 2 = 1 Sometimes we like to think of i = 1 We can treat
More informationB.Sc. MATHEMATICS I YEAR
B.Sc. MATHEMATICS I YEAR DJMB : ALGEBRA AND SEQUENCES AND SERIES SYLLABUS Unit I: Theory of equation: Every equation f(x) = 0 of n th degree has n roots, Symmetric functions of the roots in terms of the
More informationSTAAR STANDARDS ALGEBRA I ALGEBRA II GEOMETRY
STANDARDS ALGEBRA I ALGEBRA II GEOMETRY STANDARDS ALGEBRA I TEKS Snapshot Algebra I (New TEKS 2015-16) Mathematical Process Standards A.1 Mathematical process standards. The student uses mathematical processes
More informationRAVEN S MANITOBA GRADE 10 INTRODUCTION TO APPLIED AND PRE CALCULUS MATHEMATICS (20S)
RAVEN S MANITOBA GRADE 10 INTRODUCTION TO APPLIED AND PRE CALCULUS MATHEMATICS (20S) LINKED DIRECTLY TO NEW CURRICULUM REQUIREMENTS FROM THE WESTERN PROTOCOLS FOR 2008 AND BEYOND STUDENT GUIDE AND RESOURCE
More informationPre-Algebra (6/7) Pacing Guide
Pre-Algebra (6/7) Pacing Guide Vision Statement Imagine a classroom, a school, or a school district where all students have access to high-quality, engaging mathematics instruction. There are ambitious
More informationMathematics 1 Lecture Notes Chapter 1 Algebra Review
Mathematics 1 Lecture Notes Chapter 1 Algebra Review c Trinity College 1 A note to the students from the lecturer: This course will be moving rather quickly, and it will be in your own best interests to
More informationSolutions with relevant marking scheme to Board Question papers available in downloadable PDF format at
15 Model Question Papers 12 Board Question Papers Salient Features Comprises a total of 27 Test Papers: (15 Model Question Papers + 12 Board Question Papers) Provides Model Question Papers with solutions
More informationA-Level Maths Induction Summer Work
A-Level Maths Induction Summer Work Name:. (Show workings for every question in this booklet) This booklet contains GCSE Algebra skills that you will need in order to successfully complete the A-Level
More informationPOLYNOMIALS ML 5 ZEROS OR ROOTS OF A POLYNOMIAL. A real number α is a root or zero of polynomial f(x) = x + a x + a x +...
POLYNOMIALS ML 5 ZEROS OR ROOTS OF A POLYNOMIAL n n 1 n an n 1 n 1 + 0 A real number α is a root or zero of polynomial f(x) = x + a x + a x +... + a x a, n n an n 1 n 1 0 = if f (α) = 0. i.e. α + a + a
More informationPrecalculus Workshop - Equations and Inequalities
Linear Equations To solve a linear equation, we may apply the rules below. The values a, b and c are real numbers, unless otherwise stated. 1. Addition and subtraction rules: (a) If a = b, then a + c =
More informationAnswers to Sample Exam Problems
Math Answers to Sample Exam Problems () Find the absolute value, reciprocal, opposite of a if a = 9; a = ; Absolute value: 9 = 9; = ; Reciprocal: 9 ; ; Opposite: 9; () Commutative law; Associative law;
More informationPre AP Algebra. Mathematics Standards of Learning Curriculum Framework 2009: Pre AP Algebra
Pre AP Algebra Mathematics Standards of Learning Curriculum Framework 2009: Pre AP Algebra 1 The content of the mathematics standards is intended to support the following five goals for students: becoming
More informationNote: A file Algebra Unit 09 Practice X Patterns can be useful to prepare students to quickly find sum and product.
Note: This unit can be used as needed (review or introductory) to practice operations on polynomials. Math Background Previously, you Identified monomials and their characteristics Applied the laws of
More informationFind two positive factors of 24 whose sum is 10. Make an organized list.
9.5 Study Guide For use with pages 582 589 GOAL Factor trinomials of the form x 2 1 bx 1 c. EXAMPLE 1 Factor when b and c are positive Factor x 2 1 10x 1 24. Find two positive factors of 24 whose sum is
More informationRevision Materials. Functions, Quadratics & Polynomials Skills Builder
Mathematics Higher Revision Materials Functions, Quadratics & Polynomials Skills Builder Layout and content of the Unit Assessment will be different. This is not meant to be a carbon copy of the Unit Assessment.
More informationAlgebra II/Geometry Curriculum Guide Dunmore School District Dunmore, PA
Algebra II/Geometry Dunmore School District Dunmore, PA Algebra II/Geometry Prerequisite: Successful completion of Algebra 1 Part 2 K Algebra II/Geometry is intended for students who have successfully
More informationΠ xdx cos 2 x
Π 5 3 xdx 5 4 6 3 8 cos x Help Your Child with Higher Maths Introduction We ve designed this booklet so that you can use it with your child throughout the session, as he/she moves through the Higher course,
More informationMATHEMATICS (IX-X) (Code No. 041)
MATHEMATICS (IX-X) (Code No. 041) The Syllabus in the subject of Mathematics has undergone changes from time to time in accordance with growth of the subject and emerging needs of the society. The present
More information3 COMPLEX NUMBERS. 3.0 Introduction. Objectives
3 COMPLEX NUMBERS Objectives After studying this chapter you should understand how quadratic equations lead to complex numbers and how to plot complex numbers on an Argand diagram; be able to relate graphs
More informationAssignment #1 MAT121 Summer 2015 NAME:
Assignment #1 MAT11 Summer 015 NAME: Directions: Do ALL of your work on THIS handout in the space provided! Circle your final answer! On problems that your teacher would show work on be sure that you also
More informationA quadratic expression is a mathematical expression that can be written in the form 2
118 CHAPTER Algebra.6 FACTORING AND THE QUADRATIC EQUATION Textbook Reference Section 5. CLAST OBJECTIVES Factor a quadratic expression Find the roots of a quadratic equation A quadratic expression is
More information4 Unit Math Homework for Year 12
Yimin Math Centre 4 Unit Math Homework for Year 12 Student Name: Grade: Date: Score: Table of contents 3 Topic 3 Polynomials Part 2 1 3.2 Factorisation of polynomials and fundamental theorem of algebra...........
More informationTo solve a radical equation, you must take both sides of an equation to a power.
Topic 5 1 Radical Equations A radical equation is an equation with at least one radical expression. There are four types we will cover: x 35 3 4x x 1x 7 3 3 3 x 5 x 1 To solve a radical equation, you must
More informationChapter 7 Quadratic Equations
Chapter 7 Quadratic Equations We have worked with trinomials of the form ax 2 + bx + c. Now we are going to work with equations of this form ax 2 + bx + c = 0 quadratic equations. When we write a quadratic
More informationStudy Guide for Benchmark #1 Window of Opportunity: March 4-11
Study Guide for Benchmark #1 Window of Opportunity: March -11 Benchmark testing is the department s way of assuring that students have achieved minimum levels of computational skill. While partial credit
More informationsum(seq(1/(2^x), x, 3, 6, 1)= 15 64
SEQUENCE & SERIES Summation Notation sum(seq(1/(2^x), x, 3, 6, 1)= 15 64 Arithmetic- SUBTRACT to find d (common difference) Geometric- DIVIDE to find r (common ratio) EXAMPLES REAL/COMPLEX NUMBERS Complex
More informationCHAPTER 1. Review of Algebra
CHAPTER 1 Review of Algebra Much of the material in this chapter is revision from GCSE maths (although some of the exercises are harder). Some of it particularly the work on logarithms may be new if you
More informationINDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC
INDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC Surds Page 1 Algebra of Polynomial Functions Page 2 Polynomial Expressions Page 2 Expanding Expressions Page 3 Factorising Expressions
More information5. Introduction to Euclid s Geometry
5. Introduction to Euclid s Geometry Multiple Choice Questions CBSE TREND SETTER PAPER _ 0 EXERCISE 5.. If the point P lies in between M and N and C is mid-point of MP, then : (A) MC + PN = MN (B) MP +
More informationGeneral Physics I, Spring Vectors
General Physics I, Spring 2011 Vectors 1 Vectors: Introduction A vector quantity in physics is one that has a magnitude (absolute value) and a direction. We have seen three already: displacement, velocity,
More informationHOSTOS COMMUNITY COLLEGE DEPARTMENT OF MATHEMATICS INTENSIVE INTEGRATED ARITHMETIC/ALGEBRA. Placement score of 25 or above on the COMPASS M1
1 HOSTOS COMMUNITY COLLEGE DEPARTMENT OF MATHEMATICS MAT 15 INTENSIVE INTEGRATED ARITHMETIC/ALGEBRA CREDIT HOURS: 0 EQUATED HOURS: 6.0 CLASS HOURS: 6.0 ` PREREQUISITE: REQUIRED TEXTS: DESCRIPTION: EXAMINATIONS:
More informationInstructional Units Plan Algebra II
Instructional Units Plan Algebra II This set of plans presents the topics and selected for ACT s rigorous Algebra II course. The topics and standards are arranged in ten units by suggested instructional
More informationMr. Kelly s Algebra II Syllabus
Algebra II Syllabus The following is a general description and an outline of the topics covered in a full year of algebra II. The outline is broken into parts A and B to correspond to our trimesters. It
More informationYEAR 9 MATHS SEMESTER 1 EXAM REVISION BOOKLET 2018
YEAR 9 MATHS SEMESTER 1 EXAM REVISION BOOKLET 2018 Topics Examined Chapter 12 Measurement (Exercises 12.2 12.7; 6.2-6.3) o Unit Conversions o Perimeter, Area, Total Surface Area, Volume and Capacity of
More informationAQA Level 2 Certificate in Further Mathematics. Worksheets - Teacher Booklet
AQA Level Certificate in Further Mathematics Worksheets - Teacher Booklet Level Specification Level Certificate in Further Mathematics 860 Worksheets - Teacher Booklet Our specification is published on
More informationCorrelation of 2012 Texas Essential Knowledge and Skills (TEKS) for Algebra I and Geometry to Moving with Math SUMS Moving with Math SUMS Algebra 1
Correlation of 2012 Texas Essential Knowledge and Skills (TEKS) for Algebra I and Geometry to Moving with Math SUMS Moving with Math SUMS Algebra 1 ALGEBRA I A.1 Mathematical process standards. The student
More informationNCERT SOLUTIONS OF Mensuration Exercise 2
NCERT SOLUTIONS OF Mensuration Exercise 2 1 Question 1 The shape of the top surface of a table is a trapezium. Find its area if its parallel sides are 1 m and 1.2 m and perpendicular distance between them
More informationObjective Mathematics
Multiple choice questions with ONE correct answer : ( Questions No. 1-5 ) 1. If the equation x n = (x + ) is having exactly three distinct real solutions, then exhaustive set of values of 'n' is given
More informationSolving Quadratic Equations
Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic
More informationKing Fahd University of Petroleum and Minerals Prep-Year Math Program Math (001) - Term 181 Recitation (1.1)
Recitation (1.1) Question 1: Find a point on the y-axis that is equidistant from the points (5, 5) and (1, 1) Question 2: Find the distance between the points P(2 x, 7 x) and Q( 2 x, 4 x) where x 0. Question
More information1Add and subtract 2Multiply radical
Then You simplified radical expressions. (Lesson 10-2) Now 1Add and subtract radical expressions. 2Multiply radical expressions. Operations with Radical Expressions Why? Conchita is going to run in her
More informationCourse Number 420 Title Algebra I Honors Grade 9 # of Days 60
Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number
More informationAlgebra II Syllabus CHS Mathematics Department
1 Algebra II Syllabus CHS Mathematics Department Contact Information: Parents may contact me by phone, email or visiting the school. Teacher: Mrs. Tara Nicely Email Address: tara.nicely@ccsd.us Phone Number:
More informationLINEAR INEQUALITIES. Chapter Overview
LINEAR INEQUALITIES Chapter 6 6.1 Overview 6.1.1 A statement involving the symbols >, 3, x 4, x + y 9. (ii) (iii) (iv) (v) Inequalities which do not involve
More information