not to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions


 Paula Armstrong
 1 years ago
 Views:
Transcription
1 POLYNOMIALS (A) Min Concepts nd Results Geometricl mening of zeroes of polynomil: The zeroes of polynomil p(x) re precisely the xcoordintes of the points where the grph of y = p(x) intersects the xxis. Reltion between the zeroes nd coefficients of polynomil: If α nd β re the zeroes of qudrtic polynomil x + bx + c, then α + β b, αβ If α, β nd γ re the zeroes of cubic polynomil x + bx + cx + d, then α+β+γ b, α β + β γ + γ α c nd α β γ d. The division lgorithm sttes tht given ny polynomil p(x) nd ny nonzero polynomil g(x), there re polynomils q(x) nd r(x) such tht p(x) = g(x) q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x). (B) Multiple Choice Questions CHAPTER Choose the correct nswer from the given four options: Smple Question 1: If one zero of the qudrtic polynomil x + x + k is, then the vlue of k is (A) 10 (B) 10 (C) 5 (D) 5 Solution : Answer (B) c.
2 POLYNOMIALS 9 Smple Question : Given tht two of the zeroes of the cubic polynomil x + bx + cx + d re 0, the third zero is b b c (A) (B) (C) (D) d Solution : Answer (A). [Hint: Becuse if third zero is α, sum of the zeroes b = α = ] EXERCISE.1 Choose the correct nswer from the given four options in the following questions: 1. If one of the zeroes of the qudrtic polynomil (k 1) x + k x + 1 is, then the vlue of k is 4 4 (A) (B) (C). A qudrtic polynomil, whose zeroes re nd 4, is (A) x x + 1 (B) x + x + 1 (C) x x 6 (D) x + x 4. If the zeroes of the qudrtic polynomil x + ( + 1) x + b re nd, then (A) = 7, b = 1 (B) = 5, b = 1 (C) =, b = 6 (D) = 0, b = 6 4. The number of polynomils hving zeroes s nd 5 is (A) 1 (B) (C) (D) more thn 5. Given tht one of the zeroes of the cubic polynomil x + bx + cx + d is zero, the product of the other two zeroes is (A) c c (B) (C) 0 (D) b 6. If one of the zeroes of the cubic polynomil x + x + bx + c is 1, then the product of the other two zeroes is (A) b + 1 (B) b 1 (C) b + 1 (D) b 1 (D)
3 10 EXEMPLAR PROBLEMS 7. The zeroes of the qudrtic polynomil x + 99x + 17 re (A) both positive (B) both negtive (C) one positive nd one negtive (D) both equl 8. The zeroes of the qudrtic polynomil x + kx + k, k 0, (A) cnnot both be positive (C) re lwys unequl (B) cnnot both be negtive (D) re lwys equl 9. If the zeroes of the qudrtic polynomil x + bx + c, c 0 re equl, then (A) c nd hve opposite signs (C) c nd hve the sme sign (B) c nd b hve opposite signs (D) c nd b hve the sme sign 10. If one of the zeroes of qudrtic polynomil of the form x +x + b is the negtive of the other, then it (A) hs no liner term nd the constnt term is negtive. (B) hs no liner term nd the constnt term is positive. (C) cn hve liner term but the constnt term is negtive. (D) cn hve liner term but the constnt term is positive. 11. Which of the following is not the grph of qudrtic polynomil? (A) (B) (C) (D)
4 POLYNOMIALS 11 (C) Short Answer Questions with Resoning Smple Question 1: Cn x 1 be the reminder on division of polynomil p (x) by x +? Justify your nswer. Solution : No, since degree (x 1) = 1 = degree (x + ). Smple Question : Is the following sttement True or Flse? Justify your nswer. If the zeroes of qudrtic polynomil x + bx + c re both negtive, then, b nd c ll hve the sme sign. Solution : True, becuse b = sum of the zeroes < 0, so tht b of the zeroes = c > 0. EXERCISE. > 0. Also the product 1. Answer the following nd justify: (i) Cn x 1 be the quotient on division of x 6 + x + x 1 by polynomil in x of degree 5? (ii) Wht will the quotient nd reminder be on division of x + bx + c by px + qx + rx + s, p 0? (iii) If on division of polynomil p (x) by polynomil g (x), the quotient is zero, wht is the reltion between the degrees of p (x) nd g (x)? (iv) If on division of nonzero polynomil p (x) by polynomil g (x), the reminder is zero, wht is the reltion between the degrees of p (x) nd g (x)? (v) Cn the qudrtic polynomil x + kx + k hve equl zeroes for some odd integer k > 1?. Are the following sttements True or Flse? Justify your nswers. (i) If the zeroes of qudrtic polynomil x + bx + c re both positive, then, b nd c ll hve the sme sign. (ii) If the grph of polynomil intersects the xxis t only one point, it cnnot be qudrtic polynomil. (iii) If the grph of polynomil intersects the xxis t exctly two points, it need not be qudrtic polynomil. (iv) If two of the zeroes of cubic polynomil re zero, then it does not hve liner nd constnt terms.
5 1 EXEMPLAR PROBLEMS (v) (vi) (vii) If ll the zeroes of cubic polynomil re negtive, then ll the coefficients nd the constnt term of the polynomil hve the sme sign. If ll three zeroes of cubic polynomil x + x bx + c re positive, then t lest one of, b nd c is nonnegtive. The only vlue of k for which the qudrtic polynomil kx + x + k hs equl zeros is 1 (D) Short Answer Questions Smple Question 1:Find the zeroes of the polynomil x x, nd verify the reltion between the coefficients nd the zeroes of the polynomil. Solution : x x = 1 6 (6x + x 1) = 1 6 [6x + 9x 8x 1] = 1 6 [x (x + ) 4 (x + )] = 1 6 (x 4) (x + ) Hence, 4 nd re the zeroes of the given polynomil. The given polynomil is x x. The sum of zeroes = Coefficient of x = 6 Coefficient of x the product of zeroes = 4 EXERCISE. Constnt term Coefficient of x = Find the zeroes of the following polynomils by fctoristion method nd verify the reltions between the zeroes nd the coefficients of the polynomils: 1. 4x x 1. x + 4x 4 nd
6 POLYNOMIALS 1. 5t + 1t t t 15t 5. x + 7 x x + 5 x 7. s (1 + )s + 8. v + 4 v y + 5 y y 11 y (E) Long Answer Questions Smple Question 1: Find qudrtic polynomil, the sum nd product of whose zeroes re nd, respectively. Also find its zeroes. Solution : A qudrtic polynomil, the sum nd product of whose zeroes re nd is x x x x = 1 [x x ] Hence, the zeroes re = 1 [x + x x ] = 1 [ x ( x + 1) ( x + 1)] = 1 [ x + 1] [ x ] 1 nd. Smple Question : If the reminder on division of x + x + kx + by x is 1, find the quotient nd the vlue of k. Hence, find the zeroes of the cubic polynomil x + x + kx 18.
7 14 EXEMPLAR PROBLEMS Solution : Let p(x) = x + x + kx + Then, p() = + + k + = 1 i.e., k = 7 i.e., k = 9 Hence, the given polynomil will become x + x 9x +. Now, x ) x + x 9x +(x + 5x +6 x x 5x 9x + 5x 15x 6x + 6x 18 So, x + x 9x + = (x + 5x + 6) (x ) + 1 i.e., x + x 9x 18 = (x ) (x + 5x + 6) So, the zeroes of x x kx 1 = (x ) (x + ) (x + ) re,,. EXERCISE.4 1. For ech of the following, find qudrtic polynomil whose sum nd product respectively of the zeroes re s given. Also find the zeroes of these polynomils by fctoristion. (i) 8, 4 (iii), 9 (ii) (iv) 1 8, , 1. Given tht the zeroes of the cubic polynomil x 6x + x + 10 re of the form, + b, + b for some rel numbers nd b, find the vlues of nd b s well s the zeroes of the given polynomil.
8 POLYNOMIALS 15. Given tht is zero of the cubic polynomil 6x + x 10x 4, find its other two zeroes. 4. Find k so tht x + x + k is fctor of x 4 + x 14 x + 5x + 6. Also find ll the zeroes of the two polynomils. 5. Given tht x 5 is fctor of the cubic polynomil x 5x + 1x 5, find ll the zeroes of the polynomil. 6. For which vlues of nd b, re the zeroes of q(x) = x + x + lso the zeroes of the polynomil p(x) = x 5 x 4 4x + x + x + b? Which zeroes of p(x) re not the zeroes of q(x)?
NUMBER SYSTEMS CHAPTER 1. (A) Main Concepts and Results
CHAPTER NUMBER SYSTEMS Min Concepts nd Results Rtionl numbers Irrtionl numbers Locting irrtionl numbers on the number line Rel numbers nd their deciml expnsions Representing rel numbers on the number line
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More information4.0 5Minute Review: Rational Functions
mth 130 dy 4: working with limits 1 40 5Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationAn Insight into Quadratic Equations and Cubic Equations with Real Coefficients
An Insight into Qurti Equtions n Cubi Equtions with Rel Coeffiients Qurti Equtions A qurti eqution is n eqution of the form x + bx + =, where o It n be solve quikly if we n ftorize the expression x + bx
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More information2 If a branch is prime, no other factors
Chpter 2 Multiples, nd primes 59 Find the prime of 50 by drwing fctor tree. b Write 50 s product of its prime. 1 Find fctor pir of the given 50 number nd begin the fctor tree (50 = 5 10). 5 10 2 If brnch
More informationChapter 9: Quadratic Equations
Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.
More informationRational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and
Rtionl Functions Rtionl unctions re the rtio o two polynomil unctions. They cn be written in expnded orm s ( ( P x x + x + + x+ Qx bx b x bx b n n 1 n n 1 1 0 m m 1 m + m 1 + + m + 0 Exmples o rtionl unctions
More informationWritten Homework 6 Solutions
Written Homework 6 Solutions Section.10 0. Explin in terms of liner pproximtions or differentils why the pproximtion is resonble: 1.01) 6 1.06 Solution: First strt by finding the liner pproximtion of f
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationDETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.
Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of
More informationUniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
More informationChapter 6 Solving equations
Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationLinear Equations in Two Variables
Liner Equtions in Two Vribles In this chpter, we ll use the geometry of lines to help us solve equtions. Liner equtions in two vribles If, b, ndr re rel numbers (nd if nd b re not both equl to 0) then
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationSequences and Series
Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationOn the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding
Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl
More informationQuadratic Equations  1
Alger Module A60 Qudrtic Equtions  1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions  1 Sttement of Prerequisite
More informationSquare & Square Roots
Squre & Squre Roots Squre : If nuber is ultiplied by itself then the product is the squre of the nuber. Thus the squre of is x = eg. x x Squre root: The squre root of nuber is one of two equl fctors which
More informationIntroduction to polynomials
Worksheet 4.5 Polynomials Section 1 Introduction to polynomials A polynomial is an expression of the form p(x) = p 0 + p 1 x + p 2 x 2 + + p n x n, (n N) where p 0, p 1,..., p n are constants and x os
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter 11 11.1 Overview 11.1.1 Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig. 11.1). Fig. 11.1 Suppose we
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: x n+ x n n + + C, dx = ln x + C, if n if n = In prticulr, this mens tht dx = ln x + C x nd x 0 dx = dx = dx = x + C Integrl of Constnt:
More informationRandom Variables and Cumulative Distribution
Probbility: One Rndom Vrible 3 Rndom Vribles nd Cumultive Distribution A probbility distribution shows the probbilities observed in n experiment. The quntity observed in given tril of n experiment is number
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More information1 PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE
PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the smples, work the problems, then check your nswers t the end of ech topic. If you don t get the nswer given, check your work nd look
More informationAddition and subtraction of rational expressions
Lecture 5. Addition nd subtrction of rtionl expressions Two rtionl expressions in generl hve different denomintors, therefore if you wnt to dd or subtrct them you need to equte the denomintors first. The
More information5.6 POSITIVE INTEGRAL EXPONENTS
54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section
More informationhas the desired form. On the other hand, its product with z is 1. So the inverse x
First homework ssignment p. 5 Exercise. Verify tht the set of complex numers of the form x + y 2, where x nd y re rtionl, is sufield of the field of complex numers. Solution: Evidently, this set contins
More informationFormal Languages and Automata Exam
Forml Lnguges nd Automt Exm Fculty of Computers & Informtion Deprtment: Computer Science Grde: Third Course code: CSC 34 Totl Mrk: 8 Dte: 23//2 Time: 3 hours Answer the following questions: ) Consider
More informationTests for One Poisson Mean
Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationSolution: Let x be the larger number and y the smaller number.
Problem The sum of two numbers is 00 The lrger number minus the smller number is Find the numbers [Problem submitted by Vin Lee, LACC Professor of Mthemtics Source: Vin Lee] Solution: Let be the lrger
More informationNCERT INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS. Trigonometric Ratios of the angle A in a triangle ABC right angled at B are defined as:
INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: side opposite to A BC sine of A = sin A = hypotenuse
More informationLecture 3 Basic Probability and Statistics
Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The
More informationArea Between Curves: We know that a definite integral
Are Between Curves: We know tht definite integrl fx) dx cn be used to find the signed re of the region bounded by the function f nd the x xis between nd b. Often we wnt to find the bsolute re of region
More informationNumerical integration
Chpter 4 Numericl integrtion Contents 4.1 Definite integrls.............................. 4. Closed NewtonCotes formule..................... 4 4. Open NewtonCotes formule...................... 8 4.4
More information11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
More information1. 1 m/s m/s m/s. 5. None of these m/s m/s m/s m/s correct m/s
Crete ssignment, 99552, Homework 5, Sep 15 t 10:11 m 1 This printout should he 30 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. The due time
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationPolynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if
1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c
More informationQuadratic Equations. Math 99 N1 Chapter 8
Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
More informationMathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
More informationGeneralized Inverses: How to Invert a NonInvertible Matrix
Generlized Inverses: How to Invert NonInvertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax
More informationFUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation
FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationName: Lab Partner: Section:
Chpter 4 Newton s 2 nd Lw Nme: Lb Prtner: Section: 4.1 Purpose In this experiment, Newton s 2 nd lw will be investigted. 4.2 Introduction How does n object chnge its motion when force is pplied? A force
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More information1. Inverse of a tridiagonal matrix
PréPublicções do Deprtmento de Mtemátic Universidde de Coimbr Preprint Number 05 16 ON THE EIGENVALUES OF SOME TRIDIAGONAL MATRICES CM DA FONSECA Abstrct: A solution is given for problem on eigenvlues
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More information1 of 7 9/14/15, 10:27 AM
Stuent: Dte: Instructor: Doug Ensle Course: MAT117 1 Applie Sttistics  Ensle Assignment: Online 6  Section 3.2 1 of 7 9/14/15, 1:27 AM 1. 2 of 7 9/14/15, 1:27 AM The t on the right shows the percent
More informationWorksheet 4.7. Polynomials. Section 1. Introduction to Polynomials. A polynomial is an expression of the form
Worksheet 4.7 Polynomials Section 1 Introduction to Polynomials A polynomial is an expression of the form p(x) = p 0 + p 1 x + p 2 x 2 + + p n x n (n N) where p 0, p 1,..., p n are constants and x is a
More information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
More informationTwo special Righttriangles 1. The
Mth Right Tringle Trigonometry Hndout B (length of )  c  (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Righttringles. The
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More informationIn the following there are presented four different kinds of simulation games for a given Büchi automaton A = :
Simultion Gmes Motivtion There re t lest two distinct purposes for which it is useful to compute simultion reltionships etween the sttes of utomt. Firstly, with the use of simultion reltions it is possile
More informationA new algorithm for generating Pythagorean triples
A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf
More informationAlgorithms Chapter 4 Recurrences
Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering
More informationFor the Final Exam, you will need to be able to:
Mth B Elementry Algebr Spring 0 Finl Em Study Guide The em is on Wednesdy, My 0 th from 7:00pm 9:0pm. You re lloed scientific clcultor nd " by 6" inde crd for notes. On your inde crd be sure to rite ny
More informationAnswer, Key Homework 8 David McIntyre 1
Answer, Key Homework 8 Dvid McIntyre 1 This printout should hve 17 questions, check tht it is complete. Multiplechoice questions my continue on the net column or pge: find ll choices before mking your
More informationNumber Systems & Working With Numbers
Presenting the Mths Lectures! Your best bet for Qunt... MATHS LECTURE # 0 Number Systems & Working With Numbers System of numbers.3 0.6 π With the help of tree digrm, numbers cn be clssified s follows
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls : The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N):  counting numers. {,,,,, } Whole Numers (W):  counting numers with 0. {0,,,,,, } Integers (I): 
More informationQuadratic Functions. Analyze and describe the characteristics of quadratic functions
Section.3  Properties of rphs of Qudrtic Functions Specific Curriculum Outcomes covered C3 Anlyze nd describe the chrcteristics of qudrtic functions C3 Solve problems involving qudrtic equtions F Anlyze
More informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More informationASS.PROF.DR Thamer Information Theory 4th Class in Communication. Finite Field Arithmetic. (Galois field)
Finite Field Arithmetic (Galois field) Introduction: A finite field is also often known as a Galois field, after the French mathematician Pierre Galois. A Galois field in which the elements can take q
More informationTHE RATIONAL NUMBERS CHAPTER
CHAPTER THE RATIONAL NUMBERS When divided by b is not n integer, the quotient is frction.the Bbylonins, who used number system bsed on 60, epressed the quotients: 0 8 s 0 60 insted of 8 s 7 60,600 0 insted
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationMath 22B Solutions Homework 1 Spring 2008
Mth 22B Solutions Homework 1 Spring 2008 Section 1.1 22. A sphericl rindrop evportes t rte proportionl to its surfce re. Write differentil eqution for the volume of the rindrop s function of time. Solution
More informationIn this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.
Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix
More informationRational Expressions
C H A P T E R Rtionl Epressions nformtion is everywhere in the newsppers nd mgzines we red, the televisions we wtch, nd the computers we use. And I now people re tlking bout the Informtion Superhighwy,
More informationEntry Test: Mathematics for Management and Economics
Entry Test: Mthemtics for Mngement nd Economics BM.0 Introduction BM.1 Sets BM. Assertions 4 BM. Propositionl Formuls 5 BM.4.1 Algebric Lws 6 BM.4. Equivlence of Equtions 8 BM.4. Trnsforming Equtions 9
More informationEducation Spending (in billions of dollars) Use the distributive property.
0 CHAPTER Review of the Rel Number System 96. An pproximtion of federl spending on eduction in billions of dollrs from 200 through 2005 cn be obtined using the e xpression y = 9.0499x  8,07.87, where
More informationwww.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
More informationExponents base exponent power exponentiation
Exonents We hve seen counting s reeted successors ddition s reeted counting multiliction s reeted ddition so it is nturl to sk wht we would get by reeting multiliction. For exmle, suose we reetedly multily
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More informationJackson 2.23 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson.3 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: A hollow cube hs conducting wlls defined by six plnes x =, y =, z =, nd x =, y =, z =. The wlls z =
More information15. Let f (x) = 3x Suppose rx 2 + sx + t = 0 where r 0. Then x = 24. Solve 5x 25 < 20 for x. 26. Let y = 7x
Pretest Review The pretest will onsist of 0 problems, eh of whih is similr to one of the following 49 problems If you n do problems like these 49 listed below, you will hve no problem with the pretest
More informationPolynomials and Vieta s Formulas
Polynomials and Vieta s Formulas Misha Lavrov ARML Practice 2/9/2014 Review problems 1 If a 0 = 0 and a n = 3a n 1 + 2, find a 100. 2 If b 0 = 0 and b n = n 2 b n 1, find b 100. Review problems 1 If a
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More informationPrealgebra 7* In your group consider the following problems:
Prelger * Group Activit # Group Memers: In our group consider the following prolems: 1) If ever person in the room, including the techer, were to shke hnds with ever other person ectl one time, how mn
More informationMathematics in Art and Architecture GEK1518K
Mthemtics in Art nd Architecture GEK1518K Helmer Aslksen Deprtment of Mthemtics Ntionl University of Singpore slksen@mth.nus.edu.sg www.mth.nus.edu.sg/slksen/ The Golden Rtio The Golden Rtio Suppose we
More information