PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "PAIR OF LINEAR EQUATIONS IN TWO VARIABLES"

Transcription

1 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,, re rel numers, suh tht + 0, + 0. A pir of liner equtions in two vriles n e represented, nd solved, y the: (i) grphil method (ii) lgeri method. Grphil Method : The grph of pir of liner equtions in two vriles is represented y two lines. (i) If the lines interset t point, then tht point gives the unique solution of the two equtions. In this se, the pir of equtions is onsistent. (ii) If the lines oinide, then there re infinitely mny solutions eh point on the line eing solution. In this se, the pir of equtions is dependent (onsistent). (iii) If the lines re prllel, then the pir of equtions hs no solution. In this se, the pir of equtions is inonsistent. 4. Algeri Methods : We hve disussed the following methods for finding the solution(s) of pir of liner equtions : (i) Sustitution Method (ii) Elimintion Method (iii) Cross-multiplition Method. If pir of liner equtions is given y x + y + 0 nd x + y + 0, then the following situtions n rise : (i) In this se, the pir of liner equtions is onsistent. (ii) In this se, the pir of liner equtions is inonsistent send your queries to

2 (iii) In this se, the pir of liner eqution is dependent nd onsistent.. There re severl situtions whih n e mthemtilly represented y two equtions tht re not liner to strt with. But we lter them so tht they re redued to pir of liner equtions. EXERCISE. Aft tells his dughter, Seven yers go, I ws seven times s old s you were then. Also, three yers from now, I shll e three times s old s you will e. Represent this sitution lgerilly nd grphilly. Answer: Let us ssume tht seven yers go Aft s ge ws x yers nd his dughter s ge ws y. So, s per the question x 7y x 7 y (i) The eqution gives us the following tle: X Y Now, three yers from now mens 0 yers from 7 yers k Aft ge will e x + 0 Dughter s ge will e y + 0 As per question x + 0 ( y + 0) x + 0 y + 0 x y + 0 x y (ii) send your queries to

3 The eqution gives us following tle: X Y Fther's Age x y 0 0 x 7 y Dughter's Age. The oh of riket tem uys ts nd lls for Rs 900. Lter, she uys nother t nd more lls of the sme kind for Rs 00. Represent this sitution lgerilly nd geometrilly. Answer: Let us ssume numer of ts x nd tht of lls y. As per the question we get following equtions: x + y 900 x 900 y x 00 y (i) x + y 00 x 00 y (ii) As oth equtions re similr so there oinident lines in the grph, mening there n e infinite numer of solutions to this prolem send your queries to

4 . The ost of kg of pples nd kg of grpes on dy ws found to e Rs 0. After month, the ost of 4 kg of pples nd kg of grpes is Rs 00. Represent the sitution lgerilly nd geometrilly. Answer: Let us ssume tht ost of pple x nd tht of grpes y As per the question we get following equtions: x + y 0 y 0 x (i) The eqution gives us following tle: X Y x + y 00 y 00 4x y 0 x (ii) The eqution gives us following tle: X Y send your queries to

5 0 Grpes y0-x y0-x 0 4 Apples Sine we get prllel lines in this grph so there will e no solution s equtions re inonsistent. EXERCISE. Form the pir of liner equtions in the following prolems, nd find their solutions grphilly. (i) 0 students of Clss X took prt in Mthemtis quiz. If the numer of girls is 4 more thn the numer of oys, find the numer of oys nd girls who took prt in the quiz. Answer: Let us ssume numer of girls to e x nd tht of oys to e y. Then we get following equtions: x y (i) The eqution will give following tle: X Y x + y 0 x 0 y (ii) send your queries to

6 The eqution will give following tle: X Y Girls 8 4 xy+4 x0-y 0 4 Boys As oth lines interset t 7, so Numer of Girls 7 nd tht of Boys (ii) penils nd 7 pens together ost Rs 0, wheres 7 penils nd pens together ost Rs 4. Find the ost of one penil nd tht of one pen. Answer: x + 7y 0 x 0 7y (i) The eqution will give following tle: y X send your queries to

7 7 x + y 4 7x 4 y (ii) The eqution gives following tle y X Pen Penil x0-7y 7x4-y Both lines re interseting t, so x Putting the vlue of x in either of the equtions we n get the vlue of y, whih is equl to. Also, is the only vlue of y whih gives similr vlues for x in oth the tles. So ost of one penil Rs. nd one pen Chek: x + 7y On ompring the rtios, nd, find out if the lines representing the following pirs of liner equtions interset t point, re prllel or oinident: (i) x 4y x + y send your queries to

8 7 7 x x 4 4 y y As it is ler tht, so lines will e interseting. (ii) 9x + y + 0 8x + y It is ler tht, so lines re oinidentl. (iii) x y x y It is ler tht, so lines will e prllel send your queries to

9 . On ompring the rtios, nd find out whether the following pir of liner equtions re onsistent, or inonsistent. (i) x + y ; x y 7 Here,, so the pir of liner equtions is onsistent. (ii) x y 8 ; 4x y Here,, so the pir of liner equtions is inonsistent. (iii) 7 + y x, 9x 0y Here,, so the pir of liner equtions is onsistent send your queries to

10 (iv) x y ; 0x + y 0 Here,, so the pir of liner equtions is dependent nd onsistent. (v) y x ; x + y 4 8 In this se,, so the pir of liner eqution is dependent nd onsistent. 4. Whih of the following pirs of liner equtions re onsistent/inonsistent? If onsistent, otin the solution grphilly: (i) x + y, x + y send your queries to

11 Both equtions will give the sme tle s follows: y x x' y x x+y x+y0 As the lines re oinident so there n e infinitely mny solutions. In this se,, so the pir of liner eqution is dependent nd onsistent send your queries to

12 (ii) x y 8, x y 8 Here,, so the pir of liner equtions is inonsistent. (iii) x + y 0, 4x y Here,, so the pir of liner equtions is inonsistent. (iv) x y 0, 4x 4y 0 4 Here,, so the pir of liner equtions is inonsistent send your queries to

13 . Hlf the perimeter of retngulr grden, whose length is 4 m more thn its width, is m. Find the dimensions of the grden. Answer: Let us ssume Length x nd redth y Then xy+4 Perimeter (x+y) Pr imeter x + y x + x + 4 x + 4 x 4 x Hene, y-4 So, Length m nd Bredth m. Given the liner eqution x + y 8 0, write nother liner eqution in two vriles suh tht the geometril representtion of the pir so formed is: (i) interseting lines (ii) prllel lines (iii) oinident lines Answer: (i) As you know tht if, then the lines will e interseting, so let us put numer for nd in suh wy whih stisfies this ondition. Next possile eqution n e x + y 7 0 (ii) As you know tht if eqution n e s follows: 4 x + y 8 0 (iii) As you know tht if, then the lines will e prllel, so the possile, then the lines will e oinidentl, so the possile eqution n e s follows: 4 x + y send your queries to

14 EXERCISE. Solve the following pir of liner equtions y the sustitution method. (i) x + y 4, x-y4 Answer: x + y 4 x 4 y Sustituting this vlue of x in the seond eqution we get 4 y y 4 4 y 4 y y x 8 s (ii) s t, + t Answer: s +t Sustituting the vlue of s in the seond eqution we get + t t + + t + t + t t 0 t Hene, s +9 (iii) x y, 9x y 9 Answer: yx- Putting this vlue of y in the seond eqution we get 9x-(x-)9 9 x 9x It is ler tht this eqution will hve infinitely mny solutions. It is importnt to note tht here send your queries to

15 (iv) 0.x + 0.y. 0.4x + 0.y. Answer: x+y (fter multiplying the eqution y 0) y-x x Or, y Putting this vlue of y in the seond eqution we get x 4 x + (eqution is lso multiplied like eqution ) x + 0x 9 x + 9 x 4 x 4 y (v) x + y 0, x 8y 0 Answer: x y x y Putting this vlue of x in eqution we get y y y 8y 0 y 8y 0 0 y 4y 0 y 4y 0 y 4y Sine oeffiients of y re different on oth LHS nd RHS, so only possile vlue of y0 Hene, x0. Solve x + y nd x 4y 4 nd hene find the vlue of m for whih y mx +. Answer: x y Sustituting in eqution we get y 4y 4 7y 4 7 y + 4 y x 4 x send your queries to

16 Now putting vlues of x nd y in eqution we get m + m m. Form the pir of liner equtions for the following prolems nd find their solution y sustitution method. (i) The differene etween two numers is nd one numer is three times the other. Find them. Answer: Let us ssume one of the numers x nd nother numer y Then s per question x y (i) x y (ii) Sustituting the vlue of x in eqution (i) y y y y x 9 (ii) The lrger of two supplementry ngles exeeds the smller y 8 degrees. Find them. x + y (i) x y (ii) x y + 8 Sustituting in eqution (i) we get y + y y 80 8 y 8 x (iii) The oh of riket tem uys 7 ts nd lls for Rs 800. Lter, she uys ts nd lls for Rs 70. Find the ost of eh t nd eh ll. Answer: 7 x + y (i) x + y (ii) x 70 y 70 y x Sustituting in eqution (i) we get 7(70 y) + y send your queries to

17 0 y + 8y y y y x 00 Hene, Cost of Bt Rs. 00 Cost of Bll Rs. 0 (iv) The txi hrges in ity onsist of fixed hrge together with the hrge for the distne overed. For distne of 0 km, the hrge pid is Rs 0 nd for journey of km, the hrge pid is Rs. Wht re the fixed hrges nd the hrge per km? How muh does person hve to py for trvelling distne of km? Answer: Let us ssume the fixed hrge x And vrile hrge y As per question: x + 0 y (i) x + y (ii) x y Sustituting in eqution (i) we get y + 0y 0 y 0 y 0 0 y 0 x 0 So for km the fir + 0 (v) A frtion eomes 9, if is dded to oth the numertor nd the denomintor. If, is dded to oth the numertor nd the denomintor it eomes. Find the frtion. Answer: Let us ssume the numertor x nd denomintor y As per question: x + 9 y + x + 9y y x (i) x + y + x + 8 y send your queries to

18 y x (ii) y x + y x + Sustituting in eqution (i) we get 4x + 7 x 4 4 x + 7 x 0 7 x 0 x 7 y Hene, required frtion 9 (vi) Five yers hene, the ge of Jo will e three times tht of his son. Five yers go, Jo s ge ws seven times tht of his son. Wht re their present ges? Answer: Let us ssume five yers go son s ge x nd Jo s ge y Then y 7x (i) Son s ge five yers from now x+0 nd Jo s ge will e y+0 So, y + 0 ( x + 0) x + 0 y x (ii) Sustituting in eqution (i) x + 0 7x 4 x 0 x y Present ge of Jo 40 yers Present ge of son 0 yers send your queries to

19 EXERCISE 4. Solve the following pir of liner equtions y the elimintion method nd the sustitution method: (i) x + y nd x y 4 Answer: Multiplying eqution (i) y nd sutrting eqution (ii) from this s follows: x + y 0 x y y y, sustituting in eqution (i) x + 9 x (ii) x + 4y 0 nd x y Answer: Multiplying eqution (i) y nd eqution (ii) y nd sutrting ltter from former s follows: x + 8y 0 x y 0 + 4y 4 y, sustituting in eqution (i) x x x (iii) x y 4 0 nd 9x y + 7 Answer: Multiplying eqution (i) y nd sutrting eqution (ii) s follows: 9x y 4 0 9x y y y y, sustituting in eqution (i) 0 x 4 0 x send your queries to

20 x x x y (iv) + nd x y Answer: Simplify eqution (i) x y + x + 4y x + 4y (iii) Simplify eqution (ii) x y x y (iv) Sutrting eqution (iv) from (iii) s follows: x + 4y x y y y, sustituting in eqution (iii) x x x. Form the pir of liner equtions in the following prolems, nd find their solutions (if they exist) y the elimintion method : (i) If we dd to the numertor nd sutrt from the denomintor, frtion redues to. It eomes, if we only dd to the denomintor. Wht is the frtion? x + Answer: y + x + y + x y (i) x y send your queries to

21 x y (ii) Sustituting vlue of x from eqution (i) in eqution (ii) y y + y x (ii) The sum of the digits of two-digit numer is 9. Also, nine times this numer is twie the numer otined y reversing the order of the digits. Find the numer. Answer: Let two digits of the numer re x nd y. x + y 9 x 9 y (i) Then First numer 0 x + y ( x is t 0s ple nd y is t unit s ple) Reversing the numer we get numer whih n e written s 0y + x As per question: 9(0 x + y) (0 y + x) 90 x + 9y 0y + x 90(9 y) + 9y 0y + (9 y) 80 90y + 9y 0y + 8 y 80 8y 8y y y x 9 8 Smller numer 8 nd lrger numer (iii) Meen went to nk to withdrw Rs 000. She sked the shier to give her Rs 0 nd Rs 00 notes only. Meen got notes in ll. Find how mny notes of Rs 0 nd Rs 00 she reeived. Answer: x + y x y (i) 0 x + 00y ( y) + 00y y + 00y y y Numer of hundred rupees notes x 0 Numer of fifty rupees notes send your queries to

This enables us to also express rational numbers other than natural numbers, for example:

This enables us to also express rational numbers other than natural numbers, for example: Overview Study Mteril Business Mthemtis 05-06 Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers

More information

QUADRATIC EQUATION EXERCISE - 01 CHECK YOUR GRASP

QUADRATIC EQUATION EXERCISE - 01 CHECK YOUR GRASP QUADRATIC EQUATION EXERCISE - 0 CHECK YOUR GRASP. Sine sum of oeffiients 0. Hint : It's one root is nd other root is 8 nd 5 5. tn other root 9. q 4p 0 q p q p, q 4 p,,, 4 Hene 7 vlues of (p, q) 7 equtions

More information

Introduction to Olympiad Inequalities

Introduction to Olympiad Inequalities Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................

More information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr

More information

QUADRATIC EQUATION. Contents

QUADRATIC EQUATION. Contents QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

More information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

More information

m A 1 1 A ! and AC 6

m A 1 1 A ! and AC 6 REVIEW SET A Using sle of m represents units, sketh vetor to represent: NON-CALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in north-esterl diretion. Simplif:

More information

Chapter Gauss Quadrature Rule of Integration

Chapter Gauss Quadrature Rule of Integration Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to

More information

Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx

Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous Anti-Derivtive : An nti-derivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.

More information

The Ellipse. is larger than the other.

The Ellipse. is larger than the other. The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)

More information

Pythagoras theorem and surds

Pythagoras theorem and surds HPTER Mesurement nd Geometry Pythgors theorem nd surds In IE-EM Mthemtis Yer 8, you lernt out the remrkle reltionship etween the lengths of the sides of right-ngled tringle. This result is known s Pythgors

More information

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point. PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

More information

MCH T 111 Handout Triangle Review Page 1 of 3

MCH T 111 Handout Triangle Review Page 1 of 3 Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle:

More information

2.1 ANGLES AND THEIR MEASURE. y I

2.1 ANGLES AND THEIR MEASURE. y I .1 ANGLES AND THEIR MEASURE Given two interseting lines or line segments, the mount of rottion out the point of intersetion (the vertex) required to ring one into orrespondene with the other is lled the

More information

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a. Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining

More information

along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate

along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate L8 VECTOR EQUATIONS OF LINES HL Mth - Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne

More information

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272. Geometry of the irle - hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion

More information

Math 259 Winter Solutions to Homework #9

Math 259 Winter Solutions to Homework #9 Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658-659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier

More information

6.5 Improper integrals

6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

More information

are coplanar. ˆ ˆ ˆ and iˆ

are coplanar. ˆ ˆ ˆ and iˆ SMLE QUESTION ER Clss XII Mthemtis Time llowed: hrs Mimum Mrks: Generl Instrutions: i ll questions re ompulsor. ii The question pper onsists of 6 questions divided into three Setions, B nd C. iii Question

More information

8.3 THE HYPERBOLA OBJECTIVES

8.3 THE HYPERBOLA OBJECTIVES 8.3 THE HYPERBOLA OBJECTIVES 1. Define Hperol. Find the Stndrd Form of the Eqution of Hperol 3. Find the Trnsverse Ais 4. Find the Eentriit of Hperol 5. Find the Asmptotes of Hperol 6. Grph Hperol HPERBOLAS

More information

Solutions to Assignment 1

Solutions to Assignment 1 MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove

More information

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique? XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

More information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

More information

Vectors. Chapter14. Syllabus reference: 4.1, 4.2, 4.5 Contents:

Vectors. Chapter14. Syllabus reference: 4.1, 4.2, 4.5 Contents: hpter Vetors Syllus referene:.,.,.5 ontents: D E F G H I J K Vetors nd slrs Geometri opertions with vetors Vetors in the plne The mgnitude of vetor Opertions with plne vetors The vetor etween two points

More information

Quadratic Equations. Brahmagupta gave. Solving of quadratic equations in general form is often credited to ancient Indian mathematicians.

Quadratic Equations. Brahmagupta gave. Solving of quadratic equations in general form is often credited to ancient Indian mathematicians. 9 Qudrtic Equtions Qudrtic epression nd qudrtic eqution Pure nd dfected qudrtic equtions Solution of qudrtic eqution y * Fctoristion method * Completing the squre method * Formul method * Grphicl method

More information

MATH STUDENT BOOK. 10th Grade Unit 5

MATH STUDENT BOOK. 10th Grade Unit 5 MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY

More information

Section 4.4. Green s Theorem

Section 4.4. Green s Theorem The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

More information

Naming the sides of a right-angled triangle

Naming the sides of a right-angled triangle 6.2 Wht is trigonometry? The word trigonometry is derived from the Greek words trigonon (tringle) nd metron (mesurement). Thus, it literlly mens to mesure tringle. Trigonometry dels with the reltionship

More information

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion

More information

MTH 505: Number Theory Spring 2017

MTH 505: Number Theory Spring 2017 MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c

More information

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2 1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

More information

Arrow s Impossibility Theorem

Arrow s Impossibility Theorem Rep Voting Prdoxes Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Voting Prdoxes Properties Arrow s Theorem Leture Overview 1 Rep

More information

Chapter 4 State-Space Planning

Chapter 4 State-Space Planning Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 Stte-Spe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different

More information

Section 6.1 Definite Integral

Section 6.1 Definite Integral Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB/ Clulus Exm Review Sheet Solutions A Prelulus Type prolems A1 A A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f( x) Set funtion equl to Ftor or use qudrti eqution if qudrti Grph to

More information

NON-DETERMINISTIC FSA

NON-DETERMINISTIC FSA Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

More information

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers. 8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

More information

UNCORRECTED SAMPLE PAGES. Australian curriculum NUMBER AND ALGEBRA

UNCORRECTED SAMPLE PAGES. Australian curriculum NUMBER AND ALGEBRA 7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K Chpter Wht ou will lern 7Prols nd other grphs Eploring prols Skething prols with trnsformtions Skething prols using ftoristion Skething ompleting the squre Skething using

More information

Interpreting Integrals and the Fundamental Theorem

Interpreting Integrals and the Fundamental Theorem Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

More information

More Properties of the Riemann Integral

More Properties of the Riemann Integral More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl

More information

Accuplacer Elementary Algebra Study Guide

Accuplacer Elementary Algebra Study Guide Testig Ceter Studet Suess Ceter Aupler Elemetry Alger Study Guide The followig smple questios re similr to the formt d otet of questios o the Aupler Elemetry Alger test. Reviewig these smples will give

More information

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri

More information

Determinants Chapter 3

Determinants Chapter 3 Determinnts hpter Specil se : x Mtrix Definition : the determinnt is sclr quntity defined for ny squre n x n mtrix nd denoted y or det(). x se ecll : this expression ppers in the formul for x mtrix inverse!

More information

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

More information

20 MATHEMATICS POLYNOMIALS

20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

More information

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ). AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

More information

Linear Systems with Constant Coefficients

Linear Systems with Constant Coefficients Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

More information

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1) Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

More information

Reflection Property of a Hyperbola

Reflection Property of a Hyperbola Refletion Propert of Hperol Prefe The purpose of this pper is to prove nltill nd to illustrte geometrill the propert of hperol wherein r whih emntes outside the onvit of the hperol, tht is, etween the

More information

Solving Radical Equations

Solving Radical Equations Solving dil Equtions Equtions with dils: A rdil eqution is n eqution in whih vrible ppers in one or more rdinds. Some emples o rdil equtions re: Solution o dil Eqution: The solution o rdil eqution is the

More information

3 x x 3x x. 3x x x 6 x 3. PAKTURK 8 th National Interschool Maths Olympiad, h h

3 x x 3x x. 3x x x 6 x 3. PAKTURK 8 th National Interschool Maths Olympiad, h h PAKTURK 8 th Ntionl Interschool Mths Olmpid,.9. Q: Evlute 6.9. 6 6 6... 8 8...... Q: Evlute bc bc. b. c bc.9.9b.9.9bc Q: Find the vlue of h in the eqution h 7 9 7.. bc. bc bc. b. c bc bc bc bc......9 h

More information

What else can you do?

What else can you do? Wht else cn you do? ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. tht form stright line dd to 180c. lculte the size of + M, if L is stright line M + L = 180c( stright

More information

1.3 SCALARS AND VECTORS

1.3 SCALARS AND VECTORS Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd

More information

Proving the Pythagorean Theorem

Proving the Pythagorean Theorem Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or

More information

The Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5

The Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5 The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle

More information

Operations with Matrices

Operations with Matrices Section. Equlit of Mtrices Opertions with Mtrices There re three ws to represent mtri.. A mtri cn be denoted b n uppercse letter, such s A, B, or C.. A mtri cn be denoted b representtive element enclosed

More information

8 Measurement. How is measurement used in your home? 8E Area of a circle 8B Circumference of a circle. 8A Length and perimeter

8 Measurement. How is measurement used in your home? 8E Area of a circle 8B Circumference of a circle. 8A Length and perimeter 8A Length nd perimeter 8E Are of irle 8B Cirumferene of irle 8F Surfe re 8C Are of retngles nd tringles 8G Volume of prisms 8D Are of other qudrilterls 8H Are nd volume onversions SA M PL E Mesurement

More information

dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.

dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω. Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd

More information

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx... Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting

More information

Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!

Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite! Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:

More information

I 3 2 = I I 4 = 2A

I 3 2 = I I 4 = 2A ECE 210 Eletril Ciruit Anlysis University of llinois t Chigo 2.13 We re ske to use KCL to fin urrents 1 4. The key point in pplying KCL in this prolem is to strt with noe where only one of the urrents

More information

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014 SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.

More information

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year 1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.

More information

CIT 596 Theory of Computation 1. Graphs and Digraphs

CIT 596 Theory of Computation 1. Graphs and Digraphs CIT 596 Theory of Computtion 1 A grph G = (V (G), E(G)) onsists of two finite sets: V (G), the vertex set of the grph, often enote y just V, whih is nonempty set of elements lle verties, n E(G), the ege

More information

Pythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides

Pythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides Pythgors theorem nd trigonometry Pythgors Theorem The hypotenuse of right-ngled tringle is the longest side The hypotenuse is lwys opposite the right-ngle 2 = 2 + 2 or 2 = 2-2 or 2 = 2-2 The re of the

More information

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

More information

(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.

(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P. Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time

More information

Core 2 Logarithms and exponentials. Section 1: Introduction to logarithms

Core 2 Logarithms and exponentials. Section 1: Introduction to logarithms Core Logrithms nd eponentils Setion : Introdution to logrithms Notes nd Emples These notes ontin subsetions on Indies nd logrithms The lws of logrithms Eponentil funtions This is n emple resoure from MEI

More information

z TRANSFORMS z Transform Basics z Transform Basics Transfer Functions Back to the Time Domain Transfer Function and Stability

z TRANSFORMS z Transform Basics z Transform Basics Transfer Functions Back to the Time Domain Transfer Function and Stability TRASFORS Trnsform Bsics Trnsfer Functions Bck to the Time Domin Trnsfer Function nd Stility DSP-G 6. Trnsform Bsics The definition of the trnsform for digitl signl is: -n X x[ n is complex vrile The trnsform

More information

1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automata 1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

More information

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

More information

Chapter 2. Random Variables and Probability Distributions

Chapter 2. Random Variables and Probability Distributions Rndom Vriles nd Proilit Distriutions- 6 Chpter. Rndom Vriles nd Proilit Distriutions.. Introduction In the previous chpter, we introduced common topics of proilit. In this chpter, we trnslte those concepts

More information

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra The Fundmentl Theorem of Alger Jeremy J. Fries In prtil fulfillment of the requirements for the Mster of Arts in Teching with Speciliztion in the Teching of Middle Level Mthemtics in the Deprtment of Mthemtics.

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

Symmetrical Components 1

Symmetrical Components 1 Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent

More information

Algorithm Design and Analysis

Algorithm Design and Analysis Algorithm Design nd Anlysis LECTURE 8 Mx. lteness ont d Optiml Ching Adm Smith 9/12/2008 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov, K. Wyne Sheduling to Minimizing Lteness Minimizing

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS 33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

More information

Section 3.1: Exponent Properties

Section 3.1: Exponent Properties Section.1: Exponent Properties Ojective: Simplify expressions using the properties of exponents. Prolems with exponents cn often e simplied using few sic exponent properties. Exponents represent repeted

More information

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral. Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

More information

expression simply by forming an OR of the ANDs of all input variables for which the output is

expression simply by forming an OR of the ANDs of all input variables for which the output is 2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output

More information

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES

MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES CHARLIE COLLIER UNIVERSITY OF BATH These notes hve been typeset by Chrlie Collier nd re bsed on the leture notes by Adrin Hill nd Thoms Cottrell. These

More information

Similarity and Congruence

Similarity and Congruence Similrity nd ongruence urriculum Redy MMG: 201, 220, 221, 243, 244 www.mthletics.com SIMILRITY N ONGRUN If two shpes re congruent, it mens thy re equl in every wy ll their corresponding sides nd ngles

More information

ENGI 3424 Engineering Mathematics Five Tutorial Examples of Partial Fractions

ENGI 3424 Engineering Mathematics Five Tutorial Examples of Partial Fractions ENGI 44 Engineering Mthemtics Five Tutoril Exmples o Prtil Frctions 1. Express x in prtil rctions: x 4 x 4 x 4 b x x x x Both denomintors re liner non-repeted ctors. The cover-up rule my be used: 4 4 4

More information

u(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.

u(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C. Lecture 4 Complex Integrtion MATH-GA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex

More information

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions. Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene

More information

ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS

ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu

More information

Non Right Angled Triangles

Non Right Angled Triangles Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit

More information

Functions. mjarrar Watch this lecture and download the slides

Functions. mjarrar Watch this lecture and download the slides 9/6/7 Mustf Jrrr: Leture Notes in Disrete Mthemtis. Birzeit University Plestine 05 Funtions 7.. Introdution to Funtions 7. One-to-One Onto Inverse funtions mjrrr 05 Wth this leture nd downlod the slides

More information

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230 Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

More information

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are: (x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

More information

LINEAR ALGEBRA APPLIED

LINEAR ALGEBRA APPLIED 5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order

More information

Lecture 2 : Propositions DRAFT

Lecture 2 : Propositions DRAFT CS/Mth 240: Introduction to Discrete Mthemtics 1/20/2010 Lecture 2 : Propositions Instructor: Dieter vn Melkeeek Scrie: Dlior Zelený DRAFT Lst time we nlyzed vrious mze solving lgorithms in order to illustrte

More information

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR 6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

More information

Add and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions.

Add and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions. TEKS 8. A..A, A.0.F Add nd Subtrct Rtionl Epressions Before Now You multiplied nd divided rtionl epressions. You will dd nd subtrct rtionl epressions. Why? So you cn determine monthly cr lon pyments, s

More information

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,

More information