1 d Solving xy = 12 and. 2 a Solving xy = 9 and y= or x= 3 and y = 3 (point P)
|
|
|
- Raymond Wood
- 5 years ago
- Views:
Transcription
1 Coni Seions D a d Solving y and y + b Solving y and y ( )( ) So y or y So he oordinaes are, Q, P and Using he quadrai formula: ± 6 ± 5 ± ± a Solving y 9 and y 9 So and y (poin Q) or and y (poin P) The midpoin of PQ is given by M, M, The gradien of PQ is 9 The gradien of he perpendiular biseor of PQ is herefore The equaion of he biseor is herefore y 5 5 ( ) y So y + Pearson Eduaion Ld. Copying permied for purhasing insiuion only. This maerial is no opyrigh free.
2 b Solving y + 6 and y ( )( ) + a poin P. Therefore a S, and y 9 So S has oordinaes S (,9) Solving y 6 and y ( )( ) 9 + a poin P. Therefore a T, 9 and y So T has oordinaes T ( 9,) Therefore ST ( 9 ) + ( 9) The midpoin of ST is M, M( 5,5) Therefore M lies on he line y 6 6 and y, so y 6 Solving y 6 and + y + simulaneously gives: ( )( ) + + When hen When hen 6 y 9 6 y Solving y 6 and y simulaneously gives: 6 Using he quadrai formula: ± ± When hen y 9 (poin Q) 7 6 When hen y (poin R) Boh he lines + y + and y inerse he parabola y 6 a he poin (, 9), so Q is he poin (, 9) 7 6 Hene, P is (, ) and R is, There are wo approahes you an ake o find he area of riangle PQR: Pearson Eduaion Ld. Copying permied for purhasing insiuion only. This maerial is no opyrigh free.
3 oninued Mehod : Using Area sin ab C PR QR PQ PR + QR PQ Cosine rule: osprq PR QR Area PRQ 9.67 PR QR sinprq 9.6 unis ( )( ) ( )( ) sin( 9.67) Mehod : Geomerial approah Area ( 6 ) P p, p and Q q, q The gradien of PQ is q p mpq q p q p q p p q pq q p p q pq q p pq ( ) ( q p) ( ) pq q p The equaion of he line PQ is herefore y p p pq pq y p pqy q p ( p) + pqy p+ q So + pqy ( p+ q) as required Pearson Eduaion Ld. Copying permied for purhasing insiuion only. This maerial is no opyrigh free.
4 5 Solving y a () and y () From (), y y Subsiuing ino equaion (): a a a ( a) y Therefore () ( ) y a ( a) Therefore C and H inerse ealy one, and he poin of inerseion is,( a) ( a) 6 y A P, Hene y, so y So he oordinaes of P are P (,) A Q,, so y Hene y So he oordinaes of Q are Q(, ) Therefore lengh P Q ( ) a Subsiue 9 and y in he equaion y + 69 : 9 ( 9) b ( )( ) When and y When and y 7 Therefore he required poins are (, ) and (,7 ) a and y, so y b When, hen 6 and y, so P 6, he oordinae of P is When 6, hen 7 and y, so he oordinae of Q is Q ( 7,) PQ ( 7 6) + ( ) 66 + Pearson Eduaion Ld. Copying permied for purhasing insiuion only. This maerial is no opyrigh free.
5 PQ has midpoin M, M 9, 9 b The gradien of PQ is The gradien of he perpendiular biseor of PQ is herefore The equaion of he perpendiular biseor of PQ is herefore: y 9 y 9 y 7 y 9 a Solving y and + y As shown in he diagram, Area R (Area of riangle PQS) + (Area of reangle T ) (Area under y beween and ) R 6+ ( 6 ) d 5 ln [ ] 5 ln ln 5 ln ( )( ) y y Therefore he required oordinaes are, Q, P and Pearson Eduaion Ld. Copying permied for purhasing insiuion only. This maerial is no opyrigh free. 5
6 Challenge C is a hyperbola of he form y roaed hrough an angle 5 anilokwise abou he origin. The ransformaion mari required is: os 5 sin 5 sin5 os5 Applying his o a general poin on he hyperbola p, p : p p p p p + p y p p + p p p p p p So y So k and k Pearson Eduaion Ld. Copying permied for purhasing insiuion only. This maerial is no opyrigh free. 6
Differentiation 9G. 1 a. 2t 1 2t 1 dx 2 dy 2 t(2t 1) 2t. t 3t. d 2 2. dx 2(1 t ) 4t 2(1 t ) y t t t t t d 2 (1 ) 2 (1 ) 4.
Differeniaion 9G a c d e x, y, x, y 6, 6 d 6 6 x, y 6, d 6 5 x, y, 5 d 5 5, x y, 6 d 6 f x, y ( ), d ( ) d ( ) ( ) y d ( ) d g x, y ( ) ( ) d ( ) ( ) d ( ) ( ) y d ( ) ( ) ( ) h x e, y e e, d e e ( )e
Yimin Math Centre. 4 Unit Math Homework for Year 12 (Worked Answers) 4.1 Further Geometric Properties of the Ellipse and Hyperbola...
4 Uni Mah Homework for Year 12 (Worked Answers) Suden Name: Grade: Dae: Score: Table of conens 4 Topic 2 Conics (Par 4) 1 4.1 Furher Geomeric Properies of he Ellipse and Hyperbola............... 1 4.2
Solutionbank Edexcel AS and A Level Modular Mathematics
Page of 4 Soluionbank Edexcel AS and A Level Modular Mahemaics Exercise A, Quesion Quesion: Skech he graphs of (a) y = e x + (b) y = 4e x (c) y = e x 3 (d) y = 4 e x (e) y = 6 + 0e x (f) y = 00e x + 0
Learning Enhancement Team
Learning Enhancemen Team Model answers: Exponenial Funcions Exponenial Funcions sudy guide 1 i) The base rae of growh b is equal o 3 You can see his by noicing ha 1b 36 in his sysem, dividing boh sides
Variable acceleration, Mixed Exercise 11
Variable acceleraion, Mixed Exercise 11 1 a v 1 P is a res when v 0. 0 1 b s 0 0 v d (1 ) 1 0 1 0 7. The disance ravelled by P is 7. m. 1 a v 6+ a d v 6 + When, a 6+ 0 The acceleraion of P when is 0 m
Answers to 1 Homework
Answers o Homework. x + and y x 5 y To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x.. x and y, x y x To eliminae he parameer, solve for. Subsiue ino y s equaion o ge x y, x. (Noe:
12 TH STD - MATHEMATICS
e e e e e e e e e e a TH STD - MATHEMATICS HALF YEARLY EXAMINATION 7 b) d) ANSWER KEY (8 - - 7) ONE MARK QUESTIONS : X k n ( a d j I ) d) no soluion 5 d) d) has onl rivial soluion onl if rank of he oeffiien
( ) 2. Review Exercise 2. cos θ 2 3 = = 2 tan. cos. 2 x = = x a. Since π π, = 2. sin = = 2+ = = cotx. 2 sin θ 2+
Review Eercise sin 5 cos sin an cos 5 5 an 5 9 co 0 a sinθ 6 + 4 6 + sin θ 4 6+ + 6 + 4 cos θ sin θ + 4 4 sin θ + an θ cos θ ( ) + + + + Since π π, < θ < anθ should be negaive. anθ ( + ) Pearson Educaion
15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
Check in: 1 If m = 2(x + 1) and n = find y when. b y = 2m n 2
7 Parameric equaions This chaer will show ou how o skech curves using heir arameric equaions conver arameric equaions o Caresian equaions find oins of inersecion of curves and lines using arameric equaions
The equation to any straight line can be expressed in the form:
Sring Graphs Par 1 Answers 1 TI-Nspire Invesigaion Suden min Aims Deermine a series of equaions of sraigh lines o form a paern similar o ha formed by he cables on he Jerusalem Chords Bridge. Deermine he
Ch.1. Group Work Units. Continuum Mechanics Course (MMC) - ETSECCPB - UPC
Ch.. Group Work Unis Coninuum Mechanics Course (MMC) - ETSECCPB - UPC Uni 2 Jusify wheher he following saemens are rue or false: a) Two sreamlines, corresponding o a same insan of ime, can never inersec
1. VELOCITY AND ACCELERATION
1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under
Three Dimensional Coordinate Geometry
HKCWCC dvned evel Pure Mhs. / -D Co-Geomer Three Dimensionl Coordine Geomer. Coordine of Poin in Spe Z XOX, YOY nd ZOZ re he oordine-es. P,, is poin on he oordine plne nd is lled ordered riple. P,, X Y
MEI STRUCTURED MATHEMATICS 4758
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Cerificae of Educaion Advanced General Cerificae of Educaion MEI STRUCTURED MATHEMATICS 4758 Differenial Equaions Thursday 5 JUNE 006 Afernoon
Let us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
(π 3)k. f(t) = 1 π 3 sin(t)
Mah 6 Fall 6 Dr. Lil Yen Tes Show all our work Name: Score: /6 No Calculaor permied in his par. Read he quesions carefull. Show all our work and clearl indicae our final answer. Use proper noaion. Problem
Energy Momentum Tensor for Photonic System
018 IJSST Volume 4 Issue 10 Prin ISSN : 395-6011 Online ISSN : 395-60X Themed Seion: Siene and Tehnology Energy Momenum Tensor for Phooni Sysem ampada Misra Ex-Gues-Teaher, Deparmens of Eleronis, Vidyasagar
Mark Scheme (Results) January 2011
Mark (Resuls) January 0 GCE GCE Furher Pure Mahemaics FP (6667) Paper Edexcel Limied. Regisered in England and Wales No. 4496750 Regisered Office: One90 High Holborn, London WCV 7BH Edexcel is one of he
4.5 Constant Acceleration
4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),
Differential Geometry: Revisiting Curvatures
Differenial Geomery: Reisiing Curaures Curaure and Graphs Recall: hus, up o a roaion in he x-y plane, we hae: f 1 ( x, y) x y he alues 1 and are he principal curaures a p and he corresponding direcions
In this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper The moion of projeciles In his chaper he model of free moion
The Special Theory of Relativity Chapter II
The Speial Theory of Relaiiy Chaper II 1. Relaiisi Kinemais. Time dilaion and spae rael 3. Lengh onraion 4. Lorenz ransformaions 5. Paradoes? Simulaneiy/Relaiiy If one obserer sees he eens as simulaneous,
From String Art to Caustic Curves: Envelopes in Symbolic Geometry
From Sring r o ausic urves: nvelopes in Symbolic eomery Philip Todd phil@salire.com Salire Sofware, PO 565, eaveron, OR 97223, U.S.. bsrac: In his paper, we use he symbolic geomery program eomery xpressions
SMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.
SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a
MA6151 MATHEMATICS I PART B UNIVERSITY QUESTIONS. (iv) ( i = 1, 2, 3,., n) are the non zero eigen values of A, then prove that (1) k i.
UNIT MATRICES METHOD EIGEN VALUES AND EIGEN VECTORS Find he Eigen values and he Eigenvecors of he following marices (i) ** 3 6 3 (ii) 6 (iii) 3 (iv) 0 3 Prove ha he Eigen values of a real smmeric mari
Ground Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan
Ground Rules PC11 Fundamenals of Physics I Lecures 3 and 4 Moion in One Dimension A/Prof Tay Seng Chuan 1 Swich off your handphone and pager Swich off your lapop compuer and keep i No alking while lecure
Some Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
Applicable Mathematics 2A
Applicable Mahemaics A Lecure Noes Revised: Augus 00 Please noe ha hese are my lecure noes: hey are no course noes. So I will be following hese noes very closely, supplemened by he homework eamples ec
EXERCISE - 01 CHECK YOUR GRASP
UNIT # 09 PARABOLA, ELLIPSE & HYPERBOLA PARABOLA EXERCISE - 0 CHECK YOUR GRASP. Hin : Disnce beween direcri nd focus is 5. Given (, be one end of focl chord hen oher end be, lengh of focl chord 6. Focus
10.1 EXERCISES. y 2 t 2. y 1 t y t 3. y e
66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES SLUTIN We use a graphing device o produce he graphs for he cases a,,.5,.,,.5,, and shown in Figure 7. Noice ha all of hese curves (ecep he case a ) have
a. Show that these lines intersect by finding the point of intersection. b. Find an equation for the plane containing these lines.
Mah A Final Eam Problems for onsideraion. Show all work for credi. Be sure o show wha you know. Given poins A(,,, B(,,, (,, 4 and (,,, find he volume of he parallelepiped wih adjacen edges AB, A, and A.
Peakon, pseudo-peakon, and cuspon solutions for two generalized Camassa-Holm equations
JOURNAL OF MATHEMATICAL PHYSICS 54 13501 (013) Peakon pseudo-peakon and uspon soluions for wo generalized Camassa-Holm equaions Jibin Li 1a) and Zhijun Qiao 3a) 1 Deparmen of Mahemais Zhejiang Normal Universiy
Lecture 10: Wave equation, solution by spherical means
Lecure : Wave equaion, soluion by spherical means Physical modeling eample: Elasodynamics u (; ) displacemen vecor in elasic body occupying a domain U R n, U, The posiion of he maerial poin siing a U in
DE55/DC55 ENGINEERING MATHEMATICS-II DEC 2013
DE55/DC55 ENGINEERING MATHEMATICS-II DEC Q. a. Evaluae LT sin x x sin 5x + sin 6x sin x Here we have, L x sin x + sin 6x sin 5x sin x L x L x x + 6x x 6x sin + cos 5x + x 5x x cos.sin sin 4x cos( x) cos
Module MA1132 (Frolov), Advanced Calculus Tutorial Sheet 1. To be solved during the tutorial session Thursday/Friday, 21/22 January 2016
Module MA113 (Frolov), Advanced Calculus Tuorial Shee 1 To be solved during he uorial session Thursday/Friday, 1/ January 16 A curve C in he xy-plane is represened by he equaion Ax + Bxy + Cy + Dx + Ey
The Paradox of Twins Described in a Three-dimensional Space-time Frame
The Paradox of Twins Described in a Three-dimensional Space-ime Frame Tower Chen E_mail: chen@uguam.uog.edu Division of Mahemaical Sciences Universiy of Guam, USA Zeon Chen E_mail: zeon_chen@yahoo.com
UNIVERSITY COLLEGE DUBLIN. An Coláiste Ollscoile, Baile Átha Cliath MOCK SEMESTER 1 EXAMINATION 2008/2009 MST30070
UNIVERSITY COLLEGE DUBLIN An Coláise Ollscoile, Baile Áha Cliah MOCK SEMESTER EXAMINATION 8/9 MST37 Inroducion o Differenial Geomery Professor P.J. Rippon Professor S. Dineen Dr. J. B. Quigley Time Allowed:
Additional Exercises for Chapter What is the slope-intercept form of the equation of the line given by 3x + 5y + 2 = 0?
ddiional Eercises for Caper 5 bou Lines, Slopes, and Tangen Lines 39. Find an equaion for e line roug e wo poins (, 7) and (5, ). 4. Wa is e slope-inercep form of e equaion of e line given by 3 + 5y +
Congruent Numbers and Elliptic Curves
Congruen Numbers and Ellipic Curves Pan Yan pyan@oksaeedu Sepember 30, 014 1 Problem In an Arab manuscrip of he 10h cenury, a mahemaician saed ha he principal objec of raional righ riangles is he following
UNIVERSITY OF CALIFORNIA AT BERKELEY
Homework #10 Soluions EECS 40, Fall 2006 Prof. Chang-Hasnain Due a 6 pm in 240 Cory on Wednesday, 04/18/07 oal Poins: 100 Pu (1) your name and (2) discussion secion number on your homework. You need o
236 CHAPTER 3 Torsion. Strain Energy in Torsion
36 CHAPER 3 orsion Srain Energy in orsion Problem 3.9-1 A solid circular bar of seel (G 11. 1 6 psi) wih lengh 3 in. and diameer d 1.75 in. is subjeced o pure orsion by orques acing a he ends (see figure).
From Complex Fourier Series to Fourier Transforms
Topic From Complex Fourier Series o Fourier Transforms. Inroducion In he previous lecure you saw ha complex Fourier Series and is coeciens were dened by as f ( = n= C ne in! where C n = T T = T = f (e
Math 116 Practice for Exam 2
Mah 6 Pracice for Exam Generaed Ocober 3, 7 Name: SOLUTIONS Insrucor: Secion Number:. This exam has 5 quesions. Noe ha he problems are no of equal difficuly, so you may wan o skip over and reurn o a problem
This is an example to show you how SMath can calculate the movement of kinematic mechanisms.
Dec :5:6 - Kinemaics model of Simple Arm.sm This file is provided for educaional purposes as guidance for he use of he sofware ool. I is no guaraeed o be free from errors or ommissions. The mehods and
Complete solutions to Exercise 14(b) 1. Very similar to EXAMPLE 4. We have same characteristic equation:
Soluions 4(b) Complee soluions o Exercise 4(b). Very similar o EXAMPE 4. We have same characerisic equaion: 5 i Ae = + Be By using he given iniial condiions we obain he simulaneous equaions A+ B= 0 5A
15. Bicycle Wheel. Graph of height y (cm) above the axle against time t (s) over a 6-second interval. 15 bike wheel
15. Biccle Wheel The graph We moun a biccle wheel so ha i is free o roae in a verical plane. In fac, wha works easil is o pu an exension on one of he axles, and ge a suden o sand on one side and hold he
Math 221: Mathematical Notation
Mah 221: Mahemaical Noaion Purpose: One goal in any course is o properly use he language o ha subjec. These noaions summarize some o he major conceps and more diicul opics o he uni. Typing hem helps you
Elements of Computer Graphics
CS580: Compuer Graphics Min H. Kim KAIST School of Compuing Elemens of Compuer Graphics Geomery Maerial model Ligh Rendering Virual phoography 2 Foundaions of Compuer Graphics A PINHOLE CAMERA IN 3D 3
ADDITIONAL MATHEMATICS PAPER 1
000-CE A MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 000 ADDITIONAL MATHEMATICS PAPER 8.0 am 0.0 am ( hours This paper mus be answered in English. Answer
Two Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
Position, Velocity, and Acceleration
rev 06/2017 Posiion, Velociy, and Acceleraion Equipmen Qy Equipmen Par Number 1 Dynamic Track ME-9493 1 Car ME-9454 1 Fan Accessory ME-9491 1 Moion Sensor II CI-6742A 1 Track Barrier Purpose The purpose
Chapters 6 & 7: Trigonometric Functions of Angles and Real Numbers. Divide both Sides by 180
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Chapers & : Trigonomeric Funcions of Angles and Real Numbers - Angle Measures Radians: - a uni (rad o measure he size of an angle. rad
Honors Solutions. Honors Lesson 1. Honors Lesson = 2. substitute 3 for ( r = 90
Honors Lesson. A. Muliplying by : x x x x. subsiue for ( r s): + 8 9 + 9. since i is a square, we know all sides are equal, herefore: +9 9. A ( + 9) square unis using from # A ( + 9)( ) A square unis.
Math 4600: Homework 11 Solutions
Mah 46: Homework Soluions Gregory Handy [.] One of he well-known phenomenological (capuring he phenomena, bu no necessarily he mechanisms) models of cancer is represened by Gomperz equaion dn d = bn ln(n/k)
The Special Theory of Relativity
The Speial Theor of Relaii The Speial Theor of Relaii Chaper I. Conradiions in phsis?. Galilean Transformaions of lassial mehanis 3. The effe on Mawell s equaions ligh 4. Mihelson-Morle eperimen 5. insein
1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
![1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.](/thumbs/71/65799522.jpg "1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.")
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
Chapter 7: Solving Trig Equations
Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions
Amit Mehra. Indian School of Business, Hyderabad, INDIA Vijay Mookerjee
RESEARCH ARTICLE HUMAN CAPITAL DEVELOPMENT FOR PROGRAMMERS USING OPEN SOURCE SOFTWARE Ami Mehra Indian Shool of Business, Hyderabad, INDIA {Ami_Mehra@isb.edu} Vijay Mookerjee Shool of Managemen, Uniersiy
PMT. Version 1.0: abc. General Certificate of Education. Mathematics MPC4 Pure Core 4. Mark Scheme examination - January series
Version.: 8 abc General Cerificae of Educaion Mahemaics 66 MPC Pure Core Mark Scheme 8 eaminaion - January series Mark schemes are prepared by he Principal Eaminer and considered, ogeher wih he relevan
Physics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs.
Physics 180A Fall 2008 Tes 1-120 poins Name Provide he bes answer o he following quesions and problems. Wach your sig figs. 1) The number of meaningful digis in a number is called he number of. When numbers
Applications of the Basic Equations Chapter 3. Paul A. Ullrich
Applicaions of he Basic Equaions Chaper 3 Paul A. Ullrich paullrich@ucdavis.edu Par 1: Naural Coordinaes Naural Coordinaes Quesion: Why do we need anoher coordinae sysem? Our goal is o simplify he equaions
Teaching parametric equations using graphing technology
Teaching parameric equaions using graphing echnology The session will sar by looking a problems which help sudens o see ha parameric equaions are no here o make life difficul bu are imporan and give rise
Time: 1 hour 30 minutes
Paper Reference(s) 6666/0 Edexcel GCE Core Mahemaics C4 Silver Level S4 Time: hour 30 minues Maerials required for examinaion papers Mahemaical Formulae (Green) Iems included wih quesion Nil Candidaes
CHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
Parametrics and Vectors (BC Only)
Paramerics and Vecors (BC Only) The following relaionships should be learned and memorized. The paricle s posiion vecor is r() x(), y(). The velociy vecor is v(),. The speed is he magniude of he velociy
Chapter 10 INDUCTANCE Recommended Problems:
Chaper 0 NDUCTANCE Recommended Problems: 3,5,7,9,5,6,7,8,9,,,3,6,7,9,3,35,47,48,5,5,69, 7,7. Self nducance Consider he circui shown in he Figure. When he swich is closed, he curren, and so he magneic field,
Chapter 14 Homework Answers
4. Suden responses will vary. (a) combusion of gasoline (b) cooking an egg in boiling waer (c) curing of cemen Chaper 4 Homework Answers 4. A collision beween only wo molecules is much more probable han
Directional Tubular Surfaces
Inernaional Journal of Algebra, Vol. 9, 015, no. 1, 57-535 HIKARI Ld, www.m-hikari.com hp://dx.doi.org/10.1988/ija.015.5174 Direcional Tubular Surfaces Musafa Dede Deparmen of Mahemaics, Faculy of Ars
Week 1 Lecture 2 Problems 2, 5. What if something oscillates with no obvious spring? What is ω? (problem set problem)
Week 1 Lecure Problems, 5 Wha if somehing oscillaes wih no obvious spring? Wha is ω? (problem se problem) Sar wih Try and ge o SHM form E. Full beer can in lake, oscillaing F = m & = ge rearrange: F =
Chapter 8 The Complete Response of RL and RC Circuits
Chaper 8 he Complee Response of R and RC Ciruis Exerises Ex 8.3-1 Before he swih loses: Afer he swih loses: 2 = = 8 Ω so = 8 0.05 = 0.4 s. 0.25 herefore R ( ) Finally, 2.5 ( ) = o + ( (0) o ) = 2 + V for
Chapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
u(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x
. 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih
Review - Quiz # 1. 1 g(y) dy = f(x) dx. y x. = u, so that y = xu and dy. dx (Sometimes you may want to use the substitution x y
Review - Quiz # 1 (1) Solving Special Tpes of Firs Order Equaions I. Separable Equaions (SE). d = f() g() Mehod of Soluion : 1 g() d = f() (The soluions ma be given implicil b he above formula. Remember,
Mathematics Paper- II
R Prerna Tower, Road No -, Conracors Area, Bisupur, Jamsedpur - 8, Tel - (65789, www.prernaclasses.com Maemaics Paper- II Jee Advance PART III - MATHEMATICS SECTION - : (One or more opions correc Type
Distance Between Two Ellipses in 3D
Disance Beween Two Ellipses in 3D David Eberly Magic Sofware 6006 Meadow Run Cour Chapel Hill, NC 27516 eberly@magic-sofware.com 1 Inroducion An ellipse in 3D is represened by a cener C, uni lengh axes
1 Radian Measures. Exercise 1. A --- 1Vc. MAT Worksheet 10 Sections 7.1, 8.1 Name:
MAT 012 55218 1 Radian Measures Consider he following figure. corresponding o he angle 0. Exercise 1 The shaded porion of he circle is called he secor of he circle K V 1. Suppose know he radian measure
4.6 One Dimensional Kinematics and Integration
4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of
Solution of Integro-Differential Equations by Using ELzaki Transform
Global Journal of Mahemaical Sciences: Theory and Pracical. Volume, Number (), pp. - Inernaional Research Publicaion House hp://www.irphouse.com Soluion of Inegro-Differenial Equaions by Using ELzaki Transform
Chapter 3 Kinematics in Two Dimensions
Chaper 3 KINEMATICS IN TWO DIMENSIONS PREVIEW Two-dimensional moion includes objecs which are moing in wo direcions a he same ime, such as a projecile, which has boh horizonal and erical moion. These wo
3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
APPM 2360 Homework Solutions, Due June 10
2.2.2: Find general soluions for he equaion APPM 2360 Homework Soluions, Due June 10 Soluion: Finding he inegraing facor, dy + 2y = 3e µ) = e 2) = e 2 Muliplying he differenial equaion by he inegraing
9231 FURTHER MATHEMATICS
CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Level MARK SCHEME for he May/June series 9 FURTHER MATHEMATICS 9/ Paper, maximum raw mark This mark scheme is published as an aid o eachers and candidaes,
3, so θ = arccos
Mahemaics 210 Professor Alan H Sein Monday, Ocober 1, 2007 SOLUTIONS This problem se is worh 50 poins 1 Find he angle beween he vecors (2, 7, 3) and (5, 2, 4) Soluion: Le θ be he angle (2, 7, 3) (5, 2,
A 1.3 m 2.5 m 2.8 m. x = m m = 8400 m. y = 4900 m 3200 m = 1700 m
PHYS : Soluions o Chper 3 Home Work. SSM REASONING The displcemen is ecor drwn from he iniil posiion o he finl posiion. The mgniude of he displcemen is he shores disnce beween he posiions. Noe h i is onl
Displacement ( x) x x x
Kinemaics Kinemaics is he branch of mechanics ha describes he moion of objecs wihou necessarily discussing wha causes he moion. 1-Dimensional Kinemaics (or 1- Dimensional moion) refers o moion in a sraigh
Challenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
Chapter 16: Summary. Instructor: Jean-François MILLITHALER.
Chaper 16: Summary Insrucor: Jean-François MILLITHALER hp://faculy.uml.edu/jeanfrancois_millihaler/funelec/spring2017 Slide 1 Curren & Charge Elecric curren is he ime rae of change of charge, measured
ME 391 Mechanical Engineering Analysis
Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of
Topic 1: Linear motion and forces
TOPIC 1 Topic 1: Linear moion and forces 1.1 Moion under consan acceleraion Science undersanding 1. Linear moion wih consan elociy is described in erms of relaionships beween measureable scalar and ecor
Solutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
Vector Calculus. Chapter 2
Chaper Vecor Calculus. Elemenar. Vecor Produc. Differeniaion of Vecors 4. Inegraion of Vecors 5. Del Operaor or Nabla (Smbol 6. Polar Coordinaes Chaper Coninued 7. Line Inegral 8. Volume Inegral 9. Surface
UNIT # 01 (PART I) BASIC MATHEMATICS USED IN PHYSICS, UNIT & DIMENSIONS AND VECTORS. 8. Resultant = R P Q, R P 2Q
J-Phsics UNI # 0 (PAR I) ASIC MAHMAICS USD IN PHYSICS, UNI & DIMNSIONS AND VCORS XRCIS I. nclosed area : A r so da dr r Here r 8 cm, dr da 5 cm/s () (8) (5) 80 cm /s. Slope d d 6 9 if angen is parallel
Be able to sketch a function defined parametrically. (by hand and by calculator)
Pre Calculus Uni : Parameric and Polar Equaions (7) Te References: Pre Calculus wih Limis; Larson, Hoseler, Edwards. B he end of he uni, ou should be able o complee he problems below. The eacher ma provide
Electrical and current self-induction
Elecrical and curren self-inducion F. F. Mende hp://fmnauka.narod.ru/works.hml mende_fedor@mail.ru Absrac The aricle considers he self-inducance of reacive elemens. Elecrical self-inducion To he laws of
Section A: Forces and Motion
I is very useful o be able o make predicions abou he way moving objecs behave. In his chaper you will learn abou some equaions of moion ha can be used o calculae he speed and acceleraion of objecs, and
Reading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.
PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence
BEng (Hons) Telecommunications. Examinations for / Semester 2
BEng (Hons) Telecommunicaions Cohor: BTEL/14/FT Examinaions for 2015-2016 / Semeser 2 MODULE: ELECTROMAGNETIC THEORY MODULE CODE: ASE2103 Duraion: 2 ½ Hours Insrucions o Candidaes: 1. Answer ALL 4 (FOUR)
AP Calculus BC Chapter 10 Part 1 AP Exam Problems
AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a