III. (x 2) 2 = (2 x) 2. A. a = 1, b = 3. B. a = 3, b = 1. C. a = 1, b = 3. D. a = 3, b = 1. E. a = 1, b = 3.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "III. (x 2) 2 = (2 x) 2. A. a = 1, b = 3. B. a = 3, b = 1. C. a = 1, b = 3. D. a = 3, b = 1. E. a = 1, b = 3."

Transcription

1 Fom 5 HKEE 1 Mthemtics II 1 If = h 7 1 : 8 1 : 1 : 1 : 1 : = ( ) 1 ( ) ( ) ( + ) 1 ( + ) ( + ), then the tio : h is 5 6 III ( ) = ( ) I onl II onl III onl I nd II onl I nd III onl Given the identit + 1 b + + =, 1 ( 1) ( 1 ) ( 1) find the vlues of the constnts nd b = 1, b = =, b = 1 = 1, b = =, b = 1 = 1, b = If α nd β e the oots of the eqution + + =, find the vlue of α α α + β If 1 + ( ) ( 1) ( ) (1 ) = ( ), then = 7 It cnnot be detemined 1 1 Which of the following is n identit/e identities? I ( + 1)( 1) = + 1, II + 1 =, -E-MTHS II 1

2 The figue shows n infinite numbe of sques The length of side of the fist sque is 1 The side of the fist sque is equl to hlf of the side of the peceding one Find the sum of the es of the infinite numbe of sques 1 I II III 5 In the figue, is stight line with = = Thee cicles I, II nd III e dwn espectivel on, nd s dimetes es of cicle I : e of cicle II : e of cicle III = 8 Find the el vlue of such tht = k 7, whee k is constnt 1 = k 1 : : 1 : : 1 : : 1 : : 16 1 : 8 : mn dives c t 5 km/h fo hous nd then t 5 km/h fo hous His vege speed fo the whole joune is θ 1 θ 7 km/h 75 km/h 8 km/h 85 km/h km/h The figue shows two sectos with dii nd If these two sectos e equl in e, then θ 1 : θ = 1 R : 1 : 1 : 1 5 : 1 6 : 1 Sphee Right cicul clinede In the figue, if Volume of the sphee = Volume of the ight cicul clinde R, then = -E-MTHS II

3 1 If the compound inteest on $1 fo two es t % p, p hlf-el is $, find 1 1 1(1 + ) 1 5 1(1 + ) 1 1(1 + ) π π π π Which of the following functions m be epesented b the bove gph in the intevl to π? = cos = cos 1 = cos = sin = sin sin θ cos θ = 1 5 1(1 + 1 ) 1 If sin θ cos θ = 1, then (sin θ + cos θ) = 1 1 cos θ sin θ cos θ sin θ cos θ sin θ In the figue, : : = : : 1 : : 1 : : 1 : : 1 -E-MTHS II

4 18 1 : : The being of lighthouse s obseved fom n ocen line is N7oE, the being of the ocen line s obseved fom the light house is θ In the figue, if the e of the secto is, then c = π π 1 N7 o E N5 o W S7 o E S7 o W S5 o W Which of the following epesents cicle? = + = = = = L = - 5 L o 1 1 o In the figue, = = 1 = o nd = 1 o, find cos o 1 o sin o sin cos o sin o In the figue, L 1 nd L e two stight lines pependicul to ech othe nd intesecting t P on the -is If the eqution of L 1 is = 5, then the eqution of L is 1 = 5 1 = + 5 = 5 = = P -E-MTHS II

5 8 o 115 o In the figue,,,, nd E lie on cicle intesects E t K = 1 o nd E = 1 o If E //, then = In the figue, = 8 o 5 o o 6 o o o 5 77 o 8 o 6 o 1 o 115 o F E In the figue,,, E nd EF e ight ngled isosceles tingles If = = 1, how long is F? 5 5 E 1 o 1 o In shooting gme, the pobbilities fo John nd M to hit tget e 5 nd espectivel When both shoot t the 5 tget, wht is the pobbilit tht the both miss Given two goups of numbes + 1, +, + nd b + 1, b +, b +, whee > b m 1 nd m e espectivel the mens of the two goups, nd s 1 nd s e espectivel thei stndd devitions Which of the following is tue? m 1 > m nd s 1 > s m 1 > m nd s 1 = s m 1 = m nd s 1 > s m 1 = m nd s 1 = s m 1 > m nd s 1 < s -E-MTHS II 5

6 8 f If (1 ) =( z )(5 z ), then which of the following must be tue The figue shows the fequenc cuve of cetin distibution Which of the following cn be the distibution s cummultive fequenc cuve? = z = z = z = z = z + = cf cf ( + ) cf cf 1 The LM of 1 b nd 18b c is 6b 6 b c 6b 6 b c 16 b c (, ) (, ) cf (, ) + = 6 + = -E-MTHS II 6

7 Let p = + Unde the following constints + 6 wht is the getest vlue of p? If log + log = log z, whee, nd z e positive numbes, which of the following must be tue? I + = z II log + log = log z III = z I onl II onl III onl I nd II onl II nd III onl Let F() = Given tht F() = nd F( ) =, then F() cn be fctoized s ( + ) ( ) ( + 1) ( + ) ( ) ( 1) ( ) ( + ) ( + 1) ( ) ( ) ( + 1) ( ) ( + ) ( 1) 5 If, b nd c e positive numbes, which of the following is possible gphicl epesenttion of = + b + c -E-MTHS II 7

8 6 If > nd b <, which of the following is/e negtive? I II III 1 1 b b + b b b EFGH is cube of side cm tethedon H is cut w long the plne H The volume of the emining solid is 6 cm cm 15 cm 18 cm 5 cm 7 I onl III onl I nd II onl I nd III onl II nd III onl If < < nd < <, then the nge of vlues of is 1 < < 1 < < 1 < < < < 1 < < The mked pice of n ticle is oiginll P The mked pice is then incesed so tht when discount of 1% is mde on the new mked pice, the selling pice is still P Wht is the new mked pice? P 1 1 P 1 11 P P 1 1 P 8 cm E F H G The totl sufce e of egul tethedon of side cm is cm cm 7 cm cm -E-MTHS II 8

9 1 cm nnot be detemined 1 Ten lites of mitue contin 6% of lcohol nd % of wte b volume How mn lites of wte should be dded so tht it contins % of lcohol b volume? If,, nd cn finish unning the sme distnce in, nd 5 minutes espectivel, then s speed : s peed : s speed = : : 5 5 : : : 8 : 7 : 15 : 1 5 : 16 : In, =, = nd sin =, then cos = If the five inteio ngles of conve pentgon fom n P with common diffeence of 1 o, then the smllest inteio ngle of the pentgon is 6 5 o 7 o 88 o 8 o 18 o Let p be positive constnt such tht p sin θ = nd p sin θ = 1 Find ll the vlues of θ in the intevl to π In the figue, : = sin : sin cos : cos tn : tn sin : sin cos : cos π π 6 π π, π 7π, 6 7 -E-MTHS II

10 In the figue, is sque cs nd e dwn with centes nd espectivel, intesecting t c : c = o 8 1 : 1 : 1 : 1 : : E H F G 5 1 o nd e equl chods of the cicle = o nd = 1 o = o 5 o o 5 o o 5 o In the figue, nd EFGH e two sques of side 1 The e plced one upon the othe with thei centes both t to fom st with 16 sides, ech of length Find In the figue, nd e tngents to the cicle If = 5 o, then = o 1 o 85 o 8 o 5 o E 6 -E-MTHS II 1

11 5 In, P =, Q = 6 nd Q = If PQ =, then P = cicle, cente, touches the secto intenll t, E nd F = 6 o nd = 18 Find the dius of the cicle 6 5 T P R θ Q Thee cicles, centes, nd touch ech othe s shown in the figue The dii of the two cicles with cente nd e both 1 cm nd dius of the cicle with cente is cm Find the e of the shded pt in cm π 6 π π 6 π In the figue, PQ is dimete nd PT is tngent of the cicle QT cuts the cicle t R Let Q = θ nd PQ =, then TR = cosθ sinθ sinθ tnθ sin θ tn θ cos θ tn θ It cnnot be detemined 5 F 6 o E -E-MTHS II 11

Intro to Circle Geometry By Raymond Cheong

Intro to Circle Geometry By Raymond Cheong Into to Cicle Geomety By Rymond Cheong Mny poblems involving cicles cn be solved by constucting ight tingles then using the Pythgoen Theoem. The min chllenge is identifying whee to constuct the ight tingle.

More information

r (1+cos(θ)) sin(θ) C θ 2 r cos θ 2

r (1+cos(θ)) sin(θ) C θ 2 r cos θ 2 icles xmple 66: Rounding one ssume we hve cone of ngle θ, nd we ound it off with cuve of dius, how f wy fom the cone does the ound stt? nd wht is the chod length? (1+cos(θ)) sin(θ) θ 2 cos θ 2 xmple 67:

More information

Summary: Vectors. This theorem is used to find any points (or position vectors) on a given line (direction vector). Two ways RT can be applied:

Summary: Vectors. This theorem is used to find any points (or position vectors) on a given line (direction vector). Two ways RT can be applied: Summ: Vectos ) Rtio Theoem (RT) This theoem is used to find n points (o position vectos) on given line (diection vecto). Two ws RT cn e pplied: Cse : If the point lies BETWEEN two known position vectos

More information

(Ch. 22.5) 2. What is the magnitude (in pc) of a point charge whose electric field 50 cm away has a magnitude of 2V/m?

(Ch. 22.5) 2. What is the magnitude (in pc) of a point charge whose electric field 50 cm away has a magnitude of 2V/m? Em I Solutions PHY049 Summe 0 (Ch..5). Two smll, positively chged sphees hve combined chge of 50 μc. If ech sphee is epelled fom the othe by n electosttic foce of N when the sphees e.0 m pt, wht is the

More information

Exam in physics, El-grunder (Electromagnetism), 2014-03-26, kl 9.00-15.00

Exam in physics, El-grunder (Electromagnetism), 2014-03-26, kl 9.00-15.00 Umeå Univesitet, Fysik 1 Vitly Bychkov Em in physics, El-gunde (Electomgnetism, 14--6, kl 9.-15. Hjälpmedel: Students my use ny book(s. Mino notes in the books e lso llowed. Students my not use thei lectue

More information

G.GMD.1 STUDENT NOTES WS #5 1 REGULAR POLYGONS

G.GMD.1 STUDENT NOTES WS #5 1 REGULAR POLYGONS G.GMD.1 STUDENT NOTES WS #5 1 REGULAR POLYGONS Regul polygon e of inteet to u becue we begin looking t the volume of hexgonl pim o Tethedl nd to do thee type of clcultion we need to be ble to olve fit

More information

Orbits and Kepler s Laws

Orbits and Kepler s Laws Obits nd Keple s Lws This web pge intoduces some of the bsic ides of obitl dynmics. It stts by descibing the bsic foce due to gvity, then consides the ntue nd shpe of obits. The next section consides how

More information

Curvature. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015

Curvature. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015 Cuvtue Com S 477/577 Notes Yn-Bin Ji Oct 8, 205 We wnt to find mesue of how cuved cuve is. Since this cuvtue should depend only on the shpe of the cuve, it should not be chnged when the cuve is epmetized.

More information

Random Variables and Distribution Functions

Random Variables and Distribution Functions Topic 7 Rndom Vibles nd Distibution Functions 7.1 Intoduction Fom the univese of possible infomtion, we sk question. To ddess this question, we might collect quntittive dt nd ognize it, fo emple, using

More information

Chapter 22 The Electric Field II: Continuous Charge Distributions

Chapter 22 The Electric Field II: Continuous Charge Distributions Chpte The lectic Field II: Continuous Chge Distibutions Conceptul Poblems [SSM] Figue -7 shows n L-shped object tht hs sides which e equl in length. Positive chge is distibuted unifomly long the length

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. 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 information

N V V L. R a L I. Transformer Equation Notes

N V V L. R a L I. Transformer Equation Notes Tnsfome Eqution otes This file conts moe etile eivtion of the tnsfome equtions thn the notes o the expeiment 3 wite-up. t will help you to unestn wht ssumptions wee neee while eivg the iel tnsfome equtions

More information

Chapter 23 Electrical Potential

Chapter 23 Electrical Potential hpte Electicl Potentil onceptul Polems [SSM] A poton is moved to the left in unifom electic field tht points to the ight. Is the poton moving in the diection of incesing o decesing electic potentil? Is

More information

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period: Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A

More information

Geometry 7-1 Geometric Mean and the Pythagorean Theorem

Geometry 7-1 Geometric Mean and the Pythagorean Theorem Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the

More information

Binary Representation of Numbers Autar Kaw

Binary 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 information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

More information

GRAVITATION 1. BASIC FORCES IN NATURE

GRAVITATION 1. BASIC FORCES IN NATURE GRAVITATION. BASIC ORCES IN NATURE POINTS TO REMEMBER. Bsing on the ntue nd eltive stength the bsic foces in ntue e clssified into fou ctegoies. They e ) Gvittionl foce ) Electomgnetic foce 3) Stong Nucle

More information

Skills Needed for Success in Calculus 1

Skills Needed for Success in Calculus 1 Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use 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 information

Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Work, Power and Kinetic Energy Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

More information

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to . Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

More information

32. The Tangency Problem of Apollonius.

32. The Tangency Problem of Apollonius. . The Tngeny olem of Apollonius. Constut ll iles tngent to thee given iles. This eleted polem ws posed y Apollinius of eg (. 60-70 BC), the getest mthemtiin of ntiquity fte Eulid nd Ahimedes. His mjo wok

More information

Formulas and Units. Transmission technical calculations Main Formulas. Size designations and units according to the SI-units.

Formulas and Units. Transmission technical calculations Main Formulas. Size designations and units according to the SI-units. Fomuls nd Units Tnsmission technicl clcultions Min Fomuls Size designtions nd units ccoding to the SI-units Line movement: s v = m/s t s = v t m s = t m v = m/s t P = F v W F = m N Rottion ω = π f d/s

More information

AMPERE S LAW. by Kirby Morgan MISN-0-138

AMPERE S LAW. by Kirby Morgan MISN-0-138 MISN-0-138 AMPERE S LAW by Kiby Mogn 1. Usefullness................................................ 1 AMPERE S LAW 2. The Lw................................................... 1. The Integl Reltionship...............................

More information

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is

More information

VEHICLE PLANAR DYNAMICS BICYCLE MODEL

VEHICLE PLANAR DYNAMICS BICYCLE MODEL Auptions -Do, VEHE PANA DYNAMS BYE MODE o tel, (esued o instntneous cente o ottion O) o Yw, (wt Globl Ais) ongitudinl elocit is ued to be constnt. Sll slip ngles, i.e. ties opete in the line egion. No

More information

SHORT REVISION SOLUTIONS OF TRIANGLE

SHORT REVISION SOLUTIONS OF TRIANGLE FREE Download Study Package fom website: wwwtekoclassescom SHORT REVISION SOLUTIONS OF TRINGLE I SINE FORMUL : In any tiangle BC, II COSINE FORMUL : (i) b + c a bc a b c sin sinb sin C o a² b² + c² bc

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs 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

UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

More information

Math 135 Circles and Completing the Square Examples

Math 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 information

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360! 1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

More information

Maths Word Searches. List of Contents. Word Search 1. Word Search 2. Word Search 3. Word Search 4. Word Search 5. Word Search 6.

Maths Word Searches. List of Contents. Word Search 1. Word Search 2. Word Search 3. Word Search 4. Word Search 5. Word Search 6. Maths Word earches List of Contents Word earch 1 Word earch 2 Word earch 3 Word earch 4 Word earch 5 Word earch 6 Word earch 7 Word earch 8 Maths Word earch 1 The Word List at the bottom (from CE to TWO)

More information

Displacement, Velocity And Acceleration

Displacement, Velocity And Acceleration Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

More information

Review Problems for the Final of Math 121, Fall 2014

Review 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 information

(1) continuity equation: 0. momentum equation: u v g (2) u x. 1 a

(1) continuity equation: 0. momentum equation: u v g (2) u x. 1 a Comment on The effect of vible viscosity on mied convection het tnsfe long veticl moving sufce by M. Ali [Intentionl Jounl of Theml Sciences, 006, Vol. 45, pp. 60-69] Asteios Pntoktos Associte Pofesso

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

Vectors 2. 1. Recap of vectors

Vectors 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 information

Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapter 3 Savings, Present Value and Ricardian Equivalence Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

More information

Adaptive Control of a Production and Maintenance System with Unknown Deterioration and Obsolescence Rates

Adaptive Control of a Production and Maintenance System with Unknown Deterioration and Obsolescence Rates Int J of Mthemtic Sciences nd Appictions, Vo, No 3, Septembe Copyight Mind Rede Pubictions wwwjounshubcom Adptive Conto of Poduction nd Mintennce System with Unknown Deteiotion nd Obsoescence Rtes Fwzy

More information

Open Economies. Chapter 32. A Macroeconomic Theory of the Open Economy. Basic Assumptions of a Macroeconomic Model of an Open Economy

Open Economies. Chapter 32. A Macroeconomic Theory of the Open Economy. Basic Assumptions of a Macroeconomic Model of an Open Economy Chapte 32. A Macoeconomic Theoy of the Open Economy Open Economies An open economy is one that inteacts feely with othe economies aound the wold. slide 0 slide 1 Key Macoeconomic Vaiables in an Open Economy

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS 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 point-direction nd twopoint

More information

Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.

Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere. Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming

More information

Gauss Law. Physics 231 Lecture 2-1

Gauss Law. Physics 231 Lecture 2-1 Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

More information

Valuation of Floating Rate Bonds 1

Valuation of Floating Rate Bonds 1 Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned

More information

Example 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

Example 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 information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial 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 information

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.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 information

AP Physics Gravity and Circular Motion

AP Physics Gravity and Circular Motion AP Phyic Gity nd icul Motion Newton theoy i ey iple. Gity i foce of ttction between ny two object tht he. Two object itting on dektop ttct ech othe with foce tht we cll gity. They don t go flying togethe

More information

Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006

Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006 dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.

More information

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6

Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6 Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe

More information

Screentrade Car Insurance Policy Summary

Screentrade Car Insurance Policy Summary Sceentde C Insunce Policy Summy This is summy of the policy nd does not contin the full tems nd conditions of the cove, which cn be found in the policy booklet nd schedule. It is impotnt tht you ed the

More information

Operations with Polynomials

Operations 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 information

Problem Set # 9 Solutions

Problem Set # 9 Solutions Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

More information

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a. Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles

More information

GFI EventsMnge vs Netikus.net EventSenty GFI Softwe www.gfi.com GFI EventsMnge vs Netikus.net EventSenty GFI EventsMnge EventSenty Who we e Suppot fo MS SQL Seve Suppot fo MSDE / MS SQL Expess Suppot fo

More information

Lesson 4.1 Triangle Sum Conjecture

Lesson 4.1 Triangle Sum Conjecture Lesson 4.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q 2., y 3., b 31 82 p 98 q 28 53 y 17 79 23 50 b 4. r, s, 5., y 6. y t t s r 100 85 100 y 30 4 7 y 31 7. s 8.

More information

Coordinate Systems L. M. Kalnins, March 2009

Coordinate Systems L. M. Kalnins, March 2009 Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL 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 information

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

2.016 Hydrodynamics Prof. A.H. Techet

2.016 Hydrodynamics Prof. A.H. Techet .016 Hydodynmics Reding #5.016 Hydodynmics Po. A.H. Techet Fluid Foces on Bodies 1. Stedy Flow In ode to design oshoe stuctues, suce vessels nd undewte vehicles, n undestnding o the bsic luid oces cting

More information

GFI MilEssentils & GFI MilSecuity vs Bcud Spm Fiewll GFI Softwe www.gfi.com GFIMilEssentils & GFI MilSecuity vs Bcud Spm Fiewll GFI MilEssentils 12 & GFI MilSecuity 10 Bcud Spm Fiewll Who we e Integtes

More information

FI3300 Corporate Finance

FI3300 Corporate Finance Leaning Objectives FI00 Copoate Finance Sping Semeste 2010 D. Isabel Tkatch Assistant Pofesso of Finance Calculate the PV and FV in multi-peiod multi-cf time-value-of-money poblems: Geneal case Pepetuity

More information

Factoring Polynomials

Factoring 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 information

Lectures 8 and 9 1 Rectangular waveguides

Lectures 8 and 9 1 Rectangular waveguides 1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.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 information

Unit 6: Exponents and Radicals

Unit 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 information

SOLUTIONS TO CONCEPTS CHAPTER 5

SOLUTIONS TO CONCEPTS CHAPTER 5 1. m k S 10m Let, ccelertion, Initil velocity u 0. S ut + 1/ t 10 ½ ( ) 10 5 m/s orce: m 5 10N (ns) 40000. u 40 km/hr 11.11 m/s. 3600 m 000 k ; v 0 ; s 4m v u ccelertion s SOLUIONS O CONCEPS CHPE 5 0 11.11

More information

CHAPTER 10 Aggregate Demand I

CHAPTER 10 Aggregate Demand I CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income

More information

PHY 140A: Solid State Physics. Solution to Homework #2

PHY 140A: Solid State Physics. Solution to Homework #2 PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.

More information

Fluids Lecture 15 Notes

Fluids Lecture 15 Notes Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2-D, this velocit

More information

ON THE CHINESE CHECKER SPHERE. Mine TURAN, Nihal DONDURMACI ÇİN DAMA KÜRESİ ÜZERİNE

ON THE CHINESE CHECKER SPHERE. Mine TURAN, Nihal DONDURMACI ÇİN DAMA KÜRESİ ÜZERİNE DÜ Fen Bilimlei Enstitüsü Degisi Sı 9 Ağustos 9 On The Chinese Cheke Sphee M. Tun N. Donumı ON THE CHINESE CHECKER SHERE Mine TURAN Nihl DONDURMACI Deptment of Mthemtis Fult of Ats n Sienes Dumlupin Univesit

More information

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes. LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.

More information

Lecture 5. Inner Product

Lecture 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 information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 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 information

SOEPpapers on Multidisciplinary Panel Data Research

SOEPpapers on Multidisciplinary Panel Data Research Deutsches Institut fü Witschftsfoschung www.diw.de SOEPppes on Multidiscipliny Pnel Dt Resech 136 Thoms Conelissen John S. Heywood Uwe Jijhn S, Pefomnce Py, Risk Attitudes nd Job Stisfction Belin, Octobe

More information

Carter-Penrose diagrams and black holes

Carter-Penrose diagrams and black holes Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right. Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

More information

CURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line.

CURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line. CURVES ANDRÉ NEVES 1. Problems (1) (Ex 1 of 1.3 of Do Crmo) Show tht the tngent line to the curve α(t) (3t, 3t 2, 2t 3 ) mkes constnt ngle with the line z x, y. (2) (Ex 6 of 1.3 of Do Crmo) Let α(t) (e

More information

The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges

The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee

More information

Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

More information

v T R x m Version PREVIEW Practice 7 carroll (11108) 1

v T R x m Version PREVIEW Practice 7 carroll (11108) 1 Version PEVIEW Prctice 7 crroll (08) his print-out should he 5 questions. Multiple-choice questions y continue on the next colun or pge find ll choices before nswering. Atwood Mchine 05 00 0.0 points A

More information

Things to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.

Things to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request. Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to

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

. 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 information

NURBS Drawing Week 5, Lecture 10

NURBS Drawing Week 5, Lecture 10 CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu

More information

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities. Gaphs of Equations CHAT Pe-Calculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such

More information

Brillouin Zones. Physics 3P41 Chris Wiebe

Brillouin Zones. Physics 3P41 Chris Wiebe Brillouin Zones Physics 3P41 Chris Wiebe Direct spce to reciprocl spce * = 2 i j πδ ij Rel (direct) spce Reciprocl spce Note: The rel spce nd reciprocl spce vectors re not necessrily in the sme direction

More information

Controlling the Money Supply: Bond Purchases in the Open Market

Controlling the Money Supply: Bond Purchases in the Open Market Money Supply By the Bank of Canada and Inteest Rate Detemination Open Opeations and Monetay Tansmission Mechanism The Cental Bank conducts monetay policy Bank of Canada is Canada's cental bank supevises

More information

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

More information

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to

More information

Answer, Key Homework 6 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 6 David McIntyre 45123 Mar 25, 2004 1 Answe, Key Homewok 6 vid McInye 4513 M 5, 004 1 This pin-ou should hve 0 quesions. Muliple-choice quesions my coninue on he nex column o pge find ll choices befoe mking you selecion. The due ime is Cenl

More information

12. Rolling, Torque, and Angular Momentum

12. Rolling, Torque, and Angular Momentum 12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

More information