Instructions for Section 1

Size: px
Start display at page:

Download "Instructions for Section 1"

Transcription

1 Instructions for Sction 1 Choos th rspons tht is corrct for th qustion. A corrct nswr scors 1, n incorrct nswr scors 0. Mrks will not b dductd for incorrct nswrs. You should ttmpt vry qustion. No mrks will b givn if mor thn on nswr is compltd for ny qustion. QUESTION 1 3 If 66 6 is divisibl by D. -1 E. 1 3, thn is qul to QUESTION Lt f ( ) 1 3 nd g( ). Th vlus of for which th grphs of y f () nd y g() hv two uniqu intrsction points r D. E. 0 nd 0 Th School For Ecllnc 017 Mthmticl Mthods Emintion Qustions Pg 1

2 QUESTION 3 Th linr fctors of r D. E QUESTION Th fctors of y y 6 r ( y )( y 3) ( y )( y 3) ( y )( y 3) D. ( y 3)( y ) E. ( y )( y 3) QUESTION P( ) ( )( b)( c), whr Th qution P ( ) 0 hs ctly, b nd c r thr diffrnt positiv rl numbrs. 1 rl solution distinct rl solutions 3 distinct rl solutions D. distinct rl solutions E. distinct rl solutions QUESTION 6 If 0 whr is positiv rl constnt, thn 0 only 0 nd 1 1 nd D. 0 nd E. 0 nd log Th School For Ecllnc 017 Mthmticl Mthods Emintion Qustions Pg

3 QUESTION 7 Th simultnous linr qutions 3y 0 ( 1) y 0 whr is rl constnt, hv infinitly mny solutions for R 3, R\ 3, D., 3 E. R\, 3 QUESTION 8 Th systm of simultnous qutions my 1 nd y 18 whr m R will hv no uniqu solution for m m Thr r no vlus of m D. m R E. m R\ Th School For Ecllnc 017 Mthmticl Mthods Emintion Qustions Pg 3

4 QUESTION Two functions f nd g r dfind s follows: Th function f ( g( )) B : f : A R with rul f ( ) 3 g : B R with rul will b dfind if g ( ) 3 A : B : 3 D. A : E. B : QUESTION 10 Considr th two rltions h: R /{ 1) R whr Th composit function g h( ) is dfind s log 1 for R log 1 for (1, ) log 1 for R D. 1 log 1 for /{ 1} R E. log 1 for R/{ 1} 1 h ( ) 1 1 g:(1, ) R whr g( ) log ( 1) nd Th School For Ecllnc 017 Mthmticl Mthods Emintion Qustions Pg

5 QUESTION 1 QUESTION -coordint of intrsction points: f ( ) g( ) Discriminnt: Two distinct solutions: Howvr, if 0 Two distinct solutions: ( 3) (1) ( )(1) qution (1) bcoms which hs only on solution., 0. QUESTION 3 QUESTION Answr is B Th School For Ecllnc 017 Mthmticl Mthods Emintion Qustions Pg

6 QUESTION Answr is B QUESTION 6 Lt m in th qution 0. Thn m m 0 nd so ( m )( m 1) 0. Thrfor Also m m 1 nd so nd so 1 which mns tht log which mns tht log 1 0. QUESTION 7 Answr is B For infinitly mny solutions, th qutions must b idnticl. Equting cofficints givs. Howvr, th solution lon is not providd. Substitut in you will gt: 3 nd 3 (th othr nswrs) into both qutions nd whn 3, 3 3y 0 y 0 Multiply th first qution by - nd th scond by 3. This will rsult in two idnticl qutions, mning tht 3 will lso produc n infinit numbr of solutions. Ths typs of qustions r hrd to gt corrct unlss th nswrs r givn to you s in this qustion, or unlss you us th discriminnt or mtrics, which will b covrd ltr. QUESTION 8 Answr is A No uniqu solution mns no solution or infinitly mny solutions. For no solution to ist: Lins must b prlll (grdints must b th sm) nd th Y intrcpts must b diffrnt. my 1 y 18 my 60 10y 36 my y m Th School For Ecllnc 017 Mthmticl Mthods Emintion Qustions Pg 6

7 For infinitly mny solutions to ist: Lins must b prlll (grdints must b th sm) nd th Y intrcpts must b th sm. Thrfor: Th rtio of th cofficints of th nd trms must b qul. Th constnts indpndnt of nd must b th sm. y y Thr r no vlus of m for which infinitly mny solutions ist. QUESTION Rquir rn g dom f. 3 dom f : 3 0. It is thrfor rquird tht: 3 QUESTION 10 g h( ) 1 log log 1 log 1 log ( 1) log 1 Th domin of g (h()) is th sm s th domin of h() which is R / { 1}. Th School For Ecllnc 017 Mthmticl Mthods Emintion Qustions Pg 7

INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)

INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x) Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..

More information

CONTINUITY AND DIFFERENTIABILITY

CONTINUITY AND DIFFERENTIABILITY MCD CONTINUITY AND DIFFERENTIABILITY NCERT Solvd mpls upto th sction 5 (Introduction) nd 5 (Continuity) : Empl : Chck th continuity of th function f givn by f() = + t = Empl : Emin whthr th function f

More information

Chapter 16. 1) is a particular point on the graph of the function. 1. y, where x y 1

Chapter 16. 1) is a particular point on the graph of the function. 1. y, where x y 1 Prctic qustions W now tht th prmtr p is dirctl rltd to th mplitud; thrfor, w cn find tht p. cos d [ sin ] sin sin Not: Evn though ou might not now how to find th prmtr in prt, it is lws dvisl to procd

More information

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS SEMESTER TWO 2014 WEEK 11 WRITTEN EXAMINATION 1 SOLUTIONS

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS SEMESTER TWO 2014 WEEK 11 WRITTEN EXAMINATION 1 SOLUTIONS MASTER CLASS PROGRAM UNIT SPECIALIST MATHEMATICS SEMESTER TWO WEEK WRITTEN EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/MC-UPDATES QUESTION () Lt p ( z) z z z If z i z ( is

More information

Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000

Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000 Highr Mthmtics UNIT Mthmtics HSN000 This documnt ws producd spcilly for th HSN.uk.nt wbsit, nd w rquir tht ny copis or drivtiv works ttribut th work to Highr Still Nots. For mor dtils bout th copyright

More information

Walk Like a Mathematician Learning Task:

Walk Like a Mathematician Learning Task: Gori Dprtmnt of Euction Wlk Lik Mthmticin Lrnin Tsk: Mtrics llow us to prform mny usful mthmticl tsks which orinrily rquir lr numbr of computtions. Som typs of problms which cn b on fficintly with mtrics

More information

Ch 1.2: Solutions of Some Differential Equations

Ch 1.2: Solutions of Some Differential Equations Ch 1.2: Solutions of Som Diffrntil Equtions Rcll th fr fll nd owl/mic diffrntil qutions: v 9.8.2v, p.5 p 45 Ths qutions hv th gnrl form y' = y - b W cn us mthods of clculus to solv diffrntil qutions of

More information

TOPIC 5: INTEGRATION

TOPIC 5: INTEGRATION TOPIC 5: INTEGRATION. Th indfinit intgrl In mny rspcts, th oprtion of intgrtion tht w r studying hr is th invrs oprtion of drivtion. Dfinition.. Th function F is n ntidrivtiv (or primitiv) of th function

More information

PH427/PH527: Periodic systems Spring Overview of the PH427 website (syllabus, assignments etc.) 2. Coupled oscillations.

PH427/PH527: Periodic systems Spring Overview of the PH427 website (syllabus, assignments etc.) 2. Coupled oscillations. Dy : Mondy 5 inuts. Ovrviw of th PH47 wsit (syllus, ssignnts tc.). Coupld oscilltions W gin with sss coupld y Hook's Lw springs nd find th possil longitudinl) otion of such syst. W ll xtnd this to finit

More information

UNIT # 08 (PART - I)

UNIT # 08 (PART - I) . r. d[h d[h.5 7.5 mol L S d[o d[so UNIT # 8 (PRT - I CHEMICL INETICS EXERCISE # 6. d[ x [ x [ x. r [X[C ' [X [[B r '[ [B [C. r [NO [Cl. d[so d[h.5 5 mol L S d[nh d[nh. 5. 6. r [ [B r [x [y r' [x [y r'

More information

4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.

4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16. . 7 7 7... 7 7 (n )0 7 (M) 0(n ) 00 n (A) S ((7) 0(0)) (M) (7 00) 8897 (A). (5a b) 7 7... (5a)... (M) 7 5 5 (a b ) 5 5 a b (M)(A) So th cofficint is 75 (A) (C) [] S (7 7) (M) () 8897 (A) (C) [] 5. x.55

More information

Lecture 11 Waves in Periodic Potentials Today: Questions you should be able to address after today s lecture:

Lecture 11 Waves in Periodic Potentials Today: Questions you should be able to address after today s lecture: Lctur 11 Wvs in Priodic Potntils Tody: 1. Invrs lttic dfinition in 1D.. rphicl rprsnttion of priodic nd -priodic functions using th -xis nd invrs lttic vctors. 3. Sris solutions to th priodic potntil Hmiltonin

More information

CIVL 8/ D Boundary Value Problems - Rectangular Elements 1/7

CIVL 8/ D Boundary Value Problems - Rectangular Elements 1/7 CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS - In som pplictions, it m mor dsirl to us n lmntl rprsnttion of th domin tht hs four sids, ithr rctngulr or qudriltrl in shp. Considr

More information

General Notes About 2007 AP Physics Scoring Guidelines

General Notes About 2007 AP Physics Scoring Guidelines AP PHYSICS C: ELECTRICITY AND MAGNETISM 2007 SCORING GUIDELINES Gnral Nots About 2007 AP Physics Scoring Guidlins 1. Th solutions contain th most common mthod of solving th fr-rspons qustions and th allocation

More information

1997 AP Calculus AB: Section I, Part A

1997 AP Calculus AB: Section I, Part A 997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6

More information

CBSE 2015 FOREIGN EXAMINATION

CBSE 2015 FOREIGN EXAMINATION CBSE 05 FOREIGN EXAMINATION (Sris SSO Cod No 65//F, 65//F, 65//F : Forign Rgion) Not tht ll th sts hv sm qustions Onl thir squnc of pprnc is diffrnt M Mrks : 00 Tim Allowd : Hours SECTION A Q0 Find th

More information

, between the vertical lines x a and x b. Given a demand curve, having price as a function of quantity, p f (x) at height k is the curve f ( x,

, between the vertical lines x a and x b. Given a demand curve, having price as a function of quantity, p f (x) at height k is the curve f ( x, Clculus for Businss nd Socil Scincs - Prof D Yun Finl Em Rviw vrsion 5/9/7 Chck wbsit for ny postd typos nd updts Pls rport ny typos This rviw sht contins summris of nw topics only (This rviw sht dos hv

More information

1 Introduction to Modulo 7 Arithmetic

1 Introduction to Modulo 7 Arithmetic 1 Introution to Moulo 7 Arithmti Bor w try our hn t solvin som hr Moulr KnKns, lt s tk los look t on moulr rithmti, mo 7 rithmti. You ll s in this sminr tht rithmti moulo prim is quit irnt rom th ons w

More information

CONIC SECTIONS. MODULE-IV Co-ordinate Geometry OBJECTIVES. Conic Sections

CONIC SECTIONS. MODULE-IV Co-ordinate Geometry OBJECTIVES. Conic Sections Conic Sctions 16 MODULE-IV Co-ordint CONIC SECTIONS Whil cutting crrot ou might hv noticd diffrnt shps shown th dgs of th cut. Anlticll ou m cut it in thr diffrnt ws, nml (i) (ii) (iii) Cut is prlll to

More information

1973 AP Calculus AB: Section I

1973 AP Calculus AB: Section I 97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=

More information

SOLVED EXAMPLES. be the foci of an ellipse with eccentricity e. For any point P on the ellipse, prove that. tan

SOLVED EXAMPLES. be the foci of an ellipse with eccentricity e. For any point P on the ellipse, prove that. tan LOCUS 58 SOLVED EXAMPLES Empl Lt F n F th foci of n llips with ccntricit. For n point P on th llips, prov tht tn PF F tn PF F Assum th llips to, n lt P th point (, sin ). P(, sin ) F F F = (-, 0) F = (,

More information

Note If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.

Note If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark. . (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold

More information

Objective Mathematics

Objective Mathematics x. Lt 'P' b a point on th curv y and tangnt x drawn at P to th curv has gratst slop in magnitud, thn point 'P' is,, (0, 0),. Th quation of common tangnt to th curvs y = 6 x x and xy = x + is : x y = 8

More information

MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A, B and C.

MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A, B and C. MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Tim: 3hrs Ma. Marks.75 Not: This qustion papr consists of thr sctions A, B and C. SECTION -A Vry Short Answr Typ Qustions. 0 X = 0. Find th condition

More information

Present state Next state Q + M N

Present state Next state Q + M N Qustion 1. An M-N lip-lop works s ollows: I MN=00, th nxt stt o th lip lop is 0. I MN=01, th nxt stt o th lip-lop is th sm s th prsnt stt I MN=10, th nxt stt o th lip-lop is th omplmnt o th prsnt stt I

More information

(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely

(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely . DETERMINANT.. Dtrminnt. Introution:I you think row vtor o mtrix s oorint o vtors in sp, thn th gomtri mning o th rnk o th mtrix is th imnsion o th prlllppi spnn y thm. But w r not only r out th imnsion,

More information

Integration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals

Integration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals Intgrtion Continud Intgrtion y Prts Solving Dinit Intgrls: Ar Undr Curv Impropr Intgrls Intgrtion y Prts Prticulrly usul whn you r trying to tk th intgrl o som unction tht is th product o n lgric prssion

More information

Multi-Section Coupled Line Couplers

Multi-Section Coupled Line Couplers /0/009 MultiSction Coupld Lin Couplrs /8 Multi-Sction Coupld Lin Couplrs W cn dd multipl coupld lins in sris to incrs couplr ndwidth. Figur 7.5 (p. 6) An N-sction coupld lin l W typiclly dsign th couplr

More information

10. The Discrete-Time Fourier Transform (DTFT)

10. The Discrete-Time Fourier Transform (DTFT) Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!

More information

Einstein Equations for Tetrad Fields

Einstein Equations for Tetrad Fields Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for

More information

First derivative analysis

First derivative analysis Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points

More information

Errata for Second Edition, First Printing

Errata for Second Edition, First Printing Errt for Scond Edition, First Printing pg 68, lin 1: z=.67 should b z=.44 pg 71: Eqution (.3) should rd B( R) = θ R 1 x= [1 G( x)] pg 1: Eqution (.63) should rd B( R) = x= R = θ ( x R) p( x) R 1 x= [1

More information

Functions and Graphs 1. (a) (b) (c) (f) (e) (d) 2. (a) (b) (c) (d)

Functions and Graphs 1. (a) (b) (c) (f) (e) (d) 2. (a) (b) (c) (d) Functions nd Grps. () () (c) - - - O - - - O - - - O - - - - (d) () (f) - - O - 7 6 - - O - -7-6 - - - - - O. () () (c) (d) - - - O - O - O - - O - -. () G() f() + f( ), G(-) f( ) + f(), G() G( ) nd G()

More information

1997 AP Calculus AB: Section I, Part A

1997 AP Calculus AB: Section I, Part A 997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs x for which f (x) is a ral numbr.. (4x 6 x) dx=

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!

More information

CSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018

CSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018 CSE 373: Mor on grphs; DFS n BFS Mihl L Wnsy, F 14, 2018 1 Wrmup Wrmup: Disuss with your nighor: Rmin your nighor: wht is simpl grph? Suppos w hv simpl, irt grph with x nos. Wht is th mximum numr of gs

More information

Answers & Solutions. for MHT CET-2018 Paper-I (Mathematics) Instruction for Candidates

Answers & Solutions. for MHT CET-2018 Paper-I (Mathematics) Instruction for Candidates DATE : /5/8 Qustion Booklt Vrsion Rgd. Offic : Aakash Towr, 8, Pusa Road, Nw Dlhi-5 Ph.: -75 Fa : -77 Tim : Hour Min. Total Marks : Answrs & Solutions for MHT CET-8 Papr-I (Mathmatics) Instruction for

More information

MA 262, Spring 2018, Final exam Version 01 (Green)

MA 262, Spring 2018, Final exam Version 01 (Green) MA 262, Spring 218, Final xam Vrsion 1 (Grn) INSTRUCTIONS 1. Switch off your phon upon ntring th xam room. 2. Do not opn th xam booklt until you ar instructd to do so. 3. Bfor you opn th booklt, fill in

More information

Calculus concepts derivatives

Calculus concepts derivatives All rasonabl fforts hav bn mad to mak sur th nots ar accurat. Th author cannot b hld rsponsibl for any damags arising from th us of ths nots in any fashion. Calculus concpts drivativs Concpts involving

More information

2013 Specialist Mathematics GA 3: Written examination 2

2013 Specialist Mathematics GA 3: Written examination 2 0 0 Spcialist Mathmatics GA : Writtn xamination GENERAL COMMENTS Th 0 Spcialist Mathmatics xamination comprisd multipl-choic qustions (worth marks) and fiv xtndd qustions (worth 8 marks). Th papr smd accssibl

More information

Problem Set 6 Solutions

Problem Set 6 Solutions 6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr

More information

Errata for Second Edition, First Printing

Errata for Second Edition, First Printing Errt for Scond Edition, First Printing pg 68, lin 1: z=.67 should b z=.44 pg 1: Eqution (.63) should rd B( R) = x= R = θ ( x R) p( x) R 1 x= [1 G( x)] = θp( R) + ( θ R)[1 G( R)] pg 15, problm 6: dmnd of

More information

Condensed. Mathematics. General Certificate of Education Advanced Level Examination January Unit Pure Core 3. Time allowed * 1 hour 30 minutes

Condensed. Mathematics. General Certificate of Education Advanced Level Examination January Unit Pure Core 3. Time allowed * 1 hour 30 minutes Gnral Crtificat of Education Advancd Lvl Eamination January 0 Mathmatics MPC Unit Pur Cor Friday 0 January 0.0 pm to.00 pm For this papr you must hav: th blu AQA booklt of formula and statistical tabls.

More information

HIGHER ORDER DIFFERENTIAL EQUATIONS

HIGHER ORDER DIFFERENTIAL EQUATIONS Prof Enriqu Mtus Nivs PhD in Mthmtis Edution IGER ORDER DIFFERENTIAL EQUATIONS omognous linr qutions with onstnt offiints of ordr two highr Appl rdution mthod to dtrmin solution of th nonhomognous qution

More information

Limits Indeterminate Forms and L Hospital s Rule

Limits Indeterminate Forms and L Hospital s Rule Limits Indtrmint Forms nd L Hospitl s Rul I Indtrmint Form o th Tp W hv prviousl studid its with th indtrmint orm s shown in th ollowin mpls: Empl : Empl : tn [Not: W us th ivn it ] Empl : 8 h 8 [Not:

More information

4 x 4, and. where x is Town Square

4 x 4, and. where x is Town Square Accumulation and Population Dnsity E. A city locatd along a straight highway has a population whos dnsity can b approimatd by th function p 5 4 th distanc from th town squar, masurd in mils, whr 4 4, and

More information

First order differential equation Linear equation; Method of integrating factors

First order differential equation Linear equation; Method of integrating factors First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial

More information

Derangements and Applications

Derangements and Applications 2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir

More information

Engineering 323 Beautiful HW #13 Page 1 of 6 Brown Problem 5-12

Engineering 323 Beautiful HW #13 Page 1 of 6 Brown Problem 5-12 Enginring Bautiful HW #1 Pag 1 of 6 5.1 Two componnts of a minicomputr hav th following joint pdf for thir usful liftims X and Y: = x(1+ x and y othrwis a. What is th probability that th liftim X of th

More information

Lecture contents. Bloch theorem k-vector Brillouin zone Almost free-electron model Bands Effective mass Holes. NNSE 508 EM Lecture #9

Lecture contents. Bloch theorem k-vector Brillouin zone Almost free-electron model Bands Effective mass Holes. NNSE 508 EM Lecture #9 Lctur contnts Bloch thorm -vctor Brillouin zon Almost fr-lctron modl Bnds ffctiv mss Hols Trnsltionl symmtry: Bloch thorm On-lctron Schrödingr qution ch stt cn ccommo up to lctrons: If Vr is priodic function:

More information

Multiple-Choice Test Introduction to Partial Differential Equations COMPLETE SOLUTION SET

Multiple-Choice Test Introduction to Partial Differential Equations COMPLETE SOLUTION SET Mltipl-Choic Tst Introdction to Partial Diffrntial Eqations COMPLETE SOLUTION SET 1. A partial diffrntial qation has (A on indpndnt variabl (B two or mor indpndnt variabls (C mor than on dpndnt variabl

More information

2008 AP Calculus BC Multiple Choice Exam

2008 AP Calculus BC Multiple Choice Exam 008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl

More information

Higher order derivatives

Higher order derivatives Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY HAYSTACK OBSERVATORY WESTFORD, MASSACHUSETTS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY HAYSTACK OBSERVATORY WESTFORD, MASSACHUSETTS VSRT MEMO #05 MASSACHUSETTS INSTITUTE OF TECHNOLOGY HAYSTACK OBSERVATORY WESTFORD, MASSACHUSETTS 01886 Fbrury 3, 009 Tlphon: 781-981-507 Fx: 781-981-0590 To: VSRT Group From: Aln E.E. Rogrs Subjct: Simplifid

More information

TEMASEK JUNIOR COLLEGE, SINGAPORE. JC 2 Preliminary Examination 2017

TEMASEK JUNIOR COLLEGE, SINGAPORE. JC 2 Preliminary Examination 2017 TEMASEK JUNIOR COLLEGE, SINGAPORE JC Prliminary Eamination 7 MATHEMATICS 886/ Highr 9 August 7 Additional Matrials: Answr papr hours List of Formula (MF6) READ THESE INSTRUCTIONS FIRST Writ your Civics

More information

The graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the

The graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th

More information

Lecture 4. Conic section

Lecture 4. Conic section Lctur 4 Conic sction Conic sctions r locus of points whr distncs from fixd point nd fixd lin r in constnt rtio. Conic sctions in D r curvs which r locus of points whor position vctor r stisfis r r. whr

More information

b. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s?

b. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s? MATH 3012 Finl Exm, My 4, 2006, WTT Stunt Nm n ID Numr 1. All our prts o this prolm r onrn with trnry strings o lngth n, i.., wors o lngth n with lttrs rom th lpht {0, 1, 2}.. How mny trnry wors o lngth

More information

COHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.

COHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim. MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function

More information

FSA. CmSc 365 Theory of Computation. Finite State Automata and Regular Expressions (Chapter 2, Section 2.3) ALPHABET operations: U, concatenation, *

FSA. CmSc 365 Theory of Computation. Finite State Automata and Regular Expressions (Chapter 2, Section 2.3) ALPHABET operations: U, concatenation, * CmSc 365 Thory of Computtion Finit Stt Automt nd Rgulr Exprssions (Chptr 2, Sction 2.3) ALPHABET oprtions: U, conctntion, * otin otin Strings Form Rgulr xprssions dscri Closd undr U, conctntion nd * (if

More information

(Upside-Down o Direct Rotation) β - Numbers

(Upside-Down o Direct Rotation) β - Numbers Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg

More information

y = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b)

y = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b) 4. y = y = + 5. Find th quation of th tangnt lin for th function y = ( + ) 3 whn = 0. solution: First not that whn = 0, y = (1 + 1) 3 = 8, so th lin gos through (0, 8) and thrfor its y-intrcpt is 8. y

More information

MATH 1080 Test 2-SOLUTIONS Spring

MATH 1080 Test 2-SOLUTIONS Spring MATH Tst -SOLUTIONS Spring 5. Considr th curv dfind by x = ln( 3y + 7) on th intrval y. a. (5 points) St up but do not simplify or valuat an intgral rprsnting th lngth of th curv on th givn intrval. =

More information

Section 3: Antiderivatives of Formulas

Section 3: Antiderivatives of Formulas Chptr Th Intgrl Appli Clculus 96 Sction : Antirivtivs of Formuls Now w cn put th is of rs n ntirivtivs togthr to gt wy of vluting finit intgrls tht is ct n oftn sy. To vlut finit intgrl f(t) t, w cn fin

More information

Prelim Examination 2011 / 2012 (Assessing Units 1 & 2) MATHEMATICS. Advanced Higher Grade. Time allowed - 2 hours

Prelim Examination 2011 / 2012 (Assessing Units 1 & 2) MATHEMATICS. Advanced Higher Grade. Time allowed - 2 hours Prlim Eamination / (Assssing Units & ) MATHEMATICS Advancd Highr Grad Tim allowd - hours Rad Carfull. Calculators ma b usd in this papr.. Candidats should answr all qustions. Full crdit will onl b givn

More information

Chapter 10. The singular integral Introducing S(n) and J(n)

Chapter 10. The singular integral Introducing S(n) and J(n) Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don

More information

MATHEMATICS (B) 2 log (D) ( 1) = where z =

MATHEMATICS (B) 2 log (D) ( 1) = where z = MATHEMATICS SECTION- I STRAIGHT OBJECTIVE TYPE This sction contains 9 multipl choic qustions numbrd to 9. Each qustion has choic (A), (B), (C) and (D), out of which ONLY-ONE is corrct. Lt I d + +, J +

More information

Integration by Parts

Integration by Parts Intgration by Parts Intgration by parts is a tchniqu primarily for valuating intgrals whos intgrand is th product of two functions whr substitution dosn t work. For ampl, sin d or d. Th rul is: u ( ) v'(

More information

SOLVED EXAMPLES. Ex.1 If f(x) = , then. is equal to- Ex.5. f(x) equals - (A) 2 (B) 1/2 (C) 0 (D) 1 (A) 1 (B) 2. (D) Does not exist = [2(1 h)+1]= 3

SOLVED EXAMPLES. Ex.1 If f(x) = , then. is equal to- Ex.5. f(x) equals - (A) 2 (B) 1/2 (C) 0 (D) 1 (A) 1 (B) 2. (D) Does not exist = [2(1 h)+1]= 3 SOLVED EXAMPLES E. If f() E.,,, th f() f() h h LHL RHL, so / / Lim f() quls - (D) Dos ot ist [( h)+] [(+h) + ] f(). LHL E. RHL h h h / h / h / h / h / h / h As.[C] (D) Dos ot ist LHL RHL, so giv it dos

More information

Chapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7.

Chapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7. Chaptr Binomial Epansion Chaptr 0 Furthr Probability Chaptr Limits and Drivativs Chaptr Discrt Random Variabls Chaptr Diffrntiation Chaptr Discrt Probability Distributions Chaptr Applications of Diffrntiation

More information

Case Study VI Answers PHA 5127 Fall 2006

Case Study VI Answers PHA 5127 Fall 2006 Qustion. A ptint is givn 250 mg immit-rls thophyllin tblt (Tblt A). A wk ltr, th sm ptint is givn 250 mg sustin-rls thophyllin tblt (Tblt B). Th tblts follow on-comprtmntl mol n hv first-orr bsorption

More information

SAFE HANDS & IIT-ian's PACE EDT-15 (JEE) SOLUTIONS

SAFE HANDS & IIT-ian's PACE EDT-15 (JEE) SOLUTIONS It is not possibl to find flu through biggr loop dirctly So w will find cofficint of mutual inductanc btwn two loops and thn find th flu through biggr loop Also rmmbr M = M ( ) ( ) EDT- (JEE) SOLUTIONS

More information

www.onlineamhlp.com www.onlineamhlp.com UIVERSITY OF CAMBRIDGE ITERATIOAL EXAMIATIOS GCE Advancd Lvl MARK SCHEME for th Octobr/ovmbr qustion papr for th guidanc of tachrs 9 FURTHER MATHEMATICS 9/ Papr,

More information

BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results

BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim and n a positiv intgr, lt ν p (n dnot th xponnt of p in n, and u p (n n/p νp(n th unit part of n. If α

More information

SECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.

SECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero. SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain

More information

This Week. Computer Graphics. Introduction. Introduction. Graphics Maths by Example. Graphics Maths by Example

This Week. Computer Graphics. Introduction. Introduction. Graphics Maths by Example. Graphics Maths by Example This Wk Computr Grphics Vctors nd Oprtions Vctor Arithmtic Gomtric Concpts Points, Lins nd Plns Eploiting Dot Products CSC 470 Computr Grphics 1 CSC 470 Computr Grphics 2 Introduction Introduction Wh do

More information

Function Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0

Function Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0 unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr

More information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat

More information

Abstract Interpretation: concrete and abstract semantics

Abstract Interpretation: concrete and abstract semantics Abstract Intrprtation: concrt and abstract smantics Concrt smantics W considr a vry tiny languag that manags arithmtic oprations on intgrs valus. Th (concrt) smantics of th languags cab b dfind by th funzcion

More information

The Angular Momenta Dipole Moments and Gyromagnetic Ratios of the Electron and the Proton

The Angular Momenta Dipole Moments and Gyromagnetic Ratios of the Electron and the Proton Journl of Modrn hysics, 014, 5, 154-157 ublishd Onlin August 014 in SciRs. htt://www.scir.org/journl/jm htt://dx.doi.org/.436/jm.014.51415 Th Angulr Momnt Diol Momnts nd Gyromgntic Rtios of th Elctron

More information

Differential Equations

Differential Equations UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs

More information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th

More information

Coupled Pendulums. Two normal modes.

Coupled Pendulums. Two normal modes. Tim Dpndnt Two Stat Problm Coupld Pndulums Wak spring Two normal mods. No friction. No air rsistanc. Prfct Spring Start Swinging Som tim latr - swings with full amplitud. stationary M +n L M +m Elctron

More information

1 1 1 p q p q. 2ln x x. in simplest form. in simplest form in terms of x and h.

1 1 1 p q p q. 2ln x x. in simplest form. in simplest form in terms of x and h. NAME SUMMER ASSIGNMENT DUE SEPTEMBER 5 (FIRST DAY OF SCHOOL) AP CALC AB Dirctions: Answr all of th following qustions on a sparat sht of papr. All work must b shown. You will b tstd on this matrial somtim

More information

MAXIMA-MINIMA EXERCISE - 01 CHECK YOUR GRASP

MAXIMA-MINIMA EXERCISE - 01 CHECK YOUR GRASP EXERCISE - MAXIMA-MINIMA CHECK YOUR GRASP. f() 5 () 75 f'() 5. () 75 75.() 7. 5 + 5. () 7 {} 5 () 7 ( ) 5. f() 9a + a +, a > f'() 6 8a + a 6( a + a ) 6( a) ( a) p a, q a a a + + a a a (rjctd) or a a 6.

More information

Practice Final Exam. 3.) What is the 61st term of the sequence 7, 11, 15, 19,...?

Practice Final Exam. 3.) What is the 61st term of the sequence 7, 11, 15, 19,...? Discrt mth Prctic Fl Em.) Fd 4 (i ) i=.) Fd i= 6 i.) Wht is th 6st trm th squnc 7,, 5, 9,...? 4.) Wht s th 57th trm, 6,, 4,...? 5.) Wht s th sum th first 60 trms th squnc, 5, 7, 9,...? 6.) Suppos st A

More information

The Theory of Small Reflections

The Theory of Small Reflections Jim Stils Th Univ. of Knss Dt. of EECS 4//9 Th Thory of Smll Rflctions /9 Th Thory of Smll Rflctions Rcll tht w nlyzd qurtr-wv trnsformr usg th multil rflction viw ot. V ( z) = + β ( z + ) V ( z) = = R

More information

4037 ADDITIONAL MATHEMATICS

4037 ADDITIONAL MATHEMATICS CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Lvl MARK SCHEME for th Octobr/Novmbr 0 sris 40 ADDITIONAL MATHEMATICS 40/ Papr, maimum raw mark 80 This mark schm is publishd as an aid to tachrs and candidats,

More information

Solution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers:

Solution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers: APPM 6 Final 5 pts) Spring 4. 6 pts total) Th following parts ar not rlatd, justify your answrs: a) Considr th curv rprsntd by th paramtric quations, t and y t + for t. i) 6 pts) Writ down th corrsponding

More information

Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.

Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J. Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7

More information

Chapter 11 Calculation of

Chapter 11 Calculation of Chtr 11 Clcultion of th Flow Fild OUTLINE 11-1 Nd for Scil Procdur 11-2 Som Rltd Difficultis 11-3 A Rmdy : Th stggrd Grid 11-4 Th Momntum Equtions 11-5 Th Prssur nd Vlocity Corrctions 11-6 Th Prssur-Corrction

More information

Search sequence databases 3 10/25/2016

Search sequence databases 3 10/25/2016 Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an

More information

Engineering Mathematics I. MCQ for Phase-I

Engineering Mathematics I. MCQ for Phase-I [ASK/EM-I/MCQ] August 30, 0 UNIT: I MATRICES Enginring Mathmatics I MCQ for Phas-I. Th rank of th matri A = 4 0. Th valu of λ for which th matri A = will b of rank on is λ = -3 λ = 3 λ = λ = - 3. For what

More information

Chapter 13 GMM for Linear Factor Models in Discount Factor form. GMM on the pricing errors gives a crosssectional

Chapter 13 GMM for Linear Factor Models in Discount Factor form. GMM on the pricing errors gives a crosssectional Chaptr 13 GMM for Linar Factor Modls in Discount Factor form GMM on th pricing rrors givs a crosssctional rgrssion h cas of xcss rturns Hors rac sting for charactristic sting for pricd factors: lambdas

More information

6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved.

6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved. 6.1 Intgration by Parts and Prsnt Valu Copyright Cngag Larning. All rights rsrvd. Warm-Up: Find f () 1. F() = ln(+1). F() = 3 3. F() =. F() = ln ( 1) 5. F() = 6. F() = - Objctivs, Day #1 Studnts will b

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1

More information

( ) Differential Equations. Unit-7. Exact Differential Equations: M d x + N d y = 0. Verify the condition

( ) Differential Equations. Unit-7. Exact Differential Equations: M d x + N d y = 0. Verify the condition Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of

More information

BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES

BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit

More information

Revisiting what you have learned in Advanced Mathematical Analysis

Revisiting what you have learned in Advanced Mathematical Analysis Fourir sris Rvisiing wh you hv lrnd in Advncd Mhmicl Anlysis L f x b priodic funcion of priod nd is ingrbl ovr priod. f x cn b rprsnd by rigonomric sris, f x n cos nx bn sin nx n cos x b sin x cosx b whr

More information