# GCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4

Size: px
Start display at page:

Download "GCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4"

## Transcription

1 GCE Further Mathematics (660) Further Pure Uit (MFP) Tetbook Versio: 4

2 MFP Tetbook A-level Further Mathematics 660 Further Pure : Cotets Chapter : Comple umbers 4 Itroductio 5 The geeral comple umber 5 The modulus ad argumet of a comple umber 6 4 The polar form of a comple umber 8 5 Additio, subtractio ad multiplicatio of comple umbers of the form + iy 9 6 The cojugate of a comple umber ad the divisio of comple umbers of the form +iy 0 7 Products ad quotiets of comple umbers i their polar form 8 Equatig real ad imagiary parts 9 Further cosideratio of z z ad arg(z z ) 4 0 Loci o Argad diagrams 5 Chapter : Roots of polyomial equatios Itroductio Quadratic equatios Cubic equatios 4 4 Relatioship betwee the roots of a cubic equatio ad its coefficiets 7 5 Cubic equatios with related roots 8 6 A importat result 7 Polyomial equatios of degree 8 Comple roots of polyomial equatios with real coefficiets Chapter : Summatio of fiite series 8 Itroductio 9 Summatio of series by the method of differeces 40 Summatio of series by the method of iductio 45 4 Proof by iductio eteded to other areas of mathematics 48 Chapter 4: De Moivre s theorem ad its applicatios 5 4 De Moivre s theorem 54 4 Usig de Moivre s theorem to evaluate powers of comple umbers 56 4 Applicatio of de Moivre s theorem i establishig trigoometric idetities Epoetial form of a comple umber The cube roots of uity 7 46 The th roots of uity The roots of z, where is a o-real umber 77 cotiued overleaf

3 MFP Tetbook A-level Further Mathematics 660 Further Pure : Cotets (cotiued) Chapter 5: Iverse trigoometrical fuctios 85 5 Itroductio ad revisio 86 5 The derivatives of stadard iverse trigoometrical fuctios 89 5 Applicatio to more comple differetiatio 9 54 Stadard itegrals itegratig to iverse trigoometrical fuctios Applicatios to more comple itegrals 96 Chapter 6: Hyperbolic fuctios 0 6 Defiitios of hyperbolic fuctios 0 6 Numerical values of hyperbolic fuctios 04 6 Graphs of hyperbolic fuctios Hyperbolic idetities Osbore s rule 0 66 Differetiatio of hyperbolic fuctios 67 Itegratio of hyperbolic fuctios 4 68 Iverse hyperbolic fuctios 5 69 Logarithmic form of iverse hyperbolic fuctios 6 60 Derivatives of iverse hyperbolic fuctios 9 6 Itegrals which itegrate to iverse hyperbolic fuctios 6 Solvig equatios 5 Chapter 7: Arc legth ad area of surface of revolutio 7 Itroductio 7 Arc legth 7 Area of surface of revolutio 7 Aswers to the eercises i Further Pure 4

4 MFP Tetbook A-level Further Mathematics 660 Chapter : Comple Numbers Itroductio The geeral comple umber The modulus ad argumet of a comple umber 4 The polar form of a comple umber 5 Additio, subtractio ad multiplicatio of comple umbers of the form iy 6 The cojugate of a comple umber ad the divisio of comple umbers of the form iy 7 Products ad quotiets of comple umbers i their polar form 8 Equatig real ad imagiary parts 9 Further cosideratio of z z ad arg( z z) 0 Loci o Argad diagrams This chapter itroduces the idea of a comple umber Whe you have completed it, you will: kow what is meat by a comple umber; kow what is meat by the modulus ad argumet of a comple umber; kow how to add, subtract, multiply ad divide comple umbers; kow how to solve equatios usig real ad imagiary parts; uderstad what a Argad diagram is; kow how to sketch loci o Argad diagrams 4

5 MFP Tetbook A-level Further Mathematics 660 Itroductio You will have discovered by ow that some problems caot be solved i terms of real umbers For eample, if you use a calculator to evaluate 64 you get a error message This is because squarig every real umber gives a positive value; both ( 8) ad ( 8) are equal to 64 As caot be evaluated, a symbol is used to deote it the symbol used is i i i It follows that i The geeral comple umber The most geeral umber that ca be writte dow has the form i y, where ad y are real umbers The term iy is a comple umber with beig the real part ad y the imagiary part So, both i ad 4i are comple umbers The set of real umbers, (with which you are familiar), is really a subset of the set of comple umbers, This is because real umbers are actually umbers of the form 0i 5

6 MFP Tetbook A-level Further Mathematics 660 The modulus ad argumet of a comple umber Just as real umbers ca be represeted by poits o a umber lie, comple umbers ca be represeted by poits i a plae The poit P(, y) i the plae of coordiates with aes O ad Oy represets the comple umber iy ad the umber is uiquely represeted by that poit The diagram of poits i Cartesia coordiates represetig comple umbers is called a Argad diagram y r P(, y) O θ If the comple umber iy is deoted by z, ad hece z i, y z ( mod zed ) is defied as the distace from the origi O to the poit P represetig z Thus z OP r The modulus of a comple umber z is give by z y The argumet of z, arg z, is defied as the agle betwee the lie OP ad the positive -ais usually i the rage (π, π) The argumet of a comple umber z is give y by arg z, where ta You must be careful whe or y, or both, are egative 6

7 MFP Tetbook A-level Further Mathematics 660 Eample Fid the modulus ad argumet of the comple umber i Solutio The poit P represetig this umber, z, is show o the diagram y P, z ( ) ad ta θ Therefore, arg z π O Note that whe ta, θ could equal π or π However, the sketch clearly shows that θ lies i the secod quadrat This is why you eed to be careful whe evaluatig the argumet of a comple umber Eercise A Fid the modulus ad argumet of each of the followig comple umbers: (a) i, (b) i, (c) 4, (d) i Give your aswers for arg z i radias to two decimal places Fid the modulus ad argumet of each of the followig comple umbers: (a) i, (b) 4i, (c) 7i Give your aswers for arg z i radias to two decimal places 7

8 MFP Tetbook A-level Further Mathematics The polar form of a comple umber y I the diagram alogside, y rsi rcos ad P(, y) If P is the poit represetig the comple umber z i, y it follows that z may be writte i the form rcos irsi This is called the polar, or modulus argumet, form of a comple umber O θ r A comple umber may be writte i the form z r(cos isi ), where z r ad arg z For brevity, r(cos isi ) ca be writte as (r, θ) Eercise B Write the comple umbers give i Eercise A i polar coordiate form Fid, i the form i y, the comple umbers give i polar coordiate form by: (a) z cosπ π isi, (b) 4 4 π π 4cos isi 8

9 MFP Tetbook A-level Further Mathematics Additio, subtractio ad multiplicatio of comple umbers of the form + iy Comple umbers ca be subjected to arithmetic operatios Cosider the eample below Eample 5 Give that z 4i ad z i, fid (a) z z, (b) z z ad (c) zz Solutio (a) zz ( 4i) ( i) 4i (b) z z ( 4i) ( i) 6i (c) zz ( 4i)( i) 4i6i8i i 8 (sice i ) i I geeral, if z ai b ad z a i b, z z ( a a ) i( b b ) z z ( a a ) i( b b ) zz aa bb i( abab) Eercise C Fid z z ad z z whe: (a) z i ad z i, (b) z 6i ad z i 9

10 MFP Tetbook A-level Further Mathematics The cojugate of a comple umber ad the divisio of comple umbers of the form + iy The cojugate of a comple umber z iy (usually deoted by z * or z ) is the comple umber z* i y Thus, the cojugate of i is i, ad that of i is i O a Argad diagram, the poit represetig the comple umber z * is the reflectio of the poit represetig z i the -ais The most importat property of z * is that the product zz * is real sice zz* ( i y)( i y) iy iy i y y * zz z Divisio of two comple umbers demads a little more care tha their additio or multiplicatio ad usually requires the use of the comple cojugate Eample 6 z Simplify z, where z 4i ad z i Solutio 4i (4i)( i) i (i)(i) 4i6i8i i i 4i 50i 5 i z multiply the umerator ad deomiator of z by z *, ie ( i) so that the product of the deomiator becomes a real umber Eercise D z For the sets of comple umbers z ad z, fid z where (a) z 4 i ad z i, (b) z 6i ad z i 0

11 MFP Tetbook A-level Further Mathematics Products ad quotiets of comple umbers i their polar form If two comple umbers are give i polar form they ca be multiplied ad divided without havig to rewrite them i the form i y Eample 7 Fid z Solutio z if z π π cos isi ad z π π 6 6 π π π π zz cos isi cos isi π π π π 6 6 π π 6 6 cos isi π π π π π π π π 6 cos cos isi cos isi cos i si si cos πcos π si πsi π i si πcos π cos πsi π cos isi 6cos isi Notig that arg z is π, it follows that the modulus of zz is the product of the modulus of 6 z ad the modulus of z, ad the argumet of zz is the sum of the argumets of z ad z Usig the idetities: cos( A B) cos Acos Bsi Asi B si( A B) si Acos Bcos Asi B Eercise E z (a) Fid cos isi z z (b) What ca you say about the modulus ad argumet of z? if z cosπ π isi ad z π π 6 6

12 MFP Tetbook A-level Further Mathematics 660 Eample 7 If z r, ad z r,, show that zz rr Solutio zz r(cos isi ) r(cos isi ) cos( ) isi( ) rr coscos i(sicos cossi ) i sisi rr (cos cos si si ) i(si cos cos si ) rr cos( ) isi( ) If z ( r, ) ad z ( r, ) the zz ( rr, ) with the proviso that π may have to be added to, or subtracted from, if is outside the permitted rage for There is a correspodig result for divisio you could try to prove it for yourself z r (, ) ad (, ) the z, r same proviso regardig the size of the agle If z r z r with the

13 MFP Tetbook A-level Further Mathematics Equatig real ad imagiary parts z Goig back to Eample 6, z ca be simplified by aother method 4i Suppose we let a ib The, i ( i)( ai b) 4i abi( b a) 4i Now, a ad b are real ad the comple umber o the left had side of the equatio is equal to the comple umber o the right had side, so the real parts ca be equated ad the imagiary parts ca also be equated: ab ad ba 4 Thus b ad a, givig i as the aswer to a ib as i Eample 6 While this method is ot as straightforward as the method used earlier, it is still a valid method It also illustrates the cocept of equatig real ad imagiary parts If aibc i d, where a, b, c ad d are real, the a c ad b d Eample 8 Fid the comple umber z satisfyig the equatio Solutio ( 4i) z( i) z* i Let z ( a i b), the z* ( a i b) Thus, ( 4i)( ai b) ( i)( ai b) i Multiplyig out, a4iaib4i baiaibi b i Simplifyig, abi( 5a4 b) i Equatig real ad imagiary parts, ab, 5a4b So, a ad b Hece, z i Eercise F If z π z π 6, ad,, fid, i polar form, the comple umbers (a) zz, (b) z z, (c) z, (d) z, (e) z z Fid the comple umber satisfyig each of these equatios: (a) ( i) z i, (b) ( z i)(i) 7i, (c) z iz*

14 MFP Tetbook A-level Further Mathematics Further cosideratio of z z ad arg( z z) Sectio 5 cosidered simple cases of the sums ad differeces of comple umbers Cosider ow the comple umber z z z, where z iy ad z i y The poits A ad B represet z ad z, respectively, o a Argad diagram y A, ) B, ) ( y ( y O C The z z z ( ) i( y y) ad is represeted by the poit C (, y y) This makes OABC a parallelogram From this it follows that z z OC ( ) ( y y), that is to say z z is the legth AB i the Argad diagram Similarly arg( z z) is the agle betwee OC ad the positive directio of the -ais This i tur is the agle betwee AB ad the positive directio If the comple umber z is represeted by the poit A, ad the comple umber z is represeted by the poit B i a Argad diagram, the z z AB, ad arg( z z) is the agle betwee AB directio of the -ais ad the positive Eercise G Fid z z ad arg( z z) i (a) z i, z 7 5i, (b) z i, z 4 i, (c) z i, z 4 5i 4

16 MFP Tetbook A-level Further Mathematics 660 This coditio implies that the agle betwee OP ad O is fied ( ) so that the locus of P is a straight lie arg z represets the half lie through O iclied at a agle to the positive directio of O Note that the locus of P is oly a half lie the other half lie, show dotted i the diagram above, would have the equatio arg z π, possibly π if π falls outside the specified rage for arg z I eactly the same way as before, the locus of a poit P satisfyig arg( zz), where z is a fied comple umber represeted by the poit A, is a lie through A arg( z z ) iclied at a agle to the positive directio of O represets the half lie through the poit z y P A α O Note agai that this locus is oly a half lie the other half lie would have the equatio arg( zz ) π, possibly π Fially, cosider the locus of ay poit P satisfyig arg( zz) This idicates that the agle betwee AP ad the positive -ais lies betwee ad, so that P ca lie o or withi the two half lies as show shaded i the diagram below y A β α O 6

17 MFP Tetbook A-level Further Mathematics 660 Eercise H Sketch o Argad diagrams the locus of poits satisfyig: (a) z, (b) arg( z ) π, (c) z i 5 4 Sketch o Argad diagrams the regios where: (a) z i, (b) π arg( z 4 i) 5π 6 Sketch o a Argad diagram the regio satisfyig both z i ad 0arg z π 4 4 Sketch o a Argad diagram the locus of poits satisfyig both z i z i ad z i 4 7

18 MFP Tetbook A-level Further Mathematics 660 Miscellaeous eercises Fid the comple umber which satisfies the equatio ziz* 4 i, where z * deotes the comple cojugate of z [AQA Jue 00] The comple umber z satisfies the equatio ziz (a) Fid z i the form a i b, where a ad b are real (b) Mark ad label o a Argad diagram the poits represetig z ad its cojugate, z * (c) Fid the values of z ad z z* [NEAB March 998] The comple umber z satisfies the equatio zz* z z* i, where z * deotes the comple cojugate of z Fid the two possible values of z, givig your aswers i the form a i b [AQA March 000] 4 By puttig z i y, fid the comple umber z which satisfies the equatio z z* i, i where z * deotes the comple cojugate of z [AQA Specime] 5 (a) Sketch o a Argad diagram the circle C whose equatio is z i (b) Mark the poit P o C at which z is a miimum Fid this miimum value (c) Mark the poit Q o C at which arg z is a maimum Fid this maimum value [NEAB Jue 998] 8

19 MFP Tetbook A-level Further Mathematics (a) Sketch o a commo Argad diagram (i) the locus of poits for which z i, (ii) the locus of poits for which arg z π 4 (b) Idicate, by shadig, the regio for which z i ad arg z π 4 [AQA Jue 00] 7 The comple umber z is defied by z i i (a) (i) Epress z i the form a i b (ii) Fid the modulus ad argumet of z, givig your aswer for the argumet i the form p π where p (b) The comple umber z has modulus ad argumet 7π The comple umber z is defied by z zz (i) Show that z π 4 ad arg z 6 (ii) Mark o a Argad diagram the poits P ad P which represet z ad z, respectively (iii) Fid, i surd form, the distace betwee P ad P [AQA Jue 000] 9

20 MFP Tetbook A-level Further Mathematics (a) Idicate o a Argad diagram the regio of the comple plae i which 0arg π z (b) The comple umber z is such that 0arg π z ad π arg z π 6 (i) Sketch aother Argad diagram showig the regio R i which z must lie (ii) Mark o this diagram the poit A belogig to R at which z has its least possible value (c) At the poit A defied i part (b)(ii), z z A (i) Calculate the value of z A (ii) Epress z A i the form a i b [AQA March 999] 9 (a) The comple umbers z ad w are such that z 4i i ad w 4 i i Epress each of z ad w i the form a i b, where a ad b are real (b) (i) Write dow the modulus ad argumet of each of the comple umbers 4 i ad i Give each modulus i a eact surd form ad each argumet i radias betwee π ad π (ii) The poits O, P ad Q i the comple plae represet the comple umbers 0 0i, 4 i ad i, respectively Fid the eact legth of PQ ad hece, or otherwise, show that the triagle OPQ is right-agled [AEB Jue 997] 0

21 MFP Tetbook A-level Further Mathematics 660 Chapter : Roots of Polyomial Equatios Itroductio Quadratic equatios Cubic equatios 4 Relatioship betwee the roots of a cubic equatio ad its coefficiets 5 Cubic equatios with related roots 6 A importat result 7 Polyomial equatios of degree 8 Comple roots of polyomial equatios with real coefficiets This chapter revises work already covered o roots of equatios ad eteds those ideas Whe you have completed it, you will: kow how to solve ay quadratic equatio; kow that there is a relatioship betwee the umber of real roots ad form of a polyomial equatio, ad be able to sketch graphs; kow the relatioship betwee the roots of a cubic equatio ad its coefficiets; be able to form cubic equatios with related roots; kow how to eted these results to polyomials of higher degree; kow that comple cojugates are roots of polyomials with real coefficiets

22 MFP Tetbook A-level Further Mathematics 660 Itroductio You should have already met the idea of a polyomial equatio A polyomial equatio of degree, oe with as the highest power of, is called a quadratic equatio Similarly, a polyomial equatio of degree has as the highest power of ad is called a cubic equatio; oe with 4 as the highest power of is called a quartic equatio I this chapter you are goig to study the properties of the roots of these equatios ad ivestigate methods of solvig them Quadratic equatios You should be familiar with quadratic equatios ad their properties from your earlier studies of pure mathematics However, eve if this sectio is familiar to you it provides a suitable base from which to move o to equatios of higher degree You will kow, for eample, that quadratic equatios of the type you have met have two roots (which may be coicidet) There are ormally two ways of solvig a quadratic equatio by factorizig ad, i cases where this is impossible, by the quadratic formula Graphically, the roots of the equatio a b c 0 are the poits of itersectio of the curve y a b c ad the lie y 0 (ie the -ais) For eample, a sketch of part of y 8 is show below y ( 4, 0) (, 0) (0, 8) The roots of this quadratic equatio are those of ( )( 4) 0, which are ad 4

23 MFP Tetbook A-level Further Mathematics 660 A sketch of part of the curve y 4 4 is show below y (0, 4) O (, 0) I this case, the curve touches the -ais The equatio ( ) 0 ad, a repeated root 44 0 may be writte as Not all quadratic equatios are as straightforward as the oes cosidered so far A sketch of part of the curve y 4 5 is show below y (0, 5) O (, ) This curve does ot touch the -ais so the equatio 45 0 caot have real roots Certaily, 4 5 will ot factorize so the quadratic formula b b 4ac a has to be used to solve this equatio This leads to 4 60 ad, usig ideas from Chapter, this becomes 4 i or i It follows that the equatio 45 0 does have two roots, but they are both comple umbers I fact the two roots are comple cojugates You may also have observed that whether a quadratic equatio has real or comple roots depeds o the value of the discrimiat b 4 ac The quadratic equatio a b c 0, where a, b ad c are real, has comple roots if b 4ac 0

24 MFP Tetbook A-level Further Mathematics 660 Eercise A Solve the equatios (a) 60 0, (b) 06 0 Cubic equatios As metioed i the itroductio to this chapter, equatios of the form a b c d 0 are called cubic equatios All cubic equatios have at least oe real root ad this real root is ot always easy to locate The reaso for this is that cubic curves are cotiuous they do ot have asymptotes or ay other form of discotiuity Also, as, the term a becomes the domiat part of the epressio ad a (if a 0), whilst a whe Hece the curve must cross the lie y 0 at least oce If a 0, the as, ad a as ad this does ot affect the result a 4

25 MFP Tetbook A-level Further Mathematics 660 A typical cubic equatio, below y y a b c d with a 0, ca look like ay of the sketches The equatio of this curve has three real roots because the curve crosses the lie y 0 at three poits O y y O O I each of the two sketch graphs above, the curve crosses the lie y 0 just oce, idicatig just oe real root I both cases, the cubic equatio will have two comple roots as well as the sigle real root Eample (a) Fid the roots of the cubic equatio 0 (b) Sketch a graph of y Solutio y 4 (a) If f( ), the f() 0 Therefore is a factor of f() f( ) ( )( 4) ( )( )( ) Hece the roots of f() = 0 are, ad (b)

26 MFP Tetbook A-level Further Mathematics 660 Eample Fid the roots of the cubic equatio Solutio Let f ( ) 4 6 The f () Therefore is a factor of f(), ad f ( ) ( )( 6 ) The quadratic i this epressio has o simple roots, so usig the quadratic formula o 6 0, 4 a i i b b ac Hece the roots of f ( ) 0 are ad i Eercise B Solve the equatios (a) 5 0, (b) (c) 4 0, 0 0 6

27 MFP Tetbook A-level Further Mathematics Relatioship betwee the roots of a cubic equatio ad its coefficiets As a cubic equatio has three roots, which may be real or comple, it follows that if the geeral cubic equatio a b c d 0 has roots, ad, it may be writte as a ( )( )( ) 0 Note that the factor a is required to esure that the coefficiets of are the same, so makig the equatios idetical Thus, o epadig the right had side of the idetity, a b c d a( )( )( ) a a( ) a( ) a The two sides are idetical so the coefficiets of ad ca be compared, ad also the umber terms, ba( ) ca( ) d a If the cubic equatio a b c d 0 has roots, ad, the b, a c, a d a Note that meas the sum of all the roots, ad that meas the sum of all the possible products of roots take two at a time Eercise C Fid, ad for the followig cubic equatios: (a) 7 5 0, (b) The roots of a cubic equatio are, ad If, the cubic equatio 7 ad 5, state 7

28 MFP Tetbook A-level Further Mathematics Cubic equatios with related roots The eample below shows how you ca fid equatios whose roots are related to the roots of a give equatio without havig to fid the actual roots Two methods are give Eample 5 The cubic equatio 4 0 has roots, ad Fid the cubic equatios with: (a) roots, ad, (b) roots, ad, (c) roots, ad Solutio: method From the give equatio, 0 4 (a) Hece From which the equatio of the cubic must be or 6 0 (b) 4 ( ) 66 ( )( ) (4) 00 ( )( )( ) Hece the equatio of the cubic must be 0 8

29 MFP Tetbook A-level Further Mathematics 660 (c) So that the cubic equatio with roots, ad is or 4 0 9

30 MFP Tetbook A-level Further Mathematics 660 The secod method of fidig the cubic equatios i Eample 5 is show below It is ot always possible to use this secod method, but whe you ca it is much quicker tha the first Solutio: method (a) As the roots are to be, ad, it follows that, if X, the a cubic equatio i X must have roots which are twice the roots of the cubic equatio i As the equatio i is 4 0, if you substitute X the equatio i X becomes as before or X X 4 0, X 6X 0 (b) I this case, if you put X i 4 0, the ay root of a equatio i X must be less tha the correspodig root of the cubic i Now, X gives X ad substitutig ito 4 0 gives ( X ) ( X ) 4 0 which reduces to X X 0 (c) I this case you use the substitutio X or X For 4 0 X X O multiplyig by X, this gives X 4X 0 or 4X X 0 as before 4 0 this gives Eercise D The cubic equatio 47 0 has roots, ad Usig the first method described above, fid the cubic equatios whose roots are (a), ad, (b), ad, (c), ad Repeat Questio above usig the secod method described above Repeat Questios ad above for the cubic equatio 6 0 0

31 MFP Tetbook A-level Further Mathematics A importat result If you square you get ( ) ( )( ) So,, or for three umbers, ad This result is well worth rememberig it is frequetly eeded i questios ivolvig the symmetric properties of roots of a cubic equatio Eample 6 The cubic equatio has roots, ad Fid the cubic equatios with (a) roots, ad, (b) roots, ad [Note that the direct approach illustrated below is the most straightforward way of solvig this type of problem] Solutio (a) ( ) Hece the cubic equatio is (b) 5 6 usig the same result but replacig with with, ad with Thus 6 (5) 46 ( ) Hece the cubic equatio is 46 0

32 MFP Tetbook A-level Further Mathematics Polyomial equatios of degree The ideas covered so far o quadratic ad cubic equatios ca be eteded to equatios of ay degree A equatio of degree has two roots, oe of degree has three roots so a equatio of degree has roots Suppose the equatio a b c d k 0 has roots,,, the b, a c, a d, a ( ) k util, fially, the product of the roots a Remember that is the sum of the products of all possible pairs of roots, is the sum of the products of all possible combiatios of roots take three at a time, ad so o I practice, you are ulikely to meet equatios of degree higher tha 4 so this sectio cocludes with a eample usig a quartic equatio Eample 7 4 The quartic equatio has roots,, ad Write dow (a), (b) (c) Hece fid Solutio (a) (b) (c) 4 6 ( Now ( This shows that the importat result i Sectio 6 ca be eteded to ay umber of letters Hece ( ) ( ) 0

33 MFP Tetbook A-level Further Mathematics 660 Eercise E 4 The quartic equatio 58 0 has roots,, ad (a) Fid the equatio with roots,, ad (b) Fid 8 Comple roots of polyomial equatios with real coefficiets Cosider the polyomial equatio f( ) a b c k Usig the ideas from Chapter, if p ad q are real, f( pi q) a( pi q) b( pi q) k ui v, where u ad v are real Now, f( i ) ( i ) ( i ) p q a p q b p q k uiv sice i raised to a eve power is real ad is the same as i raised to a eve power, makig the real part of f ( p i q) the same as the real part of f ( p i q) But i raised to a odd power is the same as i raised to a odd power multiplied by, ad odd powers of i comprise the imagiary part of f ( p i q) Thus, the imagiary part of f ( p i q) is times the imagiary part of f ( p i q) Now if p iq is a root of f ( ) 0, it follows that u iv 0 ad so u 0 ad v 0 Hece, uiv 0 makig f ( p i q) 0 ad p iq a root of f ( ) 0 If a polyomial equatio has real coefficiets ad if p i q, where p ad q are real, is a root of the polyomial, the its comple cojugate, p i q, is also a root of the equatio It is very importat to ote that the coefficiets i f ( ) 0 must be real If f ( ) 0 has comple coefficiets, this result does ot apply

34 MFP Tetbook A-level Further Mathematics 660 Eample 8 The cubic equatio k 0, where k is real, has oe root equal to i Fid the other two roots ad the value of k Solutio As the coefficiets of the cubic equatio are real, it follows that i is also a root Cosiderig the sum of the roots of the equatio, if is the third root, ( i) ( i), To fid k, k ( i)( i)( ) 5, k 5 Eample 8 The quartic equatio roots has oe root equal to i Fid the other three Solutio As the coefficiets of the quartic are real, it follows that i is also a root Hece (i) ( i) is a quadratic factor of the quartic Now, (i) (i) (i) (i) (i)(i) 5 4 Hece 5 is a factor of Therefore 45 ( 5)( a b) Comparig the coefficiets of, a a 4 Cosiderig the umber terms, 5b 5 b Hece the quartic equatio may be writte as ( 5)( 4) 0 ( 5)( )( ) 0, ad the four roots are i,i, ad 4

35 MFP Tetbook A-level Further Mathematics 660 Eercise F A cubic equatio has real coefficiets Oe root is ad aother is i Fid the cubic equatio i the form a bc 0 The cubic equatio roots has oe root equal to i Fid the other two The quartic equatio three roots has oe root equal to i Fid the other 5

36 MFP Tetbook A-level Further Mathematics 660 Miscellaeous eercises The equatio 4 0, p where p is a costat, has roots, ad, where 0 (a) Fid the values of ad (b) Fid the value of p [NEAB Jue 998] The umbers, ad satisfy the equatios ad (a) Show that 0 (b) The umbers, ad are also the roots of the equatio 0, p q r where p, q ad r are real (i) Give that 4i ad that is real, obtai ad (ii) Calculate the product of the three roots (iii) Write dow, or determie, the values of p, q ad r [AQA Jue 000] The roots of the cubic equatio are, ad 4 0 (a) Write dow the values of, ad (b) Fid the cubic equatio, with iteger coefficiets, havig roots, ad [AQA March 000] 4 The roots of the equatio are, ad (a) Write dow the value of (b) Give that i is a root of the equatio, fid the other two roots [AQA Specime] 6

37 MFP Tetbook A-level Further Mathematics The roots of the cubic equatio 0, p q r where p, q ad r are real, are, ad (a) Give that, write dow the value of p (b) Give also that 5, (i) fid the value of q, (ii) eplai why the equatio must have two o-real roots ad oe real root (c) Oe of the two o-real roots of the cubic equatio is 4i (i) Fid the real root (ii) Fid the value of r [AQA March 999] 6 (a) Prove that whe a polyomial (b) The polyomial g is defied by 5 f is divided by a, g 6 p q, where p ad q are real costats Whe remaider is (i) Fid the values of p ad q [AQA Jue 999] (ii) Show that whe the remaider is a f g is divided by i, where i, the g is divided by i, the remaider is 6i 7

38 MFP Tetbook A-level Further Mathematics 660 Chapter : Summatio of Fiite Series Itroductio Summatio of series by the method of differeces Summatio of series by the method of iductio 4 Proof by iductio eteded to other areas of mathematics This chapter eteds the idea of summatio of simple series, with which you are familiar from earlier studies, to other kids of series Whe you have completed it, you will: kow ew methods of summig series; kow which method is appropriate for the summatio of a particular series; uderstad a importat method kow as the method of iductio; be able to apply the method of iductio i circumstaces other tha i the summatio of series 8

### Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

### Soving Recurrence Relations

Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

### Sequences and Series

CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

### In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

### Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

### Chapter 5: Inner Product Spaces

Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

### Incremental calculation of weighted mean and variance

Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

### 1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

### CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

### Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

### I. Chi-squared Distributions

1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

### THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

### SEQUENCES AND SERIES

Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say

### INFINITE SERIES KEITH CONRAD

INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal

### NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

### SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

### Section 8.3 : De Moivre s Theorem and Applications

The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =

### WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This

### Department of Computer Science, University of Otago

Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

### Infinite Sequences and Series

CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

### Basic Elements of Arithmetic Sequences and Series

MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

### Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

### Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have

Comple Numbers I spite of Calvi s discomfiture, imagiar umbers (a subset of the set of comple umbers) eist ad are ivaluable i mathematics, egieerig, ad sciece. I fact, i certai fields, such as electrical

### 2-3 The Remainder and Factor Theorems

- The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

### Asymptotic Growth of Functions

CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

### FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10

FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.

### Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

### MATHEMATICS P1 COMMON TEST JUNE 2014 NATIONAL SENIOR CERTIFICATE GRADE 12

Mathematics/P1 1 Jue 014 Commo Test MATHEMATICS P1 COMMON TEST JUNE 014 NATIONAL SENIOR CERTIFICATE GRADE 1 Marks: 15 Time: ½ hours N.B: This questio paper cosists of 7 pages ad 1 iformatio sheet. Please

### Laws of Exponents Learning Strategies

Laws of Epoets Learig Strategies What should studets be able to do withi this iteractive? Studets should be able to uderstad ad use of the laws of epoets. Studets should be able to simplify epressios that

### Chapter 7 Methods of Finding Estimators

Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

### 3. Greatest Common Divisor - Least Common Multiple

3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

### Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

### .04. This means \$1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

### Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

### CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

### Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

### Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

### Building Blocks Problem Related to Harmonic Series

TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite

### How To Solve The Homewor Problem Beautifully

Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log

### CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

### SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

### 7.1 Finding Rational Solutions of Polynomial Equations

4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?

### Math 114- Intermediate Algebra Integral Exponents & Fractional Exponents (10 )

Math 4 Math 4- Itermediate Algebra Itegral Epoets & Fractioal Epoets (0 ) Epoetial Fuctios Epoetial Fuctios ad Graphs I. Epoetial Fuctios The fuctio f ( ) a, where is a real umber, a 0, ad a, is called

### Properties of MLE: consistency, asymptotic normality. Fisher information.

Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

### Mathematical goals. Starting points. Materials required. Time needed

Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios

### Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

### Convexity, Inequalities, and Norms

Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

### Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <garrett@math.umn.edu>

(March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1

### Section 11.3: The Integral Test

Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

### Theorems About Power Series

Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

### CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

### Fast Fourier Transform

18.310 lecture otes November 18, 2013 Fast Fourier Trasform Lecturer: Michel Goemas I these otes we defie the Discrete Fourier Trasform, ad give a method for computig it fast: the Fast Fourier Trasform.

### Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

### Class Meeting # 16: The Fourier Transform on R n

MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

### S. Tanny MAT 344 Spring 1999. be the minimum number of moves required.

S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + \$ T is a sequece (f. o itegers). Solve for T? * is a recurrece,

### Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

### BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

### 3. If x and y are real numbers, what is the simplified radical form

lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y

### NATIONAL SENIOR CERTIFICATE GRADE 11

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 007 MARKS: 50 TIME: 3 hours This questio paper cosists of pages, 4 diagram sheets ad a -page formula sheet. Please tur over Mathematics/P DoE/Exemplar

### Modified Line Search Method for Global Optimization

Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

### Factors of sums of powers of binomial coefficients

ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the

### Notes on exponential generating functions and structures.

Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a -elemet set, (2) to fid for each the

### Hypothesis testing. Null and alternative hypotheses

Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

### http://www.webassign.net/v4cgijeff.downs@wnc/control.pl

Assigmet Previewer http://www.webassig.et/vcgijeff.dows@wc/cotrol.pl of // : PM Practice Eam () Questio Descriptio Eam over chapter.. Questio DetailsLarCalc... [] Fid the geeral solutio of the differetial

### MATH 083 Final Exam Review

MATH 08 Fial Eam Review Completig the problems i this review will greatly prepare you for the fial eam Calculator use is ot required, but you are permitted to use a calculator durig the fial eam period

### Solving equations. Pre-test. Warm-up

Solvig equatios 8 Pre-test Warm-up We ca thik of a algebraic equatio as beig like a set of scales. The two sides of the equatio are equal, so the scales are balaced. If we add somethig to oe side of the

### Elementary Theory of Russian Roulette

Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some

### 5: Introduction to Estimation

5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

### FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if

### Escola Federal de Engenharia de Itajubá

Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica Pós-Graduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José

### 4.3. The Integral and Comparison Tests

4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece

### Ekkehart Schlicht: Economic Surplus and Derived Demand

Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/

### Lecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k.

18.409 A Algorithmist s Toolkit September 17, 009 Lecture 3 Lecturer: Joatha Keler Scribe: Adre Wibisoo 1 Outlie Today s lecture covers three mai parts: Courat-Fischer formula ad Rayleigh quotiets The

### Normal Distribution.

Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued

### Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:

Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network

### Solutions to Exercises Chapter 4: Recurrence relations and generating functions

Solutios to Exercises Chapter 4: Recurrece relatios ad geeratig fuctios 1 (a) There are seatig positios arraged i a lie. Prove that the umber of ways of choosig a subset of these positios, with o two chose

### 1 Computing the Standard Deviation of Sample Means

Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

### 5.3. Generalized Permutations and Combinations

53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible

### NATIONAL SENIOR CERTIFICATE GRADE 11

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a -page formula sheet. Please tur over Mathematics/P DoE/November

### Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2

74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is

### Analysis Notes (only a draft, and the first one!)

Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................

### EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND 1 OBTAINED WITH ENGEL SERIES

EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND OBTAINED WITH ENGEL SERIES ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Abstract. The aciet Egyptias epressed ratioal

### The Stable Marriage Problem

The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,

### Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

### Lesson 17 Pearson s Correlation Coefficient

Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

### Integer Factorization Algorithms

Iteger Factorizatio Algorithms Coelly Bares Departmet of Physics, Orego State Uiversity December 7, 004 This documet has bee placed i the public domai. Cotets I. Itroductio 3 1. Termiology 3. Fudametal

### Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley

Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these

### Lesson 15 ANOVA (analysis of variance)

Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

### where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

### hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation

HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics

### Output Analysis (2, Chapters 10 &11 Law)

B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

### *The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.

Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.

### THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

### 5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso

### MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

### Part - I. Mathematics

Part - I Mathematics CHAPTER Set Theory. Objectives. Itroductio. Set Cocept.. Sets ad Elemets. Subset.. Proper ad Improper Subsets.. Equality of Sets.. Trasitivity of Set Iclusio.4 Uiversal Set.5 Complemet

### Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries