Draft. Complex numbers. Orientation

Transcription

1 Comple umers Oriettio Wht ou eed to kow KS4 Use the qudrtic formul. Mths Ch Solve simulteous equtios. Mths Ch Work with sie d cosie fuctios. Comple umers re crucil tool i the desig of moder electricl compoets, such s motherords. Electricl egieers use them to simplif m of their clcultios. This helps them to lse vrig currets d voltges i electricl circuits. Whe the curret flows i oe directio costtl, the resistce c ofte e clculted usig simple formul. However, whe the directio of the curret is ltertig, this formul does t work. Therefore, egieers epress the qutities s comple umers, which mkes it esier to uderstd the processes ivolved d perform the ecessr clcultios. Comple umers re used i m other fields such s chemistr, ecoomics, d sttistics. Their uique properties provide powerful ws of solvig d iterpretig complicted equtios tht c e pplied i wide vriet of cotets. From some of the deepest msteries i umer theor to the sigl processig used i plig digitl music, these so-clled imgir umers hve led to surprisig rr of dvces i the rel world. Wht ou will ler To clculte with comple umers i the form + i To uderstd d use the comple cojugte. To solve qudrtic, cuic d qurtic equtios with rel coefficiets. To covert etwee modulus-rgumet form d the form + i d clculte with umers i modulus-rgumet form. To sketch d iterpret Argd digrms. Wht this leds to Ch6 Comple umers Epoetil form The Euler formul e iq = cos q + i si q De Moivre s formul Roots of uit Creers Electricl egieerig

2 . Properties d rithmetic Fluec d skills Some equtios, icludig those derivig from rel-world situtios, hve o rel solutios. For emple, up to this poit, ou hve ee ule to solve equtios such s =. You kow tht the solutios re =±, ut this is ot rel umer d is difficult to mipulte. I order to solve this prolem, mthemticis deoted the squre root of egtive oe i. This mes tht the solutios to the equtio = c e writte s ±i. Although this umer is kow s imgir, it mes tht ll polomil equtios do ideed hve solutios. A good log is egtive umers: these c e hrd to grsp i isoltio. For emple, the cocept of mius oe pple is difficult oe, ut it s useful i sums such s pples pple = pples. I ectl the sme w, imgir umers re essetil i m clcultios d hve m rel-world pplictios. The imgir umer i is defied s i = Ke poit You c solve the equtio = 9 squre-rootig oth sides to give =± 9 The, ecuse 9 c e writte s 9, ou c use the fct tht i = to give = ±i Notice tht oce ou hve defied the squre root of mius oe, the squre root of ll other egtive umers c e writte i terms of i For emple, to solve the equtio ( ) = 5, ou first squre-root oth sides to give = ± 5, d sice 5= 5 i, this gives the solutios = ± 5i Numers with oth rel d imgir prts re clled comple umers. Ke poit Comple umers c e writte i the form + i where,. The set of comple umers is deoted. Comple umers c e dded, sutrcted d multiplied costt i the sme w s lgeric epressios. Emple Simplif the epressio (4 7 i) ( ) i (4 7 i) ( ) i = ( ) i (6 4) i = 6 7i Multipl rel d imgir prts the costt. Simplif rel prts: 6 = 6 Simplif imgir prts: i + 4i = 7i 4 Comple umers Properties d rithmetic

3 Whe multiplig comple umers together, ou will use the fct: i = ( sice i = ) Ke poit PURE Emple Solve the equtio ( + 7) = 6 + 7=± 6 = 7± 6 = 7± 6 = 7± 4i Squre-root oth sides of equtio. Sice = i Clcultor Emple AC + / % = To rtiolise the deomitor i surd form such s, ou multipl the umertor + d the deomitor You c use similr method to simplif frctios with comple deomitors. Doig this will lws chge the deomitor ito positive rel umer. To simplif + i Simplif i Tr it o our clcultor Some clcultors ele ou to mipulte comple umers. Ke poit, multipl the umertor d deomitor i + i + i (+ )( i + ) i = i ( )( i + ) i 5i 6i = + + 4i 5 5i = + = + i 5 ( i ) 8 6i Activit Fid out how to work out ( i ) o our clcultor. Use our clcultor to fid these epressios. ( + i)(5 7i) 4i( 8i) c (0 + 5i) ( i) Multipl the umertor d deomitor the sme comple umer (clled the comple cojugte, see Sectio.). The imgir prts of the deomitor ccel ech other out. Use i = Write i the form + i 5

4 Strteg Emple 4 Eercise.A Fluec d skills Solve these equtios. = 5 c = 0 e z = 9 = d + 8= 0 f z + = 0 Simplif these epressios, givig our swers i the form + i where,. ( + ) i + (5 9) i (5 7) i (+ i) c (6 9 i) d (+ 0) i + 5(4 i) e 4 9(7i + 5) f (6 i) (i 5) Write ech of these epressios i the form + i where,. ( + )( i i+ 5) (7 i)(6 i) c i(8 ) i d (9 4) i Resoig d prolem-solvig To solve equtios ivolvig imgir umers Write ll the umers d epressios i the form + i Fid rel umers d such tht ( + 5 i)( i) = 9+ i ( + 5 i)( i) = 9+ i i+ 0i 5i = 9+ i (+ 5) + (0 ) i = 9+ i Re : + 5= 9 = Im : 0 = = 8 4 Full simplif ech of these epressios. i i 4 c i 5 d ( i) e ( i) 4 f i (5i 9) 5 Simplif these frctios, givig our swers i the form + i where,. i + 7i c +i 5i i i + 6+ i i 6 d e f i i i 6 You re give tht z = i, z = 4+ i Clculte these epressios, full simplifig our swers. z+ z zz z z c d z z Equte rel prts d imgir prts o oth sides of the equtio or idetit. Solve the equtios simulteousl. Epd the rckets. Simplif usig i = d collect together rel d imgir terms. Equte rel prts. Equte imgir prts. 6 Comple umers Properties d rithmetic

5 Emple 5 Fid the comple umers z such tht z = 5+ i Let z = + i where,, the ( + i) = 5+ i + i + i = 5+ i ( ) + i = 5+ i Rel: = 5 Imgir: = 6 5 = 6 4 = = 0 Fid the rel umers d such tht ( + i) = 5 i Fid the vlues, tht stisf the equtio ( i)(+ 4 i) = i Solve the equtio z( 5) i = 8i for z. 4 Solve the equtio ( + i)(0 7) i = 4+ 6i for comple umer 5 Fid the comple umers z d z such tht z+ z = i d z z = 5+ 7i 6 Fid the comple umers z d w tht stisf the equtios z+ w= 6 i d z 4w= 0+ i 7 Fid z, z such tht z z = 0+ 8i d 5z z = 4+ 6i 8 Fid the comple umers w i ech of these cses. w = 0i 6 w = 4i c w = 0( i) = 6 = 4 or 9 =± 6 = =± z = + i or z = i Eercise.B Resoig d prolem-solvig 9 Clculte the squre roots of 4 i 0 Solve these simulteous equtios, where w d z re comple umers. w + z = 0 z w= 0 Solve ech of these equtios to fid z. z z= 50 z ( + i) = 7 7i Simplif ech of these epressios, givig our swer i the form + i i ech cse. (+ ) i 4 ( 5) i 5 c (i ) ( + i) Fid, such tht 4 ( + i) = 8+ i Simplif, usig i = You eed to fid the comple umer z such tht z = 5 + i Equte rel prts d equte imgir prts. Sustitute for d solve simulteousl. Solve the qudrtic i is rel umer so = 4 Give oth possile swers. Full A-level PURE 7

6 . Solvig polomil equtios Emple Fluec d skills Usig comple umers, the qudrtic formul = ± 4c c e used to solve ll qudrtic equtios of the form + + c = 0, with, c, R. Whe the discrimit ( 4 c) is egtive the solutios will coti imgir umers. Becuse the ± i the formul gives two solutios, ou c see tht: If z= + i is solutio to equtio the z* = i will lso e solutio. We cll z* the comple cojugte of z. Give tht z = 5 i, fid z* zz* c z + z* z* = 5+ i zz* = (5 i)(5+ i) = 5+ 5i 5i 9i = 5+ 9 = 4 c z+ z* = 5 i+ 5+ i = 0 Ke poit Write the comple cojugte of z Sice i = zz * will lws e rel umer. z + z * will lws e rel umer. If ou kow the roots of qudrtic equtio, the ou c fid the origil equtio multiplig the fctors together. Emple Fid the rel qudrtic equtio tht hs oe root of + 7i Roots re + 7i d 7i So the fctors re (+ 7) i d ( 7) i Equtio is ( (+ 7 i))( ( 7)) i = 0 ( 7) i (+ 7) i + (+ 7)( i 7) i = = 0 The comple cojugte will lso e root. Esure ou kow the differece etwee root d fctor. Sice 49i = 49 8 Comple umers Solvig polomil equtios

7 Emple Emple 4 Comple roots for rel polomil equtio will lws occur i comple cojugte pirs. Therefore, polomil equtio will lws hve eve umer of comple solutios. This implies tht cuic equtio will lws hve t lest oe rel root (s there cot e three comple roots). You c use these fcts to help ou solve cuic d qurtic equtios. Fid ll the solutios to the cuic equtio = 0, give tht oe solutio is + 4i 4i is lso solutio ( (+ 4 i))( ( 4)) i = = ( )( 6+ 5) = + 4 i, 4 i, re the three possile solutios. Fid ll the solutios to the qurtic equtio = 0give tht i is root. ( i)( + ) i = = ( + 9)( + + 5) + + 5= 0 = ± i d =± i Eercise.A Fluec d skills Write the comple cojugte of z i ech cse. z= 5 i z= 8+ i c z= 5i 6 d z= i e z= + i 4 f z= i 5 Give tht z= 9 i, clculte zz* z+ z* c z z* d z z* e ( z*)* f z* z Give tht w= 6+ i, clculte ww* w+ w* c w w* d w w* e w + ( w*) f (w + w*) Some clcultors c e used to solve qudrtic d cuic equtios. 4i is the comple cojugte of + 4i Use log divisio or equte coefficiets. Sice + i is lso root. Use log divisio or equte coefficiets. Use qudrtic formul or complete the squre. Give ll four possile swers. AC + / % = 4 Solve ech of these qudrtic equtios = 0 + 5= 0 c = 0 d 0+ 9= 0 5 Fid qudrtic equtio with solutios = 7 d = 4 = + 5i d = 5i c = 9i d = + 9i d = 5 + 4i d = 5 4i 6 A qudrtic equtio hs solutio of z= + i Write dow the other solutio to the equtio. Fid possile equtio. 7 Fid qudrtic equtio where oe solutio is give s +i 4 i c 7i d 5 i e + i f 5 i PURE See Ch.4 For remider of usig clcultor to fid roots of equtio. 9

8 Strteg 8 Fid cuic equtio with solutios = 5, = + i d = i =, = i d = i c = 0, = + i d = i d =, = i d = + i 9 A cuic equtio hs solutios of z = d z = 6 i Write dow the other solutio to the equtio. Fid possile equtio tht hs these solutios. 0 You re give tht f( ) = d f( 5) = 0 Show tht the other solutios to the equtio f( ) = 0stisf = 0 Solve the equtio f( ) = 0 Give tht + 8 i is root of = 0 show tht + 6+ = 0 Resoig d prolem-solvig You c fid polomil equtio whe give its roots. To derive polomil equtios whe give its roots Use the fct tht if z is root of f(z ) = 0 the z * will lso e root. Multipl the fctors together. Simplif d write the equtio i descedig powers of or z fid ll the solutios to the equtio = 0 Solve these cuic equtios usig the root give = 0, = 5 + 9= 0, = c = 0, = 6i d = 0, = 5+ 4i Solve these qurtic equtios usig the root give = 0, = i = 0, = + 5i 4 c = 0, = i 4 Use the fct tht 7 i is root to show tht the 4 qurtic equtio = 0 c e writte i the form ( + A)( + B)( + C) = 0 where A, B d C re costts to e foud. Emple 5 A qudrtic equtio hs roots d Show tht the equtio is ( + ) + = 0 Roots re d so equtio is ( )( ) = 0 which ecomes + = 0 ( + ) + = 0 You c ppl this result to other prolems tht ivolve qudrtics. If d re roots the d must e fctors, so multipl these together. Epd rckets the simplif. You c lso use this equtio d the fctor theorem to prove its fctors re d 0 Comple umers Solvig polomil equtios

9 Emple 6 Two of the roots of the equtio c + d + e = 0 re i d 4i Fid the vlues of,, c, d d e + i d 4i re lso roots. Roots i d + i give the qudrtic fctor 6 + Similrl, roots 4i d 4i give the qudrtic fctor If i d 4i re roots, the so re their comple cojugtes. Multipl the fctors together or use the result from Emple 5 PURE 4 ( 6+ )( + + 7) = So =, = 4, c = 8, d = 76, e = Now multipl the two qudrtic fctors together d simplif the equtio. Strteg Emple 7 You c solve cuic equtio usig the fctor theorem to fid the first solutio, the dividig the fctor foud d solvig the remiig qudrtic equtio. To fid solutios to polomil equtios Use the fctor theorem to fid oe root of the equtio. Use the fct tht if z is root of f(z ) = 0 the z * will lso e root. Divide f(z ) the fctor ou kow. 4 Alws write comple umers i the form + i Solve the cuic equtio = 0 f() = = 0 so is fctor = ( )( + + 7) = = 0 = ± i You could lso solve this usig clcultor. Tr sustitutig differet vlues for to fid fctor. Use log divisio or equte coefficiets. AC + / % = Comple umers should lws e writte i this form. 4

10 Emple 8 Solve the equtio z 4 z 7z + 8z 8 = 0, give tht oe of the solutios is i i d + i re oth solutios. Therefore, z z + is fctor of z 4 z 7z + 8z 8 z 4 z 7z + 8z 8 = (z z + )(z 9) z 9 = 0 z = ± So the solutios to the equtio re z = ± i d z = ± Sice i is solutio, + i will lso e solutio. Multipl fctors or use the result tht z ( + )z + = 0 hs solutios d Use log divisio or equte coefficiets. You c prove results ivolvig comple umers mipultig them i the form + i Strteg Emple 9 To prove results ivolvig comple umers Write comple umers i the form + i Use the fct tht if z = + i the z * = i Mipulte i the usul w to prove the result. Prove tht (z + w)* z* + w * for ll z, w. Let z = + i d w = c + di with,, c, d (z + w)* = (( + c) + ( + d )i )* Prove these equtios for ll zw,. ( zw) * = z* w* ( z*)* = z c z * z* = w w* Prove tht, for ll z, c = ( + c) ( + d )i = ( i ) + (c di ) = z * + w * s required. z + z* is rel, z z* is imgir, zz* is rel. Write i the form + i Fid comple cojugte. Eercise.B Resoig d prolem-solvig Rerrge equtio. Fid the possile comple umers z such tht z + z* = 6 d zz* = 58 4 Fid the possile comple umers w such tht w w* = 8i d ww* = 85 5 Fid the comple umer z such tht z z+ z* = d = i z* 6 Fid the comple umer w such tht w w* = 4i d w 4 = + i w* 5 5 Comple umers Solvig polomil equtios

11 7 Give tht = 7 is solutio of the equtio k = 0 fid the vlue of k solve the equtio full. 8 Give tht is fctor of the equtio k= 0, fid the vlue of k d solve the equtio. 9 Fid ll the solutios to the equtio = 0 give tht 4 + is fctor of The curve of = is show. O 6 7 Fid ll the solutios to the equtio = 0 The equtio 4 + k + k 0 = 0 hs root of i 6 Fid the vlue of k. Solve the equtio. Give tht 5 i is root of the equtio 4 + A = 0, fid the vlue of A d solve the equtio. Fid the qurtic equtio with repeted root + i i 4 f( ) = Write f( ) s the product of lier d qudrtic fctor. Solve the equtio f( ) = 0 c Sketch the grph of = f( ) 5 Use the fct tht the qurtic equtio = 0 hs ectl two roots to solve the equtio = Sketch the grph of = The curve of qurtic equtio = p() is show. 5 O Wht does the grph tell ou out the ture of the roots of p() = 0? Give tht p() = solve the equtio p() = 0 7 If the roots of cuic equtio re, d c, d the coefficiet of is, write the equtio i terms of, d c 8 The solutios of the equtio A + = 0re, d g Write the vlue of α+ β+ γ αβγ 9 A qurtic equtio hs roots,, g d d Give tht the coefficiet of 4 is, write epressio i terms of,, g d d for the coefficiet of the costt term. 0 Fid ll the solutios to 4 = 64 = 7 c = 7i Three solutios of polomil equtio re, 4 i d i Stte the lowest possile order of the equtio. Fid equtio of this order with these solutios. Full A-level PURE

12 . Argd digrms Emple Emple Fluec d skills You represet rel umers visull s poits o umer lie, ut imgir umers will ot fit oto the rel umer lie. Therefore, we eed ew umer lie of imgir umers. Comiig these two umer lies together gives ple. So comple umers c e represeted visull s poits o ple. This ple is clled Argd digrm. Argd digrms re used to represet comple umers grphicll. The comple umer z = + i c e represeted s the poit (, ) o Argd digrm. Ke poit I Argd digrm, the horizotl is is the rel is d the verticl is is the imgir is. Give tht z = + i, show z d z* o Argd digrm. O Im z = (, ) z * = (, ) Re Give tht z= 4 + i d w= + i, show z, w d z + w o Argd digrm. O Im z = + i The rel is is lelled Re d the imgir is is lelled Im. z * is the poit z reflected i the rel is. Comple umers c lso e show s positio vectors. The sum c e show s vector dditio o the Argd digrm. Esure ou lel ech vector clerl. Re 4 Comple umers Argd digrms

13 Emple Give tht z= + i, show z, z d z o Argd digrm. Im 7 6 z 5 4 z O z 4 Re The three poits lie o stright lie. PURE Eercise.A Fluec d skills Write the comple umers represeted the vectors OA, OB, OC, OD, OE d OF, s show i the Argd digrm, i the form + i Im F A D C E 5 B 6 The Argd digrm shows the comple umers u, v, w d z Im u 5 4 w z 4 O 7 v Re Write the comple umers u, v, w d z i Crtesi form. z = 5 8i, z = + 4i Show z, z d z+ z o Argd digrm. 4 z= 7 i, w= 6i 4 Show z, w d z w o Argd digrm. 5 Show the vector dditio z= (+ 5) i + ( 7) i o Argd digrm. 6 Show the vector sutrctio z= (6 ) i ( 5) i o Argd digrm. Re 7 Give tht w= + i, drw w d w* o Argd digrm d descrie the geometric reltioship etwee them. 8 Give tht z* = 7+ i, drw z d z* o the sme Argd digrm d descrie the geometric reltioship etwee them. 9 Solve ech of these equtios d plot the solutios o Argd digrm. Descrie the geometric reltioship etwee the two solutios i ech cse. = 6 = 80 0 Solve the equtio ( z + ) = Plot the solutios o Argd digrm. c Descrie the geometric reltioship etwee the two solutios. Solve the equtio ( z) = 5 Plot the solutios o Argd digrm. c Descrie the geometric reltioship etwee the two solutios. For ech vlue of z, show z d iz o Argd digrm. Wht trsformtio mps z to iz i ech cse? z= 4i z= 6i For the comple umers give, drw w d i wo the sme Argd digrm d descrie the geometricl reltioship etwee them. w= 5 7i w= + i 5

14 Resoig d prolem-solvig Strteg To solve geometric prolems ivolvig comple umers Drw Argd digrm. Use rules for clcultig re, legths d gles. Use the fct tht the product of grdiets of perpediculr lies is 4 Full defie trsformtios. Emple 4 f( ) = Fid ll the roots of the equtio f( ) = 0 give tht + 5 is fctor of f( ) Fid the re of the qudrilterl formed the poits represetig the four roots. c Prove tht the qudrilterl cotis two right gles = ( + 5)( ) = = 0 5 Solve to give roots = ± i, Are = = squre uits c Grdiet of AB = = 6 Grdiet of BC = =.5 = so ABC is right gle. Similrl for ADC Use log divisio or ispectio method to fctorise. Some clcultors hve equtio solver. This is the esiest method to use to fid the three remiig roots. Show the roots o Argd digrm. The qudrilterl formed is kite which we c split ito two trigles. The product of the grdiets is AC + / % = 6 Comple umers Argd digrms

15 Emple 5 You re give z= i The poits A, B d C re represeted the comple umers z, z d z 4z respectivel. Fid the comple umer z 4z i the form + i Descrie the trsformtio tht mps lie segmet OA to CB PURE ( ) i 4( i) = + 6i z = ( ) i = 8 6i OA d CB re prllel ut CB is four times the legth of OA CB = z ( z 4 z) = 4z = 4OA So OA hs ee elrged scle fctor 4, cetre the origi, the trslted the vector 6 Or ltertivel, OA hs ee elrged scle fctor 4, cetre (4, ). Eercise.B Resoig d prolem-solvig 0+ i z = 5 i, z = z Fid z i the form + i Show the poits A d B represetig z d z respectivel o Argd digrm. c Show tht AOB is right gle. Show the three roots of the equtio = 0 o Argd digrm. Wht tpe of trigle is formed the poits represetig the three roots? c Fid the ect re of the trigle. The poits A, B d C o Argd digrm represet the solutios to the cuic equtio = 0 Clculte the re of trigle ABC 4 Give tht the epressio c e writte i the form ( + + )( + A+ B) of A d B 4 Solve the equtio = 0 You could check this usig clcultor tht hs comple mode. Drw Argd digrm. You c use either lger or the Argd digrm to help swer the questio. Iclude ll ecessr iformtio to full defie the trsformtio. AC + / % = 4 c Show the roots of = 0 o Argd digrm. d Wht tpe of qudrilterl is formed the poits represetig the roots? e Fid the re of the qudrilterl. 5 The qurtic equtio = 0 hs repeted rel root. Show tht the repeted root is = Clculte the other solutios to the equtio. c Show ll the solutios o Argd digrm. 6 The poits A, B, C d D represet the solutios to the qurtic equtio = 0 Use the fctor theorem to fid two rel solutios. Clculte the re of qudrilterl ABCD 4 7

16 7 The solutios of the cuic equtio + 9 9= 0 re represeted the poits A, B d C o Argd digrm. Prove tht trigle ABC is isosceles. Clculte the re of trigle ABC 8 You re give tht w= 5 i The poits A, B d C represet the comple umers w, w + iw d iw respectivel. Prove tht OABC is squre. Fid the re of the squre. 9 The poits P, Q d R represet the comple umers z, z * d z + z *, where z= 4 i Show the poits P, Q d R o Argd digrm. Descrie the trsformtio tht mps OP to QR 0 The poits A, B d C re represeted the comple umers z, z d z z where z= + i Fid the comple umer z z i the form + i Descrie the trsformtio tht mps lie segmet OA to CB The solutios of qurtic equtio re z= ± i d w= c ± di, where,, c, d C, 0, d 0 Fid the re of the qudrilterl formed the roots o Argd digrm. Give tht z= 8+ i, show the comple umers z, zi, zi d zi o Argd digrm. Descrie the trsformtios tht mp z to ech of the other poits. The poits A d B represet the comple umers z d iz respectivel. Prove tht OA is perpediculr to OB 4 The poits P d Q represet the comple umers 4+ 6 i d + 4 i respectivel. Fid the re of the trigle OPQ 5 Fid the re of the shpe formed the solutios to the equtio = f( z) = z 4z + 56z 04z The equtio f( z) = 0hs comple roots z d z which re represeted o Argd digrm the poits A d B. Give tht f( z) = 0hs o rel roots, clculte the legth of AB 7 The comple umer z is such tht z= + i where w= 5i Show tht z= i Mrk o Argd digrm the poits A d B represetig the umers z d z w, respectivel. c Show tht trigle OAB is right-gled. d Hece clculte the re of trigle OAB 4 8 f( z) = z z + 47z z + 90 where, re costts. Give tht z= 5 i is root of the equtio f( z) = 0, show ll the roots of f( z) = 0o Argd digrm. 4 9 f( z) = z + z + cz + dz+ e where,, c, d, e R re costts. The solutios to f( z) = 0re plotted o Argd digrm. Descrie the shpe formed the poits represetig the solutios to f( z) = 0 whe the equtio hs i precisel two (distict) rel roots, ii precisel oe rel root, iii o rel roots. Give tht the poits represetig the solutios to f( z) = 0form squre, clculte the vlues of the costts,, c, d d e whe i z= i is root of f( z) = 0, ii z= + i is root of f( z) = 0 Full A-level 8 Comple umers Argd digrms

17 .4 Modulus-rgumet form d loci Fluec d skills Emple You hve see how the comple umer z = + i c e represeted o Argd digrm. The legth of the vector represetig z is kow s the modulus of z d writte z The gle etwee the positive rel is d the vector represetig z is kow s the (pricipl) rgumet of z d is writte rg (z) The modulus of the comple umer z= + i is give z = + Ke poit Write rg( z) = θ where π< θ π Ke poit Fid the rgumet d modulus of the comple umer w= + i w = ( ) + = tθ = θ = π 4 rgw=π π 4 = π 4 θ q Use w = + A Argd digrm will help ou to fid the correct gle. We lws use rdis o Argd digrms: π rdis is equivlet to 80, so π rdis is 45 4 You eed the gle with the positive rel is. Usig w= + i from the previous emple, if ou wished to fid the modulus of w the it is cler from the digrm tht w = w The rgumet of w c lso e see to e give rg( w) = rg( w) +πwhich is rgw + rg( ) θ 9

18 I fct, these results c e geerlised for two comple umers. zz = z z d z z z = for ll z, z. z rg( zz ) = rg( z) + rg( z) d z rg rg( z) rg( z) z = for ll z, z. Ke poit Ke poit You c quote these results d will ler how to prove them i the et sectio. AC + / % = Tr it o our clcultor Some clcultors c e used to fid the rgumet of comple umer. Work out how to fid the rgumet of 5 6i o our clcultor. rg(5-6i ) Emple Give tht z = + i d zz = 4 4i, fid the rgumet d modulus of z ( ) z = + = zz = ( 4) + ( 4) = 4 4 = z z = π rg( z) = d rg( zz ) = 6 4 π z 4 π = π + 6 rg( ) rg( z ) π = Use z = + Sice zz = z z Usig comple umer mode o clcultor. Sice rg( zz ) = rg( z ) + rg( z ) Isted of the form z= + i, sometimes kow s Crtesi form, ou c write comple umers i modulus d rgumet form. You c see i the digrm tht the rel compoet of r is r cos q d the imgir compoet is r si q Ke poit The modulus-rgumet form of the comple umer z= + i is give z= r(cosθ+ isi θ) where r is the modulus of z d θ is the rgumet. q q q 0 Comple umers Modulus-rgumet form d loci

19 Emple Write the umer z= 7 i i modulus-rgumet form. z = 7 + ( ) = 5 t 0.4 c 7 = (sf) so rgz = 0.4 So the modulus-rgumet form is z = 5 (cos( 0.4) + isi( 0.4)) θ The rgumet is mesured i rdis, where π rdis is equl to 80 o Esure ou fid the gle with the positive -is. Rememer tht π < θ π d tht gle mesured clockwise will e egtive. PURE Clcultor Emple 4 AC + / % = Tr it o our clcultor Some clcultors c e used to covert to d from modulus-rgumet form. Fid out how to + i to modulus-rgumet form o our clcultor. covert Also fid out how to eter umer i modulus-rgumet form d covert ck to the form + i You c drw loci i Argd digrm. These re the set of poits tht oe give rule. Sketch the locus of poits tht stisf z = 4 This will e circle, cetre the origi d rdius 4 +i r θ π 6 z is the distce from the origi to the poit z. As the poit z vries, circle is formed. This circle is formed from ll the poits tht re distce of 4 from the origi. If we hve fied poit z, the The locus of poits stisfig z z = r will e circle cetre z d rdius r Ke poit

20 Emple 5 Sketch the locus of poits tht stisf z + i = z ( i) = First write i the form z z = r The locus of z is circle of rdius d cetre (, ) Emple 6 Emple 7 Sketch the locus of poits stisfig rg( z) = π This will e lie tht mkes gle of π with the positive rel is. π Sketch the locus of z where rg( z + + i) = π 6 rg( z ( )) i = π 6 ( ) The locus of poits stisfig rg z z = θ is hlf-lie from the poit z t gle of θ to the positive rel is. π Ke poit rg(z ) is the gle etwee the positive rel is d the lie represetig z Notice tht the lie eds t the origi. First write i the form rg( z z ) = θ Drw horizotl lie i the positive rel directio from our poit z The locus of z is hlflie from (, ) tht π mkes gle of 6 with this lie. Comple umers Modulus-rgumet form d loci

21 The locus of poits stisfig z z = z z is the Ke poit perpediculr isector of the lie joiig z d z This is ecuse the locus icludes ll poits tht re equidistt from the fied poits z d z PURE Emple 8 Sketch the locus of poits stisfig z = z+ i We hve z = z ( ) i First write the equtio i the form z z = z z Drw the poits (, 0) d (0, ) d joi with dotted lie. Eercise.4A Fluec d skills Fid the modulus d rgumet of ech comple umer. Sketch ech oe o Argd digrm. + 5 i 4 i c e i d 6 8i + 7 i f i g i h 6i Verif i ech cse tht zw = z w d z z = w w z= + i, w= 5+ i z= i, w= 5i c z= + 6 i, w= i I ech cse, verif tht rg(zw) = rg z + rg w z d rg = rg z rg w w z= + i, w= + i z= i, w= i c z= i, w= i 4 Give tht z= i d zw = + i, fid the modulus d the rgumet of w z 5 Give tht z = i d = + 6i, fid z the modulus d the rgumet of z w 6 Give tht z= + 6i d = i, fid z the modulus d the rgumet of w 7 Write ech of these comple umers i Crtesi form. π π cos + isi 5(cos( π ) + isi( π)) c π π 0 + cos 5 isi The locus of z is the perpediculr isector of this lie. d π + π cos isi 8 Write ech of these umers i modulusrgumet form. z= + i z= i c z= i d z= 4+ 9i 9 Sketch the locus of z i ech cse. z = 7 z = 5 c z i = d z (+ ) i = e z + 5i = 5 f z+ 4 i = 4

22 0 Sketch the locus of z i ech cse. rg z = π rg( z ) = π 4 c rg( z+ i) = π d rg( z ) i = π e rg( z + i) = π f rg( z 4 i) = π 6 g rg( z+ 5 7 i) = π Sketch the locus of z i ech of these cses. z = z+ 4 z i = z c z i = z + d z+ 6+ i = z + 6 e z+ 4 i = z 5+ i Resoig d prolem-solvig Strteg Emple 9 z z You eed to e le to prove the results: = = w w, zw z w, z rg = rg z rg w d w rg( zw) = rg z+ rgw To prove results out modulus d rgumet Write comple umers i modulus-rgumet form. Simplif powers of i Split ito rel d imgir prts. 4 Use the dditio formule for sie d for cosie. z Prove tht = w z w d z rg = rg z rg w for ll z, w. w Let z = z (cos A+ isi A) d w = w (cos B+ isi B) The z z (cos A+ isi A) = w w (cos B+ isi B) z (cos A+ isi A)(cos B isi B) = w (cos B+ isi B)(cos B isi B) z (cos AcosB i cos AsiB+ i cos BsiA i si Asi B) = w (cos B i si B) z (cos AcosB+ si Asi B+ i(cosbsi A cosasi B)) = w (cos B+ si B) z (cos ( A B) + isi( A B)) = w () z = (cos ( A B) + isi( A B)) w Therefore the umer z w hs modulus z d rgumet A B w So z z = d z rg w w = A w B = rg z rgw Write oth umers i modulus-rgumet form. Epd the rckets. Use the fct tht i = Seprte rel d imgir prts. Use cos (A B ) = cos A cos B + si A si B d si (A B ) = cos A si B si A cos B 4 4 Comple umers Modulus-rgumet form d loci

23 You kow how to fid the Crtesi equtio of certi loci drwig the grph d usig kow equtios of circles or lies. However, it is possile to fid the Crtesi equtio of locus settig z = + i d fidig reltioship etwee d PURE Strteg To fid the Crtesi equtio of locus Write z s + i Clculte the modulus. Use t to form equtio from the rgumet. 4 Rerrge to the required form. See Mths Ch.5 For remider of the equtios of lies d circles. Emple 0 Fid the Crtesi equtios of these loci. z + 4i = 5 rg( z+ i) = π 6 Let z = + i + i + 4 i = 5 ( ) + ( + 4) = 5 ( ) + ( + 4) = 5 So the locus is circle with cetre (, 4) d rdius 5 Let z = + i rg( + i+ i) = π i 6 so rg( + ( + )) = π 6 + t 6 so = π + = + + = 0 As the locus is hlf-lie, we ol eed the prt where > 0 d < Write i Crtesi form. Fid the modulus of the left-hd side. Write i Crtesi form. Sice t q = opposite djcet Whe rerrged, ou c see this is the equtio of lie. 4 Strteg To fid regio ouded locus Sketch the locus of the oudr of the regio. Test poit to see if it is iside the regio or ot. Shde the correct re. 5

24 Emple Shde these sets of poits. { z : z 5 } { } z :0< rg( z 4) i < π 4 { z : z 5 } is the set of comple umers z such tht z 5 Sketch the locus of z See Mths Ch.7 For remider of set ottio. If z = 0 the z 5 = 5 which is greter th or equl to π Eercise.4B Resoig d prolem-solvig Prove tht zw = z w d rg( zw) = rg z+ rgw for ll z, w. Show tht the locus of poits stisfig z+ i = 4 is circle d sketch it. Show tht the locus of poits stisfig z i = is circle. The circle touches the imgir is t the poit A d crosses the rel is t B d C Clculte the ect re of trigle ABC 4 Fid the Crtesi equtio of these loci. z 5 = z z+ = z i c z 4i = z + d z+ i = z If z = 0 the rg( z 4) i = rg( 4) i = π which is outside the regio required. Test poit. Shde regio ot icludig the poit z = 0 e z 5+ i = z+ i f z 7+ 4i = z+ 6 i 5 I ech cse, fid the Crtesi equtio of the lie o which the hlf-lie lies. rg( z ) i = π rg( z + 5) = π 4 c rg( z+ i) = π d rg( z 4 + i) = π 6 Fid the Crtesi equtio of the locus of poits stisfig z = z+ i 7 Fid the Crtesi equtio of the locus of z + i = z 8 Fid the Crtesi equtio of the locus of z = 4i z The tested poit stisfies the iequlit z 5, so shde outside the circle. z 5 represets the oudr d the outside of the circle. Test poit. Sketch the locus of z. Use dotted lies s the iequlities re strict. 6 Comple umers Modulus-rgumet form d loci

25 9 Shde the regio tht stisfies z 4i z+ i c 4 z 0 d < z 5+ i < 5 0 Fid the re of the regio tht stisfies 7 z+ 7i 7 Shde the regio represeted ech iequlit. 0 rg z π 5 π < rg( z ) i < 0 6 c π rg( z i) π < 4 Fid the re of the regio tht stisfies z 5+ i < 8 d 0 rg ( z 5+ ) i π Shde the regio tht stisfies z z+ 5 z i < z i c z 4 i z+ 8 4i d z i > z 5+ i e z + i z i d z > z 4 Sketch d shde the regio tht stisfies oth z + i d π z 4 rg 0 5 Sketch d shde the regio tht stisfies oth z z 8i d z i 8 6 Sketch d shde the regio tht stisfies oth π z rg( ) π < < d z i < z 4 i 7 Shde o Argd digrm the set of poits { z : z+ i } 9 Shde o Argd digrm the set of poits { z : z 4 } { z : π < z rg( ) < π } 0 Shde o Argd digrm the set of poits z : z i > z+ 5 i z : π rg( z) π { } { } Use lger to show tht the locus of poits stisfig z+ = z 6 i is circle, the sketch it. Shde the regio tht stisfies z+ z 6 i d z i 0 The poit P represets comple umer z o Argd digrm such tht z + i = Stte the Crtesi equtio of the locus of P The poit Q represets comple umer z o Argd digrm such tht z+ i = z + i c Fid the Crtesi equtio of the locus of Q Fid the comple umer tht stisfies oth z + i = d z+ i = z + i, givig our swer i surd form. Fid the comple umer tht stisfies oth z i = 4 d rg( z ) i = π 4 Full A-level PURE 8 Shde o Argd digrm the set of poits { z : z+ i > z } 054, 055, 057 SEARCH 7

26 Summr d review Chpter summr The imgir umer i is defied s i = Comple umers writte i the form + i c e dded, sutrcted d multiplied i the sme w s lgeric epressios. Powers of i should e simplified: i =, i = i d so o. The comple cojugte of the umer z= + i is z* = i Frctios with comple umer i the deomitor c e simplified multiplig the umertor d the deomitor the comple cojugte of the deomitor. Comple roots of polomil equtios occur i cojugte pirs. The comple umer z= + i c e represeted the poit (, ) o Argd digrm. The modulus of comple umer z= + i is give z = + The (pricipl) rgumet of comple umer is the gle etwee the vector represetig it d the positive rel is. Write rg z = θ where π < θ π z z zz = z z d = for ll z, z z z rg( zz ) = rg( z) + rg( z) for ll z, z The modulus-rgumet form of the comple umer z= + i is give z= r(cosθ+ i si θ) where r is the modulus of z d θ is the rgumet. The locus of poits stisfig z z = r will e circle, cetre z d rdius r The locus of poits stisfig rg( z z) = θ is hlf-lie from the poit z t gle of θ to the positive rel is. The locus of poits stisfig z z = z z is the perpediculr isector of the lie joiig z d z Check d review You should ow e le to... Tr Questios Add, sutrct, multipl d divide comple umers i the form + i Uderstd d use the comple cojugte. Solve qudrtic, cuic d qurtic equtios with rel coefficiets. 4 Clculte the modulus d the rgumet of comple umer. 5 Covert etwee modulus-rgumet form d the form + i 6, 7 Multipl d divide umers i modulus-rgumet form. 8 Sketch d iterpret Argd digrms. 9 Costruct d iterpret loci i the Argd digrm Comple umers Summr d review

27 Give tht z= 5 4 i d w= i, fid ech of these i the form + i z+ w w c z w d zw e z + w f ( z+ w) g z* h w* z w i j w z k z l w* i Solve these equtios. 4+ 0= = 0 c 0+ 7 = 0 Solve these equtios usig the root give = 0, = = 0, = 5 c = 0, = 9+ i 4 d + 4= 0, = + i 4 Solve the qurtic equtio f( ) = 0i ech cse usig the comple root give. 4 f( ) = , = ( 5 + ) is root. 4 f( ) = 4 + 4, ( ) root. 5 Clculte the modulus d the rgumet of these comple umers. + 9 i i c 7 i d i e + 4i f 4i 6 Write these comple umers i modulus-rgumet form i + 5i c i d i e π π 5 cos isi 7 Write these i the form + i π π + cos i si 6 6 π + π cos i si Give tht = π + π z 6 cos isi d π π w= + cos isi 6 6 fid these i modulus-rgumet form. z zw w w c z 9 Give tht z = 5 i d z = + i, drw these comple umers o the sme Argd digrm. z z c z+ z d z z 0 Sketch these loci o seprte Argd digrms d give the Crtesi equtio of ech. z = 7 z 8 = 5 c z+ i = d z i = Sketch these loci o seprte Argd digrms. rg z = π rg( z ) = π 6 c rg( z+ ) i = π d rg( z+ i) = π 4 Sketch these loci o seprte Argd digrms d give the Crtesi equtio of ech. z = z 6 z+ i = z 8 i c z i = z + 4 d z i = z+ + i Sketch d shde the regio stisfig ech iequlit. π z 5 + i z i rg( ) π < 56 c z + 5i 5 d z 6i < z Sketch d shde the regio stisfig oth of these iequlities. z 7 7 d π rg( z 7) 0 PURE 9

28 Eplortio Goig eod the ems Histor Je-Roert Argd ws Swiss mthemtici who, erl i the 9th cetur, moved to Pris where he mged ookshop. He pulished ess o the represettio of imgir qutities i 8 d, lthough others efore him hd writte out similr ides, it ws Argd s ess tht gied ttetio d resulted i the mig of the Argd digrm we cotiue to use tod. ellipse Note A coic sectio is curve tht is foud itersectig ple with doule coe plced verte to verte. There re three tpes of coic sectio: ellipse, prol d hperol. A circle is tpe of ellipse. Ivestigtio prol circle hperol The equtio z = plotted i the comple ple gives circle with uit rdius cetred t the origi. Choose vlues of, d c such tht c > ( ) d > B vrig these vlues, ivestigte the grph of the locus of poit z tht moves so tht z + z = c, c > ( ) d > Wht is the shpe of the grph tht results? The rel prt of comple umer z c e deoted Re(z) B vrig the vlues of d, ivestigte grphs of the locus of poit z tht moves so tht Re(z z + ) = z Wht is the shpe of the grph tht results? MATERIAL TO COME Reserch The Mdelrot set provides eutiful imges. It is the set of ll the comple umers, c, for which the itertive sequece z = z + c, + where z 0 = 0 does ot ted to ifiit. Computer imges of the Mdelrot set c e developed d, o mtter how closel ou zoom i o the surfce, it still remis icredil itricte. The Mdelrot set is emple of frctl. Fid out if the comple umers i d i re i the Mdelrot set. Reserch the properties of the Mdelrot set. = z + c, + 0 Comple umers Summr d review

29 Assessmet The comple umers z d w re give s z= 9 i d w= 7+ i Clculte these epressios, givig the swers i the form + i w i w z ii wz iii z Clculte the modulus d the rgumet of z, givig our swers to sigifict figures. The qudrtic equtio + + 5= 0 hs comple roots d Fid d Show d o Argd digrm. [] c Descrie the trsformtio tht mps to [] 5 i z = + + i Show tht z= + i where d re costts to e foud. [] c Clculte the vlue of i z + z* ii z z* iii zz* [4] i Drw z d iz o the sme Argd digrm. ii Descrie the geometric reltioship etwee z d iz [] 4 Fid the vlues of d such tht ( + i) = 5+ 8i [6] 5 Fid the vlues of d such tht ( + i) = + i [6] 6 f( ) = Give tht = i is solutio to f( ) = 0, fid ll the roots of f () [] Fid the re of the trigle formed the three roots of f () [] 7 Give tht 5 i is root of the equtio + + 8= 0 fid other two roots, clculte the vlues of d [] 8 Solve the equtio z z* = 6+ 7i 9 Solve the simulteous equtios z w= 8+ 5i z w= + i 0 Give tht z= + i d w= + i clculte the modulus d rgumet of z [] clculte the modulus d rgumet of w [] [5] [] [] [4] [] PURE 054, 055, 057 SEARCH

30 c Hece, or otherwise, fid the vlue of z i zw ii w iii rg( zw) iv Give tht z= 4+ i d zw = 5 i z rg w [6] Fid w i modulus-rgumet form. [6] The poits A d B represet z d zw o Argd digrm. Clculte the distce AB, givig our swer s surd. [] Write the comple umer z= i i modulus-rgumet form. [] The comple umer w hs rgumet π Fid the rgumet of z i zw ii [4] w c Fid w give tht zw = 0 [] Sketch the locus of poits tht stisf i z 4 i = ii z+ = z i [4] Fid the Crtesi equtio of the locus of poits drw i prt. [] 4 Show tht the locus of poits stisfig z+ 4i = 4 is circle. [] Sketch the locus of poits stisfig z+ 4i = 4 [] c Shde i the regio tht stisfies z+ 4i 4 [] 5 For the locus of poits stisfig z+ 5 = z i sketch locus, fid the Crtesi equtio. [] 6 Fid the squre roots of the comple umer 9 80i [6] 7 Solve the equtio z = 4 5i [6] 4 8 The solutios of the qurtic equtio = 0 re represeted o Argd digrm the poits P, Q, R d S Oe solutio to the equtio is w= 7+ i Clculte the re of the qudrilterl PQRS [6] 9 Two solutios of cuic equtio re = d = i 4 Stte the third solutio to the equtio. [] Fid possile equtio with these solutios. [] 0 Two solutios of qurtic equtio re z = + i d z 5i Stte the two other solutios of the equtio. [] Fid possile equtio with these solutios. [] = [] Comple umers Assessmet Comple umers Summr d review

31 4 A qurtic equtio + + c + d + e = 0 hs ectl two distict solutios, oe of which is + 5i Fid the vlues of,, c, d d e [4] Solve the simulteous equtios z w = 6 z+ w= Solve these simulteous equtios. z + w = 0 z+ w= 0 4 The equtio = 0 hs repeted qudrtic root. Show tht = i is solutio to the equtio. [] Full fctorise the equtio. 50 [] 4 5 The curve of = is show. Fid ll the solutios to the equtio = 0 0 [4] 6 A qudrtic equtio hs roots d, 0 5π where hs rgumet d modulus O 4 6 Fid i modulus-rgumet form. [] Clculte α i αβ ii iii rg( αβ ) [6] β c Fid the qudrtic equtio with roots d [] 7 The comple umers z d z re give z = + i d z = i where is iteger. Fid z i terms of z Give tht z = 8 z fid the possile vlues of [5] π 8 Give tht π w= i cos si write w i modulus-rgumet form, [] fid the comple umer z such tht rg( wz) = π d w = 6 Re z Write our swer i ect Crtesi form. [] 9 Descrie the locus show i terms of z [] Im 40 0 [6] [4] [] PURE.5.0

32 0 Sketch the locus of poits tht stisf rg( z ) i = π 4 [] Fid the Crtesi equtio of the locus drw i prt. [] Sketch d shde the regio stisfig 0< rg( z+ + ) i π Sketch d shde the regio stisfig z i d z+ i [5] [4] The comple umers 4 i d i represet the poits A d B respectivel. Fid the re of the trigle OAB [5] 4 The locus of the comple umer z is hlf-lie s show. Descrie the locus i terms of z [] 5 The poit P represets comple umer z o Argd digrm such tht z+ = z i Show tht, s z vries, the locus of P is circle d stte the rdius of the circle d the coordites of the cetre. 6 The poit P represets comple umer z o Argd digrm such tht z = The poit Q represets comple umer z o Argd digrm such tht rg( z i) = π 4 Sketch the loci of P d Q o the sme es. [4] Fid the comple umer tht stisfies oth z = d rg( z i) = π 4 [6] 7 Fid the possile comple umers, z, tht stisf oth z+ i = z i d z + i = 4 [4] Sketch the regio tht stisfies oth z+ i > z i d z + i 4 [5] 8 Shde the regio tht stisfies z i [4] Fid the ect re of the shded regio. [] 9 Shde the regio tht stisfies oth z+ i d π rg( z+ i) 0 [4] Fid the ect re of the shded regio. [] O Im Re [6] 4 Comple umers Assessmet Comple umers Summr d review

33 Alger d series Oriettio Wht ou eed to kow KS4 Iequlities. Proof. Mths Ch Argumet d proof. Qudrtic fuctios. Mths Ch Epdig d fctorisig polomils. Polomils hve m uses, d oe use of prticulr iterest is i progrmmig the movemet of rootic rms. These rms re used commol i idustr, for emple, i the mufcturig of crs. Polomils re used to determie the smooth pths tht the rm joits should follow, to esure tht the mechicl compoets do ot wer too quickl. The trjector of joit is determied s the comitio of severl polomil fuctios, which re stitched together t vrious poits. Algeric methods re used to esure tht the pth of the joit is smooth d cotiuous. Aother, perhps surprisig, pplictio of polomils is i i vrious spects of computer grphics, d prticulrl i the desig of fots. Although the lger d umericl methods required for good fot desig re quite comple, the re developed from our sic uderstdig of simple polomils such s qudrtics d cuics. Wht ou will ler To relte the roots of polomil to its coefficiets. To fid ew polomil whose roots re lier trsformtio of the roots of other polomil. To evlute epressios ivolvig roots. To solve iequlities up to qurtic polomils. To use the sums of itegers, squres d cues to sum other series. To uderstd d use the method of differeces. To use proof iductio to prove divisiilit. To use Mcluri series epsios. Wht this leds to Ch Series Summig series usig prtil frctios. More Mcluri series. Creers Cr mufcturig. Grphic desig. 5

34 . Roots of polomils Fluec d skills See Mths Ch. For remider of the fctor theorem. You lred kow how to fid the roots of polomils usig the fctor theorem d log divisio. You re ow goig to ler how to trsform oe polomil ito other polomil with roots tht re relted i some w. Qudrtic equtios Let the qudrtic equtio + + c = 0 hve roots = d = Dividig through gives + + c = 0 Sice = d = re the roots of this qudrtic, ou c write the equtio i the form ( )( ) = 0 Epdig the rckets gives ( + ) + = 0 Comprig the two versios of the qudrtic equtio gives + + c ( + ) + = 0 So, comprig the coefficiets for d the costt gives ( + ) = d = c For qudrtic equtio: The sum of the roots = + = d the product of the roots = = c This lso shows tht ll qudrtics c e writte i the form (sum of the roots) + (product of the roots) = 0 Ke poit Emple The roots of 7 + = 0 re = d =. Without fidig the vlues of d seprtel, Write dow the vlues of + d Hece fid the qudrtic equtios whose roots re i d ii =, = 7 d c =, so + = ( 7) = 7 d = d α β ( + ) = d = c (Cotiued o the et pge) 6 Alger d series Roots of polomils

35 i The ew equtio must e i the form ii ( + ) + = 0 Now + = ( + ) So + = 7 = 5 d = ( )² = = 44 Hece the ew equtio is = 0 The ew equtio must e + α β + = 0 αβ α+ β 7 + α β = = αβ d αβ = Hece the ew equtio is 7 + = 0 or 7 + = 0 (sum of roots) + (product of roots) = 0 Write + d i terms of + d Sustitute usig + = 7 d = Write + α β d αβ i terms of + d Cuic equtios There re similr reltioships etwee the coefficiets of d the roots of the equtio for higher-order equtios (cuics, qurtics, d so o). Let the cuic equtio + + c + d = 0 hve roots =, = d = g Let + + c + d = 0 ( )( )( g ) = 0 ( + )( g ) = 0 + g + g + g g = 0 ( + + g ) + ( + g + g) g = 0 Dividig + + c + d = 0 gives + + c + d = 0 Hece PURE ( + + g ) = ( + g + g) = c d g = d Emple The roots of the equtio ³ 7² + + = 0 re, d g. Fid the vlues of ² + ² + g ² ³ + ³ + g ³ ( + + g )² = + + g + ² + g + g ² = ( ² + ² + g ²) + ( + g + g ) 7² = ( ² + ² + g ²) + () ² + ² + g² = 49 6 = 4 Cosider the epressio ( + + g )² to fid w to write ² + ² + g ² Sustitute + + g = 7 d + g + g = d rerrge. (Cotiued o the et pge) 7

36 is root of the equtio, which mes ³ 7² + + = 0, so rerrgig ³ = 7 ² The sme c e doe with the other roots. ³ = 7 ² g ³ = 7g ² g Now fid the sum of these epressios. ³ + ³ + g ³ = 7² + 7² + 7g ² g = 7(² + ² + g ²) ( + + g ) 6 = 7(4) (7) 6 = 74 Sustitute ² + ² + g ² = 4 d + + g = 7 Emple Qurtic equtios The sme method c lso e used to fid the reltioships etwee the roots,,, g d d, d the coefficiets of the qurtic equtio c + d + e = 0 ( + + g + d ) = ( + g + d + g + d + gd ) = c (g + gd + gd + d) = d gd = e i.e. Σ = i.e. Σ = c i.e. Σd = d Ke poit + + g + d is ofte revited to Σ + g + gd + d is ofte revited to Σ g + gd + gd + d is ofte revited to Σg Fid the cuic equtio whose roots re oe less th the roots of the equtio + 4 = 0 Solve our ew equtio. Let the roots of + 4 = 0 e, d g So, + + g = ( ) = + g + g = g = 4 The roots of the ew equtio re, d g ( ) + ( ) + (g ) = + + g = = 0 ( )( ) + ( )(g ) + (g )( ) = ( + ) + (g g + ) + (g g + ) = ( + g + g) ( + + g ) + = + = 5 Sice Σ = Sice Σ = c Sice Σ g = d Fid Σ, d write it i terms of the origil roots. Sice + + g = ( ) = Fid Σ Sice ( + g + g) = d ( + + g ) = (Cotiued o the et pge) 8 Alger d series Roots of polomils

37 ( )( )(g ) = ( + )(g ) = (g g + g + + g ) = g ( + g + g) + ( + + g ) = 4 ( ) + () = 0 Hece the ew equtio is + Σ() + Σ( ) + Σ(g ) = 0 5 = 0 5 = 0 ( 5)( + 5) = 0 = 5, 0 or 5 Eercise.A Fluec d skills Fid the sum d product of the roots of these equtios c 8 d e + 8 f These qudrtic equtios hve roots d 4 + = = 0 c + + = 0 d 7 8 = 0 e = 0 f = 0 Without fidig d, fid the equtios whose roots re i d ii d α β iii + d + iv d α v β d β α Write the sum, the sum of the products i pirs d the product of the roots of these equtios = = 0 c + 6 = 0 Fid Σg d = 0 Sice ( + g + g) = d ( + + g ) = From ove result, sustitute Σ = 0, Σα = 5 d Σg = 0 e = 0 f = 0 4 These equtios hve roots, d g = = 0 c 7 8 = 0 d = 0 Without fidig, d g, fid the equtios whose roots re i, d g ii, d α β γ iii +, + d g + 5 These equtios hve roots,, g d d = = 0 Fid the vlues of i + + g + d ii α β γ δ PURE 9

38 6 B sustitutig for, solve the equtio = 0 7 Show tht the equtio = 0 hs ol two rel roots, d fid them. 8 Prove tht the equtio = 0 hs o rel roots. 9 Solve the equtio 4 + = 0, give tht oe root is hlf oe of the other roots. Resoig d prolem-solvig Strteg Emple 4 Sometimes it is useful to trsform equtio ito other whose roots re relted i simple w to the roots of the origil equtio. If the roots re trsformed i lier w, so tht = m + c, Ke poit c the ou trsform the equtio sustitutig = m If the ew roots re reciprocls, so tht =, the ou trsform the equtio sustitutig = To solve questio ivolvig the trsformtio of oe polomil ito other Rewrite the trsformtio = m + c s = c m Sustitute c for i the origil polomil d simplif to produce the trsformed equtio. m Fid the equtio whose roots re oe less th, doule the roots of = 0 The ew equtio hs roots oe less th the first, so = = + ( + ) 4( + ) ( + ) + 0 = = 0 Let e the ew root. Write i terms of d the rerrge to give i terms of Sustitute for i the origil equtio. The ew root = = = = 0 Multipl out d simplif. 40 Alger d series Roots of polomils

39 Emple 5 The roots of the equtio + = 0 re, d g Write the vlue of g i Use the sustitutio = to fid ew cuic equtio i with iteger coefficiets. ii B cosiderig the product of the roots of this ew equtio, fid the vlue of g ( + g)( + g)( + ) Sice Σ g = d PURE g = i = so = + ( + ) ( + ) ( + ) + = = 0 ii Roots for the cuic i re, d g ( )( )(g ) = 0 ( + g )( + g )(g + g ) = 0 ( + g )( + g )g ( + ) = 0 g ( + g )( + g)( + ) = 0 Eercise.B Resoig d prolem-solvig Fid the polomil whose roots re reciprocls of, triple the roots of = 0 A cuic equtio is + = 0 Use the sustitutio = to fid cuic equtio i with iteger coefficiets. Solve this cuic equtio i c Use the sustitutio = to fid the roots of the origil equtio i Trsform = 0 usig the sustitutio = 4 Trsform 4 + 6³ = 0 usig the sustitutio = 5 Solve the equtio = 0 usig the sustitutio = 4 6 Solve the equtio = 0 mkig the sustitutio = + 7 Solve + = 0 mkig the sustitutio = + Write i terms of d rerrge. Sustitute ito origil equtio. Epd rckets d simplif. Product of roots = d = 0 Sustitute g = from prt 8 Solve = 0 icresig the roots oe. 9 Fid the vlue of if the equtios = 0 d + 4 = 0 hve commo root tht is ot equl to 0 0, d g re the roots of = 4 + B sustitutig =, = d = g ito the equtio, prove tht + + g = 9, d g re the roots of 6 = 0. B cosiderig the sum of the roots, fid suitle sustitutio to trsform this equtio ito polomil with roots +, + g d g + The equtio = 0 hs roots, d g Write dow the vlue of + + g Use the sustitutio = 4 to fid cuic equtio i c B cosiderig the product of the roots for this equtio, fid the vlue of ( + g)( + g)( + ) Full A-level 4

40 . Iequlities Fluec d skills You lred kow some of the sic fcts out iequlities. The rules for solvig iequlities re ver similr to those for solvig equtios, ut there re two differeces. Emple Emple Aswers to iequlities re rge of vlues Ke poit rther th idividul vlues. Whe ou multipl or divide egtive umer ou reverse the iequlit sig. Solve the iequlit > > 4 > < + 5 You eed to e prticulrl creful to void icorrect ssumptios whe workig with iequlities tht ivolve egtive umers. If <, is <? Epli our swer. Not lws. If = d = the < ut if = d = the ( ) > eve though <. Rememer to reverse the iequlit sig whe multiplig egtive umer. The solutio is rge of vlues ot sigle vlue. Sice 4 < 9 Sice 9 > 4 If epressio c e positive or egtive, ou must lws cosider seprte cses. 4 Alger d series Iequlities

41 Emple 8 5 Solve the iequlit < Cse : > so is positive 8 5 < ( ) 8 5 < 6 7 < 4 <. Sice cse ol cosiders >, this is lred tighter restrictio th > 4 So, > Cse : < so is egtive 8 5 > ( ) 8 5 > 6 7 > 4 > So, < Hece the solutio to < is < 4 or > c e positive or egtive so cosider two seprte cses. is positive so simpl multipl o oth sides. Solve the iequlit s ou would equtio. Reverse the iequlit s ( ) is egtive for < Comie the results from the two cses. 8 5 If the iequlit i Emple hd ee < 0, ou c multipl oth sides ( ), d use the sme rgumet s i Emple. Cse : > Solve 8 5 < 0 to get < 7.5 Cse : < Solve 8 5 > 0 to get > 7.5, which is t possile. 8 5 So, < 0 whe < < Altertivel, to work out whe is egtive, ou could cosider the sigs of the umertor d deomitor for differet vlues of : PURE < < < 7.5 > ve ve +ve ve +ve +ve 8 5 +ve ve +ve 8 5 So, < 0 whe < < 7.5 If ou eed to solve iequlit of the form f() 0 or f() 0, rememer to iclude vlues where f() = 0 (where the grph itersects the -is). 4

42 8 5 You could lso sketch grph of the fuctio = d fid the vlues of where < 0 d the grph lies elow the -is. Emple 4 Emple 5 Use grphicl clcultor or computer to sketch grph of f() = ( + )( + )( 4) Use our sketch to solve i f() = 0 ii f() > 0 iii f() < i ( + )( + )( 4) = 0 whe = or = or = 4 ii ( + )( + )( 4) > 0 whe < < or > 4 iii ( + )( + )( 4) < 0 whe < or < < For the fuctio f() =, use lgeric method to fid the vlues of where i f() = 0 ii f() > 0 iii f() < 0 i Usig grph plottig softwre, sketch grph of = f() to cofirm our swers. ii = 0 whe = 0 so whe = > 0 whe umertor d deomitor hve the sme sig. If >, umertor is positive For deomitor to lso e positive, > So > If <, umertor is egtive For deomitor to lso e egtive < So < > 0 whe > or < Use where the grph itersects, is ove d is elow the -is to solve the iequlities. f() = 0 whe umertor = 0 must stisf > d > must stisf < d < Comie the results. 44 Alger d series Iequlities (Cotiued o the et pge)

43 5 + 5 iii For < 0, umertor d deomitor must hve opposite sigs. 6 If >, umertor is positive For deomitor to e egtive, < So < < If <, umertor is egtive For deomitor to e positive > So o solutios < 0 whe < < There is o where > d < Comie the results. From the grph ou c see tht: the curve itersects the -is so f() = 0 t = the curve is ove the -is d so f() > 0 for < or > the curve is elow the is d so f() < 0 for < < I Emple 5, for < the umertor is egtive d for > it is positive so = is kow s criticl vlue. Similrl, = is the criticl vlue for the deomitor. Eercise.A Fluec d skills B trig vlues for,, d to show tht ech of these sttemets re sometimes, ut ot lws, true. If < the < c If < d < the < If < d < the < Solve these iequlities < 5 > 4 c + d ( ) 0 e 4(6 + ) < 5( ) f 5 ( + ) > (4 ) g ( 5) 0 h ( + )( ) 0 Solve these iequlities lgericll. + > 7 < 0 c ( + 4) 5 d ( + 7)( ) < 8 e 0 f 5 > 4 PURE 45

44 Strteg 4 Solve these iequlities lgericll < c > ( + 7) d e < f g + h + 5 > 5 Solve these iequlities sketchig grph i ech cse. ( + 4)( 5) > 0 ( + 5)( + 4)( 5) < 0 c ( + )( 4)(5 9) 0 d ( + 4)( ) 0 e ( 5)( + ) > 0 f ( 5)( + 7) < 0 g ( )( ) 0 6 Solve these iequlities. ( + )( )( 9) > 0 ( + )( + )( )( 7) < 0 c ( + )(4 5)( 4)( ) 0 Resoig d prolem-solvig To solve iequlit lgericll d ( ) ( ) 0 e ( 7) ( + ) > 0 f ( 49)( 5) < 0 g ( 0)( ) 0 7 For wht vlues of re the followig fuctios positive? + ( + )( + ) c d 5 ( + )( 4) ( )(+ 4) e f ( + 5) ( 7) (4 )(5+ ) ( )(+ 4) g h ( + ) 8 For wht vlues of re the followig fuctios egtive? ( + )( 5) c d + ( + 7)( + 4) (4 )( + 5) e f ( + 6) ( + 6) (9 )(6 + 5 ) ( + 5)( ) g h ( ) + Fctorise d simplif usig the sme rules s for equtios, ut rememer to chge the directio of the iequlit sig if ou multipl or divide egtive vlue. Use grphs or criticl vlues to ivestigte where fuctios re positive or egtive. Solve the iequlit cosiderig ll possiilities d cses. Emple 6 Solve the iequlit lgericll. Usig grphicl clcultor or grph sketchig softwre o our computer, sketch grph to cofirm our swer. f() is qudrtic i f() ( )( 6) ( )( + )( 6)( + 6) Fctorise the polomil. 46 Alger d series Iequlities (Cotiued o the et pge)

45 Emple 7 Solve ( )( + )( 6)( + 6) 0 ( + 6)( + )( )( 6) 0 The criticl vlues re = 6,, d 6 Whe < 6, f() is positive. Whe 6 < <, f() is egtive. Whe < <, f() is positive. Whe < < 6, f() is egtive. Whe > 6, f() is positive. Hece whe 6 or or The fuctio f() = itersects the stright lie = k. Form qudrtic equtio i d k. Hece fid the vlues of k for which f() hs rel roots. c Use grphicl clcultor or grph sketchig softwre o our computer to cofirm our swer. It might e helpful to reorder the rckets ito order of size of the root. At these vlues oe fctor = 0 This mes f() = 0 d the grph crosses the is. All four fctors re egtive. Three egtive d oe positive fctor. Two egtive d two positive fctors. Oe egtive d three positive fctors. All four fctors re positive. Iclude the criticl poits s the iequlit is 0 d so icludes whe the curve itersects or is ove the -is. You c see tht this is whe 6 or or 6 Iequlities re sometimes ecessr to fid whe fuctios re rel. PURE 4+ 4 k = k = (k + 4) + 4 = 0 Multipl oth sides d simplif to form qudrtic equtio i. (Cotiued o the et pge) 47

46 Emple 8 The fuctio hs rel roots if 4c 0 (k + 4) 4 4 k + 8k k + 8k 0 k(k + 8) k = hs rel roots if k 0 or k 8 c Fid the vlues of for which 0 < 4 + Cosider 0 Criticl vlues re = 4 d = 4 + For < 4, is egtive. 4 + For 4 < <, is positive. 4 + For >, is egtive. 4 + So 0 whe 4 < 4 + Now cosider < Whe < 4 + < 5 0 You c solve doule iequlities usig the sme methods. The discrimit must e 0 Sustitute vlues d solve the iequlit. k d (k + 8) must either oth e positive or oth e egtive The grph of = shows tht does ot eist if 8 0 So c ol hve rel roots if 0 or 8 Cosider the sigs of the umertor d deomitor for ech set of vlues. Do ot iclude s 4 + is ot defied t = ( ) is positive. < Whe > 4 + > >, ut we hve the more restrictive coditio > 4 + So < whe < or > 4 + Full solutio: 0 < whe 4 < ( ) is egtive so reverse the iequlit. Fid the vlues which stisf oth iequlities. 48 Alger d series Iequlities

47 Eercise.B Resoig d prolem solvig The digrm shows the grph of cuic fuctio, f() = A + B + C + D Write dow the coordites of ll the itercepts with the coordite es. Hece write dow d epd the equtio of the fuctio. c Write dow the vlues of A, B, C d D. d Estimte d write dow the coordites of the turig poits. e Write dow the rges of vlues of where f() < 0 The digrm shows the grph of qurtic fuctio, f() = A 4 + B + C + D + E Write dow the coordites of ll the itercepts with the coordite es. Epli wh there re ol three -itercepts eve though this qurtic hs four rel roots. Hece write dow d epd the equtio of the fuctio. c Write dow the vlues of A, B, C, D d E. d Estimte d write dow the coordites of the turig poits. e Write dow the rges of vlues of where f() > 0 Solve the iequlit Use grphicl clcultor to cofirm our swer with sketch. 4 Solve the iequlit Use grphicl clcultor to cofirm our swer with sketch. 5 Solve these iequlities. + < > 4 5 c 4 5 d I eperimet, three studets recorded the temperture of their liquid s ( + ) C, ( ) C d ( 7) C. For wht vlues of is the product of these tempertures positive? 7 f() Fid the rge of vlues of for which f() 0 8 Fid the rge of vlues for which 8 0 < 9 Fid the rge of vlues of for which + the fuctio = hs rel roots. Use grphicl clcultor or grph sketchig softwre o our computer to cofirm our results. 0 Fid the rge of vlues for which the 5 fuctio = is positive + Use grphicl clcultor or grph sketchig softwre to cofirm our swer. For ll vlues of q, si q Hece fid the vlues of such tht si q = hs solutio i q + Cofirm our swer usig sketch. If is positive umer, prove tht the sum of the umer d its reciprocl cot e less th. 49 Full A-level PURE

48 . Summig series d the method of differece Fluec d skills If ou wt to write the sum of series of terms u, u, u r, u, u, ou c use sigm ottio, Σ r= r = u r or u r mes fid the sum of ll the terms from u to u. u r is the geerl term d, if ll terms re defied lgericll, u r is fuctio of the vrile. If S = u + u + + u r + + u + u, the S = u If u r = r the S = r + = r r. Ke poit You eed to kow the formule for the sums of itegers, squres d cues. Fidig the sum of itegers from to r = ( ) + ( ) + Write the sme sequece i reverse: r = + ( ) + ( ) Add the two together: r = ( + ) + ( + ) + ( + ) + + ( + ) + ( + ) + ( + ) Ever term is equl to +. Tr it with some umers. + ( ) There re lots of ( + ) so r = ( + ) d so r = Ke poit + ( ) = = r Σ mes fid the sum of ll these terms. S mes the sum of these terms. Emple Fid the vlue of = ( ) 0 = r 0(0+ ) 0 = = = 40 Rewrite s sum of itegers. Sustitute = 0 ito the formul. 50 Alger d series Summig series d the method of differece

49 Sum of squres You c derive the formul for the sum of squres usig the method of differeces. You will see how to do this lter i this sectio. r ( + )( + ) 6 Ke poit PURE Sum of cues Compre sums of r d r for differet vlues of. Emple r r + = + 8 = = = = = = = 5 For ech vlue of, Σr = (Σr) ( + ) Hece Σ = (Σ) = = ( + ) 4 Ke poit ( + ) r = 4 Evlute rr ( + 4) 5 rr ( + 4) = 5(9) + 6(0) + + ( + 4) 5 = rr ( + 4) r( r+ 4) 4 = r + 4 r ((5) + (6) + (7) + 4(8)) ( ) = ( + )( + ) = ( + ) 6 ( ) = ( + ) 6 70 ( + )( + ) 40 = 6 ( 4)( ) = 6 You will e epected to rememer d e le to quote without proof, the formule for Σr, Σr d Σr. Write out some of the sequece to help ou uderstd it. This is the sme s rr ( + 4) without the first four terms. Use the stdrd formule. Simplif. 5

50 Emple Fid formul for the sum to terms of the series Fid the sum of the first 0 terms d check our formul. The geerl term is r(r + )(r + ) rr ( + )( r+ ) ( r + 4r + ) r r + 4 r + r ( + ) + 4 ( + )( + ) ( ) ( + ) ( + ) + 8( + ) + 8 ( + ) ( + )( + )(+ ) 0 0 r( r+ )( r + ) ( ) = 470 [ ] ( 4) + ( 5) + + (0 ) = = 470 Rewrite i terms of sums ou kow. Σ4 4Σ d Σ Σ Sustitute formule for Σ, Σ d Σ, d simplif. Fid the sum usig the formul. Clculte the first 0 terms d dd to check our formul. If series hs ltertive positive d egtive terms, Σf() c e writte with ( ) multiple i it. For emple, for terms, is epressed s ( ) r. The method of differeces There is o geerl method for summig series. Sometimes it is ot possile to epress Σ s lgeric epressio For emple However, if the geerl term of fuctio c e epressed s f(r + ) f(r), ou c fid the sum of the series usig the method of differeces. 5 Alger d series Summig series d the method of differece

51 Emple 4 Fid the sum to terms of the series r r + = rr ( + ) PURE Write the differeces for successive terms verticll d elimite terms wherever possile. Emple 5 + ( + ) Thus = rr ( + ) ( + ) = ( + ) Collect up the remiig terms d simplif. 8 The epressio (r )(r+ )(r + 4) c e writte s (r ) (r + ) + (r + 4) 8 Fid the sum to terms of the series (r )(r+ )(r + 4) 8 (r )(r + )(r + 4) (r ) (r+ ) + (r + 4) Hece (r )(r + )(r + 4) Write the differeces for successive terms verticll d elimite terms wherever possile. Collect up the remiig terms d simplif. 5

52 Eercise.A Fluec d skills Write out these series. 5 r c ( r r) 6 r 4 d m + e ( ) r r f rr ( + ) Use Σ ottio to write these sums c d e for terms f for terms. How m terms re there i the series (r r + )? Write d simplif the ( + )th term. 4 Fid the sums of these series ( ) c ( ) d ( ) e ()( + ) f ( ) 5 Fid formul for the sum to terms of the series ( ) Fid the sum of the first 0 terms d check our formul. 6 Fid the rth term d the sum to terms of these series. c d e 7 Evlute (r + ) 5 c + ( r + r) ( r r) 8 Fid the sum of ech series. r r r+ c r r+ d r r+ e r r+ f + r+ r+ r Alger d series Summig series d the method of differece

53 Strteg Resoig d prolem-solvig To solve prolems ivolvig sums of series: Rerrge the epressio ito multiples of itegers, squres d cues, or ito the form f( + ) f() Sustitute stdrd formule or terms s required. Simplif the resultig epressio for the sum. PURE Emple 6 Simplif the epressio (r + ) (r ) ( + )( + ) Hece, use the method of differeces to show tht r 6 (r + ) (r ) (8r + r + 6r +) (8r r + 6r ) 4r + 4r + (r + ) (r ) r 4 [(r + ) (r ) ] r 4 ( ) + 4 (5 ) + 4 (7 5 ) + 4 (9 7 ) + 4 [( ) ( ) ] = 4 [( + ) ( ) ] Hece r 4 [( + ) + ( )] 4 [ ] 4 [ ] 6 [ + + ] Write the differeces for successive series of terms verticll d elimite terms wherever possile. Collect up the remiig terms d simplif. ( + )( + ) 6 55

54 Emple 7 Show tht rr ( + )( r + ) c e writte s r r + + r + Hece or otherwise fid the sum to terms of the series ( + ) ( ) c Use our result to deduce ( ) + ( + ) r r + + ( r+ )( r+ ) ( r r+ ) + r( r+ ) r + r( r+ )( r + ) r r r r r r r( r+ )( r + ) = r ( r + )( r + ) S = Hece S = = c c e writte s rr ( + ( ) r + ) Therefore, whe = = r( r + )( r + ) Hece S = = = Form sigle frctio d simplif. Write the differeces for successive series of terms verticll d elimite terms wherever possile. Collect up the remiig terms d simplif. You could lso fid S sig: As gets ver lrge, ( + ) ( + )( + ) = ( + ) Hece, lim = ( + )( + ) 56 Alger d series Summig series d the method of differece

55 Eercise.B Fluec d skills The sum to terms of series is + 4 B cosiderig S, S d S fid the first three terms of the series. B cosiderig S d S, fid the th term. Admirl Nelso s ship hs trigulr pile of colls stcked o deck. The top co ll is supported three colls i the ler udereth it. These three re supported si colls i the ler udereth them, d so o. There re lers of co lls. c How m co lls re there i the ottom ler? How m colls re there ltogether? Clculte the weight of the pile of si pouder lls cotied i stck of 0 lers. A si pouder co ll weighs pproimtel.7 kg. The Spish Armd s co lls re i squre prmids. The se is squre with colls i ech side, the ler ove hs ( ) colls i ech side, d so o. There re lers i totl. How m co lls re there i this prmid? 4 Write dow the first four terms d the th term of the series whose geerl term is r ( r+ ) Fid r ( r+ ) d hece fid the sum to ifiit. 5 Write dow the geerl term i the series S = (m) + (m ) + (m ) + + (m)() m d fid S 6 Show tht (r ) (r + ) 8r (( r) ) 8r Hece fid (( r) ) d deduce r the vlue of (( r) ) 7 Show tht r(r + ) r(r ) r Use this iformtio to show + ( ) tht r 8 Simplif the epressio (r + ) (r ) ( + )( + ) Hece show tht r 6 9 Use stdrd results to fid formul for the sum of the series (r ) Cofirm our result fidig the formul i differet w. 4( ) 0 Show tht c e writte ( + )( + 4) s ( ) Hece fid ( + )( + 4) Show tht 4 r + r + r r + r + 5r + r + d rr ( + ) 4 hece fid + r 5r + r + rr ( + ) Full A-level PURE 57

56 .4 Proof iductio See Mths Ch. For remider of methods of proof. Emple Fluec d skills There re umer of differet methods of directl provig mthemticl sttemet. Proof iductio is method tht is geerll used to prove mthemticl sttemet for ll turl umers (positive itegers). It is powerful method tht c e used i m differet cotets. The priciple ehid proof iductio is tht if ou prove sttemet is true whe =, d if ou c prove it is true for = k + ssumig it is true for = k, the ou c deduce tht the sttemet must e true for ll N. The three ke steps to proof iductio re: Ke poit Prove the sttemet is true for = Assume the sttemet is true for = k d use this to prove the sttemet is true for = k + Write coclusio. You must lws epli the steps full d ed our proof with coclusio. r Prove iductio tht 4 = (4 ) for ll N. r= r 0 Whe =, 4 = 4 = r= d = (4 ) (4 ) = () = Other methods of direct proof re proof deductio d proof ehustio. N mes the set of turl umers: these re the positive itegers,,, Clculte the LHS of the sttemet whe = Clculte the RHS of the sttemet whe = Sice LHS = RHS, the sttemet is true for = So the sttemet is true whe = Assume the sttemet is true for = k d cosider = k + k+ r= k r r 4 = r= k+ k = (4 ) + 4 k Write the sum to the (k + )th term s sum to the k th term plus the (k + )th term. Sice ou re ssumig the sttemet is true for = k, k ou c replce 4 with r = (4 k ) (Cotiued o the et pge) 58 Alger d series Proof iductio

57 k k k = (4 + (4 )) = (4(4 ) ) k+ = (4 ) So the sttemet is true whe = k + The sttemet is true for = d ssumig it is true for = k it is show to e true for = k + Therefore, mthemticl iductio, it is true for ll N. Collect like terms d the use ide lws. This is (4 ) with replced k + You must lws write coclusio. PURE Emple Prove iductio tht r = ( + ( ) + ) for ll N. 6 r= Whe =, r = = r= d + + = ( )( ) 6 ( )( ) = 6 ()() = So the sttemet is true whe = Assume sttemet is true for = k d sustitute = k + ito the formul: k+ k r r = + ( k + ) r= r= = k( k+ )(k+ ) + ( k+ ) 6 = ( k+ )[ k(k+ ) + 6( k+ )] 6 = ( k+ )(k + k+ 6k+ 6) 6 = ( k+ )(k + 7k + 6) 6 = ( k+ )( k+ )(k + ) 6 = ( k+ )(( k+ ) + )(( k + ) + ) 6 So sttemet is true whe = k + The sttemet is true for = d ssumig it is true for = k it is show to e true for = k + Therefore, mthemticl iductio, it is true for ll N. Clculte the LHS of the sttemet whe = Clculte the RHS of the sttemet whe = Sice LHS = RHS, the sttemet is true for = Write the sum to the (k + )th term s sum to the kth term plus the (k + )th term. Sice ou re ssumig the sttemet is true for = k, k ou c replce r with r = k( k + ( ) k + ) 6 Look for commo fctors void multiplig out rckets uless ecessr. ( ) This is + ( + ) with 6 replced k + You must lws write coclusio. 59

58 Eercise.4A Fluec d skills See Ch4. To see how to ppl iductio to mtri proofs. Use proof iductio to prove these sttemets for ll N. = r= r= ( + ) r= c (r+ ) = ( + 4) r= d rr ( + ) = ( + )( + ) r= e ( r ) = ( )( ) r= 6 f ( r+ )( r ) = (+ 5)( ) 6 r= Use iductio to prove these sums for ll turl umers, ( + ) = 6 ( ) ( ) = 4 (5 ) Prove iductio tht r = ( + ) r= 4 for ll N. 4 Prove iductio tht r= ( + ) for ll r= N. Resoig d prolem-solvig 5 Prove iductio tht r = (+ )(4+ ) for ll N. r = 6 Use iductio to prove these sttemets for ll N. r r = ( ) = ( ) r= r= r 4 r c 4 = (4 ) d = r= r= r e = ( ) r= r f r = = 7 Prove iductio tht = for r= rr ( + ) + ll N. 8 Prove iductio tht = for r= rr ( ) ll N. 9 Prove iductio tht (+ 5) = for ll N. r + r 4( + )( + ) As well s provig sums of series, ou c use proof iductio i other cotets such s to prove lgeric epressio is divisile give iteger. r= Strteg To prove epressio is divisile prticulr iteger Sustitute = ito the epressio d show the result is divisile the iteger ou re usig. Assume the epressio is divisile the iteger for = k d sustitute i = k + Seprte the prt ou kow to e divisile the iteger give. 4 Write the coclusio. 60 Alger d series Proof iductio

59 Emple Prove tht + is divisile for ll N. Whe =, + = + = = so divisile So sttemet is true whe = Assume sttemet is true for = k d cosider = k + : ( k+ ) + ( k+ ) = k + k + k+ + k+ = k + k + 5k + = ( k + k) + (k + k+ ) k + k is divisile, so write it s A for some iteger A = A+ ( k + k + ) = ( A+ k + k + ) which is divisile The sttemet is true for = d ssumig it is true for = k it is show to e true for = k + Therefore, mthemticl iductio, it is true for ll N. Eercise.4B Resoig d prolem-solvig Use proof iductio to prove these sttemets for ll N. + is divisile 5 is divisile c 8 + 4is divisile d + 4 is divisile Prove iductio tht is divisile for ll N. Prove iductio tht is divisile for ll, N. 4 Prove iductio tht is divisile 6 for ll N. 5 Use proof iductio to prove these sttemets for ll N is divisile 5 c is divisile 8 Clculte vlue of the epressio whe = Sice ou c write it s ( iteger). + is divisile 7 Sustitute k + d epd rckets. Split up so the first prt is the epressio ou re ssumig is divisile Rememer the coclusio. 6 Prove iductio tht is multiple of 4 for ll N. 7 Prove iductio tht is multiple of 4 for ll N. 8 Prove iductio tht is multiple of for ll N. 9 Use proof iductio to prove these sttemets for ll N. r = (+ )(7+ ) r= + 6 r = (5+ )( + ) r= 4 c 8 5 is divisile for ll N 4 PURE 6

60 .5 Mcluri Series Fluec d skills It is sometimes useful to e le to epress fuctio s series of terms. The Mcluri series for fuctio f() is sed o three ssumptios: f() c e epded s coverget ifiite series of terms ech of the terms i f() c e differetited ech of the differetited terms hs fiite vlue whe = 0 With these ssumptios, ou c write f() r r + where 0,,, re costts. Differetitig repetedl d sustitutig = 0, gives f() r r + So f(0) 0 f () r r r +... So f (0) f () r(r ) r r + So f (0)! f () r(r )(r ) r r + So f (0)! Cotiuig this process gives the defiitio of the Mcluri series. f() f(0) + f (0) + f (0) + f (0) + + r Ke poit f r (0) +!! r! This is the defiitio of Mcluri series. A Mcluri series c e foud for fuctio tht meets the coditios give ove. You should e le to recogise d use the Mcluri series for the followig fuctios d kow the vlues of for which the re vlid. ( ) ( ) ( + r) ( + ) = r + for < <, R! r! e = r + for ll! r! r l ( + ) = + +( ) r + + for < r 5 r+ si = + +( ) r + for ll! 5! (r + )! 4 r cos = + +( ) r + for ll! 4! ( r)! A coverget series is oe where ifiite umer of terms hs fiite sum. The proof of these ssumptios is outside the scope of this chpter. You do ot eed to e le to prove the coditios for vlidit of these Mcluri series. Ke poit 6 Alger d series Mcluri Series

61 Emple Use the Mcluri epsio of ( + ) to fid the first five terms of the series for f() = ( + ) = + + ( ) + + ( ) ( + r ) r +! r! ( ) ( )( ) ( )( )( 4) ( ) 6 ( )( )( 4)( 5) ( ) This epsio will e give i our formule ooklet. For the epsio of, = d is ( ) replced. PURE Emple If is ot positive iteger, the epsio of ( + ) is vlid for < < This mes tht i Emple, the epsio for ( ) is vlid whe < < This c e rerrged to < < Write dow the rge of vlues of tht mke these series vlid. l ( ) l ( + ) c + + l ( ): < > So < l ( + ): < However 0, so 0 So c is vlid whe < < So is vlid whe < < + l ( + ) is vlid for < < Replce i the iequlit. Rerrge to give the rge for Replce Sice = ± ( + ) is vlid for < < Replce Eercise.5A Fluec d skills Use the Mcluri epsio to fid the first five terms of these fuctios. e e c e d e e e Use the Mcluri epsio to i fid the first four terms of these fuctios, ii fid the rge of vlues of for which ech epsio is vlid. l ( ) l ( + ) c l ( ) d l + e l Use the Mcluri epsio to fid the first four terms of these fuctios. si si d si ( ) e si c si ( ) 6

62 4 Use the Mcluri epsio to fid the first four terms of these fuctios. cos 4 cos d cos ( ) e cos 5 For ech fuctio i c cos ( ) fid the first four terms of the Mcluri epsio, ii fid the rge of vlues of for which the epsio is vlid. c e ( + ) ( ) ( + ) d (+ ) 5 ( ) f g + 6 Use the Mcluri series to write dow the first four terms i these epsios. si ( ) si ( ) c cos ( ) d cos ( ) Strteg Emple Resoig d prolem-solvig To fid series epsios for compoud fuctios: Use stdrd series to epd ech prt of the fuctio. Comie the series to give the epsio of the compoud fuctio. Cosider the vlidit of the vlues of. Epd si ( + ) s fr s the term i 5 d give the rge of for which the series is vlid. ( + ) ( + ) ( + ) = + + ( ) + ( )( ) +!! ( + ) = + + 4! + 5 8! ! ! + = si = 5 +! 5! si (si ) ( + ) ( + ) 5 = = Replce i the epsio for ( + ) Use the stdrd epsio for si Multipl the epsios to fid the epsio for the compoud fuctio. Ol multipl out the terms tht give ou powers up to 5 (Cotiued o the et pge) 64 Alger d series Mcluri Series

63 Hece si ( + ) The epsio for ( + ) is vlid for < < The epsio for si is vlid for ll vlues of si So, the epsio for is vlid for < < ( + ) Choose the rge tht is vlid for oth series. PURE Eercise.5B Resoig d prolem solvig Use the Mcluri series to fid the first three o-zero terms of these fuctios. e si e si c si + cos d cos Fid the first three o-zero terms i the Mcluri epsio of cos. Give the vlues of for which the epsio is vlid. Fid the first three o-zero terms i these Mcluri epsios. Fid the rge of vlues of for which ech epsio is vlid. (+ ) l + c cos d + 4 Fid the first four o-zero terms i the epsio of i + ii l + Fid the rge of vlues of for which the epsio of l + is vlid. 5 Mke use of kow series epsios to oti these epsios. i (e e ) (up to the term i 5 ) ii l ( + cos ) (up to the term i 4 ) Fid the rge of vlues of for which l ( + cos ) is vlid. 6 Use the Mcluri series to pproimte e 0.6 (to 4 sf) i B writig.5 s + 6 d usig the epsio of l ( + ), fid the vlue of l (.5) correct to 5 sf. ii Epli wh just sustitutig =.5 does ot give vlid swer. 7 At poit, the grdiet of epoetil fuctio e k is ke k. This mes tht the derivtive of e k is ke k. c Use this to derive the Mcluri series for e from first priciples usig the defiitio of the Mcluri series. Check our swer sustitutig ito the stdrd formul for the epsio of e Sustitute = i our first five terms d 4 hece oti pproimtio for e. 8 The differetil of si is cos d the differetil of cos is si. Use these fcts d the defiitio of the Mcluri series to fid the first four terms of the fuctio f() = si. Sustitute = π ito our epsio d fid the percetge error etwee tht d the ect vlue of si π. Full A-level 65

64 Summr d review Chpter summr For the qudrtic equtio + + c = 0, the sum of the roots = d the product of the roots = c If the roots of the cuic equtio + + c + d = 0 re, β d g, the ( + + g) = ( + g + g) = c d g = d If the roots of the qurtic equtio c + d + e = 0 re,, g d d, the ( + + g + d) = ( + g + gd + d) = c (g + gd + gd + d) = d d gd = e c If roots re trsformed lier mout so = m + c, the sustitute = m To solve iequlities, follow the sme rules s i solvig equtios, ut rememer tht swers to iequlities come s rge of vlues whe ou multipl or divide egtive umer ou reverse the iequlit sig. u mes fid the sum of ll the terms u, u,, u, u, u r r + ( ) r = r ( + )( + ) 6 ( + ) r 4 If series hs lterte positive d egtive terms, iclude ( ) r + i Σf(r). r + For emple, c e writte s ( ) If ou c fid fuctio f() such tht the geerl term, u r c e epressed s f(r + ) f(r), ou c fid the sum of the series usig the method of differeces. To prove sttemet iductio ou must first prove the result for = the ssume the sttemet is true for = k d use this fct to prove the sttemet is true for = k +. Alws rememer to write coclusio. You should recogise the followig Mcluri series d their rge of vlidit: ( ) ( ) ( + r) ( + ) = r + for ll if is positive iteger,! r! otherwise < <, R e = for ll! r! r l ( + ) = + +( ) r + r + for < 5 r+ si = + +( ) r + for ll! 5! ( r + )! 4 r cos = + + for ll! 4! r! 66 Alger d series Summr d review r +( ) r ( )

65 Check d review You should ow e le to... Tr Questios Relte the roots of polomil to its coefficiets. PURE Fid ew polomil whose roots re lier trsformtio of the roots of other polomil. 5 Evlute epressios ivolvig roots. 7 Solve iequlities up to qurtic polomils. 8, 9 Use the formule for the sums of itegers, squres d cues to sum other series. 0 Uderstd d use the method of differeces. 6 Use proof iductio to prove divisiilit 7 9 Use Mcluri series epsios 0 Write dow the sum, the sum of the products i pirs d the product of the roots of these equtios = = 0 Fid the polomil whose roots re doule the roots of = 0 Solve the equtio = 0 mkig the sustitutio = 4 Fid the roots of the equtio which re 5 less th the roots of the equtio + 0 = 0 5 The roots of the equtio + = 0 re, d g Write dow the vlue of g i Use the sustitutio = to fid cuic equtio i with iteger coefficiets. ii B cosiderig the product of the roots of this ew equtio, fid the vlue of g ( + g )( + g )( + ) 6 The equtio ³ 6² + = 0 hs roots, d g Fid the vlues of + + α β γ ( )( )( g) c ² + ² + g ² 7 Solve these iequlities lgericll < 0 c + > 4 d e < f Solve these iequlities. ( + )( ) < 0 ( + 6)( + )( 8) > 0 c ( + 8)( )( 5)( 8) 0 d ( )( 0 + ) 0 9 Solve these iequlities

66 0 Write out these sequeces. 0 r + 4 r c ( r + 5r) d m 0 + e ( ) r r f ( r r ) Fid the sums of these series to terms. r r 4r c 5r + r Fid + ( r r + ) Verif our solutio usig = 6 Fid the sums of these series to terms. r + r + r + r + 4 Use the method of differeces to fid the sum to terms of the series Hece write dow the sum to ifiit. 5 Show tht t (t + ) t (t ) 4t Hece show tht t = ( + ) The epressio c e writte s ( + ). Use the method of differeces ( + ) o ( + ) to show tht ( + ) ( + ) is ( + ) 7 Prove iductio tht + + is divisil 4 for ll N. 8 Prove iductio tht + is divisile for ll N. 9 Prove iductio tht 6 is divisile 5 for ll N. 0 Fid the vlidit of ech of these fuctios. ( + ) l Fid the first three o-zero terms i these Mcluri epsios. l( + ) ( + ) si Use the Mcluri epsio of e to fid the first four o-zero terms of e d hece fid pproimte vlue of e to 4 dp. 68 Alger d series Summr d review

67 Eplortio Goig eod the ems Histor Je-Roert Argd ws Swiss mthemtici who, erl i the 9th cetur, moved to Pris where he mged ookshop. He pulished ess o the represettio of imgir qutities i 8 d, lthough others efore him hd writte out similr ides, it ws Argd s ess tht gied ttetio d resulted i the mig of the Argd digrm we cotiue to use tod. ellipse PURE Note A coic sectio is curve tht is foud itersectig ple with doule coe plced verte to verte. There re three tpes of coic sectio: ellipse, prol d hperol. A circle is tpe of ellipse. Ivestigtio prol circle hperol The equtio z = plotted i the comple ple gives circle with uit rdius cetred t the origi. Choose vlues of, d c such tht c > ( ) d > B vrig these vlues, ivestigte the grph of the locus of poit z tht moves so tht z + z = c, c > ( ) d > Wht is the shpe of the grph tht results? The rel prt of comple umer z c e deoted Re(z) B vrig the vlues of d, ivestigte grphs of the locus of poit z tht moves so tht Re(z z + ) = z Wht is the shpe of the grph tht results? MATERIAL TO COME Reserch The Mdelrot set provides eutiful imges. It is the set of ll the comple umers, c, for which the itertive sequece z = z + c, + where z 0 = 0 does ot ted to ifiit. Computer imges of the Mdelrot set c e developed d, o mtter how closel ou zoom i o the surfce, it still remis icredil itricte. The Mdelrot set is emple of frctl. Fid out if the comple umers i d i re i the Mdelrot set. Reserch the properties of the Mdelrot set. = z + c, + 69

68 Assessmet The qudrtic equtio = 0 hs roots d. Work out the vlue of +. Select the correct swer. 4 4 c d [] The qurtic equtio = 0 hs roots,, g d d. Work out the vlue of αβδ. Select the correct swer. c d [] Solve the iequlit ( )( + ) > 0. Select the correct solutio. 0 < < < c < < d < < 0 [] or < or 0 < < or > 0 or > 4 Give tht ( ) ( ) ( + r) r ( + ) ! r! Fid the epsio of i scedig order up to the term i [4] (+ 5 ) Stte the rge of vlues of for which this epsio is vlid. [] 5 Use the Mcluri epsio of e to fid the first 4 terms i the epsio of e Use our epsio to estimte the vlue of e 0.05 [] 6 Epress the sum (6r + ) r s polomil of [] r= 7 Prove iductio tht + r= ( + 5) for ll N. [6] r= 8 Prove iductio tht ( r ) 4 ( ) = for ll N. [6] r= 9 A cuic equtio is give 4 + = + 6 Use the sustitutio u = to form cuic i u [] Solve this cuic equtio i u [] c Hece, or otherwise, solve the origil equtio i [] 0 Solve the iequlit + + > [] 5 Epress r r s fctorised polomil i [4] r= Hece evlute the sum to = 6 [] The th term of sequece is defied u = ( )( + ) Epress the sum to terms of the sequece s fuctio of [4] Evlute the sum of the first 5 terms. [] c Fid the sum of the 6th to the 0th term. [] [] 70 Alger d series Assessmet

69 Form polomil fuctio, f(), whose roots re, d 4 d whose curve psses through the poit (, 5) [] The fuctio = f() udergoes the trsformtio descried = +. Fid the equtio of the trsformed polomil, g(). [] 4 The roots of the cuic + = 0 re, d g Write dow the vlues of g, αβ d α [] Fid the cuic equtio with roots, d g [4] 5 Use the Mcluri epsio of l( + ) to fid the epsio of l( ) up to the term i 4 Stte the rge of vlues of for which this series is vlid. [] si 6 Fid the first four o-zero terms i the series epsio of [5] 7 A cuic fuctio is give f() = Evlute f() d write dow lier fctor of f() [] Fctorise f() full. c Hece solve the iequlit f() < 0 [] r 8 Prove iductio tht 4 = (4 ) for ll N. [6] r= 9 Prove iductio tht 4 + 8is divisile for ll N. [6] 0 Prove iductio tht 4 + is multiple of for ll N. [6] ( )( + 4) A fuctio is give f() = 5 Sketch the curve = f() d fid the set of vlues for for which the fuctio is positive. [] Fid the set of vlues for which f() 8 [4] Show tht = + ( + ) [] Fid epressio i for [] r= rr ( + ) c Hece evlute the sum [] d Evlute [] r= 50 rr ( + ) Prove tht = + > ; [] Fid epressio i for [4] r= r c Evlute [] 99 d Fid [] r r= e [] [] 7

70 4 Give tht =, tht r = ( ) d tht ( r+ ) r = r + r+, r= r= fid epressio for r r= 5 The qurtic equtio = 0 hs four roots,, g d d i terms of [6] Fid the qurtic equtio with iteger coefficiets which hs roots,, g d d [] B cosiderig the sustitutio u =, or otherwise, fid the qurtic equtio with roots,, g d d [] + 6 Solve the iequlit + [7] The fuctio f() = itersects the stright lie = k Form qudrtic equtio i d k d hece fid the set of vlues of k for which f() hs rel roots. [4] Hece show tht the equtio siθ = hs o solutios. [] 8 Epd l + 4 up to the term i [5] Stte the rge of vlues of for which this epsio is vlid. [] 9 Prove tht = ( )( ) [] Hece fid epressio i for r = rr ( + )( r + ) [4] c Evlute [] Show tht + = + ( + ) [] r Hece fid, s sigle frctio, epressio for + 4r + 6 rr ( + ) [6] Usig the stdrd formul for, r, r, r 5 5 d cosiderig [( r+ ) r ], fid full fctorized formul i terms of for r 4 [5] Prove iductio tht u = + for ll N. [5] Prove iductio tht = for ll N. [8] r= r 4 ( + ) Epli wh does ot eist. [] r r= r= r= 7 Alger d series Assessmet

71 Curve sketchig Oriettio Wht ou eed to kow KS4 Qudrtic fuctios. Crtesi Coordites Mths Ch Curve sketchig. It is importt tht ircrft pilots d ir trffic cotrollers re le to locte the positio of ircrft precisel, s well s their directio of trvel d speed. To do so, the use modified sstem of polr coordites. The sstem tkes the directio of mgetic orth s 60, d mesures gles clockwise from this. Just like i Crtesi coordites, i polr coordites two vlues c e used to determie the precise loctio of poit ccurtel: distce from the origi d the gle mesured from mgetic orth. Polr coordites re useful i m situtios whe workig with pheome o or er the Erth s surfce. For emple, meteorologists use them to model eistig wether sstems d predict future wether, pilots of oce-goig tkers use them for vigtio, d crtogrphers use them to mp the Erth. Wht ou will ler To sketch grphs of lier d qudrtic rtiol fuctios, fidig smptotes, turig poits, limits, etc. To solve iequlities usig grph sketches. To use polr coordites d to covert etwee crtesi d polr coordites. To use the equtios of prols, ellipses d hperole. To use the defiitios d grphs of hperolic fuctios d their iverses. Wht this leds to Ch8 Curve sketchig Reltioships etwee zeros d smptotes. Comied trsformtios. Hperolic fuctos. Ch9 Itegrtio Itegrtig hperolic fuctios. Creers Aircrft pilot. Crtogrph. 7

72 . Lier Rtiol Fuctios Emple Fluec d skills Whe solvig prolems ivolvig rtiol fuctios, ou m fid it helpful to sketch grph. Whe sketchig grph of fuctio of the form + c + d fid the itercepts where the curve crosses the es, fid smptotes, cosider wht hppes whe d ecome ver lrge d pproch ifiit. Whe sketchig grph ou might ot eed ll of these techiques. Just use the relevt oes for ech questio. For = f(), verticl smptotes occur t vlues where the deomitor is zero. To fid horizotl smptotes, rerrge to = f() d fid the vlues of which mke the deomitor zero. ( + ) Fid the ke fetures (itercepts d smptotes) d sketch the grph of = ( 4) ( + ) = ( 4) Whe = 0, = d whe = 0, = 4 So the curve crosses the es t 0, d (, 0) 4 Whe 4 = 0, = 4 So = 4 is the verticl smptote. As ecomes ver lrge. So = is the horizotl smptote. Puttig = 000 gives vlue just greter th, so the curve is ove the smptote. Puttig = 000 gives vlue just less th, so the curve is elow the smptote. As ecomes ver lrge 4 = 4 (This is other idictio tht = 4 is smptote.) 4 Fid the itercepts sustitutig = 0 d = 0 The verticl smptote is where the deomitor is zero. Cosider lrge vlues for Lookig t specific lrge d smll vlues tells ou o which side of the smptote the curve lies. Cosider lrge vlues for Sketch the grph usig the ke fetures to help ou. Asmptotes re usull mrked with dshed or dotted lies. Check our sketch plottig the curve usig grphicl clcultor or grph plottig softwre. 74 Curve sketchig Lier Rtiol Fuctios

73 Emple You c of course drw the grph usig grphicl clcultor d use this to red off the ke fetures directl. However, ou should e le to work them out lgericll ecuse ou m ot e le to plot ll curves usig clcultor. ( + ) Fid the ke fetures d sketch the grph of = where > 0 ( ) ( + ) = ( ) Itercepts with es 0, d (, 0) Verticl smptote = Horizotl smptote As ecomes ver lrge, 000+ Whe = 000, = 998 so the curve pproches the smptote from ove. Whe = 000, = so the curve pproches the smptote from elow. 0 The lgeric costt mes ou cot plot the curve o grphicl clcultor. Sustitute = 0 d = 0 Fid whe deomitor = 0 (000 + ) > 0, so > 998 (000 + ) > 0, so < 998 Sketch the grph. Lel the itercepts i terms of PURE Emple Idetif the ke fetures d sketch the grphs of = Fid the poits of itersectio. For = ( 5) : whe = 0, = 0 d whe = 0, = 0 Whe 5 = 0, = 5, so = 5 is verticl smptote d = 5 4 ( 5) Sustitute = 0 d = 0 Fid the verticl smptote. (Cotiued o the et pge) 75

74 = ( 5) = ( 5) 5 So, = Whe = 0, =, so = is horizotl smptote. As ecomes ver lrge = = 5 4 is stright lie with grdiet 5 d itercept 4 Rerrge to the form = to fid the horizotl smptote. You c sustitute lrge vlues of d to see from which side the curve pproches the smptote. Eercise.A Fluec d skills Wherever possile, ou should use grphicl clcultor or grph sketchig softwre o our computer to check our swers to the questios i this eercise. For ech of these grphs i ii iii c e 4 0 determie itercepts with the es, fid verticl d horizotl smptotes, drw sketch. = + = + 5 = The curve d the stright lie itersect whe = 5 4 ( 5) = 0 ( 4)( 6) = 0 = d = f = 0 76 Curve sketchig Lier Rtiol Fuctios 5 So the curve d lie itersect whe = 6 d = 4 Poits of itersectio re (4, 4) d (6, 6) Rerrge the equtio to form qudrtic. Fctorise. Drw sketch d lel the ke fetures. You c use grphicl clcultor to drw the grph d check our swer. Fid smptotes d itercepts with the es for ech of followig grphs. = = + c = d = 5 + Idetif the ke fetures, sketch ech pir of equtios d fid the poits of itersectio. Show ll our workig. = + 4 d = = d = 6

75 c = + d = d = d = 8 e = + 5 d = f = d = g = d = h = d = B sketchig curve d lie, solve these iequlities grphicll. Show ll our workig d lel ll the ke fetures. + 5 < c + 4 < d + 5 PURE Resoig d prolem-solvig Strteg Emple 4 Sometimes, solvig iequlit lgericll c e quite complicted. You c use grphicl method to solve the iequlit or to check our lgeric result. Whe solvig iequlities grphicll: Ivestigte the ke fetures to fid the shpe of the grph of the fuctios. Sketch the grphs d fid the poits of itersectio. Use the grphs to write dow the rges of the vrile which solve the iequlit. 8 5 Solve the iequlit > grphicll. 8 5 = hs smptotes t = d = 8 d itercepts t (0, 45) d (7.5, 0) = is stright lie with grdiet d -itercept ( 6, 7) (, 9) Poits of itersectio re ( 6, 7) d (, 9) 8 5 So > whe 6 < < d > Ivestigte the ke fetures of the fuctios. Sketch the grph d fid the poits of itersectio. You c use grphicl clcultor to do this, ut mke sure ou lso kow how to do it hd. Use the grph: 8 5 is greter th where the curve is ove the stright lie. 77

76 Emple 5 Followig eperimet, studet A s results fit the grph of ( ) =, whilst studet B s results fit with ( + ) = The studets plotted their grphs of gist o the sme grid. Fid the poit of itersectio of their grphs. Usig grphicl clcultor or grph sketchig softwre o our computer, fid the rge of vlues of where the grph of studet A gives higher results th tht of studet B. c Cofirm our results lgericll. = = ( + ) = ( ) = 9 = The poit of itersectio is (, 0.) = hs smptotes t = d = 0 = hs smptotes t = d = The results of studet A re higher th studet B whe the lue grph is ove the red grph. > whe < or whe < < + Ivestigte the ke fetures. Sketch the grphs. The lue grph is = The red grph is = + See Ch. For remider o solvig iequlities lgericll. c Cse : oth deomitors positive or oth egtive ( + ) > ( ) if ( + )( ) > 0 i.e. if > or < + 4 > 9 > So, < or < < (Cotiued o the et pge) 78 Curve sketchig Lier Rtiol Fuctios

77 Cse : oe deomitor positive d oe egtive ( + ) < ( ) if ( + )( ) < 0 i.e. if < < + 4 < 9 < So o solutio sice < < Rememer to reverse the iequlit sig whe ou multipl egtive umer. PURE Eercise.B Resoig d prolem solvig Use grphicl clcultor or grph plottig softwre to swer the questios i this eercise. Solve these iequlities grphicll, lellig ll the itercepts, smptotes d 7 Use grphicl clcultor or grph sketchig softwre to sketch the grph poits of itersectio. of = < + 5 Show tht the grph crosses the -is t = c c Show tht the smptotes of the grph re =, = 5 d = e f > Whe is + +? Solve these iequlities grphicll, 9 You c solve 5 = 0 sketchig, lellig ll the itercepts, smptotes d o the sme grid, the grph of = d poits of itersectio. pproprite curve. + < 4 < Give the equtio of pproprite curve d drw sketch to solve the 5 c equtio. e f 5 d < > 5 g + 4 Joh ss tht the rtio of to + is lws less th. Use grphs to epli wh Joh is wrog. 4 Sketch two pproprite grphs to solve the iequlit 0 5 For ll vlues of q, si q Fid the vlues of for which si q = is stisfied rel vlues of q. 6 For ll vlues of q, cos q Fid the vlues of for which cos q = + is stisfied rel vlues of q. Verif our solutio solvig 5 = 0 lgericll. 0 Solve the equtio 6 = 0 usig grphicl clcultor or grph sketchig softwre to sketch, o the sme grid, the grph of = d pproprite curve of A the form + B Stte the vlues of A d B Three frctios re, d. Sketch the + grphs o the sme grid d clculte the ect rges of vlues of whe ech grph is the lrgest of the three. Full A-level 79

78 . Qudrtic Rtiol Fuctios Emple Fluec d skills You c sketch grphs of qudrtic rtiol fuctios usig similr techiques to those ou lered i Sectio c Whe sketchig grph of fuctio of the form d + e + f fid smptotes, fid the itercepts where the curve crosses the es, cosider wht hppes whe d pproch ifiit, fid limittios o the possile rges of solvig the fuctio for d/or d use this to fid mimum or miimum poits o the curve. For the grph of = fid the itercepts where the curve crosses the es, You eed to e le to fid the ke fetures of grph usig lgeric techiques, ut mke sure ou lso kow how to plot grphs o grphicl clcultor to check our swers. show tht there is o verticl smptote, ut tht there is horizotl smptote t = Whe = 0, = 6 4 = 4 Whe = 0, 6 = 0 (provided + 4 0) So the curve crosses the es t (0, 4) (4, 0) d ( 4, 0) for rel vlue of so there is o verticl smptote. Rerrgig: ( + 4) = = ( ) = There is horizotl smptote t = Altertivel, As gets ver lrge = So = is horizotl smptote. 4 0 Fid the itercepts sustitutig = 0 d = 0 A verticl smptote occurs whe the deomitor is zero. Rerrge to = d fid whe the deomitor is zero to fid horizotl smptote. You c lso fid horizotl smptotes cosiderig lrge vlues of Sketch the grph usig grphicl clcultor to cofirm our vlues for the itercepts d smptote. 80 Curve sketchig Qudrtic Rtiol Fuctios

79 Emple Fid the rge of d vlues for which the grph of = eists. Use this iformtio to fid turig poits. Check our swers usig grphicl clcultor. vlues The grph eists for ll vlues of ecept for verticl smptotes. Verticl smptotes occur whe the deomitor = 0 Sice + + hs o rel solutios, there is o verticl smptote. This mes the grph eists for ll rel vlues of vlues Rerrge the equtio ( + + ) = ( ) + ( ) + ( ) = 0 c ol tke rel vlues whe 4c 0 9( ) 4( )()( ) ( 4) 0 So the curve ol eists for 0 4 Turig poits As gets ver lrge =, so = is horizotl smptote. = is the ol horizotl smptote d 0 4 So = 0 d = 4 must e either miimum or mimum poits. Whe = 4, = 0 so ( + ) = 0 Hece the mimum poit is (, 4) Whe = 0, = 0 (provided + + 0) So, ( + ) = 0 d the miimum poit is (, 0) (, 4) 0 Solve for to fid the limits o the rge of Chge the directio of the iequlit sice ou hve multiplied Sustitute = 4 ito the equtio d solve. Check our swers with grphicl clcultor. Notice tht, i some circumstces, curve c cross smptote. PURE 8

80 Emple Idetif the ke fetures d sketch the grph of = 6 Itercepts 6 Whe = 0, = = Whe = 0, 6 = 0 (provided 0) ( )( + ) = 0 = or = (Check: ( ) ( ) 0 d ( ) ( ) 0) So, the curve crosses the es t 0,, (, 0) d (, 0) Asmptotes = 0 ( + )( 4) = 0 So, =, d = 4 re verticl smptotes. As ecomes ver lrge = so = is horizotl smptote. Sustitutig = ±000, s, shows the grph pproches the smptote from ove o oth sides of the grph. Rge Rerrgig the equtio: ( ) = 6 ( ) ( ) + (6 ) = 0 c ol tke rel vlues whe ( ) 4( )(6)( ) 0 So, 5 49 or Turig poits Whe = (49 5)( ) 0 You hve lred show tht = is smptote, so = ± Sustitute = 0 d = 0 The deomitor is zero t verticl smptote. Cosider lrge vlues of. Solve for to fid the limits o the rge of does ot eist i the rge 5 49 < < (Cotiued o the et pge) 8 Curve sketchig Qudrtic Rtiol Fuctios

81 Whe = = = = 0 ( ) = 0 PURE So there is repeted root t = 5 This shows tht 49, is mimum poit. 5 49, 0 4 Eercise.A Fluec d skills Use these fetures to sketch the grph. You should use grphicl clcultor to check our sketch. Use grphics clcultor or grph sketchig softwre to check our swers i this eercise. Fid smptotes d poits of itersectio with the es for ech grph. = 9 = c = d = e = + 4 f = For ech of these fuctios, determie the rges of d vlues for which the grph eists. = = c = 4 d = + + For ech of these curves, fid locl mimum d miimum poits. = = c = d = ( + 4) ( + 4) 4 Sketch these curves. = c = = 5 6 d =

82 Resoig d prolem-solvig Strteg + + c To sketch curve of the form = d + e + f Fid poits of itersectio with the es d smptotes. Ivestigte the res where d hve rel vlues d determie mim/miim. Sketch the curve d use grphicl clcultor to check our swer. Emple 4 Sketch the curve = Whe = 0, = 6 = Whe = 0, the + 6 = 0 (provided 6 0) ( + )( ) = 0 (provided 6 0) So = or = (Check: ( ) ( ) 6 0 d () () 6 0) So the curve crosses the es t (0, ), (, 0) d (, 0) The deomitor of the fuctio is zero whe 6 = 0 ( + )( ) = 0 Hece = d = re verticl smptotes. As ecomes ver lrge, = So = is horizotl smptote. Sustitutig = 000, gives = Sustitutig = 000 gives = Fid the itercepts with the es. Cosider the deomitor to fid verticl smptotes. Cosider lrge to fid horizotl smptotes. So the grph pproches the smptote from ove for lrge positive d from elow for lrge egtive. Rerrgig the fuctio: ( 6) = + 6 Solve for to fid the limits o the rge of ( ) ( + ) + 6( ) = 0 tkes rel vlues whe ( + ) 4( )(6)( ) Rerrge to completed squre form to show this is true for ll vlues of (Cotiued o the et pge) 84 Curve sketchig Qudrtic Rtiol Fuctios

83 Hece is cotiuous d tkes ll vlues. Sice is cotiuous the curve must cross the smptote = It does this where = + 6 tht is t (0, ) 6 Use this iformtio to sketch the grph. PURE 0 You c see from the grph tht lthough = is smptote (d is pproched s gets ver lrge, positivel or egtivel) it is lso crossed t = 0 Eercise.B Resoig d prolem-solvig Use grphicl clcultor or grph sketchig softwre to swer the questios i this eercise. Prove tht + d hece sketch Fid the ect vlues of t the poits of itersectio of the grphs d hece the grph of the fuctio. 4 solve the iequlit Prove tht 0 d use this to + > 4 sketch the grph of the fuctio. c Use our sketch to epli wh 4 Prove tht = cot lie etwee 0 = hs o vlue etwee = 0 d = d d use this to sketch the grph. 4 Prove tht = Show tht = + A hs rel solutios hs miimum vlue of B 0 d mimum vlue of, d epli ecept for the rge A. Hece B this ppret cotrdictio usig sketch sketch the grph of = + 4 d fid the grph. + 5 Sketch the grph of the fuctio +. coordites of the mimum turig poit. A 6 Fid the gretest vlue of c for which the 0 Prove tht = hs rel vlue of the fuctio 4 4 A + 4 roots if A 0 is 5 + c i Sketch the grph whe A = 4 d Sketch the grph of the fuctio for this show there is o turig poit. vlue of c. 4 7 Clculte the rge of vlues of C for which ii Solve the equtio 4 4 = 0 the equtio = C hs rel roots. Hece k If =, fid the miimum d + + fid the coordites of the mimum d mimum vlues of i terms of k. miimum poits o the curve. Hece sketch the curve whe k = 6 8 Sketch o the sme grid the grphs 4 of = d 4 =. Full A-level 85

84 . Polr coordites Fluec d skills Emple As well s usig the d Crtesi coordites to descrie the positio of poit, ou c lso use sstem kow s polr coordites. Polr coordites re mesured from origi O, clled the pole d the positive -is, OX, clled the iitil lie. The digrm shows the poit P(, ) with polr coordites P(r, q). r is the legth of OP, so r 0 Agles mesured ti-clockwise from the iitil lie re positive d gles mesured clockwise re egtive. If poit P hs polr coordites (r, q) the r is the legth of OP so r 0 q is the gle etwee OP d the iitil lie. Ke poit You c use trigoometr to epress d i terms of r d q d vice vers. = r cos q r = + = r si q tθ = Ke poit Whe clcultig the gle q, drw digrm to esure ou fid the gle from the iitil lie to the correct qudrt. O q Y r q r P(, ) P(r, q) Alws use the positive squre root for r, so r= + Covert these Crtesi coordites ito their equivlet polr form. Use π < q π A(, ) B(, ) r = + r = + = 4 = tθ = θ = t A = π 6 I polr coordites: A π, 6 O q A A is i the st qudrt, so q is cute. (Cotiued o the et pge) X 86 Curve sketchig Polr coordites

85 r = + r = + ( ) = 4 = t B ( ) = π ut B is i the third qudrt so q is otuse. 6 PURE π 6 q Drw digrm to fid the gle required. B Emple π π 5 6 = π 6 5 Therefore, θ = π 6 I polr coordites: B 5, π 6 Poit A hs polr coordites, π Show tht the equivlet Crtesi coordites re (, ) π = cos = = π = si = = I Crtesi form: A (, ) = r cos q = r cos q Clcultor AC + / % = Tr it o our clcultor You c use grphics clcultor to covert etwee Crtesi d polr coordites. Imge to come Activit Fid out how to covert the Crtesi coordites (, ) ito the equivlet polr form o our clcultor. 87

86 Some grphs re simple to sketch from their polr form. For emple, the grph of r = is circle, cetre the pole, with rdius. Isted of drwig es, ou just eed to show the positio of the iitil lie. π q The grph of θ = π π is hlf-lie t gle of from the iitil lie. A hlf-lie is stright lie tht eteds ifiitel Ke poit from poit. q π t = gives = So the hlf-lie θ = π lies o the lie = Emple Emple 4 Drw the grph of θ = π d fid the Crtesi equtio of the lie it lies o. 4 π q t π 4 = so = = So the hlf-lie θ = π lies o the lie = 4 You c plot polr grphs usig grphicl clcultor or completig tle of vlues. Plot the curve with polr equtio r = ( si q) Your digrm should show the iitil lie d the hlf-lie. π π π 4π 5π 6π 7π 8π 9π 0π π π q r q q Plot the poits d joi them i order. This grph is hert shped d is clled crdioid. 88 Curve sketchig Polr coordites

87 You c lso sketch polr curve cosiderig its geerl shpe d the mimum d miimum poits. For emple, the curve r = 5+ cosθ will hve mimum of r = 7 whe q = 0 d miimum of r = whe q = π. 7 Whe q = π or, cosθ = 0so r = 5. π O q = 0 PURE Emple 5 Plot the curve with polr equtio r = 5siθ for 0 θ < π The mimum vlue of r is 5 This occurs whe θ = π,5 π,9 π So θ = π 6,5 π 6, π The miimum vlue of r is 0 This occurs whe θ = 0, π, π, π, 4 π, 5π π So 0,, π,, 4 π θ = π,5 π 5π q = 6 O q = π π q = 6 5 q = 0 You ol eed to cosider the vlues of q tht led to positive vlues of r Sice 0 q π, fid where si q = for 0 q < 6π All these vlues of q give positive vlues for r The curve hs loops d rottiol smmetr of order Emple 6 θ Plot the curve with polr equtio r =, for 0 θ π The miimum vlue of r is 0, which occurs whe θ = 0 The mimum vlue of r i the give domi is π, which occurs whe θ = π O q = 0 p r icreses with q so the curve is spirl. 89

88 Eercise.4A Drw polr chrt similr to the oe show d plot the give poits. Covert ech set of polr coordites to their Crtesi equivlet. 5π 6 π 7π 6 π 4π π π π π π 5π π 6 π, 4, 4 π c, 5 π d, π e 5, π 6 f, π Covert these Crtesi coordites to polr form (r, q) where 0 q π. Give our swers to sf where pproprite. (, ) (, ) c ( 4, 4 ) d ( 5, ) Covert these Crtesi coordites to polr form (r, q) where π < q π. Give r s surd d gles to sf where pproprite. (, ) (5, 5 ) c (, 4) d (, 6) 4 Stte the polr equtio of the curve + = 5 d descrie the curve. 5 Descrie the curve r = 4 d stte its Crtesi equtio. 6 Sketch the grphs for ech of these polr equtios d fid the Crtesi equtio of the lie the lie o. θ = π θ = π c θ = π d θ = π π 7 Plot the curve r = q i the domi 0 q 6π usig step-sizes of. 4 0 Plot r = i the domi 0 q π + cosθ 8 Sketch these polr curves for 0 q π d stte the mimum d miimum vlue of r i ech cse. r = cos q r = si 4q c r = cos q d r = 4 si q e r = + cos q f r = 4 + si q g r = cos q h r = 5 si q i r = q j r = 4 si q 90 Curve sketchig Polr coordites

89 Resoig d prolem-solvig Whe sketchig grphs of polr equtios, it is sometimes helpful to covert the equtios to Crtesi form. I order to do this, ou m eed to rerrge the sustitutios to epress cos q d si q i terms of d. PURE = r cos q cosθ = = r = r si q siθ = = r + + Ke poit Strteg Emple 7 To covert etwee Crtesi d polr equtios: Use sustitutios to write r d q i terms of d or vice vers. Simplif the epressio; use trigoometric idetities where required. Covert these Crtesi equtios to polr form. A stright lie pssig through the origi with equtio = A circle, cetre the origi d rdius uits, with equtio + = 4 = rsiθ = rcosθ siθ = cosθ Sustitute = r si q d = r cos q Rerrge d simplif. So, tθ = + = 4 (cos r θ) + ( rsi θ) = 4 Sustitute = r si q d = r cos q r (cos θ+ si θ) = 4 r = 4 r = Use cos q + si q = d tke the positive squre root s r 0 9

90 Emple 8 Epress these polr equtios i Crtesi form d descrie the grphs. r = siθ cosθ r = 6cosθ+ 8siθ r = siθ cosθ rsiθ rcosθ = = = + This is stright lie pssig through (0, ) with grdiet Sustitute for r si q d r cos q r = 6cosθ 8siθ = = = 0 ( ) + ( 4) 9 6= 0 ( ) + ( 4) = 5 This is circle cetre (, 4) d rdius 5 You c fid the poit of itersectio etwee two polr curves. The method is essetill the sme s for fidig the poit of itersectio etwee curves i Crtesi form, ut sice our curves re ol for r = 0 ou disregrd poits with egtive vlues of r Resoig d prolem-solvig Sustitute for r, cos q d si q Complete the squres. Strteg To fid the poits of itersectio etwee two polr curves: Use oe equtio to sustitute for r i the other equtio d solve to fid the solutios for q Sustitute ck ito oe of the origil equtios to fid the solutios for r Select ol the solutios with r 0 9 Curve sketchig Polr coordites

91 Emple 9 Fid the poits of itersectio etwee the curves r = siθ d r = cosθ for r 0 d 0 θ < π siθ = cosθ siθ = cosθ tθ = θ = π 4,7 π 4, π 4, 5π 4 θ = π 8,7 π 8, π 8, 5π 8 Whe θ = π 8 r si = π 8 = 7 Whe θ = π 8 r si 7 = π 8 = π Whe θ = 8 r = d whe θ = 5 8 π r = So ol poits of itersectio re, π 8,, π 8 Solve simulteousl, usig the idetit si θ = tθ cosθ Sustitute to fid the vlues of r. Sice r 0 these re the ol vlid solutios. PURE O q = 0 You might fid it helpful to sketch grph of the two curves to see where the itercept. 9

92 Emple 0 Fid the poits of itersectio etwee the curves r cosθ = siθ cosθ = siθ cosθ cos θ(siθ ) = 0 cosθ = 0 or si θ = Therefore, θ = π, π or θ = π 6,5 π 6 θ = π r = 0 θ = π r = 0 θ = π 6 r = 5 θ = π 6 r = so ot vlid solutio. So poits of itersectio re t 0, ± π,, π 6 O q = 0 = cosθ d r = siθ Solve simulteousl. Use the idetit si q = si q cos q Sustitute ech vlue of q to fid vlue for r. r cot e egtive. You might fid it helpful to sketch grph of the two curves whe fidig the poits of itersectio. 94 Curve sketchig Polr coordites

93 Eercise.4B Fid the polr equtio for c d the lie pssig through origi with grdiet, the lie pssig through the origi with grdiet, the positive -is, the positive -is, e = 5, f =, g = +, h = m + c, i the circle cetre the origi, rdius 5, j the circle with cetre (5, ) d rdius, k l the circle with cetre (, 0) pssig through the origi, the circle with cetre (0, 4) pssig through the origi, m the circle with cetre (5, 5) pssig through the origi. Covert these polr equtios ito Crtesi form d descrie the grphs. 5 r = r = siθ+ cosθ siθ cosθ c r = d r = cosθ si θ e r = 4siθ f r = 8cosθ+ 6siθ Fid the poits of itersectio etwee these pirs of curves for π < θ π d r 0. Give gles i terms of π. r=, r= siθ r= cos4 θ, r= c r= si θ, r= cosθ 4 Show lgericll tht spirl of the form r= θ will itersect circle cetred o the origi precisel oce. 5 For ech pir of equtios, for r 0 d 0 θ π. i ii sketch the curves o the sme grph fid their poits of itersectio. r= si θ, r= cosθ r= θ, r= θ si θ c r = + cosθ, r = cos θ d r= si θ, r= cos θ e r = si θ, cos θ 6 The fmil of curves r= + 4cosθ produces differet shpes depedig o the choice of vlue for. c d Show tht if is greter th 4 the the curve ever psses through the pole. Prove tht the curve is smmetricl out the iitil lie for vlue of Hece, sketch the curve i the domi π q π whe i = 0 ii 0 < 4 iii 4 < < 8 iv 8 Stte, i terms of, the mimum d miimum vlues of r for the curves i prt c. 7 Show tht the grphs of r = secθ d r = t θ itersect ol oce d fid their poit of itersectio. 8 Fid the poits of itersectio etwee these pirs of curves for π < θ π d r 0. Give our coordites to sf. r= cos θ, r= + siθ r = 4cos θ, r= cosθ 9 Show tht the poits of itersectio etwee the curves r = si θ d r = cosθ form rectgle d fid its re. PURE 95

94 .4 Prols, ellipses d hperols Emple Fluec d skills Coic sectios re specil ctegor of curve. The re formed the itersectio of ple d coe. You c thik of this s slicig through the coe. The gle t which the ple itersects the coe determies the tpe of curve which is formed. For emple, ple prllel to the se of the coe will form circle d ple prllel to the curved fce will cut through the se d the curved fce d form prol. You re lred fmilir with the prol = A d with trsformtios of this curve. Prols c lso e of the form = A which c e rerrged to mke the suject. = A = A A equtio of the form = k, for k > 0 descries prol with its verte t the origi. A curve hs equtio = 4. Ke poit Sketch the curve, givig the coordites of itercepts with the coordite es. Fid the poits of itersectio etwee the curve d the lie with equtio = 9 = 4 circle prol You m lso see the stdrd equtio of prol writte s = 4 0 (9 ) = = = 0 Solve the equtios simulteousl. 96 Curve sketchig Prols, ellipses d hperols (Cotiued o the et pge)

95 7 =, = 8, 6 7 So the curve d lie itersect t, 8 d, 6 Fctorise or use our clcultor to solve the qudrtic. PURE Aother tpe of coic sectio is ellipse. The ellipse is formed the itersectio of coe with ple tht cuts through ol the curved surfce of the coe. If this ple is prllel to the se of the coe the circle if formed, ut i geerl ellipse will e formed. circle ellipse Emple You kow tht the equtio of circle with cetre the origi is + = r. This c e writte + =. r r I cotrst, ellipse does ot hve costt rdius. Its equtio c e writte + = where, > 0. A equtio of the form Ke poit + = descries ellipse cetred o the origi. The ellipse will pss through the poits ( ±, 0) d (0, ± ). Sketch ech of these ellipses, clerl lellig their poits of itersectio with the coordite es. + = + = 4 9 You c see tht = 4 so = Check lettig = 0, this gives 4 d = 9 so = So the grph looks like this Check lettig = 0, this gives = 9 = ± 9 r r r r + = r + r = Rerrge the equtio to + = So = d = Tr to mke our grph pproimtel to scle. Writig the equtio i the usul form it mke it esier to idetif the vlues of d. 97

96 Emple A circle with equtio + = 9 is stretched scle fctor i the -directio to form ellipse. Sketch the ellipse d stte its equtio. 6 Mke verticl stretch of scle fctor, so the -itercept is ow = 6 The equtio will e + 9 = or + = 9 6 The fil coic sectio is hperol, this c e formed the itersectio of coe with ple tht is steeper th the curved fce of the coe. A doule-coe is show s the ple will itersect oth coes. A hperol hs equtio = for, > 0. Notice tht this is ver similr to the equtio of the ellipse. Ol the mius sig is differet. You c see tht whe = 0, = so the curve crosses the -is t ( ±,0). However, if ou sustitute = 0 ito the equtio ou get = which is ot possile sice must e positive. Therefore the hperol will ot cross the -is. I fct, sice the equtio c e rerrged to give = +, ou c see tht, therefore is ol defied whe. As d ecome lrge, = + ecomes So the grph hs smptotes t =, i.e. =±. = hperol = A equtio of the form Ke poit = descries hperol cetred o the origi with smptotes =± 98 Curve sketchig Prols, ellipses d hperols

97 Emple 4 Sketch the hperol = 4 d stte the equtios of the smptotes. I this equtio, = 4 = d = = =, so the equtios of the smptotes re =± = = PURE The -itercepts re t (±, 0) Emple 5 A hperol whose smptotes re the coordite es is clled rectgulr hperol. The equtio of rectgulr hperol is = c Ke poit A equtio of the form = c descries rectgulr hperol cetred o the origi with smptotes = 0 d = 0. A curve, C, hs equtio = 5. Sketch C, givig the equtios of the smptotes. The lie with equtio = + 5itersects C t the poits A d B. Clculte the legth of the lie segmet AB. = 5 = 5 represets rectgulr hperol. = c Asmptotes re = 0 d = 0 5 = 5 = 5 = = 0 =.5, 5 = 50, 5 So A is ( 5, 5) d B is (.5, 50) Legth of AB = = 45.6 Solve simulteousl. Use Pthgors theorem to fid the legth of AB 99

98 Eercise.4A Fluec d skills Sketch ech of these curves, clerl lellig their poits of itersectio with the coordite es. = = c + = d + = e + = f + 9 = 6 7 g = 6 h 4 + = 8 Sketch the curves with these equtios, lel the poits of itersectio with the -is d stte the equtio of the smptotes i ech cse. = = c = 4 d = e = f = 6 g 5 = 5 h 5 = 0 A prol hs equtio = 0 Sketch the curve. Fid the poits of itersectio etwee the prol d the lie with equtio = 4 A ellipse hs equtio + = 4 5 Sketch the curve. Fid the poits of itersectio etwee the ellipse d the lie = + 5 A ellipse hs equtio + = 7 Sketch the curve. Show tht the lie with equtio + = 7 is tget to the ellipse + = 7 6 For ech grph, stte wht tpe of coic sectio the re d give possile equtio. = 6 6 = 7 The smptotes of the hperol = re =± Fid the equtio of the hperol give tht it psses through the poit (, ) 8 A curve hs equtio = 6 Wht me is give to this tpe of curve? Sketch the curve, givig the equtios of the smptotes. The curve itersects the lie = + t the poits A d B c Fid the legth of the lie segmet AB 9 A hperol hs equtio = k Sketch the curve. Show tht the smptotes of the hperol re t right-gles to ech other. 00 Curve sketchig Prols, ellipses d hperols

99 Resoig d prolem-solvig You kow from AS Mths tht replcig f( ) f( c) i equtio trsltes the curve c uits i the -directio. Similrl, replcig g( ) g( c) trsltes the curve c uits i the -directio. To sketch trsltios of coic sectios, idetif the tpe of curve ivolved d fid the verte (for prol) or cetre (for hperol or ellipse) of the trsformed curve. PURE Strteg To sketch trsltio of coic sectio: Idetif the curve covertig the equtio to stdrd form. Fid the cetre or verte of the trsformed curve. Sketch the trsformed curve, sttig the vlues of itercepts d the equtios of smptotes. Emple 6 Emple 7 Sketch the curve of = + 0 The equtio c e writte s = ( + 5) The cetre of the curve hs moved from the origi to ( 5, 0) = ( + 5) A prol with equtio ( ) = 4 ( ) will hve its verte o the poit (, ) Sketch the curve of + ( 5) = 5 4 ( 5) The equtio c e writte s = The cetre of the curve hs moved from the origi to (0, 5) ( 5) = 5. Ke poit This is the equtio of prol. The curve hs ee trslted 5 uits left. You c check the -itercept is correct sustitutig = 0 to give 0= ( + 5) = 5 To fid the -itercepts, sustitute = 0 to give = 0 =± 0 This is the equtio of ellipse with rdius of 0 i the -directio d rdius of 5 i the -directio. The curve hs ee trslted 5 uits up. You c check the -itercepts re correct sustitutig = 0 to give 5= ± 5 = 0, 0 0

100 Ke poit ( ) ( ) A ellipse with equtio + = will e cetred o the poit (, ) d hve rdius of i the -directio d i the -directio. Emple 8 Sketch the curve of + = 4 The equtio c e writte s ( + ) = 4 The cetre of the curve hs moved from the origi to (0, ) This is the equtio of rectgulr hperol. The curve hs ee trslted uit dow. Emple = 4 Asmptotes re = 0 d = Ke poit A rectgulr hperol with equtio ( )( ) = c will e cetred o the poit (, ) d hve smptotes =, = Sketch the curve of ( ) = 48 The equtio c e writte s ( ) = 4 6 The cetre of the curve hs moved from the origi to (, 0) 0 5 ( ) = 48 You c check the -itercept is correct sustitutig = 0 to give = 4 This is the equtio of hperol. The curve hs ee trslted uits right. You c check the -itercepts re correct sustitutig = 0 to give =, 5 0 Asmptotes re =± ( ) The -itercepts re foud whe = 0 so 6 = 6 = 0 so =± 0 Ke poit ( ) ( ) A hperol with equtio = will e cetred o the poit (, ) d hve smptotes ( ) =± 0 Curve sketchig Prols, ellipses d hperols

101 You kow from AS Mths tht replcig f( ) f k i equtio will stretch the curve scle fctor k i the -directio. Similrl, replcig g( ) g k will stretch the curve scle fctor k i the -directio. PURE Strteg To sketch stretch or reflectio i coordite is of coic sectio: Idetif curve covertig the equtio to stdrd form. Cosider the directio to stretch i or the is to reflect i. Sketch the trsformed curve, sttig the vlues of itercepts d the equtios of smptotes. Emple 0 Emple Sketch the curve of = The equtio c e writte s = ( ) The curve hs ee reflected i the lie = 0 = 0 This is the equtio of prol. Note tht i this cse reflectio i the lie = 0 would hve o effect sice ( ) = The -itercept remis t the origi. The curve with equtio + 5 = 5is stretched scle fctor i the -directio. Write the equtio of the trsformed curve d stte its poits of itersectio with the coordite es. Equtio of origil curve is + = 5 A stretch of scle fctor i the -directio mes equtio ecomes + 5 = Which simplifies to + = 5 9 This is ellipse which itersects the coordite es t ( ± 5, 0), (0, ± ) Notice tht stretch or reflectio i coordite is will ot ffect the verte. This is the equtio of ellipse. 0

102 Emple The curve with equtio = 8is stretched scle fctor i the -directio. Sketch the trsformed curve d stte the equtios of its smptotes. = 8 4 ( ) Equtio ecomes = 8 4 = 4 so smptotes re =± = 4 This is the equtio of hperol. To stretch scle fctor i the -directio, replce Emple You c reflect i the lie = reversig the roles of d i the equtio. A prol of the form = k will ecome = k A ellipse of the form + = will ecome + = A hperol of the form = will ecome = with smptotes t =± A rectgulr hperol of the form = c will ecome = c so o chge! To reflect i the lie =, ou eed to replce with d with i the equtio. The curve with equtio ( ) + 6 = 9is reflected i the lie = Sketch the trsformed curve d stte its equtio. ( ) + 4 = 9 ( ) Curve ecomes + 4( ) = 9 ( + ) This c e rewritte s + 4 = 9 Which is ellipse with cetre (0, ) 0 This is the equtio of ellipse. Fid the -itercepts lettig = 0 the + 4 = = ± Curve sketchig Prols, ellipses d hperols

103 Eercise.4B Sketch ech of these curves, clerl lellig the poits of itersectio with the coordite es i ech cse. = ( ) ( ) = c ( + ) = 8( ) d = 4+ 8 ( + ) e + = f + = 5 4 ( + ) g + = h + 9( ) = i ( 4) + 4( + 4) = 6 j = 0 Sketch ech of these curves d stte the equtios of the smptotes i ech cse. ( + ) = 9 ( 4) = 5 c ( + )( ) = 0 d = 4 ( + 5) e 5 = f = 9 ( + ) g = 6 5 h 4( 7) = 6 i ( 5) ( + ) = 6 j = 47 Sketch ech of these curves, clerl lellig the poits of itersectio with the coordite es i ech cse. = 5 = 7 4 Write dow the equtio of the imge of ech of these trsformtios of the curve = 8 i the form = f( ) c d Stretch scle fctor i the -directio. Stretch scle fctor i the -directio. 4 Reflectio i the -is. Reflectio i the -is. 5 Write dow the equtio of the imge of ech trsformtio of the curve = 4 Trsltio the vector Reflectio i the lie = 6 Write the equtio of the imge of ech trsformtio of the curve + = 4 Write the equtios i the form + = Stretch verticll scle fctor Stretch horizotll scle fctor 5 c Reflectio i the lie = 0 d Reflectio i the lie = 7 Write the equtio of the curve formed trsltig the ellipse + = 8 the vector 8 For ech of these trsformtios of the curve ( ) = 4 i ii write dow the equtio of the imge, stte the equtios of the smptotes. Reflectio i the lie = 0 Trsltio the vector 5 c Reflectio i the lie = d Reflectio i the lie = 9 For ech of these trsformtios of the curve 9 = 8 i write the equtio of the imge, ii stte the equtios of the smptotes. Stretch verticll scle fctor Reflectio i the -is c Trsltio the vector 4 d Reflectio i the lie = ( k) 0 A curve hs equtio ( + k) = where d k re costts,, k > 0 Stte the equtios of the smptotes to the curve. Sketch the reflectio of the curve i the lie = d stte the equtios of the smptotes of the reflected curve. PURE 05

104 .5 Hperolic fuctios Fluec d skills Emple If ou plot ll the poits of the form (si q, cos q) where 0 q π, ou produce circle (show the red curve). There eists other pir of fuctios, deoted sih q d cosh q. If ou plot the poits (sih q, cosh q), ou produce the curve show i lue. This curve is clled hperol d so the fuctios re referred to s hperolic fuctios d defied usig epoetils. e e e e sih = cosh = sih e e th = = cosh e + e Fid the ect vlue of th0 th0 e 0 e 0 = 0 0 e + e = + = 0 + sih(l) Ke poit l l e e sih(l) = e e = = 4 = l l 0 These fuctios re commol red s, shie, cosh d tch or th. Use the fct tht l = l Use the defiitio of th Clcultor Tr it o our AC / % clcultor +. = You c use clcultor to evlute hperolic fuctios d iverse hperolic fuctios. +.. si h (l5) 5 Activit Fid out how to evlute sih (l5) o our clcultor 06 Curve sketchig Hperolic fuctios

105 The grph = cosh The curve hs miimum poit t (0, ) So The curve is smmetricl out the -is PURE The grph = sih The curve hs rottio smmetr out the origi Emple The grph = th As + e 0 th ± The curve hs smptotes t = d = So < < The curve hs rottiol smmetr out the origi. Grphs of hperolic fuctios c e trsformed i the sme w s other fuctios so for = f( ) : = f( ) is = f( ) stretched verticll scle fctor = f( ) is = f( ) stretch horizoll scle fctor = + f( ) is = f( ) trslted verticll uits = f( ) is = f( ) trslted horizotll uits. Sketch the grph of = + cosh Ke poit Trslte the grph of = cosh up two uits O 07

106 Emple Sketch the grphs of = sih = sih d = sih = sih o the sme es. The grph of = sih O is = sih stretched horizotll scle fctor. Esure ou lel ech curve. Emple 4 Emple 5 If ou re give the vlue of sih, cosh or th ou c use their defiitios to fid the vlue of Solve the equtio th = e e e + e = (e e ) = e + e e = e e = = l = l Use the defiitio of th Multipl oth sides e Tke logrithms o oth sides. This emple could lso e doe o clcultor usig th, however our clcultor will give deciml ot ect swer. Solve the equtio sih + cosh = e e (e + e ) + = e e + e + e = 4 e 4+ e = 0 e 4e + = 0 (e )(e ) = 0 e = or e = = l= 0or = l Use the defiitios of sih d cosh Multipl through e to give qudrtic equtio i e 08 Curve sketchig Hperolic fuctios

107 Emple 6 Prove tht cosh( A+ B) cosh( A)cosh( B) + sih( A)sih( B) Eercise.5A Work out the vlue of sih 0 cosh ( ) c sih (l ) d cosh ( l ) e cosh 0 f th (l ) Sketch these grphs. = sih( ) = 0+ sih c = th( ) d = cosh( + ) e = + th( ) f = cosh( ) Give tht f() = th ( + ), > 0 sketch the grph of = cosh( A)cosh( B) + sih( A)sih( B) e + e e + e e + f( ) write dow the equtios of the smptotes. 4 Give tht g( ) = th, > 0 e sketch the grph of = + f( ) e e A A B B A A B B A+ B ( A+ B) A+ B ( A+ B) e + + e e + e A+ B ( A+ B) e + e 4 A e + B A B + e ( + ) cosh( A+ B) s required Use the defiitios of cosh d sih Epd the rckets, usig lws of idices. PURE write dow the equtios of the smptotes. 5 Give tht f( ) = cosh, R, sketch the grphs of = f( ), for > 0 = f( ) +, for > 0 c = f( ), for < 0 6 Solve ech of these equtios, givig our swers to sf. sih = 5 cosh = c th = d cosh( + ) = e sih( ) = 4 f th( ) + = 09

108 7 Solve these equtios, levig our swer s logrithm i its simplest form where pproprite. sih = 0 cosh = c sih = d cosh = e th = f th = Solve these equtios, levig our swer s logrithm i its simplest form where pproprite. cosh sih = sih + cosh = c sih = cosh d sih cosh =4e Strteg Emple 7 Resoig d prolem-solvig Hperolic fuctios hve idetities which re ver similr to the trigoometric oes ou lred kow. You c prove these usig the defiitios. Ler this importt idetit: cosh sih To solve prolems usig hperolic fuctios Use the defiitios of sih, cosh d th Use lws of idices d logrithms s ecessr to simplif epressios. Prove tht cosh sih e cosh sih + e e e + + e s required. e e + e 4 Ke poit Notice how the sig is differet to the trigoometric idetit cos + si Use the epoetil defiitios. Simplif the umertor. 0 Curve sketchig Hperolic fuctios

109 Emple 8 Derive the logrithmic form of the iverse fuctio cosh Let = cosh e The = cosh = = e + e e e + = 0 Solve to give e = ± = l ± + e ( ) ( ) ( ) Cosider = ( + ) = + ( ) ( ) c e writte l( + ) = l( + ) =± l( + ) So l So To void miguit the iverse fuctio is defied s the positive vlue: ( ) cosh = l + for The other iverse hperolic fuctios c e derived i similr w. ( ) ( ) sih = l + + cosh = l + ; th l + = ; < < Ke poit Use the defiitio of cosh Multipl e to give qudrtic i e Tke logs of oth sides of the equtio. Sice + ( )( ) = ( ) = If <, the < 0, therefore cosh ol eists whe You c lso refer to the iverse fuctios s rsih, rcos d rth PURE Strteg To solve qudrtic equtios ivolvig hperolic fuctios: Use idetities to write the equtio i terms of sigle hperolic fuctio. Solve the qudrtic equtio to fid the vlue of sih, cosh or th Use the defiitios of the iverse hperolic fuctios to fid the ect vlues of

110 Emple 8 Solve the equtio sih + cosh = Eercise.5B Solve ech of the equtios. Give our swers s ect logrithms where pproprite. sih + cosh = cosh + sih = 5 c sih + cosh = d th + th = Use the defiitios of sih d cosh to prove tht cosh sih Use the defiitios of cosh d sih to prove these idetities. sih + (+ sih ) = sih + sih = 0 sih =, = rsih, rsih() rsih l = + + l + = 5 rsih( ) = l( + ( ) + ) = l( + ) So l + 5 =,l( ) + sih (A + B) = sih (A) cosh (B) + sih (B) cosh (A) cosh (A B) = cosh (A) cosh (B) sih (A) sih (B) e 4 Use the defiitios of sih d cosh to prove tht th = e + th Hece show tht th () = + th 5 Use the defiitios of cosh d sih to prove tht cosh () cosh Hece fid the ect solutios to the equtio cosh () + cosh = 5 ( ) 6 Use similr method s i Emple 8 to prove tht rsih = l + + ( ) Epli clerl wh rsih l + Use the idetit cosh sih Use clcultor or other method to fid vlues of sih Use the fct tht rsih = l( + + ) Curve sketchig Hperolic fuctios

111 7 Prove tht rth l + = Epli wh the formul i prt is ol vlid for < < 8 Solve these equtios, givig our swers s logrithms. sih = cosh = 4 c th = 9 Give tht sih =, clculte the ect vlue of cosh th 0 Give tht cosh =, clculte the ect vlue of sih th Give tht th =, clculte the ect vlue of sih cosh Use the defiitios to fid the derivtives of sih d cosh. Fid the grdiet of oth curves whe = 0. c Show tht for > 0 the curve = cosh is lws steeper th = sih d Fid the grdiet of the tget to the curve = th t = 0 PURE

112 Summr d review Chpter summr Whe sketchig the grph of fuctio of the form + c + d F fid the itercepts d smptotes, F cosider wht hppes whe d ecome ver lrge d pproch ifiit. + + c Whe sketchig the grph of fuctio of the form d + e + f F use the ove techiques d lso fid limittios o the possile rges of F use the results of these techiques to fid turig poits. You c use sketches of grphs to solve iequlities. For poit P with polr coordites (r, q ) Y F r is the legth of OP so r 0 P(, ) F q is the gle etwee OP d the iitil lie OX. P(r, q) To covert coordites from polr to Crtesi use = r cos q d = r si q r q To covert coordites from Crtesi to polr use O r = + d tθ = To covert polr equtio ito Crtesi form use cos θ = d si θ = + + A hlf-lie is stright lie tht eteds ifiitel from poit. You c sketch polr curves clcultig r for differet vlues of q. The equtio of prol cetred t the origi is 4 The equtio of ellipse cetred t the origi is + = The equtio of hperol cetred t the origi with smptotes t = A rectgulr hperol cetred t the origi with smptotes = 0 d = 0 hs equtio = c The hperolic fuctios re defied usig epoetils d hve grphs s show. F e e sih = sih F e + e cosh = F th e e = e + e cosh sih X cosh 4 Curve sketchig Summr d review

113 The iverse hperolic fuctios re ( ) ( ) F sih = l + + F cosh = l +, F th l + =, < < th PURE Check d review You should ow e le to... Tr Questios + Grph fuctios of the form c + d d + + c, d solve ssocited d + e + f iequlities. Use qudrtic theor to fid the rge of possile vlues of fuctio d fid sttior poits. Covert etwee polr d Crtesi coordites d equtios. 6 9 Sketch polr curves d fid poits of itersectio. 0 Sketch prol with verte t the origi d ellipse cetred t the origi, givig the coordites of the poits of itersectio with the coordite es. 4, 5 4, 5 Sketch hperol cetred t the origi, givig the equtios of the smptotes. 6, 7 Sketch sigle trsformtios of coic sectios. 8 0 Fid the equtio of coic sectios reflected i the lies = or = Kow the defiitio of sih, cosh d th i terms of epoetils; clculte ect vlues d solve equtios., 5, 6 Sketch hperolic grphs d trsformtios. Recll d use the idetit cosh sih 4 Derive the logrithmic form of the iverse hperolic fuctios d clculte ect vlues. 6 7 ( ) Sketch the grph of = ( + 4) Solve these iequlities grphicll, mig itercepts d verticl d/or horizotl smptotes. + < 4 < Sketch the grph of = =

114 4 4 Determie the rge of vlues of the fuctio = where is urel Fid the loclised miimum turig of the fuctio d fid mimum vlue Covert the sets of polr coordites to Crtesi coordites. π 5, π 5, 7 Covert the sets of Crtesi coordites to polr coordites. ( ) (,) ( ) d (, ), c, 8 Epress these Crtesi equtios i polr form. = = + c + = 6 d ( ) + ( ) = 9 Epress these polr equtios i Crtesi form. The circle r = 4 si q sttig its cetre d rdius. The grph r = 4 si q 0 For ech polr equtio i sketch the hlf-lie it represets, ii give the Crtesi equtio of the lie it lies o. θ = π θ = π 4 For ech of the polr equtios i sketch the grph for r 0 d 0 θ < π, ii stte the mimum d miimum vlues of r. r = r = 7cos θ c r = 4cos θ d r = + si θ e r = 4 cos θ f r = θ Fid the poits of itersectio etwee the polr curves r = d r = cos θ Fid the poits of itersectio etwee the polr curves r = siθ d r = cos θ 4 Sketch the prol = 4 Fid the poits of itersectio etwee this curve d the lie = 5 Sketch ech of these ellipses, clerl lellig where the cross the coordite es. + = = 6 Sketch the hperol =, givig the equtios of the smptotes Curve sketchig Summr d review

115 7 A rectgulr hperol hs equtio = 5 Sketch the curve d stte the equtios of the smptotes. Show tht the hperol does ot itersect the lie with equtio + = 8 Sketch these prols, clerl lellig where the cross the coordite es. = ( ) = 9 Sketch these ellipses, clerl lellig where the cross the coordite es. ( ) ( 5) + = + ( + ) = 4 5 PURE 0 Sketch these hperols d stte the equtio of the smptotes. ( + 5) ( + ) ( ) = = c = 7 d ( + )( 4) = The prol + = is reflected i the lie =. Sketch the curve d write dow its 8 equtio. The hperol 6 = 6 is reflected i the lie =. Sketch the curve d write dow the equtios of its smptotes. Clculte the ect vlue of cosh(l5) th( l) Sketch these grphs for. = cosh = sih( ) c = th d = cos 4 Solve the equtio cosh + sih =. 5 Solve these equtios, givig our swers s logrithms. sih + cosh = cosh + = e 6 Use the defiitio of cosh to prove tht rcosh( ) = l+ 4 ( ) 7 Fid the ect solutio to the equtios sih = th = 7

116 Eplortio Histor Je-Roert Argd ws Swiss mthemtici who, erl i the 9th cetur, moved to Pris where he mged ookshop. He pulished ess o the represettio of imgir qutities i 8 d, lthough others efore him hd writte out similr ides, it ws Argd s ess tht gied ttetio d resulted i the mig of the Argd digrm we cotiue to use tod. ellipse Note A coic sectio is curve tht is foud itersectig ple with doule coe plced verte to verte. There re three tpes of coic sectio: ellipse, prol d hperol. A circle is tpe of ellipse. Ivestigtio prol circle hperol The equtio z = plotted i the comple ple gives circle with uit rdius cetred t the origi. Choose vlues of, d c such tht c > ( ) d > B vrig these vlues, ivestigte the grph of the locus of poit z tht moves so tht z + z = c, c > ( ) d > Wht is the shpe of the grph tht results? The rel prt of comple umer z c e deoted Re(z) B vrig the vlues of d, ivestigte grphs of the locus of poit z tht moves so tht Re(z z + ) = z Wht is the shpe of the grph tht results? MATERIAL TO COME Reserch The Mdelrot set provides eutiful imges. It is the set of ll the comple umers, c, for which the itertive sequece z = z + c, + where z 0 = 0 does ot ted to ifiit. Computer imges of the Mdelrot set c e developed d, o mtter how closel ou zoom i o the surfce, it still remis icredil itricte. The Mdelrot set is emple of frctl. Fid out if the comple umers i d i re i the Mdelrot set. Reserch the properties of the Mdelrot set. = z + c, + 8

117 Assessmet 5 Sketch the grph of = +, clerl lellig itercepts with the coordite es. 7 [4] Write dow the equtios of the smptotes to the curve. [] 5 c Fid the coordites of the poits where the curve meets the lie = [] 5 7 Sketch the curve = [] + 5 Fid the poits of itersectio etwee the curve = d the lie + ( ) Epress the Crtesi coordites, s polr coordites, givig the ect vlue of r d writig q i terms of π where π < θ π. [4] Sketch the polr grph with equtio i r = 9 [] ii θ = π 6 [] c Fid the Crtesi equtios of the curves i prt. [5] 4 5 Epress the polr coordites 4, π 6 s ect Crtesi coordites. [] Fid the Crtesi equtio of the polr curve i r = siθ [] ii r = secθ [] c Sketch the grphs of the curves i prt o the sme es. [] d Stte the polr coordites of the poit of itersectio etwee r = siθ d r = secθ [4] 5 Sketch the grph of = d stte the me give to this tpe of curve. [] The lie with equtio = + c is tget to = Clculte the vlue of c 6 Give tht k >, sketch the curve with these equtios. Lel the poits of itersectio with the coordite es d give the equtios of smptotes. + = [] = [4] 9 k 9 k 7 Sketch these polr curves. r = 4q for 0 q π [] r = cos q [] c r = + siθ [] d r = 8+ cosθ [] 8 Use the epoetil defiitio sih to show tht sih(l) =. [] Solve the equtio sih 4 =, givig our solutio to deciml plce. [] 9 Sketch the grph of = th( + ). [] Write dow the equtios of the smptotes to the curve i prt. [] e c Use the epoetil defiitios of sih d cosh to show tht th = [4] e + d Hece solve the equtio th =. Give our emple s logrithm. [] 0 Sketch the grph of = + + [] Solve the iequlit [5] + 4 [4] 9 PURE

118 5+ 4 Sketch the curve = [] Solve the iequlit > [5] A rtiol fuctio is defied f( ) = + 6 A sketch of the curve = f() is show. For wht vlues of is the fuctio udefied? = Leve our swer s surd. [] Where does the curve of the fuctio cut the coordite es? c For wht vlues of is f() > [5] Show tht the equtio 5+ 9= 0 c e writte s ( 5)( + ) = k where k is costt to e foud. Hece sketch the curve of 5+ 9= 0, givig the equtios of the smptotes. [] ( + ) 4 Sketch the curve with equtio + ( ) =, lel poits of itersectio 4 with the coordite es. [] 5 Show tht the curve = hs polr equtio r = cosec θ [4] 6 Give the Crtesi equtio of the curve with polr equtio r = si θ [4] Sketch the polr curve foud i prt. [] c Stte the mimum vlue of r = si θ d the vlues of q t which it occurs. [] 7 Stte the mimum d miimum vlues of r = 4+ cos θ where 0 θ < π [] Sketch the curve r = 4+ cos θ [] 8 Sketch the grphs of = sih d = sih o the sme digrm. [] If = sih, use the epoetil defiitio of sih to show tht = l + +. [5] 9 The hperol with equtio = is reflected i the lie = 0 5 Write dow the equtio of the imge. [] 0 Curve sketchig Assessmet ( ) Sketch the imge d stte the equtios of the smptotes. [] 0 Sketch ech trsformtio of the prol with equtio = d write dow its equtio. A stretch of scle fctor i the -directio. [] A trsltio the vector 8 [] c A reflectio i the lie = [] Sketch these curves o the sme digrm: i r = si4θ [] ii r = siθ [] Fid the poits of itersectios etwee the two curves. Give our solutios s polr coordites. [8] Use the defiitios of cosh d sih to prove tht sih () sih cosh [] Hece fid the ect solutios to these equtios. i sih () sih = 0 ii sih cosh cosh () + 4 = 0 [0] O [] []

119 4 Itegrtio Oriettio Wht ou eed to kow Mths Ch Fidig solutios of simulteous equtios. Mths Ch4 Fidig defiite itegrls. Itegrl clculus tht llows us to clculte the volume of revolutio of two-dimesiol shpe out fied is c e used to support the desig of ojects i ll mer of idustries. For emple, such clculus techiques c e used to help desig icoic rchitecturl uildigs. I this cse the cross-sectio, or prts of the cross-sectio, of the uildig c e modelled mthemticl fuctio which c the e cosidered to e rotted out verticl is, to geerte the three-dimesiol outlie of the uildig. Developig this further, clculus c the e used to fid the volume iside the uildig. Such clcultios c help mke sure tht the uildig complies with helth d sfet regultios. Architecture is just oe re where such techiques hve ee used i desig recetl ow tht the desig of comple curved uildigs hs ee mde possile ecuse of the use of computers d the developmet of ew uildig mterils. The uderlig priciples c e used i wide rge of desig res icludig i techicl res such s the desig of crs through to developig pulic sculpture pieces. Wht ou will ler To clculte the me vlue of fuctio. To fid the re eclosed curve d lies. To clculte the volume of revolutio whe rotted roud the -is. To clculte the volume of revolutio whe rotted roud the -is. To clculte more complicted volumes of revolutio ddig or sutrctig volumes. Wht this leds to Ch9 Itegrtio Surfce re d volumes of revolutio. Creers Architecture

120 4. Me vlues Fluec d skills You lred kow tht ou c use itegrtio to fid the re, A, eclosed the curve -is d the lies = d = : A= f( ) d = f( ), the The Me Vlue Theorem is sed o the ide tht for cotiuous fuctio there is lws rectgle with ectl the sme re, A. The se of this rectgle will e etwee d. The height, h, of the rectgle is kow s the me vlue of the fuctio f() h O A = f() = f() A O You c clculte the height of the rectgle dividig its re, foud itegrtio, its width. So h = f d ( ) The me vlue of fuctio f( ) i the rge is give f d ( ) Ke poit Itegrtio Me vlues

121 Emple Emple Clculte the me vlue of the fuctio f( )= + 6 i the itervl [, 5] 5 Me vlue = ( 6 )d = + 5 = (8 ) = So the me vlue of + 6 i the itervl [, 5] is 4 (or.5) Use the defiitio of the me vlue s f( ) d Sustitute i limits. Clculte the me vlue with respect to t of the fuctio t t+ ( t ) for t ( ) t ( t+ )( t ) = t t 5t = t t t 4 Me vlue = 4 (t t t )dt 4 5 t = t t = = 5 So the me vlue of t ( t+ ( ) t ) for t is (or 5.9) ( ) Epd the rckets d use ide rules to simplif. Use the defiitio of the me vlue s f() t d t sice our fuctio is i terms of the vrile t Sustitute i limits. PURE Clcultor AC + / % = Tr it o our clcultor Defiite itegrls c e worked out o our clcultor. 4 -d Activit Fid out how to work out 4 d o our clcultor.

122 Eercise 4.A Fluec d skills The grph of = + 6is show. Clculte the re ouded the curve, the coordite es d the lie = Work out the me vlue of + 6 for O f( )= + Sketch the grph of = f( ), lellig where the curve crosses the coordite es. Clculte the re ouded the curve = f( ) d the -is. c Work out the me vlue of f( ) for ech of these itervls. i g( )= + 5 ii 0 Sketch the grph of = g( ), lellig where the curve crosses the coordite es. Clculte the re ouded the curve = g( ) d the -is. c Work out the me vlue of f( ) for ech of these itervls. i 5 0 ii 0 4 Clculte the me vlue of the fuctio f( )= 5 4 i ech of these itervls. [0, 4] [, ] c [, ] d, 5 Clculte the me vlue of + for ech of these rges of vlues of 0 6 c d 6 Clculte the me vlue of 8 for etwee d 4 0 d c d d d 7 Show tht the me vlue of for 5 is 0 8 Show tht the me vlue of for 4 is Show tht the me vlue of for 4 9 is Give tht f( )= Write f( ) i the form A c + B, for costts A, B d c d stte their vlues. Show tht the me vlue of f( ) i the itervl [, 9] is Give tht g( )= 5 g( ) i the form A B c d Write + for costts A, B, c d d d stte their vlues. Clculte the ect me vlue of g( ) for 4 Fid the me vlue of ech of these fuctios of t for the rge give. ( ) t 5t 8t + for t 0 t t t for t 4 4 c for t 9 t t d ( ) t t for 4 t 64 4 Itegrtio Me vlues

123 Fid epressio for the me vlue of t cross the itervl ( ) 0 t T T t T + Give our swers s polomils i T 4 Clculte the me vlue of the fuctio 4+ for 8 Give our swer i the form A 5 Derive epressio i terms of for the me vlue of the fuctio for i the itervl [, ]. Give our swer i its simplest form. 6 Fid epressio i terms of X for the me vlue of the fuctio, for i the 4 itervl 0,. Simplif our swer. X PURE Resoig d prolem-solvig Strteg Emple To solve prolems o me vlues Use the formul for the me vlue of fuctio. Form oe or more equtios usig the iformtio provided. Solve qudrtic or simulteous equtios. The velocit of prticle fter t secods is give v= t ms t Show tht the me velocit for 0 t 4is.9 ms so sigifict figures. Clculte the me ccelertio of the prticle over the first 4 secods. 4 Me velocit = t t t d 4 9 = t t t 4 8 d = t t = = =.9 ms s required dv Accelertio = dt 4 dv So me ccelertio = 4 0 dt dt 0 t = t 0 = (9 ) 0 4 = m s Use the formul for the me vlue of fuctio. Simplif ito form suitle for itegrtig. As this questio sttes to show the result, ou must write dow ll our workig. Use the formul for the me vlue of fuctio. dv Sice v = d t ou do dt ot ctull eed to itegrte here. See Mths Ch7 For remider o the lik etwee differetitio d kiemtics 5

124 Emple 4 The me vlue of the fuctio i the itervl [, ] is d i the itervl [, ] is. Clculte the vlues of d. Me vlue = 4+ 7 d 7 + = Usig me vlue theorem for first itervl. Form equtio. ( ) = ( )( + + 7) = Fctorise left-hd side = + 7 = ( ) = ( )(4+ + 7) = = Gives = = = Eercise 4.B Resoig d prolem-solvig The velocit of prticle fter t secods is give v= t t (ms ) 5 5 Clculte the me velocit for t Show tht the me ccelertio for t is.8 m s The velocit of prticle fter t secods is t 5 give v = ms 5 Show tht the me velocit over the first 5 secods is 4 ms Clculte the me ccelertio over the first 5 secods. Usig me vlue theorem for secod itervl. Fctorise LHS. Sutrctig st equtio from d equtio. B sustitutig vlue of ito either of the equtios. The displcemet of prticle fter t secods is give s= t t Show tht the me ccelertio over the rge t is 6 ms 4 The ccelertio of prticle fter t secods t 0 is give = for t > 0 0 t Show tht the me ccelertio for 9 t is m s 45 Give tht fter secod the prticle is trvellig t 5 m s, clculte the me velocit for t 6 Itegrtio Me vlues

125 5 Show tht the me vlue of stright lie = m i the itervl [, ] is give m ( + ) + c 6 Show tht the me vlue of the curve = for 0 is give 7 g( )= Sketch the grph of = g( ), lellig where the curve crosses the coordite es. c Clculte the re ouded the curve = g( ) d the -is. Fid the me vlue of g() for vlues of i ech of these itervls i [ 5, 0] ii [0, ] 8 The me vlue of the fuctio f( )= 5 for 0 is.5 Clculte the vlue of 9 The me vlue of the fuctio g( )= + for 0 X is 7 Clculte the possile vlues of X 0 Give h( )= + The me vlue of h( ) for is 75 d the me vlue of is 84. h( ) for Clculte the vlues of d Is the me vlue of the fuctio greter i the rge [, ] or the rge [, ]? Show our workig d stte how much igger the greter me vlue is. Prticle A hs speed t time t secods give t + t d prticle B hs speed t time t secods give t Which prticle hs the fstest verge speed over the rge 0 t? Show ll our workig d write dow precisel how much quicker this prticle s verge speed is. The grph of = A+ is show. B O Clculte the me vlue of the fuctio for Is it possile to clculte the me vlue of the fuctio for 0 7? Epli our swer. 4 Clculte the me vlue of ech of these fuctios for the itervl give. si for π, 0 cos for π π 6, c e for [, 0] d e for [ 0, ] Full A-level PURE 7

126 4. Volumes of revolutio Fluec d skills You c use clculus to fid the volumes of D shpes. You c lso geerte the formule for shpes like coes d spheres. = f() Emple Look t this grph of = f( ). If the sectio of this curve etwee = d = is rotted 60 roud the -is it will form solid. Sice the curve hs rotted through 60, the cross-sectio of this solid will lws e circle with rdius. Now cosider thi strips of thickess δ. Ech of these strips is pproimtel clider so it will hve volume of δv =π δ The volume of the whole solid is the sum of these thi strips. Usig itegrtio to sum these strips gives the formul V = π d The volume of the solid formed rottig the curve = f () etwee = d = full tur roud the -is is give V = π d Ke poit Becuse this is itegrl with respect to, ou eed the epressio i terms of ol, so replce the with the fuctio i terms of. Fid the volume formed whe the re eclosed the curve with equtio = +, the coordite es d the lie = is rotted 60 roud the -is, =π ( + ) V d 0 =π d =π =π = π cuic uits 7 80 roud the -is. Rottig through 80 produces ectl of the shpe i so the volume i this cse is 98 7 π = 99 7 π cuic uits. O d Usig V = π d where = + Epd d simplif efore ttemptig to itegrte. 8 Itegrtio Volumes of revolutio

127 You could lso e sked to clculte the volume whe curve is rotted roud the -is. I this cse, the roles of d re swpped. Therefore ou eed the fuctio to e = d the limits to e = d =. The the volume is V = π d f ( ) PURE = f() O Emple The volume of the solid formed rottig the curve = f( ) etwee = d = full tur roud the -is is give V = π d Ke poit The regio A is eclosed the curve with equtio = 5, the -is d the lies = 0.5 d = 0.5 O A = 5 Clculte the re whe regio A is rotted π rdis roud the -is. Re-rrgig gives ou = = π V 5 d =π d =π =π =.9 cuic uits 5 Alws rerrge ito the form = f( ) Usig V = π d, recll π rdis is the sme s 60. Give our swer to sigifict figures. 9

128 Eercise 4.A Fluec d skills Clculte the volume formed whe the regio ouded the curve =, the -is d the lies = d = 4 is rotted 60 roud the -is. The regio, R is eclosed the curve = 8 d the coordite es. Show tht the re of R is squre uits. The regio R is rotted 60 roud the -is. Clculte the volume of the solid formed O R The shded regio is eclosed the curve 4 = d the -is. Clculte the re of the shded regio. Clculte the volume of the solid formed whe the shded regio is rotted π rdis roud the -is. O 4 Fid the volume of the solid formed whe the regio ouded the curve = 4 5, the -is d the lies = 0.5 d = is rotted 80 roud the -is. 5 Clculte the volume of the solid formed whe the regio ouded the curve = d the lies = d = is rotted π rdis roud the -is. 6 The regio R is eclosed the curve =, the -is d the lies = d = Show tht the volume of the solid formed whe R is rotted π rdis roud the 7 -is is π 7 The regio A is eclosed the curve = +, the -is d the lie = 5 d = Show tht the re of A is 0 5 Clculte the volume of the solid formed whe the regio A is rotted i 60 roud the -is, ii 90 roud the -is. Give our swers to sigifict figures. 8 Clculte the volume of the solid formed whe the re ouded the curve with equtio = 4 d the lies = 0 d = 0 is rotted 60 roud the -is, the -is. 9 The re ouded the curve with equtio = 4 4 d the lie = d = 8 is rotted 60 roud the -is. Show tht the volume of the solid formed is A π where A is costt to e foud. 0 Clculte the ect volume of the solid formed whe the regio i the first qudrt ouded the curve = d the -is is rotted 80 roud the -is. 0 Itegrtio Volumes of revolutio

129 Resoig d prolem-solvig Strteg To solve prolems ivolvig volumes of revolutio Sketch grph. Note crefull which is ou re rottig the grph out. Choose the correct versio of the formul for volume of revolutio, either V = π d or V = π d Add or sutrct other volume where ecessr. PURE Emple The regio R is eclosed the curve = d the lies = 6, = 9 d = 0 Clculte the volume of the solid formed whe R is rotted 60 roud the -is. O 9 =π ( ) V d 0 9 =π d 0 9 =π 0 =π 8 0 R 6 9 = 8 Volume of coe = πrh = π = 9π 8 Volume required = π 9π 6 = π Sice it s more complicted situtio, it is helpful to sketch grph. First fid the volume formed whe the curve is rotted. Use the -coordite of the poit of itersectio for the upper limit. You c derive this from the volume of revolutio formul. Sutrct the volume of the coe to fid the volume required.

130 You c use volumes of revolutio to prove some stdrd volume formule. Emple 4 B rottig the curve = r etwee r d r roud the -is, 4 show tht the volume of sphere is πr r O r A grph will help ou uderstd wht the solid of rottio will look like. The solid of revolutio will e sphere of rdius r. r ( ) V =π r d r r =π r d r r =π r r Commet: Rememer r is just costt i this epressio. r =π π + r r r r 4 =π r = πr s required. Eercise 4.B Resoig d prolem-solvig The shded regio is ouded the curve = 9 8, the -is d the lie =.5 Use the formul. Itegrte with respect to The shded regio show is ouded the curve = 0, the -is d the lie = O Clculte the re of the shded regio. Clculte the volume of the solid formed whe the shded regio is rotted 60 roud the -is. Give our swer i terms of π 5 O Clculte the re of the shded regio. Clculte the volume of the solid formed whe the shded regio is rotted 60 roud the -is. Give our swer i terms of π Itegrtio Volumes of revolutio

131 The regio R is ouded the curve with 4 equtio = 6, the lie = 7 d the -is. Show tht the re of R is 6 squre 5 uits. Clculte the volume of the solid formed whe A is rotted 60 roud the -is. 4 The regio A is ouded the curve with equtio = 8+, the lie = 4 d the -is. Show tht the re of A is 04 squre uits. Clculte the volume of the solid formed whe A is rotted 80 roud the -is. 5 B rottig the lie = etwee = 0 d = 60 roud the -is, show tht the volume of coe is π rh, where r is the rdius of the se d h is the height. 6 Use volume of revolutio to show tht the volume of clider is π rh, where r is the rdius d h is the height. 7 Use volume of revolutio to derive formul for the volume of hemisphere of rdius r 8 The sectio etwee = d = of the A curve with equtio = is rotted π rdis roud the -is d the volume of the solid formed is 8 π Clculte the vlue of A Clculte the volume whe the sme sectio of the curve is rotted π rdis roud the -is. 9 Clculte the volume of the solid formed whe ech of these curves is rotted 60 roud the -is etwee the limits show. Give ech of our swers i its simplest form. =, = d = 5 π π = sec, = d = 6 ( ) c = 5, = d =.5 d = e, = 4 d = 0 0 Fid the volume of the solid formed whe the regio i questio is rotted 80 roud the -is. Give our swer i terms of π Full A-level PURE

132 4 Summr d review Chpter summr The me vlue of fuctio f () i the rge is give f( ) d The re eclosed etwee the curve with equtio = f (), the -is d the lies = d = is give d The volume of the solid formed rottig the curve = f () etwee = d = full tur roud the -is is give V = π d The volume of the solid formed rottig the curve = f () etwee = d = full tur roud the -is is give V = π d Check d review You should ow e le to... Tr Questios Clculte the me vlue of fuctio. 5 Fid the re eclosed curve d lies. 7, 0, Clculte the volume of revolutio whe rotted roud the -is. 6 8,, Clculte the volume of revolutio whe rotted roud the -is. 9 Clculte more complicted volumes of revolutio ddig or sutrctig volumes., Fid the me vlue of the fuctio f( )= for Show tht the me vlue of the fuctio g( )= + i the itervl [, 8] is A where A is costt to e foud. 6 Work out the me vlue of 4 i the itervl 4 Show tht the me vlue of the fuctio for i the itervl [, ] is 5 The velocit of prticle fter t secods is give v= t t 0 (5 4 ) ms Clculte the me velocit over the first 4 secods. Clculte the me ccelertio over the first 4 secods. 6 Clculte the volume of the solid formed whe the lie with equtio = etwee = d = 5 is rotted π rdis roud the -is. 4 Itegrtio Summr d review

133 7 The regio R is eclosed the curve with equtio =, the -is d the lie = Show tht the re of R is 4 squre uits. The regio R is rotted 60 roud the -is. Clculte the volume of the solid formed. O R 8 Clculte the volume of the solid formed whe the curve with equtio = 5 etwee = d = is rotted 80 roud the -is. 9 Show tht the volume of revolutio whe = 4 etwee = 0 d = is rotted π rdis roud the -is is π cuic uits. 0 0 The shded regio is ouded the curve 4 with equtio = +, 0, the coordite es d the lie with equtio = The regio A is ouded the curve with equtio =, the lie = d the -is is rotted 60 roud the -is. Sketch grph showig the regio A. c Clculte the volume of revolutio whe regio A is rotted π rdis roud the -is. Clculte the volume of revolutio whe regio A is rotted π rdis roud the -is. The shded regio is ouded the curve with equtio = 5 4, the lie = d the -is. O Clculte the re of the shded regio. Clculte the volume of the solid formed whe the shded regio is rotted 80 roud the -is. PURE O Show tht the re of the shded regio is 4 squre uits. 5 The regio is rotted 60 roud the -is. Clculte the volume of the solid formed. 5

134 4 Eplortio Goig eod the ems Histor Je-Roert Argd ws Swiss mthemtici who, erl i the 9th cetur, moved to Pris where he mged ookshop. He pulished ess o the represettio of imgir qutities i 8 d, lthough others efore him hd writte out similr ides, it ws Argd s ess tht gied ttetio d resulted i the mig of the Argd digrm we cotiue to use tod. ellipse Note A coic sectio is curve tht is foud itersectig ple with doule coe plced verte to verte. There re three tpes of coic sectio: ellipse, prol d hperol. A circle is tpe of ellipse. Ivestigtio prol circle hperol The equtio z = plotted i the comple ple gives circle with uit rdius cetred t the origi. Choose vlues of, d c such tht c > ( ) d > B vrig these vlues, ivestigte the grph of the locus of poit z tht moves so tht z + z = c, c > ( ) d > Wht is the shpe of the grph tht results? The rel prt of comple umer z c e deoted Re(z) B vrig the vlues of d, ivestigte grphs of the locus of poit z tht moves so tht Re(z z + ) = z Wht is the shpe of the grph tht results? MATERIAL TO COME Reserch The Mdelrot set provides eutiful imges. It is the set of ll the comple umers, c, for which the itertive sequece z = z + c, + where z 0 = 0 does ot ted to ifiit. Computer imges of the Mdelrot set c e developed d, o mtter how closel ou zoom i o the surfce, it still remis icredil itricte. The Mdelrot set is emple of frctl. Fid out if the comple umers i d i re i the Mdelrot set. Reserch the properties of the Mdelrot set. = z + c, + 6 Itegrtio Summr d review

135 4 Assessmet Clculte the me vlue of these fuctios for the limits show. + + for 0 ( ) for [8] You re give tht f( )= m Write + where A, B, m d re costts to e foud. [] f( ) i the form A B f( ) i the itervl. [, 4] Clculte the me vlue of Fid the me vlue of the fuctio g( )= + for Give our swer i the form + where d re costts to e foud. [4] 4 The velocit of prticle fter t secods is give v= 5+ 6t (m s ). Clculte the me velocit for t 5 [4] Clculte the me ccelertio for t 5 [4] 5 The me vlue of the fuctio + i the itervl [0, ] is 09. Evlute 6 The shded regio is ouded the curve =, the -is d the lie = 4 Fid the vlue of Clculte the re of the shded regio. [] The shded re is rotted oe full tur roud the -is. c Clculte the volume of the solid formed. [4] 7 Clculte the volume of revolutio whe the regio ouded the curve =, the -is d the lie = is rotted 60 out the -is. Give our swer i terms of π [4] 8 The regio R is ouded the curve = +, the -is d the lies = d = Clculte the volume of revolutio whe R is rotted 60 roud the -is. [4] 9 Clculte the volume of revolutio whe the regio ouded the curve = (5 ) is rotted 80 roud the -is. Give our swer to sigifict figures. [5] 0 The ccelertio of prticle is give = t t Give tht the fter secod the prticle is trvellig t m s, clculte the me velocit over the first 4 secods. [5] 5 f( )= + The me vlue of the fuctio f () i the itervl [, ] is 5 d the me vlue of f () i the itervl [0, ] is 98 Clculte the vlues of d Sketch the grphs of = 5 d = 5 o the sme es. [4] The eclosed regio ouded = 5 d = 5 is rotted 60 roud the -is. Clculte the volume of the solid formed. [5] O 4 [5] [] [5] 7

136 The curve with equtio = k etwee = 0 d = is rotted π rdis roud the -is d the volume of revolutio is π. Clculte the possile vlues of k [5] 5 4 g( )= e + + Show tht the me vlue of the fuctio g( ) i the itervl [0, ] is k 4 e4 +, where k is costt to e foud. [4] 5 Clculte the me vlue of the fuctio f( )= for i the rge Give our swer i its simplest form i terms of [4] 6 The me vlue of f( )= A( ) etwee i the itervl, is 5 Clculte the vlue of A [4] 7 Clculte the me vlue of ech of these fuctios for the limits show si for 0 π si for π π [9] 6 t 8 Clculte the me vlue of the fuctio f()= t t 4 l for t i the regio t 4 Give our swer i the form A+ B l where A d B re costts to e foud. [5] 8 Itegrtio Assessmet 9 The regio R is ouded the curve with equtio = e, the -is d the lie = Clculte the re of R Clculte the volume of revolutio whe R is rotted 60 roud the -is. [6] 0 The curve C is defied the prmetric equtios = t +, = 5t, The regio ouded C, the -is d the lies = 5 d = 0 is rotted π rdis roud the -is. Clculte the volume of revolutio of the solid formed. [6] Sketch the grph of = cosh [] The regio R is ouded the curve of = cosh, lies = ± Clculte volume of the solid formed whe R is rotted 90 out the -is. [6] The shded regio show is prt of ellipse with equtio + = 9 4 Write the vlues of d i the digrm. [] Clculte the volume whe the shded re is rotted π rdis out the -is. [5] The regio R is ouded the curve with equtio = t, the -is d the lie = π 4 R is rotted k π rdis roud the -is d the volume of the solid formed is π π Clculte the vlue of k 4 The curve C is defied the prmetric equtios = si θ, = siθ, 0 q π Clculte, i terms of π, the volume of the solid formed whe the regio ouded C d the -is is rotted 60 roud the -is. [8] [4] [6]

137 5 Mtrices Oriettio Wht ou eed to kow KS4 Trsformig ojects reflectio, rottio d elrgemet. Mths Ch Solvig simulteous equtios. Mths Ch6 Usig vectors. Mtrices hve m differet pplictios. Oe re of their use tht hs grow tremedousl i recet ers is i computer grphics, for emple i the video gmig idustr. The mthemtics of mtrices is used progrmmers to mke ojects move with relism i three dimesios. Trsformtio mtrices re used to trslte, rotte d scle ojects s well s provide relistic shdig or rederig, shift viewpoits d trsform colours. As well s i video gmig, these techiques re used i the film idustr where ccurte codig d rtistic tlet llow us to merge relit d grphics i ws tht mke it difficult to kow wht is rel d wht is virtul. Other sectors tht require ccurte visulistios lso rel o the mthemtics of mtrices. For emple the c e used i developig softwre to support desig work i rchitecture d i the utomotive idustr, d eve customers desigig ew kitche for their home. Wht ou will ler To idetif the order of mtri. To dd, sutrct d multipl mtrices sclr or coformle mtri. To ppl lier trsformtios give s mtrices, descrie trs formtios give s mtrices d write lier trsformtios s mtri. To fid ivrit poits d lies of lier trsformtios. To clculte determits d iverses of mtrices. To use mtrices to solve sstems of lier equtios. Wht this leds to Ch Further Mtrices Eigevlues d eigevectors. sstems. Mtri digolistio. Ch Grphs d etworks Mtri formultio of Prim s lgorithm. Ch4 Lier progrmmig d gme theor Zero sum gmes. 9

138 5. Properties d rithmetic Fluec d skills See Mths Ch6 For remider of vectors. Mtrices re w of represetig iformtio i form tht c e mipulted mthemticll. You will hve lred used vectors, which re prticulr sort of mtri. A mtri with rows d m colums hs order m Ke poit Emple For emple, the mtri 5 is sid to hve order The mtri 7 is sid to hve order 5 Computers use mtrices to crr out opertio, such s dditio or sutrctio, d to crr it out o multiple umers simulteousl. This is prticulrl used i the processig of computer grphics. If two mtrices hve the sme order the the c e dded or sutrcted ddig or sutrctig their correspodig elemets. You c thik of puttig the two mtrices o top of ech other d ddig or sutrctig the elemets tht mtch. To multipl mtri costt, ou should multipl ech of its elemets tht costt. 5 You re give tht = 6 5 A 0 4, B = d C = 0. Fid, if possile, A + B A + C c B d C A Not possile s A d B re ot of the sme order A+ C= = = Ke poit Ke poit Add correspodig elemets of A d C. 40 Mtrices Properties d rithmetic (Cotiued o the et pge)

139 c B = 0 5 = 0 5 Multipl ever elemet of B PURE 4 6 = d C A = = = The zero mtri, 0, is mtri, of order, with ll elemets equl to zero. Multipl ever elemet of A the sutrct from the correspodig elemet i C. This is clled zero mtri s ll its elemets re 0 Ke poit Mtrices c ol e multiplied together if the umer of colums i the first mtri is the sme s the umer of rows i the secod mtri. If two mtrices c e multiplied, the it is sid tht the re coformle for multiplictio. Ke poit Whe multiplig mtrices, to fid the elemet i the th row d mth colum, ou must multipl the first term of the th row i the first mtri the first term of the mth colum i the secod mtri, the the secod term of the th row the secod term of the mth colum d so o, the dd these terms together. The product of m mtri d m p mtri hs order p Ke poit To fid the first term i mtri multiplictio, look t the first row of the first mtri d the first colum of the secod mtri. The cosider the first row d secod colum. The move o to usig the secod row of the first mtri. Cotiue i this w util the lst term which is foud cosiderig the fil row of the first mtri d the fil colum of the secod. c c c d e f d e f d e f w w w w + d z = w + d + dz z = w + d + dz z = w + + ez cw + f c + fz 4

140 Emple 0 If = 5 A, B = fid, if possile, the products AB BA c B 0 AB = 7 4 c 5 0 ( 5) + (0 ) ( ) + (0 0) = ( 5) + ( ) ( ) + ( 0) (7 5) + ( 4 ) (7 ) + ( 4 0) 5 = 47 7 Not possile s B hs colums ut A hs rows. B 5 = (5 5) + ( ) (5 ) + ( 0) 5 = = ( 5) + (0 ) ( ) + (0 0) 5 It is ol possile to fid A if A is Ke poit squre mtri, tht is, it hs the sme umer of rows s colums. To fid the term i the rd row d the d colum: multipl the first term from the rd row of mtri A the first term from the d colum of mtri B. The multipl the secod term from the rd row of mtri A the secod term i the d colum of mtri B. Fill, dd these vlues together. Notice how the product of mtri d mtri is mtri. B mes B ë B It does ot me squre ech elemet of the mtri. Clcultor AC + / % = Tr it o our clcultor Some clcultors c e used to dd, sutrct d multipl mtrices. MtA MtB Activit Fid out how to work out 0 5 o our clcultor. 4 Mtrices Properties d rithmetic

141 Eercise 5.A Fluec d skills Write dow the order of these mtrices. 5 9 i 0 ii 9 iii 7 0 iv Which of the mtrices i prt is squre mtri? Clculte c d If A= B= 0, 6, = 0 C 7 0, D= 5 4 Clculte if possile or, if ot, epli wh. i A + D ii A + B iii B C iv B Show tht i 45 5A D = 5 4 ii C + 7B = Clculte these mtri products c d e f (9 ) ( 5) Show tht = If A B 7, 0 = =, 9 4 C= D 0 5, 5, = fid, if possile, or if the clcultio is ot possile, epli wh. AB CD c BCD d CB e B A f C 7 Clculte these mtri products, simplifig ech elemet where possile. 4 c d ( ) ( 0) Give A =, show tht A = k 7, sttig the vlue of k PURE 4

142 Resoig d prolem-solvig Strteg To solve prolems ivolvig mtri rithmetic If two mtrices re of equl order the equte correspodig elemets. Solve equtios simulteousl to fid the vlues of ukows. Use suscript ottio for the elemets of geerl mtrices, for emple,, etc. You c solve prolems ivolvig ukow elemets usig sic lger. Emple Emple 4 Give tht 5 =, fid the vlues of d 5 5 = = = d + 5 = 5 Rerrge + 5 = to give = 5 Sustitute this ito other equtio to give + 5 = 5 4 = 8 = = 5= 7 Multipl the mtrices o the left-hd side of the equtio. Sice the mtrices re equl we c equte correspodig elemets. Solve simulteousl. You could ow use mtri multiplictio o our clcultor to check the swer. Altertivel, ou m e give sstem of lier equtios d hve to write it i mtri form. You will e show lter how to use mtrices to solve sstems of equtios. Write mtri equtio for ech of these sstems of simulteous equtios. + 7= 5 5z = = z = 0 5 9z= 7 = = z Check equtig the idividul elemets. Notice tht the umer of vriles determies the umer of colums i the mtri, d the umer of equtios determies the umer of rows. 44 Mtrices Properties d rithmetic

143 You c ppl the method of proof iductio to mtrices. Mtri dditio is oth ssocitive, so A + (B + C) = (A + B) + C, d commuttive, so A + B = B + A Mtri multiplictio is ssocitive, tht is, (AB)C = A(BC), ut it is ot commuttive sice, i geerl, AB ñ BA Mtri multiplictio is lso distriutive, tht is, A(B + C) = AB + AC PURE Emple 5 Prove the ssocitive propert for mtri multiplictio of mtrices. Let A =, B = 4 c c d C = 4 c c4 The ( AB) C = 4 4 c c + + = c c 4 c c c c 4 ( + ) c+ ( + 4) c ( + ) c+ ( + ) c 4 4 = ( + 4 ) c+ ( + 44) c ( + 4 ) c+ ( + 44) c4 c + c + c+ c 4 c + c + c4+ c 4 4 = c + 4c + c + 4c 4 c + 4c + c 4+ 4c 4 4 ( c+ c) + ( c + c 4 ) ( c+ c4) + ( c + c) 4 4 = ( c+ c) + 4( c + c 4 ) ( c+ c4) + 4( c + c 4 4) c c c c 4 = c + c 4 c + c 4 4 = 4 = A(BC) s required. 4 c c c c 4 Use suscript ottio for the elemets of geerl mtrices. First fid AB. Now multipl C. Epd the rckets. Fctorise out terms from mtri A. See Ch.4 For remider of proof iductio. Strteg To solve iductio prolems ivolvig mtrices Check the se cse. Assume the sttemet is true for = k Use A k + = A A k to show sttemet is true for = k + 4 Write coclusio. 45

144 Emple 6 Strteg Prove iductio tht 0 = 0 for ll positive itegers 0 0 Whe = : = 0 0 d = so true for = Assume true for = k k+ The 0 0 = k = k 0 = k + 0 = ( k + ) s required Sice true for = d ssumig true for = k implies true for = k +, therefore true for ll positive itegers To solve prolems ivolvig tles usig mtrices Covert from tulr form ito mtri of suitle order. Idetif relevt vectors. Multipl the mtri the vector to perform the ecessr dt lsis. 4 Write coclusio, if required. Check the se cse. Assume the sttemet is true for = k, the check it is true for = k + k+ k Use A = A A Sice ou re ssumig tht k 0 0 = k Multipl the mtrices. Write the coclusio. 4 Emple 7 The tle shows the proilities of sprig d eig ri i four UK cities. Mrch April M Lodo Ediurgh Crdiff Belfst Write mtri, A, to represet the tle of iformtio d vector,, to represet the umer of ds i ech moth. Show how ou c use our mtrices to clculte the totl umer of ri ds epected i ech cit over the three moths. (Cotiued o the et pge) 46 Mtrices Properties d rithmetic

145 0. = 0.8 A Fid the vlues of,, c d d such tht + 4 = c 7 5 d Fid the vlues of, d c such tht 5 = 9 4 c Fid the vlues of d i ech cse. 5 = = 0 4 Solve ech of these equtios to fid the possile vlues of = (7 4) 0 (0 7 6 ) = 5 c d 5 0 = = = = = So we epect totl of 8 ri ds i Lodo, i Ediurgh, 5 i Crdiff d 7 i Belfst. Eercise 5.B Resoig d prolem-solvig 5 Clculte the vlues of, d c i this mtri equtio = c 9 6 Clculte the vlues of, d z i this mtri equtio = 0 z 8 7 Clculte the vlues of, d c i this mtri equtio c 5 = 5 8 Write mtri equtio for ech of these sstems of simulteous equtios. + 5= 6 6 = + z = z = z = 0 The colums of the mtri represet the moths d the rows represet the cities. Sice Mrch d M hve ds ut April hs 0 Multipl the vector the mtri. Write coclusio. Give swers to the erest d. 4 PURE 47

146 9 Prove iductio tht 5 = for ll positive itegers 0 Prove iductio tht = for 0 ll positive itegers Prove iductio tht = for ll positive itegers Prove iductio tht = 4 4( ) for 0 0 ll positive itegers Prove tht A + (B + C) = (A + B) + C for mtrices A, B d C 4 Prove tht mtri dditio is commuttive for mtrices. 5 Prove couter-emple tht mtri multiplictio is ot commuttive. 6 Prove tht A(B + C) = AB + AC for mtrices A, B d C 0 7 Prove tht = A 0 A for 0 0 mtri A Prove tht B 0 0 = 0 0 B for mtri B 9 The profit (i 000) mde three emploees for comp over the four qurters of er is show i the tle. Q Q Q Q4 Emploee A Emploee B Emploee C The comp pls to p i ouses 5% of profit from Q, % from Q, % from Q d % from Q4 c Write mtri to represet the tle of iformtio d vector to represet the percetge pid i ouses. Show how ou c use our mtrices to clculte the mout of ous pid to ech of the emploees. Write dow the totl mout of ouses pid to ll three emploees. 0 The sttioer requiremets of group of Mths techers is give i the tle. Red Pecils Rulers Bord Pper Pes Pes Clips Techer Techer Techer Red pes cost p ech, pecils cost 8p, rulers 8p, ord pes 0p d pper clips p Show how ou c use mtrices to clculte the epediture o ech tpe of sttioer ech techer. Write dow the totl cost of ll the sttioer required. Give tht e f = c d 0 g h 0, d verif tht e = d c d = c g d c fid similr epressios for f d h i terms of,, c d d Prove iductio tht = 0 for ll. Full A-level 48 Mtrices Properties d rithmetic

147 5. Trsformtios Fluec d skills Mtrices c e used to represet certi trsformtios such s some rottios, reflectios d elrgemets. Trsformtios tht c descried i this w re kow s lier. I order to fid the imge of vector uder trsformtio T = c d, we pre-multipl the vector the mtri, so = c d A poit (, ) c lso e represeted the vector 0 To fid the imge of poit A(, ) uder the trsformtio T = 0, multipl the vector T 0 = 0 = This is reflectio i the -is. O A (, ) (, ) A Ke poit 0 The mtri 0 represets reflectio i the -is. 0 The mtri 0 represets reflectio i the -is. The mtri 0 0 represets reflectio i the lie = 0 The mtri 0 represets reflectio i the lie = Ke poit 49

148 You c fid mtrices tht represet prticulr trsformtios cosiderig the effect the trsformtio will hve o pir of vectors. Emple Fid the mtri tht represets rottio of 90 ticlockwise out the origi. Let the mtri e give c d Choose two poits d their imges fter the rottio. (, 0) moves to (0, ) d (0, ) moves to (, 0) 0 So c d 0 = d 0 c d = 0 So from the first equtio: = 0, c = From the secod equtio: = d d = 0 0 Therefore the trsformtio mtri is 0 The mtri c d represets the trsformtio tht mps 0 to c d 0 to d (, 0) O (0, ) (, 0) Ke poit It follows from this fct tht the mtri 0 0 mps ll poits to themselves. I umer multiplictio the idetit is sice = = The mtri = Ke poit I 0 0 is kow s the idetit mtri d hs the propert tht AI = IA = A for ll mtrices We chose 0 d 0 s these re simple to visulise d will give uique vlues for,, c d d Notice tht the first colum of the mtri is the imge of 0 d the secod colum is the 0 imge of 0 0 This c e geerlised for squre mtri, i prticulr I = To fid the mtri tht represets stretch of scle fctor prllel to the -is, look t wht hppes to the vectors 0 d 0 A stretch prllel to the -is c e thought of stretch log the -is so ol the -coordites re ffected. The imge of 0 uder this trsformtio is 0 0 The imge of uder this trsformtio is 0 0 Therefore the trsformtio mtri is 0 50 Mtrices Trsformtios

149 The mtri Ke poit k 0 0 represets stretch of scle fctor k prllel to the -is. The mtri 0 0 k represets stretch of scle fctor k prllel to the -is. PURE Emple Fid the imge of the poits (, ), ( 4, 5) d (, 0) uder the trsformtio descried the mtri 0 0. Descrie the trsformtio geometricll = 0 0 This is elrgemet of scle fctor, cetre the origi. The mtri k 0 Trsformtios c e pplied to severl poits t oce formig mtri from their positio vectors. Esure ou lws pre-multipl the trsformtio mtri. Ke poit 0 k represets elrgemet of scle fctor k, cetre the origi. To fid the mtri tht represets 60 rottio ticlockwise out the origi, look t wht hppes to the vectors 0 d 0 cos60 The imge of 0 uder this trsformtio is si60 0 d the imge of is si60 cos60 cos60 si60 Therefore the mtri is si60 cos60 = This result c e geerlised for gle. Ke poit cosθ siθ The mtri siθ cos θ represets ticlockwise rottio gle q out the origi. Positive gles lws give ticlockwise rottios O See Mths Ch. For remider of ect vlues of trig fuctios. Trsformtios c lso e comied d ou c use the product of the mtrices to represet the comied trsformtio. 5

150 If the trsformtio represeted mtri A is followed the Ke poit trsformtio represeted mtri B, the BA is the comied trsformtio. Notice tht if ou wish to ppl trsformtio A to vector first, ou should multipl A the B hece BA See Ch 6. For vectors i Further Mths. Emple Fid the sigle mtri tht represets rottio of 70 ticlockwise followed reflectio i the -is. Rottio of 70 : 0 is trsformed to 0 d 0 is trsformed to 0 0 Therefore A = 0 Reflectio i the -is: 0 is trsformed to 0 d 0 is trsformed to 0 0 Therefore B = The order of the mtrices is importt. 0 So comied trsformtio = BA 0 0 = = 0 This is reflectio i the lie = O = You m recll lerig out D coordites i GCSE Mths. Poits i D spce hve D positio vectors, d ou will ler more out these i our Further Mths course. You c use mtrices to represet trsformtios i D. To reflect i the ple = 0, the mirror is ple through the -is d eteded i the positive d egtive z-directios. The imge of the vector 0 will e The vectors d 0 re i the mirror ple so re uchged whe 0 reflected i 0 0 Therefore the trsformtio mtri is z O The mtrices for reflectio i the ples = 0 d z = 0 c e foud usig the sme method. 5 Mtrices Trsformtios

151 The D reflectio mtrices ou eed to use re s follows. 0 0 Reflectio i = 0: Ke poit PURE 0 0 Reflectio i = 0: Reflectio i z = 0: You lso eed to kow out D rottio roud oe of the coordite es. This digrm illustrtes rottig the poit (, 0, 0) gle q ticlockwise roud the z-is. Notice how the z-coordite of the poit is uchged so this is ctull the sme s rottig roud the origi i the D cse. Therefore ou c represet ticlockwise rottio roud the z-is the mtri Notice how this is the mtri for ticlockwise rottio roud the origi. cosθ siθ 0 siθ cosθ Cosider ticlockwise rottio roud the -is or the -is i similr w. The D rottio mtrices ou eed to use re s follows. 0 0 Rottio roud the -is: 0 cosθ siθ 0 siθ cosθ Ke poit z O q cosθ 0 siθ Rottio roud the -is: 0 0 siθ 0 cosθ cosθ siθ 0 Rottio roud the z-is: siθ cosθ

152 Emple 4 Fid the mtri, T, tht represets rottio of 45 ticlockwise roud the -is d fid the imge of the poit P(, 4, ) uder T 0 0 T = 0 cos45 si45 0 si45 cos = P = 4 0 = + = So the ctul poit is (,, ) Eercise 5.A Fluec d skills Pre-multipl the vector represetig poit P the mtri T Sice ou re trsformig poit ou should give the fil swer s coordites. The mtri B represets stretch of scle fctor 5 prllel to the -is d stretch of scle fctor prllel to the -is. Use digrm to show the 0 trsformtio of the vectors d 0 uder mtri B Write dow mtri B The mtri A represets rottio of 80 Use digrm to show the trsformtio of the vector 0 uder this trsformtio. Write dow the imge of the vector 0 uder this trsformtio. c Write dow the mtri A 54 Mtrices Trsformtios

153 Fid the mtri which represets ech of these trsformtios. c d Reflectio i the lie = Rottio of 0 ticlockwise out the origi. Elrgemet of scle fctor 5, cetre the origi. Stretch prllel to the -is of scle fctor 4 4 A trigle hs vertices (0, ), (, ) d ( 4, ) Fid the imge of these poits uder the trsformtio represeted the mtri = 0 T 0 Descrie geometricll the trsformtio represeted mtri T. 5 The mtri represets ticlockwise rottio of q degrees, cetre the origi. Fid the vlue of q 6 A trsformtio is give = 0 A Fid the imge of the vectors d 4 uder the trsformtio represeted mtri A Descrie geometricll the trsformtio represeted mtri A 7 Write the trsformtio represeted ech of these mtrices c d 0 cos 0 si 0 0 e 0 f si0 cos 0 8 Give the trsformtio mtrices = A 0 d = 0 B, fid the mtri represetig i ii trsformtio A followed trsformtio B, trsformtio B followed trsformtio A Commet o our swers to prt. 9 Give the trsformtio mtrices 0 A = 0, B = 0 0 c descrie the trsformtios A d B geometricll. Fid the mtri represetig i trsformtio A followed B, ii trsformtio B followed A. Descrie oth trsformtios i prt geometricll s sigle trsformtio. 0 Fid the sigle mtri tht represets ech of these comitios of trsformtios. A rottio of 45 ticlockwise out the origi followed reflectio i the -is. c A reflectio i the lie = followed elrgemet of scle fctor out the origi. A stretch prllel to the -is of scle fctor followed clockwise rottio of 90 out the origi. A squre hs vertices t (, 5), (, ), (, ) d (, 5) The squre is rotted 5 ticlockwise out the origi d the stretched scle fctor of i the -directio. Fid the vertices of the imge of the squre followig these trsformtios. Write mtri A tht represets rottio of 5 ticlockwise out the origi. 0 Show tht A = 0 c Show tht A represets clockwise rottio of 90 out the origi. PURE 55

154 d 56 Descrie the trsformtio represeted A The rottio represeted the mtri A is such tht A = I d A I Write dow possile mtri A d descrie the trsformtio full. The rottio represeted the mtri B is such tht B = I d B I Write dow possile mtri B d descrie the trsformtio full. 4 Mtri P represets reflectio i the -is d mtri Q represets stretch of scle fctor prllel to the -is. Fid the sigle mtri tht represets trsformtio P followed trsformtio Q Descrie geometricll the sigle trsformtio tht could replce this comitio of trsformtios. 5 The mtri M represets elrgemet roud the origi of scle fctor k, k > 0, followed ticlockwise rottio of θ, 0 θ 80 roud the origi. Write M i terms of k d θ The imge of the poit (, ) uder M is (, 6) Fid the vlues of k d θ 6 A trsformtio is represeted the 0 0 mtri T = Fid the imge of the poits (4,, 0) d (, 5, ) uder T Descrie this trsformtio geometricll. 7 Fid the mtri tht represets ech of these trsformtios. A rottio of 0 ticlockwise roud the -is. A reflectio i z = 0 c A rottio of 45 ticlockwise roud the -is. Mtrices Trsformtios The mtri represets 0 ticlockwise rottio of θ degrees roud oe of the coordite es. Fid the vlue of θ d stte which is the rottio is roud. Fid the imge of the vector i + 4j k uder this trsformtio. 9 The mtri 0 0 represets 0 0 ticlockwise rottio of q roud oe of the coordite es. Fid the vlue of q d stte which is the rottio is roud. 0 The poit A(, 7, ) is trsformed trsformtio mtri M to the poit A (, 7, ) Descrie this trsformtio geometricll. Write dow the trsformtio mtri M Descrie the trsformtio represeted ech of these mtrices c d The mtri P represets clockwise rottio of 5 roud the -is. Fid P i its simplest form. Fid the imge of the poit (, 0, ) uder the trsformtio represeted P The mtri T represets ticlockwise rottio of 0 roud the z-is. Fid T i its simplest form. Fid the imge of the poit (, 0,) uder the trsformtio T

155 Resoig d prolem-solvig A poit which is uffected trsformtio is kow s ivrit poit. Give trsformtio mtri T d positio vector, if T = the represets ivrit poit. Ke poit PURE You c e sked to fid ivrit poits for specific trsformtio. Strteg Emple 5 To fid ivrit poits d lies Write vectors i geerl form. Pre-multipl vector trsformtio mtri to form sstem of simulteous equtios. Solve simulteous equtios to fid either ivrit poit or lie of ivrit poits. 4 Equte coefficiets to fid equtios of ivrit lies. Fid ivrit poits uder the trsformtio give Let e the ivrit poit. The = + = + = 0 d + = = Therefore = 0 = 0 d = 0 So the ivrit poit is (0, 0) For lier trsformtio, (0, 0) is ivrit poit. Ke poit Form equtios. Write geerl vector to represet the ivrit poit. I this cse ol the poit (0, 0) is ivrit. For some trsformtio, ll poits stisfig certi propert will e ivrit, i which cse ou c hve ivrit lie or lies. There re two ws i which lie c e ivrit. For emple, cosider reflectio i the lie = All poits o the lie will e uffected the trsformtio. Therefore = is lie of ivrit poits. O If ever poit o lie is mpped to itself uder trsformtio the it is kow s lie of ivrit poits. Ke poit 57

156 Emple 6 Fid the ivrit poits uder the trsformtio give Let e ivrit poit. The = + = = d = = So there is lie of ivrit poits give = Write geerl vector to represet the ivrit poit. Form equtios. If the two equtios simplif to the sme thig the there is lie of ivrit poits. Emple 7 I some cses, ever poit o lie will mp to other poit o the sme lie. For emple, cosider rottio of 80 roud the origi. The poit (0, 0) is ivrit ut if ou tke lie through the origi the ever other poit o tht lie will mp to differet poit o the sme lie. So lie through the origi will e ivrit lie. If ever poit o lie is mpped to other poit o the sme lie the it is kow s ivrit lie. Lies of ivrit poits re suset of ivrit lies. To fid the equtios of ivrit lies use = m+ c to rewrite our geerl vector d for the imge of tht vector. You should the use = m + c to fid the possile vlues of m d c Fid the equtios of the ivrit lies of the trsformtio give 0 0 m + c = + m+ c= ( m + c) = So ( m + c) = m( + m+ c) + c ( m m m ) + c mc c = 0 (m m ) + c mc = 0 m m = 0 m = 0 or c mc = 0 Therefore whe m = 0, c = 0 d whe m = ou hve c c= 0 so c c e vlue. Therefore the ivrit lies re = 0 d = + c (for c). O Write the vector i geerl form i terms of. Form simulteous equtios. Usig = m + c Collect terms i Equte coefficiets of Equte costt terms. Stte the ivrit lies. 4 4 Ke poit 58 Mtrices Trsformtios

157 Eercise 5.B Resoig d prolem-solvig Give the equtio of the lie of ivrit poits uder the trsformtio give ech of these mtrices Show tht the origi is the ol ivrit poit uder the trsformtio give ech of these mtrices I ech of these cses, decide whether or ot the order the trsformtios re pplied ffects the fil imge. Justif our swers. A reflectio i the -is d stretch prllel to the -is. c A rottio out the origi d elrgemet with cetre the origi. A reflectio i the lie = d stretch log the -is. 4 Descrie the comitio of trsformtios tht will e represeted ech of these trsformtio mtrices Fid the equtios of the ivrit lies uder ech of these trsformtios Suggest two differet trsformtios tht hve ivrit lie give the equtio = 0 Which of our trsformtios i prt hs lie of ivrit poits give = 0? 7 The ivrit lies uder trsformtio A re give the equtios + = c where c c e vlue. Descrie possile trsformtio A geometricll. 8 The ivrit lies uder trsformtio B re give the equtios = k where k c e vlue. Descrie trsformtio B geometricll. 9 Fid the equtios of the ivrit lies uder the trsformtio give ech of these mtrices c For ech of these trsformtios, fid either the ivrit poit or the equtios of the ivrit lies s pproprite. i ii Reflectio i the -is. Rottio of 90 roud the origi. iii Stretch of scle fctor prllel to the -is. iv Reflectio i the lie = For ech of the ivrit lies i prt, epli whether or ot it is lie of ivrit poits. A trsformtio is represeted the mtri = T 4 Fid the lie of ivrit poits uder trsformtio T Show tht ll lies of the form = + c re ivrit lies of the trsformtio T The mtri represetig trsformtio A followed trsformtio B is give 0 0. Fid the mtri tht represets the trsformtio B followed the trsformtio A Give either the ivrit poit or the equtio of ple of ivrit poits for ech of these trsformtios c d Full A-level PURE 59

158 5. Sstems of lier equtios Fluec d skills Whe ou re workig with squre mtrices ou c clculte vlue kow s the determit. The determit hs severl uses. Most sigifictl, it eles ou to fid the iverse of mtri. If A = c d the the determit of A is d c d is deoted A or det(a) Ke poit Clcultor 5 To fid the determit of the mtri A = use the formul det (A) = d c det (A) = (5 ) ( ) AC + / % = = 5 = Tr it o our clcultor You c use some clcultors to fid the determit. det(mta) If det(a) = 0 the A is clled sigulr mtri. 6 Ke poit Activit Fid out how to work 8 out det 4 o our clcultor. Emple Fid the vlue of k for which the mtri k is sigulr. det ( k) ( ) k = if mtri is sigulr the k = 0 k = The determit is give d c A sigulr mtri hs determit of 0 60 Mtrices Sstems of lier equtios

159 Cosider squre with vertices t (0, 0), (, 0), (0, ) d (, ). Uder the trsformtio = 0 T 0 the squre is elrged scle fctor cetre the origi. So the vertices of the imge re (0, 0), (, 0), (0, ) d (, ). The re of the origil squre is d the re of the imge is, which is lso the determit of T. This result eteds to ll lier trsformtios. PURE For trsformtio represeted mtri T, re of imge = re of origil det(t) Ke poit O Emple The trigle with vertices (, ), (4, ) d (0, ) is stretched scle fctor prllel to the -is d stretched scle fctor prllel to the -is. Fid the re of the imge uder this trsformtio. 0 The trsformtio is give T = 0 det(t) = = 6 The re of the origil trigle is 4 = 6 Therefore re of imge is 6 6 = 6 Notice tht i the emple ove, the determit is egtive ecuse the oriettio of the trigle chges. I other words the trigle hs ee flipped s well s elrged. If det(t) > 0 the the trsformtio represeted T preserves the oriettio of the origil shpe. I order to fid the determit of mtri, ou eed to fid the three miors of row of elemets i our mtri. Give mtri, the mior of elemet is the determit of the mtri remiig whe the row d colum of the elemet re crossed out. This mes mtri hs ie miors, oe for ech elemet i the mtri. c For emple, to fid the mior of i the mtri d e f, ou cross out the row d the colum g h i e ivolvig the fid the determit of the mtri h Therefore, the mior of is ei fh O 4 f tht remis. i A sketch will help with the re. Usig re of origil det(t) Ke poit c d e f det e f d f c d = + h i g i g g h i e h Ke poit 6

160 Emple Fid the determit of the mtri det = = 0 ( ) 0 ( 6) (0 ) = 66 + = 66 Use the formul e f d f + c d e h i g i g h Tke cre with egtive sigs. Emple 4 Recll tht = 0 I 0 is the idetit mtri. The iverse of mtri, A, is A where AA = A A = I If A = c d the the iverse mtri is give = d A det( A) c Ke poit Notice how this defiitio will ot work for mtri which is sigulr. Sigulr mtrices do ot hve iverse. Therefore, ol o-sigulr mtrices will hve iverse. 5 Fid the iverse of the mtri B = 4 det (B) = ( 4) (5 ) = 8 5 = 4 5 B = Checkig swer: BB = = 0 = 0 0 = I First clculte the determit usig d c d Use B = det( B) c You c check our swer verifig tht BB = I 6 Mtrices Sstems of lier equtios

161 Emple 5 Clcultor AC + / % = Tr it o our clcultor Some clcultors c e used to fid the iverse of mtri. A mtri A is self-iverse if A = A Ke poit To fid the iverse of mtri, P First fid the mtri of miors, M. To do this, replce ech elemet i A its mior. c A B C So if P = d e f, the the mtri of miors is M = D E F where A is the mior of, B g h i G H I is the mior of d so o. Trspose the mtri (swp rows d colums), d chge the sig of ltertig elemets to A D E give B E H C F I Divide this ew mtri the determit. A D G The iverse of P is P = B E H det( P) C F I Fid the iverse of A = 0 ( ) ( ) ( ) ( 0) ( ) ( 0) M = ( ) ( ) ( ) ( 0) ( ) ( 0) ( ) ( ) ( ) ( ) ( ) ( ) 6 = det( A) = ( ) ( ) + ( 6) = 9 5 Therefore A = MtA - 4 Ke poit This could e worked out o clcultor ut ou do eed to kow the method. Fid the mior of ever elemet. Use the miors from the top row of the mtri s det( A) = 0 0 Use A Activit Fid out how to work out the iverse of 4 o our clcultor. A D G = B E H det( A) C F I PURE 6

162 Eercise 5.A Fluec d skills Show tht det = Clculte the determits of ech of these mtrices c d + Show tht det + = ( ) 4 Fid the possile vlues of k give tht 7 k det k k = 5 Decide whether ech of these mtrices is sigulr or o-sigulr. You must show our workig c 64 4 Mtrices Sstems of lier equtios d 6 6 Fid the vlues of for which ech of these mtrices is sigulr. 5 4 c d Give tht A =, clculte the possile vlues of for which the mtri A hs o iverse. 8 Give the mtrices = A 0 5 d 0 B = 4 Clculte the vlues of i det( A) det( B) ii det( A) + det( B) Show tht i det( AB) = 0 iii det( A+ B) = 7 9 Clculte the determit of ech of these mtrices Fid the possile vlues of k give tht 4+ k 5 k det k = Fid the vlues of for which ech of these mtrices is sigulr A rectgle hs re 7 squre uits. Fid the re of the imge of the rectgle uder ech of these trsformtios. i ii 5 Epli whether or ot ech of the trsformtios i prt preserves oriettio. A trigle is trsformed the mtri 4 The re of the imge is 5 squre uits. Clculte the re of the origil trigle. Epli whether or ot the oriettio of the trigle hs chged. 4 A trigle with re 5 squre uits is trsformed the mtri 0 Clculte the re of the imge. Epli whether or ot the oriettio of the trigle hs chged. 5 Fid the iverse of ech of these mtrices c 5 d

163 6 6 Show tht the iverse of 9 is 7 = A Fid the iverse of A, writig ech elemet i its simplest form. Show tht det( A ) = for the det( A) mtri A give. 8 Show tht the mtri is self-iverse. 4 9 Fid the vlues of such tht the mtri is self-iverse. 0 Fid the vlue of such tht the mtri 5 is self-iverse. + B = Give tht the determit of B is 0, fid the possile vlues of Write the iverse of B Fid the iverse of ech of these mtrices Fid the iverse of the mtri k B = k 0 i terms of k 0 k 4 = 0 T 0 Show tht T = I Descrie the trsformtios represeted T d T geometricll. 5 A trsformtio is represeted the mtri T = The imge of the poit A uder T is A (0, 7). Use the iverse of T to fid the coordites of A 6 Write the mtri represetig rottio of 5 ticlockwise out the origi. Show tht the iverse of this mtri is 7 A trsformtio is represeted the mtri T = 0 The imge of the poit A uder T is A (,, ). Use the iverse of T to fid the coordites of poit A 8 For ech of these trsformtios, epli whether it is lws, sometimes or ever self-iverse. For those which re sometimes self-iverse, stte whe this occurs. Reflectio i coordite is. Elrgemet cetre the origi. c Rottio out the origi. d Reflectio i the lie =±. PURE Resoig d prolem-solvig Mtrices c e used to solve sstems of lier equtios. Tke, for emple, the sstem of equtios + = c + d = If M = c d the M = = + c d c + d so the equtios c e writte usig mtrices s = M 65

164 The ou c pre-multipl oth sides M to give = M M M As M M= I, this c e rewritte s = M d ou c clculte the vlues of d Emple 6 Strteg To solve prolems ivolvig sstems of lier equtios Rewrite sstem of lier equtios usig mtrices. Pre-multipl or post-multipl mtri its iverse. Use the fct tht AA = A A = I Use mtrices to solve the simulteous equtios + = 4 = 6 You c write these equtios i the form = 4 6 = 4 4 = 4 4 So = = = Therefore = d = 4 Use AA = A A = I Rewrite the sstem of lier equtios usig mtrices. Usig fct tht the iverse of A = c d is d det( A) c A equtio of the form + + cz = d c represet ple. If ou hve three differet ples the oe of these will ppl. There re o poits tht lie o ll three ples (their equtios re sid to e icosistet). This could e ecuse two or three of the ples re prllel or ecuse the three ples form trigulr prism. Pre-multipl oth sides of the origil equtio the iverse mtri. 66 Mtrices Sstems of lier equtios

165 There re ifiitel m solutios s the three ples meet log lie (clled shef). PURE The itersect t sigle poit, so there is ectl oe solutio. Emple 7 Give tht there is uique solutio, use mtrices to solve this sstem of equtios: + + z = z = + z = Write the equtios i the form = z det = ( 6 ) (4 ) + ( 4 9) = = = z So =, =, z = = 0 z = Rewrite the sstem of lier equtios usig mtrices. Clculte the determit. Fid the iverse. Pre-multipl oth sides of the origil equtio the iverse mtri. Use AA = A A = I 67

166 Emple 8 Epli wh these sstems of equtios hve o solutios. z = + z = z = + = 0 8 6z = + + z = 7 The equtio 4 6 z = is multiple of z = ecept for the costt term. Therefore these represet prllel ples. Prllel ples do ot itersect so there re o solutios. Addig the first d third equtios together gives + = However, the secod equtio is + = 0 Therefore the equtios re icosistet d there re o solutios. The sstem of equtios i Emple 9 c e writte i mtri form s 5 0 = 0 z 7 Also, otice tht 8 6z = is multiple of 4 6 z = so these represet the sme ple. Noe of the ples re the sme or prllel so we eed to strt trig to simplif the prolem elimitig vrile. The ples form trigulr prism. You c ttempt to solve these pre-multiplig oth sides of the equtio the iverse of 0. However, for this prticulr sstem of equtios det 0 ( 0) ( 0) ( ) 0 = = This implies tht there is either o solutio (s i this cse) or ifiite umer of solutios. Strteg To decide o the ture of sstem of simulteous equtios Rewrite the sstem of lier equtios usig mtrices. Clculte determit o-zero determit implies uique solutio. Write two of the vriles i terms of the third. 4 Check for icosistecies. 68 Mtrices Sstems of lier equtios

167 Emple 9 Emple 0 You re give sstem of simulteous equtios: + 7z = 8, z=, + z = 4 Decide whether there is uique solutio, ifiite umer of solutios or o solutios. Descrie the geometric sigificce. Rewrite the sstem of lier equtios usig mtrices = z 4 Sice the determit is zero, must either e o 7 solutio or ifiite umer det 0 (0 ) ( ) 7( 0) = of solutios. = 0 This implies there is ot uique solutio. Sutrctig the third equtio from the first equtio gives 4 8z = = + z The secod equtio gives = z Verif there re o icosistecies: ( z ) + (+ z) 7z = 8 ( z ) z = ( z ) (+ z) + z = 4 So there is ifiite umer of solutios. These lie o the lie with equtios = + z, = z The ples meet i lie: the form shef. You c costruct mtri proofs pre-multiplig or post-multiplig oth sides of equtio. Prove tht AB ( ) = ( AB) ( AB) = I ( AB) AB = I ( AB) ABB = IB ( AB) A= B ( AB) AA = B A B A ( AB) = B A s required. Sice AA = I Here d re epressed i terms of z, ut ou could hve chose of the three vriles to write the other two i terms of the third. You c verif there re o icosistecies sustitutig ck ito the origil equtios. You will ler more out writig the equtio of lies i D lter i the course. Sice mtri multiplictio is ssocitive. Post-multipl oth sides of the equtio B Sice BB = I d IB = B 4 Post-multipl oth sides of the equtio A PURE ( ) = Ke poit If A d B re o-sigulr mtrices the AB B A 69

168 Eercise 5.B Resoig d prolem-solvig Use mtrices to solve these pirs of simulteous equtios. 4 = 5= 0 + = 5 8= c + 6= d 4+ 6= = 8+ = Use mtrices to fid solutio to these sstems of lier equtios. + z = 4+ 6 z = + 5z = 4 = 5+ + z = 8+ 4z= 0 5 Give tht, 0 fid epressios i terms of for d Simplif our swers. 4 You re give the equtios ( k+ ) = k d k + = k Use mtrices to fid epressios i terms of k for d 5 Give tht = A , fid the mtri A 5 6 Give tht B = 4 0, fid the mtri B Give tht A 0 4 = 5 0 5, fid the mtri A Give tht 0 B = 4 8 6, 4 0 fid the mtri B 9 Give tht P =, fid epressio for det( P) i terms of d. Full simplif our swer. 0 If = 5 A 4 d = 6 8 AB 0, fid B If B = d = 0 5 AB, fid the mtri A If A = 4 d AB = 0 5, fid the mtri BA For ech sstem of equtios, show tht the ples tht the equtios represet form shef. + z= + + 4z = z = z = 0 + z = z = 9 4 Fid the vlues of for which this sstem of simulteous equtios does ot hve uique solutio. + z = z= + = 5 Show tht ech of these sstems of lier equtios is icosistet d epli the geometric sigificce. + z = z = z = + z = z = z = 6 A fmil keeps rits, hmsters d fish s pets. The iitill hve 7 pets i totl d oe more rit th hmsters. Two of the fish die. The the hve 8 more fish th hmsters. Write mtri equtio to represet this situtio. Solve the mtri equtio d stte how m of ech pet the fmil hd iitill. 70 Mtrices Sstems of lier equtios

169 7 A propert developer ows homes which hve mi of two, three d four edrooms. The totl umer of edrooms i ll her houses is 6 d she hs the sme umer of two-edroom homes s four-edroom homes. c Write mtri equtio to represet this situtio. Show tht there is o uique solutio to this prolem. How m possile solutios re there? Epli our swer. 8 If A = I, write the mtri A 9 Simplif ech of these epressios ivolvig the o-sigulr mtrices A d B ( AB) A BA ( B) 0 If A= C BC, prove tht B= CAC Prove tht if ABA = I the B= I Prove tht ( ABC) C = B A for osigulr mtrices A, B d C Prove tht if P is self-iverse the P = P 4 Prove tht if PQP = I for o-sigulr mtrices P d Q, the Q= ( P ) 5 A poit P is trsformed the mtri T = 4 d the coordites of the imge of P re (9, ) Fid the coordites of P 6 The trigle ABC is stretched scle fctor k prllel to the -is the reflected i the lie =. The imges of vertices A, B d C re give A (4, 6), B (0, 0) d C (4, ) Work out the re of the origil trigle i terms of k c d Stte the rge of vlues of k for which the trsformtio preserves the oriettio of the trigle. Fid the coordites of A, B d C i terms of k If the coordites of C re (, 4), fid the vlue of k 7 Prove tht det( AB) = det( A)det( B) for o-sigulr mtrices A d B 8 Prove tht det( A ) = [ det( A) ] for osigulr mtri A 9 Write the mtri tht represets i ii rottio of gle q ticlockwise roud the origi, rottio of gle q clockwise roud the origi. Hece, deduce the idetit si θ+ cos θ 0 B cosiderig squre with oe verte t the origi, show tht, uder lier elrgemet, T, the imge of the squre will e squre. the re of the imge will e the re of the origil squre multiplied the determit of T B cosiderig squre with oe verte t the origi, show tht, uder stretch, T, i the - or -directio (or oth), the imge of the squre will e rectgle, the re of the imge will e the re of the origil squre multiplied the determit of T Full A-level PURE 7

170 5 Summr d review Chpter summr A mtri with m rows d colums hs order m Mtrices of the sme order c e dded ddig correspodig elemets. A mtri c e multiplied costt multiplig ech of its elemets tht costt. Two mtrices re coformle for multiplictio if the umer of colums i the first mtri is the sme s the umer of rows i the secod mtri. The product of m mtri d m p mtri hs order p Uder the trsformtio give the mtri c d the vector 0 is trsformed to c d the vector 0 is trsformed to d The mtri represetig trsformtio A followed trsformtio B is give the product BA The idetit mtri is mtri with oes log the ledig digol d zeros elsewhere, so i D, = 0 I 0 The zero mtri is mtri with 0 s ever elemet, so i D, 0 = The ivrit poits of the mtri T c e foud solvig = T The ivrit lies of the mtri T c e foud solvig T + = m c m + c to fid the vlues of m d c The determit of mtri is det c d = d c c The determit of mtri is det d e f = e f d f + c d e g h i h i g i g h A mtri is sigulr if its determit is zero. Uder trsformtio T, re of imge = re of origil det(t) If det( T) > 0 the the trsformtio represeted T preserves oriettio. If A = c d the its iverse is d A = det( A) c c A D G If = d e f P the its iverse is P = B E H where A is the mior of, B is det( P) g h i C F I the mior of B d so o. For o-sigulr mtri A, AA = A A= I 7 Mtrices Summr d review

171 Check d review You should ow e le to... Tr Questios Idetif the order of mtri. PURE Add d sutrct mtrices of the sme order d multipl mtrices costt., Multipl coformle mtrices., Appl lier trsformtio give s mtri to poit or vector., 4 Descrie trsformtios give s mtrices geometricll., 4 Write lier trsformtio give geometricll s mtri. 5, 6 Fid the mtri tht represets comitio of two trsformtios. 7 Fid ivrit poits d lies. 8, 9 Clculte the determit of mtri d of mtri. 0, Uderstd wht is met sigulr mtri., Fid the iverse of mtri d of mtri. 4 9 Use mtrices to solve sstems of lier equtios. 0 4 Descrie the geometricl sigificce of solutios., 4, 4 6 A = 4, B 5 0 = 5 4 0, C 4 0 = d = D Stte the order of ech of the mtrices. Fid ech of these mtrices. i B + D ii C iii D B iv A c Clculte these mtri products where possile. If it is ot possile, epli wh ot. i AB ii BA iii BC iv CA v AC vi CD vii A viii C Give tht = A B = 0 d C = 0, 0 fid ech of these mtrices i terms of CA BC c A d B Appl ech of these trsformtios to the poit (, ) d descrie the effect of the trsformtio geometricll i 0 ii 0 iii 0 iv A trigle hs re 5 squre uits. Give the re of the imge of the trigle uder ech of the trsformtios give i prt. c Epli which of the trsformtios i prt preserves oriettio. 7

172 4 Appl ech of these trsformtios to the poit (,, 4) d descrie the effect of the trsformtio geometricll Write dow mtri to represet ech of these trsformtios. Reflectio i the lie = 0 c Rottio of 5 ticlockwise roud the origi. Elrgemet of scle fctor, cetre the origi. 6 Write dow mtri to represet ech of these D trsformtios. Rottio of 45 ticlockwise roud the -is. Reflectio i the ple = 0 7 Write dow the sigle mtri tht represets ech of these trsformtios. c Rottio of 70 ticlockwise roud the origi followed stretch of scle fctor prllel to the -is. Reflectio i the lie = followed reflectio i the -is. Reflectio i the -is followed rottio of 50 ti-clockwise out the origi. 8 Fid the ivrit poits uder ech of these trsformtios Fid ll of the ivrit lies uder ech of these trsformtios Work out the determit of ech of these mtrices. 4 Work out the determit of ech of these mtrices Clculte the vlues of for which these mtrices re sigulr Clculte the vlues of for which these mtrices re sigulr Mtrices Summr d review

173 4 Fid the iverse, if it eists, of ech of these mtrices. If it does ot eist, epli wh ot c 4 8 d 7 5 Fid the iverse, if it eists, of ech of these mtrices. If it does ot eist, epli wh ot Give tht A = 4 d AB 0 =, work out mtri B 9 7 Give tht C = 5 d BC 7 = 60 7, work out mtri B 0 8 Give tht A = 4 d AB = 5 8, work out mtri B Give tht C = 0 d BC = 9 5, work out mtri B 4 0 Use mtrices to solve ech pir of simulteous equtios. + 6= 7 5 7= 8 = 7 0+ = Use mtrices to fid solutio to ech of the sstems of simulteous equtios. + z = z = 6 + z = + 6z = 5 + z = z = Epli wh ech of these sstems of equtios does ot hve solutios z = 4 z= 5 + z = 5 + z= + z = 7 + z = A sstem of equtios is give + z = 4 z= z = 0 Show tht there is ot uique solutio to the sstem of equtios. Show tht the ples descried the three equtios form trigulr prism. 4 A ivestor us shres i three differet compies: A, B d C. The totl mout ivested is 000 d he ivests three times s much i comp A s i comp B. After oe er he hs mde totl of 555 profit. His shres i compies A d B oth icresed i vlue 0% ut his shres i comp C lost 5% of their vlue. Represet this situtio s sstem of lier equtios Solve these equtios to fid the mout iitill ivested i ech of the compies. c Iterpret the solutio geometricll. PURE 75

174 5 Eplortio Goig eod the ems Histor The Russi mthemtici Adre Adreevich Mrkov (856 9) used mtrices i his work o stochstic processes.. Stochstic processes re collectios of rdom vriles. The c e used to show how sstem might chge over time. There re ow m pplictios of Mrkov processes icludig rdom wlks, the Gmler s rui prolem, queues rrivig t irport, echge rtes d the PgeRk lgorithm. ICT Here is emple of Mrkov process tht helps to predict log term wether proilities. I simplified model the wether o oe d c either e dr or wet. Assume tht the wether tod is dr so the proilities of dr d wet re give W0 = [ 0] P(dr tomorrow dr tod) P(dr tomorrow wet tod) Let P = [ ] where P = [ ] It follows tht the wether tomorrow is give W = W0P = [ ] [ ] ] 0 = [ The wether the et d would e give [ 0.5] 0 W = WP = [ ] [ ] = [ ] W = WP = [ [ ] [ ] = [ ] 0 [ P(wet tomorrow dr tod) P(wet tomorrow wet tod) 0.75 Usig spredsheet, fid (to 4 deciml plces) the mtri tht the wether proilities ted towrds. MATERIAL Dr TO COME Wet 0.4 Reserch Mtrices c e used to solve sstems of lier equtios usig method ow kow s Gussi elimitio. The method ws first iveted i Chi efore eig reiveted i Europe i the 700s. It is med fter Crl Friedrich Guss fter the doptio, professiol computers, of specilised ottio tht Guss devised. Fid out out Gussi elimitio. Tr usig it to solve the equtios + + z =, z =, + + z = 76