Eempla Pape Specime Pape Pages Eempla Pape Specime Pape Pages Eempla Pape Specime Pape Pages Abedee Gamma School Mathematics Depatmet CfE Advaced Highe Mathematics Leaig Itetios ad Success Citeia BLOCK BLOCK BLOCK Topic Topic Topic Patial Factios 5(a) 9- Itegatio,5, 5 9 Gaussia Elimiatio - Diffeetiatio,,6 Volumes of Revolutio - Matices 6 7, 5 Diffeetiatio Related Rates 7 - Sequeces ad Seies 9 Euclidea Algoithm 5 Diffeetiatio Rectiliea Motio 7 5 McLaui Seies 8 8 Methods of Poof 9 6 Biomial Theoem 6-7 Popeties of Fuctios, - Vectos 6 7-8 Comple Numbes 7 5 8 Summatio Poof b Idicatio 6 5-6 Diffeetial Equatios 5(b) 7,8 Uit Assessmet Stadad Couse Assessmet Stadad Page of 8
CfE Advaced Highe Mathematics Fomulae List Stadad deivatives Stadad itegals Aithmetic seies f f f f d si cos ta ta sec a ta( a) a si a a ta a a a sec cot cosec a e l e a a c c c c c Geometic seies Summatios Biomial theoem Maclaui epasio S a d a S ( ) ( ),, 6 a b a b whee! C!( )! iv f ( ) f ( ) f ( ) f ( ) f ( ) f ( )!!! p De Moive s theoem (cos isi ) cos p isi p p sec cosec l e sec ta cosec cot e i j k a a a a a a Vecto poduct a b a b si ˆ a a a i j k b b b b b b b b b Mati Tasfomatio Ati-clockwise otatio though a agle, about the oigi,o cos si si cos Uit Assessmet Stadad Couse Assessmet Stadad Page of 8
Topic Methods i Algeba ad Calculus Assessmet Stadad Applig Algebaic Skills to patial factios p I kow that a atioal fuctio is a fuctio which ca be epessed i the fom ( ) q ( ) whee p ( ) ad qae ( ) polomial fuctios I kow that a pope atioal fuctio is a factio whee the degee of the umeato is LESS tha the degee of the deomiato I ca epess a pope atioal fuctio as a sum of patial factios whose deomiato is of most degee ad easil factoised Epess each of the followig i patial factios b cosideig the tpe of deomiato Distict Liea factos ) 7 ) 6 ) 8 ( )( )( ) Repeated Facto ) ( ) 9 5) ( )( ) 6) 6 ( ) 7 Repeated Facto NOT factoised 7) 5 8) 5 Liea facto ad Ieducible Quadatic Facto 9) 9 ( )( ) ) 7 ( )( ) ) 5 6 I kow that a impope atioal fuctio is a factio whee the degee of the umeato is MORE tha o EQUAL to the degee of the deomiato I kow how to educe a impope atioal fuctio to a polomial ad a pope atioal fuctio usig algebaic divisio Epess the followig impope atioal fuctios as a polomial ad a pope atioal fuctio which is give as patial factios ) 6 ( )( ) ) ) Uit Assessmet Stadad Couse Assessmet Stadad Page of 8
Topic Methods i Algeba ad Calculus Assessmet Stadad Applig calculus skills though techiques of diffeetiatio I ca udestad the method of diffeetiatio fom fist piciples usig f '( ) lim f ( h) f ( ) h h I ca diffeetiate a epoetial fuctio ad I kow that if f ) ( e the f ) '( e I ca diffeetiate a logaithmic fuctio ad I kow that if l the d d I ca diffeetiate a fuctio usig the chai ule: ( f ( g( ))' f '( g( )) g ( ) I ca diffeetiate a fuctio usig the poduct ule: ( f ( ) g( ))' f '( ) g( ) f ( ) g '( ) I ca diffeetiate a fuctio usig the quotiet ule: f ( ) '( ) ( ) - ( ) '( ) ' f g f g g( ) ( g( )) I ca use the deivative of ta If f ( ) ta the I kow that the ecipocal tigoometic fuctios ae d I ca deive ad use the deivatives: (cot ) d Diffeetiate ) e ) 5 cos ec f '( ) sec sec, cos cosec d, (sec ) sec ta d e ) 5) f ( ) si5 6) f ( ) si ( ) 7) 9) si ) f ( ) l, ) ad si cot ta d ad (cosec ) cossec cot d e ) 5 l 5 ) sec e ta ) f ( ) l si 5) e f ( ) l( ) 5 8) ) cos e ta 6) Uit Assessmet Stadad Couse Assessmet Stadad Page of 8
Topic Methods i Algeba ad Calculus Assessmet Stadad Applig calculus skills though techiques of diffeetiatio I kow that d d d d I kow that si, cos ad ta ae ivese tigoometic fuctios I ca diffeetiate a ivese fuctio usig f ( ) f ( ) ( f ( ))' f '( ) ( f ( ))' f '( ) I kow that d d si d d, cos ad d d ta I kow usig the chai ule that d f ( ) si ( f( ) d ( f( )) d f ( ), cos ( f( ) d ( f( )) ad d ( ) ta ( f( ) f d ( f ( )) Diffeetiate 7) si ( ) 8) si 9) cos (5 ) ) ta ) ta ) si ) ( ) ta ( ) ) ta I ca fid the fist ad secod deivative of a implicit fuctio 5) Fid the fist deivative of usig implicit diffeetiatio 6) Fid the equatio of the taget to the cuve, at the poit (, ) d d 7) (a) Give, use implicit diffeetiatio to obtai i tems of ad (b) Hece obtai d d i tems of ad Uit Assessmet Stadad Couse Assessmet Stadad Page 5 of 8
Topic Methods i Algeba ad Calculus Assessmet Stadad Applig calculus skills though techiques of diffeetiatio I ca fid the fist ad secod deivative of a paametic fuctio 8) Give that l( t ), l( t ) use paametic diffeetiatio to fid d i tems oft d 9) Give t ad cot t, t obtai d d ) (a) Give (b) Show that 5 t t ad t fo d at bt, d I ca appl paametic diffeetiatio to motio i a plae i tems oft t use paametic diffeetiatio to epess d i tems of t i simplified fom d detemiig the values of the costats a ad b ) At time t, the positio of a movig poit is give b t ad ) The motio of a paticle i the - plae is give b Calculate the speed whe t I ca use logaithmic diffeetiatio whe wokig idices ivolvig the vaiable t Fid the speed whe t t 5 t, t 8 t, whee t is measued i secods ad, I ca use logaithmic diffeetiatio whe wokig with eteded poducts ad quotiets Use logaithmic diffeetiatio to diffeetiate each of the followig: ) (a) ta (b) (c) ) Give that 7 6,, use logaithmic diffeetiatio to fid a fomula fo d i tems of d ae measued i metes Uit Assessmet Stadad Couse Assessmet Stadad Page 6 of 8
Topic Methods i Algeba ad Calculus Assessmet Stadad Applig calculus skills though techiques of diffeetiatio I ca appl diffeetiatio to elated ates i poblems whee the fuctioal elatioship is give eplicitl 5) 6) 7) The adius of a clidical colum of liquid is deceasig at the ate of ms, while the height is iceasig at the ate ms Fid the ate of chage of the volume whe the adius is 6 metes ad the height is metes [Recall volume of a clide: V h ] Ai is pumped ito a spheical balloo at a ate of 8 cm / s Fid the ate at which the adius is iceasig whe the volume of the balloo is (a) A cicula ipple speads acoss a pod If the adius iceases at ms cm, at what ate is the aea iceasig whe the adius is 8 cm? (b) If the aea cotiues to icease at this ate, aw what ate will the adius be iceasig whe it is 5 metes? Topic Applicatios of Algeba ad Calculus Assessmet Stadad 5 Applig algebaic ad calculus skills to poblems I ca appl diffeetiatio to poblems i cotet ) ) ) A bod moves alog a staight lie so that afte t secods its displacemet fom a fied poit O o the lie is metes If t ( t) fid (a) the iitial velocit ad acceleatio (b) the velocit ad acceleatio afte secods A motobike stats fom est ad its displacemet metes afte t secods is give b: Calculate the iitial acceleatio ad the acceleatio at the ed of the d secod t t 6 A clidical tak has a adius of metes ad a height of h metes The sum of the adius ad the height is metes (a) Pove that that the volume i m, is give bv ( ) (b) Fid the maimum volume Uit Assessmet Stadad Couse Assessmet Stadad Page 7 of 8
Topic Applicatios of Algeba ad Calculus Assessmet Stadad (a) Applig algebaic skills to the biomial theoem I kow ad ca use the otatio! ad C whee! ( )( )( ) ad! C = =!( )! I kow Pascal s tiagle up to 7 ad ca appl the esults ad I kow ad ca use the Biomial Theoem ( a b) a b fo, N to epad a epessio of the fom a b whee 5, a, b Z I kow that the geeal tem i a biomial epasio ca be used to fid a paticula tem i a biomial epasio p q I ca epad a epessio of the fom ( a b ), whee a, b Q; p, q Z; 7 ) Calculate 5! 5) Solve, fo N, 5 ) Calculate 6) Epad 9) (a) Wite dow the biomial epasio of ) Show that ) Fid the coefficiet of 8 5 ) Simplif ( )! ( )! ) Wite dow 9 9 as a biomial coefficiet 5 7) Epad ( u v) 8) Epad ( ) ad simplif ou aswe 5 ( ) (b) Hece show that 5 9 is 599 6 whee the itege is geate tha o equal to ) Epad 7 i ( ) ) Wite dow the geeal tem of the biomial epasio of ) Fid the tem idepedet of i the epasio of ( ) 6 Use ou epessio to fid the coefficiet of 9 Uit Assessmet Stadad Couse Assessmet Stadad Page 8 of 8
Topic Applicatios of Algeba ad Calculus Assessmet Stadad (b) Applig algebaic skills to comple umbes I kow the defiitio of i as a solutio of, so i I kow the defiitio of the set of comple umbes as C { a bi : a, b R} whee a is the eal pat ad bi is the imagia pat I kow that z a bi is the Catesia fom of a comple umbe ad that z a bi is the cojugate of z I ca pefom additio, subtactio, multiplicatio ad divisio opeatios o comple umbes ) Solve z 9 ) Solve z z ) Solve 5 z z ) Calculate (a) i 7i (b) 5 i i (c) 7i i (d) Divide 5 i b i 5) Evaluate i I kow the fudametal theoem of algeba ad the cojugate oots popet I ca fid the oots of a quatic whe oe comple oot is give I ca factoise polomials with eal coefficiets I ca fid the squae oot of a comple umbe I ca solve equatios ivolvig comple umbes b equatig eal ad imagia pats 6) Show that z i is a oot of the equatio z 8z 8 ad obtai the emaiig oots of the equatio 7) Give that z i is a oot of the polomial equatio z z 8z, fid the othe oots 8) Fid the squae oots of 5 i 9) Calculate 8 6i ) Solve z i z ) Solve ) Give the equatio z iz 8 7i, epess z i the fom a ib z z Uit Assessmet Stadad Couse Assessmet Stadad Page 9 of 8
Topic Geomet, Poof ad Sstems of Equatios Assessmet Stadad Applig geometic skills to comple umbes I ca fid the modulus ad picipal agumet of a comple umbe give i Catesia fom I kow that (cos isi ) is the pola fom of a comple umbe I ca covet a give comple umbe fom Catesia to pola fom o fom pola to Catesia fom ) Fid the modulus ad agumet of : (a) i (b) i (c) 5 5i ) Wite z i i pola fom ) Wite i Catesia fom ) Give the equatio z i, wite dow z ad epess I kow ad ca use De Moive s theoem with positive itege idices ad factioal idices I ca appl De Moive s theoem to multiple agle tigoometic fomulae I ca appl De Moive s theoem to fid th oots of uit 5) Wite the comple umbe z ( i ) i pola fom ad veif that z satisfies the equatio z 6 z i pola fom 6) Let Fid b usig De Moive s Theoem the modulus ad agumet of 7) Evaluate 8) Epess i i the fom cos i si, whee Hece fid the fouth oots of i 9) Solve 6 z I ca plot comple umbes i the comple plae o a Agad Diagam I ca itepet geometicall equatios o iequalities i the comple plae of the fom z ; z a b; z i z ; z a b ) Show the comple umbes z iad its cojugate, z, o a Agad diagam ) Epess z ( i) 7 i i the fom a ib, whee a ad b ae eal umbes Show z o a Agad diagam ad evaluate z ad ag( z ) ) Give a geometic itepetatio ad the Catesia equatio fo each locus (a) z i (b) z i 5 (c) z z i Uit Assessmet Stadad Couse Assessmet Stadad Page of 8
Topic 5 Methods i Algeba ad Calculus Assessmet Stadad Applig calculus skills though techiques of itegatio I ca itegate epessios usig stadad esults ( ) ( ) f () e f d e f C f '( ) d l f ( ) C f( ) a d si a C d ta a a a C Itegate the followig: ) e d ) d ) 6 d ) 5 d 5) 9 6 d I ca Itegate b substitutio whee the substitutio is give 6) Use the substitutio t 9) Itegate si to obtai 8 d 7) Itegate cos cos d usig the substitutiou si 9) Use the substitutio si d usig the substitutiou cos u to obtai d ) Fid the value of 9 d usig the substitutio u ) Itegate 6si cos d usig the substitutiou cos ) Use the substitutio siu to obtai d ) Use the substitutio ta u to obtai d ) Use the substitutio si to evaluate 6 d Uit Assessmet Stadad Couse Assessmet Stadad Page of 8
Topic 5 Methods i Algeba ad Calculus Assessmet Stadad Applig calculus skills though techiques of itegatio I ca use patial factios to itegate pope atioal fuctios whee the deomiato has distict liea factos 5) Itegate 6) Show that 7) Evaluate (a) 5 ( )( ) ( )( ) d l l (b) 6 d ( )( ) 5 d epessig ou aswe i the fom l a, whee a ad b ae iteges ( )( )( ) b (c) d 8 I ca use patial factios to itegate pope atioal fuctios whee the deomiato has a epeated liea facto 8) Itegate (a) d ( )( ) 9) Fid the eact value of 5 7 ( ) ( ) d (b) d ( )( ) d ( ) I ca use patial factios to itegate impope atioal fuctios whee the deomiato has distict liea factos 6 ) Itegate (a) d (b) ( )( ) 5 d ) Fid the eact value of 7 7 d I ca use patial factios to itegate pope ad impope atioal fuctios whee the deomiato has a liea facto ad a ieducible quadatic of the fom a ) Fid (a) ( )( ) d (b) ( )( ) ) Epess i patial factios Hece evaluate ( 5) d, d ( 5) Uit Assessmet Stadad Couse Assessmet Stadad Page of 8
Topic 5 Methods i Algeba ad Calculus Assessmet Stadad Applig calculus skills though techiques of itegatio I ca Itegate b pats usig oe applicatio 6 ) Use itegatio b pats to fid: (a) e d (b) si d 5) Evaluate (a) cos d (b) e d I ca Itegate b pats usig a epeated applicatio 6) Use itegatio b pats to fid: (a) e d (b) cos d 7) Evaluate (a) 8) (a) Wite dow the deivative of 9) Let l d (b) e si d si (b) Use itegatio b pats to obtai si d I e d fo (a) Fid the value of I (b) Show that I I e fo (c) Evaluate I Topic 5 Applicatios of Algeba ad Calculus Assessmet Stadad 5 Applig algebaic ad calculus skills to poblems I ca appl itegatio to poblems i cotet ) t t The velocit, v, of a paticle P at time t is give b v e e (a) Fid the acceleatio of P at time t (b) Fid the distace coveed b P betwee t adt l ) A object acceleates fom est ad poceeds i a staight lie At time, secods, its acceleatio is give b cm/s (a) Calculate the velocit of the object afte secods (b) How fa did the object tavel i the fist 8 secods of motio? Uit Assessmet Stadad Couse Assessmet Stadad Page of 8
Topic 5 Applicatios of Algeba ad Calculus Assessmet Stadad 5 Applig algebaic ad calculus skills to poblems I ca appl itegatio to the evaluatio of aeas icludig itegatio with espect to ) The sketch o the ight shows the gaph of 9 ) Calculate the aea betwee the fuctio i the iteval 9 ad the -ais Calculate the shaded aea o I ca appl itegatio to volumes of evolutio 5) Sketch the cuve showig the oots The egio bouded b the cuve ad the -ais is otated though 6 about the -ais Show that the volume of the solid geeated 8 6) 7) The shaded egio i the diagam, bouded b the cuve e ad the -ais ad the lie is otated though 6 about the -ais Show that the volume of the solid geeated 5 ( ) e The aea lig i the fist quadat ad bouded b the cuve adias about the ais, calculate the volume of the solid fomed, the ais ad the lies ad 5 is otated 8) (a) Fid the equatio of the chod PQ which jois the poits P(-, 8) ad Q(, 6) o the cuve 6 (b) Show that the fiite aea eclosed betwee the cuve 6 ad the chod PQ is 8 squae uits (c) Show that the volume geeated whe this aea is otated though 6 about the -ais is 5 cubic uits Uit Assessmet Stadad Couse Assessmet Stadad Page of 8
Topic 6 Applicatios of Algeba ad Calculus Assessmet Stadad Applig algebaic skills to sequeces ad seies I kow the fomulae u a ( ) d ad S [a ( ) d] fo the a( ) I kow the fomula u a ad S, fo the I ca appl the above ules o sequeces ad seies to fid: The th th tem ad the sum to tems of a aithmetic seies tem ad the sum to tems of a geometic seies th tem The sum to tems The commo diffeece of aithmetic sequeces The commo atio of geometic sequeces I kow ad ca use the fomula S a fo the sum to ifiit of a geometic seies whee I ca epad as a geometic seies ad eted to a b ) The sum S, ( ) of the fist tems of the sequece, u, u, u, is give b S( ) 8, Calculate the values of u, u, u ad state what tpe of sequece it is Obtai a fomula fo u i tems of, simplifig ou aswe ) Give that uk k, ( k ), obtai a fomula fo S u Fid the values of fo which S k k ) A geometic seies has the fist tem ad thid tem Fid the value(s) of, the commo atio, ad a associated sum(s) to ifiit ) The aithmetic sequece a, a, a, a ad the geometic sequece b, b, b, b both have thei fifth tem 8 ad thei eighth tem 5 Fid a 5 Calculate 5 b, coect to two decimal places 5) Give that thee ae two solutios fid the thid tem of the geometic sequece whose secod tem is ad whose sum to ifiit is 5 Uit Assessmet Stadad Couse Assessmet Stadad Page 5 of 8
Topic 6 Applicatios of Algeba ad Calculus Assessmet Stadad Applig algebaic skills to sequeces ad seies 6) Afte a udetected leak at a uclea powe situatio, a techicia was eposed to adiatio as follows: O the fist da he eceived a dosage of 5 cuie-hous O the secod da he eceived a futhe dosage of 6 cuie-hous O the thid da he eceived a futhe dosage of 88 cuie-hous (a) Show that these values could fom the fist tes of a Geometic sequece ad calculate how ma cuie-hous he was eposed to o the ith da, assumig the patte cotiues i the same wa (b) What was the total adiatio eceived b him b da 5? (c) If the leak had cotiued udetected i this wa, what would have bee the fial total log tem eposue b the techicia 7) (a) The sum of the fist tems of a aithmetic seies is 5 The fist tem is (i) Calculate the commo diffeece betwee tems (ii) Whe did the sum fist eceed? (b) ae the fist tems of a geometic sequece (i) Wite dow a epessio fo the commo atio i two was (ii) Hece epess i tems of (iii) Fo what values of will the sequece have a sum to ifiit? (iv) Epess the sum to ifiit i tems of (v) Fo what value of does this sum to ifiit equal? 8) Epad the followig as geometic seies ad state the ecessa coditio o fo each seies to be valid (a) (b) (c) 9) If deotes the sum of the fist tems of the geometic seies whee pove that ) Fid the commo atio of the geometic sequece Pove that fo the seies has a sum to ifiit ad that the sum to ifiit is Uit Assessmet Stadad Couse Assessmet Stadad Page 6 of 8
Topic 6 Applicatios of Algeba ad Calculus Assessmet Stadad Applig algebaic skills to sequeces ad seies I kow that a powe seies is a epessio of the fom: f ( ) a a a a a a whee a, a, aa, a, ae costats ad is a vaiable I udestad ad ca use the Maclaui seies: ( ) ( ) f () f! to fid a powe seies fo a simple o-stadad fuctio I ecogise ad ca detemie the Maclaui seies epasios of the fuctios : e!!! l( ) si l( ) 5 7! 5! 7! e, si, cos, l( ), kowig thei age of validit 6 cos!! 6! ) Fid the Maclaui seies epasios of the composite fuctios : (a) cos (b) e (c) si e ) (a) Obtai the Maclaui seies fo (b) Hece wite dow a seies fo f ( ) si up to the tem i cos up to the tem i ) Fid the Maclaui epasio of f ( ) l cos,, as fa as the tem i ) Wite dow the Maclaui epasio of e as fa as the tem i Deduce the Maclaui epasio of Hece, o othewise, fid the Maclaui epasio of e as fa as the tem i e as fa as the tem i 5) Fid the McLaui epasio fo e up to the tem i Uit Assessmet Stadad Couse Assessmet Stadad Page 7 of 8
Topic 7 Applicatios of Algeba ad Calculus Assessmet Stadad Applig algebaic ad calculus skills to popeties of fuctios I kow the meaig of the tems fuctio, domai / age, ivese fuctio, statioa poit, poit of ifleio ad local maima ad miima I kow the meaig of the tems global maima ad miima, citical poit, cotiuous, discotiuous ad asmptote I ca use the fist deivative test fo locatig ad idetifig statioa poits ad hoizotal poits of ifleio I ca use the secod deivative test fo locatig ad idetifig statioa poits ad o-hoizotal poits of ifleio I ca sketch the gaphs of si,cos,ta, e l ad thei ivese fuctios o a suitable domai I kow ad ca use the elatioship betwee the gaph of f () ad kf ( ), f () k, f ( k ), f ( k) whee k is a costat I kow ad ca use the elatioship betwee the gaph of f () ad f ( ), f ( ) I ca detemie whethe a fuctio is odd o eve o eithe, ad smmetical ad use these popeties i gaph sketchig I ca sketch gaphs of eal atioal fuctios usig ifomatio, deived fom calculus, zeos, asmptotes, citical poits ad smmet I kow that the maimum ad miimum values of a cotiuous fuctio o a closed iteval [a,b] ca occu at statioa poits, ed poits o poits whee is ot defied ) Sketch the gaph of: (a) si (b) 9 6 6 ) Detemie whethe f ( ) cos is odd, eve o eithe ) 7 The fuctio f is defied o the eal umbes b f si Detemie whethe f is odd, eve o eithe ) The fuctio f is defied b f e si whee Fid the coodiates of the statioa poits of f ad detemie thei atue 5) A fuctio is defied b g( ), (a) Fid the coodiates ad atue of the statioa poits of the cuve with equatio g() (b) Hece state the coodiates of the statioa poit pf the cuve with equatio h ( ) 5 Uit Assessmet Stadad Couse Assessmet Stadad Page 8 of 8
Topic 7 Applicatios of Algeba ad Calculus Assessmet Stadad Applig algebaic ad calculus skills to popeties of fuctios 6) The diagam shows pat of the gaph of, (a) Wite dow the equatio of the vetical asmptote (b) Fid the coodiates of the statioa poits of the gaph of (c) Wite dow the coodiates of the statioa poits of the gaph of 7) 8) A fuctio f is defied fo suitable values of b f( ) (a) Decide whethe f is odd, eve o eithe (b) Wite dow the equatios of a vetical asmptotes (c) Fid algebaicall the equatio of a o-vetical asmptote (d) Show that f has ol oe statioa poit ad justif its atue (e) Sketch the gaph of f, showig cleal what happes as The fuctio f is defied b f ( ), (a) Wite dow a equatio fo each of the asmptotes of the gaph of f (b) The gaph of f has a statioa poit whe a Fid the coodiates of this statioa poit ad justif its atue (c) Sketch the gaph of f (d) Fid the volume of evolutio fomed whe the egio betwee f ( ), a, ad is otated 6 about the -ais Uit Assessmet Stadad Couse Assessmet Stadad Page 9 of 8
Topic 8 Applicatios of Algeba ad Calculus Assessmet Stadad Applig algebaic skills to summatio ad mathematical poof I kow ad ca use the followig sums of seies: ( ), ( )( ) ad 6 ( ) ) Fid a fomula fo each of the followig usig the sum of seies (a) ( ) (b) ( 6 ) (c) (5 ) (d) ( ) ) Evaluate each of the followig usig the sum of seies: (a) ) (b) ( 7 5 (c) ( ) k (d) ) Epess i patial factios Hece evaluate, epessig ou aswe as a sigle factio 6 8 6 8 ) (a) Pove b iductio that, fo all atual umbes ( ) ( ) ( ) (b) Hece evaluate ( ) 5) Use Iductio to pove that 7 fo all positive iteges 6 6) Use Iductio to pove that fo all positive iteges 7) Pove b iductio that is divisible b 7 fo all iteges 8) Pove b iductio that (cos isi ) cos isi fo all iteges 9) If A, pove b iductio that A, whee is a positive itege Uit Assessmet Stadad Couse Assessmet Stadad Page of 8
Abedee Gamma School Advaced Highe Mathematics Topic 9 Geomet, Poof ad Sstems of Equatios Assessmet Stadad (a) Leaig Itetios ad Success Citeia Applig algebaic skills to sstems of equatios I ca use Gaussia elimiatio to solve a sstem of liea equatios ad show that it has eithe: (a) a uique solutio (b) has o solutios (icosistec) o (c) has a ifiite umbe of solutios (edudac) I ca compae the solutios of elated sstems of two equatios i two ukows ad ecogise ill-coditioig ) Use Gaussia elimiatio to solve the sstem of equatios: (a ) z z ) z z (b) z z 5 z z (c) z 9 z 9 (a) Appl the method of Gaussia elimiatio to z ad show that thee is a ifiite umbe of solutios 5 z 5 (b) If a solutio has z, show that ad ad will the sstem of equatios z 6 z 85 pz 5 (a) be icosistet ad have o solutios (b) be edudat ad have ifiitel ma solutios? ) Fo what values of ) A ca maufactue is plaig futue poductio pattes Based o estimates of time, cost ad labou, he obtais a set of thee equatios fo the umbes,, ad z of thee ew tpes of ca These equatios ae (a) (b) (c) 5) z 6 z 85 ( whee the itege is a paamete such that ) z 5, Use Gaussia elimiatio to fid a epessio fo z i tems of Give that z must be a positive itege, what ae the possible values fo z? Fid the coespodig values of ad fo each value of z Detemie if these sstems of equatios ae ill coditioed Uit Assessmet Stadad Couse Assessmet Stadad (a) 7 5 9 ad (b) 7 5 Page of 8
Topic 9 Geomet, Poof ad Sstems of Equatios Assessmet Stadad (b) Applig algebaic skills to matices I kow the meaig of the tems mati, elemet, ow, colum, ode, idetit mati, ivese, detemiat, sigula, o-sigula ad taspose I ca pefom mati additio, subtactio, multiplicatio b a scala ad multiplicatio I kow that: A B B A AB BA i geeal ( AB) C A( BC ) A( B C) AB AC I kow ad ca appl popeties of the taspose mati: ( A) A ( A B) A B ( AB) B A I kow ad ca appl popeties of the idetit ad ivese mati: AA A A I ( AB) B A I ca calculate the detemiat of ad matices I ca detemie whethe a mati is sigula I kow ad ca appl det (AB) = det A det B a b d b I kow ad ca fid the ivese of a o-sigula mati, A usig A c d A c a I ca fid the ivese, whee it eists, of a mati usig elemeta ow opeatios o the adjoit method I ca fid the solutio to a sstem of equatios AX B whee A is a mati ad whee the solutio is uique I kow ad ca use the followig matices to ca out sigle geometic tasfomatios i the plae Reflectio i the -ais Reflectio i the -ais Rotatio of adias i a positive diectio about the oigi cos si si cos Reflectio i the lie Reflectio i the lie A dilatatio about O, whee the scale facto is k k I ca idetif ad use the coect tasfomatio matices, i the coect ode, to ca out composite geometic tasfomatios i the plae k ) Give the matices A 8 6, B 5 7, C ad D m 8 Fid (a) A B C (b) CB (c) A (d) Detemie the value(s) of m fo which D is sigula Uit Assessmet Stadad Couse Assessmet Stadad Page of 8
Topic 9 Geomet, Poof ad Sstems of Equatios Assessmet Stadad (b) Applig algebaic skills to matices ) Calculate the ivese of the mati Fo what value of is this mati sigula? ) Let A be the mati 5 9 Show that A A I whee is a itege ad I is the idetit mati ) The mati A is such that A A I whee I is the coespodig idetit mati Fid iteges p ad q such that A pa qi 5) 6) 7) (a) Give that (b) Give that a X whee a is a costat ad a, fid X i tems of a X X I, whee I is the the idetit mati, fid the value of a Matices A ad B ae defied b A ad B (a) Fid the poduct AB (b) Obtai the detemiats of A ad of AB Hece, o othewise obtai a epessio fo det B 5 5 A (a) Fid 5 5 A the ivese of A (b) X ad B ae two matices such that AX B Pove that X A B 8) A mati is defied as A Show that mati A has a ivese, A, ad fid the ivese mati 9) Wite dow the mati A epesetig a otatio of adias about the oigi i a aticlockwise diectio ad the mati B epesetig a eflectio i the -ais Hece, show that the image of the poit (, ) ude the tasfomatio A followed b the p p tasfomatio B is,, statig the value of p Uit Assessmet Stadad Couse Assessmet Stadad Page of 8
Topic a Geomet, Poof ad Sstems of Equatios Assessmet Stadad Applig algebaic skills to umbe theo I ca use Euclid s algoithm to fid the geatest commo diviso of two positive iteges ) Use the Euclidea algoithm to obtai the geatest commo diviso of 9 ad 69 I ca epess the geatest commo diviso of the two positive iteges as a liea combiatio of the two ) Use the Euclidea Algoithm to fid iteges ad such that: (a) (b) (c) I ca use the divisio algoithm to wite iteges i tems of bases othe tha ) Use the divisio algoithm to epess: (a) i base 7 (b) i base 6 (c) i base 6 Topic b Geomet, Poof ad Sstems of Equatios Assessmet Stadad 5 Applig algebaic ad geometic skills to methods of poof I udestad ad make use of the otatios, ad, kow the coespodig temiolog implies, implied b, equivalece I kow the tems atual umbe, pime umbe, atioal umbe, iatioal umbe I kow the tems if ad ol if, covese, egatio ad cotapositive I ca use diect poof I ca use idiect poof b povidig a coute-eample, usig poof b cotadictio o b usig poof b cotapositive ) Fid a couteeample to dispove the cojectue that fo all eal values of ) Coside the statemets A ad B: A Fo a itege k, if 7k + is eve, the k is odd B Thee is o lagest eve itege (a) Pove statemet A b cosideig its cotapositive (b) Pove statemet B b cotadictio ) Pove that the poduct of a odd ad eve itege is eve ) Pove b cotadictio that if is a iatioal umbe, the is iatioal 5) Pove b cotapositive that if the is odd 6) Give that coside the statemets: A is alwas eve B is alwas a multiple of Fo each statemet, pove it is tue, o othewise, dispove it 7) Use the method of poof b cotadictio to show that is a iatioal umbe Uit Assessmet Stadad Couse Assessmet Stadad Page of 8
Topic Geomet, Poof ad Sstems of Equatios Assessmet Stadad Applig algebaic ad geometic skills to vectos I kow the meaig of the tem: Uit vecto, Diectio atios, Diectio cosies, Vecto poduct, Scala tiple poduct I ca evaluate the vecto poduct a b usig i j k a a a a a a a b a a a i j k b b b b b b b b b ad I kow that ( a b) ( b a) I kow that the magitude of the vecto poduct a b a b si which is the aea of a paallelogam with sides a, b ad icluded agle ) Give, ad calculate : (a) (b) (c) ) Give, ad calculate : (a) (b) ) Thee vectos OA, OB ad OC ae give b ad whee, ad Calculate Itepet ou esult geometicall I ca fid the equatio of a lie i paametic, smmetic o vecto fom I ca fid the agle betwee two lies i thee dimesios I ca detemie whethe o ot two lies itesect ad, whee possible, fid the poit of itesectio ) Fid, i vecto, paametic ad smmetic fom a equatio fo the lie which passes though the poits (,, ) ad (, 5, ) 5) Fid the acute agle betwee the lies 7 ad 5 6 5 6) Let L ad L be the lies L : 8 t, t, z t ad L z 9 : (a) Show that L ad L itesect ad fid thei poit of itesectio (b) Veif that the acute agle betwee them is cos 9 Uit Assessmet Stadad Couse Assessmet Stadad Page 5 of 8
Topic Geomet, Poof ad Sstems of Equatios Assessmet Stadad Applig algebaic ad geometic skills to vectos I ca fid the equatio of a plae i vecto fom, paametic fom o Catesia fom I ca fid the poit of itesectio of a plae with a lie which is ot paallel to the plae I ca detemie the itesectio of o plaes I ca fid the agles betwee a lie ad a plae o betwee plaes 7) Fid, i Catesia fom, the equatio of the plae which has omal vecto ad passes though the poit (, 7, ) 8) Fid a equatio of the plae which cotais the poits A (,, ), B(,, -) ad C(,,-) 9) Fid the poit of itesectio of the lie z ad the plae with equatio z ) Fid a equatio fo the lie of itesectio of the plae with equatio ad the plae with omal vecto though the poit (,,) ) Fid the agle betwee the lie t, t, z t 5 ad the plae z ) Fid the agle betwee the lie z 5 ad the plae z ) Fid the agle betwee the two plaes with equatios z 5 ad z, espectivel ) (a) Fid the equatio of the plae cotaiig the poits A(,, ), B(,, ) ad C(,, ) (b) Fid the agle betwee this plae ad the plae with equatio z (c) Fid the poit of itesectio of the plae cotaiig A, B, ad C ad the lie with equatio z t Uit Assessmet Stadad Couse Assessmet Stadad Page 6 of 8
Topic Methods i Algeba ad Calculus Assessmet Stadad Applig calculus skills to solvig fist ode diffeetial equatios d I ca solve fist ode diffeetial equatios of the fom g( ) h( ) d I ca fid geeal ad paticula solutios give suitable ifomatio d ) Solve ( ) d ) Solve d d o d d ( ) d ) Fo a diffeetial equatio ( )( ), whe, t, show that dt Hece epess the solutio eplicitl i the fom f () t 5) Solve d si d give that whe, g( ) b sepaatig the vaiables h( ) Ae kt d ) Solve e d, statig the values of Aad k d 6) Solve the diffeetial equatio, give whe epessig eplicitl i tems of d e d 7) Solve the diffeetial equatio givig i tems of : cos cosec, give that whe d, d i ca solve fist ode liea diffeetial equatios give o eaaged i the fom d p( ) f ( ) usig the itegatig facto method I ca fid geeal ad paticula solutios give suitable ifomatio d e 8) Solve d ) Solve d d d ) Solve ta sec d give that whe d 9) Solve cot cos d give that whe, ) Solve d d si Uit Assessmet Stadad Couse Assessmet Stadad Page 7 of 8
Topic Methods i Algeba ad Calculus Assessmet Stadad Applig calculus skills to solvig secod ode diffeetial equatios I kow the meaig of the tems homogeeous, o-homogeeous, auilia equatio, complemeta fuctio ad paticula itegal d d I ca fid the geeal solutio of a secod ode homogeeous odia diffeetial equatio a b c with costat coefficiets d d whee the oots of the auilia equatio ae (a) eal ad distict (b) eal ad equal (c) ae comple cojugates ) Solve the equatios: (a) d d 6 d d (b) d d 9 d d d d (c) 6 d d I ca solve iitial value poblems fo secod ode homogeeous odia diffeetial equatio with costat coefficiets ) ) d d d Solve 6 with ad d d d d d d Solve with ad d d d whe whe d d I ca solve secod ode o-homogeeous odia diffeetial equatio with costat coefficiets a b d d c f ( ) usig the auilia equatio ad paticula itegal method ) 5) 6) 7) Obtai the geeal solutio of the diffeetial equatio d (a) Fid the geeal solutio to the followig diffeetial equatio: d (b) Hece fid the paticula solutio fo which ad 8 d d d Solve the secod ode diffeetial equatio d d d d Solve the secod ode diffeetial equatio e d d d 7 si cos d d d d 5 d d whe, give that whe,, give that whe, d ad d d ad d Uit Assessmet Stadad Couse Assessmet Stadad Page 8 of 8