National Quali cations SPECIMEN ONLY

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1 AH Natioal Quali catios SPECIMEN ONLY SQ/AH/0 Mathematics Date Not applicable Duratio hours Total s 00 Attempt ALL questios. You may use a calculator. Full credit will be give oly to solutios which cotai appropriate workig. State the uits for your aswer where appropriate. Write your aswers clearly i the aswer booklet provided. I the aswer booklet, you must clearly idetify the questio umber you are attemptig. Use blue or black ik. Before leavig the eamiatio room you must give your aswer booklet to the Ivigilator; if you do ot, you may lose all the s for this paper. *SQAH0*

2 FORMULAE LIST Stadard derivatives Stadard itegrals f ( ) f ( ) f ( ) f ( ) d si cos a a + si a + c ta c a a + ta + sec ( a) a ta( a) + c ta sec a e a e a + c l, > 0 e e Summatios (Arithmetic series) S a + ( ) d (Geometric series) S ( ) a r r ( )( + ) ( + ) + ( + ) r, r, r 6 r r r Biomial theorem ( a+ b) r 0 r a r b r where Cr r! r!( r)! Maclauri epasio De Moivre s theorem Vector product iv f ( ) f ( ) f ( ) f( ) f( ) + f ( ) !!! [ (cos si )] r θ + i θ r ( cosθ + isiθ) i j k a a a a a a a b absiθˆ a a a i j + k b b b b b b b b b Page two

3 Total s 00 Attempt ALL questios MARKS +. Give f( ), show that f ( ) + ( + ).. State ad simplify the geeral term i the biomial epasio of. Hece, or otherwise, fid the term idepedet of. 6. Fid 9 6 d.. Show that the greatest commo divisor of 87 ad 79 is. Hece fid itegers ad y such that y.. Fid e d. 6. Fid the values of the costat k for which the matri k k 0 is sigular. 7. A spherical balloo is beig iflated. Whe the radius is 0 cm the surface area is icreasig at a rate of 0π cm s. Fid the rate at which the volume is icreasig at this momet. (Volume of sphere r, surface area r ) 8. (a) Fid the Maclauri epasios up to ad icludig the term i, simplifyig the coefficiets as far as possible, for the followig: (i) f( ) e ( ) (ii) g ( ) + e (b) Give that h ( ) ( + ) value of h. use the epasios from (a) to approimate the Page three

4 9. Three terms of a arithmetic sequece, u, u 7 ad u 6 form the first three terms of a geometric sequece. Show that a 6 d, where a ad d are, respectively, the first term ad commo differece of the arithmetic sequece with d 0. Hece, or otherwise, fid the value of r, the commo ratio of the geometric sequece. MARKS 0. Usig logarithmic differetiatio, or otherwise, fid dy d y ( + ) e e, >. ( ). Fid the eact value of + d +. ( ) ( ) give that 7. (a) Give that m ad are positive itegers state the egatio of the statemet: m is eve or is eve. (b) By cosiderig the cotrapositive of the followig statemet: if m is eve the m is eve or is eve, prove that the statemet is true for all positive itegers m ad.. Cosider the curve i the ( y, ) plae defied by the equatio y 8. (a) Idetify the vertical asymptotes to this curve ad justify your aswer. Here are two statemets about the curve: () It does ot cross or touch the -ais. () The lie y 0 is a asymptote. (b) (i) State why statemet () is false. (ii) Show that statemet () is true. Page four

5 . The lies L ad L are give by the followig equatios. MARKS L : L : + 6 y z + y+ z (a) Show that the lies L ad L itersect ad state the coordiates of the poit of itersectio. (b) Fid the equatio of the plae, π, cotaiig L ad L. A third lie, L, is give by the equatio y+ 7 z. (c) Calculate the acute agle betwee L ad the plae, π.. (a) Give that f( ) sec, show that f secta. ( ) (b) Solve the differetial equatio cos cos dy + yta sec d e give that y whe, epressig your aswer i the form y f( ) Let S r r+ r ( ) where is a positive iteger. (a) Prove that, for all positive itegers, S +. (b) Fid (i) the least value of such that S + S < 000 (ii) the value of for which S S S S. 8 Page five

6 MARKS 7. (a) Give z cosθ + isiθ, use de Moivre s theorem to epress z i Cartesia form. (b) Usig the biomial theorem, or otherwise, show that the real part of z is give by: cos θ 6cos θsi θ+ si θ ad fid a similar epressio for the imagiary part. (c) Hece show that taθ ta θ taθ. 6ta θ + ta θ (d) Fid algebraically the solutios to the equatio ta θ+ ta θ 6ta θ taθ+ 0 π i the iterval 0 θ. [END OF SPECIMEN QUESTION PAPER] Page si

7 AH Natioal Quali catios SPECIMEN ONLY SQ/AH/0 Mathematics Markig Istructios These Markig Istructios have bee provided to show how SQA would this Specime Questio Paper. The iformatio i this publicatio may be reproduced to support SQA qualificatios oly o a o-commercial basis. If it is to be used for ay other purpose, writte permissio must be obtaied from SQA s Marketig team o permissios@sqa.org.uk. Where the publicatio icludes materials from sources other tha SQA (ie secodary copyright), this material should oly be reproduced for the purposes of eamiatio or assessmet. If it eeds to be reproduced for ay other purpose it is the user s resposibility to obtai the ecessary copyright clearace.

8 Geeral Markig Priciples for Advaced Higher Mathematics This iformatio is provided to help you uderstad the geeral priciples you must apply whe ig cadidate resposes to questios i this Paper. These priciples must be read i cojuctio with the Detailed Markig Istructios, which idetify the key features required i cadidate resposes. (a) Marks for each cadidate respose must always be assiged i lie with these Geeral Markig Priciples ad the Detailed Markig Istructios for this assessmet. (b) Markig should always be positive. This meas that, for each cadidate respose, s are accumulated for the demostratio of relevat skills, kowledge ad uderstadig: they are ot deducted from a maimum o the basis of errors or omissios. (c) Cadidates may use ay mathematically correct method to aswer questios ecept i cases where a particular method is specified or ecluded. (d) Workig subsequet to a error must be followed through, with possible credit for the subsequet workig, provided that the level of difficulty ivolved is approimately similar. Where, subsequet to a error, the workig is easier, cadidates lose the opportuity to gai credit. (e) Where trascriptio errors occur, cadidates would ormally lose the opportuity to gai a processig. (f) Scored-out or erased workig which has ot bee replaced should be ed where still legible. However, if the scored-out or erased workig has bee replaced, oly the work which has ot bee scored out should be judged. (g) Uless specifically metioed i the Detailed Markig Istructios, do ot pealise: workig subsequet to a correct aswer correct workig i the wrog part of a questio legitimate variatios i solutios repeated errors withi a questio Defiitios of Mathematics-specific commad words used i this Specime Questio Paper Determie: determie a aswer from give facts, figures, or iformatio. Epad: multiply out a algebraic epressio by makig use of the distributive law or a compoud trigoometric epressio by makig use of oe of the additio formulae for si( A ± B) or cos( A ± B). Epress: use give iformatio to rewrite a epressio i a specified form. Fid: obtai a aswer showig relevat stages of workig. Hece: use the previous aswer to proceed. Hece, or otherwise: use the previous aswer to proceed; however, aother method may alteratively be used. Prove: use a sequece of logical steps to obtai a give result i a formal way. Show that: use mathematics to show that a statemet or result is correct (without the formality of proof) all steps, icludig the required coclusio, must be show. Page two

9 Sketch: give a geeral idea of the required shape or relatioship ad aotate with all relevat poits ad features. Solve: obtai the aswer(s) usig algebraic ad/or umerical ad/or graphical methods. Page three

10 Detailed Markig Istructios for each questio Questio Epected respose (Give oe for each ) Ma As: demostrate result Additioal guidace (Illustratio of evidece for awardig a at each ) kow ad start to use ( + )... quotiet rule complete differetiatio ( + ) ( + ) ( ) ( + ) simplify umerator ( + ) ( + ) As: 6000 correct substitutio ito geeral term 6 6 r ( ) r simplify idetify r ad fid 6 r( ) r 6 6 r coefficiet ( ) ( ) r r 6 6 r Accept ( ) r or correct equivalet for. 6 If coefficiet is foud by epadig the epressio, oly is available. As: si + c evidece of idetifyig a appropiate method eg idetify stadard itegral d a re-write i stadard form d or equivalet fial aswer with costat of itegratio si + c si + c Note: For accept ay appropriate evidece eg usig substitutio u. Page four

11 Questio Epected respose (Give oe for each ) Ma As:, y 6 Additioal guidace (Illustratio of evidece for awardig a at each ) start correctly show last o-zero remaider evidece of two correct back substitutios usig 80 or 87 or , GCD ( ) ( ) ( ) carefully check for equivalet alteratives values for ad y So,, y 6 As: e e e + +c 9 7 evidece of applicatio of itegratio by parts d ( e d e. d d) d correct choice of u ad v ' u v' correct first applicatio start secod applicatio fial aswer with costat of itegratio e e e d e e e d 9 or equivalet or equivalet e e e + + c or equivalet As: k, starts process for workig out determiat k + 0 k k 0 completig process correctly ( ) k k + 8 k simplify ad equate to 0 k + k 0 Page five

12 Questio Epected respose (Give oe for each ) fid values of k Ma Additioal guidace (Illustratio of evidece for awardig a at each ) k, k Note: Accept aswer arrived at through row ad colum operatios. 7 As: dv dt 600 πcm s iterprets rate of chage da da dr 0 π dt dr dt correct epressio for da dr A π r, da 8 πr dr fid dr dt dr π dt 0 π 80 correct epressio for dv dt dv dv dr πr dt dr dt evaluates dv dt dv π 0 600πcm s dt ( ) 8 a i 9 9 As: f( ) state Maclauri epasio for ( ) ( ) e up to correct epasio 8 a ii As: g b ( ) correct differetiatio of Page si f( ) !!! 9 9 f( ) g( ) ( + ), g ( ) ( + ), g() g ( ) 6( + ), g ( ) ( + ) correct evaluatios of g fuctios correct epasio As: h coectio betwee h(), f() ad g() g( 0), g ( 0), g ( 0) 8 g ( 0) g ( ) h ( ) f( g ) ( )

13 Questio Epected respose (Give oe for each ) 7 approimate f ad g 8 evaluate h Note: Accept aswer give as a fractio. Ma Additioal guidace (Illustratio of evidece for awardig a at each ) 7 f 87, g h As: proof, r 9 create term formulae form ratios for r complete proof evaluate r u a+ d u a+ 6d 7 u a+ d 6 a+ d a+ d 6 a+ 6d a+ d ( a+ d)( a+ d) ( a+ 6d) a + 7ad + 0d a + ad + 6d a 6d 6 a d 6 d + 6d r 9 6 d + d 0 As: dy + d + itroductio of log e epress fuctio i differetiable form differetiate ( ) + y l ( ) e y l + + l dy + d + Note: The use of modulus sigs is ot required for the award of ad. As: l 6 7 correct form of partial fractios A B C + ( ) Page seve

14 Questio Epected respose (Give oe for each ) Ma st coefficiet correct A Additioal guidace (Illustratio of evidece for awardig a at each ) d coefficiets correct B rd coefficiets correct C itegrate ay two terms + ( + ) d l l + + ( + ) 6 itegrate all three terms 6 l l + + ( + ) 7 evaluate 7 l 6 a As: m is odd ad is odd correct statemet m is odd ad is odd b As: proof cotrapositive statemet If m ad are both odd the m is odd begi proof Let m p, q where p, q are positive m pq p q + where complete proof itegers. The, ( ) pq p q is clearly a iteger therefore m is clearly odd. Ad so the cotrapositive statemet is true ad it follows that the origial statemet, if m is eve the m is eve or is eve, that is equivalet to the cotrapositive, is true. Note: For accept a equivalet statemet, eg either m or is eve but do ot accept ay other aswer, eg It is ot true to say that m is eve or is eve. a As:, with eplaatio correct asymptotes 8 0 or suitable eplaatio y teds towards ± as ad - b i As: false with eplaatio Page eight

15 Questio Epected respose (Give oe for each ) Ma Additioal guidace (Illustratio of evidece for awardig a at each ) suitable eplaatio The statemet is false because the graph meets the -ais whe. b ii As: proof method complete proof Note: eg ( ) f 8 0 ± 0 horizotal asymptote As, f ( ) Graph of fuctio. For ig guidace ot required by cadidate. ie the lie y 0 is a a As: (, -, 8) write lies i parametric form create equatios for itersectio solve a pair of these equatios (eg the first two) for p ad t t 6 p y t+ ad y p z t+ z p p t 6 p t + p t+ t ad p Page ie

16 Questio Epected respose (Give oe for each ) check that the third equatio is satisfied state coordiates of poit of itersectio Ma b As: 6 y+ 7z 6 Additioal guidace (Illustratio of evidece for awardig a at each ) eg ( ) ( ) + evidece of substitutio ito third equatio ad (,, 8) 6 use vector product to fid ormal to the plae 6 i j k 7 evaluate ormal vector 7 6i j+ 7 k 8 form equatio of plae 8 6 y+ 7z 6 c As: 9 9 select correct vectors ad 0 complete calculatios of a, b ad a b , ad 6 7 evaluate acute agle betwee ormal to plae ad lie 0 calculate agle betwee lie ad plae a As: proof b recogisig sec as cos ad start to differetiate d d cos complete differetiatio ad state result si sec ta cos As: + e π 7 y sec e epress i stadard form form of itegratig factor fid itegratig factor IF dy ta + y d cos e e IF e sec ta d cos sec Page te

17 Questio Epected respose (Give oe for each ) 6 state modified equatio Ma Additioal guidace (Illustratio of evidece for awardig a at each ) 6 d ( ye sec ) d 7 itegrate both sides 7 e sec y + c 8 substitute i for ad y ad fid c 9 state particular solutio 8 sec e π π + c, c e π 9 + e π y sec e 6 a As: proof strategy use partial fractios A B + r r+ r r+ ( ) fid A ad B A, B - state result ad start to write out series strategy complete proof 6 a As: proof (alterative) Note that successive terms cacel out (telescopic series) cacels terms ad + + state hypothesis ad cosider k+ Assume k r k true for some r r+ k + ( ) k, ad cosider k+ k+ k ie + r r+ r r+ k + k + ( ) r ( ) ( )( ) r start process for k + k + k + k + k + ( )( ) ( ) k k + + ( k + )( k + ) ( k + )( k + ) k + k + ( k + )( k + ) Page eleve

18 Questio Epected respose (Give oe for each ) complete process Ma Additioal guidace (Illustratio of evidece for awardig a at each ) ( k + ) ( )( ) k + k + ( k + ) ( k + ) + show true for For LHS + ( ) RHS + state coclusio b i As: LHS RHS so true for Hece, if true for k, the true for k +, but sice true for, the by iductio true for all positive itegers. 6 set up equatio ad start to solve 6 eg + < ad evidece of strategy 7 process > 0 8 obtai solutio 8 b ii As: 9 set up equatio 0 solve for is oly available for iductio hypothesis ad statig that k + is goig to be cosidered. is oly awarded if fial lie shows results required i terms of k + ad is arrived at by appropriate workig, icludig target/desired result approach, from the stage. is oly awarded if the cadidate shows clear uderstadig of the logic required. 7 a As: cosθ + isi θ 7 b As: proof use de Moivre s theorem cosθ + isi θ start process complete epasio ( ) cosθ+ isiθ cos θ θ( i θ) 6 θ( i θ) Page twelve + cos si + cos si +... cos θ + cos θ( isi θ) 6cos θsi θ + cos θ( isi θ) + si θ

19 Questio Epected respose (Give oe for each ) Ma Additioal guidace (Illustratio of evidece for awardig a at each ) idetify real terms real part is cos θ 6 cos θsi θ+ si θ idetify imagiary terms imagiary part is cos θsiθ cosθsi θ 7 c As: proof 7 d 6 strategy 7 divide umerator ad deomiator by cos 8 complete As: π π θ ad strategy 0 complete process ad fid π a solutio for θ 0 ta θ, θ fid both solutios si θ ta θ cos θ cos θsiθ cosθsi θ cos θ 6cos θsi θ + si θ cos θsiθ cosθsi θ cos θ cos θ cos θ 6 cos θsi θ si θ + cos θ cos θ cos θ siθ si θ cosθ cos θ taθ ta θ si θ si θ ta θ ta θ + cos θ cos θ ta θ+ ta θ 6ta θ ta θ+ 0 9 taθ ta θ 6ta θ+ ta θ taθ ta θ 6 ta θ+ ta θ π π θ ad 6 6 [END OF SPECIMEN MARKING INSTRUCTIONS] Page thirtee

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1 September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright