National Quali cations SPECIMEN ONLY

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Harvey O’Neal’
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1 AH Ntiol Quli ctios SPECIMEN ONLY SQ/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite workig. Stte the uits for your swer where pproprite. Write your swers clerly i the swer booklet provided. I the swer booklet, you must clerly idetify the questio umber you re ttemptig. Use blue or blck ik. Before levig the emitio room you must give your swer booklet to the Ivigiltor; if you do ot, you my lose ll the mrks for this pper. *SQAH0*

2 FORMULAE LIST Stdrd derivtives Stdrd itegrls f ( ) f ( ) f ( ) f ( ) d si cos sec ( ) t( ) + c si + c t + + t c + t sec l + c cot cosec e e + c sec sect cosec l coseccot e e Summtios (Arithmetic series) S + ( ) d (Geometric series) S ( ) r r ( + ) ( + )( + ) ( + ) r, r, r 6 r r r Biomil theorem ( + b) r 0 r r b r where Cr r! r!( r)! Mcluri epsio iv f ( ) f ( ) f ( ) f( ) f( ) + f ( ) !!! Pge two

3 FORMULAE LIST (cotiued) De Moivre s theorem Vector product [ (cos si )] r θ + i θ r ( cosθ + isiθ) i j k b bsiθˆ i j + k b b b b b b b b b Mtri trsformtio Ati-clockwise rottio through gle, θ, bout the origi, cosθ siθ si θ cosθ Pge three

4 Totl mrks 00 Attempt ALL questios +. Give f( ), show tht f ( ) + ( + ).. Stte d simplify the geerl term i the biomil epsio of. Hece, or otherwise, fid the term idepedet of. 6. Fid 9 6 d.. Show tht the gretest commo divisor of 87 d 79 is. Hece fid itegers d y such tht y.. Fid e d. 6. Fid the vlues of the costt k for which the mtri k k 0 is sigulr. 7. A sphericl blloo is beig iflted. Whe the rdius is 0 cm the surfce re is icresig t rte of 0π cm s. Fid the rte t which the volume is icresig t this momet. (Volume of sphere r, surfce re r ) 8. () Fid the Mcluri epsios up to d icludig the term i, simplifyig the coefficiets s fr s possible, for the followig: (i) f( ) e ( ) (ii) g ( ) + e (b) Give tht h ( ) ( + ) vlue of h. use the epsios from () to pproimte the Pge four

5 9. Three terms of rithmetic sequece, u, u 7 d u 6 form the first three terms of geometric sequece. Show tht 6 d, where d d re, respectively, the first term d commo differece of the rithmetic sequece with d 0. Hece, or otherwise, fid the vlue of r, the commo rtio of the geometric sequece. 0. Usig logrithmic differetitio, or otherwise, fid dy d y ( + ) e e, >. ( ). Fid the ect vlue of + d +. ( ) ( ) give tht 7. () Give tht m d re positive itegers stte the egtio of the sttemet: m is eve or is eve. (b) By cosiderig the cotrpositive of the followig sttemet: if m is eve the m is eve or is eve, prove tht the sttemet is true for ll positive itegers m d. 8. () Idetify the verticl symptotes to this curve d justify your swer.. Cosider the curve i the ( y, ) ple defied by the equtio y Here re two sttemets bout the curve: () It does ot cross or touch the -is. () The lie y 0 is symptote. (b) (i) Stte why sttemet () is flse. (ii) Show tht sttemet () is true. Pge five

6 . The lies L d L re give by the followig equtios. L : L : + 6 y z + y+ z () Show tht the lies L d L itersect d stte the coordites of the poit of itersectio. (b) Fid the equtio of the ple cotiig L d L. A third lie, L, is give by the equtio y+ 7 z. (c) Clculte the cute gle betwee L d the ple. Give your swer i degrees correct to deciml plces.. () Give tht ( ) l + f, fid '( ) f, epressig your swer s sigle frctio. (b) Solve the differetil equtio cos cos dy + yt sec d e give tht y whe. Epress your swer i the form y f( ) Let S r r+ r ( ) where is positive iteger. () Prove tht, for ll positive itegers, S +. (b) Fid (i) the lest vlue of such tht S + S < 000 (ii) the vlue of for which S S S S. 8 Pge si

7 7. () Give z cosθ + isiθ, use de Moivre s theorem d the biomil theorem to show tht: cos θ cos θ 6cos θsi θ+ si θ d si θ cos θsiθ cosθsi θ. (b) Hece show tht tθ t θ tθ. 6t θ + t θ (c) Fid lgebriclly the solutios to the equtio t θ+ t θ 6t θ tθ+ 0 π i the itervl 0 θ. [END OF SPECIMEN QUESTION PAPER] Pge seve

National Quali cations AHEXEMPLAR PAPER ONLY

National Quali cations AHEXEMPLAR PAPER ONLY Ntiol Quli ctios AHEXEMPLAR PAPER ONLY EP/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite