St John s College. UPPER V Mathematics: Paper 1 Learning Outcome 1 and 2. Examiner: GE Marks: 150 Moderator: BT / SLS INSTRUCTIONS AND INFORMATION

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1 St Joh s College UPPER V Mthemtcs: Pper Lerg Outcome d ugust 00 Tme: 3 hours Emer: GE Mrks: 50 Modertor: BT / SLS INSTRUCTIONS ND INFORMTION Red the followg structos crefull. Ths questo pper cossts of pges, cludg swer Sheet (pge ) d Iformto Sheet (pges d ). Plese check tht our pper s complete.. Red the questos crefull. 3. swer ll the questos. 4. The grphs of Questos 4() d () must be swered o the swer sheet provded d ths sheet must be hded wth our swer booklet. 5. Number our swers ectl s the questos re umbered. 6. You m use pproved o-progrmmble d o-grphcl clcultor, uless otherwse stted. 7. Roud off our swers to oe decml dgt where ecessr. 8. ll the ecessr workg detls must be clerl show. 9. It s our ow terest to wrte legbl d to preset our work etl. Pge of

2 NSWER SHEET NME: QUESTION 4() QUESTION () Pge of

3 Questo Solve for ll rel vlue(s) of : () correct to decml plces (3) 3 ( )( 8)( 7) 0 () log 4 () (d) 94 (3) (e) 0 (6) 6 mrks Questo () Gve the epresso: () Show tht ( ) s fctor () () Hece fctorse the epresso full () I scetfc lss the followg formul defes the reltoshp betwee two vrbles d : log() = log() + log4. Solve for terms of (3) Fd vlue for whch stsfes both of the followg equtos: (4) (d) Smplf: () () (4) (4) 9 mrks Pge 3 of

4 Questo 3 It s possble to epress ll frctos s the sum of two ut frctos. ut frcto s frcto wth umertor of. The tble below llustrtes sequece of some of these frctos frcto dett () Wrte dow the et frcto dett the sequece () Wrte formul for the th frcto dett the sequece (5) Hece, or otherwse, epress 999 s the sum of two ut frctos () 9 mrks Questo 4 The followg fuctos re gve: f( ) g( ) 3 3 () Sketch f d g o the es gve o the NSWER SHEET. Show ll turg pots, tercepts d smptotes (8) Show tht the equto below c be solved usg the grphs bove 3 6 (3) For wht vlues of k wll the equto g() = k hve o rel solutos () Pge 4 of 3 mrks

5 Questo 5 Moré Ste tkes pelt rugb mtch for the Sprgboks, s llustrted the dgrm below. The bll follows ectl the pth of prbol wth equto: p q. The org s tke to be the pot t whch the bll s kcked. The bll reches ts mmum heght of 8 metres whe t s 5 metres w from where the bll ws kcked. The crossbr s t heght of 3 metres. NOT TO SCLE 8 m 5 m 3 m () Determe the vlues of, p d q (4) If the bll s struck t dstce of 4 metres from the gol, show tht the bll clers the cross-br () Wth the sme kck, wht s the mmum dstce from whch the bll could hve bee struck such tht the bll would hve clered the cross-br? (3) 9 mrks Pge 5 of

6 Questo 6 () The grph below shows the deprectg vlue of cr over perod of tme: () Wht s the cost of the cr ew? () () Wht tpe of deprecto s llustrted? () () Use the formto o the grph to fd the rte of deprecto. (3) (v) Clculte, the vlue of the cr fter 6 ers. (3) Cse : Sphwe vests lump sum of R5 000 mmedtel. The vestmet ers compoud terest. Cse B: Dvd decdes to wt for 4 ers d the strts mothl ut of R500, the pmets beg mde t the ed of ech moth. I both cses the effectve rte s 5% per um d the vestmets mture 0 ers from ow. () Clculte the per um oml terest rte for Dvd s vestmet. () () Showg ll ecessr workg, determe who wll hve more csh sved fter 0 ers? (6) 6 mrks Pge 6 of

7 Questo 7 I the dgrm below (ot to scle), the shded re, R, represets the fesble rego ler progrmmg problem. The equto of the le jog C d D s 0 The coordtes (0; ), B(0; 5) d E(8; 0) re gve. BC CD d DE OE. () Wrte dow the set of equltes tht descrbes R. (6) Fd the coordtes of pots D d C. (4) objectve fucto, P = + b, s to be mmsed () If = d b = 3, t whch pot s P mmsed? () () If = 4, wrte dow the vlue of b such tht P s mmsed t pot o CD. () 4 mrks Questo 8 () Fd f '( ) f 3 f ( ) 3 (3) Fd If d d f g '( ) wrte dow possble fucto, g ( ) () (d) Fd f '( ) from frst prcples f: f ( ) 3 (4) (5) 3 mrks Pge 7 of

8 Questo 9 Prt of rll trck follows the pth of cubc curve. pl vew of ths secto of the trck s show the dgrm below. t cert stt, crs d B re t the turg pots of the curve. Cr C s t the pot where = 50. The rce strts t the org, O. 3 The fucto whch descrbes ths prt of the trck s: f ( ) () Determe the co-ordtes of crs d B. (6) Fd the verge grdet of the curve betwee cr d cr C. (4) Fd the equto of the tget to the trck t the strtg flg. () (d) t whch pot betwee d B does cr stop turg to the rght d strt turg to the left? () 4 mrks Questo 0 The gve shpe cossts of sem-crcle of rdus r cm d rectgle wth heght of h cm. () Wrte epresso for the re of the shpe. () If the shpe s subject to the costrt tht the permeter of the rectgle BCD must be 0 cm, determe the vlue of r tht wll eld the mmum possble re of the shpe. (5) Prove tht the vlue foud s mmum. () Pge 8 of 9 mrks

9 Questo () The grph of f( ) s gve below: O the es gve the NSWER SHEET, sketch: () f( ) () () f( ) () Gve the two fuctos: f( ) g ( ) Prove tht g( ) f ( ) (5) 9 mrks Pge 9 of

10 Questo The followg formul m be used ths questo: t Vodcom World of Golf, the golf blls re rrged the form of squre bsed prmd, s show Fgure below. The umber of blls ech ler follows the ptter, 4, 9, 6,.. etc. fter umber of complete lers of blls hve bee used the shpe would the resemble wht s kow s frustum of prmd, s show Fgure. Fgure Fgure () I Fgure, how m golf blls hve bee used t ths stge? () How m blls would there be the 40 th ler of the golf bll prmd () If there re m blls cert ler, how m would there be the ler mmedtel bove? (Gve our swer terms of m) () Determe the umber of blls betwee d cludg the 4 st d 80 th lers (4) 9 mrks END OF PPER Pge 0 of

11 Pge of INFORMTION SHEET c b b 4 ) ( d d r r r ; r r r ; r, 0 r c b T s f T T where f s the frst term of the frst dfferece d s s the secod dfferece h f h f f h 0 lm P P P P F P ) ( ) ( d ; M c m ) ( m m t m ) ( ) ( r b

12 I BC: s b s B c s C b c bc. cos re BC b. s C s ( ) s.cos cos. s s ( ) s.cos cos. s cos ( ) cos.cos s. s cos ( ) cos.cos s. s cos cos s s cos s s. cos cos s ; cos s ; f vr vr s. d P ( ) ( ) ( S) P( or B) P( ) P( B) P( d B) Pge of