St John s College. UPPER V Mathematics: Paper II. Learning Outcomes 3 and 4. Examiner: SLS / BH Marks: 150 Moderator: DG

Transcription

1 St Joh s College St Joh s College UPPER V Mthemtcs: Pper II Lerg Outcomes 3 d 4 ugust 00 Tme: 3 hours Emer: SLS / BH Mrks: 50 Modertor: DG PLESE RED THE FOLLOWING INSTRUCTIONS CREFULLY. Ths questo pper cossts of pges: Ths emto scrpt of 9 pges. oe pge NSWER SHEET Formul Sheet. Plese check tht our pper s complete.. swer LL the questos. 3. Plese ote tht dgrms re ot ecessrl drw to scle. 4. ll ecessr workg detls must be show. 5. pproved o-progrmmble d o-grphcl clcultors m be used, uless otherwse stted. 6. swers must be rouded off to two decml dgts, uless otherwse stted. 7. It s our ow terest to wrte legbl d to preset our work etl. Pge of

2 QUESTION ( 4; ), R (; 3) d C(6; 3) re pots the Crtes ple. Y R ;3 St Joh s College O X 4; C 6; 3 ) Clculte the legth of R, levg the swer smplest surd form. () b) Clculte the coordtes of F, the mdpot of RC. () c) Determe the sze of, correct to oe decml plce. (3) d) Determe the equto of the le R. (3) e) Prove tht ΔRC s rght-gled trgle. () f) Fd pot S, f RCS forms prllelogrm. (3) 5 QUESTION ) If 5cos 4 0 d t 0, use dgrm to clculte the vlue of s( ) (4) b) Evlute the followg epresso wthout usg clcultor d showg ll workg: cos 35 s5.cos95 cos 5 s5.cos5.t5 c) Prove the followg dett: s.cos cos s t (6) (3) d) Solve for f 5 t 5 where [ 90;80] (3) 6 Pge of

3 QUESTION 3 St Joh s College Zkum ws the offcl mscot for the 00 FIF World Cup. He s leoprd tht loves to pl footbll d hs m prort ws to tur the 00 World Cup to oe joful d uforgettble prt! ssume the Zkum stcker used o the es below hs the detcl frot d bck. ) Wrte dow the rule for the trsformto from to. () b) Wrte dow the rule for the trsformto from to. (3) c) The lowest pot of the 00 emblem les t B (; 3). Wrte dow the coordtes of B f the emblem s rotted 90 t-clockwse bout the org. () d) The bottom left pot of the South frc flg s gve b C( 3; ) d the re of the flg s,5 squre uts. The flg s the elrged through the org b scle fctor of 3. Determe the coordtes of C d the re of the elrged flg. (4) 0 QUESTION 4 Below re the ges of the strtg fftee Sprgbok plers for the frst 00 tr-tos rugb mtch whch we lost 3- gst New Zeld ) Clculte the me ge of the strtg fftee. () b) Clculte the stdrd devto of the strtg fftee. (3) Pge 3 of

4 I ddto, the ges of the NZ strtg leup for the frst tr-tos mtch of 00 produces the followg mesures: 8,,88 St Joh s College c) How m of the NZ rugb plers would ou epect to be older th 3? (3) d) Drw bo-d-whsker plot of the ges of the Sprgbok rugb tem bove the NZ bo-d-whsker o the gve NSWER SHEET. (4) e) NZ comfortbl bet the Sprgboks the frst mtch of the tr-tos ths er. Commet o the potetl fluece of the two tems ge dstrbuto o ther chces of wg the World Cup 0. () 4 QUESTION 5 The followg trgoometrc fuctos hve bee drw o the tervl [ 90 ; 360 ]. f ( ) cos b g( ) s( 30 ) ) Determe the vlue of d b. () b) Demostrte tht f d g do ot tersect t 35. (3) c) Stte the perod of f. () d) Determe the equto of h(), the reflecto of f the -s. () e) Determe the vlues of [0 ; 360 ], where f ( ) g( ) 0. (3) f) If f() s shfted 60 to the rght, gve the equto of the fucto thus creted. () 3 Pge 4 of

5 Frequec St Joh s College QUESTION 6 locl gm s trg to decde whch of two weght loss eercse progrmmes t wll ru for ts clets. The gm coducts plot stud wth Progrmme d Progrmme B to vestgte the weght lost b 60 me ech progrmme. The cumultve frequec grph below s drw gst the weght lost b ech m the two groups over the course of moth. Use the grph to swer the followg questos. Weght lost (kg) ) How much weght dd the four me t pot M lose? () b) Estmte the med weght loss of Progrmme. () c) Estmte the lower qurtle of Progrmme B. () d) Estmte how m me lost more th 6 kgs o Progrmme B. () e) I whch progrmme dd greter umber of me lose more th 8kg the moth? () f) Wrte bref sttemet dscussg whch progrmme s more successful. Justf our clms (3) Pge 5 of

6 St Joh s College QUESTION 7 Below s mge of strfsh set o Crtes es. The cetre of the strfsh hs bee plced t the org. You m ssume the strfsh s perfectl smmetrcl. ) Solve the gle of clto of le O, where O s the org. Gve ths swer correct to decml plce. () b) Clculte the co-ordtes of pot B, s represeted the dgrm bove. (5) c) How m es of reflecto est the strfsh? () d) Below s dgrm tht shows the strfsh s regulr petgo. () Determe the legth of O. () () Hece, or otherwse, determe the re of the petgo tht cots the strfsh. (4) 5 Pge 6 of

7 QUESTION 8 I rugb mtch, Br Hb scores tr t pot T. coverso kck must the be tke b More Ste t pot o TC, where TC s perpedculr to TP. I the dgrm, TP = d, PQ =, TCP ˆ d PCQ ˆ. St Joh s College ) Determe epresso for gle ˆ CPQ terms of () b) Prove tht cos( ) s (4) CP c) If d = 0m, 5,6m d the dstce TC = 6m, solve (5) (Solve gles ccurte to decml plce d dstces to 3 decml plces) QUESTION 9 ) Prove tht s( B) s( B) cos.s B (3) 3 b) Prove tht s 3 3s 4s (6) c) Choose oe of the dettes, ether questo () or (b) bove to ssst ou solvg the geerl soluto of s3s 0 (7) 6 Pge 7 of

8 St Joh s College QUESTION 0 Two of the cogs tht form prt of the clock the St Joh s College tower re represeted the dgrm below. We wll represet the two cogs b smller crcle wth cetre O, the org, d lrger crcle wth cetre M. The pot of cotct of the two crcles s t pot P(-3; ). The rdus of the lrger crcle s 3. ) Determe the equto of the smller crcle cetred t the org. () b) Determe the equto of the le OM. () c) Determe the equto of the commo tget, t, to both crcles. (4) d) If t pot M, wrte b terms of. () e) Determe the equto of the lrger crcle. (8) f) Whch cog wll Mr. Youg hve to replce frst? () 8 QUESTION Isted of wrtg fucto (or relto) s terms [usull f ( ) ], we wrte both d terms of ew vrble, whch we cll t. These re clled prmetrc equtos. Jules Lssjous, eteeth cetur Frech phscst, creted prmetrc equtos d used ther grphs to help hm stud hrmoc moto. The grphs creted hve become kow s Lssjous fgures. Lssjous fgures re wdel used to stud sgl stregth wreless commuctos. Pge 8 of

9 St Joh s College EXMPLE: Two rther fetchg Lssjous fgures re llustrted below. s(3 t) cos(,5 t) s(4 t) cos(7 t) We crete Lssjous fgures usg the followg prmetrc equtos s( t K) cos( bt) Use the Lssjous fgure represeted the grph below to swer the questos tht follow: Note tht the curve s smmetrcl bout the -s. ) Wht s the vlue of t 0, 90 t (0; 0,707)? () b) Fd the vlue of the mssg vrbles for () ; () d (3) the tble below. (4) t 0 () 0, () (3) 70 c) Solve, b d K (4) 0 Pge 9 of

10 NME: St Joh s College SET: NSWER SHEET QUESTION 4 (d) Pge 0 of

11 St Joh s College Pge of MTHEMTICS 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 St Joh s College 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