04 Bored of Studies Tril Emitios Mthemtics Etesio Writte b Crrotsticks & Trebl Geerl Istructios Totl Mrks 00 Redig time 5 miutes. Workig time 3 hours. Write usig blck or blue pe. Blck pe is preferred. Bord-pproved clcultors m be used. tble of stdrd itegrls is provided t the bck of this pper. Show ll ecessr workig i Questios 6. Sectio I Pges 5 0 mrks ttempt Questios 0 llow bout 5 miutes for this sectio. Sectio II Pges 6 8 90 mrks ttempt Questios 6 llow bout hours 45 miutes for this sectio.
Totl mrks 0 ttempt Questios 0 ll questios re of equl vlue Shde our swers i the pproprite bo i the Multiple Choice swer sheet provided. The followig digrm shows the sketch of f. f? Which of the followig curves is possible sketch of 3 - - -3 () 3 (C) 3-3 - -3 - -3-3 (B) (D) 3 3-3 - -3 - -3
Cosider ellipse with the equtio b d hperbol with the equtio for some positive costts,, b d b. The ellipse d hperbol shre b the sme foci. Which of the followig sttemets is true? () (B) (C) (D) The -itercepts of the ellipse re greter th those of the hperbol The -itercepts of the ellipse re less th those of the hperbol The ellipse d hperbol shre the sme -itercepts The ellipse d hperbol both do ot hve -itercepts d 3 The implicit derivtive of curve is. d Which of the followig sketches is possible grph of the curve? () (B) (C) (D)
4 Suppose tht P ( ) is polomil of degree over the set of comple umbers where 0. Wht does the Fudmetl Theorem of lgebr stte bout this polomil? () P ( ) hs t lest oe comple root (B) P ( ) hs t lest oe rel root (C) P ( ) hs ectl comple roots (D) P ( ) hs ectl rel roots 5 Which of the followig iequlities best defie the regio show below? () z i d i zz z (B) z i Re d i zz z (C) z i d i z z (D) z i d i z z Re Re Re z z 3
6 Which of the followig is the correct simplifictio of si i cos? si i cos 4 () (B) (C) (D) si 4 i cos4 si 4 i cos4 cos 4 isi4 cos 4 isi4 7 Two prticles d B trvel log itersectig circles with equl rdii, where the cetre of oe circle lies o the other circle s show i the digrm below. Prticle is relesed from poit P d moves ti-clockwise towrds poit Q log the mjor rc of oe circle with costt gulr velocit. Simulteousl, prticle B is relesed from poit Q d moves clockwise towrds P log the mior rc of the other circle with costt gulr velocit B. P B Q If the prticles collide t poit P, which of the followig best describes the reltioship betwee their gulr velocities? () (B) B (C) 3 (D) 3 B B B 4
8 Let P ( ) be o-zero polomil with degree. Which of the followig sttemets is lws true? () P ( ) hs rel qudrtic fctors if is eve. (B) P ( ) hs t lest oe rel root if is odd. (C) P ( ) hs ol rel roots if it is rel polomil. (D) P ( ) hs t lest oe rel qudrtic fctor if it is rel polomil. 9 Which of the followig iequlities holds for ll rel? () si (B) t e (C) si si (D) t 0 Let w be th root of, where w. Which of the followig is the ect vlue of 3 3 w w w w w w w w 3 4...? () (B) (C) (D) 5
Totl mrks 90 ttempt Questios 6 ll questios re of equl vlue swer ech questio i SEPRTE writig booklet. Etr writig booklets re vilble. Questio (5 mrks) Use SEPRTE writig booklet. () Let z ib d w i b, where b 0. (i) Clculte zw. (ii) Deduce tht t b t b. (b) d Evlute 0 cos si cos. 3 (c) Let F be curve such tht F 3 3. (i) Sketch the curve F (ii) Hece, sketch possible grph of F.. 4 3 (d) Cosider the polomil P b c, where, b d c re 3 o-zero costts. Prove tht if P hs triple root, the b 4c. (e) Fid the equtio of the locus of z if rg z rg z Ed of Questio. 3 6
Questio (5 mrks) Use SEPRTE writig booklet. () Let z d z be two o-zero comple umbers o the uit circle such tht zz. B usig vectors, or otherwise, prove tht 3 is rel umber. z z zz 3 (b) The rel polomil P p q hs roots, d. 3 Fid the polomil with roots, d Questio cotiues o pge 8 7
(c) sectio of flt rod hs circulr curve of rdius r. cr moves log this 4 curve d the tres provide frictiol force F. The vlue of F is mimum whe the cr trvels with velocit V 0. Suppose the sme rod surfce is ow bked t some gle from the horizotl, where 0. Whe the sme cr trvels log the bk with velocit V, it eperieces o lterl forces. The vlue of the frictiol force from the tres is mimum whe the cr trvels with velocitv. Show tht V cos. V 0 V (d) Let z, z d z 3 be three comple umbers with modulus. Prove tht zz zz3 zz3 z z z3. Questio cotiues o pge 9 8
(e) The digrms below show the grphs of f d g. 3 Sketch oe-third pge grph of f g. Ed of Questio 9
Questio 3 (5 mrks) Use SEPRTE writig booklet. () Let P z k z k0 where 0 Pi ( ). k be moic o-costt polomil with rel coefficiets, (i) Suppose P z hs p rel roots. Let r j be rel root. Show tht r i, for j,,3,..., p. j (ii) Suppose P z hs q o-rel roots k c, where k,,..., q. Prove tht c i c i... cq i (iii) Deduce tht t lest oe o-rel root stisfies i. Questio 3 cotiues o pge 0
(b) prticle is dropped from wether blloo. The equtio of motio of prticle is g kv where g is the ccelertio due to grvit, k is positive costt d v is the velocit of prticle. T secods lter, ideticl prticle B is projected dowwrds from the sme wether blloo with iitil velocit where v B is the velocit of prticle B. u ms. The equtio of motio of prticle B is g kv B (i) Show tht 4 T g ku g kv l. k g g kvb (ii) Show tht prticle B s displcemet B is give b 3 g g ku B u vb l. k k g kvb (iii) Deduce tht if prticle B ctches up with prticle, the prticle B 3 must hve bee relesed o more th u g secods fter prticle. Ed of Questio 3
Questio 4 (5 mrks) Use SEPRTE writig booklet. () The followig digrm shows the curve h e r, where r 0. h R r Use clidricl shells to fid the volume V of the solid formed whe the regio bouded b the curve, the coordite es d the lie r R is rotted bout the verticl is. Questio 4 cotiues o pge 3
(b) The followig digrm shows squre bse o the ple bouded b R R d R R. h R R For give vlue of o the is, thi slice is tke perpediculr to the squre bse d prllel to the is. The shpe of the slice is determied b the equtio h ke lies o the curve, for some positive costt k, d is mde such tht its mimum poit h e s show i the digrm bove. Let R t I e dt. R (i) Show tht the re of this slice is give b e I. (ii) Slices re tke cotiuousl log the is d the dded together, formig solid. Show tht this solid hs volume VB I. (iii) Cosider the solid formed i prt () with volume V. Suppose tht the rotted regio is ow bouded b the coordite is d the lie r R. Write dow the volume V C of the solid formed. (iv) Epli wh V VB VC. (v) Deduce tht lim R R R t e dt. Questio 4 cotiues o pge 4 3
(c) (i) Show tht for 3.... ( ) (ii) gme is pled where the probbilit of wiig is p. Let Pk ( ) represet the probbilit of wiig for the first time o the k th gme. Write dow epressio for Pk ( ) i terms of p d k. (iii) Hece, prove tht lim kpk k p Ed of Questio 4 4
Questio 5 (5 mrks) Use SEPRTE writig booklet. () Evlute si cos cos si d. 4 (b) The digrm below shows poit P o the first qudrt of the ellipse where b. Let be the positive -itercept of the ellipse., b Let d be the distce from P to the origi, d let PS d PS where S is the focus of the ellipse. (i) Show tht e PS ecos (ii) Show tht e si d si. ecos (iii) Show tht e cos d cos. ecos (iv) Hece, prove tht 3 d. si cos e Questio 5 cotiues o pge 6 5
(c) The digrm below shows clider tht hs bee sliced t some gle from the horizotl, formig solid with its top surfce beig ellipse. Cosider poit P o the edge of the top circulr surfce of the clider, which is directl bove poit P o the ellipse. The dimeter of the top circulr surfce o the clider is directl bove the mjor is of the ellipse. Let be the gle subteded t the circle s cetre b the mior rc P. (i) Usig prt (b), show tht cos cos. si cos e where e is the eccetricit of the ellipse t the top surfce of the solid (ii) Hece, or otherwise, prove tht t t cos. Ed of Questio 5 6
Questio 6 (5 mrks) Use SEPRTE writig booklet. () chord XY with midpoit M is costructed i circle with cetre. Two rbitrr chords B d CD re drw to pss through M. Let P d Q be the itersectio poits of D d CB oto XY respectivel. Let F d G be the foot of the perpediculrs from oto D d CB respectivel. C X F P M Q G Y D B (i) Prove tht FM GMC. 3 (ii) Hece, or otherwise, prove tht M is the midpoit of PQ. 3 Questio 6 cotiues o pge 8 7
(b) Let I e d, where is positive iteger. 0 (i) Use mthemticl iductio to prove tht 3 I! e r0 r! r (ii) Prove tht 0 0 e d. e (c) Cosider group of people sittig i chirs lbelled,, 3,, d pile of crds lso lbelled,, 3,,. Ech perso picks crd rdoml without returig it to the pile. perso hs mtchig crd if the umber o the crd is the sme s the umber o their chir. Let the umber of ws where obod receives mtchig crd be (i) Epli wh D D D D, where.. (ii) Show tht the epressio D! r0 r! r stisfies the equtio i (i). (iii) Use prt (b) to prove tht whe is lrge, the probbilit of obod pickig their ow crd is pproimtel e. Ed of Em 8
STNDRD INTEGRLS d, ; 0, if 0 d l, 0 e d e, 0 cos d si, 0 si d cos, 0 d 0 sec t, sec t d sec, 0 d t, 0 d si, 0, d l, 0 d l NTE: l log, 0 e 9 Bored of Studies NSW 04