HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time)

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1 HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 4 UNIT (ADDITIONAL) Time llowed Three hours (Plus 5 miutes redig time) DIRECTIONS TO CANDIDATES Attempt ALL questios ALL questios re of equl vlue All ecessr workig should be show i ever questio Mrks m be deducted for creless or bdl rrged work Stdrd itegrls re prited o pge Bord-pproved clcultors m be used Aswer ech questio i SEPARATE Writig Booklet You m sk for extr Writig Booklets if ou eed them 586

2 QUESTION Use SEPARATE Writig Booklet Mrks 6 () Evlute 9 + x dx 0 (b) Fid x l x dx si x (c) Fid cos x dx (d) Usig the substitutio u = 4 x, or otherwise, evlute x 4 x dx 0 4 (e) Fid the remider whe x + 6 is divided b x + x 6 4 x + 6 Hece, fid dx x + x 6

3 QUESTION Use SEPARATE Writig Booklet Mrks () Evlute i i (b) Let z = i 5 ( ) Simplif 8 + 4i i Express z i the form + ib, where d b re rel umbers Hece, or otherwise, fid z d rg( z) (c) Sketch the regio i the complex ple where the iequlities z + i d Im( z) 0 both hold (d) Q P O The poits P d Q i the complex ple correspod to the complex umbers z d w respectivel The trigle OPQ is isosceles d POQ is right gle Show tht z + w = 0 (e) B solvig the equtio z + = 0, fid the three cube roots of 6 Let λ be cube root of, where λ is ot rel Show tht λ = λ Hece simplif ( λ) 6

4 4 QUESTION Use SEPARATE Writig Booklet Mrks () Let f( x)= x 4 Provide seprte hlf-pge sketches of the grphs of the x followig fuctios 6 = f( x) = f( x) = e f ( x ) Lbel ech grph with its equtio (b) Let I = ( x) dx l e 4 Show tht I = e I for =,,, Hece evlute I 4 (c) The popultio P of tow decreses t rte proportiol to the umber b which the popultio exceeds 000 Thus 5 dp dt = kp 000 Show tht P = Ae kt, where A d k re costts, is solutio of this equtio Iitill the popultio of the tow ws 500 Te ers lter, it hd flle to 900 Whe will the popultio be 500? Wht does this mthemticl model predict bout the popultio of the tow i the log term?

5 5 QUESTION 4 Use SEPARATE Writig Booklet Mrks () Suppose tht k is double root of the polomil equtio f( x)= 0 Show tht f ( k)= 0 7 Wht feture does the grph of polomil hve t root of multiplicit? 7 6 The polomil P( x)= x + bx + is divisible b x Fid the coefficiets d b 4 x x x (iv) Let Ex = + x+ + + Prove Ex = 0 hs o double roots 6 4 (b) 5 P S R A plet P of mss m kilogrms moves i circulr orbit of rdius R metres roud str S Coordite xes re tke i the ple of the motio, cetred t S The positio of the plet t time t secods is give b the equtios t x = Rcos π T t d = Rsi π, T where T is costt Show tht the plet is subject to force of costt mgitude, F ewtos It is kow tht the mgitude of the grvittiol force pullig the plet towrds the str is give b F = GMm, R where G is costt d M is the mss of the str S i kilogrms Fid expressio for T i terms of R, M d G Questio 4 cotiues o pge 6

6 6 QUESTION 4 (Cotiued) Mrks (c) A ur cotis red blls d w white blls Sue drws two blls together from the ur The probbilit tht the hve the sme colour is Bill dopts differet procedure He drws oe bll from the ur, otes its colour d replces it He the drws secod bll from the ur d otes its colour The probbilit tht both blls hve the sme colour is ow 5 8 Fid ll possible vlues of w QUESTION 5 Use SEPARATE Writig Booklet () 6 R Q = mx + b P O x C S D + = + The digrm shows the circles C : x+ b d D : ( x ) + = + b, which meet t the poits R(0, b) d S(0, b) The stright lie = mx + b meets the circles t P, Q d R, s show i the digrm + mb Show tht the x coordite of the poit P is + m Fid the x coordite of the poit Q Hece fid the equtio of the locus of the midpoit of PQ s the slope of the stright lie through R vries Describe this locus geometricll Questio 5 cotiues o pge 7

7 7 QUESTION 5 (Cotiued) Mrks (b) S R 9 Z Y P Q C W x X x b A B The digrm shows sdstoe solid with rectgulr bse ABQP of legth b metres d width metres The ed PQRS is squre, d the other ed ABC is equilterl trigle Both eds re perpediculr to the bse Cosider the slice of the solid with fce WXYZ d thickess x metres, s show i the digrm The slice is prllel to the eds d AW = BX = x metres Fid the height of the equilterl trigle ABC Give tht the trigles CRS d CYZ re similr, fid YZ i terms of, b d x Let the perpediculr height of the trpezium WXYZ be h metres Show tht h = + x b (iv) Hece show tht the cross-sectiol re of WXYZ is give b 4b [ + ] + x b b x (v) Fid the volume of the solid

8 8 QUESTION 6 Use SEPARATE Writig Booklet Mrks () Cosider the followig sttemets bout polomil Q(x) If Qx is eve, the Q x is odd If Q x is eve, the Qx is odd Idicte whether ech of these sttemets is true or flse Give resos for our swers (b) The probbilit tht ccidets occur t give itersectio durig er is 6 P = e 6 6!, = 0,,, Fid the probbilit tht o ccidets occur t the itersectio i give er Give our swer correct to three deciml plces Wht is the probbilit tht, i give te-er period, there re t lest ers i which o ccidets occur t the itersectio? Give our swer correct to three deciml plces P + B cosiderig vlues of for which, determie the most P likel umber of ccidets i give oe-er period Questio 6 cotiues o pge 9

9 9 QUESTION 6 (Cotiued) Mrks (c) 7 T H O U Px (, ) Se,0 x The poit Se,0 is focus of the hperbol H : x = The tget to the hperbol t poit P(x, ) meets the smptotes of H i T d U, s show i the digrm Show tht the equtio of the tget TU is x x = Show tht the grdiet of SU is ex ( + ) Let UST = θ Show tht t θ =

10 0 QUESTION 7 Use SEPARATE Writig Booklet Mrks () Let Pz ()= z z + The complex umber w is root of P(z) = 0 Show tht iw d w re lso roots of P(z) = 0 6 Fid oe of the roots of P(z) = 0 i exct form Hece fid ll the roots of P(z) = 0 (b) Differetite si ( u) u α + u Hece show tht du = si α + α for 0 < α < u 0 (c) A bed of mss m slides log wire i the shpe of the curve 6 = x, where 0 x At time t, the bed is t xt (), t (), d its velocit is xt, ( t) The motio of the bed is govered b the equtios where E d g re costts, d mx + m + mg = E, x Whe t = 0, the bed is relesed from rest t the poit the wire towrds the origi, where it rrives t time t g Fid E, d show tht ẏ = + Fid ẋt d ẏt x = (, ) It ccelertes log Usig the result of prt (b), or otherwise, fid the time it tkes for the bed to trvel from to the origi ( 8, 8 )

11 QUESTION 8 Use SEPARATE Writig Booklet Mrks () The umbers p, q d s re fixed d positive Also p >, q > d 8 q p = q Wht positive vlue of t miimises the expressio f t p s p q t q ()= + st? Show tht for ll t >0, p s p q t + q st Prove b iductio tht for ll x,,x >0 ( xx x ) x+ x + + x (iv) Deduce tht, for ll,,, >0, (b) D C 7 E B F ABCDEF is cclic hexgo A Show tht DAB + BCD = ABC + CDA Show tht FAD + DEF = EFA + ADE Deduce tht ABC BCD + CDE DEF + EFA FAB = 0 (iv) (v) Stte d prove similr result for cclic octgo Formulte similr result for cclic -go Ed of pper

12 STANDARD INTEGRALS x dx + = x, ; x 0, if < 0 + x dx = l x, x > 0 e x dx e x =, 0 cosx dx = si x, 0 si x dx = cos x, 0 sec x dx = t x, 0 sec x t x dx = sec x, 0 x dx x = t, 0 + x dx x = si, > 0, < x < > > dx = l x + x, x x dx = l x + x + x + NOTE : l x = log x, x > 0 e 0 Bord of Studies NSW 998

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/ UNIT (COMMON) Time llowed Two hours (Plus 5 miutes redig time) DIRECTIONS TO CANDIDATES Attempt ALL questios. ALL questios