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 999 MATHEMATICS UNIT (ADDITIONAL) Time llowed Three hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions re of equl vlue All necessry working should be shown in every question my be deducted for creless or bdly rrnged work Stndrd integrls re printed on pge Bord-pproved clcultors my be used Answer ech question in SEPARATE Writing Booklet You my sk for extr Writing Booklets if you need them 586

2 QUESTION Use SEPARATE Writing Booklet x () Evlute xe dx (b) Using the substitution u = e x edx, or otherwise, find x e 3 x x + (c) Find dx x x 3 (d) Find constnts, b nd c such tht x + x x b c ( x + ) ( x ) = + x + + x x + x Hence find dx x + x ( )( ) π (e) Use integrtion by prts to evlute x sin xdx

3 3 QUESTION Use SEPARATE Writing Booklet () Let z = 3 + i nd w = + i Express the following in the form + ib, where nd b re rel numbers: 3 zw iw (b) Let α = + i 3 Find the exct vlue of α nd rg α Find the exct vlue of α in the form + ib, where nd b re rel numbers (c) Sketch the region in the Argnd digrm where the two inequlities π z i nd rg( z + ) both hold 3 (d) Consider the eqution z 3z + 8z+ = Given tht 3i is root of the eqution, explin why + 3i is nother root Find ll roots of the eqution (e) P A z B z O The points A nd B in the complex plne correspond to complex numbers z nd z respectively Both tringles OAP nd OBQ re right-ngled isosceles tringles Explin why P corresponds to the complex number ( + i) z Let M be the midpoint of PQ Wht complex number corresponds to M? Q

4 QUESTION 3 Use SEPARATE Writing Booklet () y 6 y = O 3 y = f(x) x The digrm shows the grph of the function y = f(x) The grph hs horizontl symptote t y = Drw seprte hlf-pge sketches of the grphs of the following functions: y y = f( x) = f ( x ) (iii) y ln f x = ( ) (b) Consider the ellipse E with eqution x y = nd let P = (x, y ) be n rbitrry point on E Clculte the eccentricity of E 9 (iii) (iv) Find the coordintes of the foci of E nd the equtions of the directrices of E Show tht the eqution of the tngent t P is xx yy = Let the tngent t P meet directrix t point L Show tht PFL is right ngle where F is the corresponding focus

5 5 QUESTION Use SEPARATE Writing Booklet () y 3 y = x x O 3 x The shded re shown on the digrm between the curve y = x x, the x xis, x = nd x = 3, is rotted bout the y xis to form solid Use the method of cylindricl shells to find the volume of the solid (b) Suppose the polynomil P(x) hs double root t x = α Prove tht P'(x) lso hs root t x = α 6 The polynomil A(x) = x + x + bx + 36 hs double root t x = Find the vlues of nd b (iii) Fctorise the polynomil A(x) of prt over the rel numbers (c) Determine the domin of the function sin (3x + ) 5 Sketch the grph of the function y = sin (3x + ) (iii) Solve sin (3x + ) = cos x

6 6 QUESTION 5 Use SEPARATE Writing Booklet () The roots of x 3 + 5x + =, re α, β nd γ 3 Find the polynomil eqution whose roots re α, β nd γ Find the vlue of α + β + γ (b) A θ h T O P mg A conicl pendulum consists of bob P of mss m kg nd string of length metres The bob rottes in horizontl circle of rdius nd centre O t constnt ngulr velocity of ω rdins per second The ngle OAP is θ nd OA = h metres The bob is subject to grvittionl force of mg newtons nd tension in the string of T newtons Write down the mgnitude, in terms of ω, of the force cting on P towrds centre O By resolving forces, show tht ω = g h Question 5 continues on pge 7

7 7 QUESTION 5 (Continued) (c) At time t wsp popultion consists of w(t) workers nd r(t) reproductives For the first s dys of the wsp seson the popultion produces workers only nd fter s dys the popultion produces reproductives only 8 For t s, suppose tht the equtions determining the number of workers re dw dt where k is positive constnt Find n expression for w(s) = kw nd w( )=, For t s, suppose tht the equtions determining the number of reproductives re dr = kws () dt nd rs ()= where k is positive constnt ks ()= ( ) Show tht rt ke t s for t s, (iii) If k =, find the vlue of s which mximises r() QUESTION 6 Use SEPARATE Writing Booklet () Let x be fixed, non-zero number stisfying x > Use the method of mthemticl induction to prove tht ( + x) n > + nx for n =, 3, Deduce tht > for n =, 3, n n Question 6 continues on pge 8

8 8 QUESTION 6 (Continued) (b) A bll of unit mss is projected verticlly upwrds from ground level with initil speed U Assume tht ir resistnce is kv, where v is the bll s speed nd k is positive constnt We wish to consider the bll s motion s it flls bck to ground level Let y be the displcement of the bll mesured verticlly downwrds from the point of mximum height, t be the time elpsed fter the bll hs reched mximum height, nd g be the ccelertion due to grvity dv Explin why v() =, nd = g kv while the bll is in motion dt g Deduce tht v = ( e kt ) for t k dv (iii) By writing = v dv, deduce from prt tht dt dy g k log g kv v ky g + = v ky (iv) Using prts nd (iii), deduce tht t = + g e (v) You re given tht the bll reches mximum height h = k U g log k e g+ ku g g+ ku in time th = log e k g (Do NOT prove these results) U + V Deduce tht the totl time T tht the bll is in the ir is T = g, where V is the finl speed tht the bll reches when returning to ground level (vi) If ir resistnce is ignored, the totl time T tht the bll is in the ir U + V is T = g, where V is the finl speed the bll then reches when returning to ground level By considering V nd V, determine which is lrger: T or T

9 9 QUESTION 7 Use SEPARATE Writing Booklet () Grph y = lnx nd drw the tngent to the grph t x = By considering the pproprite re under the tngent, deduce tht 3 ln xdx 8 (iii) By considering the grph of y = ln x, explin why k+ k ln xdx ln k for k =, 3,, (iv) Deduce tht n ln xdx + ln + ln3+ + ln( n )+ ln n 8 for n =, 3,, n (v) Assuming tht ln xdx = nln n n+, deduce tht 7 8 n n! e n ne n for n =, 3,, Question 7 continues on pge

10 QUESTION 7 (Continued) (b) A plyer hs one token nd needs exctly five tokens to win prize He plys gme where he cn vry the number of tokens he bets At ech stge he either doubles the number of tokens he bets or loses the tokens he bets The probbility tht he doubles the number of tokens he bets is p nd the probbility tht he loses the number of tokens he bets is q = p His strtegy is to rech his gol of exctly five tokens s quickly s possible The digrm shows the possible outcomes in terms of number of tokens nd the probbilities ssocited with ech stge 5 q q p p q q 3 p p 5 Strting with one token, wht is the probbility tht he loses ll of his tokens without ever hving four tokens? Wht is the probbility tht he obtins four tokens once nd then loses ll of his tokens without ever hving four tokens gin? (iii) If p =, find the probbility tht he wins prize

11 QUESTION 8 Use SEPARATE Writing Booklet π π () Let ρ = cos + isin The complex number α = ρ + ρ + ρ is root 7 7 of the qudrtic eqution x + x + b =, where nd b re rel 8 Prove tht + ρ + ρ + + ρ 6 = The second root of the qudrtic eqution is β Express β in terms of positive powers of ρ Justify your nswer (iii) Find the vlues of the coefficients nd b (iv) Deduce tht π π 3π sin + sin + sin = (b) P P P 7 O C C In the digrm, is circle, centre C, nd O is fixed point outside the circle The point P is vrible point on nd P is the other point of intersection of OP with The point P is on OP such tht OP OP =k where k is constnt The point C is on OC nd PC PC Explin why OP OP is constnt Deduce tht OP is constnt OP (iii) (iv) Show tht C is fixed point Describe fully the locus of P End of pper

12 STANDARD INTEGRALS n x dx n+ = x, n ; x, if n< n + x dx = ln x, x > e x dx e x =, cosx dx = sin x, sin x dx = cos x, sec x dx = tn x, sec x tn x dx = sec x, x dx x = tn, + x dx x = sin, >, < x < ( ) > > dx = ln x + x, x x ( ) dx = ln x + x + x + NOTE : ln x = log x, x > e Bord of Studies NSW 999