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

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1 HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two 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 etr Writing Booklets if you need them 585

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3 3 QUESTION Use SEPARATE Writing Booklet () Differentite tn (b) Find the cute ngle between the lines 3y = + 8 nd y = 5 9 (c) Evlute lim sin (d) Given tht log 7 = 807 (to three deciml plces), find log 4 (e) Let α, β nd γ be the roots of the polynomil 3 4 = 0 Find αβγ π (f) Evlute 3 sin d 4 0

4 4 QUESTION Use SEPARATE Writing Booklet () Find the quotient, Q(), nd the reminder, R(), when the polynomil P() = 4 + is divided by + 3 (b) Pul plns to contribute to retirement fund He will invest $500 on ech birthdy from ge 5 to 64 inclusive Tht is, he will mke 40 contributions to the fund The retirement fund pys interest on the investments t the rte of 8% per nnum, compounded nnully How much money will be in Pul s fund on his 65th birthdy? 4 (c) C 5 b A D c B The tringle ABC hs sides of length, b nd c, s shown in the digrm The point D lies on AB, nd CD is perpendiculr to AB Show tht sinb = bsina Show tht c = cosb + b cosa (iii) Given tht c =4b cosa cosb, show tht = b

5 5 QUESTION 3 Use SEPARATE Writing Booklet () Use the method of mthemticl induction to prove tht 4 n + 4 is multiple of 6 for n 3 (b) Epress sin 4t + 3cos4t in the form Rsin 4t + α, where α is in rdins 4 Hence, or otherwise, find the generl solution of the eqution sin 4t + 3cos4t = 0 in ect form (c) A prticle moves in stright line nd its position t time t is given by = + sin 4t + 3cos4t 5 Prove tht the prticle is undergoing simple hrmonic motion bout = Find the mplitude of the motion (iii) When does the prticle first rech mimum speed fter time t = 0?

6 6 QUESTION 4 Use SEPARATE Writing Booklet () y y h h O O The left-hnd digrm shows the lower hlf of the circle + (y 5) = 5 The shded re in this digrm is bounded by the semicircle, the line y = h, nd the y is Show tht the volume V formed when the shded re is rotted round the y is is given by πh V = 5πh 3 A semicircle is rotted round the y is to form hemisphericl bowl of rdius 5 cm, s shown in the right-hnd digrm 3 The bowl is filled with wter t constnt rte of 3 cm 3 s Find the rte t which the wter level is rising when the wter level is 6 cm 3 (b) Consider the function f( )= + for > 5 (iii) Give the equtions of the horizontl nd verticl symptotes for y = f () Find the inverse function f () Stte the domin of f () Question 4 continues on pge 7

7 7 QUESTION 4 (Continued) (c) A NOT TO SCALE D E B C ABC is n cute-ngled tringle D is point on AC, E is point on AB, nd BEC = BDC, s shown in the digrm Sony ws sked to prove tht AED = ACB She provided two-step proof but did not give resons Stte reson for her correct sttement tht EDCB is cyclic qudrilterl Stte reson why she could then correctly conclude tht AED = ACB

8 8 QUESTION 5 Use SEPARATE Writing Booklet () Write n 3 using Σ nottion (b) The polynomil P() = hs one rel root in the intervl < <0 6 (iii) Sketch the grph of y = P() for between nd Clerly lbel ny sttionry points Let = 4 be first pproimtion to the root Apply Newton s method once to obtin nother pproimtion to the root Eplin why the ppliction of Newton s method in prt ws NOT effective in improving the pproimtion to the root (c) A tnk contins 0 tgged fish nd 50 untgged fish On ech dy, 4 fish re selected t rndom from the tnk nd plced together in seprte tnk for observtion Lter the sme dy the 4 fish re returned to the originl tnk 5 (iii) (iv) Wht is the probbility of selecting no tgged fish on given dy? Wht is the probbility of selecting t lest one tgged fish on given dy? Clculte the probbility of selecting no tgged fish on every dy for 7 given dys Wht is the probbility of selecting no tgged fish on ectly of the 7 dys?

9 9 QUESTION 6 Use SEPARATE Writing Booklet () y V 8 45 O A prticle is projected from the point (0, ) t n ngle of 45 with velocity of V metres per second The equtions of motion of the prticle re = 0 nd y = g Using clculus, derive the epressions for the position of the prticle t time t Hence show tht the pth of the prticle is given by y = + g V A volleybll plyer serves bll with initil speed V metres per second nd ngle of projection 45 At tht moment the bottom of the bll is metre bove the ground nd its horizontl distnce from the net is 9 3 metres The bll just clers the net, which is 3 metres high (iii) Show tht the initil speed of the bll is pproimtely 0 3 metres per second (Tke g = 9 8ms ) Wht is the horizontl distnce from the net to the point where the bll lnds? (b) The ccelertion ms of prticle P moving in stright line is given by = 3 where metres is the displcement of the prticle to the right of the origin Initilly the prticle is t the origin nd is moving with velocity of 4 m s, 4 Show tht the velocity v ms of the prticle is given by v = Will the prticle ever return to the origin? Justify your nswer

10 0 QUESTION 7 Use SEPARATE Writing Booklet () Use the binomil theorem to obtin n epnsion for n ( + ) + where n is positive integer n, Hence evlute + 0 C + 0 C C 0 (b) Use the substitution y = to find 7 d Use the substitution z = to find nother epression for d (iii) Use the results of prts nd to epress sin in terms of sin ( ) for 0 < < (c) Find ll rel such tht 4 > 3 End of pper

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12 STANDARD INTEGRALS n d n+ =, n ; 0, if n< 0 n + d = ln, > 0 e d e =, 0 cos d = sin, 0 sin d = cos, 0 sec d = tn, 0 sec tn d = sec, 0 d = tn, 0 + d = sin, > 0, < < > > d = ln +, d = ln NOTE : ln = log, > 0 e 0 Bord of Studies NSW 998