05 Bored of Studies Tril Emitios Mthemtics Etesio Writte by 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 my be used. A tble of stdrd itegrls is provided t the bck of this pper. Show ll ecessry workig i Questios 6. Sectio I Pges 6 0 mrks Attempt Questios 0 Allow bout 5 miutes for this sectio. Sectio II Pges 7 90 mrks Attempt Questios 6 Allow bout hours 45 miutes for this sectio.
Totl mrks 0 Attempt Questios 0 All questios re of equl vlue Shde your swers i the pproprite bo i the Multiple Choice swer sheet provided. The equtio 0 hs root of i? Wht is the vlue of k? i k (A) i. (B) i. (C). (D) 5. Let e be the eccetricity of coic sectio with both foci o the is. Which of the followig is NOT lwys true? (A) (B) (C) For the hyperbol, s e, the symptotes pproch the y is. If two ellipses hve equl eccetricity, the they hve the sme equtio. If two hyperbole hve equl eccetricity, the they shre the sme symptotes. (D) For the ellipse, s e 0 the foci pproch the origi., the directrices move further wy from the origi whilst
3 The hyperbole d hve equtios y d is positive costt. y respectively, where Which of the followig is poit of itersectio of oe of the directrices of of the symptotes of? d oe (A),. (B),. (C),. (D),. 4 Sophi is give four optios to trsform y give curve. (I) Trslte uit to the left, the double the coordites. (II) Double the coordites, the trslte uit to the left. (III) Reflect i the lie, the multiply the coordites by. (IV) Multiply the coordites by, the reflect i the lie. She is the give sketch of the curve y f. Which of the bove optios should she select to obti the grph of y f? (A) (B) Optios Optios I or III. I or IV. (C) Optios (D) Optios II or III. II or IV.
5 Let P be poit o the Argd digrm with coordites,. Let Q be other poit such tht the comple umber represeted by vector PQ is give by bi. y The vector PQ is rotted bout the poit P ti-clockwise by gle of to form the vector PR. Wht re the coordites of R? (A) b,. (B) b,. (C) b,. (D) b,. 3
6 Let be o-zero comple umber d P be moic polyomil with rel coefficiets such tht P P 0. Which of the followig sttemets is lwys true? (A) The comple umber must be rel umber. (B) The polyomil P hs sttiory poit t,0. (C) If is o-rel, the the polyomil P must hve eve degree. (D) If is o-rel, the the smllest possible degree of P is 4. 7 A solid of revolutio is formed by rottig the regio bouded by y, the lie d the is bout the lie. By usig the method of cylidricl shells, which of the followig itegrls is equl to the volume of the solid? (A) 0 d. (B) d. 0 (C) (D) 0 d. d. 0 4
8 A prticle moves i uiform circulr motio bout the origi with gulr velocity. Let be the horizotl displcemet of the prticle. Which of the followig is the correct epressio for the horizotl compoet of the prticle s ccelertio? (A). (B). (C) (D).. 9 Let, b d c be the legths of the sides of ABC, where the logest side hs legth. If BAC is obtuse, which of the followig iequlities is lwys true? (A) bc. (B) (C) (D) b c. bc. b c. 5
0 Let d k be positive itegers. Which of the followig is equivlet to (A) si cos 0 (B) si cos k d. k d. (C) si cos 0 k d. (D) si cos 0 k d. 0 si cos k d? 6
Totl mrks 90 Attempt Questios 6 All questios re of equl vlue Aswer ech questio i SEPARATE writig booklet. Etr writig booklets re vilble. Questio (5 mrks) Use SEPARATE writig booklet. () Fid si d. cossi 3 (b) The digrm below shows right pyrmid with height H d irregulrly 3 shped bse with re A B. Cross sectios tke prllel to the groud t height h hve re A h. H h Show tht the volume of the pyrmid is A H. 3 B (c) The polyomil hs roots,,. 4 3 P p p Fid the vlues of p such tht 0. Questio cotiues o pge 8 7
Questio (cotiued) (d) Poits A, B d C represet the comple umbers, b d c respectively o the Argd digrm, s show below. y A O O B C Let A, B d C be the midpoits of BC, AC d AB respectively. Let the itersectio of AA d BB be P, represetig the comple umber p. You my ssume, without proof, tht AP BP k AA BB, where 0k. (i) Show tht y poit P o the itervl AA c be epressed s k p k b c. (ii) Use (i) to show tht bc p. 3 3 (iii) Deduce tht P lso lies o CC. Ed of Questio 8
Questio (5 mrks) Use SEPARATE writig booklet. () Cosider the curve implicitly defied by the equtio 3 3 y 3l y. (i) Show tht, is criticl poit. (ii) Let, y be y poit o. 0 0 Show tht y, lso lies o. 0 0 (iii) By usig the fct tht l lim 0 3, or otherwise, show tht if, the y. (iv) By cosiderig the epressio 3 3 y 3l 3 3, y 3l y or otherwise, show tht y is symptote. (v) Hece, sketch the grph of the curve. Questio cotiues o pge 0 9
Questio (cotiued) (b) Cosider the equtio z with roots k,,3,...,. k k k cos isi for (i) Show tht 3.... (ii) Hece, show tht 3 si 3 si si...si. Questio cotiues o pge 0
Questio (cotiued) (c) The digrm below shows cr of mss m o the cross sectio of bed i highwy, which is prt of circle of rdius r d cetre O. The rod surfce is bked t gle to the horizotl. The cr eperieces orml force N, lterl force F directed up the bed d grvittiol force g. N F O mg (i) Fid the mimum possible velocity of the cr before it leves the surfce of the coe. (ii) Hece, or otherwise, show tht the mimum mout of lterl force the cr will eperiece is mg F. si Ed of Questio
Questio 3 (5 mrks) Use SEPARATE writig booklet. () A prticle A of mss m is projected verticlly with iitil velocity u d reches mimum height H. It eperieces grvittiol force mg d resistive force mkv A, where v A is the velocity of the prticle d k is costt. At the sme time, prticle B of equl mss is dropped from height H bove the groud. It eperieces grvittiol force mg d resistive force B mkv B. H A Let w be the termil velocity of prticle B. (i) Show tht the displcemet of prticle A from the origi is 3 A w u l k w va, where v A is the velocity of prticle A. (ii) The two prticles collide whe prticle A reches velocity w. The displcemet of prticle B from the poit of relese is give by B w l k w vb (Do NOT prove this) Use this result to show tht whe the prticles collide, prticle B w hs velocity vb. Questio 3 cotiues o pge 3
Questio 3 (cotiued) (b) I PAB, the poit R is chose so tht PR bisects APB. P A R B (i) Prove tht AP AR. BP BR It c similrly be show tht if PS bisects the eterior gle of APB, the AP AS. (Do NOT prove this) BP BS P A B S Questio 3 cotiues o pge 4 3
Questio 3 (cotiued) (ii) The digrm below shows fied poits A d B, d poit P so tht for some fied k. PA k PB. P T A R B S The itervl AP is produced to T d the poits R d S re chose such tht PR d PS bisect APB d BPT respectively. Use (i), or otherwise, to epli why s P vries, R d S remi fied. (iii) Deduce tht the locus of P is circle with dimeter RS. Questio 3 cotiues o pge 5 4
Questio 3 (cotiued) (c) The digrm below shows fied comple umber strictly iside the uit circle o the Argd digrm. y O Copy the digrm ito your writig booklet. (i) Use prt (b), or otherwise, to epli why the locus of z stisfyig z z r, where r, is circle. You do ot eed to fid the cetre d rdius of the circle. (ii) Fid the vlue of r such tht the locus of z touches the uit circle. Ed of Questio 3 5
Questio 4 (5 mrks) Use SEPARATE writig booklet. () The digrm below shows qudrilterl ABCD, where BC CD d BCD BAD. A circle is drw to pss through poits A, B d D. Let the cetre of this circle be O d ssume tht the poit C is distict from O. A O C B D Copy the digrm ito your writig booklet. (i) Show tht COB COD. (ii) Hece, prove tht the poit C cot be distict from O. Questio 4 cotiues o pge 7 6
Questio 4 (cotiued) (b) The digrm below shows poit P cos, bsi hyperbol o the positive brch of the y with foci S d S. From P, tget with equtio b cos ysi is drw to itersect the is t T. Let Q d Q be poits o b such tht QS d QS re perpediculr to. y Q T S (i) Show tht TS PS TS PS. 3 (ii) Show tht QS PS QS PS. (iii) Deduce tht the lie bisects SPS. Questio 4 cotiues o pge 8 7
Questio 4 (cotiued) (c) It c be show tht for positive iteger vlues of, si cos cos cos3... cos si (Do NOT prove this) (i) Show tht 0 si... si d. si 3 5 (ii) Show tht si cos d si si d. (iii) Hece, show tht s, 0 si si d. (iv) Deduce tht... 4 3 5 7 Ed of Questio 4 8
Questio 5 (5 mrks) Use SEPARATE writig booklet. () A sequece of letters is mde usig copies of the letter U d b copies of the letter D i some order, where b. The sequece c be epressed grphiclly by mtchig the letters U d D to series of movemets. From the curret positio P, y, the letter U moves P to, y wheres the letter D moves P to, y. The digrm below shows the correspodig grph of the sequece UDDUUUDU. y O (i) Epli why if the sequece strts from the origi, it will the T b, b. termite t the poit (ii) Write dow the umber of possible pths from the origi to T. (iii) Epli why the umber of pths from, to T tht touches or crosses the is is equl to the umber of pths from, to T. (iv) Hece, show tht the umber of pths from, to T tht do NOT touch or cross the is is b b b. Questio 5 cotiues o pge 0 9
Questio 5 (cotiued) (b) Two cdidtes A d B re polled gist ech other i vote. Ech vote is writte o ideticl crds d plced i bg. The tlly is doe by pullig crd rdomly from the bg, recordig it, the discrdig the crd. Cdidte A hs p votes d Cdidte B hs q votes, where p q. Usig prt (), or otherwise, fid the probbility tht t ll times durig the vote cout, Cdidte A hs higher tlly th Cdidte B. (c) Sketch the grph of y, lbellig y symptotes, sttiory poits d itercepts. (d) Let 3 P k k be cubic polyomil with rel d o-zero coefficiets. (i) Epress the polyomil P i the form b P k, for pproprite vlues of d b. (ii) By cosiderig the sketch from prt (c), epli why P hs ectly oe rel root. (iii) Describe the behviour of the rel root of P s k. (iv) Hece describe the behviour of the modulus d rgumet of 3 the o-rel roots of P s k. Ed of Questio 5 0
Questio 6 (5 mrks) Use SEPARATE writig booklet. () Let f be o-lier cotiuous fuctio i the itervl b, where b. Defie b E f f b f d. b (i) Show tht b b E f d f b f. (ii) Show tht b E b f d. (iii) Suppose f is cocve up for b. Prove tht E 0 d epli the sigificce of this result. Questio 6 cotiues o pge
Questio 6 (cotiued) (b) A cotiuous fuctio f hs the property tht for y i j, i j f f. (i) Epli, with the use of grph, why for y i j, i j f f f i f j. i j (ii) Let be costt such tht 0 d suppose tht. Show tht. (iii) Let be costt such tht 0. Use prt (i), or otherwise, to show tht for y d, f f f. (iv) Let,, 3,, be y set of positive umbers such tht 4.... Use mthemticl iductio to prove tht for...... f f f f. (v) By choosig pproprite fuctio for f, show tht......, for positive vlues of,,...,. Ed of Em
STANDARD INTEGRALS 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 NOTE: l log, 0 e 3 Bored of Studies NSW 05