Mathematics Extension Two

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Ophelia Phillips
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1 Student Number 04 HSC TRIAL EXAMINATION Mthemtics Etension Two Generl Instructions Reding time 5 minutes Working time - hours Write using blck or blue pen Bord-pproved clcultors my be used Write your Student Number t the top of this pge nd on ll writing booklets used. All necessry working should be shown in every question Totl mrks 00 Attempt ll 6 questions. Question numbers -0 re ll multiple choice questions worth mrk ech. Answers to question numbers -0 re to be nswered in the multiple choice nswer sheet provided. Questions -6 re ech worth 5 mrks nd re to be nswered in seprte writing booklets clerly mrked Question, Question etc.

2 This pge hs been left intentionlly blnk

3 Totl Mrks 0 Attempt Questions 0 All Questions re Mrk ech. Answer questions on the multiple choice nswer sheet provided.. If z 5i find z epressed with rel denomintor. 4 (A) 5 i) 9 (B) 5 i) 9 (C) 5 i) 4 (D) 5 i). A cr hs no tendency to slip when trvelling t speed of v ms round section of trck of 0 rdius 90 m which is bnked t n ngle of 0. Tking g 9.80 ms, the speed of the cr is (A) (B) (C) (D) 0.ms.86ms 5.ms.47ms. The polynomil eqution 7 0 hs roots, nd. Which polynomil eqution hs roots, nd. (A) (B) (C) (D)

4 4. The digrm below shows the grph of the function y f (). Which of the following is the grph of (A) y f()? (B) (C) (D)

5 5. Find d (A) 4 tn c (B) tn c (C) tn c (D) sin c 6. An ellipse hs its centre st the origin nd its foci on the - is. The distnce between the foci is 4 units nd the distnce between the directrices is 6 units. The eqution of the ellipse is: (A) (B) (C) (D) y 8 0 y 6 0 y 6 y 0 4

6 7. A prticle t B is ttched to string AB tht is fied t A. The prticle rottes in horizontl circle with rdius of r. Let T be the tension in the string nd BOA. Which of the following sttements is correct? (A) (B) T cos mg m T sin mr (C) T mg (D) T mg m 8. The region bounded by the curve y log e, the - is nd the line e is rotted bout the y - is to form solid. Using the method of cylindricl shells, the volume generted could be clculted by: y (A) 0 e e dy (B) log y (C) 0 e e d e e dy (D) loge d. 5

7 9. The point P cp, c p lies on the rectngulr hyperbol y c. The eqution of the norml to the hyperbol t P is: (A) pqy c p q (B) p y cp (C) p y cp p p (D) py c p cp 0. A mss of 0.05 tonnes is ttched to light inetensible string which is fied to ceiling, s shown in the digrm below. A force of 400 N, pplied horizontlly, keeps the mss t rest, t n ngle to the verticl. ceiling T 400 N mg Which of the following represent the correct ppliction of the eqution of motion? (A) Tcos 0.05g 0 (B) Tsin 50g (C) Tcos 0.05g 400 (D) Tcos 50g 0 6

8 Totl Mrks 90 Attempt Questions 6 All Questions re of equl vlue Answer ech question in SEPARATE writing booklet. Etr writing booklets re vilble. Question ( 5 Mrks ) Use SEPARATE writing booklet. Mrks () (i) Write i in modulus/rgument form (ii) i Hence, epress z in modulus/rgument form nd find the 4 i 7 ect vlue of cos in its simplest form. (b) (i) Find the constnts A, B, C such tht 4 B C A (ii) Hence find 4 d (c) Sketch the region on the Argnd digrm where the inequlities z 4 nd 0 rg( z+) both hold. 4 (d) Using the substitution sin, evlute d End of Question 7

9 Question ( 5 Mrks ) Use SEPARATE writing booklet. Mrks () (i) Given tht i is root of the eqution 0 0, write down the other roots. (ii) 4 Given the polynomil P( ) hs root of multiplicity, solve P ( ) 0. (b) The digrm below shows the ellipse y. b The point Pcos, bsin lies on the ellipse. The norml to the ellipse t P meets the mjor nd minor es of the ellipse in G nd G respectively. N nd N re the feet of the perpendiculrs from P to the mjor nd minor es respectively. / N P( cos, bsin ) O G N / G (i) Show tht the eqution of the norml t P is sec by cosec b (ii) Show tht the rtio OG : ON e : (iii) Find the rtio of the re of PNG : PN G Question No. continues on pge 9 8

10 (c) A prticle of mss m moves in horizontl stright line. The prticle is resisted by constnt force mk nd vrible force mv, where k is positive constnt nd v is the speed. Initilly v u nd 0. (i) (ii) Show tht the distnce trvelled is ln k u k v Show tht the time tken for the prticle to be brought to rest is given by tn u t k k End of Question 9

11 Question ( 5 Mrks ) Use SEPARATE writing booklet. Mrks () Epress i i in the form r cis 4 (b) The grph of y f () is shown below. Sketch the following curves on seprte hlf pge digrms. (i) y f () (ii) y f d d (iii) y f () (iv) y f () (c) Prove tht. cis cis cis (d) Find the Crtesin eqution of the following curve nd sketch it on n Argnd Digrm. z i z i End of Question 0

12 Question 4 ( 5 Mrks ) Use SEPARATE writing booklet. Mrks () A mss m t P is freely joined to two equl light rods PQ nd PR of length. The end Q of PQ is pivoted to fied point Q nd the end R of PR is freely joined to ring of mss m tht slides on smooth verticl pole. P rottes in horizontl circle with uniform ngulr velocity. T nd T re tensions in the rods nd N is the norml rection of QR on the ring R. (i) Write four equtions of motion connecting the tensions nd the norml rection, with,r nd. 4 (ii) Comment on the tensions between the rods PQ nd PR. (iii) r Show tht the ngle is given by tn g (iv) Determine the vlue of the norml rection on the ring. where z 0, (b) Given z rcos isin (i) Use De Moivre s Theorem to show tht for positive integers n z n cos n z (ii) (iii) Epnd z z Hence evlute 5 nd show tht cos cos5 5cos 0cos cos d 0 End of Question 4

13 Question 5 ( 5 Mrks ) Use SEPARATE writing booklet. Mrks () Let In sin n d, where n is non negtive number. 0 I n sin cos d where n. (i) Show tht n n 0 (ii) Deduce tht I n n In, where n n (iii) Evlute I 4 (b) Find the vlue of e sin d 0 (c) A cr (which is represented by point of mss m), trvels round circulr bend which is bnked t n ngle of to the horizontl. The cr trvels t line on the rod where the rdius of the curve is r metres, t constnt speed. F is the frictionl force cting on the cr nd N is the Norml rection. Both F nd N re in Newtons. (i) By resolving the horizontl nd verticl components of the forces cting on the cr, find epressions for N cos nd N sin. Question No.5 continues on pge

14 (ii) At wht velocity (V ) should the cr trvel if there is to be no sidewys frictionl force cting on it. (iii) If r = 0m nd the rod is bnked to llow crs to trvel t 00km/h without ny sidewys frictionl force, find the vlue of. (Use g = 0m/s.) (e) Given tht 5 5t 0t t tn5, where t tn, 4 0t 5t Show tht if tn5 0, then 4 tn tn tn tn End of Question 5

15 Question 6 ( 5 Mrks ) Use SEPARATE writing booklet. Mrks () The region between the curve y 6 4, the -is, 0 nd, is rotted bout the line. By using the method of cylindricl shells, show tht the volume of the solid is given by V d (You do not hve to perform this integrtion) (b) A solid hs s its bse, the ellipse 9 + y 4 =. Find the volume of the solid if every cross section perpendiculr to the mjor is is n isosceles right ngled tringle with the hypotenuse on the bse of the solid. 5 Question 6 continues on pge 5 4

16 (c) y The digrm shows the hyperbol where b 0. b The points P ( sec, btn ) nd Q ( sec, btn ) lie on the hyperbol nd the chord PQ subtends right ngle t the origin. y Q O P (i) Use the prmetric representtion of the hyperbol to show tht sinsin b (ii) Hence show tht the grdient of the curve t P( sec, btn ) is dy b sin d End of Emintion 5

17 STANDARD INTEGRALS, ; 0, if 0 n n n d n 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 tn d, 0 d sin, 0, d ln, 0 d ln NOTE : ln log, 0 e 6

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Student Number Section I Multiple Choice Answer Sheet Allow bout 5 minutes for this section Select the lterntive A, B, C or D tht best nswers the question. Fill in the response ovl completely.

19 Student Number Section I Multiple Choice Answer Sheet Allow bout 5 minutes for this section Select the lterntive A, B, C or D tht best nswers the question. Fill in the response ovl completely. Smple: + 4 = (A) (B) 6 (C) 8 (D) 9 A B C D If you think you hve mde mistke, put cross through the incorrect nswer nd fill in the new nswer. A B C D If you chnge your mind nd hve crossed out wht you consider to be the correct nswer, then indicte the correct nswer by writing the word correct nd drwing n rrow s follows. A B C D Strt Here. A B C D. A B C D. A B C D 4. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D 9. A B C D 0. A B C D 8

Year 12 Mathematics Extension 2 HSC Trial Examination 2014

Year 12 Mathematics Extension 2 HSC Trial Examination 2014 Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used A tble of