Year 12 Mathematics Extension 2 HSC Trial Examination 2014

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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.

1 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 stndrd integrls is provided on the bck pge of this pper In questions 6, show ll relevnt resoning nd/or clcultions Totl mrks 00 0 mrks Attempt Questions 0 Allow bout 5 minutes for this section 90 mrks Attempt Questions -6 Section I Section II Allow bout hours nd 45 minutes for this section DO NOT REMOVE THIS PAPER FROM THE EXAMINATION ROOM

2 Section I 0 mrks Attempt Questions 0 Allow bout 5 minutes for this section Use the multiple-choice nswer sheet for Question 0. An object moving in circulr pth of rdius 4 metres trvels 48 metres in seconds. The ngulr speed of the object is: (A) (B) (C) (D) rd/s 4 rd/s rd/s 6 rd/s Wht is the grdient of the tngent to the circle t the point 0,? (A) y 9 (B) (C) (D) 6 6 The number of wys tht 6 items cn be divided between people so tht ech person receives items is: (A) 6 (B) 7 (C) 90 (D) 60

3 4 Which of the following is the epression for sin d? (A) cos cos C (B) cos cos C (C) sin sin C (D) sin sin C 5 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) Tcos mg m (B) Tsin mr (C) T mg (D) T mg m

4 6 The polynomil eqution + 7 = 0 hs roots, nd. Which polynomil eqution hs roots, nd? (A) = 0 (B) = 0 (C) 7 49 = 0 (D) = 0 7 The point, lies on the rectngulr hyperbol =. The eqution of the norml to the hyperbol t P is: (A) (B) (C) + = ( + ) + = = (D) = ( ) 8 Wht is the eccentricity for the hyperbol y? 5 64 (A) (B) (C) (D)

5 9 The Argnd digrm below shows the comple number z. Which Argnd digrm best represents? (A) (B) (C) (D) 5

6 0 The grph of = () is drwn below. Which of the following grphs represents the grph of = [()]? (A) (B) (C) (D) 6

7 Section II 90 Mrks Attempt Questions - 6. Allow bout hours nd 45 minutes for this section. Answer ech question in SEPARATE writing booklet. Etr writing booklets re vilble. In Questions - 6, your responses should include relevnt mthemtics resoning nd/ clcultions. Question (5 MARKS) Use SEPARATE writing booklet Mrks () Let = + nd =. (i) Find (ii) Epress in modulus-rgument form. (iii) Write in simplest Crtesin form. (b) (i) Find the vlues of A, B, C nd D such tht: = (ii) Hence find d. (c) Find ll solutions to the eqution + 8 = 0, given tht i is root of the eqution. (d) Find 7

8 Question (5 mrks) Use SEPARATE writing booklet Mrks () Using clculus, show tht ln( ) for. (b) Consider the ellipse + = (i) Show tht the point P(4 cos, sin ) lies on the ellipse. (ii) Clculte the eccentricity of the ellipse nd hence find the foci nd the directrices of the ellipse. (iii) Find the eqution of the tngent t P(4, ). (iv) Find the eqution of the norml t P(4, ). (v) Show tht the tngent t P cuts the positive directri t 67 7 cos,. 7 7 (vi) Hence show tht PSM = 90, if S is the positive focus. 8

9 Question (5 mrks) Use SEPARATE writing booklet Mrks () Epress () in the form. 4 (b) The grph of = () is shown below. Sketch the following curves on seprte hlf pge digrms. (i) = () (ii) y = () (iii) = [()] (iv) = () (c) Prove tht ( + ) = (d) Find the Crtesin eqution of the following curve nd sketch it on n Argnd Digrm. + + = + 9

10 Question 4 (5 mrks) Use SEPARATE writing booklet () (i) Let I (ln ) n d for n 0,,,,... n n n Show tht In (ln ) In for n (ii) Hence, or otherwise, find (ln ) d. Mrks (b) If T 8, T 0 nd Tn 4Tn 4Tn for n prove by mthemticl induction tht: Tn ( n ) n for n (c) Using the method of cylindricl shells, find the volume of the solid of revolution formed when the re bounded by =, the -is, between = 0 nd =, is rotted bout the y-is. (d) A ldder reches from the ground, over wll 8 metres high, to the side of building metre behind the wll. (i) By using similr tringles, show tht ldder, where l l l, is given by: y 8 nd hence the length l of the 64 l 64. (ii) Hence find the length of the shortest ldder which will stisfy the conditions described bove. 0

Question 5 (5 mrks) Use SEPARATE writing booklet Mrks () Use the substitution u e to find e e d. e 4 (b) The point P cp, c p with p 0 lies on the rectngulr hyperbol y c with focus S.

11 Question 5 (5 mrks) Use SEPARATE writing booklet Mrks () Use the substitution u e to find e e d. e 4 (b) The point P cp, c p with p 0 lies on the rectngulr hyperbol y c with focus S. The point T divides the intervl PS in the rtio :. (i) Show tht the coordintes of T re: c c cp c p T, (ii) Show tht the Crtesin eqution of the locus of T cn be written s: 4c c y c. HINT: find n epression for p in terms of, nd in terms of y. p

12 (c) E B A C D The dimeter AB of circle centre O is produced to E. EC is tngent touching the circle t C, nd the perpendiculr to AE t E meets AC produced t D. BAC Show tht CDE is isosceles. (d) A prticle P of mss 5kg is ttched by two chins, ech of length m, to two fied points A nd B, which lie on verticl plne. P revolves with constnt ngulr velocity bout AB. AP mkes n ngle of with the verticl. The tension in AP is T nd the tension in BP is T where T nd T 0. 0 A l P B (i) Resolve the forces on P in the horizontl nd verticl directions (ii) If the object is rotting in circle of rdius.5m t m/s, find the tension in both prts of the string. (Use g = 0 m/s )

13 Question 6 (5 mrks) Use SEPARATE writing booklet Mrks () Find the first derivtive of (b) (i) The displcement (from fied point) of body moving in stright line is given by, nd its velocity is v. d dv Show tht v v. d d (ii) A prticle of mss one kg is moving in stright line. It is initilly t the origin nd is trvelling with velocity ms. The prticle is moving ginst resisting force v v, where v is the velocity. A Briefly eplin why the ccelertion of the prticle is given by dv v v. dt B C Show tht the displcement of the prticle from the origin is given by v tn. v Show tht the time t which hs elpsed when the prticle is trvelling V with velocity V is given by t log e 4V 4 4 D Find V s function of t. E Hence find the limiting position of the prticle s t. End of Emintion

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16 6 STANDARD INTEGRALS 0, log ln ln 0, ln 0,, sin 0, tn 0, sec tn sec 0, tn sec 0, cos sin 0, sin cos 0, 0, ln 0 if 0, ;, Note d d d d d d d d e d e d n n n d e n n

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18 04 Yer Tril Emintion Student Nme: Mthemtics Etension Section I Multiple-Choice Answer Sheet 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

19 04 Mthemtics Etension HSC Tril Emintion Solutions MULTIPLE CHOICE 4 5 6

20 04 Mthemtics Etension HSC Tril Emintion Solutions

21 04 Mthemtics Etension HSC Tril Emintion Solutions QUESTION ) b) c) Since i is one root, i i 4 is fctor re fctors s coeffs re rel, so

22 04 Mthemtics Etension HSC Tril Emintion Solutions d) 4

23 04 Mthemtics Etension HSC Tril Emintion Solutions QUESTION ) b) 5

24 04 Mthemtics Etension HSC Tril Emintion Solutions 6

25 04 Mthemtics Etension HSC Tril Emintion Solutions QUESTION ) b) 8 7

26 04 Mthemtics Etension HSC Tril Emintion Solutions c) 8

27 04 Mthemtics Etension HSC Tril Emintion Solutions d) 9

28 04 Mthemtics Etension HSC Tril Emintion Solutions QUESTION 4 ) by prts b) (ln ) ln 4 I (ln ) d C mrk correct T k k k,, nk c) lim 0 sin 0

29 04 Mthemtics Etension HSC Tril Emintion Solutions d) (i) Eqution for l (ii) Correct derivtive

30 04 Mthemtics Etension HSC Tril Emintion Solutions QUESTION 5 ) b) (i) (ii)

31 04 Mthemtics Etension HSC Tril Emintion Solutions c) d) (cn be without numericl sub) (cn be without numericl sub)

32 04 Mthemtics Etension HSC Tril Emintion Solutions 4

33 04 Mthemtics Etension HSC Tril Emintion Solutions QUESTION 6 ) ln ( + ) b) (i) mrk correctly estblishes result (ii) A mrk correct eplntion B dv v vv d dv d d dv v v tn v c v c when 0, tn tn tn v v v tn tn tn tn tn tn tn tn tn tn tn tn v v tn v v v mrk some correct progress towrds result d mrk dv v mrk tn tn v mrk correctly estblishes result 5

34 04 Mthemtics Etension HSC Tril Emintion Solutions C dv vv dt dt dv v v v v v v V v t v v dv v log e v log e log e v log e V v loge v V 4 loge log e V V log e 4V D V E t log e 4V V t e 4V 4V e V V t 4e t V t 4e As t, v0 hence tn v V v V dt mrk dv v v mrk correct prtil frctions mrk correct integrtion mrk correct result mrk V e t 4V mrk correct result mrk correct both prts 6

Mathematics Extension Two

Mathematics Extension Two 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