Mathematics Extension 2

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1 004 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard itegrals is provided at the back of this paper All ecessary workig should be show i every questio Total marks 0 Attempt Questios 8 All questios are of equal value 4

2 Total marks 0 Attempt Questios 8 All questios are of equal value Aswer each questio i a SEPARATE writig booklet Etra writig booklets are available Questio (5 marks) Use a SEPARATE writig booklet (a) Use itegratio by parts to fid e d π 4 si (b) Evaluate cos d 0 d (c) By completig the square, fid 5+ 4 (d) (i) Fid real umbers a ad b such that 7+ 4 a b + + ( ) + ( ) ( ) 7+ 4 Hece fid d ( + ) ( ) (e) Use the substitutio = siθ to fid d 4 4 0

3 Questio (5 marks) Use a SEPARATE writig booklet (a) Let z = + i ad w = i Fid, i the form + iy, (i) zw 0 z (b) Let α = + i ad β = + i (i) (iii) Fid α, i the form + iy β Epress α i modulus-argumet form Give that β has the modulus-argumet form π π β = cos + isi 4 4 fid the modulus-argumet form of α β (iv) Hece fid the eact value of si π (c) Sketch the regio i the comple plae where the iequalities z+ z ad z i hold simultaeously Questio cotiues o page 4

4 Questio (cotiued) (d) The diagram shows two distict poits A ad B that represet the comple umbers z ad w respectively The poits A ad B lie o the circle of radius r cetred at O The poit C represetig the comple umber z + w also lies o this circle B C A O Copy the diagram ito your writig booklet (i) Usig the fact that C lies o the circle, show geometrically that AOB = π Hece show that z = w (iii) Show that z + w + zw = 0 Ed of Questio 4

5 Questio (5 marks) Use a SEPARATE writig booklet 4 (a) Sketch the curve y = showig all asymptotes 9 (b) The diagram shows the graph of y = f() y 4 y = f() 5 5 Draw separate oe-third page sketches of the graphs of the followig: (i) y y = f( ) = ( f( ) ) (iii) y = f ( ) (c) Fid the equatio of the taget to the curve defied by y + y = 5 at the poit (, ) Questio cotiues o page 6 5

6 Questio (cotiued) (d) The base of a solid is the regio i the y plae eclosed by the curves y = 4, y = 4 ad the lie = Each cross-sectio perpedicular to the -ais is a equilateral triagle y y = 4 O y = 4 (i) Show that the area of the triagular cross-sectio at = h is h 8 Hece fid the volume of the solid Ed of Questio 6

7 Questio 4 (5 marks) Use a SEPARATE writig booklet (a) Let α, β, ad γ be the zeros of the polyomial p() = (i) Fid α βγ + αβ γ + αβγ Fid α + β + γ (iii) Usig part, or otherwise, determie how may of the zeros of p() are real Justify your aswer (b) The vertices of a acute-agled triagle ABC lie o a circle The perpediculars from A, B ad C meet BC, AC ad AB at D, E ad F respectively These perpediculars meet at H The perpediculars AD, BE ad CF are produced to meet the circle at K, L ad M respectively A L M F H E B D K C (i) (iii) (iv) Prove that AHE = DCE Deduce that AH = AL State a similar result for triagle AMH Show that the legth of the arc BKC is half the legth of the arc MKL Questio 4 cotiues o page 8 7

8 Questio 4 (cotiued) (c) y P O Q S T y The poit P lies o the ellipse The chord through P ad the focus + = a b S(ae, 0) meets the ellipse at Q The tagets to the ellipse at P ad Q meet at the yy poit T( 0, y 0 ), so the equatio of PQ is 0 0 (Do NOT prove this) + = a b (i) Usig the equatio of PQ, show that T lies o the directri The poit P is ow chose so that T also lies o the -ais PS What is the value of the ratio? ST (iii) Show that PTQ is less tha a right agle (iv) Show that the area of triagle PQT is b e e Ed of Questio 4 8

9 Questio 5 (5 marks) Use a SEPARATE writig booklet (a) (i) Let a > 0 Fid the poits where the lie y = a ad the curve y = ( a) itersect Let R be the regio i the plae for which ( a) y a Sketch R (iii) A solid is formed by rotatig the regio R about the lie = a Use the method of cylidrical shells to fid the volume of the solid 4 (b) (i) I how may ways ca studets be placed i two distict rooms so that either room is empty? I how may ways ca five studets be placed i three distict rooms so that o room is empty? Questio 5 cotiues o page 0 9

10 Questio 5 (cotiued) (c) A smooth sphere with cetre O ad radius R is rotatig about its vertical diameter at a uiform agular velocity, ω radias per secod A marble is free to roll aroud the iside of the sphere R O θ N r P mg Assume that the marble ca be cosidered as a poit P which is acted upo by gravity ad the ormal reactio force N from the sphere The marble describes a horizotal circle of radius r with the same uiform agular velocity, ω radias per secod Let the agle betwee OP ad the vertical diameter be θ (i) Eplai why mrω = Nsiθ ad mg = Ncosθ Show that either cosθ = g or θ = 0 Rω (iii) Hece, or otherwise, show that if θ 0 the ω > g R Ed of Questio 5 0

11 Questio 6 (5 marks) Use a SEPARATE writig booklet (a) (i) Show that si cos d π = By makig the substitutio = π u, fid π 0 + π si cos d 0 + (b) A particle is released from the origi O with a iitial velocity of A ms directed vertically dowward The particle is subject to a costat gravitatioal force ad a resistace which is proportioal to the velocity, v ms, of the particle Let be the displacemet i metres of the particle below O at time t secods after the release of the particle, so that the equatio of motio is ẋ = g kv where g ms is the acceleratio due to gravity, (i) The termial velocity of the particle is B ms g Show that k = B d kt kt Verify that v satisfies the equatio ( ve )= ge dt (iii) Hece show that the velocity of the particle is give by ( ) v = B B A e gt B B (iv) Deduce that Bt g B A gt = ( ) e B Questio 6 cotiues o page

12 Questio 6 (cotiued) At the same time as the particle is released from O, a idetical particle is released from the poit P which is h metres below O The secod particle has a iitial velocity of A ms directed vertically upward B Its displacemet below O is give by h Bt g B A gt e = + ( + ) B (Do NOT prove this) (v) (vi) Suppose that the two particles meet after T secods Show that T = B g AB log e AB gh The value of A ca be varied What coditio must A satisfy so that the two particles ca meet? Ed of Questio 6

13 Questio 7 (5 marks) Use a SEPARATE writig booklet (a) (i) Let a be a positive real umber Show that a + a Let be a positive iteger ad a, a,,a be positive real umbers Prove by iductio that ( a+ a + + a ) K + + K+ a a a (iii) Hece show that cosec θ + sec θ + cot θ 9cos θ 4 (b) Let α be a real umber ad suppose that z is a comple umber such that (i) z + = cosα z By reducig the above equatio to a quadratic equatio i z, solve for z ad use de Moivre s theorem to show that z + = cos α z Let w = z+ Prove that z (iii) w + w w = z+ z z z + + z + + z Hece, or otherwise, fid all solutios of cosα + cosα + cosα = 0, i the rage 0 α π

14 Questio 8 (5 marks) Use a SEPARATE writig booklet (a) Let P p, ad Q q, be poits o the hyperbola y = with p > q > 0 Let p q P be the poit (p, 0) ad Q be the poit (q, 0) The shaded regio OPQ i Figure is bouded by the lies OP, OQ ad the hyperbola The shaded regio Q QPP i Figure is bouded by the lies QQ, PP, P Q ad the hyperbola y Q y Figure Q Figure P P O P O Q P (i) Fid the area of triagle OPP Prove that the area of the shaded regio OPQ is equal to the area of the shaded regio Q QPP Let M be the midpoit of the chord PQ ad R r, be the itersectio of the r lie OM with the hyperbola Let R be the poit (r, 0), as show i Figure y Q R M P Figure O R (iii) (iv) (v) By usig similar triagles, or otherwise, prove that r = pq By usig itegratio, or otherwise, show that the lie RR divides the shaded regio Q QPP ito two pieces of equal area Deduce that the lie OR divides the shaded regio OPQ ito two pieces of equal area Questio 8 cotiues o page 5 4

15 Questio 8 (cotiued) (b) Let I = d ad let ta π 4 0 (i) Show that I + I = + + Deduce that J J = for π (iii) Show that Jm = + 4 m = J = I for 0,,, K ( ) ( ) u (iv) Use the substitutio u = ta to show that I = + u du (v) Deduce that 0 I ad coclude that J 0as + ( ) = 0 Ed of paper 5

16 STANDARD INTEGRALS d + =, ; 0, if < 0 + d = l, > 0 e a d a e a =, a 0 cosa d = si a, a 0 a si a d = cos a, a 0 a sec a d = ta a, a 0 a sec a ta a d = sec a, a a 0 a d = a ta, 0 + a a a d = si, a> 0, a< < a a ( ) > > d = l + a, a a ( ) d = l + + a + a NOTE : l = log, > 0 e 0 6 Board of Studies NSW 004

Mathematics Extension 2

Mathematics Extension 2 009 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard