Mathematics Extension 2

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1 009 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 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. l (a) Fid d. (b) Fid e d. (c) Fid d (d) Evaluate d (e) Evaluate d. + 4

3 Questio (5 marks) Use a SEPARATE writig booklet. (a) Write i 9 i the form a + ib where a ad b are real. (b) Write +i + i i the form a + ib where a ad b are real. (c) The poits P ad Q o the Argad diagram represet the comple umbers z ad w respectively. P(z) Q(w) Copy the diagram ito your writig booklet, ad mark o it the followig poits: (i) the poit R represetig iz the poit S represetig w (iii) the poit T represetig z + w. (d) Sketch the regio i the comple plae where the iequalities z ad π π arg ( z ) hold simultaeously. 4 4 (e) (i) Fid all the 5th roots of i modulus-argumet form. Sketch the 5th roots of o a Argad diagram. (f) (i) Fid the square roots of + 4i. Hece, or otherwise, solve the equatio z + iz i = 0.

4 Questio (5 marks) Use a SEPARATE writig booklet. (a) The diagram shows the graph y = ƒ ( ). y O 4 Draw separate oe-third page sketches of the graphs of the followig: (i) y = ƒ ( ) y ƒ = ( ) ( ) (iii) y = ƒ. (b) Fid the coordiates of the poits where the taget to the curve + y + y = 8 is horizotal. (c) Let P() = + a + b + 5, where a ad b are real umbers. Fid the values of a ad b give that ( ) is a factor of P(). Questio cotiues o page 5 4

5 Questio (cotiued) (d) The diagram shows the regio eclosed by the curves y = + ad y = ( ). y y = + y = ( ) O The regio is rotated about the y-ais. Use the method of cylidrical shells to fid the volume of the solid formed. Ed of Questio 5

6 Questio 4 (5 marks) Use a SEPARATE writig booklet. (a) The ellipse + y = has foci S(ae,0) ad S ( ae,0) where e is the a b a a eccetricity, with correspodig directrices = ad =. The poit e e ( ) P 0, y 0 is o the ellipse. The poits where the horizotal lie through P meets the directrices are M ad M, as show i the diagram. = a e y = a e M α P β M S N S (i) Show that the equatio of the ormal to the ellipse at the poit P is y 0 a y y. 0 = ( b ) 0 0 The ormal at P meets the -ais at N. Show that N has coordiates (e 0,0). (iii) Usig the focus-directri defiitio of a ellipse, or otherwise, show that PS NS = PS NS. (iv) Let α = S PN ad β = NPS. By applyig the sie rule to rs PN ad to rnps, show that α = β. Questio 4 cotiues o page 7 6

7 Questio 4 (cotiued) (b) A light strig is attached to the verte of a smooth vertical coe. A particle P of mass m is attached to the strig as show i the diagram. The particle remais i cotact with the coe ad rotates with costat agular velocity ω o a circle of radius r. The strig ad the surface of the coe make a agle of α with the vertical, as show. α T r P N mg The forces actig o the particle are the tesio, T, i the strig, the ormal reactio, N, to the coe ad the gravitatioal force mg. (i) Resolve the forces o P i the horizotal ad vertical directios. Show that T = m ( g cos α + rω si α ) ad fid a similar epressio for N. (iii) Show that if T = N the ω = g ta α. r ta α + (iv) For which values of α ca the particle rotate so that T = N? Ed of Questio 4 7

8 Questio 5 (5 marks) Use a SEPARATE writig booklet. (a) I the diagram AB is the diameter of the circle. The chords AC ad BD itersect at X. The poit Y lies o AB such that XY is perpedicular to AB. The poit K is the itersectio of AD produced ad YX produced. K D A Y X C B Copy or trace the diagram ito your writig booklet. (i) (iii) Show that AKY = ABD. Show that CKDX is a cyclic quadrilateral. Show that B, C ad K are colliear. Questio 5 cotiues o page 9 8

9 Questio 5 (cotiued) (b) For each iteger 0, let + I = e d. 0 (i) Show that for I = e I. Hece, or otherwise, calculate I. (c) Let ƒ ( ) e e =. (i) Show that ƒ ( ) > 0 for all > 0. Hece, or otherwise, show that ƒ ( ) > 0 for all > 0. (iii) Hece, or otherwise, show that e e > for all > 0. Ed of Questio 5 9

10 Questio 6 (5 marks) Use a SEPARATE writig booklet. (a) The base of a solid is the regio eclosed by the parabola = 4 y ad the y-ais. The top of the solid is formed by a plae iclied at 45 to the y-plae. Each vertical cross-sectio of the solid parallel to the y-ais is a rectagle. A typical cross-sectio is show shaded i the diagram. y 4 O 45 4 = 4 y Fid the volume of the solid. (b) Let P() = + q + q +, where q is real. Oe zero of P() is. (i) Show that if α is a zero of P() the is a zero of P(). α Suppose that α is a zero of P() ad α is ot real. () Show that α =. () Show that Re ( ) q α =. Questio 6 cotiues o page 0

11 Questio 6 (cotiued) (c) The diagram shows a circle of radius r, cetred at the origi, O. The lie PQ is taget to the circle at Q, the lie PR is horizotal, ad R lies o the lie = c. y P(, y) R Q r O c (i) Fid the legth of PQ i terms of, y ad r. The poit P moves such that PQ = PR. Show that the equatio of the locus of P is y = r + c c. (iii) Fid the focus, S, of the parabola i part. (iv) Show that the differece betwee the legth PS ad the legth PQ is idepedet of. Ed of Questio 6

12 Questio 7 (5 marks) Use a SEPARATE writig booklet. (a) A bugee jumper of height m falls from a bridge which is 5 m above the surface of the water, as show i the diagram. The jumper s feet are tied to a elastic cord of legth L m. The displacemet of the jumper s feet, measured dowwards from the bridge, is m. = 0 = L L m m 5 m NOT TO SCALE = 5 The jumper s fall ca be eamied i two stages. I the first stage of the fall, where 0 L, the jumper falls a distace of L m subject to air resistace, ad the cord does ot provide resistace to the motio. I the secod stage of the fall, where > L, the cord stretches ad provides additioal resistace to the dowward motio. (i) The equatio of motio for the jumper i the first stage of the fall is = g rv where g is the acceleratio due to gravity, r is a positive costat, ad v is the velocity of the jumper. () Give that = 0 ad v = 0 iitially, show that g g v = l r. g rv r () Give that g = 9.8 m s ad r = 0. s, fid the legth, L, of the cord such that the jumper s velocity is 0 m s whe = L. Give your aswer to two sigificat figures. I the secod stage of the fall, where > L, the displacemet is give by t = e 0 ( 9si t 0 cos t ) + 9 where t is the time i secods after the jumper s feet pass = L. Determie whether or ot the jumper s head stays out of the water. 4 Questio 7 cotiues o page

13 Questio 7 (cotiued) (b) Let z = cos θ + i si θ. (i) Show that z + z = cos θ, where is a positive iteger. Let m be a positive iteger. Show that m m cos θ cos m θ cos θ ( m ) + m cos m +. + cos θ m + m m. ( ) = + ( m 4 ) θ (iii) Hece, or otherwise, prove that where m is a positive iteger. π cos m π m θ dθ = m + m 0 Ed of Questio 7

14 Questio 8 (5 marks) Use a SEPARATE writig booklet. (a) (i) Usig the substitutio t = ta θ, or otherwise, show that θ θ cot θ + ta = cot. Use mathematical iductio to prove that, for itegers, ta = cot cot. r r r = (iii) Show that lim ta r r r = = cot. (iv) Hece fid the eact value of π π ta + ta + ta π Questio 8 cotiues o page 5 4

15 Questio 8 (cotiued) (b) y y = O Let be a positive iteger greater tha. The area of the regio uder the curve y = from = to = is betwee the areas of two rectagles, as show i the diagram. Show that e < < e. (c) A game is beig played by people, A, A,..., A, sittig aroud a table. Each perso has a card with their ow ame o it ad all the cards are placed i a bo i the middle of the table. Each perso i tur, startig with A, draws a card at radom from the bo. If the perso draws their ow card, they wi the game ad the game eds. Otherwise, the card is retured to the bo ad the et perso draws a card at radom. The game cotiues util someoe wis. Let W be the probability that A wis the game. Let p = ad q =. (i) Show that W = p + q W. Let m be a fied positive iteger ad let W m be the probability that A wis i o more tha m attempts. Use part (b) to show that, if is large, to e m. W m W is approimately equal Ed of paper 5

16 STANDARD INTEGRALS + d =, ; 0, if < 0 + d = l, > 0 a a a e d = e, a 0 cosa d = sia, a 0 a sia d = cosa, a 0 a sec a d = ta a, a 0 a seca ta a d = seca, a 0 a d = ta, a 0 a + a a d = si, a > 0, a < < a a a a d = l( + a ), > a > 0 + a d = l( + + a ) NOTE : l = log, > 0 e 6 Board of Studies NSW 009

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