Mathematics Extension 2

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1 00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Extension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors m be used A tble of stndrd integrls is provided t the bck of this pper All necessr working should be shown in ever question Totl mrks 0 Attempt Questions 8 All questions re of equl vlue 4

2 Totl mrks 0 Attempt Questions 8 All questions re of equl vlue Answer ech question in SEPARATE writing booklet Extr writing booklets re vilble Question (5 mrks) Use SEPARATE writing booklet () B using the substitution u = secx, or otherwise, find sec xtnxdx dx (b) B completing the squre, find x + x+ xdx (c) Find x+ x ( )( ) (d) B using two pplictions of integrtion b prts, evlute π 0 e x cos xdx 4 (e) Use the substitution t = tn θ to find 4 π 0 dθ + cosθ

3 Question (5 mrks) Use SEPARATE writing booklet () Let z = + i nd w = + i Find, in the form x + i, zw w (b) On n Argnd digrm, shde in the region where the inequlities both hold 0 Re z nd z + i (c) It is given tht + i is root of where r nd s re rel numbers P(z) = z + rz + sz + 0, Stte wh i is lso root of P(z) Fctorise P(z) over the rel numbers (d) Prove b induction tht, for ll integers n, (cosθ isinθ) n = cos(nθ) isin(nθ) (e) Let z = (cosθ + isinθ) Find z Show tht the rel prt of is cosθ z 5 4cosθ Express the imginr prt of in terms of θ z

4 Question (5 mrks) Use SEPARATE writing booklet () (, ) = f(x) O (, 0) x (, ) The digrm shows the grph of = f(x) Drw seprte one-third pge sketches of the grphs of the following: = f ( x ) = f( x) = f( x ) (iv) = ln( f( x) ) Question continues on pge 5 4

5 Question (continued) (b) P T Q H O x The distinct points P cp, c nd Q cq, c re on the sme brnch of the p q hperbol H with eqution x = c The tngents to H t P nd Q meet t the point T Show tht the eqution of the tngent t P is x + p = cp Show tht T is the point cpq c, p+ q p+ q Suppose P nd Q move so tht the tngent t P intersects the x xis t (cq, 0) Show tht the locus of T is hperbol, nd stte its eccentricit End of Question 5

6 Question 4 (5 mrks) Use SEPARATE writing booklet () = x P = x + x x = O x The shded region bounded b = x, = x + x nd x = is rotted bout the line x = The point P is the intersection of = x nd = x + x in the first qudrnt Find the x coordinte of P Use the method of clindricl shells to express the volume of the resulting solid of revolution s n integrl Evlute the integrl in prt Question 4 continues on pge 7 6

7 Question 4 (continued) (b) R A B D S C T In the digrm, A, B, C nd D re concclic, nd the points R, S, T re the feet of the perpendiculrs from D to BA produced, AC nd BC respectivel Show tht DSR = DAR Show tht DST = π DCT Deduce tht the points R, S nd T re colliner (c) From pck of nine crds numbered,,,, 9, three crds re drwn t rndom nd lid on tble from left to right Wht is the probbilit tht the number formed exceeds 400? Wht is the probbilit tht the digits re drwn in descending order? End of Question 4 7

8 Question 5 (5 mrks) Use SEPARATE writing booklet () The eqution 4x 7x + k = 0 hs double root Find the possible vlues of k (b) Let α, β, nd γ be the roots of the eqution x 5x + 5 = 0 Find polnomil eqution with integer coefficients whose roots re α,β, nd γ Find polnomil eqution with integer coefficients whose roots re α, β, nd γ Find the vlue of α + β + γ (c) P(x, ) T(x 0, 0 ) O S Q R Directrix D x E x The ellipse E hs eqution + =, nd focus S nd directrix D s shown b in the digrm The point T (x 0, 0 ) lies outside the ellipse nd is not on the x xis The chord of contct PQ from T intersects D t R, s shown in the digrm Show tht the eqution of the tngent to the ellipse t the point P(x, ) is xx + = b Show tht the eqution of the chord of contct from T is xx = b Show tht TS is perpendiculr to SR 8

9 Question 6 (5 mrks) Use SEPARATE writing booklet () A prticle of mss m is suspended b string of length l from point directl bove the vertex of smooth cone, which hs verticl xis The prticle remins in contct with the cone nd rottes s conicl pendulum with ngulr π velocit ω The ngle of the cone t its vertex is α,where α >,nd the string 4 mkes n ngle of α with the horizontl s shown in the digrm The forces cting on the prticle re the tension in the string T, the norml rection to the cone N nd the grvittionl force mg α α α l T N NOT TO SCALE mg Show, with the id of digrm, tht the verticl component of N is N sinα mg Show tht T + N =, nd find n expression for T N in terms of sinα m, l nd ω The ngulr velocit is incresed until N = 0, tht is, when the prticle is bout to lose contct with the cone Find n expression for this vlue of ω in terms of α, l nd g Question 6 continues on pge 0 9

10 Question 6 (continued) (b) For n = 0,,, let I n π 4 = d tn θ θ 0 n Show tht I = ln Show tht, for n, (iv) + I n = n For n, explin wh I n < I n, nd deduce tht I n n + B using the recurrence reltion of prt, find I 5 nd deduce tht I < ln < n n ( ) < < 4 ( ) End of Question 6 0

11 Question 7 (5 mrks) Use SEPARATE writing booklet () 0 Drining wter The digrm represents verticl clindricl wter cooler of constnt cross-sectionl re A Wter drins through hole t the bottom of the cooler From phsicl principles, it is known tht the volume V of wter decreses t rte given b dv dt = k, where k is positive constnt nd is the depth of wter Initill the cooler is full nd it tkes T seconds to drin Thus = 0 t = 0, nd = 0whent = T when d k Show tht dt = A dt B considering the eqution for, or otherwise, show tht d 4 t = 0 T for 0 t T Suppose it tkes 0 seconds for hlf the wter to drin out How long does it tke to empt the full cooler? Question 7 continues on pge

12 Question 7 (continued) π (b) Suppose 0 < α, β < nd define complex numbers z n b z n = cos(α + nβ) + i sin(α + nβ) for n = 0,,,, 4 The points P 0, P, P nd P re the points in the Argnd digrm tht correspond to the complex numbers z 0, z 0 + z, z 0 + z + z nd z 0 + z + z + z respectivel The ngles θ 0, θ nd θ re the externl ngles t P 0, P nd P s shown in the digrm below P θ P θ P θ 0 O P 0 x (iv) Using vector ddition, explin wh θ 0 = θ = θ = β Show tht P 0 OP = P 0 P P, nd explin wh OP 0 P P is cclic qudrilterl Show tht P 0 P P P is cclic qudrilterl, nd explin wh the points O, P 0, P, P nd P re concclic Suppose tht z 0 + z + z + z + z 4 = 0 Show tht π β = 5 End of Question 7

13 Question 8 (5 mrks) Use SEPARATE writing booklet () Let m be positive integer B using De Moivre s theorem, show tht m+ m m+ m sin( m + ) θ = cos θsin θ cos θsin θ + K m m sin θ + ( ) + Deduce tht the polnomil m + px ( ) x hs m distinct roots m + x K m m m + +( ) α k kπ = cot where k =,, K, m m + Prove tht cot π π mπ cot cot m+ + m+ + K + m + m m = ( ) π (iv) You re given tht cotθ < for 0 < θ < θ Deduce tht π ( ) ( ) 6 < m + K m m m Question 8 continues on pge 4

14 Question 8 (continued) (b) C E D L K M N x P B A F In the digrm, AB nd CD re line segments of length in horizontl plnes t distnce prt The midpoint E of CD is verticll bove the midpoint F of AB, nd AB lies in the South North direction, while CD lies in the West Est direction The rectngle KLMN is the horizontl cross-section of the tetrhedron ABCD t distnce x from the midpoint P of EF (so PE = PF = ) B considering the tringle ABE, deduce tht KL = x, nd find the re of the rectngle KLMN Find the volume of the tetrhedron ABCD 4 End of pper 4

15 BLANK PAGE 5

16 STANDARD INTEGRALS n x dx n+ = x, n ; x 0, if n< 0 n + x dx = ln x, x > 0 e x dx e x =, 0 cosx dx = sin x, 0 sin x dx = cos x, 0 sec x dx = tn x, 0 sec x tn x dx = sec x, 0 x dx x = tn, 0 + x dx x = sin, > 0, < x < ( ) > > dx = ln x + x, x x ( ) dx = ln x + x + x + NOTE : ln x = log x, x > 0 e 0 6 Bord of Studies NSW 00

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