Mathematics Extension 2

Transcription

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

2 Totl mrks 0 Attempt Questions 8 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. π () Find tn sec d. 0 d (b) By completing the squre, find. + (c) Use integrtion by prts to evlute e ln d. (d) Use the substitution u = to evlute + d. (e) Find rel numbers nd b such tht 5 + ( + )( ) + b (ii) Find d. ( + )( )

3 Question (5 mrks) Use SEPARATE writing booklet. () Let z = + i nd w = + i. Find zw nd w in the form + iy. (b) Epress + i in modulus-rgument form. ( ) (ii) Hence evlute + i 0 in the form + iy. (c) Sketch the region in the comple plne where the inequlities π π z+ i nd rg z both hold. (d) Find ll solutions of the eqution z =. Give your nswers in modulus-rgument form. (e) y D A C B O In the digrm the vertices of tringle ABC re represented by the comple numbers z, z nd z,respectively. The tringle is isosceles nd right-ngled t B. Eplin why (z z ) = (z z ). (ii) Suppose D is the point such tht ABCD is squre. Find the comple number, epressed in terms of z, z nd z, tht represents D.

4 Question (5 mrks) Use SEPARATE writing booklet. y () Consider the hyperbol H with eqution =. 9 6 Find the points of intersection of H with the is, nd the eccentricity nd the foci of H. (ii) Write down the equtions of the directrices nd the symptotes of H. (iii) Sketch H. (b) The numbers α, β nd γ stisfy the equtions α + β + γ = α + β + γ = + + = α β γ. Find the vlues of αβ + βγ + γα nd αβγ. Eplin why α, β nd γ re the roots of the cubic eqution + = 0. (ii) Find the vlues of α, β nd γ. (c) The re under the curve y = sin between = 0 nd = π is rotted bout the y is. Use the method of cylindricl shells to find the volume of the resulting solid of revolution.

5 Question (5 mrks) Use SEPARATE writing booklet. () y Grph of y = ƒ () y = O The digrm shows sketch of y = ƒ (), the derivtive function of y = ƒ(). The curve y = ƒ () hs horizontl symptote y =. (ii) Identify nd clssify the turning points of the curve y = ƒ(). Sketch the curve y = ƒ() given tht ƒ(0) = 0 = ƒ() nd y = ƒ() is continuous. On your digrm, clerly identify nd lbel ny importnt fetures. (b) r h O y b A cylindricl hole of rdius r is bored through sphere of rdius R. The hole is perpendiculr to the y plne nd its is psses through the origin O, which is the centre of the sphere. The resulting solid is denoted by. The cross-section of shown in the digrm is distnce h from the y plne. Show tht the re of the cross-section shown bove is π(r h r ). (ii) Find the volume of, nd epress your nswer in terms of b lone, where b is the length of the hole. (c) Use differentition to show tht tn tn is constnt for + > 0. Wht is the ect vlue of the constnt? 5

6 Question 5 (5 mrks) Use SEPARATE writing booklet. () y Q (0, b) P (cos θ, bsin θ) O (, 0) R E Consider the ellipse E, with eqution y + =, nd the points P (cosθ, bsinθ), b Q (cos(θ + ϕ), bsin(θ + ϕ)) nd R (cos(θ ϕ), bsin(θ ϕ)) on E. Show tht the eqution of the tngent to E t the point P is cosθ ysinθ + =. b (ii) Show tht the chord QR is prllel to the tngent t P. (iii) Deduce tht OP bisects the chord QR. Question 5 continues on pge 7 6

7 Question 5 (continued) (b) A submrine of mss m is trvelling underwter t mimum power. At mimum power, its engines deliver force F on the submrine. The wter eerts resistive force proportionl to the squre of the submrine s speed v. Eplin why ( ) dv dt = m F kv where k is positive constnt. (ii) The submrine increses its speed from v to v. Show tht the distnce trvelled during this period is m k F loge F kv kv. (c) A clss of students is to be divided into four groups consisting of, 5, 6 nd 7 students. In how mny wys cn this be done? Leve your nswer in unsimplified form. (ii) Suppose tht the four groups hve been chosen. In how mny wys cn the students be rrnged round circulr tble if the students in ech group re to be seted together? Leve your nswer in unsimplified form. End of Question 5 7

8 Question 6 (5 mrks) Use SEPARATE writing booklet. () N F α Cross-section of rod mg A rod contins bend tht is prt of circle of rdius r. At the bend, the rod is bnked t n ngle α to the horizontl. A cr trvels round the bend t constnt speed v. Assume tht the cr is represented by point of mss m, nd tht the forces cting on the cr re the grvittionl force mg, sidewys friction force F (cting down the rod s drwn) nd norml rection N to the rod. By resolving the horizontl nd verticl components of force, find epressions for Fcosα nd Fsinα. ( ) mv grtnα (ii) Show tht F = cosα. r (iii) Suppose tht the rdius of the bend is 00 m nd tht the rod is bnked to llow crs to trvel t 00 kilometres per hour with no sidewys friction force. Assume tht the vlue of g is 9.8 m s. Find the vlue of ngle α, giving full resons for your nswer. Question 6 continues on pge 9 8

9 Question 6 (continued) (b) A C E F T D B In the digrm, is circle with eterior point T. From T, tngents re drwn to the points A nd B on nd line TC is drwn, meeting the circle t C. The point D is the point on such tht BD is prllel to TC. The line TC cuts the line AB t F nd the line AD t E. Copy or trce the digrm into your writing booklet. Prove tht TFA is similr to TAE. (ii) Deduce tht TE. TF = TB. (iii) (iv) Show tht EBT is similr to BFT. Prove tht DEB is isosceles. End of Question 6 9

10 Question 7 (5 mrks) Use SEPARATE writing booklet. () Suppose tht z = ( cos + isin ) where θ is rel. θ θ Find z. (ii) (iii) Show tht the imginry prt of the geometric series + z + z + z + K = z sinθ is. 5 cosθ Find n epression for + cosθ + cosθ + cosθ + in terms of cosθ. (b) Consider the eqution =0, which we denote by (*). p Let = where p nd q re integers hving no common divisors other q thn + nd. Suppose tht is root of the eqution + b = 0, where nd b re integers. Eplin why p divides b nd why q divides. Deduce tht (*) does not hve rtionl root. (ii) Suppose tht r, s nd d re rtionl numbers nd tht d is irrtionl. Assume tht r+ s d is root of (*). Show tht r s + s d s = 0 nd show tht of (*). r s d must lso be root Deduce from this result nd prt, tht no root of (*) cn be epressed in the form r+ s d with r, s nd d rtionl. (iii) Show tht one root of (*) is cos π. 9 (You my ssume the identity cosθ = cos θ cosθ.) 0

11 Question 8 (5 mrks) Use SEPARATE writing booklet. () Show tht b + b for ll rel numbers nd b. Hence deduce tht (b + bc + c) ( + b + c) for ll rel numbers, b nd c. (ii) Suppose, b nd c re the sides of tringle. Eplin why (b c). Deduce tht ( + b + c) (b + bc + c). (b) Eplin why, for α > 0, (You my ssume e <.) 0 α e d <. α + (ii) Show, by induction, tht for n = 0,,, there eist integers n nd b n such tht 0 n e d = + b e n n. p (iii) Suppose tht r is positive rtionl, so tht r = where p nd q re q positive integers. Show tht, for ll integers nd b, either + br = 0 or + br. q (iv) Prove tht e is irrtionl. End of pper

12 STANDARD INTEGRALS n d n+ =, n ; 0, if n< 0 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 d = tn, 0 + d = sin, > 0, < < ( ) > > d = ln +, ( ) d = ln NOTE : ln = log, > 0 e 0 Bord of Studies NSW 00