SPECIALIST MATHEMATICS

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1 Victorin Certificte of Eduction 006 SUPERVISOR TO ATTACH PROCESSING LABEL HERE STUDENT NUMBER Letter Figures Words SPECIALIST MATHEMATICS Written exmintion Mondy 30 October 006 Reding time: 3.00 pm to 3.5 pm (5 minutes) Writing time: 3.5 pm to 5.5 pm ( hours) QUESTION AND ANSWER BOOK Section Number of questions 5 Structure of book Number of questions to be nswered 5 Number of mrks 58 Totl 80 Students re permitted to bring into the exmintion room: pens, pencils, highlighters, ersers, shrpeners, rulers, protrctor, set-squres, ids for curve sketching, one bound reference, one pproved grphics clcultor or pproved CAS clcultor or CAS softwre nd, if desired, one scientiþc clcultor. Clcultor memory DOES NOT need to be clered. Students re NOT permitted to bring into the exmintion room: blnk sheets of pper nd/or white out liquid/tpe. Mterils supplied Question nd nswer book of 3 pges with detchble sheet of miscellneous formuls in the centrefold. Answer sheet for multiple-choice questions. Instructions Detch the formul sheet from the centre of this book during reding time. Write your student number in the spce provided bove on this pge. Check tht your nme nd student number s printed on your nswer sheet for multiple-choice questions re correct, nd sign your nme in the spce provided to verify this. All written responses must be in English. At the end of the exmintion Plce the nswer sheet for multiple-choice questions inside the front cover of this book. Students re NOT permitted to bring mobile phones nd/or ny other unuthorised electronic devices into the exmintion room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 006

2 006 SPECMATH EXAM SECTION Instructions for Section I Answer ll questions in pencil on the nswer sheet provided for multiple-choice questions. Choose the response tht is correct for the question. A correct nswer scores, n incorrect nswer scores 0. Mrks will not be deducted for incorrect nswers. No mrks will be given if more thn one nswer is completed for ny question. Tke the ccelertion due to grvity to hve mgnitude g m/s, where g = 9.8. Question ( x ) The hyperbol with eqution 9 A. x + 3y =, nd x 3y = 7 B. 3x + y =, nd 3x y = C. x 3y =, nd x + 3y = 7 D. 3y x = 7, nd y + 3x = E. 4x 9y = 5, nd 4x + 9y = 3 ( y ) 4 = hs symptotes given by Question The grph of the function with rule f ( x)= ( x 4) x+ ( ) over its mximl domin hs A. symptotes x = 4 nd x = nd turning point t (, 5) B. symptotes x = 4 nd x = nd turning point t, 5 C. symptotes x = 4 nd x = nd turning point t, 9 D. symptotes x = 4 nd x = nd turning point t (, 9) E. symptotes x = 4 nd x = nd turning point t, 9 Question 3 The position vector of prticle t time t 0 is given by r = ( + t) i + ( t ) j. The pth of the prticle hs eqution A. y = x B. y = x + C. y = x D. y = x + E. y = x SECTION continued

3 3 006 SPECMATH EXAM Question 4 The complex number + bi, where nd b re rel constnts, is represented in the following digrm. Im(z) O Re(z) All xes below hve the sme scle s in the digrm bove. The complex number i ( + bi) could be represented by A. B. Im(z) Im(z) O Re(z) O Re(z) C. D. Im(z) Im(z) O Re(z) O Re(z) E. Im(z) O Re(z) SECTION continued TURN OVER

4 006 SPECMATH EXAM 4 Question 5 One of the complex solutions to z 5 =, where is positive rel constnt, is 5 cis π 5. One of the other solutions is rel number nd is equl to A. cis B. 5 C. 5 π D. cis π E. cis π Question 6 Im(z) required region + i O Re(z) The region represented on the bove Argnd digrm, where is rel constnt, could be deþned by A. z ( + i) B. z ( + i) C. z ( + i) D. z ( + i) E. z+ i SECTION continued

5 5 006 SPECMATH EXAM Question 7 Which one of the following reltions does not hve grph tht is stright line pssing through the origin? A. z+ z =0 B. 3 Re(z) = Im(z) C. z = iz D. Re(z) Im(z) = 0 E. Re(z) + Im(z) = Question 8 The slope of the curve x 3 y = 7 t the point where y = 3 is A. 4 B. C. D. 4 E. 7 Question 9 Using suitble substitution, xx ( +) 5 is equl to A. B. C. b D. E. b 5 u du 5 u du b + + b + + b 5 u du 5 u du 6 6 b SECTION continued TURN OVER

6 006 SPECMATH EXAM 6 Question 0 A chemicl dissolves in pool t rte equl to 5% of the mount of undissolved chemicl. Initilly the mount of undissolved chemicl is 8 kg nd fter t hours x kilogrms hs dissolved. The differentil eqution which models this process is A. B. C. D. E. dt dt dt dt dt x = 0 = 8 x 0 x = 8 0 x = 0 x = 8 0 Question y x The direction (slope) Þeld for certin Þrst order differentil eqution is shown bove. The differentil eqution could be A. B. C. D. E. dy dy x = + y y = x + dy x = y dy y = x dy x = y SECTION continued

7 7 006 SPECMATH EXAM Question A prticle moves in stright line such tht its velocity v is given by v = sin(x), when t displcement x from the origin O. The ccelertion of the prticle is given by A. cos(x) B. sin(x) cos(x) D. sin(4x) C. cos( x) E. cos(x) Question 3 Two prticles, R nd S, hve position vectors r = ( t 0) i + 3j nd s= i + ( t ) j respectively t time t seconds, t 0. Then A. R nd S re in the sme position when t =. B. R nd S re in the sme position when t = 4. C. R nd S re in the sme position when t = 5. D. R nd S re in the sme position when t = 6. E. R nd S re never in the sme position. Question 4 The position vector of prticle t time t seconds, t 0, is given by r() t = ( 3 t) i 6 t j+5k.! The direction of motion of the prticle when t = 9 is A. 6 i 8 j+5k B. i j!! C. 6 i j!! D. i j+ 5k E. 3. 5i 08 j+ 45k SECTION continued TURN OVER

8 006 SPECMATH EXAM 8 Question 5 In the prllelogrm shown, = b.!! d c b Which one of the following sttements is true? A. = b!! B. + b= c+ d! C. b d= 0 D. +c=0 E. b=c d! Question 6 A unit vector perpendiculr to 5i + j k is A. ( ) 4 5 i+j k B. i 4j+3k C. D. E. ( ) i 4j + 3k ( ) i 4j + 3k ( ) 5i + j k Question 7 Let u = i + j nd v= i + j+ k.! The ngle between the vectors u nd v is!! A. 0 B. 45 C. 30 D..5 E. 90 SECTION continued

9 9 006 SPECMATH EXAM Question 8 A block of mss 0 kg lies on plne inclined t n ngle of 60 to the horizontl. The norml rection force of the plne on the block is R newtons. R is equl to A. 0 g B. 0 3 g C. D. g 3 g E. 0 g Question 9 A block of mss 0 kg is pulled long smooth horizontl plne by force. Under which one of the following sets of conditions will the mss hve the lrgest ccelertion? A. 30 N B. 0 N C. 0 0 N D. 0 0 N E. 45 N SECTION continued TURN OVER

10 006 SPECMATH EXAM 0 Question 0 F 50 F F 3 If three co-plnr forces, F, F nd F 3, ct on prticle which is in equilibrium s shown in the bove digrm, then A. F = F = 3 F 3 B. F = F 3 = 3 F C. F = F 3 = 3 F D. F = F 3 = E. F = F 3 = 3 3 F 3 3 F Question A block of mss 8 kg is t rest on plne inclined t n ngle of 30 to the horizontl. N F 30 8 g In the digrm, N newtons is the norml rection of the plne on the block, nd F newtons is the frictionl force on the block up the plne. For equilibrium to be mintined, the coefþcient of friction between the plne nd the block must be A. t lest B. less thn C. t lest D. less thn E. less thn 3 3 g 3 g 3 g 3 SECTION continued

11 006 SPECMATH EXAM Question A light inextensible string psses over smooth pulley. Prticles of mss 5 kg nd kg re ttched to ech end of the string, s shown. kg 5 kg The ccelertion of the 5 kg mss downwrds is A. 3g B. C. D. E. 5g 7 g 7 5g 3 3g 7 END OF SECTION TURN OVER

12 006 SPECMATH EXAM SECTION Instructions for Section Answer ll questions in the spces provided. A deciml pproximtion will not be ccepted if n exct nswer is required to question. In questions where more thn one mrk is vilble, pproprite working must be shown. Unless otherwise indicted, the digrms in this book re not drwn to scle. Tke the ccelertion due to grvity to hve mgnitude g m/s, where g = 9.8. Question The top prt of wine glss, while lying on its side, is constructed by rotting the grph of y = x = 0 to x = 5 bout the x-xis s shown below. All lengths re mesured in centimetres. 6x from 3 + x y O x 4. Write down deþnite integrl which represents the volume, V cm 3, of the glss. mrks b. Use the substitution u = + x 3 to write down deþnite integrl which represents the volume of the glss in terms of u. c. Find the vlue of V correct to the nerest cm 3. mrks mrk SECTION Question continued

13 3 006 SPECMATH EXAM At time t = 0 seconds wine begins to be poured into the upright glss so tht its depth (x cm in the grph opposite) is incresing t rte of cm/sec. d. Given tht dy 3 6 3x = 3 ( + x ) the wine with respect to time, in terms of x. 3, Þnd n expression for dy, the rte of chnge of the rdius of the surfce of dt mrk e. Hence Þnd n expression for the rte of chnge of the re, A cm, of the surfce of the wine in the upright glss with respect to t, in terms of x. Give your nswer in the form da b 3 x ( 6 3x ) = dt 3 c where, b nd c re constnts. ( + x ) 3 mrks f. Find the exct vlue of the depth of the wine for which the re of its surfce, A cm, is mximum. mrk Totl 0 mrks SECTION continued TURN OVER

14 006 SPECMATH EXAM 4 Question Point A hs position vector = i 4 j, point B hs position vector b= i 5 j, point C hs position vector c= 5 i 4j, nd point D hs position vector d= i + 5 j reltive to the origin O.. Show tht AC nd BD re perpendiculr. b. Use vector method to Þnd the cosine of ADC, the ngle between DA nd DC. mrks c. Find the cosine of ABC, nd hence show tht ADC nd ABC re supplementry. 3 mrks mrks SECTION Question continued

15 5 006 SPECMATH EXAM Point P hs position vector. d. Use the cosine of APC nd n pproprite trigonometric formul to prove tht APC = ADC. 3 mrks Totl 0 mrks SECTION continued TURN OVER

16 006 SPECMATH EXAM 6 Question 3. A pssenger jet of mss kg moves from rest with constnt ccelertion long runwy due to totl thrust of newtons supplied by its engines. Assume tht ir resistnce nd other frictionl forces re negligible. i. Show tht the mgnitude of the ccelertion of the jet is. ms. ii. How mny seconds, correct to one deciml plce, does it tke the jet to rech its lift-off speed of 70 ms? iii. Wht distnce is needed, correct to the nerest metre, for the jet to tke off? + + = 5 mrks SECTION Question 3 continued

17 7 006 SPECMATH EXAM b. After lift-off the pilot eses bck the thrust of the engines to newtons nd the plne climbs t n ngle of 0 to the horizontl direction t constnt velocity for short time. During this stge of the scent the jet is subject to the following forces, ll mesured in newtons. the thrust T of the engines the lifting force L supplied by the wings the weight force W the drg on the plne R due to ir resistnce 0 i. On the digrm bove, lbel clerly the forces cting on the jet. ii. By resolving forces into perpendiculr components, write down pir of equtions which would enble R nd L to be found. iii. Find L, the lift supplied by the wings, correct to the nerest newton. + + = 4 mrks SECTION Question 3 continued TURN OVER

18 006 SPECMATH EXAM 8 c. After short time, the jet touches down on the runwy with horizontl speed of 80 ms. The speed of the jet s it slows down is v ms t time t seconds fter touchdown. The jet is slowed by reverse thrust of newtons supplied by the engines, force of 5v newtons, where v is the speed of the jet in ms, supplied by the brking effect of the wing ßps nd other frictionl forces, nd force of 500(80 v) newtons supplied by the brking of the wheels. i. Assuming tht the mss of the jet is unchnged, write down the eqution of motion of the jet while it is being slowed by the three forces listed bove. ii. Hence write down deþnite integrl which gives the distnce the jet tkes to slow down to speed of 0 ms. iii. Find this distnce, correct to the nerest metre = 5 mrks Totl 4 mrks SECTION continued

19 9 006 SPECMATH EXAM Question 4 A bll rolling long horizontl plne hs position vector r() t = x() t i + y() t j, t 0, y > 0 nd velocity vector i r() t = i + ( yt ()) j.! yt ()!!. The component of velocity in the j direction gives the differentil eqution! dy = y. dt Use clculus to show tht the solution to this differentil eqution is log e y = c t, where c is the constnt of integrtion. b. Write down n eqution for dt which the bll rolls is given by dy mrks nd hence use clculus to show tht the grdient of the curve long = y( y). c. i. Show tht d y = ( yy ) ( y ). mrks SECTION Question 4 continued TURN OVER

20 006 SPECMATH EXAM 0 ii. Hence, for 0 < y <, Þnd the y-coordinte of ny points of inßection on the curve long which the bll rolls nd verify tht they re points of inßection. + = 3 mrks d. If the position of the bll t prticulr time is given by r = 05. j, sketch the pth of the bll on the direction (slope) Þeld below.!!.5 y x mrk e. Given tht y = when x = 0, use Euler s method with step-size of 4 to estimte the vlue of y when x =. mrks Totl 0 mrks SECTION continued

21 006 SPECMATH EXAM Question 5. i. Let z = cis π. Plot nd lbel crefully the points z, z nd z on the Argnd digrm below. 4 Im(z) 3 O 3 Re(z) ii. Write down the complex eqution of the stright line which psses through the points z nd z, in terms of z. + = 3 mrks b. Use double ngle formul to show tht the exct vlue of cos π 8 = +. Explin why ny vlues re rejected. 3 mrks SECTION Question 5 continued TURN OVER

22 006 SPECMATH EXAM c. Hence show tht the exct vlue of sin π 8 =. mrks d. Evlute i, giving your nswer in polr form. mrks e. For wht vlues of n is + + i n rel number? mrks SECTION Question 5 continued

23 3 006 SPECMATH EXAM f. Plot the roots of z 8 = on the Argnd digrm below. Im(z) 3 O 3 Re(z) mrks Totl 4 mrks END OF QUESTION AND ANSWER BOOK

24 SPECIALIST MATHEMATICS Written exmintions nd FORMULA SHEET Directions to students Detch this formul sheet during reding time. This formul sheet is provided for your reference. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 006

25 SPECMATH Specilist Mthemtics Formuls Mensurtion re of trpezium: curved surfce re of cylinder: volume of cylinder: volume of cone: volume of pyrmid: volume of sphere: re of tringle: sine rule: cosine rule: ( + b) h π rh π r h π r h 3 3 Ah 4 3 π r 3 bcsin A b c = = sin A sin B sinc c = + b b cos C Coordinte geometry ellipse: ( x h) ( y k) + b = hyperbol: ( x h) ( y k) b = Circulr (trigonometric) functions cos (x) + sin (x) = + tn (x) = sec (x) cot (x) + = cosec (x) sin(x + y) = sin(x) cos(y) + cos(x) sin(y) cos(x + y) = cos(x) cos(y) sin(x) sin(y) tn( x) + tn( y) tn( x+ y) = tn( x) tn( y) sin(x y) = sin(x) cos(y) cos(x) sin(y) cos(x y) = cos(x) cos(y) + sin(x) sin(y) tn( x) tn( y) tn( x y) = + tn( x) tn( y) cos(x) = cos (x) sin (x) = cos (x) = sin (x) tn( x) sin(x) = sin(x) cos(x) tn( x) = tn ( x) function sin cos tn domin [, ] [, ] R rnge π π, [0,!] π, π

26 3 SPECMATH Algebr (complex numbers) z = x + yi = r(cos θ + i sin θ) = r cis θ z = x + y = r π < Arg z π z r z z = r r cis(θ + θ ) = cis θ z r z n = r n cis(nθ) (de Moivre s theorem) Clculus d x n ( )= nx n ( θ ) n n+ x= x + c, n n + d e x e x x ( )= e = + e x c d ( log e( x) )= = + x x log x c e d ( sin( x) )= cos( x) sin( x) = cos( x) + c d ( cos( x) )= sin( x) cos( x) = sin( x) + c d ( tn( x) )= sec ( x) d sin ( ( x) )= x d cos ( ( x) )= x sec ( x) = tn( x) + c x x x = sin + c, > 0 x = cos + c, > 0 d ( tn ( x) )= = + x + x x tn + c product rule: quotient rule: chin rule: Euler s method: ccelertion: d ( uv)= u dv + v du v du u dv d u v = v dy dy du = du If dy = f ( x), x 0 = nd y 0 = b, then x n + = x n + h nd y n + = y n + h f(x n ) d x dv v dv d = = = = v dt dt constnt (uniform) ccelertion: v = u + t s = ut + t v = u + s s = (u + v)t TURN OVER

27 SPECMATH 4 Vectors in two nd three dimensions r = xi + yj + zk ~ ~ ~ ~ r ~ = x + y + z = r ~ r. r ~ = r r cos θ = x x + y y + z z dr ~ dy dz!r = = i+ j+ k ~ dt dt ~ dt ~ dt ~ Mechnics momentum: p= mv ~ ~ eqution of motion: R = m ~ ~ friction: F µn END OF FORMULA SHEET