2013 Specialist Mathematics GA 3: Written examination 2

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Leonard McDonald
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1 0 0 Spcialist Mathmatics GA : Writtn xamination GENERAL COMMENTS Th 0 Spcialist Mathmatics xamination comprisd multipl-choic qustions (worth marks) and fiv xtndd qustions (worth 8 marks). Th papr smd accssibl to studnts, with most making substantial attmpts at all qustions in Sction. Qustion a. was th only show that qustion on th 0 xam, and studnts wr rquird to vrify by substitution th solution to a givn diffrntial quation. Many studnts attmptd to mploy mthods othr than substitution, and a lack of convincing algbraic dtail charactrisd rsponss. A small numbr of studnts did not kp in mind th instruction for Sction that statd, In qustions whr mor than on mark is availabl, appropriat working must b shown. A numbr of studnts simply wrot answrs for Qustions a., b. and c., without giving any indication of how thir answrs wr obtaind. A numbr of studnts did not kp in mind th othr important instruction for Sction that statd, Unlss othrwis spcifid, an xact answr is rquird to a qustion. Ths studnts obtaind corrct xact answrs, and thn rplacd thm with dcimal approximations. This happnd mainly in Qustions dii., f. and c. Th xamination rvald aras of strngth and waknss in studnt prformanc. Aras of strngth includd th us of CAS tchnology to prform diffrntiation and to valuat dfinit intgrals Qustions dii. and a., although th complicatd form of th rsultant xprssion for dn smd to lad to som studnts not dt bing sur of how to procd th us of CAS tchnology to solv quations Qustions 4c., c. and. an improvd facility with complx numbr qustions Qustion th ability to idntify and sktch rlationships, and, to a lssr xtnt, rgions in th complx plan Qustion c. th ability to st up a dfinit intgral to rprsnt a volum of rvolution Qustion di. th us of pncil to sktch graphs fwr graphs wr compltd in pn compard to prvious yars. Aras of waknss includd untidy working, and lack of clarity about what th studnt intndd to b thir final answr work bing don lightly in pncil, making it difficult to rad som poor prsntation of sktch graphs: llipss drawn with points at th x-axis intrcpts (Qustion c.), circls and straight lins drawn roughly (Qustion a.) and graphs drawn inaccuratly at th nd points of th domain (Qustion.) lack of propr vctor notation, in particular th confusion of scalar 0 with null vctor 0 Qustion 4iii. uncrtainty of th signs of quantitis whn daling with constant acclration formulas Qustions c. and. Spcialist Maths GA Exam VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 04

2 0 SPECIFIC INFORMATION Th statistics in this rport may b subjct to rounding rrors rsulting in a total lss than 00 pr cnt. Sction Th tabl blow indicats th prcntag of studnts who chos ach option. Th corrct answr is indicatd by shading. Qustion % A % B % C % D % E % No Answr Commnts x x, so option D was corrct. Th dnominator has th form x, th vrtx is, 8 a x, a, which gav option E. At th origin, th gradints of th graphs ar a and b rspctivly, so option D was corrct. All studnts wr awardd mark for this qustion. z cis k 4 / 4 z cis k 8, Args ar 7, 8 8, so option C was corrct. x / 4 y, option D. / V y dy 0 which givs x 0zro gradints y 0infinit gradints, and positiv gradints in first quadrant, so option E was corrct. u w.v Option E as b was th only vctor with an i componnt, so option B was corrct option C. t t, which solvs to giv Spcialist Maths GA Exam Publishd: April 04

3 0 Qustion % A % B % C % D % E % No Answr mx mg kv Commnts which lads to option E. dv mv mg kv dx, Sction Qustion a. Marks 0 Avrag % x y cos( t), x y sin() t, givs 9 4 This qustion was wll don, with narly all studnts ralising that t had to b liminatd. Of th fw studnts who gav a rsult with y as th subjct, most omittd th from thir answr. Qustion b. Marks 0 Avrag % dy dy dt 7 cos( t), cot () t, t, dx dt dx sin( t) 6 6 This qustion was fairly wll don; howvr, a numbr of studnts gav only on answr for t, and othrs gav xtra solutions outsid th spcifid domain. Som studnts usd implicit diffrntiation and thn attmptd to solv on quation for both x and y. Th gnral solution for t appard occasionally Qustion c. Marks 0 Avrag % 8.8 y 4 4 O 4 x 4 4 Th llips is shown abov. Th y-axis intrcpts ar. This qustion was wll don. Many llipss wr drawn roughly, with points at th x-intrcpts and imprcis smi-axis lngths. Som studnts gav dcimal approximations for th y-intrcpts instad of xact valus. Qustion di. Marks 0 Avrag % Spcialist Maths GA Exam Publishd: April 04

4 x dx This qustion was wll don. Th main rrors wr th omission of, incorrct rarrangmnt of th quation of th llips for th intgrand and som intgrals missing th dx. Qustion dii. Marks 0 Avrag % This qustion was fairly wll don, with th most common rror bing th dcimal approximation instad of th xact valu. Qustion a. Marks 0 Avrag % y + i * * + i O x i i * * Th majority of studnts did not draw th lin y x. Most got th circl and th lin y x. Som studnts drw mor than on circl, and straight lins wr somtims drawn roughly. Qustion b. Marks 0 Avrag % Solving y xand x y 4 givs x, and so z i. This qustion was only modratly wll don. Most studnts found only two solutions for. z, z i and z i Qustion c. Marks 0 Avrag % 7 0. Othr roots ar i, i, i. All four roots ar plottd and lablld on th diagram in Qustion a. Major rrors involvd not plotting th roots, not lablling th plottd roots, and plotting th roots on th incorrct circl and in othr locations. Spcialist Maths GA Exam Publishd: April 04 4

5 0 Qustion d. Marks 0 Avrag % z i z i z i z i A numbr of studnts confusd factors and roots. Som wrot down th corrct factors but not as a product as rquird. Qustion. Marks 0 Avrag % Sgmnt as shadd on th diagram in Qustion a. Frqunt rrors includd sgmnts shadd in othr quadrants, th major sgmnt shadd, shading of an annulus, and inaccurat bordrs and shading of th dfind rgion. Qustion f. Marks 0 Avrag % 44 4 A, A 4 This qustion was don rasonably wll by studnts who shadd th corrct rgion in Qustion. Som studnts gav a dcimal approximation. A frqunt rror was A, whr only half th sgmnt was sktchd, or whr intgration was usd without doubling th intgral. Qustion a. Marks 0 Avrag % Diffrntiation of th solution with rspct to t givs diffrntial quation givs 0.4 t 0.4 t lft sid = dn. N dt 0.4t, and substituting into th lft sid of th 0.4 t 0.4 t lft sid =..4..4, lft sid = 0 = right sid. Th qustion was not don wll. Many studnts mployd a varity of approachs diffrnt to th rquird mthod of substitution. Qustion b. Marks 0 Avrag % ( ) N t 0 log, N, N 0 This qustion was quit wll don; howvr, a numbr of studnts lft thir answr as, instad of giving an answr to th narst intgr. Qustion c. Marks 0 Avrag %. ( ) As t, log N 6, limiting numbr is 40 Spcialist Maths GA Exam Publishd: April 04

6 0 This qustion was fairly wll don, with still som studnts giving th answr as 40. A numbr of studnts did not show thir working. Qustion di. Marks 0 Avrag % log ( N) 0.4N d N d dn dn dn dt dn dt dt N dt d N 6, rathr than th narst intgr,, 0.4log ( N) 0.4N6 log ( N) This qustion rquird th us of both th chain and product ruls. Vry fw studnts answrd this qustion corrctly. d N d t A numbr of studnts found in trms of t, whil othrs found dt dn and attmptd to invrt th rsult, which d dn showd littl undrstanding of th proprtis of scond drivativs. A common rror was to find dn dt. Qustion dii. Marks 0 Avrag % d N 0 0.4log ( ) 0 N, solving givs N 48, so th point of inflction is at t.7, N 48 dt dt Th majority of studnts ralisd that th scond drivativ ndd to b quatd to zro. Howvr, thr was a small group who quatd th first drivativ to zro. Som studnts obtaind th corrct rsult indpndntly of thir fforts in part i. Qustion. Marks 0 Avrag % N O 0 t This qustion was modratly wll don. Major rrors includd inaccurat position of ndpoints, inaccurat placmnt of th point of inflction and lack of chang in concavity. Spcialist Maths GA Exam Publishd: April 04 6

7 0 Qustion 4a. Marks 0 Avrag % ˆb i j k 4 This qustion was wll don, with most studnts knowing how to find th unit vctor. Most rrors rlatd to finding th magnitud of b. Qustion 4b. Marks 0 Avrag % 40.7 Th paralll rsult a.bˆˆ 7 b = 4 i j+ k = i j k 4 4 Th prpndicular rsult a a.bˆˆ b = i j Most studnts knw th formulas to apply; howvr, th surd calculations provd to b a problm for many. A small numbr attmptd to find th rsoluts from first principls. A fw studnts had th rsoluts th wrong way around, finding th rsolut of b in th dirction of a, tc. Qustion 4c. Marks 0 Avrag % mi j k. i j k m 4 cos solvs to giv m Most studnts wr abl to st up th formula, and thn solv for m using thir calculator. A small numbr of studnts did not liminat th solution which gav vctor a. Qustion 4d. Marks 0 Avrag % i j k. i j k cos( ), 7. Most studnts applid th corrct mthod, but a larg numbr mad rrors in computation along th way. Som studnts gav th answr in radians, and othrs did not round corrctly to on dcimal plac. Qustion 4i. Marks 0 Avrag % AN u v This qustion was vry wll don. Th main rror was to giv th ngativ of th corrct rsult. Spcialist Maths GA Exam Publishd: April 04 7

8 0 Qustion 4ii. Marks 0 Avrag % 6. u u v v CM u, BP u v This qustion was rasonably wll don, with th main rrors rlating to signs. Qustion 4iii. Marks 0 Avrag % This qustion was not wll don. Th most common rror was th omission of th tild for this null vctor. Th majority of studnts had scalar 0 as th rsult of th vctor calculation, rathr than th vctor 0. This is a concptual rror a sum of vctors is a vctor. Qustion a. Marks 0 Avrag % r t t 7.i cos j, Min spd = 7. m/s, Max spd = 9. m/s 6 Common rrors includd th omission of j, and not idntifying which wr th minimum and maximum spds. Som studnts gav only on spd and nglctd to say whthr it was th minimum or th maximum. Qustion b. Marks 0 Avrag % r t t sin 8 6 j, acclration is zro for t 6 n, whr n Z ( n Z {0} was also accptd) Many studnts had n Z. Corrct chain rul diffrntiation was also a problm for som studnts. Qustion c. Marks 0 Avrag % sin(0 ) t 4.9t, t =.0 (s) Th major rror with this qustion rlatd to consistncy of signs. A numbr of studnts brok th motion into two stags, causing mor work for thmslvs and making th qustion mor complx. Som studnts confusd th 0 m lngth of th ramp with th spd of m/s in thir calculations. Qustion d. Marks 0 Avrag % cos(0 ).0 6 m Th main rror with this qustion was th incorrct horizontal componnt of vlocity. Spcialist Maths GA Exam Publishd: April 04 8

9 0 Qustion. Marks 0 Avrag % 9 7 9g 9g mg cos(0 ) mg sin(0 ) ma, a, 0 s, s 9. m This qustion was rasonably wll don by studnts who attmptd it. A numbr had th wight forc down th plan and friction acting in opposit dirctions. Othrs took th dirction down th plan as positiv, and had difficulty daling with th ngativ valu of s obtaind. Qustion f. Marks 0 Avrag % mg sin(0 ) mg cos(0 ) 0, or Ngativ valus for wr somtims sn, and occasionally unsimplifid xprssions for wr givn. Spcialist Maths GA Exam Publishd: April 04 9

Note If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.

Note If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark. . (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold