TEMASEK JUNIOR COLLEGE, SINGAPORE JC Prliminary Eamination 7 MATHEMATICS 886/ Highr 9 August 7 Additional Matrials: Answr papr hours List of Formula (MF6) READ THESE INSTRUCTIONS FIRST Writ your Civics group and nam on all th work that you hand in. Writ in dark blu or black pn on both sids of th papr. You may us a soft pncil for any diagrams or graphs. Do not us stapls, papr clips, highlightrs, glu or corrction fluid. Answr all th qustions. Giv non-act numrical answrs corrct to significant figurs, or dcimal plac in th cas of angls in dgrs, unlss a diffrnt lvl of accuracy is spcifid in th qustion. You ar pctd to us a graphic calculator. Unsupportd answrs from a graphic calculator ar allowd unlss a qustion spcifically stats othrwis. Whr unsupportd answrs from a graphic calculator ar not allowd in a qustion, you ar rquird to prsnt th mathmatical stps using mathmatical notations and not calculator commands. You ar rmindd of th nd for clar prsntation in your answrs. Th numbr of marks is givn in brackts [ ] at th nd of ach qustion or part qustion. At th nd of th amination, fastn all your work scurly togthr. This documnt consists of printd pags. TJC/MA 886/PRELIMINARY EXAMS 7 [Turn ovr
Sction A: Pur Mathmatics [4 marks] dy k Th gradint of a curv C is givn by k d ( ), whr k. Find th st of valus of k for which C has distinct turning points. [4] (a) Diffrntiat. [] (b) Find ( )d. [] Th curv C has quation y ln,. (i) Find th quation of th tangnt to C at th point whr, laving your answr in trms of. [] Sktch th graph of C and th tangnt to C at th point whr on th sam diagram, stating th coordinats of intrsction with th as and th quation of any asymptot(s). [] (iii) By using th rsult ln d ln c, whr c is an arbitrary constant, find th ara of th rgion boundd by C, th tangnt to C at th point whr, th -ais and th y-ais. Giv your answr in trms of. [] 4 A prism with a cross-sction in th shap of a right-angld triangl has dimnsions (in cm) as shown in th diagram blow. y Th volum of th prism is 7 cm. Show that th surfac ara of th prism is givn by 7 S 6 cm. [] Without using a calculator, find in surd form th valu of that givs a stationary valu of S. Hnc stat, with a rason, whthr S is a maimum or minimum. [4] It is also givn that is dcrasing at. cm/s, find th rat at which th surfac ara is dcrasing whn cm. [] TJC/MA 886/PRELIMINARY EXAMS 7
Th profit P (in thousands dollars) of a company aftr th start of a promotion Claranc Sal can b modlld by th quation Pk( b rt ), whr t is th numbr of days lapsd sinc th start of th promotion and b, r and k ar positiv constants. P (i) Eprss ln in trms of b, r and t [] k P Th graph of ln against t is givn blow. k t By using th graph abov, show that b and r. [] (iii) Using diffrntiation, show that d P dt for t. Hnc plain why th maimum profit occurs at t =. Givn that th maimum profit is $47, find th valu of k, corrct to narst intgr valu. [] Th Claranc Sal nds aftr a wk and anothr promotion Happy Sal taks plac immdiatly aftr. Th Happy Sal lasts for wks and th profit during Happy Sal can b modlld by th quation (iv) ( 4) P a t, for 7 t 8. 6 Givn that th company s maimum profit during Happy Sal is $8, find th valu of a. [] (v) Find th total profit of th company in 4 wks, corrct to th narst dollars. [] Sction B: Statistics [6 marks] 6 Find th numbr of 6-lttr passwords that can b formd using th lttrs from th word SINGAPORE if (i) rptitions of lttrs ar not allowd. [] at last two vowls must b chosn and rptition of lttrs ar not allowd. [] (iii) thr distinct vowls and thr distinct consonants ar chosn, and vowls and consonants must altrnat? [] TJC/MA 886/PRELIMINARY EXAMS 7 [Turn ovr
4 7 A fid numbr, n, of tais ntring VICOM is obsrvd and th numbr of thos tais that fails inspction is dnotd by X. (i) Stat, in contt, two assumptions ndd for X to b wll modlld by a binomial distribution. [] Assum now that X has th distribution B( n, p ). Givn that n and p., find P( X or ). [] (iii) Givn that n and p., find P( X 6). [] (iv) Givn that n and P( X )., show that 4p 6p. Hnc find th valu of p. [] 8 Th numbr of hours,, spnt daily on rvision for mathmatics and th marks, y, obtaind for th mathmatics yar-nd amination ar rcordd for randomly slctd studnts. Th rsults ar givn in th following tabl......7...4.. y 68 74 64 76 7 66 8 8 86 7 (i) Giv a sktch of th scattr diagram for th data, as shown on your calculator. [] Find th product momnt corrlation cofficint and commnt on its valu in th contt of th data. [] (iii) Find th quation of th rgrssion lin of y on, in th form y m c, giving th valus of m and c corrct to 4 significant figurs. Sktch this lin on your scattr diagram. [] (iv) Us th quation of your rgrssion lin to stimat th marks obtaind by a studnt who spnds. hours a day on rvision for mathmatics. Commnt on th rliability of your stimat. [] 9 A bag contains black balls and whit balls. Paul draws a ball at random from th bag and nots th colour. If a black ball is slctd, Paul rturns it to th bag and adds an additional black ball into th bag. If a whit ball is slctd, Paul dos not rturn it to th bag but adds black balls into th bag. Paul thn draws a ball at random from th bag again and nots th colour. Draw a tr diagram to rprsnt th information of th two draws. [] (i) Show that th probability that Paul slcts a black ball on both draws is 6. [] (iii) Find th probability that Paul slcts a whit ball ithr on his first or scond draw, or both. [] Find th probability that Paul slcts a whit ball on his first draw, givn that h slcts a black ball on his scond draw. [] TJC/MA 886/PRELIMINARY EXAMS 7
A company claims that thir lctric-powrd V cars is dsignd to travl a man distanc of km on on full charg. To tst this claim, a random sampl of 8 V cars is takn and th distanc, km, travlld on on full charg ar summarisd by ( ) 46 and ( ) 46. (i) Find th unbiasd stimats of th population man and varianc. [] Suggst a rason why, in this contt, th givn data is summarisd in trms of ( ) instad of. [] (iii) Tst, at th % lvl of significanc, whthr th company s claim is valid. [4] (iv) Stat, with a rason, whthr it is ncssary to assum a normal distribution for th tst to b valid. [] Th company introducs a nw lctric-powrd V car and claims that th V can travl mor than th man distanc of th V on on full charg. Th population varianc of th V cars is known to b km. A random sampl of V cars is takn. (v) Find th st of valus within which th man distanc of this sampl must li, such that thr is not nough vidnc from th sampl to support th company s claim at th % lvl of significanc. [4] Thr frinds Anand, Bng and Charli gos racing rgularly at th Tmask Circuit, which offrs a standard rout on Track or a mor challnging rout on Track. Th tim takn, in minuts, takn by thm to complt a round on Tracks and hav indpndnt normal distributions with mans and standard dviations as shown in th following tabl. (i) (iii) Track Man Standard dviation Anand.. Bng.8. Charli 4.. Find th probability that Bng taks lss than. minuts to complt a randomly chosn round on Track. [] Find th probability that Anand and Bng ach taks lss than. minuts to complt a randomly chosn round on Track. [] Find th probability that Bng is fastr than Anand in complting a randomly chosn round on Track. [] (iv) Find th probability that out of 8 complt rounds on Track, thr ar mor than 4 rounds in which Charli taks lss than 4. minuts to complt. [] Tmask Circuit chargs customrs $8 pr minut on Track and $ pr minut on Track. On a particular day, Bng complts rounds on Track and Charli complts 8 rounds on Track. (v) Find th probability that Bng and Charli pay a total of lss than $. [] End of Papr TJC/MA 886/PRELIMINARY EXAMS 7
www.ampapr.biz Sction A: Pur Mathmatics [4 marks] dy k Th gradint of a curv C is givn by k d ( ), whr k. Find th st of valus of k for which C has distinct turning points. [4] dy k k d ( ) (k)( ) k ( ) (k) (k) k ( ) At turning points: (k) (k) ( k) Sinc curv has distinct turning points, th quation (k) (k) ( k) has distinct ral roots 4(k ) 4(k )( k ) k k (k)( k) k or k [B] [M] corrct inquality [A] corrct cofficints [A] (a) Diffrntiat. [] (b) Find ( )d. [] (a) (b) d d d d d d d () d () d ( ) c ( ) c [B] [B] [B] [M] [A]
Th curv C has quation y ln,. (i) Find th quation of th tangnt to C at th point whr, laving your answr in trms of. [] Sktch th graph of C and th tangnt to C at th point whr on th sam diagram, stating th coordinats of intrsction with th as and th quation of any asymptot(s). [] (iii) By using th rsult ln d ln c, whr c is an arbitrary constant, find th ara of th rgion boundd by C, th tangnt to C at th point whr, th -ais and th y-ais. Giv your answr in trms of. [] (i) y ln d y d [B] dy Whn, y ln, d Equation of tangnt at th point whr : y ( ) [M] i.. y [A] y [B] Graph of y ln, including -intrcpt and vrtical asymptot [B] Graph of y, including y-intrcpt and intrsction with th graph of y ln (iii) Rquird ara ( ) ln d [B] ln d [M] formulat ara ( ) ln ( ) ln [A]
4 A prism with a cross-sction in th shap of a right-angld triangl has dimnsions (in cm) as shown in th diagram blow. y Th volum of th prism is 7 cm. Show that th surfac ara of th prism is givn by 7 S 6 cm. [] Without using a calculator, find in surd form th valu of that givs a stationary valu of S. Hnc stat, with a rason, whthr S is a maimum or minimum. [4] It is also givn that is dcrasing at. cm/s, find th rat at which th surfac ara is dcrasing whn cm. [] 4 Givn volum of prism = 7 W hav 4 ( )( ) y 7 y Surfac ara of prism S ( )( ) y y y 6 y 4 6 7 6 ds 7 7 d At minimum S, d S d 7 7 6 Using First Drivativ Tst: 6 6 6 6 ds d -v +v S is minimum at 6 [B] [M][A] [B] [M] [A] for surd form only [B]
Givn that is dcrasing at. cm/s, whn d. dt ds ds d () 7. 6 dt d d Thrfor, surfac ara is dcrasing at 6 cm / s [M] [A] Th profit P (in thousands dollars) of a company aftr th start of a promotion Claranc Sal can b modlld by th quation Pk( b rt ), whr t is th numbr of days lapsd sinc th start of th promotion and b, r and k ar positiv constants. P (i) Eprss ln in trms of b, r and t [] k P Th graph of ln against t is givn blow. k t By using th graph abov, show that b and r. [] (iii) Using diffrntiation, show that d P dt for t. Hnc plain why th maimum profit occurs at t =. Givn that th maimum profit is $47, find th valu of k, corrct to narst intgr valu. [] Th Claranc Sal nds aftr a wk and anothr promotion Happy Sal taks plac immdiatly aftr. Th Happy Sal lasts for wks and th profit during Happy Sal can b modlld by th quation (iv) ( 4) P a t, for 7 t 8. 6 Givn that th company s maimum profit during Happy Sal is $8, find th valu of a. [] (v) Find th total profit of th company in 4 wks, corrct to th narst dollars. [] (i) rt P Pk(b ) b k rt 4
P rt Thn, ln ln( b ) ln b rt k From graph, y-intrcpt = ln bb Gradint of lin r r. (iii) d d t d t t P k( ) k( ) k( ) dt dt dt t Sinc t for t, thn k( ) whr k is a constant i.. d P dt P is dcrasing for t Hnc, P is maimum at t. Maimum profit occurs at t. Maimum profit = 47 47 P () k( ) 4.7 k.999996498 k (iv) Maimum profit occurs at t 4, [B] [M][A] [M][A] [B] [B] for d P dt [M] [A] (v) 8 (4 4) a 6 a 8 Total profit 7 8 t ( ) dt 8 ( t4) dt 6 7 4.8 Total profit of company in 4 wks is $486 Sction B: Statistics [6 marks] [M] [A] [B] [B] [B] 6 Find th numbr of 6-lttr passwords that can b formd using th lttrs from th word SINGAPORE if (i) rptitions of lttrs ar not allowd. [] at last two vowls must b chosn and rptition of lttrs ar not allowd. [] (iii) thr distinct vowls and thr distinct consonants ar chosn, and vowls and consonants must altrnat? [] 6(i) 6 6(iii) No. of ways No. of ways No. of ways 9 6! 648 6 4 648 6! 76 4!! 88 [B] for answr [M] for complmnt mthod [A] [B] for choosing vowls and consonants [B] for cass of altrnating [B] for answr
7 A fid numbr, n, of tais ntring VICOM is obsrvd and th numbr of thos tais that fails inspction is dnotd by X. (i) Stat, in contt, two assumptions ndd for X to b wll modlld by a binomial distribution. [] Assum now that X has th distribution B( n, p ). Givn that n and p., find P( X or ). [] (iii) Givn that n and p., find P( X 6). [] (iv) Givn that n and P( X )., show that 4p 6p. Hnc find th valu of p. [] 7(i) Probability that a tai fails inspction is constant for all th tais Whthr a tai fails inspction or not is indpndnt of anothr tai [B] [B] 7 X B(,.) P( X or ) P( X ) P( X ).476.47 [M][A] 7(iii) X B(,.) P( X 6) P( X ) P( X ).6986.7 [B] [B] 7(iv) X B(, p ) P( X ). ( ) ( ). p p p 6. p p p p p p. p p 4 6 (shown) p p Using GC, p. or.66 (rjctd p ) or.66 (rjctd p ) [M] for us of formula [A] 8 Th numbr of hours,, spnt daily on rvision for mathmatics and th marks, y, obtaind for th mathmatics yar-nd amination ar rcordd for randomly slctd studnts. Th rsults ar givn in th following tabl. [A].....7...4.. y 68 74 64 76 7 66 8 8 86 7 (i) Giv a sktch of th scattr diagram for th data, as shown on your calculator. [] Find th product momnt corrlation cofficint and commnt on its valu in th contt of th data. [] (iii) Find th quation of th rgrssion lin of y on, in th form y m c, giving th valus of m and c corrct to 4 significant figurs. Sktch this lin on your scattr diagram. [] (iv) Us th quation of your rgrssion lin to stimat th marks spnt by a studnt who spnds. hours a day on rvision for mathmatics. Commnt on th rliability of your stimat. [] 6
8(i) [B] for shap [B] for as corrctly lablld 8 r.98 ( s.f.) This indicats a strong positiv linar corrlation btwn rvision hours and th marks obtaind, i.. as th hours for rvision incrass, th marks obtaind incrass linarly. 8(iii) [B] [B] [B] for rgrssion lin y 8.8.89 m 8.8 (4 s.f.) and c.89 (4 s.f.) 8(iv) Whn., y 68. ( s.f.) Th stimat is rliabl sinc r-valu is clos to and. is within th data rang [B] for m and c [B] [B] [B] 9 A bag contains black balls and whit balls. Paul draws a ball at random from th bag and nots th colour. If a black ball is slctd, Paul rplacs it in th bag and adds an additional black ball into th bag. If a whit ball is slctd, Paul dos not rplac it in th bag but adds black balls into th bag. Paul thn draws a ball at random from th bag again and nots th colour. Draw a tr diagram to rprsnt th information of th two draws. [] (i) Show that th probability that Paul slcts a black ball on both draws is 6. [] (iii) Find th probability that Paul slcts a whit ball ithr on his first or scond draw, or both. [] Find th probability that Paul slcts a whit ball on his first draw, givn that h slcts a black ball on his scond draw. [] 9 4 9 B 8 B 9 W 8 W 9 B 4 9 W 7
9(i) 4 P(slcts B on both draws) (shown) 8 9 6 9 P(slcts W on first or scond draw) = 8 8 9 = 6 or 4 8 9 6 9(iii) P(slcts W on first draw slcts B on scond draw) 8 9 4 8 9 8 9 7 [B] [M][A] [M][A] [A] A company claims that thir lctric-powrd V cars is dsignd to travl a man distanc of km on on full charg. To tst this claim, a random sampl of 8 V cars is takn and th distanc, km, travlld on on full charg ar summarisd by ( ) 46 and 8 ( ) 46. (i) Find th unbiasd stimats of th population man and varianc. [] Suggst a rason why, in this contt, th givn data is summarisd in trms of ( ) instad of. [] (iii) Tst, at th % lvl of significanc, whthr th company s claim is valid. [4] (iv) Stat, with a rason, whthr it is ncssary to assum a normal distribution for th tst to b valid. [] Th company introducs a nw lctric-powrd V car and claims that th V can travl mor than th man distanc of th V on on full charg. Th nw population varianc of th V cars is known to b km. A random sampl of V cars is takn. (v) Find th st of valus within which th man distanc of this sampl must li, such that thr is not nough vidnc from th sampl to support th company s claim at th % lvl of significanc. [4] (i) 46 [B].7 8 46 s 46.4879 79 8 [B] Valus may b too larg [B] (iii) H : H: Lvl of significanc: % [B] for hypothss X Tst statistic: Z N(,) appro. by CLT [B] for application of CLT s and s instad of n Undr H and using GC, w hav p.8 ( s.f.) [B] for p-valu
Sinc p., w rjct H. Thr is sufficint vidnc at th % lvl of significanc to conclud that th company s claim is invalid. (iv) Not ncssary sinc n is larg and X is still approimatly normal by CLT (v) H : H: Lvl of significanc: % X Tst statistic: Z N(,) appro. by CLT n Sinc company s claim is not accptd. H is not rjctd.6.8 [B] for conclusion [B] [B] for hypothss [B] for tst statistic with and CLT sn or implid [M] [A] Thr frinds Anand, Bng and Charli gos racing rgularly at th Tmask Circuit, which offrs a standard rout on Track or a mor challnging rout on Track. Th tim takn, in minuts, takn by thm to complt a round on Tracks and hav indpndnt normal distributions with mans and standard dviations as shown in th following tabl. (i) (iii) Track Man Standard dviation Anand.. Bng.8. Charli 4.. Find th probability that Bng taks lss than. minuts to complt a randomly chosn round on Track. [] Find th probability that Anand and Bng ach taks lss than. minuts to complt a randomly chosn round on Track. [] Find th probability that Bng is fastr than Anand in complting a randomly chosn round on Track. [] (iv) Find th probability that out of 8 complt rounds on Track, thr ar mor than 4 rounds in which Charli taks lss than 4. minuts to complt. [] Tmask Circuit chargs customrs $8 pr minut on Track and $ pr minut on Track. On a particular day, Bng complts rounds on Track and Charli complts 8 rounds on Track. (v) Find th probability that Bng and Charli pay a total of lss than $. [] (i) Lt X and Y b th tim takn by Anand and Bng to complt a round on Track X N(.,. ) and Y N(.8,. ) P( Y.).88.8 ( s.f.) [B] Rquird probability..88.94.9 [M][A] 9
(iii) Y X N(.8.,.. ) i.. Y X N(.,.4) P( Y X ).44868.449 ( s.f.) (iv) Lt W b th tim takn by Charli to complt a round on Track W N(4.,. ) P( W 4.).44696 Lt S b th numbr of rounds which Charli taks lss than 4. minuts to complt S B(8,.44696) P( S 4) P( S 4).46. ( s.f.) (v) Lt C b th total cost C 8( YY Y) ( WW W8) E( C) (8.8) (84.). Var( C) (8. ) ( 8. ) 9. C N(., 9.) P( C ).87.8 ( s.f.) [B] for E( Y X) and Var( Y X) or quivalnt [B] [B] for s.f. [M][A] [B] for man and varianc of tim [B] for a Var( Y ) [B] for E( C ) and Var( C ) [B] End of Papr