7' The growth of yeast, a microscopic fungus used to make bread, in a test tube can be

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Claude Bruce
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1 N Sction A: Pur Mathmatics 55 marks] / Th rgion R is boundd by th curv y, th -ais, and th lins = V - +7 and = m, whr m >. Find th volum gnratd whn R is rotatd through right angls about th -ais, laving your answr in trms ofm. / _ (+ Pa) : Th compl numbr z is givn by z, whr p <0 Givn that I zi = ', show that p = -. Hnc show that arg (z) I 7C Find th smallst positiv intgr n such that z' is purly imaginary and z'>000 7' Th growth of yast, a microscopic fungus usd to mak brad, in a tst tub can b modlld by th diffrntial quation d =k ),.. whr is th numbr of yast clls at tim, t hours. Show that f--- A kt whr k and A ar constants.. k Givn that k =000 and th primnt startd with 00 yast clls, find th tim takn for th numbr of yast clls to rach 900, laving your answr to th narst scond. Find th limiting numbr of yast clls in th tst tub / Givn that y=+ tan', show.that.-. By furthr diffrntiation, obtain th Maclaurin sris for y, up to and including th trm Hnc find th sris pansion of as far as th trm in.. F+ L-. 5 Using intgration by parts, show that J cos d=k ( sin cos ) + c, whr k is a constant to b rmind. A curv C is dfind by th paramtric quations i = cos t, y = -',whr ---<t< 7r -- Sktch th curv C, and find th act ara of th rgion boundd by C and th y-ais... Anglo-Chins Junior Collg H Mathmatics 9758: 07 JC Mid-Yar Eamination Pag oj'6./ + [Turn Ovr

2 (6 Th quations of thr plans 7, and [[ ar +py+z, +ay-z=7 and +by z= rspctivly, whr a, b andp ar constants. Th lin has quation r = 6i - j + k + u( - j - k), whr u is a paramtr. Givn that l dos not intrsct fl,. show that p = and hnc find th distanc btwn th lin l and [T I. [] Find th coordinats of a point that lis on and. [] Givn that th lin, lis on both and, find a vctor paralll to, laving your answr in trms of a and b. [] i Givn that and 7 intrsct at a point and a = b, find th valu of a. [] 7 (a) A filtr funnl is mad from a thin flat sht of mtal in th shap of a sctor of a circl with radius cm and angl 0 radians (s Diagram ). Th two straight sids of th mtal sctor ar thn joind togthr, without ovrlap, to form a con with hight Hcm and radius R cm (s Diagram ). ',, 0 NV Diagram I " -c5 Diagram Eprss R in trms of 0 and 7c. [] Hnc, show that th volum, V, of th filtr funnl is givn by v = 58 ( m _9 ). [] Find, in act form, th valu of 0 such that th volum of th filtr funnl is a maimum. You nd not show that th volum is a maimum. [] [Volum of con = -- itrh, whr r is th bas radius and h is th hight of th con.] Anglo-Chins Junior Collg H Mathmatics 9758: 07C Mid-Yar Eamination Pag of 6

3 (b) ri A café prpars consommé soup by pouring broth through th filtr funnl into a hmisphrical bowl with radius 5 cm, as shown in th figur abov. Givn that at tim t minuts, soup fills th initially mpty bowl at a rat of 800 m pr minut, find th volum, S, of soup in th bowl aftr 0t+l minuts, laving your answr corrct to dcimal placs. [] Th volum of soup in th bowl is givn by S =.( l5h - h), whr his th dpth of th soup, as shown in th diagram Find th rat of chang of th dpth of th soup aftr minuts. [ Sction B: Statistics 0 marks] Th random variabl Xis th numbr of succsss inn indpndnt trials of an primnt in which th probability of a succss at any trial is p. Dnoting P(X=k) byp,ts givn that (n k. + I)p k=l,,, n Pk- k(l - p) A distribution is said to b bimodal if it has two mods. Find th last valu of h, and th corrsponding mods of X, givn that Xis bimodal and that p= [] 9 Th discrt random variabl X has probability distribution function givn by k( ) =0,,, P(X=)= k(-) =,, 5, 0 othrwis. Find th valu of k, [] p( Var(X). [] [Turn Ovr nglochins Junior Collg H Mathmatics 9758: 07.TC Mid-Yar Eamination Pag 5of6

4 0 In a duck farm, a duckling infctd with a particular virus has a 9% of survival., i?f In a random sampl of 0 ducklings infctd with th virus, find th probability 0 that at last two of thm will not surviv. [] In 80 such sampls of 0 ducklings, find th probability that th man numbr of ducklings that do not surviv is mor than.5. [] Vhicls manufacturd by automobil companis undrgo tsting at th National Vhicl Inspction Cntr whr thy ar givn scors basd on svral critria. Th highr th scor, th bttr th vhicl prformd at th tst. Scors obtaind by th cars, trucks, and motorcycls manufacturd by AMW Automobil ar modlld by indpndnt normal distributions. Th mans and standard dviations of th scors ar givn in th tabl blow. Typ of vhicl Man Standard dviation Car 70 9 Truck 69 5 Motorcycl 6 Find th probability that a truck prformd bttr than a motorcycl in th tst. [] t) Find th probability that th avrag tst scor of thr cars is lss than 65. [] Evry yar, ach automobil company snds thr cars, on truck and on motorcycl for compulsory tsting, and an ovrall scor is givn to th company. Th avrag tst scor of th constituts 70% of th ovrall tst scor, whil th tst scors for th truck an otorcycl ach constitut 5% of th ovrall tst scor. An automobil manufacturr rcivs an "Ecllnt" award if its ovrall tst scor is mor than k Find th minimum valu of k, a whol numbr, such that thr is only at most a 5% chanc of AMW Automobil rciving th "Ecllnt" award. [] End of Papr - Anglo-Chins Junior Collg H Mathmatics 9758: 07 JC Mid-Yar Eamination Pag 6of6

5 07 JC H Math MYE Marking Schm m Volum = d 7 m d 7 m d m tan m tan tan m tan 6 (i) Thrfor (ii) (iii) For to b purly imaginary, By obsrvation, Hnc smallst n is 0. d k d ( k ) d kt c k ln ln k kt c ln kt c k ktc kt A k Givn that k = 000 and whn t = 0, = A A

6 Whn = 900, t 000t ln8 t sc (narst sc) t t t As t, Limiting numbr of yast is 000. y tan y tan Diffrntiat w.r.t., dy y d Diffrntiat w.r.t., d y dy d d y y d y dy d d Diffrntiat w.r.t., d y dy d y dy d y (shown) 000t y d d d d d dy d y d y 5 d d d 8 Whn 0, y,,, y!! 8 8 tan tan tan 8 8!

7 5(i) 5(ii) cos d cos d ( sin ) d cos d (sin ) d cos d (sin ) d (cos ) d sin cos cos d 5 cos d sin cos c ' cos d 5 k 5 sin cos y c Ara dy t cos t ( ) t cos t sin cos 5 sin cos sin( ) cos( ) 5 ( ) (i) dos not intrsct, thrfor thy ar paralll. Hnc Distanc btwn and = 6(ii) Lt th point b. Thn sub into quation of and. and Solving givs. Hnc th point is. 6(iii) 6(iv) Equat and and sub

8 Thrfor Hnc 7(a)(i) Obsrv that th arc lngth of th sctor in Diagram is also th circumfrnc of th circl in Diagram. 6 Thus, R, i.., R. 7(a)(ii) 7(a)(iii) 7(b)(i) V R R H 6 R V 58 6 Diffrntiating V ( ) w.r.t. dv 58 5 V 6 6 d dv 58 V 8 d Whn d V 0 d, (N.A.) or (rjct v) Thus,. ds 800 0t 800 S d t 0ln 0 t C 0t Whn t 0, S 0 C 0 S 0 ln 0t Whn t, S 0 ln cm ( dp)

9 7(b)(ii) S 5h h Whn t, S 0 ln 8 By GC, h.886 or.6 (rjctd sinc h5) S 5h h ds dh 0hh 800 dh () dh 0. cm/min (sf) 8 For X to b a bimodal Binomial distribution, pk p k ( n k ) p k( p) np kp p k kp k p n 8 Givn p 5 k 8 n 5 8 n 5k Sinc both n and k ar intgrs, th last valu of n k 8 Mods ar 7 and P(X = ) 9k k k k k 9k 9k k k k k 9k 8k k 8 5 E X (by obsrvation) OR E 5 9 X k k k k k 5 70k E X k k k k 5 9k 06k 5

10 Var X E X E X Lt X dnot th numbr of ducklings, out of a random sampl of 0, that do not surviv. X B(0, 0.08) P( X ) P( X ) (sf) X X X80 Lt X 80 X n is larg, by Cntral Limit Thorm,.08 N(., ) approimatly 80 PX (.5) (sf) C T S ~ N(70,9 ) ~ N(69,5 ) ~ N(6, ) T S ~ N 69 6,5 6, 6 P( T S) P( T S 0) (sf) C ~ N 70, 9 C C C 9 C ~ N70, 70, 7 P( C 65) ( s.f.) Lt X 0.7C 0.5T 0.5S E( X ) 0.7(70) 0.5(69) 0.5(6) 68.8 Var( X ) 0.7 (7 ) 0.5 (5 ) 0.5 ( ) 6.55 X ~ N(68.8, 6.55) Lt th minimum mark of obtaining a distinction b k P( X k) 0.05 P( X k) 0.95 k 75.8 Hnc, last valu of k is 76.

Chapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7.

Chapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7. Chaptr Binomial Epansion Chaptr 0 Furthr Probability Chaptr Limits and Drivativs Chaptr Discrt Random Variabls Chaptr Diffrntiation Chaptr Discrt Probability Distributions Chaptr Applications of Diffrntiation