f ( x) dx SAJC 2017 JC2 Preliminary Exam H2 MA 9758/01

Transcription

1 SAJC 07 JC Prelimiary Eam H MA 9758/0 The volume of a spherical bubble is icreasig at a costat rate of λ cm per secod. Assumig that the iitial volume of the bubble is egligible, fid the eact rate i terms of λ at which the surface area of the bubble is icreasig whe the volume of the bubble is 0 cm. [5] 4 [The volume of a sphere, V π r ad the surface area of a sphere, the radius of the sphere.] A 4π r where r is The diagram shows the triagle ABC. It is give that the height AD is h uits, ABD π ad ACD π 4 +. A h B D C Show that if is sufficietly small for ad higher powers of to be eglected, the h h BC + h ( p + q + r ) π ta + 4, for costats p, q, r to be determied i eact form. [5] It is give that b for a < a a f ( ) ( a) a for a a < a ad that f ( + 4 a) f ( ) for all real values of, where a ad b are real costats ad 0 < a < b. (i) Sketch the graph of y f ( ) for a 8 a. [] Use the substitutio a cosθ to fid the eact value of ad π. 4a a f ( ) d i terms of a [5]

2 SAJC 07 JC Prelimiary Eam H MA 9758/0 4 (i) State a sequece of trasformatios that would trasform the curve with equatio y e oto the curve with equatio y f ( ) ad b >., where f ( ) e a b, a > 0 [] Sketch the curve y f ( ) ad the curve y. f ( ) You should state clearly the equatios of ay asymptotes, coordiates of turig poits ad aial itercepts. [5] 5 It is give that u + v w is perpedicular to u v + w, where u, v ad w are uit vectors. (i) Show that the agle betwee v ad w is 60. [4] Referred to the origi O, the poits U, V ad W have positio vectors u, v ad w respectively. Fid the eact area of triagle OVW. [] (iii) Give that u ad v w are parallel, fid the eact volume of the solid OUVW. [] [The volume of a pyramid is bh, where b is the base area ad h is the height of the pyramid.] 6 (a) (i) Fid e cos d, where is a positive iteger. [4] π Hece, without the use of a calculator, fid e cos d whe is odd. π [] (b) The regio bouded by the curve y, the y-ais ad the lie y is 6 rotated π radias about the -ais. Fid the eact volume of the solid obtaied. [5] 7 (i) i Show that for ay comple umber z re θ, where r > 0, ad π < θ π, z θ cot i z r [] π i Give that z e is a root of the equatio z z State, i similar form, the other root of the equatio. [] (iii) Usig parts (i) ad, solve the equatio 4w 4w ( w ) w [4]

3 SAJC 07 JC Prelimiary Eam H MA 9758/0 8 I a traiig sessio, a athlete rus from a startig poit S towards his coach i a straight lie as show i the diagrams below. Whe he reaches the coach, he rus back to S alog the same straight lie. A lap is completed whe he returs to S. At the begiig of the traiig sessio, the coach stads at A which is 0 m away from S. After the first lap, the coach moves from A to A ad after the secod lap, he moves + from A to A ad so o. The distace betwee A i to A i + is deoted by Ai A i +, i Z. Figure (i) For traiig regime (show i Figure ), the coach esures that the distace + Ai A i + m for i Z. Fid the least umber of laps that the athlete must complete so that he covers a total distace of more tha 000 m. [] Figure For traiig regime (show i Figure ), after the first lap, the coach esures that the distaces A A m, A A 6 m ad the distace Ai + Ai + Ai Ai + where + i Z. Show that the distace the coach is away from S just before the athlete r + 9 m. completed r laps is ( ) Hece fid the distace ru by the athlete after complete laps. Also fid how far the athlete is from the coach after he has ru 8 km. [6]

4 SAJC 07 JC Prelimiary Eam H MA 9758/0 9 The diagram below shows the curve C with parametric equatios + si θ, y 4+ cos θ, for π < θ π. y C P 0 The poit P is where π θ. 6 (i) Usig a o-calculator method, fid the equatio of the ormal at P. [4] The ormal at the poit P cuts C agai at poit Q, where θ α. Show that 8siα cosα ad hece deduce the coordiates of Q. [] (iii) Fid the area of the regio bouded by the curve C, the ormal at poit P ad the vertical lie passig through the poit Q. [4] 0 A populatio of 5 foes has bee itroduced ito a atioal park. A zoologist believes that the populatio of foes,, at time t years, ca be modelled by the Gompertz equatio give by: d 40 c l dt where c is a costat. 40 (i) Usig the substitutio u l, show that the differetial equatio ca be writte as d u cu. dt [] Hece fid u i terms of t ad show that 40e Be ct, where B is a costat. After years, the populatio of foes is estimated to be 0. [5] (iii) Fid the values of B ad c. [] (iv) Fid the populatio of foes i the log ru. [] (v) Hece, sketch the graph showig the populatio of foes over time. [] 4

5 SAJC 07 JC Prelimiary Eam H MA 9758/0 A computer-cotrolled machie ca be programmed to make plae cuts by keyig i the equatio of the plae of the cut, ad drill holes i a straight lie through a object by keyig i the equatio of the drill lie. A 0cm 0 cm 0 cm cuboid is to be cut ad drilled. The cuboid is positioed relative to the -, y-ad z-aes as show i Figure. z G F G Drill lie F D E 0 cm D P R T S C B 0 cm C Q B O 0 cm A y Figure Figure First, a plae cut is made to remove the corer at E. The cut goes through the poits P, Q ad R which are the midpoits of the sides ED, EA ad EF respectively. 0 0 (i) Show that PQ 5 ad PR [] Fid the cartesia equatio of the plae, p that cotais P, Q ad R. [] (iii) Fid the acute agle betwee p ad the plae DEFG. [] A hole is the drilled perpedicular to triagle PQR, as show i Figure. The hole passes through the triagle at the poit T which divides the lie PS i the ratio of 4:, where S is the midpoit of QR. (iv) Show that the poit T has coordiates ( 4, 9, 4). [] (v) State the vector equatio of the drill lie. [] (vi) If the computer program cotiues drillig through the cuboid alog the same lie as i part (v), determie the side of the cuboid that the drill eits from. Justify your aswer. [4] O A y Ed Of Paper 5

6 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 4 V π r dv dr 4π r dt dt Whe V0, 4 0 π r 5 r π Whe 5 r π, d V dt 5 dr λ 4π π dt dr λ π dt 4π 5 Surface Area, A 4π r da dr 8π r dt dt λ. 5 Whe r π, dr λ π dt 4π 5. da da dr dt dr dt 5 λ π 8π π 4 π 5 π λ 5 cm /s 6

7 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) (i) BC BD + DC h h + π ta π ta + 4 BC h h + π ta + ta 4 π ta ta 4 h h( ta ) + + ta h h( ) + + h + h( )( + ) h ( )( ) + h( )[ + ( ) + ) +...]! h + h( )[ ] h + h( ) h + + 7

8 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) (i) 4a a f ( ) d 0 b d a a π b ( asi θ ) d π π ab π si θ d ab π π a cos θ dθ ab si θ θ ab π π π ab 4 cos θ a θ π π θ 8

9 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 4 (i) ( ) a y e b e a b If f ( ) e, the ( ) ( ) f a ad so e a ( ) ( ) ( ) y f y f a y f a + b Hece the sequece of trasformatios are:. Scale by a factor of parallel to the -ais, a. Traslate the resultig curve by b uits i the egative y-directio. y yf() O (0,-b) y y/f() y0 O 9

10 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 5 (i) Sice u + v w is perpedicular to u v + w, u + v w u v + w 0 ( ) ( ) u u u v + u w + v u v v + v w w u + w v w w 0 Sice u u u, v v v, w w w, ad u v v u, u w w u, v w w v, u v w + v w 0 Sice u, v, w are uit vectors, u, v, w, + v w 0 v w v w cosθ cosθ Hece, θ 60 W O V Area of OVW ( OV )( OW ) si 60 ( )( ) uits 4 0

11 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) (iii) U W O V Sice u ad v w are parallel, we have OU OV, OU OW. Volume of OUVW ( Area of OVW )( OU ) ( ) 4 uits

12 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 6 (a) (i) Usig itegratio by parts, e cos d dv u e cos d du si e v d si e e si d dv u e si d du cos e v d si e cos e cos e + d si e cos e e cos d Rearragig, + + e cos d si e cos e + si e cos e cos d e c + + where c is a costat + π e cos d π si e cos e + + si cos si cos e e π π π π π π For ay positive iteger, si π 0 ad cos π If is odd, si π 0 ad cos π π π π e cos d π π π e 0 + e π π ( e e ) (As)

13 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) (b) y 6 y ( 6 ) Hece volume required π r h π y d 0 π ( ) π d 0 6 ( ) π π ( ) d 6 ( ) 0 ( ) ( 6 ) π π ( ) + 4 π π π uits 88 0

14 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 7 (i) iθ iθ re e iθ θ θ θ re r i i i e e e θ i e θ isi θ θ cos + isi θ isi θ + cot i θ cot i π i z e (iii) 4w 4w ( w ) w w w w w w Let z w, the z z π π i i From the solutios are z e or z e Sice w z w zw z w w( z ) z z w z π i π π Part (i) result ca be used as z e, where r with θ, θ. π π w cot i or w i cot 6 6 w i or w + i 4

15 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 8 (i) Distace travelled per lap is i AP: a (0) 60, d 6. Give total distace travelled > 000 [ (60) + ( )6] > > 0 ( + 4.5)(.5) > 0 < 4.5 or >.5 Sice Z +, least 4 Distace of the coach from S just before the ruer completes the rth lap 0 r 0 + ( ) + ( ) + ( ) ( ) r + r ( ) r 0 + r 0 + ( ) 9 Distace covered by the athlete after laps r r ( + 9) r ( ) r r r 58 r ( ) + 58 Whe D 8000m ( ) From GC, 8.54 Hece the athlete has ru 8 complete laps. The athlete has completed 704 m Hece he still have m 9 O the 9th lap, the coach is m from S. Hece the athlete would be m away from the coach oce he fiishes 8 km. 5

16 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 9 (i) d cosθ dθ, dy siθ dθ dy siθ ta θ d cosθ π Whe θ,, 6 y, d y d Equatio of ormal : y ( ) y + + si θ () y 4 + cos θ () Substitute equatio () ad () ito 4 + cosθ ( + si θ ) + + cosθ 4siθ 8siθ cosθ y + At Poit Q, θ α 8siα cosα (show) Usig GC: α or π α (Reject, same as,poit P) 6 Hece, usig GC coordiates of Q (0.405,.4) Q (0.4,.4) (iii) y Req 6

17 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) whe siθ siθ θ or.8479 ( at poit Q) y d y d π 4 + cos θ cosθ dθ d Required Area ( ) ( ) uits ( s.f.) 7

18 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 0 (i) d 40 c l dt 40 u l l(40) l( ) du d du du d dt d dt 40 c l cu du cu dt d u c d t u l u ct + d u e ct+ d d u ± e e ct ct d Be, B ± e 40 Replace u with l 40 l Be ct 40 Be e ct 40 Be ct e ct Be 40e (iii) Whe t 0, 5, 5 40 e B B e 8 8 B l Whe t, 0 8

19 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 0 40e e c Be B c Be c e l c l ( e ) l 8 l c l l 8 0.6t 0.98e 40e (iv) The populatio of foes i the log ru is 40. (v) t 9

20 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) (i) OP 5 OQ 0 OR PQ OQ OP A ormal to p: PR OR OP Equatio of plae: 0 r r y + z 90 Or ay equivalet equatio of plae (iii) 0 A ormal to the plae EFGH 0 (or ay equivalet vector) cosθ θ (iv) OS OQ OR

21 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) OT Hece the coordiates of T are ( 4, 9, 4). (v) 4 Equatio of the drill lie: r 9 + λ 6, λ R. 4 (vi) Shortlist the possible plaes: ODGC, GCBF, OABC 0 Equatio of Plae ODGC: ri Equatio of Plae OABC: ri 0 0 Equatio of Plae GCBF: ri If the lie of the drill eits from the cuboid, all of the followig coditios must be satisfied: 0 0; 0 y 0; 0 z 0. The itersectio of plae ODGC 4 + λ λ i 0 4 λ λ 0 λ Positio vector is

22 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 7 Hece the poit of itersectio has coordiates, 0,. Hece the drill lie will eit from the side ODGC. The itersectio of plae OABC 4 + λ λ i λ λ 0 λ 4 + ( ) 40 Positio vector is 9 + 6( ) ( ) 0 Hece the poit of itersectio has coordiates ( 40, 6, 0) Hece the drill lie will ot eit from the side OABC. The itersectio of plae GCBF ri λ 9 + 6λ i λ λ 0 λ Positio vector is Hece the poit of itersectio has coordiates 0,,. Hece the drill lie will ot eit from the side GCBF.

23 SAJC 07 JC Prelimiary Eam H MA 9758/0 Sectio A: Pure Mathematics [40 marks] Without the use of a calculator, fid the comple umbers z ad w which satisfy the simultaeous equatios z wi z w i 0 [6] (i) A + B Show that +, where A ad B are costats to be foud. [] + r + 6 Hece fid. r r r [4] (iii) r + 0 Use your aswer to part to fid r + r + r +. [] r ( )( )( ) The fuctio f is defied by f :, R, >. (i) Fid f ( ) ad write dow the domai of f. [] O the same diagram, sketch the graphs of y f ( ), y f ( ) ad y f f ( ) statig the equatios of ay asymptotes ad showig the relatioships betwee the graphs clearly. [4] (iii) State the set of values of such that ff ( ) f f ( ). [] 4 Referred to the origi O, the poit A has positio vector 5i + j + k ad the poit B has positio vector i + j k. The plae π has equatio: r ( + λ µ ) i + ( λ ) j + ( µ ) k where λ, µ R (i) Fid the vector equatio of plae π i scalar product form. [] Fid the positio vector of the foot of perpedicular, C, from A toπ. [] The lie l passes through the poits A ad B. The lie l is the reflectio of the lie l about the plaeπ. Fid a vector equatio of l. []

24 SAJC 07 JC Prelimiary Eam H MA 9758/0 5 A θ D G a B E H F C It is give that DEFG is a square with fied side a cm ad it is iscribed i the isosceles triagle ABC with height AH, where AB AC ad agle BAH θ. (i) Takig t taθ, show that the area of the triagle ABC is give by S a 4 + 4t + t [] Fid the miimum area of S i terms of a whe t varies. [4] (iii) Hece sketch the graph showig the area of the triagle ABC as θ varies. [] Sectio B: Statistics [60 marks] 6 (a) There are three yellow balls, three red balls ad three blue balls. Balls of each colour are umbered,, ad. Fid the umber of ways of arragig the balls i a row such that adjacet balls do ot sum up to two. [] (b) I a restaurat, there were two roud tables available, a table for five ad a table for si. Fid the umber of ways eleve frieds ca be seated if two particular frieds are ot seated et to each other. [4] 7 For the evets A ad B, it is give that P( A B ') 0.6, P( A B ') 0.8 ad P( A B ') 0.8 Fid, (i) P( B ) [] P( A B) [] (iii) P( B A ') [] Hece determie whether A ad B are idepedet. [] 4

25 SAJC 07 JC Prelimiary Eam H MA 9758/0 8 A fairgroud game ivolves tryig to hit a movig target with a gushot. A roud cosists of a maimum of shots. Te poits are scored if a player hits the target. The roud eds immediately if the player misses a shot. The probability that Lida hits the target i a sigle shot is 0.6. All shots take are idepedet of oe aother. (i) Fid the probability that Lida scores 0 poits i a roud. [] The radom variable X is the umber of poits Lida scores i a roud. Fid the probability distributio of X. [] (iii) Fid the mea ad variace of X. [4] (iv) A game cosists of rouds. Fid the probability that Lida scores more poits i roud tha i roud. [] 9 Si cities i a certai coutry are liked by rail to city O. The rail compay provides the iformatio about the distace of each city to city O ad the rail fare from that city to city O o its website. Charles copied the table below from the website, but he had copied oe of the rail fares wrogly. City A B C D E F Distace, km Rail fare, $y (i) Give a sketch of the scatter diagram for the data as show o your calculator. O your diagram, circle the poit that Charles has copied wrogly. [] For parts, (iii) ad (iv) of this questio you should eclude the poit for which Charles has copied the rail fare value wrogly. Fid, correct to 4 decimal places, the product momet correlatio coefficiet betwee (a) l ad y, (b) ad y. [] (iii) Usig parts (i) ad, eplai which of the cases i part is more appropriate for modellig the data. [] (iv) By usig the equatio of a suitable regressio lie, estimate the rail fare whe the distace is 0 km. Eplai if your estimate is reliable. [] 0 A factory maufactures roud tables i two sizes: small ad large. The radius of a small table, measured i cm, has distributio N (0, ) ad the radius of a large table, measured i cm, has distributio N (50,5 ). (i) Fid the probability that the sum of the radius of 5 radomly chose small tables is less tha 60 cm. [] Fid the probability that the sum of the radius of radomly chose small tables is less tha twice the radius of a radomly chose large table. [] (iii) State a assumptio eeded i your calculatio i part. [] A shipmet of large tables is to be eported. Before shippig, a check is doe ad the shipmet will be rejected if there are at least two tables whose radius is less tha 40 cm. (iv) Fid the probability that the shipmet is rejected. [] The factory decides ow to maufacture medium sized tables. The radius of a medium sized table, measured i cm, has distributio N ( µ, σ ). It is kow that 0% of the medium sized tables have radius greater tha 44 cm ad 0% have radius of less tha 40 cm. 5

26 SAJC 07 JC Prelimiary Eam H MA 9758/0 (v) Fid the values of µ adσ. [4] The Kola Compay receives a umber of complaits that the volume of cola i their cas are less tha the stated amout of 500 ml. A statisticia decides to sample 50 cola cas to ivestigate the complaits. He measures the volume of cola, ml, i each ca ad summarised the results as follows: 470, (i) 46. Fid ubiased estimates of the populatio mea ad variace correct to decimal places ad carry out the test at the % level of sigificace. [6] Oe director i the compay poits out that the compay should test whether the volume of cola i a ca is 500 ml at the % sigificace level istead. Usig the result of the test coducted i (i), eplai how the p-value of this test ca be obtaied from p-value i part (i) ad state the correspodig coclusio. [] The head statisticia agrees the compay should test that the volume of cola i a ca is 500 ml at the % level of sigificace. He iteds to make a simple rule of referece for the productio maagers so that they will ot eed to keep comig back to him to coduct hypothesis tests. O his istructio sheet, he lists the followig:. Collect a radom sample of 40 cola cas ad measure their volume.. Calculate the mea of your sample, ad the variace of your sample, s.. Coclude that the volume of cola differs from 500 ml if the value of lies.. (iii) Usig the above iformatio, complete the decisio rule i step i terms of. s [4] A party orgaiser has cas of cola ad packets of grape juice. Assume ow that the volume of a ca of cola has mea 500 ml ad variace 44 ml, ad the volume of a packet of grape juice has mea 50 ml ad variace 5 ml. She mies all the cola ad grape juice ito a mocktail, which she pours ito a 0-litre barrel. Assume that is sufficietly large ad that the volumes of the cas of cola ad packets of grape juice are idepedet. (iv) Show that if the party orgaiser wats to be at least 95% sure that the barrel will ot overflow, must satisfy the iequality , [4] Ed Of Paper 6

27 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) Method z wi z w i zi --- () i Substitute () ito z w i 0 z (i zi) i 0 z + zi ( ) i ± i 4()(6) i ± 4 z i ± 5i z i or z i w i (i)i w i ( i)i + i + i Method z wi z + wi. () Substitute () ito z w i 0 ( w ) + i w i wi w w i 0 w ( 6i) w i 0 w w + + w ( 6i) 5 i 0 + ( 6i) w 5 i 0 ( 6i) ( 6i) 4()( 5 i) w ± + 6i ± i i + 6i ± 5 w + i or w + i z + ( + i)i i z + ( + i)i i Method wi z w iz + i z ( iz + i) i 0 z + iz Let z a + bi where a, b R ( a + bi) + i( a + bi) a b + abi + ai b a b b ( ab + a)i 0 By comparig the real ad imagiary parts, 7

28 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) a b b () ab + a 0 () From (), a 0 or b Whe a 0, b + b 6 0 ( b )( b + ) 0 b or b Hece z i, w i(i) + i + i or z i, w i( i) + i + i 5 Whe b, a There is o real solutio for a. (i) ( )( ) ( )( ) ( )( ) ( )( )( ) ( + ) ( ) + ( ) + r + 6 r r r r r + r r + r r r r

29 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) (iii) r + 0 r + r + r + r ( )( )( ) Let r + p r p p p + p 4 p + 6 ( p )( p)( p + ) p + 6 p p + p + 6 p + 6 p p p p p p (i) f : Let y + y ± + y Sice >, + y f ( )

30 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) y 0 From graph of f, R f ( 0, ) D ( ) ( 0, ). f y 0 y f() y 0 (iii) ff ( ) f f( ) have the same rule, we ivestigate the domai D ( ) D ( ) Sice f f, ff 0, Takig the itersectio of these domais, Rage of values is >. 0

31 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 4 (i) Equatio of plae is r + λ + µ 0, λ, µ R 0 A ormal vector to plae is Hece vector equatio of the plae is ri i 4 4 ri 4 5 lac : r + s, s R 4 5 Thus OC + s for some s R. 4 Sice C lies o the plae: 5 + s i 4 4 ( 5+ s)+(+ s) + 4( + 4 s) s 5 7 Thus OC

32 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) (iii) Usig mid-poit theorem OA' OC OA B is the poit of itersectio of l ad π. BA' OA' OB l : r + 9, 7 t t R or l : r + t 9, t R 0 5 (i) a a The height of triagle ADG is ta θ t. a Hece AH a + a + t t. BH BE + EH a ta θ + a a(t + ) Area S ( AH )( BC ) a S + ( a(t + ) ) t S a + (t + ) t S a 4 + 4t + t ds a 4 dt t Whe d S 0 dt, t 4

33 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) t ± Reject t ta θ as θ is acute d S a dt t d S Whe t, a 6a > 0. dt Hece the miimum value of S occurs whe S a a Miimum ( ) (iii) To sketch the graph of S a taθ + taθ S t. ta,8a θ 0 π θ θ 6 (a) Sice adjacet balls do ot sum up to two, balls umbered eeds be separated. Number of ways of arragig the other balls with o restrictio 6! Slottig i the balls umbered, permutatio is doe as balls are of differet colour 7 C! No of ways 7 6! C! 500

34 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) (b) Method frieds X X Table of 5 Table of 6 (6-)! X Case frieds are seated together at table of 5 No. of ways to select other frieds ad arrage them at the table of 5 No. of ways to arrage the frieds! No. of ways to sit the remaiig frieds at the table of 6 (6-)! 5! 0 Total o. of ways 9 C (4 )!! 5! 0960 C (4 )! 9 Case frieds are seated together at table of 6 frieds (5-)! X X X X Table of 5 Table of 6 No. of ways to select 4 other frieds ad arrage them at the table of 6 9 C 4 (5 )! 04 No. of ways to sit the frieds at the table of 6! No. of ways to sit the remaiig frieds at the table of 5 (5-)! 4! 4 Total o. of ways 9 C 4 (5 )!! 4! 455 No of ways to arrage frieds without restrictios C 5 (5 )! (6 )! 0560 Total o. of ways of arragig people such that particular frieds are ot seated together Method Alterative Method Case : Two particular frieds seated at table of 5 No of ways 9 C! 5!

35 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 9 C : Selectio of frieds to be seated at table of 5. This automatically selects frieds to be seated at table of 6. (-)!: Arragig the other frieds i table of 5. P : Slottig i the particular frieds 5!: Arragig the 6 other frieds i table of 6. Case : Two particular frieds seated at table of 6 No of ways 9 C 4!! C 4 : Selectio of frieds to be seated at table of 5. This automatically selects frieds to be seated at table of 6. (5-)!: Arragig the 5 frieds i table of 5. 4!: Arragig the 5 frieds i table of 6. 4 P : Slottig i the particular frieds Case : Two particular frieds seated at separate tables No of ways 9 C 4! 5! C 4 : Selectio of frieds to be seated at table of 5. This automatically selects frieds to be seated at table of 6. (5-)!: Arragig the 5 frieds i table of 5. (6-)!: Arragig the 6 frieds i table of 6. : The particular frieds ca switch tables Total o. of ways

36 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 7 (i) Give P( A B ') 0.8 P( A B ') 0.8 P( B ') P( B) P( B) Let P( A B ) A B P( A B) P( A B ') + P( B) P( A B)' Sice P( A B ') P( A B ) (iii) P( B A') P( B A') P( A') ( ) Sice P( B A ') P( B) B is ot idepedet of A A ad B are ot idepedet. 6

37 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 8 (i) P(Lida scores 0 poits) P ({hit, hit, hit}) 0.6 (0.6) Let X be the umber of poits scored by Lida i a roud. X P(X) (iii) E(X) E(X ) Var(X) E(X ) [E(X)] (iv) Let X be the umber of poits scored by Lida i Roud ad let X be the umber of poits scored by Lida i Roud. P(Lida scores more i roud tha i roud ) P( X 0 & X 0) + P( X 0 & X 0) + P( X 0 & X 0) P( X 0) P( X 0) + P( X 0) P( X 0) + P( X 0) P( X 0) 0.4 ( 0.4) ( ) ( s.f.) 7

38 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) 9 (i) 8.8 y 7.6 (a) Product momet correlatio coefficiet, r (b) Product momet correlatio coefficiet, r (iii) From the scatter diagram, as icreases, the value of y icreases at a decreasig rate, that seems to fit model (a) better. Also, the value of r for model (a) is closer to as compared to model (b). (iv) We use the regressio lie y o l y 6.69 l l 7. ( ) Whe 0, y 6.69 l ( ) As the value of r is close to ad 0 is withi the give data rage, the estimatio may be reliable. 0 (i) Let S be the radom variable radius of a small table i cm. Let L be the radom variable radius of a large table i cm. S ~ N (0, ) L ~ N (50, 5 ) S S S S S ~ N(5 0, 5 ) 4 5 ( ) S + S + S + S + S ~ N 50, 0 P( S+ S+ S+ S4 + S 5 < 60) S + S + S L ~ N( 0 50, + 5 ) ( ) S + S + S L ~ N 0, P( S + S + S < L) P( S + S + S L < 0)

39 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) (iii) The radii of the large ad small roud tables are idepedet of oe aother. (iv) Let X be the radom variable umber of large tables, out of, with radius less tha 40 cm. X ~ B(, P( L < 40 )) X ~ B(, ) ( X ) P( X ) P (v) Let Y be the radom variable radius of a medium sized table i cm P( Y 44 ) 0.0 P( Y < 44 ) µ P Z < 0.80 σ 44 µ σ µ σ P( Y < 40 ) 0.0 ( ) 40 µ P Z < 0.0 σ 40 µ σ µ σ ( ) Solvig () ad (), σ σ 4.660σ σ.98.9 µ (i) Ubiased estimate of populatio mea, Ubiased estimate for populatio variace, 470 s Let X be the volume of beer i oe beer ca i ml ad µ be the populatio mea volume of beer of the beer cas. 9

40 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios) H 0 : µ 500 H : µ < 500 Uder H 0, sice 50 is large, by the Cetral Limit Theorem, s X N ~ 500, approimately. 50 Use a left-tailed z-test at the % level of sigificace. X 500 Test statistic: Z ~ N (0,). s 50 Reject H 0 if p-value 0.0. From the sample, p value Sice p-value , we reject H 0. There is sufficiet evidece at the % level of sigificace to coclude that the volume of cola i a ca is less tha 500 ml. (iii) Let X be the volume of cola i oe ca i ml ad µ be the populatio mea volume of cola of the cas. H : µ H : µ 500 Ubiased estimate of populatio variace, s s 9 Uder H 0, sice 40 is large, by the Cetral Limit Theorem, 40 ( ) s X N ~ 500, approimately. 9 Use a two-tailed z-test at the % level of sigificace. X 500 Test statistic: Z ~ N (0,) s 9 Critical values: zcrit ().5758 zcrit () Reject H 0 if z.5758 or z cal Sice H 0 is rejected, cal 40

41 SAJC 07 JC Prelimiary Eam H MA 9758/0 (Solutios).5758 z or z.5758 cal or.5758 s s 9 9 s s or s or s cal s or s Hece the decisio rule should read: Coclude that the volume of cola differs from 500 ml if the value of lies withi this rage : s or s. (iv) Let X be the volume of cola i oe ca i ml. sice is large, by the Cetral Limit Theorem, X + X X ~ N ( 500,44) approimately. Let Y be the volume of grape juice i oe packet i ml. sice is large, by the Cetral Limit Theorem, Y + Y Y ~ N 500,50 approimately. ( ) X + X X + Y + Y Y ~ N(000,94 ) P( X + X X + Y + Y Y 0,000) , P Z , , ,