2015 SRJC H2 Mathematics Prelim Paper 2

Transcription

1 05 SRJC H Mthemtics Prelim Pper Section A: Pure Mthemtics [40 mrks] Functions f nd g re defined s elow. f :, g : ln( ), (i) Sketch the grph of g() nd stte its ect rnge. [] (ii) Determine whether the composite functions fg nd gf eist, justifing our nswer. Find the rnge of the composite function if it eists. [4] (iii) On the sme digrm s prt (i), sketch the grph of g (), indicting on our sketch the line in which the grph of g() must e reflected in order to otin the grph of g (). [] (i) g() (, +ln4) (+ln4, ) g () Rnge of g, R g ( + ln4, + ) (ii) From GC, the rnge of f is (0, ]. Since Rf D g (, ), the composite gf does not eist. Since R g ( + ln4, + ) [, ) D f the composite fg eists. Direct Method for finding rnge of fg (, ) ln 4 fg() ln( ) R fg Rnge of fg (0, ) ln 4

2 Indirect Method for finding rnge of fg R fg ln 4 +ln4 R g f() Rnge of fg (0, ) ln 4 A tnk contins wter which is heted n electric wter heter working under the ction of thermostt. When the wter heter is first switched on, the temperture of the wter is 5 C. The heter cuses the temperture to increse t rte r C per minute, where r is constnt, until the wter temperture hits 75 C.The heter then switches off. (i) Write down, in terms of r, the time tken for the temperture to increse from 5 C to 75 C. [] The temperture of the wter then immeditel strts to decrese. The temperture of the wter t time t minutes fter the heter is switched off is C. It is known tht the temperture of the wter decreses t vrile rte k( 5) C per minute, where k is positive constnt, until 5. (ii) Write down differentil eqution involving nd t, to represent the sitution s the temperture is decresing. [] (iii) Given tht when 55, the temperture is decresing t rte of 5Cper minute, find the totl length of time for the temperture to increse from 5 C to 75 C nd then decrese to 5 C, leving our nswer in ect form, in terms of r. [7]

3 (i) Time tken d dt (ii) k 5 40 mins r (iii) d k dt 5 ln 5 kt c, where c is constnt, since > 5 o C kt 5 e.e c 5 Ae kt, where A is n ritrr constnt. When 55, d 5, k dt 6 When t 0 nd 75, A 50 6 t 5 50e When 5, 6 t e t 6 e 5 t 6ln 6ln 5 5 Totl length of time 40 6ln 5 r mins

4 4 [It is given tht the volume of sphere of rdius r is 4 π right squre prmid with squre se of length nd height h is r nd tht the volume of h.] In the digrm elow, hemisphere of fied rdius cm lies on the se of right prmid such tht its curved surfce is in contct with ll four fces of the prmid. The prmid hs squre se of length cm nd height cm. Given tht the volume of region inside the prmid tht is not prt of the hemisphere is denoted V, (i) show tht V π 4. [] (ii) use differentition to find, in terms of, the minimum vlue of V ectl, proving tht it is minimum. [7] (i) Method Consider the similr tringles ABC nd ACD BC CD AB AC 4 A D B C

5 ( 4 ) 4 4 or 4 (rejected > 0) V π π 4 Method Considering the re of ABC, we hve B A C 4 4 V π π 4 (ii) V π 4

6 6 dv d Let d V d or 0 (Reject 0) 6 or 6 (Reject 0) Method d V d d 6 d When 6, d V d > 0

7 7 Method dv d since 6 -ve 4 0for < 6 0 +ve since for > 6 Slope \ / Hence, V is minimum vlue when 6. The minimum vlue of V π 6 4 π

8 8 4 () The comple numer z is given z re i, where r > 0 nd 0. (i) Given tht w ( i) z, find w in terms of r nd rg w in terms of. [] (ii) For fied vlue of r, drw on the sme Argnd digrm the the loci of z nd w s vries. [] (iii) If r.5 units, clculte (in terms of ) the length of the locus of w for Im(w) 0 s vries. [] () Sketch on single Argnd digrm the set of points representing ll comple numers v stisfing oth of the following inequlities: v 5 8i 5 nd v 8i v 0i. Hence find (in rdins) the lest vlue of rg ( v 5 i). [6] () (i) w i z ( ) r r rg( ) rg rg( ) w i z 6 (ii) Im r r Locus of z Locus of w 0 r r 6 Re (iii) Length of the locus of w ()( )() units 6 () Im(v) i +8i 0 5 Re(v)

9 9 From the digrm, 5 4 [M] Therefore, the lest vlue of v 4 tn rd [M] rg 5 i rd [A] Section B: Sttistics [60 mrks] 5 A compn with eight hundred emploees wishes to find out how much time its emploees tke to trvel to work. It is given tht the emploees go to work either cr or us/trin nd tht ech of them tkes the sme form of trnsport to work ever d. The following tle shows the numer of emploees going to work cr nd the numer of emploees going to work us/trin. Cr Bus/Trin Men Women 8 9 The compn wnts to tke rndom smple of 80 emploees. (i) Eplin wht is ment, in this contet, the term rndom smple. [] (ii) Descrie how rndom strtified smple cn e otined. [] (iii) Give reson wh quot smpling is not s suitle in this contet compred to strtified smpling. [] (i) (ii) (iii) It mens tht ever emploee hs n equl chnce of eing selected nd tht the selection of ech emploee is independent of one nother. Rndoml ssign numers to ech emploee in ech strtum: Men tking cr from to 65; Men tking us/trin from to 60; Women tking cr from to 8; Women tking us/trin from to 9. Rndoml select the required numers for ech strtum (s shown elow) to form smple of 80 emploees. Quot smpling does not produce rndom smple, therefore it is not suitle in this contet s compred to strtified smpling which produces rndom smple s required the compn.

10 0 6 Slt is pcked in gs to e sold. The mnufcturer clims ech g contins t lest 00 g of slt. To test this clim, rndom smple of 5 gs of slt is emined nd the mss, g, of the contents of ech g is determined. It is found tht the smple hs men of 99. g nd vrince of.864 g. (i) Test t the 0% significnce level whether the mnufcturer s clim is vlid. [5] (ii) Stte n ssumption necessr for the test in (i) to e vlid. [] (i) 99. nd unised estimte for the popultion vrince is s 5 (.864) Let denote the men mss of slt in rndoml chosen g. To test H o : 00 Aginst H : < 00 Using tiled test t 0% level of significnce X 00 S Under H o, test sttistic T t4 Using GC, p vlue Since p vlue < 0.0, we reject H o nd conclude tht there is sufficient evidence to tht the men mss of slt in g is less thn 00 g, thus the mnufcturer s clim is not vlid t 0% level of significnce. (ii) The mss of slt in rndoml chosen g is ssumed to e following norml distriution. 7 A g contins w white lls nd lck lls. One ll is selected t rndom from the g, its colour noted nd it is then returned to the g long with n dditionl lls of the sme colour. A second ll is then rndoml selected from the g. (i) Construct proilit tree showing this informtion. [] (ii) Show tht the proilit tht the second ll selected is lck is independent of n. [] It is now known tht the second ll drwn is lck. Show tht the proilit tht the first w ll drwn is white is. [] w n

11 w n w n White w w White w n Blck w (ii) P(second ll is lck) Blck w w n n w n White Blck w ( n) ( w )( w n) ( w )( w n) ( w n) ( w )( w n) w is independent of n Required proilit P(first ll is white nd second ll is lck) P(second ll is lck) w ( w )( w n) w w w n (shown)

12 8 The tle elow shows Singpore s GDP per cpit over the ers from 965. Yer, GDP per cpit (in thousnds), Source: Deprtment of Sttistics Singpore () () Using the dt ville, (i) drw the sctter digrm, lelling the es clerl, [] (ii) find the lest squre regression line of on, [] (iii) estimte the GDP per cpit in the er 05, correct to the nerest whole numer. Comment on the reliilit of the vlue otined. [] It is suggested tht the dt from 965 to 00 cn e modelled 965 insted of liner model. Find the vlue of the product moment correltion coefficient for ech of the proposed models nd determine which is the etter model. [] (i) (00,6.498) (965,.58) (ii) The regression line of on is (iii). The GDP per cpit in 05 is $6 94. [ccept vritions] Since this is n etrpolted vlue nd the liner model m not hold eond the dt rnge, the vlue clculted is not relile.

13 () Using liner model, the product moment correltion coefficient, r Using the qudrtic model r , the product moment correltion coefficient, Since the r vlue for the qudrtic model is closer to, this is etter model. 9 In fctor mnufcturing clcultors, it is found tht.5% of the clcultors produced re defective. In rndom smple of 90 clcultors, find the proilit tht (i) there re ectl defective clcultors, [] (ii) the 90 th clcultor is the second defective clcultor given tht there re ectl defective clcultors in tht smple. [] The clcultors re pcked in oes of 90. (iii) Using suitle pproimtion, find the proilit tht not more thn o of clcultors, out of 60 oes, contin more thn defective clcultors. [5] (i) Let X e the r.v. numer of defective clcultors in smple of 90. X ~ B(90, 0.05) P X 0.8 Using GC, 0.8 (to s.f.) (ii) Let Y e the r.v. numer of defective clcultors in smple of 89. Y ~ B(89, 0.05) Using GC, required proilit P 90 th clcultor is defective nd X P X Y th P P 90 clcultor is defective P X P Y 0.05 P X (s.f.) 0.8 Alterntivel, Required proilit P 90 th clcultor is defective nd X P X

14 C (s.f.) (iii) Let W e the r.v. numer of oes, out of 60 oes, with more thn defective clcultors. W ~ B(60, P X ) W ~ B(60,0.58) Since n 60 is lrge, np nd nq , W ~ N(9.08,7.79) pproimtel. cc. P P (to s.f.) Using GC, W W 0 Seven men nd seven women, including Sll nd And, prticipted in speed-dting session t communit centre. All prticipnts re to sit in w such tht no two persons of the sme gender sit net to ech other. How mn ws cn the prticipnts e rrnged if (i) the re seted t round tle of 4 sets? [] (ii) the re seted t similr round tles of 6 sets ech without Sll nd And? [] (iii) the re seted on oth sides of rectngulr tle with 7 sets on ech side, such tht Sll nd And sit net to ech other on the sme side? [] (i) No of ws 7!7! [rrnge 7 men in circle 7! or 7! 7 ] [slot in the 7 women etween the men 7! ] (ii) No of ws 6 C! 6! 8800 [out of 6 men, choose to plce t one tle 6 C ; note tht the tles re similr, nd hence there s doule counting nd thus need to divide ] [rrnge men in circle! ; note tht this vlue is squred] [rrnge 6 women in the remining sets t oth tles 6!]

15 5 (iii) No of ws 6 C 6! 6! [on one side, choose slot for Sll nd And 6 C ] [there re two possile sides t tle ] [Sll nd And cn swop plces ; when this hppens ll other couple will swop plces s well ecuse no two persons of sme gender re to sit eside one nother hence, there s doule counting, thus need to dived ] [there re 6 possile ws left to slot the rest of 6 women6!] [there re 6 possile ws then to slot the rest of 6 men6!] () A frm in the west of Singpore grows turnips for sle to the locl mrket. (i) (ii) 5 turnips re rndoml chosen. Find the proilit tht ectl one turnip weighs less thn the lower qurtile weight nd ectl two turnips weigh more thn the medin weight. [] The mss of rndoml chosen turnip hs men 40 g nd stndrd devition of g. If the proilit tht the men mss of lrge smple of n turnips is greter thn 9.6 g eceeds 0.95, find the lest vlue of n. [] () A rndom vrile X hs the distriution X ~ N(40, ). The rndom vrile Y is relted to X the formul Y X, where nd re constnts nd > 0. Given tht P( Y 85) P( Y 55) 0.075, find the vlues of E( Y) nd Vr( Y ), nd hence find the vlues of nd. [5] () (i)p(t < q L ).P(T > m) 5!.[P( ql T m)].!! 5! 4 4!! (0)

16 6 (ii) Let T denotes the mss of rndoml chosen turnip. Since n is lrge, T P( T 9.6) 0.95 ~ N(40, ) n pproimtel Centrl Limit Theorem. P( T 9.6) 0.95 P( T 9.6) 0.05 Let T 40 Z ~ N(0, ) n PZ 0.05 n n n.675 n 5.86 The lest vlue of n is 5. () P( Y 85) Since Y is liner function of X, X is norml implies tht Y is norml. Since the norml distriution is smmetricl out the men nd P( Y 85) P( Y 55) 0.075, E( Y) P( Z ) Using GC, P( Z c) c

17 nd Vr( Y) Vr( Y) Vr( X ) Vr( ), where Vr( ) 0 Vr(Y) (reject negtive since > 0) E( Y) E( X ) (8.0449) (to s.f.) () A techer discovered tht the proilit tht rndoml chosen student is lte two ds in month is three times the proilit tht the student is lte four ds in month. If the numer of ds student is lte in month follows Poisson distriution, find the non-zero vrince of the Poisson distriution. [] () Records indicte tht certin hospitl delivers n verge of 650 ies ech er. Ech d, there re shifts of equl durtion in the hospitl. It is ssumed tht the numer of deliveries in d cn e modelled Poisson distriution. Tking er to consist of 65 ds, (i) show tht the men numer of deliveries per shift is 0 nd find the most likel numer of deliveries per shift, [] (ii) find the epected numer of shifts with t lest 5 deliveries in week, [] (iii) eplin wh Poisson distriution m not e suitle model for the numer of deliveries in d for hospitl with significnt numer of plnned deliveries (e.g. cesren section). []

18 8 () Let X denote the rndom vrile representing the numer of times student is lte in month. X ~ Po() where Vr(X) P(X ) P(X 4) e λ e λ! 4! λ λ 4 λ 4 (since > 0) () (i) Men numer of deliveries per shift ech d Let X denote the rndom vrile representing the numer of deliveries in shift. Then, X ~ Po 0 From the GC, P(X ) most likel numer of deliveries per shift is. [M, A] (ii) P(X 5) P(Y 4) Let Y denote the rndom vrile representing the numer of shifts with t lest 5 deliveries. i.e. Y ~B(, 0.45) the epected numer of shifts with more thn 5 deliveries in week (iii) Plnned deliveries (e.g. cesren section) re not nturl child irths nd re less likel to e scheduled t night. Hence, Poisson distriution m not e suitle model for the numer of deliveries in d s the verge numer of deliveries per shift m not e constnt.