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CAT 2006 Section I Answer Questions 1 to 5 on the basis of the information given below:
In a Class X Board examination, ten papers are distributed over five Groups - PCB, Mathematics, SocialScience, Vernacular and n!lish ach of the ten papers is evaluated out of #$$ %he final score of a student iscalculated in the follo&in! manner 'irst, the Group Scores are obtained b( avera!in! mar)s in the papers&ithin the Group %he final score is the simple avera!e of the Group Scores %he data for the top ten studentsare presented belo& *+ipans score in n!lish Paper II has been intentionall( removed in the table
ote: !
or G a!ainst the name of a student respectivel( indicates &hether the student is a bo( or a !irl # .o& much did +ipan !et in n!lish Paper II/*#01*203 4*506*107*4002 Students &ho obtained Group Scores of at least 04 in ever( !roup are eli!ible to appl( for a pri8e 9mon!those &ho are eli!ible, the student obtainin! the hi!hest Group Score in Social Science Group is a&arded this pri8e %he pri8e &as a&arded to:*lShre(a*2 ;am*5 9(esha*1 +ipan*4 no onefrom the top ten5 9mon! the top ten students, ho& man( bo(s scored at least 04 in at least one paper from each of the!roups/*##*22*55*11*441 ach of the ten students &as allo&ed to improve his<her score in exactl( one paper of choice &ith theob=ective of maximi8in! his<her final score ver(one scored #$$ in the paper in &hich he or she chose toimprove 9fter that, the topper amon! the ten students &as:*l;am*29!m*5 Pntam*1 9(esha*4 +ipan4 .ad >oseph, 9!m, Pntam and %irna each obtained Group Score of #$$ in the Social Science Group, thentheir standin! in decreasin! order of final score &ould be:*# Pntam, >oseph, %irna, 9!m*2 >oseph, %irna, 9!m, Pntam*5 Pntam, 9!m, %irna, >oseph*1 >oseph, %irna, Pntam, 9!m*4 Pntam, %irna, 9!m, >oseph
Answer Questions 6 to 10 on the basis of the information given below:
Mathematicians are assi!ned a number called rdos number *named after the famous mathematician, Paulrdos ?nl( Paul rdos himself has an rdos number of 8ero 9n( mathematician &ho has &ritten a research paper &ith rdos has an rdos number of # 'or other mathematicians, the calculation of his<her rdosnumber is illustrated belo&:Suppose that a mathematician X has co-authored papers &ith several other mathematicians 'rom amon!them, mathematician @ has the smallest rdos number Aet the rdos number of @ be ( %hen X has an rdosnumber of (# .ence an( mathematician &ith no co-authorship chain connected to rdos has an rdosnumber of infinit( In a seven da( lon! mini-conference or!ani8ed in memor( of Paul rdos, a close !roup of ei!htmathematicians, call them 9, B, C, +, , ', G and ., discussed some research problems 9t the be!innin! of the conference, 9 &as the onl( participant &ho had an infinite rdos number obod( had an rdos number less than that of ' ?n the third da( of the conference ' co-authored a paper =ointl( &ith 9 and C %his reduced the avera!erdos number of the !roup of ei!ht mathematicians to 5 %he rdos numbers of B, +, , G and . remainedunchan!ed &ith the &ritin! of this paper 'urther, no other co-authorship amon! an( three members&ould have reduced the avera!e rdos number of the !roup of ei!ht to as lo& as 5 9t the end of the third da(, five members of this !roup had identical rdos numbers &hile the other threehad rdos numbers distinct from each other ?n the fifth da(, co-authored a paper &ith ' &hich reduced the !roups avera!e rdos number b( $ 4 %herdos numbers of the remainin! six &ere unchan!ed &ith the &ritin! of this paper o other paper &as &ritten durin! the conference 3 %he person havin! the lar!est rdos number at the end of the conference must have had rdos number *atthat time:*#4*26*50*1#1*4#46 .o& man( participants in the conference did not chan!e their rdos number durin! the conference/*#2*25*51*14*4 cannot be determined7 %he rdos number of C at the end of the conference &as:*##*22*55*11*440 %he rdos number of at the be!innin! of the conference &as:*#2*24*53*16*47#$ .o& man( participants had the same rdos number at the be!innin! of the conference/*#2*25*51*14*4 cannot be determined
Answer Questions 11 to 15 on the basis of the information given below:
%&o traders, Chetan and Michael, &ere involved in the bu(in! and sellin! of MCS shares over five tradin!da(s 9t the be!innin! of the first da(, the MCS share &as priced at ;s #$$, &hile at the end of the fifth da( it&as priced at ;s ##$ 9t the end of each da(, the MCS share price either &ent up b( ;s #$, or else, it camedo&n b( ;s #$ Both Chetan and Michael too) bu(in! and sellin! decisions at the end of each tradin! da( %he be!innin! price of MCS share on a !iven da( &as the same as the endin! price of the previous da( Chetan and Michael started &ith the same number of shares and amount of cash, and had enou!h of both Belo& are some additional facts about ho& Chetan and Michael traded over the five tradin! da(s ach da( if the price &ent up, Chetan sold #$ shares of MCS at the closin! price ?n the other hand, eachda( if the price &ent do&n, he bou!ht #$ shares at the closin! price If on an( da(, the closin! price &as above ;s ##$, then Michael sold #$ shares of MCS, &hile if it &as belo& ;s 0$, he bou!ht #$ shares, all at the closin! price ## If Chetan sold #$ shares of MCS on three consecutive da(s, &hile Michael sold #$ shares onl( oncedurin! the five da(s, &hat &as the price of MCS at the end of da( 5/*l;s0$*2;sl$$*5;sll$*1 ;s #2$*4 ;s #5$#2 If Michael ended up &ith ;s #$$ less cash than Chetan at the end of da( 4, &hat &as the difference in thenumber of shares possessed b( Michael and Chetan *at the end of da( 4/
(1)
Michael had #$ less shares than Chetan
(2)
Michael had l$ more shares than Chetan
(3)
Chetan had #$ more shares than Michael
(4)
Chetan had 2$ more shares than Michael
(5)
Both had the same number of shares #5 If Chetan ended up &ith ;s #5$$ more cash than Michael at the end of da( 4, &hat &as the price of MCSshareattheendofda(1/*l;s0$*2;sl$$*5;sll$*1 ;s #2$*4otuniDuel( determinable#1 Ehat could have been the maximum possible increase in combined cash balance of Chetan and Michael atthe end of the fifth da(/*l;s56$$*2;s1$$$*5 ;s 16$$*1 ;s 4$$$*4 ;s 3$$$#4 If Michael ended up &ith 2$ more shares than Chetan at the end of da( 4, &hat &as the price of the shareattheendofda(5/*l;s0$*2;sl$$*5;sll$*1 ;s #2$*4 ;s #5$
Answer Questions 16 to 20 on the basis of the information given below:
9 si!nificant amount of traffic flo&s from point S to point % in the one-&a( street net&or) sho&n belo& Points 9, B, C, and + are =unctions in the net&or), and the arro&s mar) the direction of traffic flo& %he fuelcost in rupees for travellin! alon! a street is indicated b( the number ad=acent to the arro& representin! thestreet Motorists travellin! from point S to point % &ould obviousl( ta)e the route for &hich the total cost of travellin! is the minimum If t&o or more routes have the same least travel cost, then motorists are indifferent bet&een them .ence, the traffic !ets evenl( distributed amon! all the least cost routes %he !overnment can control the flo& of traffic onl( b( lev(in! appropriate toll at each =unction 'or example if a motorist ta)es the route S-9-% *usin! =unction 9 alone, then the total cost of travel &ould be ;s #1 *i e F;s 0 ;s 4 plus the toll char!ed at =unction 9 #3 If the !overnment &ants to ensure that all motorists travellin! from S to % pa( the same amount *fuelcosts and toll combined re!ardless of the route the( choose and the street from B to C is under repairs *andhence unusable, then a feasible set of toll char!ed *in rupees at =unctions 9, B, C, and + respectivel( toachieve this !oal is:*#2,4,5,2*2$,4,5,#*5#,4,5,2*12,5,4,# *4#,5,4,##6 If the !overnment &ants to ensure that no traffic flo&s on the street from + to %, &hile eDual amount of traffic flo&s throu!h =unctions 9 and C, then a feasible set of toll char!ed *in rupees at =unctions 9, B, C,and + respectivel( to achieve this !oal is:*##,4,5,5*2#,1,1,5*5#,4,1,2*1$,4,2,5 *4$,4,2,2#7 If the !overnment &ants to ensure that all routes from S to % !et the same amount of traffic, then afeasible set of toll char!ed *in rupees at =unctions 9, B, C, and + respectivel( to achieve this !oal is:*#$,4,2,2*2$,4,1,#*5#,4,5,5*1#,4,5,2 *4#,4,1,2#0 If the !overnment &ants to ensure that the traffic at S !ets evenl( distributed alon! streets from S to 9 from S to B, and from S to +, then a feasible set of toll char!ed *in rupees at =unctions 9, B, C, and +respectivel( to achieve this !oal is:*#$,4,1,#*2$,4,2,2*5#,4,5,5*1#,4,5,2 *4$,1,5,22$ %he !overnment &ants to devise a toll polic( such that the total cost to the commuters per trip isminimi8ed %he polic( should also ensure that not more than 6$ per cent of the total traffic passes throu!h =unction B %he cost incurred b( the commuter travellin! from point S to point % under this polic( &ill be:*l;s6*2;s0*5;sl$*1;sl5*4 ;s #1

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