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ISC Prelims 2009 : Mathematics (Cathedral & John Connon School, Mumbai)

6 pages, 40 questions, 10 questions with responses, 10 total responses
Indian Institute of Management (IIM), Bangalore
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Downloaded from | Be a part of the growing student community online D ate:l2 Jan 2009 THE CATHEDRAL & JOHN CONNON SCHOOL Preliminarv Examination 2008-09 CLASS XII {nI'l[arks: 100 T ime : 3 h p u r s MATHEMATICS Section A: Answer Question I (compulsory) and FIVE other questions. Section B & Section C: Answer any TWO questionsfrom eilher Section B or Section C. AII essential working must he clearly shown on the same sheet as the rest of the answer. SectionA (Answer Question I and ony other FIVE questions) Questionl: I l0 x 3 :30 marksI (a) [-r -tanel f r showthatl lx | ftanO (b) lf | _l I ,urrol lcos2e -sin2ol | -l I l_-tanO lJ !in20 cosr0 | rt for a, I e, +. . I is a Booleanalgebra all the elements b, c of B, TJ provethat(a + b) (b'+ c) + b(a'+ c') : ab'* ac + b. (c) Find the equationof the tangentto the parabola!' : l6x which is pcrpendicular to the line 3r- 4y + 5: 0. Also find the point ol'contact. (d) In a singlethrow of two dice, what is the probability of getting a total o1' at most 9 ? (e) Determine so that the line y:2x + I may touchthe ellipse3x'+ 5y: l" (0 Solvethe dilfcrentialequatio ", (g) # "r** 15. : e'-v Find the angle betweenthe lines and the equation of the bisectorsof the angle betweenthe pair of lines given by 2x2* 7xy + 3 )F:0. Continued on page 2

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Downloaded from | Be a part of the growing student community online ( h) Findfo when y:logltt * c o s e c 2 x ] dx ( i) E valuate 0) E valuate: (tan2xs e c z x ) t *tan6 x dx n/4 tog(1+ t a n e )d / 0 f J 0 Answer any FIW questions [5 x I0: 50 marksl Question 2 : (a) Using Cramer'srule, solve the following systemof equations: t5l x-y:l x+z:-6 x I !-22:3 (b) Vcrifu L.M.V. theorem forthe functionf(r): x (x - l) (x -2) t/, . in the interval 0< x < tsl Question3 : (a) Find the equationsof the pair of straight lines represented by 8x'+ 8 xy+2y+26 x + l3 y + 1 5 : 0 andshow that the angle betweenthem is zero. t5l tsl Question 4 : (a) Write the Booleanexpression corresponding the following to switching circuit network. Use the laws of BooleanAlgebra to simplify the circuit. Constructthe network lor the simplified circuit. l 5l (4 /" -r 1u' continued on page 3 --trJ1J'.

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Downloaded from | Be a part of the growing student community online ( b) -r -r Solvetheequation:sin 6x * sin 1Orl3 x; : - nl2 tsl Question5 : (a) Find the probabilityof drawinga onenrpeecoin from a pursewith two one compartments, of which contains3 fifty paisecoins and2 one rupee coins and the other contains2 fifty paisecoins and 3 one rupeecoins? (b) t5l A certainteamwins with probability 0.7, loseswith probability0.2 and Find the probability ties with probability 0.l. The team plays threegames. 15l that the team wins at leasttwo of the gamcs,but not lose. Question 6 : (a) of A oylinderis suchthat the sum of its height and the circumference volume of the cylinder. its baseis l0 metres.Find the greatest (b) tsl Evaluate: f log 1v + I (a2+ x2 )l dx l5l I Qu e stion7: (a) betweenthe two curvesl' Find the areaenclosed (b) :9x andx' : 9y t5l prove that of [Jsing properties dcterminants, lxx2 :( r - f) ' xtlx t5l xx'I Question 8 : (a) formula to obtain the correlation coefficient Use Karl Pearson's t5l for the followins data: X 16 l8 21 20 22 26 27 l5 Y 22 25 24 26 25 30 JJ l4 Continued on page 4

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Downloaded from | Be a part of the growing student community online ( b) The lollowing is the distributionof 100 membersof Parliament. Find the standard deviation. 30-40 Age No. of members t J I5l 40-50 50-60 60-70 51 25 5 70-80 80-90 9 Question 9 : (a) Find the locusof the complexnumber z.: x t iy satisfying relation thc lz-zi r l ' .+' -; Illustratethe locus of z in the Argand plane. (b) t5l Solvethe diffbrcntialequation: + y) dy+ (x*y) dx - 0 giventhar (x y: lwhenx: l. 15] Section B Answerany TWOquestions 1 2x l0 : 2 0 ma rk s l Que stionl0: (a) Find the equation thc planeperpendicular the planes + y +22:3 y of to and 3x + 2y + z: 4 andpasscs through point(1,2,3). the (b) f 5] Find the equations the perpendicular of from the point (3, -1, 1 l) to the line x yQ z-3 234 Also find the foot of the perpendicular and the lengthof the perpendicular. t5l Question I l : (a) (b) nnn Findthe sineofthe anglebefweenthe vectors 2 i -j + 3 k ancl nnnt5l i +3j+2k. If A, B, C havepositionvectors I, l), (3, l, 5) and (0, 3, 3) (0, respectively,then show that A ABC is right angled at C. I5l continued on page 5 4

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Downloaded from | Be a part of the growing student community online Q uestion12 : (a) A factory has three machinesA, B and C producing 1000, 2000 and defectivebolts, B A 3000 bolts per day respectively. produces1olo produces l.5o/o defectivebolts and C produces2o/odefectivebolts. What is the At the end of a day, abolt is chosenand found def-ective. tsl probability it comesfrom machineA ? (b) In a binomial distribution,the sum of the meanand varianceis 42 I5l and their product is 360. Determinethe distribution' SectionC AnsweranYTllO questions = [2 x 10 20 marksl Question13: 'l'he differencebetweenthe true and Banker'sdiscounton a bill (a) interestper annumis Rs.10.Find due after 3 monthsat 5%o (i) (ii) the Banker'sdiscount,and (iii) (b) the truc discount the facevalueof the bill' Is] 000 for a numberof yearsis The amountof an annuity of Rs.1 ten. Find the numberof yearsthe Rs.44500correctto the nearest annuity is to go if the rate of compoundinterestis 5% per annum. tsl Que stion14: (a) Price Relative Given the following data and usingweightedaverage method,constructindex number for 2004basedon 1994prices. t{l Goods Weight 1994Price(Rs) 2004hice (Rs) Wheat 40 100 200 Rice 30 2A0 800 Milk l5 0.s0 4 Ghee l0 8 40 Sugar 5 I 6 Continued on Page 6

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Downloaded from | Be a part of the growing student community online (b) The number of traffic offencescommitted in a certain city over a period of 3 yearsis given in the table: Year Jan-Mar Apr-Jun Jul-Sept Oct-Dec 1980 74 56 48 69 1981 83 52 49 81 t982 94 60 48 79 Calculatelbur quarterlymoving average and illustratethcseand the original figures on the samegraphusingthe sameaxesfbr both. t6l Question15: (a) Suppose manufacturer sellx itemsa weekat a priccp -.20 - 0.001x a can Rupeeseachwhen it costsy -. 5x +2000 Rupeesto procluce items. x Determinethe numberof itemshe should produceper week fbr Maximum profit. (b) t4l Maximize Z: - x + 2y subjectto the constraints *x + 3 Y < 1 0 x+ y <6 x- <2 x)0 , v y>0. End rt'question paner 6 t6l


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