HOMEWORK FOR CLASS XII 8-9 Show tht the reltion R on the set Z of ll integers defined R,, Z,, is, divisile,, is n equivlene reltion on Z Let f: R R e defined if f if Is f one-one nd onto if If f, g : R R re defined respetivel f, g, find i fog ii gof iii fof iv gog If f, show tht fof= for ll Wht is the inverse of f 6 Let * e inr opertion on Z defined *, for, ll,, Z i Show tht * is oth ommuttive nd ssoitive ii Find the identit element in Z 6 Show tht the opertion * on Q-{}, defined * = + stisfies the ssoitive nd ommuttive lws, find the identit element, for eh element find the inverse 7 Let T e the set of ll tringles in plne with R s reltion in T given R T, T : T T Show tht R is n equivlene reltion 8 If f 7, nd, g 7, R, find, gof 7 9 Is the inr opertion *, defined on N, given *, for, ll,, N ommuttive nd ssoitive? Let * e inr opertion on Q defined * Show tht * is ommuttive s well s ssoitive Also find its identit element, if eists i os e ii tn os sin iii se sin sin Prove tht:- ot,, sin sin 8 Solve for :- tn tn tn Prove tht:- tn tn os 9 Prove tht :- tn tn tn 7 99 os 6 Prove tht:- tn tn os os 7Find the Prinipl vlue of sin 8 Solve: tn tn sin os os
9 Prove tht : 7 tn 7 tn tn Solve for : tn - + tn - = /,, A nd d A then prove tht Find, If :- O If A = tn tn nd I = show tht I + A = os sin sin os A I Use the mtri method to solve the sstem liner equtions 6 Given tht,, 7 B nd A, find AB Use this result to solve the following sstem of liner equtions: 9 Using properties of determinnts find the vlue of If A is squre mtri of order nd A find A 6 7 If A find A Using A,solve the following sstem of equtions 6 8 If A is squre mtri of order nd A find dja A 9 If A is squre mtri of order suh tht dja = 8 find A A mtri A of order hs determinnt Wht is the vlue of dja Using mtri,solve the following sstem of liner equtions
X+-=-, ++=, --= If A is squre mtri of order nd A then find the vlue of dj A nd A dj A If A = [ ] nd B = [ ] find AB Hene solve the sstem of equtions: =, + + = 7 nd + = 7 Disuss the ontinuit of the funtion f given f={ if + if > Find the vlues of & suh tht the funtion defined if F={ + if < < is ontinuous funtion + if Prove tht the funtion f defined f =, R is not differentile t = 6 Differentite sin + sin os wrto 7 If = tn Show tht + + + = 8 If os= os+ with os ± prove tht d = os + d sin 9 If = ost +t sint nd = sint tost, find d Differentite: ot +sin + sin [ +sin sin ] d w r to where < <π/ Find the vlue of nd so tht following if funtions f = if is ontinuous t = 7 if For wht vlue of K the funtion is ontinuous t =, f k,, If = Sinm sin - then prove tht - + m = Snd is pouring from pipe t the rte of m /se the flling snd forms one on the ground in suh w tht the height of the one is lws one- sith of the rdius of the se How fst is the height of the snd-one inresing, when height is m Find the intervl in whih the funtion given f= -6 +6+ is stritl inresing or deresing 6 Find the intervl in whih the funtion given f= - +6+7 is stritl inresing or deresing 7 Find the intervls in whih the funtion f given f=sin + os, π is stritl inresing or stritl deresing 8 Find the slope of the tngent to the urve = -+ t the point whose oordinte is 9 Find the points t whih the tngent to the urve = - -9+7 is prllel to the is Find the eqution of the tngents nd Norml to the urve = - 6 + - + t, Find the slope of the norml to the urve =os θ,=sin θ t θ= π Find the re ounded the prol =,, the is of nd lines = nd =
Find re inluded etween the urves = nd =, Find re of the region : {, + + } Using Integrtion find the re of the tringle ABC whose verties re A,,B, nd C6, 6 Find the re of the region enlosed etween the two irles + = nd + = 7 Sketh the grph of = + nd evlute + d Wht does this integrl 6 represent? 8 Drw rough sketh of the region enlosed etween the irles + = 9 nd + = 9 Using integrtion find the re of the enlosed region 9 Find the re of the irle + = 9 whih is interior to the prol = 6 Solve the following differentil eqution: d log, d 6 Solve the following differentil eqution: d os tn, d 6 Solve the following differentil eqution: d d if = when = 6 Solve the following differentil eqution: d tn sin, giventht t d 6 Solve the following differentil eqution: d d 6 Find the differentil eqution of the fmil of ll irles touhing the -is t the origin 66 Form the differentil eqution of fmil of prols hving verte t the origin nd is long positive -is 67 Find the vlue of λ for whih the vetor = i + j k nd = i + λj k re perpendiulr to eh other 68 If p is unit vetor nd p + p = 8 then find 69 If = i + j k nd = i + j k, find unit vetor in the diretion of 7 Find the ngle etween two vetors &, if =, = nd = 6 7 Find the projetion of + on, where = i j + k, = i + j k nd = i j + k 7 If d nd d Show tht d is prllel to where d & 7 Using vetor find the re of the tringle with verties A,,, B,, 8 And C, 7, 8 7 Find the vlue of λ whih mkes the vetors oplnr, where = i 6j k, = i + j + k nd = 8i j + λk 7 If, nd re unit vetors suh tht + + =, find the vlue of + + 76 Show tht the vetors i j + k, i j k, i j k form the verties of right ngled tringle 77 Find unit vetor perpendiulr to eh of the vetor + nd - where = i + j + k nd = i + j + k
78 Find the re of prllelogrm whose djent sides re given the vetors = i + j + k nd = i j + k 79 Find the distne of the point,, from the plne + = 9 8 Find α when vetors i - j + k & i+ α j + k re i Perpendiulr ii Prllel 8 Find the eqution of line pssing through the point -,, - nd prllel to the 6 line 8 8 Find the eqution of the plne whih ontins line of intersetion of plnes, r i j k r i j k nd whih psses through the Point,,- 7 8 Find the vlue of P so tht the lines nd p re perpendiulr to eh other 7 7 6 p 8 Find the projetion of the vetor iˆ ˆj kˆ onthevetor iˆ ˆj kˆ 8 Find the shortest distne etween the lines nd 7 6 7 86 Find the eqution of the plne pssing through the line of intersetion of the plnes + = nd + + = 8 nd prllel to the line with diretion rtios,, Also find the perpendiulr distne of the point P,, from the plne 87 Find the imge of, 6, in the line = = 88 Show tht the lines r ˆ i 7 ˆj kˆ ˆ i ˆj kˆ nd r 8ˆ i ˆj kˆ 7ˆ i ˆj kˆ Interset Find the point of intersetion 89 Find the eqution of the plne pssing through the point,, nd perpendiulr to eh of the plnes + + = nd + + = 9 Find the vetor eqution of the line pssing through the point,, nd perpendiulr to the two lines: 9 Find the eqution of the plne tht psses through three points,,, 6,,,,, 9 Find the ngle etween the plnes whose vetor equtions re nd 9A lol television network is fed with the following prolem It hs een found tht progrmme A with minutes of musi nd minute of dvertisement drws viewers while progrmme B with minutes of musi nd minute of dvertisements drws 7 viewers Within one week the dvertiser visits tht t lest 7 minutes e devoted to his dvertisement nd TV network n fford not more thn 9 minutes of musi How mn times/weeks should eh progrmme e given in order to otin the mimum numer of viewers? Formulte the prolem s liner progrmming Solve it grphi method 9 A deler in rurl re wishes to purhse numer of sewing mhines He hs onl `,76 to invest nd hs spe for t most items An eletroni sewing mhine osts
him `6 nd mnull operted sewing mhine ` He n sell n Eletroni Sewing Mhine t profit of ` nd mnull operted sewing mhine t profit of `8 Assuming tht he n sell ll the items tht he n u how should he invest his mone in order to mimie his profit Mke it s liner progrmming prolem nd solve it grphill 9A to ompn mnuftures two tpes of dolls A nd B Mrket tests shows tht the omined prodution level should not eeed dolls per week nd the demnd of dolls B is t most hlf of dolls A Further the prodution level of tpe A n eeed times the prodution of tpe B t most 6 units If the ompn mkes profit of Rs nd Rs6 per doll respetivel on dolls A nd B How mn of eh should e produed weekl in order to mimie the profit? 9A ftor owner purhsed two tpes of mhines, eletroni nd mnull operted requirements nd the limittions for the mhines re s follows: Are oupied the Lor fore on eh Dil mhine mhine output Eletroni Sqm men 6 Mnull Sqm 8 men He hs mimum re of 8 m ville, nd 88 skilled lors who n operte oth the mhines of eh tpe should he u to mimie the dil output? Keeping unemploment in mind justif the vlues to e promoted for seletion of the mnull operted mhine 9 An ero plne n rr mimum of pssengers A profit of Rs is mde on eh first lss tiket nd profit of Rs is mde on eh seond lss tiket The irline reserves t lest sets for first lss However t lest four times s mn pssengers prefer to trvel seond lss thn first lss Determine how mn tikets of eh tpe must e sold to mimie profit for the irline Form n LPP nd solve it grphill 96 Given two independent events A & B suh tht PA= & PB=6 find i P A &B PA or B iii PA nd not B iv Pneither A nor B 97 If fmil hs two hildren Wht is the onditionl proilit tht oth re girls, if given tht i The oungest is girl ii At lest one is girl 98 There re workers in ftor who re prtiipting in strike for inrese in wges Out of them workers re sinere to mngement nd elieve tht mngement know eh thing nd do the needful Mngement selets workers t rndom out of them Write the proilit distriution for the seleted worker who is sinere Also find the men of the distriution Write the importnt of sinerit in servie 99 In g A there re red lls nd white lls nd in g B there re red lls nd white lls If ll is drwn from one of these two gs nd found to e red, find the proilit tht it is drwn from g A A nd B toss oin lterntel till one of them gets hed nd wins the gme If A strts the gme, find their respetive proilities of winning If fir oin is tossed times, find the proilit of Etl si heds At lest si heds At most si heds Evlute: Evlute: d os os + d Evlute: e d Evlute: +log +log d
6 Evlute: 7 Evlute: + d ++ 8 Evlute: e sin + os d 9 Evlute: e log + d Evlute: d, using integrl s limit of sum d +log +log Evlute: + ++ d d π eos π e os +e os π os + sin Evlute: + Evlute: + d, using integrl s limit of sum Evlute: Evlute: log + tn 6 Evlute: d, using the propert of definite integrl d, using the propert of definite integrl d, using the propert of definite integrl 7Evlute: 6 d se 8 Evlute d tn 9 Evlute d / sin Evlute d sin os