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1 SML QUSTION Clss XII Mthemtis Time llowed: hrs Mimum Mrks: Generl Instrutions: i ll questions re ompulsor ii The question pper onsists of 6 questions divided into three Setions, B nd C iii Question No to 6 in Setion re er Short nswer Tpe Questions rring one mrk eh iv Question No 7 to 9 in Setion Bre Long nswer I Tpe Questions rring four mrks eh v Question No to 6 in Setion Cre Long nswer II Tpe Questions rring si mrks eh vii Use of lultor is not permitted You m sk for logrithmi tles, if required SCTION - Find the ngle etween the lines z nd 6 z Find if the vetors i j k, i j k, j k re oplnr If m nd n re the order nd degree respetivel of the differentil eqution then write the vlue of m + n Find the differentil eqution representing the fmil of urves ritrr onstnts e nd i j k d d, Be d d where nd B re Find unit vetors perpendiulr to the vetors i j k 6 On epnding first row, the vlue of third order determinnt is Write the epression for its vlue on epnding nd olumn, where ij is the o-ftor of element ij SCTION - B 7 To promote the mking of toilets for women n orgnistion tried to generte wreness through i House s, ii s iii s The numer of ttempts mde in three villges X, Y, Z re given elow House Clls Letters nnounements X Y Z 7 Find the totl ost inurred the orgnistion for the three villges seprtel ug mtries Write one vlue generted the orgnistion in the soiet Ug properties of determinnts, prove tht: 9 vlute: d ot ~ Let,, 6 e point in spe nd Q e point on the line, r i j k i j k find the vlue of for whih vetor Q is prllel to the plne z O Find the distne of the point -,, - from the line plne z z z, then mesured prllel to the Show tht the lines nd z interset Find the point of intersetion lso os, prove tht os Sin If os O

2 rove: tn os vlute:- d vlute: vlute: Sin n d Find the inverse of mtri If tn os d O, ug elementr row trnsformtions O, Find f if f 6 If, prove tht: 7 Show tht e e d d d d d d is solution of differentil eqution Two die re thrown simultneousl Let X denote the numer of Sies, find the proilit distriution of X lso find men nd vrine of X ug proilit distriution tle O mn tkes step forwrd with proilit & kwrd with proilit 6 Find the proilit tht t the end of steps, he is one step w from the strting point 9 Show tht funtion f is ontinuous ut not differentile t SCTION - C Let X e non-empt set X e its power set Let * e n opertion defined on elements of X * B B, B X Then, i rove tht * is inr opertion in X ii Is * ommuttive? iii Is * ssoitive? iv Find the identit element in X wrt * v Find ll the invertile elements of X vi If o is nother inr opertion defined on X s o B B, then verif tht o distriutes itself over * O Let If,,,9 nd e the reltion in defined,, d if + d + for,,, d If rove tht is n equivlene reltion lso otin the equivlene lss [, ] Find the solute mimum nd solute minimum vlues of the funtion f given f Cos Sin,, furniture firm mnuftures hirs nd tles, eh requiring the use of three mhines, B nd C rodution of one hir requires hours on mhine, hour on mhine B nd hour on mhine C h tle requires hour eh on mhine nd B nd hours on mhine C The profit otined selling one hir is s while selling one tle the profit is s 6 The totl time ville per week on mhine is 7 hours, on mhine B is hours nd on mhine C is 9 hours How mn hirs nd tles should e mde per week so s to mimize profit? Formulte the prolem s L nd solve it grphill info@newprmeterom

3 given quntit of metl is to e st into hlf irulr linder ie with retngulr se nd semiirulr ends Show tht in order tht the totl surfe re m e minimum, the rtio of the length of the linder to the dimeter of its irulr ends is : letter is known s hve ome from either TTNG or CLCUTT On the envelope just two onseutive letters T re visile Wht is the proilit tht the letter hs ome from i TT NG ii CLCUTT Sketh the grph of: vlute: f,, f d Wht does the vlue of this integrl represent on the grph? 6 Solve the following differentil eqution, given when Find the prtiulr solution of the differentil eqution e, given tht, when d d O d d info@newprmeterom

4 SML Solutions qution of lines 6 z nd z ngle etween the lines Cos 6 6 Cos Cos Cos Cos ; So, 9 Let k j k nd j i k j i, for oplnr k j i 6 k j i k j i 7 nd 7 k j k j i ; ; 7 d d d d Squring oth sides

5 d d d d d d d d d Order, m, degree n So, m + n + m + n Given e Be d Differentiting twie wrt, we get d e Be nd d d d d e Be d d d d d e Be d d d d d d d d d d i j k nd i j & i j k Let ug ie, whih is the required differentil eqution Let k etor perpendiulr to oth i j k i k i k i k Unit vetor i k 6 On epnding nd Column Let 7 7, B Totl ost the orgnistion for eh villge is represented olumn mtri X C Y, Here C B Z X Y 7 Z 9 There orgnistion is helping women in the soiet info@newprmeterom

6 We hve Multipling,,,, respetivel nd tking out,, ommon from C, C nd C, we hve Now ppling C C C nd C C C, we hve B tking out ++ ommon oth from C nd C we hve ppling Now, epnding long C, we hve H S 9 Let I ot d tn d tn d tn tn ot tn n e written s d tn tn tn tn d tn d tn d tn [ ] d f f d tn d tn d ] tn d [log info@newprmeterom

7 log log log [ log ] r i j k i j k,, Line Coordintes of point Q on the line Given point,, 6 D s of Q,, 6,, Sine Q is to the plne + z Norml of given plne i j k s Q 6 ; Co-ordintes of Q 7, -, - O qution of plne: z So norml to the plne will nd Be, i j k qution of line n z,, i j i 9 j k rile point Q on the line etor Q k Sine Q lne Q 9 i j k i j k 6 ; So Co-ordintes of Q,, Distne Q Q,, 6 6 Distne Q units 9 7 For the lines to interset shortest distne etween them must e zero l z ; : l 7 z info@newprmeterom

8 i j k, i j k, i j i j i j k i j k k k i j k ; i j k Distne d Sine the shortest distne is zero for point of intersetion Let z z nd z z On ompring nd On solving &, So point of intersetion -, -, - B Let Cos nd Cos B, it gives Cos, CosB So Cos Cos +B Cos Cos B Sin Sin B Cos Cos Squring oth sides Cos Cos Cos Cos Cos Cos Cos Sin Let Cos O info@newprmeterom

9 So, Cos LHS tn tn tn tn tn tn tn tn tn tn B tn tn tn tn tn tn tn tn tn tn tn tn Cos ; ; ; tn tn d Dividing Numertor nd Denomintor d d d ; d d v u where v dv du & log tn v v C log tn O d Let B d d B B On ompring

10 B +B; d d d d dt t d t C Sin C Sin 9 Here for for f Therefore d d d d d Integrting oth integrls on right hnd side, we get os os d Let I then, Operte Operte Operte

11 Operte Operte Operte Operte I B, where B Hene, - B 6 Tking log on oth sides Log log log log log

12 7 d d d d d d d d e e d e e d d d d e e d d d d d [{ e e d d d d d d Clerl, X n tke vlues, nd We hve, X roilit of not getting si on n die 6 X roilit of getting one si 6 X roilit of getting two sies 6 Thus, the proilit distriution of X is given X: X: X i i X i i i i i p i i 6 p i i 6 We hve, 7 p i i nd pi i 6 p i i p i i pi i X nd, r X 7 9 info@newprmeterom

13 9 Hene, X nd r X O The mn is one step w from the strting point fter steps This n hppen in two ws i He tkes steps forwrd nd steps kwrd ii He tkes steps kwrd nd steps forwrd roilit of first se C 6 C roilit of seond se Hene totl required proilit C 6 C if f if if if f lim f lim nd Here lim f lim Sine, lim f lim f f info@newprmeterom The funtion if ontinuous t for differentiilit test LHD f HD f f h f h h lim lim lim h h h h h k f h f h lim lim h h h h Sine, LHD + HD f is not differentile t Given * B B, B X X B X, B X i Sine,, B Hene, *B is inr opertion ii * B B B B*, B X So, * is ommuttive iii * B* C B* C B C B C B* C * B* C, B, C X iv v So, * is ssoitive Let e the identit element in X with respet to * Then * * For ll X For ll X, X X Thus, X is the identit element with respet to * on X Let e n invertile element of X nd let S e its inverse Then, *S X S* S X S X, S S X [ X ]

14 vi Thus, X is the onl invertile element of X with respet to * nd it is the inverse of itself We hve, the reltion * B B,, B X o B B,, B X Now, o B*C B C B C o B * o C i lso, B*C o B C B C B o *C o ii Hene, from i nd ii, we onlude tht o distriutes over * f Cos Sin, f f Cos Sin Cos Cos Sin For ritil points, f Sin Cos Cos, Sin, nd, 6 6 ll points re,,,, 6 6 f Cos Sin f Cos 6 f Cos Sin 6 6 Sin f Cos Sin f Cos Sin Hene, solute mimum solute minimum Let hirs nd tles e mde per week Now, ording to question we hve Mhines B C Chirs Tles Here, totl time ville per week on mhine is 7 hours 7 Tht of mhine B is hours nd mhine C is ville for 9 hours 9 Its profit, Z + 6 So, the L is Z m + 6 Sujet to the onstrints 7 i ii info@newprmeterom

info@newprmeterom 9---77-9 iii, iv On plotting i to iv, nd shding ording to onstrints we hve Here, OBCD is the required fesile region show in

firm mde hirs nd tles per week given quntit of metl is st into hlf linder with retngulr se nd semiirulr ends Let the length of retngulr se nd its

15 iii, iv On plotting i to iv, nd shding ording to onstrints we hve Here, OBCD is the required fesile region show in figure whih is ounded Now, we find the vlue of Z t eh orner point Corner point Z, B, C, D, 9 So, the furniture firm hs mimum profit of s 9, if the firm mde hirs nd tles per week given quntit of metl is st into hlf linder with retngulr se nd semiirulr ends Let the length of retngulr se nd its redth Dimeter of the semiirle dius Let the volume of metl, whih is known nd is onstnt i To minimise the totl surfe re, S, we hve S d ds d ds d ds d S d

16 When d S d Totl surfe re, S, is minimum When v nd ] [ i from : : tio of length of the linder to its dimeter : + Let e the event tht the letter me from Clutt nd e the event tht the letter me from Ttngr Let denote the event tht two onseutive letters visile on the envelope re T Sine the letters hve ome either from Clutt or Ttngr, therefore, If hs ourred, then it mens tht the letter me from Clutt In the word CLCUTT there re 7 onseutive lphet ie, {C, L, LC, CU, UT, TT nd T} nd T ours onl one time Therefore, 7 If hs ourred, then the letter me from TTNG In the word TTNG there re onseutive letters ie, {T, T, T, N, N, G, G, } in whih T ours twie Therefore, B Be s theorem: i 7 7 ii 7 7 Given,,, f For f, X F For, f X Y 7 The grph of the funtion is shown in the figure

Now, f d f d f d d d d 6 6 6 6 sq units info@newprmeterom 9---77-7 The ove vlue of the integrl represent the re of the shded region on the

17 Now, f d f d f d d d d sq units info@newprmeterom The ove vlue of the integrl represent the re of the shded region on the grph d d 6, given, d d It is liner eqution of the form Here, d Q d, Q Now, IF e Its Solution is d e I F Q I F d d d d d - d d log log e e e d d

18 C C Now, when, we hve C equired solution is

are coplanar. ˆ ˆ ˆ and iˆ

are coplanar. ˆ ˆ ˆ and iˆ SMLE QUESTION ER Clss XII Mthemtis Time llowed: hrs Mimum Mrks: Generl Instrutions: i ll questions re ompulsor. ii The question pper onsists of 6 questions divided into three Setions, B nd C. iii Question