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CBSE Class XII 2014 : MATHEMATICS

8 pages, 72 questions, 13 questions with responses, 13 total responses,

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:;l*, Series : OSR/I -)-i +fi=i. qtq{rqT st's st rR if,qrqfrnil Roll No. 651u1 rm-gk-*r *.5qr Candidates must write the Code on the title page of the answer-book. o !ffir cirqe.cdtogs qlcr-rr+ gtrd ysS t o trrr-rlrr { qrk+ aq ftt ett iqq rcqis;wc +} on wr-gfisr*r *. g*-U*qc flmri o !ffir qtq s{ d t*'ge nQEr-rr il zq ,rfi t o 1FttrttltFTqrvmftrstvJs,a,T+*qrd, vrqq, fiqiq,^ wq{qffi o {s qrr-Er q,i q6i + tdq 15 ftffie sT {rrrq flEqr .r.n t r vra-qx er ka<or Y3Rq it to. ts qi fuqr qT+rTr I 10.15 E-Q t 10.30 qQ ,ro om il-+er }rq';r-Er qtn e+{- es erqFT + d{lT a r r r r etn-giwsT qr *tt srr ffin r "fi o Please check that this question paper contains 8 printed pages. o o o o Code number given on the right hand side of the question paper should be written on the title page of the answer-book by the candidate. Please check that this question paper contains 29 questions. Please write down the Serial Number of the question before attempting it. 15 minutes time has been allotted to read this question paper. The question paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question paper only and will not write any answer on the answer-book during this period. MATHEMATICS funftd Wpt : 3 qo+j Timeallowed:3hours felfeffiaq eiq 1 : 100 t MaximumMarks:l00 IttqTql;Itrt; (0 wfi*efiladf t (ii) wsnr-w*zg yfitql fqwslf (iii) (iv) (v) 6snn fuittrdff; of qwT tr / qry-rrif to s{a#Fnc CrF+a- gaaio+r & I q'ea iitz yw iFfiC ps# Erreiaw * r qrc silz sw SFil-+c'6sdar' dawt r rrJwTr&gqr fi} sre,rr# wfrvrrt#mr Ww-q qtqTqq asrqrqw fr fu t fvr rt? w od ard t rr'+ # Trqr d'3H'a7i+ 2 wit etafufqq@ff I qH w?vnitf Cailqmi Wd\fuw acr* r Zof yw-q* it ildr'd t *m*de{ *yqhTa? erlqfu %?* tqfr era?qffdraldl eilqdgtTUrqTq T{MaTsot Sr 1 [P.T.O. General Instructions : (i) (ii) All (iii) All questions in Section - A are to be answered in one word, one questions are compulsory. The question paper consists of 29 questions divided into three sections A, B and C. Section - A comprises of 10 questions of one mark each, Section * B comprises of 72 questions of four mnrks each and Section - C comprises of 7 questions of six marks each. sentence or a.s per the exact requirement of the question. (iv) There is no overall choice. However, internal choice has been provided in 4 questions of four marlcs each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. You may askfor logarithmic tables, if required. Etug- st SECTION w{ {iqr 1 + 10 rordq;sqr f eioer t Question numbers 1 to 10 carry 1. I -A r mark each. rtrfi qr ko{qr it *'ngqq it, qmr * qotrem*rt {hqr t qi H-t a, b e R - {0} *ftrqu *U=*rnrwf,t tfr2 x (x* 5) = 10t, nirsTq-{wcdfrnq {-fr r Let * be a binary operation, on the set of all non-zero real numbers, given by a * b = + for all 2. qk a, beR / sin Gtr-' - t0l. Find the value of x, given that} *. (x * 5) = 10. f * .or-t ,) = , t, n), irr qFr am otlqq r If sin (sin-'f * .or-'*)= ,, then find the vatue of x. 3 *,[ l 1 ].[ ; I ]=[ ; ! ]nd(,-y)srqnff6eiruq t-3 or_[1yl_[7 0t Ir2L s, l' L o r l-L ro, J,Rno(x-v)' 4. fqq sTrq'6qdr+.rur +r x *-flaqra +Tqq ,, t, rr Solve the following maffix equarion forx:, k [ _; lf [ o 3 .] = _1, 3 ]= " s *I? ;l=I; ? l*Hrr*rrrnfttus If 6. l2x 5l l6 -zl I S ,l=1, , l,writethevalueofx. (r^f .fJ*sfr- ffisrqftfiqq Write the antiderivative " (rrE . r 1f,) 3 i. qFTnkTqikq ,l# 0 3 (dx - Evaluate, )ffi 0 8. fq{r i + tj + zt or efu1 2i - 3i * of q{yqiq rm dfeq Find the projection of the vector g. qi( A nqr B qi tt qlrqrrrR{r tn +1&q i + :i + r 7t on the vector Zi -$ + et. tfu a + B qt w'qffs'nflE{r t, H]' ? aqr d +.+q er *tur r If 7 and B *. angle between 10. g-{T JAAA qrkrm ? .ti +i *r two unit vectors such that ? ? + B is also a unit vector, then find the ana B. efEvr Hrflq.{ur ma + ii) = 2 *'uqient *Tqq qt frE (a, b, c) t t't*-r qror t f,erT qrrtrm r Write the vector equation of the plane, passing through the point (a, b, c) and parallel to the plane 6snn ? fi +j +ty = z. 3 IP.T.O. Etug:- E SECTION. B t 22 nq'r&oyqq a eis.or t Question numbers ll ta 22 carry 4 marks each. 11. qHr A - {1, 2, 3,.....,9} Hqr A x A it nf'o{Giq t, * R x A il 1a, b), (c, d) *.ftrq (a, b) R (c, d) qfr a + d = b + cERr qfuflisd t firc*Torqfu pqs ${rf,r {dq t I Wrrf,r srq {qr +t [(z, s)] qt rno *1Hq 11 r r r Let A = {1, 2, 3,....., 9} and R be the relation in A x A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A x A. Prove that R is an equivalence relation. Also obtain the equivalence class l(2,5)J. 12. flsq*llhqfr'corl =l; *. (0,f;) oIerEr k +1FKr6 2 'urrt(}) *,".-, ffi + 2 ,un-t(,l = f; =f;*. Prove that2"r-' (}) 13. {Rfri$f +Wqt mr + sec-l (9 + (o,f) 2 ,urrt(,,) = f rdrr $!frftr *1iqqfu 2y y-z-x 2y 2z 2z z- x-y =(x*y+D3 x-y -z 2x 2x Using properties of determinants, prove that 2y 2z x-y -z 14. ,ur-, 1"/rJ\ [*Jsr Differentiate tan-r 65lllt y -z-x 2y 2z z-x-y =(x*y+43 cos-r (9 (zr'[G) +qriqT ifl?FrercT with respect to cos-l 4 f,rd dfrq, qqflo,r * (zrt[G), when x * o 0. t r ls. qfr y = t' t, dtuqft1r$Etu#-i ffi1f r, prove ,nurffi-i rry - (*I-f = o. = o. 16. Tf ,iffinm nm *lPsrq kqif q'em f(x) = 3f - 4f * t2x2 + 5 (a) frtd{*iqmt (b) ftt-nr 6mq1q $ I 1 Oter tf E[!F-x AN = a sin30 iTe[ y = a cos30 +"ftT+ e = f, EFtlcrq wwt *.v*fiq flf, I Find the intervals in which the function f(x) = 3# strictlyincreasing (a) (b) ter nqt qfii?iq - 4f - l2i + 5 is strictlydecreasing OR Find the equations of the tangent and normal to the curve x = a sin30 and y = a cos30 at -|t =7. + 0 t7 . q-{ nrn qfri'q, u, [+F4 Srn'x cos'x J . ENf,i[T crt f,rd drmq Evaluate, , I a- :;fi2 a 3, - 1s 6, [#4* Sln-,r. cos--r J OR f Evaluate , ) A- 3n[r[3* - 1g 66 18. i?rrrirwts{q x=0t diF * *Idy = 0 q;rfrlvre6fif,rfre1Pqq, tflrtfuy = 1qe t Find the particular solution of the differential equation e-{1 -tz * *!*dy = 0, given thaty= l whenx=0. 65nfl 5 IP.T.O. Z 19. frq er++m {+fiur +} m sltw : rf - r>ff.r*=h. Solve the following differential equation : tl-rr***=*. zo. ffi+{HGaii ?, B, ?+feqnsm+itqqfu' [? * B, B * ?, ? * ?] =zlid, ?] STTET ntgr ?, B aqr d t+ tt*. ? * B + ? = d aqr I ? I = 3, I B I = 5 iqt I ? I = z t r ? aqr d *.etq or otor arc qifqq I Prove that, for any three vectors ?, B, a [? * B, B * ?, ? * ?] =zlid', ?] OR vectors r-9 d, b and ? are such that ?*B+?=d ana l?l = 3, ldl = 5 ana l?l = z. Find the angle between ?anA B. 2l- x-2 v-4 2-6 ^ ' Htgqtrt{srt+=I*] . 3 s -L+5 - flaur':,=?=?qicr E*-mrirs;rcrt nftrdqqfr5$ura*1tqq Show that the rines $ r = # =+ *d+ =+ =+intersect. Also find their point of intersection. 22. qm aT fl*'qe+ ti qrd qd mr irgs, qr m-g'st 6hr qq{irrTq t r qkffi qkqr+ d) +{i wit +re-+i Et+ st qqftraiq qrksf,r rn *1kq, qfr.r fqqr rrqr tfu' (i) qe+ sta wqr rrg+t (ii) oq * s'rT t Ei wn t, r gs' ilEqr cwt t r Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls ? Given that (i) (ii) 65nlt the youngest is a girl. atleast one is a girl. 6 EtEg-v SECTION qrt fur 23 + 29 rordq.sw o eioe.r t - C r Question numbers 23 to 29 carry imarks each. 23' ffiii kflmq P den *t+ g+ gq *t erlvne-+, lvreilr inr v,rq q.r qrcfr Eti + T t 9 q{ 5{sR +{r qE+ r fufleq P sTq+ xq1. i, 2 iTen r +t er dr {. i +ftrq Hrr{T' { x,7y esr t z**qmntwh.gqry,t6Tsdrq t 1,000 t rkaroqa s{q+ !F-rr{r. 4, t ak 3 q{ *t Tiqi + tiTq W 1,500 g{rftr{ rqtFq tm qrear Cdqr rd:d f+srim q+ fi ilq Ti+ *. mt gcs* odn *r U- r qn q{ d+ faqt T{ tqq rrq \rs'-qs'gtffiT +t gm "uufrr r ooo d) oTrEFf mr rdrr s.r+ q+-s qF +-ik+ qt r* grw,rcrflvrwo*ikq q) t ffiii ffiqt i t t r 3'Rlffi fr Tiqi + eTfdFffi g;n-e+q qw WWcil grmrti.*fuqwftm o1qr qrf6q r Two schools P and Q want to award their selected students on the values of Discipline, Politeness andPunctuality. The school Pwants to award txeach, {y each and ( zeach for the three respective values to its 3,2 and 1 students with a total award money of < 1,000. school e wants to spend < 1,500 to award its 4, I and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is t 600, using matrices, find the award money for each value. Apart from the above three values, suggest one more varue for awards. 24. qfllEq fu * gi fr{q. ffi eir slFtftm eirrrtr{ ed yrgi sr 'G6kna ilfuftd. +iur cos -rl.s I show that the semi-vertical angre of the cone of the maximum volume and of given slant height zs. qH Ern i, ,or-trfu. e _ ) l+{cot"r sitEq | .fdx I trviuuate. ) I +{cotx El6 - p 26' urrvqqf${gn * y2 =32, tqr y =x\rd.r-Brtr+frt txor qiilffi ;noqiHq r Find the area of the region in the first quadrant enclosed by the x-axis, the line y x = and the circle * + y2 - 32. 65mt lP.T.O. 27. ft*Ssif A(2,5, -3),8(-2,-3, Enaqifqq 5) dqr c(5, 3, -3) Enr frffi|frT qrrrm +t eS (7,2, 4) t d r sTq.ff (-t, -s, -10)*tqr ? =2i-i +zt+ tqfrdfi k' +dq dr Et fliT *1fqq fug )" (31+ +i + zfl)aqtv,krm ?' tl -i + t) = s I Find the distance between the point (7 , 2,4) and the plane determined by the points A(2,5, *3),8(-2, -3, 5) and C(5, 3, -3). OR Find the distance of the point (*1, -5, -10) from the point of intersection of the line ? = 21,- j + zt 28. + ). (3i + 4 +zt) ano the plane ?' (i - j + t) = s. fu *r qm'qqrqt p firort qd+ rqfuT qroar t I e-s+ qrs ft+vr + frq affim ( 5,760 tnqr *snur +H qftr*.t 3lftrfi20 {rrl +ffi qrrt t l\1*.{*qrks-fisdr$ qrqlur qyfr{ ftT {iq < 360 {tdk*.qqfrq *t ErFr rr-r t, qqfs qrs-6rer t qdr} qrdr wfr{ HT Wq < 240 t < zz apr wdq fu' qa rrfr+ Tn q,ft q'r iq r qo q+' rro.n taqr 6rq Tiri etd q{fr{ fti t 18 HFT q{ I z16 v*m t, qe nlvr m Fr&r tqtq vmn et to Yd glftq-ffi t ertr d ? Bqiffi *l qo\k*'utffrqr qrrFIT slfs-tuTrErtr 6 +1ka r A dealer in rural area wishes to purchase a number of sewing machines. He has only ? 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him { 360 and a manually operated sewing machine < 240. He can sell an electronic sewing machine at a profit of 7 22 and a manually operated sewing machine at a profit of 18. Assuming that he can selI all the items that he can buy, how should he invest his money in order to maximize his profit ? Make it as a LPP and solve it graphically. ( 29. t qs rdl d qror r qiq Yni tt t (tr q+ Fffid =n+ t Fr{gqT qrksilr t t qftr nqqT {feq qi erfr Egt +qr+ qT+ r d.rq T+ * g5u +E}+ e1 wr HRr + 52 qdf *1 qo'r$ ii t t EIEFil q t, q-f-w'*-rh qlil nqqr qt6fr + rcif +r qo yffi frsTmT rrqr I Zflz{ui qwtdr liqr *T Hk*f,T da rna qikq I wfr: de{ sT qtzl FIIT 1s ffiif *'r'+ if, t *, ffii qiriqq s Zftqui r A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade. OR From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution. 6st1ft

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