CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find

Transcription

1 BSE SMLE ER SOLUTONS LSS-X MTHS SET- BSE SETON Gv tht d W d to fd 7 7 Hc, Lt, W ow tht Thus, osd th vcto quto of th pl z - + z = - + z = Thus th ts quto of th pl s - + z = Lt d th dstc tw th pot,, - to th pl

2 BSE Thus, d cz c d d d d d uts 7 Gv tht of = s Susttut = d = Thus, = s = s osd th quto, =, wh s th pt Thus, th ov quto psts th fl of ls whch pss though th og = Dffttg th ov quto wth spct to, = d d d d d [ fo quto ] d d d Thus w hv ltd th tt, Th qud dfftl quto s d d osd th gv dfftl quto: d log log d Dvdg th ov quto log, w hv,

3 BSE log log d d log log log d d log osd th gl l dfftl quto d Q wh d Q fuctos of d opg quto d th gl quto, w hv, d Q log Th tgtg fcto s gv th foul d Thus, d osd log Susttut g Thus Hc, F dt t d F d log t; dt log t loglog d log d log loglog log SETON-B 7

4 BSE osd - - Now -- = -= = - ostultpl - -= -

5 BSE OR =----- =-- = Hc - sts - = R R pplg

6 BSE R R R R R pplgr R pplgr R R pplgr R R pplgr

7 BSE Lt log Epdg R R pplgr R R d R R R pplgr d pplg Lt d f - fd Usg opt, s s s s d d

8 BSE l l l cot l cot l s s s s s s s d, ddg, s cot l c c d c d d d d d d c OR

9 BSE h f h d h f H d l dfto B,, Now pplgth lt w gt l l l l h h h h h h h h f h h h h h h h d B D B d d d Usg ptl fcto, Equtg th coffcts fo oth th utos w gt, +B+= -B+D= +B-= -B-D= Solvg th ov qutos w gt,

10 BSE =, B= -, =, D= Ou tgl, d d t log log t log log Lt E, E d th vts dfd s follows: E = Slctg co hvg hd o oth th sds E = Slctg co ot hvg hd o oth th sds = Gttg ll hds wh co s tossd fv ts W hv to fd E Th hvg hds o oth th sds E Th ot hvg hds o oth th sds E E E B B's Tho, w hv E E E E E E E OR Lt p dots th polt of gttg hds Lt q dots th polt of gttg tls p = ½

11 BSE q = ½ = ½ Suppos th co s tossd ts Lt X dot th u of ts of gttg hds tls,,,, X X X X q p X X X X X X X X =,, So th f co should tossd fo o o ts fo gttg th qud polt osto vcto of O osto vcto of OB osto vcto of O osto vcto of OD 7 O OD D O O O OB B 7 Th ov th vctos copl

12 BSE 7 7 D B Lt th quto of th l H, Equto of th l s Lt L th foot of th ppdcul d th qud pot fo whch w hv to fd th lgth of th ppdcul L = osto vcto of L- posto vcto of ] [ ppdcul to ] Ls [Sc Now, L L L fo L Lgth of th ppdcul dw o th l fo = s - - s - =

13 BSE s - - = + s - -= s s = s - = - = = = =, = ½ OR 7 s t L H S, t t t t t t Hc ovd = t 7 7 t 7 t 7 7 t

14 BSE d s d d s d d d s d d d d s d d d d d [Susttutg d d d d d d d d d d d d d Hc ovd s fo ] + + = Lt u = log u = log du log u d du log d Lt v = log v = log dv d log v d d dv d Lt w = Log w = log dw w d dw d d log d d log d d d c wtt s u + v + w = log

15 BSE du d d d d d dv d log dw d log log d log d d log d log log log log log log log log d d 7 = s t + t, = t t d =t t + st st dt =t+ t s t =t + t d = st t + tst dt = st + stt + t st = st + stt = st +st d d log d d log

16 BSE t t t t dt d t t t t dt d dt d dt d dt d dt d dt d t t t t s s s s s s s s d d d d Ths s of th fo [f +f ']d= f+ d

17 BSE X Y Z Th fuds collctd X = Rs, Y = Rs, Z = Rs Totl fuds collctd = Rs 7 Vlu gtd: t wo SETON Lt = Q Q, wh Q s th st of tol us Gv tht * s th opto o dfd, * c, d = c, + d fo,, c, d Î W d to fd th dtt lt of th opto * Lt, th dtt lt Thus,, *, =, *, =,, fo ll,, + =, = d + = = d = Thfo,, s th dtt lt wth spct to th opto * W d to fd th vtl lts of Lt p, q th vs of th lt, Thus,, * p, q =, p, + q =, p = d + q = p d q Thus th vs lts of, s, Now lt us fd th vs of, d,

18 BSE Hc, vs of, s, d vs of, s, =, - OR Lt f: W W dfd s, f s odd f, f s v W d to pov tht 'f' s vtl od to pov tht 'f' s vtl t s suffct to pov tht f s cto fucto f: B s o-o fucto o cto, f f = f = fo ll, s : f d odd Lt f = f = = s : f d v, Lt f = f + = + = Thus, oth th css, w hv, f = f = fo ll, W Hc f s cto Lt t lt of W f s odd whol u, th sts v whol u W such tht f = + = f s v whol u, th th sts odd whol u + W such tht f + = + = lso, f = d f = Thus, v lt of W co-do hs ts p-g W do So f s oto fucto Thus, t s povd tht f s vtl fucto Thus, fucto g: B whch ssocts ch lt B to uqu lt such tht f = s clld th vs of f Tht s, f = g = Th vs of f s gll dotd f - Now lt us fd th vs of f Lt, W such tht f =

19 BSE + =, f s v d =, f s odd, f s odd, f s v, f s odd f, f s v tchg, d, w hv,, f s odd f, f s v Rwtg th ov w hv,, f s v f, f s odd Thus, f - =f osd th gv quto Ths quto psts sccl wth ct t th og d dus = uts Gv tht th go s oudd th ov sccl d th l = - Lt us fd th pot of tscto of th gv cuv ts th l = - Squg oth th sds, w hv, - = - - = = --= --= -+-= -+-= +-= =-, = Wh =-, = Wh =, = osd th followg fgu Thus th tscto pots -, d, osd th followg stch of th oudd go

20 BSE squts d d d d d d d d s s Rqud s s, Rqud s s - d = + d d d Lt = v, d dv v d d Susttutg, w gt

21 BSE dv v v v d dv v v v d dv v v d dv d v v tgtg oth sds dv d v v v - v dv vv log v log v log log v log v log d log log uttg d, [Rovg logth oth sds], whch s th gl soluto =+,whch s th ptcul soluto OR

22 BSE t t t d d d d d d

23 BSE th ov quto,, d utg, Susttutg, Susttutg t Lt t Thus th soluto s, t t t t t t t t t t dz dz d dz d z d d d d Q F Q z z d d d d

24 BSE t t t t th DEs soluto of tcul f = s, f = s + s = s + Equtg f to zo f = s + = s = =,π + = = - = π f = s = s s f fπ = s π π = Of ths vlus, th u vlu s, d th u vlu s Thus, th solut u d solut u vlus of f d, whch t tts t = d = π ovt g to cts fo,

25 z,, z,, z,, z,,,, c odto fo th ls to copl s th ls copl tscto of th two ls s Lt th quto - cz- z = Dcto to of th pl s + c = + c = Solvg coss-ultplcto, c = -λ, = - λ, c = λ Sc th pl psss though,, fo l + + c z = - λ λ + λz = z = - + z = - + z = =, ovtg th qutos to qutos, w ot th ls X = + = - + = X =, = BSE

26 BSE Fo th gph, w gt th co pots s,, B, 7,,, D, 7, Th vlus of th octv fucto : ot, Vlus of th octv fucto Z = +, + = B, = Mu, + = D, = O, + = Mu Th u vlu of Z s d ts u vlu s Fst s postv tgs {,,,,, } No, of ws of slctg us fo us wthout plct = = X dots th lg of th two us, so X c t th vlus,,,, olt dstuto of X: oputto of M d Vc: X = px px

27 BSE 7 M px 7 px 7 7 Vc = px px px 7