# CBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane.

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1 CBSE CBSE SET- SECTION. Gv tht d W d to fd 7 7 Hc, Lt,. W ow tht.. Thus,. Cosd th vcto quto of th pl.. z. - + z = - + z = Thus th Cts quto of th pl s - + z = Lt d th dstc tw th pot,, - to th pl. Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

2 CBSE Thus, d cz c d d d d d uts 7. Gv tht of = s Susttut = d = Thus, = s = s. Cosd th quto, = m, wh m s th pmt. Thus, th ov quto psts th fml of ls whch pss though th og. =m. m Dffttg th ov quto wth spct to, =m d m d d m d d [ fom quto ] d d d Thus w hv lmtd th tt, m. Th qud dfftl quto s d d. Cosd th gv dfftl quto: d log log d Dvdg th ov quto log, w hv, Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

3 CBSE log log d d log log log d... d log Cosd th gl l dfftl quto. d Q.wh d Q fuctos of d Compg quto d th gl quto, w hv, d Q log Th tgtg fcto s gv th fomul d Thus, d Cosd I log Susttut g Thus Hc, I. F. I dt t d I. F. d log t; dt log t loglog d log d log loglog log SECTION-B 7. Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

4 CBSE Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct: Cosd - - I Now --I = -=I = I - ostmultpl - -I= -

5 CBSE Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct: OR =----- =-- = Hc - sts. - =I R R pplg

6 CBSE Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct: R R R R R pplgr R pplgr R R pplgr R R pplgr

7 CBSE. Lt pplgc pplgr pplgr C 7 R R Epdg log C C R R d C d R C R 7 C R. Lt I I s d... s s d. s Usg opt, fd f - d Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

8 CBSE Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct: l l l cot l cot l s.s. s. s s s s d, ddg,... s cot l I I c c I I d c I d I d I d I d I d I d I c OR

9 CBSE Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct: h f h d h f H d lm dfto B,, Now pplgth lmt w gt lm lm.. lm. lm h h h h h h h h f h h h h h h h. d C B D C B d d d Usg ptl fcto, Equtg th coffcts fom oth th umtos w gt, +B+C=.. -B+D= +B-C=.. -B-D= Solvg th ov qutos w gt,

10 CBSE Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct: =, B= -, C=, D= Ou Itgl coms, C C 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 tms W hv to fd E. Th hvg hds o oth th sds. C C E Th ot hvg hds o oth th sds. E E C C E B B's Thom, 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 CBSE Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct: q = ½ = ½ Suppos th co s tossd tms. Lt X dot th um of tms of gttg hds tls....,...,,, X X X X C C q p C X C X X X X X X X... =,,. So th f co should tossd fo o mo tms fo gttg th qud polt.. osto vcto of O osto vcto of OB osto vcto of OC osto vcto of OD 7 O OD D O OC C O OB B 7 Th ov th vctos copl

12 CBSE Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct: 7 7 D C B. Lt th quto of th l H, Equto of th l s Lt L th foot of th ppdcul d th qud pot fom 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 fom L Lgth of th ppdcul dw o th l fom =. s - - s - =

13 CBSE 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 Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

14 CBSE 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 fom ]. + + =. 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 Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

15 CBSE 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 Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

16 CBSE Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct: 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 fom [f +f ']d= f+ C d C.

17 CBSE X Y Z Th fuds collctd X = Rs., Y = Rs., Z = Rs. Totl fuds collctd = Rs. 7 Vlu gtd: tm wo SECTION C. Lt = Q Q, wh Q s th st of tol ums. Gv tht * s th opto o dfd, * c, d = c, + d fo,, c, d Î. W d to fd th dtt lmt of th opto *. Lt, th dtt lmt. Thus,, *, =, *, =,, fo ll,, + =, = d + = = d = Thfo,, s th dtt lmt wth spct to th opto *. W d to fd th vtl lmts of. Lt p, q th vs of th lmt, Thus,, * p, q =, p, + q =, p = d + q = p d q Thus th vs lmts of, s, Now lt us fd th vs of, d, Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

18 CBSE 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. I 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,. Cs : If d odd. Lt f = f = = Cs : If d v, Lt f = f + = + = Thus, oth th css, w hv, f = f = fo ll, W. Hc f s cto. Lt t lmt of W. If s odd whol um, th sts v whol um W such tht f = + =. If s v whol um, th th sts odd whol um + W such tht f + = + =. lso, f = d f = Thus, v lmt of W co-dom hs ts p-mg W dom. So f s oto fucto. Thus, t s povd tht f s vtl fucto. Thus, fucto g: B whch ssocts ch lmt B to uqu lmt 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 = Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

19 CBSE + =, f s v d =, f s odd, f s odd, f s v, f s odd f, f s v Itchg, 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. Cosd th gv quto Ths quto psts smccl wth ct t th og d dus = uts Gv tht th go s oudd th ov smccl d th l = - Lt us fd th pot of tscto of th gv cuv mts th l = - Squg oth th sds, w hv, - = - - = = --= --= -+-= -+-= +-= =-, = Wh =-, = Wh =, = Cosd th followg fgu. Thus th tscto pots -, d, Cosd th followg stch of th oudd go. Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

20 CBSE Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct: 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 CBSE dv v v v d dv v v v d dv v v d dv d v v Itgtg oth sds. dv d v v v - v. dv vv log v log v log log C v log v log C C C C C C d log C log C uttg d, [Rmovg logthm oth sds], whch s th gl soluto. C =+,whch s th ptcul soluto. OR Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

22 CBSE Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct: t t t d d d d d d m m m

23 CBSE I. F. Thus th soluto s Lt m m m t z m d t t utg d m t m m t Susttutg d d m t Susttutg t d m t, Q d m t m d... d dz Q dz z dz,, d z m t. t t d.... d C... m, th ov quto, Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

24 CBSE Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct: t t t t th D.E.s soluto of tcul m m m C C m C m m m m m m. 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 mmum vlu s, d th mmum vlu s. Thus, th solut mmum d solut mmum vlus of f d, whch t tts t = d = π..... Covt g to cts fom,

25 z,, z,,... z,, z,,,, c Codto fo th ls to copl s th ls copl. Itscto of th two ls s Lt th quto - cz- z =... Dcto to of th pl s + c = + c = Solvg coss-multplcto, c = -λ, = - λ, c = λ Sc th pl psss though,, fom l + + c z = - λ λ + λz = z = - + z = - + z = =, Covtg th qutos to qutos, w ot th ls X = = = X =, = CBSE Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

26 CBSE Fom th gph, w gt th co pots s,, B.,.7, C,, D.,.7,. Th vlus of th octv fucto : ot, Vlus of th octv fucto Z = +, + = B., = Mmum C, + = D., = O, + = Mmum Th mmum vlu of Z s d ts mmum vlu s.. Fst s postv tgs {,,,,, } No, of ws of slctg ums fom ums wthout plcmt = C = X dots th lg of th two ums, so X c t th vlus,,,,. olt dstuto of X: Computto of M d Vc: X = px px Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

27 CBSE 7 M px. 7 px 7 7 Vc = px px px 7 Bu s Clsss. lh + ls o. Id s most ld Eductol Comp Cotct:

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