ikb~;øe l= ek/;fed f'k{kk e.my] e/;izns'k] Hkksiky d{kk 12oha xf.kr ikb~;øe vk/kkfjr v/;kiu izf'k{k.k funszf'kdk

Transcript

1 ikb~;øe l= ikb~;øe vk/kkfjr v/;kiu if'k{k.k funszf'kdk d{kk oha f.kr ek/;fed f'k{kk e.my] e/;ins'k] Hkksiky lokzf/kdkj lqjf{kr ek/;fed f'k{kk e.my] e/;ins'k] Hkksiky

2 i'u&i= CywfiUV BLUE PRINT OF QUESTION PAPER d{kk %& XII ijh{kk % gk;j lsd.mjh iw.kk±d %& 00 fo"k; %& mpp f.kr le; %?k.vs l- bdkbz bdkbz ij olrqfu"b vadokj i'uksa dh la[;k dqy i'u Ø- vkoafvr i'u vad vad vad 5 vad 6 vad - vkaf'kd fhkuu & & 0 - ifrykse Qyu & & 0 - f=foeh; T;kferh; - lery 5 0 & ljy js[kk,oa ksyk 6- lfn'k 7- lfn'kksa dk q.kuqy 5 0 & lfn'kksa dk f=foeh; T;kferh; esa vuqi;ks 9- Qyu] lhek rfkk lkarr; 05 & & 0 & 0 0- vodyu & & 0 - dfbu vodyu - vodyu dk vuqi;ks & & 0 - lekdu 5 05 & 0 & 0 - dfbu lekdyu 5- fuf'pr lekdyu 6- vody lehdj.k 0 & & 0 & 0 7- lglaca/k & & 0 8- lekj;.k & & 0 9- ikf;drk 05 & & 0 & 0 0- vkafdd fof/k;k & & & & ;ks = funsz'k % - i'u Øekad ds ik p Hkk gksas ir;sd Hkk esa 5 i'u ¼iR;sd,d&,d vad ds olrqfu"b i'u gksas ftlds vurzr fjdr LFkkuksa dh iwfrz],d 'kcn esa mùkj] lgh tksm+h] lgh fodyi rfkk lr; vlr; dk p;u vkfn ds i'u gksasa½ - olrqfu"b i'uksa dks NksM+dj lhkh i'uksa esa fodyi dk iko/kku j[kk tk;sa ;g fodyi leku bdkbz ls rfkk ;Fkk lahko leku dfbukbz Lrj okys gksus pkfg,a - dfbukbz Lrj & 0% ljy i'u] 5% lkeku; i'u] 5% dfbu i'u - vkafdd fof/k;ksa esa vk dm+s n'keyo esa,d LFkku ls vf/kd u gksaa

3 vuqøef.kdk l-d- fo"k;ka'k i`"b Ø-. vkn'kz i'u&i=. vkn'kz mùkj

4 () vkn'kz i'u&i= i'u. (A) lgh fodyi pqudj fyf[k, % ¼5 vad½ (i) O;atd (a) L NM L NM (c) ( + ) + O QP + (ii) ;fn Tan = 5 O QP dks vkaf'kd fhkuu ds :i esa fy[k ldrs gsa % θ gks rks cot θ dk eku gksk % (b) (d) L NM L NM + + O QP O QP (a) cot 5 (b) 5 (c) 5 (d) (iii) fcunq (,, ) dh y-v{k ls nwjh gksh % (a) 0 (b) (c) (d) 0 (iv) lery y + + = 0 dh ewyfcunq ls nwjh gs % (a) 5 (b) (c) (d) (v) js[kk y = = + vksj lery + y + = 0 ds e/; dks.k gs % (a) 0 (b) 0 (c) 5 (d) 90 B. lr;&vlr; crkb, % ¼5 vad½ (i) lg&lecu/k q.kkad dk eku ls vf/kd gksrk gsa (ii) lg&lecu/k q.kkad lekj;.k q.kkadksa dk lekurj ek/; gksrk gsa (iii) sin + cos dk mfpp"b eku gsa (iv) ;fn ksys dk dsuæ c rfkk mldk flfkfr lfn'k c r c a rfkk f=t;k a gks rks ksys dk lehdj.k gksk =. (v) vpj jkf'k dk vodyu q.kkad 'kwu; gksrk gsa C.,d 'kcn esa mùkj nhft, % ¼5 vad½ (i) nks lfn'kksa dk vfn'k q.kuqy fdl fu;e dk ikyu ugha djrk \ (ii) fleilu dk fu;e fdl fl)kur ij vk/kkfjr gs \

5 (iii) dk vodyu q.kkad D;k gksk \ (iv) dk;z fdl idkj dh jkf'k gs \ (v) a a () f( ) dk lekdyu D;k gksk tcfd f () fo"ke Qyu gks \ D. lgh tksm+h cukb;s % ¼5 vad½ LrEHk A LrEHk B (a) cot (b) lery r. (i j + k) = (ii) π rfkk r. ( i + j) = ds chp dk dks.k (i) l + my + n = p (c).6e E05 (iii) (cot + ) π (d) 0 sin (iv).057e09 sin + cos (e) vfhkyec :i esa lery dk lehdj.k (v) cos E. fjdr LFkkuksa dh iwfrz dhft;s % ¼5 vad½ (i) dh ifke vko`fùk dk eku gsa (ii) sin dk eku gsa (iii) ;fn -v{k ij leku vurjky ij fn,, ekuksa dh lar Å pkbz Øe'k% a, a, a, a, a 5, a 6, a 7 gks rks leyech fu;e ls {ks=qy gkska (iv) e 5 dk nok vodyt gkska (v) U;wVu&jsQlu fof/k esa ifke vko`fùk dh flfkfr esa ;fn f ( 0 ) =, f ' ( 0 ) =, 0 = rks dk eku gkska 5 58 i'u. + ( ) i'u. fl) dhft, fd dks vkaf'kd fhkuuksa esa fohkdr dhft,a dks vkaf'kd fhkuuksa esa fohkdr dhft,a sin 5 + cos = sin ¼ vad½ ¼ vad½

6 (5) fueukafdr dks ljyre :i esa fyf[k, % Tan Fcos sin I HGcos+ sin K J. i'u. cot dk ifke fl)kur ls vodyu Kkr dhft;sa ¼ vad½ ds lkis{k sin ( ) dk vodyu Kkr dhft,a i'u 5.,d o`ùk dh f=t;k ls-eh-@lsd.m dh,d leku nj ls c<+ jgh gsa {ks=qy esa o`f) fdl nj ls gksh tcfd f=t;k 0 ls-eh- gksa ¼ vad½ Qyu f () = dh vurjky [, ] esa jksys ies; dh tk p dhft,a i'u 6. fueu vk dm+ksa ls dkyz&fi;lzu lg&lecu/k q.kkad dh.kuk dhft,a ¼ vad½ : y : fl) dhft;s fd nks Lora= pjksa ds fy, dkyz&fi;lzu dk lg&lecu/k q.kkad 'kwu; gksrk gsa D;ksa \ i'u 7. fueukafdr vk dm+s fn,, gsa % ¼ vad½ l-ek ieki fopyu 8 lg&lecu/k q.kkad = 0.66 buds vk/kkj ij nksuksa lekj; lehdj.k Kkr dhft,a fueukafdr vk dm+ksa ds vk/kkj ij y dh ij lekj;.k js[kk Kkr dhft, % 5 y i'u 8. ;fn y = a cos p + b sin p gks rks fl) dhft, % ¼ vad½ y dy + py = 0.

7 i'u 9. (6) ;fn + y + y + = 0 gks rks fl) dhft, fd dy = ( + ). ml lery dk lehdj.k Kkr dhft, tks fcunqvksa (,, ) rfkk (,, ) ls gksdj tkrk gks rfkk lery + y + = 5 ij yec gksa ¼5 vad½,d lery v{kksa dks A, B, C ij feyrk gs f=hkqt ABC dk dsuæd (a, b, c) gs rks fl) dhft, fd lery dk lehdj.k a + b + = gsa i'u 0. lfn'k fof/k ls fl) dhft, ¼5 vad½ i'u. fl) dhft, fd lim 0 ( ) (sin ) sin (α + β) = sin α cos β + cos α sin β. y c [ a b b c c a] = [ a b c] dh.kuk dhft;sa ¼5 vad½ fueufyf[kr Qyu dh = 0 ij lkarr; dh foospuk dhft;sa f () = R S T cos,, 0 = 0. i'u. dk eku Kkr dhft,a ¼5 vad½ sin dk eku Kkr dhft,a i'u. n'kkzb, fd π π 6 cot tan + cot = π - ¼5 vad½

8 (7) ijoy; y = a vksj js[kk y = m ds chp f?kj {ks= dk {ks=qy Kkr dhft,a i'u. vody lehdj.k cos + ysin = dks gy dhft,a ¼5 vad½ dy + + vody lehdj.k dy 5y y = dks gy dhft,a i'u 5. cppksa ds rhu lewgksa esa Øe'k% ym+fd;ksa vksj ym+dk] ym+fd;k vksj ym+ds] ym+dh vksj ym+ds gsaa ir;sd lewg ls ;n`pn;k,d cppk pquk tkrk gsa fn[kkb;s fd ;fn rhu pqus gq, cppksa esa ym+dh vksj ym+ds gksa rks bldh ikf;drk gsa ¼5 vad½,d lk{kkrdkj esa,d in gsrq ifr,oa iruh 'kkfey gq,] ifr ds pqus tkus dh ikf;drk gs] tcfd iruh ds pqus tkus dh ikf;drk gsa muesa ls fdlh ds Hkh u pqus tkus 6 dh ikf;drk Kkr dhft,a i'u 6. fl) dhft, fd js[kk, ¼6 vad½ = y = rfkk = y leryh; gsa js[kkvksa dk ifrpnsn fcunq Hkh Kkr dhft,a = 5 i'u 7. lekukurj js[kkvks a y = = vksj y = = 6 8 ds chp dh U;wure nwjh Kkr dhft,a mu js[kkvksa ds chp dh U;wure nwjh Kkr dhft, ftuds lfn'k lehdj.k ¼6 vad½ r = ( + λ) i + ( + λ) j + ( + λ) k r = ( + µ) i + ( + µ) j + ( + 5µ) k gsa ml lery dk lfn'k lehdj.k Kkr dhft, tks fcunq (,, ) rfkk leryksa r (i j + k) = 5 rfkk r (i + 5j k) = dh ifrpnsnh js[kk ls gksdj qtjsa

9 i'u. (A) (i) L NM O (8) vkn'kz mùkj ¼5 vad½ (ii) b 5 (iii) a + QP 0 (iv) c (v) c 5. (B) (i) vlr; (ii) vlr; (iii) lr; (iv) vlr; (v) lr; (C) (i) lkgp;z fu;e (ii) ijoy; (iii) (v) vfn'k (vi) 0 (D) (a) cot (iii) (cot + ) (b) lery r (i j + k) = (v) cos rfkk r ( i + j) = ds chp dks.k (c).6e E05 (iv).057e09 (d) π 0 sin sin + cos (e) vfhkyec :i esa lery dk lehdj.k (i) l + my + n = p (E) (i) dh ifke vko`fùk dk eku.6 gsa (ii) sin dk eku cos + cos gsa i'u. (iii) ;fn v{k ij leku vurjky ij fn,, ekuksa dh lar Å pkbz Øe'k% a, a, a, a, (iv) a 5, a 6, a 7 gks rks leyech fu;e ls {ks=qy h a a a a a a a gkska e 5 dk nok vodyt 5 n e 5 gkska (ii) π 5 58 [( + 7) + ( )] (v) U;wVu&jsQlu fof/k esa ifke vko`fùk dh flfkfr esa ;fn f ( 0 ) =, f ' ( 0 ) =, 0 = rks dk eku.87 gkska ( ) gy % ekuk = y y = ( ) = ( y ) y ¼ vad½

10 y dk eku j[kus ij (9) = y+ y y y = = + y y y y + ( ) ( ) ( ) ( ) +. gy % + = ( + )( + ) vr% ekukfd + = B C A () ¼ vad½ = A ( + ) + (B + C)(+ )...() vpjksa ds fu/kkzj.k ds fy, % lozlfedk () ds nksuksa i{kksa esa + = 0 vfkkzr~ = j[kus ij] = A ( + + ) + (B + C)( ) = A, = A vc lehdj.k () ds nksuksa i{kksa esa ds q.kkadksa dh rqyuk djus ij 0 = A + B 0 = + B = B iqu% lozlfedk () ds nksuksa i{kksa ls vpj inksa dh rqyuk djus ij] = A + C Þ C = A C =, C= vr% lozlfedk () esa A, B o C ds eku j[kus ij i'u. sin 5 + = ( + ) ( + ) + cos gy % ekuk cos = a = sin

11 cos a = (0) sin a = 5 a = sin 5 leh- () ls cos L.H.S. = sin 5 dk eku j[kus ij + cos...() sin sin. Q sin + sin y = sin { y + y } sin 5 + sin 5 = sin = sin R S T RST UVW U V W = sin 65 Fcos sin gy % Tan cos ls va'k o gj dks Hkk nsus ij tan tan F HG F G HG tan tan cos+ sin HG I K J. tan + tan I K J π tan tan π + tan tan F HG π 56 = sin. 65 I J KJ I K J π. i'u. cot dk ifke fl)kur ls vodyu q.kkad Kkr dhft;sa gy % ekuk fd f () = cot f ( + δ) = cot ( + δ) ¼ vad½

12 () vr% d (cot ) = lim δ0 cot ( + δ) cot δ ekuk fd cot = t = cot t + δ = cot (t + δt) δ = cot (t + δt) = cot (t + δt) cot t rfkk ;fn δ 0 δt 0 lozlfedk () esa eku j[kus ij] d (cot ) = lim δt0 cot cot( t+ δt) t cot( t+ δt) cot t...() = lim δt0 = lim δt0 = lim δt0 = lim δt0 t+ δt t cot( t+ δt) cot t δt cot( t+ δt) cot t sin( t+ δt)sin t. δt cos( t+ δt) sin t costsin( t+ δt) sin( t+ δt).sin t. δt sin( t t δt) = lim δt0 sin( t+ δt)sin t. δt sinδt δt = sin t lim lim δt0 sinδt δt0 sin (t + δt) = sin t sin t = sin t = cosec t = + cot t d sin ( ) = d ekuk = sin A = sin B + sin ( ( ) ).

13 () = d sin (sina sin B sin B sin A) = d sin (sin A cos B cos A sin B) = d sin sin (A B) = d (A B) i'u 5. gy % Kkr gs % Kkr djuk gs da dt =? ge tkurs gsa A = πr = d (sin sin ) =. dr = c.m/sec....() dt r = 0 c.m. da dr leh- () vksj () dk q.kk djus ij dr dt = πr...() da = πr dr da dt da dt = πr = π 0 = 0π oz lsa-eh-@ls- gy % ;gk ge ns[krs gsa fd % (i) f (), dk,d cgqinh; Qyu gksus ds dkj.k vurjky [, ] esa larr gsa (ii) f' () = + ftldk vflrro [, ] ds lhkh ekuksa ds fy;s gsa blfy, f(), foo`r vurjky (, ) ds fy;s vodyuh; gkska (iii) f () = = 0, f () = = 0. f () = f ()

14 () vr% lhkh flfkfr;ksa ds fy;s jksys ies; larq"v gksrh gsa vr% (, ) dk vflrro bl idkj gksk fd f' (c) = 0. f' (c) = 0 c c + = 0. c = ± 6 F I HG K J. = ± Li"Vr;k c ds nksuksa eku vurjky (, ) esa flfkr gsa vr% jksys ies; fl) gqbza i'u 6. ;gk = 68, y = 69. y y y u i v i u i v i (u i ) (v i ) ( )(y y ) ( ) (y y ) Σ = 76 Σy = 8 Σu i = 0 Σv i = 0 Σu i v i = 0 Σu i = 8 Σv i = ρ (u, v) = = nσuivi ΣuiΣvi nσu ( Σu ) nσv ( Σv ) i i i i 0 96 = ;fn,oa y nks Lora= pj gsa rks Cov (, y) = 0. ge tkurs gsa fd r (, y) = Cov (, y ) σσ Cov (, y) = 0. = ( 0) 7 ( 0) ¼ vad½ = = ¼ vad½ = 0 = 0. ¼ vad½ σσ y y i'u 7. gy % fn;k ;k = 6, y = 85, σ =, σ y = 8, ρ = y dk ij lekj;.k js[kk dk lehdj.k y y = ρ σ σ y ( ) ¼ vad½

15 () y 85 = ( 6) y 85 = 0.8 ( 6) y 85 = y = ,oa dk y ij lekj;.k js[kk dk lehdj.k = ρ σ σ y ( y) 6 =.66 (y 85) 8 6 = 0.90 (y 85) 6 =.90y 76.5 =.90y lkj.kh y y y S = 5 Sy = 9 Sy = 8 S = 55 Sy 09 b y = n Σ y Σ Σ y nσ ( Σ) = = = 0 50 = 0. = Σ = 5 5 y dk ij lekj;.k js[kk dk lehdj.k =, y = Σy = 9 5 = 5.8. y y = ρ σ y σ y 5.8 = 0. ( ) ( )

16 (5) y = = i'u 8. gy % y = a cos p + b sin p...() dy = d dy dy (a cos p + b sin b) = ap sin p + b p cos p = ap cos p bp sin p dy = p (a cos p + b sin p) leh- () ls a cos p + b sin p dk eku y j[kus ij dy dy = p y + p y = 0. + y + y + = 0 y + = + y nksuksa vksj oz djus ij y ( + ) = ( + y) y ( + ) y = 0 y = ( ) ± ( ) ( + ) ( + ) y = ± ( + + ) ( + ) y = ± ( + ) ( + ) y = ± ( + ) ( + ) y = b± b ac a

17 + fpug ysus ij (6) y = + + ( + ) y = + ( + ) fpug ysus ij y = ( + ) ( + ) y = dy =. y = ( + ) y = ( + ) i'u 9. gy % y = + dy = [ + ] = ( + ) ( + ). fcunq (,, ) ls gksdj tkus okyk lery dk lehdj.k a ( ) + b (y + ) + c ( ) = 0...(i) lery (i) ij fcunq (,, ) flfkr gs a ( ) + b ( + ) + c ( ) = 0 a + b = 0...(ii) lery (i) ds vfhkyec ds fnd~&vuqikr a, b, c gsaa lery + y + = 5...(iii) ds vfhkyec ds fnd~&vuqikr (,, ) gsaa pw fd (i) (iii) ij.a + b + + c = 0...(iv) leh- (ii) o (iv) ls a = b = c 6 = k a = k, b = k, c = 6k ;s eku leh- (i) esa j[kus ij

18 i'u 0. gy % (7) k ( ) + k (y + ) 6k ( ) = 0 (k dk Hkk nsus ij½ + y + = 0. ekuk lery dk lehdj.k gs % y + + =...() α β γ ;fn ;g lery funsz'kka{kksa dks fcunqvksa A, B, C ij feyrk gs rks os fcunq Øe'k% A (α, 0, 0), B (0, β, 0) rfkk C (0, 0, γ) gksasa F HG ABC dk dsuæd α β ,, + γ vfkkzr~ α, β, γ i'ukuqlkj] dsuæd (a, b, c) gsa α = a; β = b; γ = c α = a; β = ; γ = c vr% vhkh"v lery dk lehdj.k ¼vUr%[k.M :i esa½ gs % y + + = a b c a + y b + c =. I K J F HG I K J gsa ¼fp= ij vad½ ekuyks dkt ds lery esa OX vksj OY nks yecor~ js[kk, gsa tks fcunq O ij dkvrh gsaa ekuk OX o OY ds vuqfn'k ek=d lfn'k Øe'k% i o j gsaa ekuk OX ds uhps,d js[kk OA gs tks OX ls dks.k β cukrh gsa ekuk OX ds Åij,d js[kk OB gs tks OX ls α dks.k cukrh gsa rc] AOB = α + β OA ij,d fcunq P bl idkj fy;k fd OP =, PM OX [khapka OB ij,d fcunq Q bl idkj fy;k fd OQ = rfkk QN OX [khapka rc] Y j Q B X A O C Z B Y O α+β α β N M P i X A ¼fp= ij vad½

19 (8) OQ = PN+ NQ = ( ON+ NQ) = (OQ cos α) i + (OQ sin α) j =.cos α i +.sin α j OQ = i cos α + j sin α rfkk OP = OM+ MP = ( OM ) i + ( MP ) ( j) = (OP cos β) i + OP sin β ( j) OP = cos β i sin β j OP OQ = (cos β i sin β j) (cos α i + sin α j) OP OQ = [sin α cos β + cos α sin β] k OP OQ sin (α + β) k = (sin α cos β + cos α sin β) k sin (α + β) = sin α cos β + cos α sin β gy % [ a b, b c, c a] [Q OQ = [( a b) ( b c)].( c a) = rfkk OP = ]. ¼ vad½ = [ d ( b c)].( c a) ekuk a b = d = [( dc. ) b ( db. ) c].( c a) = [{( a b) c} b {( a b). b} c].( c a) = [[ abc] b [ abb] c].( c a) = [[a b c] b 0].( c a) = [[a b c] b ].( c a) = [ abc][ bca] abc = [ abc][ ] = [ abc]. ¼ vad½

20 i'u. gy % lim 0 ( ) (sin ) (9) ekuk sin = q. lim θ0 = sin θ = sin θ = cos θ. 0 θ 0. sin θ( cos θ) cosθθ = lim θ0 = lim θ0 tanθ θ. sin θ( + sin θ/ ) cosθθ sin θ / θ. = lim tan θ θ0. θ. lim θ0 F HG sin θ / θ / I K J.. =. = gy % L.H. Limit : lim 0 f () = lim h0 f (0 h) = lim = lim h0 = lim h0 cos( h) ( h) h0 cosh = lim sin h / h h0 h F sinh/ I. HG h / K J = R.H. Limit : lim + 0 f () = lim h0 f (0 + h) = lim = lim h0,oa f (0) =. h0.. cosh h sin h / h =.. lim h0 = F sinh/ I HG h / K J =. i'u. gy % bl idkj lim vr% Qyu = 0 ij lrr gsa + 5 f () = f (0) = lim f ()

21 = (0) ( ) 5 5 = / ( 5 ) / ( 6) ( ) =. (5 ) / sin 6 = 5 gy % I = = 5+ sin 6 sin. ¼ vad½ F I 5 cos + sin sin cos H K + = sec F I 5+ tan 8tan H K + ekuk tan t = sec dt I = 5 ( + t ) + 8 t = dt = dt 5 8 t + t+ 5 = dt t + t+ + 5 H = 5 F t + 5 dt I K +F H I 5K

22 i'u. gy % ekuk fd I = π π (i) o (ii) dks tksm+us ij () = tan 5 / 5 = 5t + tan / 6 t + 5 / 5 = tan L NM 5tan + / cot...(i) tan + cot π π cot + π/ 6 I = π H K / 6 Fπ π I Fπ π tan + cot H 6 K + H + 6 cot π/ = π / 6 = π π/ 6 tan F H F H I / tan cot + tan F π I K F π π cot K + H π/ tan + cot I = π / 6 tan + cot π/ π/ = π = [ ] / 6 π/ = π π 6 I I K I K O QP....(ii) 6 = π 6 I = π. gy % ijoy; dk lehdj.k y = a...() js[kk dk lehdj.k y = m...()

23 () i'u. ijoy; js[kk ds X-mHk;fu"V fcunqvksa ds fy;s () ls y dk eku () esa j[kus ij m = a Þ (m a) = 0. = 0, a m. vhkh"v {ks=qy = ijoy; ds Hkk PON dk {ks=qy js[kk }kjk cus Hkk PON dk {ks= am / = 0 ( y y) L NM am / = 0 am / = 0 = = ( a m) ( a m) O QP / a / m am / 8 a a a m 0 m 6a m. = a m = 8 a m 8a m oz bdkbza ¼ vad½ gy % cos + ysin =. ¼ vad½ tks fd dy dy dy + y tan = sec. + py = Q ds :i dk gsa ¼ vad½

24 gy % fn;k ;k lehdj.k gs % () ;g,d le?kkrh vody lehdj.k gsa vr% y = v rfkk dy = v + dv dy = + 5 y + y j[kus ij v + dv = + 5 v + v v + dv = + 5v + v ¼ dv = + v + v = (v + ) dv ( v + ) = vad½, ¼pj i`fkd~ djus ij½ dv = ( v + ) ( v+ ) = log + c F H I K. y + = log + c ( + y) = log + c tks vhkh"v gy gsa i'u 5. gy %,d ym+dh,oa ym+dksa dks fueu idkj ls pquk tk ldrk gs % (i) ifke lewg ls ym+fd;ksa f}rh;,oa r`rh; lewg ls ym+dksa dks pqus tkus dh ikf;drk (ii) = 9 =. ifke lewg ls ym+dk,oa nwljs lewg ls ym+fd;ksa,oa rhljs lewg ls ym+dksa dks pquus dh ikf;drk = =.

25 (iii) () ifke lewg ls ym+dk nwljs lewg ls ym+ds,oa rhljs lewg ls ym+fd;ksa dks pqus tkus dh ikf;drk = 9 =. ir;sd lewg ls ym+dh o ym+ds pqus tkus dh ikf;drk = 9 =. ¼ vad½ gy % ifr ds pqus tkus dh ikf;drk P (A) = 7, iruh ds pqus tkus dh ikf;drk P (B) = 5 ifr ds pqus tkus dh ikf;drk PA ( ) = [ P(A)] i'u 6. = 6 = 7 7 ifru ds pqus tkus dh ikf;drk PB ( ) = [ P(B)] = 5 nksuksa esa ls fdlh ds Hkh u pqus tkus dh ikf;drk P( A B) gy % js[kk, l ;fn = y y m = n y y l m n l m n = 6 7 vksj = 0 Þ 5 = 5 5 Ç = PA ( ) PB ( ) = ¼ vad½ l = y y m = n = 0 Þ 5 leryh; gksaha 5 (5 6) + ( 0) + (8 9) = 0 + = 0, tks lr; gsa vr% nh gqbz js[kk, leryh; gsaa vc ifrpnsnh fcunq ds fy;s nh bz js[kk esaa ekuk = y = = r = r +, y = r +, = r +. ;fn nksuksa js[kk ifrpnsnh gsa rks, y, ds ;s eku nwljh js[kk ds lehdj.k dks larq"v djsha = 0

26 r+ (5) = r+ = r 5 + r gy djus ij r =. vr% ifrpnsnh fcunq = ( ) + = y = ( ) + = = ( ) + = ¼ vad½ (, y, ) (,, ). fn;k ;k lekurj js[kk dk lehdj.k - y- - = = vksj y - = - = ekuk fd nh bz js[kk l vksj m gs ftu ij Øe'k% A (,, ) rfkk B (,, ) fcunq flfkr gsa A (,, ) l B (,, ) C m ¼ vad½ fcunq A ls js[kk m ij yec AC cukb,a gesa AC dh yeckbz Kkr djuk gsa js[kk m dh fnd~ dkst;k, (direction cosines) gsa % ,, ;k 6, 6 8 6, 6 Þ 9,, 9 9 AB = ( ) + ( ) + ( ) = BC = AB dk m ij i{ksi (projection) = = 9 ( ) + 9 ( ) ( ) AC = AB BC = 8 = 9 vr% nh gqbz lekurj js[kkvksa ds chp dh nwjh = ¼ vad½ 9

27 (6) i'u 7. gy % fn;s ;s lehdj.k dks fueu idkj ls fy[k ldrs gsa % ;gk a = i + j + k, b r = i + j + k + λ (i + j + j) r = i + j + k + µ (i + j + 5k) = i + j + k, a = i + j + k, b = i + j + 5k a a = i + j + k i i k = i + j + k i j k b b = 5 b b b b = i (5 6) + j ( 0) + k (8 9) = + + = 6 vr% U;wure nwjh = ( a a ).( b b ) b b gy % = ( i+ j+ k).( i+ j k) 6 = + 6 = 0. ¼ vad½ ml lery dk lehdj.k tks fn;s ;s leryksa dh ifrpnsnh js[kk ls gksdj qtjrk gks r (i j + k) 5 + λ { r (i + 5j k) } = 0....() pw fd ;g (,, ) ;k (i j + k) ls gksdj tkrk gsa (i j + k). (i j + k) 5 + λ {(i j + k). (i + 5j k) } = l ( 5 ) = 0 5 5λ = 0 λ =. λ dk eku leh- () esa j[kus ij r (i j + k) 5 + r. (i + 5j k) = 0 r {( + ) i + ( + 5) j + ( ) k} 6 = 0. tks fd vhkh"v lery dk lehdj.k gsa r (i + j k) 6 = 0 ¼ vad½