izkn'kz&iz'u i= d{kk & ckjgoha fo"k; & xf.kr

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Ezra Bruce
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1 izkn'kz&iz'u i 0 04 [MODEL QUESTION PAPER] Set-D d{kk & ckjgoh Clss - Th fo"k; & f.kr Sub - Mthemtics le; &?kuvs iw.kkzd & 00 funsz'k& - lhkh iz'u g djuk vfuok;z gsa - iz'u&i es nks [k.m gs ^v*,o ^c* - [k.m ¼v½ es ls 5 rd olrqfu"b iz'u gs o izr;sd es vd fu/kkzfjr gsa 4- iz'u Ø- 6 ls 0 rd izr;sd iz'u ij vd fu/kkzfjr gsa 5- iz'u Ø- ls 7 rd izr;sd iz'u ij 4 vd gsa 6- iz- Ø- 8 ls rd izr;sd iz'u ij 5 vd gsa 7- iz'u Ø- o 4 ij 6 vd fu/kkzfjr gsa Instruction :. All question re compulsor.. Question pper hs two section 'A' nd 'B'. In section 'A' Q.No. to 5 is objective tpe ech question crries mrk. 4. Q.No. 6 to 0 crries mrk. 5. Q.No. to 7 crries 4 mrk. 6. Q. No. 8 to crries 5 mrk. 7. Q.No. nd 4 crries 6 mrk. [k.m & ^v* Section 'A' iz- izr;sd iz'u es fn;s fodyiks es lgh mrrj ff[k,a Choose the correct Answer. JAB-P- ()

2 ¼v½ ( + ) dh vkf'kd fhkuu gksh % (i) + (ii) + + (iii) + (iv) (A) Prtil frction of ( + ) is : (i) + (ii) + + (iii) + (iv) ¼c½ tn + tn cjkcj gs % (i) tn 6 (ii) π (iii) π 4 (iv) π 6 (B) tn + tn is equl to : (i) tn 6 (ii) π (iii) π 4 (iv) ¼l½?ku ds fdugh nks fod.kksz ds dks.k dh dkst;k gsa π 6 (i) (ii) (iii) (iv) 5 (C) Cosine of the ngle between the n two dligonls of cube is : (i) (ii) JAB-P- ()

3 (iii) 5 ¼n½ log sin dk vod q.kkd gksk % (iv) (i) cosec (ii) tn (iii) sec (iv) cot (D) Wht will be the differentil coefficient of log sin : (i) cosec (ii) tn (iii) sec (iv) cot ¼bZ½ cosec d dk eku gksk % (i) log tn (ii) log (cosec + cot) (iii) log (sec tn) (iv) log tn π + 4 (E) The vlue of cosec d is : (i) log tn (ii) log (cosec + cot) (iii) log (sec tn) (iv) log tn iz- fjdr LFkku dh iwfrz dhft, % Fill in the blnks : π + 4 ¼v½ b f ()d ds f, fleilu fu;e gsa (A) The simpson's rule for b f ()d is... ¼c½ lehdj.k dk ew vurjk es flfkr gsa (B) The root of eqution is lines on...intervl. ¼l½ iw.kz lglecu/k gksus ij nksuks lekj;.k js[kk, gksrh gsa (C) In perfct correltion both regression lines be... ¼n½ nks pjks ds chp izdkj dk lecu/k gksrk gsa (D) There re... tpes of reltionship between the two vrible. JAB-P- ()

4 ¼b½ cos dk n ok vodt gkska (E) The n th derivtive of cos is... iz- lgh tksm+h cukbz;s %. Mke the right mtch : v c A B (i) d () log + (ii) d (b) + + log + + d (iii) (c) + sin d (iv) (d) + log + (v) + d (e) iz- 4 lr;@vlr; ff[k,a log + Stte true nd flse : ¼v½ v{k dh fnd~ dkst;k, (, 0, 0) gksrh gsa (A) Direction cosine of -is re (, 0, 0) ¼c½ fcunq (,, z) z ler ls nwjh z gksrh gsa (B) Distnce of z plne from the point (,, z) is z. ¼l½ lfn'k r dh fn'kk es,dkd lfn'k r gsa (C) Unit vector in the direction of vector r is r r ¼n½ d (sec) dk vodu q.kkd sec tn gsa d (D) Defferencil coeffecient of sec is sec tn ¼b½ lfn'k $ $ $ rfkk $ $ $ ds chp dks.k 'kwu; gsa i+ j+ k i j k (E) The ngle between the vectors i $ + $ j+ k$ nd i $ $ j k $. r JAB-P- (4)

5 iz- 5,d okd; ds mrrj nhft, % Give the nswer in one sentence ¼v½ leec prqhkqzt dk fu;e dk lw ff[k,a (A) Write the formul for Trpezoidl Rule. ¼c½ U;wVu jsqlu fof/k ls l[;k N dk ozew Kkr djus dk lw ff[k,a (B) Write the formul for finding squre roof of n number N b Rphson's method ¼l½ fleilu dk fu;e fdl fl)kur ij vk/kkfjr gs \ (C) The simpson's rule is bsed on which principle : ¼n½ fdlh d.k dh vf/kdre Å pkbz ij os lnso fdruk gksrk gs \ (D) Wht will be the velocit of n prticle t mimum height? ¼b½ fcunq (,, z) dh v{k ls nwjh D;k gksrh gs \ (E) Wht will be the distnce from is to the point (,, z)? [k.m & ^c* Section - 'B' iz- 6 fl) dhft, fd Øe ls h bz fhkqt dh rhu Hkqtkvks ls fu:fir lfn'kks dk ;ks 'kwu; lfn'k gksrk gsa Prove tht ddition of vectors repersented b three sides of tringle is zero. vfkok (Or) ;fn fdlh prqhkqzt ABCD ds fod.kz AC rfkk BD gks rks fl) dhft, A uuur uuur uuur uuur AB + DC AC + DB AC nd BD re the digonls of qudrilterl ABCD prove tht - uuur uuur AB + DC uuur uuur AC + DB iz- 7 lfn'k fof/k ls fcunq (,, ) rfkk (,, ) ds chp dh nwjh Kkr dhft,a B using vector method find the distnce between the pint (,, ) nd (,, ) vfkok (Or) lfn'k 6 i $ $ j $ k dh fnd~&dkst;k, Kkr dhft,a Find the direction cosine of vector 6 i $ $ j $ k JAB-P- (5)

6 iz- 8 lerks r r. ( i $ $ j + 4 $ k ) rfkk r r. ( i $ $ j ) 4 ds chp dk dks.k Kkr dhft,a Find the ngle between the plnes r r. ( i $ $ j + 4 $ k ) nd r r. ( i $ $ j ) 4. vfkok (Or) ml kss dk lfn'k rfkk dkrhz; lehdj.k Kkr dhft, ftldk dsunz (,, 4) rfkk ft;k 5 gsa Find the vector nd crstesin eqution of the sphere whose centre (,, 4) nd rdius is 5. iz- 9 d + sin dk eku Kkr dhft,a d Solve + sin vfkok (Or) sec sec tn d dk eku Kkr dhft,a Evlute sec sec tn d. iz- 0 tn + d dk eku Kkr dhft,a Evlute tn + d vfkok (Or) e cos d dk eku Kkr dhft,a Evlute e cos d. iz- + dks vkf'kd fhkuu es fohkdr dhft,a Seprte + in to prtil frction. JAB-P- (6)

7 vfkok (Or) ( ) 4 dks vkf'kd fhkuu es O;Dr dhft,a Resolve ( ) 4 into prtil frction. iz- fl) dhft, fd& cos 5 Prove tht cos 5 + cos + cos fl) dhft, fd& sin 6 65 sin 6 65 vfkok (Or) tn b + tn + b b c + bc Prove tht - + tn c + c 0 tn b + b + tn b c + bc + tn c + c 0 iz- log tn π + 4 dk ds lkis{k vodu dhft,a Find the differentil coefficient oflog tn π + 4 vfkok@(or) ;fn sin gks] rks d d Kkr dhft,a iz- 4 If sin, then find d d (sin) log dk ds lkis{k vod q.kkd Kkr dhft,a Find the differentil coefficient of (sin) log w.r.t.. JAB-P- (7)

8 iz- 5 iz- 6 vfkok (Or) ;fn log + log + log +... gks rks fl) dhft, fd d d ( ) If log + log + log +... then prove tht - d d ( ),d d.k fueukfdr fu;e ls lj js[kk es freku gs % S 5e t cost tc t π gks] rks bldk os o Roj.k D;k gksk \ A prticle is moving in stright line ccording to lw S 5e t cost. find its velocit nd ccelertion when t π vfkok (Or) Qu f () dh vrjk [, ] es jkss izes; dh tk p dhft,a Verifl the Rolle's theorem for the function f () on [, ] fueu vk dm+ks ls lglecu/k q.kkd Kkr dhft,a Find the correltion coeficient of the following dt: vfkok (Or) nks pj jkf'k;ks vksj dk lglecu/k q.kkd r gs] rks fl) dhft,a r σ +σ σ σσ tgk σ ] σ vksj σ Øe'k%, rfkk ds izlj.k q.kkd gsa If r is coefficient of correltion of two vrible nd then proved tht, r σ +σ σ σσ nd respectivel., where σ ] σ nd σ re coefficient of vrinces of, iz- 7 fl) dhft, fd lekj;.k q.kkdks dk lekurj ek/; lglecu/k q.kkd ls cm+k gksrk gsa JAB-P- (8)

9 Prove tht rithmetic men of the regression coefficient is greter thn the coefficient of correltion. vfkok (Or) nks lekj;.k js[kkvks + vksj + 7 ds vk/kkj ij vksj ds chp lglecu/k q.kkd Kkr dhft,a 4 ds f, ds eku dh.kuk dhft,a Find the correltion coefficient between nd on the bsis of two regression lines + nd + 7 clculte the vlue of then 4 iz- 8,d pj ler ew fcunq ls P nwjh ij jgrk gs rfkk v{kks dks fcunqvks A, B o C ls funsz'kkd lerks ds lekurj ler [khps tkrs gsa fl) dhft, fd muds izfrpnsn fcunq dk fcunqifk + + z p A verible plne is t constnt distnce p from the origin nd meets the coordinte es in A, B, C, Through A, B, C the plnes re chrwn prllel to the coordinte plnes. Prove tht the locus of point of their intersecting point is. + + z p vfkok (Or) fcunq (,, ) ls tkus okk ml ler dk lehdj.k Kkr dhft,] tks lerks + z vksj z 5 ij Ec gksa Find the eqution of the plne pssing through the point (,, ) nd perpendiculr to the plnes + z nd z 5. gsa iz- 9 ;fn f () log e + gks] rks fl) dhft, fd f () + f (b) f + + b If f () log e + then prove tht f () + f (b) f + + b vfkok (Or) cos 4 0 ;fn f () 4 0 rks f () ds 0 ij lkrr; dh foospuk dhft,a iz- 0 cos 4 0 If f () 4 0 fl) dhft, fd& then discuss the continuit of f () t 0 JAB-P- (9)

10 π/ 0 Prove tht : sin sin + cos d 4 π π/ 0 sin sin + cos d 4 π vfkok (Or) nh?kzo`rr + b dk {ksq Kkr dhft,a find the re of the ellipse + b iz- vod lehdj.k ( + ) d d + 4 dks g dhft,a Solve the differentil eqution ( + ) d d + 4 vfkok (Or) vod lehdj.k cos d d + cos sin dks g dhft,a Solve the differentil eqution cos d d + cos sin. iz- nks?kukdkj ik ls,d lkfk Qsds tkrs gsa igs ik ls ij fo"ke l[;k vfkok nksuks ik lks ds Åijh l[;kvks dk ;ks 9 izkir djus dh izkf;drk Kkr dhft,a Two cubicl dice re thrown simultneousl. find the probbilit of getting n odd number on the first dice or the sum of 9'. vfkok (Or),d flddk nks ckj mnkk tkrk gsa 'kh"kksz dh l[;k dk izkf;drk cvu Kkr dhft,a A Coin is tossed twice. find the probbilit distribution of the number of hed. iz-,d kss dk lehdj.k + + z + z 5 0 gs blds,d O;kl AB ds fljs A funsz'kkd dks (, 4, ) gsa fljs B ds funsz'kkd Kkr dhft,a JAB-P- (0)

11 AB is the dimeter of the sphere + + z + z 5 0. the coordinte of A re (, 4, ). find the coordinte of point B. vfkok (Or) fl) dhft, fd js[kk, z + 5 7,o izfrpnsn djrh gsa izfrpnsn fcunq ds funsz'kkd Kkr dhft,a 4 z 6 5 ijlij Prove tht the lines z + 5 nd 7 to ech other. find their point of intersection. iz- 4 lfn'k fof/k ls fl) dhft, fd cos (A + B) cosa cosb sina sinb Prove tht b vector method cos (A + B) cosa cosb sina sinb vfkok@(or) 4 z 6 5 re intersecting ;fn D, E, F Øe'k% fhkqt ABC dh Hkqtkvks BC, CA, AB ds e/; fcunq gks] rks lfn'k fof/k ls fl) dhft, fda DEF 4 ABC If D, E, F re the mid point of the sides BC, CA, AB of the tringle ABC then prove b vector method tht. DEF 4 ABC JAB-P- ()

12 d{kk & oh vd ;kstuk Mrk Dirsbution 0-4 sgk;j lsds.mjh iw.kkzd & 00 fo"k; % f.kr le; & -00?k.V Ø- bdkbz,o fo"k; olrq bdkbz ij vk- vd olrqfu"b vd vdokj iz'uks dh l[;k dq - vkf'kd fhkuu 5 vd & vd vd & vd iz'u & - izfrkse Qu 5 & & & - 4- ffoeh; T;kferh; ler 5 4 & & 5- lj js[kk,o ksk 6-7- lfn'k lfn'kks dk q.kuq 5 & & 4 8- lfn'kks dk ffoeh; T;kes vuqiz;ks 9- Qu] lhek] lkrr; 5 & & & & 0- vodu - dfbu vodu 0 & & & - vodu dk vuqiz;ks 5 & & & - lekdu 4- dfbu lekdu 5- fuf'pr lekdu 5 6 & & 6- vodu lehdj.k 05 & & & & 7- lglc/k 05 & & & 8- lekj;.k 05 & & & 9- izkf;drk 05 & & & & 0 vkfdd fof/k;k 05 5 & & & & & ;ks funsz'k % iz'ui fuekz.k gsrq fo'ks"k funsz'k - iz'u Ø- ls 5 rd 5 izdkj ds olrqfu"b iz'u gkssa ftlds vrzr,d 'kcn es mrrj esfp] lgh fodyi rfkk fjdr LFkkuks dh iwfrz ds iz'u gkssa izr;sd iz'u ds f, vd fu/kkzfjr gsa ( 5 5 5) ;g iz'u izr;sd Nk dks g djuk vfuok;z gsa - iz'u Ø- 6 ls 4 izr;sd izdkj ds iz'uks dh mrrj lhek fu- gksh vfr?kqmrrjh; iz'u 0 vd Hk 0 'kcn?kqmrrjjh; iz'u 04 vd Hk 75 'kcn nh?kzmrrjh; iz'u 05 vd Hk 0 'kcn nh?kzmrrjh; iz'u 06 vd Hk 50 'kcn fuc/kkred iz'u 07 vd Hk 50 ls 50 'kcn - 4- olrqfu"b iz'uks dks NksM+dj 'ks"k lhkh iz'uks es fodyi ;kstuk jgsha fodyi ds iz'u mlh bdkbz ls] leku dfbukbz Lrj oks rfkk ikb~;øe vuqlkj gksuk pkfg,a 5- dfbukbz Lrj& 40% lj iz'u] 45% lkeku; iz'u] 5% dfbua JAB-P- ()

13 Answer Sheet Set-D mrrj iqflrdk Higher Mthmetics mpp f.kr (XII) iz- g% lgh fodyi pqudj ff[k,a 5 5 v- (A) (iii) c- (B) (iii) + π 4 l- (C) (i) n- (D) (iv) cot b- (E) (i) log tn iz- g% fjdr LFkku dh iwfrz dhft,a 5 5 Fill in the blnks h (i) [ + n ( ) + ( n )] (ii) 5.5 (iii) Coinsident (iv) lj lg Simple Correltion (v) cos n + iz- g% lgh tksm+h cukbz,a 5 5 Mke the right mtch (c) (d) () 4 (e) + log + log + sin log [ + ] JAB-S- ()

14 5 (b) + + log [ + + ] iz- 4 g% ff[k,a 5 5 Stte true nd flse. v- (i) lr; c- (ii) vlr; l- (iii) vlr; n- (iv) lr; b- (v) vlr; iz- 5 g%,d okd; es mrrj nhft,a 5 5 Give the nswer in one sentence. (i) b f () d h [0 + ( n ) + n ] (ii) n + N n + n (iii) ijo;@ Prbol (iv) 'kwu;@ Zero iz- 6 (v) + z A g% vd ekuk fd fhkqt ABC es uuur r uuur r AB, BC b, CA rc ges fl) djuk gs fd r + b r + c r o r ABC es lfn'k ;ks ds fhkqt fu;e ls] B uuur uuur uuur AB + BC CA nksuks i{kks es lfn'k CA uuur tksm+us ij uuur c r C JAB-S- ()

15 uuur uuur uuur uuur uuur r AB + BC + CA CA + CA o r + b r + c r o r ;gh fl) djuk FkkA vfkok vd D C A ekuk ABCD,d prqhkqzt gsa ABC es lfn'k ;ks ds fhkqt fu;e ls uuur uuur uuur AB + BC AC uuur uuur uuur AB AC BC iqu% BCD es lfn'k ;ks ds fhkqt fu;e ls] leh- () vksj leh- () dks tksm+us ij uuur uuur uuur DC DB + BC B uuur uuur uuur uuur uuur uuur AB + DC AC BC + DC + BC uuur AC uuur + DB + ( uuur uuur BC BC ) uuur uuur r AC + DB + o uuur uuur AC + DB vd...()...() vd ;gh fl) djuk gsa iz- 7 g% ekuk nks fcunq A o B gs ftuds funsz'kkd Øe'k% (,, ) rfkk (,, ) gsa uuur OA $ i + $ j k $ uuur rfkk OB $ i $ j + k $ uuur uuur uuur AB OB OA uuur (i $ $ j + $ k ) ( i $ + $ j $ k ) AB i $ 4 $ j + 4 k $ vd uuur AB AB $ i 4 $ j + 4 k $ + ( 4) + ( 4) vd JAB-S- ()

16 ekuk vfkok r r 6i $ $ j $ k iz- 8 g% ;gk r r r 6 + ( ) + ( ) r7 vd ;gk 6, b, c nh bz lfn'k dh fnd~ dkstt;k, gs& r, b r, c r 6 7, 7, 7 h i $ $ j + 4 $ k h i $ + $ j ekuk lerks ds chp dk dks.k θ gsa rc () vr% lehdj.k gksk% (b) nn cosθ n n $ $ ( $ i j+ 4 k $ ) $ i+ j θ cos 58 vfkok lfn'k lehdj.k c r r r r c r r $ $ ( $ i j+ 4k) dkrhz; lehdj.k& ( α) + ( β) + (z γ) dsunz ds funsz'kkd (α, β, γ) (,, 4) rfkk ft;k 5 mi;qdr lw es eku j[kus ij ( ). 9 vd vd vd i r j r + 4k r 5 vd JAB-S- (4)

17 iz- 9 ( ) + ( + ) + (z 4) z 8z z z + 40 vd d g% ekuk + sin v'k vksj gj es ( sin) ls q.kk djus ij iz- 0 sin ( + sin )( sin ) d sin d sin sin d { sin cos } cos cos d sin cos d vd sec d sec d cos. sin cos d sec.tn d tn sec + C vfkok sec ( sec tn ) d v'k vksj gj es (sec + tn) ls q.kk djus ij sec ( sec + tn ) ( sec tn )( sec + tn ) d sec ( sec + tn ) d sec tn sec ( sec + tn ) d + tn tn sec + sec.tn d sec tn d tn + sec + C sec d + vd vd vd g% e cos d JAB-S- (5)

18 cos t j[kus ij d et dt e t dt vd cos e vfkok tn d + tn t j[kus ij d dt + t dt t ( tn ) iz- g% ekuk fd + ( + )( + ) vd vd vd A + + B + c + ( ) ( )( + + ) ( A + ) + ( B+ c)( + ) ( + )( + )...() vd A ( + ) + (B + C) ( + )...() leh- () es + 0 j[kus ij A [( ) + ( ) + ] + 0 A A vd leh- () ls A A + A + B + B + C + C A ( + ) + (B + C) ( + ) (A + B) + ( A + B + C) + (A + C) ds q.kkdks dh rquk djus ij A + B0 JAB-S- (6)

19 B A B B ds q.kkdks dh rquk djus ij A + B + C C + A B iz- C C A, B, C ds eku leh- () es j[kus ij ekuk fd ( + ) ( + ) vfkok ( ) ( + ) ( + ) ( ) ( ) 4 + ( ) + ( + ) vd vd vd vd g% cos 5 + cos cos 5 5 JAB-S- (7)

20 cos cos cos JAB-S- (8) cos 65 6 cos 65 blh izdkj sin 6 65 ( 65) ( 6) sin ( 65) ( ) ( 65 6) sin ( 65) sin ( ) sin sin 65 vfkok tn tn tn + vd cos sin nk;k i{k vd rfkk b j[kus ij b tn tn btn + b b c tn b tn c tn + bc c tn c tn tn + c b b c L.H.S. tn + tn + b + bc leh- (), () o () ls c + tn + c vd...()...() vd...()

21 iz- π g% log tn + 4 π vc tn + 4 tn tn b + tn c + tn c tn 0 R.H.S. vd d d d d log tn π + 4 t j[kus ij d d d d log t d d d dt log t. dt d d d d dt. dt d t d π tn + d 4 vd vr% π + 4 u j[kus ij π tn + 4 d d tnu π tn + 4 d du tnu du d vd sec u π tn + 4 d π + d 4 π sec + 4 π tn + 4. π cos + 4 π cos + 4 π sin + 4 π π sin + cos JAB-S- (9)

22 sin π + 4 π sin + cos.sec vd vfkok fn;k gs sin [ ] sinθ j[kus ij θ sin...() leh () ls ds lkis{k vodu djus ij iz- 4 g% ekuk log sus ij sin sin θ sin θ sin sinθ cos θ sin [ sinθcosθ] sin [ ] sin θ [ sin θ.cos θ sin θ] θ sin ( sin ) vd sin d d d d sin d d (sin) log loglog (sin) log loglog. log(sin) ds lkis{k vodu djus ij ekuk sin t d d log d d ( ) log.log sin d d log d d log d d log (sin) + log(sin) d d log d d log d d logt + log( sin ) vd vd JAB-S- (0)

23 fn;k ;k gs& d d d dt log ( sin ) log log t + dt d d d (sin)log ( ) log d log sin sin + t d d cos log ( sin ) d (sin)log log + sin d log( sin ) d (sin)log cot.log + vfkok log + log + log +... vd iz- 5 g% log + nksuks i{kks dk oz djus ij log + log nksuks i{kks dk ds lkis{k vodu djus ij d d ( ) d d log d d d d d d d d d d vd d log d vd d d d d ( ) d d d d ( ) vd t π S5e cost os ds dt 5 d dt (e t cost) v5 t d d e cost+ cost e dt dt t JAB-S- ()

24 v5 t t e sin t e cost v 5e t [sint + cost] vd tc t π dk rc os v v π 5e π 5e π π sin + cos [ + 0] π v 5e bdkbz Roj.k dv dt 5 d t e ( sin t+ cost) dt vd d dt d dt f 5 ( sin cos ) t t t+ t e + e ( sin t,cos t) f 5 ( sin cos ) t t t t e + + e ( cos t sin t) t π ij d.k dk Roj.k f 5e t [ sint cost + cost sint] f ( 5) ( )e t [sint] f 0e t sint bdkbz π e f0 π sin π f0 e vfkok f () fn;k ;k Qu cgqinh; Qu gsa vr% ;g vurjk [, ] es lrr gkska d d f () d d [ 6 + 6] f' () + fn;k ;k Qu vurjk [, ] es voduh; gsa f () () 6() + 6 f ()0 f () 6 () vd vd f ()0 vr% f ()f () vd JAB-S- ()

25 vurjk [, ] es C bl izdkj gs fd f' (c)0 f' (c)c c + 0 ± 4 C ( ) ( ) C C ± 44 6 ± 6 C ± 6 C 6 ± C ± Li"V gs fd c ds nksuks eku vurjk [, ] es gs vr% C izdkj gs fd f' (c) 0 jkss izes; lr; gqvka iz- 6 g% ± d 5 d dd d d ge tkurs gs fd [, ] bl vd d 0 d 0 d.d 57 d 60 d 69 r n dd d d ( ) ( ) n d d n d d vd JAB-S- ()

26 ge tkurs gs fd& r ( ) ( ) vfkok σ n ( ) ( ) n + ( ) ( ) n vd 0.95 vd ( ) + ( ) ( )( ) n n ( ) + n ( ) σ σ + rσ σ σ + σ σ vd ( ) n ( ) ( )( ) σ rσ σ r nσσ σ +σ σ r σσ iz- 7 g% dk ij lekj;.k q.kkd σ b r σ dk ij lekj;.k q.kkd σ b r σ ge fl) djss fd b + b >r b + b >r vd...()...() vd JAB-S- (4)

27 leh- (),o () ls σ r σ σ + r σ σ >r vd σ + σ +σ σσ σ σ > > σ + σ >.σ σ σ + σ.σ σ >0 (σ σ ) >0 vd vfkok +...() () dk ij lekj;.k js[kk gs] b vd dh ij lekj;.k js[kk gsa ge tkurs gs fd () r b. b vd 6 b ve b ve r v e r 6 leh- ¼½ ls [ nksuks lekj;.k q.kkd _.kkred gs ] + 7 JAB-S- (5)

28 4 rc iz- 8 g% ekuk ler dk lehdj.k gs& + b + z c ler () ij ew fcunq lks Mks ;s c dh EckbZ P gsa vd...() vd P b c + + b c b c oz P + + vd nksuks i{kks dk oz djus ij P + + b c P + b + c ler () v{kks A (, 0, 0), B (0, b, 0), C (0, 0, c) ij dkvrk gs fcunqvks A, B, C ls funsz'kkd lerks ds lekurj [khps ;s lerks ds lehdj.k gkssa, b, z c, b, c ds eku leh- ¼½ es j[kus ij vhkh"v fcunq ifk gkska + + z p vfkok fcunq (,, ) ls gksdj tkus oks ler dk lehdj.k gksk& A ( + ) + B ( + ) + C (z )0 vd...() vd fn;s ;s lerks ds lehdj.k gs& + z...() JAB-S- (6)

29 iz-9 rfkk z5...() ler () vksj () Ecor~ gs blf, A +B C0...(4) ler () vksj () Ecor~ gs] blf, 5A 4B + C0...(5) lehdj.k ¼4½ vksj ¼5½ ls A + B C0 5A 4B + C0 A 0 B 8 C A B C ekuk k 5 9 A5k, B 9k, C k A, B, C ds eku lehdj.k ¼½ es j[kus ij 5k ( + ) + 9k ( + ) + k (z ) z 80 0 vd vd g% fn;k gs& f () log e...() + leh- ¼½ es j[kus ij f () log e...() + leh () es b j[kus ij b f (b) log e...() vd + b leh () vksj leh () dks tksm+us ij b f () + f (b)log e + log + e + b ( )( b) f () + f (b)log e ( + )( + b) vd b+ b f () + f (b)log e + + b+ b + b leh- () es j[kus ij + b + b f + b + b log + b e + b + + b...(4) vd JAB-S- (7)

30 + b b + b f + b log + b e + b + + b + b + b f + b log + b b e+ b+ + b leh- (4) o (5) ls + b f () + f (b)f + b vfkok cos 4 fn;k gs f () 0 + h j[kus ij tc 0 rc h 0 Rf (0 + h) lim h 0 lim 0 ( ) cos4 0+ h h cos4h h h lim h 0 lim h 0 lim h 0 lim h 0 sin h 4 h sin h h sinhsinh h sin h h (5) vd sin cos 8 8 vd 0 h j[kus ij tc 0 rc h 0 Lf (0 h) lim h 0 lim 0 ( ) cos4 0 h h cos 4h h h lim h 0 lim h 0 lim h 0 sin h h sin h.sinh h h sin h h vd JAB-S- (8)

31 fn;k gs f (0)4 Rf (0 + h)f (0 h ) f (0) vr% fn;k ;k Qu 0 ij lrr ugh gsa iz- 0 g% ekuk π/ 0 sin sin + cos vd d...() π/ 0 π sin d π π sin + cos leh- () vksj () dks tksm+us ij I + I I I nh?kz o`rr dk lehdj.k π/ 0 π/ 0 π/ 0 π/ 0 f ( ) d ( ) d 0 0 cos d...() cos + sin sin cos + sin + cos cos + sin d sin + cos d sin + cos π d [] 0 vfkok π 0 π vd vd vd B A C (0, 0) B A vd + b vd b b ( ) JAB-S- (9)

32 iz- leh- () dh rquk d d ekuk + d dt lehdj.k dk g gkska b vhkh"v {ksq4 CAB dk {ksq 4 4b 4b 4b b 0 d + sin sin sin ( 0+ 0) 4b ( + ) d d + 4 Pd d d + e P Q ls djus ij P, Q + I.F. e + d e dt d e logt t I.F. + 0 vd 0 sin 0 π πb vd 4 + I.F. I.F. Qd ( + ) ( + 4 ). + ( + ) 4 d d...() vd vd vd JAB-S- (0)

33 ( + ) g% fn;k ;k vod lehdj.k 4 vfkok (OR) + C vd cos d d + cos sin d d + cos sin cos d d + sec tn sec vd bl lehdj.k dh rquk d d + pq ls djus ij P sec Q tn θ. sec I.F I.F e e pd sec d vd lehdj.k dk g gksk& I.F tn e. I.F IF θ d vd e tn e tn. tn. sec d tn t j[kus ij d dt tn d d sec ddt. e tn e t t dt. e tn t e t dt dt d t e t dt dt. e tn t. e t e t dt. e tn t. e t e t + C. e tn tn. e tn e tn + C. e tn e tn (tn ) + C vd JAB-S- ()

34 g% rc ekuk izfrn'kz lef"v s gs (tn ) + C e tn n (s)6 igs ik ls ij fo"ke l[;k vkus dh?kvuk A {(, ), (, ), (, ), (, 4), (, 5), (, 6), (, ), (, ), (, ), (, 4), (, 5), (, 6), (5, ), (5, ), (5, ), (5, 4), (5, 5), (5, 6)} n (A)8 fo"ke l[;k vkus dh izkf;drk na P(A) ( ) ns () vd P(A) 8 6 l[;kvks dk ;ks 9 vkus dh?kvuk B {(, 6), (6, ), (4, 5) (5, 4)} n (B)4 l[;kvks dk ;ks 9 izkir djus dh izkf;drk nb P(B) ( ) ns () 4 6 mhk;fu"v?kvuk, A B{(, 6), (5, 4)} n (A B) vd vd vhkh"v izkf;drk na P (A B) ( B ) ns () 6 P(A B)p(A) + p(b) p(a B) vfkok (OR) g% fldds dks,d ckj mnkus ij 'kh"kz izkir djus dh izkf;drk vd 'kh"kz izkir u djus dh izkf;drk p(a) p( A ) p(a) JAB-S- ()

35 p( 0)0 ¼dksbZ 'kh"kz ugh½ p(a) p( A ) 4 vd vd p( )p ¼,d 'kh"kz½ p(a) p( A ) + p( A ) p(a) p( )p ¼nks 'kh"kz½ p(a) + p(a) 4 vd 'kh"kz dh l[;k dk izkf;drk cvu fueu gksk i 0 p i 4 4 vd g% kss dk lehdj.k gs& + + z + z () lehdj.k dh rquk + + z wz + d 0 ls djus ij u, v, w, d 5 u, v, w, d 5 vd u, v, w, d 5 kss dk dsunz JAB-S- ()

36 ( u, v, w),, A ds funs Z'kkd (,, z ) (, 4, ) gs] ekukfd B fljs ds funs Z'kkd (,, z ) gs] vd rc +, + z, z + z +, 4 +,, + 4, z 4,, z mùkj& B fljs ds funs Z'kkd (4,, ) gkss] vfkok (OR) g% nh bz js[kkvks ds lehdj.k gs& + z vd vd z () 7 4 z 6...() 5 ;gk,, z 5 l, m 5, n 7, 4, z 6 l +, m, n 5 z z l m n l m n vd R ds lkis{k folrkj djus ij (5 ) 7(5 7) + (9 5) vd vr% js[kk, ijlij izfrpnsn djrh gs js[kk () ij flfkr fcunq z r JAB-S- (4)

37 vfkkzr~ r, 5r, z 7r 5 js[kk () ij flfkr dksbz fcunq 4 r +, r + 4, z 5r + 6 izfrpnsn fcunq ds f, z 6 5 r r +, 5r r + 4, 7r 5 5r + 6 r r, 5r r 7, 7r 5r r r, r r 0 5r r 7, 5r r 7 0 ctz q.ku[km fof/k ls g djus ij r vd r 7 9 r r r 6 4 r, r vr% r dk eku izfrpnsn fcunqvks es j[kus ij r 5r z 7r mùkj& vr% izfrpnsn fcunq ds funsz'kkd (,, ) gkssa vd g%4 ekuk OX ds vuqfn'k ekd lfn'k i $ rfkk OY ds vuqfn'k ekd lfn'k $ j gsa JAB-S- (5)

38 Y P ( ) A O A B M N X vd Q ( ) ekuk OA ij fcunq uuur OP AOX A BOX B AOB A + B P(, ) bl izdkj f;k ;k fd fcunq P ls OX ij Ec PM gs OM uuuur uuur, MP OPM es] OM uuuur $ uuur i, MP $ j uuur uuuur uuur OP OM + MP vd i $ + $ j cos A OM OP, sin A MP OP, cos A sin A uuur OP $ i cos A + $ j sin A OB ij fcunq Q(, ) bl izdkj gs fd OQ uuur fcunq Q ls OX ij Ec QN gs vd uuur ON, uuur NQ OQ uuur ON uuur + NQ OQ uuur + OQ uuur i + j JAB-S- (6)

39 rc cos B ON OQ, cos B, sin A OQ uuur $ i cos B $ j sin B NQ sin B OQ vd uuur uuur OP. OQ uuur OP. OQ uuur ( $ i cos A + $ j sin A) ( $ i cos B $ j sin B) cos (A + B) cos A cos B sin A sin B g% $ i $ j cos (A + B) cos A cos B sin A sin B cos (A + B) cos A cos B sin A sin B vd fl) gqvka vfkok (OR) fn;k gs& ABC dh Hkqtkvks BC, CA o AB ds e/; fcunq Øe'k% D, E o F r r gs] 'kh"kz A dks ew fcunq ekudj 'kh"kz B o C ds flfkfr lfn'k Øe'k% b o c gsa uuur r rc] AB b ] uuur r AC c r r b+ c e/; fcunqd, E o F ds flfkfr lfn'k Øe'k%, c r, b r gkssa A F E vd B D C ABC dk {ksq uuur AB uuur AC b r c r...() DEF dk {ksq uuur DE uuur DF...() JAB-S- (7)

40 uuur DE E dh flfkfr D dk flfkfr lfn'k r r C b c uuur ur rfkk DF F r r r uuur c b c DE b r dk flfkfr lfn'k D dk flfkfr lfn'k uuur b r r r b c DF ur ur r b b c DEF dk {ksq c r vd r b r c r r b c 4 b r c r 4 b r c r 4 ABC dk {ksq [leh- () ls] ;gh fl) djuk FkkA vd JAB-S- (8)

mpp xf.kr Set C ( Higher Mathematics) (Hindi & English Version) 3 GB (i) A = 5, B = 7 (ii) A = 5, B = 7 (iii) A = 5, B = 7 (iv) A = 3, B = 5 uuur

mpp xf.kr Set C ( Higher Mathematics) (Hindi & English Version) 3 GB (i) A = 5, B = 7 (ii) A = 5, B = 7 (iii) A = 5, B = 7 (iv) A = 3, B = 5 uuur mpp f.kr Set C ( Higher Mathematics) (Hindi & English Version) le; % 3?k.Vs vf/kdre vad % 00 Time : 3 Hours Maimum Marks: 00 funsz'k& - lhkh iz'u gy djuk vfuok;z gsa - iz'u ls iz'u 5 rd olrqfu"b iz'u gsa