çfrykse f=dks.kferh; Qyu (Inverse trigonometric functions)
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1 Download FREE Stud Package from & Learn on Video Phone : , fo/u fopkjr Hkh# tu] ugha vkjehks dke] foifr ns[k NksMs rqjar e/;e eu dj ';kea iq#"k flag ladyi dj] lgrs foifr vusd] ^cuk^ u NksMs /;s; dks] j?kqcj jk[ks VsdAA jfpr% ekuo /kez iz.ksrk ln~q# Jh j.knksmnklth egkjkt çfrkse fdks.kferh; Qu (Inverse trigonometric functions) There are three sides to an argument: our side, m side and the right side. Introduction : ge fdks.kehfr; Quksa tsls, cos,, cosec, sec, cot ds ckjs esa i< pqds gsa] ftuds fd izkur Øe k% R, R, R - {(n ) / g}r - { R-(-, ), R- (-, ), R gsa n } R - {(n ) / }, R - { n } vksj ifjlj Øe k% [-, ], [-, ] R, Sin - : izrhd - ;k arc dks.k θ dks bl rjg fu:fir dgrs gs fd θ. bldk lh/kk vfkz gs fd -,d Qu ugha gs D;ksafd ;g Qu gksus ds f;s vko ;d fu;eksa dk larq V ugha djrk gs] sfdu mi;qdr fodyi ds :i esa [-, ] dks izkur ds :i esa vksj ekud leqpp; [ /, / ] dks blds ifjlj :i esa sus ij -,d,d eku Qu (gle valued function) cu tkrk gsa vr% - dks izkur [-, ] vksj ifjlj [ /, / ] esa,d Qu ds :i esa f;k tk ldrk gsa - dk vks[k uhps n kkz;k ;k gs] ftls ds / izfrfcec sdj izkir fd;k ;k gsa ls / rd ds Hkk dks js[kk ij niz.k cos - : Åij fn;s ;s dfkuksa ds vuqlkj cos - ;k arccos dks,d Qu dks rjg f;k tkrk gs ftldk izkur [-, ] vksj ifjlj [, ] gsa cos - dk vks[k Hkh blh rjg cos ds ls rd ds Hkk dks js[kk ij niz.k izfrfcec sdj izkir fd;k ;k gsa Page
2 Download FREE Stud Package from & Learn on Video Phone : , : an - ;k arc dks,d Qu ds :i esa sus ds f;s mldk izkur R vksj ifjlj ( /, / ) j[krs gsa - dk vks[k cosec - : cosec - ;k arccosec,d Qu gs ftldk izkur R - (-, ) vksj ifjlj [ /, / ] { } srs gsa cosec - dk vks[k sec - : sec - ;k arcsec,d Qu gs ftldk izkur R - (-, ) vksj ifjlj [, ] { / } gsa sec - dk vks[k cot - : cot - ;k arccot,d Qu gs ftldk izkur R vksj ifjlj (, ) gsa cot - dk vks[k Page
3 Download FREE Stud Package from & Learn on Video Phone : , Propert : - -, -, cosec - ds vks[k ewfcunq ds lkis{k lefer gsa vr% ge dg ldrs gs fd - (-) (-) - - cosec - (-) - cosec -. rfkk cos -, sec -, cot - ds vks[k fcunq (, / ) ds lkis{k lefer gs] blls ge dg ldrs gsa fd cos - (-) - cos - sec - (-) - sec - cot - (-) - cot -. Propert : T(T - ) (i) ( - ), - Proof : ekuk θ -. rks [, ] vksj [ /, /.] θ, izrhd ds vfkz }kjk ( - ) blh rjg ge fueu dk fl) dj ldrs gsa (ii) cos (cos - ). (iii) ( - ), R (iv) cot (cot - ), R (v) sec (sec - ),, (vi) cosec (cosec - ), ( - ) - cos (cos - ) dk vks[k ( - ) cot (cot - ) dk vks[k Page
4 Download FREE Stud Package from & Learn on Video Phone : , cosec (cosec - ) sec (sec - ) dk vks[k Propert : T - (T) (i) - ( ) n, [ n /, n / ] n ), [( n ) /,( n )] /,n Z Proof : ;fn [ n /, n / ], rks n [ /, / ] vksj ( n ). vr% ( ) n for [ n /, n / ]. nd Hkk dk lr;kiu Lo;a djsa - ( ) dk vks[k n, (ii) cos - (cos ) n, cos - (cos ) dk vks[k [n,(n ) ] [(n ), n ], n Ι Page
5 Download FREE Stud Package from & Learn on Video Phone : , (iii) - ( ) - n, / < < n /,n Z - ( ) dk vks[k (iv) cosec - (cosec ) dk vks[k Hkh - ds leku fg gsa cosec - (cosec ) dk vks[k (v) sec - (sec ) dk vks[k cos - (cos ) ds leku gsa sec - (sec ) dk vks[k Page
6 Download FREE Stud Package from & Learn on Video Phone : , (vii) cot - (cot ) n, (n,(n ) ),n Z cot - (cot ) dk vks[k Remark : ( - ), cos (cos - ),... cot (cot - ) vuko`rhd (non periodic) Qu gs tcfd - ( ),..., cot - (cot ) vko`fr Qu gsa Propert : / (i) cosec - () - (/), g% ekuk cosec - θ / θ - (/) - ( θ ) θ (as θ [, / ]) cos ec (ii) sec - cos - {/}, (iii) cot - { / ), > ( / ), < Propert : / (i) - cos - - g% ekuk A - vksj B cos - A vksj cos B A cos B A ( / B) A / - B D;ksafd A vksj / B [ /, / ] A B /. blh rjg ge dg ldrs gs fd (ii) - cot -, R (iii) cosec - sec -, Propert 6: ;ks o vurj dh lozlfedk, (Identities on additioin and substractioin): (i) - - -,,,oa ( ),,,oa 6 Page 6
7 Download FREE Stud Package from & Learn on Video Phone : , Page 7 g% ekuk A - vksj B - tgk ], [, (A B) - (A B) - - B A / B) (A / B A B A tc tc ), (, (ii) ], [, ; (iii) cos - cos - cos - ], [, ' (iv) cos - - cos - < ; cos ], [, ; cos (v) - - > < < > &, &, &, / &, / fn ; (vi) ,, uksv % uksv % uksv % uksv % (i) &, vksj &, (ii) < vksj, > < ; vksj < <, (iii) < ;k > ds f, bu lozlfedkvksa dks q.k/kez - dh lgk;rk ls iz;qdr dj ldrs gsa vfkkzr~ dks -,oa dks - esa cnrs gs tks fd /kukred gs Propert 7 : dqn mi;ks lozlfedk, dqn mi;ks lozlfedk, dqn mi;ks lozlfedk, dqn mi;ks lozlfedk, (More useful identities) (i) ( ) < < dk zkq
8 Download FREE Stud Package from & Learn on Video Phone : , (ii) cos ( ) cos cos cos - ( - ) dk zkq if if < (iii) ( ) < < > dk zkq (iv) ( ) < > dk zkq (v) cos > cos dk zkq (vi) ;fn z rks z (vii) ;fn z rks z (viii) (i) Page 8
9 Download FREE Stud Package from & Learn on Video Phone : , Eercise - -A ¼cgqfodYih; iz u½ dso,d fodyi lgh - dk eku gs& 7 (B) (D) - f() cos - cos - cosec - dk izlr gs& [-, ] (B) R (, ] [, ) (D) {-, } - f() - - sec dk ifjlj gs&, (B),, (D) bueas ls dksbz ugha - cosec - (cos ) oklrfod gksk ;fn& [, ] (B) R - ;fn cos[ {(cot )}] gks rks& (B), dk fo ke q.kt gs(d), dk q.kt gsa (D) 6- dk eku gs& 6 7 (B) (D) 6 7- ;fn gks] rks cos (cos ) (B) (D) 8- ;fn gks] rks cos cos (B) 6 (D) 9- ;fn rfkk θ cos gks] rks & 9 Page 9
10 Download FREE Stud Package from & Learn on Video Phone : , θ (B) θ (D) θ - ;fn < gks rks () dk eku gs& (B) (D) buesa ls dksbz ugha - a b, ¼tgk a >, b >, ab > ) dk eku gs& a b ab a b ab a b (B) ab a b (D) ab - dk eku gs& (B) (D) buesa ls dksbz ugha - cos cos dk ekugs& cos (B) 6 cos cos (D) buesa ls dksbz ugha lehdj.k cos cos dk & dksbz g ugha gsa (B) vf}rh; g gsa vuur g gsa (D) buesa ls dksbz ugha - ;fn cos gks] rks (B) (D) 6- lehdj.k dk g gs& 6 (B) - (D) buesa ls dksbz ugha n 7- ;fn cos αi gks] rks α i i n i Page
11 Download FREE Stud Package from & Learn on Video Phone : , n (B) -n (D) buesa ls dksbz ugha 8- lq ( ) ds lr; gksus sd f, ds ekuksa dk leqpp; gs& (-, ) (B) [, ], (D), _,d ls vf/kd fodyi lgh 9- ;fn z gks] rks z (B) z z z z z (D) 9 (z) - ;fn α vlfedk - - > dks larq V djrk gks] rks fueuff[kr esa ls fdldk eku fo eku gksk& α (B) cos α sec α (D) cos ec α gs ;fn& (B) (D) B ¼fo k;kred iz u½ - fueuff[kr ds eku Kkr dhft,& (i) (iii) (v) (ii) cos cos ec (iv) sec ( ) cos - fueuff[kr ds eku Kku dhft,& (i) (iii) (ii) cos cos - fueuff[kr ds zkq [khafp,& (i) - ( ) (ii) cos - () Page
12 Download FREE Stud Package from & Learn on Video Phone : , (iii) - ( - ) - fueuff[kr vlfedkvksa dks g dhft,& (i) > (ii) cos < (iii) cos < - fueuff[kr dk eku Kkr dhft,& (i) cos (ii) cos (iii) (v) 6 cos ec sec (iv) cos ec 6 cos (vi) cos cos 6 6- sec dk eku Kkr dhft,& 7- fueuff[kr ds eku Kkr dhft,& (i) cos (ii) (iii) 7 cos cos (iv) sec sec 8- fueuff[kr ds eku Kkr dhft,& (i) - () (ii) cos - (cos) (iii) - ((-6)) (iv) cot - (cot(-)) (v) cos 9 9 cos 9- ( θ),cos (cos θ), ( θ) cot (cot θ ) dk θ, ds f, eku Kkr dhft,& - fueuff[kr dk eku Kkr dhft,& (i) cot( a cot a) (ii) ( cos ), Page
13 Download FREE Stud Package from & Learn on Video Phone : , (iii) cos sec - fl) dhft, fd& (i) (ii) (iii) (iv) cos 8 7 cos cos cot cos( ) dks ds f, g dhft,& - lehdj.k cot dks g dhft,& - fueuff[kr lehdj.kksa dk g dhft,& (i) (ii) - fueuff[kr lehdj.kksa dks dhft,& (i),( > ) (ii) 6- ;fn > > gks] rks cos dk eku Kkr dhft,& 7- - (cos( - ) cos - ( (cos - ) dk eku Kkr dhft,a Eercise - -A ¼cgqfodYih; iz u½ dso,d fodyi lgh Page
14 Download FREE Stud Package from & Learn on Video Phone : , cos cos, dk eku gs& (B) (D) - ;fn, gks] rks [cos{cos (cos ) ( )}] dk eku gs & (B) (D) - ;fn gks] rks& (B) (/) (D) 8 - cot, < <, dk eku gs & (B) (D) - lehdj.k cos ( ) ( ) ds gksa dh la[;k gs& (B) (D) ls vf/kd 6-, dk U;wure,oa vf/kdre eku gs&, (B),, (D), n 7- ;fn cos >,n N gks] rks n dk vf/kdre eku gs& 6 (B) 9 (D) buesa ls dksbz ugha 8- vlfedk [cot ] 6[cot ] 9 tgk [.] egrre iw.kkzad Qu dks n kkzrk gs] ds leiw.kz gksa dk leqpp; gs& (B) 9 (D) buesa ls dksbz ugha 9- ;fn θ gks] rks θ dk eku gs& cos θ / (B) (D) - _,d ls vf/kd lgh Page
15 Download FREE Stud Package from & Learn on Video Phone : , cos cos cos dk eku gs& 7 cos (B) cos (D) cos - ds fdu ekuks ds f, > cos lr; gs& ] ds lhkh ekuks ds f, (B),, (D).7 - ;fn < < gks] rks cos dk eku gs& (B) cos (D) - ;fn cos ] rks & (B) (cos ) (D) (cos ) - n n n n dk eku gs& - - (B) - / (D) sec ( ) -B ¼fo k;kred iz u½ - ;fn X cos ec cos cot sec a vksj Y sec cot cos ec cos a; tgk a gks rks X,oa Y ds e/; lecu/k Kkr dhft,a mldk a ds inksa eas folrkj dhft,a - ;fn (i) f () cos cos gks] rks fueuff[kr ds eku Kkr dhft,& f (ii) f : Page
16 Download FREE Stud Package from & Learn on Video Phone : , fueuff[kr dks fl) dhft,& (i) cot cos when <. (ii) cos sec cot when < < - ;fn a b cos c gks] rks b cos dk eku Kkr dhft,a - fueuff[kr vflfedkvksa dks g dhft,& (i) cos - > cos - (ii) - > cot -. (iii) arccot arccot 6 > 6- lehdj.k ( ( )) dks larq V djus oks ds ekuksa dh la[;k Kkr dhft,a 7- fueuff[kr lehdj.k dks g dhft,& sec sec a sec b b sec a a ;b,a b. 8- fl) dhft, fd α < ds f, lehdj.k ( ) (cos ) α dk dksbz ew fo eku ugha gsa 9- fueuff[kr Jsf.k;ksa dk ;ksq Kkr dhft,& (i) (ii) 9... n n 7... n 7 (iii)... 6 n n... n(n ) - (i) lehdj.k cot ds lhkh /kukred iw.kkzad g Kkr dhft,a (ii) ;fn k /kukred iw.kkzad gks] rks iznf kzr dhft, fd lehdj.k k dk dksbz v kwu; iw.kkzad g ugha gksrk gsa 6 Page 6
17 Download FREE Stud Package from & Learn on Video Phone : , ;fn < < ds f, α rfkk rks α β dk eku D;k gksk \ β gks] rks fl) dhft, fd α β. ;fn > gks] - ;fn cos cos cos z, tgk,, gks] rks z z dk eku Kkr dhft,a Eercise - -A ¼LrEHk feku½ - LrEHk feku dhft,& LrEHk & ΙΙ LrEHk & Ι (B) (D) cos dk eku gs & (p) dk lkaf[;d eku gs & (q) cos 8 cos dk eku gs & (r) cos cot dk eku gs & (d) LrEHk feku dhft,& LrEHk & Ι LrEHk & ΙΙ cos ij vf/kdre gs & (p) (B) (D) ( ( ) ) ) (cos ) ij vf/kdre gs & (q) (cot ) ij U;wure gs & (r) (cot ) ij U;wure gs & (d) -B ¼dFku@dkj.k½ - dfku&% ;fn α, β lehdj.k 6 ds ew gs] rks cos α fo eku gs sfdu cos β ugha ( α > β). dfku&% cos dfku& lr; gs] dfku& lr; gs ; dfku&] dfku& dk lgh Li Vhdj.k gsa 7 Page 7
18 Download FREE Stud Package from & Learn on Video Phone : , (B) dfku& lr; gs] dfku& lr; gs ; dfku&] dfku& dk lgh Li Vhdj.k ugha gsa dfku& lr; gs] dfku& vlr; gsa (D) dfku& vlr; gs] dfku& lr; gsa - dfku&% (sec ) cot (cos ec ). dfku &% θ sec θ cot θ cos ec θ. dfku& lr; gs] dfku& lr; gs ; dfku&] dfku& dk lgh Li Vhdj.k gsa (B) dfku& lr; gs] dfku& lr; gs ; dfku&] dfku& dk lgh Li Vhdj.k ugha gsa dfku& lr; gs] dfku& vlr; gsa (D) dfku& vlr; gs] dfku& lr; gsa a b - dfku& % ;fn a >,b >, ab. m m n m dfku& % ;fn m,n N,n m gks] rks. n n n m dfku& lr; gs] dfku& lr; gs ; dfku&] dfku& dk lgh Li Vhdj.k gsa (B) dfku& lr; gs] dfku& lr; gs ; dfku&] dfku& dk lgh Li Vhdj.k ugha gsa dfku& lr; gs] dfku& vlr; gsa (D) dfku& vlr; gs] dfku& lr; gsa -C ¼vuqPNsn½ 6- vuqpnsn izfrkse o`rh; Quksa ds izkur vksj ifjlj uhps ifjhkkf kr fd; ;s GSA izkur ifjlj - [-, ], cos - [-, ] [, ] - R, cot - R (, ) cosec - (, ] [, ) sec - (, ] [, ). { } [, ] 6- <, rks ds gksa dk leqpp; gsa, (B),, (D) buesa ls dksbz ugha 8 Page 8
19 Download FREE Stud Package from & Learn on Video Phone : , ij cos ec cjkcj gs& (B) (D) 6- ;fn [, ], rks () dk ifjlj gsa 7, (B), [, ] (D), 7- vuqpnsn θ, < θ < ( θ) θ, < θ <, ( θ) θ, < θ < q θ θ,,, θ < θ, < θ θ, θ cos (cosθ ) θ, o θ θ, < θ mijksdr ifj.kkeksa ds vk/kkj ij fueu ds mrrj nhft, & 7- cos cjkcj gs& 7- ;fn < < (B) ;fn < < ;fn < < (D) ;fn < < cjkcj gs& cos ;fn < < (B) cos ;fn < < cos ;fn < < (D) cos ;fn < < 7- cos cjkcj gs& ;fn < < (B) ;fn < < ;fn < < (D) ;fn < < -D ¼lR;@vlR; dfku½ 8- dks.kksa A ( ) vksj B esa ls cmk dks.k A gsa 9 Page 9
20 Download FREE Stud Package from & Learn on Video Phone : , O;atd A (cot A) (cot A) dk eku < A < ( / ) ds f, A ls Lora gsa - vlfedk ( ) > lhkh > 9 ds f; lr; gsa n n(n ) - ;fn i n rks i n i - ;fn > rks. -E ¼fjDr LFkku dh iwfrz½ - ekuk a,b,c rhu /kukred oklrfod la[;k, gs vksj a(a b c) b(a b c) c(a b c) θ rks θ cjkcj gs& bc ca ab - ;fn cos 7 f () e ] rks f dk eku cjkcj gsa - cos dk eku arc ds inksa esa gksk A 6- dk g gsa 7- lehdj.k cos ( ) dk ds f, oklrfod gksa dh la[;k gsa Eercise - -A ¼iwoZorhZ JEE ijh{kk iz u½ IIT - JEE - 8 / - ;fn < <, rc [{ cos(cot ) (cot )} } (B) (D) IIT-JEE-7 Page
21 Download FREE Stud Package from & Learn on Video Phone : , ekuk fd (, ) bl izdkj gs fd (a) cos () cos (b ) dkwe (Column) ΙΙ esa fn;s ;s odro;ksa dks dkwd Column - ΙΙ esa fn;s ;s odro;ksa ls lqes djk;saa viuk mrrj ORS esa f;k ;k esfvªdl ds mfpr cqyksa (bubbles) dk dkk djds n kkz,aa dkwe - I dkwe -II ;fn a rfkk b, rc (, ) (p) o`rr flfkr gs (B) ;fn a rfkk b, rc (, ) (q) ( - ) ( - ) ij flfkr gs ;fn a rfkk b, rc (, ) (r) ij flfkr gs (D) ;fn a rfkk b, rc (, ) (s) ( - ) ( - ) ij flfkr gsa IIT-JEE- - dk og eku Kkr dhft, ftlds f, (cos ( )) cos( ) gs& / (B) (D) -/ IIT-JEE- - fl) dhft, fd cos cos. IIT-JEE- - ;fn < < ds f, 6... cos... gks rks dk eku gs & (B) (D) - IIT-JEE ( ) ds oklrfod gksa dh la[;k gs& (B) (D) vuur IIT-JEE ;fn ge izfrks fdks.kfefr; Quksa ds dso eq[; ekuksa dks gh s rks cos dk eku cjkcj 7 gksk& 9 (B) 9 9 (D) 9 -B ¼iwoZorhZ AIEEE/DCE ijh{kk iz u½ 8- ;fn cos ec, rks dk eku gs& (B) (D) Page
22 Download FREE Stud Package from & Learn on Video Phone : , ;fn cos cos α gks] rks cos α cjkcj gs& α (B α (D) α - dk eku tks ( ) cos dks larq V djrk gs&, (B), - (D) buesa ls dksbz ugha t t d - tc, tgk < <, rks cjkcj gs& t t d (B) - (D) Answers EXERCISE # - A. B. D. C. D. B 6. D 7. D (iii) 8. B 9. D. B. C. A. B. B. B 6. C 7. A 8. D 9. AB. CD. BD EXERCISE # - B. (i) < (ii) cos < (iii) no solution.. (i) (ii) (iii) 6 (iv) 6. (i) (ii) 6 6 (iii) (iv) (v) (v) 8 (vi) 6. (i) (ii) (iii) (i) (ii) 6 (iii) (iv). (i) (ii) 8. (i) (ii) (iii) 6 (iv) 7 (v) Page
23 Download FREE Stud Package from & Learn on Video Phone : , θ, θ 9. ( θ) ; θ, < θ.. X Y a. (i) ab c(a b) a b (ii) cos cos (cos θ) θ, θ < ; θ, θ. (i) [-, ) (ii) > (iii) (,cot ) (cot, ) θ, ( θ) θ, < θ < ; < θ 6. Infinite 7. ab 9. (i) ( n) (ii) (iii) cot (cot θ) θ, θ < θ, < θ <. Two solutions (, ) (, 7).. EXERCISE #. (i) (ii) (iii). ±. (p), (B) (q) (r) (D) (s). (p), (B) (p) (q)... (i) ± (ii). A. D. B 6. A 6. C 6. B 7. D 7. C 7. D 8. lr; 9. lr;. vlr;. (i) (ii) 6.. vlr;. vlr;.. e / 7. EXERCISE # -A. C. D. D. B. B 6. B 7. B 8. A 9. B. BCD. CD. ABC. AC. AD EXERCISE # B EXERCISE #. C. (p), (B) (q) (p) (D) (s). D. B 6. C 7. D 8. B 9. B. B. C Page
24 Download FREE Stud Package from & Learn on Video Phone : , MQB EXERCISE # ¼cgqfodYih; iz u½ dso,d fodyi lgh - cos. cos cos lurq V gksrk gs ;fn& (B) R (D) - lehdj.k ( ) ( ) dk,d g gs& (B) - (D) - a ds ekuksa dk leqpp; ftuds f, a ( ) cos ( ) dk de ls de,d g gks] gs& (, ] [, ) (B) (, ) (, ) R (D) buesa ls dksbz ugha - p,oa q ds lhkh lahko eku ftuds f, cos p cos p cos q larq V gks] rs& p, q (B) q >, q p, q (D) buesa ls dksbz ugha - fueuff[kr esa ls dksu lk,d lgh gs \ > (B) < (D) buesa ls dksbz ugha 6- lehdj.k ( ) ds gksa dh la[;k gs& (B) (D) nks ls vf/kd 7- ;fn [cot ] [cos ] tgk [.] egrre iw.kkzad Qu dks iznf kzr djrk gks] rks ds ekuksa dk leiw.kz leqpp; gs& (cos, ] (B) (cot, cos ) (cot, ] (D) buesa ls dksbz ugha 8- cot cos ec dk eku gs& (B) (D) 6 u 9- ;fn u cot α α gks] rks dk eku gs& α (B) cot α α (D) cot α Page
25 Download FREE Stud Package from & Learn on Video Phone : , a b - b c c a (B) dk eku gs ;fn fhkqt ABC esa C 9 gks& (D) (B) (D) - lehdj.k ds gksa dh la[;k gs& - cos cos dk eku gs& 8 / (B) -/ /6 (D) /,d ld vf/kd fodyi lgh - ;fn cos b cot gks] rks & a b 7 / (B) a b 7 / a b 7 / (D) a b / - rhu dks.k α, β,oa γ fueu izdkj fn, tkrs gs& β a ( ),,oa γ cos, rc & α > β (B) β > γ α < γ (D) α > γ - ekuk f() cos. rc f (B) f(k k ),k R f,k R (D) f(-) k 6- lehdj.k ( - a) ( - a - a ) ds f, fueuff[kr esa ls dksulk veku; gs\ a a (B) a a (D) a -, 7- ;fn (cos - (/) - (/) dk la[;kred eku a/b gks] rks & a b (B) a - b b a (D) a b 8- ;fn cos ec gks] rks dk eku gks ldrk gs& Page
26 Download FREE Stud Package from & Learn on Video Phone : , (B) (D) EXERCISE # ¼fo k;kred iz u½ - n kkzb;s fd f?kkrh; lehdj.k ( - ) ds ew r, s vksj t oklrfod vksj /kukred gs rfkk (r) (s) (t) dk eku Hkh Kkr dhft,a - ;fn ( - ),oa gks] rks,oa esa lecu/k Kkr dhft,a - ;fn arc arc arc, (,, > ) z gks] rks iznf kzr dhft, fd& (i) z z z (ii) z z ( z z z ) - ;fn c gks] rks iznf kzr dhft, fd b a b c a c a b ab - fl) dhft, fd & (i) cos cos cos 6 cos (ii) (iii) cos (cos ec cot ) ¼tgk ½ 6 6 (iv) ( z) cot(cot cot cot z) 6- ;fn X ( ) ( ) ( ); Y gks] rks ( - Y) dk eku Kkr dhft,a 7- fueuff[kr vlfedkvksa dks g dhft, & (i) (arc ) > (ii) - - (iii) fudk; arc arc > vksj [ ] > [cos ] tgk [] egrre iw.kkzad o Qu dks iznf kzr djrk gsa 8- uhps nh bz lfeej Jsf.k;ksa dk n inksa rd dk ;ks Kkr dhft, vksj rc vuur inksa dk ;ks Hkh Kkr dhft,a 6 Page 6
27 Download FREE Stud Package from & Learn on Video Phone : , (i) cot 9 9 cos cot... cot n (ii) n θ 9- ;fn θ ( θ) gks] rks θ dk O;kid eku Kkr dhft,a cos θ - eq[; ekuksa dk mi;ks djds dks,d dks.k esa ff[k,a a b -, tgk a b c, c dks g dhft,a c cos - fl) dhft, fd cos cot tgk (, ) cos Answers 6 EXERCISE #. C. C. D. C. A 6. B 7. C 8. C 9. A. A. B. A. AC. BC. AC 6. BC 7. ABC 8. ACD EXERCISE #. X Y n 8. (i) n inksa dk ;ksq cot vuur n inksa dk ;ksq n (ii) n inksa dk ;ksq vksj vuur n inksa dk ;ksq ( ) 9. θ n,n ( / ), n where n Ι 7. (i),, (ii).. {,, } for 9 Yrs. Que. of IIT-JEE & Yrs. Que. of AIEEE we have distributed alread a book 7 Page 7
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