g{g<g! ye<hkil<!ugh<h! kq{<mijl!yv!hius<osbz<! kq{<mijl!yv!ohvr<gx<xl<! kq{<mijl!leqkk<!ke<jlbx<x!osbz<!

Transcript

1 g{g<g ye<hkil<ugh<h kq{<mijlyvhius<osbz< kq{<mijlyvohvr<gx<xl< kq{<mijlleqkk<ke<jlbx<xosbz< klqp<fim<mh< himf~z<gpgl< gz<z~iqs<sijz?ose<je

2 klqp<fimnvs Lkz<hkqh<H.00 kqvk<kqbhkqh<h.004 lxhkqh<h.006 GPk<kjzuI LjeuIsi. dkbhi^<gve< g{qk-j{h<ohvisqiqbi? lifqzg<gz<z~iq (ke<eim<sq), ose<je Olzib<uitIgt< kqvw/nis<see< kqvgq. kr<gouz OkIUfqjzg<g{qkuqiqUjvbitI? LKfqjzg<g{qkuqiqUjvbitI? d/fi/nvsgz<z~iq? hs<jsbh<he<gz<z~iq? ohie<oeiq ose<je NsqiqbIgt< kqv uq. >vil< kqv g. nxqupge< hm<mkiiqnsqiqbi)g{qkl<*? ht<tqdkuqnsqiqbi)g{qkl<*? h/os/g/g/nvsqeiolz<fqjzh<ht<tq? nvsqein{<gt<olz<fqjzh<ht<tq? Ogiml<hig<gl<? ose<je dtf<k~iohm<jm kqvlkq vi. fl<hqg<jgo\bvi\< kqvlkq Os. uq\bi LKfqjzh<hm<mkiiqNsqiqjb)g{qkl<*? LKfqjzh<hm<mkiiqNsqiqjb)g{qkl<*? Heqkne<eit<oh{<gt<Olz<fqjzh<ht<tq? hib<zihqti{<m<oh{<gt<olz<fqjzh<ht<tq? -vibhvl<? ose<je jgzishvl<?kqvs<sq

3 ohivtmg<gl< hg<gr<gt<. w{<ljxgt< -. w{<ljxgt<. olb<ob{<ofig<ogim 8. ntjugt< hvh<hlx<xl<sx<xtu. %m<mdvur<gt< 7. sqzlg<gqbgxqbqm nxquqbz<gxqbqm 49. lmg<jgbqe<gxqbqm 5. g{g<gxqbqm bx<g{qkl< 9-4. hz<zxh<hg<ogijugt< bx<g{qklx<oxivjlgt< giv{qh<hmk<kz< yvhz<zxh<hg<ogijujblx<oxivhz<zxh<hg< Ogijubiz<uGk<kz< 6 5. kqiugt<gi[l<ljxgt< nelier<gt<lx<xl<fq'h{r<gt< 5. g{qklikqiqgt< 6. nxqljxucuqbz< siqhiik<kzg<giqbokx<xr<gt< 5 6. kvg<giqkqbigfq'hqg<gou{<cbokx<xr<gt< hgljxucuqbz< giicsqbe<ns<sk~vljx Ogim<ce<sib<U 7 7. (x, y ) lx<xl<(x, y ) Ngqb-VHt<tqgTg<G -jmobbt<tokijzu 79

4 8. Lg<Ogi{uqbz< Lg<Ogi{uqbz<uqgqkr<gt< Lg<Ogi{uqbz<Lx<oxiVjlgt< fqvh<hg<ogi{r<gtg<gielg<ogi{uqbz<uqgqkr<gt< 0 9. osb<ljxucuqbz< yvlg<ogi{k<kqz<yvht<tqupqg<ogimgt< svisiqgtqe<ucuqbz<uqtg<gl< 0. uquvr<gjtg<jgbitkz< jlbfqjzh<ohig<gntjugt< 9. ujvhmr<gt< OfIg<OgiMujvhmr<gt< 9. OfIg<OgiMujvhmr<gtqe<hbe<hiM lmg<jgh<hm<cbz< 4-4 wkqilmg<jgh<hm<cbz< 4-44

5 . w{<ljxgt< fl< ne<xim uip<uqz< w{<gt< wr<gl< Okie<Xgqe<xe/ njek<kqzl< w{<gt< dt<te we hpl<ohvl< gqovg<g g{qk uz<zfi< hqkigv^< %xqbt<tii</ w{<gjth< hx<xq fqjxb nxqf<k ogi{<omioleqz<, g{qkk<kqje fil< fe<g gx<xuigtiouil</ giiz<h<vm<iqs<gi^<wel<o\ileqfim<jms<siif<kg{qkuz<zfi<, uqr<rier<gtqe< nvsq g{qkl< lx<xl< g{qkr<gtqe< nvsq w{<gtqe< ogit<jg weg< %xqbt<tii</ w{<gt<lqgdbiqbh{<hgjth<ohx<xt<te/njuuqr<riek<kqz<wpl<hz uqeig<gtg<guqjmgi{dkugqe<xe/-f<kqbg<g{qkoljk-vilie\e< w{<gt< wekf{<higt< weg<%xqbt<tii</o\ileqfim<mgoviofg<giwel<uz<zfi -bz< w{<gjt gmut< hjmk<kii; lx<xju leqkeqe< kqxjlbiz< Wx<hm<mju we uqbf<k %xqbt<tii/ Lf<jkb ugh<hgtqz< -bz< w{<gt<, LPg<gt<, uqgqklx w{<gt<, uqgqklxi w{<gt< lx<xl< olb<ob{<gt< Ngqbux<jxh< hx<xq nxqf<k ogi{<omil</ w{<[ukx<g dku -bz< w{<gt< Wx<hm<me/ OkjugTg<Ogx<h lx<x w{<gt< dvuibq<e/-h<himk<kqz<w{<gtqe<h{<hgjth<hx<xqnxqf<kogit<ouil</.. w{<ljxgt<.. -bz<w{<gt< (Natural Numbers),,,. we<he -bz< w{<gtigl</ -ju w{<{qm dkul< w{<gt< weul< njpg<gh<hml</ -bz< w{<gtqe< okigkqjb N weg< Gxqh<Ohil</ N e< dxh<hgt< njek<jkbl<hm<cbzqmlcbiuqm<mizl<, N {,,, } wewpkouil</-r<g " " wel< GxqbQM N e< lx<x dxh<hgt<,,, e< Ljxjb yx<xq hm<cbzqmh<hmgqe<xe we<hjkd{i<k<kgqe<xk/-r<g,, 5, we<heyx<jxh<hjm-bz<w{<gt<weul<,, 4, 6, we<he -vm<jmh<hjm -bz< w{<gt<weul<njpg<gh<ohxl</ x 9 0, x 6 0, x 54 0 Ohie<x sle<himgtg<gk< kqiu okigh<h N zqvf<k ohxzil</ Neiz<? x + 5 5, x Ngqb sle<himgtqe< kqiugt<< N z< -z<jz/ Woeeqz<? -s<sle<himgt< Nz<-z<zikw{<0 Nz<fqjxUosb<bh<hMgqe<xe/.. LPw{<gt< (Whole Numbers) 0,,, we<he LP w{<gtigl</ LP w{<gtqe< okigkqjb W weg< Gxqh<Ohil</NgOuW {0,,, }. -h<ohik, x + 5 5, x Ohie<xsle<gtqe< kqiu?w{<0 NGl<; -KW z<dt<tk/-bz<w{<gt<njek<kl<lpw{<gtigl<; Neiz<, 0 wel<lpw{<-bz<w{<nz<z/nokofvk<kqz< x + 5 5, x + 9

6 we<he W-z<kQIUgjtg<ogi{<cvi/Woeeqz<?nux<xqe<kQIUgtie 0 lx<xl< we<helpw{<gtigi... LPg<gt< (Integers) 0,,,,, we<hju LPg<gtiGl</ -ux<xqz<,,, we<he lqjg LPg<gt<;,,, we<hegjxlpg<gtigl</ LPg<gtqe< okigh<h Z weg< Gxqh<hqmh<hMgqe<xK/NgOu, Z {,,,, 0,,,, }. njek<klpw{<gtl< LPg<gtiGl</ Neiz< Gjx LPg<gtie,,, we<he LP w{<gtigi/ -h<ohik x + 5 5, x + 9 wel<sle<gtqe<kqiugtie 0 lx<xl< we<he Z z< dt<tkiz<, ns<sle<himgt< Z z<kqiugjth<ohx<xt<te/neiz<? x + 5, x NgqbeZ z<kqiugjth<ohx<xqvg<gik/woeeqz<?-ux<xqe<kqiugtie lx<xl< Z z<njlbik/..4 uqgqklxw{<gt< (Rational Numbers) p, q LPg<gtigUl<, q 0 weul<ogi{<m q p wel<ucuqz<wpkh<hml<w{<gt< uqgqklx w{<gtigl</ uqgqklx w{<gtqe< okigkqjb Q weg<gxqh<ohil<. q p wel< uqgqklx w{<{qz<, q lqjg LP w{<{igul<, p lx<xl< q gtg<g -Jk< kuqik<k Ouoxf<k ohikg< giv{qblqz<jz weqz<, q p NeK OfIucU nz<zk kqm<mucuqz< -Vg<gqe<xK we<ohil</ wmk<kg<gim<mig, w{<gtigl</ lixig, 5 lx<xl< 4 7, 5 we<he OfIucuqZt<t uqgqklx we<he OfIucuqz< -z<zik uqgqklx w{<gtigl</neiz<yu<ouivuqgqklxw{<[l<keqk<kyvsllieofiucu njlh<jhh< ohx<xqvg<gl</ wmk<kg<gim<mig, lx<xl< 4 we<he OfIucuk<kqz<njlgqe<xe/yu<ouiVLPUl<uqgqkLXw{<NGl</wMk<Kg<gim<mig, 9 9 we<x LPju we wpkzil<; -r<g 9, Ngqb w{<gt< Z z< dt<te lx<xl< 0. weou 9 NeKuqgqkLXw{<NGl</weOunjek<KLPg<gTl<

7 uqgqklx w{<gtigl</ Neiz< LPg<gtx<x uqgqklx w{<gtl< d{<m/ wmk<kg<gim<mig? 5 4 NeKuqgqkLXw{<; Neiz<, -KLPnz<z/..5 N, W, Z lx<xl< Q okigh<hgtqz< +,,, Ngqbux<xqe<sqzh{<Hgt< %m<mz< + lx<xl< ohvg<gz< Ngqb -v{<ml< Q z< hqe<uvl< h{<hgjth< ohx<xqvh<hjkw{<gtme<dt<tflknehuk<jkg<ogi{<mnxqbzil</. x, y Ngqbe uqgqklx w{<gt< weqz<, x + y l< uqgqklx w{<{igl</ wmk<kg<gim<mig, lx<xl< -v{<ml< Qz<dt<te/nux<xqe<%Mkzie + + ( ) +. -KUl< Q z<dt<tk/-h<h{<h Q z<%m<mzg<gienjmuh<h{<hweh<hml</. x, y NgqbeuqgqkLXw{<gt<weqz<, x + y y + x. wmk<kg<gim<mig, 4 ( ) + ( ) 4 lx<xl< 4 ( ) + ( ) 4 +. weou K%m<mZg<giehiqlix<Xh<h{<Hwenjpg<gh<hMl</. x, y, z NgqbeuqgqkLXw{<gt<weqz<, x + (y + z) (x + y) + z. wmk<kg<gim<mig, 4 Ngqbju Q z<dt<te;,, ( 8) + ( 0) ( 4) 44, ( ) ( 4) + ( 0) K Q-z<%m<mZg<gieOsIh<Hh<h{<Hwenjpg<gh<hMl</ 4.0NeKuqgqkLXw{</OlZl< 0 + x x + 0 x. wz<ziuqgqklxw{< x-g<gl<-kohivf<kl</wmk<kg<gim<mig, lx<xl< uqgqklxw{<0 ju Q z<%m<mzqe<lx<oxivjldxh<hwe<gqoxil</ 5. yu<ouiv uqgqklx w{; x g<gl< x wel< uqgqklx w{<{qje x + ( x) ( x) + x 0 we njlblix gi{zil</ x J x e< Gjx nz<zk x-e< %m<mzqe<ofilixweg<%xouil</wmk<kg<gim<mig, we<gqxuqgqklxw{<{qx<g, 0

8 ( ) + 0 we<x uqgqklx w{<{qje + 0 LcgqxK/weOu NeK e<%m<mzqe<ofilixigl</ we -Vg<GliX gi{ 6. x, y NgqbeuqgqkLXw{<gt<weqz<, x y uqgqklxw{<{igl</x y J xy we<oxwpkouil</wmk<kg<gim<mig, 5, we<heuqgqklxw{<gt</olzl<, 5 ( 5) ( 5) ohvg<gzg<gienjmuh<h{<hweg<%xouil</ 0, yvuqgqklxw{</-f<kh<h{<hqjeq z< 7. x, y Ngqbe uqgqklx w{<gt< weqz<, xy yx NGl</ wmk<kg<gim<mig,, we<heuqgqklxw{<gt</ ( ), ( ) weou? ( ) ( ). -f<kh<h{<hqjeq z<ohvg<gzg<giehiqlix<xh<h{<h 7 7 wenjph<ohil</ 8. x, y, z we<he uqgqklxw{<gt<weqz< x(yz) (xy)z. wmk<kg<gim<mig,,, 7 we<heuqgqklxw{<gt</ 7 4 ( ) ( ) (), [() ( )] ( 6). weou 7 7 ( ) ( ) [()( ) ] f<kh<h{<hqjeq z<ohvg<gzg<gieosih<hh<h{<hwenjph<ohil</ 9. w{< NeK uqgqklx w{<{igl</ x x x we<hk njek<k uqgqklx x-g<gl< d{<jlbigl</wmk<kg<gim<mig, (). fil<gueqh<hk? ( x) y [( )(x)] y ( )xy xy; ( x) ( )[( )x] [( )( )x] x x. 0. yu<ouivh,s<sqblqz<zikuqgqklxw{< x-g<gl<, NeKuqgqkLXw{<NGl</ x OlZl<? x x. x x wmk<kg<gim<mig, x, yvuqgqklxw{</-r<g, x 0, 4 4 4, yvuqgqklxw{<ngl</olzl< x 4 x w{< J Q z< x e<kjzgqpqnz<zkohvg<gzqe<gqp< x 4 84 x OfIliXwe<gqOxil</ 5 7 4

9 . x, y, z we<heuqgqklxw{<gt<weqz<? x(y + z) xy + xz, (x+y) z xz+ yz NGl</ wmk<kg<gim<mig, x, y, z 5 weqz<, x (y + z) + 5, 0 ( ) xy + xz + (5) +. weou, x(y+z) xy + xz. 4 + ( ) 5 -u<uiox, (x + y)z , ( 5) 5 xz + yz NgOu, (x+y) z xz + yz f<kh<h{<hqje Q z<ohvg<gzg<gie%m<mzqe<hr<gqm<mh<h{<hwenjph<ohil</,,, 6, 7, 8 lx<xl< Nl< h{<hgt< Q z< dt<t Gxqh<hqm<m dxh<hg<gjts< siif<kqvg<guqz<jz/ weou? OlOz osie<e h{<hgt< N, W lx<xl< Z gtqe< dxh<hgtg<gl< d{<jlbigl</ Neiz<h{<H 4 NeK0 jus<siif<kt<tk/0 we<xw{<n z<-z<jzbikziz<, h{<h 4, N z<d{<jlbz<z/lixig 0 we<xw{< W lx<xl< Zz<dt<tkiz<, h{<h 4 NeK W, Z gtqz<d{<jlbigl</ h{<h 5, Gjx w{<gjts< siif<kt<tk/ weou N lx<xl< W gtqz< -K d{<jlbz<z/neiz<?-h<h{<h Z z<ohivf<kl</ h{<h 9 NeKw{<-Js<siIf<kqVh<hkiZl<, N, W, Z gtqz<, dxh<hwe<hkizl< N, W, Z gtqz<h{<h 9d{<jlbigqxK/ h{<h 0 NeK, H,s<sqblx<x w{<gtqe< kjzgqp< w{<gjth< ohivk<k njlukizl<, N, W, lx<xl< Z gtqz< dt<t H,s<sqblx<x dxh<hg<gtqe< kjzgqpqgt< nux<xqz<-vg<gikwe<hkizl<n, W, Z gtqz<h{<h0 d{<jlbigik/ gpqk<kz<, wel< osbzqjb %m<mz< + &zl< ujvbxg<gzil</ Q z< x, y dt<teoueqz<, x y x + ( y) NGl</ osbzq, Q z< hiqlix<xh< h{<hqje fqjxu osb<bik/ wmk<kg<gim<mig, 4 5, 5 4. weou? ugk<kz< wel<osbzqjbohvg<gz<gxq &zl<hqe<uvlixujvbxg<gzil</ Q z<x, y dt<telx<xl< y 0 weqz<, x y x. osbzq NeK Q z<hiqlix<xh< y h{<hqjefqjxuosb<bik/wmk<kg<gim<mig, , weou gpqk<kz<osbzq NeK Q z<osih<hh<h{<hqjefqjxuosb<bik/wmk<kg<gim<mig,,, we<x Q uqzt<tw{<gjtg<gvkg/ ( ) 9 40 ( ) ( ), 5

10 ( 4) 44 [( ) ] ( 4). weou, ( ) [( ) ]. -u<uiox, ( ) ( ) ( ), [( ) ] 9. weou, ( ) [( ) ]. gpqk<kz< osbzq NeK N z<, njmuh<h{<hqjefqjxuosb<bik/woeeqz<, 5, 7 we<x N e<dxh<hgjtg<gvkg/ 5 7, N e<dxh<hz<z/-u<uioxosbzq NeK N z<njmuh<h{<hqjefqjxuosb<bik/woeeqz<, , N e<dxh<hz<z. N, W, Z lx<xl< Q gtqe< h{<hgjth< hbe<hmk<kq yv sleqe< kqiu Gxqh<hqm<m w{<gtqe< okigkqbqz< dt<tki we nxqbzil</ wmk<kg<gim<mig, sle< 5x 0 0 e< kqiu x NGl<., N e<dxh<hikziz<, 5x 0 0 we<xsleqe<kqiun z<dt<tk/ 5x 0 we<xsleqe<kqiu x 0NGl</w{< 0NeK N z<-z<jz/neiz<?w{< 0 NeK W z<dt<tk/weou? 5x 0 wel<sle<nek N z< kqiu ogi{<cvg<guqz<jz/ lixig,nke<kqiu W z<dt<tk/ 5x we<xsljeg<gvkouil</-ke<kqiu x. w{< NeK N lx<xl< W -v{<czl<-z<jz;neiz?< NeK Z z<dt<tk/ weou, 5x we<xsleqe< kqiu N lx<xl< W uqz<-z<jz;neiz<, Z z<dt<tk/ 5 x we<xlx<oxivsljeg<gvkouil</-ke< kqiu x. w{< NeK Nz< z<jz, NeK W uqzl<-z<jz lx<xl< NeK Z zqzl<-z<jz/weou, N, W lx<xl< ZNgqbw{<okiGkqgtqz<sle< x NeKkQIuqjeh<ohx<xqVg<guqz<jz/ 5 Neiz<? w{< NeK Q uqz< -Vh<hkiz<, sle< x NeK Q uqz< kqiuqjeh< ohx<xqvg<gqxk/ -Vh<hqEl<,, π, we<he uqgqklxi w{<gt< we nxqf<kqvh<hkiz<, x 0, x 0, x π 0 Ohie<xsle<hiMgt< Qz<kQIUgjth< 6

11 ohx<xqvg<gik/ uqgqklxi w{<gjth< hx<xq nxqf<k ogit<ukx<g Le<hig uqgqklx w{<gjtksl ucuqz<wpkl<ljxjblq{<ml<nvib<ouil</..6 uqgqklxw{<gjtkslucuk<kqz<gxqk<kz< fqt<ugk<kz< Ljxbqz< uqgqklx w{<gjt kslg< GxqbQm<cz< wpkl< 5 5 LjxbqjenxqOuil</wMk<Kg<gim<mig, lx<xl< gtqe<kslhqe<enjlh<hgt< 7 LjxOb Olx<Gxqh<hqm<Mt<t fqt< ugljxbqz< hqe<hx<xqb uqkqjb fil< nxqf<k ogit<ouil</ -kx<g LPg<gjt wu<uix Gxqg<gqOxil< we<hjkg< gi{<ohil</ wmk<kg<gim<mig 4 J F~Xgt<, hk<kgt<, 4 ye<xgt< Ngqbeux<xqe< %Mkzigg< gi{zil</nkiuk, u<uiox weou 0,,,, 9 wel< w{< -zg<gr<gjtg< ogi{<m nux<xqe< Lglkqh<Hg<gjt, 0 0, 0, 0, gtqe< lmr<ggtig LcU osb<k wpkgqe<oxil</ -K OhizOu hqe<er<gjtbl<, 0,,,, 9 Jg< ogi{<m, 0 -, 0 -, 0 -, Ngqbeux<xqe< lmr<ggtigh<ohxzil</wmk<kg<gim<mig, weou? 8 5 J0.65 weg<gxqh<ohil</-r<ght<tqbqe<uzkhg<gk<kqz<lkz<w{<{ie 6 6 e< Lglkqh<H, -v{<miuk w{< e< Lglkqh<H 0 00 we nxqbzil</ Ht<tqjb hkqe<lieh< Ht<tq nz<zk kslh<ht<tq we njph<hi/ kslh<ht<tqbqe< -mh<hxk<kqz< dt<t 0 e<lkqh<h NGl</weOukslh<Ht<tqNeKkslucuk<kqz< J 8 wpkl<ohik LPjlh< hgkq, hqe<eh<hgkq nz<zk kslh<hgkq -ux<jxh< hqiqk<k wpkqmh<hbe<hmk<kh<hmgqxk/ 7

12 Olx<%xqbGxqbQm<cz<,.05e<ohiVt< h<ohipk, we<xhqe<ek<jkwmk<kg<ogit<ouil</-kqz<? (osb<ljxlx<xh<ohxgqxk) nmk<k, hqe<uvl<osb<ljxjbg<gvkouil</ (A)

13 gi{zil< (B) osb<ljxbiek(a)-zqvf<k lq{<ml<nvl<hligqxk/ 5 e< ksl ucul< wpkl<ohik fqt<ugk<kz< Ljx Lx<Xh<ohXujkg< (lqkq 0 uvgqe<xhcbiz<). weou 5 Lx<Xh<ohXl< ksl uqiqjuh< ohx<xt<tk/ Neiz< 7 5 e< kslucug< Gxqk<kzqz<? fqt<ugk<kz< Ljx wf<fqjzbqzl< Lx<Xh<ohxuqz<jz (lqkq 0 we wf<k fqjzbqzl< dt<tk). nok Ofvk<kqz< (B) gm<mk<kqz< ohxh<hml< lqkll< (A) gm<mk<kqzt<t lqkll< ye<ox we nxqg/ weou? gm<ml< (A), gm<ml< (B) -ux<xqx<gqjmob uvgqe<x -zg<gr<gt<, 8, 5, 7,, 4 uiqjs lixilz<kqvl<hk<kqvl<h gqjmg<gl</weou 7 5 e<kslucug<gxqk<kjzlx<xh<ohxi lx<xl<lqt<okimi<ljx (Non-terminating and Recurring)we<Ohil</-kje , 7 we wpkouil</ -r<g 8574 e< lqk -mh<hm<mt<t OgiM (bar)? 7 5 e< fqt<ug Ljxbqz< 8574 kqvl<hk< kqvl<h nok uiqjsbqz< -ml<ohxujkg< Gxqg<Gl</ 5 g<gie Lx<Xh<ohXl< ksl uqiquie J Lx<Xh<ohxik lqt< okimvigul< gvkzil</woeeqz<, OlZl<nOk g<g, 9

14 ( 4 + ) ( 0) ( 9 + ) ( 0) ( 9 + ) NgOu, yu<ouiv uqgqklx w{<[l< Lx<Xh<ohXl< nz<zk Lx<Xh<ohxik lqt< okimv< Ljxbqz< ksl uqiquig njlgqxk/ fqt< ugljxbqz< ohxh<hml< lqkr<gt< Gjxbqz<zi LPg<gtigUl<? ug w{<gjt uqm GjxuigUl< -Vh<hkiz<kie<, uqgqklx w{<gt< -k<kjgb sqxh<hh< h{<hqjeh< ohx<xt<te/ Lf<jkb fqjzbqz< ohxh<hm<m lqkliek OuoxiV fqjzbqz< lqklig lxhcbl<< uvujkg< gi{zil</ weou? uqz< dt<t -zg<gr<gt< kqvl<hul< uv Nvl<hqg<gqe<xe/ -ke< lxkjz d{<jlbiwenvib<ouil</lx<xh<ohxl<nz<zklx<xh<ohxikokimiujgksl uqiqugt<wkjeg<gxqg<gqe<xe@-f<kuqeijuwmk<kg<gim<mgt<&zl<nvib<ouil<. (i) ksluqiqu 0.45 Jg<gVKg (ii) ksluqiqu Jg<gVKg/ x we<g/weou 00x x x ( ) ( ) nz<zk 99x 45 nz<zkx (iii) ksluqiqu 0.49 Jg<gVKg/ x we<g/ngou00x x x 00x ( ) ( ) x 5 nz<zk x

15 weou, yu<ouivlx<xh<ohxl<nz<zklx<xh<ohxiklqt<okimiujgksluqiqu,yv uqgqklx w{<{qjeg< Gxqg<Gl</ nkiuk, yvlx<xh<ohxl<nz<zklx<xh<ohxik lqt<okimiujgksluqiqju LPuqe<gQp<LP we<xucuk<kqz<njlg<g-bzl</..7 uqgqklxiw{<gt< Lx<Xh<ohxik Neiz< okimi uiqjsbx<x ksl uqiqugjtg< gvkouil</ wmk<kg<gim<mig, u<uqiquqz< 0 lx<xl< got dt<te/ OlZl< gt< NeK H,s<sqbl<, H,s<sqbr<gt<, H,s<sqbr<gt<, 4 H,s<sqbr<gt<, Ngqbeux<xiz< hqiqg<gh<hm<mt<tjk nxqg/ -kqz< kqvl<hk< kqvl<h uvl< okimi ujg nmg<ggt<-z<jz/weou?-k<kjgbksluqiqugt<uqgqklxw{<gjtg<gxqg<gik/ nk<kjgb ksl uqiqugt< uqgqklxi w{<gjtg< Gxqg<Gl< we<gqoxil</ uqgqklx lx<xl< uqgqklxi w{<gjtob olb<ob{<gt< we<hv</ yv ksl uqiquiek hqe<uvueux<xt<wokel<yvucuqz<lm<mol-vg<gl</ (i) Lx<Xh<ohXl<ucuqz<-Vg<Gl</ (ii) Lx<Xh<ohxikNeiz<kqVl<hk<kqVl<huVl<okimIujgksluqiqU/ (iii) Lx<Xh<ohxiklx<Xl<kqVl<hk<kqVl<huvikokimIujgksluqiqU/ NgOuyu<ouiVksluqiqUl<yVolb<w{<{iGl</yu<ouiVolb<w{<{qx<Gl<ksl uqiqud{<mwefil<%xzil</yvolb<ob{< x-e<ksluqiquqz<0 n e<g{gr<gtqz< Gjxf<khm<sl<ye<xiuKlqjgLPw{<Ng-Vf<kiz<? nkjelqjgolb<w{< we<ohil</ -u<uiox, olb<ob{< x NeK lqjgbigoui nz<zk h,s<sqbligoui -z<zikqvh<hqe<nkgjxolb<ob{<weh<hml</wmk<kg<gim<mig, we<hklqjgolb<ob{</neiz<, Gjxolb<ob{<{iGl</ olb<ob{<gtqe< okigkqjb R weg<gxqh<ohil</ NgOu R NeK uqgqklx lx<xl< uqgqklxi w{<gtqe< okigh<higl</ -bz< w{<gt<, LP w{<gt<, LPg<gt<, uqgqklx lx<xl< uqgqklxi w{<gt< njek<kl< olb<ob{<gtigl</ wf<kouiv uqgqklx w{<[l< uqgqklxi w{<{igik/ -jkh<ohizou? wf<kouiv uqgqklxi w{<[l< uqgqklx w{< NgiK/ R z< upg<glie %m<mz<, gpqk<kz<, ohvg<gz< lx<xl< ugk<kz< uqkqgt< biul< d{<jlbigl</ Gxqh<hig, R z< dt<t hiqlix<x lx<xl< hr<gqm<mh< h{<hgtiue;x, y lx<xl< z WOkEl<&e<Xolb<ob{<gTg<G? (i) x + y y + x, xy y x; (ii) x(y + z) xy + xz; (iii) (x + y)z xz + yz,

16 Olx<%xqBt<th{<HgtqzqVf<Khqe<uVueux<jxg<gi{zil</ (i) x (y z)x [y + ( z)]xy + x( z) xy xz. (ii) (x + y) (x + y)(x + y) (x + y) z, -r<g z x + y weh<hqvkqbqm xz + yz x (x + y) + y (x + y) xx + xy + yx +yy x + xy + xy + y x + xy + y. (iii) (x y) (x y)(x y) (x y) z, -r<g z x y weh<hqvkqbqm xz yz x (x y) y (x y) xx xy (yx yy) x xy xy + y x xy + y. (iv) (x + y) (x y) (x + y) z -r<g z x yweh<hqvkqbqm xz + yz x (x y) + y (x y) xx xy + yx yy x y. wmk<kg<gim<m : Nz<Gxqg<gh<hMl<uqgqkLXw{<{qjeg<gi{<g/ kqiu: x we<g/ x x x x ( ) ( ) x 75 nz<zk x. 99 wmk<kg<gim<m : we<hjkfqbibh<hmk<kul</ 9 9 kqiu: -r<g. 0 (0) (9 + ) + ( 0) ( 0) OlOzdt<tLjxbieKLcuqe<xqose<Xogi{<OmbqVg<Gl</NgOu, nz<zk ( 9 + ) + ( + ) wmk<kg<gim<m : JLPuqe<gQp<LPucuqz<gi{<g/ kqiu: x. 5 we<g/ x x x x nz<zk 99x 50 nz<zk x. 99 y.5 we<g/ y.5.. 0y 5.

17 00y y 0y nz<zk 90y 7 nz<zk y weou? wmk<kg<gim<m 4: lx<xl< Ngqb uqgqklxi w{<gjtg< %m<mg/ uqjm yv uqgqklx w{<{ig -Vg<Glieiz<,nkjeLPuqe< gqp<lpucuqz<gi{<g/ kqiu: x lx<xl< y we<g/ x + y r<g x + y e<ksluqiqulx<xh<ohxikneiz<kqvl<hk<kqvl<huvl<okimiujgjbs< siv<f<kk/ weou, x + y yvuqgqklxw{</a 0. we<g/hqe<h0a. 0a a nz<zk 9a nz<zk a 9. x + y 9. Gxqh<H; Olx<g{<m wmk<kg<gim<czqvf<k, -V uqgqklxi w{<gtqe< %Mkz< yv uqgqklxi w{<{ig-vg<gou{<cbnusqblqz<jz/-jkh<ohizou, -VuqgqkLxi w{<gtqe< ohvg<gx<hzel< yv uqgqklxi w{<{ig -Vg<gOu{<cb nusqblqz<jz/ ( yvuqgqklxi w{</neiz<, yvuqgqklxw{<). we<x uqgqklxi w{<j{g< gvkouil</ -uqe< ksl uqiqju gi[l< LjxjbfqjeUogit<Ouil</ weou filxquk? e< ksluqiqu Lx<Xh<ohxilZl<, kqvl<hk< kqvl<huvl<okimiujg-z<zilzl<dt<tk/weou yvuqgqklxiw{<{igl</ -u<uiox,, 5, 7, uqgqklxiw{<gt<wefqxuzil</

18 g{qkk<kqz< π lx<xl< e wel< keqs<sqxh<hlqg<g -V uqgqklxi w{<gt< -ml<ohxgqe<xe/ wf<kouiv um<mk<kqzl< nke< Sx<xtU lx<xl< uqm<ml< Ngqbux<xqx<gqjmOb dt<t uqgqkl< yv lixqzq NGl</ -l<lixqzqjb gqovg<g wpk<kie π Nz< Gxqh<Ohil</ π e< ksl uqiqu.4596 we<hkigl</ -K Lx<Xh<ohxik, kqvl<hk< kqvl<h uvik okimvig dt<tk/ wel< uqgqklx 7 w{<{qjeπ e<okiviblkqh<higg<g{g<gqmzqz<hbe<hmk<kqbt<otil</-f<kqbg<g{qk Oljk -vilie\e< π J dt<tmg<gqb hz $k<kqvr<gjt dvuig<gqbt<tii/ 97 l< N{<Mujvbqz<, π g<gieksluqiquqz<,000,000 -zg<gr<gt<-ml<ohxgqxujgbqz< gi{h<hm<mt<tk/ hz Ogic -mk< kqvk<kk<kme< π e< uqiqjug< gi{<hk we<hk kx<ohik fjmohx<x uvgqx gcelie Neiz< uqbk<kg h{qbigl</ nmk<kkig? ,,,, we<x Ljxob{<gtqe< lkqh<hgjtg< g{g<gqmjgbqz< 4 5 -jubjek<kl< yv Gxqh<hqm<<m olb<ob{<{qx<g lqg lqg nvgijlbqz< njlujk nxqbzil</ nf<k w{<{qje e weg< Gxqh<Ohil</ e e< ksluqiqu e KUl< Lx<Xh<ohxik lx<xl< kqvl<hk< kqvl<h uvik okimvig dt<tk/ -f<k uqgqklxiw{< ejh<hx<xqolzl<dbiugh<hgtqz<fqr<gt<hck<kxqbzil</..8 R-z<uiqjsdxU sqz ohivm<gjt njugtqe< h{<hgtg<ogx<h uiqjs hmk<kl<ohik, nju uiqjsh<hmk<kh<hm<mju weh<hmgqe<xe/ wmk<kg<gim<mig, li{uigjt nuvui dbvk<kqx<ogx<h, lqgg< Gjxf<k dbvk<kqzqvf<k lqg nkqghm<s dbvl< wel< nch<hjmbqz<fqx<gs< osb<kiz<, nl<li{uigt<uiqjsbqz<fqx<hkigg<%xgqe<oxil</ x lx<xl< y olb<ob{<gt< we<g/ y x NeK lqjg w{< weqz<, xnek y-j uqmg< Gjxf<kw{< nz<zky NeK x Juqmh<ohiqbw{<weh<hMl</ xnek y Juqmg< GjxuieK we<hjkg<gxqg<g x < y we<xgxqbqm<jmbl<, y NeK x Juqmh<ohiqbK we<hjkg<gxqg<g y > x we<x GxqbQm<jmBl<hbe<hMk<KOuil</NgOu, x < y lx<xl< y > x we<he y x yv lqjg w{< we<x yov ohivt< kvue/ y x Gjxw{<weqz<, (y x) x y lqjgbigl</ weou? y < x nz<zk x > y/wmk<kg<gim<mig, 9 lx<xl< 7 we<x-vw{<gjtwmk<kg<ogit<ouil</9 7 NeKlqjgwe<hkiz<, 7 < 9 nz<zk 9 > 7 weg< gqjmg<gqxk/ nmk<k, 9 lx<xl< 7 gjtg< gvkouil</ filxquk? ( 9) ( 7) 9 + 7, yvgjxw{</weou, 9 < 7 nz<zk 7 > 9. x lx<xl< y we<heolb<ob{<gt<we<g/hqe<uvl<&e<xqz<wokel<ye<xlm<mol d{<jl/ (i) x y yvgjxw{<; nkiuk, x < y (ii) x y yvlqjgw{<; nkiuk, x > y (iii) x y 0; nkiuk, x y. x < y nz<zk x y we<hjk xnek y J uqms<sqxqb nz<zkslw{<we<ohil</-kje x y wewpkouil</-k?xnekyjuqm 4

19 lqjgbz<z we<x Gxqh<hjk nxqf<k ogit<g/ -u<uiox, x > y nz<zk x y weqz<, x NeK y Juqmh<ohiqbnz<zKslw{<we<Ohil</-kjex ywewpkouil</ -K?xNeKyJuqmGjxf<kkz<zwe<XGxqh<hjknxqf<Kogit<g/ x NeKlqjg w{<weqz<? x > 0 NGl<. x NeKGjxnz<zweqz<? x 0 NGl</ x Gjxw{<weqz<, x < 0 NGl</ x NeK lqjg w{< nz<z weqz<? x 0 NGl</NgOu wu<uqv olb<ob{<gt< x lx<xl< y gtg<g, hqe<uvl<&e<xqz<ye<xlm<mold{<jlbigl</ x < y, x y, x > y. -kjeolb<ob{<gtg<gielh<hqvqh<h (trichotomy) uqkqwe<ohil</ x, y lx<xl< z we<he x < y lx<xl< y < z, welix dt<t WOkEl< &e<x olb<ob{<gt< weqz<, y x lx<xl< z y gt<lqjgw{<gtigl</weou, (y x) + (z y) lqjgngl</nkiuk, z x lqjg NGl</ weou? x < z. x < y lx<xl< y < z weqz<, x < z d{<jlbigl</ -h<h{<hqje lix<xh<h{<hwe<ohil</ x < y lx<xl< y < zwe<hjk x < y < zweg<gxqh<ohil</-r<g y NeK x lx<xl< zgtg<g-jms<osvgzigdt<tkwe<ohil</ olb<ob{< x lx<xl< 0 Jg< gvkg/ hqe<uvhjugtqz< ye<x lm<mol d{<jlbigl</ x < 0, x 0, x > 0. x < 0 weqz<, x x x Gjxw{< Gjxw{< lqjgw{</ x 0 weqz<, x x x x > 0 weqz<, x x x lqjgw{< lqjgw{< lqjgw{</ weou wf<kouiv olb<ob{< x-g<gl< x 0/ Gxqh<hig H,s<sqblqz<zi wu<ouiv olb<ob{<x (> 0 nz<zk < 0)-g<Gl<x > 0. WOkEoliVolb<ob{<x Jg<gVKg/x 0 we<xohipk?x e<lm<mlkqh<hx we<xl<;x < 0 we<xohipk?x e<lm<mlkqh<h x we<xl< ujvbxg<gh<hmgqxk/ x e< lm<mlkqh<hqje x we<x GxqbQm<ceiz< Gxqh<Ohil</ weou? x x, x 0 we<xohipk; x x, x < 0 we<xohipk/ filxquk? x 0, x x lx<xl< x x. wmk<kg<gim<m 5: π lx<xl< -ux<jxuiqjsh<hmk<k/ 7 kqiu: π.458, π π NeKlqjgw{</ π < 7. wmk<kg<gim<m 6: uqgqklx w{<gt<.0 lx<xl<.0 -ux<xqx<g -jmbqz< WOkEl< fie<gw{<gjt-jms<osvgzigs<osig<gul</ 5

20 kqiu: -r<g > 0 weou.0 <.0..0,.0,.0,.04 Ngqb w{<gjtg< gvkg/ -jugt< Lx<Xh<ohXl< ksl uqiqu ucuqzt<tkiz<, njek<kl<uqgqklxw{<gtigl</ > 0. weou.0 < > 0, we<hkiz<.0 < > 0,we<hkiz<.0 < > 0, we<hkiz<.0 < > 0, we<hkiz<.04 <.0..0 <.0 <.0 <.0 <.04 <.0. wmk<kg<gim<m 7:.0 lx<xl<.0 gtg<gqjmob fie<g uqgqklxi w{<gjt -jms<osvgzigs<osig<gul</ kqiu: > 0. weou?.0 <.0. a , b , c , d , wel< fie<g w{<gjtg< gvkg/ -jubiul< Lx<Xh<ohxik, kqvl<hk< kqvl<h uvl< okimiujgjbs<sivikouu<ouxieksluqiqugjtg<ogi{<mt<tkiz<?ouu<ouxie uqgqklxiw{<gtigl</olzl<, a > 0, b a > 0, c b > 0, d c > 0,.0 d > 0..0 < a < b < c < d <.0. wmk<kg<gim<m 8:. 00 lx<xl<. 00 uqgqklxi w{<gtg<gqjmob WOkEl< -V uqgqklxw{<gjt-jms<osvgzigs<osig<gul</ kqiu: lx<xl< (Okiviblig*> 0. 6

21 . 00 < lx<xl<.446 Ngqb w{<gjtg< gvkg/ Lx<Xh<ohXl< ksl uqiqjug< ogi{<mt<tkiz<,-jubqv{<ml<uqgqklxw{<gtigl</olzl< > < > < > < <.446 <.446 <.00. Olx<%xqBt<t wmk<kg<gim<mg<gtqzqvf<k, a, b wel< wu<uqv ouu<oux olb<ob{<gt< a < b we -Vh<hqe<, r we<x uqgqklx w{<{qje a < r < b welix gi{zil<. a lx<xl< b ye<xg<ogie<x lqg nvgijlbqz< -Vh<hqEl<,OlOz%xqBt<t h{<h d{<jl/ -h<h{<hqje R z< Q uqe< nmik<kqh< h{<h we<gqoxil</ yv OfIg<Ogim<cz< njlf<k -V ouu<oux Ht<tqgTg<gqjmOb OuoxiV keqk<kh< Ht<tq -Vg<Gl<< we<x d{<jlbme< R e< nmik<kqh< h{<hqje yh<hqmzil</ -k<kjgb yh<hqm? olb<ob{<gjtofig<ogimye<xqeiz<gxqg<gupqugg<gqxk/ njek<kuqgqklxw{<gtl<olb<ob{<gotwe<hjkfqjeuqz<ogit<g/ Neiz<? uqgqklx nz<zik olb<ob{<gtl< d{<m we<hjk nxqouil</ wmk<kg<gim<mig,,, π, Ngqbe olb<ob{<gt<;neiz<uqgqklxw{<gt< nz<z/-h<ohik, sle<gt< x 0, x 0, x π 0, R z<kqijuh<ohx<xt<te/ Neiz<? x + 0 we<xsleqx<g R z<kqiu-z<jz/woeeqz<? x wewf<kouiv olb<ob{< x JBl< ohx -bzik/ -k<kjgb sle<himgjt flk ugh<hqz< gvkh< Ohiukqz<jz/ hbqx<sq.. hqe<uvueux<xqx<gksluqiquqjeg<gi{<g. 7 (i) (ii) (iii) hqe<uvl<uqgqklxw{<gjtlpuqe<gqp<lpucuqz<gi{<g: (i) (ii) 0. 7 (iii) 0.5 (iv) 0.5 (v) siqbinz<zkkuxiweg<%xg/ (i) we<hkyvlpw{<. 7 (ii) we<hkyvlp/ (iv) 4 7

22 (iii) we<hkyvuqgqklxw{</ (iv) 0.6 we<hkyvuqgqklxw{</ (v) yu<ouivksluqiqul<yvolb<ob{<ngl</ (vi) yu<ouivolb<ob{<[l<yvuqgqklxw{<ngl</ (vii) 0. yvuqgqklxw{</. olb<ob{<ofig<ogim Le< ugh<hgtqz< olb<ob{< OfIg<Ogim<cje nxqouil</ olb<ob{<gt< wu<uix OfIg<Ogim<ce< Ht<tqgtigg< Gxqg<gh<hMgqe<xe we<hjk lq{<ml< nxqf<k ogit<ouil</ yvofig<ogim<cz<wokel<yvht<tqjbg<gvkg/-h<ht<tqjb O weh<ohbiqmouil</ w{< 0 J -h<ht<tq Gxqh<hkigg< ogit<ouil</ Ht<tq O-g<G uzh<hxlig w{< Jg< Gxqg<Gl<Ht<tqbigOfIg<Ogim<cz< A Ht<tqjbg<gVKg/OfIg<Ogim<Mk<K{<M OA NeK YvzG fqtljmbk we<g/ Ht<tq A-jbh< ohivk<k fqtl< OA -Vg<Gl<< we<hjk fqjeuqz<ogit<g/ht<tq A,w{< Jg<Gxqg<gqxKwe<Xogi{<mhqe<H A fqjzbie Ht<tqbiGl<; OAe< fqtl< nzgigl</ OA J ntuogizigg< ogi{<m yu<ouiv olb<ob{<{qjebl< ng<ogim<cz< yv Ht<tqbigg< Gxqg<gzil</ -k<kjgb OfIg<Ogim<jm olb<ob{< OfIg<OgiMnz<zKolb<g<OgiM (Real Number Line) we njph<ohil</ olb<g<ogim<cz< wu<uix lqjg LPg<gt< Ht<tqgtigg< Gxqg<gh<hMgqe<xe we<hjklq{<ml<fqjeuogit<ouil</ O lx<xl< A LjxObw{<gt< 0 lx<xl< gjtg< Gxqg<gqe<xe/ O g<g uzh<hg<gk<kqz< nzgk< okijzuqzt<t Ht<tq (OA-uqe< fqtk<jkh< Ohiz lmr<g), nzgk< okijzuqz< dt<t Ht<tq,, -ju LjxOb w{<gt<,, Ngqbux<jxg<Gxqg<Gl<(hml<. Jh<hiIg<gUl<). hml<. -u<uiox, O-g<G-mh<Hxlig,,, w{<gjtg<gxqg<gl<ht<tqgjtljxob nzg, nzggt<<, nzggt<, okijzugtqz<gxqg<gzil</ uqgqklx w{<gjt olb<g<ogim<cz< Gxqg<Gl< Ljxjb kx<ohik okiqf<kogit<ouil</ wmk<kg<gim<cx<gig, uqgqklx w{< Jg< gvkouil</ uqgqklx w{< e< hgkqbqzt<t (Denominator of ) -Jg< Gxqg<Gl< Ht<tq P we<g/ OP-g<G 8

23 osr<gk<kig PQ we<el< Ogim<cje e< okigkq (numerator of ) nzg fqtk<kqz< ujvg/oq-jus<osig<g/ PQ-g<G-j{big A-upqs<osz<Zl<Ofi<g<Ogim<cje ujvg/-g<ogim OQ J R we<xht<tqbqz< sf<kqg<gqxk we<g/ AR nzggt< NGl</ Woeeqz<? OAR lx<xl< OPQ we<he ucouik<k Lg<Ogi{r<gt</ weou, PQ OP nz< zk nz< zk AR. AR OA AR O-ju jlbligul<> AR-g<G sllie fqtk<jk NvligUl< ogi{<m yv um<ml< ujvg/ -u<um<ml< olb<g<ogim<jm O-g<G hml<. uzh<hxk<kqz<yvht<tqbqz<oum<mgqe<xk/ -h<ht<tqbiek uqgqklx w{< Jg< Gxqg<gqe<xK (hml<.jh<hii<g<gul<*/nok um<ml< O-g<G -mh<hxlig olb<g<ogim<jm oum<ml< Ht<tpbieK wel< uqgqklx w{<{qjeg< Gxqg<gqe<xK/ -Ok Ljxbqz<? wf<kouiv uqgqklx w{<{qjebl< olb<g<ogim<cz< Gxqg<gzil</ olb<g<ogim<cz< P, Q we<he ouu<oux, nvgijlh< Ht<tqgt< weqz< nux<xqx<gqjmbqz<, P lx<xl< Q gjtk< kuqv OuoxiV Ht<tqbqjeBl< gi{zil</ nkiuk olb<g<ogim<cz< wf<kuqv Ht<tqgTg<Gl< -jmobbl< -jmoutq -z<jz we<hjk nxqgqe<oxil</ weou? olb<g<ogim okimis<sqh< Ht<tqgtiz< (continuum of points) NeK/yu<ouiVuqgqkLXw{<{qx<Gl<olb<g<Ogim<cz<yV keqk<k Ht<tq d{<m we<hjk -Kujvbqz< hiik<okil</ Neiz< olb<g<ogim<czt<t njek<kh<ht<tqgtl< uqgqklx w{<gjt lm<mol Gxqg<Gli we uqei wph<houil</ -kx<giqb uqjm -z<jz we<hkkie</ wmk<kg<gim<mig, we<x uqgqklxi w{<j{g<gvkouil</ hml<. hg<gl< OA weg<ogi{<mskvl< OABC ujvg/hqkigv^<okx<xh<hc? OB OA + AB +. weouob. O-jujlbligUl< OB JNvligUl<ogi{<M yvum<ml<ujvg )hml<. Jh< hiig<gul<*/ -u<um<ml<, olb<ob{< Ogim<cje O g<g -mh<hxl< Ht<tq Q uqzl<, O-g<Guzh<Hxl<Ht<tqP-bqZl<sf<kqg<Gl</ OP OB, OQ OB we<hkiz<ht<tqgt< P, Q uqgqklxw{<gjtg<gxqg<gik/ngouht<tq P, uqgqklxiw{<{qjebl<, Ht<tq Q, wel< wel<uqgqklxiw{<{qjebl<gxqg<gl</ 9

24 wmk<kg<gim<m 9:, 5,, 5 Ngqb uqgqklxi w{<gjt olb<ob{< Ogim<cz< Gxqg<gUl</ kqiu: olb<ob{< g<giqb Ht<tq P-J olb<g<ogim<cz<gxqg<gul<. lx<xl< hqe<uvlix gi{zil<; OP lx<xl< PD welix osu<ugl< OPDC (hml<.4 Jh< hiig<gul<* ujvg/ hqkigv^<okx<xh<hc, OD OP + PD ( ) +. hml<.4 weou OD. O-jujlbligUl<, OD J NvligUl<ogi{<MyVum<ml<ujvg/ -u<um<ml< olb<g<ogim<cje Q lx<xl< Q gtqz< sf<kqg<gm<ml</ hmk<kqzqvf<k filxquk? OQ OQ OD. Q lx<xl< Q wel; Ht<tqgt< O-g<Guzlx<Xl<-mh<Hxr<gtqz< LjxOb njlukiz<, Ht<tq Q NeK JBl<? Ht<tq Q NeK JBl< Gxqg<gqe<xe/ 5, 5 -ux<xqx<g{<mie Ht<tqgjtg< Gxqg<g Olx<%xqb dk<kqjbg< jgbit<ouil</ olb<ob{< Ogim<ce<lQKO uqzqvf<k nzggt< okijzuq<z<o uqx<guzh<hg<gk<kqz<r we<x Ht<tqjbg< Gxqg<gUl< (Wx<geOu uqgqklx w{<gtg<g d{<mie Ht<tqgjt olb<ob{< Ogim<cz< wu<uix gi{<hk we<hjkg< hml<.5 g{<cvg<gqe<oxil<)/osu<ugl< ORFC J fqtl<or, ngzl<rf we<xuixujvg ( hml<.5 Jh<hiIg<gUl<). hqkigv^<okx<xh<hc, OF OR + RF weou? OF 5. O J jlbliul<, OF J NvligUl< ogi{<m yv um<ml< ujvg/ nu<um<ml< olb<ob{<ogim<cje O uqx<guzh<hg<gk<kqz<s we<xht<tqbqzl<, O uqx<g -mh<hg<gk<kqz< S we<xht<tqbqzl<oum<mgqe<xk/ -r<gos OS OF 5 we<hkiz<, S NeK 5 JBl<?S NeK 5 JBl<Gxqg<gqe<xe/ Olx<gi[l< Nvib<uqzqVf<K, wf<kouiv olb<ob{<[l<, olb<ob{< Ogim<ce< lqk keqk<k yv Ht<tqBme< ohivk<kh<hmgqxk we nxqgqe<oxil</ lxkjzbig? olb<ob{< Ogim<ce< lqkt<tyu<ouivht<tqbl<keqk<kyvolb<ob{<{qjeg<gxqg<gl<weul< nxqgqe<oxil</ O uqx<g uzh<hg<gk<kqz< dt<t Ht<tqgt< lqjg olb<ob{<gjtbl<? O uqx<g -mh<hg<gk<kqzt<t Ht<tqgt< Gjx olb<ob{<gjtbl< Gxqg<gqe<xe/ 0

25 olb<ob{<gt< a lx<xl< b -ux<xqx<gie olb<ob{< Ogim<ce< Ht<tqgt< LjxOb P lx<xl<q we<g/a < b weqz<p NeKQ e<-mh<hg<glig-vh<hjknxqf<kogit<tul</ nkiuk? Q uiek P g<g uzh<hg<gk<kqz< njlf<k Ht<tq we nxqf<k ogit<tul< )hml<.6 Jh< hiig<gul<*/ OlZl<? a < x < b we<xuixt<t x we<x olb<ob{<{qx<g olb<ob{< hml<.6 Ogim<ce< lqk ohivk<kh<hm<m Ht<tq R weqz<? R NeK P g<gl< Q g<gl< -jmh<hm<m Ht<tqbig-Vh<hjknxqgqe<Oxil</.. uqgqklxiw{<gtqz<g{g<gqmz<,, 5, Ohie<xuqgqkLxiw{<gt<wu<uiXdVuibqewe<hjkfqjeU %IOuil</ fil< x 0 we<x sle<him<cjek< kqiu gi{ uqjpouioleqz<? flg<g x kqiuigg<gqjmk<kk/-jkh<ohizoux n r we<xsle<him<cx<golb<lkqh<h x kqiuigh<ohxuqjpgqoxil<we<g/-r<gr yvuqgqklxw{<?n yvlqjglpuigl</ ujg: r < 0, n, 4, 6, weqz<? x n r we<xuixyvlqjgolb<ob{<x gi{ LcbiK/Woeeqz<?yVolb<ob{< x x<g? x x x > 0, x 4 x x > 0,, x n > 0; Neiz< r < 0. ujg: r > 0, n, 4, 6, weqz<?x n r we<xuixyvlqjgolb<ob{<x gi{ -bzl</wmk<kg<gim<mig? x 4 65 g<gx 5 gi{zil</ ujg: r > 0, n,, 5, weqz<?x n r we<xuixyvlqjgolb<ob{<x gi{ LcBl</wMk<Kg<gim<mig? x 64 g<gx 4 gi{zil</ ujg4: r < 0, n,, 5, weqz<?x n r we<xuixyvlqjgolb<ob{<x gi{ LcbiK/Woeeqz<?yVlqjgolb<ob{<x g<gx > 0, x x x > 0, x x x > 0, x n > 0; Neiz<r < 0. Olx<gi[l< Nvib<uqzqVf<K? fil< nxquk? n yv lqjg LP (n,,, ) lx<xl< ryvlqjguqgqklxw{<weqz<?x n r we<xuix yv lqjg olb<ob{< x gi{ -bzl</ -f<fqjzbqz< ohxh<hml< lqjg olb<ob{< x J n r we wpkgqoxil<; nkiuk?x n r x n r ; OlZl<x Jr e<n uk&zl<we<gqoxil</-r<g n r yv uqgqklx w{<{igoui nz<zk uqgqklxi w{<{igoui -Vg<Gl</ wmk<kg<gim<mig? 64 4, 4 4. Gxqh<hig?fil<gueqh<hK?

26 ujg : r NeK q p we<xyvlqjguqgqklxw{<{qe<n uk &zlig n p p -Vh<hqe<, nkiuk? r weqz<? n r yvuqgqklxw{<{igl</ q q ujg : r NeK wf<kouiv lqjg uqgqklx w{<{qe< n uk nmg<gig -z<jzobeqz<? n r yv uqgqklx w{<{ig -Vg<g LcbiK; Woeeqz<? n r yv n p p uqgqklx w{< weqz<? r? yv uqgqklx w{<o{e Lv{<himig q q gqjmg<gqxk/ weou? yv lqjg uqgqklx w{< r NeK OuX wf<k yv uqgqklx w{<{q<e<n NuKnMg<gig-Vg<guqz<jzobeqz<? n r NeKyVuqgqkLXw{<{ig -Vg<giK; nkiuk? n r yvuqgqklxiw{<ngl</ r yvlqjguqgqklxw{<{ig-vf<k? n r we<x lqjg olb<ob{<? yv uqgqklxi w{<{ig -Vh<hqe<? n r J yv LVm<om{< )surd) nz<zk yv uqgqklxi &zl<we<gqoxil</ Gxqh<H: yv LVm<om{< we<hk Gxqh<hqm<m ucuqz< njlf<k uqgqklxi w{<{igl</ r NeKwf<kouiVuqgqkLXw{<{qe<nNuKnMg<gig-z<jzwe<xOhiK? n r yv uqgqklxi&zligl</-r<gnwe<hkuqgqklxi&zl< n r e<gxqob{<we<gqoxil</ n we<xgxqbiek&zg<gxq (Radical) we<gqoxil</gxqob{<n weqz<, r Jr e< uig<g&zl<we<xjpk<knkjes<svg<glig r Jre<ge&zl<we<gqOxil</ r we<ox wpkgqe<oxil</ n weqz<, LVm<om{<gt<olb<ob{<gt<we<hkiz<?nux<Xme<-bx<g{qkosbzqgt<+,,, -ux<jxs< osbz<hmk<kzil</ -V uqgqklxi &zr<gjtg< %m<m yv uqgqklxi w{< gqjmh<hqe<? nf<k uqgqklxi w{<{qjebl< yv LVm<om{< we<ox njpg<gqe<oxil</-u<uiox?yvlvm<om{<{qje?yvuqgqklxw{<{qeiz<ohvg<g yv uqgqklxi w{< gqjmh<hqe<? nf<k uqgqklxi w{<{qjebl<? yv LVm<om{< we<ohil</ wmk<kg<gim<mig?? 5 5 7? we<he LVm<om{<gt</ + 7, we<heuqgqklxiw{<gt</-ux<jxbl<fil<lvm<om{<gt<we<xjpg<gqe<oxil</ a lx<xl< b we<he -V ouu<ouxie uqgqklx w{<gt< we<g/ a, b -v{<moluqgqklxi&zr<gt<)uig<g&zr<gt<*we<g/a + b, a b, a + b, a b Ngqbju uqgqklxi &zr<gtigl</ -r<g a b NeK a + b e< Kj{bqbuqgqkLxi&zl<we<gqOxil</-u<uiOx? a + b NeK a b e<kj{bqb &zl<, a + b NeK a b e<kj{bqb&zl<? a b NeK a + b e< Kj{bqb&zl<wenxqgqe<Oxil</

27 uqgqklxi &zk<jkbl<? nke< Kj{bqb uqgqklxi &zk<jkbl< ohvg<gg< gqjmh<hkyvuqgqklxw{<{igl</wmk<kg<gim<mig? ( )( ) ( ) + ( ) 4 ( + 5)( 5) ( ) ( 5) 5. + lx<xl< uqgqklxi&zr<gjtg<jgbit<ukx<ghqe<uvl<uqkqgjth<hbe<hmk<kouil</ a, b we<helqjguqgqklxw{<gt<<;m, n we<helqjglpg<gt<weqz<? n (i) ( ) a. a n n n n (ii) ( a ) ( b ) ab. n a a (iii) n. n b b (iv) p n r + q n r ( p + q) n r, -r<gp lx<xl< q -v{<mololb<ob{<gt</ (v) n m mn a a. (vi) n n m m a a. (vii) a < b weqz<, n a < n b NGl</ fil<gueqh<hk?uqkqgt< (ii), (iii) lx<xl< (iv) z<, uqgqklxi&zr<gt<njek<kl< sllie Gxqob{< ogi{<mjubigl</ weou? ouu<oux Gxqob{<gt< ogi{<m uqgqklxi &zr<gt< kvh<hm<m nux<jxg< ogi{<m fie<g nch<hjms< osbzqgjt fqgp<k<k Ou{<Moleqz<? nf<k uqgqklxi &zr<gjt sl Gxqob{< djmb &zr<gtig,uqkq(vi) Jg<<ogi{<Mlix<xqbhqxGosbz<hmOu{<Ml</uqkq(vi) NeK uqkq(i), uqkq)v*-ux<xqe<-j{h<ohbigl<we<hjkhqe<uvlixnxqbzil<: uqkq(i) e<hc?a m a m n n m m. a a. uqkq (vii) Jg<ogi{<M?sllieGxqob{<djmbuqgqkLxi&zr<gjtyh<hqmzil</ Gxqh<H: n a J a n weg<gxqh<ohil</ wmk<kg<gim<m 0: hqe<uvueuqgqklxi&zr<gti?-z<jzbiwe<hjkgiv{r<gtme< uqjmbtqg<gul</ kqiu: (a) (d) (b) (e) (c) 7 5 (f) 4 6 (a), yv uqgqklx w{</ &zle<x/, yv uqgqklxi 5

28 (b), yv uqgqklx w{</ &zle<x/ 7 (c) yv uqgqklxi &zligl</ Woeeqz<? 5 w{<{qekuig<glz<z/ 44 (d) yvuqgqklxi&zligl</ 7 (e) , yv uqgqklxi 5 7 NeK yv uqgqklx 5 yvuqgqklxiw{<we<hkiz<, yvuqgqklxi&zligl</ 6 (f) , yvuqgqklxw{<. 4 6 yvuqgqklxi&zlz<z/ uqgqklxi &zr<gjt? +,,, ogi{<m jgbitl<ohipk gqjmg<gl< uqjtuiekyvuqgqklxw{<{ig-vg<gzil</ wmk<kg<gim<m : hqe<uvueuqgqklxi&zr<gti?-z<jzbiwe<hjkgiv{r<gtme< uqjmbtq: kqiu: (i) ( 5 + ) ( ) (ii) ( + ) ( + ) (iii) ( + 4 )( ) (iv) (i) ( 5 + ) ( ) (5 + 4) + ( ) ( ) 9 + ( ) 9. -K yv uqgqklx w{<? nkiuk? , 4 Ngqb -v{<ml< uqgqklxi &zr<gtig-vh<hqel<nux<xqe< %Mkz<yVuqgqkLxi&zlz<z/weOuuqgqkLxi &zr<gt<%m<mjzh<ohivk<knjmuh<h{<hqjefqjxuosb<buqz<jz/ (ii) ( + ) ( + ) ( ) + ( ) + 0, yvuqgqklxw{</ kvh<hm<mogijuyvuqgqklxi&zlz<z/ (iii) ( + 4 )( ) ( + 4 ) ( 6 ) ( + 4 ) ( 4 ) KyVuqgqkLXw{</weOu?kvh<hm<mOgijubieKyVuqgqkLxi&zle<X/ 4

29 (iv) KyVuqgqkLXw{</weOu?kvh<hm<mOgijubieKyVuqgqkLxi&zlz<z/ wmk<kg<gim<m : hqe<uvueux<xt< yu<ouie<jxbl< yv keqk<k uqgqklxi &zlig wpkul<; (i) (ii) 40 5 (iii) (iv) kqiu: (i) , , (ii) lx<xl< ( 6) (iii) 6 7 lx<xl< (iv) ( 4 ) 0 4 lx<xl< ( 0). wmk<kg<gim<m : hqe<uvueux<xt< yu<ouie<jxbl< yv keqk<k uqgqklxi &zlig wpkul<: (i) 7 6 (ii) 5 4. (iii) (iv) 8 ( ) kqiu: (i) -r<g 7 NeKGxqob{<djmbK/ 6 NeKGxqob{<djmbK/, -ux<xqe< lq/ohi/l 6. weou? 7, 6 Ngqb -v{<jmbl< Gxqob{< 6 djmb uqgqklxi&zr<gtigwpkouil</-r<g , 5

30 weou, (ii) (iii) (iv) 8 ( 5 0) wmk<kg<gim<m 4: 4, 0, 5 we<heux<jx nux<xqe< lkqh<hgtqe< WXuiqjsbqz< njlg<gul</ kqiu: Lkzqz<? kvh<hm<m uqgqklxi w{<gtg<gie ohikuie Gxqob{< gi{<ohil</ kvh<hm<m uqgqklxi &zr<gtqe< Gxqob{<gt< 4, 6,. -ux<xqe< lq/ohi/l NGl</ weou kvh<hm<m uqgqklxi &zr<gjt Gxqob{< dt<t uqgqklxi &zr<gtig lix<xou{<ml</. kvh<hm<muqgqklxi&zr<gt<. -h<ohipkuqkq(vii) Jh<hbe<hMk<k? 6 nkiuk? , 00, 5 5 < 7 < < < 0. sqzslbr<gtqz<?uqgqkl< y x z<dt<thgkqy NeK yvuqgqklxi&zlig -Vg<gzil</nh<hch<hm<muqgqkr<gtqz<?yVkGf<kLjxjbg<ogi{<MhGkqjbyV uqgqklx w{<{ig lix<xzil</ -l<ljxjb? hgkqjbuqgqklx w{<{ig<gl<ljx we<xjpg<gqe<oxil</-l<ljxjbhqe<uvlixkvgqe<oxil<: (i) hgkq y NeK a we<x njlh<hqz<-vf<k? a yv uqgqklx w{< weqz<? okigkq?hgkq-v{<jmbl< a Nz<ohVg<gUl</wMk<Kg<gim<mig, (ii) hgkq y NeK a + b we<x njlh<hqz< -Vf<K? a lx<xl< b uqgqklx w{<gt<weqz<?okigkq?hgkq-v{<jmbl< a b Nz<ohVg<gOu{<Ml</-h<ohiPK? hgkq (a + b ) (a b ) a a b a b ( b ) a b +, yv uqgqklx w{</ wmk<kg<gim<mig, ( ) ( + )( ) ( ) 9 7 (iii) hgkq y NeK a b we<x njlh<hqz< -Vf<K? a lx<xl< b uqgqklx w{<gt<weqz<?okigkq?hgkq-v{<jmbl<a + b Nz<ohVg<gOu{<Ml</-h<ohiPK? hgkq (a b ) (a + b ) a a b a b ( b ) a b +., yvuqgqklxw{</ 6

31 + wmk<kg<gim<mig? (iv) hgkq y NeK ( + ) ( + ( )( + weqz<?okigkq?hgkq-v{<jmbl< + + ( ) ) ( + ) ) ( ) a + b we-vf<k?a lx<xl< b -juuqgqklxw{<gt< a b Nz<ohVg<gOu{<Ml</-h<ohiPK? hgkq ( a + b)( a b) ( a) ( b) a b, yvuqgqklxw{</ wmk<kg<gim<mig? ( 5) ( 5) + 5 ( + 5)( 5) (v) hgkq y NeK ( ) ( ) 5 a b we<x njlh<hqz< -Vf<K? a, b -ju uqgqklx ( ) ( ) ( 5) 5. w{<gt< weqz<? okigkq? hgkq -v{<jmbl< a + b Nz< ohvg<g Ou{<Ml</ -h<ohipk? hgkq ( a b)( a + b) ( a) ( b) a b, yv uqgqklx w{</ wmk<kg<gim<mig? wmk<kg<gim<m 5: kqiu: Gxqh<H: ( ) ( 5 7 )( ) ( 5) ( 5 + 5) ( 5) ( 7 ) 5 7 e<hgkqjbuqgqklxw{<{iglix<xul</ ( 5 + 4)( 5 4) ( 5) ( 4) ( ) ( 4 5) x we<x uqgqkk<kqe< okigkq x NeK a nz<zk a + b nz<zk a b y nz<zk a + b nz<zk a b we<x njlh<hqz< dt<tk/ x J uqgqklx w{<{ig lix<x Ou{<Moleqz<? okigkqbqe< Kj{bqb w{<{qeiz< okigkq, hgkq -v{<jmbl<ohvg<gou{<ml</ wmk<kg<gim<m 6: kqiu: 4 e<okigkqjbuqgqklxw{<{ig<gg/ ( + ) ( ) ( ( ) 4 + ) ( ) ( ) ( ) ( ). 7

32 wmk<kg<gim<m 7: + a + b + c 6 weqz<, a + b + c Jg<g{<Mhqcg<gUl</ kqiu: + ( ) ( )( ) + ( ) ( ) ( ) ( ) ( ) + ( ) ( ) c b a.. 4, 4, c b a c b a Gxqh<H; p, q uqgqklx w{<gt<? n a yv uqgqklxi &zl< weqz<? p+ n a q we<x uqgqklxi&zk<kqz<, p JuqgqkLXw{<hGkqwe<Xl<? n a q JuqgqkLxiw{<hGkq we<xl< njpg<gqe<oxil</ -V uqgqklxi &zr<gt< sllibqvg<g? nux<xqe< uqgqklx w{< hgkqgt< sllibqvg<g Ou{<Ml< lx<xl< nux<xqe< uqgqklxi w{< hgkqgt< sllibqvg<gou{<ml</ wmk<kg<gim<m 8: y x weqz<?x + y Jg<g{<Mhqc/ kqiu: ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

33 y x x 4, y 0 x + y wmk<kg<gim<m 9: x + weqz<? x x + Jg<g{<Mhqcg<gUl</ kqiu: x + ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) + + x ( ) ( ) ( ) ( ) ( ) ( ) x x wmk<kg<gim<m 0: a + weqz<?a (a 6) e<lkqh<hqjeg<g{<mhqc/ kqiu: a ( ) ( ) ( ) ( ) ( ) ( ) a ( ) a (a ( ) ( ) ) ( )( ) [ ] + ( ) ( ). 9 8 wmk<kg<gim<m :.44 weqz<? + e<okiviblkqh<hqjeg<g{<mhqcg<gul</ kqiu: ( ) ( ) + ( ) ( ) ( )

34 hbqx<sq.. hqe<uvueuqgqklxi&zr<gti-z<jzbiwe<hjkgiv{r<gtme<uqjmbtq/ (i) 6 (ii) 4 (iii) 5 79 (iv) 4 4 (v) 50. hqe<uvueyu<ouie<jxbl<svg<gucuqz<wpkul<; (i) 5 + (ii) 0 40 (iii) ( 5 ) ( 4 + ) (iv) ( 0 ) ( ) (v) 5 5 (vi) 6 (vii) (viii) 5. WXuiqjsbqz<njlg<gUl<; (ix) 5 5 (i), 5, 6 (ii) 5, 7, 4 9 (iii), 5, 4 4. hqe<uvue yu<ouie<xqe<hgkqjbuqgqklxw{<{iglix<xul<: (i) (ii) (iii) (iv) (v) (vi) hqe<uvueyu<ouie<xqzl<x lx<xl<y -ux<jxg<gi{<g/ (i) x + y (ii) x + y (iii) x + y 5 (iv) x + y a + weqz<?a (a 4) e<lkqh<jhg<g{<mhqcg<gul</ + 7. a weqz<?a + e<lkqh<jhg<g{<mhqcg<gul</ a weqz<? e<okiviblkqh<jhg<g{<mhqcg<gul</ 9..44,.7 weqz<? e<okiviblkqh<jhg<g{<mhqcg<gul</ + 0

35 uqjmgt< hbqx<sq.. (i) 0.85 (ii).8 (iii) 0.5 (iv) (i) (ii) (iii) (iv) (i) kux (ii) siq (iii) siq (iv) siq (v) siq (vi) kux (vii) kux hbqx<sq. (v) 9. (i) LVm<om{< (ii) LVm<om{<nz<z (iii) LVm<om{< (iv) LVm<om{< (v) LVm<om{<nz<z. (i) 9 (ii) 5 (iii) (iv) 4 5 (v) 6 5. (i) (vi) (vii) 5 7 (viii) 65 6, 5, (ii) 4 9, 7, 5 (iii) (ix) 6 5, 4, 5. 4 (ii) ( ) (iii) ( + 5) 4. (i) 6 (iv) ( 5 ) (v) (i) x 7, y 4 (ii) x, y 59 (iii) x 7, y 0 (iv) x 0, y 9 (vi) ( 6 0) 6. a (a 4)

36 . ntjugt< fl<ne<ximuip<uqz<hz<ouxfqjzjlgtqz<fil<ntjugt<osb<gqe<oxil</ wmk<kg<gim<mig? fil< K{q jkh<hkx<gig fqtk<jkbl<? out<jtbch<hkx<gig Sux<xqe< hvh<htuqjebl<? OuzqbqMukx<gigfqzk<kqe<Sx<xtuqjeBl<? fqvh<hukx<gighik<kqvk<kqe< ogit<ttuqjebl< ntg<gqe<oxil</ ntjugjt nch<hjmbigg< ogi{<m fl< Okjug<Ogx<h OlZl< g{g<gqmgt< osb<gqe<oxil</ kt lx<xl< ge dvur<gtqe< fqtr<gt<? Ogi{r<gt<? hvh<htugt< Sx<xtUgt< lx<xl< ge ntugt< hx<xqb g{qkh< hqiquqje ntuqbz< we<xjpg<gqe<oxil</ Lf<jkb ugh<hgtqz< fil<? Lg<Ogi{r<gt<? fix<gvr<gt< lx<xl< um<mr<gt< Ohie<x sqz kt ucu dvur<gjth< hx<xq gx<xxqf<kt<otil< )wz<zi ucu dvur<gtl< ktk<kqz< ujvbh<hm<mjubigg< ogit<ouil<*/ -f<k nk<kqbibk<kqz<? -v{<m nz<zk nkx<g Olx<hm<m Lg<Ogi{r<gt<? fix<gvr<gt< nz<zk um<mr<gt< -ux<jx ye<xe<olz< ye<xig fqxk<kq gqjmg<gh<ohxl< sqz kt %m<m dvur<gjth< hx<xq gx<ohil</ fil< gvkl< njek<k dvur<gtl< ktk<kqz<njlf<kjuwe<hkiz<?ktdvuk<jkdvul<we<xnjph<ohil<.. hvh<hlx<xl<sx<xtu fil<?ktucuqbz<dvur<gtqe<sx<xtugt<lx<xl<hvh<hgtqe<uib<h<himgjt lqt<hiijubqmouil</.. osu<ugl;: hvh<h l bskvnzggt< Sx<xtU (l + b) nzggt< d l + b nzggt< hml<... -j{gvl;: hvh<h b h skvnzggt< Sx<xtU (a + b) nzggt< hml<... ogimg<gh<hm<mnch<hg<gl<lx<xl<dbvk<jkbjmblg<ogi{l<; hvh<h b h skvnzggt<. hml<.

37 ..4 osr<ogi{lg<ogi{l<; hvh<h b h skvnzggt< hml<.4 Sx<xtU b + h + dnzggt< d b + h nzggt<..5 slhg<glg<ogi{l<; dbvl< h anzggt< hvh<h a skvnzggt< 4 Sx<xtU anzggt< hml< Vslhg<gLg<Ogi{l<: hvh<h h a Sx<xtU ( a a h ) h skvnzggt< + nzggt< hml<.6..7 nslhg<glg<ogi{l;: hvh<h s( s a)( s b)( s c) s/n a + b + c -r<g s nzggt< hml<.7 Sx<xtU a + b + c nzggt<..8 siqugl<; hml<.8 hvh<h (a + b) h skvnzggt<..9 fix<gvl<; hvh<h d (h + h ). skvnzggt< hml<.9..0sib<skvl<; hvh<h d d s/nzggt< hml<.0 Sx<xtU d d + 4a nzggt<

38 ..um<ml<: um<mk<kqe<hvh<h πr skvnzggt< um<mk<kqe<sx<xtu πrnzggt< njvum<mk<kqe<hvh<h πr skvnzggt< njvum<muqz<zqe<fqtl< πrnzggt< giz<um<mk<kqe<hvh<h 4 πr skvnzggt< hml<. giz<um<muqz<zqe<fqtl< πrnzggt< Gxqh<H: A lx<xl< B Ht<tqgjt -j{g<gl< Ofi<g<Ogim<Mk< K{<cje AB nz<zkab weg<gxqg<gqe<oxil</olzl< ABwe<Ox AB e<fqtk<jkbl<gxqg<gqe<oxil</ wmk<kg<gim<m : yv osu<ug ucu SuI? nch<hg<g fqtl< 6lQ weul< dbvl< lq weul< ogi{<mt<tk/ ns<sux<xqx<g u{<{l< H,s skv lqm<mvg<g '/7 oszuigoleqz<? olik<ksux<xqx<gl<u{<{l<h,sngl<oszjug<g{<mhqc/ kqiu: b 5? h 0NGl</ osu<ugh<hvh<h b h s/lq. s/lqu{<{l<h,soszu '/ 6 50 s/lqu{<{l<h,soszu 6 50 ' hml<. wmk<kg<gim<m : yv osu<ugk< kgm<ce< ntugt< 5lQ 4lQ/nk<kgm<cje5os/lQ hg<g fqtl< ogi{<m skvk< kgmgtig oum<m Ou{<Ml</ skvk< kgmgtqe< w{<{qg<jgjbg<g{<mhqc/ kqiu: osu<ugk<kgm<ce<hvh<h ?0000os/lQ. yvskvk<kgm<ce<hvh<h os/lq.,0000 skvk<kgmgtqe<w{<{qg<jg hml<. wmk<kg<gim<m ; 40os/lQ hvh<hjmb -j{gvk<kqe< dbvl< 5 os/lq weqz<? nke< nch<hg<gl<g{<mhqc/ kqiu:hvh<h b h. 40 b b. 5 nch<hg<gl< 8 os/lq. hml<.4 4

39 wmk<kg<gim<m 4: os/lq? 60 os/lq lx<xl< 7 os/lq hg<g fqtr<gt< ogi{<m yv Lg<Ogi{k<kqe<hvh<jhBl<? Sx<xtjuBl<g{<Mhqc/ kqiu:hvh<h s( s a)( s b)( s c). -r<g s a + b + c s 66? s a 66 55? s b ? s c hvh<h s/os/lQ. Sx<xtU a + b + c os/lq. hml<.5 wmk<kg<gim<m 5: hml<.6 z< ogimg<gh<hm<mt<t fix<gvl< ABCD e< hvh<hqjeg< g{<mhqc/ kqiu:hvh<h d ( h + h ) 50 (0 + 0) lq / hml<.6 wmk<kg<gim<m 6:hml</8z<ogiMg<gh<hm<Mt<tsiqugl<ABCDe<hvh<Hg{<Mhqc/ kqiu: hvh<h ( a + b) h ( + 5) 4 4 skvnzggt</ hml<.7 wmk<kg<gim<m 7: fqzk<jk slh<hmk<k skv lqm<mvg<g '/ oszuigl</ siqug ucu fqzk<kqe< -j{ hg<gr<gtqe< fqtr<gt< 8lQ lx<xl< lq/ lx<x -V hg<gr<gt< yu<ouie<xl< 6lQ fqtljmbk weqz<? olik<k fqzk<jk slh<hmk<k NGl< oszjug< g{<mhqc/ kqiu: ABCD ogimg<gh<hm<m siqugl</ -r<g AB 8lQ? CD lq? AD BC 5 lq/ DA g<g -j{bigce J ujvg)hml<.8 Jh< hii<g<gul<*/ EBC YI-Vslhg<g Lg<Ogi{l</nke<dbvl<h siqugl<abcdbqe<hvh<h 5 6 4lQ/ (a + b) h (8 + ) s/lq/ s/lqslh<hmk<kngl<oszu '/. olik<kfqzk<jkslh<hmk<kngl<oszu 60 '. 70. hml<.8 5

40 wmk<kg<gim<m8: yvsib<skvk<kqe<sx<xtuos/lq/nke<yv&jzuqm<mk<kqe<fqtl< 9os/lQ/nke<lx<oxiV&jzuqm<mk<kqe<fQtk<jkBl<hvh<jhBl<g{<Mhqc/ kqiu: d? d we<he&jzuqm<mr<gt<weg<ogit<g/ Sx<xtU d + d /Neiz< Sx<xtU os/lq weg<ogimg<gh<hm<mt<tk/ d + d 0 os/lqnz<zk d d -r<gyv&jzuqm<mk<kqe<fqtl<8os/lq/ d 8 we<g/ 64 + d 00 nz<zkd 6. d 6os/lQ/ hml<.9 sib<skvk<kqe<hvh<h d d os/lq. wmk<kg<gim<m 9: 64 os/lq fqtlt<t gl<hq -V sl higr<gtigh< hqiqg<gh<hmgqxk/ yv higk<jk um<mligul< lx<oxie<jx slhg<g Lg<Ogi{ligUl< ujtk<k osb<bh<hmgqxk/njudt<tmg<gl<hvh<hgtqe<uqgqkk<kqjeg<g{<mhqc (π 7 we<x hbe<hmk<k). 64 kqiu: um<mk<kqe<sx<xtu os/lq/ Neiz<um<mk<kqe<Sx<xtU πr. 7 r nz<zk r os/lq/ um<mk<kqe<hvh<h πr 7 86 os/lq. hml<.0 slhg<glg<ogi{k<kqe<sx<xtu a Neiz<Sx<xtU os/lq/ a nz<zka 44os/lQ/ slhg<glg<ogi{k<kqe<hvh<h a os/lq um<ml< lx<xl< slhg<g Lg<Ogi{k<kqe< hvh<hgtqe< uqgqkl< 86 : 484 : hbqx<sq. hml<.. gqp<g<g{<mlg<ogi{r<gtqe<hvh<hgjtg<gi{<g/ (i) nch<hg<gl< 8os/lQ?dbvl< os/lq/ (ii) Lg<Ogi{k<kqe<&e<Xhg<gr<gtqe<fQtr<gt<os/lQ? 48os/lQ?6os/lQ/ (iii) slhg<glg<ogi{k<kqe<hg<gfqtl<9os/lq/ 6

41 . ABCD we<xfix<gvk<kqe<&jzuqm<ml< AC e<fqtl; 44 os/lqlx<xl<b? DzqVf<K ACg<G ujvbh<hml< Gk<Kbvr<gt< LjxOb os/lqlx<xl< os/lq weqz<? nf<fix<gvk<kqe<hvh<hg{<mhqc.. yv fix<gvk<kqe< yv &jzuqm<mk<kqe< fqtl< 6 os/lq lx<xl< nl<&jzuqm<mk<kqx<g ujvbh<hml< osr<gk<kg<ogimgtqe< fqtr<gt< 4 os/lq lx<xl<6os/lqweqz<?nf<fix<gvk<kqe<hvh<hg{<mhqc/ 4. 60lQ Sx<xtUt<t yv sib<skvk<kqe< yv &jzuqm<ml< 66lQ/ lx<oxiv &jzuqm<mk<kqe<fqtl<gi{<g/olzl<ns<sib<skvk<kqe<hvh<hgi{<g/ 5. yv-j{gvk<kqe<hvh<h7os/lq lx<xl<nke<dbvl<9os/lqweqz<?nke< nch<hg<gk<jkg<g{<mhqc/ 6. 8 os/lq lx<xl< 8 os/lq -j{ hg<gr<gtigg< ogi{<m siqugk<kqe< hvh<h 4 os/lq. -j{hg<gr<gtg<gqjmh<hm<mk~vl<g{<mhqc/ 7. yvsiqugk<kqe<-j{hg<gr<gtg<g-jmh<hm<mk~vl<5os/lqlx<xl< yv -j{h<hg<g fqtl< 8 os/lq/ns<siqugk<kqe< hvh<h 45 os/lq weqz<? lx<oxiv -j{hg<gk<kqe<fqtl<gi{<g/ 8. os/lq 4 os/lq ntut<tyvosu<ugfqzk<kqe<hvh<hqx<gs<sllie hvh<hjmbyvum<mfqzk<kqe<sx<xtugi{<g/. %m<m dvur<gt< yv fix<gvl< ABCD Jg< gvkg (hml<. Jh< hii<g<gul<*/ BD jb -j{g<gul</ kx<ohipk fix<gvliek ABD lx<xl< BCD we -v{<m Lg<Ogi{r<gtigh<hqiqg<gh<hMgqe<xK/-v{<MLg<Ogi{r<gTg<Gl< BD hg<gl< ohikuiek/hqe<oeig<gqh< hii<g<gl<ohik? Lg<Ogi{r<gt< ABD lx<xl< BCDJ BD Jh< ohikh<hg<gligg< ogi{<m ye<xe< Olz< ye<xig jug<g fix<gvl< ABCD ohxh<hmgqe<xk/weou?fix<gvl< ABCD NeK-V Lg<Ogi{r<gtqe<Osi<g<jgbiGl< hml<. hml<. nz<zk ABCD yv%m<mdvuligl</-u<uiox?siqugl<yv%m<mdvuligl<<nk yv osu<ugk<kqe< -V hg<gr<gtqz< -V osr<ogi{ Lg<Ogi{r<gt< Osi<k<K dvuig<gh<hm<mk)hml<. Jh<hiIg<gUl<*/fil<nxqf<Kogi{<mK we<eoueqz<?-v dvur<gjt ye<xe< hg<gk<kqz< ye<jx juk<k yv %m<m dvul< dvuig<g Ou{<Moleqz< Lkz< dvuk<kqe< WkiuokiV hg<gl< -v{<miuk dvuk<kqe< 7

42 WkiuokiVhg<gk<kqx<Gh<ohiVf<Kukig-Vg<gOu{<Ml</sqz%m<MdVur<gt<? hml<.4lkz<hml<.5 ujv ogimg<gh<hm<mt<te/ hml<.4 hml<.5 hml<.6 hml<.9 hml<.7 hml<.8 hml<.0 hml<. hml<. hml<. 8

43 hml<.4 hml<.5 ye<xqe< Olz< ye<xig -Vg<Gl< hg<gr<gjt Ht<tqg< OgiMgtiz< gim<cbqvg<gqe<oxil</ hml<.4 zqvf<k hml<.5 ujvbt<t %m<m dvur<gjt wtqkig nxqbzil</ wmk<kg<gim<mig? hml<.5 NeKyVLg<Ogi{l<? Yi<njvum<ml<Ngqbux<xqe<%m<M dvuligl</nkyvhl<hvk<kqe<fqjzg<gk<kgxg<goum<mk<okix<xligk<okiqgqxk/ hml<.7 NeKyVosu<ugl<lx<Xl<Yi<njvum<ml<Ngqbux<xqe<%m<MdVuliGl</ -u<uvul<? Yi< njvum<mk<kqje Olz< Wx<Gl< osu<ug se<ez< Ohiz< Okix<xltqg<gqe<xK/ hml<.9 NeK yv osu<ugl< lx<xl< -V sllie giz<um<mr<gjtg< ogi{<m %m<m dvuligl<. hml<. NeK yv osu<ugl<? yv Lg<Ogi{l<? yv siqugl< Ngqbux<jxg< ogi{<m %m<m dvuligl<; -K YI WUgj{bqjeh< Ohiz< okiqgqxk/ %m<m dvur<gtiue? Lg<Ogi{r<gt<? fix<gvr<gt<? um<mr<gt< Ngqbux<xqe< %m<mig -Vh<hkiz<? nux<xqe< Sx<xtUgt<? hvh<hgt< Ngqbux<jx Lf<jkbuGh<hqz<gx<x$k<kqvr<gjtg<ogi{<Mg{g<gqmzil</ %m<mk<ktdvur<gtiesiqugr<gt<lx<<xl<hz<ogi{r<gjth<hx<xq-h<ohik fil<hii<g<gzil</.. siqugl< siqugl< we<hk fie<g hg<gr<gjtbjmb yv kt dvul</ -kqz< -V hg<gr<gt<-j{bieju)hml<.6 Jh<hii<g<gUl<*/ ABCD we<xsiqugk<kqz< AB Bl< DC Bl<-j{bieju/ AB a? CD bwe<g/ -j{ogimgtg<g -jmh<hm<m K~vl< h we<g/ fil< siqugl< ABCD J Lg<Ogi{r<gt< ABC lx<xl< ACD gtqe< %m<m dvuligg< ogit<tzil</ Lg<Ogi{l< ABC z< nch<hg<gl< AB?Gk<Kbvl< h. NgOu? glk;.6 ABC bqe<hvh<h a h ACD bqe<hvh<h b h siqugk<kqe<hvh<h a h + b h (a + b)h skvnzggt</ 9

44 .. hz<ogi{l< hz<ogi{l< we<hk n Ofi<g<Ogim<Mk< K{<Mgtiz< ucujlg<gh<hm<m kt dvul< NGl</hzLg<Ogi{r<gtqe<%m<MdVuOlhz<Ogi{l<we<hjkfil<nxqgqOxil</ hz<ogi{k<kqe<hg<gr<gt<slligul<? Ogi{r<gt< slligul< -Vh<hqe<? nh<hz<ogi{l< Yi< ypr<g hz<ogi{l< weh<hml</ NX hg<gr<gjtg< ogi{<m ypr<g hz<ogi{l< Yi< ypr<g nxogi{l< weh<hml</ Yi< ypr<g nxogi{k<kqz< njek<k hg<gr<gtl< slligul<? dm<ogi{r<gt< yu<ouie<xl< 0 g<gs< slligul< -Vg<Gl< (hml<.7 Jh< hii<g<gul<). -f<k Gxqh<hqm<m ypr<g hz<ogi{l< ncg<gc hbe<hmk<kh<hmukiz<? nke< Sx<xtU lx<xl<hvh<htuhx<xqg<gi{<ohil</ ABCDEF< glk;.7 we<hkyi<ypr<gnxogi{l<we<g/hg<gr<gt< AB? BC? CD? DE? EF lx<xl< FA sllie ntujmbju/yu<ouivhg<gk<kqe<fqtll<a nzggt<we<g/ypr<gnxogi{k<kqe< Sx<xtuieK a + a + a + a + a + a 6a nzggt</ fil< -h<ohik ypr<g nxogi{k<kqe<hvh<hgi[l<$k<kqvk<kqjeuvuqh<ohil</&jzuqm<mr<gt< AD? BE? CF sf<kqg<gl<ht<tqbqje? O we<g/lg<ogi{r<gt< OAB? OBC? OCD? ODE? OEF? OFA slhg<g Lg<Ogi{r<gt< NGl</ weou? yu<ouiv Lg<Ogi{k<kqe< hvh<h a weou? ypr<gnxogi{k<kqe<hvh<h 6 4 wmk<kg<gim<m0:hml<.8e<hvh<hqjeg<g{<mhqc/ kqiu;hml<abcde NeKhg<gl< BD bqjeh< ohikfqjzbigg<ogi{<m ABDE lx<xl< BCD e<%m<migl</ ABDE we<xsiqugk<kqz< -j{hg<gr<gtie AE? BD gtqe< fqtr<gt<ljxob 0 os/lq?6 os/lqngl</ -j{hg<gr<gtg<g-jmh<hm<mk~vl<9 os/lq/ NgOu? siqugl; ABDEe<hvh<H ( a + b) h (0 + 6) 9 7os/lP. a skv nzggt</ hml<.8 a NGl</ 4 BCD yvlg<ogi{l</-ke<nch<hgkq BD bqe<fqtl<6os/lq? Gk<Kbvl< 8 os.lq. NgOu-ke<hvh<H b h nrkP. 40

45 %m<mdvul< ABCDEe<hvh<H ABDE e<hvh<h + BCD e<hvh<h os/lq. wmk<kg<gim<m : yv fqzl< nth<hui yv fqzk<kqe< ntugjth< hqe<uvlix Gxqk<Kt<tii/fqzk<kqe<hvh<hqjeg<g{<Mhqc/ hml<.9 kqiu: A bqzqvf<k D ujv dt<t fqzlth<huiqe< Gxqgt< P? Q? R? S we<g/ hqe<h AP 5 lq? AQ 7 lq? AR lq? AS 5 lq? AD 7 lq? BP 0 lq? FQ 8 lq? CR 8 lq? ES 9 lq. ogimg<gh<hm<m fqzliek siqugr<gt< PRCB? FESQ lx<xl< Lg<Ogi{r<gt< AQF? APB? DSE? CRD -ux<xqe<%m<migl< )hml<.40 Jh<hiIg<gUl<). siqugl< PRCB e< hvh<h : BP? CR -j{ hg<gr<gt</gk<kbvl< PR. fil<nxquk? BP 0 lq? CR 8 lq?lx<xl< PR AR AP 5 7lQ. NgOu? PRCB e<hvh<h (BP + CR) PR (0 + 8) 7 6 lq hml<.40 siqugl< QFES e< hvh<h; ES? FQ -j{ hg<gr<gt</ Gk<Kbvl< QS. fil<nxquk ES 9 lq? FQ 8 lq? QS AS AQ 5 7 8lQ/NgOu? QFES e<hvh<h (ES + FQ) QS (9 + 8) lq Lg<Ogi{l<AQFe<hvh<H AQ FQ lq Lg<Ogi{l< DSE e<hvh<h DS ES (AD AS) 9 (7 5) lq. Lg<Ogi{l< CRD e<hvh<h RD CR (AD AR) 8 4 (7 ) lq. Lg<Ogi{l<APB e<hvh<h AP BP 5 0 5lQ fqzk<kqe<hvh<h lq. 4

46 wmk<kg<gim<m : hml<.4 z<gi{<hqg<gh<hm<mucuk<kqe<hvh<hg{<mhqcg<gul<(π weh<hbe<hmk<kul<). 7 hml<. 4 kqiu;osu<ugl< ABDE? njvum<ml< AFE lx<xl<slhg<glg<ogi{l< BCD Ngqbux<xqe< -j{h<ohogimg<gh<hm<mljewefil<nxqgqoxil</ osu<ugl< ABDE e<hvh<h os/lq. njvum<ml< AFE e<hvh<h π r os/lq slhg<glg<ogi{l<bcde<hvh<h 7 a os/lq. 4 ljebqe<hvh<h os.lq wmk<kg<gim<m : hml<.4 z<gi{<hqg<gh<hm<mucuk<kqe<hvh<hg{<mhqcg<gul< (π weh<hbe<hmk<kul<*/ 7 glk;.4 kqiu; osu<ugl< ABCD? njvum<ml< CDE lx<xl< giz< um<mh< hgkqgt< AFD? BCG e< -j{h<oh ogimg<gh<hm<mljewefil<nxqgqoxil</ osu<ugl<abcde<hvh<h 4 48os/lQ. njvum<ml< CDEe<hvh<H π os/lq giz<um<mh<hgkq AFD e<hvh<h π giz<um<mh<hgkq BCG e<hvh<h 7 4 os.lq. 4 7 ljebqe<hvh<h os.lq os.lq

47 wmk<kg<gim<m 4:hml<.4 Nz<$ph<hm<Mt<thvh<Hg{<Mhqcg<gUl</ hml<.4 kqiu; -h<hml<? osu<ugl< CDFG? njvum<ml< DEF lx<xl< siqugl< ABCG Ngqbux<xqe< Osi<g<jgbiGl</ osu<ugl< CDFG e<hvh<h 8 64 os.lq. siqugl< ABCGe<hvh<H (6 + 8) os.lq. njvum<ml< DEF e<hvh<h os.lq 7 ogimg<gh<hm<mnjlh<hqe<hvh<h os.lq. sqz slbr<gtqz< fil< ohiqb kt dvur<gtqzqvf<k oum<c fqg<gqb lqkq dvur<gjtg< gvk Ou{<cBt<tK/ nux<xqe< hvh<hgjtg< gi{ Olx<g{<muiX hvh<hgjt%m<mukx<ghkqzigkvh<hm<mdvuk<kqe<hvh<hqzqvf<k?oum<cwmg<gh<hm<m dvur<gtqe<hvh<hgjtg<gpqk<klqkqdt<tdvur<gtqe<hvh<jhg<gi{zil</ um<mucuujtbk<kqe<hvh<h yv um<m ujtbl< we<hk ohik jlbk<jkbjmb -V um<mr<gtg<g -jmh<hm<mhgkqbigl<)hml<.44 Jh<hii<g<gUl<*/um<mujtbk<kqe<hvh<hieKoutq um<mh<hvh<hqzqvf<kdt<um<mk<kqe<hvh<jhg<gpqk<kkx<gs<slligl</nkiuk? ujtbhvh<h π R π r π (R r ). hml<.44 njvum<mujtbk<kqe<hvh<h π (R r ) skvnzggt</ 4

48 wmk<kg<gim<m5: hml<.45 z<fqpzqmh<hm<mhgkqbqe<hvh<hg{<mhqcg<gul</-f<khgkq giz<um<mr<gtiz<$ph<hm<mt<tk(π weg<ogit<tul<). 7 hml<.45 kqiu; ogimg<gh<hm<m fqpzqmh<hm<mh< hgkq? 8 os/lq hg<glt<t skvk<kqz< fie<g &jzgtqzl<4 os.lq NvLt<tgiz<um<mh<hGkqgjtfQg<gqbkx<Gs<slliGl</ skvk<kqe<hvh<h os.lq. yvgiz<um<mk<kqe<hvh<h π os.lq Okjubiehvh<H 784 4(54) sKvos/lQ/ wmk<kg<gim<m 6: 7lQ ngzlt<t YMhijk hml<.46 z< gim<cbuix njlg<gh<hm<mt<tk/ -ke< dt<um<ms< Sx<xtU 70lQ lx<xl< yu<ouiv OfIg<OgiMh< hgkqbqe< fqtl< 40lQ. ujtuh< hgkqgt< njvum<m ucuk<kqz< dt<te/ YMhijkbqe< hvh<jhg<g{<mhqc (π weh<hbe<hmk<kul<). 7 glk;.46 kqiu;dt<njvum<mk<kqe<nvl<r we<g/weoudt<sx<xtu 40 + (π r) nz<zk 80 + πr.neiz<-ke<lkqh<h 70lQweogiMg<gh<hm<Mt<tK πr 70 nz<zkπr 440 nz<zkr lq. π weoudt<njvum<mk<kqe<nvl< r 70lQ. outqnjvum<mk<kqe<nvl< R lQ. -h<ohipk YM hijkbqe< hvh<hiek njvum<m ujtbr<gtqe< hvh<h lx<xl< osu<ug ucuhijkbqe<hvh<hgtqe<%mkzg<gs<slligl</neiz<?yvnjvum<mhijkbqe< hvh<h π (R r ) (77 70 ) skv lq/

49 yvosu<ugucuhijkbqe<hvh<h skv lq. hijkbqe<hvh<h skv lq. wmk<kg<gim<m 7: 4.lQ/hg<gntUt<tyVsKvucugm<cmk<kqe<yVoutqh<Hx &jzbqz< yv hs gm<mh<hm<mt<tk/ gbqx<xqe< fqtl< 4.9 lq weqz<? hs Olbg<%cb hgkqbqe<hvh<hg{<mhqc(π weh<hbe<hmk<kul<). 7 kqiu;skvk<kqe<awe<xljebqz<hsgm<mh<hm<mdt<tk (hml<.47 Jh<hii<g<gUl<*/ gbqx<xqe< fqtl< 4.9lQ lx<xl< Sux<xqe< fqtl< 4.lQ/ hsuiek Sux<jxg< gmg<g -bzikkiz<?nke<gbqxskvk<kqe<b lx<xl< ELjegjtk<ki{<c Dlx<Xl< G ujv fqt<gqxk/ weou? hsuiek 4.9lQ NvLt<t Lg<giz< um<mh<hgkq lx<xl< lQNvLt<t-Vgiz<um<mh<hGkqgt<Ngqbux<jxOlbLcBl</ hsolbg<%cbhvh<h π π lQ. hml<.47 wmk<kg<gim<m 8: hml<.48 z< fqpzqmh<hm<m hgkqbqe< hvh<hqjeg< g{<mhqc (π weg<ogit<tul<*/ 7 hml<.48 kqiu;fqpzqmh<hm<mhgkqbqe<hvh<h (4 os.lq NvLt<tnjvum<mk<kqe<hvh<H) (7 os.lq. NvLt<tnjvum<mk<kqe<hvh<H* π (4) π (7) os.lq. 7 45

50 hbqx<sq.. fqzlth<huiqe< Ofim<Mh< Hk<kgk<kqZt<t hqe<uvl< Gxqh<HgtqzqVf<K dkuqh<hml< ujvf<knux<xqe<hvh<hgjtg<g{<mhqc; (i) (ii) (iii) hml<.49 hml<.50 hml<.5. yv uqjtbim<m jlkiel< hml<.5 z< gim<cbuix -V OfIOgim<Mk< K{<Mgt< lx<xl< -V njvum<m uqz<gtiz< dvuig<gh<hmou{<ml</ yu<ouiv njv um<mk<kqe< Nvl< lq. yu<ouiv glk;.5 Ofi<g<Ogim<Mk<K{<ce<fQtl< 85lQ. uqjtbim<mk<kqmzqe<hvh<jhg{<mhqc (π weg<ogit<tul</). 7. yvsiqugk<kqe<-v-j{hg<gr<gtqe<fqtr<gt< os.lq lx<xl< os.lq. nke< lx<x-vhg<gr<gt<yu<ouie<xl<0 os.lq fqtl<weqz<?nke<hvh<hgi{<g/ 4. YIyPr<GnXr<Ogi{k<kqe<hvh<H50 os.lq weqz<?nke<hg<gk<jkg< g{<mhqcg<gul</ 5. hml<.5 z<fqpzqm<mh<hgkqbqe<hvh<hqjeg<g{<mhqc/ hml<.5 46

51 6. gqp<g<gi[l<hmr<gtqe<fqpzqm<mh<hgkqgtqe<hvh<hgjtg<g{<mhqcg<gul</ (i) (ii) hml<.54 (iii) (iv) hml<.55 hml<.56 hml< yvum<mk<kqe<uqm<ml< 54 os.lq.> nke<yvuqm<ml< AB OfIOgim<Mk<K{<M AB e< lqk C we<x Ht<tq? BC 0 os.lq we<xuix dt<tk/ AC Juqm<mligg< ogi{<m yv um<ml< ujvbh<hmgqxk/ nu<um<mr<gtg<gqjmh<hm<m hvh<jhg< g{<mhqcg<gul</ π weg<ogit<tul</ 7 8. hml<.58 z<fqpzqm<mhgkqbqe<hvh<hlx<xl<sx<xtjug<g{<mhqc/ hml< lq hg<glt<t skv ucu ubzqe< fie<g &jzgtqzl< fie<g hsg<gt< gm<mh<hm<mt<te/ yu<ouiv hsul< lx<x -V hsg<gjt sf<kqg<gqe<x ntuqx<gg< gm<mh<hm<mt<tk/ -h<hsg<gt<? nux<xqe< wz<jzg<gm<hm<m fqzh<hgkqbqe< Hz<jz Olb<gqe<xe/hSg<gt<Olbikfqzh<hGkqbqe<hvh<hqjeg<g{<Mhqcg<gUl</π weg<ogit<tul</ 0. ABCD? 6lQ 4lQ ntut<tyvosu<ugucufqzligl</fie<g&jzgtqzl< fie<g Gkqjvgt< yu<ouie<xl< 0lQ. fqtg< gbqx<xqeiz< gm<mh<hm<mt<te/ yu<ouiv GkqjvBl< nkx<g wm<cb ujvbqz< Olb<gqxK/ Gkqjvgt< Olbik fqzh<hgkqbqe<hvh<hqjeg<g{<mhqc. π weg<ogit<tul</

52 uqjmgt; hbqx<sq.. (i) 7 os.lq (ii) 480 os.lq (iii) 7.7 os.lq. 704 os.lq. 60os.lQ 4. lq? 696 lq 5. 4 os.lq 6. 4 os.lq 7. 0 os.lq 8. 6.os.lQ hbqx<sq.. (i) 7?00 s/lq (ii) 5?00 s/lq (iii) 7?55 s/lq. 4?956s/lQ. 47. os.lq 4. 0 os.lq lq 6. (i) 6.9s/os.lQ (ii) 5 lq (iii) 7.7 os.lq (iv) 40.8 os.lq os.lq os.lq? 94 os.lq 9. 4 lq lQ 48

53 . sqzlg<gqbgxqbqm. nxquqbz<gxqbqm uieqbz<? -bx<hqbz<? Oukqbqbz<? dbqiqbz< lx<xl< ohixqbqbz< Ohie<x himr<gtqz< fil< lqgh<ohiqb w{<gjtbl<? lqgs< sqxqb w{<gjtbl< g{<cvg<gqe<oxil</ wmk<kg<gim<mig? (i) H,lqbqzqVf<Kgkqvue<dt<tokijzU9,900,000 jlz<gt</ (ii) yvsvisiqbieosz<ziek00, 000,000,000,000 &zg<%xgjtogi{<cvg<gqxk/ (iii) nch<hjmk<kgt<ye<xqe<uip<fitiek uqficgt</ (iv) yvlqe<e[uqe<uqm<ml<slivig ose<clQm<mviGl</ -jkh<ohie<x w{<gjt nh<hc wpkkzl<? jgbitkzl< nu<utu wtqkz<z/ -Vh<hqEl<? nux<jx nmg<gg< Gxq uqkqgjth< hbe<hmk<kq wpkq jgbit LcBl</ nmg<gg<gxquqkqgjtfqjeu%i<ouil</-bz<w{< m, olb<ob{< a -ux<xqx<g a m a a. m giv{qgt</wmk<kg<gim<mig?a 5 a a a a a. -r<ga we<hk ncliel< we<xl<? m NeK hc nz<zk nmg<g we<xl< njpg<gh<hmgqe<xe. a m we<x GxqbQm<cjea e<nmg<g m nz<zk m hcdbi<k<kqb aweh<hcg<gqe<oxil</ nmg<gg<gxquqkqgt<hqe<uvlixkvh<hmgqe<xe: (i) a m a n a m+ n (ohvg<gz<uqkq) m a (ii) a m n, a 0, m > n (ugk<kz<uqkq) n a (iii) (a m ) n a mn (nmg<guqkq) (iv) a m b m (a b) m (Osi<g<jguqkq) a 0 -Vg<Gl< OhiK J a m weg< Gxqh<hqMgqe<Oxil</ a 0 we m a ujvbxg<gqe<oxil</ nmg<gg<gxq uqkqgjth< hbe<hmk<kq? wf<kouiv lqjg olb<ob{<{qjebl< a 0 n we<xnjlh<hqz<wpklcbl<a-r<g a < 0, n yvlp NGl</wMk<Kg<gim<mig? (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

54 -eq?w{<we<xiz<lqjgw{<j{lm<molgxqg<gl</yvw{<{qjea 0 n wenxquqbz<gxqbqm<cz<wpkqeiz<? a e<ksluqiquqz<uvl<lph<hgkqbieknek zqvf<k 9 ujvbqzie w{<{ig -Vg<Gl</ OlZl< n yv LPuiGl</ OlZl< kvh<hm<m Yi<w{<{qjenxquqbz<GxqbQm<cz<lix<xqwPKl<OhiK?fil<kvh<hm<mw{<{qz<dt<t kslh<ht<tqbqje-mkhg<guig<gqz<r -mr<gt<fgi<k<kqeiz<njkmosb<b0 r Nz< ohvg<ggqoxil</ nu<uiox? kslh< Ht<tqjb uzkhg<guig<gqz< r -mr<gt< fgik<kqeiz< njkmosb<b0 r Nz<ohVg<GgqOxil</ wmk<kg<gim<mig? lqgh< ohiqb nz<zk lqgs< sqxqb w{<gjtnxquqbz<gxqbqm<cz<njlg<g?nux<jx nf<knjlh<hqz<ohvg<gouinz<zkugg<gouiwtqjlbig-vg<gl</ wmk<kg<gim<m : hqe<uvl<w{<gjtnxquqbz<gxqbqm<cz<wpkg; (i) 749 (ii) 0500 (iii) (iv) (v) (vi) kqi<u: wmk<kg<gim<m : hqe<uvl<w{<gjtkslucuqz<wpkg; (i) (ii) (iii) (iv) (v) (vi)

55 kqi<u: (i) (ii) (iii) (iv) (v) (vi) wmk<kg<gim<m : g{g<gqmjz fqgp<k<kq hqe<uvueux<xqe< uqjmbqje nxquqbz< GxqbQm<cz<wPKg; (i) (000000) (ii) (4000) 5 (00) (iii) ( ) 4 (iv) (000) (0.000) 4 kqi<u : (i) (000000) ( ) (.0) (0 6 ) (ii) , (4000) 5 (00) (4.0 0 ) 5 (.0 0 ) (4.0) 5 (0 ) 5 (.0) (0 ) (iii) ( ) 4 ( ) 4 (5.0) 4 (0 5 ) (iv) , (000) (0.000) 4 (. 0 0 ( ) ) (. 0) (0). (0 (0 ) ) ( 6)

56 hbqx<sq.. hqe<uvl<w{<gjtnxquqbz<gxqbqm<cz<njlg<gul<; (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) hqe<uvl<w{<gjtksluqiquqz<wpkul<; (i) (ii) (iii) (iv).4 0 (v) (vi) (vii) (viii).4 0. hqe<uvueux<xqe<lkqh<hgjtnxquqbz<gxqbqm<cz<g{<mhqcg<gul<; (i) (00) (40) 5 (ii) (000) (0.00) 4 (iii) (8000) 4 (0000) (iv) (0.00) 8 (0.000) (0.0) 4 (v) (0000) (0.0005) (400000). lmg<jgbqe<gxqbqm \ie< Ofh<hqbi< we<el< Nr<gqOzb g{qk Oljkbieui< g{g<gqmjz wtqjlbigul<? uqjvuigul< osb<k Lcg<g lmg<jgbqe< GxqbQm<cje nxqlgh<hmk<kqeii</ lmg<jg jbg<gxqg<gl<nr<gqzs<osiz< logarithm NeK logos lx<xl< arithmos we<x -V gqovg<gs< osix<gtqzqvf<k uvuqg<gh<hm<mkigl</ logos we<xiz< Äg{g<gQMosb<kz<} lx<xl< arithmos we<xiz< w{< we<x ohivt<hml</ NgOu logarithm we<xiz< w{< g{g<gqm osb<kz< NGl</ lmg<jgbqe< GxqbQm<cje nxqlgh<hmk<k?lkx<g{<{ignmg<gg<gxqbqm<jmh<hx<xqnxqouil</.. nmg<gg<gxqbqm a we<hk yv lqjg w{< we<g/ x yv LP weqz< a x we<x GxqbQm<cje Wx<geOunxqLgh<hMk<kqBt<Otil</x yvuqgqklxw{</nkiuk x q p, -r<gp yvlp? qyvlqjglpweqz<?a x NeKhqe<uVliXujvbXg<gh<hMgqxK/ q p x a a. wmk<kg<gim<mig, ( ) 5 8, 7 ( 7 ) 4. x we<hk yv uqgqklxi w{< we<xohipk? a x we<hjk yv olb<ob{<{qjeg< Gxqh<hkigujvbXg<gzil</Neiz<-f<k ujvbjx osb<ukx<g flg<g dbi<g{qk 5

57 gvk<kg<gt< Okjuh<hMgqe<xe/ -Vh<hqEl<? a > 0 we<xuixt<t wf<kouiv olb<ob{<{qjeg<ogi{<mxwel<yu<ouivolb<ob{<{qx<gl<?a x we<xkeqk<kyv olb<ob{<{qje ujvbxk<k u a x we wpkgqoxil</ -f<fqjzbqz< u NeK nmg<g njlh<hqz< nz<zk nmg<gg< GxqbQm<cz< wpkh<hm<mt<tk we<ohil</ -r<g a J ncliel<we<xl<?x JnMg<Gnz<zKhcwe<Xl<njph<Ohil</LPg<gTg<GwPkqb nmg<gg<gxq uqkqgjt njek<k olb<ob{< nmg<ggtg<gh< ohivf<kukigg< ogit<ouil</-u<uqkqgjt-r<ghqe<uvlixwpkgqe<oxil</ (i) (ii) (iii) x a a x y a y a x a x y a + y (iv) a x a x (v) a b ( a b) x y x xy a a (vi) a 0 x x Olx<%xqb uqkqgjt nmg<gg<gxq uqkqgt< we<xjpg<gqe<oxil</ -h<ohipk lmg<jggxqbqm<cjeujvbxg<glx<hmouil</.. lmg<jgg<gxqbqm b we<xuixb we<hkyvlqjgw{<we<g/xwe<xyvolb<ob{<{qx<gl< b x NeKkeqk<kyVolb<ob{<a Jg<Gxqg<Gl<weWx<geOunxqf<Kt<Otil</a b x we wpk? -r<gt<t nmg<g x J b ncliel< ogi{<m a e< lmg<jg we<x njpg<gqe<oxil<. -kjex log awewpkgqe<oxil</weoux log a we<hk b b a b x we<hke<sllienjlh<higl</olzl<?x log a we<xlmg<jgnjlh<hiek b x a b we<xnmg<gg<gxqnjlh<hqe<sllienjlh<hwe<gqoxil</-v{<m njlh<hqzl<ncliel<)b)ye<ox/ x log a we<xgxqbqm?lmg<jgg<gxqbqmweh<hml</ b -K a b x JGxqg<gqe<xK/ wmk<kg<gim<mig? (i) log 9 79 we<hk 9 79e<sllienjlh<H; (ii) log8 we<hk 8 e<sllienjlh<ha (iii) log we<hk e<sllienjlh<H; (iv) log 7 49 we<hk 7 49e<sllienjlh<H; (v) log 9 e<sllienjlh<h 9 nz< zk 9 ; (vi) log 4 we<hk 4 e<sllienjlh<h/ 8 8 5

58 Gxqh<H: lmg<jgg< GxqbQm<cz< ncliel< Gxqh<hqm Ou{<Ml</ y log x we<x wpkuk ohivtx<xk; Woeeqz< nkx<gs< sllie nmg<gg<gxq njlh<ohpk ncliel< okiqf<kiz<kie< LcBl</ -Vh<hqEl<? sqz fqjzgtqz<? ncliek<kqjeg< Gxqh<hqmilz< lmg<jggjt wpkl<ohik? nr<g njek<k lmg<jggtl< yov ncliek<jkh< ohikuigg<ogi{<mt<tewe<xogit<tou{<ml</ wmk<kg<gim<m 4: hqe<uvl<lmg<jgnjlh<hqjenmg<gg<gxqnjlh<higlix<xul<; (i) log 5 5 (ii) log (iii) log 6 6 (iv) log 4 9 kqi<u: ncliel<-v{<mjlh<hgtqzl<slliekigl</weou NeK ( ) 5 (i) log g<gs<slliekigl</ (ii) log NeK ( ) 4 4 g<gs<slliekigl</ (iii) log 6 6 NeK ( 6) 6 g<gs<slliekigl</ (iv) log NeK () g<gs<slliekigl</ 9 9 wmk<kg<gim<m 5: hqe<uvl<nmg<gg<gxqnjlh<hqjelmg<jgnjlh<higlix<xul<; 6 (i) 64 (ii) 9 (iii) kqi<u: -v{<mjlh<hgtqzl<ncliel<ye<xig-vh<hkiz<? 6 (i) 64 NeK log 64 g<gs<slliekigl</ 6 (ii) 9 79 (iii) (iv) 8 4 NeK NeK log 9 79 g<gs<slliekigl</ log g<gs<slliekigl</ 4 7 NeK log 7 g<gs<slliekigl</ (iv) 7 7 wmk<kg<gim<m 6: lkqh<hqmg; (i) log 9 79 (ii) log 4 8 (iii) log 9 (iv) log (4). 7 kqi<u : (i) x log 9 79 we<g/hqe<h 9 x x. (ii) x log 4 8 we<g/hqe<h 4 x 8. Neiz< 4 x ( ) x x. x. x nz<zk x 54

59 (iii) x log 9 we<g/hqe<h 9 x 7 x, x nz<zk x.. Neiz<9 x ( ) x x. 7 (iv) x log (4) we<g/ hqe<h x (4) nz<zk x 5 nz<zk x 5. wmk<kg<gim<m 7: hqe<uvl<sle<himgjtk<kqi<g<gul<; (i) log x (ii) log b 00 (iii) x log 5 (iv) x + log kqi<u: (i) log x. x nz<zk x. 9 (ii) log b 00. b b 0. (iii) x log x 8 x nz<zk x. 5 8 nz<zk (8 ) x 8 nz<zk 8 x 8. x (iv) x + log x log 7 9 nz<zk log 7 9 x x x x ( 7) 9 nz<zk ( ) nz<zk ( ). nz<zk x 4. -h<ohipk lqjg w{<gtqe< lmg<jggtqe< h{<hgjtg< %xq fq'h{l< osb<ouil</jk<kuqvnjek<klqjgw{<gjtg<gvkgqe<oxil</ (i) ohvg<gz<uqkq: a, m, n we<helqjgw{<gt<? a weqz<? log ( mn) log m + log n. a a a fq'h{l<: log m x, log n y we<g/ a a hqe<h m a x, n a y. m n a x a y nz<zk mn a x+y. -f<knmg<gg<gxqnjlh<hqjelmg<jgnjlh<hqx<glix<xqwpk? log ( mn) x + y log m + log n. a a a uig<gqbk<kqz<? Olx<%xqb uqkq %XuK? -V lqjg w{<gtqe< ohvg<gx<hzeqe< lmg<jgbiek?nu<uqvw{<gtqe<lmg<jggtqe<%mkzg<gs<slligl</ (ii) ugk<kz<uqkq: m, n, a we<helqjgw{<gt<lx<xl< a weqz<? 55

60 m log log m log n a. n a a fq'h{l<: log m x, log n y we<g/hqe<h a a m a x, n a y. x m a n a y a x y. -KnMg<Gnjlh<hqz<dt<tK/-jklmg<jgnjlh<hqz<lix<x? m log x y log m log n. a n a a uig<gqbk<kqz<? ugk<kz< uqkq %XukiuK? n m wel< uqgqkk<kqe< lmg<jgbiek log m log n we<xuqk<kqbisk<kqx<gs<slligl</ a a (iii) hcuqkq: a, m we<he lqjgw{<gt<? a lx<xl< n yvolb<ob{<weqz<? log m n n log m. a a fq'h{l<: log m x we<g/hqe<h m a x NGl</-VHxl<n nmg<gwmg<g? a m n (a x ) n a xn. -KnMg<Gnjlh<hqz<-Vg<gqe<xK/-kjelmg<jgnjlh<hqz< wpk? log m n nx nlog m. a a (iv) a lqjgw{<weqz<? fq'h{l<: log a x log 0. a we<g/hqe<h a x a 0. x 0. nkiuk)n.k* log a (v) a yvlqjgw{<weqz<? log a a. fq'h{l<: x log a we<g/hqe<h a x a a. x n.k a 0. log a. a (vi) nclielix<xz<uqkq: m, n, p lqjgw{<gt<?n lx<xl< p weqz<? log m log m ( log p) n p. n fq'h{l<: x log m, y log p we<g/ p n 56

61 hqe<hp x m, n y y xy p.-ux<xqz< p JfQg<g? ( n ) m n. K n m. -KnMg<Gucuqz<dt<tK/-kjelmg<jgnjlh<hqx<Glix<x? log m xy n.k n x ( log m) ( log p). log n m p (vi) kjzgqp<uqkq: m, n we<he fqr<gzielqjgw{<gt<weqz<? fq'h{l<: log n x we<g/hqe<h m log n m x m. log m n n n x x n n x m n. -KnMg<Gnjlh<hqz<dt<tK/-kjelmg<jgnjlh<hqx<Glix<x? x x. log m nz<zk n x log n m. log n m log a (viii) a, b we<he-vlqjgw{<gt<? b weqz< b b a. fq'h{l<: x log a we<g/hqe<hb x angl</-kqz<x x<gh<hkqzig log a Jh< b b hqvkqbqm log a b b a. (ix) m, n lx<xl< a lqjgw{<gt<? a we<g/ log m log n, weqz< m nngl</ a a fq'h{l<: x log m we<g/hqe<h x log n. a a log m a x n n.k a a n n.k m n (uqkq (viii)e<hc). Gxqh<H: njek<klmg<jggtqzl<ncliek<jkwe<xqz<zilz<hii<k<kg<ogit<gqe<oxil</ Woeeqz<? ncliel< djmb lmg<jgjbg< gvkqoeioleqz<? wmk<kg<gim<mig? log 9 Jg< gvk, -ke< lkqh<h xweqz<x log 9 nz<zk x 9 weg< gqjmg<gqxk/ Neiz<?wf<kolb<ob{<xx<Gl< x 9weg<gqjmg<giK/ ws<siqg<jg: ohikuighqe<uvl<kuxgt<fqgpuib<h<hd{<m/ m log m () log a. a n log n a () log ( m + n) log m + log n. a a a 57

62 () NeK kuxigl</ Woeeqz<? -mk hg<gk<kqz< dt<tkl< uzk hg<gk<kqz< log m dt<tkl<slliglcbik/nkiuk? log m log n a. a a log n a () NeK kux/ Woeeqz<? uzk hg<gk<kqz< dt<tkl< -mk hg<gk<kqz< dt<tkl<sllqz<jz/nkiuk? log ( mn) log ( m + n). a a wmk<kg<gim<m 8: SVg<Gg; (i) log 7 + log 79 (ii) log log kqi<u: (i) kvh<hm<m Ogiju -v{<m lmg<jggtqe< %Mkz< lx<xl< -f<k lmg<jggtqe< nclier<gt<sllibqvg<gqe<xe/weouohvg<gz<uqkqjbh<hbe<hmk<kzil</ log 7 + log 79 log (7 79) 6 log ( ) 9 log 9 log 9 9. (ii) log log 5 log log 5 5 log 5 log 5 ( 5 ) 5 ( ) log 5 ( ). 5 wmk<kg<gim<m 9: SVg<Gg: (i) log 7 98 log 7 4 (ii) log log 9 4 log kqi<u: (i) log 7 98 log 7 4 log 7 log (ii) log log 9 4 log 9 4 log log 9 4 log 9 4 log9 6 + log9 6 log9 64 log9 (6 6) log9 64 log9 96 log log 9 log wmk<kg<gim<m 0: fq'hqg<g; (i) log log0 (ii) log log 5 6 log log. kqi<u: (i) uzkhg<gl< log 4 log

63 4 log 0 log0 8 4 log0 0 log0 8 log log log0 log050 -mkhg<gl</ 8 0 (ii) uzkhg<gl< log 5 6 log log log 5 (6) log 5 (8) + 4 5log 5 log 6 log + 4log log 5 + 4log log log + 4log 5 log + log log log 5 65 log 5 ( 65) log mkhg<gl</ wmk<kg<gim<m : fq'hqg<g; log 4 log 4 5 log5 6 log 6 7 log 7 8 log8 9. kqi<u: -mkhg<gl< ( log 4 log 4 5) ( log5 6 log 6 7) ( log 7 8 log8 9) log 5 log5 7 log7 9 log 5 ( log 5 7 log 7 9) log 5 log 5 9 log 9 log log uzkhg<gl</ wmk<kg<gim<m : log 0 (x + 50) Jk<kQi<g<gUl</ kqi<u: sle<him<cjenmg<gnjlh<hqz<wpk? x nz<zk x nz<zk x 475. log 9 wmk<kg<gim<m : 8 e<lkqh<jhg<g{<mhqc/ log log 9 kqi<u: log 9 log log b a. ( Woeeqz< b a ) wmk<kg<gim<m 4: log 6 x log 6 ( x + ) 0. Jk<kQi<g<gUl</ x kqi<u: ugk<kz< uqkqjbh< hbe<hmk<k? kvh<hm<m sle<him<cje log 6 0 x + wpkzil</-kjenmg<gucuqz<wpk? x 6 0 x + nz<zk x x + nz<zk x. we 59

64 wmk<kg<gim<m 5: log (7 x) log( x) Jk<kQi<g<gUl</ 7 x kqi<u: ugk<kz<uqkqjbh<hbe<hmk<k?kvh<hm<msle<him<cje log x wewpkzil</-kjenmg<gucuqz<wpk? 7 x x nz<zk 7 x ( x) nz<zk 7 x x nz<zk x 4 nz<zk x. wmk<kg<gim<m 6: log ( log x ) Jk<kQi<g<gUl</ kqi<u: y log x we<g/ hqe<h kvh<hm<m sle<himiek log y we<xigqxk/ -kje nmg<g njlh<hqz< wpk? y 4 nz<zk log x 4. -kjebl<? nmg<g ucuqz< wpk? x 4 nz<zk x 8. wmk<kg<gim<m 7: log5 log9 x + log5 Jk<kQi<g<gUl</ kqi<u: kvh<hm<msle<him<cjelix<xqwpk? log 5 log 9 x log 5 nz<zk log 5 9 log 9 x log 5 log 5 5 log mkhg<gk<kqz<nclielix<xuqkqjbh<hbe<hmk<k? log x 5 log5 5. x. 5 hbqx<sq... hqe<uvue siqbi nz<zk kuxi we<xuqjmbtqg<gul<: (i) log 4 5. (ii) log 7. (iii) 6 6 log 4 log log 4. (iv) log (8 4) log 8 log 4. (v) log a. (vi) log a ( m + n) log a m + log a n. a. hqe<uvueux<xqx<gs<sllielmg<jgnjlh<hqjeg<gi{<g: (i) (iv) (ii) (v) (iii) (vi) hqe<uvueux<xqe<lkqh<hgi{<g: (i) log (ii) log 6 6. (iii) log 9. (iv) log 9. (v) log 8. (vi) log 4. 60

65 log (vii) 0. (viii) 5. (ix) okiqbikjkk<kqi<g<gul<: (i) log x (vi) log x. c (ii) log x 7. (vii) log (iii) log 00. (viii) x log 4 (iv) log 5. (ix) log 8 N a b log (v) log 5. (x) log 9 N. x 5. hqe<uvl< yu<ouie<xqx<gl< kvh<hm<m lix<xgtqz< siqbie uqjmjbk< Oki<f<okMk<K wpkg: (i) log x 5, weqz< x NeK (A) 5 (B) 5 (C) 5 (D) 65 (ii) log4 + log (A) 0 (B) (C) (D) (iii) log 4 04 (A) 0 (B) 8 (C) 7 (D) 5 (iv) log a 4 log a x, weqz< x (A) 0 (B) (C) (D) (v) log6 x, weqz< x (A) 4 (B) 8 (C) 6 (D) (vi) log 5 x weqz< x (A) 5 (B) 5 (C) 5 (D) hqe<uvl<yu<ouie<xqz<dt<togijujbyovlmg<jgbigs<svg<gqwpkg: (i) log0 + log0 9. (ii) log + 4log 8. (iii) 5 log + log 4 + log 6. (iv) log4 + log 5 log 5 + log 7. (v) 5log0 + log0 6log (vi) log0 5 + log0 0 log0 4 + log log a x,log a y,log a 5 z lx<xl< log a 7 t, weqz<? hqe<uvl< yu<ouie<xqe< lkqh<hqjebl<x, y, z lx<xl< t-ux<xiz<g{<mhqcg<gul</ (i) log a 6 (ii) log a 4 (iii) log a. 5 (iv) log a 7.. log 6

66 8 (v) log a (vi) log a 600 (vii) log a (viii) log a (ix) log a 5 (x) log a (xi) log a 4. 9 (xii) log a 5 8. hqe<uvl<yu<ouivsle<him<cjebl<kqi<g<gul<: (i) log5 x log5 (ii) log x + log 7 log (iii) x log (iv) log4 x log4 6 log4 56 (v) log4 ( x + ) + log4 (vi) log (x + ) log (x ) log 4 (vii) log 0x log( x + ) (viii) log (7x + ) log (5x ) log (ix) log5 (0 + x ) log5 ( + 4x) (x) log5 (5log x ) (xi) log 0x + 5 log x + 9. hqe<uvl<yu<ouie<xqz<dt<tsle<him<cjefq'hq: (i) log 5 + log 5 (ii) log log0 (iii) log log0 5 (iv) log0 5 log0 (v) log0 5 log0 (vi) log log 0 (vii) log log w{< fqr<gzie lqjg w{<gt< a, b, c weqz<? log b a log c b log a c we fq'hqg<gul</.. ohiklmg<jggt< lmg<jggjt ujvbxk<kohipk? lqjg w{<gtqe< lmg<jg ujvbxg<gh<hm Ou{<Moleqz<? nl<lmg<jgbqe< ncliel< w{< fqr<gzig lx<x wf<kouiv lqjg w{<{ig -Vg<gzil< we uzqbxk<kqbt<otil</ ncliek<jk e we<x uqgqklxi w{<{ig Oki<f<okMk<kiz<? nh<hch<hm<m lmg<jggt< -bz< lmg<jggt< NGl<. ncliek<kqje w{< 0 Ngg< ogi{<miz<? nk<kg lmg<jggt< ohik lmg<jggt< weh<hml</ -bz< lmg<jggjt \ie< Ofh<hqbi< we<hui< nxqlgh<hmk<kqeii</ ohik lmg<jggjt Ofh<hqbvK f{<hi< oae<xq hqiqg<^< we<x Nr<gqOzb g{qk uz<zfi< nxqlgh<hmk<kqeii</ -bz< lmg<jg log e x J? \ie<ofh<hqbvg<gh<ohvjlosig< <Gl< uqklig ln x we<x SVg<glig wpkouil</ ln x Jh< hx<xq uqiquig Olz< ugh<hgtqz< hcg<gzil</ -h<ohipk? ohik lmg<jggtqe< hbe<him hx<xq nxqf<kogit<t Lx<hMOuil</ ohik lmg<jg log 0 x J SVg<glig log x we<x ncliel< 0 Gxqh<hqmilz< wpkouil</ weou? log x y we<xiz< log x y we<x ohivt< ogit<t Ou{<Ml<;-Kx 0 y g<gslliekigl</ y log xz< x weh<hqvkqbqm?

67 y log log jkh<ohizou, for x,,, 0, 00, 000, weg<ogi{<miz<? nkx<ogx<xix<ohiz< 00 0 y,, 0,,,,.weh<ohXgqe<Oxil</-ux<jxh<hm<cbzqmflg<Gg<gqjmh<hK x log x 0 0 fil< gueqh<hk? x NeK olb<ob{< Ogim<ce< lqjg ns<sh< Ohig<gqz< nkqgiqg<gl< OhiK? log 0 x l< nkqgiqg<gqe<xk/ OlZl< fil< gueqh<hk? x > njlf<k njek<k x lkqh<hgtg<g log 0 x NeK lqjg w{<{ig -Vg<gqe<xK? 0 < x < we<xqvg<gl< xe<lkqh<hgtg<g log 0 x e<lkqh<hgjtg<gqp<g{<muixnm<muj{h<hmk<kzil<: xe<uqs<s log x e<-vh<hqml< log x e<lkqh<h < x < < log0 x < a a 0 4 < x < 0 4 < log0 x < b b 0 < x < 0 < log 0 x < + 0.c c 0 < x < 0 < log0 x < + 0.d d 0 < x < 0 0 < log0 x < e e 0 0 < x < 0 0 < log x < f f 0 0 < x < 0 < log x < + 0.g g 0 0 < x < 0 < log x < + 0.h h 0 < x < 0 4 < log x < i i 0 4 < x < < log x < j j Olx<gi[l<nm<muj{bqzqVf<Kfil<gueqh<hK?ohiKlmg<jg log 0 x e<lkqh<h (yvlp) + (0.r r r r 4 ) we<xwpklcbl</-f<knjlh<hqz<?lph<hgkqjb log 0 x e< Ofi<g<%X weul< kslhqe<eh<hgkqjb log 0 x e< hkqe<lieg<%x weul< njph<ohil</ ohikuig hkqe<lieg<%xqje fie<g hkqe<lier<gtqz< )kslk<kiek<kqz<* wpkouil</ log 0 x e< hkqe<lieg< %xiek 0 g<gl< g<gl< -jmh<hm<m lqjg w{< we<hjk nxqgqe<oxil</ log 0 x e<ofig<<%xiek x e<lkqh<jhh<ohivk<kyvlqjglpuigouinz<zkgjx LPuigOuinz<zKH,s<sqbligOui-Vg<Gl</x < weqz<? log 0 x e<ofi<g<%xiek yvgjxlpuig-vg<gl<ax >0 weqz<? log 0 x e< Ofi<g<%xieK yv lqjg LPuig-Vg<Gl<A < x < 0weqz<? log 0 x e<ofi<g<%xiekh,s<sqblig-vg<gl</ 0 6

68 wmk<kg<gim<m 8: hqe<uvl< w{<gtqe< ohik lmg<jggtqe< Ofi<g<%Xgjtg< g{<mhqcg<gul</ (i) 00 (ii) 00. (iii) 0.0 (iv).00 (v) 0.00 (vi) (vii) (viii) (ix) (x) kqi<u: (i) 00 NeK 0 g<gl< 0 4 g<gl<-jmbqz<-vh<hkiz<? log 0 00 e<lkqh<hiek +0. dd d d 4 we<xqvg<gl</weou log 0 00 e<ofi<g<%x; hkqe<lieg<%x 0. dd d d 4 NGl</OuXupqbig?00Jnxquqbz<GxqbQm<cz<wPk? log0 00 log0 (.00 0 ) log log0 0 log log0 0 + log dd dd 4. 0 log 0 00 e<ofig<%x. [x wel< lqjg olb<ob{<{qje a 0 n we<xuix nxquqbz< GxqbQm<cz< wpkqeiz<? -r<ggqjmg<gh<ohxl<lpnnek log0 x e<ofi<g<%xigl<a log 0 a e<lkqh<hiek log 0 x e<hkqe<lieg<%xigl</] (ii) log 00. e<ofi<g<%x ; lx<xl< log 00. e< hkqe<lieg<%x log 0.00 ( log 00. e< hkqe<lieg<%xl<? log 00 e<hkqe<lieg<%xl<sll<we<hjknxqg/* (iii) log 0.0 e<ofi<g<%x; lx<xl<log 0.0 e<hkqe<lieg<%x log.00. (iv) log.00 e<ofi<g<%x 0; log.00 e<hkqe<lieg<%x log.00. (v) log 0.00 e<ofi<g<%x ; log 0.00 e<hkqe<lieg<%x log.00. (vi) log e<ofi<g<%x ; log e<hkqe<lieg<%xlog.00. (vii) log e<ofi<g<%x ; log e<hkqe<lieg<%xlog.00. (viii) log e<ofi<g<%x 4; log e<hkqe<lieg<%xlog.00. (ix) log e<ofi<g<%x 5; log e<hkqe<lieg<%x log.00 (x) log e<ofi<g<%x 6; log e<hkqe<lieg<%x log.00. Olx<gi[l<wMk<Kg<gim<MgtqzqVf<K?fil<nxquK? 64

69 (i) lqjg w{< x e< LP w{< hgkq n -zg<gr<gt< ogi{<m w{< weqz<? log 0 x e< Ofi<g<%X n. (ii) lqjgw{< x e<lpw{<hgkq0lx<xl<xe<hkqe<lie)kslk<kie*hgkqbqz< hkqe<lie Ht<tqg<G uzh<hxk<kqz< dt<t Lkz< H,s<sqblx<x -zg<gk<kqx<g Le< n H,s<sqbr<gt<ohx<xqVh<hqe<? log 0 x e<ofi<g<%x (n + ). (iii) 00000, 00, 00.,, jubjek<kl< yov hkqe<lieg<%xqjeh< ohx<xt<te/ NgOu? kvh<hm<m uiqjsbqz< njlf<k sllie lkqh<hjmb -zg<gr<gjth<ohx<xnjek<kw{<gtg<gl<yovhkqe<lieg<%xkie<-vg<gl</ wmk<kg<gim<m 9: log kvh<hce<?hqe<uvueux<jxg<g{<mhqcg<gul<; (i) log 60 (ii) log 6. (iii) log.6 (iv) log 0.6 (v) log 0.06 (vi) log kqi<u: log (i) log (ii) log (iii) log (iv) log (v) log (vi) log Gxqh<H: JSVg<glig.095wewPKOuil<. -jkh<ohizou J 5.0 we wpkouil</ g{g<gqml<ohipk? yv ohik lmg<jgg<g Gjx w{< gqjmg<gzil</ kgf<k lqjg LPju nf<k Gjx w{<[me< %m<c gpqg<g? Gjx w{<{igg< gqjmk<k lmg<jgbqje )Gjx LP* + 0.d d d d 4 we<x njlh<hqz< wpk LcBl</ wmk<kg<gim<m 0: log lx<xl< log we<x kvh<hce<? hqe<uvueux<jxg<g{<mhqcg<gul</ 5.00 (i) log 0.05 (ii) log (iii) log (iv) log kqi<u: log log 0.5 e< hkqe<lieg< %X NGl</ -jkh<ohizou?log log 00 e<hkqe<lieg<%x (i) log (ii) log (iii) log log 5 log

70 .00 (iv) log 500 log.00 log lmg<jggtqe<nm<muj{.000 Lkz< ( hkqe<lieh< hgkqbme<) ujvbqzie lqjg w{<gtqe< ohik lmg<jggtqe< lkqh<hg<gjtg< g{g<gqm<m nm<muj{ njlh<hqz< hm<cbz< hmk<kq dt<tei</ )-f<f~zqe< gjmsq fie<g hg<gr<gjth< hii<g<gul<*/ -u<um<muj{jb lmg<jgnm<muj{ we<ohil</-f<knm<muj{jbh<hbe<hmk<kq?wf<kouivlqjg w{<{qx<gl< ohik lmg<jgbqjeg< g{g<gqmzil</ lmg<jgbqe< hbe<himgjth< hx<xq nxqf<k ogit<ukx<g Le< lmg<jg nm<muj{jb hck<kxqbl< Ljxbqjeh< hpg<gh<hmk<kqg<ogit<ouil</ wmk<kg<gim<m : log 6.78 Jg<g{<Mhqc/ kqi<u: Ofi<g<%X. hkqe<lieg<%xqjeh<ohxukx<g?.678 Jg<gVKg/lmg<jg nm<muj{bqe<-mh<hg<guqtql<hfqvzqz< (column).6 e<-mk<jknxqbul</ Mean differences (svisiquqk<kqbisl<) dt<t fqjx (row) upqbig hck<kg<ogi{<om ose<xiz< 7 e< fqvzg<gg< gqp< g< gi{<ohil</ OlZl< nok fqjxbqz< okimi<f<k ose<x svisiq uqk<kqbisk<kqz< 8 e< fqvzqe< gqp< w{<0jg< gi{zil</ -f<k w{< 0 J dme< %m<m gqjmg<gl</ -KOu.678e<ohiKlmg<jgbiGl<)hkqe<lieg<%xiGl<*/weOu? log wmk<kg<gim<m : log Jg<g{<Mhqcg<gUl</ kqi<u: Ofi<g< %X, hkqe<lieg< %X log h<ohipk.006 Jg< gvkg/ -kqz< 5 hkqe<lier<gt< dt<te/ Neiz< ohik lmg<jgbqz< hkqe<lier<gt<dt<tw{<gtg<gh<hm<cbz<-mh<hm<mt<te/weou we Okivibh<hMk<kqg<ogit<Ouil< )4uK hkqe<lie -mk<kqz< 5J uqmg< Gjxuie -Vh<hkiz<*/ -h<ohipk? lmg<jg nm<muj{jbg< ogi{<m log we nxqbzil</weou? log wmk<kg<gim<m : log 70.9Jg<g{<Mhqcg<gUl</ kqi<u:

71 0Ofi<g<%X. hkqe<lieg<%x log (Woeeqz< 4 uk hkqe<liel< 9. -K5 J uqms< sqxqbkz<z). weou lmg<jgh< hm<cbzqz< -Vf<K log log wkqi<lmg<jgnm<muj{ log x y weqz< x Jye< wkqi<lmg<jgwe<gqoxil</nkiuk? log 0 x yweqz<? x 0 y NeK y e< wkqi<lmg<jgbigl</ weou? y e< wkqi<lmg<jg 0 y. wkqi<lmg<jg ohxuk lmg<jgh< ohxukx<g Ofi<lixiGl</ wmk<kg<gim<mig? log 0.00 weqz<.00 e<wkqi<lmg<jg 0NGl</ zqvf<k (fie<g -zg<gr<gt< dt<t hkqe<lieh< hgkq) dt<t w{<gtqe<wkqi<lmg<jggtg<gienm<muj{-f<f~zqe<gjmsq-v{<mh<hg<gr<gtqz< ogimg<gh<hm<cvg<gqe<xk/ -u<um<muj{jbh< hbe<hmk<kq ogimg<gh<hm<m wf<kouiv w{<{qe<wkqi<lmg<jgbqjebl<ohxzil</ wmk<kg<gim<m 4: hqe<uvueux<xqe<wkqilmg<jgbqjeg<g{<mhqcg<gul</ (i).05 (ii).469 (iii).8658 (iv).0578 kqiu: ( y e< wkqilmg<jgjbg< g{<mhqcg<g Lkx<g{< y e< hkqe<lieg< %xqje lm<ml< gvkg/ hqe<h hkqe<lieg< %xqe< Lke< &e<x -zg<gr<gtg<g Wx<x wkqi< lmg<jgbqjeg< g{<m nkeme< svisiq uqk<kqbisk<kqz< dt<t 4 uk-zg<gk<kqx<gie lkqh<hqjeg<%m<mg<gqjmh<hkokjubiewkqilmg<jgbigl</) (i).05 e<wkqilmg<jg wkqv<lmg<jgh<hm<cbzqzqvf<k svisiquqk<kqbisl< 5 g<gwx<xk (ii).469 e<wkqilmg<jg (iii).8658 e<wkqilmg<jg

72 (iv).0578 e<wkqilmg<jg Gxqh<H: Olx<gi[l<wMk<Kg<gim<czqVf<K?fil<nxquK (i) log x e<ofig<%xiek n wel<gjxbz<zilpweqz<?log x e<wkqilmg<jgbqz< dt<thkqe<lieh<ht<tqbiek ( n + ) uk-zg<gk<kqx<gh<hqe<-mh<hmgqe<xk/ OfIg<%X hkqe<lieh<ht<tqbqe<fqjz 0 uk-zg<gk<kqx<gnmk<k uk-zg<gk<kqx<gnmk<k uk-zg<gk<kqx<gnmk<k 4 uk-zg<gk<kqx<gnmk<k 4 5 uk-zg<gk<kqx<gnmk<k (ii) log x e<ofig<%x n we<xgjxw{<weqz<?log x e<wkqilmg<jgbqz<?hkqe<lieh< Ht<tqbieK? hkqe<lieh< Ht<tqg<G nmk<kt<t n -mr<gtqz< H,s<sqbLl<? n uk -mk<kqz<h,s<sqblqz<ziw{<uvlix-mh<hmgqe<xk/ OfIg<%X hkqe<lieh<ht<tqbqe<fqjz 0.d d d d 4 0.0d d d d d d d d d d d d d d d d 4 Olx<gi[l<nm<muj{bqz<, d. d d d 4 we<hkkvh<hm<mlmg<jgbqz<dt<thkqe<lieg< %xqekwkqilmg<jgbigl</..6 lmg<jgbqjeh<hbe<hmk<kqg<g{g<gqmz< -e<jxb dzgqeqz<? g{g<gqmjz lqgougligul<? lqgk< Kz<zqbligUl< ohx lqe<e[ g{g<gqmuie< )calculator) lx<xl< g{qeq (computer) -jubqv{<ml< dt<te/ Neiz<-s<siker<gt<Wx<hMukx<GLe<eIleqkNx<xjzg<ogi{<Mg{g<gqm<Omil</ g{g<gqmjz lqg Ouglig leqk Nx<xjzg< ogi{<m g{g<gqm lmg<jggt< nxqlgh<hmk<kh<hm<me/ -kx<gig lmg<jg? wkqilmg<jg Ngqbux<xqx<gie nm<muj{gt<kbiiqg<gh<hm<me/-h<ohipk?g{g<gqmzqz<lmg<jgbqe<hbjeg<gim<m sqz wmk<kg<gim<mgjtg< gi{ Lbx<sqh<Ohil</ hqe<uvl< uib<h<himgt< Okjuh<hMgqe<xe/ 68

73 (i) log ( mn) log m + log n a a a (ii) m log log m log n a n a a (iii) log m n nlog m. a a wmk<kg<gim<m 5: gi{<g (i) (ii) kqiu; (i) x we<g/ hqe<h log x log ( ).4456 log log x.854 e<wkqilmg<jg (ii) x we<g/hqe<h log x log log log x. 7 e<wkqilmg<jg0.07. wmk<kg<gim<m 6: lkqh<hqmg; (i) (ii) kqv<u: (i) x we<g/hqe<h log x log 00 log x.09 e<wkqilmg<jg (ii) x we<g/hqe<h log x log 0.4 log ( +0.54) ( ) x e<wkqilmg<jg

74 wmk<kg<gim<m 7: hqe<uvueux<xqe<lkqh<hgjtg<g{g<gqmg/ (i) (9.76) 5 (ii) (0.749) 7 kqiu: (i) x (9.76) 5 we<g/hqe<h log x log (9.76) 5 5 log (9.76) x e<wkqilmg<jg (ii) x (0.749) 7 we<g/hqe<h log x 7 log( 0.749) ( ) x.07 e<wkqilmg<jg wmk<kg<gim<m 8: e<lkqh<hgi{<g/ kqiu: x (0.7) 5 we<g/hqe<h log x log (0.7) log x.8867e<wkqilmg<jg wmk<kg<gim<m 9: e<lkqh<hqjeg<gi{ul</ kqiu: x we<g/hqe<h log x log log.59 log Lkz< (&e<x hkqlie hqe<eh< hgkqbme<) ujvbqzie w{<gtqe< ohik lmg<jgh< hm<cbziek kbii osb<k kvh<hm<mt<tkiz<? fil< hqe<uvl< Okivibr<gjt osb<bou{<cbqvg<gqxk/ log x x 0.69e<wkqIlmg<jg.. 70

75 (76.5).98 wmk<kg<gim<m 0: e<lkqh<hg{<mhqc/ 5 (4.75) kqiu: kvh<hm<mogijubqjex we<g/hqe<h log x log (76.5) + log. 98 [log (4.75) 5 + log 0.046] log log.98 [5 log log 0.046] [ ] [ ] ( ) x.995 e<wkqilmg<jg wmk<kg<gim<m : log 4.56 e<lkqh<hgi{<g/ kqiu: -r<g ncliel< we dt<tk/ lmg<jg nm<muj{jbh< hbe<hmk<k lmg<jgbqe< ncliel< 0 we -Vg<g Ou{<Ml</ ncliek<jk lix<xl< uqkqjbh< hbe<hmk<k? log 4.56 log log 0 log log 0.69 x (0.0) + ( ) + (0.775).775 x.775 e<wkqilmg<jg we<g/hqe<h log x log.079 hbqx<sq... hqe<uvl<w{<gtqe<ohiklmg<jggtqe<ofig<%xgjtg<gi{<g; (i) 4 (ii) 7.6 (iii).65 (iv) (v) (vi) (vii) (viii) hqe<uvlix ohik lmg<jggt< ohx<xt<t w{<gtqe< LPh< hgkqbqz< dt<t -zg<gr<gtqe<w{<{q<g<jgbqjeg<g{<mhqcg<gul<; (i).45 (ii).456 (iii).4567 (iv) 0.4 (v) (vi)

76 . w{<gtqe< ohik lmg<jggt< gqop kvh<hm<mt<te/ w{<gtqe< hkqe<lieh< Ht<tqg<G nh<hiz< Lkz< H,s<sqblqz<zik -zg<gk<kqx<g Le<uVl< H,s<sqbr<gtqe< w{<{qg<jgjbg<g{<mhqc/ (i). 456 (ii).45 (iii). (iv) (v) (vi) e<ohiklmg<jgbqe<hkqe<lieg<%x 0.55 NGl</hqe<uVl<w{<gtqe< ohiklmg<jggjtwpkg; (i) 740 (ii) 74 (iii) 7.4 (iv).74 (v).74 (vi) 0.74 (vii) (viii) lmg<jgnm<muj{jbh<hbe<hmk<kqhqe<uvl<w{<gtqe<ohiklmg<jggjtg< g{<mhqcg<gul</ (i) 87 (ii) 8450 (iii) (iv) (v) (vi) hqe<uvl<ohiklmg<jggtqe<wkqilmg<jggjtg<g{<mhqcg<gul</ (i).890 (ii) 0.4 (iii).458 (iv). 46 (v) 5.50 (vi) hqe<uvlixx e<ohiklmg<jg-vf<kiz<?x e<lkqh<jhg<gi{<g/ (i) 5.07 (ii).968 (iii).04 (iv).77 (v) 0.96 (vi) 4.08 (vii). +. (viii) 5.4. (ix). 5.4 (x). (xi).4 (xii) hq<e<uvueux<jxlkqh<hqmosb<g/ (i) (ii) (iii) (iv) (v) (0.075) (vi) (50.49) 5 (vii) (55.9) 8 (viii) (ix) (x) (xi) ( 4.) (xii) (.4) (xiii) log 5.6 (xiv) log

77 . g{g<gxqbqm fuqeg{qkk<kqe<sikjeobg{g<gxqbqmigl</njek<kg{qkh<hqiqugtqzl< g{g<gxqbqmokie<xgqe<xk/g{qkk<kqjencogit<ki<g<giqkqbqz<dvuig<gg{qk uz<zfi<gt< Lbx<sqk<kke< uibqzig g{g<gxqbqmgt< Okie<xqe/ o\i<le< g{qk Oljk \ii<<\< Oge<mi< (845 98) dvuig<gqb g{r<gtqe< ogit<jgob g{qk uti<s<sqg<g jlz<gz<zig njlf<kk/ g{k<kqe< gvk<kvuqjebl<? g{g<ogit<jg hx<xqbnch<hjmg<gvk<kg<gjtbl<nxqlgh<hmk<k-h<ohipklbz<ouil</.. yvg{k<kqe<gvk<kv fe<g ujvbjx osb<bh<hm<m dxh<hgtiz< Ne okigkqjb g{l< we<ohil</ wmk<kg<gim<mig? njek<k -bz< w{<gtqe< okigh<h? yv ktk<kqzt<t njek<k slhg<g Lg<Ogi{r<gtqe< okigh<h? kqvk<k{q nvsqei< N{<gt< Olz< fqjzh< ht<tqbqe< njek<k ye<hkil< ugh<h li{ui<gtqe< okigh<h? njek<k olb<ob{<gtqe<okigh<h?nr<gqzwpk<kg<gtqzt<tnjek<k vowels.gtqe<okigh<h Ohie<xjugt< g{r<gtg<gs< sqz wmk<kg<gim<mg<gt< NGl</ OlOz Gxqh<hqm<Mt<t okigh<hg<gtqz< dt<t dxh<hg<gt<? fl<liz< wju we<x dxkqbqm<mg< %x Lcukiz<kie<njug{r<gt<Ngqe<xe/hqe<uVl<wMk<Kg<gim<Mgjtg<gVKOuil<; (i) de<ejmbugh<hqzt<tdbvlieli{ui<gtqe<g{l</ (ii) fqhck<kfz<zhk<kgr<gtqe<g{l</ dbvl<? fz<z Ohie<xosix<gTg<Gsiqbientuqz<ohiVt<ogit<tLcbikkiz< Olx<Gxqh<hqm<mwMk<Kg<gim<Mgt<fe<GujvbXg<gh<hm<mjunz<z/weOunux<xqe< dxh<hgjtfl<liz<siqbigg<%xlcbuqz<jz/weounjug{r<gt<ngik/ Gxqh<H: yvg{k<kqzt<tdxh<hgt<keqk<kjubigl</ yv g{k<kqe< ohivtqje yv keqll< nz<zk Yi< dxh<h nz<zk Yi< nr<gk<kqei<wenjph<ohil</ g{r<gjtg<gxqg<ga nz<zk B Ohie<xohiqbNr<gqz wpk<kg<gjtbl<? nux<xqe< dxh<hgjtg< Gxqg<g x,y,a,b Ohie<x sqxqb Nr<gqz wpk<kg<gjtbl<hbe<hmk<kouil</x NeKg{l<A z<yi<dxh<hwe<hjkg<gxqg<g x AwewPKOuil</ we<xgxqbqm dt<tk weh<ohivt<hml</ keqll< x NeK Az< dxh<hz<z we<hjkg< Gxqg<g? x A we wpkouil</ wmk<kg<gim<mig? A we<x g{k<kqe<keqlr<gt<,, 4, 5weqz<?-kje A, A, 4 A, 5 A. Neiz< 6 A, A, 9 A, Glii< A. yvg{k<kqjenjlg<g?hqe<uvl<-vljxgjth<hqe<hx<xouil<: () nm<muj{nz<zkuiqjsh<hm<cbz<ljx () g{g<gm<mjlh<hnz<zkuqkqljx 7

78 .. nm<muj{ nz<zk uiqjsh<hm<cbz<ljxbqz<g{k<kqjeg<gxqk<kz< njmh<hgt< { } -ux<xqx<gt< yv g{k<kqzt<t njek<k dxh<hg<gjtbl<? wel< Gxq ogi{<mh< hqiqk<kh< hm<cbzqm<m ng<g{k<kqje wpkouil</ wmk<kg<gim<mig? mathematics we<el< osiz<zqzt<t njek<k dbqi< wpk<kg<gtqe< g{l< {a, e, i}ngl</ -r<g a wel<wpk<k-vljx mathematics we<el<osiz<zqz< uf<kizl< g{k<kqz< Gxqh<hqm<omPKl<OhiK yovobiv Ljx lm<mol dxh<hig wpkouil</ Woeeqz< g{k<kqz< dt<t dxh<hgt< keqk<kjugtig -Vk<kz< Ou{<Ml</ -u<uix dxh<hgjt { } wel< njmh<hg<gt< wpkl< Ljxbqje nm<muj{ nz<zkuiqjsh<hm<cbz< Ljx we<ohil</nm<muj{ljxbqz<sqz g{r<gjthqe<uvlixwpkgqoxil<: (i) Juqmg<Gjxuiehgiw{<gtqe<g{l<{,, 5, 7, }NGl</ (ii) FOOTBALL we<el<osiz<zqzt<twpk<kgjtg<ogi{<mg{l< {F, O, T, B, A, L} (iii) njek<k -bz< w{<gtqe< g{l< {,,, }. -r<g we<x GxqbQM? okimi<f<k uvl< njek<k dxh<hgjtbl< ye<xqjebl< uqmik Gxqh<hkiGl</ -g<g{k<kqje NwEl<sqxh<ohPk<kiz<Gxqh<Ohil<AN {,,, } (iv) njek<k LP w{<gt< 0,,,, Ngqbeux<jx dxh<hg<gjtg< ogi{<mg{k<kqje W we<xsqxh<ohpk<kiz<gxqh<ohil<aw {0,,,, }. (v) njek<k LPg<gtie 0,,,,,,, Ngqbeux<jx dxh<hqei<gtigg< ogi{<m g{l< Z we<x sqxh<ohpk<kiz< Gxqg<gh<hMgqe<xKA Z{0,,,,,,, }. wmk<kg<gim<m : 5 e<lmr<ggt<?neiz< 50g<Gg<Gjxuienjek<K-bz<-vm<jmh< hjmw{<gjtg<ogi{<mg{k<kqjenm<muj{ljxbqz<wpkg/ kqi<u: 5 Nz<uGhMl<-vm<jmh<hjm-bz<w{<gt< 0, 0, 0, 40, 50, NGl</ 50g<Gg<Gjxuie5Nz<uGhMl<njek<K-vm<jmh<hjmw{<gtqe<g{l< {0, 0, 0, 40}... g{g<gm<mjlh<h nz<zkuqkqljxbqz<g{k<kqjenjlk<kz< g{k<kqjeg< g{g<gm<mjlh<h we<el< hqxqokiv LjxbqZl< wpkzil</ -l<ljxbqz<g{k<kqjewpk?lkzqz<nke<dxh<hgtg<gqjmobdt<tohikuie h{<hqje nxqb Ou{<Ml</ -h<h{<h? ng<g{k<kqz< dt<t dxh<hgtg<g lik<kqvol d{<jlbigou{<ml</wmk<kg<gim<cx<g?{6, 6, 6}we<xg{k<kqjeg<gVKOuil</ -ke< dxh<hgt< 6, 6, 6 we<he 6e<nMg<Ggt<wEl<h{<hqjeh<ohx<Xt<te/ weou x 6 n, -r<g n,, wel< fqhf<kje 6, 6, 6Ngqbw{<gjtlm<MOl kvgqe<xk/ -f<k fqhf<kjeg<gm<hm<m Ouoxf<k w{<j{bl< ohx -bzik/ NgOu? g{l< {6, 6, 6} NeK x 6 n, n,, wel<fqhf<kjejbfqjxuosb<bl< njek<kdxh<hgt<xnz<nekwe<xnxqbzil</-kje { x x 6 n, n,, } wewpkouil</-l<ljxbqz<?{}we<xnjmh<hg<gxq -ux<jxg<ogi{<mt<tg{l< we<hjkg< Gxqg<gh< hbe<hmgqxk/ we<el< (vertical bar) Ofi<Gk<Kg<Ogim<Mg<Gxq 74

79 hqe<uvueux<jxk<kvuz<z (such that)we<xohivtqz<hbe<hmk<kh<hmgqxk/ohikh< h{<hie x 6 n, n,,. we<hk g{k<kqje dvuig<gl< uqkqbig -Vh<hkiz< -l<ljx uqkqljx nz<zk g{g<gm<mjlh<h Ljx weh<hmgqe<xk/ Ouoxf<k dxh<hqx<ge<xq ogimg<gh<hm<m g{l< Ae< dxh<hqx<g lm<mol h{<h P ohivk<klx<xiz<? g{l< A JA { x h{<hp Jx ohx<xt<tk} wewpkouil</-kje?h{<hp J ohx<xt<tnjek<kdxh<hgt<x gtqe<g{l< Aweh<hch<Ohil</ wmk<kg<gim<m : hqe<uvueux<jxuqkqljxbqz<g{ligg<gxqg<gul<: (i) 6Juqmg<Gjxuienjek<K-bz<w{<gtqe<g{l</ (ii) Nr<gqzwPk<Kg<gtqZt<tnjek<Kdbqi<wPk<Kg<gtqe<g{l</ (iii), 4, 6,.Ngqbw{<gjtg<ogi{<mg{l</ kqi<u: (i) x N, x < 6 we<hkxyv-bz<w{<lx<xl<6 Juqmg<GjxUwe<hjk uquiqg<gl<%x<xigl</ { x x N, x < 6}we<hKOkjubieg{k<kqjeg<Gxqh<hkiGl</ (ii) GxqbQM x we<hkyi<nr<gqzdbqi<wpk<kqjeg<gxqh<hkigg<ogi{<miz<? {x x yvnr<gqzdbqi<wpk<k}we<hokokjubieg{ligl</ -g<g{k<kqje{ x x a, e, i, o, u} nz<zk{a, e, i, o, u} wewpkzil</ (iii), 4, 6, we<xucuqzt<tyi<w{<j{ x n, n N we<x %x<xqeiz< uquiqg<gzil</ { x x n, n N}we<hKOkjubieg{liGl</ Gxqh<H: dxh<hgjth< hm<cbzqmilz< nux<xqe< h{<hgjtg< ogi{<m g{g<gm<mjlh<h Ljxbqz< Gxqh<hqmh<hMukiz< g{g<gm<mjlh<h Ljxbqje uquiqk<kz< Ljx weul< njpg<gzil</ wmk<kg<gim<m 4: g{l< A {x x + 5 7, x N}Jnm<muj{Ljxbqz<wPKg/ kqi<u: x x r<g N. w{<jk< kuqvouoxf<kw{<[l< x + 5 7, x N we<xh{<hqjefqjxu osb<bik/ A {}. wmk<kg<gim<m 5: g{l< A,,,,,, Jg{g<gm<mjlh<HLjxbqz<wPKg/ kqi<u: g{l< Ae<dXh<Hg<gt<Lkz<WP-bz<w{<gtqe<kjzgQpqgtiGl</weOu A x x, n N lx< Xl< n 7. n Gxqh<H: x, y we<hk x lx<xl< y weg<gxqg<gl</..4 LcUXlx<Xl<Lcuqzqg{r<gt< g{l< Ae<dXh<Hg<gjtye<xe<hqe<ye<xigw{<{qmk<okimr<gq?ns<osbz< LcUohXlieiz<?g{l<AJLcUXg{l<wenjpg<gqOxil</ 75

80 nu<uixw{<[l<ljxlcuohxuqz<jzobeqz<a J Lcuqzqg{l< we<ohil</ g{l<a e<dxh<hg<gtqe<w{<{qg<jg?nkqw{<we<xnjpg<gh<hml</-kjen(a) weg<gxqh<ohil</ngoun(a) we<hkg{l< Ae<dXh<Hg<gtqe<w{<{qg<jgbiGl</A yvlcuxg{l<weqz<n(a) yvlpw{<ngl</ wmk<kg<gim<m 6: hqe<uvueux<xqzqvf<k LcUX lx<xl< Lcuqzq g{r<gjtg< Gxqh<hqMg: (i) A {x x W, x < 5}. (ii) {klqpgk<kqzt<tnjek<kh<ht<tqgt<}. (iii) {dekht<tqbqzt<tnjek<kye<hkil<ugh<hli{ui<gt<}. (iv) N. (v) W. (vi) Z. (vii) njek<khgiw{<gt<. kqi<u: (i) x W, x < 5 x 0,,,, 4. A {0,,,, 4}. A e<dxh<hg<gt<0,,,, 4 gjtljxoblkzil<?-v{<mil<? &e<xil<? fie<gil< lx<xl< Jf<kil< dxh<hg<gt< we w{<{qml< osbz< LcUXgqxK/ weou n(a) 5 lx<xl< A yvlcuxg{l<. (ii) klqpgk<kqzt<t njek<kh< ht<tqgjtbl< yu<ouie<xig w{<{qm Nvl<hqk<kiz<? ns<osbz< yv gm<mk<kqz< LcUXl</ weou {klqpgk<kqzt<t njek<kh< ht<tqgt<} wel<g{l<lcuxg{ligl<. (iii) dek ht<tqbqzt<t ye<hkil< ugh<h li{ui<gt< njeujvbl< w{<{qml< osbz< LcUXl</ weou {dek ht<tqbqzt<t njek<k ye<hkil< ugh<h li{ui<gt<} g{l<lcuxg{ligl<. (iv) N {,,, }.-g<g{k<kqe<dxh<hg<gjtyu<ouie<xigw{<{qmnvl<hqk<kiz<? -s<osbz<ljxlx<xh<ohxik/weou N yvlcuqzqg{ligl</ (v) W {0,,,, }. W e< dxh<hgjt yu<ouie<xig Lkz< dxh<h? -v{<mil< dxh<h?////wew{<[l<ljxlx<xh<ohxik/weouwyvlcuqzqg{ligl</ (vi) Z {0,,,,, } e< dxh<hg<gtie 0,,,,. J LjxOb Lkzil< dxh<h?-v{<mil<dxh<h////wew{<[l<ljxlx<xh<ohxik/ngou Z l<lcuqzq g{ligl</ (vii) hgi w{<gt<,, 5, 7,,, 7 gjt w{<[l< Ljx Lx<Xh<ohxikkiz<? njek<kh<hgiw{<gtqe<g{l<lcuqzqg{ligl<...5 oux<xg<g{l<nz<zkdxh<hqzqg{l< dxh<hgt< nx<x yv g{k<kqje oux<xg<g{l< nz<zk dxh<hqzqg{l< we njph<ohil</oux<xg<g{k<jkøweg<gxqh<ohil</ Ø { }lx<xl< n(ø) 0. wmk<kg<gim<m 7: hqe<uvueux<xt<oux<xg<g{r<gt<biju@ (i) {Nz<uGhMl<yx<jxh<hjm-bz<w{<gt<} (ii) {4JyVgiv{qbigh<ohx<xhgiw{<gt<} (iii) {x x W, x N}. (iv) {Ø}. 76

81 kqi<u: (i) Nz<uGhMl<yx<jxh<hjm-bz<w{<gt<wKUlqz<jzwe<hkiz<? { Nz<uGhMl<yx<jxh<hjm-bz<w{<gt<} Ø NGl</ (ii) Yi< w{<{qe< ug w{<gt< lx<xl< nok w{< we lm<mol -Vh<hqe<? nf<k w{<j{h<hgiw{<we<ohil</weou?4 Jgiv{qbigg<ogi{<mhgiw{<wKlqz<jz/ {4JyVgiv{qbigh<ohx<xhgiw{<gt<} ØNGl</ (iii) W g<gdxh<higul<neiz< Ng<GdXh<hig-z<zikyOvdXh<HNGl</ nkiuk? x W, x N x 0. {x x W, x N} {0}. -KYVXh<Hg{l<NGl</ weou-kyvoux<xx<xg{l<ngl</ (iv) {Ø} z<oux<xg<g{l<ø Yi<dXh<hiGl</weOu {Ø} oux<xg<g{lz<z/nkiuk {Ø} Ø. 0{Ø} YIYVXh<Hg{liGl</..6 slieg{r<gt< g{r<gt< A, B we<he yov w{<{qg<jgbqz< dxh<hg<gjtg< ogi{<cvh<hqe< njugt< slie g{r<gt<weh<hml</g{l< A NeK g{l< Bg<G slie g{lig -Vh<hqe<n(A) n(b) NGl</-kje A BwewPKOuil</wMk<Kg<gim<mig? A {,, }, B {, 9, }, n(a), n(b). weou A B...7 slg{r<gt< -V g{r<gt< A, B we<he nok dxh<hgjth< ohx<xqvh<hqe<? nux<jxs< sl g{r<gt<we<gqoxil</-u<uixibqe< A B we wpkgqe<oxil</ wmk<kg<gim<mig? A {,,, 4}, B {x x N, x < 5}weqz< A B. Woeeqz<? Bz<dt<tuqkq x N, x < 5 x,,, 4. fil< nxquk? A, B -v{<czl< nok dxh<hgt< -Vh<hOk A, B -ju sllibqvk<kjzdxkqosb<ukigl</weouyvg{k<kqz<dt<tdxh<hgjtwf<k uiqjsbqzl<uqvh<hl<ohiz<wpkzil</wmk<kg<gim<mig? {,,, 4} {4,,, } {,,, 4} {, 4,, } Gxqh<H: -V g{r<gt< sloleqz<? nju yov w{<{qg<jgbqzie dxh<hgjtg< ogi{<mt<te/weounjuslieg{r<gt</wmk<kg<gim<mig?{,,, } lx<xl< {,,, } -v{<ml<slg<g{r<gtibqvk<kozimslieg{r<gtigul< -Vg<gqe<xe/ Neiz<slieg{r<gt<slg{r<gtig-Vg<gOu{<Ml<we<hkqz<jz/wMk<Kg<gim<mig? {,,, } lx<xl< {,,, 4} we<el<g{r<gt<yovw{<{qg<jgbqziedxh<hgjtg< ogi{<mt<tkiz< nju slie g{r<gt</ Neiz< nju sl g{r<gtigik/ Woeeqz<?, {,,, } lx<xl<, {,,, 4}...8 YVXh<Hg<g{l< Yi< dxh<h lm<mol ogi{<m g{k<kqje Yi< YVXh<Hg< g{l< we<gqoxil</ wmk<kg<gim<mig? -vm<jmh< hjm hgi w{<gtqe< g{l< Yi< YVXh<Hg< g{ligl</ 77

82 Woeeqz<?hgiw{<gtqz<-vm<jmh<hjmw{<{ig-Vh<hKlm<MOlNGl</weOu {njek<k-vm<jmh<hjmhgiw{<gt<} {} lx<xl< n(a)...9 njek<kg<g{l< yv g{qk Nb<uqz< Okie<Xl< g{r<gt< njek<kqe< dxh<hgt< yv g{k<kqz< dxh<hgtig -Vg<Gl</ -g<g{k<kqje njek<kg< g{l< (universal set) we<xjpg<gqe<oxil</ njek<kg< g{k<kqje U nz<zk E we<x Gxqbiz< Gxqh<hqMgqe<Oxil</ wmk<kg<gim<mig? yv g{qk Nb<uqz< gvkl< g{r<gt< A {,, 4, 5}, B {,, 7, }weqz<?njek<kg<g{l<u uqje U {,,, 4, 5, 7,,} nz<zk U N nz<zk U W nz<zk U Zweg<ogit<tzil</..0 dm<g{l< A lx<xl< B we<he -V g{r<gt< we<g/ A e<yu<ouivdxh<hl< B e< dxh<hieiz<? A bqje B e< Yi< dm<g{l< (subset) we<xjpg<gqe<oxil</ -kjeob B NeKA e< lqjgg<g{l< (superset) we<xjpg<gqe<oxil</ -u<u{<jljb A B nz<zk B A we<xgxqbiz<wpkgqoxil</-r<g we<x Gxq dm<g{l<nz<zk dt<tmr<gqbk we<hjkg< Gxqg<gqe<xK/-u<uiOx we<x Gxq lqjgg<g{l< nz<zk ogi{<mt<tk we<hjkg< Gxqg<gqxK/ wmk<kg<gim<mig? A {,,,, }, B {,,,,, } we<xg{r<gjtg<gvkg/-r<gaz<dt<tdxh<hgtie,,,, Ngqbju B bqzl<-vg<gqe<xe/weouanekb e<dm<g{ligl<a nkiuk? A B. -r<g B, A. weou B NeK A e< dm<g{lz<za -f<k d{<jljb B A we<x Gxqbiz< Gxqh<hqMOuil</-r<G we<x GxqbieK dm<g{lz<znz<zkdt<tmr<gikkwe<xohivt<hml</ Gxqh<H: X we<xg{l< Y we<xg{k<kqe<dm<g{ligul<?g{l< Y NeKg{l< X x<g dm<g{ligul< njlf<kiz<? X l< Y l< nok dxh<hgjth< ohx<xt<te/ weou? -f<fqjzbqz<x l<y l<slg<g{r<gtigl</x Y weqz<?xe<yu<ouivdxh<hl<y e< dxh<hig-vg<gl<;olzl<y e<yu<ouivdxh<hl<x e<dxh<hig-vg<gl</x l<? Y l< g{r<gt<<<a w{<gt< nz<z/ -Vh<hqEl< w{<gtg<gqjmob hbe<hmk<kl< Gxqjb slg<g{r<gjtg< Gxqg<gh< hbe<hmk<kgqe<oxil</ Gxqjb -Vg{r<gt< X, Y -ux<xqx<gqjmob hbe<hmk<kl< OhiK? fil< nxquk? X e< dxh<hgt< njek<kl< Y z< dt<teaolzl<y e<dxh<hgt<njek<kl<x z<dt<tewe<hokankiuk?x l<y l< nokdxh<hgjtg<ogi{<mt<te/ Gxqh<H: yu<ouiv g{l< A l< A g<og dm<g{lig -Vg<gqe<xK/ Woeeqz< A e< dxh<hgt< A z< dt<te/ oux<xg<g{l< Ø NeK yu<ouiv g{l< A g<gl< dm<g{lig -Vg<gqe<xKA Woeeqz< Ø NeK A g<g dm<g{lig njlbuqz<jzobeqz<? oux<xg<g{l< Ø z< A -z< -z<zik Yi< dxh<h -Vg<g Ou{<Ml<A-KLv{<himz<zui@weOuA A lx<xl< Ø A. 78

83 4/4/ kgdm<g{l< X lx<xl< Y we<he-vg{r<gt<we<g/g{l< X NeKg{l< Y e<dm<g{lig -Vf<K? u X we<xuix Yi< dxh<h u Y weqz<? X we<x g{liek Y e< yv kg g{l< )proper subset) we<xjpg<gh<hmgqe<xk/ -kje X Y we<x Gxqh<hqMgqe<Oxil</ -f<fqjzbqz<? -kje Y X we<xl< wpkgqe<oxil</ wmk<kg<gim<mig? A {,,, 4}, B {0,,,, 4, 5} we<g/ A z< dt<t njek<k dxh<hgtl< B z< dt<tea nkiuk A NeK B e< dm<g{ligl</ Neiz< A z< -z<zik 5 we<x dxh<ohie<x B z< dt<tka nkiuk 5 A, 5 B. weou A NeK B e<kgdm<g{ligl<ankiuka B. Gxqh<H: A NeK A g<og dm<g{l< we<x Wx<geOu g{<cvf<okil</ Neiz< A NeK A g<gkgdm<g{lz<zwe<hjknxqf<kogit<g/weoua JA e<kgidm<g{l< (improper subset) we<xjpg<gqe<oxil</.. nmg<gg<g{l< yv kvh<hm<m g{l< A e< njek<k dm<g{r<gtqe< okigh<hqje A e< nmg<gg<g{l<(power set)we<gqoxil</a e<nmg<gg<g{k<kqje p(a)we<xgxqbiz< Gxqh<hqMgqe<Oxil</ wmk<kg<gim<mig? A {a, b} weqz<? A e< dm<g{r<gt< {},{ a },{ b},{ a,b}. weoup(a) {{},{ a},{ b},{ a,b}. -r<ga z< dxh<hgt<dt<te; p(a) z<4 dxh<hgt< dt<te/ weou A e< Nkq w{< A p(a) e<nkqw{< 4; nkiukn(a) ; n[p(a)] 4. -jkh<ohizou?a {a, b, c} weqz<? p(a) {{},{ a},{ b},{ c},{ a,b},{ b,c},{ c,a},{ a, b,c} }. -r<ga e<nkqw{<n(a) ; p(a)e<nkqw{<n[p(a)] 8. -kqzqvf<kfil< nxquk?n(a) mweqz< n[p(a)] m. $k<kqvl<;n[p(a)] n(a). hbqx<sq... hm<cbz<njlh<hqjeg<ogi{<mhqe<uvl<g{r<gjtlq{<ml<wpkg: (i) A { SUNDAY we<xosiz<zqz<dt<tdbqovpk<kg<gt<}. (ii) B {N{<ce<hVugizr<gt<}. (iii) C { MATHEMATICSwe<xosiz<zqz<dt<twPk<Kg<gt<}. (iv) D { TAMILNADUwe<xosiz<zqz<dt<twPk<Kg<gt<}.. hm<cbz<njlh<hqz<hqe<uvl<g{r<gjtwpkg: (i) P {x x we<hk TAMILNADUwe<xosiz<zqz<dt<twPk<K}. (ii) Q {x x yvlpw{<lx<xl< x < 7}. (iii) R {x x NeKke<-zg<gr<gtqe<%Mkz<9we<xuiXt<tYi<iqzg<gw{<}. 79

84 . g{g<gm<mjlh<hnjlh<hqz<hqe<uvl<g{r<gjtwpkg: (i) {, 6, 9, } (ii) {5, 5, 5, 65} (iii) {,, 5, } (iv) {, 4, 9, 6, 5, 6, 49, 64, 8, 00}. 4. -mkhg<gl< hm<cbz< njlh<hqz< uquiqk<kt<t g{l< yu<ouie<jxbl< nok uzh<hg<gl<g{g<gm<mjlh<hnjlh<hqz<uquiqk<kt<tnokg{k<kme<ohivk<kul<. (i) {,,, 6} (a) {x x yvhgiw{<lx<xl<nk6e<yvugh<hie<}. (ii) {, } (b) {x x NeK0 Juqmg<GjxuieYi<yx<jxh<hjm-bz< w{<}. (iii) {, 4, 6, 8} (c) {x x yvlqjglpw{<lx<xl<6e<ugh<hie<}. (iv) {,, 5, 7, 9} (d) {x x NeK0Juqmg<GjxuieYi<-vm<jmh<hjm-bz< w{<}. 5. hqe<uvl<g{r<gtqe<nkqw{<gjtg<g{<mhqcg<gul<: (i) Nr<gqzwPk<Kg<gtqz<dt<tnjek<Kdbqi<wPk<Kg<gtqe<g{l</ (ii) 00Juqmg<Gjxuienjek<Kui<g<gw{<gt<g{l</ (iii) A { x x NeK mathematics we<xosiz<zqz<dt<twpk<k}. (iv) B {x x < 0, x W}. (v) C {x x < 4, x Z}. (vi) 0 g<gl< 0g<Gl<-jmh<hm<mhgiw{<gtqe<g{l</ 6. hqe<uvl<lcuxg{r<gtqe<uqmhm<mdxh<hgjtwpkul<: (i) A {, 0, 00,,,,00,000}. (ii) B {, 5, 8,,,, 0, }. 7. hqe<uvl<lcuqzig<g{r<gtqz<nmk<k&e<xdxh<hgjtwpkul<; (i) C {, 6,, 4,,,,.}. (ii) D { 4,,,,,,, }. 8. hqe<uvl<g{r<gt<oux<xg<g{r<gtiwe<hjkg<%xg: (i) A {Nz<uGhMl<-vm<jmh<hjm-bz<w{<gt<}. (ii) B { x x R, x + 0}. (iii) C {fie<ghg<gr<gt<ogi{<mhz<ogi{r<gt<}. (iv) D {Jf<Khg<gr<gt<ogi{<mfix<gvr<gt<}. 9. A {p, q, r, s}, B {,, 5, 7}, C {q, r}, D {8, 4, 6, }, E {r, q, s, p} F {, 6, 4, 8}we<g/hqe<uVuesiqbinz<zKkuxiwewPKg: (i) A lx<xl< C slieg{r<gt</ (ii) A lx<xl< E slg{r<gt</ (iii) F lx<xl< B slieg{r<gt</ (iv) A lx<xl< B slg{r<gt</ (v) F lx<xl< D slg{r<gt</ 0. hqe<uvl<yu<ouivg{k<kqe<nmg<gg<g{k<kqjewpkg: (i) A {, } (ii) B { x, y, z} (iii) C {a, b, c, d} 80

85 . (i) n(a) 5 weqz<? n[p(a)]jg<gi{<g/ (ii) n[p(a)] 8weqz<? n(a) Jg<gi{<g/.. g{s<osbzqgt< -h<ohipk? (i) -Vg{r<gtqe<Osi<h<H (ii) -Vg{r<gtqe<oum<M (iii) yvg{k<kqe<lqjgfqvh<h Ngqbux<jxg<gx<xxqOuil</ (i) -Vg{r<gtqe<Osi<h<H A lx<xl< B we<he-vg{r<gt<we<g/a bqozinz<zk B bqozinz<zk -v{<czoli-vg<gl<njek<kdxh<hgjtbl<ogi{<mg{k<kqjea lx<xl<b e< Osi<h<H (union) we<gqoxil</a, B-ux<xqe<Osi<h<Hg{k<kqjeA U Bweg< Gxqh<hqMgqe<Oxil</weOu A U B { x x A nz<zk x B nz<zk x A lx<xl< B}. -kjea U B {x x A nz<zk x B} wes<svg<gligwpkouil</-r<gänz<zk} we<x osiz< Ängh<hMk<kq -j{f<k} we<x ohivtqz< ogit<th<hmgqe<xka nkiuk x A nz<zk x B we<hk x A nz<zk x B nz<zk x A lx<xl< B we<hjkg< Gxqh<hkiGl</ wmk<kg<gim<m 8: A {,,, 4}, B {, 4, 6} weqz<a U BJg<g{<Mhqc/ kqi<u: A lx<xl< B -ux<xqe< njek<k dxh<hgjtbl< hm<cbzqm<m? lq{<ml< uvkjz uqzg<g?,,, 4,, 4, 6. weou?a U B {,,, 4, 6}. (ii) -Vg{r<gtqe<oum<M A, B we<he -Vg{r<gt< we<g/ A, B -ju -v{<cx<gl< ohikuig njlf<k dxh<hgtiz< njlbh< ohx<x g{k<kqje A lx<xl< B -ux<xqe< oum<m (intersection) we<gqoxil</-u<oum<mg<g{k<kqjea I Bwe<xGxqbiz<Gxqh<hqMgqe<Oxil</ weou? A I B {x x A lx<xl< x B}. I B Jg<g{<Mhqc/ wmk<kg<gim<m 9: A {,, }, B {,, 4} weqz< A kqi<u: A, B-ux<xqz<dt<tnjek<KdXh<Hgt<;,,,,, 4 A, B-ux<xqe<ohiKdXh<Hgt<;,. A I B {, }...4 -j{h<hge<xg{r<gt< A lx<xl< B -jubqv{<cx<gl< ohikuie dxh<h -z<jzobeqz<? nu<uqv g{r<gjt -j{h<hge<x g{r<gt< (disjoint sets)we<gqoxil</nkiuk?a I B Ø nz<zk { } weqz<? A Bl< B Bl< -j{h<hge<x g{r<gtigl</ wmk<kg<gim<mig? A {,,, 7}, B {4, 5, 6}weqz<? A I B { }. weou ABl< B Bl< -j{h<hge<x g{r<gt</ 8

86 ..5 uqk<kqbisg<g{l< A lx<xl<< B we<he-vg{r<gt<we<g. B z< -z<zik A e<dxh<hgt< njek<jkbl< ogi{<m g{k<kqje yv uqk<kqbisg{l< (difference set) we<gqoxil</ -kje A Bweg<Gxqh<hqMgqOxil</weOu? A B {x x A, x B}. Gxqh<H: B A {x x B, x A}. wmk<kg<gim<m 40: A {,,, 4, 5, 6}, B {,, 7} weqz<? A B lx<xl< B A-ux<jxg< g{<mhqcg<gul</ kqi<u: A B {, 4, 5, 6}. B A {7}. Gxqh<H: A B B A...6 yvg{k<kqe<fqvh<hq A we<hkkvh<hm<myvg{l<? U we<hknjek<kg<g{l<we<g/a z<-z<zik U e< dxh<hgjtg< ogi{<m g{k<kqje g{l< A e< fqvh<hq )complement) we<gqoxil</ -g<g{k<kqje A nz<zk A c nz<zk A we<xgxqbqeiz<gxqh<hqmgqe<oxil</ Gxqh<H: A c U A. A:,,, 4, 5, 6 B:,, 7 wmk<kg<gim<m 4: U {,,, 4, 5}, A {, 4} weqz<? A c Jg<gi{<g/ kqi<u: A c {,, 5}. U:,,, 4, 5 A :, 4..7 g{g<ogit<jgbqz<yvlx<oxivjl g{g<ogit<jgbqz<? g{r<gtqe< Osi<h<Hg< g{k<kqz< dt<t dxh<hgtqe< w{<{qg<jgjbg< g{g<gqml< hbet<t Lx<oxiVjl dt<tk/ nk hqe<uvlix %xh<hmgqe<xk; A, B we<he-vg{r<gt<weqz<? n(a U B) n(a) + n(b) n(a I B). wmk<kg<gim<m 4: A {,, 4, 5, 6, 7, 8, 9}, B {,,, 5, 7}weqz< n(a), n(b), n(a U B) lx<xl< n(a I B) Ngqbux<jxg<gi{<g/OlZl< n(a U B) n(a) + n(b) n(a I B)we<x Lx<oxiVjljbs<siqhii<g<gUl</ kqi<u: fil<nxquk? A U B {,,, 4, 5, 6, 7, 8, 9} A I B {,, 5, 7}. n(a) 8, n(b) 5, n(a U B) 9 lx<xl< n(a I B) 4. weou n(a) + n(b) n(a I B) r<g n(a U B) 9. weou n(a U B) n(a) + n(b) n(a I B) we<x Lx<oxiVjl siqhii<g<gh<hmgqe<xk/ 8

87 hbqx<sq... A U B lx<xl< A I B -ux<jxhqe<uvl<g{r<gtg<ggi{<g: (i) A {a, e, i, o, u} lx<xl< B {a, b}. (ii) A {,, 5} lx<xl< B {,, }. (iii) A {x x Yi<-bz<w{<lx<Xl< < x 6} lx<xl< B {x x Yi<-bz<w{<lx<Xl< 6 < x < 0}. (iv) A {p, q, r} lx<xl< B Ø.. hqe<uvl<g{r<gtg<g A B, A C lx<xl< B A Ngqbux<jxg<g{<Mhqcg<gUl<: (i) A {a, b, c, d, e, f, i, o, u}, B {a, b, c, d} lx<xl< C {a, e, i, o, u}. (ii) A {, 4, 5}, B {5, 6, 7, 8} lx<xl< C {7, 8, 9}.. U {a, b, c, d, e, f, g, h}, A {a, c, g} lx<xl< B {a, b, c, d, e, f}weqz<? hqe<uvueux<jxg<g{<mhqcg<gul<; (i) A c (ii) B c (iii) (A U B) c (iv) (A c I B) c (v) A c I B c (vi) A c U B c...8 oue<hml< g{k<kqe< lqk fqgp<k<kh<hml< osbzqgjt )Osi<h<Hg< g{l<? oum<mg<g{l<? fqvh<hg<g{l<ngqbux<jxnjlk<kz<*g{<upqg<gi{<hkx<gdkuqosb<bl<ujgbqz<? \ie< oue< (John Venn) we<x Nr<gqOzb g{qk uz<zfi< g{r<gjt ujvhmr<gt< &zl< njlg<gl< upqbqje nxqlgh<hmk<kqeii</ -u<uix g{r<gjtg< Gxqh<hqMl< ujvhmr<gjt oue<hmr<gt< (Venn diagrams) we<xjpg<gqe<oxil</ -l<ljxbqz< njek<kg<g{liekyvosu<ugk<kqeiz<gxqg<gh<hml<<anke<kgdm<g{r<gt<ns< osu<ugk<kqet< um<mr<gtiz< Gxqg<gh<hMl</ -h<ohipk oue<hmk<kqz< hz<oux g{r<gtqe<njlh<hqjek<kvouil</ njek<kg{l< hml<. A U B hml<. A I B hml<. 8

88 A c hml<.4 B c hml<.5 (A U B) c hml<.6 (A I B) c hml<.7 A B hml<.8 B A hml<.9 wmk<kg<gim<m 4: gqop (hml<.0jh< hii<g<gul<) kvh<hm<mt<t hmk<kqzqvf<k? hqe<uvueux<jxg<gi{<g; (i) A U B (ii) A I B (iii) (A U B) kqi<u: -r<g U {,,, 4, 5, 6, 7}. (i) A U B {,,, 4, 5, 6}. (ii) A I B {, 5}. (iii) (A U B) c {7}. hml<.0 84

89 wmk<kg<gim<m 44: oue<hmk<jkh<hbe<hmk<kq? A {,, 5, 7} lx<xl< B {, 9, } Ngqbg{r<gjtg<gim<Mg/OlZl<n(A U B) n(a) + n(b) n(a I B) we<x $k<kqvk<jks<siqhii<g<gul</ kqi<u: A U B {,, 5, 7} U {, 9, } {,, 5, 7, 9, } n(a U B) 6. () A I B {,, 5, 7} I {, 9, } {} n(a I B). fil<ohx<xqvh<hk n(a) 4, n(b). n(a) + n(b) hml<. n(a) + n(b) n(a I B) 7 6. () (), () -ux<xqzqvf<k (A U B) n(a) + n(b) n(a I B). Gxqh<H; A lx<xl< B -u<uqv{<ml<-j{h<hge<xqvh<hqe<?a I B Ø. OlZl< n(a I B) 0. weou n(a U B) n(a) + n(b). wmk<kg<gim<m 45 : Yi<Diqz<?jkbz<uGh<hqx<Gs<osz<Zl<oh{<gtqe<w{<{qg<jg45? Okim<mOujzuGh<hqx<Gs<osz<Zl<oh{<gtqe<w{<{qg<jg70/-ui<gtqz<0Ohi< -v{<m ugh<hgtg<gl< osz<gqxii<gt< weqz<? oue<hmk<kqjeg< ogi{<m? hqe< uvueux<jxg<g{<mhqcg<gul<; (i) yvgxqh<hqm<mugh<hqx<glm<ml<ose<xui<gt<wk<kjeohi<@ (ii) wk<kje Ohi< olik<kk<kqz< -u<uqv{<m ugh<hgtqz< WOkEl< ye<xqx<giuk ose<xui<gt<@ kqi<u: A, B we<he LjxOb jkbz< ugh<h? Okim<m Oujz ugh<h -ux<xqx<gs< osz<zl< oh{<gtqe< g{r<gt<weqz<? n(a) 45, n(b) 70. kvh<hm<mt<t uquvk<kqe<hc? n(a I B) 0. weououe<hmk<kqzqvf<k? (i) (a) jkbz< ugh<hqx<g lm<ml< ose<x oh{<gtqe< w{<{qg<jgn(a B) , hml<. (i) (b) Okim<m Oujz ugh<hqx<g lm<ml< ose<x oh{<gtqe<w{<{qg<jg n(b A) (ii) -u<uqv{<mugh<hgtqz<wokel<yvugh<hqx<giukose<xoh{<gtqe<w{<{qg<jg? n(a U B) kjeouxuqkligul<ohxzil<; n(a U B) n(a) + n(b) n(a I B) wmk<kg<gim<m 46: Yi< Diqz< dt<t 45 uqmgtqz<? 5 uqmgtqz<okijzg<gim<sqh<ohm<c -Vg<gqe<xKA 0 uqmgtqz<uioeizqh<ohm<c-vg<gqe<xk. wk<kje uqmgtqz< -v{<ml< -Vg<gqe<xe/ kqi<u: A {okijzg<gim<sqh<ohm<cgt<dt<tuqmgt<} 85

90 B {uioeizqh<ohm<cgt<dt<tuqmgt<} A I B {okijzg<gim<sqh<ohm<clx<xl<uioeizqh<ohm<c-v{<ml<dt<tuqmgt<}. n(a I B) x we<g/hqe<houe<hmk<kqzqvf<k? 5 x + x + 0 x 45 n.k 55 x 45 n.k x n.k x 0 n.k x 0. okijzg<gim<sqh<ohm<clx<xl<uioeizqh<ohm<c -v{<jmbl<ogi{<muqmgtqe<w{<{qg<jg 0. hml<. wmk<kg<gim<m 47: 5 li{ig<gi<gt< ogi{<m ugh<hqz<? 8 Ohi< uvzix himk<kqz< Oki<s<sqBx<xei<; g{g<gh< himk<kqz< Ohi< Oki<s<sqBx<xei<A 8 Ohi< -v{<m himr<gtqzl< Oki<s<sqBx<xei</ wk<kje li{ig<gi<gt< -v{<m himr<gtqzl< Okiz<uq ohx<xei<@ oue< hmk<jkg< ogi{<m (i) uvzix<xh< himk<kqz< lm<ml< Oki<s<sq ohx<xui<gt<wk<kjeohi< (ii) g{g<gh<himk<kqz<lm<ml<oki<s<sqohx<xui<gt<wk<kje Ohi<we<hjkg<g{<Mhqcg<gUl</ kqi<u: H, M we<he LjxOb uvzix? g{g<g himr<gtqz< Oki<s<sq ohx<x li{ui<gtqe< g{r<gt<weqz<? n(h) 8, n(m), n(h I M) 8 we<x kvh<hm<mt<tk/ H U M we<x g{l<? uvzix nz<zk g{g<g -ux<xqz< Oki<s<sqBx<x li{ui<gtqe<g{k<kqjeg<gxqg<gqxk/weou? uvzix nz<zk g{g<g -ux<xqz< ye<xqz< hml<.4 Oki<s<sqBx<xli{ui<gtqe<w{<{qg<jg? n(h U M) n(h) + n(m) n(h I M) nkiuk? n(h U M) ugh<hqz< olik<kl< 5 Ohi< -Vg<gqe<xei</ weou -v{<m himr<gtqzl< Okiz<uqBx<xui<gtqe< w{<{qg<jg 5 NGl</ oue< hmk<jkg<ogi{<m? fil< gi{<hk? )i*uvzixhimk<kqz<lm<ml<oki<s<sqbx<xui<gtqe<w{<{qg<jg n(h ) n(h I M) (ii) g{g<gh<himk<kqz<lm<ml<oki<s<sqbx<xui<gtqe<w{<{qg<jg n(m) n(h I M)

91 hbqx<sq... A {,, 5, 8, 0}, B {, 7, 8, 9}, C {,, 5, 8, }kvh<hce<? (i) n(a U B) (ii) n(b I C) (iii) n(a B) (iv) n(c B) Ngqbux<jxg<gi{<g/. n(a) 0, n(b) 4 lx<xl< n(a I B) weqz<? n(a U B)Jgi{<g/. ye<hkil< ugh<h li{ui<gt< A, B Ngqb -V Oki<Ugtqz< ye<jxobel< wpk Ou{<Ml</ Oki<U A uqje 40 OhVl<? Oki<U B bqje 0 li{ui<gtl<? -v{<jmbl< 0 OhVl< wpkgqe<xei</ nu<ugh<hqz< dt<t li{ig<gi<gtqe< w{<{qg<jgjbg<g{<mhqcg<gul</ 4. yvgcbqvh<hhgkqbqz<dt<t400 uqmgtqz<wmg<gh<hm<mgvk<kg<g{qh<hqe<hc? 50 Ohi<Nr<gqzosb<kqk<kit<gjtBl<, 70 Ohi<klqp<osb<kqk<kit<gjtBl<? 65 Ohi< -v{<cjebl< uir<ggqe<xii<gt</ wk<kjeohi< wf<kuqklie osb<kqk< kijtbl<uir<gukqz<jzwe<hjkg<g{g<gqmul</ 5. Yi< Diqz< 00 GMl<hr<gt< dt<te/ nr<g A, B we<x -V Osih<H ujggt< lg<gtqjmobhvuqbt<te/60 GMl<hr<gt<A ujgosih<jhbl<? 40 GMl<hr<gt< B ujg Osih<jhBl< hbe<hmk<kgqe<xei</ njek<k GMl<hr<gTl< -u<uqv ujg Osih<Hgtqz< WOkEl< ye<xqjeh< hbe<hmk<khui<gtibqvh<hqe<? -v{<m ujg Osih<HgjtBl< hbe<hmk<kl< GMl<hr<gt< wk<kje we<hjkg< g{<mhqcg<gul</ Ohi< gzf<kogi{<m yv uqvf<kqz<? 0 Ohi< gihqbl<? 50 Ohi< CBl<? sqzi< -v{<cjebl<hvgqei</0 Ohi<gihqnz<zKC-ux<jxnVf<kuqz<jzweqz<? gihq lx<xl< C -v{<jmbl< hvgqbui<gt< wk<kje Ohi< we<hjkg< g{<mhqcg<gul</ 7. yv gz<z~iq-kpqz<?-bx<hqbz< GP? g{g<gg< GP?-jubqv{<cZl<%m<mig 50 li{ui<gt<dxh<hqei<gtig dt<tei</g{g<gg<gpuqz<70 li{ui<gt< dxh<hqei<gtib< -Vg<gqe<xei< lx<xl< 50 li{ui<gt< -v{<m GPuqZl< dxh<hqei<gtig -Vg<gqe<xei< weqz<? -bx<hqbz< GPuqz< dxh<hqevib< -Vh<hui<gtqe<w{<{qg<jgjbg<g{<Mhqc/ 8. 0 li{uqbi< dt<t ugh<hqz<? 0 li{uqbi< him<mh<ohim<cbqz< hr<ogx<xei<a 0 li{uqbi< him<m? fim<cbl< -v{<czl< hr<ogx<xei</ 5 li{uqbi< -v{<cz< 87

92 ye<xqzl<%mh< hr<ogx<guqz<jz/ wk<kjeohi< fim<cbk<kqz< lm<ml< hr<ogx<xei< we<hjkg<g{<mhqcg<gul</ 9. Ogijmgiz uqmljxbqe< OhiK? XII ugh<h li{ui<gtqz< 5 Ohi< g{q{q ugh<hgtg<gs< osz<gqe<xei<a 5 Ohi< FjpUk< Oki<uqx<gie hbqx<sq ugh<hgtg<gs< osz<gqe<xei<a 5 li{ui<gt< -v{<m ugh<hgtg<gl< osz<gqe<xei</ ugh<hqz< li{ui<gtqe< olik<k w{<{qg<jg 50 weqz<? wf<k ugh<hqzl<hr<ogx<gikli{ui<gtqe<w{<{qg<jgwe<eoue<xg{<mhqcg<gul</ uqjmgt< hbqx<sq.. (i) (ii). 0 (iii).08 0 (iv) (v) (vi) (vii).7 0 (viii) (ix) (x) (xi). 0 0 (xii) 9 0. (i) (ii) (iii) (iv) (v) (vi) (vii) 40 (viii) 4. (i) (ii) (iii) (iv) (v) hbqx<sq... (i) siq (ii) kux (iii) kux (iv) kux (v) siq (vi) kux. (i) log (ii) log 4 (iii) log (iv) log 79 6 (v) log 6 6 (vi) log (i) 4 (ii) (iii) 4 (iv) (v) 4 5 (vi) (vii) 6 (viii) 64 (ix) 4 4. (i) 0 (ii) 7 (iii) (iv) 5 (v) 0 (vi) 5 (vii) (viii) 6 (ix) (x) 5. (i) C (ii) A (iii) D (iv) C (v) A (vi) D 88

93 6. (i) log 0 8 (ii) 8 50 log (iii) (iv) log 8 (v) 7 5 log 0 (vi) log (i) x + y (ii) x (iii) y x (iv) y (v) t y (vi) x + y + z (vii) (x y) (viii) y + z (ix) z + t (x) x + y (xi) t x z (xii) x y z + t 4 (iii) (iv) (vii) 9 (viii) 9 (ix) (x) 4 hbqx<sq.. 8. (i) (ii). (i) (ii) (iii) 0 (iv) (v) (vi) (vii) 5 (viii). (i) (ii) (iii) 4 (iv) (v) (vi). (i) 0 (ii) (iii) (iv) (v) 4 (vi) 4. (i) 4.55 (ii).55 (iii).55 (iv).55 (v) 0.55 (vi).55 (vii) 4.55 (viii) (i) (ii) 5.96 (iii).58 (iv) (v)..670 (vi).7 (v) (xi) 6. (i) (ii).705 (iii) 0.87 (iv) (v) (vi) (i) (ii) 8.56 (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (i) 04 (ii) 0550 (iii) (iv) 80 (v) (vi) (vii) (viii) (ix) 0.64 (x) 4. (xi) 0.06 (xii) 0.89 (xiii) 0.75 (xiv) (vi)

94 hbqx<sq... (i) A {U, A} (ii) B { Ogijmgizl<?GtqIgizl<?-tOueqx<gizl<?-jzBkqIgizl< } (iii) C { M, A, T, H, I, C, S, E.} (iv) D { T, A, M, I, L, N, D, U}. (i) P { T, A, M, I, L, N, D, U} (ii) Q {,,, 0,,,, 4, 5, 6} (iii) R { 8, 7, 6, 45, 54, 6, 7, 8, 90}. (i) A {x x n, n,,, 4} (ii) B {x x 5 n, n,,, 4} (iii) C {x x N, x NeKyx<jxh<hjmw{<<} (iv) D {x x N, x n, 0 < n 0} 4. (i) (c) ; (ii) (a); (iii) (d); (iv) (b) 5. (i) 5 (ii) 9 (iii) 8 (iv) 0 (v) 7 (vi) 4 6. (i) 000, 0000 (ii) 4, 7 7. (i) 48, 96, 9 (ii) 0,, 8. (i) -z<jz (ii) oux<xg<g{l< (iii) -z<jz (iv) oux<xg<g{l< 9. (i) kux (ii) siq (iii) siq (iv) kux (v) siq,,,, 0. (i) p(a) {{}{ }{ }{ } (ii) p(b) {{}{ x, y}{}{, z, x, y}{, x, z}, { y, z}, { x, y, z}, { }} (iii) p(c) {{a},{b},{c},{d},{a, b}, {a, c}, {a, d}, {b, c},{b, d},{c, d}, {a, b, d}, {b, c, d},{c, d, a}, {a, b, c},{a, b, c, d},{}}. (i) (ii) 7 hbqx<sq... (i) A U B {a, e, i, o, u, b}, A I B {a} (ii) A U B {,,, 5}, A I B {, } (iii) A U B {,, 4, 5, 6, 7, 8, 9}, A I B Ø (iv) A U B {p, q, r}, A I B Ø. (i) A B {e, f, i, o, u }, A C {b, c, d, f }, B A Ø (ii) A B {, 4}, A C {, 4, 5}, B A {6, 7, 8}. (i) A c {b, d, e, f, h} (ii) B c {g, h} (iii) (A U B) c {h} (iv) (A c I B) c {a, c, g, h} (v) A c I B c {h} (vi) A c U B c {b, d, e, f, g, h } hbqx<sq... (i) 7 (ii) (iii) (iv)

95 4. -bx<g{qkl< -bx<g{qkk<jkg< Gxqg<Gl< Nr<gqzs< osiz< Algebra NeK al jabr we<x nvihqbs< osiz<zqzqvf<k uvuqg<gh<hm<mkigl</ nvihqb olipqbqz< al we<hk nf<k we<xl<? jabr we<hk djmf<k hgkqgtqe< ye<xqj{h<h we<xl< ohivt<hml</ -s<osiz<zqe< hbe<him<cje YI wmk<kg<gim<ce< &zl< Hiqf<K ogit<tzil</ x we<xsle<him<cz<-mkhg<gliekx lx<xl< 5 -u<uqv{<mhgkqgtqe<%mkzigl</fil< ( 5) Jsle<him<ce<-Vhg<gr<gtqZl<%m<ceiz<)OsIk<kiz<*?flg<G (x + 5) + ( 5) 9 + ( 5) nz<zk x + [5 + ( 5)] 9 5 nz<zk x nz<zk x 4 weg< gqjmg<gqxk/ -r<g 9 lx<xl< 5 wel< -v{<m hgkqgt< -j{g<gh<hm<m 4 ohxh<hm<mk/-u<ujgg{qkol-bx<g{qkl< weh<hmgqe<xk/-f<kqbg{qkoljkgt< NIbhm<mI? hqvl<lgh<ki? laiuqvi? >kvi lx<xl< hi^<gvi II Ohie<xuIgt< -h<himh<hqiqju ohvltuqz< utik<kt<tei/ gqovg<g g{qk Oljk cobihi{<m^< we<hui -bx<g{qkk<kqz< liohvl< utis<sqjb dvuig<gqbt<tkiz< -ujv -bx<g{qkk<kqe<kf<jkwe<xjpg<gqe<oxil</ g{qkk<kqe<-h<himh<hqiquqz<?a, b, x, y Ohie<xwPk<Kg<gjtw{<gjtg<Gxqg<gh< hbe<hmk<kouil</-g<gxqgt<?w{<gt<-ux<jxg<ogi{<m%m<mz<?gpqk<kz<?ohvg<gz<? ugk<kz<? uig<g&zl< wmk<kz< Ngqb osbx<himgtqeiz< -bx<g{qkg< Ogijugt< weh<hmujkh<ohxgqe<oxil</-bx<g{qkg<ogijugt<sqzgqopogimg<gh<hm<mt<te; 5 x + x +, (a + b)(x y),, 4 x + 9 y. 9 x + 7 -V -bx<g{qkg< Ogijugjts< slh<hmk<kukiz<? -bx<g{qks< sle<him gqjmg<gqe<xk/ -bx<g{qks< sle<himgtg<g sqz wmk<kg<gim<mgt< gqop 7 x ogimg<gh<hm<mt<te; x + x + 6, 5 x +, x x -bx<g{qkg< Ogijubqz< dt<t wpk<kg<gt< ng<ogijubqe< lixqgt< weh<hml</ wmk<kg<gim<mig? ax + b we<x Ogijubqz<? a lx<xl< b Gxqh<hqm<m w{<gt<? x Gxqh<hqmh<hmuqz<jzweqz<?ax + b g<gx lixqbigl</-u<uioxx + xy + y z< x lx<xl< y lixqgt</ yv Ogijubqz< dt<t lixqgtg<g Gxqh<hqm<m lkqh<hgt< M osb<b? ng<ogijubiek yv w{<{qjek< kvgqe<xk/ -f<k w{<{qje? ng<ogijubqe< lkqh<h we<gqoxil</wmk<kg<gim<mig?x + y we<hkyi-bx<g{qkg<ogiju?x lx<xl< y -g<ogijubqe<lixqgt</ x x<g l<?y x<g l<hqvkqbqm?x + y e<lkqh<h () + 9. x x<g l<?y x<g l< hqvkqbqm<miz<?x + y ( ) + 4. yvogijubqe<lixqbqe< sqz olb<ob{< lkqh<hgtg<g ng<ogijubiek olb<ob{< lkqh<hqjek< kvilx< Ohigzil</ wmk<kg<gim<mig? x we<x Ogijubqz< x weh<hqvkqbqce< 9

96 ?-KyVolb<ob{<nz<z; Woeeqz<?yVolb<ob{<{qe<uIg<gl<Gjx w{<{ig-vg<gik; weou?yvgjxob{<{qe<uig<g&zl<olb<ob{<{ig-vg<g LcbiK/ -f<k nk<kqbibk<kqz<? ye<x nz<zk hz lixqgjtg< ogi{<mjlf<k -bx<g{qkg< Ogijugjth< hx<xqh< hch<<ohil</ lixqgtl<? OgijugTl< olb<ob{< lkqh<hgjtlm<molgxqh<hkigg<ogit<ouil</-eqw{<we<xiozolb<ob{<{qje lm<molgxqh<hkigg<ogit<ouil</ 4. hz<zxh<hg<ogijugt< ax n we<xnjlh<hqz<dt<tyi-bx<g{qkg<ogijujbx z< njlf<k YI YVXh<Hg<Ogiju we<gqoxil</ -r<g a yv Gxqh<hqm<m w{<? x yv lixq lx<xl< n yv Gjxbz<zi LPuiGl</ w{< a ju x n e< ogp we<xl<? n J YVXh<Hg< Ogijubqe< nmg<gwe<xl<njpg<gqe<oxil</wmk<kg<gim<mig? 7x we<hknmg<g djmbx z< njlf<kyiyvxh<hg<ogiju; -r<g7, x e<ogp/-vyvxh<hg<ogijugtqe<%mkz< YI VXh<Hg<Ogiju weh<hml</ -u<uiox &e<x YVXh<Hg<Ogijugtqe< %Mkz< &UXh<Hg<Ogiju weh<hml</ wmk<kg<gim<mig? x + x we<hk YI VXh<Hg<Ogiju; x 5 x + we<hk yv &UXh<Hg<Ogiju/ LcUX w{<{qg<jgbqz<? x z< njlf<k YVXh<Hg<Ogijugtqe<%Mkz<x z<njlf<kyvhz<zxh<hg<ogiju we<xjpg<gh<hml</ hz<zxh<hg<ogijubqz< dt<t YVXh<Hg<Ogijugtqe< ogpg<gt< hz<zxh<hg<ogijubqe< ogpg<gt< we<xjpg<gh<hml</ hz<zxh<hg< Ogijubqz< dt<t njek<k ogpg<gtl< H,s<sqb w{<{ig -Vh<hqe<? nh<hz<zxh<hg<ogiju H,s<sqb hz<zxh<hg<ogiju we<xjpg<gh<hml</ -eq? hz<zxh<hg<ogiju yu<ouie<jxbl< H,s<sqblz<zik hz<zxh<hg<ogijubigg< ogit<ouil<; nkiuk -equvl< hz<zxh<hg<ogiju yu<ouie<xqzl< Gjxf<khm<sl< yv ogpuigqzl< H,s<sqblx<xkig -Vg<Gl</ hz<zxh<hg<ogijubqz< dt<t YVXh<Hg<Ogijugt< nh<hz<zxh<hg< Ogijubqe< dxh<hgt<weh<hml<; dxh<hgtqe<lqgdbif<knmg<g?hz<zxh<hg<ogijubqe<nmg<g nz<zkhcweh<hml</ yvhz<zxh<hg<ogijubqe<lqgdbif<knmg<gjmbdxh<hqe< ogpju? nh<hz<zxh<hg< Ogijubqe< kjzbibogp nz<zkupqfmk<kqs< osz<zl< ogp we<gqoxil</ wmk<kg<gim<mig? x 6 + x x 5 4x we<hk 5 hcbjmb hz<zxh<hg<ogiju; x, 6, x, x 5, 4x we<hedxh<hgt<; kjzbibogp NGl</ yv hz<zxh<hg<ogijubqe< ogpg<gt< olb<ob{<gt< we<hkiz<? nux<jx olb<ob{<gtqe< h{<hgjtg< ogi{<m x e< nmg<ggt< -xr<glglig uvlix lix<xqbjlg<gzil</ wmk<kg<gim<mig? x 5x x we<x hz<zxh<hg<ogijujb x + x 5x we nmg<ggt< -xr<glglig lix<xqbjlg<gzil</ yv hz<zxh<hg<ogiju? nke< dxh<hgtqe< nmg<ggt< -xr<glglig -Vg<GliX lix<xqbjlk<k wpkh<ohx<xiz<? nk kqm<m ucuqz< -Vh<hkigs< osiz<ouil</ yv hz<zxh<hg<ogijubqz<?x -z<zikdxh<hqje(nmg<g0 Ngdt<tdXh<hqje*keq dxh<hnz<zklixqzqdxh<hwe<gqoxil</wmk<kg<gim<mig?x 6 + x x 5 4x z< lixqzq dxh<h 6 NGl</ kip<f<k nmg<gjmb hz<zxh<hg<ogijugtg<gs< sqxh<hh< ohbigt< dt<te/ Lkx<hc hz<zxh<hg<ogijujb Ofiqb hz<zxh<hg<ogiju 9

97 we<gqoxil</ -v{<m hcbjmb hz<zxh<hg<ogijujb -Vhch< hz<zxh<hg<ogiju we<gqoxil</ &e<x hcbjmb hz<zxh<hg<ogijujb Lh<hch< hz<zxh<hg<ogiju we<gqoxil</ fie<g hcbjmb hz<zxh<hg<ogijujb fix<hch< hz<zxh<hg<ogiju we<gqoxil</ f(x) we<x yv hz<zxh<hg<ogijujbh< ohikuigg< Gxqk<kiz<? f(x)e< hcbqje deg(f(x)) we<xgxqbiz<gxqh<hqmouil</wmk<kg<gim<mig? x + 5x x we<x hz<zxh<hg<ogijujb f(x) weg< Gxqh<hqm<miz<? deg(f(x)) NGl</ yv hz<zxh<hg<ogijubqe< dxh<hgt< w{<gt< we<hkiz<? njek<k hz<zxh<hg< OgijugTl<w{<gjtg<Gxqg<gqe<xe/weOu?fil<-ux<jxg<ogi{<M%m<m?gpqg<g? ohvg<g lx<xl< ugg<g LcBl</ yv hz<zxh<hg<ogijujb lx<oxiv hz<zxh<hg< Ogijubqeiz< ugh<hjkh< hx<xq hqe<ei hch<ohil</ -h<ohipk hz<zxh<hg<ogijugjtg<%m<mujkh<hx<xqh<hch<ohil</ 4.. hz<zxh<hg<ogijugtqe<%m<mz< yk<k nmg<ggtqe< ogpg<gjtg< %m<mukeiz<? flg<g -v{<m hz<zxh<hg< Ogijugtqe<%Mkz<gqjmg<gqe<xK/ wmk<kg<gim<m : x 4 x + 5x +, 4x + 6x 6x -ux<xqe<%mkz<gi{<g/ kqiu; olb<ob{<gtqe< OsIh<Hh< h{<h? hr<gqm<mh< h{<h Ngqbux<jxh< hbe<hmk<kouil</ (x 4 x + 5x + ) + (6x 6x + 4x ) x 4 + 6x x 6x + 5x + 4x + x 4 + 6x (+6)x + (5+4)x + x 4 + 6x 9x + 9x +. gqopogimg<gh<hm<mkqm<ml<-v{<mhz<zxh<hg<ogijugjtg<%m<mukqz<dkuqosb<bl</ x 4 + 0x x + 5x + 0x 4 + 6x 6x + 4x x 4 + 6x 9x + 9x hz<zxh<hg<ogijugtqe<gpqk<kz< hz<zxh<hg<ogijugjtg<%m<mukohie<oxgpqk<kjzbl<osb<gqoxil</ wmk<kg<gim<m : x + 5x 4x 6 zqvf<kx x Jg<gpqg<gUl</ kqiu; olb<ob{<gtqe<osih<hh<h{<h?hr<gqm<mh<h{<hngqbux<jxh<hbe<hmk<kqeiz< (x + 5x 4x 6) (x x ) x + 5x 4x 6 x + x + x x + 5x + x 4x 6 + (x x ) + (5x + x ) + ( 4x) + ( 6+) x + 8x 4x 5. gqopogimg<gh<hmgqe<xljxbqzl<gpqk<kjzs<osb<bzil</ uiq (): x + 5x 4x 6. uiq (): x x. uiq () z<dt<thz<zxh<hg<ogijubqe<gxqgjtlix<xg<gqjmh<hk, uiq (): x + x +. uiq (), uiq () gtqz<dt<thz<zxh<hg<ogijugjtg<%m<mg<gqjmh<hk, x + 8x 4x 5. Olx<g{<mLjxgQp<g<gi[liXwPkh<hMl</ 9

98 x + 5x 4x 6 x x + + x + 8x 4x Vhz<ZXh<Hg<Ogijugtqe<ohVg<gz< -Vhz<ZXh<Hg<Ogijugtqe<ohVg<gz<nz<zK ohvg<gx<hze<gi{?hr<gqm<mh< h{<jhbl<?nmg<gg<gxquqkqgjtbl<hbe<hmk<kgqoxil</ wmk<kg<gim<m : x x 4 lx<xl< x + x -ux<xqe<ohvg<gx<hze<gi{<g/ kqiu; (x x 4) (x + x ) x (x + x ) + ( x ) (x + x ) + ( 4) (x + x ) (x 5 + x 4 x ) + ( 4x 4 6x + x ) + ( 8x x + 4) x 5 + x 4 x 4x 4 6x + x 8x x + 4 x 5 + (x 4 4x 4 ) + ( x 6x ) + (x 8x ) + ( x) +4 x 5 x 4 7x 6x x V hz<zxh<hg< Ogijugtqe< ohvg<gx<hze< gi[l<ohik yv hz<zxh<hg< OgijubqEjmb yu<ouii dxh<jhbl< lx<oxiv hz<zxh<hg< Ogijubqe< yu<ouii dxh<hizl< ohvg<gq hqe<ei? nu<uqv ohvg<gx<hze<gtl< %m<mh<hmgqe<xe/ gqop ogimg<gh<hm<mljxhkqkigg<gx<xg<ogit<huigtg<gdkuqbig-vg<gl</ x x 4 x + x x (x + x ) : x 5 + x 4 x x (x + x ) : 4x 4 6x + x 4(x + x ) : 8x x + 4 x 5 x 4 7x 6x x + 4 sqz Oujtgtqz< hz<zxh<hg< Ogijugtqe< ohvg<gx<hzeqz< sqz Gxqh<hqm<m dxh<hgtqe< ogpg<gt< Okjubig -Vg<Gl</ Ofvk<jkBl<? -mk<jkbl<sqg<geh<hmk<k fil< hz<zxh<hg< Ogijugjt LPuKl< ohvg<gilozob ogpg<gjtg< gi{zil</ wmk<kg<gim<mig? A, B we<x -v{<m hz<zxh<hg< Ogijugtqe< ohvg<gx<hzeqz< x e< ogpjuh< ohx Ou{<Ml< weqz< gqop uvgqe<x upqljx? Hkqkigg< gx<xg< ogit<huigtg<gdkuqbig-vg<gl</ hc gqopogimg<gh<hm<mjugtqe<ohvg<gx<hze<gi{<g: A bqz<dt<tx dxh<hqe<ogp B bqz<dt<tlixqzqdxh<h(keqdxh<h* A bqz<dt<t x dxh<hqe<ogp B bqz<dt<tx dxh<hqe<ogp A bqz<dt<tx dxh<hqe<ogp B bqz<dt<t x dxh<hqe<ogp A bqz<dt<tlixqzqdxh<h B bqz<dt<tx dxh<hqe<ogp 94

99 hc : OlOzhc z<dt<twz<zih<ohvg<gx<hze<gjtbl<%m<mg/gqjmg<gl<lkqh<h A, B Ngqb-Vhz<ZXh<Hg<Ogijugtqe<ohVg<gx<hzeqz<x e<ogpuigl</ wmk<kg<gim<m 4: x 4, x, x, x NgqbdXh<Hgtqe<ogPju7x 6x 9x + 8 lx<xl< 5x x + 5 gtqe<ohvg<gx<hzeqzqvf<klpukl<ohvg<gilozobg{<mhqc/ kqiu; A 7x 6x 9x + 8 lx<xl<b 5x x + 5 we<g/ dxh<h x 4 e<ogpjug<gi{z<; A bqz<dt<t x 4 dxh<hqe<ogp B bqz<dt<tlixqzqdxh<h A bqz<dt<t x dxh<hqe<ogp B bqz<dt<t x dxh<hqe<ogp 7. A bqz<dt<t x dxh<hqe<ogp B bqz<dt<t x dxh<hqe<ogp A bqz<dt<t x dxh<hqe<ogp B bqz<dt<t x dxh<hqe<ogp A bqz<dt<tlixqzq dxh<h B bqz<dt<t x 4 dxh<hqe<ogp weoua B z<x 4 e<ogp 0 + ( ) + ( 0) dxh<h x e<ogpjug<gi{z<; A bqz<dt<t x dxh<hqe<ogp B bqz<dt<tlixqzq dxh<h A bqz<dt<t x dxh<hqe<ogp B bqz<dt<t x dxh<hqe<ogp 6 8. A bqz<dt<t x dxh<hqe<ogp B bqz<dt<t x dxh<hqe<ogp A bqz<dt<tlixqzqdxh<h B bqz<dt<t x dxh<hqe<ogp Njgbiz< A B z<x e<ogp ( 45) dxh<h x e<ogpjug<gi{z<; A bqz<dt<t x dxh<hqe<ogp B bqz<dt<tlixqzq dxh<h A bqz<dt<t x dxh<hqe<ogp B bqz<dt<t x dxh<hqe<ogp 9 7. A bqz<dt<tlixqzqdxh<h B bqz<dt<t x dxh<hqe<ogp Njgbiz< A B z<x e<ogp ( 0) x dxh<hqe<ogpjug<gi{z<: A bqz<dt<t x dxh<hqe<ogp B bqz<dt<tlixqzq dxh<h A bqz<dt<tlixqzq dxh<h B bqz<dt<t x dxh<hqe<ogp 8 4. Njgbiz< A B z<x e<ogp ( 45) + ( 4) hzlixqgtqz<njlf<khz<zxh<hg<ogijugt< x, y z<njlf<kyiyvxh<hg<ogiju ax n y m weqz<?nkqz<a yvolb<ob{<; x, y we<hju lixqgt<; n lx<xl< m we<he lqjglpg<gt</ wmk<kg<gim<mig? 5x y we<hk x, y z< YI YVXh<Hg< Ogiju/ x, y z<njlf<klcuxw{<{qg<jgbqzieyvxh<hg< Ogijugtqe< %m<mx< hze<? x, y z< njlf<k yv hz<zxh<hg< Ogiju weh<hml</ wmk<kg<gim<mig? 5x y + x + y, x 8y, x + xy + y Ngqbju x, y z< njlf<k hz<zxh<hg< Ogijugt</ -OkOhie<X hz lixqgtqz< njlf<k hz<zxh<hg< Ogijugt< d{<m/ yv lixqbqz< njlf<k hz<zxh<hg< Ogijugtqz< %m<mz<? gpqk<kz<? ohvg<gz< osb<kkohie<oxhzlixqgtqz<njlf<khz<zxh<hg<ogijugtqzl<osb<bzil</ 95

100 wmk<kg<gim<m 5: x y + x y xy, x x y + y + 4xy Ngqbhz<ZXh<Hg<Ogijugtqe< %Mkz<gi{<g/ kqiu; (x y + x y xy ) + (x x y + y + 4xy ) x y + x y xy + x x y + y + 4xy (x y x y) + (x y ) + ( xy + 4xy ) + (x ) + (y ) x y + x y + xy + x + y. wmk<kg<gim<m 6: x + y, x xy + y Ngqbux<xqe<ohVg<gx<hze<gi{<g/ kqiu; x + y nz<zk (x + y) (x xy + y ) x xy + y x (x xy + y ) + y (x xy + y ) x(x xy + y ) : x x y + xy x x y + xy + x y xy + y y(x xy + y ) : x y xy + y x + x y xy + y x + ( x y + x y) + (xy xy ) + y x + x y xy + y. yvhz<zxh<hg<ogijujblx<oxivhz<zxh<hg<ogijubiz<ugg<gl<osbz<?hgkq 4.4 z<nvibh<hmgqxk/ hbqx<sq 4.. siqnz<zkkuxwehkqz<ogim: (i) x + x we<hkyvhz<zxh<hg<ogiju/ (ii) x + x + we<hkyi-v{<mil<hchz<zxh<hg<ogiju/ (iii) 5 x x + x z<x e<ogp. (iv) 5xy we<hkyivxh<hg<ogiju/ (v) x + y + 5z we<hkyv&uxh<hg<ogiju/ 4 ujvbqzt<tyu<ouivg{g<gqzl<%mkz<g{<mkqm<mucuqz<wpkg/. (x + x ) + (x 4x + 5). (x 4 + x + x) + (x 4 x + 7x 8) 4. (6 0x + 5x + x ) + (x x 4) 5 7 ujvbqzt<t yu<ouiv g{g<ggtqz< gpqk<kz< osbz< osb<k? kqm<m ucuqz< wpkg/ 5. (x + 5x 0x + 6) (x x 4) 6. (x 4 x + 7x 8) (x 4 + x + x) 7. (x 5 5x + 4x 7) ( x + x x ) 8 0 ujvbqz<dt<tg{g<ggtqe<ohvg<gx<hze<gi{<g/njkkqm<mucuqz<wpkg/ 8. (x 6x + ) (x 4x + 9) 9. (x 4x + 5x 7) (x x + 4) 0. (7 x x ) (x 5x + x) 96

101 Lkz< ujvbqzt<t g{g<ggtqz< LPuKl< ohvg<gilozob x, x, x Ngqb dxh<hgtqe<ogpg<gjtg<gi{<g/. (x 4x + 4) (x + x ). (x x ) ( + x x ). (7x 6x 9x ) (x x ) 4 Lkz<6 ujvbqzt<tg{g<ggtqz<ohvg<gx<hzjek<kqm<mucuqz<gi{<g/ 4. (ax + by) (cx + dy) 5. (x + y) (x xy y ) 6. (x xy + y ) (x + xy + y ) 7. (x px + 9x ), (x x x + ) Ngqbjugtqe<ohVg<gx<hzeqz< x e<ogp weqz<, pbqe<lkqh<hgi{</ 8. (x x + 5) (a x x ) e<ohvg<gx<hzeqz<x e<ogpuiek? (x + x ) (x x ) e<ohvg<gx<hzeqz<x e<ogpuqx<gs<sll<weqz<?a bqe< lkqh<hgi{<g/ 9. ( x x ) (x mx + ) we<gqxohvg<gx<hzeqz<x e<ogp?x e<ogp-ux<xqe< %Mkz< 5 weqz<, m e<lkqh<jhg<gi{<g/ 4.. -bx<g{qklx<oxivjlgt< -bx<g{qk Lx<oxiVjlgt< weg<%xh<hml<? -bx<g{qk sle<himgjth< hx<xq -r<g hch<ohil</ -bx<g{qk sle<himgjt Wx<geOu gx<xt<otil</ YI -bx<g{qk sle<himiekye<xnz<zknkx<golx<hm<mlixqgjtg<ogi{<mk/wmk<kg<gim<mig x + 6 x z< x we<hklixq/x g<ghkqzig Jh<hqvkqbqmsle<himieK 5 5 we<el< d{<jl uig<gqblig dt<tk/ x g<g OuX wf<k lkqh<hgt< ogimk<kizl<, wmk<kg<gim<mig x weg< ogi{<miz< sle<himiek 7 4 we<el< kuxie uig<gqbligl</ lixqg<gigh< hqvkqbqmh<hm<m YI w{< nf<k sle<him<jm d{<jlbie uig<gqblig<gqeiz<nf<kw{<, sle<him<ce<kqiu nz<zk&zl< we<x%xh<hml</yiw{< yvsle<him<ce<kqiuweqz<?nknf<ksle<him<jmfqjxuosb<bl</wmk<kg<gim<mig? x + 6 x we<hjk fqjxuosb<gqxk/neiz< fqjxu osb<buqz<jz/ x (x + )(x ) we<el< sle<him wf<k w{<{izl< fqjxujmgqxk/ -u<uix wz<zi w{<gtizl< fqjxu ohxl< YI -bx<g{qk sle<him? -bx<g{qklx<oxivjl weh<hml</ A B we<el< YI -bx<g{qk sle<him? -bx<g{qk Lx<oxiVjlbieiz<? fil< -kje A B wewpkouil</hg<gr<gjtlix<xg{qklx<oxivjljb? B A we<xl<wpkzil</fil<-h<ohipksqz-bx<g{qklx<oxivjlgjtg<ogi{iouil</ 4.. (x + a)(x + b) g<gie-bx<g{qklx<oxivjl w{<gtqe<hr<gqm<mh<h{<jhh<hbe<hmk<k? (x + a )(x + b ) x(x + b) + a(x + b) x + xb + ax + ab x + ax + bx + ab x + (a + b)x + ab. fil<ohxuk (x + a)(x + b) x + (a + b) x + ab 97

102 Olx<gi[l< Lx<oxiVjlg<G yv ucuqbz<uqtg<gl<kvouil</ ABCD we<gqx osu<ugk<kqe< hvh<htuiek? AHFE we<x skvk<kqe< hvh<htu? HBGF, FGCI lx<xl< EFID Ngqb osu<ugr<gtqe< hvh<htu -ux<xqe< %MkZg<Gs< sll< (hml< 4.. Jh< hiig<gul<). Njgbiz<fil<ohXuK (x + a)(x + b) x + ax + ab + xb x + (a+b)x + ab. hml< 4.. OlOz dt<t Lx<oxiVjljbh< hbe<hmk<kq sqz Lg<gqblie Lx<oxiVjlgjtg<ogi{IOuil</ (i) (x a)(x + b) [x + ( a)] (x + b) x + [( a) + b]x + ( a)b x + (b a)x ab. (ii) (x + a)(x b) (x + a) [x + ( b)] x + [a + ( b)]x + a( b) x + (a b)x ab. (iii) (x a)(x b) [x + ( a)] [x + ( b)] x + [( a) + ( b)] x + ( a) ( b) x (a + b)x + ab. (iv) (a + b) (a + b)(a + b) a + (b + b)a + b a + ab + b. (v) (a b) (a b)(a b) [a + ( b)] [a + ( b)] a + [( b) + ( b)]a + ( b) ( b) a ab + b. (vi) (a + b)(a b) a + [b + ( b)]a + (b)( b) a + 0 a b a b. weou?fil<ohxuk (x + a)(x + b) x + (a + b)x + ab (x a)(x + b) x + (b a)x ab (x + a)(x b) x + (a b)x ab (x a)(x b) x (a + b)x + ab (a + b) a + ab + b (a b) a ab + b (a + b)(a b) a b (x a)(x b) we<hke<lx<oxivjl x (a + b)x + ab. nk Nl<hchz<ZXh<Hg< OgijubiGl</ nke< x e<ogp?keqdxh<hngqbjuljxob (a + b) lx<xl< ab NGl</ OlOz ogimg<gh<hm<mt<t Lx<oxiVjlgt<? (x + a)(x + b) we<hke< ohvg<gx<hzeqe< uqiqju nch<hjmbigg< ogi{<cvh<hkiz<? -ju ohvg<gx<hze< $k<kqvr<gt< we<xl< %xh<hml</ -v{<m VXh<Hg< Ogijugtqe< ohvg<gx<hzjeg< g{<mxqbul<-f<k$k<kqvr<gt<hbe<hml</ 98

103 wmk<kg<gim<m 7: hqe<uvueux<xqe<ohvg<gx<hze<gt<gi{<g; (a) (x + ) (x + 5) (b) (p + 9) (p ) (c) (z 7) (z 5) (d) (x 8) (x + ) kqiu; (a) (x + ) (x + 5) x + ( + 5) x + 5 x + 8x + 5. (b) (p + 9) (p ) p + (9 ) p 9 p + 7p 8. (c) (z 7) (z 5) z (7 + 5) z z z + 5. (d) (x 8) (x + ) x + ( 8)x 8 x 6x 6. wmk<kg<gim<m 8: ohvg<gx<hze< $k<kqvk<jkh< hbe<hmk<kq hqe<uvueux<xqe< ohvg<gx< hze<gtqe<lkqh<hgt<gi{<g/ (a) 07 0 (b) kqiu; (a) 07 0 (00 + 7) (00 + ) 00 + (7 + ) (b) (50 + 6) (50 ) 50 + (6 ) wmk<kg<gim<m 9: hqe<uvueux<jxuqiquig<gg; (i) (x + 7y) (ii) (a 7b) (iii) (p + 5q)(p 5q) kqiu; (i) (x + 7y) (x) + (x)(7y) + (7y) 9x + 4xy + 49y. (ii) (a 7b) (a) (a) (7b) + (7b) a 54ab + 49b. (iii) (p + 5q)(p 5q) (p) (5q) 4p 5q. wmk<kg<gim<m 0: ohvg<gx<hze< $k<kqvr<gjth< hbe<hmk<kq hqe<uvueux<xqe< lkqh<h gi{<g/ (i) 0 (ii) 98 (iii) kqiu; (i) 0 (00 + ) 00 + (00)() (ii) 98 (00 ) 00 (00)() (iii) (00 + 4) (00 4) ohvg<gx<hze<$k<kqvk<kqeqe<xolzl<sqzhbet<tlx<oxivjlgjtg<ogi{<m uvouil</ (i) (a + b) + (a b) (a + ab + b ) + (a ab + b ) (a + a ) + (ab ab) + (b + b ) a + b. [(a + b) + (a b) ] [(a + b )] a + b. (ii) (a + b) (a b) (a + ab + b ) (a ab + b ) a + ab + b a + ab b 4ab. 4 [(a + b) (a b) ] 4 [4ab] ab. (iii) (a + b) ab (a + ab + b ) ab a + ab + b ab a + b. 99

104 (iv) (a + b) 4ab (a + ab + b ) 4ab a + ab + b 4ab a ab + b (a b). (v) (a b) + ab (a ab + b ) + ab a ab + b + ab a + b. (vi) (a b) + 4ab (a ab + b ) + 4ab a ab + b + 4ab a + ab + b (a + b). -u<uixflg<ghqe<uvl<hbet<tlx<oxivjlgt<gqjmk<kt<te; [(a + b) + (a b) ] a + b 4 [(a + b) (a b) ] ab (a + b) ab a + b (a + b) 4ab (a b) (a b) + ab a + b (a b) + 4ab (a + b) OlOzogiMg<gh<hm<Mt<tLx<oxiVjlgtqe<hg<gr<gjtlix<xqwPk?fil<ohXuK a + b [(a + b) + (a b) ] ab 4 [(a + b) (a b) ] a + b (a + b) ab (a b) (a + b) 4ab a + b (a b) + ab (a + b) (a b) + 4ab wmk<kg<gim<m : a + b, a b Ngqbux<xqe<lkqh<Hgt<LjxOb7, 4 weqz<? a + b, ab Ngqbux<xqe<lkqh<Hgjtg<gi{</ kqiu: a + b [(a + b) + (a b) ] ab 4 [(a + b) (a b) ] [(7) + (4) ] (49 + 6) [(7) (4) ] (49 6). 4 4 wmk<kg<gim<m : a + b, ab Ngqbux<xqe< lkqh<hgt< LjxOb, weqz<, a + b, (a b) e<lkqh<hgjtg<gi{<g/ kqiu: a + b (a + b) ab (a b) (a + b) 4 ab () () () 4() wmk<kg<gim<m : a b, ab Ngqbux<xqe< lkqh<hgt< LjxOb 6, 40 weqz<? a + b, (a + b) Ngqbux<xqe<lkqh<Hgjtg<gi{<g/ kqiu: a + b (a b) + ab (a + b) (a b) + 4ab 6 + (40) (40)

105 wmk<kg<gim<m 4: (x + p)(x + q) x 5x 00 weqz<?p + q e<lkqh<hgi{<g/ kqiu: ohvg<gx<hze<$k<kqvk<kqe<hc, (x + p) (x + q) x + (p + q)x + pq. yh<hqmjgbqz<fil<nxquk? p + q 5, pq 00. -h<ohipk? p + q (p + q) pq ( 5) ( 00) fil<-eq(a + b + c) g<gieg{qklx<oxivjljbg<g{<mxqouil</ (a + b + c) [(a + b) + c] (a + b) + (a + b) c + c (a + ab + b ) + ac + bc + c a + b + c + ab + bc + ca a + b + c + (ab + bc + ca). Njgbiz<fil<ohXl<Lx<oxiVjl (a + b + c) a + b + c + (ab + bc + ca) hg<gr<gjtlix<xqwpkqeiz<fil<ohxuk a + b + c + (ab + bc + ca) (a + b + c) OlZl<fil<gi{<hK (a + b + c) (ab + bc + ca) a + b + c + (ab + bc + ca) (ab + bc + ca) a + b + c. weou?lx<oxivg{qklx<oxivjljbh<ohxgqoxil</ (a + b + c) (ab + bc + ca) a + b + c hg<gr<gjtlix<xqwpkqeiz<fil<ohxuk a + b + c (a + b + c) (ab + bc + ca) wmk<kg<gim<m 5: hqe<uvueux<jxuqiquhmk<kg/ (i) (x + y + z) (ii) (x y + z) (iii) (p q r) (iv) (a + b c) kqiu: (i) (x + y + z) [(x) + y + (z)] (x) + y + (z) + (x)y + y(z) + (z)(x) 4x + y + 4z + 4xy + 4yz + 8zx. (ii) (x y + z) [x + ( y) + z] x + ( y) + z + x( y) + ( y)z + zx x + 4y + z 4xy 4yz + zx. (iii) (p q r) [(p) + ( q) + ( r)] (p) + ( q) + ( r) + (p) ( q) + ( q) ( r) + ( r)(p). 4p + 9q + r pq + 6qr 4rp. (iv) (a + b c) [(a) + (b) + ( c)] (a) + (b) + ( c) + (a)(b) + (b)( c) + ( c)(a) 4a + 9b + 4c + ab bc 8ca. 4.. (x + a) (x + b) (x + c) g<gie-bx<g{qklx<oxivjl (x + a)(x + b)(x + c) (x + a)[(x + b)(x + c)] (x + a)[x + (b + c)x + bc] (x + a)(x + bx + cx + bc) x(x + bx + cx + bc) + a(x + bx + cx + bc) x + bx + cx + bcx + ax + abx + acx + abc x + (a + b + c)x + (ab + bc + ca)x + abc. weou? (x + a)(x + b)(x + c) x + (a + b + c)x + (ab + bc + ca)x + abc 0

106 OlOz ogimg<gh<hm<mt<t Lx<oxiVjlbqz< a, b, c -ux<xqx<g hkqzig a, b, c Jh< hqvkqbqmflg<gg<gqjmh<hk (x a)(x b)(x c) [x + ( a)][x + ( b)][x + ( c)] x + [( a) + ( b) + ( c)]x + [( a) ( b) + ( b) ( c) + ( c) ( a)]x + ( a) ( b) ( c) x (a + b + c)x + (ab + bc + ca)x abc. -u<uix (x a)(x b)(x c) x (a + b + c)x + (ab + bc + ca)x abc. -kqz<, x e<ogp (a + b + c), x e<ogp ab + bc + ca, lixqzqdxh<h abc. -jkh<ohizou?(x + a)(x + b)(x + c) z< x e<ogp a + b + c, x e<ogp ab + bc + ca, lixqzqdxh<h abc. wmk<kg<gim<m 6 : uqiqugi{<g/ (i) (x + 4)(x + )(x + 5) (ii) (x + )(x )(x + 5) (iii) ( x)(x + 7)(x + ) (iv) (x a)(x a)(x a) kqiu: (i) (x + 4)(x + )(x + 5) x + ( )x + [ ]x x + x + [ ]x + 60 x + x + 47x (ii) (x + )(x )(x + 5) (x + )[x + ( )](x + 5) (x) + [ + ( ) + 5](x) + [ ( ) + ( ) ](x) + ( ) 5 8x + (4x ) + [ 5 + 5](x) 5 8x + x 6x 5. (iii) ( x)(x + 7)(x + ) [ (x )](x +7 )(x + ) [x + ( )](x + 7)(x + ) [(x) + {( ) }(x) + {( ) ( )}(x) + ( ) 7 ] [8x + (5)(4x ) + ( + 7 )(x) ] [8x + 0x 4x ] + 4x 0x 8x. (iv) (x a)(x a)(x a) [x+( a)][x+( a)][x+( a)] x + {( a) + ( a)+( a)}x +{( a) ( a) + ( a) ( a) + ( a) ( a)}x + ( a) ( a) ( a) x + { 6a}x + {a + 6a + a }x 6a x 6ax + a x 6a. wmk<kg<gim<m 7 : ohvg<gx<hze< $k<kqvr<gjth< hbe<hmk<kq? x dxh<hqe< ogp? x dxh<hqe<ogp?lixqzqdxh<hngqbux<jxg<gi{<g/ (i) (x + )(x + 5)(x + 6) (ii) (x 7)(x + )(x + 4) (iii) (x 5)(x )(x + 4) (iv) (x )(x 5)(7 x) kqiu: (i) (x + )(x + 5)(x + 6) J (x + a)(x + b)(x +c ) Bme<yh<hqMjgbqz<? a, b 5, c 6. x e<ogp a + b + c , x e< ogp ab + bc + ca ( 5) + (5 6) + (6 ) , lixqzqdxh<h abc (ii) (x 7)(x + )(x + 4) [x + ( 7)](x + )(x + 4) -jk(x + a)(x + b)(x + c) Bme<yh<hqMjgbqz<? a 7, b, c 4. x e<ogp a + b + c ( 7)

107 x e<ogp ab + bc + ca ( 7) ( 7) ( 4) ( 8) ( 4) + 8 4, lixqzqdxh<h abc ( 7) (iii) (x 5)(x )(x + 4) [x + ( 5)][x + ( )](x + 4) -jk (x + a)(x + b)(x + c) Bme<yh<hqMjgbqz<?a 5, b, c 4. x e<ogp a + b + c ( 5) + ( ) + 4 ( 7) + 4, x e<ogp ab + bc + ca ( 5) ( ) +( ) ( 5) , lixqzqdxh<h abc ( 5) ( )4 40. (iv) (x )(x 5)(7 x) we<hkawe<gqx-bx<g{qkogijuwe<g/-kqz< yx weg<ogi{<miz<, A (y )(y 5)(7 y) (y )(y 5) [ (y 7)] [(y )(y 5 )(y 7)] [y + {( ) + ( 5) + ( 7)}y + {( )( 5) + ( 5)( 7) + ( 7)( )}y + ( )( 5)( 7)] [y 5y + ( )y 05] y + 5y 7y + 05 (x) + 5(x) 7(x) x + 60x 4x + 05 x e<ogp 60, x e<ogp 4, lixqzqdxh<h 05 lx<oxivljx;(x )(x 5)(7 x) 5 7 x x x x + x + x + x e<ogp 8(a + b + c) , x e<ogp 8( ab + bc + ca) ( ) (7) 4, 4 lixqzqdxh<h 8(abc) wmk<kg<gim<m 8 : (x + a)(x + b)(x + c) x 6x + x 6 weqz<?a + b + c e<lkqh<h gi{<g/ kqiu; ohvg<gx<hze<$k<kqvk<kqzqvf<kfil<nxquk? (x + a)(x + b)(x + c) x + (a + b + c)x + (ab + bc + ca)x + abc. yh<hqmjgbqz<flg<gg<gqjmh<hk?a + b + c 6, ab + bc + ca, abc 6. a + b + c (a + b + c) (ab + bc + ca) ( 6) () 6 4. (x + a)(x + b)(x + c) e<lx<oxivjlbqzqvf<ksqzlx<oxivjlgjtk<kvuqh<ohil</ (i) (a + b) g<gielx<oxivjl (a + b) (a + b)(a + b)(a + b) a + (b + b + b)a + (b b + b b + b b)a + b b b a + a b + ab + b (a + b) a + a b + ab + b 0

108 hg<gr<gjtlix<xqobpk? a + a b + ab + b (a + b) (ii) (a b) g<gielx<oxivjl (a b) [a + ( b)] a + a ( b) + a( b) + ( b) a a b + ab b hg<gr<gjtlix<x? (a b) a a b + ab b a a b + ab b (a b) (a + b), (a b) we<x Lx<oxiVjlgtqe< nch<hjmbqz< hqe<uvl< Lx<oxiVjlgjtk< kvuqh<ohil</ (i) (a + b) ab(a + b) a + a b + ab + b a b ab a +b. (a + b) ab(a + b) a + b hg<gr<gjtlix<x? a + b (a + b) ab(a + b) (ii) (a b) +ab(a b) a a b + ab b + a b ab a b. hg<gr<gjtlix<x? (a b) + ab(a b) a b a b (a b) + ab(a b) (iii) a + b (a + b) ab(a + b) (a + b)(a + b) ab(a + b) (a + b)[(a + b) ab] (a + b)[(a + ab + b ) ab] (a + b)(a ab + b ). a + b (a + b)(a ab + b ) hg<gr<gjtlix<x? (a + b)(a ab + b ) a + b (iv) a b (a b) + ab(a b) (a b)(a b) + ab(a b) (a b)[(a b) + ab] (a b)[(a ab + b ) + ab] (a b)(a + ab + b ). hg<gr<gjtlix<x? a b (a b)(a + ab + b ) (a b)(a + ab + b ) a b 04

109 a + b + c ab bc ca g<gielx<oxivjl fil<ohxuk? a + b + c ab bc ca (a + b + c ab bc ca) [( a ab + b ) + ( b bc + c ) + ( c ca + a )] [( a b) + ( b c) + ( c a) ]. -u<uix?fil<hqe<uvl<lx<oxivjljbh<ohxgqoxil<; a + b + c ab bc ca [( a b) + ( b c) + ( c a) ] hg<gr<gjtlix<xqobpk? [( a b) + ( b c) + ( c a) ] a + b + c ab bc ca (a + b + c) (a + b + c ab bc ca) uqx<gielx<oxivjl hr<gqm<mh<h{<jhh<hbe<hmk<kqfil<njmuk? (a + b + c) (a + b + c ab bc ca) a (a + b + c ab bc ca) + b (a + b + c ab bc ca) + c (a + b + c ab bc ca) a + ab + c a a b abc ca + a b + b + bc ab b c abc +ca + b c + c abc bc c a a + b + c abc. -u<uixfil<ohxuk? hg<gr<gjtlix<x? (a + b + c) (a + b + c ab bc ca) a + b + c abc a + b + c abc (a + b + c) (a + b + c ab bc ca). wmk<kg<gim<m 9 : hqe<uvueux<jxuqiqk<okpkg; (i) (x + y) (ii) (x y) kqiu; (i) (x + y) (x) + (x) (y) + (x)(y) + (y) 7x + (9x )(y) + (x)(4y ) + 8y 7x + 54x y + 6xy + 8y. 05

110 (ii) (x y) (x ) (x ) (y) + (x ) (y) (y) 8x 6 (4x 4 )(y) + (x )(9y ) 7y 8x 6 6x 4 y + 54x y 7y wmk<kg<gim<m 0 : a + b, ab Ngqbux<xqe<lkqh<Hg<gt<LjxOb4, weqz<? a + b e< lkqh<hgi{<g/ kqiu; a + b (a + b) ab(a + b) (4) ()(4) wmk<kg<gim<m : a b 4 lx<xl< ab weqz<? a b e<lkqh<hgi{<g/ kqiu; a b (a b) + ab(a b) (4) + ()(4) wmk<kg<gim<m : a + b, a + b 8 weqz<? a + b, a 4 + b 4 lkqh<hgi{<g/ kqiu; a + ab + b (a + b) ab (a + b) (a + b ) () (8) ab (ab) ( 4). a + b (a + b) ab(a + b) () ( )() 8 ( 4) lix<xupq; a + b (a + b)(a ab + b ) (a + b)(a + b ab) () [8 ( )] (0) 0. a 4 + b 4 (a ) + (b ) [(a ) + (b )] (a )(b ) (a + b ) a b (a + b ) (ab) (8) ( ) 64 (4) hbqx<sq 4.. ohvg<gx<hze<$k<kqvk<jkh<hbe<hmk<kqhqe<uvueux<jxg<gi{<g; (i) (x + 9) (x + ) (ii) (x + 8) (x ) (iii) (t )(t + 6) (iv) (p 4)(p ) (v) 0 06 (vi) 59 6 (vii) 4 6 (viii) ohvg<gx<hze<$k<kqvk<jkh<hbe<hmk<kqg<gi{<g; (i) (5x + 8y) (ii) (s 4t) (iii) (4p + 7q)(4p 7q) (iv) (0) (v) (98) (vi) a + b 5, a b 4 weqz<? a + b lx<xl< ab bqe<lkqh<hg<gi{<g/ 4. a + b 0, ab 0 weqz<? a + b, (a b) Jg<gi{<g/ 5. (x + l)(x + m) x + 4x + weqz<? l + m, (l m) Jg<gi{<g/ 6. hqe<uvueux<jxuqiqk<okpkg; (i) (x + y + z) (ii) (4x y + z) (iii) (p + q r) (iv) (a b c) 06

111 7. a + b + c, ab + bc + ca 8 weqz<? a + b + c gi{<g/ 8. ohvg<gx<hze<$k<kqvk<jkh<hbe<hmk<kqh<hqe<uvueux<xqe<uqiqugi{<g/ (i) (x + )(x + )(x + 4) (ii) (x + )(x + )(x 4) (iii) (x + )(x )(x + 4) (iv) (x + )(x )(x 4) (v) (x )(x )(x 4) 9. ohvg<gx<hze< $k<kqvk<jkh< hbe<hmk<kqh< hqe<uvueux<xqz< x dxh<h? x dxh<h Ngqbux<xqe<ogPg<gjtBl<?lixqzqdXh<jhBl<gi{<g; (i) (x + 0)(x )(x + ) (ii) (x )(x + 4)(x ) (iii) (6x + )(6x 5)(7 6x) 0. (x + a)(x + b)(x + c) x 9x + x 5 weqz<? a + b + c, + +, a + b + c Ngqbux<jxg<gi{<g/. ohvg<gx<hze<$k<kqvk<jkh<hbe<hmk<kquqiquig<gg; (i) (x + y ) (ii) (u 7v) (iii) x (iv) (x y + ) x. a b, ab 6 weqz<?8a 7b e<lkqh<hgi{<g/ a b c. x + weqz<? x + x x, x + x Jg<gi{<g/ 4. x + y 6, xy 8 weqz<?x + y, x + y Jg<gi{<g/ 5. p + q 6, p + q weqz<?pq, p + q, p 4 + q 4 Jg<gi{<g/ 4.. giv{qh<hmk<kz< Lf<jkb hgkqbqz< -v{<m nz<zk nkx<g Olx<hm<m hz<zxh<hg< Ogijugjt ohvg<gq? lx<oxiv hz<zxh<hg< Ogijujb wu<uix ohxzil< weg< g{<mxqf<okil</ -h<ohipk yv hz<zxh<hg< Ogijujb wu<uix -v{<m nz<zk nkx<g Olx<hm<m hz<zxh<hg< Ogijugtqe< ohvg<gzig wpkzil< weg< gx<ohil</ yv hz<zxh<hg< Ogijujb -v{<m nz<zk nkx<g Olx<hm<m wtqb hz<zxh<hg< Ogijugtqe< ohvg<gzigwpkukgiv{qh<hmk<kz< weg< %xh<hml</ ohvg<gzqz< dt<t yu<ouiv wtqb hz<zxh<hg< OgijuBl< ogimg<gh<hm<m hz<zxh<hg< Ogijubqe< yv giv{q weh<hml</wmk<kg<gim<mig? x +, x we<hju x 9 e<giv{qgtigl</ x 9 (x + )(x ) we<hkiz<?-r<gx 9 we<hk-vhchz<zxh<hg<ogiju/ x +, x we<hjuyvhc hz<zxh<hg< Ogijugt</ -u<uix giv{qh<hmk<kkz< Ogijugjts< SVg<Gukx<G dkugqxk/ giv{qh<hmk<kkz< we<el< -f<ks< osbz<? giv{qgtigh< hgk<kz< we<xl< %xh<hml</ 4... giv{qh<hmk<kz<upqljx upq : (ohikuieg<giv{qjbg<g{<mhqck<kz<) A we<gqxyi-bx<g{qkg<ogijubqe< dxh<hgtqz< B we<gqx ohikg<giv{q -Vf<kiz<? fil< A bqe<yu<ouivdxh<jhbl< B 07

112 biz<ugg<gc we<gqxogijugqjmg<gl</-h<ohipka JB C weg<giv{qh<hmk<kq wpkzil</ wmk<kg<gim<m : giv{qh<hmk<kg; 6x 4 y 4x y + 0xy. kqiu; xy NeKohiKg<giv{qwe<hjkg<gi{<gqOxil</ 6x 4 y 4x y + 0xy xy 4 6x y 4x y 0xy + xy xy xy xy (x y x + 5y). upq : (dxh<hgjtg< GPh<hMk<Kkz<) YI -bx<g{qkg< Ogijubqe< dxh<hgtg<gh< ohikuie giv{q -z<jzobeqz< nf<k dxh<hgjtk<kgf<kljxbqz<gph<hmk<kq yvohikuiegiv{qkqilieqg<gh<hml</ wmk<kg<gim<m 4 : giv{qh<hmk<kg; x xy x + y. kqiu; Ogijubqe<dXh<HgTg<Gh<ohiKuieg<giv{q-z<jz/weqEl<dXh<Hgt<gQOp g{<muixgph<hmk<kh<hmzil<weg<gi{<gqoxil</ x xy x + y (x xy) (x y) x(x y) + ( ) (x y) (x y) [x + ( )] (x y) (x ). wmk<kg<gim<m 5 : giv{qh<hmk<kg; 6x 5 y + 6x 4 y + 9x y 4 + 9xy 5. kqiu;lkz<upq? l<upq-v{<jmbl<hbe<hmk<k?fil<gi{<hk? 6x 5 y + 6x 4 y + 9x y 4 + 9xy 5 xy (x 4 + x y + xy + y ) xy [(x 4 + xy ) + (x y + y )] xy [x(x + y ) + y(x + y )] xy (x + y ) (x + y). hbqx<sq 4.. ohikuieyvgiv{qjbg<g{<mhqck<kgiv{qgtigh<hgg<gul<;. 9m n. 4a 8a + 6a. x 5 + 4x 4. 6x 5 y 5 + x y + 4xy 5. 7pq p q ohikuie yv giv{qjbg< g{<mhqck<oki nz<zk GPh<hMk<kz< Ljxjbh< hbe<hmk<kqobi?giv{qgtigh<hgg<gul<; 6. mn p pn + m 7. x x x x x ax + a 9. p p + p 0. 8x + 4x + 4x giv{qh<hmk<kz<$k<kqvr<gjth<hbe<hmk<kq?giv{qh<hmk<kz< sqz Ofvr<gtqz< yv Ogijujbg< giv{qh<hmk<kl<ohik giv{qh<hmk<kz< $k<kqvr<gjth< hbe<hmk<kouil</ -s<$k<kqvr<gt< ohvg<gz< $k<kqvr<gtqzqvf<k ohxh<hm<mjubigl</ohvg<gz<$k<kqvr<gtiue; (i) (X + Y) X + XY + Y (ii) (X Y) X XY + Y (iii) (X + Y)(X Y) X Y (iv) ( X + Y) (X XY + Y ) X + Y (v) (X Y)(X + XY + Y ) X Y (vi) (X + Y) X + Y + X Y + XY X + Y + X Y (X +Y ) (vii) (X Y) X Y X Y + XY X Y X Y (X Y ) 08

113 (viii) (X + Y + Z) X + Y + Z + XY + YZ + ZX (ix) (X + Y + Z) (X + Y +Z XY YZ ZX) X + Y + Z XYZ OlOz ogimg<gh<hm<mt<t $k<kqvr<gjt uzk hg<gk<kqzqvf<k -mkhg<glig hck<okiole<xiz< flg<g gqop ogimg<gh<hm<mt<t giv{qh<hmk<kz< $k<kqvr<gt< gqjmg<gl</ (i) X + XY + Y (X +Y) (ii) X XY + Y (X Y) (iii) X Y (X + Y) (X Y) (iv) X + Y ( X + Y) (X XY + Y ) (v) X Y (X Y) (X + XY + Y ) (vi) X + Y + X Y+ XY (X + Y) (vii) X Y X Y + XY (X Y) (viii) X + Y + Z + XY + YZ + ZX ( X + Y + Z) (ix) X +Y +Z XYZ (X + Y + Z) (X + Y + Z XY YZ ZX) X + XY + Y (X + Y) Jh<hbe<hMk<kqgiv{qh<hMk<kz< wmk<kg<gim<m 6: giv{qgtigh<hgg<gul<; 4x + xy + 9y. kqiu; ogimg<gh<hm<mogijujbh<hqe<uvlixlix<xqbjlg<gzil</ 4x + xy + 9y (x) + (x)(y) + (y) X x, Y y wewmk<kg<ogi{<miz< uzkhg<gl< X + XY + Y. weou?nk (X + Y) weg<giv{qh<hmk<kh<hml</weoufil<ohxuk 4x + xy + 9y (x + y). X XY + Y (X Y) Jh<hbe<hMk<kqgiv{qh<hMk<kz< wmk<kg<gim<m 7 : giv{qh<hmk<kg; p 8pq + 8q. kqiu; ogimg<gh<hm<mogijujbhqe<uvlixwpkzil</ p 8pq + 8q p (p)(9q) + (9q) X p, Y 9q wewmk<kg<ogi{<miz<uzkhg<gl<x XY + Y NGl</Njgbiz<nK (X Y) wegiv{qh<hmk<kh<hml</weou?fil<ohxuk p 8pq + 8q (p 9q). X Y (X + Y) (X Y) Jh<hbe<hMk<kqgiv{qh<hMk<kz< wmk<kg<gim<m 8: giv{qh<hmk<kg; 6x 4 y 5. kqiu; 6x 4 y (4x y), 5 (5), we<hkiz< 6x 4 y 5 (4x y) (5) (4x y + 5) (4x y 5). X + Y (X + Y) (X XY + Y ) Jh<hbe<hMk<kqgiv{qh<hMk<kz< wmk<kg<gim<m 9: giv{qgtigh<hg; 5a + 64b. kqiu; 5a (5a), 64b (4b) we<hkiz<?fil< X 5a, Y 4b wewmk<kg<ogit<ouil</ hqe<ei5a + 64b (5a) + (4b) X + Y (X + Y) (X XY + Y ) (5a + 4b) [(5a) (5a)(4b) + (4b) ] (5a + 4b) (5a 0ab + 6b ). 09

114 X Y (X Y) (X + XY + Y ) Jh<hbe<hMk<kqgiv{qh<hMk<kz< wmk<kg<gim<m 0 : giv{qh<hmk<kg; 6p 8q. kqiu; 6p (6p), 8q (q) wewpkzil</njgbiz<? X 6p, Y q wewmk<kg< ogi{<omilieiz< 6p 8q (6p) (q) X Y (X Y) (X + XY + Y ) (6p q) [(6p) +(6p)(q) +(q) ] (6p q)(6p +pq+4q ). X + Y + X Y + XY (X + Y) Jh<hbe<hMk<kqgiv{qh<hMk<kz< wmk<kg<gim<m : giv{qh<hmk<kg; 8x + y + x y + 6xy. kqiu; 8x + y + x y + 6xy (x) + y + (x) y + (x)y [(x) + y] (x + y). X Y X Y + XY (X Y) Jh<hbe<hMk<kqgiv{qh<hMk<kz< wmk<kg<gim<m : giv{qh<hmk<kg; 8x 7y 6x y + 54xy. kqiu; 8x 7y 6x y + 54xy (x) (y) (x) (y) + (x)(y) (x y). X + Y + Z + XY + YZ + ZX (X + Y + Z) Jh<hbe<hMk<kqgiv{qh<hMk<kz< wmk<kg<gim<m : giv{qh<hmk<kg; x + 9y 6xy + 4x y + 4. kqiu; Ogiju x + 9y + 4 we<x%mkjzg<ogi{<mt<tk/-k&e<xuig<gr<gtqe< %Mkz</Njgbiz<?fil<wPKuK x + 9y 6xy + 4x y + 4 x + (y) + + x ( y) + x() + ()( y) [x + ( y) + ] (x y + ). X + Y + Z XYZ (X + Y + Z) (X + Y + Z XY YZ ZX) Jh< hbe<hmk<kq giv{qh<hmk<kz< wmk<kg<gim<m 4: giv{qh<hmk<kg; x 8y + 7z + 8xyz. kqiu; 8y ( y), 7z (z) we<hkiz<ogimg<gh<hm<mogijujbp we<g/-k&e<x g{r<gtqe<%mkjzg<ogi{<mt<tk/weoufil<wpkuk P (x) + ( y) + (z) (x)( y)(z) [(x) + ( y) + (z)] [(x) + ( y) + (z) (x) ( y) ( y)(z) (z)(x)] (x y + z) (x + 4y + 9z + xy + 6yz zx). X + Y + Z 0 wel<ohikx + Y + Z Jg<giv{qh<hMk<kz< X + Y + Z (X + Y + Z XYZ) + XYZ (X + Y + Z) (X + Y + Z XY YZ ZX) + XYZ (0) (X + Y + Z XY YZ ZX) + XYZ XYZ. Gxqh<H; X + Y + Z 0 weqz<, X + Y Z. (X + Y) ( Z) Z. n-k X + Y + XY(X +Y ) Z. n-k, X + Y + Z + XY( Z) 0. X + Y + Z XY Z. 0

115 wmk<kg<gim<m 5: giv{qh<hmk<kg; (x y) + (y z) + (z x). kqiu; X x y, Y y z, Z z x we<g/ hqe<ei X + Y + Z (x y) + (y z) + (z x) x y + y z + z x 0. X + Y + Z XYZ X, Y, Z Ngqbux<xqx<Ghqvkqbqm<miz< (x y) + (y z) + (z x) (x y) (y z) (z x). GPF[g<gk<jkh<hbe<hMk<kqg<giv{qh<hMk<kz< wmk<kg<gim<m 6: giv{qh<hmk<kg; 4x + 0xy + 5y 0x 5y. kqiu; ogimg<gh<hm<mogiju4x + 0xy + 5y 0x 5y (x) + (x)(5y) + (5y) 5(x) 5(5y) [(x) + (5y)] 5[(x) + (5y)] (x + 5y) 5(x + 5y) (x + 5y) (x + 5y 5). wmk<kg<gim<m 7: giv{qh<hmk<kg; 4a 4ab + b a + b. kqiu; ogimg<gh<hm<mhz<zxh<hg<ogiju 4a 4ab + b a + b (a) (a)(b) + (b) (a b) (a b) (a b) (a b) (a b ). wmk<kg<gim<m 8: giv{qh<hmk<kg; 8x 8x + 5y. kqiu; dxh<hgjtg<gph<hmk<k 8x 8x + 5y [(9x) (9x)() + () ] (5y) [(9x) ()] (5y) (9x ) (5y) [(9x ) + (5y)][(9x ) (5y)] (9x + 5y ) (9x 5y ). wmk<kg<gim<m 9: giv{qh<hmk<kg; x 4 +. kqiu; x J%m<cg<gpqg<g?fil<ohXuK x 4 + (x 4 + x + ) x [(x ) + (x )() + () ] x (x + ) ( x) [(x + ) + ( x)][(x + ) ( x)] (x + x + ) (x x + ). wmk<kg<gim<m 40: giv{qh<hmk<kg; x 4 + x y + y 4. kqiu; x y Jg<%m<cg<gpqg<g?fil<ohXuK x 4 + x y + y 4 (x 4 + x y + y 4 ) x y [(x ) + (x )(y ) + (y ) ] (xy) (x + y ) (xy) [x + y + (xy)] [x + y (xy)] (x + xy + y ) (x xy + y ). wmk<kg<gim<m 4: giv{qh<hmk<kg; x 4 + 5x + 9. kqiu; x Jg<%m<cg<gpqg<g, fil<ohxuk x 4 + 5x + 9 (x 4 + 6x + 9) x (x + ) x [(x + ) + x][(x + ) x] (x + x + ) (x x + ).

116 wmk<kg<gim<m 4: x 8 x y 6 Jg<giv{qgtigh<hqiq. kqiu; x JyVohiKg<giv{qbigwMg<g? x 8 x y 6 x (x 6 y 6 ) x [(x ) (y ) ] x (x + y ) (x y ) x [(x + y)(x xy + y )] [(x y)(x + xy + y )] x (x + y)(x y)(x + xy + y ) (x xy + y ). hbqx<sq 4.. hqe<uvuesiqbikuxiweuqjmbtq/. x + x + ( + x ). x 6 4x + 4 (x + ). a b (a b) 4. a + b (a + b)(a + ab + b ) 5. a b (a b)(a + ab + b ) giv{qh<hmk<kz<$k<kqvr<gjth<hbe<hmk<kqgiv{qgjtg<g{<mhqc/ x + 9x 7. 44x 7x a b + 0abcd + 5c d 9. x + y a b + xy + ab 0. x y + 7z. (x + y) + 8y. (x + ) + (x ). x 6 y 6 4. (x + y) (x y) 5. (p + q) + (p q) + 6p(p q ) 6. 7x + y + 7x y + 9xy 7. x x +48x x 7y 6x y + 54xy 9. 4x + 9y + z + xy + 4xz + 6yz 0. a + b + 9c + ab 6ac 6bc. x y + + xy. 8x 5y + 80xy x 7y + z +8xyz 4. a 8b 5c 0 abc 5. (a b) + (b c) + (c a) 6. (x + y z) + (y + z x) + (z + x y) 7. (a b ) + (b c ) + (c a ) 8. a (b c) + b (c a) + c (a b) 9. x(x + z) y(y + z) 0. xy x y. x ax + bx + c we<el<-vhcg<ogijujbg<giv{qh<hmk<kz< ogpg<gt< a, b, c Ngqbju njek<kl< LPg<gt< we wmk<kg< ogit<ouil</ OlZl< a 0/ ogpg<gt< a, b, c sqz gm<mh<himgtg<g dme<hm<miz< ax + bx + c Jg< giv{qh<hmk<kzil</-f<kg<gm<mh<himgjtbl<giv{qgjtbl<-eqgi{<ohil</ Lkzqz<? a, b Bl< c Bl< LPg<gt< we<el< wtqb ujgjb wmk<kg< ogit<ouil</ fil< -h<ohik x + bx + c we<hjkg< giv{qh<hmk<k Ou{<Ml</ fil< lixqzq dxh<higqbc we<el<lpju? p + q b we-vg<glix?p?q we<el<-vlpg<gtqe< ohvg<gx<hzeigwpkouil</fl<ljmblbx<sqbqz<fil<oux<xqohx<xiz<? x + bx + c x + (p + q)x + pq (x + px) + (qx + pq) x(x + p) + q(x + p) (x + p) (x + q) -u<uix?fil<uqvl<hqbjks<sikqk<kt<otil</

117 uqkq : x + bx + c bqe< lixqzq dxh<h c biek? p? q we<el< -V LPg<gtqe< ohvg<gx<hzeigul<? nu<uqv LPg<gtqe< %Mkzigqb p + q uiek x e< ogpuigqb b g<gs<slligul<-vg<glixhqiqg<gh<hm<miz<?x + bx + c (x + p) (x + q). wmk<kg<gim<m 4: giv{qh<hmk<kg; x + 9x + 8. kqiu; ogimg<gh<hm<mogijujbx + XY + Y wel<ucuk<kqz<wpklcbik/weou? X + XY + Y (X + Y) we<hjkh<hbe<hmk<klcbik/weou?lixqzqdxh<h8 Jg< giv{qh<hmk<klbz<ouil</ 8 e<wz<zigiv{qs<osicgt< g<giv{qgtqe<%mkjzwpkouil</ ( 8) + ( ) ( ) + ( 8) ( ) + ( 9) ( 9) + ( ) ( ) + ( 6) ( 6) + ( ) 9. x e< ogpjubl<? giv{qgtqe< %MkjzBl< yh<hqm<mh< hiik<okiole<xiz<, 6 Ngqb giv{qgtqe<%mkz<x e<ogpuigdt<tk/weou? x + 9x + 8 (x + ) (x + 6). wmk<kg<gim<m 44: giv{qh<hmk<kg; x 5x kqiu; lixqzq 54 lx<xl< x e< ogp 5. -r<g 54 e< giv{qgjtbl<? nux<xqe< %MkjzBl<gQOphm<cbzqm<Mg<gim<MOuil</ giv{qgt< %Mkz< {, 54} 55 {, 54} 55 {, 7} 9 {, 7} 9 {, 8} {, 8} {6, 9} 5 { 6, 9} 5 weou? x 5x + 54 (x 6) (x 9). wmk<kg<gim<m 45: giv{qh<hmk<kg; 5 x x. kqiu; kqm<mucuqz<wpk? 5 x x x x + 5 ( ) (x + x 5). -r<gfil< 5 5, 5 + ( ) weg<gi{<gqoxil</ weou? 5 x x ( ) [(x + 5) {x + ( )}] ( ) (x + 5)(x ) (x + 5) ( x). wmk<kg<gim<m 46: giv{qh<hmk<kg; x x. kqiu; ( ), ( ) + wefil<gi{<gqoxil<</ weou? x x [x + ( )] (x + ) (x ) (x + ).

118 nmk<kkig? a 0, a, b, c LPg<gtigg< ogi{<m ax + bx + c we<gqx -Vhc hz<zxh<hg< Ogijubqjeg< gvkouil</ p, q Ngqb -V LPg<gjt pq ac weul<? p + q b weul<-vg<glixfl<liz<g{<mhqcg<glcf<kiz<? ax + bx + c a (a x + abx + ac) a [a x + a(p + q)x + pq] a [a x + apx + aqx + pq] a [ax (ax + p) + q(ax + p)] a (ax + p) (ax + q) -u<uix?fl<liz<ogijujbgiv{qh<hmk<klcgqxk/ uqkq : p?q wel<lpg<gjt? p q a c? p + q b we<xqvg<glixgi{lcf<kiz<? ax + bx + c a (ax + p) (ax + q). wmk<kg<gim<m 47: giv{qh<hmk<kg; x + 7x +. kqiu; -r<g a x e<ogp b x e<ogp 7 c lixqzq fil<gi{<hk a c 6 6, b. Njgbiz< x + 7x + (x + 6) (x + ) (x + )(x + ). uqkqbqe< Lcjuh< hbe<hmk<kukx<gh< hkqzig fm dxh<jhh< hqiqk<k? gqpg<gi[lixgpljxbqz<giv{qh<hmk<kzil</wmk<kg<gim<mig? x + 7x + x + (6 + )x + x + 6x + x + x(x + ) + ()(x + ) (x + ) (x + ). wmk<kg<gim<m 48: giv{qh<hmk<kg; 8a + a. kqiu; -r<g? , 6 + ( 4) weg<gi{<gqoxil</hqiqk<okpkqbhqe<? GPh<hMk<kqeiz<fil<njmuK? 8a + a 8a + 6a 4a a(4a + ) ()(4a + ) (4a + ) (a ). wmk<kg<gim<m 49: giv{qh<hmk<kg; 6 + x + x. kqiu; Ogijujblix<xqobPk? 6 + x + x (x + x + ). -r<g? 4 8, 8 +. weou?fmdxh<jhh<hqiqk<kl<?gph<hmk<kqbl<wpk? 6 + x + x (x + 8x + x + ) [x(x + 4) + (x + 4)] (x + 4) (x + ). ax + bx + c z< a, b, c Ngqbju LPg<gtig -z<zilz<? olb<ob{<gtig -Vh<hqEl< %m fm dxh<jhh< hqiqk<okpkq GPh<hMk<kqg< giv{qh<hmk<kl< Ljxjb fil<hbe<hmk<kzil</ 4

119 wmk<kg<gim<m 50: giv{qh<hmk<kg; 7 x 0x 4. kqiu; -r<g? ( 4) 4 OlZl<( 4) weou?fmdxh<jhh< hqiqk<okpkq?gpljxbqz<wpk? 7 x 0x 4 7 x 4x + 4x 4 7x ( x ) + ( x ) ( x ) (7x + ). wmk<kg<gim<m 5: giv{qh<hmk<kg; x + 5 x + 6 kqiu; -r<g? 6 8 ( ) ( ) OlZl< + 5. weou?fmdxh<jhh< hqiqk<okpkq?gpljxbqz<giv{qh<hmk<k? x + 5 x + 6 x + x + x + 6 x (x + ) + (x + ) (x + ) (x + ). wmk<kg<gim<m 5: giv{qh<hmk<kg; 5x + 7xy + 4y. kqiu; fil< , weg<gi{<gqoxil</weou?fmdxh<jhh< hqiqk<okpkq?gpljxbqjeh<hbe<hmk<kqeiz<, 5x + 7xy + 4y (5x + xy) + (5xy + 4y ) x(5x + 4y) + y(5x + 4y) (5x + 4y) (x + y). wmk<kg<gim<m 5: giv{qh<hmk<kg; 6(a ) b 5(a )b 6b kqiu; x a wewmk<kg<ogi{<miz<? 6(a ) b 5(a )b 6b 6x b 5xb 6b b(6x 5bx 6b ) -r<g? ( 9) 4, OlZl< ( 9) weg<gi{<gqoxil</ Njgbiz< b(6x 9bx + 4bx 6b ) b[x(x b) + b(x b)] b(x b) (x + b) b[(a ) b] [(a ) + b] b[(a b ) (a + b )]. giv{qh<hmk<klcbiklpg<gjtg<ogpg<gtigg<ogi{<m-vhchz<zxh<hg< OgijugTl<d{<M/ wmk<kg<gim<m 54: giv{qh<hmk<kg; x + x. kqiu; ax + bx + c Bme<yh<hqMjgbqz< a, b, c. -r<g ac. OlZl<( ) + 0 b. weou?x + x J LPg<gjtg<ogPg<gtigg<ogi{<mgiv{qgtig<gLcbiK/ Gxqh<<H;x + x x + x x + 4 x + x + + x +. 5

120 gqopuvueux<jxg<giv{qgtigh<hqiq/ hbqx<sq x + 7x +. x + 9x + 0. d + 0d + 4. z 7z a a 7 6. x + x p 8p y y y 0y t 8t + 95 gqopuvueux<jxg<giv{qh<hmk<kg;. a + a x + 8x +. 4x + x x + x 5. 6p + 7p a a m + 6m p + 9p 9. 6x + 5x y y 6. 4x x. 9a 9a +. a a x 7x x x + 7 gqp<uvueux<jxg<giv{qgtigh<hqiq/ 6. 9x + 4xy + 5y 7. 4x 6xy 9y 8. 6c + cd 0d 9. 5x xy + 6y 0. a 5ab + 8b gqp<uvueux<jxg<giv{qh<hmk<kg;. 0 8 x x. u u +. x x x 8x x x x + x + 7. x + x x + 0x x + 5x x + 4x yv hz<zxh<hg< Ogijujb lx<oxiv hz<zxh<hg< Ogijubiz< ugk<kz< f(x) we<el< hz<zxh<hg< Ogijujb g(x) we<el< hz<zxh<hg< Ogijubiz< ugg<gl<osbziekf(x) q(x) g(x) + r(x) we<x-vg<glixl<?r(x) 0 nz<zkr(x) e< hcbiek g(x) e< hcjb uqm sqxqbkigul< -Vg<GliX? q(x), r(x) we<el< -V hz<zxh<hg< Ogijugjtg< gi{<hkigl</ -r<g f(x) J okigkq we<xl<? g(x) J hgkq we<xl<? q(x) J U we<xl<? r(x) J lqkq we<xl< njpg<gqe<oxil</ f(x) J g(x) Nz< ugg<gl< osbzqz< ughml< Ogijubigqb f(x) e< hc wh<ohipkl<? ugg<gl< Ogijubigqbg(x) e<hcg<gnkqgligouinz<zkslligoui-vg<gm<ml</woeeqz< f(x) e<hcg(x) e<hcjbuqms<sqxqbkig-vf<kiz<q(x) 0 NGl<; OlZl<r(x) f(x) we<xigl</u?lqkqgi{-vljxgt<dt<te/ 6

121 Ljx : fq{<mugk<kz<ljx: -f<kljxjbfil<yiwmk<kg<gim<ce<&zl<uqtg<gouil</ wmk<kg<gim<m 55: + 5x + x + x we<hjk + x Nz<uGg<gUl</ kqiu; ughml<ogiju f(x) + 5x + x + x ugg<gl<ogiju g(x) + x, f(x) e<hc, g(x) e<hc / flkgxqg<ogit<uq(x) JBl<?lQkqr(x) JBl<gi{<hOkbiGl</ gm<ml< : f(x), g(x) Ngqbux<jxk<kqm<mucuqz<wPKOuil</ f(x) x + x + 5x + g(x) x + gm<ml< : ughml<ogijubqe<lkz<dxh<higqb (x ) J ugg<gl< Ogijubqe< Lkz< x dxh<higqb(x) Nz<uGk<kiz<flg<Guqe<Lkz<dXh<H x gqjmg<gl</ x gm<ml<: ugg<gl<ogijubigqb(x + ) Juqe<Lkz<dXh<higqb (x ) Nz<ohVg<gq uvl< ohvg<gx<hzeigqb x + x J ughml< OgijubqzqVf<K gpqg<gou{<ml</ flg<gg< gqjmg<gl< lqkq x + 5x +. lqkqbqe< hc? ugg<gl< Ogijubqe< hcjb uqmh< ohiqbk/ gm<ml< 4: OlOz gqjmk<k x + 5x + we<el< lqkqjb Hkqb ughml< Ogijubig wmk<kg< ogi{<m gm<ml< Jh< hbe<hmk<kqeiz< uqe< -v{<mil< dxh<h x x gqjmg<gl</ x gm<ml< 5: ugg<gl< Ogiju )x + ) Juqe<-v{<mil<dXh<hiz<(x) ohvg<gq? uvl< ohvg<gx<hzje (x + x) J Hkqb ughml< Ogijubqeqe<X gpqg<g Ou{<Ml</ flg<g gqjmg<gl<lqkq(4x + ). lqkqbqe<nmg<gugg<gl<ogijubqe<hcbigdt<tk/ gm<ml< 6: lqkq (4x + ) J Hkqb ughml< Ogijubig wmk<kg< ogi{<m gm<ml< Jh< hbe<hmk<kqeiz<uqe<&e<xil<dxh<h 4 x x 4 gqjmg<gl</ gm<ml< 7: fil< ugg<gl< Ogiju (x + ) J?uqe<&e<xiuKdXh<hiz<(4) ohvg<gq uvl<(4x + 4) Jh<HkqbuGhMl<Ogijubqeqe<Xgpqg<gOu{<Ml</gqjmg<Gl<lQkq. -f<klqkqbqe<hch,s<sqbl</-kugg<gl<ogijubqe<hcjbuqms<sqxqbk/weou? fil<ugg<gl<osbjzfqxk<kquqm<m?u?lqkqngqbux<jxwmk<kwpkgqoxil</ Olx<%xqbgm<mr<gt<gQPt<tucuk<kqz<ogiMg<gh<hm<Mt<te/ x + x + x + 4 x + x + 5x + x + x x + 5x + x + x 4x + 4x + 4 x x x x x x 4x 4 x 7

122 -u<uixigflg<gg<gqjmg<gl<uq(x) x + x + 4, lqkq r(x). fil< nxquk, f(x)q(x) g(x) + r(x). x + x + 5x + (x + x + 4) (x + ) + ( ) Ljx : ugk<kg< giv{qh<hmk<kz< Ljx; -f<k Ljxbqz<? f(x) J -v{<m dxh<hgtqe< %Mkzigh< hqiqk<koz Gxqg<Ogit< NGl</ Lkz< dxh<h g(x) J giv{qbigg< ogi{<cvg<gl</ -v{<miuk dxh<h g(x) J uqm nmg<gqz< Gjxf<k yv hz<zxh<hg< OgijubiGl</-jkfil<ohx?f(x) dme<okjubiedxh<hgjtg<%m<c?gpqh<ohil</ wmk<kg<gim<m 56: 9x + x 5x + 7 J x + x Nz<uG/ kqiu; 9x + x 5x + 7 [(x + x ) (x) 6x + x] + x 5x + 7 (x + x )(x) + [ 6x + x ] + [x 5x] + 7 (x + x )(x) + ( x ) + ( x) + 7 (x + x ) (x) + [(x + x ) ( ) + x ] + ( x) + 7 (x + x ) (x) + (x + x ) ( ) + [(x) + ( x)] + [( ) + 7] (x + x )(x) + (x + x ) ( ) + 6 (x + x ) (x ) + 6 weou?u x, lqkq 6. wmk<kg<gim<m 57: 4 7x x x x 4 J x x + 4 Nz<uG/ kqiu; fil<kqiju-vupqljxgtqzl<gi{<ohil</ fq{<mugk<kz<ljx x 8x 68 x x 4 x x 7x x + 4 x x 4 + 6x 8x + + x 8x 4x 7x + 4 8x 8x + 54x 7x + + x 68x + 55x x 68x + 04x 7 x x + 76 weou?u x 8x 68, lqkq 49x ugk<kg<giv{qh<hmk<kz<ljx 4 7x x x x 4 x 4 x x 7x + 4 [(x x + 4) ( x ) 6x + 8x ] x x 7x + 4. (x x + 4) ( x ) 6x + 8x x x 7x + 4 (x x + 4) ( x ) 8x 4x 7x + 4 (x x + 4) ( x ) + [(x x + 4) ( 8x) 54x + 7x] 4x 7x + 4 (x x + 4) ( x ) + (x x + 4) ( 8x) 54x + 7x 4x 7x + 4 x 8x 68 8

123 (x x + 4) ( x ) + (x x + 4) ( 8x) 68x +55x+ 4 (x x + 4) ( x ) + (x x + 4) ( 8x) + [(x x + 4) ( 68) 04x + 7] +55x + 4 (x x + 4) ( x ) + (x x + 4) ( 8x) + (x x + 4) ( 68) 04x x + 4 (x x + 4) ( x 8x 68) + ( 49x + 76). weou?u x 8x 68, lqkq 49x hbqx<sq x x + x 7 NeKgQOpogiMg<gh<hm<mOgijugtiz<uGhMl<OhiKU?lQkq gi{<g/ (i) x + (ii) x 4 (iii) x. 5 + x 4 8x NeKgQOpogiMg<gh<hm<mOgijugtiz<uGhMl<OhiKU?lQkq gi{<g/ (i) (x + )(x + ) (ii) (x ) (iii) x + x. uqjmgt< hbqx<sq 4.. (i) kux (ii) kux (iii) kux (iv) kux (v) siq. x + x x + 4. x 4 x + 0x 8 4. x + 5x x + 5. x + 5x 7x x 4 4x + 4x 8 7. x 5 + x 8x + 6x x 4 6x + 5x 66x x 5 + x 4 + 9x + x 9x 8 0. x 5 + 4x 4 + 9x 8x + x x e<ogp x e<ogp x e<ogp acx + bdy + (ad + bc)xy 5. x x y 5xy y 6. x 4 + x y + y 4 7. p 9 8. a 7 9. m hbqx<sq 4.. (i) x + x + 8 (ii) x + 6x 6 (iii) t + 4t (iv) p 7p + (v) 08 (vi) 658 (vii) 4 (viii) 95. (i) 5x + 80xy + 64y (ii) 9s 4st + 6t (iii) 6p 49q (iv) 00 (v) 9604 (vi) ,9/ , 0 5., 8 6. (i) 9x + y + 4z + 6xy + 4yz + xz (ii) 6x + 4y + 9z 6xy yz + 4xz (iii) 4p + 9q + 4r + pq qr 8pr (iv) 9a + 4b + 4c ab + 8bc ac 9

124 (i) x + 9x + 6x + 4 (ii) x + x 4x 4 (iii) x + x 0x 4 (iv) x 5x x + 4 (v) x 9x + 6x 4 x e<ogp x e<ogp lixqzq 9. (i) (ii) 0 6 (iii) ,, 5 5. (i) 8x + x y + 6xy 4 + y 6 (ii) 8u 84u v + 94uv 4v (iii) x x + x x (iv) x 6 y 9 + 6x 4 y 6 + x y , ,7 5., 80,06 hbqx<sq 4... (m n). 4a(a a + 4). x(x 4 + 4) 4. xy (6x 4 y + x + 4) 5. 7pq ( pq) 6. (m p) (n + ) 7. (x + ) (x ) (x ) 8. (x + a ) (x a ) (x ) 9. (p + ) (p ) 0. (x + ) (x + ) hbqx<sq 4... siq. kux. kux 4. kux 5. siq 6. ( + x) 7. (x ) 8. (ab + 5cd) 9. (x + y + a b) (x + y a + b) 0. ( xy + z) (x y xyz + 9z ). (x + y) (x + y ). x (x 4 + ). (x + y) (x y) (x xy + y ) (x + xy + y ) 4. y(x + y ) 5. 8p 6. (x + y) 7. (x 4) 8. (x y) 9. (x + y + z) 0. (a + b c). (x y + ) (x + y + + xy + y x). (x 5y + 6) (4x + 5y xy + 0y x). (x y + z) (4x + 9y + z + 6xy + yz zx) 4. ( a b 5c) (a + 4b + 5c + ab 0bc + 5 ca ) 5. (a b) (b c) (c a) 6. (x + y z) (y + z x) (z + x y) 7. (a + b) (a b) (b + c) (b c) (c + a) (c a) 8. abc (a b) (b c) (c a) 9. (x y) (x + y + z) 0. ( + x + y) ( x y). (x + x + ) (x x + ) 0

125 hbqx<sq 4... (x + ) (x + 4). (x + 4) (x + 5). (d + ) (d + 7) 4. (z + 7) (z 4) 5. (a + 8) (a 9) 6. (x + 0) (x 9) 7. (p ) (p 5 ) 8. (y 6) (y 7) 9. (y 9) (y ) 0. (t ) ( t 5). (a + 5) (a + ). (x + ) (x + ). (x + ) (x + ) 4. (x + )(x ) 5. (p + ) (6p + 5) 6. (a + ) (4a 5) 7. (m + ) (7m 5) 8. (p + 4) (8p ) 9. (x + ) (x ) 0. (y + ) (5y 6). (x ) (7x + ). (a )(a ). (a ) (a 9) 4. (x ) (4x ) 5. (4x ) (4x 7) 6. (x + y) (9x + 5y) 7. (x + y) (x 9y) 8. (c + 5d) (c d) 9. (x y) (5x 6y) 0. (a 4b) (a 7b). (x ) (5x +). (u ) (u 4). (4x )(4x ) (4x ) (4x 7) 5. (4x ) (6x ) 6. ( x + ) ( x + ) ( x + ) (x + ) 8. ( 5 x + ) ( 5x + 5) 9. (x + 5 ) (x + 5) 40. ( 7 x + )( 7x + ) hbqx<sq 4.4 U lqkq. (i) x x (ii) 4x + x (iii) 4x x 5. (i) x x 9x + 7 (ii) x + 4x + 4 (iii) x 0x + 5

126 5. kqiugt<gi[l<<ljxgt<<<<<<<<<<<< flg<gyvg{g<gogimg<gh<hm<m, nkx<gk< kqiu gi{s< osie<eiz<, Lkzqz<nkje LPjlbig fe<g hck<k,hiqf<kogi{<m, ng<g{g<gqz< we<e ogimg<gh<hm<mt<tk, we<eg{<mhqcg<gou{<ml<nz<zkwkjefq'hqg<gou{<ml<we<hkjenxqf<k osbz<hmkz< Ou{<Ml<. hqxg ng<g{g<gqx<gk< kqiugi{ wf<k g{qk d{<jlgjth< hbe<hmk<k Ou{<Ml<,wf<k Ljxbqz< nf<k g{qk d{<jlgjth< hbe<hmk<kou{<ml< we<hkjek<< kqilieqg<g Ou{<Ml</ g{g<gqx<gk< kqiu gi{k< Okjubie g{qk d{<jlgjts< siquvh< hbe<hmk<ks< osbz< kqxel<, hbqx<sqbl< OkjubiGl</ osbz< kqxe<&zligk<< kqiu gi{ fqjxb upqljxgjtbl<, dk<kqgjtbl< nxqbzil<. sqz Ofvr<gtqz<yVg{g<gqe<kQIUhz<OuXupqgtqz<nz<zKhz<OuXdk<kqgtqz<gi{ LcBl<.wMk<Kg<gim<miggQOpdt<tg{g<gqjewMk<Kg<ogit<Ouil</ g{g<g: a b 4, a + b 5 weqz<, b a -e<lkqh<hgi{<g/ -g<g{g<gqe<kqiugi{gqp<g{<mljxgtqz<flkw{<o{im<ml<osbz<hmgqxk/ uqei: we<eogimg<gh<hm<mt<tk? uqjm: a b 4, a + b 5. uqei: we<e g{<mhqcg<gou{<ml<? uqjm: a -e<lkqh<h/ b uqei: wh<hcs<osbz<hmkz<ou{<ml<? uqjm: dk<kq: a b 4. () 9 () + () a 9 a. a + b 5. () () () b b. b a 9 9. dk<kq: a a ( a + b) + ( a b) b b ( a + b) ( a b) 5 4 dk<kq : a b 4 () a + b 5 () a a x x we<g. hqe<ei? n.k a bx. b b

127 () bx b 4 b (x ) 4. () bx + b 5 b (x + ) 5. b( x ) 4 x 4 n.k n.k 5x 5 4x + 4 n.k 5x 4x n.k x 9. b( x + ) 5 x + 5 weou fl<ljmb F{<{xqUk<kqxe<, osbz<kqxe<, kqxjl, OlZl< nehul< Ngqbju g{g<ggjtk< kqig<g Wx<hiMgjtBl<? dk<kqgjtbl< dvuig<gukx<gk< Okju/ Olx<%xqb g{g<jgk< kqih<hkx<g &e<x dk<kqgt< lm<mol -Vh<hkig LcU osb<b Ou{<mil</ yv g{g<gqe<< kqiu gi{? Hkqb dk<kqgjt nehuk<kqe< uibqzigul<, nok g{g<gqx<g lq{<ml<< lq{<ml< kqiu gi[l< OhiKl< ohx LcBl</ OlZl<flKw{<{lieKsqf<kqg<Gl<kqxeqz<ujtf<KogiMh<hkib<-Vg<gOu{<Ml<, lx<xupqljxgjtwx<xg<ogit<tkqxf<kkib<-vg<gou{<ml<?oux<xqml<-z<zilz< hz<oux gvk<okim<mr<gjth< ohxukib< -Vg<g Ou{<Ml</ yv g{g<gqe< kqiuqjeg< gi{ YI dk<kqbiz< Lcbuqz<jz weqz<, lx<oxiv dk<kqjbh< hbe<hmk<kqk< kqiu gi[kz<ou{<ml</-u<uix? yvg{g<gqe<kqiuqjeg<gi{flg<gk<okiqf<khz<oux dk<kqgt<kqif<kohigl<ujvhbe<hmk<kqg<ogi{<om-vg<gou{<ml</ 5. nelier<gt<lx<xl<fq'h{r<gt<< gqp<gi[l<w{<gtqe<njlh<jhofig<gg/,, 7,,, fil< -f<k njlh<hqe< ohik dxh<hqjeg< gi{ uqjpgqe<oxil< we<g/ dxh<hgjtg< gqp<g{<muixhgk<kxqbzil</ h<ohipk, -f<knjlh<hqz<nmk<kw{<we<ewe<hjkfl<liz<g{<mhqcg<glcbl</ weou?6 NuKdXh<H. OlZl< + ( ). 7 NuKdXh<H ( ). 8 NuKdXh<H ( ). 9 NuKdXh<H

128 + ( ). -r<ogdxh<hgtqe<lkqh<hgqjmg<gl<ljxjbofig<gl<ohipk, njlh<hqe<ohik dxh<h)nkiukn NuKdXh<H, n,,, ) + [ (n )] wefil<%xlcgqxk/ Neiz< -f<kg< %x<xqz< (n ) we<x %Mkz< dt<tk. -g<%mkjz g{<mxqbilz< njlh<hqe< ohik ucuk<jkg< g{<muqm<omil< we<x osiz<z LcbiK/ wmk<kg<gim<mig? %Mkz< we<hjkg< gvkouil<. -K -zqvf<k 00 ujv dt<t nmk<kmk<k -bz< w{<gtqe< %Mkz< NGl<</ S we<hk -g<%mkz<weqz<, S fil<nxquk? nkiuk S , S S kqz<f~x0 gt<-vh<hkjeg<gi{lcgqxk/ 00 0 weou?s 00 0 nz<zk S Olx<gi[l<%Mkz<gqjmg<gh<ohx<xLjxjbg<%If<KOfig<gqe<, fil< ( n ) n (n ) we<x%x<jxneliel<)conjecture) osb<bzil</ n,,, weg< ogi{<m -g<%x<jxs< siqhiig<gzil</ -r<g fq'h{l< ogimg<gh<hmilz< yv %x<xqje njlk<k nkjes< siqhiig<gqoxil</ fq'hqg<gh<hmikujv ng<%x<x YI nelief<kie< )gi^< we<x o\i<lieqb g{qk Oljk ke< 0uK ubkqz< 4uK hck<kg< ogi{<cvf<kohik? ugh<hisqiqbi< li{ig<gigjt e<%mkz<gi{s<osie<eohik? dmoegi^<5050we<xuqjmbtqk<kii/nsqiqbinujvwu<uixdmoehkqztqg<g Lcf<kKwe<XOgm<ggi^<?OlOzfil<%xqbLjxjb?NsqiqbIuqbf<Khivim<MliX ntqk<kii*/flknjlh<hqe<ohikdxh<h n( n ) + + n(n ) n n +. weou?,, 7,,. we<xnjlh<hqe<ohikdxh<h n n + NGl<. Olx<g{<m Nb<uqz<, fil< YI njlh<hqjeg< %If<kib<f<K, yv %x<xqje d{<mig<gqoeil</ d{<jlbqz< -bx<g{qkl<, ucuqbz< Ohie<x hqiqugtqz< hz Okx<xr<gt<w{<gtiZl<dVur<gtiZl<njlf<knjlh<Hgjt%If<kib<Us<osb<kke< uqjtuiz<g{<mxqf<kjungl</-h<ohipkfil<g{qkk<kqz<dt<touu<oux %x<xgt<?ng<%x<xgjth<hgk<kxqukx<gh<hbe<hmk<kh<hml<hzdk<kqgt<ngqbux<jx nxqf<kogit<ouil</ 4

129 %x<xgt< weh<hmhju uzqbxk<kqg< %xh<hml< uig<gqbr<gtigl</ gqop sqz %x<xgt<ogimg<gh<hm<mt<te: (i) x + 7 5, -r<g x N. (ii) skvk<kqe<&jzuqm<<mr<gt<ye<xg<ogie<xosr<gk<kigl</ (iii) (a + b) (a b) a b, -r<g a, b R. (iv) yvuqgqklxw{<. (v) sib<skvk<kqe<&jzuqm<<mr<gt<ye<xg<ogie<xosr<gk<kigik. (vi) n( n +) n. %x<xgt< olb<biejubigoui nz<zk olb<bx<xejubigoui -Vg<gzil</ wmk<kg<gim<mig OlOz dt<t %x<xgtqz< (i), (iv) lx<xl< (v) Ngqbju olb<bx<x %x<xgt<; (ii), (iii) lx<xl< (vi) Ngqbjuolb<bie%x<Xgt<. 5.. ujvbjxwe<xiz<we<e? sqz %x<xgt< Wx<geOu dt<t gvk<kgtqzqvf<k Hkqb gvk<kg<gjt dvuig<gq d{<jlbie LcUgjth< ohx upqugg<gl</ ng<%x<xgot ujvbjx weh<hml</wx<geouwx<xg<ogit<th<hm<msqz ujvbjxgt<gqopogimg<gh<hm<mt<te/ (i) wz<zih<hg<gr<gtl<slntuogi{<mlg<ogi{l<slhg<glg<ogi{l< weh<hml</ (ii) a x b weqz<,x NeKncliel<a ogi{<mb -e<lmg<jgweh<hml</ (iii) -v{<mogi{r<gtqe<%mkz<80 weqz<,njulqjgfqvh<hogi{r<gt< weh<hml</ 5.. ncogit<gt<< sqz %x<xgjt d{<jlobe LPjlbig Wx<Xg<ogit<tzil</ -k<kjgb %x<xgt< ncogit<gt<<< nz<zk yh<hg<ogi{<m d{<jlgt< weh<hml</ ucuqbzqzl< -bx<g{qkk<kqzl<dt<tsqzyh<hg<ogi{<md{<jlgt<gqopogimg<gh<hm<mt<te; (i) -v{<mht<tqgt<upqobyovyvogimkie<-vg<gqxk/ (ii) WOkEl< -v{<m olb<ob{<gt< x, y gtg<g x + y, xy NgqbjuBl< olb<ob{<gtigl</ (iii) n yv-bz<w{<weqz<?n + dl<-bz<w{< NGl</ (iv) yvogim<mk<k{<miekyovyvjlbh<ht<tqjbk<kie<ohx<xqvg<gl</ (v) yv Ogi{lieKyOvyV-Vsloum<cjbk<kie<ohx<xqVg<Gl<. 5..,, NgqbGxqgt< g{qkl<? kvg<g vqkqbig fqjeu %If<K ohxh<hm<m yv himh<hgkqbigl</ yv g{g<gqe< kqiu nz<zk g{g<gqe< upqljxgt< nz<zk g{g<gqe< fq'h{l< hch<hcbigohxh<hm<myvnjlh<higl</upqljxbqe<yu<ouixhcbl<nkx<gle<h dt<t hcbqjes< siif<kkig njlf<kqvg<gl</ kk<ku vqkqbie upqgjt Gxqg<g fil< 5

130 we<x Gxqbqjeh< hbe<hmk<kgqoxil</ wmk<kg<gim<ce< &zlig we<x Gxqbqe< hbjenxqouil</hqe<uvl<%x<xgt<p, Q jug<gvkg/ P: x, Q: x 4. P Jd{<jlobeg<ogi{<miz< x NGl</ x x x 4. Q d{<jlbigl</ weou? P d{<jlobeqz< Q d{<jlbigl</ -jkg< GxqbQm<cz< P d{<jl Q d{<jl nz<zkwtqkig P Q wewpkzil<. GxqbQM NeKd{Ik<KgqxKwe<hkjeg<Gxqg<Gl</ -h<ohipkgxqbqm we<hkjenxqouil<. lq{<ml<nok%x<xgt<p, Q -ux<jxwmk<kg<ogit<ouil</nkiuk P: x, Q: x 4 Q d{<jlobeqz<x 4 NGl<. Neiz<-kx<Gx l< <kqiu; x l<kqiuigl</weou? P olb<bie%x<xnz<z/-f<kd{<jljbfil<q P wewpkzil</ nmk<kkig we<xgxqbqm<jmnxqf<kogit<ouil</ -v{<m%x<xgt<q, Q wmk<kg<ogit<ouil</ Q : x + y OlZl< 5x 6y. Q : x OlZl< y. Q d{<jlobeqz< x + y, () 5x 6y. () () 4x + 6y 4 () () 5x 6y. (4) () + (4) 9x 7 x. x J() -z<hqvkqbqm 6 + y nz<zk y 6 nz<zk y. nkiuk x, y. weou?q d{<jlbigl</weou? Q Q. Q d{<jlobeqz< x, y. x + y ; OlZl< 5x 6y weou Q d{<jlbigqxk/weou Q Q. 0 Q Q OlZl< Q Q. -f<k LcUgt< Q Q, Q Q Ngqb -v{<jmbl< OsIk<K Q Q we wpkzil<. -k<kjgb $p<fqjzbqz<, Q dl< Q dl< slliekig -Vg<Gl<. -ke<ohivtiuk?q olb<weqz<?q olb</ Q olb< weqz<?q olb</nkiuk Q olb<big -Vf<kiz< lx<xl< -Vf<kiz< lm<mol Q olb<bigl</ we<x GxqbQM -Vf<kiz<lx<Xl<-Vf<kiz<lm<MOl )if and only if) we<hkjeg<gxqg<gl< Okx<xl<weh<hMuKbiK? yv %x<x? siqbiek nz<zk kuxiek we fq'hqg<gh<hmikujv njk neliel< )Conjecture) we<xjpg<gqe<oxil</ yv neliel< )Conjecture) siq we fq'hqg<gh<hm<muqm<miz< nk Okx<xligquqMgqxK/ kux we fq'hqg<gh<hm<m neliel< olb<bx<x %x<x weh<hml</ weou Wx<geOu olb< we<x fq'hqg<gh<hm<muqm<m %x<x Okx<xl< weh<hml</ yv %x<x yv Gxqh<hqm<m ujgbqz< d{<jlobeqz< nf<kg< %x<x hz<oux fqjzgtqz< siqhiig<gh<hm<mk we<x osiz<gqoxil</ -bx<<g{qkk<kqzl<? ucuqbzqzl<fil<nxqf<kt<tsqzokx<xr<gt<hqe<uvlixogimg<gh<hm<mt<te; 6

131 (i)yvlg<ogi{k<kqz<-v{<mhg<gr<gt<sll<weqz<?nux<xqx<gwkqov dt<togi{r<gt<sll</ (ii) yvlg<ogi{k<kqe<ogi{ntugtqe<%mkz<-v{<m osr<ogi{ligl</ (iii) -j{gvk<kqe<&jzuqm<mr<gt<ye<jxobie<x-vslg<%xqml</ (iv) yx<jxw{<{qe<uig<gll<yx<jxw{<o{ngl</ (v) -vm<jmw{<{qe<uig<gll<-vm<jmw{<o{ngl</ (vi) yvuqgqklxiw{<ngl<. (vii) (a + b) a + ab + b. (viii) log a (mn)log a m + log a n yvokx<xk<kqe<fq'h{l<we<xiz<we<e? yvokx<xk<kqe<fq'h{l<weh<hmuknke<d{<jljbfqi{bqg<gl<uquikl< NGl</wMk<Kg<gim<mig yx<jxw{<{qe<uig<gll<yiyx<jxw{<o{ we<x%x<xqjewmk<kg<ogit<ouil</ -f<k %x<x yv Okx<xl< weqz<? -kx<g kk<kuvqkqbie fq'h{l< -Vg<g Ou{<Ml</gQp<<g{<muquikk<jkgueqg<gUl<: n YIyx<jxw{<we<g/hqe<Hn m +, -r<g m yvlpw{</ -h<ohipk? n (m + ) 4m + 4m + (m + m)+. m yvlpw{<we<hkiz<, m + m dl<yvlpw{<{igl<. weou (m + m) YI-vm<jmw{<{iGl<. Nkziz< (m +m)+ YI yx<jx w{<{igl<. weou n YI yx<jx w{<. fq'h{l< ntqg<g hz<oux dk<kqgt< dt<<te/ ohikuig nju hqe<uvlix ujgh<hmk<<kh<hm<mt<te; (i) (ii) (iii) (iv) (v) Ofvcfq'h{Ljx. ljxlgfq'h{ljxnz<zklv{<him<m<fq'h{ljx. wkqiwmk<kg<gim<mgtqe<hcfq'h{l<. ucuqbz<fq'h{dk<kq/ njlh<hljxfq'h{l</ (i) Ofvcfq'h{Ljx: P Q we fq'hqg<g Ou{<Ml< we<g. Lkzqz< P d{<jl weg<ogi{<m hch<hcbig uquikqk<k Q d{<jl weh< ohxkz< Ou{<Ml</ P we<hk olb<obeqz< Q NeK olb< NGl< we<x fq'hqg<gl< LjxOb Ofvc fq'h{ Ljx weh<hml</ a a b wmk<kg<gim<m : 5 weqz<, weg<gi{<hqg<gul<. b a + b a a b 5b b 4b kqiu: 5 we<g. hqe<ha 5b.. b a + b 5b + b 6b 7

132 a c a b c d wmk<kg<gim<m : OfvcLjxbqz<? weqz<, wefq'hqg<gul</ b d a + b c + d a c a a c c kqiu: we<g/-r<g u weh<hqvkqbqmul</ hqe<h a ub. we<hkiz<? u. b d b b d d c ud. a b ub b ( u ) b u c d ud d ( u ) d u. -jkh<ohie<ox. a + b ub + b ( u + ) b u + c + d ud + d ( u + ) d u + a b a + b c d c + d. (ii) ljxlgfq'h{ljx: %x<xp NeK%x<X Q Jd{Ik<KgqxKwefq'hqg<g Ou{<Ml<we<Xogit<Ouil<. -kx<gig%x<xp olb<?%x<xq olb<bz<zweg<ogit<g/ hqe<h hch<hcbig uquikqk<k Lv{<himie uqjtuqje njmouil<)nkiuk wmogit< %x<xqx<gofilixie%x<xohxouil<*/weouwmogit<kuxiekwe<xlcuosb<k? Q d{<jlbigk<kie< -Vg<g Ou{<Ml< we dxkq osb<gqoxil</ %x<x P olb< weqz<? %x<x Q dl< olb< we fq'hqg<gl< -f<k Ljx ljxlg fq'h{ Ljx weh<hml< )wmk<kg<ogi{<m%x<xqx<gwkqvie%x<jxnjmkz<*/ wmk<kg<gim<m : ljxlgfq'h{ljxbqz<? yvuqgqklxiw{<wefq'hq. kqiu: wmogitig? yv uqgqklx w{< we OfIlixigg< ogit<ouil<. yu<ouiv p uqgqklxw{<j{bl<kq<m<mucuqz< wewpkzil<wenxqouil</hqe<h q q p, -r<gp, q lqjgw{<gt<<a p g<gl<, q g<gl< Jk<kuqvOuoxf<kgiv{qBl< -z<jz. -h<ohipk q p p q p q p YI-vm<jmw{< weou p YI-vm<jmw{< p m, m yv LPw{< 4m q q m q YI-vm<jmw{< q YI-vm<jmw{< q n, n yv LPw{< p g<<gl<q g<<gl< giv{qbigl< Lv{<hiM )p g<<gl< q g<gl< Jk< kuqv Ouoxf<k giv{qbl<-z<jz we<hkwmogit<*/ yvuqgqklxiw{<ngl<. 8

133 wmk<kg<gim<m 4: ljxlg fq'h{ Ljxg<ogi{<M? 00 hf<kgt< 9 ohm<cgtqz< jug<gh<hm<cvf<kiz< WkiuK yv ohm<cbqz< nz<zk nkx<g OlZl< hf<kgt< jug<gh<hm<cvg<gl<wefq'hq. kqiu: 00 hf<kgjt 9 ohm<cgtqz<jug<gl<ohipkwf<kyvohm<cbqzl<< hf<kgoti nkx<g OlOzi jug<gh<hmuqz<jz we wmk<kg< ogit<ouil</ NgOu yu<ouiv ohm<cbqzl< nkqghm<slig hf<kgt< jug<gh<hm<cvg<gl<. weou 9 ohm<cgtqz< jug<gh<hm<m hf<kgtqe< %Mkz< nkqghm<slig 9 99 NGl</ -K ogimg<gh<hm<m ogit<jgg<g Lv{<himiekiGl</ Woeeqz<? 00 hf<kgjt 9 ohm<cgtqz< jug<g Ou{<Ml</ weou Gjxf<k hm<sl< yv ohm<cbqziuk hf<kgoti nkx<g OlOzi jug<gh<hm<cvg<gl<. wmk<kg<gim<m 5: P we<xyvht<tq AB we<xogim<mk<k{<jmm : n we<xuqgqkk<kqz< dt<hr<gqmosb<bl<weqz<? P keqk<kkwe?ljxlgfq'h{ljxbqz<fq'hq/ kqiu: P we<xyvht<tq AB we<xogim<jm dt<hr<gqm<cz< m : n we<xuqgqkk<kqz< hqiqh<hkigg< ogit<ouil</ ogit<jgg<g dm<hm<m? P we<hk keqk<kk -z<jz weg< ogi{<miz<? lx<oxiv Ht<tq P? AB we<x Ogim<Mk<K{<jm m : n we<x uqgqkk<kqz< dt<hr<gqmosb<ukigl< (hml<5. Jh<hiIg<gUl<). hqxg AP m AP. OlZl< m. PB n P B n hml<5. n AP m PB OlZl< n AP m P B n AP m (AB AP) OlZl< n AP m (AB AP ) n AP m AB m AP OlZl< n AP m AB m AP (m + n) AP m AB OlZl< (m + n) AP m AB mab mab AP, AP. m + n m + n AP AP P Bl< P l<yovht<tqbigl</ -Kogit<jgg<GLv{<himignjlgqxK. weoup?yvkeqk<kht<tqbigl</ (iii) wkqiwmk<kg<gim<mgtqe<hcfq'hqk<kz<: P, P -v{<m%x<xgt<we<g/ P P we<hk d{<jlbi we Nvib Ou{<Ml< we<g/ P olb<bigul< P olb<bx<xkigul< -Vg<GliX WOkEl< YI wmk<kg<gim<m gi{lcf<kiz< P P we<x LcUg<G fil< uf<kuigtiouil</nxqbh<hm<mwmk<kg<gim<m YIwkqIwMk<Kg<gim<Mweh<hMl</ 9

134 wmk<kg<gim<m 6: x we<hkyvolb<ob{<weqz<?x x wenjlbli? kqiu: P : x we<hkyvolb<ob{<. P : x x. x weqz<?p : x we<hkyvolb<ob{<we<x%x<xolb<bigqxk/neiz< x g<gx 4 < x ; nkiuk P olb<bz<zwe<xigqxk/weou? P P. -r<g x YI wkqi wmk<kg<gim<migl</ weou x we<x wz<zi olb<ob{<gtg<gl< x x we<x%x<xolb<bigikwe<xlcuqjenjmgqoxil</ wmk<kg<gim<m 7: n yv-bz<w{<weqz<?n 4n we<hkd{<jlbi@ kqiu: P : n yv-bz<w{</ P : n 4n. n we<g/ YI-bz<w{</weOu?P d{<jlbigl</ Neiz<n n n. 4n 4 n 4 4. weoun < 4n. weou?p d{<jlbz<z/weou? P P. wmk<kg<gim<m8: x we<xwz<ziolb<ob{<gtg<gl< x > 0 wenjlbli? kqiu: P : x we<hkyvolb<ob{<. P : x > 0. x weqz<? P olb<bigl<. Neiz< x < 0. weou?p olb<bz<z r<gx YIwkqIwMk<Kg<gim<MNGl</ weou x we<xwz<ziolb<ob{<gtg<gl< x > 0 olb<bx<x%x<xigl</ Gxqh<H: Olx<gi{h<hm<m wmk<kg<gim<cz< x -e< yv Gxqh<hqm<m w{< lkqh<jhg< ogi{<m yv%x<xqe<olb<bx<xke<jljbfl<liz<fqi{bqg<glcf<kt<tk/ (iv) ucuqbz<fq'h{ljx: -bx<g{qkk<kqz<sqz g{g<ggjtucuqbz<ljxh<hckqiugi{zil</ wmk<kg<gim<m 9: (a + b) (a b) a b wefq'hq/ kqiu: hmk<kqz<abcd yvosu<ugl</ AB AI a, BE DI b (hml<5. Jh<hiIg<gUl<). hqe<had AI + DI a + b, AE AB BE a b. weou osu<ugl< AEFD -e<hvh<htu (a + b) (a b) Neiz<osu<ugl<AEFD -e<hvh<htu skvl< ABGI -e<hvh<htu osu<ugl< BEHG -e<hvh<htu + osu<ugl< DFHI -e<hvh<htu hml< 5. 0

135 a ab + (osu<ugl< CDIG -e<hvh<htu skvl< CFHG -e<hvh<htu) a ab + (ab b ) a b. (a + b) (a b) a b. (v) njlh<hljxfq'h{l<<: ucuqbzqz< sqz g{g<ggtg<gk< kqiu gi{<hkx<g? kvh<hm<m dvur<gtqz< sqz OgiMgjt %Mkzig njlg<gqe<oxil<</ hz ucuqbz< g{g<ggtqz< njlk<kz<< we<hk lqgs<sqxf<k dk<kqbigl</ siqbie -mr<gtqz< njlh<hgt<< njlbou{<ml</ weou njlk<kz<<&zl< fq'h{l< gi{<hk? g{g<gqe< kqiu gi{<huvk Le<%m<cOb nxqbl< Nx<xjzOu{<cbqVg<gqxK/ wmk<kg<gim<m 0: yvlg<ogi{k<kqz<-v{<mhg<gr<gt<sll<weqz<?nux<xqx<gwkqov dt<togi{r<gt<sll<wefq'hq/ kqiu: ABC yv Lg<Ogi{l<? AB AC we<g. B C wefq'hqg<gou{<ml<. D, BC -e< jlbh<ht<tq we<g. AD Js< OsIg<g)ujvkz<Ljx*/ Lg<Ogi{r<gt< ADB, ADC Ngqbux<Xt< BD DC, AD ohik? AB AC (hml< 5. Jh< hiig<gul<*/ weou -l<lg<ogi{r<gt<< siusll</ weou yk<k Ogi{r<gt<sll</ weou B C. hml< 5. hbqx<sq 5.. neliel<we<xiz<we<e?. Okx<xl<we<xiz<we<e?. Okx<xk<kqe<fq'h{l<we<hKwe<e? 4. We<siqhiIk<kz<fq'h{k<kqx<GLe<evig-Vg<gOu{<Ml<@ 5. YIwMk<Kg<gim<Mme<ucuqbz<fq'h{Ljxjbuqtg<Gg/ 6. wkqiwmk<kg<gim<mfq'h{ljxjbwmk<kg<gim<mme<uqtg<gg/ 7. njlh<hljxwe<xiz<we<e@ 8. ljxlgfq'h{ljxbq<z<?x + y we<xuixx, y WkiuK-v{<M-bz<w{<gt< weqz<?x nz<zky wefq'hq/ 5. g{qklikqiqgt< dzgqz< fjmljxs< sqg<gz<gjth< Hiqf<K ogit<ukx<gig g{qk likqiqgt< dvuig<gh<hmgqe<xe/likqiqgt<-bx<g{qkl<?ucuqbz<ohie<xhimh<hgkqgtqe<kj{ ogi{<m dvuig<gh<hmgqe<xe/ ucuqbz< gvk<kgjtg< ogi{<m njlg<gh<hm<<m g{qk likqiqgt< ucuqbz< likqiqgt< weh<hml</ -bx<g{qk gvk<kgjtg<< ogi{<m njlg<gh<hm<m g{qk likqiqgt< -bx<g{qk likqiqgt< weh<hml</ sqz slbr<gtqz< hiqosikjeosb<ukx<gle<ohg{qklikqiqgt<hgk<kxqbh<hml</hiqosikjeosb<uk we<hk lqgf<k oszu? Ofvl< nkqglikz<? hikgih<h Ngqbux<jxs<< siif<kkigl<</

136 fjmljxbqz<dt<t g{g<ggjth<hx<xqhcg<gl<ohipk fqjxb g{qk gvk<kg<gt< dvuigqbt<te/ -h<ohipk sqz -bx<g{qk likqiqgjth<hx<xqbl<? ucuqbz< likqiqgjth<hx<xqbl<nxqf<kogit<ouil</ fil< Lkzqz< yv wmk<kg<gim<cjeg< gvkouil</ Ofim<Mh<Hk<kgk<kqe< uqjz ' weqz<? 0 Ofim<Mh<Hk<kgr<gtqe<<uqjz 0 '0 NGl</ 0 Ofim<Mh<Hk<kgr<gtqe<<uqjz 0 '40 NGl</ 5 Ofim<Mh<Hk<kgr<gtqe<<uqjz 5 '60 NGl</ Ofim<Mh<Hk<kgr<gtqe<<uqjz '6 NGl</ Ofim<Mh<Hk<kgr<gtqe< w{<{qg<jg (x) uqjz(y) Ofim<Mh<Hk<kgr<gtqe< w{<{qg<jg nkqgligl<ohipk uqjz nkqgliujkbl<? w{<{qg<jg GjxBl<ohiPK uqjz GjxujkBl< nxqbzil</ fil< wmk<kg<ogi{<m -v{<m lixqgtl< ye<xig nkqgiqg<gqe<xe nz<zk ye<xig Gjxgqe<xe/-u<uiXnjlBl<-Vlixqgt<Ofvcuqgqkk<kqz<-Vh<hkigg<%XOuil</ OlOz %xqb wmk<kg<gim<cz< Ofim<Mh<Hk<kgr<gtqe< w{<{qg<jgbl< nux<xqe< uqjzbl< OfIuqgqkk<kqz<dt<tewenxqbzil</uip<g<jg$p<fqjzbqz<dt<tyVg{g<jg -h<ohipk wmk<kg<ogit<ouil</ yv Gxqh<hqm<m Diqz<? ubk 0 zqvf<k5 g<gt< -Vh<huIgtqe< w{<{qg<jgjb x we<x lixq Gxqg<Gl< weul<? y we<x lixq nuigtt< dbifqjzh<ht<tqg<gs< osz<huigtqe< w{<{qg<jgjb Gxqg<Gl< weul< ogit<ouil</ 999-zqVf<K00 uvml< ujv gqjmk<k uquvr<gjt gqp<g<g{<muix nm<muj{h<hmk<kqbt<otil<; x y y -r<gyu<ouivuvmk<kqx<gl<uqgqkl< 0.45 wenjlukignxqbzil<. hqe<hx, y x we<el<lixqgjt-j{g<gl<sle<himigy 0.45x gqjmg<gqxk/ gmf<k N{<Mgtqe< uquvr<gjt Ofig<Gl<ohiPK uvh<ohigl< N{<MgtqZl<< uqgqkl< fqjzbig njlbl< wele<likqiqbigwmk<kg<ogit<ouil</nmk<kuvmk<kqz<nf<kdiqz<ubkuvl<h 0 zqvf<k 5 g<gt< -Vh<huIgt< w{<{qg<jg 5000 we -Vf<kiz< nuigtt< dbifqjzh<ht<tq osz<ouii w{<{qg<jg 6750wey 0.45x sle<him<miz< nxqbzil</ weou y 0.45x J ubkuvl<h 0zqVf<K5 g<gt< -Vh<huIgTg<Gl<? nuigtt< dbifqjzh<ht<tq osz<hui<gtqe< w{<{qg<jgg<gl< okimihhmk<kl< yv g{qk likqiqbig ogit<tzil</ -K yv -bx<g{qks< sle<himig njlukiz< -l<likqiqjb -bx<g{qklikqiqwe<ohil</ -f<kwmk<kg<gim<cz<ubkuvl<h0zqvf<k5g<gt<-vh<huigt<w{<{qg<jg )x* g<gl<, dbifqjzh<ht<tq osz<hui<gtqe< w{<{qg<jg )y* g<gl< -jmob njlf<k uqgqkl<yvlqjgw{<lixqzqbigl</ weou-s<$p<fqjzbqz<y NeKx dme<ofvc lixkz< ogi{<mt<tk we<gqoxil</ sle<him y 0.45x yv g{qk Ofvc lixkz< Le<likqiqbiGl</ohiKuigyVg{qkOfvcliXkz<Le<likqiqy kx wenjlbl</ -r<gk NeKlixqzqNGl</-KyVucuqbz<g{g<gig-z<jzweqEl<-kje yv hmlig ujvf<k nh<hmk<jkg<ogi{<m g{g<gqe< kqiu gi{ Lbx<sqg<gzil</

137 wmk<kg<gim<mig OlOz njlf<k -bx<g{qk likqiqjb gqp<g{<muix ucuqbz< g{qk likqiqbig njlg<gzil</ y 0.45x we<x sle<him<cx<g ujvhml< ujvbzil</ -f<k ujvhml< yv OfIg<Ogimig njlbl<)hml< 6/5 Jh< hiig<gul<*/-k Ofvc lixkzqe< ucuqbz<likqiqbigl</ hml< 5.4 nmk<kkig? hqe<uvl< wmk<kg<gim<jmg< gvkouil</ yv -vbqz< u{<c yov sqvieougk<kqz<osz<ukigg<ogit<ouil</-vbqz<u{<c 60 gq/lq/k~vk<jkg<gmg<g5 60 l{q Ofvl< NGl< weqz<? u{<cbqe< Ougl< l{qg<g 40 gq/lq/ NGl</ 60 gq/lq/ 4 60 K~vk<jkg<gmg<gl{qOfvl<NGl<weqz<?u{<cbqe<Ougl<l{qg<G 80 gq/lq/ NGl</ -kqzqvf<k fil< Ougl< )v* -V lmr<gieiz< gmg<gl< Ofvl< )t* njv lmr<gigl< we nxqbzil</ -f<k uquvk<kqzqvf<k fil< -vbqzqe< Ougl< nkqgiqg<gl<ohipkyvgxqh<hqm<mk~vk<jkgmg<gwmk<kg<ogit<tl<ofvl<gjxbl< wenxqbzil</nkiukv nkqglieiz<t GjxBl/<OlZl<tnkqglieiz<v GjxBl< we<hkiz< OugLl< )v*? OfvLl< )t* kjzgqp< ohivk<kl< ohx<xqvh<hjk nxqbzil</ weou -f<k -v{<m ntugtl< )v lx<xl< t) kjzgqp< uqgqkk<kqz< -Vh<hkig nxqbzil</ yv ntu nkqgiqg<gl<ohipk lx<oxiv ntu GjxuKl<? yv ntu GjxBl<ohiPK lx<oxiv ntu nkqgliukl< ohx<xqvh<hqe< -f<k -V ntugtl< kjzgqp<uqgqkk<kqz< njlgqe<xe we<ohil</ gqp<g<g{<m nm<muj{bqz< yv Gxqh<hqm<m K~vk<jkg< gmg<g -vbqz< u{<c wmk<kg<ogit<tl< OfvLl<? OugLl< ogimg<gh<hm<mt<te/ Ofvl< (t) l{qbqz< Ougl< (v) gq/lq/ Olx<g{<mnm<muj{bqzqVf<K, t v 80 60, t v , t v 5 60, t v we<hkjenxqbzil</nkiuk

138 60 tv 60 nz<zk v. t v Bl< t Bl< wkqiuqgqkk<kqz< dt<te/ -KOfvk<jkh<ohiXk<KOugk<jk Gxqg<Gl< YI -bx<g{qk likqiqbigl</ vt 60 g<gyvujvhml<ujvf<kiz< nf<k ujvhml< yv OfI OgimigiK/ nk hmk<kqz< dt<tk Ohiz< yv sqxqb uqz< Ohiz< njlbl< )hml< 5.5 Jh< hiig<gul<*/ -h<hmk<kqz< t Js< siif<k v -e< uqpl< Ohig<gqjeg< gi{zil</ -u<uix Ougk<kqx<Gl<? Ofvk<kqx<Gl< okimih hmk<kl< ujvhml<yvucuqbz<likqiqbigl</ hml< 5.5 -bx<hqbzqz<pv yvlixqzq? we<el<sle<him<jmnxqouil</-kyvkjzgqp< uqgqk lixkzg<g lx<xolii wmk<kg<gim<migl</ lixik ouh<h fqjzbqz< npk<kl< nkqglieiz<gentugjxbl<weul<?npk<kl<gjxf<kiz<gentunkqgligl< weul< nxqouil</ kjzgqp< uqgqk lix<xk<kqx<<g lx<xolii wmk<kg<gim<mig? Gxqh<hqm<m hvh<htuaogi{<mosu<ugk<kqz<fqtk<kqe<ntux nkqglieiz<ngzk<kqe<ntuy GjxBl<weUl<?fQtk<kqe<ntUGjxf<kiz<ngzk<kqe<ntUnkqgliGl<weUl< nxqbzil</nkiukxy A weqz<?)x fqtl<, y ngzl<) y x A nz<zk x y A. fqtl<gjxf<kiz<ngzl<nkqgiqg<gl<?ngzl<gjxf<kiz<fqtl<nkqgiqg<gl</. g{qklikqiqwe<xiz<we<e@. g{qklikqiqbqe<okjujbg<%x/. OfIuqgqklixz<uqtg<Gg/ 4. wkqiuqgqklixz<uqtg<gg/ hbqx<sq 5. 4

139 6. nxqljxucuqbz< ucuqbz< we<x Gxqg<Gl< Nr<gqzs< osiz< geometry NeK? Huq we<x ohivtjmb geo lx<xl< ntuqmz< we<x ohivtjmb metron we<x -u<uqv{<m gqovg<gs< osix<gtqzqvf<k ohxh<hm<mkigl</ ucuqbz< we<hk fqzg<g{g<gqjeg< Gxqg<Gl< himlig -keiz< ohbiqmh<hm<mk/ -h<himh< hqiquqe< okimg<g giz uti<s<sqbqz<? wgqh<kqbi<gt< ucuqbz< uqkqgjt fqz ntjugtg<gl<? Ogibqz<gt< gm<mukx<gl<? gz<zjxgt< njlh<hkx<gl<? %i<r<ogihvr<gt< )hqvlqmgt<* njlh<hkx<gl< hbe<hmk<kqei</ hqe<ei< gqovg<gi<gt< ucuqbz< d{<jlgjt fe<g kvg<g iqkqbigs< sqf<kqk<k? hqkigv^<? B,g<tqm< Ohie<x gqovg<g uz<zfi<gtqe< hjmh<hgtqe< &zlig kr<gt< ucuqbz< nxquqje g{qk dzgk<kqx<g ni<h<h{qk<kei</ -ui<gtk hjmh<hgtqe< dbifqjzob nxqljx ucuqbz< himh<hqiquqx<g Okix<Xuibig njlf<kk/-h<himh<hqiquqz<?ucuqbz<gvuqgt<kj{bqe<xq?ucuqbz<d{<jlgjt uqkqkvljxbqz< hgk<kxqukx<og Lke<jl kvh<hmgqe<xk/ SliI 0 gq/l/ gizk<kqz< uip<f<k ucuqbzqe< kf<jk we<xjpg<gh<hml< gqovg<g nxqri B,g<tqm< we<hui? ke< gizk<kqb ucuqbz< kk<kur<gjt keqjlg<%xgt< we<x ohbiqm<m hkq&e<x F~z< okigh<hgtig upr<gqbt<tii/ ucuqbjzh< hch<hk Hkqbkigs< sqf<kqh<hkx<gl< lx<x himr<gtqz<kqxjljbwx<hmk<kukx<gl<ohiqkl<dkuukigl</ 6. siqhii<k<kzg<giqbokx<xr<gt< flk Lf<jkb ugh<hgtqz<? ujvbxg<g -bzik nch<hjmbie ucuqbz< osix<gtigqbht<tq?ogim?ktl<?ogi{l<ohie<xux<jxnxqf<kt<otil</olzl<fil< Lg<Ogi{r<gtqe< ogit<jggt<? -j{ogimgt<? sqz sqxh<hie fix<gvr<gt< Ngqbux<jxBl<nxqf<Kt<Otil</weqEl<?nux<jxh<hx<xqOlZl<nxqf<Kogit<ukx<G Le<?fil<Wx<geOuhck<kux<jx-h<ohiPKfqjeU%i<Ouil<. 6.. nch<hjmbieucuqbz<osix<gt< Ht<tq we<hk outqbqz< yv fqjzbqjeg< Gxqh<hkx<G hbe<hmukigl</ upg<gk<kqz<?gigqkk<kitqe<lqokinz<zkgvl<hzjgbqe<lqokiyvht<tqbiek?yv sqxqb njmbitlqm<m Gxqg<gh<hMl</ Neiz<? nxqljxbqe<hc Ht<tqg<G ntoui nz<zk ucuoli gqjmbik/ -v{<m OgiMgt< ye<jxobie<x oum<cg<ogit<tl< fqjzbqzqvf<k Ht<tqbqje nxqf<k ogit<tzil</ Neiz<? -r<g Ht<tqbqje nxqf<k ogit<togim<jmh<hx<xqbgvk<khbe<hmk<kh<hm<mk/-v{<mktr<gt<ye<jxobie<x oum<cg<ogit<tl< -mk<kqz< njlf<k Ht<tqgtqe< okigh<hqje OgiM we nxqbzil</ Neiz<-r<GOgim<cjenxqf<Kogit<tktk<kqjeh<hx<xqbgVk<Khbe<hMk<kh<hm<mK/ 5

140 ke< lqkt<t wf<kuqv Ht<tqgjtBl< -j{g<gl< OfIg<Ogim<ce< njek<k Ht<tqgjtBl< ke<lqok ogi{<mt<tuixjlf<k? njek<k kqjsgtqzl< Lcuqz<zilz< osz<zg<%cb Olx<hvh<Oh yv ktl< wenxqf<kogit<gqoxil</weou?nch<hjm ucuqbz<osix<gtieht<tq?ogim?ktl<-ux<jxh<hx<xqnehuk<kqz<lm<molnxqf<k ogit<t LcBle<xq nux<jxs< siqbig ujvbxh<hk -bzik ye<xigl</ Ht<tqgt<? Nr<gqzh< ohiqb wpk<kgtie A, B, C, D Ohie<xux<xiz< Gxqk<Kg< gim<mh<hml</ A, B we<heyvogim<ce<lqkt<t-vht<tqgt<weqz<?ng<ogim AB nz<zk BA weg<gxqg<gh<hml</-kjeogimabweul<hcg<gzil</ -Vkjznl<Hg<Gxq? we<hkqz< dt<t -V nl<hgt<? OgiM -V kqjsgtqzl< Lcux<X osz<gqxk we<hkjeg< Gxqg<gqxK/ SVg<glig? yv OgimieK l we<x keq wpk<kizl< Gxqg<gh<hMl</ A, B we<hju yv Ogim<ce< lqkt<t -V Ht<tqgt< weqz<? A, B Ngqb Ht<tqgjtBl< Osi<k<K nh<ht<tqgtg<gqjmobnjlbl<nf<ofig<ogim<ce<higk<kqje? A, B we<x -V Ht<tqgTg<gqjmOb njlf<k OfIg<Ogim<Mk<K{<M nz<zk SVg<glig Ogim<Mk<K{<Mweh<hMl</-KGxqbQm<cz< AB weg<gxqg<gh<hml</ AB -e<fqtk<jksvg<glig AB nz<zkba wewpkzil</yvht<tqbqz<okimr<gqwokel<yvkqjsbqz<lcuqe<xqs<osz<zl<yv Ogim<ce<higl<gkqi<weh<hMl</-r<Gokimr<Gl<Ht<tqbieK?ng<gkqiqe<Nvl<hh<Ht<tq nz<zk okimg<gh<ht<tq weh<hml</ A we<hk yv gkqiqe< okimg<gh<ht<tq? BNeK ng<gkqiqe<lqkt<twokel<yvht<tqobeqz<?-g<gkqviek AB weg< Gxqg<gh<hMl</ -kje? yvkjz nl<hg<gxqbme< Gxqg<gilz<? gkqi< AB weul< Gxqg<gzil</ AB bqe<lqkjlf<kt<tyvkjznl<hg<gxqgkqi<ab bqe< kqjsjbg< Gxqg<Gl</ Gxqh<H; -eq? AB we<hkje? -Vkjz nl<hg<gxq JAB bqe< lqk Gxqg<gilz< SVg<gligOgiMABweUl<? AB we<hkjegxqbqm JABbqe<lQKGxqg<gilz< SVg<glig Ogim<Mk<K{<M AB weul<? gkqi AB J yvkjz nl<hg<gxq J AB bqe< lqk Gxqg<gilz< SVg<glig gkqi AB weul< Gxqh<Ohil</ Ht<tqgt< A, B -ux<xqx<gqjmh<hm<mokijzuqjeab we<xl<gxqh<ohil</ yvohikh<ht<tqabqz<nvl<hqg<gl<-vgkqi<gt< AB, AC Neju?Ht<tqA z< yv Ogi{k<kqje njlg<gqe<xe we<gqoxil</ Ht<tq A NeK Ogi{k<kqe< Lje weh<hml</ofig<ogim<mk<k{<mgt<abbl< AC Bl<ng<Ogi{k<kqe<-VgkqIgt<weh<hMl<. -g<ogi{liek? BAC nz<zk CAB weg<gxqg<gh<hml</ BAC nz<zk CAB we yv Ogi{lieK Gxqg<gh<hm<miz<? Ht<tq A NeK ng<ogi{k<kqe< Lje? AB lx<xl< ACng<Ogi{k<kqe<gkqIgt<we<hjkk<okiqf<Kogit<g/ Abqz<njlBl<Ogi{lieK 6

141 yv gkqiqzqvf<k okimr<gq nmk<k gkqvg<gs< osz<ukig yv sqxqb uqz<ziz< Gxqg<gh<hMl<(hml< 6. Jh<hii<g<gUl<). Awe<xLjebqZt<tyVOgi{k<kqe<gkqIgt< okiqf<k fqjzbqz< ng<ogi{k<jk wtqkig A we wpkzil</ yv Ogim<Mk< K{<jmh<Ohie<X yv Ogi{k<kqx<Gl< ntu d{<m/ A we<x Ogi{k<kqe< ntju m A weg< Gxqg<gzil</ Gph<hl< WKl< -z<jzobeqz<? m A J wtqkig A weg< Gxqh<Ohil</Neiz<?m A we<hkyintu? A we<hkyvgxqbqmwe<hjkh<hiqf<k ogit<tul</ hml< 6. hml< 6. yvogi{k<kqe<ntu?ng<ogi{k<kqe<gkqigtqe<fqtntjuh<ohivk<kkz<z/ Ogi{r<gt<? Ähijg} weh<hml< nzgqeiz< ntg<gh<hmgqe<xe/ yv ktk<kqe< lqkjlf<k yvgkqi<nke<okimg<gh<ht<tqjbg<ogi{<myvlpspx<sqjbwx<hmk<kqeiz<?nk 60 hijg ntut<t Ogi{k<kqje njlg<gqe<xk we<gqoxil</ -f<k ntu 60º we wpkh<hml<(hml<6. Jh<hii<g<gUl<). -r<gogi{k<kqe<gkqigt<ye<xqe<lqkye<xig njlf<kt<tk/lx<xogi{r<gtqe<ntugt<60º jbnch<hjmbigg<ogi{<mkigl</ yv gkqi<? yv LP Spx<sqbqe<< higk<kiz< njlg<gl< Ogi{k<kqe< ntu 4 (60º) 90º NGl<; higk<kiz< njlg<gl< Ogi{k<kqe< ntu (60º) 60º NGl<; higk<kiz<njlg<gl<ogi{k<kqe<ntu (60º) º NGl</ hml< 6.4 hml< 6. hml< 6.5 m BAC 90º weqz<? BAC yv osr<ogi{l< (hml<6. Jh<hii<g<gUl<) weh<hml<. m BAC e< ntu 90º g<gl< nkqgoleqz<? BAC yv uqiqogi{l< (hml< 6.4 Jh< hii<g<gul<) weh<hml<. m BAC e< ntu 90º g<gl< GjxU weqz<? BAC yv GXr<Ogi{l< (hml< 6.5 Jh<hii<g<gUl<) weh<hml</ 7

142 m BAC 80º weqz<? BAC yvofi<g<ogi{l< (hml< 6.6 Jh<hii<g<gUl<) weh<hml</ BAC yv Ofi<g<Ogi{oleqz<? BC yv OfIg<Ogim<Mk< K{<miGl<A A, B, C we<he yvogimjlf<k Ht<tqgtiGl<A nkiuk? -h<ht<tqgt< yvogim<cz< njlbl< Ht<tqgtiGl</ hml< 6.6 hml< 6.7 -VOgiMgt< BD Bl< CE Bl<Awe<xHt<tqbqz<oum<cg<ogit<ukiz<njlBl< Ogi{r<gt< BAC Bl< DAE Bl< Gk<okkqi< Ogi{r<gt< weh<hml</ OlZl<? CAD, BAE NgqbOgi{r<gt<Gk<okkqi<Ogi{r<gtignjlujkBl<nxqbzil<)hml< 6.7 Jh<hii<g<gUl<). m BAC Ogi{k<kqe< ntu 80º g<g nkqgligul<? 60º g<gg< Gjxuig -Vh<hqe<ng<Ogi{l<hqe<ujtOgi{l< weh<hml<(hml< 6.8 Jh<hii<g<gUl<). hml< 6.8 hml< 6.9 hml< 6.0 -VOgi{r<gTg<GohiKds<sqBl<?yVohiKg<gkqVl<njlf<K?ohiKuie gkqvg<g -V hg<gr<gtqzl< Ogi{r<gt< njlblieiz<? nu<uqv Ogi{r<gTl< nmk<kmk<kogi{r<gt<weh<hml< (hml< 6.9 Jh<hii<g<gUl<). hml< 6.9 -z< BAD, DAC Ngqbux<xqx<G ohiklje A NGl<;ohiKgkqI AD g<g -V hg<gr<gtqzl< -ju njlf<kqvh<hkiz<? BAD Bl< DAC Bl< nmk<kmk<k Ogi{r<gt< NGl</ BAC Bl< BAD Bl<ohiKuiejgABohx<xqVf<kiZl<?njunMk<kMk<kOgi{r<gt<NgiK; Woeeqz<? nju AB we<x ohik hg<gk<kqx<g yov hg<gk<kqzjlf<k Ogi{r<gtig -Vh<hKkie<we<hjknxqf<Kogit<Ouil</-VnMk<kMk<kOgi{r<gtqe<%Mkz<90º weqz<? nju fqvh<hg<ogi{r<gt< (hml< 6.0 Jh< hii<g<gul<) weh<hml</ hml< 6.0 -z< m BAD + m DAC 90º; weou? BAD Bl< DAC Bl<fqvh<Hg<Ogi{r<gtiGl<;-r<G m BAD 90º m DAC ; yv Ogi{l< lx<xkqe< fqvh<hg<ogi{ligl</ weou? -g<ogi{r<gtqz<ye<xqe<ntuokiqf<kiz<?lx<xkqe<ntuqjenxqblcbl</ 8

143 -VnMk<kMk<kOgi{r<gtqe<%Mkz< 80º weqz<?njulqjgfqvh<hg<ogi{r<gt< weh<hml<(hml< 6. Jh<hii<g<gUl<*/ hmk<kqz<? m BAD + m CAD 80 o. weou? -r<g filxquk BAD Bl< CAD Bl< lqjgfqvh<hg<ogi{r<gtigl</olzl<? m DAC 80º m BAD, m BAD 80º m DAC. nkiuk?yvogi{lieklx<xkqe<lqjgfqvh<hg< Ogi{liGl</-r<GyVOgi{l<kvh<hce<?lx<xjkg<gi{zil</ 6.. <OgiMgtqe<lQkieh{<Hgt<lx<Xl<Okx<xr<gt< hml< 6. nxqf<k ogit<ukx<gig? fil<? OfIg<OgiMgtqe< lqkie sqz h{<hgjtbl<? Okx<xr<gjtBl< -h<ohik gi{<ohil</ -r<g Okx<xr<gjt fq'hqg<gilz< osbz< uqtg<gk<kqe<&zl<siqhiik<kg<ogit<gqoxil</ osbz<; yvktk<kqz< A, B we<x-vht<tqgjtg<gxqg<g/ hml< 6. Ht<tq A e<upqbighzogimgt<ujvg/nux<xt<yvogimlm<ml<<bupqob osz<ujkg< gi{<g/ -jkh< Ohie<Ox? Ht<tq B e< upqob hz OgiMgt< ujvg/ nux<xt< yov yv OgiM lm<ml< A e< upqob osz<ujk okiqf<k ogit<g/ -g<ogimiek? A lx<xl< B gjt -j{g<gl< OfIg<OgiM AB we nxqg/ -s< osbzqzqvf<k?gqp<g<gi[l<h{<hgqjmg<gqxk/ h{<h;yvktk<kqz<?-vht<tqgt<ogimg<gh<hm<cvf<kiz<?nux<xqe<upqobyovyv OgiMkie<njlBl</ Gxqh<H;OlOz%xqbnch<hjmg<ogit<jgbqzqVf<KgQp<g<g{<mux<jxnxqbzil<; (i) yv ktk<kqz< dt<t -V ouu<oux Ht<tqgt< yov yv Ogim<jmk<kie< njlg<gl</x, Y we<he-vht<tqgoteqz<?ogim XY ye<xkie<njlbl</ (ii) &e<x nz<zk &e<xg<g Olx<hm<m Ht<tqgt< yv Ogim<cz< njlf<kiz<? nju yvogimjlf<kht<tqgt<weh<hml</ 9

144 (iii) &e<xnz<zk&e<xg<golx<hm<mht<tqgtqz<wkiuokie<xlx<xuqvht<tqgtiz< njlbl<ogim<cz<njlbiokeqz<?njuyvogimjlf<kht<tqgtigi/ osbz<;yvktk<kqz<ab, CD we<x-vouu<ouxogimgt<ujvg/-vogimgjtbl< Ou{<Mltuqx<G fqm<mg/ -V OgiMgTg<Gl< ohikuig yv Ht<tq -Vh<hjkObi nz<zk-z<zilz<ohiujkobigi{<g (hml< 6. Jh<hii<g<gUl<). hml< 6. Olx<gi[l<osbzqzqVf<KgQp<g<g{<mh{<Hgqjmg<gqxK/ h{<h : -VOuXOgiMgTg<Gl<ohiKuigyVHt<tqg<GOlz<-Vg<giK/ Gxqh<H;-VOgiMgTg<gqjmObyVohiKh<Ht<tqnjlBlieiz<njuye<jxobie<X oum<ml<ogimgt< weh<hml</ ohikh<ht<tq njlbik OgiMgt< oum<cg<ogit<tik OgiMgt< weh<hml</ -k<kjgb ohikh<ht<tq njlbik OgiMgt< -j{ OgiMgt< weh<hml</ osbz<: OgiMABujvg/OgiMAB bqz<njlbik P we<xht<tqjbwmk<kg<ogit<g/ P bqe<upqobosz<zlixogimgt<ujvg/nk<kjgbogimgtqz<yovyvogimkie< AB g<g-j{bignjlujkg<gi{zil<(hml< 6.4 Jh<hii<g<gUl<). hml< 6.4 -s<osbzqzqvf<kgqp<g<g{<mh{<hgqjmg<gqxk/ h{<h: yvogiml<?nkqz<njlbikyvht<tqbl<kvh<hce<?kvh<hm<mht<tqupqob kvh<hm<mogim<cx<g-j{bigyovyvogimkie<njlbl</ Gxqh<H; &e<xnz<zknkx<golx<hm<mogimgt<yvht<tqupqobosz<zoleqz<? ng<ogimgt<yvht<tqupqg<ogimgt<weh<hml</ 40

145 osbz<; AB, CD wel< -V ouu<oux OgiMgt< ye<jxobie<x oum<mlix ujvf<k? oum<ml< Ht<tqjb O weg< Gxqg<gUl< (hml< 6.5Jh< hii<g<gul<*/ AOC, BOD, AOD, BOC NgqbOgi{r<gjtntg<gUl</ -kqzqvf<kgqp<g<g{<mux<jxnxqgqe<oxil</ m AOD m BOC, m AOC m BOD. -r<g AOC, BOD NgqbOgi{r<gt< yvosic Gk<okkqi< Ogi{r<gtiGl</ -u<uiox? filxquk BOCBl< AOD Bl< yvosic Gk<okkqi< Ogi{r<gtiGl</ hml< 6.5 -s<osbzqzqvf<kgqp<g<g{<mh{<hgqjmg<gqxk/ h{<h 4; -VOgiMgt< ye<jxobie<x oum<cg<ogi{<miz< d{<migl< Gk<okkqi< Ogi{r<gt<sll</ Gxqh<H: m AOC xº, m AOD yº we<g/ Olx<g{<m nch<hjmg< ogit<jgbqe<hc m BOD xº, m BOC yº. Neiz< m AOC + m BOC + m BOD + m AOD 60º nz<zk xº + yº + xº + yº 60º nz<zk xº + yº 60º nz<zk xº + yº 80º. weou xº, yº lqjgfqvh<hg<ogi{r<gt<ngl</-kqzqvf<k? m AOC + m BOC 80º, m BOD + m AOD 80º, m AOC + m AOD 80º, m BOC + m BOD 80º we nxqbzil</ Osicgt< AOC? BOC Bl<A BOD, AODBl<A AOC, AOD Bl<A BOC? BOD Bl<yu<ouie<Xl<lqjgfqvh<Hg<Ogi{r<gjtnjlg<gqe<xe/ osbz<: AB, CDwe<x-j{OgiMgt<ujvg/AB g<g-j{big-z<zilz<ogimpq ujvg (hml< 6.6Jh<hii<g<gUl<). hml< 6.6 PQ NeK?-j{OgiMgt< AB jbbl< CD jbbl<oum<mujknxqg/pq NeKAB, CD -ux<xqe< yv GXg<Goum<cbiGl< we<hjk nxqg/ AB, PQ uq<x<g njlbl< ohikh<ht<tqjb L weul<? CD, PQ uqx<g njlbl< ohikh<ht<tqjb M weul< Gxqg<g/ PLB, PLA, BLM, ALM, LMD, LMC, CMQ, QMD Ngqbux<jx ntf<khiig<gul</filxquk? m PLB m LMD, m BLM m DMQ, m PLA m LMC, m ALM m CMQ. -s<osbzqzqvf<khqe<uvl<h{<hgqjmg<gqxk/ 4

146 h{<h 5: -V-j{OgiMgTg<GyVOgiMGXg<Goum<cbignjlf<kiz<?nr<gjlBl< wf<kouivosicyk<kogi{r<gtl<ntuqz<sllig-vg<gl</ Gxqh<H: (i) Ogi{l< ALM l< LMD Bl< yv Osic ye<xuqm<m dm<ogi{r<gjt njlg<gqe<xe/ogi{l< BLM l< LMC Bl<lx<oxiVOsicye<Xuqm<mdm<Ogi{r<gt< NGl</Olz<njlf<kosbzqe<hc? m BLM m LMC, m ALM m LMD wenxqbzil</-kjek<okx<xligg<%xouil</ Okx<xl< : -V -j{ogimgtg<g yv OgiM GXg<Goum<cbig njlf<kiz<? nr<gjlbl<wf<kouivosicye<xuqm<mdm<ogi{r<gtl<ntuqz<sll</ (ii) Ogi{l< PLA Ul< DMQ Ul< yvosic ye<xuqm<m outqg<ogi{r<gjt njlg<gqe<xe/ Ogi{l< BLP Bl< CMQ Ul< lx<oxiv Osic ye<xuqm<m outqg<ogi{r<gt<ngl</olz<njlf<kosbzqe<hc? m PLA m DMQ, m BLP m CMQ wenxqgqe<oxil</-kjek<okx<xligg<%xouil</ Okx<xl< : -V -j{ogimgtg<g yv OgiM GXg<Goum<cbig njlf<kiz<? nr<gjlbl<wf<kouivosicye<xuqm<moutqg<ogi{r<gtl<ntuqz<sll</ (iii) Ogi{r<gt< BLM l< LMD Bl< GXg<Goum<cbqe< yovhxl< njlf<k yvosic dm<ogi{r<gjt njlg<gqe<xe/ -jkh<ohizou ALM l< LMC Bl< GXg<Goum<cbqe< yovhxl< njlf<k lx<oxivosic dm<ogi{r<gt< NGl</ Olz< njlf<kosbzqe<hc? m BLM + m LMD 80, m ALM + m LMC 80 wenxqbzil</-f<kd{<jljbk<okx<xligg<%xzil</ Okx<xl< : -V -j{ogimgtg<g yv OgiM GXg<Goum<cbig njlf<kiz<? yov hg<gk<kqz<njlf<kyvosicdm<ogi{r<gt<lqjgfqvh<hg<ogi{r<gtigl</ osbz<: yvogimab ujvg/ogim ABg<G-j{bignjlbikOgiMPQ ujvg/ OgiMgt< AB Bl< PQ Ul< sf<kqg<gl< Ht<tqjb L weg<gxqg<gul</lnz<ziklx<oxivht<tqm J OgiM PQe<lQKGxqg<gUl</ ALM Jntg<gUl</ M -e< upqob m ALM m LMD we njlblix OgiM CD ujvg (hml< 6.7Jh< hii<g<gul<). -kqzqvf<k OgiM PQ we<hk OgiMgt< AB, CD -ux<xqx<g GXg<Goum<c weul<? AB Bl< CDBl<ye<jxobie<Xsf<kqg<gikOgiMgt<weUl< hml< 6.7 nxqblcgqe<xk/ -s<osbzqzqvf<khqe<uvl<h{<hgqjmg<gqxk/ h{<h 6: yvgxg<goum<c-vogimgjtoum<ml<ohipkwx<hmk<kl<yvosic ye<xuqm<mogi{r<gt<sllig-vh<hqe<?nu<uqvogimgt<-j{ogimgtigl</ 4

147 Olx<g{<mh{<hqzqVf<Khqe<uVl<Okx<xr<gjtfq'hqg<gzil</ Okx<xl< 4: yvgxg<goum<c-vogimgjtoum<ml<ohipkwx<hmk<kl<yvosicyk<k Ogi{r<gt<sllig-Vh<hqe<?nu<uqVOgiMgt<-j{OgiMgtiGl</ Okx<xl< 5: yvgxg<goum<c-vogimgjtoum<ml<ohipkwx<hmk<kl<ogi{r<gtqz<? GXg<Goum<cbqe< yovhxl< njlf<k yv Osic dm<ogi{r<gt< lqjgfqvh<hg< Ogi{r<gtignjlf<kiz<?nu<uqVOgiMgt<-j{OgiMgtiGl</ wmk<kg<gim<m : hml< 6.8-z<OgiM l Ul<OgiM l Ul<-j{OgiMgtiGl</OgiM l NeK OgiMgt< l, l gtg<g GXg<Goum<cbiGl</, Ngqb Ogi{r<gtqe< ntugtqe< uqgqkl< 4 : 5 weqz<?,,, 4, 5, 6, 7, 8 Ngqb Ogi{r<gtqe<ntUgjtg<gi{<g/ hml< 6.8 m 4 4 kqi<u: m : m 4 : 5 m m. Neiz<?, wel< m 5 5 lqjgfqvh<hg<ogi{r<gtqe<osicwe<hkiz<?m + m m + 5 m m + m 80 (n.k* (n.k* 9 m 80 5 (n.k* m 00. m 80 m h<ohipk? m m 80 (, we<heyvosicgk<okkqi<ogi{r<gt<), m 4 m 00 (, 4 we<heyvosicgk<okkqi<ogi{r<gt<), m 5 m 80 (, 5 we<heyvosicyk<kogi{r<gt<), m 6 m 00 (, 6 we<heyvosicyk<kogi{r<gt<), m 7 m 5 80 ( 5, 7 we<heyvosicgk<okkqi<ogi{r<gt<), m 8 m 6 00 ( 6, 8 we<heyvosicgk<okkqi<ogi{r<gt<). 4

148 wmk<kg<gim<m : hml< 6.9 -z<ab NeK CD g<g-j{ogimweul<? AC NeK BD g<g -j{ogimweul<fq'hq/ hml< 6.9 kqi<u: OgiM CD NeK OgiM AC g<gl< BD g<gl< GXg<Goum<cbiGl</ -r<g? m ACD + m CDB weou? GXg<Goum<cbqe< yvhxl< njlf<k dm<ogi{r<gtie ACD Bl< CDB Bl<lqjgfqvh<Hg<Ogi{r<gtiGl</weOu?AC Bl< BD Bl< -j{ OgiMgt</ weou? yk<k Ogi{r<gt< XCA Bl< CDB Bl< sll</ weou?m XCA 0. nmk<kkig?ab g<gl< CD g<gl<?ac GXg<Goum<cbiGl</ -r<g? m XCA m CAB 0. nkiuk ye<xuqm<m Ogi{r<gt< XCA Ul< CAB Bl< sll</weou? AB Bl< CD Bl<-j{OgiMgtiGl<. 6.. Lg<Ogi{k<jkh<hx<xqbh{<HgTl<Okx<xr<gTl< Lg<Ogi{k<jkh< hx<xq Wx<geOu fil< nxqf<kt<otil</ Lg<Ogi{l< we<hk&e<xhg<gr<gtizieyvucuqbz<dvuligl</hml< 6.0-z< DE, XY, PQ &e<xogimgt<. OgiMgt< PQ Ul< XYBl< Ht<tq A -z< oum<cg< ogit<gqe<xe/ OgiMgt< DE Bl< XY Bl< Ht<tq B -z< oum<cg< ogit<gqe<xe/ OgiMgt< PQ Ul< DE Bl< Ht<tq C -z< oum<cg< ogit<gqe<xe/ AB, BC, CA NgqbOgim<Mk<K{<Mgtiz<njlBl<ucul< Lg<Ogi{l< ABC weh<hml</ OfIg<Ogim<Mk< K{<Mgt< AB, BC, CA Lg<Ogi{k<kqe< hg<gr<gt< weh<hml</ Ht<tqgt< A, B, C Lg<Ogi{k<kqe< ds<sqgt< weh<hml</ hml< 6.0 Ogi{r<gt< BAC, ABC, BCA Lg<Ogi{k<kqe<dm<Ogi{r<gt<weh<hMl</-g<Ogi{r<gt<wtqkig A, B, C weg< Gxqg<gh<hMl</Ogi{l< PAB Lg<Ogi{k<kqe<outqg<Ogi{liGl</OlZl<Ogi{r<gt< DBA, YBC, BCQ, ECA, CAX we<helg<ogi{l<abcbqe<outqg<ogi{r<gt< wenxqf<kogit<ouil</ Le<uGh<hqz<hck<kux<xqzqVf<Khqe<uVl<ujvbjxgjtfqjeU%i<Ouil<; (i) yvlg<ogi{k<kqz<wf<kuqvhg<gr<gtl<sll<-z<jzweqz<?nl<lg<ogi{l<yv nslhg<glg<ogi{ligl</ (ii) yv Lg<Ogi{k<kqz< WkiuK -V hg<gr<gt< sll< weqz<? nl<lg<ogi{l< YI -Vslhg<gLg<Ogi{liGl</ (iii) yvlg<ogi{k<kqz<wz<zihg<gr<gtl<sll<weqz<?nl<lg<ogi{l<yvslhg<g Lg<Ogi{liGl</ 44

149 (iv) yv Lg<Ogi{k<kqz< &e<x Ogi{r<gTl< GXr<Ogi{r<gtibqe<? nl<lg<ogi{l< yvgxr<ogi{lg<ogi{ligl</ (v) yvlg<ogi{k<kqz< WkiuKyVOgi{l<uqiqOgi{l<weqz<?nl<Lg<Ogi{l<yV uqiqogi{lg<ogi{ligl</ (vi) yv Lg<Ogi{k<kqz< yv Ogi{l< osr<ogi{libqe< (Ogi{k<kqe< ntu 90 )? nl<lg<ogi{l<yvosr<ogi{lg<ogi{ligl</ Lg<Ogi{k<jkh<hx<xqbhqe<uVl<Okx<xr<gjtg<%XOuil</ Okx<xl< 6: yvlg<ogi{k<kqe<&e<xogi{r<gtqe<ntugtqe<%mkz<80 NGl</ Okx<xl< 7: yv Lg<Ogi{k<kqe< yv hg<gl< fqm<mh<hm<m< d{<migl< outqg<ogi{k<kqe< ntudt<otkqi<ogi{r<gtqe<ntugtqe<%mkzg<gs<slligl</ fq'h{l<: ABC yvlg<ogi{l</hg<gl< BC jb fqm<mg/ BC e< fqm<sqbqz< X we<x Ht<tqjbg< Gxqg<gUl</ hml< 6. -z< ACX outqg< Ogi{liGl<? A Ul< B Bl< dt<otkqi< Ogi{r<gtiGl</ m ACX m A + m B we fq'hqk<kz< Ou{<Ml</ ACX Ul< C Bl< lqjgfqvh<hg<ogi{r<gtigl</ m ACX + m C 80. hml< 6. Neiz<? m A + m B + m C 80. m ACX + m C m A + m B + m C. -Vhg<gLl<ohiK m C jbfqg<gl< osb<bm ACX m A + m B NGl</ osbz<:hml<6. -z< dt<thc Lg<Ogi{l< PQR ujvg/ hg<gr<gt< PQ, QR, RP Ngqbux<xqe< fqtr<gtigqb PQ, QR, RP J ntf<k hii<k<k PQ + QR, QR + PR, PR + PQNgqblkqh<Hgjtg<g{<Mhqcg<gUl</-kqzqVf<K? (i) PQ + QR > PR, (ii) QR + PR > PQ, (iii) PR + PQ > QR weg<g{<mxqbzil</ -s<osbzqzqvf<khqe<uvl<h{<hgqjmg<gqxk/ h{<h 7: yv Lg<Ogi{k<kqe< -V hg<gr<gtqe< fqtr<gtqe< %Mkz<&e<xiuKhg<gk<kqe<fQtk<jkuqmg<%MkziGl</ hml< 6. Gxqh<H: wf<kouivlg<ogi{l< ABC bqzl<? (i) AB < BC + CA (ii) BC < CA + AB (iii) CA < AB + BC. -julg<ogi{sleqe<jlgt<weh<hml</ osbz<: hml< 6. -z< dt<tuix Lg<Ogi{l< ABC ujvg/ Ogi{r<gt< A, B, C Ngqbux<jx ntg<gul</olzl<?ab, BC, CANgqbhg<gtUgjtBl< hml< 6. 45

150 g{<mhqcg<gul</ Ogi{r<gtqe< ntugjt yh<hqm<mh< hii<k<k nkqz< lqgh< ohiqb Ogi{l< wk we<hkjeg< g{<mxqbul</ -jkh< Ohie<Ox hg<gr<gtqe< ntugjt yh<hqm<mh<hii<k<k nkqz< lqgh<ohiqb hg<gl< wk we<hkjeg< g{<mxqbul</-kqzqvf<k ohiqb Ogi{k<kqx<G wkqovbt<t hg<gl< ohiqbk we nxqbzil</ Ogi{ ntu nkqglig dt<tjk? ohiqbogi{l< weul<? hg<g ntu nkqglig dt<tjk? ohiqb hg<gl<weul<ogit<tzil</-f<kd{<jljbh<hqe<uvl<h{<higwpkzil</ h{<h8: yvlg<ogi{k<kqz<?lqgh<ohiqbhg<gk<kqx<gwkqiqzt<togi{l<lqgh<ohiqbk/ Gxqh<H: l we<x Ogim<jmg< gvkg/ hml< 6.4-z< dt<tuix l e<lqknjlbik P we<x Ht<tqjb wmk<kg<ogit<g/ P bqzqvf<k OgiM l g<g osr<gk<kg<ogim PL ujvg/ L nz<zik OuoxiV Ht<tqM Jle<lQKwMk<Kg<ogit<g/hg<gl<PM J-j{g<gUl</Lg<Ogi{l< PLM yv osr<ogi{ hml< 6.4 Lg<Ogi{liGl</Woeeqz<hg<gl< PL,l g<gosr<gk<kigl</ weou? m PLM 90. m PLM + m LMP + m LPM 80 we<hkiz<? 90 + m LMP + m LPM 80 nz<zk m LMP + m LPM 90 m LMP < 90, m LPM < 90, m LMP 90 m LPM. nkiuk? Ogi{l< LMP Bl< LPM Bl< GXr<Ogi{r<gt< lx<xl< fqvh<hg<ogi{r<gt</ weou? PLM NeK Lg<Ogi{l< PLM-z<lqgh<ohiqbOgi{l</weOuhg<gl< PM lqgh< ohiqb hg<gl<a nkiuk? PL < PM, M we<hk Ogim<ce< lqkt<t WkiuK yv Ht<tq/weOuPbqzqVf<Kujvbh<hMl<osr<Gk<Kg<OgiM PL, P bqzqvf<kogimlg<g ujvbh<hml< lx<x Ogim<Mk<K{<Mgjt uqm lqgg< Gjxuie fqtljmbkigl</ -r<g? LM < PM. lqgh<ohiqb hg<gligqb PMJ?osr<Ogi{Lg<Ogi{l<PLM e< gi{l< we<gqoxil</ 6..4 si<usllg<ogi{r<gt< ucuqbjzh< hx<xq nxqbl<ohipk? si<usl Lg<Ogi{r<gjt ncg<gc gvk Ou{<cBt<tK/weOunux<xqe<sqzLg<gqbd{<jlgjth<hx<xqokiqf<Kogit<Ouil</ osbz<: -V oux<xk< kit<gjt wmk<kg<ogit<tul</ nux<xqx<gqjmob yv jlbs<sk< kit< ye<xqjeh< ohivk<kul</ yv kitqe< lqk yv Lg<Ogi{l< ujvbul</ nmk<k kitqz< njkh<ohie<x yk<k yv Lg<Ogi{l< njlf<kqvh<hjkg< gi{zil< )jlbs<sh< hcul<*/ -u<uqv Lg<Ogi{r<gjtBl< si<usllg<ogi{r<gt< weg< %xzil</ -u<uqv Lg<Ogi{r<gtqe<slhg<gr<gjtBl<?slOgi{r<gjtBl<ohiVk<kzil</ohiVk<kh<hm<m hm<m slhg<gr<gt< yk<k hg<gr<gt< weul<? slogi{r<gt< yk<k Ogi{r<gt< weul< %xh<hml</ yv Lg<Ogi{k<kqZt<t njek<k hg<gr<gtl<? Ogi{r<gTl< lx<oxiv 46

151 Lg<Ogi{k<kqe< yk<k hg<gr<gtg<gl<? Ogi{r<gTg<Gl< sllieiz<? nl<lg<ogi{r<gjtsi<usllg<ogi{r<gt<we<xjph<ohil</ hml< 6.5 wmk<kg<gim<mig? hml< 6.5 z< dt<t Lg<Ogi{r<gt< ABC, DPX Jg< gvkg/ -kqz< AB DX, BC PX, CA PD; B X, C P, A D we -Vh<hkiz< Lg<Ogi{l< ABC Bl< Lg<Ogi{l< DXP Bl< si<usl Lg<Ogi{r<gtiGl</ -kje ABC DXPwewPkzil</ si<usl Lg<Ogi{r<gjth< ohixk<k NX yk<k sle<himgt< dt<tk we<hjk nxqf<k ogit<g/ lxkjzbig? NX yk<k sle<himgt< ogimg<gh<hm<cvf<kiz< Lg<Ogi{r<gt< si<uslligl</ hqe<uvl< &e<x okigh<hgtqz< WOkEl< ye<x ogimg<gh<hm<cvf<kiz< yv Lg<Ogi{l< ujvbzil< we Lf<jkb ugh<hgtqz< nxqf<kt<otil</ (i) -V hg<gr<gtqe< ntugt<? nux<xqx<gqjmob dt<t Ogi{k<kqe< ntu )h.ogi.hogit<jg*/ (ii) &e<xhg<gr<gtqe<ntugt<)h.h.hogit<jg*/ (iii) -VOgi{ntUgt<?yVhg<gk<kqe<ntU)Ogi.h.Ogiogit<jg*/ gqp<g<gi[l<ohivk<kr<gtqz<wokel<ye<xqje-vlg<ogi{r<gtg<gfqi{bqk<kiz<? nl<lg<ogi{r<gjtsi<usll<weg<%xzil</ (i) hg<gl<.ogi{l<.hg<gl< (ii) Ogi{l<.hg<gl<.Ogi{l< (iii) hg<gl<.hg<gl<.hg<gl< osbz<: AB PQ, A P, AC PR we-vg<glix-vlg<ogi{r<gt<abc, PQR ujvbul</ nux<jxg< gk<kiqk<k wmk<knux<jxye<xe<lqkye<xigh< ohivk<klbx<sqg<gul<(hml< 6.6 Jh< hii<g<gul<). -l<lbx<sqbqz<? ds<sq P, ds<sq A e<lqkl<?ds<sq Q, ds<sq B e< lqkl<?ds<sq R, ds<sq Ce<lQKl<?AB, AC Ngqb hg<gr<gtiz< dt<tmg<gqb Ogi{l< A NeK PQ, PR Ngqb hml<

152 hg<gr<gtiz< dt<tmg<gqb Ogi{l< P BmEl< lqgs< siqbigh< ohivf<kujk dx<xofig<gzil</-kqzqvf<khqe<uvl<h{<hqjenxqgqoxil</ h{<h 9: yvlg<ogi{k<kqe<-vhg<gr<gtl<njudt<tmg<gqbogi{ll<?lx<oxiv Lg<Ogi{k<kqe< -V hg<gr<gtg<gl< nju dt<tmg<gqb Ogi{k<kqx<Gl< sllieiz<? nu<uqvlg<ogi{r<gtl<si<uslligl</ Gxqh<H: -f<kh{<hqjesi<usllg<ogi{k<kqe<hg<gl<.ogi{l<.hg<gl<nch<hjmk< kk<kul<nz<zksvg<gligh.ogi.hnch<hjmk<kk<kul<weg<%xzil</ osbz<: B Q, BC QR, C R we-vg<glix-vlg<ogi{r<gt<abc, PQR ujvbul</ nux<jxg< gk<kiqk<okmk<k ye<xe<lqokie<xig ohivk<kqh<hiig<jgbqz<? njubqv{<ml<siqbigh<ohivf<kgqe<xe(hml<6.7 Jh<hii<g<gUl<). hml< 6.7 -s<osbzqzqvf<kgqjmg<gl<nch<hjmk<kk<kuk<kqjeyvokx<xligg<%xgqe<oxil</ Okx<xl< 8: yv Lg<Ogi{k<kqe< -V Ogi{r<gTl< nux<xiz< -j{f<k hg<gll< lx<oxiv Lg<Ogi{k<kqe< -V Ogi{r<gTg<Gl< nux<xiz< -j{f<k hg<gk<kqx<gl< sllieiz<?nf<k-vlg<ogi{r<gtl<si<usllg<ogi{r<gtigl</ Gxqh<H: -f<kh{<hqjesi<usllg<ogi{k<kqe<ogi{l<.hg<gl<.ogi{l<nch<hjmk< kk<kul<nz<zkogi.h.oginch<hjmk<kk<kul<weg<%xzil</ osbz<: A D, B E, BC EF we-vg<glix-vlg<ogi{r<gt<abc, DEF ujvbul< (hml<6.8 Jh<hii<g<gUl<). hml< 6.8 ABC yvlg<ogi{l</weou A + B + C 80. () 48

153 DEF yvlg<ogi{l</weou D + E + F 80 Neiz<? D A, B E A + B + F 80 () (), ()e<hc? A + B + C A + B + F. C F. weoulg<ogi{l< ABC, DEF z< B, C Bl<nkEjmbhg<gl< BC Bl<? E, F x<gl< nkx<giqb hg<gl< EF x<gl< sllig -Vh<hkiz< Ogi.h.Ogi h{<hqe<hc ABC DEF. -kqzqvf<kgqp<g<g{<mokx<xk<jknxqbzil</ Okx<xl< 9: -V Lg<Ogi{r<gtqz< ye<xqe< -V Ogi{r<gTl<? nke< WkiuK yv hg<gll<? lx<oxiv Lg<Ogi{k<kqe< -V Ogi{r<gTg<Gl<? yk<k hg<gk<kqx<gl< sloleqz<?nf<k-vlg<ogi{r<gtl<si<uslligl</ Gxqh<H: si<usll< hx<xqb -f<k kk<kuk<kqje Ogi{l<. Ogi{l<. hg<gl< nch<hjmk< kk<kul<nz<zksvg<gligogi.ogi.hnch<hjmk<kk<kul<weg<%xzil</ osbz<: BC EF, CA FD, AB DE we -Vg<GliX Lg<Ogi{r<gt< ABC, DEF Ngqbux<jx ujvbul< (hml< 6.9Jh< hii<g<gul<). -kqz< Lg<Ogi{l< DEF J oum<cobmk<klg<ogi{l<abc e<lqkjug<g?njubqv{<ml<lqgs<siqbigh< hml< 6.9 ohivf<kgqe<xe/ -f<fqjzbqz<? D, A bqe< lqkl<? E, B bqe< lqkl<? F, C bqe< lqkl< njlujknxqbzil</weou ABC DEF. -s<osbzqzqvf<khqe<uvl<okx<xk<jk nxqbzil</ Okx<xl< 0: yv Lg<Ogi{k<kqe< &e<x hg<gr<gt< lx<oxiv Lg<Ogi{k<kqe< &e<x hg<gr<gtg<gs<sloleqz<?nu<uqvlg<ogi{r<gtl<si<usll<ngl</ Gxqh<H: Olx<%xqb kk<kul< hg<gl<. hg<gl<. hg<gl< nch<hjmk< kk<kul< weg< %xh<hml<asvg<glig?-kjeh.h.hnch<hjmk<kk<kul<weg<%xzil</ h.h.ogi kk<kul< nz<zk Ogi.Ogi.Ogi ogit<jg si<usl Lg<Ogi{r<gTg<Gh< ohivf<kliwe-h<ohipkosikqh<ohil</ osbz<: ABC we<xlg<ogi{l<ujvg/bc g<g-j{bigogimxy ujvg (hml< 6.0Jh< hii<g<gul<). XY Bl< AB Bl<sf<kqg<Gl<Ht<tqjbD weg<gxqg<gul<axy Bl< AC Bl< 49

154 sf<kqg<gl<ht<tqjbe wegxqg<gul</hg<gl<ab Bl<?hg<gl<AC Bl<-j{OgiMgtie BC, XYgTg<GGXg<Goum<cgtiGl</weOu D B, E C (yk<kogi{r<gt<). hml< 6.0 -r<g? Lg<Ogi{r<gt< ABC, ADE gtg<g Ogi.Ogi.Ogi ohivk<kl< dt<tk/ Neiz<? nu<uqvlg<ogi{r<gtl<si<usllg<ogi{r<gt<nz<z/woeeqz<?yk<khg<gr<gt< sllz<z/ weou? Ogi.Ogi.Ogi ohivk<kliek Lg<Ogi{r<gt< siusllibqvh<hkx<g nch<hjmk<kk<kulibqvg<gikwenxqgqoxil</ osbz<: OhiKlie ntuqx<g fqtlie yv OgiM AX ujvg/ a nzggt< fqtljmb OfIg<Ogim<Mk<K{<MAB J?Ogi{l< BAX yvgxqh<hqm<mntu-vg<glixujvf<k ogit<g/ B jb jlbligul<? b (< a) nzg NvligUl< djmb um<ml< ujvg/ nu<um<ml<ax jb-vht<tqgtqz<oum<mujkfil<gi{<gqoxil</-h<ht<tqgjtc, D weg<gxqg<gul< (hml<6. Jh<hii<g<gUl<). hml< 6. Lg<Ogi{r<gt< ABC? ABD gtg<g h.h.ogi ohivk<kl< dt<tk/ Neiz< AC AD. weou ABC NeK ABD g<gs< siusllz<z/ -s<osbzqzqvf<k h.h.ogi ohivk<kl<? Lg<Ogi{r<gt< si<usllig -Vh<hkx<G nch<hjmk< kk<kulig njlbik we nxqbzil</ osbz<: BC QR a nzggt<-vg<glixogim<mk{<mgt<bc, QR ujvg/ BC bqe< jlbh<ht<tqjb X weul<? QR -e< jlbh<ht<tqjb Y weul< Gxqg<gUl</ X, Y -ux<jx jlbbr<gtigul<? a nzggt< NvligUl< ogi{<m um<mr<gt< ujvg/ B, Q -ux<jx jlbr<gtiigul<?b(< a) nzggt< NvligUl<ogi{<Muqx<gt<ujvg/-f<kuqx<gt< 50

155 Wx<geOu ujvf<k um<mr<gjt oum<ml< Ht<tqgjt LjxOb A, P weg< Gxqg<gUl< (hml< 6. Jh<hii<g<gUl<). AB, AC, PQ, PR jb-j{g<gul</ AC, PR fqtr<gjt hml< 6. ntg<gul</ AC PR we<hjk nxqbzil</ weou ABC PQR. -s<osbzqzqvf<k gqp<g<gi[l<okx<xk<kqjeg<gi{<gqoxil</ Okx<xl< : yv osr<ogi{ Lg<Ogi{k<kqe< gi<{l<? WOkEl< yv hg<gl<? lx<oxiv osr<ogi{ Lg<Ogi{k<kqe< gi{l<? yv hg<gl< Ngqbux<xqx<G LjxOb sllig -Vh<hqe<?nu<uqVLg<Ogi{r<gTl<si<uslliGl</ Gxqh<H: -k<okx<xl< kvl< nch<hjmk< kk<kuk<kqje osr<ogi{ Lg<Ogi{r<gtqe< siu slk<kuk<kqx<gie osr<ogi{l<. gi<{l<. hg<gl< nch<hjmk< kk<kul< nz<zk SVg<gligos.g.hnch<hjmk<kk<Kul<wewPkh<hMgqxK/ j{gvk<kqe<h{<hgt< fie<g hg<gr<gtiz< njmhml< dvul< fix<gvl< we<hkje nxqouil</ wkqi<hg<gr<gt<-j{bigdt<tfix<gvl<-j{gvl<weh<hml</ hml< 6.4 hml<6. hml< 6. -z< ABCD yv fix<gvl</ hml< z< PQRS yv -j{gvligl</ -h<hmk<kqz< PQ SR (PQ NeK SR g<g -j{bigl< we<hjk -u<uix Gxqh<Ohil<); OlZl< PS QR. h{<h : YI-j{gvk<kqz<?wkqi<hg<gr<gt<slfQtLjmbju/ fq'h{l<: ABCD YI -j{gvl< we<g/ BD jb -j{g<gul< (hml< 6.5Jh< hii<g<gul<). ABD, BDC Ngqbux<jxg<gVKg/-r<GAB CD, BD NeKAB, CDNgqbux<xqx<GGXg<Goum<cbigdt<tK/weOu? m ABD m BDC. -u<uiox? AD BC, BD NeK AD, BC Ngqbux<xqx<GGXg<Goum<cwe<hkiz<? hml< 6.5 5

156 m ADB m DBC. hg<gl< BD, Lg<Ogi{r<gt< ABD, BDCg<G ohikuig dt<tk/ weou? Ogi.Ogi.h< ogit<jgbqe<hc ABD BDC. yk<khg<gr<gt<sll</weou? AB CD lx<xl< AD BC. h{<h : YI-j{gvk<kqz<?wkqi<Ogi{r<gt<<slntUjmbju/ fq'h{l<: ABCD YI-j{gvl<we<g/ BD jb-j{g<gul< (hml< 6.6Jh<hiIg<gUl<). ABD, BDC Ngqbux<Xt< AB DC, BD NeK AB, DC Ngqbux<Xg<G yv GXg<Goum<cwe<hkiz<?m ABD m BDC. OlZl< AD BC, BD NeK AD, BC Ngqbux<Xg<G yv GXg<Goum<cwe<hkiz<?m ADB m CBD. m ABC m ABD + m DBC m BDC + m ADB m ADC. hml< 6.6 -u<uiox m BAD m BCD wenxqbzil</ h{<h : -j{gvk<kqe<&jzuqm<mr<gt<ye<jxobie<xslg<%xqml</ fq'h{l<: ABCD YI -j{gvl</ AC, BD we<he &jzuqm<mr<gt</ -ju oum<cg< ogit<tl< Ht<tqjb M we<g/-r<g? BAM DCM, ABM CDM, AB CD. weou? Ogi.h.Ogi ogit<jgbqe< hc? AMB CMD (hml< 6.7Jh<hii<g<gUl<). AM CM, BM DM. weou&jzuqm<mr<gt<slg<%xqml</ hml< 6.7 h{<h 4: yv fix<gvk<kqe< wkqi<hg<gr<gt< sl ntu ogi{<cvf<kiz< nf<k fix<gvl< -j{gvligl</ fq'h{l<: ABCD we<xfix<gvk<kqz< AB CD, AD BC. AC jb-j{g<gul</ Lg<Ogi{r<gt<ACB, ADC Ngqbux<Xt<h.h.hogit<jgbqe<hc ABC CDA (hml< 6.8Jh<hii<g<gUl<). hqe<h? m BAC m ACD, m CAD m ACB. AB CD ; OlZl< AD BC. weouabcd YI-j{gvl</ hml< 6.8 h{<h 5: yv fix<gvk<kqe< wkqi<ogi{r<gt< sl ntu ogi{<cvf<kiz<? nf<k fix<gvl< -j{gvligl</ fq'h{l<: ABCD we<el< fix<gvk<kqz< (hml< 6.9 Jh< hii<g<gul<). m BAD m BCD, m ABC m ADC. BD jb -j{g<gul</ ABD, CDB Ngqb Lg<Ogi{r<gjtg<gVKg/-r<G? m + m + m BCD 80 ; m + m 4 + m BAD 80. m + m + m BCD m + m 4 + m BAD, hml< 6.9 Woeeqz<?m BCD m BAD. 5

157 m + m m + m 4 () Neiz<? m + m m ABC, m + m 4 m ADC m + m m + m 4 nkiuk? m m m 4 m ) () + () m m 4 m m 4. AD BC. () () m m m m. AB CD. weou?abcd YI-j{gvl</ h{<h 6: yvfix<gvk<kqe<&jzuqm<mr<gt<ye<jxobie<xslg<%xqmoleqz<?nf<fix<gvl< YI-j{gvliGl</ fq'h{l<: ABCD yvfix<gvl</ AC, BD &jzuqm<mr<gt</ AC Bl< BD Bl<Ht<tq M -z<oum<cg<ogit<gqe<xe (hml< 6.40Jh<hii<g<gUl<). M NeKAC, BD gtqe< fmh<ht<tq/ AM CM, BM DM. Gk<okkqIOgi{r<gt<sll<we<hkiz<? m AMB m CMD, m AMD m BMC. hml< 6.40 h.ogi.hogit<jgbqe<hc? AMB CMD, AMD CMB. weou BAM DCM, ABM CDM () AB, CD gtg<gac yvgxg<goum<c/() e<hc?ye<xuqm<mdm<ogi{r<gt<sll</ AB CD. -jkh<ohizou? AD BC. weou ABCD YI-j{gvl</ Okx<xl< : yv Osic wkqi<hg<gr<gt< -j{bigul<? slligul< dt<t fix<gvl< YI -j{gvligl</ ogit<jg: ABCD we<xfix<gvk<kqz< AB CD, AB CD. fq'hqg<g: ABCD YI-j{gvl</ ujvkz<: ACjb-j{g<gUl<(hml< 6.4Jh<hii<g<gUl<). fq'h{l<: ABC, ADC Ngqbux<Xt<? (i) AB CD (ogit<jg)a (ii) AC ohika hml< 6.4 (iii) AB CD, AB g<gl< CD g<gl< AC GXg<Goum<c m BAC m ACD)ye<Xuqm<mOgi{r<gt<slliGl<*/ h.ogi.hogit<jgbqe<hc? ABC ADC. Lg<Ogi{r<gtqz<yk<khg<gr<gt<? yk<kogi{r<gt<sll</ AD BC, m DAC m ACB. AD BC. weou ABCD YI -j{gvl</ h{<h 7: &e<xl<? &e<xqx<gl< Olx<hm<m -j{ogimgtiz< yv GXg<Goum<cbqe< lqokx<hmk<kh<hml< oum<mk<k{<mgt< slfqtljmbeoueqz<? OuX wf<k GXg<Goum<cbqe< lqkl< nf<k -j{ OgiMgt< d{<migl< oum<mk<k{<mgt< slfqtljmbeuigl</ fq'h{l<: OgiMgt< l, l, l we<he ye<xg<ogie<x-j{ogimgtigl</ PQ, XY Ngqbju l, l, l g<g GXg<Goum<cgtiGl</ AC CE we<g/ BD DF we fq'hqg<g Ou{<Ml</ AG BD NgUl< CH DFNgUl< ujvg (hml< 6.4 Jh<hiIg<gUl<). hml< 6.4 5

158 AGDB, CHFD Ngqbju-j{gvl<wenxqgqOxil</ AG BD, CH DF () ACG, CEHNgqbLg<Ogi{r<gtqz< CE AC, m GAC m HCE (yk<kogi{r<gt<), m ACG m CEH (yk<kogi{r<gt<). weouogi.h.ogiogit<jgbqe<hc? ACG CEH. AG CH. () (), () e<hcbd DF. h{<h 8: yv Lg<Ogi{k<kqz<? -Vhg<gr<gtqe< fmh<ht<tqgjts< Osi<g<Gl< OgiM &e<xiukhg<gk<kqx<g-j{bigul<?nke<ntuqz<hikqbigul<-vg<gl</ fq'h{l<: Lg<Ogi{l< ABC -z<? D, E LjxOb AB, AC Ngqb hg<gr<gtqe< fmh<ht<tqgtigl</fil<fq'hqg<gou{<cbk? DE BC, DE ( BC ). CF BD ujvg/ -K DE bqe< fqm<sqjb F-z<sf<kqg<Gl<(hml< 6.4Jh< hii<g<gul<). Lg<Ogi{l< ADE, CFENgqbux<Xt<?AD CF, AC GXg<Goum<c/ m DAE m ECF. AD CF, DF GXg<Goum<c/ m ADE m CFE. OlZl< E NeK AC bqe<fmh<ht<tqnekiz<? AE EC. weouogi.ogi.hogit<jgbqe<hc ADE CFE. AD CF OlZl< DE EF. Neiz< D, AB bqe<fmh<ht<tqwe<hkiz<? BD AD. BD CF. Wx<geOu BD CF. BCFD YI -j{gvligl</ DF BC lx<xl< DF BC n.k DE BC. DE + EF BC n.k DE + DE BC n.k DE BC. hml< 6.4 weou DE BC lx<xl< DE (BC). h{<h 9: yv Lg<Ogi{k<kqz<? yv hg<gk<kqe< fmh<ht<tq upqob lx<oxiv hg<gk<kqx<g -j{bigujvbh<hml<ogim?&e<xiukhg<gk<jkslg<%xqml</ fq'h{l<: ABC we<xlg<ogi{k<kqz< D, ABbqe< fmh<ht<tq we<g/ D upqbig BC g<g -j{big DE ujvg/ nk AC jb E z< oum<mm<ml< (hml< 6.44 Jh<hii<g<gUl<). E NeKAC bqe<fmh<ht<tq wefq'hqk<kz<ou{<ml<. -r<gbc DE, BC, DE gtqe<gxg<goum<c AB NGl</ m ADE m ABC, () AD CF, AD,CF gtqe<gxg<goum<cdf NGl</ m ADE m EFC lx<xl< () hml< 6.44 m AED m BCE. () (), ()e<hc m ABC m EFC. (4) AD CF, AD, CF gtqe< GXg<Goum<cAC NGl</ m BAC m ACF. (5) -h<ohipk?m BCF m BCE + m ECF 54

159 m AED + ACF, () e<hc m AED + m DAE, (5) e<hc m BDE (outqg<ogi{l< dt<otkqi<ogi{r<gtqe<%mkz<). nkiuk m BCF m BDE (6) (4), (6)e<hc? BCFD YI-j{gvliGl</ BD CF, BC DF. Lg<Ogi{r<gt< ADE, CFE Ngqbux<jxg<gVKg/ -ux<xqz<?cf BD AD we<hkiz<?cf AD. OlZl<? m EFC m ADE, m DAE m BAC m ACF m ECF. Ogi.h.Ogiogit<jgbqe<hc, ADE CFE. DE EF, AE EC. E, AC -e<fmh<ht<tqbigl</ 6..6 yvht<tqupqobosz<zl<ogimgt< &e<xnz<zk&e<xg<golx<hm<mogimgt<pwe<xyvht<tqupqobosz<zl< weqz<? ng<ogimgt< yv Ht<tqupqg< OgiMgt< weul< nh<ht<tq P J ng<ogimgt< sf<kqg<gl<ht<tqweul<njpg<gh<hml</ osbz< : Lg<Ogi{l< ABC ujvg/ hg<gl< BC, CA Ngqbux<Xg<G LjxOb jlbg< Gk<Kg<OgiMgt< DX, EY ujvg (hml< 6.45 Jh< hii<g<gul<). nju oum<ml<ht<tqjb S we<g/ hg<gl< AB bqe< jlbg<gk<kg<ogim FZ ujvg. FZ NeK S e<upqob osz<ujk nxqbzil</ -s<osbzqzqvf<k gqp<g<gi[l< Okx<xk<jknxqgqe<Oxil</ hml< 6.45 Okx<xl< : Lg<Ogi{k<kqe< hg<gr<gtqe< jlbg<gk<kogimgt< yv Ht<tq upqob osz<zl</ Gxqh<H: yv Lg<Ogi{k<kqe< &e<x hg<gr<gtqe< jlbg<gk<kg<ogimgt< osz<zl< Ht<tq nl<lg<ogi{k<kqe<sx<xum<mjlbl<weh<hml</-k Swe<xwPk<kiz<Gxqg<gh<hMl</ osbz<: ABC we<x Lg<Ogi{k<jkg< gvkg/ -kqz< Sx<Xum<m jlbl< S jbg< gi{<g/ SA, SB, SC Ngqbux<jx -j{g<gul</ SA, SB, SC Ngqbux<xqe< fqtr<gjt ntf<k hii<g<gul</filxquk?sa SB SC/ S jbjlbligul<? SA junvligul<ogi{<m yv um<ml< ujvg/ -u<um<ml<? Lg<Ogi{k<kqe< &e<x ds<sqh<ht<tqgtqe< upqob osz<ujkg< gi{zil</ nu<um<ml< Lg<Ogi{k<kqe< Sx<Xum<ml< weh<hml<a Nvl< SA (SB SC) Sx<Xum<mNvl<weh<hMl</ 55

160 osbz<: ABCwe<xLg<Ogi{l<ujvg/hg<gl<BC g<gosr<gk<kigad ujvg/nk BCjb D z< sf<kqg<gm<ml</hg<gl<ca Ug<Gosr<Gk<kigBEujvg/nKCA jb Ez< sf<kqg<gm<ml</ AD Bl< BE l< sf<kqg<gl< Ht<tqjb H weg< Gxqg<gUl</ AB e< osr<gk<k CF ujvg/ nk AB jb F z< sf<kqg<gm<ml< (hml< 6.46Jh< hii<g<gul<). CF NeK H e<upqobosz<ujk nxqbzil</ osr<gk<k OgiMgt< AD, BE, CF we<he Lg<Ogi{k<kqe hml< 6.46 osr<ogimgt<weh<hml</ -s<osbzqzqvf<kgqp<g<g{<mokx<xk<jknxqbzil</ Okx<xl< 4: yv Lg<Ogi{k<kqe<osr<OgiMgt<yVHt<tqupqObosz<gqe<xe/ Gxqh<H: yv Lg<Ogi{k<kqe< osr<ogimgt< sf<kqg<gl< Ht<tqbieK? nl<lg<ogi{k<kqe< osr<ogim<mjlbl<weh<hml</-k Hwe<xwPk<kiz<Gxqg<gh<hMl</ osbz<: Lg<Ogi{l< ABC ujvg/lg<ogi{k<kqz< Ogi{l< A, Ogi{l< BNgqbux<xqe<sloum<cgt< ujvg/ -jugt< sf<kqg<gl< Ht<tqjb I weg< Gxqg<gUl</ C e<sloum<cujvg/-kie< upqob osz<ujk nxqbzil< (hml< 6.47Jh< hii<g<gul<). -s<osbzqzqvf<kgqp<g<gi[l<okx<xk<jk hml<6.47 nxqbzil</ Okx<xl< 5: yvlg<ogi{k<kqe<?&e<xogi{r<gtqe<sloum<cgt<yvht<tqupqob osz<zl</ Gxqh<H: yv Lg<Ogi{k<kqz< &e<x Ogi{r<gtqe< sloum<cgt< sf<kqg<gl< Ht<tqbieK nl<lg<ogi{k<kqe<dt<um<mjlbl<weh<hml</-k Iwe<xwPk<kiz<Gxqg<gh<hMl</ osbz<: I -zqvf<k Lg<Ogi{l< ABC e< hg<gr<gtg<g osr<gk<kg< OgiMgt< ujvg/ nux<jx ntf<k hii<g<gul</ nju slfqtl< ogi{<meuig -Vh<hkjeg< gi{zil</ I jb jlbligul<? slntuie osr<gk<kg< Ogim<ce< fqtk<jk NvligUl< ogi{<m yv um<ml< ujvbul< (hml< 6.48Jh< hii<g<gul<). -u<um<ml< Lg<Ogi{k<kqe< &e<x hg<gr<gjtbl< okim<ms< osz<ujkg<gi{zil</-kjelg<ogi{k<kqe< hml<

161 dt<um<ml< we<xjpg<gqoxil</ -u<um<mk<kqe< Nv ntu Lg<Ogi{l< ABC -e< dt<um<mnvl<weh<hml</ osbz<: Lg<Ogi{l< ABC ujvg/ hg<gr<gt< BC, CA, AB Ngqbux<xqe<fMh<Ht<tqgjtLjxObD, E, F weg< Gxqg<gUl</ AD jbbl< BE jbbl< -j{g<gul</-juoum<cg<ogit<tl<ht<tqjb G weg< Gxqg<gUl</ CF jb ujvg/ nk G e< upqobosz<ujknxqg (hml< 6.49Jh<hii<g<gUl<). OgiMgt< AD, BE, CF Ngqbju Lg<Ogi{l< ABCbqe<fMg<OgiMgt<weh<hMl</ AG, GD, BG, GE, CG, GF Ngqbux<xqe<fQtr<gjtntf<KhiIg<gUl</filxquK? AG GD BG GE CG GF -s<osbzqzqvf<kgqp<g<gi[l<okx<xk<jknxqbzil</. hml<6.49 Okx<xl< 6: yv Lg<Ogi{k<kqe< fmg<ogimgt< yv Ht<tq upqob osz<zl<a -h<ht<tq yu<ouivfmg<ogim<cjebl< :we<xuqgqkk<kqz<hqiqg<gl</ Gxqh<H: yv Lg<Ogi{k<kqe< fmg<ogimgt< sf<kqg<gl< Ht<tqbieK nl<lg<ogi{k<kqe< fmg<ogim<mjlbl<weh<hml</-kgwe<xwpk<kiz<gxqg<gh<hml</ osbz<: AB we<x Ogim<Mk<K{<jm ujvg/ AB g<g -j{big l we<x OgiM ujvbul< (hml< 6.50Jh< hii<g<gul<). l e< lqk Ht<tqgt< C, DNgqbux<jx AB CD we -Vg<GliX Gxqg<gUl</ -kqzqvf<k ABCD YI -j{gvl< we nxqbzil</ l g<gs< hml< 6.50 osr<gk<k AL ujvg/ AL Jntf<Khii<g<gUl</ -j{gvl< ABCD e<hvh<htu nch<hg<gl< dbvl< AB AL. OuX-VHt<tqgt< P, Q Ngqbux<jx PQ ABwe-Vg<GliXle<lQKGxqg<gUl</ ABPQYI-j{gvl<wenxqbzil</ ABPQe<hvh<htUnch<hg<gl< dbvl< AB AL. -s<osbzqzqvf<kgqp<g<gi[l<okx<xk<jknxqbzil</ Okx<xl< 7: yovncbqe<lqkl<?-v-j{ogimgtg<glqjmbqz<njlbl<-j{gvr<gt< slhvh<hjmbju/ 57

162 osbz<: hml< 6.5 Jh< hiig<gul<. AB we<x Ogim<Mk<K{<jm ujvg/ AB g<g-j{big l we<x Ogim<cjeujvbUl</l e<lqkht<tqcjbg<gxqg<g/ l we<x Ogim<Mg<G A bqzqvf<k AL we<x osr<gk<kogim ujvg. AL e< fqtk<jk ntf<k hii<g<gul</ filxquk? ABC we<x Lg<Ogi{k<kqe< hvh<htu nch<hg<gl< dbvl< AB AL. hml<6.5 le<lqklx<oxivht<tqp jbg<gxqg<gul</ Lg<Ogi{l<ABP -e<hvh<htu AB AL weg<gi{<gqoxil</lg<ogi{k<kqe<ds<sq C, l e<lqkwr<gnjlf<kqvf<kizl<lg<ogi{r<gtqe<hvh<htugt<slolenxqgqoxil</ -s<osbzqzqvf<kgqp<g<gi[l<okx<xk<jknxqbzil</ Okx<xl< 8: yov ncbqe< lqkl<? -V -j{ogimgtg<gl< -jmbqz< njlbl< Lg<Ogi{r<gt<slhvh<Hjmbju/ hbqx<sq 6. hqe<uvl<%x<xgt< siqbi,kuxiweg<%xul</. -V-j{OgiMgjtyVGXg<Goum<coum<ceiz<?ye<Xuqm<mOgi{r<gt<sll</. -V-j{OgiMgjtyVGXg<Goum<coum<ceiz<?yk<kOgi{r<gt<sll</. -V-j{OgiMgjtyVGXg<Goum<coum<ceiz<?GXg<Goum<cbqe<yOvHxl< njlf<kdm<ogi{r<gt<sll</ 6. kvg<giqkqbigfq'hqg<gou{<cbokx<xr<gt< -Kujv fil< sqz osbz<gjts< osb<k hiik<k? sqz nch<hjmg< ogit<jggjtbl<? Okx<xr<gjtBl< siq hiik<okil</ -h<ohipk? Ogi{r<gt<? Lg<Ogi{r<gt< Ngqbux<jxh< hx<xqb sqz Okx<xr<gTg<G kvg<g vqkqbig? nkiuk? uqkqkvljxbqz<(method of Logical Reasoning)fq'h{r<gjtk<kvLx<hMOuil</ Okx<xl< 9: yv gkqi< lx<oxiv Ogim<ce< lqk fqx<glieiz<? nkeiz< Wx<hMk<kh<hMl< nmk<kmk<kogi{r<gtqe<ntugtqe<%mkz< 80 g<gs<sll</ kvu: gkqi< PQ, XYwe<xOgim<ce<lQKnjlf<Kt<tK/ fq'hqg<g: m QPX + m YPQ 80. njlh<h: XY g<gosr<gk<kigpe ujvg. fq'h{l<: m QPX m QPE + m EPX m QPE + 90 () m YPQ m YPE m QPE hml< m QPE () 58

163 () + () m QPX + m YPQ (m QPE + 90 ) + (90 m QPE) 80. weouokx<xl<fq'hqg<gh<hm<mk/ Okx<xl< 0: -VOgiMgt<ye<jxobie<Xoum<cg<ogi{<miz<?Gk<okkqi<Ogi{r<gt<sl ntut<tjubigl</ kvu: OgiMgt< AB, CD Ht<tq O uqz<oum<cg<ogit<gqe<xe(hml< 6.5 Jh<hii<g<gUl<). fq'hqg<g: m AOC m BOD, m BOC m AOD. fq'h{l<: gkqi< OB NeK OgiM CDbqe<lQKfqx<gqe<xK/ m BOD + m BOC 80. () gkqi< OC OgiM ABbqe<lQKfqx<gqe<xK/ m BOC + m AOC 80. () (), ()e<hc m BOD + m BOC m BOC + m AOC m BOD m AOC. gkqi< OA, CDbqe<lQKfqx<gqe<xK/ m AOC + m AOD 80. () hml<6.5 (), ()e<hc? m BOC + m AOC m AOC + m AOD. m BOC m AOD. weouokx<xl<fq'hqg<gh<hm<mk/ Okx<xl< : yvlg<ogi{k<kqz<&e<xogi{k<kqe<%mkz< 80 NGl</ kvu: ABCyVLg<Ogi{l<. fq'hqg<g: A + B + C 80. njlh<h: ds<sqa e<upqob XY we<xogim<cje? BCg<G-j{bigujvg(hml<6.54Jh<hii<g<gUl<)/ fq'h{l<: XY BC. -h<ohipk?xy, BCgTg<G ABGXg<Goum<c/ m XAB m ABC (ye<xuqm<mogi{r<gt<) B. () hml< 6.54 XY, BC gtg<g AC GXg<Goum<c/ m YAC m ACB (ye<xuqm<mogi{r<gt<) C. () OlZl< m BAC m A. () () + () + () m XAB + m YAC + m BAC m B + m C + m A (m XAB + m BAC) + m CAY m A + m B + m C m XAC + m CAY m A + m B + m C 80 m A + m B + m C. weou?okx<xl<fq'hqg<gh<hm<mk/ Okx<xl< : yv Lg<Ogi{k<kqz<? slhg<gr<gtg<g wkqovbt<t Ogi{r<gt< slntut<tju/ kvu: Lg<Ogi{l<ABC -z< AB AC. fq'hqg<g: B C. njlh<h: BC e< jlbh<ht<tqjb M weg< Gxqk<K AM J -j{g<g(hml<6.55 Jh<hii<g<gUl<)/ fq'h{l<: Lg<Ogi{l< AMB, AMC Ngqbux<Xt<? 59 hml< 6.55

164 (i) BM CM (ii) AB AC (iii) AM ohik/weouh.h.hogit<jgh<hc? AMB AMC. yk<kogi{r<gt<sll</gxqh<hig? B C. weouokx<xl<fq'hqg<gh<hm<mk/ Okx<xl< : yv Lg<Ogi{k<kqe< -V Ogi{r<gtqz<? ohiqb Ogi{k<kqx<G wkqiqz< njlbl< hg<gl< sqxqb Ogi{k<kqx<G wkqiqz< njlbl< hg<gk<jkuqm nkqg fqtljmbk/ kvu: Lg<Ogi{l< ABC -z< Ogi{l< B NeK Ogi{l< C ju uqm ohiqbk/ n.km B > m C (hml<6.56jh<hii<g<gul<). fq'hqg<g: AC we<x hg<gk<kqe< fqtl<? AB we<x hg<gk<kqe< fqtk<jk uqmh< ohiqbkigl</ nkiuk?ac > AB. fq'h{l<: AB, AC we<x hg<gr<gtqe< fqtr<gt< lqjg w{<gt< we<hkiz<? gqp<g<gi[l< hqiqugt< njlbl</ hml< 6.56 (i) AC < AB (ii) AC AB (iii) AC > AB hqiqu (i) AC < AB we<g/ hqxg? hg<gl< AB -e< fqtliek hg<gl< AC e< fqtk<jkuqmh< ohiqbkigl</ weou? AB g<g wkqovbt<t Ogi{l< C NeK AC g<g wkqovbt<t Ogi{l< B Juqmntuqz<ohiqbkignjlbOu{<Ml</nkiuK? m C > m B. -K kvuqz< dt<t d{<jl m B > m C g<g Lv{<himiekiGl</ weou? wmogit< AC < AB NeKolb<bz<z. hqiqu (ii) AC AB wenjlblieiz<?-vhg<gr<gt<ab Bl< AC Bl<slliGl</Neiz<? slhg<gr<gtg<g wkqovbt<t Ogi{r<gt< sll< we nxqouil</ nkiuk? B C. -KUl<kvuqz<dt<td{<jl B > Cg<GLv{<himiekiGl</weOuAC AB NeKolb<bz<z. weou?lqklt<thqiqu(iii) AC > ABwe<hOkolb<biGl</ Okx<xl< 4: &jzuqm<mr<gt< ye<xg<ogie<x osr<gk<kig dt<t YI -j{gvl< yv sib<skvligl</ kvu: ABCD we<hk YI -j{gvl</ -kqz< &jzuqm<mr<gt< AC Bl< BD Bl< ye<xg<ogie<xosr<gk<kig-vg<gqe<xe/ fq'hqg<g: ABCDyVsib<sKvl</ njlh<h: &jzuqm<mr<gt< AC, BDujvf<Knjuoum<Ml<Ht<tqjbM weg<gxqg<gul< (hml<6.57jh<hii<g<gul<). fq'h{l<: Lg<Ogi{r<gt< AMB, BMCNgqbux<xqz<? (i) AMB BMC 90 (ii) AM MC (iii) BM ohik/ weou?ogit<jgh.h.ogiuqe<hc AMB BMC. weou?yk<khg<gr<gt<sll</gxqh<higab BC. ABCDYI-j{gvl<we<hkiz<< AB CD, BC AD. AB BC CD AD. weou ABCD sib<skvligl</okx<xl<fq'hqg<gh<hm<mk/ hml<

165 wmk<kg<gim<m : gqp<g<gi[l<ogi{l<yu<ouie<xqe<fqvh<hg<ogi{l<gi{<g; (i) 0 (ii) 45 (iii) 55 (iv) 8 kqi<u: fqvh<hg<ogi{r<gtqe<%mkz< 90. weou? (i) 0 bqe<fqvh<hg<ogi{l< (ii) 45 bqe<fqvh<hg<ogi{l< (iii) 55 bqe<fqvh<hg<ogi{l< (iv) 8 bqe<fqvh<hg<ogi{l< wmk<kg<gim<m 4: gqp<g<gi[l<ogi{l<yu<ouie<xqe<lqjgfqvh<hg<ogi{l<gi{<g; (i) 70 (ii) 45 (iii) 0 (iv) 55 kqi<u: lqjgfqvh<hg<ogi{r<gtqe<%mkz<80.weou? (i) 70 e<lqjgfqvh<hg<ogi{l< (ii) 45 e<lqjgfqvh<hg<ogi{l< (iii) 0 e<lqjgfqvh<hg<ogi{l< (iv) 55 e<lqjgfqvh<hg<ogi{l< wmk<kg<gim<m 5: hqe<uvl<yu<ouie<xqzl<dt<togi{r<gjtg<g{<mhqcg<gul<; (i) -VOgi{r<gt<lqjgfqvh<Hg<Ogi{r<gt<; -ux<xt<ohiqbk?sqxqbjkh<ohiz< lmr<gigdt<tk/ (ii)-vogi{r<gt<fqvh<hg<ogi{r<gt<; -ux<xt<ohiqbk?sqxqbjkuqm0 lqjgbigdt<tk/ (iii) -VOgi{r<gt<nMk<kMk<kOgi{r<gtigUl<? 0 Ogi{l<njlh<hkigUl<< dt<te/nux<xt<ohiqbk?sqxqbjkh<ohiz<4lmr<jguqm0 Gjxuig dt<tk/ (iv) -VOgi{r<gt< Gk<okkqi<Ogi{r<gtigUl<?fqvh<Hg<Ogi{r<gtigUl<dt<te/ kqi<u: (i) sqxqb Ogi{k<jk x we<g/ 0 ohiqb Ogi{l< x NGl</ -V Ogi{r<gt< lqjgfqvh<hg<ogi{r<gt<we<hkiz<? x + x 80 n.kx 80 n.k x 60. hml< 6.58 sqxqbogi{l< 60, ohiqbogi{l< )ii* sqxqbogi{l< x we<g/0ohiqbogi{l< x + 0 NGl</-u<uqVOgi{r<gt<fqvh<Hg<Ogi{r<gtikziz<? x + (x + 0 ) 90 n.k x 70 n.k x 5. sqxqbogi{l< 5, ohiqbogi{l< hml< 6.59 (iii) sqxqbogi{l< x we<g/0ohiqbogi{l< x 0 NGl</-u<uqVOgi{r<gt<0 jbnjlh<hkiz<? x + (x 0 ) 0. 4x 40 n.k x 5. sqxqbogi{l< 5,ohiqbOgi{l< hml<

166 (iv) Gk<okkqi<Ogi{r<gt< yu<ouie<xl<x we<g/ -jufqvh<hg<ogi{r<gt<we<hkiz<? x + x 90 nz<zk x 90 nz<zk x 45. Ogi{r<gt< yu<ouie<xl< 45 NGl<. hml< 6.6 wmk<kg<gim<m 6: hml< 6.6 z< OgiM l NeK? -j{ogimgt< l, l Ngqbux<xqx<GGXg<Goum<c NGl</Ogi{r<gt<x, ygi{<g/ kqi<u: ye<xuqm<mogi{r<gt<sll</ x 0. GXg<Goum<cbqe<yOvHxl<njlf<k dm<ogi{r<gtqe<%mkz<80 o. hml<6.6 y nz<zky wmk<kg<gim<m 7: hml< 6.6 z<? l 4 NeK -j{ogimgt< l, l, l Ngqbux<xqx<<G GXg<Goum<cbiGl</Ogi{r<gt<x, y gi{<g/ kqi<u: yk<kogi{r<gt<sll</ x 75. GXg<Goum<cbqe< yovhxl< njlf<k dm<ogi{r<gtqe<%mkz<80 o. y nz<zk y hml< 6.6 wmk<kg<gim<m 8: hml<6.64z<?ogiml NeK -j{ogimgt< l, l Ngqbux<xqx<G GXg<Goum<cbiGl</Ogi{r<gt<x, ygi{<g/ kqi<u: GXg<Goum<cbqe< yovhxl< njlf<k dm<ogi{r<gt<lqjgfqvh<hg<ogi{r<gt</ 04y nz<zk 4y nz<zk y. yk<kogi{r<gt<sll< x + y 9 x hml< 6.64 x wmk<kg<gim<m 9: Lg<Ogi{k<kqe<Ogi{ntUgtqe<uqgqkl< : 4 : 5 weqz<?nux<jxg<< gi{<g/ kqi<u: Ogi{r<gjt x, 4x, 5xweg<ogi{<miz<?x + 4x + 5x 80 nz<zk x 80 n.k x 5. Ogi{r<gt< 5, 4 5, 5 5 nz<zk 45, 60, 75 NGl</ wmk<kg<gim<m 0: hml<6.65z<gxqh<hqmh<hm<<mogi{r<gt< x, ygi{<g/ kqi<u: ABC-z< x n.k x 5. BDC-z< x + y n.k 5 + y n.k y n.k y hml<

167 wmk<kg<gim<m : hqe<uvl<<hmr<gtqz< x, y gi{<g; (i) (ii) hml< 6.66 hml< 6.67 kqi<u: (i) AD BC, AB CD. ABCD YI-j{gvl</ x 4, y 60 (ye<xuqm<mogi{r<gt<sll<). x, y 0. (ii) Lg<Ogi{r<gt<ACD, ACB, -ux<xqz< AD AB, CD BC, AC NeKohiK/ ADC ABC. yk<kogi{r<gt<sll</weou x + 0 6, y 5 4 nz<zk x 6 0, y nz<zk x 6, y 47. wmk<kg<gim<m : YI -Vslhg<g Lg<Ogi{k<kqe< ds<sqogi{k<kqe< sloum<c nch<hg<gk<kqx<gjlbg<gk<kg<ogimwefq'hq/ kqi<u: ABC we<hk YI -Vslhg<g Lg<Ogi{l<< we<g/ -r<g AB AC, Ogi{l< A bqe< sloum<c AD we<g/ AD, nch<hg<gl< BC g<g jlbg<gk<kg<ogim we fq'hqk<kz< Ou{<Ml</Lg<Ogi{r<gt<ADB, ADCNgqbux<Xt< AB AC, m BAD m DAC ( AD NeK A e<sloum<c*? AD ohik/ h.ogi.hogit<jgh<hc, ABD ACD. yk<khg<gr<gt<sll</ BD DC. nkiuk?d NeK BCbqe<fMh<Ht<tqNGl</ ADC x we<g/hqe<h ADB 80 x. Lg<Ogi{l<ADC z<? ADC + C + CAD 80. Lg<Ogi{l<ADB z<? ADB + B + BAD 80. ADC + C + CAD ADB + B + BAD. n.kx 80 x n.k x 80 n.kx 90. AD NeKBC g<gs<osr<gk<kigdt<tk/ hml< 6.68 n.kad NeKBC bqe<jlbg<gk<kg<ogimigl</ wmk<kg<gim<m : ABC we<hk yv Lg<Ogi{l</ Ht<tq D NeK BC bqe< jlbh<ht<tq/ DA -j{g<gh<hmgqxk/ DA DCweqz<? BAC yvosr<ogi{l<wefq'hq/ kqi<u: DA DC wek<kvh<hm<mt<tk D NeKBCe<jlbh<Ht<tqwe<hkiz<BD DC/ weoulg<ogi{r<gt<abd?< ACD -Vslhg<gLg<Ogi{r<gt<NGl</ DAB DBA () DAC DCA () 6

168 () + () DAB + DAC DBA + DCA BAC DBA + DCA BAC CBA + BCA () Neiz< BAC + CBA + BCA 80 (4) (4) BAC + ( CBA + BCA) 80 BAC + BAC 80 (() Jh<hbe<hMk<k) BAC 80 BAC 90. hml< 6.69 wmk<kg<gim<m 4: yvfix<gvk<kqe<fie<gogi{r<gtqe<%mkz< 60 wefq'hq/ kqi<u: ABCD yv fix<gvl< we<g/ A + B + C + D 60 we fq'hqk<kz< Ou{<Ml</ -kx<g? &jzuqm<ml< AC J ujvbul</ Lg<Ogi{r<gt<ACD, ABCNgqbux<xqzqVf<K? DAC + D + ACD 80 () CAB + B + ACB 80 () hml< 6.70 () + () DAC + D + ACD + CAB + B + ACB 60 ( DAC + CAB) + B + ( ACD + ACB)+ D 60 A + B + C + D 60. wmk<kg<gim<m 5: AB AC we<xqvg<g ABC YI -Vslhg<g Lg<Ogi{l< we<g/ DBC DCB we njlblix Ht<tq D? Lg<Ogi{l< ABC -e< dt<ot njlf<kt<tk/ AD NeK A -e< sloum<cwefq'hq/ kqi<u: DBC DCB we<hkiz<? Lg<Ogi{l< DBC YI-Vslhg<gLg<Ogi{liGl</weOu BD DC. Lg<Ogi{r<gt< ADB, ADC Ngqbux<Xt<, BD DC,AB AC, ADohiK/ weou h.h.h ogit<jgh<hc? ADB ADC. 0yk<kOgi{r<gt<sll</ Gxqh<hig? BAD CAD. AD NeK Ae<sloum<cbiGl</ hml< 6.7 wmk<kg<gim<m 6: yvlg<ogi{l<abc e< osr<ogimgt<ad, BE we<he AE BD we<xuixt<te/ AD BEwefq'hq/ kqi<u: Lg<Ogi{r<gt< ADB, AEB Ngqbux<Xt< (i) ADB AEB 90 (ii) AB ohik (iii) BD AE. os.g.hogit<jgh<hc ADB AEB. yk<khg<gr<gt<sll</ad BE. hml<

169 wmk<kg<gim<m 7: ABCDwe<El<osu<ugk<kqz<BC e<jlbh<ht<tqe,ngl</ AE ED wefq'hq/ kqi<u: hml<6.7 Jh<hii<g<gUl<. Lg<Ogi{r<gt< ABE, DCE Ngqbux<xqz< (i) ABE DCE 90 (ii) BE CE (E, BC e<jlbh<ht<tq) (iii) AB CD (ABCD yv osu<ugl<) h.ogi.hogit<jgh<hc ABE DCE. 0yk<khg<gr<gt<sll</ AE ED. hml< 6.7 wmk<kg<gim<m 8: yv sib<skvk<kqz<? &jzuqm<mr<gt< yu<ouie<xl< lx<oxie<jxs< osr<gk<kig-vslg<%xqml<wefq'hq/ kqi<u: ABCD yvsib<skvl</&jzuqm<mr<gt<ac, BD ujvg/ AC Bl< BD Bl< oum<mh<ht<tqjbo weg<gxqg<g/ht<tqonekac, BD-e<jlbh<Ht<tqweUl<? ACNeKBDg<Gosr<Gk<K( ) weul<fq'hqk<kz<ou{<ml</ sib<skvl< YI -j{gvl< we<hkiz< &jzuqm<mr<gt< yu<ouie<xl< lx<oxie<jx -Vslg<%xqMl</nkiuK?OA OC, OB OD. -h<ohipk?lg<ogi{r<gt<< AOB, BOCNgqbux<Xt< (i) AB BC (ii) OB ohik(iii) OA OC h.h.hogit<jgh<hc AOB BOC. 0yk<kOgi{r<gt<sll</ AOB BOC x we<g/. Neiz<? AOB + BOC 80. hml< x + x 80 x 80 n.k x 90. &jzuqm<mr<gt<yu<ouie<xl<lx<oxie<jxs<osr<gk<kig-vslg<%xqml</ wmk<kg<gim<m 9: yv sib<skvk<kqe< &jzuqm<mr<gt< yu<ouie<xl< nju osz<gqe<x ds<sqg<ogi{r<gjtslg<%xqml<wefq'hq/ kqi<u: ABCD yv sib<skvl</ AC, BD &jzuqm<mr<gt</ AB CD, AC GXg<Goum<c we<hkiz<? BAC ACD (ye<xuqm<mogi{r<gt<). () Neiz< AD CD (ABCD sib<skvl<). ADC YI-Vslhg<gLg<Ogi{l</ slhg<gk<kqx<g wkqovbt<t Ogi{r<gt< sll< we<hkiz<? ACD DAC. () hml< 6.75 (), ()e<hc BAC DAC. weou AC, Ogi{l< Ajuslg<%xqMl</ -jkh<ohie<ox? AC NeK C jbbl<? BD NeK B lx<xl< D Ngqbux<jxBl< slg<%xqml<wefq'hqg<gzil</ 65

170 wmk<kg<gim<m 0: AB, CD -j{ogimgt</ AB, CD gtg<g -jmob Ht<tq O (hml< 6.76Jh< hii<g<gul<) APO 45 -Vg<GliXl< OQC 5 -Vg<GliXl< njlf<kt<tkweqz<? POQ g{<mhqc/ kqi<u: PO jufqm<mg/nkcd jb sf<kqg<gl< Ht<tqjb X weg< Gxqg<g/ QO ju fqm<mg/ nk AB J sf<kqg<gl< Ht<tqjb Yweg<Gxqg<g/ hml< 6.76 AB CD, PX GXg<Goum<cwe<hkiz<? OXQ OPY 45 (ye<xuqm<mogi{r<gt<). Lg<Ogi{l<OXQ z<? outqg<ogi{l< POQNeKnke<dt<otkqi<Ogi{r<gtigqb OXQ lx<xl< OQXNgqbux<xqe<%MkZg<Gsll</ 0 POQ OXQ + OQX wmk<kg<gim<m : ABC -z<? B -e< sloum<cbiek AC jb Ht<tq D -z< sf<kqg<gqe<xk/ ABC 80, BDC 95 weqz<? A, Cgi{<g/ kqi<u: hml< 6.77 Jh<hii<g<gUl</ BDC-z<? C 80 C ABC-z<? A + B + C 80. A A wmk<kg<gim<m : ABCDyVsiqugl</ -kqz< AB Bl<CD Bl<-j{OgiMgt</AD BC weqz<? ADC BCD wefq'hq/ kqi<u: hml< 6.78 Jh<hii<g<gUl</AD g<g-j{bigbe ujvg/-h<ohipkabed YI-j{gvl<we<hkiz<? BE AD. Neiz<AD BC. BC BE. weoulg<ogi{l<bec YI -Vslhg<gLg<Ogi{l<. BCE BEC. () Neiz< AD BE, AD, BE -ux<xqg<gdec yvgxg<goum<c. ADC BEC (yk<kogi{r<gt<). () weou (), ()e<hc? BCE ADC nz<zk BCD ADC. hml<6.77 hml< 6.78 hbqx<sq 6.. hqe<uvl<%x<xgtqz<wjusiq?wjukuxwewpkg; (i) yv gkqi<? yv Ogim<ce< lqk njlukiz< njlbl< nmk<kt<t Ogi{r<gtqe< %Mkz<80 NGl</ (ii) -VOgiMgt<oum<cg<ogi{<miz<?Gk<okkqi<Ogi{r<gt<sll</ (iii) yvlg<ogi{l<-vuqiqogi{r<gjth<ohx<xqg<gl</ 66

171 (iv) yvfix<gvk<kqe<ogi{r<gtqe<%mkz<80 NGl</ (v) ABC PQR weqz<? A Q. (vi) DEF XYZ weqz<? DE XY. (vii) YI-j{gvk<kqz<?&jzuqm<mr<gt<yu<ouie<Xl<lx<xjkslg<%xqMl</. gqp<g<g{<myu<ouie<xqe<fqvh<hg<ogi{l<gi{<g/ (i) 0 (ii) 65 (iii) 70 (iv) 78. gqp<g<g{<myu<ouie<xqe<lqjgfqvh<hg<ogi{l<gi{<g/ (i) 50 (ii) 0 (iii) 80 (iv) hqe<uvl<yu<ouie<xqzl<ogi{r<gjtg<gi{<g; (i) Ogi{r<gt<fqvh<Hg<Ogi{r<gt</sqxqbOgi{l<?ohiqbOgi{k<jkuqm 40 GjxuiGl</ (ii) Ogi{r<gt< fqvh<hg<ogi{r<gt</ ohiqb Ogi{l<? sqxqb Ogi{k<jkh<Ohiz< 4 lmr<g. (iii) Ogi{r<gt<lqjgfqvh<Hg<Ogi{r<gt</ohiqbOgi{l<?sqxqbjkuqm58 nkqgl</ (iv) Ogi{r<gt<lqjgfqvh<Hg<Ogi{r<gt</ohiqbOgi{l<?sqxqbOgi{k<kqe< lmr<jguqm0 GjxuiekiGl</ (v) -VnMk<Kt<tOgi{r<gt<40 Ogi{k<kqjenjlg<gqe<xe/sqxqbOgi{l< ohiqbogi{k<jkuqm8 GjxuiGl</ (vi) Ogi{r<gt< Gk<okkqi< Ogi{r<gtigUl<? lqjgfqvh<hg<ogi{r<gtigul< -Vg<gqe<xe/ 5. hqe<uvl<hmr<gtqz<x, ygi{<g; (i) (ii) hml< 6.80 hml< 6.79 (iii) (iv) hml< 6.8 hml<

172 6. hqe<uvl<hmr<gt<yu<ouie<xqzl<x, ygi{<g; (i) (ii) hml< 6.8 (iii) hml< 6.84 hml< 6.85 uqjmgt< hbqx<sq 6.. kux. siq. kux hbqx<sq 6.. (i) siq (ii) siq (iii) kux (iv) kux (v) kux (vi) kux (vii) siq. (i) 70 (ii) 5 (iii) 0 (iv). (i) 0 (ii) 50 (iii) 00 (iv) 8 4. (i) 5, 65 (ii) 8 (iii) 6, 9 (iv) 50, 0 (v) 56, 84 (vi) 90, (i) x 0, y 50 (ii) x 80, y 70 (iii) x 0, y 0 (iv) x 50, y 0 6. (i) x 9, y 8 (ii) x 48, y (iii) x 6, y. 68

173 7. hgljxucuqbz< yv OfIg<Ogim<cz< njlf<k yu<ouiv Ht<tqBl< siqbig yv olb<ob{<[me< ohivk<kh<hm<mt<tk we<hjk Lkz< nk<kqbibk<kqz< g{<omil</ -f<k nk<kqbibk<kqz< yv ktk<kqz< njlf<kt<t yv Ht<tqbieK wu<uix olb<ob{<gtiz< Sm<cg< gim<mh<hmzil< we<hjk Nvib<Ouil</ ovoe om^<giim^< we<gqx Hgp< ohx<x hqovr<s fim<m g{qk uz<zfi, Lke<Lkzig ucuqbjz hgk<kxqb? -bx<g{qk Ljxjb )w{<gt<lx<xl<fie<gnch<hjms<osbz<gt<hbe<hmk<kl<ljx*nxqlgh<hmk<kqeii/ weoukie<? -bx<g{qk Ljxjbh< hbe<hmk<kq ucuqbjz hgk<kxqbl< himh<hgkq -bx<g{qk ucuqbz< nz<zk hgljx ucuqbz< weg< %xh<hmgqe<xk/ -f<k hgljx ucuqbjz ovoe om^<giim^< nxquqk<kkiz<, nui hgljx ucuqbzqe< kf<jkweh<hmgqxii/ 7. gii<csqbe<ns<sk~vljx yv ktk<kqz< ujvbh<hm<mt<t sqz dvur<gtqe< -bz<hgjth< hx<xq gx<g uqjpgqe<oxil</yvktk<kqz<njlf<kht<tqgtqe<okigh<ohnf<kktk<kqz<njlf<k yvucul<ngl</weou?ucuqbzqe<nch<hjmg<ogit<jgyvht<tqngl</yv ktk<kqz< njlf<kt<t yu<ouiv Ht<tqObiMl< yv Osic olb<ob{<gjt -j{h<hk hx<xq-h<ohipkfil<gi{<ohil</ yv kitqe< ktk<kqje, Okjubie ktlig wmk<kg<ogi{<m, nf<k ktk<kqz< ye<xg<ogie<x osr<gk<kig njlf<k -V fqjzbie OfIg<OgiMgjt ujvg/ hml< 7. z< gim<cbt<tk Ohiz< upg<glig? yv Ofi<g<Ogicje gqjmg<ogimigul< lx<oxivofig<ogicjeosr<gk<kg<ogimigul<ujvouil</-vf<kohikqzl<hml<7. z< gim<cbt<tkohiz<njuwf<kujgbqzl<ujvbh<hmzil</-u<uqv{<mogimgtl< hml< 7. hml< 7. oum<cg< ogit<tl< Ht<tq, O weh<ohbiqmh<hm<m, Nkqh< Ht<tq we njpg<gh<hmgqxk/ -v{<m osr<gk<kg< OgiMgTl< fqjzbie OgiMgtig -Vh<hkiz< O we<gqx -f<kh< Ht<tqBl< fqjzbiek/ -h<ohipk -u<uqv OgiMgTg<Gl< O we<el< Ht<tq, w{< 0(H,s<sqbl<* jkg< Gxqh<hkigg< ogi{<m, -u<uqv Ofi<g<OgiMgjtBl< nzgqmouil</ yov 69

174 nzgqje -v{<m OgiMgTg<Gl< hbe<hmk<kouil</ -h<ohipk -V osr<gk<kg< OgiMgTl<, -V osr<gk<k w{< OgiMgtig lixgqe<xe/ gqjmg<ogim<cz< O uqe< uzkhg<gk<kqz< njlf<k w{<gt< lqjg w{<gt</ Gk<Kg<Ogim<cz< O uqe< Olz< hg<gk<kqz< njlf<k w{<gt< lqjg w{<gt</ -OkOhie<X gqjmg<ogim<cz< O uqe< -mkhg<gw{<gtl<, Gk<Kg<Ogim<cz<O uqe<gqp<h<hg<gk<kqz<njlf<kw{<gtl<gjx w{<gt</ w{<gtqe< WXuiqjs hml< 7. z<ogimgtqe<ljebqz<nl<hg<gxqgt< -mh<hm<m gim<mh<hm<mt<tk/ gqjmk<kqjs w{< Ogim<cje, x-ns<s we<xl< Gk<Kkqjs w{<ogim<cje, y-ns<swe<xl<fil<njpg<gqoxil</-u<uqv{<mogimgtl<ktk<kqje, gix<hgkqgt<wenjpg<gh<hml<fie<gslhigr<gtigh<hqiqg<gqe<xe/-juhml<7. z< gim<cbt<tkohiz<i )Lkzil<* gix<hgkq, II )-v{<mil<*gix<hgkq, III )&e<xil<*gix<hgkq lx<xl< IV )fie<gil<* gix<hgkq weh<ohbiqmh<hm<mt<te/ O we<gqx Ht<tqbieK fie<g gix<hgkqgtg<gl< ohikuiek/ ktk<kqz< P we<gqx WOkEl< yv Ht<tqjb wmk<kg< ogit<ouil</-h<ht<tqp NeKyVgix<hGkqbqz<njlf<Kt<tK/P upqbigy-ns<sg<g -j{bigyvofi<g<ogimujvg/-g<ogimx-ns<jsl we<gqxht<tqbqz<sf<kqg<gm<ml</ -u<uiox, P upqbigx-ns<sg<g-j{bigyvofi<g<ogimujvg/-g<ogimy-ns<js M we<gqx Ht<tqbqz< oum<mm<ml</ x-ns<sw{<ogim<cz<l we<x Ht<tqjbg<< Gxqg<Gl< olb<ob{< a NgUl<, y-ns<sw{<ogim<cz< M we<xht<tqjbg<gxqg<gl<olb<ob{<b NgUl<-Vg<gm<Ml</P NeKx-ns<sqz<njlf<kiz<, b 0 weg<gi{<gqoxil</pnek y-ns<sqz<njlf<kiz<, a 0 weg<gi{<gqoxil</p NeK x-ns<sqozinz<zky-ns<sqozi -z<zilz<lkz<gix<hgkqg<gt<njlf<kiz<, a > 0 lx<xl<b > 0 NGl</ a < 0 lx<xl< b > 0 Ng-Vf<kiz<P biek-v{<mil<( II ) gix<hgkqg<gt<njlbl</ P biek&e<xil< ( III ) gix<hgkqg<gt< njlf<kiz<, a < 0 lx<xl< b < 0. a > 0 lx<xl<b < 0 Ng-Vh<hqe<, P biekfie<gil<( IV ) gix<hgkqg<gt<njlbl</ P bieknkqh<ht<tq O Ng -Vh<hqe<, a 0 lx<xl<b 0 NGl</ a we<xw{< P wel< Ht<tqbqe< x-ns<s K~vl< nz<zk x-okijzu(abscissa) we<xl<, b wel<w{<p wel< Ht<tqbqe< y-ns<s K~vl< nz<zk y-okijzu (ordinate) we<xl< njpg<gh<hml< (hml< 7. Jh< hii<g<gul<*.fil<a, b wel< w{<gjt gix<ht<tqbiz< (, ) -u<uix hqiqg<gh<hm<mhqjxnjmh<hg<gt<, (a, b) we<x hml< 7. wpkouil</njk a, b we<gqxuiqjsosic we njph<<ohil</ gix<ht<tqbqe< -mkhg<glt<t w{<, P Ht<tqbqe< x-ns<s K~vk<jkBl<, gix<ht<tqbqe<uzkhg<glt<t w{<, P Ht<tqbqe<y-ns<SK~vk<jkBl<Gxqh<hkiz<?(a, b) NeKyVuiqjsOsicwe njpg<gh<hmgqxk/ P Ht<tqjbg<Gxqg<Gl<uiqjsOsic (a, b)we<hkye<ox ye<xkie</ nkiukp wel<yvht<tqg<gouxwf<kuiqjsosicbl<-vg<gik/ P wel<ht<tq P(a, b) weul< nz<zk SVg<glig (a, b) weul< Gxqg<gh<hMl</ fil< P biek (a, b) 70

175 ns<sk~vr<gjtg< ogi{<mt<tk weg< %XOuil</ -u<uix ktk<kqz< dt<t yu<ouiv Ht<tqBl<, olb<ob{<gtiz< Ne yv uiqjs Osicbiz< Gxqh<hqmh<hMl</ ovoe om^<giim^<^qe< dbif<k osbjz lkqg<gl< ohivm<m -k<ktliek gii<csqbe< ktl< we<xnjpg<gh<hmgqxk/olzl<nkosu<ugns<sktl< nz<zk xy ktl< we<xl< njpg<gh<hml</ -u<uix w{<uiqjs Osicgtiz< ktk<kqz< njlf<k Ht<tqgjtg< Gxqg<Gl<Ljxgii<Csqbe<Ljxnz<zKosu<ugLjxnz<zKxy ns<sk~vljx we<xnjpg<gh<hmgqxk/-v{<mns<sgtl<osu<ugns<sgt<nz<zkokijzu ns<sgt<weh<hmgqe<xe/ fil<nxquk, (i) Nkqh<Ht<tqO uqe<ns<sk~vr<gt< (0, 0). (ii) x-ns<sqe<lqkt<twf<kht<tqg<gl< y-ns<sk~vl<0. (iii) y-ns<sqe<lqkt<twf<kht<tqg<gl< x-ns<sk~vl< 0. (iv) olb<ob{<gtizie uiqjs Osic ye<x ogimg<gh<hm<miz< nkx<g gii<csqbe< ktk<kqz<keqk<kyvht<tqbqeiz<-ml<fqi<{bqg<gzil</ktk<kqz<nke<siqbie -mk<jk. we<xht<tqbiz<gxqg<gzil</ (v) yv Ht<tqbieK Lkz< gix<hgkqg<gt< njlf<kiz< -Vns<S K~vr<gTl< lqjg, -v{<mil< gix<hgkqg<gt< njlf<kiz<, x-ns<s K~vl< Gjx lx<xl< y-ns<s K~vl< lqjg/ &e<xil<gix<hgkqg<gt<<njlf<kiz<, -Vns<S K~vr<gTl< Gjx/ fie<gil< gix<hgkqg<gt< njlf<kiz<, x-ns<s K~vl< lqjg lx<xl< y-ns<s K~vl< Gjx NGl</ yv Ht<tqbqe< ns<s K~vr<gtqe< -bx<g{qkg< Gxqgt< hml< hml< z< gim<mh<hm<mt<te/ (vi) x-ns<sg<g-j{bigs<osz<zl<ogim<cz<njlf<kwz<zih<ht<tqgtl<sllie y-ns<sk~vk<jkg<ogi{<cvg<gl<)hml< 7.5 Jh<hii<g<gUl<). hml<7.5 hml< 7.6 (vii) y-ns<sg<g-j{bigs<osz<zl<ogim<cz<njlf<kwz<zih<ht<tqgtl<sllie, x-ns<sk~vk<jkg<ogi{<cvg<gl<)hml<7.6 Jh<hiIg<gUl<). 7

176 A(x, y ) lx<xl< B(x, y ) we<hjugii<csqbe<ktk<kqz<njlf<kwokel<-vht<tqgt< we<g/-ux<xqx<g-jmobbt<tgqjmlm<mk<okijzu, x > x weqz< x x weul<, x > x weqz< x x weul<, x x weqz< 0 weul< ujvbxg<gh<hmgqxk/ -jkh< OhizOu, A, B -ux<xqx<gqjmobbieosr<gk<kk<okijzuieky > y weqz<?y y weul<, y > y weqz<?y y weul<, y y weqz<?0 weul<ujvbxg<gh<hmgqe<xk/ wmk<kg<gim<mig, hml< 7.7 z<, A, B -ux<xqx<g -jmobbt<tgqjmlm<mk<okijzju BN l<, osr<gk<kk< okijzju AN l< Gxqg<gqe<xe/ fil<nxquk, BN LMOL + OM ( x ) + (x ) x x, AN AL + LN y + MB y + ( y ) y y. -u<uiox, hml< 7.8 z< A(x, y ) lx<xl< B(x, y ) weg<ogi{<miz< A, B -ux<xqx<g hml<7.7 -jmobbt<t gqjmlm<mk< okijzu, osr<gk<kk<okijzungqbeljxob, BN ML OM OL x ( x ) x x, AN LN AL BM ( y ) ( y ) + y y y. -k<okijzugjtljxob x x we<xl< y y we<xl<gxqh<hqmgqe<oxil</ hml< 7.8 wmk<kg<gim<m : A (, 0), B (0, ), C (4, 4), D (, ), E (.5, ), F (, ), G (, 0), H (0, 4) NgqbHt<tqgjtyVujvhmk<kitqz<Gxqg<gUl</OlZl<yu<ouiVHt<tqBl< wf<kgix<hgkqbqz<njlgqe<xkwe<hjkbl<gxqh<hqmul</ kqi<u: hml<7.9 -z< yvktk<kqz<ht<tqgt< Gxqg<gh<hm<Mt<te/ A we<gqxht<tqx-ns<sqe< lqjgh<hgkqbqzl<, B we<gqxht<tq y-ns<sqe< lqjgh<hgkqbqzl<, C we<gqxht<tqfie<gil< gix<hgkqbqet<tl<, D we<gqxht<tqlkz< gix<hgkqbqet<tl<, E we<gqxht<tq-v{<mil< gix<hgkqbqet<tl<, F we<gqxht<tq&e<xil< gix<hgkqbqet<tl<, G we<gqxht<tq x-ns<sqe< Gjxh<hg<gk<kqZl<, H we<gqxht<tq< y-ns<sqe< Gjxh<hg<gk<kqZl<LjxObnjlf<Kt<te/ hml< 7.9 wmk<kg<gim<m : (, 4), ( 9, ) we<gqx -V Ht<tqgTg<G -jmob dt<t gqjmk<okijzu, osr<gk<kk<okijzungqbux<jxg<g{<mhqc/ kqi<u: (, 4), ( 9, ) Ngqb -V Ht<tqgTg<G -jmob dt<t gqjmk<okijzu, osr<gk<kk<okijzuljxob( ) ( 9) 9 6 lx<xl< () ( 4) 5 NGl</ 7

177 hbqx<sq 7.. gqop ogimg<gh<hm<mt<t Ht<tqgjt ujvhmk<kitqz< Gxqk<K nju wf<okf<k gix<hgkqgtqz< njlgqe<xewe<hjkg<gxqh<hqmul</ (i) (, ) (ii) (7, 6) (iii) (, ) (iv) (6, ) (v) ( 9, 0) (vi) (5, 0) (vii) (0,) (viii) (, ). siqbikuxiweh<hkqz<ntqg<gul</ (i) (9, ), -v{<mil<gix<hgkqbqet<njlbl</ (ii) (, 0), y-ns<sq<e<olz<njlbl</ (iii) (,), y-ns<sqe<uzkhg<gl<njlbl</ (iv) (, ), x-ns<sqe<gqopnjlbl</ (v) (0, 0) we<hkx, y ns<sg<gt<oum<cg<ogit<tl<ht<tq/ (vi) (, ), gix<hgkq II z<njlbl</ (vii) ( π, ), gix<hgkq III z<njlbl</ (viii) (, ), gix<hgkq IV z<njlbl</ (ix) (0, ), x-ns<sqe<-mkhg<gl<njlbl</ (x) (5, 0), x-ns<sqe<gqopnjlbl</ (xi) x-ns<sg<g-j{bigdt<togim<ce<olz<wokel<-vht<tqgt<, sllie x-ns<sk~vr<gjtogi{<cvg<gl</ (xii) (a, b), (c, d) Ngqbju y-ns<sg<g-j{bieogim<ce<olz< WOkEl<-VHt<tqgt<weqz<, a c.. ogimg<gh<hm<mt<t-vht<tqgtg<g-jmobdt<tgqjmk<okijzu, osr<gk<kk< okijzungqbux<jxg<gi{<g/ (i) (, 4) lx<xl< (, 5). (ii) (, ) lx<xl< (4, 6). (iii) (, 5) lx<xl< (7,). (iv) (, ) lx<xl< ( 4, ). 7. Ogim<ce<sib<U Lkzqz<, x-ns<sg<ogi nz<zk y-ns<sg<ogi -j{big njlbik LL we<x Ofi<g<Ogim<ce< sib<uqje ujvbxh<ohil</ -kx<gig LL Ofi<g<Ogim<ce< lqkjlf<k P(x, y) we<gqx Ht<tqbig yv leqkje fqjek<kg<ogit<ouil</ -l<leqke< P Neue< ng<ogim<ce<lqkymgqe<xie<we<xfqjek<kh<hiig<gul</hml<7.0 nz<zkhml<7. Ngqbux<xqz< Gxqh<hqm<Mt<tK Ohiz< WOkEl< -V kqjsgtqz< nue< osz<zzil< we<hjkfil<nxqgqoxil</ hml< 7.0 hml< 7. P biek yv Gxqh<hqm<m kqjsbqe< upqbig fgvgqe<xohik, P bqe< x-ns<sk<k~vl< nkqgiqh<hjkfil<gi{zil< (hmr<gt<7. lx<xl<7. Jh<hiIg<gUl<). -f<k Gxqh<hqm<m 7

178 kqjsjbg< Ogim<ce< lqjgk<kqjs weg< %XgqOxil</ lx<oxiv kqjsbiek Ogim<ce< Gjxk<kqjs weh<hml</ P biek Gjxk<kqjsbqz< fgvl<ohipk nkejmb x-ns<s K~vl<Gjxgqe<xK/ hml< 7. hml< 7. P NeK P (x, y ) we<xfqjzbqzqvf<kp (x, y ) we<xfqjzg<glqjgk<kqjsbqz< fgvm<ml</ P bqe< x-ns<s K~vlieK x lkqh<hqzqvf<k x we<gqx lkqh<hqx<gl<, y-ns<s K~vlieK y lkqh<hqzqvf<k y we<gqx lkqh<hqx<gl<, lixujkg< gi{<gqoxil</ x-ns<s okijzuqz<wx<hml<lix<xl<x x. -K, fgvl<ht<tqp bqe<ym<ml<(run) weh<hml</ -Ok Ohie<X y-ns<s okijzuqz< Wx<hMl< lix<xl< )lixkz<* y y. -K fgvl< Ht<tqbqe<Wx<xl<nz<zKwPs<sq(rise) weh<hml</fil<hmr<gt< 7., 7. Ngqbux<xqz< Ym<ml< x x lqjgbig -Vh<hjkg< gi{<gqoxil</ hml< 7. z< P NeK Ogim<cz< Olz<Ofig<gq fgvgqe<xk/ nkiuk, nk Ogim<cz< Olz<Ofig<gq P zqvf<k P uqx<g wps<sq njmgqxk/ OlZl< P uiek P J uqm dbi<f<k -mk<kqz< dt<tk/ -r<g Wx< xl< y y Wx<xl< y y lqjgbigl</ weou? we<gqxuqgqkl<lqjgw{<ngl</ Ym< ml< x x hml<7. z<, Ht<tqP NeKOgim<cz<gQp<Ofig<gqfgVgqe<xK/nkiuK nk, P zqvf<k P uqx<guqp<gqe<xk/ht<tqp NeK P Juqmk<kip<f<k)Gjxf<k*-mk<kqz<dt<tK/ Wx< xl< -r<g Wx<xl< y y GjxbiGl</ weou? Ym< ml< Wx< xl< weou Ym< ml< y x y x y x y x, yvgjxw{<ngl</ we<gqxuqgqkliek kvh<hm<mofig<ogimwpgqe<xofig<ogimi nz<zk uqp<gqe<x OfIg<Ogimi we<hjk fqi{bqg<gqxk/ weou -f<k uqgqkk<jk ng<ogim<ce<sib<uwe<gqe<oxil</ nmk<khcbig, x-ns<<sg<g -j{bigs< osz<zl< Ogim<ce< sib<uqjeg< g{<mxqouil</ hml; 7.4 z<, OgiM LL NeK x-ns<sg<g -j{bigdt<tk/ogim<ce<olz<njlf<k wz<zih< Ht<tqgTl< yov y-ns<s K~vk<jkg< ogi{<mt<tewe<xl<gi{<gqoxil</ P (x, y ), P (x,y ) Ngqbju-j{g<Ogim<cz<njlf<k -V Ht<tqgt< we<g/ -r<g, y y. weou Wx<xl<y y 0. hml<

179 Wx< xl< y y NgOu?sib<U 0. Ym< ml< x x nmk<khcbig, y-ns<<sg<g -j{bigs< osz<zl< Ogim<ce< sib<uqjeg< gi{<ohil</ P (x, y ), P (x, y ) Ngqbju -j{g< Ogim<cz< njlf<k -V Ht<tqgt< we<g (hml<7.5 Jh<hiIg<gUl<). -r<g, x x. weou, Ym<ml< x x 0. P, P ouu<oux Ht<tqgt</weOu, y y. NgOu hml< 7.5 Wx< xl< y y y y sib<u. Ym< ml< x x 0 -KujvbXg<gh<hmuqz<jz/ y-ns<<sg<g -j{bigs< osz<zl< Ogim<ce<sib<U ujvbxg<gh<hmikk/ wpl<hl<ogim<ce<sib<u> 0. uqpl<ogim<ce<sib<u < 0. x-ns<sg<g-j{bigs<osz<zl<ogim<ce<sib<u 0. y-ns<<sg<g-j{bigs<osz<zl<ogim<ce<sib<u ujvbxg<gh<hmikk/ ( ) y y y y -r<g fil< gueqg<g Ou{<cbK, x x ( x x ) (x, y ) Ngqb-VHt<tqgjts<Osi<g<Gl<Ogim<ce<sib<U Wx< xl< y y y y. Ym< ml< x x x x -kqzqvf<k, sib<uiekogim<ce<kqjsjbh< ohivk<k njlukz<z we<hjk fil< nxqgqoxil</ OlZl< Ogim<ce< sib<u y y we<gqx uqgqkl< P, P we<gqx x x Ht<tqgtqe< Gxqh<hqm<m okiqf<okmk<kjzs< sii<f<kk nz<z/ -f<k d{<jljbk< okiqf<k ogit<t, nf<ofig<ogim<ce< lqk OuX -V Ht<tqgt< P (x, y ) lx<xl< P 4 (x 4, y 4 ) wmk<kg<ogit<ouil<(hml<7.6 Jh<hiIg<gUl<). P zqvf<kp 4 g<gs<osz<zl< y4 y Ogim<ce<sib<U. x4 x P zqvf<k P g<gs< osz<zl< Ogim<ce< y y sib<u. x x y x y x. -u<uix (x, y ), hml<

180 Neiz< P AP, P BP 4 we<hjuucouik<klg<ogi{r<gt</ P A AP AP nz<zk BP y 4 y y4 y.. P B BP 4 PA P B x x x4 x nkiuk, sib<uiekogim<ce<olzjlf<k-vht<tqgtqe<-mk<jks<sii<f<kkz<z/ Gxqh<H: ogimg<gh<hm<mt<t (x, y ) lx<xl< (x, y ) Ngqb-VHt<tqgtqe<upqObyOvyV Wx< xl< y y Ofi<g<OgiMkie<ujvbLcBl</nf<OfIg<Ogim<ce<sib<U. Ym< ml< x x wmk<kg<gim<m : (5, 6) lx<xl< (5, 9) NgqbHt<tqgtqe<upqObosz<Zl<Ogim<ce<sib<U gi{<g/ OlZl< ng<ogim OlOzx<xl< ohxgqxki nz<zk gqpqxg<gl< ogit<gqxki weg< %Xg/ kqi<u: (5,6)J (x, y ) we<xl<? (5, 9) J (x, y ) we<xl<ogit<g/ogim<ce<sib<u Wx< xl< y y Ym< ml< x x 9 6, yvlqjgw{</ weou?ogimiek)hml< 7.7 Jh<hiIg<gUl<* Olz<Wx<xl<ohx<Xt<tK/ hml< 7.7 wmk<kg<gim<m 4: ( 6, 9), (40, 6) we<gqxht<tqgtqe<upqobosz<zl<ogim<ce<sib<u gi{<g/ OgimieK Olz< Wx<xl< ohxgqxki nz<zk gqp< -xg<gl< ogit<gqxki weg< %Xg/ kqi<u: Wx< xl< 6 9 Ogim<ce<sib<U Ym< ml< 40 ( 6) 5 5 yvgjxw{< weou OgimieK)hml< 7.8 Jh<hiIg<gUl<* gqp<-xg<gl<ohx<xt<tk/ hml< 7.8 wmk<kg<gim<m 5: gqopogimg<gh<hm<mht<tqgjts<osi<g<gl<ogim<ce<sib<ugjth<hx<xq uqtg<gg/ (i) (6, 4) lx<xl<( 7, 4) (ii) (, 8) lx<xl< (, 7) kqi<u: (i) Ogim<ce<sib<U y x y x , 7 6 OgimieKx-ns<Sg<G-j{biGl</ y y 7 8 (ii)ogim<ce<sib<u ujvbxg<gh<hmikk/ x x + 0 OgimieK x-ns<sg<gs<osr<gk<kigdt<tk/ 76

181 wmk<kg<gim<m 6: sib<u 5 ogi{<m?(, ) we<xht<tqupqbigs<osz<zl<ogim<ce< lx<oxivht<tqjbg<gi{<g/ kqi<u: Lkzqz<? sib<u 5 we wpkouil< (nkiuk, hgkqjb lqjgbigg< ogit<ouil<). (, ) we<x Ht<tqg<G hml< 7.9 z< dt<tk Ohiz< P weh<ohbiqm/ P bqzqvf<k 5 nzggt< x-ns<sqx<<g-j{biguzkhg<gligfgif<k )Woeeqz<Ym<ml< 5) Q ( + 5, ) nkiuk, Q (, ) we<x Ht<tqjbs< ose<xjmg/ Q uqzqvf<k nzgy-ns<sqx<<g-j{big gqpqxr<gq )Woeeqz<? -r<g Wx<xl< ) R(, + ( )) nkiuk R (, 0) we<x Ht<tqjbs< ose<xjmg/ R (, 0) we<hk Ogim<ce< Olzjlf<k lx<oxiv Ht<tq/ -kjes<siqhii<g<g?p, R Js<Osi<g<Gl< 0 Ogim<ce<sib<U. 5 ( ) hml< 7.9 lix<xupq;(, ) J(x, y ) we<g/ogim<cz<lx<oxivht<tqjb(x, y ) we<g/weou, y y y sib<u /Neiz<, sib<u wek<kvh<hm<mt<tk/ x x x + 5 y nz<zk x 6 5y 5 nz<zkx + 5y 9. -kqz<y x<glkq<h<h x + 5 ogimg<gx gqjmg<gl</wmk<kg<gim<mig, y 0 weqz<, x nz<zk x 9 nz<zkx. weou, (, 0) NeKOgim<ce<lQKlx<oxiVHt<tqbiGl</ 7.. Ofi<g<Ogim<ce<sle<hiM P(x,y) we<hk ogimg<gh<hm<m Ogim<cz< njlbl< lixgqe<x Ht<tqbig -Vg<gm<Ml</ lixqgt< x, y -ux<jxs< Osi<g<Gl< -bx<g{qks< sle<him ng<ogim<ce< sle<him NGl</ Ogim<cz< njlf<k yu<ouiv Ht<tqbqe< x-ns<sk~vl<, y-ns<sk~vl< NgqbjuOgim<ce<sle<him<jmfqjxUosb<Bl</-f<k(x, y) we<xuiqjsosicgjt gii<csqbe< ktk<kqz< Gxqg<g OfIg<Ogim<ce< ujvhml< gqjmg<gl</ -eq? Ogim<ce< sle<him, OgiMwe<Oxnjpg<gh<hMl</Ogim<ce<ujvhml<? x-ns<js A we<gqxyovyv Ht<tqbqZl<, y-ns<js B we<gqx yov yv Ht<tqbqZl< sf<kqg<gqxk/ A NeK x-ns<sqz< njlukiz< nke< y-ns<s K~vl< 0 NGl</ a we<hk A e< x-ns<sk~vl< weqz<, (a, 0) we<hkogim<ce<sle<him<jmfqjxuosb<bl</x g<g a JBl<, y g<g 0 JBl<Ogim<ce< sle<him<cz< hqvkqbqm, a bqe<lkqh<jhg<gi{zil</a bqe< lkqh<h, Ogim<ce< x oum<mk<k{<m weh<hml</ nkiuk, yv Ogim<ce< x oum<mk<k{<miek, OgimieK x- ns<js sf<kqg<gl< Ht<tqbqe< x-ns<sk~vl< NGl</ -Ok Ohie<X, B biek y-ns<sqz< -Vh<hkiz<nke< x-ns<sk~vl<0 NGl</weOu, b we<hkb bqe< y-ns<sk~vl<weqz<? (0, b) we<hkogim<ce<sle<him<jmfqjxuosb<bl</weou?x g<gh<hkqzig0 jubl<, 77

182 y g<gh<hkqzig b jbbl< Ogim<ce<sle<him<cz<hqvkqbqm, b e<kqijug<gi{zil</ b e< -f<k lkqh<h, Ogim<ce< y oum<mk<k{<m weh<hml</ weou? OgimieK wf<k Ht<tqbqz< y-ns<js oum<mgqxoki, nf<kh< Ht<tqbqe< y ns<sk~vl<, Ogim<ce< y oum<mk<k{<m weh<hml<. kx<ohipk sib<u m we<xl<, y oum<mk<k{<m c we<xl< ogi{<m Ogim<ce< sle<him<jmhqe<uvlixgi{<ohil</ Ogim<ce< y oum<mk<k{<m c big -Vh<hkiz< P (0, c) we<hk, OgimieK y-ns<js oum<ml<ohik (hml< 7.0 Jh< hiig<gul<) Wx<hMl< Ht<tq/ P(x, y) we<hk Ogim<ce< OuX WOkEl< yv Ht<tqbigm<Ml</hqe<ei< Ogim<ce<sib<U y c y c (n.k). x 0 x Neiz< Ogim<ce< sib<u m we<x hml< 7.0 ogimg<gh<hm<mt<tk/ y c m (n.k) y c mx (n.k) y mx + c. x OlZt<tsle<himieKyVOgim<ce<sle<him<ce<sib<U.oum<Mk<K{<M$k<kqvliGl</ Gxqh<H: OgimieKNkqh<Ht<tq (0, 0) upqbigs<ose<xiz<, nke< y oum<mk<k{<m c 0. weou?ogim<ce<sle<him y mx + 0 nz<zk y mx. wmk<kg<gim<m 7: sib<u NgUl<, y-oum<mk<k{<m NgUl<dt<tOgim<ce<sle<hiM gi{<g/ kqi<u: sib<u.oum<mk<k{<m$k<kqvk<jkh<hbe<hmk<k, Ogim<ce<sle<hiM y x + ( ) m nz<zk y x 6 nz<zk x y 6 0. c y mx + c wmk<kg<gim<m 8: x + 4y we<xogim<ce<sib<u, y-oum<mk<k{<mngqbux<jxg< gi{<g/ 5 kqi<u: sle<him<jmlix<xqwpkjgbqz<gqjmh<hk, 4y x 5 (n.k) y x f<ks<sle<him<jm, y mx + c dme<yh<hqmjgbqz<? sib<um, y-oum<mk<k{<m c hbqx<sq 7.. ogimg<gh<hm<m-vht<tqgjts<osig<gl<ogim<ce<sib<jug<gi{<g/ (i) ( 4, ) lx<xl< ( 5, ). (ii) (4, 8) lx<xl< (5, ). (iii) ( 5, 0) lx<xl< (0, 8). (iv) (0, 0) lx<xl< (, ). (v) (a, b) lx<xl< (a, b). (vi) (a, 0) lx<xl< (0, b). 78

183 . ogimg<gh<hm<mht<tqupqbigul<, kvh<hm<msib<jubl<ogi{<mogim<ce<lx<oxiv Ht<tqjbg<gi{<g/ (i) Ht<tq(5, 6), sib<u. (ii) Ht<tq (0, 4), sib<u 4. (iii) Ht<tq (, ), sib<u. (v) Ht<tq (, 4), sib<u 7. (iv) Ht<tq (, ), sib<u4.. sib<ul<, y- oum<mk<k{<ml<ljxobogimg<gh<hm<mt<te/ogim<ce<sle<him<jmg< gi{<g/ (i) lx<xl< 7. (ii) 5 lx<xl< 9. (iii) lx<xl< 5. (iv) 6 lx<xl<. (v) lx<xl<. (vi) lx<xl< ogimg<gh<hm<mogim<ce<sib<u, y- oum<mk<k{<mgi{<g/ (i) x + y 4 (ii) x y (iii) x y 0 (iv) 5x 4y 8 7. (x, y ) lx<xl< (x, y ) Ngqb -V Ht<tqgTg<G -jmobbt<t okijzu -V Ht<tqgTg<G -jmobbt<t okijzu we<hk ucuqbzqz< yv nch<hjmg< gvk<kvuigl</-h<ohiknkx<gie-bx<g{qkg<ogijujbg<g{<mxqouil</ P (x, y ), P (x, y ) we<hju gii<csqbe< ktk<kqz< njlf<k -V Ht<tqgtigm<Ml</ P, P -ux<xqx<g -jmobbt<t okijzju d(p, P ) nz<zk P P weg<gxqh<ohil</ P P we<xogim<mk<k{<jmujvg/&e<xujggt<wpgqe<xe/ ujg (i): Ogim<Mk<K{<M PP NeK x-ns<sg<g -j{big dt<tk )hml< 7. Jh< hiig<gul<). -r<g y y. P L, P M Ngqbux<jxx-ns<Sg<G osr<gk<kigujvg/p P ML yvosu<ugligl</ weoup P LM. nkiuk, d(p, P ) we<hkl lx<xl< M we<heux<xqx<g -jmobbt<t okijzu/ Neiz<, L lx<xl< M we<x Ht<tqgt< hml< 7. x-ns<sqe<lqkt<te/-h<ht<<tqgjtljxobx, x we<gqx olb<ob{<gt< x-ns<sigqb w{<ogim<cz< Gxqg<gqe<xe/ nkeiz< LM e< fqtl< LM x x. weoud (P, P ) x x. ujg (ii): P P we<gqx Ogim<Mk<K{<M y-ns<sg<g -j{big dt<tk (hml< 7. Jh< hiig<gul<). -r<gx x. P L, P MNgqbux<jx y-ns<sg<g osr<gk<kig ujvg/ P P ML yv osu<ugligl</weoup P LM. nkiuk, hml< 7. 79

184 d(p, P ) we<hk L lx<xl< M we<heux<xqx<g-jmobbt<tokijzu/neiz<, L lx<xl< M we<x Ht<tqgt< y-ns<sqe< lqkt<te/ -h<ht<tqgjt LjxOb y, y we<gqx olb<ob{<gt<y-ns<sigqbw{<ogim<cz<gxqg<gqe<xe/nkeiz<?lm e<fqtl< y y. weou d(p, P ) y y. ujg(iii): Ogim<Mk<K{<M PP NeK x-ns<sg<gl<-j{bqz<jz; y-ns<sg<gl< -j{bqz<jz(hml< 7. Jh<hiIg<gUl<). P upqbig x-ns<sg<g -j{big yv OgiMl<, P upqbigy-ns<sg<g-j{big yv OgiMl< ujvg. -g<ogimgt< P z< oum<mm<ml</ hqe<ei< P, (x, y ) NGl</ Ogim<Mk<K{<MP P e<fqtl< x x NGl</ Ogim<Mk<K{<M P P e< fqtl< y y. P P P we<hk yv osr<ogi{ Lg<Ogi{l< we<hjk fil< gi{<gqoxil</ weouhqkigv^<okx<xk<kqe<hc, [ d( P P )] [ d( P, P )] + [ d( P P )],, x x + y y (x x ) + (y y ) (x x ) + (y y ) d(p, P ) ( x x ) + ( y y. ) hml< 7. -KokijzUgi[l<$k<kqvl<NGl</-K(x, y ) lx<xl< (x, y ) NgqbogiMg<gh<hm<m -V Ht<tqgTg<G -jmobbt<t okijzu d Jk< kvl</ gqjmg<ogim<cozi nz<zk Gk<Kg<Ogim<cOzinjlbik-VHt<tqgTg<gie$k<kqvl<ohxh<hm<Mt<tK/-r<Gfil< gueqg<gou{<cbkd(p, P ) d(p, P ). OlZl< -f<k $k<kqvl< wf<k -V Ht<tqgTg<Gl< ohivf<kl</ P lx<xl< P NgqbjuyOvgqjmg<Ogim<cz<njlf<kiz<, y y NGl</Njgbiz< d(p, P ) ( x x) + ( y y) x x + 0 x x. P lx<xl<p NgqbjuyOvGk<Kg<Ogim<cz<njlf<kiz<, x x NGl</Njgbiz< d(p, P ) ( x x ) + ( y y ) 0 + y y y y. Gxqh<H: Nkqh<Ht<tq O NeK (0, 0). P (x, y) WOkEl<yVHt<tqweqz<, OP ( x 0) + ( y 0) x + y. -f<kk<okijzu x + y we<hk Nkqh<Ht<tqbqzqVf<K, (x, y) we<gqxht<tqbqe<njv oug<mi< weh<hml</ 80

185 okijzu$k<kqvk<jkh<hbe<hmk<kq, fil< (i) &e<xht<tqgt<yovogim<czjlgqe<xeuinz<zkyvosr<ogi{lg<ogi{l< nz<zk -Vslhg<g Lg<Ogi{l< nz<zk slhg<glg<ogi{k<jkdvuig<gli we<xnvibzil</ (ii) fie<g Ht<tqgt<, -j{gvl<, osu<ugl<, skvl< nz<zk sib<skvl< Ngqbux<jx dvuig<gliwe<xnvibzil</ wmk<kg<gim<m 9: A( 5, ) lx<xl< B (7, ) -ux<xqx<gqjmobbt<tokijzjug<gi{</ kqi<u: A lx<xl< B -ux<xqx<gqjmobbt<tokijzu d we<g/ d (A, B) ( x x) + ( y y) (x, y ) ( 5, ) ( 7 + 5) + ( + ) (x, y ) (7, ) wmk<kg<gim<m 0: ( 4, 9), (, 0) lx<xl< (4, ) NgqbjuyOvOgim<czjlBl<Ht<tqgt< weg<gim<mg/ kqi<u: A, B lx<xl< Cwe<heLjxOb?ogiMg<gh<hm<mHt<tqgtig-Vg<gm<Ml</ AB ( + 4) + (0 + 9) A ( 4, 9) likqiqh<hml< B (, 0) 7 9. hml<7.4 BC ( 4 ) + ( 0) C (4, ) AC ( 4 + 4) + ( + 9) r<g + 4. AB + BC AC we<hjkg<gi{<gqoxil</ A, B lx<xl< C yovogim<czjlbl<ht<tqgt</ wmk<kg<gim<m : (, ), (, 5) lx<xl< (8, 7) Ht<tqgt<yV-Vslhg<gLg<Ogi{k<jk njlg<gl<weg<gim<mg/ kqi<u: ogimg<gh<hm<m Ht<tqgt< LjxOb P, Q lx<xl< R we<g. PQR yv-vslhg<glg<ogi{l< we fq'hqg<g nkejmb -V hg<gr<gt< slfqtlt<tjuweg<gim<mzil<. -h<ohipk? d(p,q) ) + (5 ) ( d (Q, R) (8 ) + ( 7 5) ,, hml< 7.5 d (R, P) ( 8) + ( + 7) ( 5) d (P, Q) d (R, P) d (Q, R). PQR yv-vslhg<glg<ogi{l<(hml<7.5 Jh< hiig<gul<).neiz<slhg<glg<ogi{lz<z/ 8

186 wmk<kg<gim<m : (0, ), (0,) lx<xl< (,) Ngqbju yv slhg<g Lg<Ogi{k<kqe< ds<sqgt<weg<gim<mg/ kqi<u: ogimg<gh<hm<mht<tqgt<ljxob A, B lx<xl< C Ng-Vg<gm<Ml<. ABC yvslhg<glg<ogi{l<we fq'hqg<g nke< wz<zih< hg<gr<gtl< sl fqtlieju weg< gim<mzil</-r<g d (A, B) (0 0) + ( ) 0 + ( ) 4, d (B, C) ( 0) + ( ) + 4, d (C, A) (0 ) + ( ) + 4. d (A, B) d (B, C) d (C, A). ABC yvslhg<glg<ogi{l<(hml<7.6 Jh< hiig<gul<*/ hml< 7.6 wmk<kg<gim<m : P (7, ), Q ( 4, ) lx<xl< R (4, 5) yvosr<ogi{lg<ogi{k<kqe< ds<sqgtiwenvib<g/ kqi<u: PQRyVosr<Ogi{Lg<Ogi{l<weg<gim<mYIds<sqg<Ogi{l<90 weg<gim<m Ou{<Ml</-kx<GLg<Ogi{k<kqe<hg<gr<gtqe<fQtr<gt<hqkigv^<Okx<xk<jkfqjxU osb<bou{<ml</-r<og PQ ( 4 7) + ( ) , QR (4 + 4) + (5 + ) , PR (4 7) + (5 ) hml<8.7 PQ 5, QR 00 lx<xl< PR 5. QR + PR PQ we<hjkg<gi{<gqoxil</weouhqkigv^<$k<kqvl<siqhiig<gh<hm<mk/ PQR yvosr<ogi{lg<ogi{l<? R 90. wmk<kg<gim<m 4: (, ), (, ), (5, ) lx<xl< (4, 6) we<xuiqjsbqz<njlf<kht<tqgt< YI-j{gvk<kqjenjlg<Gl<weg<gim<Mg/-KyVosu<ugli@Nvib<g/ kqi<u:ogimg<gh<hm<mht<tqgt<ljxob P, P, P lx<xl< P 4 Ng-Vg<gm<Ml</ P P P P 4 YI-j{gvl<weg<gim<Mukx<GyVupqLjx?nkEjmbwkqi<hg<gr<gtqe< fqtr<gt<sll<weg<gim<mukigl</-r<og P P ( ) + ( ) + 9 0, P P (5 ) + ( + ) , P P 4 (4 5) + (6 ) + 9 0, P 4 P ( 4) + ( 6) P P P P 4 0 lx<xl< P P P 4 P 5. hml< 7.8 P P P P 4 we<hkyi-j{gvl</-r<g? P P (5 ) + ( ) lx<xl< (P P ) + (P P ) , (P P ) 7, (P P ) + (P P ) (P P ). P P P we<hkosr<ogi{lg<ogi{lz<z/ P P P we<hkosr<ogi{lz<z/ P P P P 4 we<hkyvosu<ugl<-z<jz. 8

187 wmk<kg<gim<m 5: (0, ), (, ), (6, 7) lx<xl< (8, ) we<x uiqjsbqz< wmk<kg< ogit<th<hm<mht<tqgt<yvosu<ugk<jknjlg<gl<weg<gim<mg/ kqi<u: ogimg<gh<hm<mht<tqgt<a, B, C lx<xl< D Ng-Vg<gm<Ml;. ABCD yvosu<ugl< weg<gim<mukx<g yv upqljx? nke< wkqih<hg<gr<gt< slfqtlt<tju we<hkigl</ OlZl< nke< YI ds<sqg<ogi{ ntu 90 we<hkigl</ YI ds<sqg<ogi{l< 90 weg<gim<mukx<g yv upqljx? ABC e< hg<gr<gt< hqkigv^< Okx<xk<jk fqjxu osb<gqxkweg<gim<muok/-r<ogfil<gi{<hk AB ( 0) + ( + ) , BC (6 + ) + (7 ) , CD (8 6) + ( 7) , AD (8 0) + ( + ) AC (6 0) + (7 + ) hml< 7.9 AB CD 5, BC AD 4 5 lx<xl< AB + BC AC. ABCD yvosu<ugl</neiz<skvlz<z/ wmk<kg<gim<m 6: (0, ), (, ) (0, ) lx<xl< (, ) we<xuiqjsbqz<wmk<kg<ogit<th<hm<m Ht<tqgt<yVsKvk<kqe<ds<sqgt<weg<gim<Mg/ kqi<u:ogimg<gh<hm<mht<tqgt<ljxob A, B, C, D we<g. ABCD yvskvl<weg<gim<mukx<gnkejmb hg<gr<gt< slfqtljmbju lx<xl< &jz uqm<mr<gt< slfqtlt<tju weg<gim<muk yv upqngl</-r<g AB ( 0) + ( + ) , BC (0 ) + ( ) , CD ( 0) + ( ) , AD ( 0) + ( + ) , BD ( ) + ( ) , AC (0 0) + ( + ) hml< 7.0 AB BC CD AD lx<xl< BD AC 4weg<gi{<gqOxil</ ABCD yvskvl</ wmk<kg<gim<m 7: A(, ), B(6, 5), C(, ) lx<xl< D( 6, 7) we<xuiqjsbqz<wmk<kg< ogit<th<hm<m Ht<tqgt< yv sib<skvk<jk d{<mig<gl<;neiz< nkskvlz<zweg< gim<mg/ kqi<u: ABCD yv sib<skvligl< weg< gim<mukx<g yv upq? nkejmb wz<zi hg<gr<gtl< sll< weg< gim<mukkie</ yv sib<skvliek? skvlz<z weg< gim<mukx<g yv upq? nkejmb &jzuqm<mr<gt< slfqtlz<zikju weg< gim<mukkie</-r<g AB ( 6 ) + (5 + ) , 8

188 BC AC BD CD AD 6) + ( 5) ( +, ) + ( ) 6 6 ( + ( +, , 6 6) + ( 7 5) ) + ( 7 ) ( +, 6 ) + ( 7 ) ( + +. AB BC CD AD, AC BD. ABCD we<hkyvsib<skvl<;neiz<nkskvlz<z/ hml<7. hbqx<sq7.. gqopogimg<gh<hm<mt<tosicht<tqgtg<g-jmobbt<tokijzjug<gi{<g/ (i) (, ) lx<xl< (4, ) (vi) (a, b) lx<xl< ( b, a) (ii) (, 4) lx<xl< ( 7, ) (vii) ( +,) lx<xl< (, ) (iii) ( 7, ) lx<xl< (, ) (viii) 5, lx<xl< (, ) 4 (iv) (4, 5) lx<xl< ( 4, 5) (ix) (, 0) lx<xl< (5, 4) (v) (a, b) lx<xl< (b, a) (x) (, ) lx<xl< (, 5).gQOpogiMg<gh<hm<mHt<tqgt<yOvOgim<czjlBl<Ht<tqgt<weg<gim<Mg/ (i) (5, ), (, ) lx<xl< (8, 8) (ii) (, ), (, ) lx<xl< (0, ) (iii) (, 4), (, ) lx<xl< (, 6) (iv) ( 4, 8), (, 4) lx<xl< (, 6) (v) (8, 4), (5, ) lx<xl< (9, 6).. gqop ogimg<gh<hm<m Ht<tqgt< -Vslhg<g Lg<Ogi{k<jk d{<mig<gli we Nvib<g/ (i) (5, 4), (, 0) lx<xl< (, ). (ii) (6, 4), (, 4) lx<xl< (, 0). (iii) (, ), ( 4, ) lx<xl< (, 5). 4. gqopogimg<gh<hm<mht<tqgt<slhg<glg<ogi{k<jkd{<mig<gliwenvib<g/ (i) (, ), (, ) lx<xl< (, 4). (ii) (, ), (0, ) lx<xl< (0, ). (iii) (0, ) (0, 5) lx<xl< (, 4). 5. gqopogimg<gh<hm<mht<tqgt<osr<ogi{lg<ogi{k<kqe<ds<sqgtiwenvib<g/ (i) (4, 4), (, 5) lx<xl< (, ). (ii) (, 0), (, ) lx<xl< (, 5). 6. gqopogimg<gh<hm<mds<sqgjtg<ogi{<mlg<ogi{l<wu<ujglg<ogi{l<weg< gi{<g/ (i) (, 7), ( 4, 0) lx<xl< ( 0, 8). (ii) ( 5, ), (0, 6) lx<xl< (8, ). 84

189 7. gqopogimg<gh<hm<muiqjsbqz<njlf<kht<tqgt<-j{gvk<jknjlg<gliwe Nvib<g/ (i) (, 5), ( 5, 4), (7, 0) lx<xl< (5, 9). (ii) (5, 8), (6, ), (, ) lx<xl< (, 6). (iii) (6, ), (5, 6), ( 4, ) lx<xl< (, ). (iv) (0, ), (4, 4), (6, ) lx<xl< (, ). 8. gqopogimg<gh<hm<muiqjsbqz<njlf<kht<tqgt<yvosu<ugk<jknjlg<gliwe Nvib<g/ (i) (8, ), (0, ), (, ) lx<xl< (6, 7). (iii) (, 0), (, ), (5, 6) lx<xl< (, 8). (ii) (, 7), (5, 4), (, 0) lx<xl< ( 8, 7). (iv) (, ), (0, 0) (, ) lx<xl< (, 4). 9. gqopogimg<gh<hm<muiqjsbqz<njlf<kht<tqgt<yvskvk<kqjenjlg<gliwe Nvib<g/ (i) (, ), (, ), (, ) lx<xl< (, ). (ii) (, 8), (4, 6), (, ) lx<xl< (, ). (iii) (, ), (0, 4), (7, ) lx<xl< (8, 0). (iv) (, 9), (0, 6), (5, 4) lx<xl< (, ). (v) (, ), (, 0), (, 4) lx<xl< (, ). 0. gqopogimg<gh<hm<muiqjsbqz<njlf<kht<tqgt<yvsib<skvk<kqe<ds<sqgti wenvib<g/ (i) (0, 0), (, 4), (0, 8) lx<xl< (, 4). (ii) (, ), (6, 5), (, ) lx<xl< ( 6, 7). (iii) (, 4), (5, ), (, ) lx<xl< (, ) 85

190 uqjmgt< hbqx<sq 7.. (i) I (ii) I (iii) III (iv) IV (v) wf<kgix<hgkqbl<-z<jz (vi) wf<kgix<hgkqbl<-z<jz (vii) wf<kgix<hgkqbl<-z<jz (viii) II. (i) kux (ii) kux (iii) kux(iv) siq (v) siq(vi) siq (vii) siq (viii) kux (ix) kux(x) kux(xi) kux (xii)siq. (i) (, ) (ii) (6, 9) (iii) (0, 7) (iv) (, ) hbqx<sq 7. 8 b b. (i) (ii) 6 (iii) (iv) (v) 4 (vi) 5 a a. (i) (6, 7) (ii) (4, 5) (iii) (, ) (iv) (, ) (v) (, ). (i) x + y (ii) 5x y (iii) x + y 5 0 (iv) 6x y 0 (v) x + 5y 5 0 (vi) x + 5y (i), 5 (ii) (, 0) (iii) (, ) (iv), 4 hbqx<sq 7.. (i) 0 (ii) 6 (iii) 0 (iv) 4 (v) (a b) 48 (vi) (a + b) (vii) 6 (viii) (ix) 5 (x) 65.. (i) yovogim<czjlbl<(ii) yovogim<czjlbik(iii) yovogim<czjlbl< (iv) yovogim<czjlbik (v) yovogim<czjlbik. (i) -Vslhg<gLg<Ogi{l<(ii) -Vslhg<gLg<Ogi{l< (iii) -Vslhg<gLg<Ogi{l< 4. (i) slhg<glg<ogi{l< (ii) slhg<glg<ogi{l< (iii)slhg<glg<ogi{l< 5. (i) osr<ogi{lg<ogi{l< (ii) osr<ogi{lg<ogi{l<nz<z 6. (i) -Vslhg<gosr<Ogi{Lg<Ogi{l<(ii) -Vslhg<gosr<Ogi{Lg<Ogi{l< 7. (i) -j{gvl< (ii) -j{gvl< (iii) -j{gvl< (iv) -j{gvl< 8. (i) osu<ugl<(ii) osu<ugl< (iii) osu<ugl< (iv) osu<ugl< 9. (i) skvl< (ii) skvl<(iii) skvlz<z(iv) skvl<(v) skvlz<z 0. (i) sib<skvl< (ii) sib<skvl< (iii) sib<skvl< 86

191 8. Lg<Ogi{uqbz< hz F~x<xi{<MgTg<G Le<Oh uieuqbz< hx<xq nxqukx<g? g{qkuqbzqz< Lg<Ogi{uqbz<we<El<hqiqUOkix<Xuqg<gh<hm<mK/aqh<hiIg<g^<)Hipparchus*we<El< gqovg<g uieuqbz< lx<xl< g{qk uz<zfi<? Lg<Ogi{uqbjz uqiquig<gq nke< uqkqgjth< ohvltu hbe<hmk<kq uie<outqh< ohivm<gtqe<-br<g hijkgjtbl<? fqjzgjtbl< fqi<{bl< osb<kii</ weou -ujvob? Lg<Ogi{uqbzqe< kf<jk we<xjph<hk ohivk<kliekigl</ Lg<Ogi{uqbjzg< Gxqg<Gl< Nr<gqzs< osiz<zie Trigonometry NeK? Lg<Ogi{l< we<x ohivtjmb ÄTrigon}lx<Xl<ntUwe<x ohivtjmb ÄMetra} Ngqb -V gqovg<gs< osix<gtqzqvf<k ohxh<hm<mkigl</ weou Lg<Ogi{uqbz<we<hKyVLg<Ogi{k<kqe<hg<gr<gtqe<ntUgTg<Gl<?Ogi{r<gtqe< ntugtg<gl< -jmob njlf<k okimi<hgjth<hx<xq nxqbl< himh<hgkqbigl</ Lg<Ogi{uqbz<himk<kqjehcg<gk<okimr<Gukx<GLe<?fil<Ogi{r<gt<?nux<xqe< ntugt<ngqbjuhx<xqwx<geounxqf<kjklq{<ml<fqjeuqx<ogit<ouil</ Ogi{r<gTl<nux<xqe<ntUgTl< yv ohikh< Ht<tqbqzqVf<K okimr<gl< -V gkqi<gtiz< yv Ogi{l< njlg<gh<hmgqxk/ yv gkqi<? Ogi{k<kqe< okimg<gg<gkqi )okimg<gh<hg<gl<* weul<? lx<oxivgkqi<?ogi{k<kqe<lcug<gkqi)lcuh<hg<gl<*weul<%xh<hml</ohikh<ht<tq? Ogi{k<kqe< Lje weh<hml</ LjebqzqVf<K okimr<gl< gkqovie<x okimg<gh< hg<g fqjzbqzqvf<klcuh<hg<gfqjzujvspzukiz<ogi{l<njlgqe<xk/ hml<8. hml<8. gkqiqe< Spx<sq gcgiv Lt< Spx<sqbqe< wkqikqjsbqozi )hml< 8. Jh< hiig<gul<* nz<zk gcgiv Lt< Spx<sqbqe<< kqjsbqozi )hml< 8. Jh< hiig<gul<* njlujk nxqbzil</ gkqi<gt< OA Ul< OB Bl< LjxOb? yv Ogi{k<kqe< okimg<gg<jg? LcUg<jgweqz<?Ogi{k<jkg<GxqbQm<cz< AOBweg<Gxqh<Ohil</ 87

192 nu<uh<ohik? gii<csqbe< ktk<kqz< Nkqh<Ht<tqjb LjebigUl<? x-ns<js okimg<gg<gkqvigul< ogi{<m? hml< 8. -z< gi{<hkohiz< Ogi{k<kqje njlh<ohil</ yv Ogi{lieK Olx<%xqb upqbqz< fqjzh<hmk<kh<hce<? ng<ogi{liek kqm<m fqjzbqz< dt<tk we<ohil</ yv Ogi{k<jk ntg<g hijg we<xjpg<gh<hml<nzgqjeh<hbe<hmk<kgqe<oxil</ hijgntu hml<8. yv gkqi<? gcgivlt<spx<sqbqe< wkqi< kqjsbqz< yv LP Spx<sqjb Wx<hMk<Kl< ohipk? nr<g 60 hijggt< )-K 60 o we wpkh<hml<* ntut<t yv Ogi{l< njlukigg< %Xgqe<Oxil</ lx<x Ogi{r<gjt 60 o Ogi{k<jk nch<hjmbigg< ogi{<m ntg<gqe<oxil</ wmk<kg<gim<mig? yv gkqi< Spx<sqjb Wx<hMk<kuqz<jzobeqz<? nk 0 o ntut<t Ogi{l< njlg<gqe<xk we<ohil</ yv gkqi< gcgivlt< Spx<sqbqe< wkqi<kqjsbqz<?lpspx<sqbq<z<¼higl<spx<sqjbwx<hmk<kqeiz<?nk¼ (60 o * 90 o ntut<togi{l<njlg<gqe<xkwe<ohil</yvgkqigcgivlt<spx<sqbqe<kqjsbqz<? LPs<Spx<sqbqz<¼higl<Spx<sqjbWx<hMk<Klieiz<?nK¼ ( 60 o * 90 o ntut<t Ogi{l<njlg<gqe<xKwe<Ohil</weOu?gcgivLt<Spx<sqbqe<wkqi<kqjsbqz<njlBl< Spx<sqgt< lqjg Ogi{r<gjt d{<mig<ggqe<xe weul<? gcgivlt< Spx<sqbqe< kqjsbqz< njlbl< Spx<sqgt< Gjx Ogi{r<gjt d{<mig<ggqe<xe weul< nxqgqe<oxil</ -u<uk<kqbik<kqz< fil<? 0 o zqvf<k90 o ujvnjlbl<ogi{r<gjt lm<molgvkgqe<oxil</ 0 o zqvf<k90 o g<gt<njlbl<ogi{l<gxr<ogi{l<weh<hml</ 90 o ntuqz< njlbl< Ogi{l< osr<ogi{l< weh<hml</80 o ntuqz< njlbl< Ogi{l< Ofi<g<Ogi{l< weh<hml</ -V GXr<Ogi{r<gtqe< %Mkz< 90 o weqz<?nju fqvh<hg<ogi{r<gt<weh<hml</-vlqjgogi{r<gtqe<%mkz<80 o weqz<?njulqjg fqvh<hg<ogi{r<gt<weh<hml</ osr<ogi{lg<ogi{ll<?hqkigv^<okx<xll< yv Lg<Ogi{k<kqz< yv Ogi{k<kqe< ntu 90 o weqz<? nl< Lg<Ogi{l< yv osr<ogi{ Lg<Ogi{l< weh<hml</ hml< 8.4 z< ABC yv osr<ogi{ Lg<Ogi{l<? -kqz< ABC we<x Ogi{k<kqe< ntu 90 NGl</ hg<gl< AC gi<{l<weh<hml</-kou hml<8.4 88

193 lqgh<ohiqb hg<gl<? OlZl< -K osr<ogi{k<kqx<g wkqov njlf<kt<t hg<gligl</ hqkigv^<we<xgqovg<gg{qkuz<zfi<?gi<{k<kqe<lqknjlbl<skvk<kqe<hvh<htu? lx<x -V hg<gr<gtqe< lqkjlbl< skvr<gtqe< hvh<htugtqe< %MkZg<Gs< sll< weg< g{<mxqf<kii</ nkiuk? hml< 8.4-z<? AC AB + BC NGl</-KOuhqkigv^< Okx<xl<weh<hMl</ 8. Lg<Ogi{uqbz<uqgqkr<gt< WkiuK yv GXr<Ogi{l< AOB J wmk<kg<ogit<ouil</ -jk gqovg<g wpk<kie θ Nz< Gxqh<Ohil</ P we<el< Ht<tq gkqi< OB bqe< lqk njlf<k Ht<tq we<g/ gkqi< OA Ug<Gs<osr<Gk<kigP bqzqvf<kogim<mk<k{<m PQ ujvg/ hqxg Lg<Ogi{l< OQP yv osr<ogi{ Lg<Ogi{l<A Lje Q uqz< osr<ogi{l< njlf<kt<tk/ OQP -z< hml< 8.5 hg<gl<op gi<{l<ngl</hg<gl< PQ Ogi{l< θ uqe< wkqi<hg<gligl</ hg<gl< OQ Ogi{l< θ uqe< nmk<kt<t hg<gligl</ OP, PQ, OQ Ngqb hg<gr<gtqe< fqtr<gjt LjxOb OP, PQ, OQ we<x Gxqh<hqMgqe<Oxil</-f<fQtr<gjtg<ogi{<Mhqe<uVl<NXLg<Ogi{uqbz<uqgqkr<gjt ujvbxg<gqe<oxil</ wkqihg< gk< kqe< fqtl< PQ sine θ, gi< {k< kqe< fqtl< OP nmk< Kt< thg< gk< kqe< fqtl< OQ cosine θ, gi< {k< kqe< fqtl< OP wkqi< hg< gk< kqe< fqtl< PQ tangent θ, nmk< Kt< thg< gk< kqe< fqtl< OQ gi< {k< kqe< fqtl< OP cosecant θ, wkqi< hg< gk< kqe< fqtl< PQ gi< {k< kqe< fqtl< OP secant θ, nmk< Kt< thg< gk< kqe< fqtl< OQ nmk< Kt< thg< gk< kqe< fqtl< OQ cotangent θ. wkqi< hg< gk< kqe< fqtl< PQ Olx<gi[l<uqgqkr<gjts<?SVg<gligLjxOb?sinθ, cosθ, tanθ, cosecθ, secθ, cotθ wewpkouil</olx<gi[l<uqgqkr<gtqe<ntugt<?ogi{l<θ ju lm<mol siif<kjubigl<a osr<ogi{ Lg<Ogi{l< OQP -e< ntjus< sii<f<kkz<z/ -kje nxqb? P we<el< OuoxiV Ht<tq gkqi< OB bqe< lqk wmk<kg<ogit<g; OlZl< P Q J OA g<gosr<gk<kig ujvg (hml< 8.5 Jh<hiIg<gUl<). Lg<Ogi{l< OQP l< Lg<Ogi{l< OQ P l<ucouik<klg<ogi{r<gt<we<hkoktquigl</-kqzqvf<kfil<? PQ P Q OQ OQ OP OP 89

194 wenxqouil</-kqzqvf<kfil<ohxl<uqgqkr<gt< PQ P Q OQ OQ PQ P Q,, OP OP OP OP OQ OQ OP OP OP OP OQ OQ nz<zk,, PQ P Q OQ OQ PQ P Q wenxqbzil</weouolx<%xqbnxuqgqkr<gtl<ht<tq P, gkqi< OB bqe<lqkwf<k fqjzbqz< njlf<kqvf<kizl< lixik we nxqbzil</ Olx<gi[l< uqgqkr<gtqzqvf<k filxquk? PQ OP sin θ cosecθ, cosec θ, sin θ. OP PQ sin θ cos ecθ OQ OP cosθ secθ, sec θ, cos θ. OP OQ cosθ secθ PQ OQ tan θ cot θ. cot θ, tan θ. OQ PQ tanθ cot θ PQ sin θ PQ OP PQ OlZl< fil< gueqh<hk? OP tan θ. kjzgqpqgjt wmg<g? cos θ OQ OP OQ OQ OP cosθ sin θ cosθ fil<ohxuk cot θ. weou? tan θ, cot θ NGl</ sin θ tan θ cosθ sin θ Gxqh<H: θ yvgxr<ogi{ligul<?θ uqe<lg<ogi{uqbz<nxuqgqkr<gtqz<ye<xqe< lkqh<h lm<mol okiqf<k fqjzbqz<? OlOz %xqb $k<kqvr<gjth< hbe<hmk<kq lx<x Lg<Ogi{uqbz< uqgqkr<gjtg< gi{ -bzl</ fl< himh< hgkqbqz<? GXr<Ogi{r<gjt lm<molgvkgqe<oxil</ wmk<kg<gim<m : hml< 8.6 -z< kvh<hm<mt<t osr<ogi{ Lg<Ogi{k<kqzqVf<K θ uqe< NXLg<Ogi{uqbz<uqgqkr<gjtg<gi{<g/ kqi<u;hmk<kqz<ogi{l< θ uqx<g wkqi<hg<gfqtl< 6; nmk<kt<thg<gfqtl< 8. hqkigv^<okx<xh<hc? (gi<{k<kqe<fqtl<) gi<{k<kqe<fqtl< weou? wkqi< hg< gfqtl< 6 sin θ, gi< {k< kqe< fqtl< 0 5 nmk< Kt< thg< gfqtl< 8 4 cos θ, gi< {k< kqe< fqtl< 0 5 wkqi< hg< gfqtl< 6 tan θ, nmk< Kt< thg< gfqtl< 8 4 hml<8.6 gi< {k< kqe< fqtl< 0 5 cosec θ, wkqi< hg< gfqtl< 6 gi< {k< kqe< fqtl< 0 5 sec θ, nmk< Kt< thg< gfqtl< 8 4 nmk< Kt< thg< gfqtl< 8 4 cot θ. wkqihg< gfqtl< 6 90

195 wmk<kg<gim<m: ABC bqz< m B 90, AB 8 os/lq/, AC 7 os/lq/weqz<?ogi{l< A lx<xl< Ogi{l<C gtqe<njek<klg<ogi{uqbz<uqgqkr<gjtbl<gi{<g/ kqi<u; hmk<kqz< A m BAC lx<xl< C m BCA (hml< 8.7 Jh< hiig<gul<). hqkigv^<okx<xh<hc, AC AB + BC BC AC AB BC 5 5. weou? BC 5 AB 8 BC 5 sin A, cos A, tan A, AC 7 AC 7 AB cot A, sec A, cos ec A, tan A 5 cos A 8 sin A 5 AB 8 BC 5 AB 8 sin C, cos C, tan C, AC 7 AC 7 BC cot C, sec C, cos ec C, tan C 8 cos C 5 sin C 8 hml< 8.7 Gxqh<H: Olx<g{<m g{g<gqz<? sin C cos A, cos C sin A, tan C cot A, we Ofig<Ggqe<Oxil</-u<uixqVh<hkx<Gg<giv{l<ABl<CBl<fqvh<Hg<Ogi{r<gt<. 7 wmk<kg<gim<m: sin θ weqz<?lx<xlg<ogi{uqbz<uqgqkr<gjtg<gi{<g/ 5 wkqih< hg< gfqtl< 7 kqi<u; sin θ we<hkiz<? m ABC 90, m ACB θ? AB 7, gi{k< kqe< fqtl< 5 AC 5 we<xt<tuix yv osr<ogi{ Lg<Ogi{l< ABC Jg< gvkul< (hml< 8.8Jh< hiig<gul<).weou?hqkigv^<okx<xh<hc? AC AB + BC BC nz<zk BC. BC BC BC 4 weou? cos θ, AC 5 AB 7 5 tan θ, secθ, BC 4 cosθ 4 5 cosecθ, 4 cot θ. hml< 8.8 sin θ 7 tanθ 7 wmk<kg<gim<m4: cosec A weqz<? (i) sin A + cos A (ii) tan A + cot A Ngqbux<xqe< lkqh<hgjtg<gi{<g/ kqi<u; cosec A gi< {k< kqe< fqtl< wkqi< hg< gfqtl<. weou?m QRP A, gi{k<kqe<fqtl< PR, wkqih<hg<gk<kqe< fqtl< PQ we njlblix yvosr<ogi{lg<ogi{l< PQR Jg<gVKOuil< (hml< 8.9 Jh<hiIg<gUl<). hml<8.9 9

196 hqkigv^<okx<xh<hc, PR PQ + QR. ( ) () + QR. + QR. QR. QR. weou? (i) sin A + cos A sin A tan A PQ PR PQ QR QR, cos A, PR QR, cot A. PQ +, (ii) tan A + cot A +. Gxqh<H; WOkEl< yv sle<him<jm fq'hqg<g Ou{<Ml< weqz<? gqp<g<gi[l< upqgtqz< ye<xqjeh<hbe<hmk<kkz<ou{<ml</ upq; sle<him<ce<-mkhg<gg<ogijujbnz<zkuzkhg<gg<ogijujbsvg<gl< osb<klx<xhg<gk<kqz<dt<togijujbh<ohxkz<ou{<ml</ upq; sle<him<ce<-mkhg<gk<kqz<dt<togijujbsvg<gl<osb<kyvucul<() ogi{i<ouil</nmk<kkig?uzkhg<gk<kqz<dt<togijujbsvg<gl<osb<kyv ucul<()ogi{i<ouil</hqe<ei<ucul<()ucul<()weg<gim<mouil</ tan A+ tan B sin Acos B + cos Asin B wmk<kg<gim<m5: wefq'hq/ tan A tan B cos Acos B sin Asin B kqi<u; sin A sin B sin Acos B + cos Asin B + tan A+ tan B -mkhg<gogiju cos A cos B cos Acos B tan Atan B sin A sin B cos Acos B sin Asin B cos A cos B cos Acos B (sin Acos B + cos Asin B) cos Acos B cos Acos B cos Acos B sin Asin B sin Acos B + cos Asin B uzkhg<gogiju. cos Acos B sin Asin B wmk<kg<gim<m6: tan A+ cot B tan A wefq'hqg<gul</ cot A + tan B tan B tan A tan Atan B + + tan A+ cot B kqi<u; -mkhg<gogiju tan B tan B cot A + tan B tan B + tan Atan B + tan A tan A (tana tan B + ) tan A tan A uzkhg<gogiju. tan B ( + tan A tan B) tan B 9

197 + tan θ sin θ + tan θ wmk<kg<gim<m7: wefq'hqg<gul</ + cot θ + cosθ + tan θ + tanθ ( + tanθ) + tanθ tanθ kqi<u; -mkhg<gogiju + cot θ + tan θ + + tanθ tan θ tan θ tan θ () sin θ sinθ + tanθ sin θ + uzkhg<gogiju cosθ (sin θ cosθ + sin θ) + cosθ + cosθ cosθ ( + cosθ) sin θ (cosθ + ) sin θ tan θ. () cosθ (+ cosθ) cosθ () lx<xl< ()e<hc-mkhg<gogiju uzkhg<gogiju. sqzgxqh<hqm<mogi{r<gtqe<lg<ogi{uqbz<uqgqkr<gt< 0, 45 lx<xl< 60 < ntut<t Ogi{r<gtqe< Lg<Ogi{uqbz< uqgqkr<gtqe< lkqh<hgjtg<gi{<ohil</ (i) 0 lx<xl< 60 Ogi{r<gtqe<Lg<Ogi{uqbz<uqgqkr<gt< hg<g fqtl< nzggt< ogi{<m slhg<g Lg<Ogi{l< ABC jbwmk<kg<ogit<ouil< (hml< 8.0Jh< hii<g<gul<). hg<gl< ABg<G Lje C bqzqvf<kosr<gk<kg<ogimcdujvg. Ht<tqD, hg<gl< AB e<jlbh<ht<tqbigl</-h<ohipk AD, AC, m DAC 60, m ACD 0. ADC yv osr<ogi{ Lg<Ogi{l< (hml< 8.Jh< hii<g<gul<).hqkigv^<okx<xh<hc, AC AD + DC n.k + DC. DC n.k DC. hml<8.0 osr<ogi{lg<ogi{l<adc bqzqvf<k? sin 60 DC AC DA cos 60 AC DC tan 60 DA DA cot 60 DC AC sec 60 DA cosec 60 AC DC AD sin 0 AC DC cos 0 AC AD tan 0 DC DC cot 0 AD AC sec 0 DC AC cosec 0 AD hml<8. 9

198 (ii) 45 Ogi{k<kqe<Lg<Ogi{uqbz<uqgqkr<gt< ABC YI-Vslhg<gosr<Ogi{Lg<Ogi{l<we<g/-r<Gm B 90, AB BC (hml< 8.Jh<hiIg<gUl<). -r<gac, m CAB 45?m BCA 45 wenxqf<kogit<g/ Lg<Ogi{l<ABC bqzqvf<kfil<ohxuk? AB sin 45 AC AB cos 45 AC BC tan 45 AB AB cot 45 BC AC sec 45 AB AC cosec 45 BC hml<8. (iii) 0, 90 Ogi{r<gtqe<Lg<Ogi{uqbz<uqgqkr<gt< -g< Ogi{r<gtqe< Lg<Ogi{uqbz< uqgqkr<gjth< ohxukx<g giicsqbe< ns<s ktk<kqz< Nkqh< Ht<tqjb jlbligul<? Nv ntu r nzgl< ogi{<m yv um<mk<kqjeg< gvkg/ giicsqbe< ktk<kqe< Lkx< gix<hgkqbqz< njlf<k -u<um<mk<kqe<uqz<abbqe<lqkpwe<hkwokel< yvht<tqwe<g)hml<8.jh<hii<g<gul<*/ x ns<sqx<g PM we<x Gk<Kg<OgiM ujvg/ P we<x Ht<tqbqe< ns<s K~vr<gt< x, y we<g/ hqe<ei< OM x, PM yngl</osr<ogi{lg<ogi{l< OMP -z<?hqkigv^<okx<xh<hc? x + y r weh< ohxgqoxil</ r x + y hml<8.. MOP θ we<g/θ yvgxr<ogi{ligl</ sin[θ r y, cos θ r x. uqz<abbqe<lqkouu<ouxfqjzgtqz<p Jg<ogi{<miz<?fil<gueqh<hK?gkqi<OP NeKfqjzOA uqzqvf<kfqjzob g<g Spx<sqbjmBl<ohiPK?Ogi{l<θ NeK0 ntuqzqvf<k 90 ntuqx<g nkqgiqg<gqe<xka fqtl< x NeK r zqvf<k 0 uqx<g Gjxgqe<xKAy NeK0 uqzqvf<kr x<gnkqgiqg<gqe<xk/weou?θ NeK0 zqvf<k 90 g<gnkqgiqg<gl<ohipk? r x e<ntuzqvf<k0ug<ggjxujkbl<? r y e<ntu 0zqVf<K x<g nkqgliujkbl< nxqgqe<oxil</ nkiuk θ uqe< ntu 0 o zqvf<k 90 o g<gnkqgligl<ohipk?cos θ uqe<ntuzqvf<k0uqx<ggjxujkbl<?sin θ uqe< ntu 0 zqvf<k x<g nkqgliujkbl<? nxqgq<e<oxil</ OlZl<? yu<ouiv GXr<Ogi{l< θ uqx<g x, y l<? Lg<Ogi{uqbz< uqgqkr<gtl< keqk<kjugtig njlbl< we nxqgqe<oxil</ OP NeK OA uqe< fqjzjb njmbl<ohipk? θ 0, y 0 x r, y 0 NGl</ weou? sin 0 o o x r 0, cos 0. OP NeK OB bqe< r r r r fqjzjbnjmbl<ohipk?θ 90,x 0, y r NGl</weOu? o y r o x 0 sin 90, cos90 0. r r r r 94

199 -h<ohipk?fil<ohxuk? o sin 0 tan 0 cos 0 o cos 0 cot 0 sin 0 o sec 0 cos 0 cosec 0 tan 90 o o o o o o o sin 0 0 0,, ujvbxg<g-bzikk? 0 ; o sin 90 cos 90 o o 0 0 o o cos 90 cot 90 o sin 90 o sec 90 o cos 90 o cosec 90 o sin 90 0, ujvbxg<g-bzikk, ujvbxg<g-bzikk, 0 0,, ujvbxg<g-bzikk,. 0, 0, 45, 60, 90 e< Lg<Ogi{uqbz< uqgqkr<gjt gqp<g<g{<muix nm<muj{h< hmk<kzil</ θ sin θ 0 cos θ tan θ 0 cot θ sec θ cosec θ ujvbxg<g -bzikk ujvbxg<g -bzikk 0 ujvbxg<g -bzikk 0 ujvbxg<g -bzikk GxqbQM; (sin θ) we<hkje sin θ we SVg<glig wpkgqe<oxil</ -jkh<ohie<ox lx<xhcgtg<gl< wpkgqe<oxil</ wmk<kg<gim<mig? (tanθ ) tan θ. cos 4 θ we<hk (cos θ ) 4 Jg<Gxqh<hkiGl</sin θ Jsinθ wewpkukkuxigl</ 95

200 wmk<kg<gim<m9: lkqh<hgi{<g; cos 0 tan 60 sec 45 sin 60. kqi<u; cos 0, tan 60, sec 45, sin 60. cos 0 tan 60 sec 45 sin 60 ( ) ( ) wmk<kg<gim<m:; tan A o sin 60 + cos60 o weqz<?gxr<ogi{l<ajbg<gi{<g/ kqi<u; sin 60 tan A + Neiz< tan 0, cos A 0. wmk<kg<gim<m0: sin (A + B), cos B, weqz< A, B gi{<g/ kqi<u; sin (A + B) we<hkiz<?sin (A + B). Neiz<? sin 60. weou A + B 60. cos B, we<hkiz< cos B Neiz< cos 45 weou B 45.. (), ()Jk<kQi<g<g? A 5.. () () 96

201 hbqx<sq 8. g{g<ggt< Lkz<4 -ux<xqzt<tosr<ogi{lg<ogi{r<gtqz<gxqh<hqm<mt<t Lg<Ogi{uqbz<uqgqkr<gjtg<g{<Mhqc/.. hml< 8.4 sin B, cos C, tan B hml<8.5 sec X, cot Z, cosec Z. 4. hml< 8.6 cos Q, tan R, cot Q hml< 8.7 tan M, sec N, cosec N g{g<ggt< 5 -zqvf<k 0 ujv ogimg<gh<hm<m uqgqkl< Ohig θ uqe< lx<x Lg<Ogi{uqbz<uqgqkr<gjtg<gi{<g/ 5. cos θ 6. sin θ 5 8. cosec θ 0 9. cot θ 7 7. sec θ 0. tan θ 5 5 sec A+ tan A. cos A weqz<? Jg<g{<Mhqcg<gUl</ 7 sec A tan A cos ecθ. sin θ weqz<? Jg<g{<Mhqcg<gUl</ 5 cot θ cosθ 97

202 sin θ. cosec θ weqz<? cot θ + Jg<g{<Mhqcg<gUl</ + cosθ 4. cot θ cos θ weqz<? weg<gi{<g/ sin θ 5 sin θ + cosθ 5. cot θ 4 weqz<? secθ + cos ecθ lkqh<jhg<g{<mhqcg<gul</ 6. lkqh<hgi{<g; (i) cosec 45 cot 0 + sin 60 sec 0 (ii) cos 0 sin 0 cos 60 (iii) 8 sin 60 cos 60 (iv) o tan 45 o o tan 0 + tan hqe<uvueux<jxsiqhii<g<gul<; (i) sin 0 + cos 0 (ii) sec 60 tan 60 (iii) + cot 0 cosec 0 8. sin (A+B) sin (A B) weqz<? A, B gjtg<g{<mhqcg<gul</ 8. Lg<Ogi{uqbz<Lx<oxiVjlgt< nch<hjmbie &e<x Lg<Ogi{uqbz< Lx<oxiVjlgjt yv GXr<Ogi{l< θ uqx<gohxouil</nbqel<?wz<ziogi{r<gtg<gl<-juohivk<klxl</ θ e<ljejbnkqh<ht<tqwe<g/θ e<okimg<gg<jgjb x ns<sigogit<ouil</ P (x, y) we<hkθ e<lcug<jgbqe<lqkjlf<kwokel<yvht<tqwe<g(hml< 8.8 Jh< hii<g<gul<). PbqzqVf<K xns<sg<g? Gk<Kg<OgiM PQ ujvg/ hqe<ei< OQ x, PQ y. OP r we<g/hqkigv^<okx<xh<hc? osr<ogi{lg<ogi{l<oqp -z<? x + y r NGl</ -Vhg<gLl< r Nz<uGg<g? x + y r x y hml< 8.8 n.k + r r r r x y n.k +. r r Neiz<? sin θ r y, cos θ r x. (cos θ ) + (sin θ ). nkiuk? cos θ + sin θ. () 98

203 () e<-vhg<gll< cos θ Nz<uGg<g? cos θ + sin cos θ θ cos θ sin θ cos cos + θ θ n.k cos cos θ θ sin + cos θ θ cos n.k +(tan θ ) (sec θ ) θ nkiuk? + tan θ sec θ. () () e<-vhg<gll< sin θ Nz<uGg<g? cos θ + sin θ cos θ sin θ n.k + (cosecθ) sin θ sin θ sin θ sin θ sin θ n.k cot θ + cosec θ nkiuk? + cot θ cosec θ. () -f<k &e<x Lx<oxiVjlgt< (), () lx<xl< () Ngqbju hqkigv^< Okx<xk<jk nch<hjmbigg< ogi{<mjubigl</ -ux<xqzqvf<k OlZl< sqz Lx<oxiVjlgjt nxqf<kogit<ouil</ () Jh<hbe<hMk<kq (i) sin θ (sin θ + cos θ ) cos θ cos θ (ii) cos θ (cos θ + sin θ ) sin θ sin θ wenxqbzil</ () Jg<ogi{<M (i) tan θ ( + tan θ ) sec θ (ii) sec θ tan θ ( + tan θ ) tan θ wenxqbzil</ () Jg<ogi{<M(i) cot θ ( + cot θ ) cosec θ (ii) cosec θ cot θ ( + cot θ ) cot θ wenxqbzil</ Olx<gi[l<Lx<oxiVjlgjth<hqe<uVliXnm<muj{h<hMk<kzil</ sin θ + cos θ + tan θ sec θ + cot θ cosec θ sin θ cos θ tan θ sec θ cot θ cosec θ cos θ sin θ sec θ tan θ cosec θ cot θ OlZl<? -f<k Lx<oxiVjlgjt -mh<hg<gk<kqzqvf<k uzh<hg<gk<kqx<gl<? uzh<hg<gk<kq zqvf<k-mh<hg<gk<kqx<gl<hbe<hmk<kouil</nkiuk?wmk<kg<gim<mig? sec θ tan θ we<xl< sec θ tan θ weul< hbe<hmk<kouil</ -mkhg<g Ogijujb-eq-/h/Ogiwe<Xl<?uzKhg<gOgijujbu/h/Ogiwe<Xl<Gxqh<Ohil</ wmk<kg<gim<m : fq'hqg<g; sin 4 θ + cos 4 θ sin θ cos θ. kqi<u: -/h/ogi sin 4 θ + cos 4 θ (sin θ ) + (cos θ ) [sin θ + cos θ ] (sin θ )(cos θ ) ( a + b (a + b) ab) () sin θ cos θ sin θ cos θ u/h/ogi/ 99

204 cosθ wmk<kg<gim<m : fq'hqg<g; sec θ tan θ. + sin θ cosθ cosθ sin θ kqi<u: -/h/ogi + sin θ + sin θ sin θ cosθ ( sin θ) cosθ ( sin θ) sin θ cos θ sin θ sin θ sec θ tan θ u/h/ogi. cos θ cosθ cosθ + cos A wmk<kg<gim<m : fq'hqg<g; (cos ec A+ cot A). cos A + cos A + cos A kqi<u: -/h/ogi cos A + cos A ( + cos A) ( + cos A) + cos A cos A sin A sin A (cosec A + cot A) u/h/ogi. OuXupqbig? cos A + sin A sin A u/h/ogi (cosec A + cot A) cos A + cos A + sin sin A A sin A ( + cos A) (+ cos A) (+ cos A) sin A cos A (+ cos A)( cos A) + cos cos wmk<kg<gim<m 4: fq'hqg<g; sin 4 θ cos 4 θ sin θ cos θ. kqi<u: -/h/ogi sin 4 θ cos 4 θ (sin θ ) (cos θ ) (sin θ + cos θ ) (sin θ cos θ ) () (sin θ cos θ ) sin θ cos θ u/h/ogi. wmk<kg<gim<m 5: fq'hqg<g; sec A tan A sec A + tan A sec A tan A kqi<u: u/h/ogi sec A + tan A sec A+ tan A sec A tan A sec A tan A sec A tan A sec A tan A sec A tan A -/h/ogi/ wmk<kg<gim<m 6: fq'hqg<g; (sec θ + cosθ ) (secθ cosθ ) tan θ + sin θ. kqi<u: -/h/ogi (sec θ + cos θ ) (sec θ cos θ ) sec θ cos θ ( + tan θ ) cos θ tan θ + ( cos θ ) tan θ + sin θ u/h/ogi. A A -/h/ogi/ 00

205 wmk<kg<gim<m7: + cosec θ wefq'hqg<gul</ + cosθ cosθ ( cosθ) + (+ cosθ) kqi<u: -/h/ogi + + cosθ cosθ (+ cosθ)( cosθ) cosθ + + cosθ cos θ sin θ cosec θ u/h/ogi. wmk<kg<gim<m 8: fq'hq;sin A sin B + cos A cos B + sin A cos B + cos A sin B. kqi<u: -/h/ogi (sin A sin B + sin A cos B) + (cos A cos B + cos A sin B) sin A (sin B + cos B) + cos A (cos B + sin B) sin A() + cos A () sin A + cos A u/h/ogi/ wmk<kg<gim<m 9: m tan A + sin A, n tan A sin Aweqz<? m n 4 fq'hqg<gul</ kqi<u: -/h/ogi m n (tan A + sin A) (tan A sin A) tan A + sin A + tan A sin A (tan A + sin A tan A sin A) 4 tan A sin A u/h/ogi 4 mn 4 (tan A+ sin A)(tan A sin A) mn we 4 sin tan A sin A 4 A sin A cos A 4 sin A sin Acos A sin A( cos A) 4 cos A cos A 4 sin A tan A 4 sin A tan A/ () (), () -ux<xqe<hc?-/h/ogi u/h/ogi/ wmk<kg<gim<m 0: cos 6 θ + sin 6 θ cos θ sin θ wefq'hqg<gul</ kqi<u : -/h/ogi cos 6 θ + sin 6 θ (cos θ ) + (sin θ ) (cos θ + sin θ ) (cos 4 θ cos θ sin θ + sin 4 θ ) () (cos 4 θ + sin 4 θ cos θ sin θ ) [(cos θ ) + (sin θ ) ] cos θ sin θ [(cos θ + sin θ ) cos θ sin θ ] cos θ sin θ () cos θ sin θ cos θ sin θ u/h/ogi. wmk<kg<gim<m : fq'hqg<g; + sinθ cosθ sinθ + cosθ kqi<u: -/h/ogi sin θ + cos θ + sin θ cosθ sin θ + cos (sin θ + cosθ) sinθ + cosθ u/h/ogi/ () θ 0

206 hbqx<sq 8.. fq'hqg<gul<; sec A sin Asec A.. fq'hqg<gul<; (sin A + cos A) + (sin A cos A). cot θ cosec θ. SVg<Gg;. sec θ tan θ sec A+ tan A + sin A 4. fq'hqg<gul<;. sec A tan A sin A 5. fq'hqg<gul<; + sec θ. + sin θ sin θ 6. x r sin A sin B, y r sin A cos B, z r cos A weqz<?x + y + z e<lkqh<hgi{<g/ 7. tan A + cot A cosec A sec A weg<gi{<hqg<gul</ tan A 8. fq'hqg<gul<; cos A. + tan A 9. fq'hqg<gul<; cosec θ + cot θ. cosec θ cot θ 0. (tan A + cot A) sec A+ cosec A wefq'hqg<gul</ cos A sin A. + sin A+ cos A wefq'hqg<gul</ tan A cot A tan A+ sec A + sin A. fq'hqg<gul<;. tan A sec A+ cos A. fq'hqg<gul<; (tan A tan B) + ( + tan A tan B) sec A sec B. tan θ cot θ 4. fq'hqg<gul<; + sec θ cosec θ +. cot θ tan θ + sin θ cosθ cosθ 5. fq'hqg<gul<; sin θ cosθ cosθ 6. (sin θ + cosec θ ) + (cos θ + sec θ ) 7 + tan θ + cot θ wefq'hqg<gul</ cos θ + sin θ cos θ sin θ 7. + wefq'hqg<gul</ cosθ + sin θ cosθ sin θ 4 4 cos θ + sin θ 8. fq'hqg<gul<; tan θ. 4 4 sin θ + cos θ 8. fqvh<hg<ogi{r<gtg<gielg<ogi{uqbz<uqgqkr<gt< Wx<geOu osr<ogi{ Lg<Ogi{k<kqe< fqvh<h Ogi{r<gjth< hx<xq fil< nxqf<kt<otil</ osr<ogi{lg<ogi{l<oqp -z< (hml<8.9 Jh< hii<g<gul<), Q yvosr<ogi{l</-r<g m QOP + m OPQ hml< 8.9

207 weou? QOP, OPQ NgqbOgi{r<gt<fqvh<Hg<Ogi{r<gtiGl</m QOP θ we<g/ hqe<hm OPQ 90 θ NGl</Lg<Ogi{uqbz<uqgqkr<gtqe<ujvbjxjbh<hqe<hx<xq Ogi{l< θ uqx<ghqe<uvueux<jxwx<geounxqf<kt<otil</ PQ OQ PQ sin θ, cosθ, tan θ, OP OP OQ.. () OP OP OQ cosec θ, secθ, cot θ PQ OQ PQ -h<ohipk? 90 θ g<glg<ogi{uqbz<uqgqkr<gt<nxqblx<hmouil</hmk<kqzqvf<k? OQ o PQ o OQ sin (90 θ ), cos(90 θ ), tan(90 θ), OP OP PQ..() o OP o OP o PQ cosec(90 θ ), sec(90 θ), cot (90 θ) OQ PQ OQ (), () yh<hqmgqp<g<g{<mux<jxnxqbzil</ PQ o OQ sin θ cos (90 θ) cos θ sin (90 θ ) OP OP PQ OP tan θ cot (90 θ ) cosec θ sec (90 θ ) OQ PQ OP OQ sec θ cosec (90 θ ) cot θ tan (90 θ ). OQ PQ -kjeyinm<muj{bignjlg<gzil</ sin (90 θ ) cosθ cos (90 θ ) sin θ tan (90 θ ) cot θ o tan 65 wmk<kg<gim<m : lkqh<hgi{<g;. o cot 5 cosec (90 θ ) sec θ sec (90 θ ) cosec θ cot (90 θ ) tan θ o o tan 65 cot 5 kqi<u : tan 65 tan (90 5 ) cot 5. o o cot 5 cot 5. wmk<kg<gim<m : lkqh<hgi{<g; sin 0 tan 60 sec 70 kqi<u : sec 70 sec(90 0 ) cosec 0 o sin 0 sin 0 tan 60 sec 70 sin 0 tan 60 cosec 0 sin 0 o sin 0. wmk<kg<gim<m 4: cosec x sec 5 weqz<?x e<lkqh<hgi{<g/ kqi<u: cosec x sec(90 x ), sec(90 x ) sec x 5. x Gxqh<H: OlOzdt<tg{g<gqz<?xNeK?sle<him<ce<-Vhg<gLl<secjbfQg<gl< osb<ukiz< gqjmg<guqz<jza Neiz<? GXr<Ogi{k<kqx<gie Lg<Ogi{uqbz< uqgqkr<gtqe<keqk<ke<jlh<h{<hqjeh<hbe<hmk<kqgi{h<hm<mkwe<hjknxqg/ 0

208 o sin 6. lkqh<hgi{<g; (i) o cos54 (ii) hbqx<sq 8. o tan 5 o cot 55 (iii) sin θ sec (90 θ ) o o o o o tan sin 4 sec5. SVg<Gg; (i) o o o cot cos 48 cos ec9. (ii) sin sec o cos67 cosec 4 o. x jbg<gi{<g; (i) sin 60 cos x (ii) cosec x cos 54 (iii) sec x cosec 5 (iv) tan x tan 5. 5, 5 5, 4, tan θ 5 5, tan, cosθ. 5,, 5 uqjmgt< hbqx<sq 8.. 8, sin θ, cos ecθ, secθ, cot θ cos θ θ, cos ec θ, secθ, cot θ sin θ, tan θ, cosecθ, cot θ sin θ, cosθ, tan θ, secθ, cot θ sin θ, cosθ, tan θ 7, cosecθ, secθ sin θ, cosθ, cosecθ, secθ, cot θ (i) 7 (ii) 0 (iii) (iv) 4 8. A 60, B 0 hbqx<sq 8., ,,. 6. r hbqx<sq 8.. (i) (ii) (iii). (i) (ii) 7. (i) 0 (ii) 6 (iii) 65 (iv) 55 04

209 9. osb<ljxucuqbz< nxqljxucuqbzqz<uqkqgjtbl<?kig<gljxg<giv{r<gjtbl<hbe<hmk<kq Lg<Ogi{l<Ohie<xucuqbz<dVur<gtqe<h{<Hgjtg<%Xl<Okx<xr<gjtfq'h{l< osb<gqoxil</ -g<g{qkh<hqiquqz< uvl< ucu dvur<gjt ntugtg<gs< siqbig ujvukqz<jz/ likqiqh< hmr<gjt ujvf<k njkk< Okx<xr<gtqe< kvg<g Ljx fq'h{r<gtg<gk< Kj{bigg< ogit<gqe<<<oxil</ wf<k uqklie ucuqbz< gvuqgtl< nxqljx ucuqbjzh< hbqz<ukx<gk< OkjubqVg<giK/ wmk<kg<gim<mig? nxqljx ucuqbzqz< PQ we<x OfIg<Ogim<Mk< K{<cje? AB we<x OfIg<Ogim<Mk< K{<ce< jlbg<gk<kg<ogimig-vg<gqxkwe<hjkk<okiqbh<hmk<k?kz<zqblig PQ ju AB g<gs< osr<gk<kig fil< ujvukqz<jz/ Neiz<? AB g<gs< osr<gk<kig? PQ ju likqiqbigujvgqoxil</nh<hch<hm<mucuqbz<dvur<gjtk<kz<zqbligujvbnkqg Hk<kq %IjlBl<? kqxjlbl< Okju/ ucuqbz< dvur<gjt ujvb? hz ucuqbz< gvuqgt<dt<te/ucuqbz<gvuqgjtg<ogi{<mucuqbz<ujvhmr<gjtujvuok osb<ljx ucuqbzigl</ -h<himh<hqiquqz<? nxqljx ucuqbzqz< g{<mt<tkohiz<? nch<hjmg< ogit<jggt<? Okx<xr<gt<? kig<g Ljx fq'h{r<gt< WKl< gqjmbik/ nkx<gh< hkqzig? nxqljx ucuqbz< upqbig fq'hqg<gh<hm<m njlh<h Ljxgjtg< ogi{<m ucuqbz< dvur<gjt ujvuk lm<mol -h<hqiquqe< Gxqg<OgitiGl</ ntu Gxqg<gh<hmikOfIuqtql<H(straight edge) lx<xl< -V guvibr<gt< -ux<jx lm<mol ogi{<m ucuqbz< dvur<gjt ujvb Ou{<Mole<hK? ucuqbz< dvur<gjt dvuig<gl< g{g<ggtqz< suizig njlbl</ ucug{qk dvur<gjt dvuig<gl< g{g<ggjt Nvib<f<k ujgbqz< lqg dbif<k d{<jlgtl<? Okx<xr<gTl< dvuig<gh<hm<mt<te/ wmk<kg<gim<mig? hqkigv^< Okx<xl< nu<uiox dvuiek/ Lf<jkbuGh<Hgtqz<?fil<gQp<g<g{<mux<xqx<gieujvLjxgjtg<gx<Xt<Otil<; (i) yvofig<ogim<mk<k{<ce<jlbg<gk<kg<ogimujvkz</ (ii) yvogi{k<kqe<-vsloum<cujvkz</ (iii) yvslhg<glg<ogi{l<ujvkz</ (iv) yvofig<ogim<mk<k{<jmkvh<hm<muqgqkk<kqz<hqiqk<kz</ (v) yvosr<ogi{lg<ogi{l<ujvkz</ (vi) yv-j{gvl<ujvkz</ (vii) yvsib<skvl<ujvkz</ (viii) yvjlbum<mr<gt<ujvkz</ (ix) yvsiqugl<ujvkz</ (x) hg<gntugt<kvh<hce<?lg<ogi{k<jkujvkz</ 05

210 -f<knk<kqbibk<kqz<?gqp<g<g{<mljxgjtg<gx<ohil</ (i) yvlg<ogi{k<kqe<fmg<ogim<mjlbl<?osr<ogim<mjlbl<?sx<xum<mjlbl<? dt<um<mjlbl<ngqbux<jxg<gi{z</ (ii) %m<mlx<xl<ohvg<gz<svisiqgjtucuqbz<ujvljxupqbigh<ohxkz</ (iii) -Vw{<gtqe<uqgqkslsvisiqjbosb<Ljxucuqbz<&zl<gi{z</ 9. yvlg<ogi{k<kqz<yvht<tqupqg<ogimgt< nxqljxucuqbzq<z<?gqp<g<g{<mux<jxk<okiqf<kjuk<kt<otil</ (i) yvlg<ogi{k<kqe<fmg<ogimgt<yvht<tqbqz<sf<kqg<gl</-f<kh<ht<tq Lg<Ogi{k<kqe< fmg<ogim<m jlbl< weh<hml</ -h<ht<tq yu<ouiv fmg< Ogim<jmBl< : wel<uqgqkk<kqz<hqiqg<gl</ (ii) yv Lg<Ogi{k<kqe< hg<gr<gtqe< jlbg<gk<kg<ogimgt< yv Ht<tqbqz< sf<kqg<gl</ -h<ht<tqlg<ogi{k<kqe<sx<xum<mjlbl<weh<hml</ (iii) yv Lg<Ogi{k<kqe< &e<x Ogi{r<gtqe< sloum<cgt< yv Ht<tqbqz< sf<kqg<gl</ -h<ht<tqlg<ogi{k<kqe<dt<um<mjlbl<weh<hml</ (iv) yv Lg<Ogi{k<kqe< osr<ogimgt< yv Ht<tqbqz< sf<kqg<gl</ -h<ht<tq osr<ogim<m jlbl<weh<hml</ osb<ljxbqz<-h<ht<tqgjtg<gxqg<gl<ljxjb-eq<gi{lx<hmouil</ 9.. fmg<ogim<mjlbl< Lg<Ogi{k<kqe<yVds<sqh<Ht<tqjbBl<-ke<wkqIh<hg<gk<kqe<fMh<<Ht<tqjbBl< -j{g<gl<ofig<ogim<mk<k{<myvfmg<ogimweh<hml</&e<xds<sqh<ht<tqgt< dt<tkiz< &e<x fmg<ogimgt< yv Lg<Ogi{k<kqx<G -Vg<gqe<xe/ -f<k fmg<ogimgt< yv Ht<tqbqz< sf<kqg<gl</-h<ht<tq Lg<Ogi{k<kqe< fmg< Ogim<Mjlbl< weh<hml</ -h<ht<tq? upg<glig G wel< Nr<gqz wpk<kiz< Gxqg<gh<hMl</ G wel< -h<ht<tq yu<ouiv fmg<ogim<jmbl< : wel< uqgqkk<kqz< hqiqg<gl</ G we<hk ds<sqh<ht<tqjb uqm wkqih<hg<gk<kqx<g nvgijlbqz< -Vg<Gl</ G e< h{<hgjt nch<hjmbigg<ogi{<m?nke<-vh<hqmk<jkg<gi[l<ljxgqopkvh<hm<mt<tk/ hc: kvh<hm<mlg<ogi{l< ABC jbujvg/ hc: BC bqe<fmh<ht<tqd e<-vh<hqml<g{<m?fmg<ogim AD Jujvg/ hc: CA bqe<fmh<ht<tqe e<-vh<hqml<g{<m?fmg<ogim BE Jujvg/ hc 4: AD, BE we<he sf<kqg<gl< Ht<tqjb G we<x Gxqg<g/ G we<hk ABC e< fmg<ogim<mjlbligl</ 06

211 Gxqh<H: Olx<g{<m Ljxbqz<? G Jg< gi{ &e<xiuk fmg<ogim<jm ujvbuqz<jz/ Woeeqz<? oum<ml<ht<tqbie fmg<ogim<m jlbl< G Jg< gi{-v fmg<ogimgot OhiKlieK/ &e<xiuk fmg<ogim<jm ujvf<kiz<? nk G upqbigs< osz<ujkg< gi{zil</ wf<kouiv njlh<h nz<zk ujvhm g{g<gqzl<? Lkzqz< fil< kvh<hm<m ntugjtbl<?okjujbbl<okiqf<kogit<tou{<ml</-kx<gh<hqe<?fil<yvlikqiqh< hmk<jk ujvf<k ogit<t Ou{<Ml</ -l<likqiqh< hmk<kqz<? njlh<h nz<zk ujvkzg<g{<mieupqljxh<hcgjts<sm<cg<gim<mou{<ml</ wmk<kg<gim<m : AB 7 os/lq? AC 7.5 os/lq? BC 5.5 os/lq weqz< ABC J ujvg/ nke<fmg<ogim<mjlbl<g Jg<gi{<g/AD JGwf<kuqgqkk<kqz<hqiqg<Gl<we<hjk wpkg/ kqiu: ABC bqe< likqiqh< hmk<jk ujvf<k kvh<hm<m ntugjtg< Gxqg<gUl< (hml< 9. Jh< hiig<gul<). -h<ohik fmg<ogim<m jlbk<jkg< gi{<ohil</ hcgt< gqop kvh<hm<mt<te/ hc : BC 5.5 os/lq ntuqz< BC wel< OfIg< Ogim<Mk<K{<jmujvg/B Jjlbligg<ogi{<M likqiqh<hml< 7 os/lqnvk<kqz<yvuqz<ujvg/-okohiz<c J jlbligg< ogi{<m 7.5 os/lq Nvk<kqz< yv uqz< ujvg/ -u<uqv uqx<gtl< oum<ml< Ht<tqjb A weg< Gxqg<gUl</ AB lx<xl< AC ujvg/ -h<ohik Lg<Ogi{l< ABC ujvbh<hm<mt<tk/ hc : jlbg<gk<kg<ogim Ljxbqz<? BC e< fmh<ht<tq D JBl<? AC e< fmh<ht<tq E JBl< gi{<g/ hml< 9. hc : AD, BE ujvg/ -ju G z< sf<kqg<gl</ G we<hkkie< ABC e<fmg<ogim<mjlbl</ hc 4: AB e< jlbh<ht<tq F Jg< gi{<g/ CF ujvg/ -K G upqs< osz<ujkg< gi{zil</ hc 5: AG lx<xl< GD Ngqbux<xqe< fqt ntugjtg< gi{<g/ AG 4.5 os/lq? GD.5 os/lq/ weg< gi{zil</ weou? AG 4.5. weou G NeK AD J : GD.5 we<xuqgqkk<kqz<hqiqg<gqe<xk/ hml< 9. 07

212 9.. Sx<Xum<mjlbl< yvlg<ogi{k<kqe<hg<gr<gtqe< jlbg<gk<kg<ogimgt< yvht<tqupqosz<zl</ -h<ht<tq? Lg<Ogi{k<kqe< Sx<Xum<m jlbl< weh<hml</ -h<ht<tq upg<glig S wel< Nr<gqz wpk<kiz< Gxqg<gh<hMl</ S NeK Lg<Ogi{k<kqe< &e<x ds<sqh<ht<tqgtqzqvf<kl< slk~vk<kqz< -Vg<Gl</ -s<slk~vk<kqje R we<g/ S J jlbligul<? R J NvligUl< ogi{<m ujvbh<hml< um<ml<? Lg<Ogi{k<kqe< &e<x ds<sqh<ht<tqgt< upqob osz<zl</ -u<um<ml< Lg<Ogi{k<kqe< Sx<Xum<ml< weul<? R we<hksx<xum<mnvl<weul<%xh<hml</ Sx<Xum<m jlbk<jkbl<? Sx<Xum<mk<jkBl< gi{? gqp<g<g{<m upqljxjbh< hqe<hx<xgqe<oxil</ hc : Lg<Ogi{l<ABC Jujvg/ hc : BC lx<xl< AC Ngqbux<xqe<jlbg<Gk<Kg<OgiMgjtujvg/ hc : BC lx<xl< AC Ngqbux<xqe< jlbg< Gk<Kg< OgiMgt< oum<cg< ogit<tl< Ht<tqjbS weg<gxqg<g/ Ht<tqS NeK ABC e<sx<xum<mjlbl<we<xjpg<gh<hml</ hc 4: AB e< jlbg< Gk<Kg< Ogim<cje ujvg/ -g<ogim S upqs<osz<ujkg< gueqg<gul</ SA, SB lx<xl< SC Ngqbux<xqe< fqtr<gjt ntg<gul</ SA SB SC we<xqvh<hjkk< okiqf<k ogit<g/ S J jlbligul<? SA ju NvligUl< ogi{<m yv um<ml< ujvg/ -u<um<mf<kie< Lg<Ogi{k<kqe< Sx<Xum<ml</ -K A, B, C upqs< osz<ujkk<okiqf<kogit<tul</ hc 5: CA e< jlbg< Gk<Kg< Ogim<cje ujvg/ -K S e< upqs< osz<ujkg< gi{<gqoxil</ wmk<kg<gim<m : AB 7 os/lq, m B 45º, BC 6 os/lq ntugt< ogi{<m ABC ujvf<knke<sx<xum<mk<jkujvg/ kqiu; kvh<hm<m ntugtg<ogx<h ABC J ujvg likqiqh<hml< (h.ogi.h ogit<jg). hqe<<ei? hqe<uvl< hcgjtk< okimvul<; hc: BC lx<xl< AB Ngqbux<xqe<jlbg<Gk<Kg< OgiMgt<ujvg/ hc : -l< jlbg<gk<kg<ogimgt< oum<ml< Ht<tqjb S weg< Gxqg<g/ S NeK Sx<Xum<m jlbligl</ hc: SA, SB lx<xl< SC Ngqbux<xqe<fQtr<gjt ntg<gul</ SA SB SC.6 os/lq weg< hml< 9. gi{<gqoxil</ hc4: S JjlbligUl<?SA JNvligUl<ogi{<Mum<ml<ujvg/-u<um<ml<A, B, C Ngqbux<xqe<upqbigs<osz<gqe<xK/-KOuOkjubieSx<Xum<mliGl</ 08

213 hc5: CA e<jlbg<gk<kg<ogim<cjeujvg/-ks e<upqs<osz<ujkg<gi{<gqoxil</ hml< 9.4 Gxqh<H: yv Lg<Ogi{k<kqe< Sx<Xum<mk<kqje ujvf<kume<? nl<lg<ogi{liek Sx<Xum<mk<kiz<$ph<hm<Mt<tKwe<Ohil</ 9.. dt<um<mjlbl< yv Lg<Ogi{k<kqe< &e<x Ogi{r<gtqe< -V sloum<cgt< yv Ht<tq upqs< osz<gqe<xe/ -h<ht<tq? Lg<Ogi{k<kqe< dt<um<m jlbl< weh<hml</ -K I wel< Nr<gqz wpk<kiz< Gxqg<gh<hMl</ dt<um<m jlbk<kqe< Lg<gqblie h{<h we<eoueqz<? I bqzqvf<k Lg<Ogi{k<kqe< hg<gr<gtg<g ujvbh<hml< IL, IM, IN we<el< Gk<Kg<OgiMgtqe< fqtr<gt< slligl</ -f<k hml< 9.5 fqtr<gtqe< sllie lkqh<h? r wel< Nr<gqz wpk<kiz<gxqg<gh<hml</ I JjlbligUl<?r JNvligUl<ogi{<mum<mk<kqjeujvf<kiz<?-u<um<ml< Lg<Ogi{k<kqe<&e<Xhg<gr<gjtBl<dm<Hxligk<okiMgqe<xK/-KLg<Ogi{k<kqe< dt<um<ml< weh<hml< (hml< 9.5 Jh< hiig<gul<). Nvl<r J dt<tivl< (inradius) we<gqoxil</ dt<um<m jlbl< gi{ul<? dt<tivk<jk ntuqmul<? dt<um<mk<jk ujvbul< upqljxgt<gqopkvh<hm<mt<te/ hc : ABC Jujvg/ hc : B lx<xl< C NgqbOgi{r<gtqe<Ogi{sloum<cjbujvg/ hc : -f<k -V sloum<cgt< oum<cg< ogit<tl< Ht<tqjb I weg< Gxqh<hqMg/ -f<kh<ht<tqi we<hokokjubiedt<um<mjlbligl</ hc 4: A e<sloum<cbl<i upqs<osz<ujks<siqhiig<gul</ hc 5: I bqzqvf<khg<gl< BC g<gs<osr<gk<kg<ogimujvg/nke<fqtk<jkntg<gul</ nkdt<tivl<r Jk<kVl</ 09

214 hc 6: I jb jlbligul< r J NvligUl< ogi{<m yv um<ml< ujvg/ -KOu? Lg<Ogi{k<kqe<dt<um<mliGl</ wmk<kg<gim<m : PQ 8 os/lq, m P 50º, m Q 60º NgqbntUgTjmb PQR e< dt<um<ml<ujvg/olzl<dt<tivk<jkntg<gul</ kqiu: PQR e<likqiqhmk<kqz<?nke<ntugjtg<gxqg<gul</ likqiqh<hml< hc : Ogi.h.Ogi nch<hjmg< ogit<jgjbh< hbe<hmk<kq PQR ujvg/ hc : P e<-vsloum<cujvg/ hc : Q e<-vsloum<cujvg/ hc 4: -f<k -V sloum<cgt< oum<ml< Ht<tqjb I weg< Gxqg<gUl</ -h<ht<tq I NeK dt<um<m jlbligl</ hml<9.6 hc 5: PQ wel<hg<gk<kqx<g ID wel<osr<gk<kg<ogimujvg/ hc 6: ID e< fqtk<jk ntg<gul</ -f<k fqtl< Lg<Ogi{k<kqe< dt<tivligl</ ID os/lq. weg<gi{zil</ hc 7: I J jlbligul<? ID J NvligUl< ogi{<m yv um<ml< ujvg/ -u<um<ml< Lg<Ogi{k<kqe<dt<um<mliGl</ hml< osr<ogim<mjlbl< Lg<Ogi{k<kqe< yv osr<ogim we<hk Lg<Ogi{k<kqe< YI ds<sqh< Ht<tqbqzqVf<K wkqih<hg<gk<kqx<g ujvbh<hml< osr<gk<kg<ogim we<hjk fil< fqjeuhmk<kqg<ogit<ouil</ yvlg<ogi{k<kqx<g &e<x osr<ogimgt< dt<te we<hjk nxqouil</ -s<osr<ogimgt< yv Ht<tq upqs< osz<gqe<xe we<hjk fil< Le<Oh nxqouil</ -h<ht<tq Lg<Ogi{k<kqe< osr<ogim<m jlbl< weh<hml</ -K H wel< Nr<gqz wpk<kiz< Gxqg<gh<hMl</ hml< 9.8 z<? ABC we<hk yv Lg<Ogi{l</ AL, BM lx<xl< CN we<hju hml< 9.8 0

215 osr<ogimgt</ -ju H z< sf<kqg<gqe<xe/ -r<g H we<hk ABC e< osr<ogim<m jlbligl</ Lg<Ogi{k<kqe< osr<ogim<m jlbl< H Jg< gi{ upqljxgt< gqop kvh<hm<mt<te; hc : kvh<hm<mntugtg<glg<ogi{l<abc ujvg/ hc : AL, BM wel<osr<ogimgjtujvg/ hc : AL? BM sf<kqg<gl<ht<tqjbh weg<gxqg<gul</h NeKLg<Ogi{l<ABC e< osr<ogim<mjlbligl</ wmk<kg<gim<m 4: XY 9 os/lq, YZ 8 os/lq? ZX 7 os/lq weqz<? XYZ e< osr<ogim<m jlbl<gi{<g/ kqiu: XYZ likqiqh<hmk<jkujvf<k?ogimg<gh<hm<mt<tntugjtg<gxqg<gul</ hc: h.h.h nch<hjmg<ogitqjeh< hbe<hmk<kq? Lg<Ogi{l< XYZ Jujvg/ likqiqh<hml< hc : osr<ogimgt< XL, YM ujvg/ hc : XL, YM oum<cg< ogit<tl< Ht<tqjb H weg< Gxqg<gUl</ H NeK XYZ e< osr<ogim<m jlbligl</ hc 4: ZH J-j{k<K? XY JN z<sf<kqg<glix fqm<mg/ ZN NeK Z upqbie osr<ogim weg< gi{zil</ hml< 9.9 hml<9.0 ouu<ouxuqklie Lg<Ogi{r<gTg<G fmg<ogim<m jlbl<? Sx<Xum<m jlbl<? dt<um<mjlbl<?osr<ogim<mjlbl<ngqbux<jxg<gxqk<kiz<?nux<xqe<-vh<hqmr<gt< gqopkvh<hm<mt<thcgi{zil</

216 nm<muj{ yvljeh<ht<tq Lg<Ogi{k<kqe<ujg fmg<ogim<mjlbl< njek<kujgbie Lg<Ogi{l< GXr<Ogi{Lg<Ogi{l< Sx<Xum<mjlbl< osr<ogi{lg<ogi{l< uqiqogi{lg<ogi{l< dt<um<mjlbl< njek<kujgbie Lg<Ogi{l< GXr<Ogi{Lg<Ogi{l< osr<ogim<mjlbl< osr<ogi{lg<ogi{l< uqiqogi{lg<ogi{l< yvr<ght<tqbqe< -Vh<hqml< Lg<Ogi{k<kqe<dt<Ot Lg<Ogi{k<kqe<dt<Ot gi{k<kqe<jlbh<ht<tq Lg<Ogi{k<kqx<GoutqOb Lg<Ogi{k<kqe<dt<Ot Lg<Ogi{k<kqe<dt<Ot osr<ogi{k<kqe<ds<sqh< Ht<tq Lg<Ogi{k<kqx<GoutqOb hbqx<sq 9. g{g<ggt< Lkz< 4 ujv?kvh<hm<mt<tlg<ogi{k<kqe<fmg<ogim<mjlbl<g gi{<g/. ABC z<?bc 6 os/lq, m B 40º, m C 60º.. ABC z<?njek<kh<hg<gr<gtl<6.5 os/lq fqtl<ogi{<mju/. PQR z<? m R 90º, PQ 7 os/lq, PR 6 os/lq. 4. LMN z<? LM 6 os/lq, m L 95º, MN 8 os/lq. g{g<ggt<5 Lkz< 8 ujv?kvh<hm<mlg<ogi{k<kqe<sx<xum<ml<ujvg/sx<xum<m Nvk<kqe<ntjug<gi{<g/ 5. ABC z<? AB 8 os/lq, BC 5 os/lq, AC 7 os/lq. 6. PQR z<? PQ 5 os/lq, PR 4.5 os/lq, m P 00º. 7. XYZ z<? XY 7 os/lq, m X 70º, m Y 60º. 8. PQR z<?yu<ouivhg<gll< 5.5 os/lqfqtlt<tk/ g{g<ggt< 9 Lkz< ujv?dt<um<ml<ujvf<k?dt<tivk<kqjentuqmg/ 9. ABC z<? AB 9 os/lq, BC 7 os/lq, CA 5 os/lq. 0. XYZ z<? XY YZ ZX 8 os/lq.. PQR z<? PQ 0 os/lq, m P 90º, m Q 60º.. ABC z<? AB 5.4 os/lq? m A 50º, AC 5 os/lq. g{g<ggt< Lkz< 6 ujv?lg<ogi{k<kqe<osr<ogim<mjlbl<gi{<g/. ABC z<? BC 5.6 os/lq, m B 55º, m C 65º. 4. PQR z<? m P 90º, m Q 0º, PR 4.5 os/lq. 5. LMN z<? LM 7 os/lq, m M 0º, MN 6 os/lq. 6. XYZ z<? XY 7 os/lq, YZ 5 os/lq, ZX 6 os/lq.

217 9. svisiqgtqe<ucuqbz<uqtg<gl< -h<ohikfil<-vlqjglpw{<gtqe<%m<ms<svisiq?ohvg<gz<svisiq Ngqbux<jxucuqbz<Ljxbqz<gi{<Ohil</ 9... %m<ms<svisiq a, b wel< -V w{<gtqe< %m<ms< svisiq a + b NGl</ a + b ucuqbz<&zl<gi{gqp<g<g{<mljxjbg<jgbit<ouil</ hc : a + b fqtl<ogi{<mofig<ogim<mk<k{<m PQ JwMk<Kg<ogit<g/ wel< w{<j{ hc : PQ e<jlbg<gk<kg<ogim<cjeujvg/ hc : PQ Ul<?jlbg<Gk<Kg<OgiMl<sf<kqg<Gl<Ht<tqjbg<Gxqk<K?P nz<zk Q a + b uqzqvf<k nh<ht<tq dt<t okijzuqje ntg<gul</ -k<okijzu Jk< kvl</ -kx<gg<giv{l<we<eoueqz<? XY we<xofig<ogim<mk<k{<ce<fqtl<l weqz<?xy e< l jlbh<ht<tqbiekx? Y zqvf<k okijzuqz<-vg<gl</ wmk<kg<gim<m 5: 6 lx<xl<9 Ngqbux<xqe<%m<Ms<svisiqjbg<gi{<g/ kqiu: osb<ljxgqopkvh<hm<mt<tk; hc : OkjubiefQtk<kqx<GyVOfIg<Ogim<jmujvg/ AB 6 os/lq wel<hc AB J nkqz<oum<cg<ogit<tul<(hml<9. Jh<hiIg<gUl<)/ hc : AB e<uzh<hxk<kqz<bc 9 os/lq wel<hc BC wel<k{<jmoum<cg<ogit<g/ hc : AC e<jlbg<gk<kg<ogicjeujvg/-k AC Js<sf<kqg<Gl<Ht<tqjbGxqk<K nkjem weh<ohbiqmg/ hc 4: A nz<zk C bqzqvf<k M e< okijzju ntg<gul</ AM 7.5 os/lq weg< gi{zil</-kou6 lx<xl<9 Ngqbw{<gtqe<%m<Ms<svisiqNGl</ hml< ohvg<gz<svisiqnz<zkuqgqkslsvisiq a, b we<heux<jx -V lqjg LP w{<gt< we<g/ a lx<xl< b bqe< ohvg<gz< svisiqwe<hk ab NGl</ x ab weqz<? x a x ab nz<zk x x ab nz<zk x b nz<zk a : x x : b. weou x NeK a, b g<g -jmbqzie uqgqksl svisiq wezil< (yvw{<x NeKa : x x : b welix-vf<kiz<?x NeKa lx<xl< b NgqbjugTg<G

218 -jmbqzieuqgqkslsvisiqweh<hmgqxk). ucuqbz<ljxh<hc gqopkvh<hm<mt<tk; hc : Okjug<Ogx<h fqtlie yv OfIg<Ogim<Mk<K{<jmujvg/ hc : nkqz<? AB a, BC b wel<hc-v K{<Mgt< oum<cg< ogit<tul< (hml< 9. Jh<hiIg<gUl<)/ ab Jg<gi[l<Ljx hc : AC J uqm<mligg< ogi{<m yv um<ml<ujvg/ hc 4: AC g<gs< osr<gk<kig B upqbig DE wel<fi{<ujvg/ hc 5: BD nz<zk BE e< fqtk<kqje hml< 9. ntg<gul</-f<kfqtl<a, b g<g-jmbqzie uqgqkslsvisiqjbk<kvl</-h<ohik BD e< fqtliek AB lx<xl< BC we<heux<xqe<fqtr<gtqe<uqgqkslsvisiqjbwu<uixkvl< we<hjkh<hiqf<kogit<ouil</ AC we<hkuqm<ml<?d we<hkum<mk<kqe<olzt<tht<tq/ weou ADC 90º. BD we<hk AC g<gs<osr<gk<kiek/ B Bl<? BD we<xhg<gll< ADB lx<xl< DCB we<gqx Lg<Ogi{r<gTg<Gh< ohikuieju/ weou -u<uqv Lg<Ogi{r<gTl<ucouik<kju/weOuhg<gr<gt<OfIuqgqkk<kqz<-Vg<Gl</nkiuK? AB BD nz<zk BD AB BC nz<zk x ab nz<zk x ab. BD BC wmk<kg<gim<m 6: 9 os/lq, os/lq fqtr<ogi{<m -V K{<Mgtqe< ohvg<gz< svisiqjbg< Gxqg<Gl<fij{g<gi{<g/ kqiu: hc : AX wel< Ogim<cje Okjug<Ogx<h fqtligg<ogit<g/ hc : nkqz< AB lx<xl< BC Ngqb -V K{<Mgjt LjxOb 9 os/lq, os/lq oum<cg< ogit<g(hml<9. Jh<hiIg<gUl<)/ hc : AC e< jlbg< Gk<Kg< Ogicje ujvf<k? nk AC J oum<ml< Ht<tqjb O we<g/ hml< 9. hc 4: O J jlbligul<? OA J NvligUl< ogi{<myvum<ml<ujvg/ hc 5: B upqbig DE wel<fij{ AC g<gs<osr<gk<kigujvg/ hc 6: BD nz<zk BE we<he AB, BC g<gqjmbqzieohvg<gz<svisiqjbg<gxqg<gl</ BD nz<zk BE e<fqtr<gjtntg<gul</-f<kfqtol9, e<ohvg<gz<svisiqngl</ BD fqtk<jkntg<g? BD 5. os/lq weg<gi{zil</ 4

219 wmk<kg<gim<m 7: 4 lx<xl< 9 g<g-jmobuqgqkslsvisiqjbg<gi{<g/ kqiu: 4 lx<xl< 9 Ngqbux<xqe<< uqgqksl svisiq we flg<gk< okiqbl</ -l<lkqh<h6 Jucuqbz<Ljxbqz<gi{<Ohil</ hc : AX wel< OfIg<Ogim<Mk< K{<cje OhiKlie ntuqx<g fqtlig ujvf<k ogit<ouil</ hc : AX e< Olz< 4 os/lq fqtlt<t AB we<gqxogim<mk<k{<jmoum<mg/ hc : B g<g uzh<hxlig? B bqzqvf<k 9 os/lq fqtlt<tk{<m BC Jujvg(hml<9.4 Jh< hiig<gul<)/ hc 4: AC e< jlbh<ht<tqjb O weg< Gxqg<gUl</ hc 5: O jujlbligul<?oa ju NvligUl< hml< 9.4 ogi{<myvum<ml<ujvg/ hc 6:B upqbig? AC g<gs<osr<gk<kig DE wel<fij{ujvg/ hc 7: BD we<hk AB lx<xl< BC bqe<uqgqkslsvisiqjbg<gxqg<gl</ hc 8: BD fqtk<jkntf<kogit<g/ BD 6 os/lqweg<gi{zil</-f<kw{< 6 NeK4 lx<xl< 9 gtqe<uqgqkslsvisiqngl</ wmk<kg<gim<m 8: e<lkqh<jhucuqbz<ljxbqz<gi{<g/ kqiu: 4 weg< gi{zil</ weou we<hjk 4 lx<xl< Ngqb w{<gtqe< ohvg<gz< svisiqbigg< gvkzil</ 4 lx<xl< gtqe< uqgqksl svisiqjbg< gi{ ucuqbz< Ljxjb fil<hbe<hmk<kouil</ hc : AX wel< Ogim<ce< lqk AB 4 os/lq/, BC os/lq/wel<hc? AB? BC wel< K{<Mgjt oum<cg< ogit<g (hml<9.5 Jh<hiIg<gUl<)/ hc : AC J uqm<mligg< ogi{<m yv hml<9.5 um<ml<ujvg/ hc : AC g<gs<osr<gk<kig?b upqbig DE wel<fij{ujvg/ hc 4: BD e<fqtk<jkntg<gul</bd.4 os/lqwefil<gi{zil</weou?.4. 5

220 hbqx<sq 9. Lkz< 4 ujvbqzieg{g<ggtqz<?ogimg<gh<hm<mt<tw{<gtqe<%m<ms<svisiqjbg< gi{<g/. 6 lx<xl< 4. 0 lx<xl< 5. 9 lx<xl< 4..5 lx<xl< Lkz< 8 ujvbqzieg{g<ggtqz<?kvh<hm<mw{<gtqe<ohvg<gz<svisiqjbg<gi{<g/ 5.. lx<xl< lx<xl< lx<xl< lx<xl< 4 9 Lkz< ujvbqzieg{g<ggtqz<?kvh<hm<mw{<gtqe<uig<g&zk<kqjeucuqbz< Ljxh<hcg<gi{<g/ uqjmgt< hbqx<sq os/lq 6..7 os/lq os/lq 8.. os/lq os/lq 0.. os/lq..7 os/lq..5 os/lq hbqx<sq

221 0. uquvr<gjtg<jgbitkz< fl<ljmb ne<xim uip<g<jgbqz< Ht<tququvr<gt< we<xjpg<gh<hml< w{<{x<x w{<dvur<gjtfil<hiig<gqoxil</wmk<kg<gim<mig?fitqkp<gjth<hcg<gl<ohik?yi nj{bqe<fqik<okg<gntjubl<nz<zknj{g<gt<hibl<fqiqe<ntjubl< gi{<gqe<oxil</ -f<k w{<{tugt<? nux<xqe< wkqigizh< Ohig<Ggjth< hx<xq nxqf<k ogit<ukx<gig sqvie giz -jmoutqgtqz< hkqbh<hmgqe<xe/ -u<uix hkqbh<hml< Ht<tq uquvr<gt< nkqg ntuqz< -Vh<hkiz<? -ux<jx Ljxh<hMk<kq hbet<t uquvr<gjt nxqb nxquqbz< siif<k Ljx Okjuh<hMgqxK/ nxquqbz< siif<k Ljxbqz< kvh<hm<m Ht<tq uquvr<gjt? hgk<kxqukx<gie g{qkk<kqe< yv hqiqu Ht<tqbqbz< NGl</ -h<hqiquqe< okimg<gg< giz utis<sqbqz<? -K yv fim<ce< Nm<sqbitVg<G?ke<fim<Mlg<gtqe<ohiVtikivfqjzjbnxqf<Knkx<Ogx<huiqgt< uqkqg<g upq nxqb dkuqbk/ Neiz<? -e<x Ht<tqbqbz< lqgh<ohiqb utis<sqjb njmf<k njek<kh< hqiqugtqzl< LcouMh<hkqZl<? kqm<mlqmukqzl< lqgh< ohiqb hr<gqjeugqg<gqxk/ Ht<tqbqbz< we<hjkg<gxqg<gl<nr<gqzs<osiz< statistics we<hk zk<kqe< olipqs< osiz<zie? Status we<hkqzqvf<k dvuiek/ Status we<hkx<g nvsqbz< fqjz (political state) we<xohivtigl</fil<?lf<jkbugh<hgtqz<ht<tq uquvr<gjt Osgiqg<Gl< Ljx hx<xqbl<? nux<jx nxqlgh<hmk<kl< Ljx hx<xqbl< hck<kjk-h<ohipkfqjeu%iouil</ Ht<tq uquvr<gt< Lkz< fqjzh< Ht<tq uquvr<gt< )primary data*? -v{<mil< fqjzh< Ht<tq uquvr<gt< (secondary data) we-vujgh<hml</nb<uitiyvui< kioeosgiqg<gl<uquvr<gt<lkz<fqjzh<ht<tququvr<gt<weh<hml</sqzofvr<gtqz<? Wx<geOu Yi< Nb<uitviz< Osgiqg<gh<hm<m Lkz< fqjzh< Ht<tq uquvr<gjt lx<oxiv Nb<uiti< OuoxiV Okjug<gigh< hbe<hmk<<kqg< ogi{<miz<? nh<ohik nh<ht<tq uquvr<gjt -v{<mil< fqjzh< Ht<tq uquvr<gt< we<gqoxil</ Nb<uitviz< Osgiqg<gh<hMl< uquvr<gt< okigg<gh<hmik Ht<tq uquvr<gt< nz<zk osh<heqmh<hmik Ht<tq uquvr<gt< weh<hml</-k<kjgbht<tququvr<gt<svg<gucuqz<ljxbig okigg<gh<hm<m?nm<muj{ucuqz<kvh<hmgqxk/-u<um<muj{biek?yvfqgp<ou{<< hm<cbz< we<xjpg<gh<hmgqxk/ fqgp<ou{< hm<cbz< upqbigk< kvh<hml< Ht<tq uquvr<gjt okigg<gh<hm<m Ht<tq uquvr<gt< we<gqoxil</ wmk<kg<gim<mig? gqop kvh<hm<mt<t0 li{uigtqe<g{qklkqh<oh{<gjtg<gvkouil</

222 Olx<gi[l<Ht<tququvl<okiGg<gh<hmikfqjzbqz<dt<tK/kvh<hm<mlkqh<oh{<gjt WXuiqjsbqz<wPk?flg<Gg<gqjmh<hK? 0,,,, 7, 7, 7, 9, 9, 40, 40, 4, 4, 4, 4, 4, 4, 45, 45, 46, 46, 46, 48, 48, 48, 49, 50, 50, 56, 56. Olx<%xqb uiqjsbqz< lkqh<oh{< 0 yv LjxBl<? &e<x LjxBl<? 7 &e<x LjxBl<lx<Xl<-u<uiOxlx<xlkqh<oh{<gTl<njlf<Kt<te/nf<kLjxbqz< kvh<hm<m lkqh<oh{<gjt w{<[ukqeiz< flg<gg< gqjmg<gl< fqgp<ou{<gt< nm<muj{jb okigg<gh<hmik Ht<tq uquvr<gtqe< fqgp<uh< hm<cbz< we<gqoxil</ x we<hk yv lkqh<oh{<{qjebl<? f we<hk x lkqh<oh{< ohx<x li{uigtqe< w{<{qg<jg nz<zk x e< fqgp<ou{<{qjeg< Gxqh<hkigg< ogit<g/ fqgp<ou{<{qje njzou{<we<xl<fil<%xouil</ x f 4 -r<g? x we<hk ntjubqe< lixq nz<zk lix<xv (variate) )-u<oumk<kg<gim<cz<? x NeK lkqh<oh{<{qjeg< Gxqg<gqxK* we<xjpg<gh<hml<; f fqgp<ou{<? nkiuk? yv lixqbqe<gxqh<hqm<mlkqh<hwk<kjeljxfqgp<gqe<xkwe<hjkg<gxqg<gqxk/ -r<g?lqgh<ohiqblkqh<h 56 lx<xl<lqgs<sqxqblkqh<h 0. weou? uqs<s lqgh<ohiqblkqh<h lqgs<sqxqblkqh<h fil<? kvh<hm<cvg<gl< lkqh<<oh{<gjts< OsIk<Kg<ogit<t hqiqu -jmoutqgt<weh<hml< -jmoutqgt<njlg<gzil</lkz<ugh<h-jmoutq0-4. -f<k-jmoutq0,,, lx<xl< 4 Ngqb lkqh<oh{<gjtg< ogi{<mt<tk/ nmk<k ugh<h -jmoutq 5-9/-u<uixigk<okimv?gjmsquGh<H-jmoutq NeK 55, 56, 57, 58 lx<xl< 59 Ngqb lkqh<oh{<gjtg< ogi{<mt<tk/ -r<g -jmoutqgt< yu<ouie<xl< gqp< lx<xl< Olz< wz<jzg<giqb w{<gjt dt<tmg<gq -Vh<hkiz<? -u<uqjmoutqgt< dt<tmr<gqb-jmoutqgt<ngl</-h<ohik?fqgp<ou{<hm<cbjznjlh<ohil</ hqiqu-jmoutq w{<{qg<jggxq fqgp<ou{< olik<kl< 0 kvh<hm<cvg<gl< Ht<tq uquvk<kqzt<t lkqh<hgjt ye<xe< hqe< ye<xig hcg<g? yu<ouiv lkqh<hqx<gl< nkx<giqb -jmoutqg<g wkqov OfIg<OgiM (yh<hg< Gxq nz<zkw{<{qg<jgg< Gxq) -m Ou{<Ml</ yov hqiquqz<-ml<ohxgqe<x yu<ouiv 5 uk w{<{qx<giew{<{qg<jgg<gxqbqjewx<geounh<hqiquqx<ogkqov-ml<ohx<xqvg<gl< fie<g w{<{qg<jgg< GxqgTg<G GXg<gig ( \ ) -m Ou{<Ml</ -f<k LjxbieK? kvh<hm<cvg<gl< Ht<tq uquvr<gtqzt<t nk<kje lkqh<hgtg<gl< -m Ou{<Ml</ Olx<%xqb nm<muj{bqz<? yu<ouiv ugh<h-jmoutqg<giqb w{<{qg<jgg< Gxqgtqe< 8

223 w{<{qg<jgbiek nu<ugh<hqe< fqgp<ou{< (frequency) weh<hmgqxk/ -u<um<muj{? okigg<gh<hm<m Ht<tq uquvr<gtqe< fqgp<ou{<hm<cbz< weh<hmgqxk/ Olx<%xqb nm<muj{bqe< hqiqu -jmoutqgtg<gt< 4.5, 9.5 Ohie<x lkqh<oh{<gjts< OsIg<g -bzik we<hjk nxqg/ -k<kjgb $pzqz<? fil< ugh<h -jmoutqgtqe< Olz< wz<jzjbnf<kf<k-jmoutq<juk<kqvg<gikwe<xgvk<kme< , ,, we lix<xqnjlg<gzil</-k<kjgblix<xk<jkdjmbnm<muj{jbgqop gi{zil</ hqiqu-jmoutq w{<{qg<jggxq fqgp<ou{< olik<kl< 0 -r<g? fqgp<ou{< hm<cbziek x lixqbqe< okimis<sqbie lix<xk<jkk< okiquqg<gqxk/ yv hqiqu -jmoutqbqe< Olz< wz<jz lx<xl< gqp< wz<jzgtqe< uqk<kqbisk<jk nu<uqjmoutqbqe< ntu (size) we<xl<? Olz< wz<jz lx<xl< gqp< wz<jzgtqe< svisiqjb hqiquqe< lkqh<h (class mark) we<xl< njpg<gqoxil</ -r<g hqiqu -jmoutqbqe< ntu 5 lx<xl<hqiquqe<lkqh<h 7?-K-jmoutqbqe<jlblkq<h<H NGl</OlZt<tnm<muj{bqzqVf<K?4.5 g<gg<gqp<lkqh<oh{<ohx<xuigt< 4, 9.5 g<gg< gqp<lkqh<oh{<gt<ohx<xuigt< , 44.5 g<gg<gqp<lkqh<oh{<ohx<xuigt< , 49.5 g<gg<gqp< lkqh<oh{<gt< ohx<xuigt< , 54.5 g<gg<gqp< lkqh<oh{<gt<ohx<xuigt< lx<xl< 59.5 g<gg<gqp<lkqh<oh{<gt< ohx<xuigt< k<kjgbfqgp<ugtqe<w{<{qg<jggjtguqu fqgp<ou{<gt<(cumulative frequencies (c.f)) nz<zkguqunjzou{<gt<we<gqoxil</ GuqUfqgp<ou{<gtqe<nm<muj{gQOpkvh<hm<Mt<tK hqiqu-jmoutq jlblkqh<h x fqgp<ou{<f GuqUfqgp<ou{<c.f olik<kl< 0 0. jlbfqjzh<ohig<gntjugt< osh<heqmh<hmik Ht<tququvr<gtqe< fqgp<ou{<hm<cbjz Olx<%xqbuiX njlh<hke< &zl<? nh<ht<tq uquvr<gtqe< yv oktquie njlh<hqjeg< gi{lcgqxk/ yv Gxqh<hqm<m lkqh<hqzqvf<k Ht<tququv lkqh<hgt< njmbl< lix<xk<kqe< Ohig<gqjeh<hx<xq 9

224 OlZl< nxqf<kogit<t? Ht<tq uquvr<gjt yvlqk<klig nxqf<kjvg<gl< kqm<mum<mlie ntjugt<dt<te/nk<kjgbntjugt<jlbfqjzh<ohig<gntjugt<(measures of Central Tendency) nz<zk jlbih<h ntjugt< (Measures of Location) weh<hmgqe<xe/nk<kjgb?sqzntjugtiue:. %m<ms<svisiq(average). -jmfqjz(median). LgM(Mode) 0.. %m<ms<svisiq,, 7,, 7 we<x lkqh<hgjtg< gvkouil</ Olx<%xqb yu<ouiv lkqh<hqzqvf<kl<? 0 Jg< gpqg<g? flg<gg< gqjmh<hk 9,,,, 7. -f<k uqk<kqbisr<gjtg< %m<m flg<g gqjmh<hk 0. -kqzqvf<k? w{< 0 kvh<hm<cvg<gl< 5 lkqh<hgtqe<fmuqz<njlf<kqvg<gqxkwenxqgqoxil</-kkvh<hm<cvg<gl<lkqh<hgtqe< svisiqnz<zk%m<ms<svisiqweh<hml</ohikuig?x, x,, x n we<gqxn lkqh<hgtqe< %m<ms<svisiq(average) nz<zkw{<g{qks<svisiq(arithmetic mean) nz<zksvg<glig svisiq (mean) x we<x w{<? x zqvf<kn lkqh<hgt< x, x,, x n e< uqzg<gr<gtqe< %Mkz< 0 we<xqvg<glix ujvbxg<gh<hmgqe<xk/ nkiuk? x, x,, x n wel< n lkqh<hgtqe<svisiq x NeK (x x ) + (x x ) (x n x ) 0 nz<zk ( x + x xn ) n x 0 we<xsle<him<miz<gqjmg<gh<ohxl</ weou x x x +... n + + g{qkk<kqz<?gxqbqm, sqg<li)sigma*nek?%mkjzg<gxqg<gh<hbe<hmk<kh<hmgqxk/ -f<kg<gxqbqm<ce<&zl<?%mkz< x n. n + x +... xn NeK x i nz<zk wtqkig i x + wegxqg<gh<hml</weou?fil<ujvbxh<hk? x i x. n Ht<tququvlkqh<Hgt<fqgp<ou{<hm<cbzqz<njlg<gh<hm<cVf<kiz<?svisiq x NeK? fx + f x f n xn x f + f f n we<x ohxh<hmgqe<xk/ -r<g x, x,, x n NeK LjxOb f, f,, f n we<x fqgp<ou{<gjtg< ogi{<m keqh<hm<m lkqh<hgtigoui nz<zk f, f,, f n we<x fqgp<ou{<gjtg< ogi{<m hqiqu -jmoutqgtqe< -jmlkqh<h ntugtigoui -Vg<Gl</sqg<liGxqbQm<cz<wPk? f i xi x, -r<gn f + f f. N n kvh<hm<cvg<gl<lkqh<hgtqe<lm<mlkqh<hgt<lqgh<ohiqbjubig-vf<kiz<?svisiqbiek SVg<G.upqLjxobie<xqeiz< g{g<gqmh<hml</ A we<hk ohivk<klig OkIf<okMg<gh<hm<m xi 0

225 YI w{< we<g/ fil<? x A, x A,, x n A we<gqx lixz<gjt njlh<ohil</ -l<lixz<gtg<g c we<x ohikuie giv{q -Vh<hkigg< ogi{<miz<? hqe<uvl< uqgqkr<gjtnjlh<ohil<; x A x A xn A,,...,. c c c -ux<jxljxob d, d,, d n we<g/ -h<ohipk? x A x A xn A f idi f d + f d +..+ f n d n f + f f n c c c [( f x fa) + ( f x f A) ( f n xn f n A) ] c [( f x + f x f n xn ) A( f + f f n )] [ f xi A N ] c c i. f xi A N c f n.k i i d i f i x A N + c f i id i f i xi n.k x A f id i + c. N N Gxqh<H; x A, x A,, x n A gjt nxqf<khqe<hkie<? ohikg<giv{q c Jg< Gxqh<hqmLcBl</ wmk<kg<gim<m : 9,,, 5, 7, 9 we<gqxht<tququvr<gtqe<svisiqjbg<gi{<g/ x i kqiu : x 4. N 6 6 wmk<kg<gim<m : hqe<uvl<ht<tququvr<gtg<g%m<ms<svisiqjbg<gi{<g/ kqiu: OfvcLjx; x f f x olik<kl< N 0 fx 40 x f SVg<G.upqLjx; A 4, c weg<ogit<g/hqe<h d x A x f d f d olik<kl< N0 fd x f i xi N 0 fd 0 x A+ c 4 + N

226 wmk<kg<gim<m : hqe<uvl<ht<tququvr<gtg<g%m<ms<sivsiqgi{<g/ lkqh<oh{< li{uigtqe< w{<{qg<jg x A kqiu: A 90, c 5, d we<g/ x A gjtg<g{g<gqm<mc 5 wenxqg/ c x f d f d N 0 fd x A+ c N fd wmk<kg<gim<m 4: hqe<uvl<ht<tququvr<gtg<g%m<ms<svisiqgi{<g/ hqiqu-jmoutq lkqh<oh{< kqiu: OfvcLjx; hqiqu-jmoutq jlblkqh<h x fqgp<ou{< f f x N 00 fx 800 nm<<muj{bqzqvf<k?n olik<kfqgp<ou{< 00, fx 800. fx 800 x 8. N 00

227 SVg<G.upqLjx;A 0, c 5, d x A we<g/ c hqiqu-jmoutq jlblkqh<h x d fqgp<ou{< f f d N 00 fd x A+ c fd N 00 hbqx<sq 0... svisiqjbg<gi{<g/ 7,, 8, 4, 9, 0.. yv ugh<hqzt<t 5 li{uigtqz<? 4 li{uigt< 66 lkqh<oh{<gtl<? 5 li{uigt< 67 lkqh<oh{<gtl<?6 li{uigt<68 lkqh<oh{<gtl<wmk<kiz<?nu<ugh<hqe<%m<ms< svisiqjbg<gi{<g/. hqe<uvl<ht<tququvr<gtg<g%m<ms<svisiqjbg<g{g<gqmg/ x f hqe<uvl<ht<tququvr<gtg<g%m<ms<svisiqjbg<g{g<gqmg/ lkqh<oh{<gt< li{uigtqe< w{<{qg<jg hqe<uvl<ht<tququvr<gtqzqvf<k%m<ms<svisiqjbg<gi{<g/ lixq fqgp<ou{< hqe<uvl<nm<muj{jbg<ogi{<m%m<ms<svisiqg<gi{<g hqiqu-jmoutq fqgp<ou{<

228 0.. -jmfqjzntu kvh<hm<cvg<gl< osh<heqmh<hmik Ht<tq uquvr<gjt WX nz<zk -xr<g uiqjsbqz< wpkl<ohik? fl<liz< ypr<gh<hmk<kq -Vg<Gl< uiqjsbqz< fmfqjzbig njlf<kqvg<gl<yvlkqh<jhg<g{<mxqblcbl</-f<kfmlkqh<hnz<zklk<kqbqz< dt<tlkqh<hht<tququvr<gtqe<-jmfqjzntuweh<hml</ wmk<kg<gim<mig? 4, 8, 0, 9, 8, 5, 6, 7, 6 we<gqx Ht<tq uquvr<gjtg< gvkouil</nux<jxwxuiqjsbqz<wpk?4, 7, 8, 0, 5, 6, 8, 9, 6 weflg<gg< gqjmg<gqxk/ -u<uiqjsbqz<? 5 NeK? Ht<tq uquvr<gtqe< fmfqjzbig njlf<kqvg<gqxk/ weou 5 NeK? Ht<tq uquvr<gtqe<-jmfqjz ntu NGl</ -r<g? kvh<hm<cvg<gl< lkqh<hgtqe< w{<{qg<jg yx<jxh<hjmbqz< njlf<kt<tkiz<? uiqjsbqz<njlf<kt<tlkqh<hig-jmfqjzntu-vg<gqxk/ 85, 79, 57, 59, 66, 6, 40,, 48, 5 we<gqx Ht<tq uquvr<gjtg< gvkouil</ nux<jxypr<ghmk<kqwxuiqjsbqz<wpk? 6,, 40, 48, 5, 57, 59, 66, 79, 85 weg< gqjmg<gqxk/ -r<g -vm<jmh<hjm w{<{qg<jgbqz< lkqh<hgt< njlf<kt<tkiz<? ypr<ghmk<kqb uiqjsbqe< fmuqz< 5 lx<xl< 57 njlf<kt<te/ weou? fil< -u<uqv w{<gtqe< svisiqjbg< g{g<gqm? 55 NeK uiqjsg<g jlb lkqh<hig -Vg<gqxK/ lkqh<h 55 NeK? kvh<hm<m Ht<tq uquvr<gtqz< yv lkqh<hig -z<jzobe<xizl<? nkou -jmfqjz lkqh<hig njlgqxk/ NgOu? kvh<hm<m osh<heqmik Ht<tq uquvr<gtqe< -jmfqjz ntuqjeh< ohx hqe<uvl< upqgjt Olx<ogit<Ouil</ Lkzqz<? kvh<hm<m olik<k Ht<tq uquvr<gjt WXuiqjsbqOzi nz<zk -xr<guiqjsbqozi wpk Ou{<Ml</ N we<hk lkqh<hgtqe< w{<{q<g<jg we<g/ N yx<jxh<hjm w{<{ig -Vf<kiz<? Olx<g{<m uiqjsbqz< yv lkqh<h fmfqjzbig N + -Vg<Gl</ -K kvh<hm<m lkqh<hgtqe< WX nz<zk -xr<guiqjsbqz< uk dxh<hig njlbl</ -KOu -jmfqjz lkqh<h NGl</ N -vm<jmh<hjm w{<{ig njlf<kiz<? nr<g -V lkqh<hgt< fmfqjzbig njlbl</ nju Ht<tq uquv N N lkqh<hgtqe< WX nz<zk -xr<guiqjsbqz< uk lx<xl< + uk dxh<hgtigl</weou?-u<uqvdxh<hgtqe<svisiqob-jmfqjzngl</ kvh<hm<mht<tququvr<gt<?fqgp<ou{<hm<cbzqz<njlf<kqvf<kiz<-jmfqjzjb gi{h<hqe<uvl<ljxjbg<jgbitzil</ N Lkzqz<?GuqUfqgp<ou{<fqvjznjlk<K? e<lkqh<jhg<gi{ou{<ml</ -r<g N we<hk olik<k fqgp<ou{< (N f + f f n ) NGl</ hqxg 4

229 nm<muj{bqzqvf<k? N g<gs< sllie nz<zk N g<g sx<ox nkqglie GuqU fqgp<ou{< lkqh<hqje djmb lixqbqe< lkqh<h nz<zk hqiqu -jmoutqjbg< gi{ Ou{<Ml< )-K hvuzqe< -jmfqjz hqiqu -jmoutq weh<hml<*/ kvh<hm<m Ht<tq uquvr<gtg<g? -jmfqjz hqiqu -jmoutqg<giqb lixqbqe< lkqh<oh kvh<hm<m Ht<tq uquvk<kqx<gie-jmfqjzlkqh<higl</ wmk<kg<gim<m 5:, 5, 9, 0, 9 Ngqbux<xqe<-jmfqjzgi{<g/ kqiu: kvh<hm<cvg<gl< lkqh<hgt<wx<geouwxuiqjsbqz<njlf<kt<te/ N lkqh<hgtqe<w{<{qg<jg 5. -KYIyx<jxh<hjmw{</NgOu? N jmfqjz ukdxh<h ukdxh<h ukdxh<h 9. -jmfqjz 9. wmk<kg<gim<m 6:, 4, 0,, 7, 60, 55, 49, 50, 4,, 6, 7, 75, 80e<-jmfqjz gi{<g/ kqiu: kvh<hm<cvg<gl<lkqh<hgjtwxuiqjsbqz<njlg<g?, 4, 0,, 7,, 4, 49, 50, 55, 60, 6, 7, 75, 80. N lkqh<hgtqe<w{<{qg<jg 5, YIyx<jxh<hjmw{</ N jmfqjz ukdxh<h ukdxh<h 8 ukdxh<h 49. wmk<kg<gim<m 7:9,, 5, 9, 0, 5, 8 -ux<xqe<-jmfqjzgi{<g/ kqiu: kvh<hm<cvg<gl<lkqh<hgjtwxuiqjsbqz<wpk?, 5, 5, 8, 9, 9, 0. N lkqh<hgtqe<w{<{qg<jg 7, YIyx<jxh<hjmw{</ N jmfqjz ukdxh<h ukdxh<h 4uKdXh<H 8. wmk<kg<gim<m 8:6, 5, 9,, 5, 9, 0, 5, 8, 0 e<-jmfqjzgi{<g/ kqiu: kvh<hm<cvg<gl<ht<tququvr<gjtwxuiqjsbqz<wpk?, 5, 5, 5, 6, 8, 9, 9, 0, 0. N lkqh<hgtqe<w{<{qg<jg 0, YI-vm<jmh<hjmw{</ N N -jmfqjz ukdxh<hlx<xl< + ukdxh<hngqbux<xqe<svisiq 5 uklx<xl<6 ukdxh<hgtqe<svisiq lx<xl<8 gtqe<svisiq 7. 5

230 wmk<kg<gim<m 9: hqe<uvl<nm<muj{g<g-jmfqjzntjug{g<gqmg/ kqiu: lixq ( x) fqgp<ou{<( f ) x f GuqU fqgp<ou{< olik<kfqgp<ou{< N f. Njgbiz< N 6. N -jmfqjz uklkqh<h 6 ukdxh<hqe<lkqh<h/neiz<6uklkqh<hiek? GuqU fqgp<ou{< 9 dt<t hqiqu -jmoutqbqz< njlf<kt<tk/ -kx<gh< ohivk<klx<xlixqbqe<lkqh<h5ngl</weou?-jmfqjz 5. hbqx<sq 0... hqe<uvl<lixqgtqe<okigh<hqx<g-jmfqjzgi{<g; (i) 66, 6, 55, 60, 46, 0 (ii) 5, 9, 6, 4, 8, 7, 45, 4 (iii) 60, 6, 60, 58, 57, 59, 70 (iv) 4, 45, 6, 7, 4, 45, 4, li{uigtqe<lkqh<oh{<gtg<gie-jmfqjzgi{<g; lkqh<oh{<gt< li{uigtqe< w{<{qg<jg hqe<uVl<Ht<tququvr<gTg<G-jmfqjzgi{<g; x f

231 4. 4 okipqzitigtqe< %zqgt< kvh<hm<mt<te/ -ux<xqe< -jmfqjz lkqh<hqjeg< gi{<g/ %zq okipqzitigtqe< w{<{qg<jg LgM jlbfqjzntjugtqz<lgml<yintuigl</ (i) keqk<okigkqbig njlf<kt<t lkqh<hgtqe< g{k<kqz< wf<k yv lkqh<hiek nkqg w{<{qg<jgbqz< -Vg<gqxOki nk kvh<hm<m Ht<tq uquvr<gtqe< LgM weh<hml</ (ii) kvh<hm<m Ht<tq uquvr<gjt ypr<ghmk<kq yv fqgp<ou{< hm<cbzqz< njlk<kiz<? nkqg fqgp<ou{<j{ ogi{<m hqiqu? LgM hqiqu weh<hmgqxk/ -h<hqiquqz<dt<tlixqbqe<-jmh<hm<mlkqh<h?kvh<hm<mht<tququvk<kqe<lgm weh<hml</ wmk<kg<gim<m 0: 7, 4, 5,, 7,, 4, 6,7 e<lgmgi{<g/ kqiu: Ht<tququvr<gjtWXuiqjsbqz<wPk?,, 4, 4, 5, 6, 7, 7, 7. Olx<gi[l<Ht<tququvr<gtqz<7, nkqgkmjugi{h<hmgqxk/weou?lgm 7. wmk<kg<gim<m : 9, 0,, 4, 7, 0 e<lgmgi{<g/ kqiu: Ht<tq uquvr<gt< Wx<geOu WXuiqjsbqz< dt<te/ yu<ouiv lkqh<hl< yov yv kmju Olx<g{<m uiqjsbqz< njlf<kqvg<gqxk/ weou? kvh<hm<m Ht<tq uquvr<gtg<g LgM-z<jz/ wmk<kg<gim<m :, 5,,, 9, 5, 4, 7, 0,, 9, 5 e<lgmgi{<g/ kqiu: kvh<hm<mht<tququvr<gjtwxuiqjsbqz<wpk?,,,, 5, 5, 5, 9, 9, 0, 4, 7. -r<g&e<xljxbl<?5 &e<xljxbl<njlf<kqvg<gqxk/ lx<xl< 5 -v{<ml<kvh<hm<mht<tququvr<gtg<glgmgtignjlgqe<xe/-r<g? kvh<hm<mht<tququvr<gtg<g-vlgmgt<-vg<gqe<xewe<x%xgqe<oxil</ wmk<kg<gim<m : hqe<uvl<fqgp<ou{<hm<cbzqzqvf<klgmgi{<g/ %zq okipqzitigtqe< w{<{qg<jg kqiu: hm<cbzqzqvf<k fil< nxquk? lqg dbif<k fqgp<ou{< 4 NGl</ -kx<g -jsuie lixq %zqbqe< lkqh<h 55 NGl</ -KOu kvh<hm<m uquvk<kqe< LgmiGl</ nkiuk?lgm 55. 7

232 hbqx<sq0... hqe<uvl<ht<tququvr<gtg<glgmgi{<g; (i) 84, 9, 7, 68, 87, 84 (ii) 65, 6, 7, 8, 5, (iii) 8,,, 0,, 6, 5, 0 (iv) 5,, 8,,, 9,. hqe<uvl<hvuzqx<glgmgi{<g; x f hqe<uvl<nm<muj{bqzqvf<klgmgi{<g/ x f uqjmgt< hbqx<sq hbqx<sq 0... (i) 57.5 (ii) 5.5 (iii) 60 (iv) hbqx<sq0... (i) 84 (ii) LgM-z<jz (iii) 0, (iv)

233 . ujvhmr<gt< nxquqbz<? ohixqbqbz< lx<xl< uik<kgl< Ohie<x Kjxgtqz< fil< olb<ob{< lkqh<hgt< ohxl< hz<oux lixqgjtg< g{<cvg<gqoxil</ wmk<kg<gim<mig? u{qgk<kqz<? yv svg<gqe< kvu )s* lx<xl< uqjz )p* -V olb<lixqgtigl</ -l<lixqgjt yv sle<him<ce< upqbig -j{g<gzil</ -s<sle<him<jmh< hbe<hmk<kq? yu<ouiv s -e< lkqh<hqx<gl< p -e< lkqh<jh fil< ohxzil<; lx<xl< olb<ob{<gtiz< Ne uiqjs Osicgt< )s, p* -e< g{k<jkh< ohxzil</ wz<zi uiqjs OsicgjtBl<? gqjmbs<s s-ns<sigul<? Gk<ks<S p-ns<sigul< ogi{<m giicsqbe< ktk<kqz< Ht<tqgtigg< Gxqg<gzil</ -h<ohipk -h<ht<tqgt<? lixqgtg<gqjmobbt<t dxuqe< ujvhmk<kqje ujvbxg<gqxk we<x osiz<gqoxil</ -u<ujvhml< lixqgtqe< dxuqe< ke<jljbg< gim<mgqxk/ lqglg<gqblie? fil< ncg<gc hbe<hmk<kl< yv ujvhml< OfIg<OgiM ujvhmligl</ fil< -h<ohipk OfIg<OgiM ujvhmr<<gjth< hx<xqbl<? nju wu<uix ujvbh<hmgqe<xe we<hjkh< hx<xqbl<? nux<jxh< hbe<hmk<kqs< sqz sle<himgjtk< kqih<hkgxqk<kl<gi{lx<hmouil</. OfIg<OgiMujvhmr<gt< x lx<xl< y -Vlixqgt<we<g/njuy mx + c we<x sle<him<m ucuk<kqz< -j{g<gh<hm<miz<? nju Ofiqb Ljxbqz< -j{g<gh<hm<mt<te we<x %XOuil</ fil< Wx<geOuhGLjxucuqbz<himk<kqz<?giICsqbe<ktk<kqz<y mx + cwe<xsle<him yv OfIg<Ogicjeg< Gxqg<gqxK weg< g{<mt<otil</ -g<giv{k<kiz<kie< x lx<xl< y -ux<xqx<gqjmobbie dxju yv Ofiqb dxuwe<gqoxil</yu<ouivx lkqh<hqx<gl<? y mx + cwe<xsle<himy-e<yvlkqh<jhg<ogimg<gqxk;lx<xl<x, y w{<gtiz< njlbh<ohx<x yvuiqjsosic)x, y* gqjmg<gqxk/ -u<uixigh< ohxh<hm<m uiqjs Osicgtqe<g{l<?y mx + c -e<ujvhmk<kqjeujvbjxosb<gqxk/olzl<-jk Ofiqbujvhml<wenjpg<gqOxil</sle<hiMy mx + cz<?m NeKnf<OfIg<Ogice< sib<u? c NeK nke< y-oum<mk<k{<m we<hjk fil< fqjeu %IOuil</ x0 wel<ohik? y -e< lkqh<h y-oum<mk<k{<migl</ sqz slbr<gtqz< y-oum<mk<k{<ce< lkqh<h 0 Ng -Vg<Gl</ -s<$p<fqjzbqz<? sle<him y mx we<xigqxk; lx<xl< -f<ofig<ogim Nkqh<Ht<tq upqs<osz<gqxk/ fil< -h<ohipk hz<oux $p<fqjzgtqz< OfIg<Ogice< ujvhml< ujvuk Gxqk<Kh< hiih<ohil</ yv OfIg<OgiM ujvukx<gie nch<hjmg<gxqg<ogit<?nf<ofig<ogice<lqk-vht<tqgt<flg<gk<okjuwe<hkigl</ hqe<uvl<<ljx?ofig<ogimujvukx<gigh<hqe<hx<xh<hmgqe<xk; hc : x -e< -V ouu<oux lkqh<hgt< x, x -ux<jxs< sle<him y mx + c -z<< hqvkqbqm<m?y g<g-vlkqh<hgt<y, y fil<ohxgqe<oxil</-h<ohipkfil<-v Ht<tqgt<)x, y *lx<xl< (x, y ) -ux<jxnf<ofig<ogice<lqkohx<xt<otil</ 9

234 hc : x-ns<s lx<xl< y-ns<s -ux<jx yv ujvhmk<kitqz< ujvf<k nux<xqe< lqk Okjug<Ogx<h ntuqmgjtg< Gxqk<Kg<ogit<Ouil</ hc -z< ohxh<hm<m ns<s K~vr<gtqe< ntjuh< ohixk<ok -V ns<sgjt nzgqmkz< osb<gqoxil</ ns<s K~vr<gt<nkqglkqh<Hgt<djmbeweqz<?ns<Sgtqz<os/lQ/ntjuohiqb lkqh<jhg<gxqg<gl<nzgigg<ogit<gqoxil</ hc : -V Ht<tqgt< (x, y ) lx<xl< (x, y ) J ujvhmk<kitqz< gvkqb giicsqbe< ktk<kqz<gxqg<gul</ hc 4: -VHt<tqgjtBl<yVOfIg<Ogim<Mk<K{<miz<-j{k<K-Vhg<gLl<fQm<mUl</ -KOuOkjubieOfIg<OgiMujvhmliGl</ wmk<kg<gim<m : (, ) lx<xl< ( 4, ) J-j{g<Gl<Ogim<ce<ujvhml<ujvg. kqiu: ujvhmk<kitqz< x, y-ns<sgjt ujvf<k nux<xqe< lqk os/lq/ nzg weg< Gxqg<gUl</ A(, ) lx<xl< B ( 4, ) we<he kvh<hm<m -V Ht<tqgt< we<g. fil< -u<uqv Ht<tqgjtBl< ujvhmk<kitqz< Gxqg<gqOxil</ fil<? A lx<xl< B J OfIg<Ogimiz< -j{k<k -V HxLl< fqm<mouil</ -h<ohipk Okjubie ujvhml< ohxh<hm<mk (hml<. Jh<hiIg<gUl<). hml<. wmk<kg<gim<m : y x -e<ujvhml<ujvg. kqiu: y x we<hkofig<ogice<sle<himwe<hkiz<? OfIg<OgiMNkqh<Ht<tqupqbigs<osz<gqxK/ x, 0, wes< sle<him<cz< hqvkqbqm? LjxOb fil< y, 0, weh< ohxgqoxil</ -kjeg< gqop nm<muj{bigk<kf<kt<otil<. x 0 hml<. y 0 ujvhmk<kitqz< x-ns<s? y-ns<s ujvf<k? nux<xqe< lqk os/lq/ nzg weg< ogit<gqe<oxil</ fil< -h<ohipk (, ), (0, 0), (, ) NgqbHt<tqgjt ujvhmk<kitqz< Gxqg<gqOxil<? -h<ht<tqgjt -j{k<k -Vhg<gLl< fqm<mgqoxil</ -h<ohipk flg<gk< Okjubie OfIg<OgiM ujvhml< gqjmk<kt<tk (hml<. Jh< hiig<gul<). wmk<kg<gim<m : y x -e<ujvhml<ujvg/ kqiu: x, 0, wes< sle<him<cz< hqvkqbqm LjxOb fil< y 4,, weh< ohxgqoxil</ ujvhmk<kqz< (, 4), (0, ) lx<xl< (, ) Jg< Gxqg<gUl</Ht<tqgjtOfIg<Ogimiz<-j{k<K-V HxLl< fqm<mul</ flg<gk< Okjubie OfIg<Ogice< ujvhml<gqjmk<kt<tk(hml<. Jh<hiIg<gUl<). hml<. 0

235 wmk<kg<gim<m 4: sib<u NgUl<? y-oum<mk<k{<m NgUl< ogi{<m OfIg<Ogice< ujvhml<ujvg. kqiu: Ogice<sle<hiM y mx + c nz<zk y x + ( ) nz<zk y x. x, 0, weh< hqvkqbqm LjxOb y 0,, 6 wefil<ohxgqoxil</ (, 0), (0, ) lx<xl< (, 6) Ht<tqgjt ujvhmk<kitqz<gxqg<gul</ht<tqgjt x 0 y 0 6 -j{k<k -V HxLl< fqm<mul</ -h<ohipk flg<gk< Okjubie OfIg<Ogice<ujvhml<gqjmg<gqxK)hml<.4Jh<hiIg<gUl<*/ hml<.4 wmk<kg<gim<m 5: x + y we<xofig<ogim<ce<ujvhml<ujvg. kqiu: ogimg<gh<hm<ms<sle<him<jmhqe<uvlixwpkzil</ y x + nz<zky x + 4. x, 0, weh< hqvkqbqm LjxOb y 6, 4, we fil<ohxzil</ (, 6), (0, 4) lx<xl< (, ) Ngqb Ht<tqgjt ujvhmk<kitqz<gxqg<gul</ x 0 y 6 4 Ht<tqgjt -j{k<k -V HxLl< hml<.5 fqm<mul</ -KOu Okjubie OfIg<OgiM ujvhmligl<)hml<.5jh<hiig<gul<*/ wmk<kg<gim<m 6: x -e<ujvhml<ujvg. kqiu: x we<x sle<him<cz< y Jh< hx<xq Gxqh<hqmh<hmuqz<jz we<hjk fil< gueqg<gqoxil</ weou? y -e< wf<k yv lkqh<hqx<gl<x NGl<. y g<glx<xl<wek< OkIf<okMg<g?-VHt<tqgt<(, ) lx<xl< (, ), x we<x Ogim<ce< lqk -Vg<gqxK weh< ohxgqoxil<. -h<ht<tqgjt ujvhmk<kitqz< Gxqg<gUl</-VHt<tqgjtOfIg<Ogimiz< hml<.6

236 -j{k<k-vhxll<fqm<mul</flg<gk<okjubieofig<ogimujvhml<gqjmg<gqxk )hml<.6 Jh< hiig<gul<*/ -r<g OfIg<O<giM y-ns<sg<g -j{big dt<tjk gueqg<gul</ wmk<kg<gim<m 7: y 4 -e<ujvhml<ujvg. kqiu: y -e< lkqh<h lixilz< 4 weul<? sle<him<cz< x -e< lkqh<h Gxqh<hqmh<hmilZl< -Vh<hjkg<gi{<gqOxil</weOu?fil<x g<g-v lkqh<hgt<, wek< OkIf<okMg<gqOxil<. -h<ohipkflg<g-vht<tqgt< (, 4) lx<xl< (, 4), OfIg<OgiM y 4 -e< lqkqvh<hkigh< ohxgqoxil</ x y 4 4 hml<.7 -u<uqvht<tqgjtbl<ujvhmk<kqz<gxqk<kyvofig<ogiceiz< -j{g<gul</-kje -VHxLl< fqm<mul</ -KOu flg<gk< Okjubie ujvhmligl< )hml<.7 Jh< hiig<gul<*/-r<gofig<ogimiekx-ns<sg<g-j{bigdt<tjkgueqg<gul</ hbqx<sq.. gqp<g<g{<mht<tqgt<upqs<osz<zl<ofig<ogimujvhml<ujvg; (i) (, ) lx<xl< (4, 6) (ii) (, 0) lx<xl< (, 5) (iii) (, ) lx<xl<(5, ) (iv) (, ) lx<xl< (5, 4). hqe<uvueux<xqe<ujvhml<ujvg: (i) y x (ii) y x (iii) x 5y (iv) x 4y. hqe<uvueux<xqe<ujvhml<ujvg: (i) x (ii) y 5 (iii) x 5 (iv) y 4 (v) x + 0 (vi) + y 0 4. y mx + c we<hkx<gieujvhmk<kqjeh<hqe<uvl<lkqh<hgjtg<ogi{<mujvg; (i) m lx<xl<c 4 (ii) m lx<xl< c (iii) m lx<xl<c 4 (iv) m lx<xl< c 5 5. hqe<uvl<sle<himgtqe<ujvhml<ujvg: (i) x + y (ii) x 5y 0 (iii) y + x 5 0 (iv) x y + 0

237 . OfIg<OgiMujvhmr<gtqe<hbe<hiM -bx<g{qks< osbz<himgtqe<xq -V yvr<gjl OfIg<OgiMs< sle<himgtg<g? nux<xqe< ujvhmr<gjt ye<xig ujvf<k kqiu gi{zil</ x, y -z< njlf<k yv Ofiqbsle<hiM ax + by + c 0 we<xucuqz<-vg<gl<. -s<sle<himgiicsqbe<ktk<kqz< yvofig<ogijmg<gxqg<gl</weou?-vyvr<gjls<sle<himgtqe<kqiugi{<hk-v OfIg<OgiMgtqe<ohiKh<Ht<tqjbg<g{<Mhqch<hkx<Gs< slligl</-r<g?&e<x ujggt< d{<migqe<xe/ Lkziukig?-VOfIg<OgiMgTl<ye<Oxiomie<xignjlgqe<xe;nkiuK?-V ujvhmr<gtl< ye<xigqe<xe/ -s<$p<fqjzbqz< Lcuqzi Ht<tqgt< -V ujvhmr<gtg<gl< ohikuieju we nxqgqoxil</ weou, ogimg<gh<hm<m sle<himgtg<gw{<{x<xkqiugt<d{<m/ -v{<miukig? -V OfIg<OgiM ujvhmr<gt< ye<oxiomie<xig njlukqz<jz; Neiz<? nju -j{bieju/ -s<$p<fqjzbqz< -V OfIg<OgiM ujvhmr<gtl< ye<jxobie<x oum<cg< ogit<ukqz<jz/ weou? -V OgiMgTg<Gl< ohikuie Ht<tq gqjmbik/weou?-u<ouivr<gjls<sle<himgtg<gk<kqiugqjmbik/ &e<xiukig? -V OfIg<OgiMgt< yov yv Ht<tqbqz< oum<cg<ogit<gqe<xe/ -s<$p<fqjzbqz<? ogimg<gh<hm<mt<t yvr<gjls< sle<himgtg<gk< keqk<k kqiugt< d{<m/ -ju? nf<ofig<ogimgt< oum<cg<ogit<tl< Ht<tqbqe< ns<s K~vr<gtiGl</ weou?-u<ouivr<gjls<sle<himgtg<gyovobivkqiud{<m/ wmk<kg<gim<m 8: x + y lx<xl< 4x + y we<x yvr<gjls< sle<himgjt ujvhml<&zl<kqi. kqiu: OgiM : y x + OgiM : y 4x + i.e., y x + x y x y yov ujvhmk<kitqz< -V sle<himgtqe< Ht<tqgjtg< Gxqg<gUl</ Ht<tqgjt )Ogim<Mk<K{<miz<* -j{k<k -VHxLl< fqm<mul</ -h<ohipk -V ye<oxiomie<x -j{f<k OfIg<OgiMgt< gqjmg<gqe<xe )hml<.8 Jh< hiig<gul<*/ yv Ogim<ce< lqkt<t wf<k yv Ht<tqBl< lx<oxiv Ogim<ce<lQKdt<tK/weOu?w{<{x<x Ht<tqgt< -V OfIg<OgiMgTg<Gl< ohikuig dt<te/ weou? w{<{x<x kqiugt<? ogimg<gh<hm<m -V yvr<gjls< sle<himgtg<gd{<m/ hml<.8

238 wmk<kg<gim<m 9: x y 4 lx<xl< x y 6 -e<ujvhml<ujvf<knke<&zl< -s<sle<himgtqe<kqiugi{<g/ kqiu: OgiM : x y 4 nz<zk y x 4 nz<zky x. x 0 y OgiM : x y 6 nz<zk y x + 6 nz<zk y x +. x 0 y 4 hml<.9 fil<(0, ) lx<xl< (, ) we<xht<tqgjtujvhmk<kitqz<gxqk<knux<xqe<upqob OfIg<OgiM ujvouil</ nmk<k? fil< (0, ) lx<xl< (, 4) Ngqb Ht<tqgjt nok ujvhmk<kitqz< Gxqk<K nux<xqe< upqob OfIg<OgiM ujvouil</-r<g OfIg<OgiMgt< -j{bigs<osz<ujkfil<gi{zil</weou?njuoum<cg<ogit<tuqz<jz/weou? -u<ouivr<gjls<sle<himgtg<gk<kqiugqjmbik(hml<.9jh<hiig<gul<*/ wmk<kg<gim<m 0: yvr<gjls<sle<himgt< x + y 5, &zl<kqig<gul<. kqiu: OgiM : x + y 5 nz<zk y x + 5 () x y -ux<jx ujvhml< OgiM : x y nz<zky x () x y 7 6 (, 7), (, 6) lx<xl< (, ) Ngqb Ht<tqgjt ujvhmk<kitqz< Gxqg<gUl</ -h<ht<tqgt< upqob OfIg<OgiM ujvbul</-kou sle<him)*-e< Ofi<g<OgiMujvhmliGl</nMk<K (, ), (0, ) lx<xl< (, 0) Ngqb Ht<tqgjt nok ujvhmk<kitqz< Gxqg<gUl</ -h<ht<tqgt< upqob OfIg<OgiM ujvbul</ -KOu sle<him () -e< OfIg<<OgiM ujvhmligl</ -u<uqv OfIg<OgiM ujvhmr<gtl< yv Ht<tq P(4, ) -z< oum<mgqxk (hml<.0 Jh< hiig<gul<*/ -h<ht<tq? -V OgiMgtqe<< lqkl< njlukiz<? yvr<gjls<sle<himgtqe<kqiu x 4, y we nxqgqe<oxil<. x 0 y 0 hml<.0 4

239 hbqx<sq. gqop ogimg<gh<hm<mt<t Lkz<0 ujvbqzie sle<himgtqe< okigkqg<g ujvhml< &zl<kqiugi{<g;. x + y 0, x x + y, 4x + y.. x y 0, y. 7. x + y 4, x + y 6.. x + y, x y. 8. x y 4, x + y. 4. x y 6, x + y x + y, 6x y x + y 5, x y. 0. x + 0, 4x + y uqjmgt< hbqx<sq.. (i). (ii). (iii). (iv) (i) x 0 y 0. (ii) x 0 y 0 5

240 . (iii). (iv) x y 0 x y 0. (i) (ii) x y x y (iii) x y 0 (iv) x 0 y

241 (v) x y 0 (vi) x 0 y (i) (ii) x 0 y 4 7 x 0 y 5 (iii) x 0 y 4 7 (iv) x 0 y 5 7

242 5. (i) (ii) x 0 y 6 4 x y (iii) x 0 y 7 5 (iv) x 5 y. x 0 y 0 hbqx<sq. x y 0 kqiux 4; y 4.. x 0 y 0 x 0 y kqiu x ; y. 8

243 . x 0 y 0 x 4 5 y kqiu x ; y x 4 5 y x 4 y 5 kqiu x 5; y. 5. x 4 y x 4 y kqiu x ; y. 6. x 0 y x 0 y 7. w{<{x<xkqiugt/< x 0 4 y 0 x 0 4 y kqiu-z<jz/ 9