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. 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) 4.7877 (v) 5.745 (vi) 4.0 4. 740 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) 0.00074 (viii) 0.074 0 5 5. lmg<jgnm<muj{jbh<hbe<hmk<kqhqe<uvl<w{<gtqe<ohiklmg<jggjtg< g{<mhqcg<gul</ (i) 87 (ii) 8450 (iii) 0.0049 (iv) 0.00004 (v) 0.00468 (vi) 0.56 6. hqe<uvl<ohiklmg<jggtqe<wkqilmg<jggjtg<g{<mhqcg<gul</ (i).890 (ii) 0.4 (iii).458 (iv). 46 (v) 5.50 (vi).090. 7. 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) 5.5 8. hq<e<uvueux<jxlkqh<hqmosb<g/ (i) 5.46 80.7 (ii) 7.4 86. (iii). 0.0084 (iv) 86. 0.065 (v) (0.075) (vi) (50.49) 5 (vii) (55.9) 8 (viii) 45. 5 (ix) 0. 08745 (x) 0.495 45.7 0.054 (xi) ( 4.) 7.5 4.7 84. (xii) (.4) 6.94. (xiii) log 5.6 (xiv) log 9 6. 8 7

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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

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

. 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

. (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

..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) 8 + 5 4 9. -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

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

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

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) 4 + 7. 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) 45 0 5, hml<. (i) (b) Okim<m Oujz ugh<hqx<g lm<ml< ose<x oh{<gtqe<w{<{qg<jg n(b A) 70 0 40 (ii) -u<uqv{<mugh<hgtqz<wokel<yvugh<hqx<giukose<xoh{<gtqe<w{<{qg<jg? n(a U B) 5 + 0 + 40 85. -kjeouxuqkligul<ohxzil<; n(a U B) n(a) + n(b) n(a I B) 45 + 70 0 85. 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

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 45 55 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) 8 + 8. 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) 8 8 0 (ii) g{g<gh<himk<kqz<lm<ml<oki<s<sqbx<xui<gtqe<w{<{qg<jg n(m) n(h I M) 8 4. 86

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</ 6. 50 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

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6. (i) log 0 8 (ii) 8 50 log (iii) (iv) log 8 (v) 7 5 log 0 (vi) log 0 5 48 7. (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) 6 7 7 (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) 7.55 5. (i) 4.977 (ii) 5.96 (iii).58 (iv). 5.505 (v)..670 (vi).7 (v) (xi) 6. (i) 776.7 (ii).705 (iii) 0.87 (iv) 0.00668 (v) 0.0000 (vi) 0.009 7. (i) 00700 (ii) 8.56 (iii) 0.0099 (iv) 0.0006 (v) 0.4059 (vi) 0.00007789 (vii) 0.000098 (viii) 0.00050 (ix) 0.00050 (x) 0.098 (xi) 0.0000000698 (xii) 0.06. 8. (i) 04 (ii) 0550 (iii) 0.004 (iv) 80 (v) 0.0000079 (vi) 85000 (vii) 585 0 8 (viii) 7.670 (ix) 0.64 (x) 4. (xi) 0.06 (xii) 0.89 (xiii) 0.75 (xiv).89. 0 7 (vi) 5 6 89

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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

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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 + 7 5 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) 56 48 kqiu; (a) 07 0 (00 + 7) (00 + ) 00 + (7 + ) 00 + 7 0000 + 0 00 + 0000 + 000 + 0. (b) 56 48 (50 + 6) (50 ) 50 + (6 ) 50 6 500 + 00 688. 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< $k<kqvr<gjth< hbe<hmk<kq hqe<uvueux<xqe< lkqh<h gi{<g/ (i) 0 (ii) 98 (iii) 04 96 kqiu; (i) 0 (00 + ) 00 + (00)() + 0000 + 600 + 9 0609. (ii) 98 (00 ) 00 (00)() + 0000 400 + 4 9604. (iii) 04 96 (00 + 4) (00 4) 00 4 0000 6 9984. ohvg<gx<hze<$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

(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) 65. 4 [(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 () () 44 64 80. () 4() 44 8 6. 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) 6 + 80 6. 6 + 4(40) 6 + 60 96. 00

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) 5 + 600 65. 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

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 + (4 + + 5)x + [4 + 5 + 5 4]x + 4 5 x + x + [ + 5 + 0]x + 60 x + x + 47x + 60. (ii) (x + )(x )(x + 5) (x + )[x + ( )](x + 5) (x) + [ + ( ) + 5](x) + [ ( ) + ( ) 5 + 5 ](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) + {( ) + 7 + }(x) + {( ) 7 + 7 + ( )}(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 + 5 + 6 4, x e< ogp ab + bc + ca ( 5) + (5 6) + (6 ) 5 + 0 + 8 6, lixqzqdxh<h abc 5 6 90. (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) + + 4. 0

x e<ogp ab + bc + ca ( 7) + 4 + 4 ( 7) ( 4) + 8 + ( 8) ( 4) + 8 4, lixqzqdxh<h abc ( 7) 4 56. (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) ( ) +( ) 4 + 4 ( 5) 0 8 0 8, 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 + (5 + 5 + )y 05] y + 5y 7y + 05 (x) + 5(x) 7(x) + 05 8x + 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 5 7 8 x + x + x + x 5 7 5 e<ogp 8(a + b + c) 8 8 60, x e<ogp 8( ab + bc + ca) 5 5 7 7 8 + + 5 5 5 + 5 + 8 + + 8 4 4 4 ( ) (7) 4, 4 lixqzqdxh<h 8(abc) 5 7 05 8 8 05. 8 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

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

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

(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) 64 5. 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) 64 + 4 88. 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) 4 8 4. ab (ab) ( 4). a + b (a + b) ab(a + b) () ( )() 8 ( 4) 8 + 0. 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) 64 8 56. hbqx<sq 4.. ohvg<gx<hze<$k<kqvk<jkh<hbe<hmk<kqhqe<uvueux<jxg<gi{<g; (i) (x + 9) (x + ) (ii) (x + 8) (x ) (iii) (t )(t + 6) (iv) (p 4)(p ) (v) 0 06 (vi) 59 6 (vii) 4 6 (viii) 5 55. ohvg<gx<hze<$k<kqvk<jkh<hbe<hmk<kqg<gi{<g; (i) (5x + 8y) (ii) (s 4t) (iii) (4p + 7q)(4p 7q) (iv) (0) (v) (98) (vi) 0 98. 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

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

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 + 4 8. x x ax + a 9. p p + p 0. 8x + 4x + 4x + 4... 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

(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

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

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 + ).

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/ 6. + 6x + 9x 7. 44x 7x + 9 8. 4a 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 64 8. 8x 7y 6x y + 54xy 9. 4x + 9y + z + xy + 4xz + 6yz 0. a + b + 9c + ab 6ac 6bc. x y + + xy. 8x 5y + 80xy + 6. 8x 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 4 + 4 4... 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</

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< 8 8 8 8 8 8 9 9 9 9 8 6 6 6 6 -g<giv{qgtqe<%mkjzwpkouil</ 8 + + 8 9 ( 8) + ( ) ( ) + ( 8) 9 + 9 9 + ( ) + ( 9) ( 9) + ( ) + 6 6 + 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 + 54. 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 + ).

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, 6 + 7 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? 8 4 6 4, 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

wmk<kg<gim<m 50: giv{qh<hmk<kg; 7 x 0x 4. kqiu; -r<g? 7 4 56 ( 4) 4 OlZl<( 4) + 4 0. 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< 5 4 60 5, 7 + 5 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? 6 6 6 ( 9) 4, OlZl< ( 9) + 4 5 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 + 9 4 9 4 x + 4 x + x + + x +. 5

gqopuvueux<jxg<giv{qgtigh<hqiq/ hbqx<sq 4.... x + 7x +. x + 9x + 0. d + 0d + 4. z 7z 98 5. a a 7 6. x + x 90 7. p 8p + 5 8. y y + 4 9. y 0y + 99 0. t 8t + 95 gqopuvueux<jxg<giv{qh<hmk<kg;. a + a + 5. 4x + 8x +. 4x + x + 9 4. 6x + x 5. 6p + 7p + 0 6. 4a a 5 7. 7m + 6m 5 8. 8p + 9p 9. 6x + 5x 6 0. 5y y 6. 4x x. 9a 9a +. a a + 8 4. x 7x + 5. 6x 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 + 0 0 6 4. 7 5 4x 8x + 5. 4x x + 4 6 6. x + x + 7. x + x + 6 8. 5 5x + 0x + 5 9. x + 5x + 5 40. 7x + 4x + 4.4 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

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

-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 + 4 4 x + 4 x x 4 + 6x 8x + + x 8x 4x 7x + 4 8x 8x + 54x 7x + + x 68x + 55x + 4 68x 68x + 04x 7 x + + 49x + 76 weou?u x 8x 68, lqkq 49x + 76. 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

(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 + 7 + 55x + 4 (x x + 4) ( x 8x 68) + ( 49x + 76). weou?u x 8x 68, lqkq 49x + 76. hbqx<sq 4.4. 4x 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 + 0 6. x 4 4x + 4x 8 7. x 5 + x 8x + 6x 8 8. 6x 4 6x + 5x 66x + 7 9. 0x 5 + x 4 + 9x + x 9x 8 0. x 5 + 4x 4 + 9x 8x + x x e<ogp x e<ogp x e<ogp. 7 0. 6 0. 8 9 9 4. 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) 9898. 4,9/4 4. 60, 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

7. 45 8. (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) 9 6 60 (ii) 0 6 (iii) 96 8 5 0. 9,, 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 + 8. 04. 7, 8 4. 0,7 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 + 6 + 0xy + 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

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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

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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

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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

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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

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< 6.5 90 m QPE () 58

() + () 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

(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< 6.57 60

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< 90 0 60. (ii) 45 bqe<fqvh<hg<ogi{l< 90 45 45. (iii) 55 bqe<fqvh<hg<ogi{l< 90 55 5. (iv) 8 bqe<fqvh<hg<ogi{l< 90 8 9. 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< 80 70 0. (ii) 45 e<lqjgfqvh<hg<ogi{l< 80 45 5. (iii) 0 e<lqjgfqvh<hg<ogi{l< 80 0 60. (iv) 55 e<lqjgfqvh<hg<ogi{l< 80 55 5. 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< 60 0. )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< 5 + 0 55. 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< 5 0 05 0 85. hml< 6.60 6

(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 + 0 80 nz<zky 80 0 50. 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 + 75 80 nz<zk y 80 75 05. 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 + 9 80 nz<zk 4y 80 9 88 nz<zk y. yk<kogi{r<gt<sll< x + y 9 x + 44 9 hml< 6.64 x 9 44 48. 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 + 65 + 90 80 n.k x 5. BDC-z< x + y + 90 80 n.k 5 + y + 90 80 n.k y + 5 80 n.k y 80 5 65. hml< 6.65 6

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 4 + 5 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

() + () 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< 6.7 64

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< 6.74 80 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

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 45 + 5 80. 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<? 40 + 95 + C 80 C 80 5 45. ABC-z<? A + B + C 80. A + 80 + 45 80 A 80 5 55. 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

(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) 5. 4. 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< 6.8 67

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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< 7.6 75

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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 + 5 0 9 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

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 + 5 0 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 +. 4 4 -f<ks<sle<him<jm, y mx + c dme<yh<hqmjgbqz<? sib<um, y-oum<mk<k{<m c. 4 45 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

. 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< 8. 5 5 5 4. 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

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

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, ) + 4 484 + 6 500 0 5. 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< 6 + 9 6 + 8 B (, 0) 7 9. hml<7.4 BC ( 4 ) + ( 0) C (4, ) + 4 + 9. AC ( 4 + 4) + ( + 9) 8 + 64 + 44 08 6 4. -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 ) + 7 + 49 ( + 50 5 d (Q, R) (8 ) + ( 7 5) 6 + 6 + 44 80 6 5,, hml< 7.5 d (R, P) ( 8) + ( + 7) ( 5) + 5 5 + 5 50 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

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) + ( ) + 4 5 5 5, QR (4 + 4) + (5 + ) 64 + 6 00 0, PR (4 7) + (5 ) 9 + 6 5 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 ) + ( + ) 9 + 6 5 5, P P 4 (4 5) + (6 ) + 9 0, P 4 P ( 4) + ( 6) 9 + 6 5 5. 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 ) + ( ) 6 + 7 lx<xl< (P P ) + (P P ) 0 + 5 5, (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

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) + ( + ) 4 + 6 0 5, BC (6 + ) + (7 ) 64 + 6 80 4 5, CD (8 6) + ( 7) 4 + 6 0 5, AD (8 0) + ( + ) 64 + 6 80 4 5. AC (6 0) + (7 + ) 6 + 64 00 0. hml< 7.9 AB CD 5, BC AD 4 5 lx<xl< AB + BC 0 + 80 00 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) + ( + ) 4 + 4 8, BC (0 ) + ( ) 4 + 4 8, CD ( 0) + ( ) 4 + 4 8, AD ( 0) + ( + ) 4 + 4 8, BD ( ) + ( ) 6 + 0 6 4, AC (0 0) + ( + ) 0 + 6 6 4. 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 + ) 6 + 64 80, 8

BC AC BD CD AD 6) + ( 5) 64 6 80 ( +, ) + ( ) 6 6 ( + ( +, 44 + 88, 6 6) + ( 7 5) 44 6 + ) + ( 7 ) 6 64 80 ( +, 6 ) + ( 7 ) 64 6 80 ( + +. 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

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

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 + 7 0 (ii) 5x y + 9 0 (iii) x + y 5 0 (iv) 6x y 0 (v) x + 5y 5 0 (vi) x + 5y 8 0 4. (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

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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 7 8 89 64 5. BC 5 5. weou? BC 5 AB 8 BC 5 sin A, cos A, tan A, AC 7 AC 7 AB 8 8 7 7 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 5 5 7 7 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 5 7 + BC nz<zk 65 49 + BC. BC 65 49 576. BC 576 4. 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

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

+ 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

(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

-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 0 0 45 60 90 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

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 ( ) ( ) 4 4 9. 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 60... 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

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

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 60 7. 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

() 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

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

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

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 90. 0 hml< 8.9

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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 57 + + cos 48 cos ec9. (ii) sin sec47 + 4. 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, 7 8 5 4 4 5 5 5. sin θ, cos ecθ, secθ, cot θ. 4 5 6. cos θ θ, cos ec θ, secθ, cot θ. 5 5 7. sin θ, tan θ, cosecθ, cot θ. 0 8. sin θ, cosθ, tan θ, secθ, cot θ. 0 0 7 5 9. sin θ, cosθ, tan θ 7, cosecθ, secθ 5. 5 5 7 5 9 9 5 0. sin θ, cosθ, cosecθ, secθ, cot θ. 9 9 5 49 5 4... 5. 5 8 75 6. (i) 7 (ii) 0 (iii) (iv) 4 8. A 60, B 0 hbqx<sq 8., 8 5 4.,,. 6. r hbqx<sq 8.. (i) (ii) (iii). (i) (ii) 7. (i) 0 (ii) 6 (iii) 65 (iv) 55 04

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(v) x.5.5.5 y 0 (vi) x 0 y 0.5 0.5 0.5 4. (i) (ii) x 0 y 4 7 x 0 y 5 (iii) x 0 y 4 7 (iv) x 0 y 5 7

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

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