Transcript

1

2 ntw;wpf;f top (Way to Success) ⓬ fzpjk; murg; nghjj;njh;t rpwg;g ifnal jahupg;g jpu. f. jpnd\; M.Sc., M.Phil., P.G.D.C.A., Ph.D., ghlrk;ke;jkhd tpsf;fk; ngw kpd;dq;ry; : dinesh.skksv9@gmail.com & ways00@gmail.com njhlh;gf;f : (ghlg;nghus; rhh;ghf kl;lk;)> (Gj;jfq;fs; thq;f) tiyjsk; : ghl cjtpf; Fwpg;Gfis vkj,izajsj;jpypue;j,ytrkhf gjptpwf;fpf;nfhs;syhk; --

3 murj;nju;t tpdhj;jhs; tbtikg;g tpdh Nfl;fg;gLk; voj kjpg; gpupt tpdh tif vz; tpdhf;fs; Ntz;bait ngz;fs; gpupt-m - 40 njupttpdh (xu kjpg;ngz; tpdhf;fs;) gpupt-m kjpg;ngz; tpdhf;fs; fl;lha 6 kjpg;ngz; tpdh 6 gpupt-, kjpg;ngz; tpdhf;fs; fl;lha 0 kjpg;ngz; tpdh ,ay; vz;,ay; tpdhj;jhs; - gfg;gha;t xu kjpg;ngz; tpdhf;fs; MW kjpg;ngz; tpdhf;fs; gj;j kjpg;ngz; tpdhf;fs; mzpfs; kw;wk; mzpf;nfhitfspd; gad;ghlfs; 4 ntf;lh;,aw;fzpjk; 6 fyg;ngz;fs; 4 4 gfkiw tbtf;fzpjk; 4 5 tif Ez;fzpjk; : gad;ghlfs; I 4 6 tif Ez;fzpjk; : gad;ghlfs; II 7 njhif Ez;fzpjk; : gad;ghlfs; 4 8 tiff;nfor;rkd;ghlfs; 4 9 jdpepiy fzf;fpay; 4 0 epfo;jftg; guty; 4 $Ljy; Fwpg;G:,e;j Gj;jfk; jukhf tu Ntz;Lk; vd;w Nehf;fpy;,ad;wtiu jtwfspd;wp njhfj;j toq;fpas;nshk;. mtw;iwak; kpwp rpy jtwfs; cq;fsf;fj; njd;glyhk;. mt;thw VNjDk; jtwfs;,ue;jhy; vq;fsj kpd;dq;ry; Kftupf;F (ways00@gmail.com) clnd njuptpf;ftk;. Gj;jfj;jpy; cs;s jtwfsf;fhd jpuj;jq;fs; mt;tg;nghj vq;fsj tiyjsj;jpy; ntspaplg;gl;l mit mt;tg;nghj update nra;ag;glk; vd;gijak;> mlj;jlj;j gjpg;gfspy; mit rup nra;ag;gl;l tplk; vd;gijak; njuptpj;jf;nfhs;fpnwhk;. Way to Success Gj;jfq;fs; Ntz;LNthu; , , Mfpa vz;fisj; njhlu;gnfhs;sq;fs; --

4 md;ghd khztu;fsf;f tzf;fk;. k; tfg;g fzpjf; ifnal jw;nghj cq;fs; ifapy; jto;fpwj. tof;fkhd topfhl;b E}y; Nghd;W,J vojg;gltpy;iy. kw;w Fwpg;NgLfSf;Fk;,jw;Fk; kp;fe;j NtWghL cz;l. khztu;fs; fzpj ghlj;ij Gupe;J nfhz;l> vspa Kiwapy; vt;thw tpil mspg;gj> mnj rkaj;jpy; murj; Njh;tpy; mjpf kjpg;ngz; ngwk; tifapyk;> nky;yf; fw;fk; khzth;fspd; gaj;ij Nghf;fp fzpjg; ghlj;jpy; ntw;wp ngwk; tifapyk;,e;j ifnal tbtikf;fg;gl;ls;sj. nky;yf; fw;nghuf;fhd MNyhridfs;: Kaw;rp nra;jhy; fzpjg;; ghlj;jpy; Rygkhf Nju;r;rpngw;W ey;y kjpg;ngz;fsk; ngwkbak; vd;gij Kjypy; ek;gq;fs;. gapw;rpfis nra;jghu;j;jy; kpf mtrpak;. fzpjg; ghlj;jpy; Mh;tk; kpf Kf;fpak;. gpd;tuk; ghlg;gfjpfsf;f Kf;fpaj;Jtk; nfhlj;j ed;f gapw;rp nra;atk; xu kjpg;ngz; tpdhf;fs; - njhfjp ( tpdhf;fs;)> njhfjp (50 tpdhf;fs;) MW kjpg;ngz; tpdhf;fs; -. mzpfs; kw;wk; mzpf;nfhitfspd; gad;ghlfs;. fyg;ngz;fs; 9. jdpepiyf; fzf;fpay; gj;j kjpg;ngz; tpdhf;fs; -. ntf;lh;,aw;fzpjk;. fyg;ngz;fs; 4. gfkiw tbtf;fzpjk; 6. tif Ez;fzpjk; - gad;ghlfs; II 8. tiff;nfor;rkd;ghlfs; (gad;ghlfs; tpdhf;fs; 0 kl;lk;) 9. jdpepiyf; fzf;fpay; rpwg;ghf nray;gl;lhy; ntw;wp cwjp. murj;nju;tpy; 00f;F 00 ngw tho;j;jfpnwhk;. - ntw;wpf;f top FO cs;slf;fk;,ay; jiyg;g gf;fk; vz; kjpg;ngz; Gj;jf tpdhf;fs; (njhfjp I kw;wk; II) 4 6 kjpg;ngz;fs; mzpfs; kw;wk; mzpf;nfhitfspd; gad;ghlfs; 6 6 kjpg;ngz;fs; fyg;ngz;fs; 6 kjpg;ngz;fs; jdpepiyf; fzf;fpay; 4 0 kjpg;ngz;fs; ntf;lh;,aw;fzpjk; 47 0 kjpg;ngz;fs fyg;ngz;fs; 55 0 kjpg;ngz;fs gfkiw tbtpay; 60 0 kjpg;ngz;fs tif Ez;fzpjk; - gad;ghlfs; II 78 0 kjpg;ngz;fs tiff;nfor;rkd;ghlfs; 8 0 kjpg;ngz;fs jdpepiyf; fzf;fpay; 85 -,f;fwpaplg;gl;l tpdhf;fs;> Gj;jfj;jpw;F ntspapy;,ue;j Nfl;fg;gl;lit --

5 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top. mzpfs; kw;wk; mzpf;nfhitfspd; gad;ghlfs;. 4 vd;w mzpapd; juk; fhz;f () () () (4) 4 (OCT-). vd;w %iytpl;l mzpapd; juk; fhz;f (MAR-0, JUN-6) ()0 () () (4)5. A = 0 vdpy; AA T,d; juk; fhz;f (OCT-06,MAR-08,JUN-09,MAR-, OCT-5) () () () (4)0 4. A =, vdpy; AA T,d; juk; fhz;f (MAR-09, JUN-,OCT-4, OCT-6) () ()0 () (4) 5. λ 0 0 λ vd;w mzpapd; juk; vdpy;> λ d; kjpg;g (JUN-08,OCT-09,JUN-,OCT-5) 0 λ () () () (4) VNjDk; xu nka;naz; 6. xu jpirapyp mzpapd; thpir > jpirapyp k 0, vdpy; A vd;gj (OCT-07,MAR-08,JUN-08,OCT-08,MAR-0,MAR-4,JUN-4,OCT-4) () k I () k () I k (4)kI 7. k vd;w mzpf;f Neh;khW cz;l vdpy; k d; kjpg;gfs; 4 5 (OCT-06,OCT-09,MAR-,OCT-,MAR-5) () k VNjDk; xu nka;naz; () k = 4 () k 4 (4) k 4 8. A = vd;w mzpf;f (adj A)A= (MAR-07,6, JUN-07,5, OCT-08,0,,5) 4 () () 0 0 () xu rju mzp A,d; thpir n vdpy; adj A vd;gj (MAR-06,JUN-06,MAR-,JUN-4) 0. () A () A n () A n (4) A () vd;w mzpapd; Neh;khW () () (4) (4) (JUN-,JUN-,JUN-6) A vd;w mzpapd; thpir vdpy; det (ka) vd;gj (OCT-06, 07,JUN-09,JUN-0,JUN-,MAR-6,7) ()k det(a) () k det(a) () k det(a) (4) det(a) ways00@gmail.com

6 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top. myf mzp I,d; thpir n, k 0 xu khwpyp vdpy; adj(ki )= (OCT-0,MAR-,JUN-,MAR-,OCT-,JUN-5,JUN-6) () k n (adj I) () k(adj I) () k (adj I) (4) k n (adj I). A, B vd;w VNjDk;,U mzpfsf;f AB=O vd;w,ue;j NkYk; A xu G+r;rpakw;w Nfhit mzp vdpy;> (MAR-07,OCT-07,MAR-08,MAR-09,MAR-,MAR-,OCT-4,MAR-5) ()B=0 ()B xu G+r;rpaf;Nfhit mzp () B xu G+r;rpakw;w Nfhit mzp (4)B=A 4. A = , vdpy;> A vd;gj (JUN-07,JUN-09,JUN-0, OCT-6, MAR-7) () () 0 60 () (4) vd;gjd; Neh;khW (MAR-06, OCT-07, OCT-08,OCT-09,OCT-,OCT-, MAR-4,JUN-4) 5 () 5 () 5 () 5 (4) 5 6. kjpg;gpl Ntz;ba %d;w khwpfspy; mike;j %d;w Nehpa mrkgbj;jhd rkd;ghl;lj; njhfg;gpy; = 0 kw;wk; x = 0, y 0, z = 0, vdpy;> njhfg;gf;fhd jph;t (JUN-06,07,, MAR-0,) () xnu xu jph;t (),uz;l jph;tfs; () vz;zpf;ifaw;w jph;tfs; (4 ) jph;t,y;yhik 7. ax + y + z = 0; x + by + z = 0; x + y + cz = 0 Mfpa rkd;ghlfspd; njhfg;ghdj xu ntspg;gilaw;w jph;it ngw;wpug;gpd; + + = a b c (MAR-07,MAR-09,JUN-0,OCT-,MAR-4,JUN-5,MAR-6) () () () (4)0 8. ae x + be y = c; pe x + qe y = d kw;wk; = a b p q, = c b d q, = a c p d vdpy; (x, y),d; kjpg;g (JUN-08,OCT-0,MAR-, OCT-6, MAR-7) (), () log, log () log, log (4) log, log 9. x + y + z = l, x y + z = m, x + y z = n, vd;w rkd;ghlfs; l + m + n = 0 vdkhw mikakhapd; mj;njhfg;gpd; jph;t (MAR-06,JUN-06,JUN-,OCT-,JUN-,OCT-,MAR-5) () XNu xu G+r;rpakw;w jph;t () ntspg;gilj; jph;t () vz;zpf;ifaw;w jph;t (4) jph;t,y;yhik ngw;w,uf;fk;. ntf;lh;,aw;fzpjk;. a xu G+r;rpakw;w ntf;luhftk; m xu G+r;rpakw;w jpirapypahftk;,ug;gpd; ma MdJ XuyF ntf;lh; vdpy; (MAR-07,OCT-,MAR-4,MAR-6) () m = ± () a = m () a = m (4) a =. a kw;wk; b,uz;l XuyF ntf;lh; kw;wk; θ vd;gj mtw;wpw;f,ilg;gl;l Nfhzk; (a + b) MdJ XuyF ntf;luhapd; (OCT-06,OCT-07,MAR-08,OCT-09,JUN-0,MAR-,MAR-,JUN-5, OCT-5) () θ = π () θ = π 4 () θ = π (4) θ = π ways00@gmail.com

7 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top. a f;fk; b f;fk;,ilg;gl;l Nfhzk; 0 NkYk; mtw;wpd; vz;zstfs; KiwNa, vdpy; a.b MdJ (JUN-07,JUN-09,JUN-4) () () () (4) 4. u = a b c + b c a + c a b vdpy; (JUN-06,MAR-07,JUN-08,OCT-08,MAR-,MAR-5) () u xu XuyF ntf;lh; () u = a+b + c () u = 0 (4)u 0 5. a+b + c=0, a =, b = 4, c = 5 vdpy;> a f;fk; b f;fk;,ilg;gl;l Nfhzk; (JUN-,MAR-,MAR-4, OCT-6) () π 6 () π () 5π (4) π 6. i+j+4k, ai + bj + ck Mfpa ntf;lh;fs; nrq;fj;j ntf;lh;fshapd;> (JUN-07,JUN-,JUN-4) () a=, b=, c= 4 ()a=4, b=4, c=5 () a=4, b=4, c= 5 (4)a=, b=, c=4 7. i + j k vd;w ntf;liu xu %iy tpl;lkhftk; i j+4k I xu gf;fkhftk; nfhz;l,izfuj;jpd; gug;g (JUN-08,OCT-09,MAR-0,JUN-,OCT-,OCT-4,MAR-6) ()0 ()6 0 () 0 (4) 0 8. a + b = a b vdpy; (MAR-06,JUN-06,MAR-07,JUN-09,MAR-5) () a k; b k;,izahfk; () a k; b k; nrq;fj;jhfk; () a = b (4) a kw;wk; b XuyF ntf;lh; 9. p, q kw;wk; p + q Mfpait vz;zst λ nfhz;l ntf;lh;fshapd; p q MdJ (OCT-08,MAR-09,OCT-) () λ () λ () λ (4) 0. a b c + b c a + c a b = x y vdpy;> (JUN-,JUN-,OCT-5) () x = 0 () y = 0 () x k; y k;,izahfk; (4) x = 0my;yJ y = 0my;yJ x k; y k;,izahfk;. PR = i + j + k, QS = i + j + k vdpy; ehw;fuk; PQRS,d; gug;g (OCT-06,OCT-07,OCT-0,MAR-,MAR-5, MAR-7) ()5 ()0 (). OQ vd;w myf ntf;lh; kpjhd OP,d; tpoyhdj OPRQ vd;w,izfuj;jpd; gug;ig Nghd;W Kk;klq;fhapd; POQ MdJ ( JUN-06,MAR-09,JUN-0, MAR-, JUN-6) ()tan ()cos () sin (4) sin 0 0. b,d; kpj a,d; tpoy; kw;wk; a,d; kpj b,d; tpoyk; rkkhapd; a + b kw;wk; a b f;f,ilg;gl;l Nfhzk; (JUN-07,OCT-09,MAR-,OCT-,JUN-4) 5 (4) () π () π () π 4 (4) π 4. a, b, c vd;w xu jskw;w ntf;lh;fsf;f a b c = a b c vdpy;> (MAR-09,OCT-) () a MdJ b f;f,iz () b MdJ c f;f,iz () c MdJ af;f,iz (4) a+b + c=0 ways00@gmail.com

8 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 5. xu NfhL x kw;wk; y mr;rfsld; kpif jpirapy; 45, 60 Nfhzq;fis Vw;gLj;JfpwJ vdpy; z mr;rld; mj cz;lhf;fk; Nfhzk; (JUN-,MAR-4,OCT-4, MAR-7) () 0 () 90 () 45 (4) a b, b c, c a = 64 vdpy; a, b, c,d; kjpg;g (MAR-06,MAR-08,OCT-08,MAR-,OCT-6) () () 8 () 8 (4)0 7. a + b, b + c, c + a = 8 vdpy; a, b, c,d; kjpg;g (JUN-07,MAR-0,MAR-,OCT-,MAR-6) () 4 () 6 () (4)-4 8. i + j, j + k, k + i,d; kjpg;g (MAR-09,MAR-5, OCT-5, MAR-7) ()0 () () (4)4 9. (,0,) vd;w Gs;spf;Fk; r. i j + 4k = 6 vd;w jsj;jpw;fk;,ilg;gl;l kpff; Fiwe;j J}uk; (MAR-07,MAR-08,JUN-09,OCT-09,MAR-,JUN-5) () 6 () 6 () (4) 6 0. a b (c d) vd;gj (OCT-,OCT-5) () a, b, c kw;wk; d f;f nrq;fj;j () a b kw;wk; (c d) vd;w ntf;lh;fsf;f,iz () a, bi nfhz;l jskk; c, d I nfhz;l jskk; ntl;bf;nfhs;sk; Nfhl;bw;F,iz (4) a, b I nfhz;l jskk; c,di nfhz;l jskk; ntl;bf; nfhs;sk; Nfhl;bw;F nrq;fj;j.. a, b, c vd;gd a, b, c Mfpatw;iw kl;lf;fshff; nfhz;l tyf;if mikg;gpy; xd;wf;nfhd;w nrq;fj;jhd ntf;lh;fs; vdpy; a, b, c d; kjpg;g (JUN-08,JUN-, JUN-6) ()a b c () 0 () abc (4) abc. a, b, c vd;gd xu jsk; mikah ntf;lh;fs; NkYk; a b, b c, c a = a + b, b + c, c + a vdpy; a, b, c,d; kjpg;g () () () (4)0 (OCT-,OCT-4). r = si + tj vd;w rkd;ghl Fwpg;gJ (JUN-08,OCT-0) () i kw;wk; j Gs;spfis,izf;Fk; Neh;f;NfhL () xoy jsk; () yoz jsk; (4) zox jsk; 4. i + aj k vdk; tpir i + j vdk; Gs;sptopNar; nray;glfpwj. j + k vdk; Gs;spiag; nghwj;j 5. mjd; jpug;gj; jpwdpd; mst 8 vdpy; a,d; kjpg;g (OCT-0,JUN-,OCT-, OCT-6) () () () (4)4 x = y+ = z 5 5 rkd;ghl f;f,izahftk; (,,5) Gs;sp topahftk; nry;yf;$ba Nfhl;bd; ntf;lh; () r = i + 5j + k + t(i + j + 5k) () r = i + j + 5k + t(i + 5j + k) () r = i + 5j + k + t(i + j + 5k) (4) r = i + j + 5k + t i + 5j + k (MAR-7) 6. r = i k + t(i + j + 7k) vd;w NfhLk; r. i + j k = 8 vd;w jskk; ntl;bf;nfhs;sk; Gs;sp (MAR-07,MAR-08, JUN-6) ()(8,6,) ()( 8, 6, ) ()(4,,) (4)( 4,, ) ways00@gmail.com

9 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 7. (,, ) vd;w Gs;sp topahftk;> jsq;fs; r. i + j k = 0 ; r. j + k = 0 ntl;bf; nfhs;sk; Nfhl;il cs;slf;fpajkhd jsj;jpd; rkd;ghl (MAR-0,OCT-0,MAR-,JUN-5) ()x + 4y z = 0 () x + 9y + z = 0 () x + y z + 5 = 0 (4) x y + z = 0 8. F = i + j + k vd;w tpir xu Jfis A(,,) vdk; epiyapypue;j B(4,4,4) vdk; epiyf;f efh;j;jpdhy; mt;tpir nra;ak; NtiyasT. (MAR-06, MAR-,JUN-) () myffs; () myffs; ()4 myffs; (4)7 myffs; 9. a = i j + k kw;wk; b = i + j + k vdpy; a f;fk; b f;fk; nrq;fj;jhf cs;s xu XuyF ntf;lh; (OCT-06, JUN-6) 0. x 6 6 i+j +k () i j +k () i+j +k () (4) i j k = y+4 = z 4 kw;wk; x+ = y+ = z+ vd;w NfhLfs; ntl;bf; nfhs;sk; Gs;sp (OCT-07,JUN-09,MAR-,MAR-,OCT-,MAR-4,JUN-4, OCT-6, MAR-7) ()(0,0,-4) ()(,0,0) ()(0,,0) (4)(,,0). r = i + j + k + t( i + j + k) kw;wk; r = i + j + 5k + s(i + j + k)vd;w NfhLfs;.. 4. ntl;bf;nfhs;sk; Gs;sp (OCT-06, JUN-0, JUN-,JUN-,MAR-5,MAR-6) () (,,) ()(,,) ()(,,) (4)(,,) x = y = z kw;wk; x = y 4 = z () x = y = z 5 4 njhiyt () 6 kw;wk; x = y = z 4 vd;w NfhLfSf;fpilNaAs;s kpff; Fiwe;j njhiyt () (MAR-06,OCT-,OCT-4) (4) 6 vd;w,iz NfhLfSf;fpilNaAs;s kpff; Fiwe;j ( OCT-07, JUN-, OCT-,MAR-6) () () () (4)0 x x 5 Mfpa,U NfhLfSk; (),iz () ntl;bf;nfhs;git () xu jsk; mikahjit (4) nrq;fj;j (JUN-06,MAR-0) 5. x + y + z 6x + 8y 0z + = 0 vd;w Nfhsj;jpd; ikak; kw;wk; Muk; ( OCT-08,JUN-0,OCT-,JUN-,MAR-4,JUN-4) () (,4, 5),49 ()( 6, 8, 0), ()(, 4,5),7 (4)(6, 8,0), i + i 00,d; kjpg;g. fyg;ngz;fs; () ()0 () (4) (JUN-0,JUN-,MAR-6). e iπ 4 vd;w fyg;ngz;zpd; kl;l tpr;r KiwNa (JUN-07,08, MAR-08,0,5,OCT-09,5) ()e 9, π () e 9, π () e 6, π 4 (4) e 9, π 4. m + + i(n ) vd;w fyg;ngz;zpd;,iznad; m 5 + i(n + 4) vdpy; (n, m) vd;gj (MAR-07,OCT-0,JUN-6, MAR-7) (), 8 (), 8 (), 8 (4), 8 ways00@gmail.com

10 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 4. x + y = vdpy; +x+iy +x iy,d; kjpg;g (JUN-09,MAR-,MAR-,JUN-4, OCT-6) () x iy () x () iy (4) x + iy 5. + i vd;w fyg;ngz;zpd; kl;l (JUN-) () () () 7 (4)7 6. A + ib = (a + ib )(a + ib )(a + ib ) vdpy; A + B,d; kjpg;g (JUN-5) ()a + b + a + b + a + b ()(a + a + a ) + (b + b + b ) ()( a + b )( a + b )( a + b ) (4)( a + a + a )( b + b + b ) 7. a = + i kw;wk; z = i vdpy; cs;s az, az kw;wk; az vd;gd xu Mh;fd; jsj;jpy; (OCT-06,JUN-08,OCT-4) () nrq;nfhz Kf;Nfhzj;jpd; Kidg;Gs;spfs; () rkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs; (),U rkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs; (4) xnu Nfhliktd 8. fyg;ngz; jsj;jp;y; z, z, z, z 4 vd;w Gs;spfs; KiwNa xu,izfuj;jpd; Kidg; Gs;spfshf,Ug;gjw;Fk; mjd; kwjiyak; cz;ikahf,ug;gjw;fk; cs;s epge;jid (OCT-5,MAR-6) () z + z 4 = z + z () z + z = z + z 4 () z + z = z + z 4 (4) z z = z z 4 9. z xu fyg;ngz;izf; Fwpg;gnjdpy; arg z + arg(z ) vd;gj (OCT-08,MAR-09,MAR-, OCT-6) () π 4 () π ()0 (4) π 4 0. xu fyg;ngz;zpd; tpr;r π vdpy; me;j vz; (OCT-0,OCT-,OCT-,MAR-4) () Kw;wpYk; fw;gid vz; () Kw;wpYk; nka; vz; ()0 (4) nka;aky;y fw;gidaky;y. iz vd;w fyg;ngz;iz Mjpiag; nghwj;j π Nfhzj;jpy; fbfhu vjph;jpirapy; Row;Wk;NghJ me;j vz;zpd; Gjpa epiy (JUN-,OCT-,MAR-,OCT-5) () iz () iz () z (4) z. fyg;ngz; i 5,d; Nghyhh; tbtk; (MAR-06,OCT-06,MAR-07,OCT-07,JUN-09,JUN-5) () cos π + i sin π ()cos π + i sin π () cos π i sin π (4) cos π i sin π. P MdJ fyg;g vz; khwp z I Fwpf;fpd;wJ z = z vdpy; P,d; epakg;ghij (JUN-06,MAR-0,MAR-,JUN-,OCT-4, MAR-7) 4. () x = 4 vd;w Neh;f;NfhL () y = vd;w Neh;f;NfhL 4 () z = vd;w Neh;f;NfhL (4) x + y 4x = 0 vd;w tl;lk; +e iθ +e iθ = (JUN-07,MAR-,JUN-) () cos θ + i sin θ () cos θ i sin θ () sin θ i cos θ (4) sin θ + i cos θ 5. z n = cos nπ + i sin nπ vdpy; z z. z 6 is vd;gj (JUN-08,OCT-,JUN-4,JUN-6) () () () i (4) i 6. z %d;whk; fhy;gfjpapy; mike;jhy; mikak; fhy;gfjp (OCT-,MAR-4,MAR-6) () Kjy; fhy;gfjp (),uz;lhk; fhy;gfjp ( ) %d;whk; fhy;gfjp (4) ehd;fhk; fhy;gfjp ways00@gmail.com

11 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 7. x = cos θ + i sin θ vdpy; x n + n,d; kjpg;g (MAR-08,OCT-08,MAR-09,MAR-5) x () cos n θ () i sin nθ () sin nθ (4) i cos n θ 8. a = cos α i sin α, b = cos β i sin β, c = cos γ i sin γ vdpy; (a c b ) abc vd;gj ()cos (α β + γ) + i sin (α β + γ) () i sin (α β + γ) 9. z = 4 + 5i, z = + i vdpy; () i () z z vd;gj + i () () cos(α β + γ) (4) cos(α β + γ) (MAR-4,JUN-4) (OCT-06,OCT-07,OCT-,JUN-,JUN-6) i (4) + i 0. i + i + i + i 4 + i 5,d; kjpg;g vd;gj (MAR-06,JUN-06,JUN-07,JUN-09) () i () i () (4). i + i 4 + i 5 + i 6,d;,iz fyg;ngz; () () ()0 (4) i. i + vd;gj ax bx + c = 0vd;w rkd;ghl;bd; xu %ynkdpy; kw;nwhu jph;t () i () i () + i (4) i + i (MAR-08,OCT-09,MAR-0,MAR-). ±i 7 vd;w jph;tfisf; nfhz;l,ugbr; rkd;ghl (OCT-09,JUN-0,MAR-) () x + 7 = 0 () x 7 = 0 () x + x + 7 = 0 (4) x x 7 = i kw;wk; 4+i vd;w %yq;fisf; nfhz;l rkd;ghl (MAR-07) ()x + 8x + 5 = 0 () x + 8x 5 = 0 () x 8x + 5 = 0 (4) x 8x 5 = 0 5. ax + bx + = 0 vd;w rkd;ghl;bd; xu jph;t i, a Ak; b Ak; nka; vdpy; a, b vd;gj +i (OCT-07,JUN-) () (,) ()(, ) () (0,) (4)(,0) 6. x 6x + k = 0 vd;w rkd;ghl;bd; xu %yk; i+ vdpy; k,d; kjpg;g (OCT-08,MAR-) ()5 () 5 () 0 (4)0 7. ω vd;gj,d; Kg;gb %yk; vdpy; ( ω + ω ) 4 + ( + ω ω ) 4,d; kjpg;g (MAR-06,JUN-06,OCT-0,JUN-,OCT-,OCT-4,JUN-5) ()0 () () 6 (4) 8. ω vd;gj,d; nmk; gb %yk; vdpy; (JUN-0,JUN-,MAR-5) () + ω + ω 4 + = ω + ω + ω 5 + () ω n = 0 () ω n = (4)ω = ω n 9. ω vd;gj,d; Kg;gb %yk; vdpy;> ω ω ω 4 ( ω 8 ),d; kjpg;g (MAR-09,OCT-,OCT-, OCT-6, MAR-7) () 9 () 9 ()6 (4) ways00@gmail.com

12 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 4. gfkiw tbtf;fzpjk;. y y + 8x = 0vd;w gutisaj;jpd; mr;r (OCT-08,MAR-, OCT-,JUN-, OCT-6) () y = () x = () x = (4) y =. 6x y x y 44 = 0 vd;gj (JUN-08,OCT-0) () xu eps;tl;lk; () xu tl;lk; () xu gutisak; (4) xu mjpgutisak;. 4x + y = c vd;w NfhL y = 6x vd;w gutisaj;jpd; njhlnfhl vdpy; c,d; kjpg;g (JUN-09,MAR-0,JUN-,OCT-,MAR-5) () () ()4 (4) 4 4. y = 8x vd;w gutisaj;jpy; t = t kw;wk; t = t vd;w Gs;spfspy; tiuag;gl;l njhlnfhlfs; ntl;bf;nfhs;sk; Gs;sp (MAR-08,OCT-09,JUN-4) () (6t, 8t) () (8t, 6t ) () (t, 4t) (4) (4t, t ) 5. y 4x + 4y + 8 = 0 vd;w gutisaj;jpd; nrt;tfyj;jpd; epsk; (MAR-07,MAR-09,MAR-4,JUN-5,JUN-6) ()8 ()6 ()4 (4) 6. y = x + 4 vd;w gutisaj;jpd;,af;ftiuapd; rkd;ghl (JUN-0) ()x = 5 () x = 5 () x = 7 (4) x = (, ) vd;w Kid x = 4 vd;w,af;ftiuiaf; nfhz;l gutisaj;jpd; nrt;tfy epsk; (OCT-07,MAR-6) () () 4 () 6 (4) 8 8. x = 6y vd;w gutisaj;jpd; Ftpak; (OCT-5) () (4,0) () (0,4) ()(-4,0) (4)(0,-4) 9. x = 8y vd;w gutisaj;jpd; Kid (MAR-) () 8, 0 () 8, 0 () 0, 8 (4) 0, 8 0. x + y + 9 = 0 vd;wf; NfhL y = 8x vd;w gutisj;ij njhlk; Gs;sp (MAR-06,MAR-) ()(0,-) ()(,4) () 6, 9 (4) 9, 6. y = x vd;w gutisaj;jpd; Ftpehzpd;,Wjpg;Gs;spfspy; tiuag;glk; njhlnfhlfs; re;jpf;fk; Gs;sp mikak; NfhL (JUN-07,OCT-) () x = 0 () x + = 0 () y + = 0 (4) y = 0. (-4,4) vd;w Gs;spapypUe;J y = 6x f;f tiuag;glk;,u njhlnfhlfsf;f,ilnaas;s Nfhzk; (JUN-08) ()45 () 0 () 60 (4) 90. 9x + 5y 54x 40y + 6 = 0 vd;w $k;g tistpd; ikaj; njhiyj;jft (e),d; kjpg;g (MAR-07,OCT-, OCT-6, MAR-7) () () () 4 9 (4) 5 4. x 44 + y 69 = vd;w eps;tl;lj;jpd; miu-nel;lr;r kw;wk; miu-fw;wr;r epsq;fs; (JUN-0,MAR-5) () 6, (),4 (),6 (4), ways00@gmail.com - -

13 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 5. 9x + 5y = 80 vd;w eps;tl;lj;jpd; Ftpaq;fSf;fpilNa cs;s njhiyt (JUN-09,JUN-,MAR-,JUN-,OCT-,MAR-4) () 4 () 6 () 8 (4) 6. xu eps;tl;lj;jpd; nel;lr;r kw;wk; mjd; miu Fw;wr;Rfspd; epsq;fs; KiwNa 8, mjd; rkd;ghlfs; y 6 = 0 kw;wk; x + 4 = 0 vdpy;> eps;tl;lj;jpd; rkd;ghl () x+4 + y 6 x+4 y 6 = () + = () x+4 y 6 = (4) x+4 y 6 = x y + c = 0 vd;w Neh;f;NfhL4x + 8y = vd;w eps;tl;lj;jpd; njhlnfhl vdpy; c,d; kjpg;g (OCT-08,OCT-0,MAR-,OCT-) ()± ()±6 ()6 (4)±4 8. 4x + 9y = 6 vd;w eps;tl;lj;jpd; kpjs;s VNjDk; xu Gs;spapypUe;J 5, 0 kw;wk; ( 5, 0) vd;w Gs;spfSf;fpilNa cs;s njhiytfspd; $Ljy; () 4 () 8 () 6 (4) 8 (MAR-06,OCT-09) 9. 9x + 6y = 44 vd;w $k;g tistpd;,af;f tl;lj;jpd; Muk; (OCT-06,JUN-08,MAR-,OCT-4,MAR-6) () 7 () 4 () (4) x + 5y = 400 vd;w tistiuapd; Ftpaj;jpypUe;J xu njhlnfhl;lf;f tiuag;glk; nrq;fj;jf; NfhLfspd; mbapd; epakg;ghij (MAR-09,JUN-6) () x + y = 4 () x + y = 5 () x + y = 6 (4) x + y = 9. y 4x 4x + 48y 7 = 0 vd;w mjpgutisaj;jpd; ikaj;njhiyj;jft (OCT-09,MAR-0,OCT-5) () 4 () () (4)6. nrt;tfyj;jpd; epsk;> Jizar;rpd; epsj;jpy; ghjp vdf; nfhz;ls;s mjpguisaj;jpd; ikaj; njhiyj; jft (JUN-06,JUN-07,MAR-5). () () 5 () (4) 5 x y a b = vd;w mjpgutisaj;jpd; kpjs;s VNjDk; xu Gs;spapypUe;J Ftpaj;jpw;F,ilNaAs;s njhiytfspd; tpj;jpahrk; 4 kw;wk; ikaj;njhiyj;jft vdpy; mjpgutisaj;jpd; rkd;ghl (MAR-07,JUN-) () x y x = () y x = () y = (4) x y = x 4(y ) = 6 vd;w mjpgutisaj;jpd;,af;ftiu (MAR-06,OCT-09, MAR-6) ()y = ± 8 5 () x = ± 8 5 () y = ± 5 8 (4) x = ± x y = 6 f;f 5x y + 4k = 0 vd;w NfhL xu njhlnfhl vdpy; k,d; kjpg;g (OCT-06,JUN-5) () 4 9 () () 9 4 (4) x = vd;w mjpgutisaj;jpw;f (,)vd;w Gs;spapypUe;J tiuag;glk; njhlnfhlfspd; 6 9 njhlehz; (JUN-) () 9x 8y 7 = 0 () 9x + 8y + 7 = 0 ()8x 9y 7 = 0 (4) 8x + 9y + 7 = 0 ways00@gmail.com - -

14 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 7. x y = vd;w mjpgutisaj;jpd; njhiyj;njhlnfhlfsf;fpilnaas;s Nfhzk; 6 9 (MAR-08,MAR-,OCT-) () π tan 4 () π tan 4 () tan 4 (4) tan y 5x = 0 vd;w mjpgutisaj;jpd; njhiyj;njhlnfhlfs; (MAR-7) ()y = ± 6 x () y = ± x () y = ± x (4) y = ± x (8,0)vd;w Gs;spapypUe;J x y = vd;w mjpgutisaj;jpd; njhiyj;njhlfsf;f tiuag;glk; 64 6 nrq;fj;j J}uq;fspd; nguf;fy; gyd; (JUN-0,JUN-4,JUN-6) () () () x y = vd;w mjpgutisaj;jpd; nrq;fj;jj;njhlnfhlfspd; ntl;lk; Gs;spapd; epakg;ghij 6 9 (OCT-08,JUN-) (4) 5 6 () x + y = 5 () x + y = 4 () x + y = (4) x + y = 7. x + y 5 = 0,x y + 5 = 0 vd;w njhiyj;;;;njhlnfhlfisf; nfhz;l mjpgutisaj;jpd; ikaj;njhiyj;jft (OCT-06,OCT-07) () () () (4). xy = 8 vd;w nrt;tf gutisaj;jpd; miu FWf;fr;rpd; epsk; (OCT-0,OCT-,MAR-4) () () 4 () 6 (4) 8. xy = c vd;w nrt;tf mjpgutisaj;jpd; njhiyj;njhlnfhlfs; (MAR-) ()x = c, y = c () x = 0, y = c () x = c, y = 0 (4) x = 0, y = 0 4. xy = 6 vd;w nrt;tf mjpgutisaj;jpd; Kidapd; Maj;njhiyTfs; (MAR-07,OCT-) () (4,4),(-4,-4) () (,8),(-,-8) ()(4,0),(-4,0) (4) (8,0),(-8,0) 5. xy = 8 vd;w nrt;tf mjpgutisaj;jpd; xu Ftpak; (MAR-09, MAR-7) ()(6,6) ()(,) ()(4,4) (4)(5,5) 6. xy = vd;w nrt;tf mjpgutisaj;jpd; nrt;tfyj;jpd; epsk; (MAR-08,0, JUN-, OCT-4,5,6) ()8 () ()8 (4)6 7. xy = 7 vd;w jpl;l nrt;tf mjpgutisaj;jpd; kpjs;s VNjDk; xu Gs;spapypUe;J tiuag;glk; njhlnfhl mjd; njhiyj;njhlnfhlfsld; cz;lhf;fk; Kf;Nfhzj;jpd; gug;g (MAR-,JUN-5) ()6 ()8 ()7 (4)44 8. xy = 9 vd;w nrt;tf mjpgutisaj;jpd; kpjs;s 6, vd;w Gs;spapypUe;J tiuag;glk; nrq;fj;j> tistiuia kpz;lk; re;jpf;fk; Gs;sp (JUN-,JUN-4) (), 4 () 4, 8 8 (), 4 8 (4) 4, 8 5. tif Ez;fzpjk; : gad;ghlfs; - I. x =,y; y = x + x + 5 vd;w tistiuapd; rha;t (OCT-6) () 0 ()7 () 6 (4). r Muk; nfhz;l xu tl;lj;jpd; gug;g A,y; Vw;gLk; khwk; tpjk; (OCT-08) ()πr () πr dr dt ()πr dr dt (4) π dr dt ways00@gmail.com - -

15 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top. Mjpap;ypUe;J xu Neh;f;Nfhl;by; x njhiytp;y; efuk; Gs;spapd; jpirntfk; v vdtk; a + bv = x vdtk; nfhlf;fg;gl;ls;sj.,q;f a kw;wk; b khwpypfs;. mjd; KLf;fk; MdJ (MAR-09) () b x () a x 4. xu cufk; gdpf;fl;bf; Nfhsj;jpd; fd mst nr.kp / epkplk; vdf; Fiwfpd;wJ. mjd; tpl;lk; 0 nr.kp vd,uf;fk; NghJ tpl;lk; FiwAk; Ntfk; MdJ (MAR-0,MAR-,JUN-4) () nr.kp / epkplk; () nr.kp / epkplk; 50π 50π () () x b 75π (4) x a nr.kp / epkplk; (4) 75π nr.kp / epkplk; 5. y = x + sin x vd;w tistiuf;f x = 0 tpy; njhlnfhl;bd; rha;t (MAR-07,JUN-08,JUN-09,MAR-,JUN-4) () () () (4)- 6. y = x vd;w tistiuf;f x,d; Maj;njhiyT vdf; nfhz;ls;s Gs;spapy; nrq;nfhl;bd; rha;thdj (MAR-06,OCT-06,JUN-07,OCT-09,JUN-,MAR-5) () () 4 () (4) 7. y = x 6x 4 vdk; tistiuap;y; x mr;rf;f,izahfts;s njhlnfhl;bd; ; njhlgs;sp (OCT-0,JUN-5) () 5, 7 () 5, 7 () 5, 7 (4), 7 8. y = x vdk; tistiuf;f (, ) vd;w Gs;spapy; njhlnfhl;bd; rkd;ghl (MAR-08) 5 5 ()5y + x = () 5y x = ()x 5y = (4) x + y = 9. θ = t vdk; tistiuf;f Gs;sp, vd;w Gs;spapy; nrq;nfhl;bd; rkd;ghl ()θ = 7t 80 () 5θ = 7t 80 ()θ = 7t + 80 (4) θ = t (OCT-,MAR-4) 0. x + y x = kw;wk; y = vdk; tistiufsf;f,ilg;gl;l Nfhzk; (JUN-07,MAR-09,OCT-, OCT-6) () π 4 () π () π 6 (4) π. y = e mx kw;wk; y = e mx > m > vd;dk; tistiufsf;f,ilg;gl;l Nfhzk; (OCT-,MAR-6) m ()tan m m () tan m m () tan +m. x + y = a vdk; tistiuapd; Jiz myfr; rkd;ghlfs; ()x = a sin θ ; y = a cos θ () x = a cos θ ; y = a sin θ ()x = a sin θ ; y = a cos θ (4) x = a cos θ ; y = a sin θ m (4) tan m +. x + y = a vd;w tistiuapd; nrq;nfhl x mr;rld; θ vd;dk; Nfhzk; Vw;gLj;Jnkdpy; mr;nrq;nfhl;bd; rha;t (MAR-,MAR-) () cot θ () tan θ () tan θ (4) cot θ 4. xu rjuj;jpd; %iytpl;lj;jpd; epsk; mjpfhpf;fk; tpjk; 0. nr.kp / tpdhb vdpy; gf;f mst 5 nr.kp Mf,Uf;Fk;NghJ mjd; gug;gst mjpfhpf;fk; tpjk; (OCT-,JUN-6) (). 5 nr.kp / tpdhb () nr.kp / tpdhb () nr.kp / tpdhb (4) 0.5 nr.kp / tpdhb ways00@gmail.com

16 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 5. xu Nfhsj;jpd; fd mst kw;wk; Muj;jpy; Vw;gLk;;;; khwtpjq;fs; vz;zstpy; rkkhf,uf;fk;nghj Nfhsj;jpd; tisgug;g (JUN-06,MAR-0) () () () 4π (4) 4π π 6. x x + x+8 mjpfhpf;fk; tpjkhdj x mjpfhpf;fk; tpjj;ij Nghy;,Uklq;F vdpy; x,d; kjpg;gfs; (JUN-08,JUN-) (), (), (), (4), 7. xu cuisapd; Muk; nr.kp tpdhb vd;w tpjj;jp;y; mjpfhpf;fpd;wj. mjd; cauk; nr.kp tpdhb vd;w tpjj;jpy; Fiwfpd;wJ. Muk; nr.kp kw;wk; cauk; 5 nr.kp Mf,Uf;Fk;NghJ mjd; fd mstpd; khw tpjk; (JUN-07) ()π () π () 4π (4) 5π 8. y = 6x x NkYk; x MdJ tpdhbf;f 5 myffs; tpjj;jpy; mjpfhpf;fpd;wj. x = vdk; NghJ mjd; rha;tpd; khwtpjk; (MAR-) () - 90 myffs; tpdhb ()90 myffs; tpdhb ()80 myffs; tpdhb (4)-80 myffs; tpdhb 9. xu fdr;rjuj;jpd; fd mst 4 nr.kp tpdhb vd;w tpjj;jpy; mjpfhpf;fpd;wj. mf;fdr;rjuj;jp;d; fd mst 8 f.nr.kp Mf,Uf;Fk; NghJ mjd; Gwg;gug;gsT mjpfhpf;fk; tpjk; () 8 nr.kp tpdhb () 6 nr.kp tpdhb () nr.kp tpdhb (4) 4 nr.kp tpdhb 0. y = 8 + 4x x vd;w tistiu y-mr;ir ntl;;lk; Gs;spap;y; mikak; njhlnfhl;bd; rha;t ()8 () 4 ()0 (4)-4. y = x kw;wk; x = y vd;w gutisaq;fsf;fpilna Mjpapy; mikak; Nfhzk; (JUN-06,JUN-0,MAR-4,OCT-4) () tan 4 () tan 4 () π (4) π 4. x = e t cos t; y = e t sin t vd;w tistiuapd; njhlnfhl x-mr;rf;f,izahfts;sj vdpy; t,d; kjpg;g (JUN-) () π 4 () π 4 ()0 (4) π. xu tistiuapd; nrq;nfhl x - mr;rpd; kpif jpirapy; θ vd;dk; Nfhzj;ij Vw;gLj;JfpwJ. mr;nrq;nfhl tiuag;gl;l Gs;spapy; tistiuapd; rha;t (OCT-07) () cot θ () tan θ () tan θ (4) cot θ 4. y = e x kw;wk; y = a e x vd;dk; tistiufs; nrq;fj;jhf ntl;bf;nfhs;fpd;wd vdpy; a,d; kjpg;g () () () (4) (OCT-0) 5. s = t 4t + 7 vdpy; KLf;fk; G+r;rpakhFk; NghJs;s jpirntfk; (OCT-06,MAR-07,OCT-09,JUN-5) () m/sec () 6 m/sec () 6 m/sec (4) m/sec 6. xu Neh;f;Nfhl;by; efuk; Gs;spapd; jpirntfkhdj> mf;nfhl;by; xu epiyg;gs;spapypue;j efuk; Gs;sp;f;F,ilapy; cs;s njhiytpd; th;f;fj;jpw;f Neh; tpfpjkhf mike;js;sj vdpy; mjd; KLf;fk; gpd;tuk; xd;wpdf;f tpfpjkhf mike;js;sj. (OCT-, JUN-6) ()s () s () s (4) s 4 ways00@gmail.com

17 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 7. y = x vd;w rhh;gpw;f [,],y; Nuhypd; khwpyp () () 0 () (4) 8. a = 0, b = vdf;nfhz;l f x = x + x vd;w rhh;gpw;f nyf;uhq;rpapd;,ilkjpg;gj; Njw;wj;jpd;gbAs;s c,d; kjpg;g (MAR-09,OCT-,OCT-4) () () ()0 (4) 9. f x = cos x vd;w rhh;gpw;f [π, π],y; Nuhy; Njw;wj;jpd;gb mike;j c,d; kjpg;g (MAR-06, 08,, 7) ()0 () π () π (4) π 0. a = kw;wk; b = 4 vdf;nfhz;l> f x = x vd;w rhh;gpw;f nyf;uhq;rpapd;,ilkjpg;gj; Njw;wj;jpd;gb mikak; c,d; kjpg;g (JUN-0,JUN-, OCT-5, OCT-6) (OCT-07,OCT-08) () 9 4 () () (4) 4 lim x. x e x d; kjpg;g () () 0 () (4). lim x 0 a x b x c x d x d; kjpg;g (MAR-07,OCT-09) () () 0 ()log ab. f a = ; f a = ; g a = ; g a = vdpy; lim x a cd g x f a g a f(x) x a (4) log(a/b) log(c/d),d; kjpg;g () 5 () 5 () (4) (JUN-08,MAR-6) 4. gpd;tutdtw;ws; vj (0, ),y; VWk; rhh;g? (OCT-06,OCT-,OCT-5) () e x () x () x (4)x 5. f x = x 5x + 4 vd;w rhh;g VWk;,ilntsp (MAR-,JUN-) ()(, ) ()(,4) ()(4, ) (4) vy;yh Gs;spfsplj;Jk; 6. f x = x vd;w rhh;g,wq;fk;,ilntsp (JUN-09,MAR-5) ()(, ) ()(, 0) ()(0, ) (4) (, ) 7. y = tan x x vd;w rhh;g () 0, π,y; VWk; rhh;g () 0, π,y;,wq;fk; rhh;g (JUN-4) () 0, π 4,y; VWk; π 4, π,y;,wq;fk; (4) 0, π 4,y;,wq;Fk; π 4, π,y; rhh;g 8. nfhlf;fg;gl;ls;s miu tl;lj;jpd; tpl;lk; 4 nr.kp. mjds; tiuag;glk; nrt;tfj;jpd; nguk gug;g (MAR-06,OCT-4) () ()4 ()8 (4) kp gug;g nfhz;ls;s nrt;tfj;jpd; kpr;rpw Rw;wsT (OCT-07,OCT-08,OCT-0,OCT-,JUN-5) ()0 ()0 ()40 (4) f x = x 4x + 5vd;w rhh;g [0,],y; nfhz;ls;s kpg;ngu nguk kjpg;g (MAR-,7, OCT-5) () () ()4 (4)5 ways00@gmail.com

18 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 4. y = e x vd;w tistiu (MAR-0,MAR-) () x > 0 tpw;f Nky;Nehf;fpf; FopT () x > 0 tpw;f fpo;nehf;fpf; FopT () vg;nghjk; Nky;Nehf;fpf; FopT (4) vg;nghjk; fpo;nehf;fpf; FopT 4. gpd;tuk; tistiufss; vj fpo;nehf;fp FopT ngw;ws;sj? (MAR-08,JUN-09,OCT-) ()y = x () y = x ()y = e x (4) y = x + x 4. y = x 4 vd;w tistiuapd; tist khw;wg;gs;sp (JUN-,MAR-5) ()x = 0 () x = ()x = (4) vq;fkpy;iy 44. y = ax + bx + cx + d vd;w tistiuf;f x =,y; xu tist khw;wg;gs;sp cz;nldpy; ()a + b = 0 () a + b = 0 ()a + b = 0 (4) a + b = 6. tif Ez;fzpjk; : gad;ghlfs; - II. u = x y vdpy; u x f;fr; rkkhdj (OCT-08,OCT-0,OCT-5, OCT-6) ()yx y ()u log x () u log y (4) xy x. u = sin x 4 +y 4 x +y kw;wk; f = sin u vdpy;> rkgbj;jhd rhh;g f,d;gb ()0 () () (4) 4. u = u u x +y, vdpy; x + y = x y () u () u () u (4) u (MAR-08,JUN-09,JUN-0,JUN-) 4. y x = x ( + x) vd;w tistiuf;f (OCT-09,MAR-, MAR-7) () x-mr;rf;f,izahd xu njhiyj;njhlnfhl cz;l () y- mr;rf;f,izahd xu njhiyj;njhlnfhl cz;l (),U mr;rfsf;fk;,izahd njhiyj;njhlnfhlfs; cz;l (4) njhiyj; njhlnfhlfs;,y;iy 5. x = r cos θ, y = r sin θ vdpy; r x = ()sec θ ()sin θ ()cos θ (4) cosec θ (JUN-09,MAR-,MAR-4) 6. gpd;tutdtw;ws; rhpahd $w;wfs;: (MAR-,JUN-6) (i) xu tistiu Mjpiag; nghwj;j rkr;rph; ngw;wpug;gpd; mj,u mr;rfisg; nghwj;jk; rkr;rph; ngw;wpuf;fk; (ii) xu tistiu,u mr;rfisg; nghwj;j rkr;rph; ngw;wpug;gpd; mj Mjpiag; nghwj;jk; rkr;rph;; ngw;wpuf;fk; (iii) f x, y = 0 vd;w tistiu y = x vd;w Nfhl;ilg; nghwj;j rkr;rph; ngw;ws;sj vdpy; f x, y = f(y, x) (iv) f x, y = 0 vd;w tistiuf;f f x, y = f( y, x),cz;ikahapd; mj Mjpiag; nghwj;j rkr;rph; ngw;wpuf;fk; () (ii),(iii) ()(i),(iv) ()(i),(iii) (4)(ii),(iv) 7. u = log x +y vdpy; x u u + y vd;gj (JUN-06,MAR-07,OCT-07,MAR-0,OCT-,JUN-5) xy x y () 0 () u () u (4)u ways00@gmail.com

19 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 8. 8,d; Mk; gb%y rjtpfpjg; gpio Njhuhakhf 8,d; rjtpfpjg; gpioiag; Nghy; klq;fhfk; (MAR-06,OCT-06,MAR-,OCT-4) () 8 () () (4)8 9. a y = x (a x ) vd;w tistiu (OCT-07,OCT-09,MAR-0,JUN-,OCT-,JUN-5) () x = 0 kw;wk; x = a f;f,ilapy; xu fz;zp kl;lnk nfhz;ls;sj () x = 0kw;Wk; x = a f;f,ilapy;,u fz;zpfs; nfhz;l cs;sj () x = a kw;wk; x = a f;f,ilapy;,u fz;zpfs; nfhz;l cs;sj (4) fz;zp VJkpy;iy 0. y a + x = x (a x) vd;w tistiuapd; njhiyj; njhlnfhl (JUN-06,07,08,, OCT-,, 6) () x = a () x = a/ () x = a/ (4) x = 0. y a + x = x (a x) vd;w tistiu gpd;tutdtw;ws; ve;jg; gfjpapy; mikahj? (MAR-09,JUN-0,,4,6) () x > 0 () 0 < x < a () x a kw;wk; x > a (4) a < x < a. u = y sin x, vdpy; u = x y () cos x ()cos y ()sin x (4)0. u = f y x vdpy;> x u u + y,d; kjpg;g x y (OCT-08,MAR-09,JUN-) ()0 () ()u (4) u (JUN-07,JUN-08) 4. 9y = x (4 x ) vd;w tistiu vjw;f rkr;rph;? (MAR-06,OCT-06,OCT-0,MAR-5,OCT-5) () y mr;r () x mr;r () y = x (4),U mr;rfs; 5. ay = x (a x) vd;w tistiu y mr;ir ntl;lk; Gs;spfs; (MAR-08,MAR-5,MAR-6) () x = a, x = 0 ()x = 0, x = a () x = 0, x = a (4) x = 0 7. njhif Ez;fzpjk; : gad;ghlfs;.. π/ 0 () π π/ 0 cos 5/ x cos 5/ x+sin 5/ x sin x cos x +sin x cos x dx,d; kjpg;g () π 4 dx,d; kjpg;g ()0 (4) π (MAR-,JUN-,JUN-5) (JUN-0, MAR-7) () π () 0 () π 4 (4) π. x( x) 4 dx,d; kjpg;g 0 (MAR-06,MAR-09,JUN-0,MAR-,OCT-,JUN-4,MAR-5,OCT-5) () 4. π/ π/ sin x +cos x () 0 dx,d; kjpg;g () 4 (4) 0 (JUN-07,OCT-07,OCT-0, OCT-6) ()0 () ()log (4)log 4 π 5. sin 4 x dx,d; kjpg;g (OCT-06,OCT-09,JUN-,MAR-4) 0 () π 6 () 6 ()0 (4) π 8 ways00@gmail.com

20 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top π/4 6. cos x dx 0 () () π 7. sin x cos xdx,d; kjpg;g 0,d; kjpg;g (MAR-07, 08, 0, 4, 7, JUN-09, OCT-08,, 4) ()0 (4) π (JUN-08,MAR-,OCT-, JUN-6) ()π () π () π (4) y = x vd;w Nfhl;bw;Fk; x-mr;r> NfhLfs; x = kw;wk; x = Mfpatw;wpw;Fk;,ilg;gl;l muq;fj;jpd; gug;g (JUN-07,OCT-08,MAR-09,JUN-,OCT-5) () () 5 () (4) 7 9. x = 0,ypUe;J x = π 4 gug;g tiuapyhd y = sin x kw;wk; y = cos x vd;w tistiufspd;,ilg;gl;l (JUN-06,OCT-07,MAR-0,OCT-,JUN-4,MAR-6) () + () () (4) + 0. x + y a b = vd;w eps;tl;lj;jpw;fk; mjd; Jiz tl;lj;jpw;fk;,ilg;gl;l gug;g (MAR-06,JUN-06,MAR-07,JUN-09) () πb(a b) () πa(a b) () πa(a b) (4) πb(a b). gutisak; y = x f;fk; mjd; nrt;tfyj;jpw;fk;,ilg;gl;l gug;g. x () y 6 () 6 () (MAR-08,MAR-,OCT-,JUN-5) (4) 8 = vd;w tistiuia Fw;wr;ir nghwj;j Row;wg;gLk; jplg;nghuspd; fd mst ()48π ()64π () π (4)8π (JUN-07,JUN-,MAR-,OCT-,OCT-6). y = + x vd;w tistiu x = 0 tpypue;j x = 4 tiu x- mr;ir mr;rhf itj;jr; Row;wg;gLk; jplg;nghuspd; fd mst (MAR-06,OCT-08,OCT-09,MAR-,JUN-) ()00π () 00 9 π ()00 π (4)00 4. NfhLfs; y = x, y = kw;wk; x = 0 Mfpait Vw;gLj;Jk; gug;g y-mr;ir nghwj;jr; Row;wg;gLk; jplg;nghuspd; fd mst (JUN-08,OCT-0,JUN-,JUN-4,OCT-4) () π 4 () π () π 5. x + y a b = vd;w eps;tl;lj;jpd; gug;ig nel;lr;r> Fw;wr;R,tw;iw nghwj;jr; Row;wg;gLk; jplg;nghuspd; fd mstfspd; tpfpjk; (JUN-06,MAR-09,JUN-0,JUN-,MAR-,OCT-,MAR-5, MAR-7) (4) π ()b : a ()a : b () a: b (4) b: a 6. (0,0),(,0) kw;wk; (,) Mfpatw;iw Kidg;Gs;spfshff; nfhz;l Kf;Nfhzj;jpd; gug;g x- mr;ir nghwj;jr; Row;wg;gLk; jplg;nghuspd; fd mst (OCT-06, OCT-07,JUN-09,OCT-,MAR-4,MAR-6) ()8π ()π () 6π (4)9π 7. x + y = 4 vd;w tistiuapd; tpy;ypd; epsk; (MAR-08,JUN-08,MAR-,OCT-4,MAR-5,JUN-6) ()48 ()4 () (4)96 ways00@gmail.com

21 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 8. y = x, x = 0 kw;wk; x =,tw;wpw;f,ilna Vw;gLk; gug;g x-mr;ir nghwj;jr; Row;wg;gLk; jplg;nghuspd; tisg;gug;g (OCT-06,MAR-07,OCT-0,OCT-,MAR-,JUN-5,OCT-5,MAR-6) ()8 5π () 5π () 5π (4)4 5π 9. Muk; 5 cs;s Nfhsj;ij jsq;fs; ikaj;jpypue;j kw;wk; 4 J}uj;jpy; ntl;lk;,u,izahd jsq;fsf;f,ilg;gl;l gfjpapd; tisgug;g (OCT-09,MAR-0,JUN-,JUN-6, OCT-6).. ()0π ()40π ()0π (4)0π dy 8. tiff;nfor; rkd;ghlfs; + y = dx x e4x vd;w tiff;nfor; rkd;ghl;bd; njhiff; fhuzp () log x () x () e x (4) x dy dx (JUN-09,JUN-) + Py = Q vd;w tiff;nfor; rkd;ghl;bd; njhiff; fhuzp cos x vdpy; P,d; kjpg;g () cot x () cot x () tan x (4) tan x (JUN-07,OCT-08,MAR-09,OCT-5, OCT-6). dx + xdy = e y sec ydy,d; njhiff;fhuzp (OCT-0, MAR-,OCT-4, JUN-6) 4. () e x () e x () e y (4) e y dy + dx. y = x log x x,d; njhiff;fhuzp (OCT-06,MAR-07,JUN-07,OCT-09,MAR-,JUN-5) () e x () log x () (4) e x x 5. m < 0 Mf,Ug;gpd; dx + mx = 0,d; jph;t (MAR-08,0,, 4, 7 JUN-09,0, OCT-) dy ()x = ce my () x = ce my () x = my + c (4) x = c 6. y = cx c vd;gjidg; nghjj; jph;thfg; ngw;w tiff;nfo rkd;ghl (OCT-08,MAR-4) 7. () y xy + y = 0 () y = 0 () y = c (4) y + xy + y = 0 dx + 5y / = x vd;w tiff;nfotpd; (MAR-06, JUN-08,MAR-0, MAR-7) dy () thpir kw;wk; gb () thpir kw;wk; gb () thpir kw;wk; gb 6 (4) thpir kw;wk; gb 8. xu jsj;jpy; cs;s x -mr;rf;f nrq;fj;jy;yhj NfhLfspd; tiff;nfor; rkd;ghl () dy = 0 () d y dy dx dx = 0 () = m (4) d y = m dx dx 9. Mjpg;Gs;spia ikakhff; nfhz;l tl;lq;fspd; njhfg;gpd; tiff;nfor; rkd;ghl (MAR-09,,JUN-07,5,6) ()xdy + y dx = 0 () xdy ydx = 0 () xdx + ydy = 0 (4) xdx ydy = 0 0. tiff;nfor; rkd;ghl dy + py = Q tpd; njhiff; fhuzp (MAR-06) dx () pdx () Q dx () e Q dx p dx (4) e. (D + )y = e x,d; epug;gr; rhh;g (JUN-09,JUN-,MAR-5) () (Ax + B)e x () A cos x + B sin x ()(Ax + B)e x (4) (Ax + B)e x. (D 4D + 4)y = e x,d; rpwg;gj; jph;t (P.I) (OCT-06,JUN-08,OCT-08,OCT-09,JUN-5) () x ex () xe x () xe x (4) x e x ways00@gmail.com

22 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top. y = mx vd;w Neh;f;NfhLfspd; njhfg;gpd; tiff;nfor;rkd;ghl (OCT-07) () dy = m () ydx xdy = 0 y dx ()d dx = 0 (4) ydx + xdy = dy dx / = d y dx vd;w tiff;nfor; rkd;ghl;bd; gb () () () (4)6 5. c = + dy dx / (MAR-07,OCT-,MAR-,MAR-5,OCT-5,MAR-6) d vd;w tiff;nfor; rkd;ghl;bd; gb (,q;f c xu khwpyp) (JUN-) y dx () () ()- (4) 6. xu fjphpaf;f nghuspd; khwtpj kjpg;g> mk;kjpg;gpd; (p)neh;tpfpjj;jpy; rpijtwfpwj.,jw;f Vw;w tiff; nfor; rkd;ghl (k Fiw vz;) (OCT-07,JUN-,JUN-) () dp dt = k p () dp dt = kt dp () = kp dt dp (4) = kt dt 7. xy jsj;jpys;s vy;yh Neh;f;NfhLfspd; njhfg;gpd; tiff;nfor;rkd;ghl (MAR-6, OCT-6) () dy = xu khwpyp () d y dy dx dx = 0 () y + = 0 (4) d y + y = 0 dx dx 8. y = ke λx vdpy; mjd; tiff;nfor; rkd;ghl (JUN-06,OCT-0,JUN-,JUN-4, OCT-6, MAR-7) () dy = λy dy () = ky dy () + ky = 0 dy (4) = dx dx dx dx eλx 9. y = ae x + be x vd;w rkd;ghl;by; a iaak; b iaak; epf;fpf; fpilf;fk; tiff;nfor; rkd;ghl (OCT-06,MAR-09,JUN-0,OCT-,OCT-4) () d y + ay = 0 () d y dx dx 9y = 0 () d y 9 dy = 0 (4) d y + 9x = 0 dx dx dx 0. y = e x (A cos x + B sin x) vd;w njhlh;gpy; A iaak; B iaak; epf;fpg; ngwg;glk; tiff;nfor; rkd;ghl (OCT-09,OCT-,OCT-) ()y + y = 0 () y y = 0 () y y + y = 0 (4) y y y = 0. dy dx = x y x+y vdpy; ()xy + y + x = c (MAR-08,MAR-,OCT-,JUN-, JUN-6) () x + y x + y = c () x + y xy = c (4) x y xy = c. f x = x kw;wk; f = vdpy; f x vd;gj (OCT-07, MAR-0,JUN-5,OCT-5) () (x x + ) () (x x + ) () (x x + ) (4) x( x + ). x dy + y x + y dx = 0 vd;w rkgbj;jhd tiff;nfo rkd;ghl;by; y = vx vd gpujpapl nra;ak; NghJ fpilg;gj (JUN-06, MAR-, JUN-4,OCT-4) ()xdv + v + v dx = 0 ()vdx + x + x dv = 0 ()v dx x + x dv = 0 (4)vdv + x + x dx = 0 4. dy y tan x = cos x vd;w tiff;nfor; rkd;ghl;bd; njhiff; fhuzp dx (MAR-08,JUN-08,JUN-0,OCT-,MAR-,OCT-,OCT-,MAR-4,JUN-4, MAR-7) ()sec x ()cos x ()e tan x (4)cot x 5. (D + D 4)y = e x d; rpwg;g jph;t (MAR-06,MAR-07,JUN-,MAR-) () 6xe x () xe x () xe x (4)x /e x ways00@gmail.com - -

23 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 6. f D = D a g D, g(a) 0 vdpy; tiff;nfor; rkd;ghl f D y = e ax,d; rpwg;gj; jph;t (JUN-06,OCT-0,JUN-,MAR-6) () me ax () e ax g(a) () g(a)e ax (4) xeax g(a) 9. jdpepiy fzf;fpay;. fpo;f;fz;ltw;ws; vit $w;wfs;? (MAR-,JUN-,MAR-6 ) (i) flts; cd;id Mrph;tjpf;fl;Lk; (ii) Nuhrh xu G+ (iii) ghypd; epwk; ntz;ik (iv) xu gfh vz; ()(i),(ii),(iii) ()(i),(ii),(iv) ()(i),(iii),(iv) (4)(ii),(iii),(iv). xu $l;lf; $w;w %d;w jdpf;$w;wfisf; nfhz;ljhf,ug;gpd;> nka;al;ltizapys;s epiufspd; vz;zpf;if (OCT-09,OCT-,MAR- ) () 8 () 6 () 4 (4). p apd; nka;kjpg;g T kw;wk; q,d; nka;kjpg;g F vdpy; gpd;tutdtw;wpy; vit nka;kjpg;g T vd,uf;fk;? (OCT-07,MAR-0,OCT-,MAR-4,MAR-0,JUN-6, MAR-7 ) (i) p q (ii) p q (iii) p q (iv) p q ()(i),(ii),(iii) ()(i),(ii),(iv) ()(i),(iii),(iv) (4)(ii),(iii),(iv) 4. ~[ p (~q)] d; nka; ml;ltizapy; epiufspd; vz;zpf;if (JUN-06, 08,OCT-0,, 6 ) () () 4 () 6 (4) 8 5. epge;;jidf; $w;w p q f;fr; rkhdkhdj (MAR-06,MAR-09,MAR-,OCT-,JUN-4,OCT-4,JUN-5,OCT-5) () p q () p ~q () ~p q (4) p q 6. gpd;tutdtw;ws; vj nka;ikahfk;? (JUN-07,MAR-08,MAR-09,JUN-09,JUN-0,MAR-,MAR-,OCT-) () p q () p q () p ~p (4) p ~p 7. gpd;tutdtw;ws; vj Kuz;ghlhFk;? (MAR-06,OCT-06,OCT-08,MAR-4,MAR-6) () p q () p q () p ~p (4) p ~p 8. p q f;fr; rkhdkhdj (MAR-07,JUN-,MAR-5) ()p q () q p () p q (q p) (4) p q (q p) 9. fpo;fz;ltw;wpy; vj R,y; <UWg;Gr; nrayp my;y? (OCT-07,JUN-08,JUN-09,MAR-0,MAR-,MAR-5) ()a b = ab ()a b = a b () a b = ab (4) a b = a + b 0. rkdpaila miuf;fyk;> Fykhtjw;F G+h;j;jp nra;a Ntz;ba tpjpahtj (MAR-08,JUN- ) () milg;g tpjp () Nrh;g;G tpjp () rkdp tpjp (4) vjph;kiw tpjp. fpo;f;fz;ltw;ws; vj Fyk; my;y? (OCT-08,JUN-0,JUN-6 ) ()(Z n, + n ) () (Z, +) () (Z,. ) (4) (R, +). KOf;fspy; * vd;w <UWg;Gr; nrayp a b = a + b ab vd tiuawf;fg;glfpwj vdpy; *(4 *5),d; kjpg;g (JUN-06,OCT- ) ()5 ()5 ()0 (4)5 ways00@gmail.com - -

24 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top. (Z 9, + 9 ),y; [7],d; thpir (OCT-06,4, MAR-07, 09,,7, JUN-08,0,4) ()9 ()6 () (4) 4. nguf;fiyg; nghwj;j Fykhfpa xd;wpd; Kg;gb %yq;fspy;> ω,d; thpir (JUN-07,OCT-08,JUN-5,OCT-5) ()4 () () (4) 5. + ([5]+ [6]),d; kjpg;g (MAR-06,JUN-09,JUN-) ()[0] ()[] ()[] (4)[] 6. nka;naz;fspd; fzk; R,y; *vd;w <UWg;Gr; nrayp a b = a + b. vd tiuawf;fg;glfpwj vdpy; ( * 4) * 5,d; kjpg;g (JUN-07,OCT-0,JUN-5, JUN-6) ()5 ()5 ()5 (4)50 7. fpo;f;fz;ltw;ws; vj rhp? (OCT-06,JUN-,OCT-4) () xu Fyj;jpd; xu cwg;gpw;f xd;wpw;f Nkw;gl;l vjph;kiw cz;l () Fyj;jpd; xt;nthu cwg;gk; mjd; vjph;kiwahf,uf;fnkdpy; mf;fyk; xu vgpypad; FykhFk; () nka;naz;fis cwg;gfshff; nfhz;l vy;yh mzpf;nfhitfsk; nguf;fy; tpjpapy; FykhFk; (4) vy;yh a, b G f;fk; a b = a b 8. nguf;fy; tpjpiag; nghwj;j Fykhfpa xd;wpd; ehyhk; %yq;fspy;> i,d; thpir (JUN-, JUN-,OCT-5, MAR-6, OCT-6) ()4 () () (4) 9. nguf;fiy nghwj;j Fykhfpa xd;wpd; nmk; gb %yq;fspy; ω k,d; vjph;kiw ( k < n ) (JUN-06,MAR-08,OCT-09,OCT-,MAR-,OCT-,JUN-4, MAR-7) () ω /k () ω () ω n k (4) ω n/k 0. KOf;fspy; * vd;w <UWg;Gr; nrayp a b = a + b vd tiuawf;fg;glfpwj vdpy; rkdp cwg;g (MAR-07,MAR-0,OCT-0,JUN-,MAR-,MAR-4, OCT-6) ()0 () () a (4) b. f x = k x, 0 < x < 0, kw;nwq;fpyk; () 0. epfo;jftg; guty; vd;gj epfo;jft mlh;j;jpr; rhh;g vdpy; k,d; kjpg;g (JUN-08, 09,OCT-,6) () 6 () 9 (4). f x = A π 6+x, < x < vd;gj X vd;w njhlh; rktha;g;g khwpapd; xu epfo;jft mlh;j;jpr; rhh;t ( p.d.f. ) vdpy; A,d; kjpg;g (MAR-06,MAR-07,OCT-08,OCT-09,MAR-,OCT-,MAR-4,JUN-4,JUN-5) ()6 () 8 () 4 (4). X vd;w rktha;g;g khwpapd; epfo;jftg; guty; gpd;tukhw: X P(X=x) /4 a a 4a 5a /4 P( x 4),d; kjpg;g () 0 () 7 () 4 (JUN-0,JUN-, MAR-7) (4) ways00@gmail.com - -

25 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 4. X vd;w rktha;g;g khwpapd; epfo;jft epiwr;rhh;g guty; gpd;tukhw X λ λ λ P(X= x) 6 4 λ tpd; kjpg;g (JUN-6 ) () () () (4)4 5. X vd;w xu jdpepiy rktha;g;g khwp 0,, vd;w kjpg;gfisf; nfhs;fpwj. NkYk; P X = 0 = 44, P X = = vdpy; P X =,d; kjpg;g (MAR-09,OCT-0,JUN- ) () () 4 69 () 69 (4) xu rktha;g;g khwp X,d; epfo;jft epiwr; rhh;g(p.d.f.) gpd;tukhw (MAR-6 ) X P(X=x) 0 k k k k k k 7k + k k,d; kjpg;g () 8 () 0 () 0 (4) or 7. E X + c = 8 kw;wk; E X c = vdpy; c,d; kjpg;g (OCT-06,OCT-07,JUN-09,MAR-,OCT-,MAR-5,OCT-5, MAR-6 ) () () 4 () 4 (4) 8. X vd;w rktha;g;g khwpapd;, 4 kw;wk; Mfpa kjpg;gfs; KiwNa, kw;wk; 5 Mfpa 4 epfo;jftfisf; nfhs;snkdpy; E X,d; kjpg;g (OCT-08,JUN-6, OCT-5) () 5 () 7 () 6 (4) 9. X vd;w rktha;g;g khwpapd; gutw;gb 4 NkYk; ruhrhp vdpy; E(X ),d; kjpg;g(jun-07,, MAR-09) () () 4 () 6 (4) 8 0. xu jdpepiy rktha;g;g khwp X f;f μ = 0. NkYk; μ = 76 vdpy; X,d; ruhrhpapd; kjpg;g (OCT-09,MAR-0,JUN-,MAR-4 ) ()6 ()5 () (4). Var(4X + ),d; kjpg;g (MAR-06, JUN-06,MAR-08,JUN-08,JUN-5 ) ()7 ()6 Var(X) ()9 (4)0. xu gfilia 5 Kiw tprk; NghJ> my;yj fpilg;gj ntw;wpnadf; fujg;glfpwj> vdpy; ntw;wpapd; ruhrhpapd; kjpg;g (OCT-06,JUN-,MAR-) 0 () 5 () 5 () 5 9 (4) 9 5. xu <UWg;Gg; gutypd; ruhrhp 5 NkYk; jpl;ltpyf;fk; vdpy; n kw;wk; p,d; kjpg;gfs; (OCT-,OCT-,OCT-5, MAR-7) () 4 5, 5 () 5, 4 5 () 5, 5 (4) 5, 5 4. xu <UWg;Gg; gutypd; ruhrhp kw;wk; jpl;ltpyf;fk; vdpy; gz;gsit p,d; kjpg;g (OCT-0,MAR-,MAR-4,OCT-4 ) () () 5. xu gfilia 6 Kiwfs; tprk; NghJ>,ul;ilg;gil vz; fpilg;gj ntw;wpahfk; vdpy; ntw;wpapd; gutw;gb (JUN-07,MAR-0,MAR-,MAR-6) () 4 () 6 () (4) 56 () (4) 4 ways00@gmail.com

26 k; tfg;g fzf;f xu kjpg;ngz; tpdhf;fs; ntw;wpf;f top 6. xu ngl;bapy; 6 rptg;g kw;wk; 4 nts;isg; ge;jfs; cs;sd. mtw;wpypue;j ge;jfs; rktha;g;g Kiwapy; jpug;gp itf;fhky; vlf;fg;gl;lhy;> nts;isg; ge;jfs; fpilf;f epfo;jft (JUN-0,OCT-,OCT-4, JUN-6) () 0 () 8 5 () 4 5 (4) 0 7. ed;f fiyf;fg;gl;l 5 rpl;lfs; nfhz;l rpl;lf;fl;bypue;j rpl;lfs; jpug;gp itf;fhkhy; vlf;fg;glfpd;wd.,uz;lk; xnu epwj;jpy;,uf;f epfo;jft (JUN-4,MAR-5 ) () () 6 5 () 5 5 (4) xu gha;]hd; gutypy; P X = 0 = k vdpy; gutw;gbapd; kjpg;g (MAR-07,OCT-07,OCT-08,MAR-09,JUN-,OCT-,JUN-4) () log k ()log k ()e λ (4) k 9. xu rktha;g;g khwp X gha;]hd; gutiyg; gpd;gw;wfpwj. NkYk; E X = 0 vdpy; gutypd; gutw;gb (JUN-08,MAR-, JUN-6, OCT-6 ) ()6 ()5 ()0 (4)5 0. rktha;g;g khwp X,d; guty; rhh;g F(X) xu (MAR-08, JUN-,OCT-4 ) (),wq;fk; rhh;g () Fiwah (,wq;fh) rhh;g () khwpypr; rhh;g (4) Kjypy; VWk; rhh;g gpd;dh;,wq;fk; rhh;g. gha;;]hd; gutypd; gz;gsit λ = 0.5 vdpy;,uz;lhtj tpyf;fg; nguf;fj; njhif (OCT-09,OCT-,OCT-,MAR-,MAR-5 ) ()0.5 ()0.5 ()0.065 (4)0.05. xu gha;]hd; gutypy; P X = = P X = vdpy;> gz;gsit λ,d; kjpg;g (MAR-06, JUN-06,MAR-08,JUN-09,MAR-,JUN-5) ()6 () () (4)0. xu,ay;epiyg; gutypd; epfo;jft mlh;j;jpr; rhh;g f(x),d; ruhrhp μ vdpy; f(x)dx,d; kjpg;g (OCT-06,MAR-07,JUN-07,OCT-0 ) () ()0.5 ()0 (4)0.5 (x 00 ) 4. xu rktha;g;g khwp X >,ay;epiyg; guty; f x = ce 5 I gpd;gw;wfpwj vdpy; c,d; kjpg;g (JUN-06,MAR-0,JUN-,MAR-,MAR-7 ) () π () π ()5 π (4) 5. xu,ay; epiy khwp X,d; epfo;jft mlh;j;jpr; rhh;g f(x) kw;wk; X~N(μ, σ ) vdpy; μ f(x)dx,d; kjpg;g () tiuawf;f KbahjJ () (). 5 (4).5 5 π (OCT-6) khzth;fs; vojpa fzpjj; Njh;tpd; kjpg;ngz;fs;,ay;epiyg; gutiy xj;jpuf;fpwj.,jd; ruhrhp 65. NkYk; 0 khzth;fs; 85 kjpg;ngz;fsf;f Nky; ngw;wpug;gpd;> kjpg;ngz;fs; 45,ypUe;J 65f;Fs; ngwk; khzth;fspd; vz;zpf;if (OCT-07,JUN-0,JUN-, MAR-6 ) ()0 ()0 ()80 (4)60 ways00@gmail.com

27 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top 6 kjpg;ngz; tpdhf;fs;. mzpfsk; mzpf;nfhitfsk;. xu G+r;rpakw;w Nfhit mzpahapd; A T = A T vd;gij epwtf (OCT-07) AA = I = A A. AA = I,d;,UGwKk; epiu epuy; khw;w fhz AA T = I T epiu epuy; khw;wf;fhpa thpir khw;wg; gz;gg;gb> A T A T = I...,Nj Nghy; A A = I,d;,UGwKk; epiu epuy; khw;w fhz A T A T = I. () kw;wk; (),ypue;j A T A T = A T A T = I vdnt A T MdJ A T,d; Neh;khwhFk; mjhtj A T T = A. Neh;khWfSf;Fhpa thpirkhw;w tpjpia vojp epwtf. ( JUN-,OCT-4) A kw;wk; B Mfpait xnu thpir nfhz;l VNjDk;,U G+r;rpakw;w Nfhit mzpfs; vd;f. mt;thwhapd; AB Ak; xu G+r;rpakw;w Nfhit mzpahfk;. NkYk; (AB) = B A mjhtj nguf;fypd; Neh;khW mzpahdj Neh;khW mzpfspd; thpir khw;wg; nguf;fyf;fr; rkkhfk;. ep&gzk;: A kw;wk; B G+r;rpakw;w Nfhit mzpfs; vd;f. A 0 kw;wk; B 0 MFk; AB = A B A 0, B 0 A B 0 AB 0 vdnt AB Ak; xu G+r;rpakw;w Nfhit mzpahfk; AB Neh;khW fhzj;jf;fj. (AB)( B A ) = A(BB ) A = A I A = A A = I,t;thNw ( B A ) AB = I vd epwtyhk; (AB)( B A ) = ( B A ) AB = I AB,d; Neh;khW B A MFk; (AB) = B A. A = 5 6 vdpy; A T = A T vd;gijr; rhpghh;f;f ( MAR-0 ) A = 5 6 A = = + 5 = adj A = A = A A T = 7 adj A = A T = 5 6, A T = A T = 6 = + 5 = 7 0 adj A T = 6 5 adj AT AT 5 A T = 7.. () kw;wk; () ypue;j A T = A T vd;gj rhpghh;f;fg;gl;lj vd;w mzpapd; Neh;khW mzpiaf; 4 fhz;f. (OCT-07) A =, vdpy; A = 4 4 = 0 A xu G+r;rpakw;w Nfhit mzp. vdnt Neh;khW fhzj;jf;fj.,izf;fhuzpfspd; mzpahdj A ij = 4 adj A = 4 5. A = 0 mzpiaf; fhz;f. A KbAk; A = adj A A = 4 = vd;w mzpapd; Neh;khW (OCT-09,OCT-, MAR-7) A = 0 A = 0 = 0 G+r;rpakw;w Nfhit mzp. vdnt A fhz A = A adj A A ij = adj A = ways00@gmail.com

28 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top = = A = kw;wk; B = vdpy;> (AB) = B A vd;gij rhpghh; ( JUN-09,JUN-0) A = = = 0 B = 0 = 0 + = 0 A kw;wk; B G+r;rpakw;w Nfhit mzpfs;. vdnt A kw;wk; B fhz KbAk; AB = 0 = AB = = 0 AB G+r;rpakw;w Nfhit mzp. vdnt (AB) fhz KbAk; adj AB = (AB) = AB adj AB =.. A = adj A A adj A = A = B = adj B B adj B = 0 B = 0 B A = 0 =. () kw;wk; () ypue;j (AB) = B A vd;gj rhpghh;f;fg;gl;lj 5 kw;wk; B = 7, vdpy; (AB) = B A vd;gij rhpghh;. ( JUN-06,JUN-,JUN-,JUN-5) 7. A = A = 5 ; B = 7 AB = 5 7 A If; fhz A = 5 7 = 8 4 = 5 4 = 0 adj A = 7 5 A = A adj A A = 7 5 B If; fhz B = = = 0 adj B = B = adj B B B = B A = 7 5 = 4 8. () (AB) If; fhz AB = 8 = + = 0 4 adj AB = 4 8 (AB) = AB adj AB (AB) = 4 8 () () kw;wk; ()ypue;j (AB) = B A rhpghh;f;fg;gl;lj 5 8. A = kw;wk; B = 7, vdpy; (AB) T = B T A T vd;gij rhpghh; (JUN-4) A = 5 kw;wk; B = 7 AB = 8 4 (AB) T = 8 4 B T =, AT = 5 7 B T A T = = = ()kw;wk; () ypue;j (AB) T = B T A T vd;gj rhpghh;f;fg;gl;lj 9. A = vd;w mzpapd; Nrh;g;igf; fz;lgpbj;j 5 A adj A = adj A A = A. I vd;gij rhpghh;f;f (MAR-07,MAR-09,MAR-) A = 5 A = = 5 6 = 5 adj A = 5 A adj A = 5 5 ways00@gmail.com

29 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top = 0 0 = 0 0 = A. I.. adj A A = 5 5 = 0 0 = 0 0 = A. I... () kw;wk; ()ypue;j A adj A = adj A A = A. I 4 0. A = 0 d; Nrh;g;G mzp A vd epwtf. 4 4 (MAR-08,MAR-,MAR-6, OCT-6) 4 A = A ij = adj A = (A ij ) T = 0 = A 4 4. A = 4 4 vdpy;> A = A vdf; fhl;lf ( MAR-06,MAR-4) A = A = = = = 4 4 A ij = adj A = A = (adj A) = 4 4 A = A A = A.. Neh;khW mzpfhzy; Kiwapy; jph;f;f x + y =, x + y = 8 (JUN-08,OCT-08,0,) jug;gl;ls;s rkd;ghlfis gpd;tukhw voj x y = 8 A X = B,q;F, A = = 0 A G+r;rpakw;w Nfhit mzp Mjyhy; A fhzkbak; A = X = A B vd;gj jph;thfk; x y = 8 = x =, y =. Neh;khW mzp fhzy; Kiwapy; jph;f;f x y = 7, x y = ( JUN-07,MAR-,5) 4. x y = 7 x y = x y = 7 A X = B,,q;F A = ; X = x y ; B = 7 X = A B vd;gj jph;thfk; A If;fhz A = = 4 + = 0 A ij = adj A = vdnt X = A B = A = ~ ~ ~ ~ A = adj A A A = = = x =, y = vd;w mzpapd; juk; fhz;f C C 5 ( MAR-5) R R 4R R 5 R R R + R filrp rkhd mzpahdj VWgb tbtpy; cs;sj.,jpy;,uz;l G+r;rpakw;w epiufs; cs;sjhy; ρ A = ways00@gmail.com

30 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top A = ~ ~ ~ vd;w mzpapd; juk; fhz;f (OCT-06,MAR-07, OCT-6) R R R R R R R R R R R filrp rkhd mzpahdj VWgb tbtpy; cs;sj.,jpy; %d;w G+r;rpakw;w epiufs; cs;sjhy; ρ A = 0 4 A = ~ ~ ~ vd;w mzpapd; juk; fhz;f R R R R + R R R 7R (OCT-0) filrp rkhd mzpahdj VWgb tbtpy; cs;sj.,jpy; %d;w G+r;rpakw;w epiufs; cs;sjhy;, ρ A = 0 A = ~ ~ ~ vd;w mzpapd; juk; fhz;f R R R R R R R R R R R (JUN -08 ) filrp rkhd mzpahdj VWgb tbtpy; cs;sj.,jpy;,uz;l G+r;rpakw;w epiufs; cs;sjhy;, ρ A = 0 0 A = ~ ~ ~ vd;w mzpapd; juk; fhz;f ( JUN-,OCT-, MAR-7) R R R R R R 5R + R filrp rkhd mzpahdj VWgb tbtpy; cs;sj.,jpy; %d;w G+r;rpakw;w epiufs; cs;sjhy; ρ A = 4 6 A = ~ ~ vd;w mzpapd; juk; fhz;f (OCT-08, OCT-5) R R R R R R R R R filrp rkhd mzpahdj VWgb tbtpy; cs;sj.,jpy;,uz;l G+r;rpakw;w epiufs; cs;sjhy;. ρ A = A = ~ vd;w mzpapd; juk; fhz;f 0 0 (JUN-07) R R R R R R filrp rkhd mzpahdj VWgb tbtpy; cs;sj.,jpy; xnu xu G+r;rpakw;w epiu cs;sjhy;, ρ A = vd;w mzpapd; juk; fhz;f ways00@gmail.com

31 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top. A = ~ ~ (MAR-06,JUN-0) R R + R R R + R R R R filrp rkhd mzpahdj VWgb tbtpy; cs;sj.,jpy;,uz;l G+r;rpakw;w epiufs; cs;sjhy; ρ A = A = 4 = ~ ~ ~ vd;w mzpapd; juk; fhz;f(jun-) R R R R + R R R R filrp rkhd mzpahdj VWgb tbtpy; cs;sj.,jpy; %d;w G+r;rpakw;w epiufs; cs;sjhy; ρ A =. mzpf;nfhit Kiwapy; gpd;tuk; rkd;ghlfspd; njhfg;gfis jph;f;f (OCT-5) (i) x y = (ii) x + y + z = 0 y = x 7 x + y z = 0 x + y + z = 0 (i) x y = y = x 7 = = 0 x = 7 = = 0 kw;wk; x 0 vd;gjhy;> njhfg;g xuq;fikt mw;wj>,jw;f jph;t fpilahj. (ii) x + y + z = 0 x + y z = 0 x + y + z = 0 = = 0, Mjyhy; njhfg;g xnu xu jph;tpidf; nfhz;buf;fk;. vdnt Nkw;fz;l rkgbj;jhd njhfg;g ntspg;gilj; jph;t kl;lnk ngw;wpuf;fk; x, y, z = (0,0,0) 4. mzpf;nfhit Kiwapy; gpd;tuk; rkd;ghl njhfg;gpid jph;f;f x + y = 8, 4x + 6y = 6 ( JUN-06, MAR-) x + y = 8.. 4x + 6y = 6... = = = x = 8 = = y = 8 = = = 0 kw;wk; x = y = 0 vd;gjhyk;,d; Fiwe;jJ xu nfo a ij MtJ G+r;rpakw;W,Ug;gjhy;> njhfg;g xuq;fikt cilajhfk;. vz;zpf;ifaw;w jph;tfs; fpilf;fk;. vy;yh rpw;wzpf; Nfhitfs; G+r;rpaq;fshfTk;> Fiwe;jJ xu ( ) rpw;wzpf;nfhit G+r;rpakw;wJ Mjyhy; njhfg;g xnu xu jdpr;rkd;ghl;bw;f FiwAk;. x (my;yj y)f;f VNjDk; xu kjpg;gspj;j y (my;yj x),d; kjpg;igf; fhzyhk;. x = t vdj; ju y = (8 t) vdnt jph;t fzkhdj x, y = t, (8 t) t R 5. mzpf;nfhit Kiwapid gad;glj;jp x + y + z = 4; x + y + 4z = 8; x + y + 6z = 0 vd;w njhfg;gpid jph;f;f = y = ( JUN-, MAR-6) 4 = 0, x = 8 4 = = 0, z = = 0 = x = y = z = 0 NkYk; d; vy;yh rpw;wzpf;nfhitfspd; kjpg;gfs; G+r;rpakhtjhYk; x, y kw;wk; z,d; rpy rpw;wzpf; Nfhitfs; G+r;rpakw;wjhAs;sjhy; njhfg;g xuq;fikt mw;wj. vdnt mjw;f jph;t fpilahj. 6. x + y + z = 4; x + y + 6z = 7; x + y + z = 0 vd;w rkd;ghl;l njhfg;gpid mzpf;nfhit Kiwapid gad;glj;jp jph;t fhz;f. (MAR-) = 6 = 0 ways00@gmail.com

32 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top 4 x = = (7 0) = = 0 = 0, x 0 vdnt njhfg;g xuq;fikt mw;wj. jph;t fpilahj 7. gpd;tuk; rkd;ghlfspd; njhfg;gfis mzpf;nfhit Kiwapy; jph;f;f. x + y + z = 0; x y z = 5; x + z = 6 (MAR-4) = 0 = = + 4 = 0 0 x = = =0 + = 0 = 0 kw;wk; x 0 vdnt njhfg;g xuq;fikt mw;wj. jph;t fpilahj 8. mzpf;nfhit Kiwapy; jph;f;f x y = ; x 4y = 4 ( JUN-4) = 4 = = 0 x = 4 4 = = 0 y = 4 = 4 4 = 0 = 0, x = 0, y = 0 jug;gl;l rkd;ghl njhfg;g xuq;fikt cilaj NkYk; vz;zpf;ifaw;w jph;tfs; fpilf;fk;. y = k vdj;ju x k = x = + k vdnt jph;t fzkhdj ( + k, k),k R 9. gpd;tuk; rkd;ghlfspd; njhfg;gpid mzpf;nfhit Kiwapy; jph;f;f x + y = 4, 4x + 8y = 6 ( JUN-5) x + y = 4. 4x + 8y = 6 = 4 8 = 8 8 = 0 x = 4 = = y = 4 = 6 6 = = 0 kw;wk; x = y = 0 vd;gjhyk;,d; Fiwe;jJ xu nfo a ij MtJ G+r;rpakw;W,Ug;gjhy;> njhfg;g xuq;fikt cilajhfk;. vz;zpf;ifaw;w jph;tfs; fpilf;fk;. vy;yh rpw;wzpf; Nfhitfs; G+r;rpaq;fshfTk;> Fiwe;jJ xu ( ) rpw;wzpf;nfhit G+r;rpakw;wJ Mjyhy; njhfg;g xnu xu jdpr;rkd;ghl;bw;f FiwAk;. x (my;yj y)f;f VNjDk; xu kjpg;gspj;j y (my;yj x),d; kjpg;igf; fhzyhk;. x = t vdj; ju y = 4 t y = (4 t) vdnt jph;t fzkhdj x, y = t, (4 t) t R 0. mzpf;nfhit Kiwapy; x + y + z = 5, x y + z =, x + y + z = 4 vd;w rkd;ghl;lj; njhfg;gpid jph;f;ftk; (MAR-08,MAR-09,MAR-,OCT-) = = 0 5 x = = = 0 kw;wk; x 0 vdnt njhfg;g xuq;fikt mw;wj. jph;t fpilahj. 4x + 5y = 9, 8x + 0y = 8 vd;w njhfg;gpid mzpf;nfhit Kiwapy; jph;f;f (OCT-06,OCT-09) = = = 0 x = 9 5 = = y = 4 9 = 7 7 = = 0 kw;wk; x = y = 0 vd;gjhyk;,d; Fiwe;jJ xu nfo a ij MtJ G+r;rpakw;W,Ug;gjhy;> njhfg;g xuq;fikt cilajhfk;. vz;zpf;ifaw;w jph;tfs; fpilf;fk;. y = k vd ju 4x = 9 5k x = 9 5k jph;t 9 5k 4, k, k R. mzpf;nfhit Kiwapy; gpd;tuk; rkd;ghlfspd; njhfg;igj; jph;f;f x y = 7, 4x 6y = 4 (JUN-09) = = + = x = 7 = = y = 7 = 8 8 = = 0 kw;wk; x = y = 0 vd;gjhyk;,d; Fiwe;jJ xu nfo a ij MtJ G+r;rpakw;W 4 ways00@gmail.com - -

33 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top,ug;gjhy;> njhfg;g xuq;fikt cilajhfk;. vz;zpf;ifaw;w jph;tfs; fpilf;fk;. vy;yh rpw;wzpf; Nfhitfs; G+r;rpaq;fshfTk;> Fiwe;jJ xu ( ) rpw;wzpf;nfhit G+r;rpakw;wJ Mjyhy; njhfg;g xnu xu jdpr;rkd;ghl;bw;f FiwAk;. x (my;yj y)f;f VNjDk; xu kjpg;gspj;j y (my;yj x),d; kjpg;igf; fhzyhk;. x y = 7,y; y = k vdj; ju x k = 7 x = 7 + k x = 7+k vdnt jph;t fzkhdj x, y = 7+k, k, k R. α d; vk;kjpg;gfsf;f αx + y + z = 0, 4x + y + 8z = 0 4x + y + 4z = 0 vd;w njhfg;g (i) ntspg;gilahd jph;t kl;lk; ngw;wpuf;fk; (ii) ntspg;gilahd kw;wk; ntspg;gilaw;w jph;t ngw;wpuf;fk; (mzpf;nfhit Kiwapid gad;glj;jf) (JUN-6) αx + y + z = 0 4x + y + 8z = 0 4x + y + 4z = 0 α = α (8 ) = α ( 4) = 4α + 6 = 4α + 4 (i) ntspg;gilahd jph;t α 0 vdnt njhfg;g xnu xu jph;t kl;lk; nfhz;buf;fk;. njhfg;g ntspg;gilj; jph;t kl;lnk ngw;wpuf;fk; x, y, z = (0,0,0) (ii)ntspg;gilahd kw;wk; ntspg;gilaw;w jph;t α = = 0 = 0 vd;gjhy; vz;zpf;ifaw;w jph;tfs;,uf;fk;. NkYk; tpd; Fiwe;jJ xu rpw;wzpf;nfhit G+r;rpakw;wjha;,Ug;gjhy;,j;njhFg;ghdJ,uz;L rkd;ghlfshff; FiwAk;. vdnt VNjDk; xu khwp VNjDk; xu kjpg;gk; kw;w,u khwpfspd; kjpg;gpid,jd; %yk; fhzyhk; z = k vd ju x + y = k 4x + y = 8k = 4 = 4 = 0 x = k = 9k + 8k = k 8k y = k = 8k + k = 4k 4 8k fpnukhpd; tpjpg;gb x = k, y = 4k jph;thdj x, y, z = (k, 4k, k) 4. x + y + z = 7, x + y + z = 8, y + z = 6 vd;w rkd;ghlfspd; njhfg;g xuq;fikt cilajh vd;gij ju Kiwapy; Muha;f (OCT-07,MAR-0, JUN-6) x + y + z = 7 x + y + z = 8 y + z = 6 jug;gl;l rkd;ghlj;njhfg;gpid gpd;tukhw mzpr; rkd;ghlhf khw;wp vojyhk; 0 x y z A X = B tphptg;glj;jg;gl;l mzpahdj 7 [A, B] = ~ ~ = R R R R R R filrp rkhd mzpahdj VWgb tbtpy; cs;sj. ρ A, B = kw;wk; ρ A = ρ A, B ρ A vdnt jug;gl;l rkd;ghl;lj; njhfg;g xuq;fikt,y;yhjj. KbahJ vd;gjhy; jph;t fhz 5. ju Kiwapy; gpd;tuk; rkd;ghlfspd; njhfg;gfisj; jph;f;f x y + z = ; x + y z = 7; x + y = (OCT-4) [A, B] = 7 0 ~ ~ R R R R R R R R R filrp rkhd mzpahdj VWgb tbtpy; cs;sj. mj %d;w G+r;rpakw;w epiufisg; ngw;ws;sjhy; ρ A, B = > kw;wk; ρ A =, ρ A, B ρ A vdnt jug;gl;l rkd;ghl;lj; njhfg;g xuq;fikt mw;wj. vdnt jph;t fhz KbahJ ways00@gmail.com - -

34 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top 6. jukiwia gad;glj;jp x 4y + 7z = 4, x + 8y z =, 7x 8y + 6z = 5 vd;w njhfg;gpw;f xuq;fiktj;jd;ikia Muha;f. xuq;fikt cilajhapd;> jph;f;f (OCT-,OCT-) jug;gl;l rkd;ghlj;njhfg;gpid gpd;tukhw mzpr; rkd;ghlhf khw;wp vojyhk; x y z A X = B tphptg;glj;jg;gl;l mzpahdj [A, B] = ~ ~ = 4 5 R R R R R 7R R R R filrp rkhd mzpahdj VWgb tbtpy; cs;sj. ρ A, B = and ρ A = ρ A, B ρ A vdnt jug;gl;l rkd;ghl;lj; njhfg;g xuq;fikt mw;wj. vdnt jph;t fhz KbahJ. fyg;ngz;fs;. nka;> fw;gid gfjpfis fhz; +i (OCT-08) = i = i = + i +i +i i. nka; gfjp =. fw;gidg; gfjp = +i i n = vdpy; n,d; kpr;rpw kpif KO vz; kjpg;igf; fhz;f +i = +i +i = +i i i +i +i i = i = i n = i n = n,d; kpr;rpw kpif KO vz; 4. (MAR-6). gpd;tuk; rkd;ghl;bid epiwt nra;ak; x kw;wk; y apd; nka; kjpg;gfisf; fhz;f i x + + i y = i (JUN-) x + y + i( x + y) = i nka;> fw;gidg; gfjpfis xg;gpl x + y =, x + y =,r;rkd;ghlfisj; jph;f;f x =, y = 4. gpd;tuk; rkd;ghl;bid epiwt nra;ak; x kw;wk; y apd; nka; kjpg;gfisf; fhz;f x + x x + 4 i = y( + i) (OCT-0,JUN-4) x + x x + 4 i = y( + i) x + x + 8 = y + i(y x 4) nka;> fw;gidg; gfjpfis xg;gpl x + x + 8 = y x + x + 8 = 4y.. y x 4 = 0 x + 4 = y () I (),y; gpujpapl x + x + 8 = 4(x x) x + 9x + 56 = 0 x + 7 x + 8 = 0 x = 7 my;yj x = 8 x = 7 vdpy; y = = x = 8 vdpy; y = = 4 x = 7, y = NkYk; x = 8, y = 4 5. fyg;ngz;fspy; Kf;Nfhz rkdpypia vojp ep&gpf;f (OCT-09,JUN-0,MAR-,MAR-4),U fyg;ngz;fspd; $Ljypd; kl;l mt;tpu vz;fspd; kl;lfspd; $LjYf;Ff; FiwthfNth my;yj rkkhftnk,uf;fk;. z + z z + z ep&gzk;: z kw;wk; z,u fyg;ngz;fs; vd;f. z + z = (z + z )(z + z ) z = zz = z + z (z + z ) = z z + z z + z z + z z = z z + z z + z z + z z = z + z + Re (z z ) z + z + z z (Re z z ) = z + z + z z = z + z z + z z + z,ugwkk; kpif th;f;f%yk; vlf;f z + z z + z ways00@gmail.com - -

35 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top 6. z kw;wk; z vd;w,u fyg;ngz;fsf;f, (i) z z = z z (ii)arg z. z = arg z + arg z (OCT-07,JUN-08,JUN-) z = r (cos θ + i sin θ ) kw;wk; z = r (cos θ + i sin θ ) vd;f z. z = r r (cos θ + i sin θ )(cos θ + i sin θ ) = r r [ cos θ. cos θ sin θ sin θ + i sin θ. cos θ + cos θ sin θ ] = r r [cos(θ + θ ) + i sin ( θ + θ ) z z = r r = z. z kw;wk; arg z. z = θ + θ = arg z + arg z 7. z, z vd;w VNjDk;,U fyg;ngz;fsf;f = r r (i) z z = z z,(z 0) (ii) arg z = arg z z arg z (JUN-4) z = r (cos θ + i sin θ ) kw;wk; z = r (cos θ + i sin θ ) vd;f z = r, arg z = θ and z = r, arg z = θ z = r (cos θ + i sin θ ) z r (cos θ + i sin θ ) = r cos θ +i sin θ cos θ i sin θ r cos θ +i sin θ cos θ i sin θ cos θ. cos θ + sin θ sin θ + i sin θ. cos θ cos θ sin θ cos θ + i sin θ = r [cos(θ r θ ) + i sin ( θ θ )] z = r = z and z r z arg z z = θ θ = arg z arg z 8. a + ib a + ib (a n + ib n ) = A + ib vdpy; ep&gp: (i) a + b a + b a n + b n = (A + B ) (ii) tan b a + tan b a + + tan b n a n = kπ + tan B A, k Z (OCT-) nfhs;if: a + ib a + ib (a n + ib n ) = A + ib a + ib a + ib (a n + ib n = A + ib a + ib a + ib (a n + ib n = A + ib a + b a + b a n + b n = A + B th;f;fg;glj;j a + b a + b a n + b n = A + B NkYk;, arg[ a + ib a + ib (a n + ib n )] = arg (A + ib) arg a + ib + arg a + ib + + arg(a n + ib n ) = arg (A + ib) tan b a + tan b a nghjthf> tan b a + tan b a + + tan b n a n = tan B A + + tan b n a n = kπ + tan B A, k Z 9. fyg;ngz;fs; 7 + 9i, + 7i, ( + i) vdk; fyg;ngz;fs; Mh;fd; jsj;jpy; xu nrq;nfhz Kf;Nfhzj;ij mikf;fk; vd epwtf. ( JUN-07,MAR-0,OCT-4, MAR-7) A, B, C vdk; Gs;spfs; KiwNa 7 + 9i, + 7i, + i vdk; fyg;ngz;fis Mh;fd; jsj;jpy; Fwpf;fl;Lk; AB = 7 + 9i ( + 7i) = 0 + i = 0 + = 04 BC = + 7i ( + i) = 6 + 4i = = = 5 CA = + i (7 + 9i) = 4 6i = = = 5 AB = BC + CA BCA = 90 vdnt ABC xu,u rkgf;f nrq;nfhz Kf;NfhzkhFk.; 0. fyg;ngz; jsj;jpy; fyg;ngz;fs; 0 + 8i + 4i kw;wk; ( + i) mikf;fk; Kf;Nfhzk; xu nrq;nfhz Kf;Nfhzk; vd epwtf (OCT-0) ways00@gmail.com

36 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top A, B, C vdk; Gs;spfs; KiwNa 0,8,,4, (,) vdk; fyg;ngz;fis Mh;fd; jsj;jpy; Fwpf;fl;Lk; AB = 0 + 8i ( + 4i) = + 4i = = 60 BC = + 4i ( + i) = 9 7i = = = 80 CA = + i (0 + 8i) = + i = + = 970 AB + BC = CA = 970 nfhlf;fg;gl;l Gs;spfs;> fyg;ngz; jsj;jpy; xu nrq;nfhz Kf;Nfhzj;ij mikf;fk; i, 5 + i, (4 + 7i) kw;wk; + 4i vdk; fyg;ngz;fs; xu,izfuj;ij mikf;fk; vd epwtf. (OCT-6) A, B, C kw;wk; D vdk; Gs;spfs; KiwNa, 4, 5,, (7, 5) kw;wk; (4, 7) AC d; elg;gs;sp = +7, 4+5, +7 = 9, 9 BD d; elg;gs;sp = 5+4 = 9, 9 AC d; elg;gs;spak;> BD apd; elg;gs;spak; xd;nw. vdnt> nfhlf;fg;gl;l Gs;spfs; xu,izfuj;ij mikf;fpd;wd.. ( 8 6i),d; th;f;f%yk; fhz;f x + iy = 8 6i vd;f,ugwkk; th;f;fg;glj;j> x y + xyi = 8 6i (MAR-06, OCT-06,JUN-5) nka; > fw;gidg; gfjpfis xg;gpl, x y = 8. xy = 6.. x + y = x y + (xy) = 8 + ( 6) = 00 x + y = ()+() x =,, x = ± x = vdpy; y = x = vdpy; y = 8 6i = i my;yj + i. ( 7 + 4i),d; th;f;f%yk; fhz;f (MAR-07,JUN-09,MAR-5) x + iy = 7 + 4i vd;f,ugwkk; th;f;fg;glj;j> x y + xyi = 7 + 4i nka;> fw;gidg; gfjpfis xg;gpl, x y = 7. xy = 4.. x + y = x y + (xy) = 7 + (4) = 65 x + y = x = 9, y = 6 x = ±, y = ±4 xy kpif vz; vd;gjhy; x -k; y -k; xnu Fwpahf nfhs;s Ntz;Lk; x =, y = 4 my;yj x =, y = i = + 4i my;yj ( 4i) 4. P vd;gj xu fyg;ngz; khwp z vdpy; Re z+ z i = 0 vd;w epge;jidf;f P d; epakg;ghij fhz;f. z = x + iy, z = x iy vd;f z + = x iy + = x + iy z i = x iy i = x i(y + ) z + z i Re x+ iy = x+i(y+) x i(y+) x+i(y+) = (OCT-) x+ x+i x+ y+ iyx +y(y+) x + y+ z + z i = 0 x+ x+y(y+) x + y+ = 0 x + x + y + y = 0 x + y + x + y = 0 ways00@gmail.com

37 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top 5. P vd;gj xu fyg;ngz; khwp z vdpy; Re z+ z i = 0 vd;w epge;jidf;f P d; epakg;ghij fhz;. (MAR-08) z = x + iy vd;f z + = x + iy + = x + + iy z i = x + iy i = x + i(y ) z+ z i x+ +iy = x i(y ) x+i(y ) x i(y ) x+ x i x+ y +iyx +y(y ) = x + y Re z+ = 0 z i x+ x+y(y ) x + y = 0 x + x + y y = 0 x + x + y y = 0 x + y + x y = 0 x + (y 5) = x + (y + 5),UGwKk; th;f;fg;glj;j, x + (y 5) = x + (y + 5) (y 5) = (y + 5) y 0y + 5 = y + 0y + 5 0y = 0 y = 0 6. P vdk; Gs;sp fyg;ngz; khwp z If; Fwpj;jhy; P -,d; epakg;ghijia Z 5i = Z + 5i vdk; fl;lg;ghlfsf;f cl;gl;l fhz;f. (MAR- 5) z = x + iy vd;f x + iy 5i = x + iy + 5i x + i(y 5) = x + i(y + 5) 7. P vdk; Gs;sp fyg;g vz; khwp z If; Fwpj;jhy;> P,d; epakg; ghijia z 5 = z + vd;dk; fl;lg;ghl;lf;f cl;gl;l fhz;f. (JUN- ) z = x + iy (x + iy) 5 = x + iy + x + iy 5 = x + + iy x 5 + iy = x + + iy x 5 + (y) = x + + y,ugwkk; th;f;fg;glj;j, 9x + 5 0x + 9y = 9[ x + + x + y ] 9x + 5 0x + 9y = 9x x + 9y 0 = 8x + 0x = 48x 6 48x = 6 x = 8. P vdk; Gs;sp> fyg;g vz; khwp Z If; Fwpj;jhy;> P,d; epakg; ghijia Z i = Z + i vd;dk; epge;jidf;f cl;gl;l fhz;f. (MAR-09) z = x + iy vd;f x + iy i = x + iy + i x + i(y ) = x + i(y + ) x + (y ) = x + (y + ),UGwKk; th;f;fg;glj;j, x + (y ) = x + (y + ) (y ) = (y + ) y 6y + 9 = y + 6y + 9 y = 0 y = 0 P d; epakg;ghij y = 0 9. P vdk; Gs;sp fyg;ngz; khwp Z If; Fwpj;jhy; P d; epakg;ghijia Z 4i = Z + 4i vd;w epge;jidf;fl;gl;l fhz;f. (OCT-5) z = x + iy vd;f x + iy 4i = x + iy + 4i x + i(y 4) = x + i(y + 4) x + (y 4) = x + (y + 4),UGwKk; th;f;fg;glj;j, x + (y 4) = x + (y + 4) (y 4) = (y + 4) y 8y + 6 = y + 8y + 6 6y = 0 y = 0 0. P vd;dk; Gs;sp fyg;g vz; khwp z If; Fwpj;jhy; P,d; epakg; ghijia z = z vd;w epge;jidf;f cl;gl;l fhz;f (MAR-06) z = x + iy vd;f (x + iy) = x + iy x + iy = x + iy x + (y) = x + (y) ways00@gmail.com

38 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top,ugwkk; th;f;fg;glj;j, 4x + 4x + 4y = x + 4 4x + y x + y = x + y = P d; epakg;ghij xu tl;lkhfk;.. nka;naz; Fzfq;fisf; nfhz;l P x = 0 vd;w gy;ywg;gf; Nfhitr; rkd;ghl;bd; %yq;fs;,iznaz;,ul;ilahfj;jhd;,lk;ngwk; vd ep&gpf;f. (OCT-) P x = a n x n + a n x n + + a x + a 0 = 0 vd;gj nka; Fzfq;fSila xu gy;ywg;gf; Nfhit. P x = 0 f;f z xu %yk; P x = 0f;F z k; xu %yk; vd fhl;l Ntz;Lk;. P x = 0f;F z xu %yk; Mjyhy; P z = a n z n + a n z n + + a z + a 0 = 0,UGwKk; fyg;ngz;,iznaz; fhz> P z = a n z n + a n z n + + a z + a 0 = 0,U fyg;ngz;fspd; $Ljypd;,iznaz;> mtw;wpd; jdpj;jdp,izfspd; $LjYf;Fr; rkkhtjhy; a n z n + a n z n + + a z + a 0 = 0 a n z n + a n z n + + a z + a 0 = 0,q;F z n = z n kw;wk; a 0, a, a a n nka;naz;fs; Mjyhy; mit xt;nthd;wk; jdf;fj;jhnd fyg;ngz;,izahfpd;wd. a n z n + a n z n + + a z + a 0 = 0 P z = 0 P x = 0 f;f z k; xu %yk; vd;gnj,jd; nghushfk;. + i I xu jph;thf nfhz;l x 4 4x + 8x + 5 = 0 vdk; rkd;ghl;ilj; jph; (MAR-6, JUN-6) + i xu %yk;> vdnt i kw;nwhu %yk;. %yq;fspd; $Ljy; = 4 %yq;fspd; nguf;fk; =( + i)( i) = 4 + = 7 x (%yq;fspd; $Ljy;)x+%yq;fspd; nguf;fk; = x 4x + 7 x 4 4x + 8x + 5 x 4x + 7 (x + px + 5) x,d; nfoit xg;gpl> 8 = 7p 0 p = 4 x + 4x + 5 vd;gj kw;nwhu fhuzp x + 4x + 5 = 0 x = ± i vdnt %yq;fs; ± i kw;wk; ± i MFk;. ( i)i xu jph;thf nfhz;l x 4x + 6x 4 = 0 vdk; rkd;ghl;il jph;. ( JUN-07 ) i xu %yk;> + i kw;nwhu %yk;. %yq;fspd; $Ljy; = + i + i = %yq;fspd; nguf;fk; = + i i = + = x (%yq;fspd; $Ljy;)x+%yq;fspd; nguf;fk;= 0 x x + = 0 x 4x + 6x 4 = x x + (x ) x = 0 { + i, i, } 4. + i I xu jph;thf nfhz;l x 4 8x + 4x x + 0 = 0 vdk; rkd;ghl;il jph;. (MAR-09,MAR-) + i xu %yk;> i kw;nwhu %yk;. %yq;fspd; $Ljy; =6 %yq;fspd; nguf;fk; = 0 x (%yq;fspd; $Ljy;)x+%yq;fspd; nguf;fk; = x 6x + 0 x 4 8x + 4x x + 0 x 6x + 0 (x + px + ) x,d; nfoit xg;gpl 0p = p = x x + vd;gj kw;nwhu fhuzp x x + = 0 x = ± i vdnt %yq;fs; ± i, ± i 5. + i I xu jph;thff; nfhz;l x 4 4x + x 4x + 0 = 0 vdk; rkd;ghl;bd; jph;tfisf; fhz;f (JUN-09,MAR-) + i xu %yk;, i kw;nwhu %yk;. %yq;fspd; $Ljy; = %yq;fspd; nguf;fk; =5 x (%yq;fspd; $Ljy;)x+%yq;fspd; nguf;fk; = x x + 5 x 4 4x + x 4x + 0 x x + 5 (x + px + ) ways00@gmail.com

39 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top x,d; nfoit xg;gpl 5p 4 = 4 p = x x + vd;gj kw;nwhu fhuzp x x + x = ± i vdnt %yq;fs; ± i, ± i 6. + ii xu jph;thff; nfhz;l x = 0 vdk; rkd;ghl;bd; jph;tfisf; fhz;f. (JUN-06) + i xu %yk;> i kw;nwhu %yk;. %yq;fspd; $Ljy; = + i + i = %yq;fspd; nguf;fk; = + i i = + = x (%yq;fspd; $Ljy;)x+%yq;fspd; nguf;fk;= 0 x x + = 0 x x + 0 x + 0x + 4 = x x + x + px + = 0 x,d; nfoit xg;gpl 0 = p 4 p = 4 p = x + x + = 0 vd;gj kw;nwhu fhuzp x = 7. RUf;Ff : ± 4 8 = ± 4 = ±i = ± i { + i, i, + i, i} cos θ+i sin θ 4 5 (OCT-06,,6) sin θ+i cos θ cos θ+i sin θ 4 = cos θ+i sin θ 4 sin θ+i cos θ 5 cos π θ +i sin π θ 5 = cos 4θ 5 π θ +i sin 4θ 5 π θ + sin θ + i cos θ + sin θ i cos θ 9. RUf;Ff : n + z = + = z n z = (sin θ + i cos θ) n n = cos π θ + i sin π n θ = cos n π θ + i sin n π θ cos θ i sin θ 7 cos θ+i sin θ 5 cos 4θ+i sin 4θ cos 5θ i sin 5θ 6 cos θ i sin θ 7 cos θ+i sin θ 5 cos 4θ+i sin 4θ cos 5θ i sin 5θ 6 ( JUN-) = (cos θ+i sin θ) 4 (cos θ+i sin θ) 5 (cos θ+i sin θ) 48 (cos θ+i sin θ) 0 = (cos θ + i sin θ) = (cos θ + i sin θ) 07 = cos 07θ i sin 07 θ 0. n xu kpif KO vz; vdpy; ( + cos θ + i sin θ) n + ( + cos θ i sin θ) n = n+ cos n θ ( + cos θ + i sin θ) n cos nθ = cos θ + i sin θ cos θ = cos θ cos θ + i sin θ vd epwtf( JUN-,MAR-4) = n cos n θ cos n θ + i sin n θ.. (),y; i f;fg; gjpy; i I gpujpapl> ( + cos θ i sin θ) n = n cos n θ cos n θ i sin n θ. ()+() + cos θ + i sin θ n + + cos θ i sin θ n n n = cos 9θ 5π +i sin 9θ 5π = n cos n θ cos nθ = cos 5π 9θ i sin 5π 9θ = cos π 9θ i sin π 9θ = sin 9θ i cos 9θ 8. n vd;gj kpif KO vz; vdpy; +sin θ+i cos θ n π = cos n θ + i sin n π θ +sin θ i cos θ vd ep&gpf;f (MAR-) z = sin θ + i cos θ vd;f z = sin θ i cos θ = n+ cos n θ cos nθ. n vd;gj kpif KO vdpy;> ( + i) n + ( i) n = n+ cos nπ vd ep&gp. 4 (OCT-07,MAR-08,MAR-0,OCT-5) + i = r(cos θ + i sin θ) nka; kw;wk; fw;gidg; gfjpfis xg;gpl> r cos θ =, r sin θ = r = + = NkYk;, cos θ =, sin θ = θ = π 4 + i = cos π 4 + i sin π 4 ways00@gmail.com

40 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top + i n = n cos π 4 + i sin π 4 = n cos nπ 4 + i sin nπ 4.. (),y; i f;fg; gjpy; i I gpujpapl, i n = n cos nπ 4 i sin nπ 4 () IAk; () IAk; $l;l + i n + i n = n cos nπ 4 n + = cos nπ 4. n N vdpy; ( + i ) n + ( i ) n = n+ cos nπ epwtf (JUN-08) + i = r(cos θ + i sin θ) nka; kw;wk; fw;gidg; gfjpfis xg;gpl> r cos θ =,r sin θ = r = + ( ) = NkYk;, cos θ =, sin θ = θ = π + i = cos π + i sin π + i n = n cos nπ + i sin nπ. (),y; i f;f gjpy; i I gpujpapl, i n = n cos nπ i sin nπ.. () IAk; () IAk; $l;l + i n + i n = n cos nπ. x + = cos θ vdpy; x (i) x n + xn = cos nθ, (ii) x n xn = i sin nθ vd ep&gp x + = cos θ x x cos θ x + = 0 x = cos θ± 4 cos θ 4 n = n+ cos nπ = cos θ± cos θ (JUN-0) x = cos θ ± sin θ = cos θ ± i sin θ x = cos θ + i sin θ vdf; nfhs;f x n = cos θ + i sin θ n x n = cos nθ + i sin nθ.. x n = cos nθ i sin nθ.. ()+ () x n + x n = cos nθ () () x n x n = i sin nθ 4. x = cos α + i sin α, y = cos β + i sin β vdpy; x m y n + x m y n = cos(mα + nβ) vd ep&gp (MAR-07,OCT-) x m y n = (cos α + i sin α) m (cos β + i sin β) n = cos mα + i sin mα (cos nβ + i sin nβ) = cos(mα + nβ) + i sin(mα + nβ) x m y n = cos(mα + nβ) i sin(mα + nβ) x m y n + = cos mα + nβ + i sin mα + nβ + x m y n cos mα + nβ i sin(mα + nβ) x m y n + x m y n = cos(mα + nβ) 5. cos α + cos β + cos γ = 0 = sin α + sin β + sin γ vdpy; cos α + cos β + cos γ = cos( α + β + γ) kw;wk; sin α + sin β + sin γ = sin ( α + β + γ) vd epwtf (MAR-,OCT-) a = cos α + i sin α b = cos β + i sin β c = cos γ + i sin γ a + b + c = cos α + cos β + cos γ +i sin α + sin β + sin γ = 0 + i0 = 0 a + b + c = 0 vdpy; a + b + c = abc cos α + i sin α + cos β + i sin β + cos γ + i sin γ = (cos α + i sin α)(cos β + i sin β)(cos γ + i sin γ) cos α + i sin α + cos β + i sin β + (cos γ + i sin γ) = [cos( α + β + γ) + i sin ( α + β + γ) ] cos α + cos β + cosγ + i(sin α +sin β + i sin γ) = [cos( α + β + γ) + i sin ( α + β + γ) ] nka; kw;wk; fw;gid gfjpfis xg;gpl, cos α + cos β + cos γ = cos( α + β + γ) sin α + sin β + sin γ = sin ( α + β + γ) 6. cos α + cos β + cos γ = 0 = sin α + sin β + sin γ vdpy; cos α + cos β + cos γ = 0 kw;wk; sin α + sin β + sin γ = 0 vd epwtf (JUN-06,JUN-) ways00@gmail.com

41 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top a = cos α + i sin α = cos α i sin α a b = cos β + i sin β = cos β i sin β b c = cos γ + i sin γ = cos γ i sin γ c,q;f + + = (cos α + cos β + cos γ) a b c (sin α + sin β + sin γ) = 0 i 0 = 0 a + b + c + ab + bc + ca = (a + b + c) a + b + c + abc + + = 0 c a b a + b + c + abc 0 = 0 a + b + c = 0 cos α + i sin α + cos β + i sin β + cos γ + i sin γ = 0 cos α + i sin α + cos β + i sin β + cos γ + i sin γ = 0 cos α + cos β + cos γ + i (sin α + sin β + sin γ) = 0 nka; kw;wk; fw;gid gfjpfis xg;gpl, cos α + cos β + cos γ = 0, sin α + sin β + sin γ = 0 7. i vy;yh kjpg;gfisak; fhz;f i = cos π + i sin π i = cos π + i sin π (MAR-,6) = cos kπ + π + i sin kπ + π = cos 4k + π + i sin 4k + π = cos 4k + π 6 + i sin 4k + π, k = 0,, 6 vdnt cis π, cis 5π, cis 9π Mfpa kjpg;gfisg; ngwk; 8. ω =, vdpy; epwtf +i i = vd (OCT-08,JUN-,MAR-) ω xd;wpd; Kg;gb %yk; vdpy;> ω = +i ω = i +i 5 + i 5 = ω 5 + ω 5 = ω 5 + ω 0 = ω + ω = 9. ω vd;gj xd;wpd; Kg;gb %yk; kw;wk; x = a + b, y = aω + bω, z = aω + bω vdpy; (i) xyz = a + b (ii) x + y + z = (a + b ) vd ep&gpf;f (JUN-5) (i) xyz = a + b aω + bω aω + bω = a + b a + abω + abω + b = a + b a + ab(ω + ω) + b = a + b a ab + b = a + b (ii) x + y + z = a + b + aω + bω + aω + bω = a + ω + ω + b + ω + ω + c + ω + ω = a 0 + b 0 + c 0 = 0 x + y + z = 0 vdpy; x + y + z = xyz x + y + z = (a + b ) [ (i) d; %yk;] 40. jph;f;f x = 0 (OCT-08,OCT-09,OCT-4) x = 0 x 4 = 4 = 4( ) x = 4 4( ) 4 = ( ) 4(cos π + i sin π) 4 = (cos(kπ + π) + i sin(kπ + π)) 4 = cos k + π 4 + i sin k + π 4, k = 0,,, cis π 4, cis π 4, cis 5π 4, cis 7π 4 Mfpa kjpg;gfis ngwk;. 4. α kw;wk; β vd;git xd;wf;nfhd;w,izahdj. NkYk; α = + i vdpy; α + β αβ d; kjpg;gpidf; fhz;f. (Mar-7) α kw;wk; β Mfpad,iz vz;fs; MFk;. α = + i, β = i α = + i = + i + i = i = i β = i = + i i = + i ways00@gmail.com

42 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top = + i αβ = + i i = i = + = α + β αβ = i + + i = = 9. jdpepiyf; fzf;fpay;. (a) (p q) (~q) vd;w $w;wf;f nka; ml;ltizia mikf;f (b) p (~p) xu Kuz;ghL vd ep&gp (JUN-06) (a) (p q) (~q) f;fhpa nka; ml;ltiz p q ~q p q (p q) (~q) T T F T F T F T T T F T F T F F F T F F (b) p (~p) f;fhpa nka; ml;ltiz p ~p p ~p T F F F T F filrp epuypy; F kl;lnk cs;sjhy;> xu p (~p) Kuz;ghlhFk;. (p q) [~(p q)] vd;w $w;wf;f nka; ml;ltizia mikf;f (JUN-4, MAR-5) (p q) [~(p q)] f;fhpa nka; ml;ltiz p q p q ~(p q) (p q) [~(p q)] T T T F T T F F T T F T F T T F F F T T. ~( ~p ~q ) vd;w $w;wf;f nka; ml;ltiz mikf;f (OCT-06) ~( ~p ~q ) f;fhpa nka; ml;ltiz p q ~p ~q (~p) (~q) ~( ~p ~q ) T T F F F T T F F T F T F T T F F T F F T T T F 4. (p q) (~r) f;fhpa nka; ml;ltizia mikf;f (JUN-07, OCT-, MAR-,7) p q r p q ~ r (p q) (~r) T T T T F T T T F T T T T F T F F F T F F F T T F T T F F F F T F F T T F F T F F F F F F F T T 5. (p q) r,d; nka; ml;ltizia mikf;f (MAR-08) p q r p q (p q) r T T T T T T T F T F T F T T T T F F T F F T T T T F T F T F F F T F F F F F F F 6. (p q) r,d; nka; ml;ltizia mikf;f (OCT-08, 5) p q r p q (p q) r T T T T T T T F T T T F T F T T F F F F F T T F T F T F F F F F T F T F F F F F 7. ((~p) (~q)) p xu nka;ik vdf; fhl;lf. (OCT-09) ((~p) (~q)) p f;fhpa nka; ml;ltiz p q ~p ~q (~p) (~q) ((~p) (~q)) p T T F F F T T F F T T T F T T F T T F F T T T T filrp epuy; KOtJk; T Mjyhy; ((~p) (~q)) p xu nka;ikahfk; 8. nka; ml;ltiziaf; nfhz;l (p ~q ) ((~p) q) vd;w $w;w nka;ikah my;yj Kuz;ghlh vdf; fhz;f. (OCT-09, JUN-) ways00@gmail.com

43 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top p q ~p ~q p (~q) (~p) q (p (~q)) ((~p) q) T T F F F T T T F F T T F T F T T F F T T F F T T F T T (p ~q ) ((~p) q) xu nka;ikahfk; 9. q p ~q vd;w $w;w nka;ikah my;yj Kuz;ghlh vd;gijf; fhz;f. (MAR-7) p q ~q p (~q) q p ~q T T F T T T F T T T F T F F T F F T T T q p ~q xu nka;ikahfk; 0. ((~q) p) q xu Kuz;ghL vdf; fhl;lf. (MAR-, 6) ((~q) p) q f;fhpa nka; ml;ltiz p q ~q (~q) p ((~q) p) q T T F F F T F T T F F T F F F F F T F F filrp epuy; KOtJk; F Mjyhy; ((~q) p) q xu Kuz;ghlhFk;.. p q (p q) vdf; fhl;lf. (JUN-06,JUN-,OCT-,OCT-4) p q p q p q p q (p q) T T T T T T F T F T F T T F T F F F F T p q (p q) vd;gj xu nka;ik MFk;.. ( ~p q) (p ( q)) xu nka;ikah vd;gjid nka; ml;ltiziaf; nfhz;l jph;khdpf;f. (OCT-07) p q ~p ~q ~p q p (~q) ( ~p q) (p ( q)) T T F F T F T T F F T F T T F T T F T F T F F T T T F T filrp epuy; KOtJk; T Mjyhy; jug;gl;l $w;w xu nka;ikahfk;.. (p (~p)) ((~q) p) nka;ikah Kuz;ghlh vdf; fhz;f (JUN-08,0, MAR-) (p (~p)) ((~q) p) f;fhpa nka; ml;ltiz p q ~p ~q p (~q) (~q) p (p (~p)) ((~q) p) T T F F F F F T F F T F T F F T T F F F F F F T T F F F (p (~p)) ((~q) p) xu Kuz;ghlhFk;. 4. p q (p q) (q p) vdf; fhl;lf (OCT-06, JUN-09, JUN-5) p q f;fhpa nka; ml;ltiz p q p q T T T T F F F T F F F T (p q) (q p) f;fhpa nka; ml;ltiz p q p q q p (p q) (q p) T T T T T T F F T F F T T F F F F T T T,U ml;ltizfsnk xnu khjphpahd filrp epuy;fisg; ngw;ws;sjhy; p q (p q) (q p) 5. ~ p q (~p) (~q) vdf; fhl;lf (MAR-06) ~ p q f;fhpa nka; ml;ltiz p q p q ~ p q T T T F T F T F F T T F F F F T (~p) (~q) f;fhpa nka; ml;ltiz p q ~p ~q (~p) (~q) T T F F F T F F T F F T T F F F F T T T filrp epuy;fs; xnu khjphpahdit ~ p q (~p) (~q) ways00@gmail.com

44 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top 6. p q (~p) q vdf; fhl;lf (MAR-) p q f;fhpa nka; ml;ltiz p q p q T T T T F F F T T F F T (~p) q f;fhpa nka; ml;ltiz p q ~p (~p) q T T F T T F F F F T T T F F T T p q kw;wk; (~p) q f;fhpa ml;ltizfspd; filrp epuy;fs; xnu khjphpahapug;gjhy; p q (~p) q 7. ~ p q ~p (~q) vdf;fhl;lf (OCT-0) ~ p q f;fhpa nka; ml;ltiz p q p q ~ p q T T T F T F F T F T F T F F F T ~p (~q) f;fhpa nka; ml;ltiz p q ~p ~q (~p) (~q) T T F F F T F F T T F T T F T F F T T T,U ml;ltizfspyk; filrp epuy;fs; xnu khjphpahdit. ~ p q ~p (~q) 8. p q kw;wk; q p rkhdkw;wit vdf; fhl;lf. (MAR-09,JUN-6) p q f;fhpa nka; ml;ltiz p q p q T T T T F F F T T F F T q p f;fhpa nka; ml;ltiz p q q p T T T T F T F T F F F T p q kw;wk; q p f;fhpa epuy;fs; xnu khjphpahdit my;y. vdnt p q kw;wk; q p Mfpait jh;f;f rkhdkhdit my;y. 9. p q ~p q ( q) p vdf; fhl;lf. (MAR-07,0,4, JUN-07,, OCT-08,,) p q f;fhpa nka; ml;ltiz p q p q T T T T F F F T F F F T ~p q ( q) p f;fhpa nka; ml;ltiz p q ~p ~q ~p q ( q) p ~p q ( q) p T T F F T T T T F F T F T F F T T F T F F F F T T T T T,U ml;ltizfspyk; filrp epuy;fs; xnu khjphpahas;sd. p q ~p q ( q) p 0.,d; Mk; gb %yq;fs; xu Kbthd vgpypad; Fyj;ij nguf;fypd; fpo; mikf;fk; vdf; fhl;lf. (OCT-4) G = {, ω, ω }. Nfa;yp ml;ltizahdj. ω ω ω ω ω ω ω ω ω ω,e;j ml;ltizapypue;j> (i) ml;ltizapy; cs;s vy;yh cwg;gfsk;> G,d; cwg;gfshfk;. vdnt milg;g tpjp cz;ikahfpwj. (ii) nguf;fy; vg;nghojk; Nrh;g;G tpjpf;fl;glk;. (iii) rkdpawg;g. mj rkdp tpjpiag; G+h;j;jp nra;ak;. (iv),d; vjph;kiw ω,d; vjph;kiw ω ω,d; vjph;kiw ω kw;wk;,j vjph;kiw tpjpiag; G+h;j;jp nra;ak; (G,. ) xu FykhFk;. (v) ghpkhw;w tpjpak; cz;ikahfk;. ways00@gmail.com

45 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top (G,. ) xu vgpypad; FykhFk;. (vi) G xu Kbthd fzk;. Mjyhy; (G,. ) xu Kbthd vgpypad; FykhFk;.,d; 4Mk; gb %yq;fs; nguf;fypd; fpo; vgpypad; Fyj;ij mikf;fk; vd epwtf. (JUN-),d; 4 Mk; gb %yq;fs;, i,, i G = {, i,, i }. Nfa;yp ml;ltizahj. i i i i i i i i i i i i,e;j ml;ltizapypue;j, (i) milg;g tpjp cz;ikahfk;. (ii) C,y; nguf;fyhdj Nrh;g;G tpjpf;fl;glkhjyhy; G apyk; mj cz;ikahfk;. (iii) rkdp cwg;g G kw;wk; mj rkdp tpjpiag; G+h;j;jp nra;fpwj (iv),d; vjph;kiw ; i,d; vjph;kiw i,d; vjph;kiw ; i,d; vjph;kiw i. vjph; kiw tpjpiaak; G+h;;j;jp MfpwJ (G,. ) xu FykhFk;. (v) ml;ltizapypue;j> ghpkhw;w tpjpak; cz;ik. (G,. ) xu vgpypad; FykhFk;.. (Z, +) xu Kbtw;w vgpypad; Fyk;; vd epwtf. (OCT-08) (i) milg;g tpjp:,uz;l KO vz;fspd; $LjYk; xu KO vz;. a, b Z a + b Z (ii) Nrh;g;G tpjp: Z y; $l;ly; vg;nghojk; Nrh;g;G tpjpf;fl;glk; a, b, c Z, a + b + c = a + (b + c) (iii)rkdp tpjp: rkdp cwg;g O Z kw;wk; 0 + a = a + 0 = a, a Z I G+h;j;jp nra;fpwj. vdnt rkdp tpjp cz;ikahfk; (iv) vjph;kiw tpjp: xt;nthu a Z f;fk; a Z I a + a = a + a = 0vDkhW fhzyhk;. vdnt vjph;kiw tpjp cz;ikahfk;. (Z, +) xu FykhFk; (v) a, b Z, a + b = b + a $l;ly; ghpkhw;w tpjpf;fl;glk; (Z, +) xu vgpypad; FykhFk; (vi) Z Kbtw;w fzk; Mjyhy; (Z, +) xu KbTw;w vgpypad; FykhFk.;. G+r;rpakw;w fyg;ngz;fspd; fzk; fyg;ngz;fspd; tof;fkhd nguf;fypd; fpo; xu vgpypad; Fyk; vdf; fhl;lf. (JUN-6) (i) milg;g tpjp: G = C {0} vd;f. G+r;rpakw;w,U fyg;ngz;fspd; nguf;fy; vg;nghjk; G+r;rpakw;w fyg;ngz;zhf,uf;fk; milg;g tpjp cz;ikahfk; (ii) Nrh;g;G tpjp: fyg;ngz;fspy; nguf;fy; Nrh;g;G tpjp vg;nghjk; cz;ikahfk;. (iii) rkdp tpjp: = + i0 G, rkdp cwg;ghfk;. NkYk;. z = z. = z z G rkdp tpjp cz;ik. (iv) vjph;kiw tpjp: z = x + iy G.,q;F z 0 x kw;wk; y,uz;lnk G+r;rpakw;wit my;yj VNjDk; xd;whtj G+r;rpakw;wJ. x + y 0 z = x + iy = x iy x iy = x + iy x iy x + y = x y x +y + i G x +y NkYk; z. z = z. z = z MdJ vd;w vjph;kiwia G,y; z ngw;ws;sj. vjph;kiw tpjp cz;ikahfpwj. G,. xu FykhFk; (v) ghpkhw;wg; gz;g: z z = a + ib c + id = ac bd + i(ad + bc) = ca db + i da + cb = z z ghpkhw;wg; gz;igak; mj G+h;j;jp nra;fpwj. G MdJ fyg;ngz;fspd; tof;fkhd nguf;fypd; fpo; xu vgpypad; FykhFk;. 4. thpir nfhz;l G+r;rpakw;w Nfhit mzpfs; ahtk; Kbtw;w vgpypad; my;yhj Fyj;ij mzp nguf;fypd; fpo; mikf;fk; vdf; fhl;lf. (,q;f mzpapd; cwg;gfs; ahtk; R Ir; Nrh;e;jit) (OCT-07) G vd;gj thpir nfhz;l G+r;rpakw;w Nfhit mzpfs; ahtk; mlq;fpa fzk;. cwg;gfs; ahtk; R Ir; Nrh;e;jit ways00@gmail.com

46 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top 5. (i) milg;g tpjp:,uz;l thpir nfhz;l G+r;rpakw;w Nfhit mzpfspd; nguf;fw;gyd; xu thpir G+r;rpakw;w Nfhit mzpahfk;. vdnt milg;g tpjp cz;ikahfk;. A, B G AB G (ii) Nrh;g;G tpjp: mzpg; nguf;fy; vg;nghojk; Nrh;g;G tpjpf;fl;glk;. vdnt Nrh;g;G tpjp cz;ikahfk;. A BC = AB C, A, B, C G (iii) rkdp tpjp: rkdp cwg;g I = 0 G.,J rkdpg; 0 gz;ig G+h;j;jp nra;fpwj. (iv) vjph;kiw tpjp: A G,d; vjph;kiw A I G,y; fhz KbAk;. NkYk; mj thpir nfhz;lj. kw;wk; AA = A A = I vdnt> vjph;kiw tpjp cz;ikahfk;. vdnt G xu FykhFk;. nghjthf mzp nguf;fy; ghpkhw;w tpjpf;fl;glhj. Mjyhy; G xu vgpypad; my;yhj FykhFk;. G,y; vz;zpf;ifaw;w cwg;gfs; cs;sjhy;> mj KbTw;w vgpypad; my;yhj FykhFk; ,,, Mfpa ehd;f mzpfsk; mlq;fpa fzk; mzpg;nguf;fypd; fpo; xu vgpypad; Fyj;ij mikf;fk; vdf; fhl;lf. (MAR-,OCT-,MAR-5) I = 0 0 B = 0, A = 0 0,, C = G = {I, A, B, C} vd;f,t;tzpfis,uz;l,uz;lhfg; nguf;fp nguf;fy; ml;ltizia mikf;fyhk;. I A B C I I A B C A A I C B B B C I A C C B A I (i) milg;g tpjp: nguf;fy; ml;ltizapd; vy;yh cwg;gfsk; G,d; cwg;gfs;. G MdJ.,d; fpo; milt ngw;ws;sj. vdnt milg;g tpjp cz;ik. (ii) Nrh;g;G tpjp: mzpg;nguf;fy; nghjthf Nrh;g;G tpjpf;fl;glk;. (iii) rkdp tpjp: I I Kd; itj;j vojg;gl;ls;s epiuapd; cwg;gfs; vy;yhtw;wpw;fk; NkNyAs;s epiualdk; I I NkNy itj;j vojg;gl;ls;s epuypy; cs;s cwg;gfs;,lg;gw,wjpapy; mike;j epuyld; xd;wp tpljyhy;> I MdJ rkdp cwg;ghfk;. (iv) vjph;kiw tpjp: I. I = I I,d; vjph;kiw I A. A = I A,d; vjph;kiw A B. B = I B,d; vjph;kiw B C. C = I C,d; vjph;kiw C ml;ltizapypue;j. ghpkhw;w tpjpf;fl;glk;. vdnt G MdJ mzpg;nguf;fypd; fpo; xu vgpypad; FykhFk; 6. tiuawf;fg;gl;l FwpaPl;bd; gb (Z 5 0,. 5 ) xu Fyk; vd ep&gp. (JUN-0) G = Z 5 0 = {,,, [4]} Nfa;yp ml;ltizahj. 5 [] [] [] [4] [] [] [] [] [4] [] [] [4] [] [] [] [] [] [4] [] [4] [4] [] [] [] ml;ltizapypue;j (i) nguf;fy; ml;ltizapd; vy;yh cwg;gfsk; G-,d; cwg;gfshfk;. milg;g tpjp cz;ikahfk; (ii) 5-,d; kl;lf;fhd nguf;fy;> Nrh;g;G tpjpf;fl;glk;. (iii) rkdpawg;g[] G kw;wk;,u rkdp tpjpiag; G+h;j;jp nra;ak; (iv) [],d; vjph;kiw [], [],d; vjph;kiw [], [],d; vjph;kiw [], [4],d; vjph;kiw [4] vdnt vjph;kiw tpjp g+h;j;jpahfpwj. (Z 5 0,. 5 ) xu FykhFk;. 7. (Z 7 0,. 7 ) vd;w Fyj;jpy; cs;s xt;nthu cwg;gf;fk; thpiriaf; fhz;f (MAR-) ways00@gmail.com

47 k; tfg;g fzf;f MW kjpg;ngz; tpdhf;fs; ntw;wpf;f top nfhlf;fg;gl;l FykhdJ (,,, 4, 5, 6,. 7 ) O = ; O = O = 6 ; O 4 = O 5 = 6 ; O 6 = 8. (z 6, + 6 ) vd;w Fyj;jpd; vy;yh cwg;gfspd; thpiriaf; fhz;f (JUN-08) z 6 = { 0,,,, 4, [5]} O 0 = ; O = 6 O = ; O = O 4 = ; O 5 = 6 9. Fyj;jpd; epf;fy; tpjpfis vojp ep&gpf;f. (MAR-06,MAR-08,OCT-0, MAR-4) G xu Fyk; vd;f. a, b, c G vd;f (i) a b = a c b = c (,lj epf;fy; tpjp) (ii) b a = c a b = c (tyj epf;fy; tpjp) ep&gzk;: (i) a b = a c a a b = a a c a a b = (a a) c e b = e c b = c (ii)b a = c a b a a = c a a b a a = c (a a ) b e = c e b = c 0. Fyj;jpd; vjph;kiw tpjpapid vojp ep&gp. (my;yj) G xu Fyk; vd;f. a, b G vd;f. mt;thwhapd; a b = b a (MAR-07,0, JUN-09,,4,5, OCT-, 5) b a MdJ (a b),d; vjph;kiw vdf; fhl;bdhy; NghJkhdJ (i) a b b a = e kw;wk; (ii) b a a b = e vd ep&gpf;f Ntz;Lk; (i) a b b a = a b b a = a e a = a a = e (ii) b a a b = b a a b = b e b = b b = e a b d; vjph;kiw b a a b = b a. xu Fyj;jpd; rkdp cwg;g xuikj; jd;ik tha;e;jj - ep&gpf;f (JUN-, OCT-6) G xu Fyk; vd;f. G,d; rkdp cwg;gfis e, e vd,ug;gjhf nfhs;nthk;. e I rkdp cwg;ghff; nfhs;nthkhapd; e e = e. e I rkdp cwg;ghff; nfhs;nthkhapd; e e = e () kw;wk; (),ypue;j, e = e vdnt> xu Fyj;jpd; rkdp cwg;g xuikj; jd;ik tha;e;jjhfk;.. xu Fyj;jpd; xt;nthu cwg;gk; xnu xu vjph;kiwiag; ngw;wpuf;fk; - ep&gpf;f (JUN-, OCT-6) G xu Fyk; vd;f. a G vd;f a,d; vjph;kiw cwg;gfs; a, a vd;gjhff; nfhs;nthk; a I a,d; vjph;kiwahff; nfhs;nthkhapd; a a = a a = e a I a,d; vjph;kiwahff; nfhs;nthkhapd; a a = a a = e a = a e = a a a = (a a) a = e a = a vdnt xu cwg;gpd; vjph;kiw xuikj;jd;ik tha;e;jjhfk;.. xu Fyj;jpd; xt;nthu cwg;gk; mjd; vjph;kiwahf,uf;fnkdpy; mf;fyk; xu vgpypad; FykhFk; vd ep&gpf;ftk;. (MAR-6) G xu Fyk; vd;f a, b G vd;f nfhlf;fg;gl;lit a = a kw;wk; b = b x G, x G vd;f x = ab vd;f nfhlf;fg;gl;lj x = x ab = ab = b a = ba ghpkhw;w tpjpia epiwt nra;fpwj. G xu vgpypad; FykhFk;. ways00@gmail.com

48 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top 0 kjpg;ngz; tpdhf;fs;. ntf;lh;,aw;fzpjk;. xu Kf;Nfhzj;jpd; Fj;Jf;NfhLfs; xnu Gs;spapy; re;jpf;fk; vd;gjid ntf;lh; Kiwapy; epwtf. ( OCT-06,JUN-08,OCT-,MAR-5) ABC,y; Fj;Jf;NfhLfs; AD, BE Ak; O,y; re;jpf;fpd;wd. Fj;Jf;NfhLfs; xnu Gs;sp topnar; nry;yk; vd;gij epwt CO MdJ AB f;f nrq;fj;jhf,uf;fk; vd fhl;bdhy; NghJk;. O I Mjpahff; nfhs;f. A, B, C,d; epiy ntf;lh;fs; KiwNa a, b, c OA = a, OB = b, OC = c AD BC OA BC OA. BC = 0 a. c b = 0 a. c a. b = 0 () BE CA OB CA OB. CA = 0 b. ( a c) = 0 b. a b. c = 0.() () kw;wk; () If; $l;l a. c a. b + b. a b. c = 0 a. c b. c = 0 (a b). c = 0 BA. OC = 0 OC AB vdnt %d;w Fj;Jf; NfhLfSk; xnu Gs;spapy; re;jpf;fk; NfhLfshFk;.. a = i + j k, b = i + 5k, c = j k vdpy; a b c = (a. c)b a. b c vd rhpghh;f;f. ( MAR-07,OCT-08,OCT-09,JUN-6) i j k b c = 0 5 = 5i 6j k 0 i j k a b c = 5 6 = i + 9j + k (a. c) = i + j k. j k = (0) + () ( ) = + = 6 (a. c)b = 6 i + 5k = i + 0k a. b = i + j k. i + 5k = ( ) + (0) (5) = 4 5 = 9 a. b c = 9 j k = 9j + 7k (a. c)b a. b c = i + 0k + 9j 7k (a. c)b a. b c = i + 9j + k vdnt, a b c = (a. c)b a. b c. a = i + j + k, b = i + k, c = i + j + k, d = i + j + k vdpy; a b c d = a b d c a b c d vd;gijr; rhpghh;f;f. ( MAR-09,OCT-,OCT-,MAR-6) a b = c d = a b c d = a b d c a b c d i j k 0 i j k = i + j k = i j + k i j k = 5i j 4k.() a b c = 0 = a b d = 0 = = i + j + k (i + j + k) = 4i j k i j k = 5i j 4k..() () kw;wk; () ypue;j a b c d = a b d c a b c d ways00@gmail.com

49 k; tfg;g fzf;f ntw;wpf;f top 4. cos A B = cos A cos B + sin A sin B vd epwtf. (JUN-,JUN-) 5. cos A + B = cos A cos B sin A sin B vd epwtf( MAR-06,08,4,7, JUN-, OCT-4) 6. sin A + B = sin A cos B + cos A sin B vd epwtf (OCT-08,MAR-,,JUN-4) 7. sin A B = sin A cos B cos A sin B vd ntf;lh; Kiwapy; ep&gp ( JUN-07, MAR-, OCT-07,0, 5,6) O I ikakhff; nfhz;l myf tl;lj;jpd; ghpjpapy; P, Q vd;w,u Gs;spfis vlj;j nfhs;f. OP kw;wk; OQ Mdit x-mr;rld; Vw;gLj;Jk; Nfhzk; KiwNa A, B POQ = POx QOx = A B POQ = POx + QOx = A + B POQ = POx + QOx = A + B POQ = POx QOx = A B 4 P, Q d; Maj;njhiyfs; KiwNa (cos A, sin A) kw;wk; (cos B, sin B) P, Q d; Maj;njhiyfs; KiwNa (cos A, sin A)kw;Wk; (cos B, sin B) 5 i, j vd;w myf ntf;lh;fis x, y mr;rj; jpirfspy; vlj;jf; nfhs;f. P, Q d; mr;rj}uq;fs; KiwNa (cos A, sin A)kw;Wk;(cos B, sin B) P, Q d; Maj;njhiyfs; KiwNa (cos A, sin A) kw;wk; (cos B, sin B). 6 OP = OM + MP = cos Ai +sin A j OQ = OL + LQ = cos Bi +sin B j OP = OM + MP = cos Ai +sin A j OQ = ON + NQ = cos Bi sin B j OP = OM + MP = cos Ai +sin A j OQ = ON + NQ = cos Bi sin B j OP = OM + MP = cos Ai +sin A j OQ = OL + LQ = cos Bi +sin B j 7 OP. OQ = cos Ai +sin A j. cos Bi +sin B j = cos A cos B + sin A sin B. OP. OQ = cos Ai +sin A j. cos Bi sin B j = cos A cos B sin A sin B. OQ OP = OQ. OP sin A + B k = sin(a + B) k OQ OP = OQ. OP sin A B k = sin(a B) k 8 tiuaiwapd;gb, tiuaiwapd;gb, tiuaiwapd;gb, tiuaiwapd;gb, OP. OQ = OP. OQ cos(a B) = cos (A B).. OP. OQ = OP. OQ cos(a + B) = cos (A + B).. OQ OP = i j k cos B sin B 0 cos A sin A 0 OQ OP = i j k cos B sin B 0 cos A sin A 0 = k(sin A cos B + cos A sin B) = k(sin A cos B cos A sin B)...() 9 () kw;wk; () ypue;j cos A B = cos A cos B + sin A sin B () kw;wk; () ypue;j cos A + B = cos A cos B sin A sin B () kw;wk; () ypue;j sin A + B = sin A cos B + cos A sin B () kw;wk; () ypue;j sin A B = sin A cos B cos A sin B ways00@gmail.com

50 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top 8. x = y = z+ x 4 kw;wk; = y = z+ vd;w 0 0 NfhLfs; ntl;lk; vdf; fhl;b mit ntl;lk; Gs;spiaf; fhz;f. ( JUN-07,JUN-09,JUN-5 ) NfhLfs; ntl;bf;nfhs;tjw;fhd epge;jid x x y y z z l m n = 0 l m n x x l = y y m = z z n kw;wk; x x l = y y m = z z n cld; xg;gpl fpilg;gj x = x = 4 l = l = 9. x = y+ = z kw;wk; x = y = z vd;w NfhLfs; ntl;bf; nfhs;sk; vdf; fhl;lf. NkYk; mit ntl;lk; Gs;spiaf; fhz;f. ( JUN-06,JUN-0,JUN-) NfhLfs; ntl;bf;nfhs;tjw;fhd epge;jid x x y y z z l m n = 0 l m n x x l = y y m = z z n kw;wk; x x l = y y m = z z n cld; xg;gpl fpilg;gj, y = y = 0 m = m = 0 z = z = n = 0 n = x = x = l = l = = = ( 0) + (9 0) + 0(0 + ) = = 0 Nkw;Fwpg;gpl;l NfhLfs; xd;iwnahd;w ntl;bf; nfhs;fpd;wd. ntl;lk; Gs;sp: x = y = z+ = λ vd;f 0 x = λ y = λ z+ = λ 0 x = λ y = λ z + = 0 x = λ + y = λ + z =,e;j Nfhl;by; mike;js;s VNjDk; xu Gs;spapd; mikg;g (λ +, λ +, ) x 4 = y = z+ = μ vd;f 0 x 4 = μ y = μ z+ = μ 0 x 4 = μ y = 0 z + = μ x = μ + 4 y = 0 z = μ,e;j Nfhl;by; mike;js;s VNjDk; xu Gs;spapd; mikg;g (μ + 4,0,μ ),it ntl;bf;nfhs;tjhy; VNjDk; λ, μ f;f (λ +, λ +, ) = (μ + 4,0,μ ) λ + = μ + 4 λ + = 0 λ = = μ μ = 0 ntl;lk; Gs;sp (4,0, ) y = y = m = m = z = 0 z = n = n = = = ( 6) ( ) ( + ) = = 0 Nkw;Fwpg;gpl;l NfhLfs; xd;iwnahd;w ntl;bf; nfhs;fpd;wd. ntl;lk; Gs;sp: x = λ vd;f x y+ z x = λ y + = λ z = λ x = λ + y = λ z = λ,e;j Nfhl;by; mike;js;s VNjDk; xu Gs;spapd; mikg;g (λ +, λ, λ) x = y = z+ = μ vd;f x = μ y = μ z+ = μ x = μ y = μ z + = μ x = μ + y = μ + z = μ,e;j Nfhl;by; mike;js;s VNjDk; xu Gs;spapd; mikg;g (μ +, μ +, μ ),it ntl;bf;nfhs;tjhy; VNjDk; λ, μ f;f (λ +, λ, λ) = (μ +, μ +, μ ) λ + = μ +.() ways00@gmail.com

51 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top λ = μ +.() λ = μ..() () kw;wk; () I jph;f;f μ + = 0 μ = μ = μ = I ()y; gpujpapl λ + = + λ + = λ = 0 ntl;lk; Gs;sp (,0) 0. (,, ) topna nry;yf;$bajk; x = y = z 4 x kw;wk; = y+ = z Mfpa NfhLfSf;F,izahf cs;sjkhd jsj;jpd; ntf;lh; kw;wk; fhh;brpad; rkd;ghlfisf; fhz;f (MAR-0,OCT-,6, JUN-6) ntf;lh; rkd;ghl: Njitahd jskhdj A(,, ) topna nry;;yk;. NkYk; u = i + j 4k kw;wk; v = i j + k f;f,izahf,uf;fk; a = i j k Njitahd rkd;ghl r = a + su + tv r = i j k + s i + j 4k +t(i j + k) fhh;brpad; mikg;g: jsj;jpd; rkd;ghl x x y y z z l m n = 0 l m n x = l = l = y = m = m = z = n = 4 n = x y + z + 4 = 0 (x )(4 ) (y + )(6 + 8) +(z + )( 9 4) = 0 (x )( 8) (y + )(4) +(z + )( ) = 0 8x + 6 4y 4 z 9 = 0 8x 4y z = 0 8x + 4y + z + 7 = 0,JNt Njitahd rkd;ghl;bd; fhh;brpad; mikg;g MFk;.. (,, ) kw;wk; (,, ) Mfpa Gs;spfs; topnar; nry;yf; $bajk; x + y + z = 5 vd;w jsj;jpw;f nrq;fj;jhf miktjkhd jsj;jpd; ntf;lh; kw;wk; fhh;brpad; rkd;ghl;ilf; fhz;f (MAR-07, 09,JUN-0,OCT-4) ntf;lh; rkd;ghl: nfhlf;fg;gl;l,uz;l Gs;spfs; topnar; nry;yf;$bajk; xu ntf;luf;f,izahf cs;sjkhd jsj;jpd; ntf;lh; rkd;ghl r = ( s)a + sb + tv,,q;f a = i + j + k, b = i j + k, v = i + j + k r = ( s) i + j + k + s i j + k fhh;brpad; mikg;g: jsj;jpd; rkd;ghl x x y y z z x x y y z z l m n x = x = l = y = y = m = z = z = n = +t(i + j + k) = 0 x + y z 0 = 0 (x + )( 4 0) (y )(4 0) + (z )(4 + ) = 0 (x + )( 4) (y )(4) + (z )(6) = 0 4x 4 4y z 6 = 0 4x 4y + 6z 6 = 0 4x + 4y 6z + 6 = 0 x + y z + = 0. (,, ), (, 4, ) kw;wk; (7, 0, 6) Mfpa Gs;spfs; topnar; nry;yf; $ba jsj;jpd; ntf;lh; kw;wk; kw;wk; fhh;brpad; rkd;ghl;ilf; fhz;f (OCT-09) ntf;lh; rkd;ghl: xnu Nfhl;likahj nfhlf;fg;gl;l %d;w Gs;spfs; topnar; nry;yk; jsj;jpd; ntf;lh; rkd;ghl r = ( s t)a + sb + tc ways00@gmail.com

52 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top,q;f a = i + j k, b = i + 4j + k, c = 7i + 6k r = ( s t) i + j k +s(i + 4j + k) + t(7i + 6k) fhh;brpad; mikg;g: jsj;jpd; rkd;ghl x x y y z z x x y y z z = 0 x x y y z z x y z (x )( 6) (y )( 9) +(z )(4 9) = 0 = 0 (x )( ) (y )( 7) +(z )( 5) = 0 x y 4 5z + 5 = 0 x + 7y 5z = 0 x 7y + 5z + = 0,JNt Njitahd fhh;brpad; rkd;ghlhfk;.. x x = x = x = 7 y = y = 4 y = 0 z = z = z = 6 x y z + = (x )(4 + 6) (y )(7 5) +(z + )( 0) = 0 (x )(0) (y )( 8) +(z + )( ) = 0 0x y 6 z = 0 = y 0x + 8y z 68 = 0 4 5x + y z 7 = 0 = z cs;slf;fpajk; vd;w x+ = y = z+ Nfhl;il vd;w Nfhl;bw;F,izahdJkhd jsj;jpd; ntf;lh; kw;wk; fhh;brpad; rkd;ghlfisf; fhz;f(mar-4) ntf;lh; rkd;ghl: u = i + j + k kw;wk; v = i + j + k a = i + j + k Njitahd rkd;ghl r = a + su + tv r = i + j + k + s i + j + k + t(i + j + k) fhh;brpad; mikg;g: jsj;jpd; rkd;ghl x x y y z z l m n = 0 l m n x = l = l = y = m = m = z = n = n = 4. (,, ) vd;w Gs;sp topr; nry;tjk; x+ = y+ = z+ kw;wk; x = y+ = z+ vd;w NfhLfSf;F,izahdJkhd jsj;jpd; ntf;lh; kw;wk; fhh;brpad; rkd;ghlfisf; fhz;f (JUN-) ntf;lh; rkd;ghl: u = i j + k kw;wk; v = i + j + k a = i + j + k Njitahd rkd;ghl r = a + su + tv r = i + j + k + s i j + k + t(i + j + k) fhh;brpad; mikg;g: jsj;jpd; rkd;ghl x x y y z z l m n = 0 l m n x = l = l = y = m = m = z = n = n = x y z = 0 (x )( 6) (y )(4 ) + (z )(4 + ) = 0 (x )( 8) (y )() +(z )(5) = 0 8x + 8 y + + 5z 0 = 0 8x y + 5z + = 0 8x + y 5z = 0 ways00@gmail.com

53 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top 5. (,, ) vd;w Gs;sp topr; nry;tjk; x + y + z = 5 kw;wk; x + y + z = 8 Mfpa jsq;fsf;fr; nrq;fj;jhdjkhd jsj;jpd; ntf;lh; kw;wk; fhh;brpad; rkd;ghlfis fhz;f. (JUN-) ntf;lh; rkd;ghl: Njitahd rkd;ghl r = a + su + tv u = i + j + k kw;wk; v = i + j + k a = i + j + k r = i + j + k + s i + j + k + t(i + j + k) fhh;brpad; mikg;g: jsj;jpd; rkd;ghl x x y y z z l m n = 0 l m n x = l = l = y = m = m = z = n = n = x + y z = 0 (x + )(4 ) (y )( 6) + (z )( 6) = 0 (x + )() (y )( 4) +(z )( 5) = 0 x + + 4y 5z + 0 = 0 x + 4y 5z = 0 6. A(,, ) kw;wk; B(,, ) vd;w Gs;spfs; topnar; nry;yf;$bajk; x = y+ = z 4 vd;w Nfhl;bw;F,izahdJkhd jsj;jpd; ntf;lh; kw;wk; fhh;brpad; rkd;ghlfisf; fhz;f ( JUN-4 ) ntf;lh; rkd;ghl: Njitahd rkd;ghl r = ( s)a + sb + tv,q;f a = i j + k, b = i + j k, v = i + j + 4k r = ( s) i j + k + s i + j k fhh;brpad; mikg;g: jsj;jpd; rkd;ghl +t(i + j + 4k) x x y y z z x x y y z z = 0 l m n x = x = l = y = y = m = z = z = n = 4 x y + z 4 4 = 0 4 (x )(6 + ) (y + )( 8 + 8) + (z )( 6 8) = 0 (x )(8) (y + )(0) + (z )( 4) = 0 8x 8 4z + 4 = 0 8x 4z + 4 = 0 4 x z + = 0 7. (,, ) kw;wk; (,, ) vd;w Gs;spfs; topnar; nry;yf; $bajk; x y + 4z 5 = 0 vd;w jsj;jpw;fr; nrq;fj;jhftk; mike;j jsj;jpd; ntf;lh; kw;wk; fhh;brpad; rkd;ghlfisf; fhz;f. ntf;lh; rkd;ghl: (MAR-06,, OCT-06,07,5, JUN-08, 5) Njitahd rkd;ghl r = ( s)a + sb + tv,q;f a = i + j + k, b = i + j + k, v = i j + 4k r = ( s) i + j + k + s i + j + k fhh;brpad; mikg;g: jsj;jpd; rkd;ghl x x y y z z x x y y z z l m n x = x = l = y = y = m = z = z = n = 4 +t(i j + 4k) = 0 x y z = 0 4 (x )(4 4) (y )(4 + 6) + (z )( ) = 0 (x )(0) (y )(0) + (z )( 5) = 0 ways00@gmail.com

54 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top 8. x 0y + 0 5z + 5 = 0 0y + 5z 5 = 0 5 y + z 7 = 0 = y = z vd;w Nfhl;il cs;slf;fpajk; (,, ) vd;w Gs;sp topnar; nry;yf; $bajkhd jsj;jpd; ntf;lh; kw;wk; fhh;brpad; rkd;ghlfisf; fhz;f. (MAR-,6 ) ntf;lh; rkd;ghl: Njitahd rkd;ghl r = ( s)a + sb + tv,q;f a = i + j k, b = i + j + k, v = i + j k r = ( s) i + j k + s i + j + k fhh;brpad; mikg;g: jsj;jpd; rkd;ghl x x y y z z x x y y z z l m n x = x = l = y = y = m = z = z = n = +t(i + j k) = 0 x + y z + = 0 (x + )( 6) (y )( 6 4) + (z + )(9 ) = 0 (x + )( 8) (y )( 0) + (z + )(7) = 0 8x 8 + 0y 0 + 7z + 7 = 0 8x + 0y + 7z = 0 8x 0y 7z + = 0 9. i + 4j + k, i j k kw;wk; 7i + k Mfpatw;iw epiy ntf;lh;fshff; nfhz;l Gs;spfs; topnar; nry;yk; jsj;jpd; ntf;lh; kw;wk; fhh;brpad; rkd;ghlfisf; fhz;f ( JUN-09, MAR-,OCT-, MAR-7) ntf;lh; rkd;ghl: Njitahd rkd;ghl r = ( s t)a + sb + tc,q;f a = i + 4j + k, b = i j k, c = 7i + k r = ( s t) i + 4j + k fhh;brpad; mikg;g: +s(i j k) + t(7i + k) jsj;jpd; rkd;ghl x x y y z z x x y y z z = 0 x x y y z z x = x = x = 7 y = 4 y = y = 0 z = z = z = x y 4 z 6 = (x )(6 ) (y 4)( + ) + (z )(4 + 4) = 0 (x )( 6) (y 4)() +(z )(8) = 0 6x + 8 y z 56 = 0 6x y + 8z + 4 = 0 6x + y 8z 4 = 0 0. ntl;lj;jz;l tbtpy; xu jsj;jpd; rkd;ghl;ilj; jutpf;f (MAR-0, MAR-5) Fwpg;G: jsj;jpd; rkd;ghl;il fhh;brpad; mikg;gpy; my;yj ntf;lh; mikg;gpy; jutpf;fyhk;.,it,uz;lnk ntf;lh; KiwahFk;. fhh;brpad; mikg;g: a, b kw;wk; c vd;gd KiwNa x, y kw;wk; z d; ntl;lj;jz;lfs; jskhdj (a, 0,0), (0, b, 0), (0,0, c) vd;w Gs;spfs; topr; nry;fpwj. jsj;jpd; rkd;ghl x x y y z z x x y y z z = 0 x x y y z z x = a x = 0 x = 0 y = 0 y = b y = 0 z = 0 z = 0 z = c ways00@gmail.com

55 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top x a y 0 z 0 a b 0 0 = 0 a 0 c 0 (x a)(bc) (y 0)( ac) + (z 0)(0 + ab) = 0 (x a)(bc) + yac + zab = 0 xbc abc + yac + zab = 0 xbc + yac + zab = abc xbc abc + yac + zab = abc abc abc abc abc x + y + z = a b c fhh;brpad; mikg;g: %d;w Gs;spfs; topr; nry;yk; jsj;jpd; rkd;ghl r = ( s t)a + sb + tc r = ( s t)ai + sbj + tck xi + yj + zk = ( s t)ai + sbj + tck x = ( s t)a; y = sb; z = tc x a = s t, y b = s, z c = t x + y + z = s t + s + t a b c x + y + z = a b c. (,, ) vd;w Gs;sp topr; nry;tjk; x + y + 4z + 7 = 0 kw;wk; x y + z + = 0 Mfpa jsq;fsf;f nrq;fj;jhftk; cs;s jsj;jpd; ntf;lh; kw;wk; fhh;brpad; rkd;ghlfisf; fhz;f (JUN-06,MAR-08) ntf;lh; rkd;ghl: u = i + j + 4k kw;wk; v = i j + k a = i j + k Njitahd rkd;ghl r = a + su + tv r = i j + k + s i + j + 4k +t(i j + k) fhh;brpad; mikg;g: jsj;jpd; rkd;ghl x x y y z z l m n = 0 l m n x = l = l = y = m = m = z = n = 4 n = x + y + z 4 (x + )(6 + 4) (y + )( 8) (x + )(0) (y + )( 5) = 0 + (z )( 4) = 0 +(z )( 5) = 0 0x y + 0 5z + 5 = 0 0x + 5y 5z + 5 = 0 5 x + y z + 5 = 0. (,, ) topna nry;yf;$bajk; x+ = y+ = z 4 4 vd;w Nfhl;bw;F,izahfTk; x + y + z = 8 vd;w jsj;jpw;f nrq;fj;jhftk; cs;s jsj;jpd; ntf;lh; kw;wk; fhh;brpad; rkd;ghlfisf; fhz;f ntf;lh; rkd;ghl: u = i j 4k kw;wk;v = i + j + k a = i + j k Njitahd rkd;ghl r = a + su + tv r = i + j k + s i j 4k fhh;brpad; mikg;g: +t(i + j + k) jsj;jpd; rkd;ghl x x y y z z l m n = 0 l m n x = l = l = y = m = m = z = n = 4 n = x y z + 4 = 0 (x )( 6 + ) (y )(9 + 8) + (z + )(9 + 4) = 0 (x )(6) (y )(7) +(z + )() = 0 6x 6 7y z + 6 = 0 6x 7y + z + 54 = 0 (OCT-0) ways00@gmail.com

56 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top. x = y = z+ vd;w Nfhl;il cs;slf;fpajk; x y + z = 0 vd;w jsj;jpw;fk; nrq;fj;jhftk; mike;j jsj;jpd; ntf;lh; kw;wk; fhh;brpad; rkd;ghlfisf; fhz;f ntf;lh; rkd;ghl: Njitahd rkd;ghl r = a + su + tv (OCT-) a = i + 0j k, u = i j + k, v = i j + k r = i + 0j k + s i j + k + t(i j + k) fhh;brpad; mikg;g: jsj;jpd; rkd;ghl x x y y z z l m n = 0 l m n x = l = l = y = 0 m = m = z = n = n = x y z + (x )( 9 + ) (y)(6 ) = 0 + (z + )( 4 + ) = 0 (x )( 7) (y)(5) + (z + )( ) = 0 7x + 7 5y z = 0 7x + 5y + z 6 = 0. fyg;ngz;fs;. P vdk; Gs;sp fyg;g vz; khwp z If; Fwpj;jhy; P,d; epakg;ghijia Im. z+ / = vd;w epge;jidf;f cl;gl;l iz+ fhz;f (MAR-0,JUN-) z = x + iy vd;f z+ (x+iy )+ = = (x+)+iy iz+ i(x+iy )+ ( y)+ix = = (x+)+iy ( y) ix ( y)+ix ( y) ix (x+)( y)+xy +i[y( y) x(x+)] ( y) +x Mdhy;, Im. z+ iz+ / = y( y) x(x+) ( y) +x = y y x x = [ + y y + x ] y y x x = y + 4y x y x + = 0 P d; epakg;ghij x + y = 0. P vdk; Gs;sp fyg;g vz; khwp z If; Fwpj;jhy; P,d; epakg;ghijia Re. z / = z+i vd;w epge;jidf;f cl;gl;l fhz;f (JUN-) z = x + iy vd;f z x+iy = = (x )+iy z+i x+iy +i x+i(y+) = (x )+iy x i(y+) x+i(y+) x i(y+) = x(x )+y(y+) + i [fw;gidg; gfjp] x +(y+) Re. z / = z+i x(x )+y(y+) = x +(y+) x + y x + y = x + y + y + x y = P d; epakg;ghij x + y + = 0. P vdk; Gs;sp fyg;g vz; khwp z If; Fwpj;jhy; P,d; epakg;ghijia Im. z+i / = vd;w epge;jidf;f cl;gl;l iz fhz;f (MAR-4) z = x + iy vd;f z+i (x+iy )+i = = x+iy+i iz i(x+iy ) ix y = x+i(y+) (y+) ix (y+)+ix (y+) ix = x(y+) ix i(y+)(y+)+x(y+) (y+) +x Mdhy;, Im. z+i iz / = x (y+)(y+) (y+) +x = x y y = y y x x + y + y = 0 P d; epakg;ghij x + y + y = 0 4. P vdk; Gs;sp fyg;g vz; khwp z If; Fwpj;jhy; P,d; epakg;ghijia arg. z z+ vd;w epge;jidf;f cl;gl;l fhz;f (MAR-) ways00@gmail.com

57 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top arg(z ) arg(z + ) = π arg(x + iy ) arg(x + iy + ) = π arg((x ) + iy) arg (x + ) + iy = π y tan x tan tan y x+ = π y x y x+ = π +. y y /. x x+ / y x +y = tan π y x +y = y = (x + y ) x + y y = 0 vd;gj Njitahd epakg;ghijahfk;. 5. P vdk; Gs;sp fyg;g khwp z If; Fwpj;jhy; Re. z+ / = vd;w epge;jidf;f cl;gl;l z+i P,d; epakg;ghijia fhz;f (MAR-6) z = x + iy vd;f z + x + iy + (x + ) + iy = = z + i x + iy + i x + i(y + ) = (x+)+iy x i(y+) x+i(y+) x i(y+) = x(x+)+y(y+) + i [fw;gidg; gfjp] x +(y+) Re. z / = z+i x(x + ) + y(y + ) x + (y + ) = x + y + x + y = x + y + y + x y = P d; epakg;ghij x y = 6. α, β vd;git x x + = 0,d; %yq;fs; kw;wk; cot θ = y + vdpy; (y+α) n (y+β) n α β = sin nθ sin n θ vdf; fhl;lf. (MAR-06,OCT-,OCT-4) x x + = 0,d; %yq;fs; ± i α = + i, β = i vd;f (y + α) n =,(cot θ ) + ( + i)- n =,cot θ + i- n =,cos θ + i sin sin n θ-n θ (y + α) n = [cos nθ + i sin nθ] sin n θ,njnghy, (y + β) n = [cos nθ i sin nθ] sin n θ i sin nθ (y + α) n (y + β) n = sin n θ α β = ( + i) ( i) = i NkYk;, (y + α) n (y + β) n i sin nθ sin nθ = α β i sin n = θ sin n θ 7. x px + p + q = 0 vd;w rkd;ghl;bd; %yq;fs; α, β kw;wk; tan θ = q vdpy; (y+α) n (y+β) n α β = q n sin nθ sin n θ x px + (p + q ) = 0 x = p± 4p 4(p +q ) = p ± iq α = p + iq, β = p iq vd;f α β = qi tan θ = q y+p q vd epwtf y+p (MAR-07,OCT-09,6) vdf; nfhlf;fg;gl;ls;sj y + p = tan θ y + p = q cot θ y = q cot θ p y + α = q cot θ p + (p + iq) cos θ + i sin θ = q,cot θ + i- = q sin θ (y + α) n n (cos θ+i sin θ)n = q (y + α) n =,Nj Nghy (y + β) n = qn sin n θ qn sin n θ () () vdk; NghJ sin n θ (y + α) n (y + β) n = (y+α) n (y+β) n α β = [cos nθ + i sin nθ]..() [cos nθ i sin nθ] () q n qn sin n θ (iq ) sin n θ n sin nθ = q sin n θ [i sin nθ],i sin nθ- 8. x x + 4 = 0,d; %yq;fs; α kw;wk; β vdpy; α n β n = i n+ sin nπ vd epwtf. mjpypue;j α 9 β 9 d; kjpg;ig ngwf (OCT-06,OCT-08,MAR-09,MAR-,JUN-5) x x + 4 = 0 x = ± i α = + i, β = i α n = + i n = n.cos nπ + i sin nπ / ways00@gmail.com

58 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top β n = i n = n.cos nπ i sin nπ / α n β n = n.cos nπ + i sin nπ / n.cos nπ i sin nπ / α n β n = n.i sin nπ / α n β n = i n+ sin nπ n = 9 vdg; gpujpapl> = x 9 + x 5 x 4 = 0 vd;w rkd;ghl;ilj; jph;f;f (JUN-06,) x 9 + x 5 x 4 = 0 x 5 (x 4 + ) (x 4 + ) = 0 (x 5 )(x 4 + ) = 0 x 5 = 0; x 4 + = 0 (i) x = () 5 = (cos 0 + i sin 0) 5 α 9 β 9 = i 0 sin 9π = i 0 (sin π) = 0 9. a kw;wk; b vd;git x + x + 4 = 0 vd;w rkd;ghl;bd; %yq;fshf,ug;gpd; a n + b n d; kjpg;gpidf; fhz;f.,jpypue;j a + b d; kjpg;gpid jutpf;f (n vd;gj xu KO vz;) (JUN-6) x + x + 4 = 0 x = ± 4()(4) () = ± 6 ± 4 = = ± i ( ± i) = x = ± i a = + i, b = i a n = + i n = n cos 5π 6 + i sin 5π 6 = n.cos 5nπ 6 b n = i n = n cos 5nπ a n + b n = n.cos 5nπ 6 = n.cos 5nπ 6 6 n + i sin 5nπ 6 / i sin 5nπ 6 5nπ + i sin / + 6 n.cos 5nπ 5nπ i sin / i sin 5nπ 6 = n.cos 5nπ 6 / a n + b n = n+ cos 5nπ 6 n = vd gpujpapl + cos 5nπ 6 i sin 5nπ 6 / a + b = cos 5()π 6 = (cos 0 π) = 89() = (cos k π + i sin kπ) 5 = cos kπ kπ + i sin 5 5 k = 0,,,,4 (ii) x = ( ) 4 = (cos π + i sin π) 4 = (cos(k + ) π + i sin(k + )π) 4 (k + )π (k + )π = cos + i sin 4 4 k = 0,,,,t;thW 9 %yq;fs; ngwg;glfpd;wd.. x 7 + x 4 + x + = 0 vd;w rkd;ghl;ilj; jph;f;f (JUN-09) x 7 + x 4 + x + = 0 x 4 (x + ) + (x + ) = 0 (x 4 + )(x + ) = 0 x 4 =, x = (i) x = ( ) 4 = (cos π + i sin π) 4 = (cos(k + ) π + i sin(k + )π) 4 = cos (k+)π 4 + i sin (k+)π 4 k = 0,,, (ii) x = ( ) = (cos π + i sin π) = (cos(k + ) π + i sin(k + )π) = cos (k+)π + i sin (k+)π k = 0,,. x 4 x + x x + = 0 vd;w rkd;ghl;ilj; jph;f;f ( JUN-08,JUN-0,OCT-, MAR-7),e;j gy;ywg;gf; Nfhitapd; kjpg;gfs; nguf;fy; njhlhpy; cs;sd. r = x, a =, n = 5 x + x x + x 4 = a(rn ) = x5 + r x + ways00@gmail.com

59 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top,q;f x x 5 + = 0 Ij; jph;j;j x = vd;w %yj;ij epf;fptpl Ntz;Lk; x 5 + = 0 x = ( ) 5 = (cos π + i sin π) 5 = (cos(k + ) π + i sin(k + )π) 5 = cos (k+)π 5,q;F k = 0,,,,4 + i sin (k+)π 5 cis π 5, cis π 5, cis π, cis 7π 5, cis 9π 5 Mfpa kjpg;gfisg; ngwk;,q;f cis π = vd;w %yj;ij epf;fptpl cis π 5, cis π 5, cis 7π 5, cis 9π 5 Mfpait rkd;ghl;bd; jph;tfshfk;.. x + = cos θ, y + = cos φ vdpy; x y (i) xm y (ii) xm y yn n + x yn n x m = cos( mθ nφ) m = i sin( mθ nφ) vdf; fhl;lf x + = cos θ x x cos θ x + = 0 x = cos θ ± i sin θ x = cos θ + i sin θ vdf; nfhs;f,jnghynt, y = cos φ + i sin φ x m = cos mθ + i sin mθ y n = cos nφ + i sin nφ x m cos mθ + i sin mθ = yn cos nφ + i sin nφ (JUN-4) = (cos mθ + i sin mθ)(cos( nφ) + i sin( nφ)) x m y n = cos( mθ nφ) + i sin( mθ nφ) () y n x m = cos( mθ nφ) i sin( mθ nφ) () () + () () () x m y x m y y n n + x m = cos( mθ nφ) y n n x m = i sin( mθ nφ) 4. a = cos α + i sin α, b = cos β + i sin β, c = cos γ + i sin γ vdpy; (i) abc + abc = cos( α + β + γ) (ii) a b +c = cos( α + β γ) vd ep&gp abc (OCT-0) abc = (cos α + i sin α) (cos β + i sin β) (cos γ + i sin γ) abc = (cos α + i sin α)(cos β + i sin β)(cos γ + i sin γ) abc = cos( α + β + γ) + i sin( α + β + γ) abc = cos( α + β + γ) i sin( α + β + γ) (ii) a b +c abc ab c ab c c ab ab c abc + = cos( α + β + γ) abc = a b abc + c abc = ab c + c ab = (cos α+i sin α)(cos β+i sin β ) (cos γ+i sin γ) = cos( α + β γ) + i sin( α + β γ) = cos( α + β γ) i sin( α + β γ) + c = cos( α + β γ) + ab i sin( α + β γ) + cos( α + β γ) i sin( α + β γ) = cos( α + β γ) a b +c ab c 5. + i = cos( α + β γ),d; vy;yh kjpg;gfisak; fhz;f + i = r(cos θ + i sin θ) r cos θ =, r sin θ = r = ( ) + = cos θ =, sin θ = θ = π 6 + i =.cos π + i sin π / 6 6 kjpg;gfs; =.cos π + i sin π / (JUN-07) =.cos.kπ + π / + i sin.kπ + π // = 0cos(6k + ) π 9 + i sin(6k + ) π 9,q;F k = 0,, cis. π 9 /, cis 7π 9, cis π 9 ways00@gmail.com

60 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top 6. i,d; vy;yh kjpg;gfisak; fhz;f i = r(cos θ + i sin θ) r cos θ =, r sin θ = r = ( ) + = (OCT-) cos θ =, sin θ = θ = π + π = 5π 6 6 i =.cos. 5π / + i sin. 5π // 6 6 i =.cos. 5π / + i sin. 5π // 6 6 = cos kπ 5π 6 + i sin kπ 5π 6 =.cos(k 5) π 9 + i sin(k 5) π 9 / vdnt cis 5π, cis 7π 9 9, cis 9π 9 Mfpa kjpg;gfisg; ngwk; k = 0,, (my;yj) cis. π 9 / 7.. i / 4 d; vy;yh kjpg;gfisak; fhz;f kw;wk; mjd; kjpg;gfspd; nguf;fw;gyd; vdtk; fhl;lf. (OCT-07,MAR-08,MAR-, OCT-5) i = r(cos θ + i sin θ) vd;f r cos θ =, r sin θ = r = = cos θ =, sin θ = θ, 4tJ fhy;gfjpapys;sj θ = π i = cos. π/ + i sin. π /. i / 4 =.cos. π / + i sin. π // 4 = (cos( π) + i sin( π)) 4 = (cos(kπ π) + i sin(kπ π)) 4 = cos. k / π + i sin.k / π 4 4 k = 0,,, cis. π /, cis π, cis π, cis 5π Mfpa kjpg;gfisg; ngwk;,tw;wpd; nguf;fw;gyd; cis 0 π + π + π + 5π = cis π = cisπ 4 = cos π + i sin π = i / 4 d; vy;yh kjpg;gfisak; fhz;f kw;wk; mjd; kjpg;gfspd; nguf;fw;gyd; vdtk; fhl;lf. (MAR-5) + i = r(cos θ + i sin θ) r cos θ =, r sin θ = r = = cos θ =, sin θ = θ, Kjy; fhy;gfjpapys;sj θ = π + i = cos π + i sin π. + i / 4 π =.cos + i sin π / 4 = (cos π + i sin π) 4 = (cos(kπ + π) + i sin(kπ + π)) 4 = cos. k+ / π + i sin.k+ / π 4 4 cis. π 4 /, cis π 4, cis 5π 4, cis 7π 4 ngwk;.,tw;wpd; nguf;fw;gyd; cis 0 π 4 + π 4 + 5π 4 + 7π 4 = cis 06π 4 = cis4π k = 0,,, Mfpa kjpg;gfisg; = cos 4π + i sin 4π = ways00@gmail.com

61 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top 4. gfkiw tbtpay;. xu uapy;nt ghyj;jpd; Nky; tist gutisaj;jpd; mikg;igf; nfhz;ls;sj. me;j tistpd; mfyk; 00 mbahftk; mt;tistpd; cr;rpg;gs;spapd; cauk; ghyj;jpypue;j 0 mbahftk; cs;sj vdpy;> ghyj;jpd; kj;jpapypue;j,lg;gwk; my;yj tyg;gwk; 0 mb J}uj;jpy; ghyj;jpd; Nky; tist vt;tst cauj;jpy;,uf;fk; vdf; fhz;f. (MAR-09,JUN-6),q;F gutisak; fpo;nehf;fp jpwg;gilajhf vlj;jf; nfhs;nthk; x = 4ay,J (50, 0) topahfr; nry;fpwj = 4a( 0) a = 50 4 x = / y x = 50y gutisaj;jpd; Nky; cs;s Gs;sp B(0, y ) 00 = 50y y = = 5 AB vd;gj ghyj;jpd; ikaj;jpypue;j tyg;gwj;jpy; 0 mb njhiytpy;> ghyj;jpd; caukhfk;. AC = 0 kw;wk; BC = 5 AB = 0 5 = 9 5 mb mjhtj> Njitg;gl;l,lj;jpypUe;J ghyj;jpd; kpf cauk; 9 5 mb MFk;,J (6, 4) topr; nry;fpwj 6 = 6a a = 9 4 rkd;ghl x = 9y..() ( 6, 4) y; rha;itf; fzf;fpl> () I x-i nghwj;j tiff;nfo fhz x = 9 dy dx dy dx = 9 x ( 6, 4),y; dy = 6 = 4 dx 9 tan θ = 4 θ = tan. 4 / Njitahd vwpnfhzk; tan. 4 /. jiukl;lj;jpypue;j 7.5 kp cauj;jpy; jiuf;f,izahf nghuj;jg;gl;;l xu FohapypUe;J ntspnawk; eph; jiuiaj; njhlk; ghij xu gutisaj;ij Vw;gLj;JfpwJ. NkYk;,e;j gutisag; ghijapd; Kid Fohapd; thapy; mikfpwj. Foha; kl;lj;jpw;f.5 kp fpno ephpd; gha;thdj Fohapd; Kid topahfr; nry;yk; epiy Fj;Jf;Nfhl;bw;F kpl;lh; J}uj;jpy; cs;sj vdpy; Fj;Jf; Nfhl;bypUe;J vt;tst J}uj;jpw;F mg;ghy; epuhdj jiuapy; tpok; vd;gijf; fhz;f.(oct-09,mar-,oct-). xu uhf;nfl; ntbahdj nfhsj;jk;nghj mj xu gutisag; ghijapy; nry;fpwj. mjd; cr;r cauk; 4kP-I vl;lk;nghj mj nfhsj;jg;gl;l,lj;jpypue;j fpilkl;l J}uk; 6 kp njhiytpys;sj.,wjpahf fpilkl;lkhf kp njhiytpy; jiuia te;jilfpwj vdpy; Gwg;gl;l,lj;jpy; jiuald; Vw;gLj;jg;gLk; vwpnfhzk; fhz;f (MAR-06,JUN-09,JUN-0,JUN-,OCT-,MAR-4, MAR-7) gutisaj;jpd; rkd;ghl x = 4ay (Kidia Mjpahff; nfhs;f). nfhlf;fg;gl;l tptuq;fspd; gb gutisak; fpo;nehf;fp jpwg;gilajhf mikfpwj x = 4ay P vd;w Gs;sp gutisag; ghijapy; Foha; kl;lj;jpw;f.5 kp fpnoak;> Fohapd; Kid topna nry;yk; epiy Fj;Jf; Nfhl;bw;F kp mg;ghyk; cs;sj. P vd;gj(,.5) vdnt, 9 = 4a(.5) ways00@gmail.com

62 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top a = 9 0 gutisaj;jpd; rkd;ghl x = Fj;Jf; Nfhl;bd; mbg;gs;spapypue;j x J}uj;Jf;F mg;ghy; epuhdj jiuapy; tpotjhf nfhs;f. Mdhy; FohahdJ jiukl;lj;jpypue;j 7.5kP cauj;jpy; mike;js;sj. (x, 7.5) vd;w Gs;sp gutisaj;jpys;sj. x = 4 9 ( 7.5) = 7 0 x = vdnt> jz;zph; jiuiaj; njhlk;,lj;jf;fk; Fohapd; KidapypUe;J tiuag;glk; Fj;Jf;Nfhl;bw;Fk;,ilg;gl;l J}uk; kp. 4. xu thy; tpz;kpd; (comet) MdJ #hpaidr; (sun) Rw;wp gutisag; ghijapy; nry;fpwj. kw;wk; #hpad; gutisaj;jpd; Ftpaj;jpy; mikfpwj. thy; tpz;kpd; #hpadpypue;j 80 kpy;ypad; fp.kp njhiytpy; mike;j,uf;fk; NghJ thy; tpz;kpidak; #hpaidak;,izf;fk; NfhL ghijapd; mr;rld; Nfhzj;jpid Vw;gLj;Jkhdhy; (i) thy; tpz;kpdpd; ghijapd; rkd;ghl;ilf; fhz;f. (ii)thy; tpz;kpd; #hpadf;f vt;tst mufpy; tukbak; vd;gijak; fhz;f. (ghij tyjgwk; jpwg;gilajhf nfhs;f) (MAR-08,MAR-,JUN-,MAR-6,OCT-5,6) π FQ = x = FP. cos π = 80 = 40 VQ = a + 40 if VF = a P vd;gj (VQ, PQ) = (a + 40,40 ) P vd;gj gutisak; y = 4ax Nky;,Ug;gjhy; (40 ) = 4a(a + 40) a = 60 my;yj a = 0 a = 60 Vw;Gilajy;y. ghijapd; rkd;ghl y = 4 0 x y = 80x #hpadf;fk; thy; tpz;kpdf;fk;,ilnaas;s kpff; Fiwe;j J}uk; VF kpff; Fiwe;j J}uk; 0 kpy;ypad; fp.kp. 5. xu njhq;f ghyj;jpd; fk;gp tlk; gutisa tbtpys;sj. mjd; ghuk; fpilkl;lkhf rpuhf gutpas;sj. mij jhq;fk;,u J}z;fSf;F,ilNaAs;s J}uk; 500 mb. fk;gp tlj;ijj; jhq;fk; Gs;spfs; J}zpy; jiuapypue;j 00 mb cauj;jpy; mike;js;sd. NkYk; jiuapypue;j fk;gp tlj;jpd; jho;thd Gs;spapd; cauk; 70 mb> fk;gptlk; mb cauj;jpy; jhq;fk; fk;gj;jpw;f,ilna cs;s nrq;fj;j epsk; (jiuf;f,izahf) fhz;f. (OCT-07, OCT-, JUN-4) gutisaj;jpd; ghij tyjgf;fk; jpwg;gilaj. NkYk; Kidg;Gs;sp MjpapYs;sJ. thy;tpz;kpdpd; epiy P FP = 80 kpy;ypad; fp.kp. P apypue;j gutisaj;jpd; mr;rf;f PQ vd;w nrq;fj;j tiua FQ = x vd;f Kf;Nfhzk; FQP apypue;j PQ = FP. sin π = 80 = 40 fk;gp tlj;jpd; kpj kpfj;jho;thd Gs;sp KidahFk;.,jid Mjpahf nfhs;f. AB kw;wk; CD jhq;fk; J}z;fs;.,U J}z;fSf;F,ilNaAs;s njhiyt 500mb vd;gjhy; VA = 750 mb> AB = 00 mb A B = = 0 mb B vd;gj (750,0) gutisaj;jpd; rkd;ghl x = 4ay B vd;w Gs;sp x = 4ay,y; cs;sjhy; (750) = 4a(0) ways00@gmail.com

63 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top a = rkd;ghl x = y fk;gk; RQ tpypue;j fk;gp tlj;jpw;f nrq;fj;jhd epsk; PQ. RQ =, RR = 70 R Q = 5 VR = x Q vd;gj (x, 5) Q gutisaj;jpd; kpjs;s xu Gs;sp. x = 5 x = 50 0 PQ = x = 00 0 mb 6. xu njhq;f ghyj;jpd; fk;gp tlk; gutisa tbtpypys;sj. mjd; epsk; 40 kpl;lh; MFk;. topg;ghijahdj fk;gp tlj;jpd; fpo;kl;lg; Gs;spapypUe;J 5 kpl;lh; fpno cs;sj. fk;gp tlj;ij jhq;fk; Jhz;fspd; cauq;fs; 55 kpl;lh; vdpy;> 0 kpl;lh; cauj;jpy; fk;gp tlj;jpw;f xu Jiz jhq;fp $Ljyhff; nfhlf;fg;gl;lhy; mj;jizj;jhq;fpapd; epsj;ijf; fhz;f (JUN-06,JUN-5) njhq;f ghyj;jpd; fk;gptlk; Nkw;Gwk; jpwg;gila gutisa mikg;ig ngw;ws;sj. x = 4ay nfhlf;fg;gl;l tptuq;fspypue;j tlj;jpd; Kid topg;ghijapy;,ue;j 5 kp Nky; mike;js;sj. njhq;f ghyj;jpd; tpl;lk; 40 kpl;lh;. Gs;sp A(0,50) gutisaj;jpd; kpj mike;j cs;sj. 400 = 4a(50) a = Njitahd rkd;ghl x = 8y Q (x, 5) vd;w Gs;sp gutisaj;jpd; kpj mike;js;sj x = 8(5) = 00 = 0 PQ = x = 0 kp. jhq;fpapd; epsk; MFk;. 7. xu tist miu-eps;tl;l tbtpy; cs;sj. mjd; mfyk; 48 mb> cauk; 0 mb jiuapypue;j 0 mb cauj;jpy; tistpd; mfyk; vd;d? (OCT-06,OCT-) jiuapd; elg;gs;spia ikak; C(0,0) Mff; nfhs;s jiuapd; mfyk; 48 mb. Kidfs; A(4,0), A ( 4,0) a = 48 kw;wk; b = 0 eps;tl;lj;jpd; rkd;ghl x + y =.() 4 0 0kP cauks;s J}zpw;Fk; ikaj;jpw;fk;,ilna cs;s J}uk; x vdnt (x, 0) vd;w Gs;sp rkd;ghl () I epiwt nra;ak;. x + 0 = 4 0 x = jiuapypue;j 0 mb cauj;jpy; tistpd; mfyk; vd;gj x = 4 Njitahd tistpd; mfyk; 4 mb. 8. xu EioT thapypd; Nkw;$iuahdJ miu eps;tl;l tbtj;jpy; cs;sj.,jd; mfyk; 0mb. ikaj;jpypue;j mjd; cauk; 8 mb kw;wk; gf;fr; Rth;fspd; cauk; mb vdpy; VNjDk; xu gf;fr; RthpypUe;J 4 mb J}uj;jpy; Nkw;$iuapd; cauk; vd;dthf,uf;fk;? (MAR-07,0,7) gf;fr; RthpypUe;J 4 mb J}uj;jpy; cs;s Nkw;$iuapd; cauk; PQR. glj;jpd; %yk; PQ = mb. QR,d; epsj;ij fhz Ntz;Lk;. mfyk; 0 mbahf,ug;gjhy; Kidfs; A, A,d; Maj;njhiyfs; KiwNa (0,0), ( 0,0). glj;jpd; %yk; AA = a = 0 ways00@gmail.com

64 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top a = 0 kw;wk; b = 8 = 6 x 00 + y 6 = QR vd;gj y vdpy; R,d; Maj;njhiyfs; (6, y ) eps;tl;lj;jpd; kpj R miktjhy; S 6 + y 00 = y 6 = 4.8 PQ + QR = Njitahd Nkw;$iuapd; cauk; 6.8 mbahfk;. 9. #hpad; Ftpaj;jpypUf;FkhW G+kpahdJ #hpaid xu eps;tl;lg; ghijapy; Rw;wp tufpwj. mjd; miu-nel;lr;rpd; epsk; 9.9 kpy;ypad; iky;fs; MfTk;> ikaj;njhiyj; jft 0.07 MfTk; cs;sj vdpy; G+kpahdJ #hpadf;f kpf mufhikapy; tuk;nghj cs;s J}uKk; kpfj; njhiytpy; tuk;nghj cs;s J}uKk; fhz;f. miu nel;lr;rpd; epsk; CA a = 9.9 kpy;ypad; iky;fs; e = 0.07 vd nfhlf;fg;gl;ls;sj #hpadf;f kpf mufhikapy; tuk;nghj cs;s J}uk; = FA, kw;wk; kpfj; njhiytpy;,uf;fk; NghJ cs;s J}uk;= FA CF = ae = 9.9(0.07) FA = CA CF = (0.07) = 9.9, = = 9.07 kpy;ypad; iky;fs; FA = CA + CF = (0.07) = 9.9( ) = 9.9(.07) = kpy;ypad; iky;fs; 0. xu rkjsj;jpd; Nky; nrq;fj;jhf mike;js;s Rthpd; kpj 5kP epsks;s xu VzpahdJ jsj;jpidak; Rtw;wpidAk; njhlkhw efh;e;j nfhz;l,uf;fpwj. vdpy;> Vzpapd; fpo;kl;l KidapypUe;J 6kP J}uj;jpy; Vzpapy; mike;js;s P vd;w Gs;spapd; epakg;ghijiaf; fhz;f. (OCT-07,OCT-08,MAR-,JUN-5) AB vd;gj Vzp. Vzpapd; kpj P(x, y ) vd;w Gs;sp AP = 6kP,Uf;FkhW vlj;jf; nfhs;s x-mr;rf;f nrq;fj;jhf PD Ak; y-mr;rf;f nrq;fj;jhf PC Ak; tiua ADP kw;wk; PCB tbnthj;jit PC = PB = BC DA AP PD x = 9 = BC DA 6 y DA = 6x = x 9 BC = 9y = y 6 OA = OD + DA = x + x = 5x OB = OC + BC = y + y = 5y OA + OB = AB 5x 9 + 5y 4 = 5 x (x, y ) d; epakg;ghij x eps;tl;lkhfk; khw;wkiw: PCB,y; cos θ = x 9 ADP,y; sin θ = y 6 cos θ + sin θ = x + y 8 + y y 6 4 = 9 PAO = BPC = θ 6 = (x, y ) d; epakg;ghij x eps;tl;lkhfk;. 8 + y 6 =.,J Xh; =.,J Xh;. xu Nfh-Nfh tpiahl;l tpuh; tpisahl;lg; gapw;rpapd; NghJ mtuf;fk; Nfh-Nfh Fr;rpfSf;Fk;,ilNaAs;s J}uk; vg;nghojk; 8kP Mf,Uf;FkhW czh;fpwhh;. mt;tpu Fr;rpfSf;F,ilg;gl;l J}uk; 6kP vdpy; mth; XLk; ghijapd; rkd;ghl;ilf; fhz;f.(mar-,5) Nfh-Nfh Fr;rpfs;,uz;Lk; F kw;wk; F,y; mike;js;sd. P(x, y) vd;w Gs;spahdJ tpisahl;l tpuhpd; epiy ways00@gmail.com

65 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top F P + F P = a = 8 a = 4 F F = ae = 6 ae = 4e = e = 4 NkYk; b = a ( e ) = / b = 7 ghijapd; rkd;ghl x 6 + y 7 =. xu eps;tl;lg; ghijapd; Ftpaj;jpy; G+kp,Uf;FkhW xu Jizf;Nfhs; Rw;wp tufpwj.,jd; ikaj; njhiyj;jt MfTk; G+kpf;Fk; Jizf; NfhSf;Fk;,ilg;gl;l kpr;rpw J}uk; 400 fpnyh kpl;lh;fs; MfTk;,Uf;Fkhdhy; G+kpf;Fk; Jizf;NfhSf;Fk;,ilg;gl;l mjpfgl;r J}uk; vd;d? ( JUN-07,JUN-08,JUN-,JUN-4,OCT-4) glj;jpypue;j G+kpapd; epiy F. Jizf;NfhSf;Fk;. G+kpf;Fk;,ilg;gl;l kpr;rpw J}uk; F A = 400 fp.kp. G+kpf;Fk; > Jizf;NfhSf;Fk;,ilg;gl;l mjpfgl;r J}uk; F A fzf;fpl Ntz;Lk; CA = a, CF = ae, F A = 400 fp.kp. F A = CA CF = a ae 400 = a( e) 400 = a. / a = 800 fp.kp. CA = 800 kw;wk; CF = ae = 800 = 400fp.kP F A = F C + CA = = 00fp.kP.. #hpad; Ftpaj;jpypUf;FkhW nkh;fhp fpufkhdj #hpaid xu eps;tl;lg;ghijapy; Rw;wp tufpwj. mjd; miu nel;lr;rpd; epsk; 6 kpy;ypad; iky;fs; MfTk; ikaj;njhiyj;jft 0.06 MfTk;,Uf;Fkhapd; (i) nkh;f;fhp fpufkhdj #hpadf;f kpf mufhikapy; tuk;nghj cs;s J}uk; (ii) nkh;f;fhp fpufkhdj #hpadf;f kpfj; njhiytpy;,uf;fk;nghj cs;s J}uk; Mfpatw;iwf; fhz;f. (OCT-09,JUN-0,OCT-,MAR-6) glj;jpypue;j #hpadpd; epiy F CA = 6 kpy;ypad; iky;fs;, e = 0.06 nkh;f;fhp fpufkhdj #hpadf;f kpf mufhikapy;> kpfj; njhiytpy;,uf;fk; epiyfs; A kw;wk; A (i) kpf mufhikapy; J}uk; F A F A = CA CF = a ae = a( e) = 6( 0.06) = mufhik J}uk; = kpy;ypad; iky;fs; (ii) kpfj; njhiytpy; J}uk; F A F A = F C + CA = ae + a = a(e + ) = 6( ) =.06 6 = 4.46 kpy;ypad; iky;fs;. 4. xu ghyj;jpd; tisthdj miu eps;tl;lj;jpd; tbtpy; cs;sj. fpilkl;lj;jpy; mjd; mfyk; 40 mbahftk; ikaj;jpypue;j mjd; cauk; 6 mbahftk; cs;sj vdpy; ikaj;jpypue;j tyj my;yj,lg;gwj;jpy; 9 mb J}uj;jpy; cs;s jiug;gs;spapypue;j ghyj;jpd; cauk; vd;d? (OCT-0, JUN-, MAR-4) ghyj;jpd; elg;gs;spia ikak; C(0,0)Mf vlj;jf; nfhs;nthk;. fpilkl;lk; 40 mb vdnt Kidfs; A(0,0) kw;wk; A ( 0,0) a = 40 a = 0, b = 6 rkd;ghl x + y = ikaj;jpypue;j 9 mb tyg;gwj;jpy; cauj;ij y vd;f. vdnt (9, y ) vd;w Gs;sp rkd;ghl;by; cs;sj. 9 + y = ways00@gmail.com

66 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top y 56 = = y = / y = = ikaj;jpypue;j tyj my;yj,ljgwj;jpy; 9 mb J}uj;jpy; cs;s jiug;gs;spapypue;j ghyj;jpd; cauk; 4 9 mb 5 ehd;f tifahd gutisaq;fspd; KbTfspd; njhfg;g y + 6y = 8x 9 y + 6y + = 8x 9 (y + ) = 8x 9 (y + ) = 8x (y + ) = 8x Y = 8X,q;F X = x, Y = y + Y = 4()X a = gutisak; tyjgwk; jpwg;gilaj X, Y I nghwj;j mr;r Y = 0 Kid (0, 0) Ftpak; (a, 0) (, 0),af;Ftiu nrt;tfyk; X = a X = X = a X = x, y I nghwj;j X = x, Y = y + Y = 0 y + = 0 X = 0, Y = 0 x = 0, y + = 0 V(0, ) X =, Y = 0 x =, y + = 0 F(, ) X = x = X = x = nrt;tfyj;jpd; epsk; 4a = y 8x + 6y + 9 = 0 vd;w gutisaj;jpd; mr;r> Kid> Ftpak;>,af;Ftiu> nrt;tfyj;jpd; rkd;ghl kw;wk; nrt;tfyj;jpd; epsk; fhz;f. NkYk; mt;tistiuia tiuf y 8x + 6y + 9 = 0 (JUN-08,OCT-0,4) 6. x x + 8y + 7 = 0 vd;w gutisaj;jpd; mr;r> Kid> Ftpak;>,af;Ftiu> nrt;tfyj;jpd; rkd;ghl kw;wk; nrt;tfyj;jpd; epsk; fhz;f. NkYk; mt;tistiuia tiuf ( MAR-5) x x + 8y + 7 = 0 x x = 8y 7 x x + = 8y 7 (x ) = 8y 7 (x ) = 8y 7 + (x ) = 8y 6 ways00@gmail.com

67 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top (x ) = 8(y + ) X = 8Y,q;F X = x, Y = y + X = 4()Y a = gutisak; fpo;gwk; jpwg;gilaj X, Y I nghwj;j mr;r X = 0 Kid (0, 0) Ftpak;,af;Ftiu nrt;tfyk; (0, a) (0, ) Y = a Y = Y = a Y = x, y I nghwj;j X = x, Y = y + X = 0 x = 0 x = X = 0, Y = 0 x = 0, y + = 0 V(, ) X = 0, Y = x = 0, y + = F(, 4) Y = y + = y = 0 Y = y + = y = 4 nrt;tfyj;jpd; epsk; 4a = 8 8 X = 4()Y a = gutisak; fpo;gwk; jpwg;gilaj X, Y I nghwj;j mr;r X = 0 Kid (0, 0) Ftpak;,af;Ftiu nrt;tfyk; (0, a) (0, ) Y = a Y = Y = a Y = x, y I nghwj;j X = x, Y = y X = 0 x = 0 x = X = 0, Y = 0 x = 0, y = 0 V(,) X = 0, Y = x = 0, y = F(, 0) Y = y = y = Y = y = y = nrt;tfyj;jpd; epsk; 4a = x 4x + 4y = 0 vd;w gutisaj;jpd; mr;r> Kid> Ftpak;>,af;Ftiu> nrt;tfyj;jpd; rkd;ghl kw;wk; nrt;tfyj;jpd; epsk; fhz;f. NkYk; mt;tistiuia tiuf. (JUN-) x 4x + 4y = 0 x 4x = 4y x 4x + = 4y (x ) 4 = 4y (x ) = 4y + 4 (x ) = 4(y ) (x ) = 4(y ) X = 4Y,q;F X = x, Y = y 8. y + 8x 6y + = 0 vd;w gutisaj;jpd; mr;r> Kid> Ftpak;>,af;Ftiu> nrt;tfyj;jpd; rkd;ghl kw;wk; nrt;tfyj;jpd; epsk; fhz;f. NkYk; mt;tistiuia tiuf (OCT-06,MAR-07,6) y + 8x 6y + = 0 y 6y = 8x y 6y + = 8x (y ) 9 = 8x (y ) = 8x + 9 = 8x + 8 = 8(x ) Y = 8X,q;F X = x, Y = y Y = 4()X a = gutisak; fpo;gwk; jpwg;gilaj ways00@gmail.com

68 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top X, Y I x, y I nghwj;j nghwj;j X = x, Y = y mr;r Y = 0 Y = 0 y = 0 Kid (0, 0) Ftpak; ( a, 0) (, 0),af;Ftiu nrt;tfyk; nrt;tfyj;jpd; X = a X = X = a X = X = 0, Y = 0 x = 0, y = 0 V(,) X =, Y = 0 x =, y = 0 F(,) X = x = x = 0 X = x = x + = 0 epsk; 4a = 8 8 Kid (0, 0) Ftpak;,af;Ftiu nrt;tfyk; (0, a) (0, ) Y = a Y = Y = a Y = X = 0, Y = 0 x = 0, y + = 0 V(, ) X = 0, Y = x = 0, y + = F(,) Y = y + = y + 4 = 0 Y = y + = y = 0 nrt;tfyj;jpd; epsk; 4a = 4a = 9. x 6x y = 0 vd;w gutisaj;jpd; mr;r> Kid> Ftpak;>,af;Ftiu> nrt;tfyj;jpd; rkd;ghl kw;wk; nrt;tfyj;jpd; epsk; fhz;f. NkYk; mt;tistiuia tiuf (MAR-0) x 6x y = 0 x 6x = y + x 6x + = y + (x ) 9 = y + (x ) = y + (x ) = (y + ) X = Y,q;F X = x, Y = y + X = 4()Y a = gutisak; fpo;gwk; jpwg;gilaj X, Y I nghwj;j mr;r X = 0 x, y I nghwj;j X = x, Y = y + X = 0 x = 0 x = 0. y 8x y + 7 = 0 vd;w gutisaj;jpd; mr;r> Kid> Ftpak;>,af;Ftiu> nrt;tfyj;jpd; rkd;ghl kw;wk; nrt;tfyj;jpd; epsk; fhz;f. NkYk; mt;tistiuia tiuf ( JUN-07) y 8x y + 7 = 0 y y = 8x 7 y y + = 8x 7 (y ) = 8x 7 (y ) = 8x 7 + (y ) = 8x 6 (y ) = 8(x ) Y = 8X,q;F X = x, Y = y Y = 4()X a = gutisak; fpo;gwk; jpwg;gilaj X, Y I x, y I nghwj;j nghwj;j X = x, Y = y mr;r Y = 0 Y = 0 y = 0 Kid (0, 0) Ftpak; (a, 0) (, 0) X = 0, Y = 0 x = 0, y = 0 V(,) X =, Y = 0 x =, y = 0 F(4,) ways00@gmail.com

69 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top,af;ftiu nrt;tfyk; X = a X = X = a X = X = x = x = 0 X = x = x = 4 nrt;tfyj;jpd; epsk; 4a = 8 8 eps;tl;lj;jpd;,u tbtq;fs; x a + y b = x b + y a =. y + 4y + 4x + 8 = 0 vd;w gutisaj;jpd; mr;r> Kid> Ftpak;>,af;Ftiu> nrt;tfyj;jpd; rkd;ghl kw;wk; nrt;tfyj;jpd; epsk; fhz;f. NkYk; mt;tistiuia tiuf ( MAR- ) y + 4y = 4x 8 y + 4y + = 4x 8 (y + ) = 4x 8 (y + ) = 4x (y + ) = 4x 4 (y + ) = 4(x + ) Y = 4X,q;F X = x +, Y = y + a = gutisak; fpo;gwk; jpwg;gilaj X, Y I x, y I nghwj;j nghwj;j X = x +, Y = y + mr;r Y = 0 Y = 0 y + = 0 Kid (0, 0) Ftpak; ( a, 0) (, 0),af;Ftiu nrt;tfyk; X = a X = X = a X = X = 0, Y = 0 x + = 0, y + = 0 V(, ) X =, Y = 0 x + =, y + = 0 F(, ) X = x + = x = 0 X = x + = x + = 0 nrt;tfyj;jpd; epsk; 4a = 4 4 ikak; C(0,0) C(0,0) Kidfs; Ftpaq;fs; A(a, 0), A ( a, 0) F (ae, 0), F ( ae, 0) A(0, a) A (0, a) F (0, ae) F (0, ae). 6x + 4y 7x + y 44 = 0 vd;w eps;tl;lj;jpd; ikaj;njhiyj;jft> ikak;> Kidfs;> Ftpaq;fs; Mfpatw;iwf; fhz;f. kw;wk; mjd; tiuglk; tiuf ( MAR-06,JUN-06,OCT-5 ) 6x + 4y 7x + y 44 = 0 6x 7x + 4y + y 44 = 0 6(x x) + 4(y + 8y) 44 = 0 6(x x + ) +4(y + 8y ) 44 = 0 6*(x ) + + 4*(y + 4) 6+ = 44 6(x ) 6 + 4(y + 4) 64 = 44 6(x ) + 4(y + 4) = (x ) 44 (x ) 4 6(x ) + 4(y + 4) = (y+4) 44 + (y+4) 6 = = ways00@gmail.com

70 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top X 4 + Y 6 =,q;f X = x, Y = y + 4 nel;lr;r Y-mr;Rtopr; nry;fpwj e = b a = 6, b = 4, a = 6, b = = 4 = 6 4 = a = 6 = ae = 6 = 4 X, Y I nghwj;j ikak; (0, 0) Kidfs; Ftpaq;fs; (0, ±a) (0, ±6) (0, ±ae) (0, ±4 ) = x, y I nghwj;j X = x, Y = y + 4 X = 0, Y = 0 x = 0, y + 4 = 0 C(, 4) (0, a) (0,6) X = 0, Y = 6 x = 0, y + 4 = 6 A(,) (0, a) (0, 6) X = 0, Y = 6 x = 0, y + 4 = 6 A (, 0) (0, 4 ) X = 0, Y = 4 x = 0, y + 4 = 4 x =, y = F (, ) (0, 4 ) X = 0, Y = 4 x = 0, y + 4 = 4 x =, y = 4 4 F (, 4 4 ). x + 4y 8x 6y 68 = 0 vd;w eps;tl;lj;jpd; ikaj;njhiyj;jft> ikak;> Kidfs;> Ftpaq;fs; Mfpatw;iwf; fhz;f. kw;wk; mjd; tiuglk; tiuf (x 8x) + (4y 6y) 68 = 0 (x 8x ) + 4(y 4y + ) = 68 *(x 4) *(y ) 4+ = 68 (x 4) + 4(y ) = (x 4) + 4(y ) = 00 (x 4) (y ) 00 = (x 4) (y ) + = 00 5 nel;lr;r x- mr;rf;f,izahf cs;sj X = x 4, Y = y X + Y = 00 5 a = 00, b = 5, a = 0, b = 5 e = 5 = 00 5 = = = 5 = ae = 0 = 5 0 X, Y I nghwj;j ikak; (0,0) Ftpaq;fs; Kidfs; (±ae, 0) (±5, 0) (±a, 0) (±0,0) x, y I nghwj;j X = x 4, Y = y X = 0, Y = 0 x 4 = 0, y = 0 C(4,) 5, 0 X = 5, Y = 0 x 4 = 5, y = 0 x = 4 + 5, y = F (4 + 5, ) 5, 0 X = 5, Y = 0 x 4 = 5, y = 0 x = 4 5, y = F (4 5, ) (0, 0) X = 0, Y = 0 x 4 = 0, y = 0 x = 4, y = A(4,) ( 0,0) X = 0, Y = 0 x 4 = 0, y = 0 x = 6, y = A ( 6,) ways00@gmail.com

71 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top 4. 6x + 9y + x 6y = 9 vd;w eps;tl;lj;jpd; ikaj;njhiyj;jft> ikak;> Kidfs;> Ftpaq;fs; Mfpatw;iwf; fhz;f. kw;wk; mjd; tiuglk; tiuf (OCT-, MAR-, JUN-6) 6x + 9y + x 6y = 9 6x + x + 9y 6y = 9 6(x + x) + 9(y 4y) = 9 6(x + x + ) + 9(y 4y + ) = 9 X 6*(x + ) + + 9*(y ) 4+ = Y 6 6(x + ) 6 + 9(y ) 6 = 9 6(x + ) + 9(y ) = (x + ) 44 (x + ) 9 9(y ) + 44 = (y ) + = 6 =,q;f X = x +, Y = y nel;lr;r Y -mr;rf;f,iz e = b a = 6, b = 9, a = 4, b = = 9 = 6 9 = 7 = 7 a ae = = 7 X, Y I nghwj;j ikak; (0,0) Kidfs; (0, ±a) (0, ±4) x, y I nghwj;j X = x +, Y = y X = 0, Y = 0 x + = 0, y = 0 C(,) (0,4) X = 0, Y = 4 x + = 0, y = 4 A(,6) (0, 4) X = 0, Y = 4 x + = 0, y = 4 A (, ) Ftpaq;fs; (0, ±ae) (0, ± 7) (0, 7) X = 0, Y = 7 x + = 0, y = 7 x =, y = + 7 F (, + 7) (0, 7) X = 0, Y = 7 x + = 0, y = 7 x =, y = 7 F (, 7) 5. 6x + 9y x + 6y 9 = 0 vd;w eps;tl;lj;jpd; ikaj;njhiyj;jft> ikak;> Kidfs;> Ftpaq;fs; Mfpatw;iwf; fhz;f. kw;wk; mjd; tiuglk; tiuf ( OCT -6) 6x + 9y x + 6y = 9 6x x + 9y + 6y = 9 6(x x) + 9(y + 4y) = 9 6(x x + ) + 9(y + 4y + ) = 9 X 6*(x ) + + 9*(y + ) 4+ = Y 6 6(x ) 6 + 9(y + ) 6 = 9 6(x ) + 9(y + ) = (x ) 44 (x ) 9 9(y + ) + 44 = (y + ) + = 6 =,q;f X = x, Y = y + nel;lr;r Y -mr;rf;f,iz e = b a = 6, b = 9, a = 4, b = = 9 = 6 9 = 7 = 7 a ae = = 7 ways00@gmail.com

72 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top X, Y I nghwj;j ikak; (0,0) Kidfs; Ftpaq;fs; (0, ±a) (0, ±4) (0, ±ae) (0, ± 7) x, y I nghwj;j X = x, Y = y + X = 0, Y = 0 x = 0, y + = 0 C(, ) (0,4) X = 0, Y = 4 x = 0, y + = 4 A(,) (0, 4) X = 0, Y = 4 x = 0, y + = 4 A (, 6) (0, 7) X = 0, Y = 7 x = 0, y + = 7 x =, y = + 7 F (, + 7) (0, 7) X = 0, Y = 7 x = 0, y + = 7 x =, y = 7 F (, 7) 6. 9x + 5y 8x 00y 6 = 0 vd;w eps;tl;lj;jpd; ikaj;njhiyj;jft> ikak;> Kidfs;> Ftpaq;fs;> Mfpatw;iwf; fhz;f. kw;wk; mjd; tiuglk; tiuf (MAR-09) 9x + 5y 8x 00y 6 = 0 9x 8x + 5y 00y = 6 9(x x) + 5(y 4y) = 6 9(x x + ) +5(y 4y + ) = 6 9*(x ) + + 5*(y ) 4+ = 6 9(x ) 9 + 5(y ) 00 = 6 9(x ) + 5(y ) = 5 9(x ) + 5(y ) = 5 (x ) (y ) + = 5 9 X + Y =,q;f X = x, Y = y 5 9 nel;lr;r X -mr;r topna nry;fpwj a = 5, b = 9, a = 5, b = e = 9 = = = = ae = = 4 X, Y I nghwj;j ikak; (0,0) Ftpaq;fs; Kidfs; (±ae, 0) (±4, 0) (±a, 0) (±5,0) x, y I nghwj;j X = x, Y = y X = 0, Y = 0 x = 0, y = 0 C(,) (4, 0) X = 4, Y = 0 x = 4, y = 0 x = 5, y =, F (5,) ( 4, 0) X = 4, Y = 0 x = 4, y = 0 x =, y = F (,) (5, 0) X = 5, Y = 0 x = 5, y = 0 x = 6, y = A(6, ) ( 5,0) X = 5, Y = 0 x = 5, y = 0 x = 4, y = A ( 4,) ways00@gmail.com

73 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top mjpgutisaj;jpd;,u tbtq;fs; x a y b = x b y a = ikak; C(0,0) C(0,0) Kidfs; A(a, 0), A ( a, 0) Ftpaq;fs; F (ae, 0), F ( ae, 0) tiuglk; A(0, a) A (0, a) F (0, ae) F (0, ae) 7. 9x 6y 8x 64y 99 = 0 vd;w mjpgutisaj;jpd; ikaj; njhiyj;jft> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f NkYk; mjd; tistiuia tiuf. (OCT-) 9x 8x 6y 64y = 99 9(x x) 6(y + 4y) = 99 9(x x + ) 6(y + 4y + ) = 99 9{(x ) } 6*(y + ) 4+ = 99 9(x ) 6(y + ) = (x ) 6(y + ) = 44 9(x ) 6(y + ) = (x ) (y + ) = 6 9 X Y =,q;f X = x, Y = y FWf;fr;R X-mr;rpw;F,izahf cs;sj. a = 6, b = 9, a = 4, b = e = + b a = = = 5 6 = 5 4 ae = = 5 X, Y I nghwj;j x, y I nghwj;j X = x, Y = y + ikak; (0,0) X = 0, Y = 0 x = 0, y + = 0 C(, ) Ftpaq;fs; (±ae, 0) (5, 0) Kidfs; (±5, 0) X = 5, Y = 0 x = 5, y + = 0 x = 6, y = F (6, ) ( 5, 0) X = 5, Y = 0 x = 5, y + = 0 x = 4, y = F ( 4, ) (±a, 0) (±4,0) (4, 0) X = 4, Y = 0 x = 4, y + = 0 x = 5, y = A(5, ) ( 4,0) X = 4, Y = 0 x = 4, y + = 0 x =, y = A (, ) 8. 9x 6y + 6x + y + 64 = 0 vd;w mjpgutisaj;jpd; ikaj; ikak;> Ftpaq;fs;> cr;rpfs; njhiyj;jft> Mfpatw;iwf; fhz;f NkYk; mjd; tistiuia tiuf. ( JUN-5) 9x 6y + 6x + y + 64 = 0 9(x + 4x) 6(y y) = 64 9(x + 4x + ) 6(y y + ) = 64 9{(x + ) 4} 6*(y ) + = 64 9(x + ) 6(y ) = 44 6(y ) 9(x + ) = 44 6(y ) 9(x + ) = (y ) (x + ) = 9 6 Y X 9 6 =,q;f X = x +, Y = y FWf;fr;R Y-mr;rpw;F,izahf cs;sj. a = 9, b = 6, a =, b = 4 e = + b = + 6 = 6+9 a 9 9 = 5 9 = 5 ways00@gmail.com

74 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top X, Y I nghwj;j ikak; (0,0) Kidfs; Ftpaq;fs; (0, ±a) (0, ±) (0, ±ae) (0, ±5) ae = 5 = 5 x, y I nghwj;j X = x +, Y = y X = 0, Y = 0 x + = 0, y = 0 C(,) (0,) X = 0, Y = x + = 0, y = A(,4) (0, ) X = 0, Y = x + = 0, y = A(, ) (0, 5) X = 0, Y = 5 x + = 0, y = 5 x =, y = + 5 F (,6) (0, 5) X = 0, Y = 5 x + = 0, y = 5 x =, y = 5 F (, 4) X Y 4 =,q;f X = x +, Y = y FWf;fr;R X-mr;rpw;F,izahf cs;sj. a = 4, b =, a =, b = e = + b = + = 4+ = 5 a 4 4 ae = 5 = 5 X, Y I nghwj;j ikak; (0,0) Ftpaq;fs; Kidfs; (±ae, 0) (± 5, 0) (±a, 0) (±,0) x, y I nghwj;j X = x +, Y = y X = 0, Y = 0 x + = 0, y = 0 C(,) 5, 0 X = 5, Y = 0 x + = 5, y = 0 x = + 5, y = F ( + 5, ) 5, 0 X = 5, Y = 0 x + = 5, y = 0 x = 5, y = F ( 5, ) (, 0) X =, Y = 0 x + =, y = 0 x =, y = A(,) (,0) X =, Y = 0 x + =, y = 0 x = 5, y = A ( 5,) 9. x 4y + 6x + 6y = 0 vd;w mjpgutisaj;jpd; ikaj; njhiyj;jft> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f NkYk; mjd; tistiuia tiuf. (MAR-0,JUN-) x 4y + 6x + 6y = 0 x + 6x 4y + 6y = (x + 6x + ) 4(y 4y + ) = {(x + ) 9} 4*(y ) 4+ = (x + ) 4(y ) = 4 (x + ) (y ) = x y + 6x + 6y + 8 = 0 vd;w mjpgutisaj;jpd; ikaj; njhiyj;jft> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f NkYk; mjd; tistiuia tiuf. (MAR-08,OCT-08,OCT-09,JUN-0,MAR-,MAR-4) x y + 6x + 6y + 8 = 0 ways00@gmail.com

75 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top x + 6x y + 6y + 8 = 0 (x + 6x) (y y) = 8 (x + 6x + ) (y y + ) = 8 {(x + ) 9} *(y ) + = 8 (x + ) (y ) = (x + ) (y ) = (y ) (x + ) = Y X 4 (y ) (y ) 4 (x+) (x+) = = =,q;f X = x +, Y = y FWf;fr;R Y-mr;rpw;F,izahf cs;sj. a = 4, b =, e = + b ae = () = 4 a =, b = = + = 4+ a 4 4 X, Y I nghwj;j ikak; (0,0) Kidfs; Ftpaq;fs; (0, ±a) (0, ±) (0, ±ae) (0, ±4) = 6 4 = x, y I nghwj;j X = x +, Y = y X = 0, Y = 0 x + = 0, y = 0 C(,) (0,) X = 0, Y = x + = 0, y = A(,) (0, ) X = 0, Y = x + = 0, y = A(, ) (0, 4) X = 0, Y = 4 x + = 0, y = 4 x =, y = + 4 F (,5) (0, 4) X = 0, Y = 4 x + = 0, y = 4 x =, y = 4 F (, ). 9x 7y + 6x + 4y + 9 = 0 vd;w mjpgutisaj;jpd; ikaj; njhiyj;jft> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f NkYk; mjd; tistiuia tiuf. ( JUN- ) 9x 7y + 6x + 4y + 9 = 0 9x + 6x 7y + 4y + 9 = 0 9(x + 4x) 7(y y) = 9 9(x + 4x + ) 7(y y + ) = 9 9{(x + ) 4} 7*(y ) + = 9 9(x + ) 7(y ) = (x + ) 7(y ) = 6 7(y ) 9(x + ) = 6 Y X 9 7 7(y ) 9(x+) 6 6 (y ) (x+) 9 7 = 6 6 = =, X = x +, Y = y FWf;fr;R Y-mr;rpw;F,izahf cs;sj. e = + b a = 9, b = 7, a =, b = 7 a = = = 6 9 = 4 X, Y I nghwj;j ikak; (0,0) Kidfs; Ftpaq;fs; (0, ±a) (0, ±) (0, ±ae) (0, ±4) ae = 4 = 4 x, y I nghwj;j X = x +, Y = y X = 0, Y = 0 x + = 0, y = 0 C(,) (0,) X = 0, Y = x + = 0, y = A(,4) (0, ) X = 0, Y = x + = 0, y = A(, ) (0, 4) X = 0, Y = 4 ways00@gmail.com

76 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top x + = 0, y = 4 x =, y = + 4 F (,5) (0, 4) X = 0, Y = 4 x + = 0, y = 4 x =, y = 4 F (, ). 6x 9y x 6y 64 = 0 vd;w mjpgutisaj;jpd; ikaj; njhiyj;jft> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f NkYk; mjd; tiuglk; tiuf. ( JUN-4) 6x 9y x 6y 64 = 0 6x x 9y 6y 64 = 0 6(x x) 9(y + 4y) = 64 6(x x + ) 9(y + 4y + ) = 64 6{(x ) } 9*(y + ) 4+ = 64 6(x ) 9(y + ) = (x ) 9(y + ) = 44 6(x ) 9(y + ) = (x ) (y + ) = 9 6 X 9 Y 6 =,q;f X = x, Y = y + FWf;fr;R X-mr;rpw;F,izahf cs;sj. a = 9, b = 6, a =, b = 4 e = + b ae = 5 = 5 = + 6 = 6+9 a 9 9 X, Y I nghwj;j ikak; (0,0) Ftpaq;fs; (±ae, 0) (±5, 0) = 5 9 = 5 x, y I nghwj;j X = x, Y = y + X = 0, Y = 0 x = 0, y + = 0 C(, ) (5, 0) X = 5, Y = 0 Kidfs; (±a, 0) (±,0) x = 5, y + = 0 x = 6, y = F (6, ) ( 5, 0) X = 5, Y = 0 x = 5, y + = 0 x = 4, y = F ( 4, ) (, 0) X =, Y = 0 x =, y + = 0 x = 4, y = A(4, ) (,0) X =, Y = 0 x =, y + = 0 x =, y = A (, ). x 4y 4x + y 7 = 0 vd;w mjpgutisaj;jpd; ikaj; njhiyj;jft> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f NkYk; mjd; tistiuia tiuf. (OCT-07) x 4y 4x + y 7 = 0 x 4x 4y + y = 7 (x x) 4(y 8y) = 7 (x x + ) 4(y 8y ) = 7 {(x ) } 4*(y 4) 6+ = 7 (x ) 4(y 4) = (x ) 4(y 4) = 75 (x ) (y 4) =. 75 / / X Y. 75 / / =,q;f X = x, Y = y 4 FWf;fr;R X-mr;rpw;F,izahf cs;sj. a = 75, b = 75 4, e = + b 75 = + 4 = + = + = a 4 75 ways00@gmail.com

77 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top ae = 75 = 5 4 = 5 = x + y = 9 vd;w Neh;f;NfhL mjpgutisak; x 9y = 9 Ij; njhlfpwj vd ep&gpf;f. NkYk; njhlk; Gs;spiaAk; fhz;f. ( JUN-09,6, MAR-,OCT-, 4,6) X, Y I nghwj;j ikak; (0,0) Ftpaq;fs; Kidfs; (±ae, 0) (±5, 0) (±a, 0) ± 5, 0 x, y I nghwj;j X = x, Y = y 4 X = 0, Y = 0 x = 0, y 4 = 0 C(,4) (5, 0) X = 5, Y = 0 x = 5, y 4 = 0 x = 6, y = 4 F (6,4) ( 5, 0) X = 5, Y = 0 x = 5, y 4 = 0 x = 4, y = 4 F ( 4,4). 5, 0/ X = 5, Y = 0 x = 5, y 4 = 0 x = 7, y = 4 A 7, 4. 5, 0/ X = 5, Y = 0 x = 5, y 4 = 0 x =, y = 4 A., 4/ y = mx + c vd;w NfhL mjpgutisak; x y a b = f;f njhlnfhlhf,uf;f epge;jid c = a m b 5x + y = 9 y = 5x + 9 x 9y = 9 x y = 5x + 9 y = 5 x + 4 m = 5, c = 4 y 9 = a = 9, b = c = 9 6 ; a m b = / = 8 44 = 9 6 c = a m b mjpgutisaj;jpd; njhlnfhl;bd; rkd;ghl 5x + y = 9,J mjpgutisaj;ij njhlk; Gs;sp a m c b c njhlg;gs;sp.5, 4 /. a m c = = 5 = 4 = 4, b c / 5. x y + 4 = 0 vd;w Neh;f;NfhL eps;tl;lk; x + y = f;f njhlnfhlhf cs;sj vd ep&gpf;f. NkYk; njhlk; Gs;spiaAk; fhz;f. ( JUN-,MAR-6) ways00@gmail.com

78 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top y = mx + c vd;w NfhlhdJ x + y = f;f a b njhlnfhlhf,uf;f epge;jid c = a m + b x y + 4 = 0 m =, c = 4 y = x + 4 x + y = x 4 = a =, b = 4 c = 6 ; a m + b = () + 4 = 6 c = a m + b x y + 4 = 0 eps;tl;lj;jpd; njhlnfhl MFk; njhlg;gs;sp. a m c a m c, b c / = () 4 = 4 = b njhlg;gs;sp (,) c = 4 4 = 6. xu mjpgutisaj;jpd; ikak; (, 4) NkYk; mj (, 0) topnar; nry;fpwj.,jd; njhiyj;njhlnfhlfs; x + y = 0 kw;wk; x y + 8 = 0 Mfpatw;wpw;F,izahf,Uf;fpd;wd vdpy; mjpgutisaj;jpd; rkd;ghl;ilf; fhz;f. (MAR-06, 09,5, JUN-06,08,, OCT-5) njhiyj;njhlnfhlfspd;,izf;nfhlfs; x + y = 0 kw;wk; x y + 8 = 0 njhiyj;njhlnfhlfspd; rkd;ghlfspd; tbtk; x + y + l = 0 kw;wk; x y + m = 0,J mjpgutisaj;jpd; ikak; (,4) topahf nry;fpwj. vdnt, l = 0 l = m = 0 m = 6 njhiyj;njhlnfhlfspd; rkd;ghlfs; x + y 0 = 0 kw;wk; x y + 6 = 0 njhiyj;njhlnfhlfspd; Nrh;g;Gr; rkd;ghl (x + y 0)(x y + 6) = 0 mjpgutisaj;jpd; rkd;ghl;bd; tbtk; (x + y 0)(x y + 6) + k = 0,J (,0) topahf nry;fpwj ( 0)( + 6) + k = 0 k = 64 mjpgutisaj;jpd; rkd;ghl (x + y 0)(x y + 6) + 64 = 0 7. x + y 5 = 0 I xu njhiyj;njhl NfhlhfTk; (6, 0) kw;wk; (, 0)vd;w Gs;spfs; topna nry;yf;$bajkhd nrt;tf mjpgutisaj;jpd; rkd;ghl fhz;f. (OCT-06, 08,0,, MAR-07,08,, JUN-07) xu njhiyj; njhlnfhl x + y 5 = 0 kw;nwhu njhiyj; njhlnfhl;bd; tbtk; x y + k = 0 nrt;tf mjpgutisaj;jpd; rkd;ghl (x + y 5)(x y + k) + c = 0,J (6, 0) topahf nry;fpwj (6 5)( + k) + c = 0 k + c =.. () NkYk; (,0) topahftk; nry;fpwj. ( 5)( 6 + k) + c = 0 ( 8)( 6 + k) + c = k + c = 0 8k + c = 48 () () kw;wk; () I jph;f;f k = 4 kw;wk; c = 6 Njitahd nrt;tf mjpgutisaj;jpd; rkd;ghl (x + y 5)(x y + 4) 6 = 0 ways00@gmail.com

79 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top 6. tif Ez;fzpjk; - gad;ghlfs; - I. tifaplfisg; gad;glj;jp 4 y = kjpg;gfisf; fhz;f. y = f(x) = x x =, dx = x = 0.0 vd;f dy = x dx = () (0.0) f;f Njhuha (MAR-05) = (0.0) = f(x + x) y + dy = f() = (.0).0066 NkYk; y = f(x) = x 4,q;F x =, dx = x = 0.0 dy = 4 x 4dx = 4 () 4(0.0) = (0.0) = f(x + x) y + dy = f() = (.0) (.0) + (.0) u = sin x y x y vdpy; A+yhpd; Njw;wj;ijg; gad;glj;jp x u u + y = tan u vdf; fhl;lf x y (MAR-07, MAR-08, JUN-4) R.H.S. rkgbj;jhd rhh;g my;y. vdnt f = sin u = x y x y f(tx, ty) = tx ty = t(x y) = t (x y) tx ty t / ( x y) ( x y) = t (x y) ( x y) = t f f vd;gj> gb cila rkgbj;jhd rhh;g. A+yhpd; Njw;wj;jpd; gb x f f + y = f x y x (sin u) x (sin u) + y = (sin u) y x u u cos u + y cos u = (sin u) x y cos u.x u u + y / = (sin u) x y x u u + y x sin u = y cos u x u u + y = tan u x y. u = tan. x + y / vdpy; A+yhpd; Njw;wj;ijg; x y gad;glj;jp x u u + y = sin u vdf; fhl;lf. x y (OCT-09,, 5) u vd;gj rkgbj;jhd rhh;gy;y.mdhy; tan u vd;gj xu rkgbj;jhd rhh;ghfk;. f = tan u = x + y x y f(tx, ty) = (tx ) + (ty ) tx ty = t (x + y ) (x y) vd tiuawf;ftk; = t (x + y ) t(x y) = t f = t (x + y ) (x y) f vd;gj gb cila rkgbj;jhd rhh;ghfk; A+yhpd; Njw;wj;jpd; gb x (tan u) x x f f + y = f x y (tan u) + y = (tan u) y x. sec u u + y. x sec u u = tan u y sec u.x u x x u u tan u + y = x y = u + y / = tan u y sec u = sin u cos u cos u sin u cos u cos u = sin u cos u = sin u 4. u = sin x+y x+ y x u u + y = x y ep&gpf;f vdpy; x+y x+ y sin u = x+y cos x+ y (JUN-) u = sin. x+y x+ y / x+y x+ y = f(x, y) vd ways00@gmail.com

80 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top x = tx, y = ty f(tx, ty) = tx +ty tx + ty = t(x+y) t( x+ y ) = t = t (x+y) x+ y (x+y) ( x+ y ) = t f f d; gb, A+yhpd; Njw;wj;jpd; gb x x f f + y = nf x y x ( sin u) + y ( sin u) = n( sin u) x y u + y u = u x u y sin sin. x+y / x+ y u 5. f(x, y) = rhpghh;f;f f(tx, ty) = u u.x + y / =. x+y / x y x+ y x u u + y =. x+y / x y x+ y u =. x+y x+ y / sin. x+y x+ y / =. x+y x+ y / cos. x+y x+ y / =. x+y x+y / cos. / x+ y x+ y x +y = t f;f A+yhpd; Njw;wj;ij (JUN-06,OCT-0,MAR-4) = t x +t y t (x +y ) (x +y ) = t = t f(x, y) f(x, y) f vd;gj cila rkgbj;jhd rhh;g. vdnt A+yhpd; Njw;wj;jpd;gb rhpghh;j;jy;: f x = x f f + y x y = f x (x +y ) = x (x +y ) x f x = x (x +y ),ijg; Nghy; f y = y (x +y ) y f y = y (x +y ) = y (x +y ) x f f + y = x x y (x +y ) y (x +y ) = x +y (x +y ) = A+yhpd; Njw;wk; rhpghh;f;fg;gl;lj. x +y = f 6. u = tan. x / vdpy; u = u vd;gij y x y y x rhpghh;f;f. (MAR-0) u = tan. x y / u = = x +. x y / y = y y +x = = + x y y +x y y y y x +y u = x. y +. x y / y / = = y +x y = x x +y y /. x + x y. x y y / =. x y +x y / u = x y x. u/ =. x y x x +y / = y y +x y = 0 (x +y ). (x).x (x +y ) = 0 x +y x (x +y ) = 0 x +y (x +y ) = 0 x y (x +y )..() u = y x y. u/ = x () kw;wk; () ypue;j. y y x +y / = 0 (x +y ). (y).y (x +y ) = 0 x +y y (x +y ) = 0 x y (x +y ).() u = u x y y x ways00@gmail.com

81 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top 7. u = x y y x vd;w rhh;gf;f u x y = u y x vd;gij rhpghh;f;f (JUN-, MAR-7) u = x y y x u = x y y( )x = + y y x u = x. y ( )y = x x y x u = x y x. u/ = x. y x y x / = ( ) y x = x y.. () u = y x y. u/ =. + y x y y x / = y + x = x y.() () kw;wk; () ypue;j u x y = u y x 8. u = x y y x vd;w rhh;gf;f u = u vd;gjid rhpghh;f;f. (OCT - 4) x y y x u = x y y x u = x ( )y = x + y x y x y x u = ( )x (y) = x 4y y y x y x u = x y x. u/ = x. 4y / y x y x = x ( )4y = x + 4y y x y x = 4y x x y.() u = y x y. u/ = x y.x + y / y x = ( )x y + (y) x = x y + 4y x = 4y x x y () () kw;wk; () ypue;j u = u x y y x 9. u = sin. x y / vd;dk; rhh;gf;f u x y = u y x vd;gij rhpghh;f;f (MAR-) u = sin. x y / u x = y cos.x y / u = x cos y y.x/ y u = x y x. u/ = x. cos y x y.x// y = y x. sin x y y + cos x y () = y. x y. sin.x y / + cos.x y //.. () u = y x y. u/ = x y y cos.x y / = y. sin.x y /. x y // + cos.x y /. y / = y x y sin.x y / + cos.x y / ( ) () kw;wk; () ypue;j u = u x y y x 0. u = sin x cos 4y vd;w rhh;gf;f u = u x y y x u = sin x cos 4y u x u y vd;gij rhpghh;f;f (MAR-06) = cos 4y. cos x. = cos x cos 4y = sin x( sin 4y). 4 = 4sin x sin 4y u = x y x. u/ = ( 4sin x sin 4y) y x = 4 sin 4y cos x. = cos x sin 4y.. () u = y x y. u/ = (cos x cos 4y) x y () kw;wk; () ypue;j = cos x sin 4y. 4 = cos x sin 4y () u = u x y y x ways00@gmail.com

82 Mk; tfg;g fzf;f ntw;wpf;f top. y = x + vd;fpw tistiuia tiuf. y = x vd;fpw tistiuia tiuf. y = x vd;fpw tistiuia tiuf ( JUN-09, OCT-, JUN-6, OCT-6) (OCT-06, JUN-07,OCT-07, JUN-08, OCT-08, JUN-0, MAR-, MAR-) (MAR-06, MAR-09, JUN-, OCT-,JUN-5) rhh;gfk; epl;bg;g ntl;lj; Jz;Lfs; Mjp rkr;rph; Nrhjid njhiyj; njhl NfhLfs; Xhpay;G jd;ik x-,d; vy;yh nka; kjpg;gfsf;fk; f(x) MdJ tiuawf;fg;glfpwj. vdnt f(x),d; rhh;gfk; (, ) vd;fpw KO,ilntsp. fpilkl;l epl;bg;g < x < epiyf;fj;j epl;bg;g < y < x = 0 vdpy; y = y = 0 vdpy; x = tistiuahdj Mjp topr; nry;yhj tistiuahdj rkr;rph; jd;ikia ngwtpy;iy tistiuf;f ve;j xu njhiyj;njhlnfhlfsk;,y;iy. vy;yh x f;fk; y 0 Mjyhy;> tistiuahdj (, ) KOtJkhf VWKfkhf nry;yk; x-,d; vy;yh nka; kjpg;gfsf;fk; f(x) MdJ tiuawf;fg;glfpwj. vdnt f(x),d; rhh;gfk; (, ) vd;fpw KO,ilntsp. fpilkl;l epl;bg;g < x < epiyf;fj;j epl;bg;g < y < x = 0, vdpy; y = 0 y = 0, vdpy; x = 0 tistiuahdj Mjp topr; nry;yk; tistiuahdj Mjpia nghwj;j rkr;rpuhdj tistiuf;f ve;j xu njhiyj;njhlnfhlfsk;,y;iy. vy;yh x f;fk; y 0 Mjyhy;> tistiuahdj (, ) KOtJkhf VWKfkhf nry;yk; x 0 vd,uf;fk; NghJ y ed;f tiuawf;fg;gl;ls;sj. [0, ) fpilkl;l epl;bg;g 0 x < epiyf;fj;j epl;bg;g < y < x = 0, vdpy; y = 0 y = 0, vdpy; x = 0 tistiuahdj Mjp topr; nry;yk; tistiuahdj x mr;ir nghwj;j rkr;rpuhdj. tistiuf;f ve;j xu njhiyj;njhlnfhlfsk;,y;iy. y = x vd;w fpisapy; tistiu VWKfkhf,Uf;Fk;. y = x vd;w fpisapy; tistiu,wq;f Kfkhf,Uf;Fk;. rpwg;gg; Gs;spfs; (, 0) vd;w,ilntspapy; fpo;nehf;fp FopthfTk; kw;wk; (0, ) vd;w,ilntspapy; Nky; Nehf;fp FopthfTk;,Uf;Fk;. (0, ) vd;gj tist khw;wg; Gs;sp (, 0) vd;w,ilntspapy; fpo;nehf;fp FopthfTk; kw;wk; (0, ) vd;w,ilntspapy; Nky; Nehf;fp FopthfTk;,Uf;Fk;. (0, 0) vd;gj tist khw;wg; Gs;sp (0,0) vd;gj tist khw;wg; Gs;spay;y. tiuglk; ways00@gmail.com

83 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top 8. tiff;nfor;rkd;ghlfs;. xu,urhad tpistpy;> xu nghus; khw;wk; milak; khw tpjkhdj t Neuj;jpy; khw;wkilahj mg;nghuspd; mstpw;f tpfpjkhf cs;sj. xu kzp Neu Kbtpy; 60 fpuhkk; kw;wk; 4 kzp Neu Kbtpy; fpuhkk; kpjkpue;jhy;> Muk;g epiyapy; > mg;nghuspd; vilapidf; fhz;f. (MAR-, OCT-5) t vd;w Neuj;jpy; nghuspd;,ug;g A vd;f da da αa = ka A = cekt dt dt t = vdpy; A = 60 ce k = 60.. () t = 4 vdpy; A = ce 4k =. () () c 4 e 4k = 60 4 () () () c = 604 c = 85.5 (klf;ifiag; gad;glj;jp) Muk;gj;jpy; t = 0tpy; A = c = 85.5 fpuhk; (Njhuhakhf) Muk;gj;jpy; nghuspd; vil 85.5 fpuhk; (Njhuhakhf).. xu tq;fpahdj njhlh; $l;l Kiwapy; tl;biaf; fzf;fplfpwj. mjhtj tl;b tpjj;ij me;je;j Neuj;jpy; mrypd; khw tpjj;jpy; fzf;fplfpwj. xutuj tq;fp,ug;gpy; njhlh;r;rpahd $l;l tl;b %yk; Mz;nlhd;Wf;F 8 % tl;b nguffpwj vdpy;> mtuj tq;fpapug;gpd; xu tul fhy mjpfhpg;gpd; rjtpjj;ijf; fzf;fplf. [ e vlj;jf; nfhs;f]. (OCT-07) t vdk; Neuj;jpy; mry; A vd;f da da da αa = ka = 0.08 A,,q;F k = 0.08 dt dt dt A(t) = ce 0.08t xu tul mjpfhpg;g rjtpjk; = A() A(0) 00 A(0) = A() c. e = 00 A(0) c = 8.% vdnt xu Mz;by; mjpfhpf;fk; rjtpjk; = 8.%. xu,we;jth; cliy kuj;jth; ghpnrhjpf;fk; NghJ>,we;j Neuj;ij Njhuhakhf fzf;fpl Ntz;bAs;sJ.,we;jthpd; clypd; ntg;gepiy fhiy 0.00 kzpastpy; 9. 4 o F vd Fwpj;Jf; nfhs;fpwhh;. NkYk; kzp Neuk; fopj;j ntg;g epiy msit 9. 4 o F vdf; fhz;fpwhh;. miwapd; ntg;gepiy mst (epiyahdj) 7 o F vdpy;>,we;j Neuj;ij fzf;fplf. (xu kdpj clypd; rhjhuz c\;z epiy o F vdf; nfhs;f) 9.4 [log e = kw;wk; log e = ].4 (JUN-,JUN-6) t vd;w Neuj;jpy; clypd; ntg;gepiyapid T vd;f epa+l;ldpd; Fsph;r;rp tpjpg;gb dt α(t 7) [Vnddpy; S = dt 7o F ] dt = k(t 7) T 7 = cekt dt my;yj T = 7 + ce kt t = 0Mf,Uf;Fk; NghJ, T = = 7 + ce k(0) 9.4 = 7 + c c = c =.4 [Kjypy; Fwpf;fg;gl;l Neuk; fhiy 0 kzp vd;gj t = 0 vd;f ] /T = 7 +.4e kt [Njhuhaj;jpd; Jy;ypaj;jd;ikia mjpfhpf;f kzpahdj epkplkhf vlj;jf; nfhs;sg;glfpwj] t = 0 vdpy;, T = 9.4 e 0k = k = 0 log 9.4 e.4 = 0 ( ) t vd;gj,we;j Neuj;jpw;Fg; gpd; fhiy 0 kzpf;f cs;shd Neuk; vd;f t = t vdk; NghJ T = = 7 +.4e kt t = k log e = = 66 epkplk; Kjy; mstplhd fhiy 0 kzpf;f Kd;djhf 4 kzp 6 epkplk;.,we;j Neuk; Njhuhakhf ways00@gmail.com

84 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top 0.00 kzp 4 kzp 6 epkplk;.,we;j Neuk; Njhuhakhf 5.4 A.M. 4. Ez;Zaph;fspd; nguf;fj;jpy;> ghf;bhpahtpd; nguf;ftpjkhdj mjpy; fhzg;glk; ghf;bhpahtpd; vz;zpf;if tpfpjkhf mike;js;sj.,g;nguf;fj;jhy; ghf;bhpahtpd; vz;zpf;if kzp Neuj;jpy; Kk;klq;fhfpwJ vdpy; Ie;J kzp Neu Kbtpy; ghf;bhpahtpd; vz;zpf;if Muk;g epiyiaf; fhl;byk; 5 klq;fhfk; vdf; fhl;lf. (JUN-06, MAR-09, OCT-) t Neuj;jpy; ghf;bhpahf;fspd; vz;zpf;if A vd;f. da da αa = ka A = cekt dt dt Muk;gj;jpy; t = 0 vdk; NghJ A = A 0 /A 0 = ce o = c /A = A 0 e kt t = vdk;nghj A = A 0 A 0 = A 0 e k e k = t = 5vDk; NghJ A = A 0 e 5k = A 0 (e k ) 5 = 5. A 0 / 5 kzp Neu Kbtpy; ghf;bhpaq;fspd; vz;zpf;if 5 klq;fhfk;. 5. Nubak; rpijak; khwtpjkhdj> mjpy; fhzg;glk; mstpw;f tpfpjkhf mike;js;sj. 50 tulq;fspy; Muk;g mstpypue;j 5 rjtpjk; rpije;jpuf;fpwj vdpy; 00 tul Kbtpy; kpjpapuf;fk; mst vd;d? [A 0 I Muk;g mst vdf; nfhs;f] (MAR-06, JUN-09,MAR-0,OCT-, OCT-6) t vd;w Neuj;jpy; Nubaj;jpd; mst A vd;f A = A(t) da da αa = ka A = cekt dt dt t = 0 vdpy; A = A 0 / A 0 = ce 0 = c / A = A 0 e kt 50 tulq;fspy; Nubak; Muk;g epiyapypue;j 5 % rpijtwfpwj. t = 50 vdpy;, A =.95 A 0 / 0.95 A 0 = A 0 e 50k e 50k = 0.95 NkYk; t = 00 vdpy; A = A 0 e 00k = A 0 e 50k = A 0 (.95) = 0.905A 0 00 tul Kbtpy; kpjpapuf;fk; Nubaj;jpd; mst 0.905A 0 6. &.000 vd;w njhiff;f njhlh;r;rp $l;l tl;b fzf;fplg;glfpwj tl;b tpjk; Mz;nlhd;Wf;F 4 rjtpjkhf,ug;gpd;> mj;njhif vj;jid Mz;Lfspy; Muk;gj; njhifiag; Nghy;,U klq;fhfk;? (log e = 0. 69). (MAR-5,JUN-07,08,,OCT-06,0) t vd;w Neuj;jpy; mry; A vd;f da da da α A = ka = 0.04 t,,q;f k = 0.04 dt dt dt A = c. e.04t t = 0 vdpy; A = = ce 0 c = 000 / A = 000e.04t A = 000 Mf,Uf;Fk; NghJ t If; fhz;f 000 = 000e.04t t = log = 0.69 = 7 tulq;fs; (Njhuhakhf) C ntg;gepiy cs;s xu miwapy; itf;fg;gl;ls;s NjePhpd; ntg;gepiy 00 0 C MFk;. mj 5 epkplq;fspy; 60 0 C Mf Fiwe;J tplfpwj. NkYk; 5 epkplk; fopj;j NjePhpd; ntg;gepiyapidf; fhz;f. (OCT-09, OCT-, JUN-5, MAR-7) t vd;w Neuj;jpy; NjePhpd; ntg;gepiy T vd;f epa+l;ldpd; Fsph;r;rp tpjpg;gb dt dt α(t S) = k(t S) dt dt (T S) = ce kt T = 5 + ce kt,,q;f S = 5 0 C t = 0 vdk;nghj T = = 5 + ce k(0) 00 5 = c c = 85 T = e kt t = 5 vdk; NghJ T = = e 5k e 5k = t = 0 vdk; NghJ T = e 0k 45 = = ways00@gmail.com

85 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top 8. xu efuj;jpy; cs;s kf;fs; njhifapd; tsh;r;rptpjk; me;neuj;jpy; cs;s kf;fs; njhiff;f tpfpjkhf mike;js;sj. 960 Mk; Mz;by; kf;fs; njhif,0,000 vdtk; 990,y; kf;fs; njhif,60,000 MfTk;,Ug;gpd; 00 Mk; Mz;by; kf;fs; njhif vt;tsthf,uf;fk;? 0log e 0 6 =. 070; e.4 =. 5 (MAR-08, JUN-0, MAR-4, JUN-4) t vdk; Neuj;jpy; kf;fs; njhif A vd;f da da αa = ka A = cekt dt dt 960k; Mz;L kf;fs; njhifapid njhlf;f kf;fs; njhifahff; nfhs;f. t = 0 tpy; 990 Mk; Mz;by; A = = ce 0 = c A = 0000e kt t = 0 tpy;, A = 60,000 /60,000 = 0000 e 0k e 0k = 6 00 k; Mz;by; A If; fhz t = 60 apy; A = 0000 e 60k = 0, ~ y; kf;fs;njhif Njhuhakhf xu fjphpaf;fg; nghus; rpijak; khwtpjkhdj> mjd; vilf;f tpfpjkhf mike;js;sj. mjd; vil 0 kp.fpuhk; Mf,Uf;Fk;NghJ rpijak; khwtpjk; ehnshd;wf;f 0.05 kp.fpuhk; vdpy; mjd; vil 0 fpuhkpypue;j 5 fpuhkhff; Fiwa vlj;jf; nfhs;sk; fhy msitf; fhz;f. (log e = 0. 69) (MAR-, JUN-,OCT-4) t vdk; Neuj;jpy; fjphpaf;fg; nghuspd; vil A da dt α A da dt = ka A = cekt t = 0 vdpy; A = 0 c = 0 A = 0e kt kwgbak; da dt = ka A = 0vd,Uf;Fk; NghJ da = 0.05 vdf; dt nfhlf;fg;gl;ls;sj. [rpijtwtjhy;] 0.05 = 0k k = / A = 0e A = 5 vdk; NghJ t If; fhz 5 = 0e 0.005t = e 0.005t = e 0.005t log = 0.005t t = log ~6 ehl;fs; xu Nehahspapd; rpwephpypue;j Ntjpg;nghUs; ntspnawk; mstpid njhlh;r;rpahf Nfj;Njlh; vd;w futpapd; %yk; fz;fhzpf;fg;glfpwj. t = 0 vd;w Neuj;jpy; Nehaspf;F 0kp.fpuhk; Ntjpg;nghUs; nfhlf;fg;glfpwj.,j t / kp.fpuhk; / kzp vd;dk; tpjj;jpy; ntspnawfpwj vdpy; (i) Neuk; t > 0 vdk; NghJ> Nehahspapd; clypys;s Ntjpg;nghUspd; msitf; fhzk; nghjr; rkd;ghl vd;d? (ii) KOikahf Ntjpg;nghUs; ntspnaw vlj;jf; nfhs;sk; Fiwe;jgl;r fhy mst vd;d? (i) A vd;w Neuj;jpy; Ntjpg;nghUspd; vil t vd;f Ntjpg;nghUs; ntspnawk; tpjk; t da dt = t A = t + c t = 0 vdpy;, A = 0 c = 0 t vdk; Neuj;jpy; A = 0 t (ii) A = 0 vdpy;> Ntjpg;nghUs; ntspnawptpl;lj vdg; nghus; 0 = 0 t 5 = t t = 5 t =.9 kzp. vdnt Nehahspapd; clypue;j.9 kzp KOikahf my;yj kzp 54 epkplj;jpy; Ntjpg;nghUs; KOikahf ntspnawk;. ways00@gmail.com

86 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top 9. jdpepiyf; fzf;fpay;. (Z, ) xu Kbtw;w vgpypad; Fyk; vdf; fhl;lf.,q;f MdJ a b = a + b + vdkhw tiuawf;fg;gl;ls;sj. (OCT-08,JUN-0,MAR-6) (i) milg;g tpjp: a, b kw;wk; KO vz;fs; Mjyhy; a + b + k; xu KO vz; a b z, a, b z milg;g tpjp cz;ikahfk;. (ii) Nrh;g;G tpjp: a, b, c z vd;f (a b) c = (a + b + ) c = (a + b + ) + c + = a + b + c + 4 a (b c) = a (b + c + ) = a + (b + c + ) + = a + b + c + 4 (a b) c = a (b c) Nrh;g;G tpjp cz;ikahfk; (iii) rkdp tpjp: e rkdp cwg;g vd;f. e d; tiuaiwapypue;j a e = a,d; tiuaiwapypue;j a e = a + e + a + e + = a e = Z. rkdp tpjp cz;ikahfk;. (iv) vjph;kiw tpjp: a G vd;f. a d; vjph;kiw a vdf; nfhz;lhy; a,d; tiuaiwg;gb, a a = e =,d; tiuaiwg;gb, a a = a + a + a + a + = a = a 4 a 4 Z. vjph;kiw tpjp cz;ikahfk; (Z, ) xu FykhFk;. (v) ghpkhw;wg; gz;g: a, b G vd;f a b = a + b + = b + a + = b a ghpkhw;w tpjpf;fl;lgl;lj. (Z, ) xu vgpypad; FykhFk;. NkYk; Z xu Kbtw;w fzkhjyhy;,f;fyk; Kbtw;w vgpypad; FykhFk;. x x.. /, x x x R {0} vd;w mikg;gpy; cs;s mzpfs; ahtk; mlq;fpa fzk; G MdJ mzpg;nguf;fypd; fpo; xu Fyk; vdf; fhl;lf (JUN-, MAR-5) G =. x x / / x R *0+ x x vd;f. mzpg;nguf;fypd; fpo; G xu Fyk; vd fhl;lnthk; (i) milg;g tpjp: A =. x x y / G, B =.y x x y y / G AB = xy xy xy xy G, ( x 0, y 0 xy 0) G MdJ mzpg;nguf;fypd; fpo; milt ngw;ws;sj. (ii) mzpg;nguf;fy; vg;nghojnk Nrh;g;G tpjpf;fl;glk;. (iii) E =. e e / G vd;gjae = A, A G vd;f e e AE = A. x x e x /.e / =.x x x e e x x / xe xe x. / =.x xe xe x x / xe = x e = ( x 0) / / vdnt> E = G vd;gj / / AE = A, A G vdkhw cs;sj,nj Nghy; EA = A, A G vdf; fhl;lyhk; G,d; rkdp cwg;g E MFk;. vdnt rkdp tpjp cz;ikahfk;. (iv) A =. y y y y / G vd;gj A A = E xy xy / /,t;thwhapd; = xy xy / / xy = y = 4x A = 4x 4x 4x 4x G vd;gj A A = E vdkhw cs;sj.,nj Nghy; AA = E A d; vjph;kiw A MFk; mzpg;nguf;fypd; fpo; G xu FykhFk;. ways00@gmail.com

87 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top. Ij; jtpu kw;w vy;yh tpfpjkw vz;fsk; mlq;fpa fzk; G vd;f. G y; I a b = a + b ab a, b G vdkhw tiuawg;nghk;. (G, ) xu Kbtw;w vgpypad; Fyk; vdf;fhl;lf. (JUN-08,5, MAR-) G = Q { } vd;f a, b G. a kw;wk; b tpfpjkw vz;fs; a, b (i) milg;g tpjp: a b = a + b ab xu tpfpjkw vz; MFk;. a b G vdf; fhl;ltjw;f a b vd ep&gpf;f Ntz;Lk;. khwhf a b = vdf; nfhz;lhy; a + b ab = b ab = a b( a) = a b = ( a a 0 ),J rhj;jpakpy;iy. Vnddpy; b. / ekj jw;nfhs; jtwhdj. / a b vdnt a b G / milg;g tpjp cz;ikahfk;. (ii) Nrh;g;G tpjp: a (b c) = a (b + c bc) = a + (b + c bc) a(b + c bc) = a + b + c bc ab ac + abc (a b) c = (a + b ab) c = a + b ab + c (a + b + ab)c = a + b + c ab ac bc + abc / a (b c) = (a b) c, a, b, c G / Nrh;g;G tpjp cz;ikahfk;. (iii) rkdp tpjp: e vd;gj rkdp cwg;g vd; d; tiuaiwg;gb, a e = a + e ae e d; tiuaiwg;gb, a e = a a + e ae = a e( a) = 0 e = 0 Vnddpy; a e = 0 G / rkdp tpjp G+h;j;jpahfpwJ. (iv) vjph;kiw tpjp: a G,d; vjph;kiw a vd;f. vjph;kiwapd; tiuaiwg;gb a a = e = 0 d; tiuaiwg;gb, a a = a + a aa a = a a a + a aa = 0 a ( a) = a G, Vnddpy; a / vjph;kiw tpjp G+h;j;jpahFk;. / (G,*) xu FykhFk;. (v) ghpkhw;w tpjp: a, b G f;f a b = a + b ab = b + a ba = b a / G,y; * ghpkhw;w tpjpf;fl;glfpwj. vdnt (G, ) xu vgpypad; FykhFk;. G KbTw;wjhjyhy; (G, ) KbTw;w vgpypad; FykhFk;. 4. G+r;rpakw;w fyg;ngz;fspd; fzkhd C *0+,y; tiuawf;fg;gl;l f (z) = z, f (z) = z, f (z) =, f z 4 (z) = z C {0} vd;w z rhh;gfs; ahtk; mlq;fpa fzk; {f, f, f, f 4 } MdJ rhh;gfspd; Nrh;g;gpd; fpo; xu vgpypad; Fyk; mikf;fk; vd epwtf. (OCT-06,09,5) G = {f, f, f, f 4 } vd;f (f f )(z) = f f (z) = f (z) f f = f, f f = f, f f = f, f 4 f = f 4 NkYk; (f f )(z) = f f (z) = f ( z) = ( z) = z = f (z) f f = f,njnghy;> f f = f 4, f f 4 = f (f f )(z) = f f (z) = f ( z) = z = f 4 (z) f f = f 4,NjNghy;> f f = f, f f 4 = f (f 4 f )(z) = f 4 f (z) = f 4 ( z) = z = z = f (z) f 4 f = f,nj Nghy; f 4 f = f, f 4 f 4 = f Nkw;fz;ltw;iw gad;glj;jp nguf;fy; ml;ltizia mikf;f f f f f 4 f f f f f 4 f f f f 4 f f f f 4 f f f 4 f 4 f f f nguf;fy; ml;ltizapypue;j (i) ml;ltizapd; vy;yh cwg;gfsk; G,d; cwg;gfshjyhy;> milg;g tpjp cz;ikahfk; ways00@gmail.com

88 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top (ii) rhh;gfspd; Nrh;g;G nghjthf Nrh;g;G tpjpf;fl;glk;. (iii) G d; rkdp cwg;g f MFk;. vdnt rkdp tpjp cz;ikahfpwj. (iv) ml;ltizapypue;j f,d; vjph;kiw f ; f,d; vjph;kiw f f,d; vjph;kiw f ; f 4,d; vjph;kiw f 4 vjph;kiw tpjp cz;ikahfpwj. (G, ) xu FykhFk; (v) ml;ltizapypue;j ghpkhw;w tpjp cz;ikahfpwj. (G, ) xu vgpypad; FykhFk;. 5. (Z n, + n ) xu Fyk; vdf; fhl;lf. (JUN-, MAR-4) Z n = {,0-,,-,,-, [n ]} vd;gj n,d; kl;lf;f fhzg;ngw;w rh;trkj; njhfg;gfs; vd;f.,l-,,m- Z n 0 l, m < n vd;f (i) milg;g tpjp: tiuaiwg;gb>,l + m-, l + m < n vdpy;,l-+ n,m- =,r-, l + m n vdpy;,q;f l + m = q. n + r 0 r < n,u epiyfspyk;,,l + m- Z n kw;wk; [r] Z n milg;g tpjp cz;ikahfk; (ii) n,d; kl;lf;fhpa $l;ly; vg;nghojk; Nrh;g;G tpjpf;fl;glk;. Mjyhy; Nrh;g;G tpjp cz;ikahfk;. (iii) rkdp cwg;g [0] Z n mj rkdp tpjpiag; G+h;j;jp nra;fpwj. (iv) [l] Z n,d; vjph;kiw [n l] Z n,l-+ n,n l- = [0],n l-+ n,l- = [0] vjph;kiw tpjp cz;ikahfk;. (Z n, + n ) xu FykhFk;. 6. (Z 7 *,0-+,. 7 ) xu Fyj;ij mikf;fk; vdf; fhl;lf. (MAR-0, OCT-6) G = {,-,,-, [6]} vd;f Nfa;yp ml;ltizahdj. 7 [] [] [] [4] [5] [6] [] [] [] [] [4] [5] [6] [] [] [4] [6] [] [] [5] [] [] [6] [] [5] [] [4] [4] [4] [] [5] [] [6] [] [5] [5] [] [] [6] [4] [] [6] [6] [5] [4] [] [] [] ml;ltizapypue;j: (i) nguf;fy; ml;ltizapd; vy;yh cwg;gfsk; G,d; cwg;gfshfk; milg;g tpjp cz;ikahfk;. (ii) 7-d; kl;lf;fhd nguf;fy;> Nrh;g;G tpjpf;fl;glk;. (iii) rkdpawg;g [] G kw;wk;,j rkdp tpjpiag; G+h;j;jp nra;ak;. (iv) [],d; vjph;kiw []; [],d; vjph;kiw [4]; [],d; vjph;kiw [5];[4],d; vjph;kiw []; [5],d; vjph;kiw []; [6],d; vjph;kiw [6] vdnt vjph;kiw tpjp G+h;j;jpahfpwJ. (Z 7 *,0-+,. 7 ) xu Fyj;ij mikf;fk; 7. tof;fkhd nguf;fypd; fpo;,d; nmk; gb %yq;fs; Kbthd Fyj;ij mikf;fk; vdf;fhl;lf (MAR-),d; nmk; gb %yq;fshtd vd;f, ω, ω ω n G = {, ω, ω ω n } vd;f.,q;f ω = cis π n (i) milg;g tpjp: ω l, ω m G, 0 l, m (n ) ω l ω m = ω l+m G vd ep&gpf;f Ntz;Lk; epiy (i) l + m < n vd;f l + m < n vdpy; ω l+m G epiy(ii) l + m n vd;f tfj;jy; Nfhl;ghl;bd;gb> l + m = (q. n) + r, 0 r < n, kpif KO vz;. ω l+m = ω qn +r = (ω n ) q. ω r = () q. ω r = ω r G 0 r < n milg;g tpjp cz;ikahfk;. (ii) Nrh;g;G tpjp: fyg;ngz;fspd; fzj;jpy; nguf;fyhdj vg;nghojk; Nrh;g;G tpjpia cz;ikahf;fk;. ω l. (ω p. ω m ) = ω l. ω (p+m) = ω l+(p+m) = ω (l+p)+m = ω l. ω p. ω m ω l, ω p, ω m G (iii) rkdp tpjp: rkdp cwg;g G kw;wk; mj. ω l = ω l. = ω l ω l G vd;gij G+h;j;jp nra;fpwj. (iv) vjph;kiw tpjp: ω l G f;f ω n l G kw;wk; ways00@gmail.com

89 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top ω l. ω n l = ω n l. ω l = ω n =,t;thwhf vjph;kiw tpjp cz;ikahfpwj. (G,. ) xu FykhFk;. (v) ghpkhw;w tpjp: ω l. ω m = ω l+m = ω m +l = ω m. ω l ω l, ω m G (G,. ) xu vgpypad; FykhFk;. G,y; n cwg;gfs; kl;lnk cs;sjhy; (G,. )MdJ n thpir nfhz;;l Kbthd vgpypad; FykhFk;. 8.,d; kl;lf;f fhzg;ngw;w nguf;fypd; fpo; *,-,,-,,4-,,5-,,9-+ vd;w fzk; xu vgpypad; Fyj;ij mikf;fk; vdf; fhl;lf. (MAR-07, JUN-09) G = *,-,,-,,4-,,5-,,9-+ vd;f nfa;ypapd; ml;ltizahdj.,-,-,4-,5-,9-,-,-,-,4-,5-,9-,-,-,9-,-,4-,5-,4-,4-,-,5-,9-,-,5-,5-,4-,9-,-,-,9-,9-,5-,-,-,4- Nkw;fz;l ml;ltizapypue;j (i) nguf;fy; ml;ltizapd; vy;yh cwg;gfsk; G,d; cwg;gfshfk;. / milg;g tpjp cz;ikahfk;. (ii),d; kl;lf;fhd nguf;fy; vg;nghojk; Nrh;g;G tpjpf;fl;glk;. (iii) rkdpawg;g,- G (iv),-,d; vjph;kiw [] [],d; vjph;kiw [4] [4],d; vjph;kiw [] [5],d; vjph;kiw [9] [9],d; vjph;kiw,5- / vdnt jug;gl;l fzk; f;fl;gl;l fhzg;ngw;w nguf;fypd; fpo; xu Fyj;ij mikf;fk;. (v) ml;ltizapd; %yk; ghpkhw;w tpjpak; cz;ikahfpwj. / vdnt xu vgpypad; FykhFk;. 9. G vd;gj kpif tpfpjkw vz;fspd; fzk; vd;f. a b = ab vdkhw tiuawf;fg;gl;l nrayp,d; fpo; G xu Fyj;ij mikf;fk; vdf;fhl;lf. (MAR-06, JUN-06, OCT-07,0,,4) (i) milg;g tpjp: a, b G vd;f. a, b xu kpif tpfpjkw vz;fs; Mjyhy; ab k; xu kpif tpfpjkw vz;. vdnt ab k; xu kpif tpfpjkw vz; ab G, a b G milg;g tpjp cz;ik. (ii) Nrh;g;G tpjp: a, b, c G vd;f (a b) c = ab c =.ab /c = abc 9 a (b c) = a bc = a.bc / = abc 9 (a b) c = a (b c) vdnt Nrh;g;G tpjp cz;ikahfk;. (iii) rkdp tpjp: rkdpawg;g e vd;f. e,d; tiuaiwg;gb, a e = a,d; tiuaiwg;gb, a e = ae ae = a e = e = G vdnt rkdp tpjp cz;ikahfk;. (iv) vjph;kiw tpjp: a G vd;f. a,d; vjph;kiw a a,d; tiuaiwg;gb, a a = e =,d; tiuaiwg;gb, a a = aa aa = a a = 9 a = 9 a G vdnt vjph;kiw tpjp cz;ikahfk;. (G, ) xu FykhFk; , 0ω 0 0 ω, ω 0, 00, 0 ω 0 ω, 0 0 ω 0 ω 0 ω 0 vd;fpw fzk; mzpg;nguf;fypd; fpo; xu Fyj;ij mikf;fk; vdf;fhl;lf. (ω = ), (JUN-,MAR-,7) I = 0 0 0, A = 0ω 0 0 ω, B = ω 0 0 ω, C = 0 0, D = 0 ω 0 ω, E = 0 0 ω 0 ω 0 G = {I, A, B, C, D, E} vd;f,t;tzpfis,uz;l,uz;lhf vlj;j nguf;fp> nguf;fy; ml;ltizia gpd;tukhw mikf;fyhk;: ways00@gmail.com

90 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top I A B C D E I I A B C D E A A B I E C D B B I A D E C C C D E I A B D D E C B I A E E C D A B I (i) nguf;fy; ml;ltizapy; cs;s vy;yh cwg;gfsk; G,d; cwg;gfshfk;. vdnt G MdJ mzpg;nguf;fypd; fpo; milt ngw;ws;sj. mjhtj Nrh;g;G tpjp cz;ikahfk;. (ii) nghjthf> mzp nguf;fy; Nrh;g;G tpjpf;fl;glkhjyhy;,q;f. MdJ Nrh;g;G tpjpf;fl;glk;. (iii) ml;ltizapypue;j njspthf I MdJ rkdp cwg;g MFk;. (iv) I. I = I I,d; vjph;kiw I A. B = B. A = I A Ak; B Ak; xd;wf;nfhd;w vjph;kiwahfk;. C. C = I C,d; vjph;kiw C D. D = I D,d; vjph;kiw D. E. E = I E,d; vjph;kiw E vdnt mzpg;nguf;fypd; fpo; GxU FykhFk;. z = vdkhw cs;s fyg;ngz;fs; ahtk; mlq;fpa fzk; M MdJ fyg;ngz;fspd; nguf;fypd; fpo; xu Fyj;ij mikf;fk; vdf; fhl;lf. (OCT-,JUN-4) M = *z C/ z = + (i) milg;g tpjp: z, z M vd;f z z = z z =. = z, z M / milg;g tpjp cz;ikahfk;. (ii) Nrh;g;G tpjp: fyg;ngz;fspd; nguf;fy; vg;nghojk; Nrh;g;G tpjpf;fl;glk; z. (z. z ) = (z. z ). z (iii) rkdp tpjp: xt;nthu z M f;fk; = C I vdkhw fhzyhk; z. =. z = z rkdp cwg;g. vdnt rkdp tpjp cz;ikahfk;. (iv) vjph;kiw tpjp: z M vd;f z =,q;f z = z = = z M kw;wk; z. z = z. z = / z,d; vjph;kiw z M vdnt vjph;kiw tpjp cz;ikahfk;. / vdnt fyg;ngz;fs; nguf;fypd; fpo; M xu FykhFk;.. I jtpu kw;w vy;yh tpfpjkw vz;fsk; cs;slf;fpa fzk; G MdJ a b = a + b + ab vdkhw tiuawf;fg;gl;l nrayp *,d; fpo; xu vgpypad; Fyj;ij mikf;fk; vdf; fhl;lf. (JUN-07, MAR-09,JUN-6) G = Q { } vd;f. a, b G vd;f. vdnt a kw;wk; b tpfpjkw vz;fs; a, b. (i) milg;g tpjp: a b = a + b + ab vd;gj xu tpfpjkw vz;. a b G vd epwt a b > vdtk; fhl;l Ntz;Lk;. khwhf a b = vdf; nfhs;nthk;.,t;thwhapd; a + b + ab = b + ab = a b( + a) = ( + a) b = ( a + a 0) Mdhy; b vdnt>,j rhj;jpakpy;iy. / vdnt> ek; jw;nfhs; jtwhdjhfk;. / a b a b G / milg;g tpjp cz;ikahfk;. (ii) Nrh;g;G tpjp: a (b c) = a (b + c + bc) = a + (b + c + bc) + a(b + c + bc) = a + b + c + bc + ab + ac + abc) (a b) c = (a + b + ab) c = a + b + ab + c + (a + b + ab)c = a + b + c + ab + ac + bc + abc / a (b c) = (a b) c, a, b, c G / Nrh;g;G tpjp cz;ikahfk;. (iii) rkdp tpjp: e vd;gj rkdp cwg;g vd;f.,d; tiuaiwg;gb, a e = a + e + ae e,d; tiuaiwg;gb, a e = a a + e + ae = a e( + a) = 0 e = 0 a Mjyhy; e = 0 G / rkdp tpjp G+h;j;jpahfpwJ. ways00@gmail.com

91 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top (iv) vjph;kiw tpjp: a G,d; vjph;kiw a vd;f vjph;kiwapd; tiuaiwg;gb> a a = e = 0,d; tiuaiwg;gb a a = a + a + aa a + a + aa = 0 a ( + a) = a a = a G, [Vnddpy; a ] +a / vjph;kiw tpjp G+h;j;jpahfpwJ. / (G,*) xu FykhFk;. (v) ghpkhw;w tpjp: VNjDk; a, b G f;f a b = a + b + ab = b + a + ba = b a / G,y; * ghpkhw;w tpjpf;fl;gltjhy; (G, ) xu vgpypad; FykhFk;. a o.. /, a R *0+ mikg;gpy; cs;s vy;yh o o mzpfsk; mlq;fpa fzk; mzpg;nguf;fypd; fpo; xu vgpypad; Fyj;ij mikf;fk; vdf; fhl;lf. (MAR-08) a o G vd;gj. /, a R *0+ vd;w o o mikg;gpy; cs;s mzpfs; ahtk; mlq;fpa fzk; vd;f. (i) milg;g tpjp: A =. a o o / G, B =.b o o o o / G AB =. ab o / G( a 0, b 0 ab 0) o o G MdJ mzpg;nguf;fypd; fpo; miltg; ngw;ws;sj. (ii) Nrh;g;G tpjp: mzpg; nguf;fyhdj vg;nghojk; Nrh;g;G tpjpf;fl;glk;. (iii) rkdp tpjp: E =. e o / G MdJ xt;nthu A G f;fk; o o AE = A vd;gjhf mikfpwj vdf; nfhs;f. AE = A. a o o o /.e / =.a o o o o o o /. ae o o / =.a o o o o / ae = a e = vdnt> E =. o / G MdJ xt;nthu o o A G f;fk; AE = A vd;gjhf mikfpwj.,nj Nghy; xt;nthu A G f;fk; EA = A vdf; fhl;lyhk; vdnt G,y; E MdJ rkdp cwg;g MFk; vdnt rkdp tpjp cz;ikahfpwj. (iv) vjph;kiw tpjp: A =. x o o o / G MdJ A A = E x o o o. /.a / =. o o o o o o / xa o o. / =. o o o o / xa = x = a A =. /a o o o / G MdJ A A = E vdkhw cs;sj.,nj Nghy; AA = E vdtk; fhl;lyhk;. vdnt A MdJ A,d; vjph;kiw MFk;. vdnt vjph;kiw tpjp cz;ikahfpwj. mzpg;nguf;fypd; fpo; G xu FykhFk; (v) ghpkhw;w tpjp: A, B G AB =. ab o o / =.ba o o o o / = BA mzpg;nguf;fypd; fpo; G xu vgpypad; FykhFk;. 4. G = { n / n Z} vd;w fzkhdj nguf;fypd; fpo; xu vgpypad; Fyj;ij mikf;fk; vdf; fhl;lf. (OCT-) G = { n / n Z} (i) milg;g tpjp: x = r, y = s G, r, s Z vd;f. xy = r. s = r+s G ( r, s Z r + s Z) milg;g tpjp cz;ikahfk;. ways00@gmail.com

92 k; tfg;g fzf;f gj;j kjpg;ngz; tpdhf;fs; ntw;wpf;f top (ii) Nrh;g;G tpjp: x = r, y = s, z = t G, r, s, t Z vd;f (x. y). z = ( r. s ) t = r+s. t = (r+s)+t = r+(s+t) = r ( s. t ) = x. (y. z) Nrh;g;G tpjp cz;ikahfk;. (iii) rkdp tpjp: xt;nthu x = r G f;fk; = 0 G MdJ x. = r. = r = x vdkhwk;. x =. r = r = x vdkhwk; cs;sjhy; MdJ rkdp cwg;g MFk;. vdnt rkdp tpjp cz;ikahfk;. (iv) vjph;kiw tpjp: xt;nthu x = r G f;fk; x = r G MdJ x. x = r. r = r+( r) = 0 = vdtk; x. x = r. r = ( r)+r = 0 = vdtk; cs;sj r,d; vjph;kiw r G,t;thwhf vjph;kiw tpjpak; cz;ikahfpwj. vdnt> G xu FykhFk;. (v) ghpkhw;w tpjp: x = r, y = s G x. y = r. s = r+s = s+r = s r = y. x (Vnddpy; Z,y; $l;ly; ghpkhw;w tpjpf;fl;glk; ) (G,. ) xu vgpypad; FykhFk;. ways00@gmail.com

93