+2 (HSC) Maths Tamil Medium Study Materials 84 Pages

1

Weightage to Content: XIIXIIXIIXII Mathematics

Chapter

No Name

Marks Total

Marks Sec-A Sec-B Sec-C

1 mzpfs; kw;Wk; mzpf; Nfhitfspd; mzpfs; kw;Wk; mzpf; Nfhitfspd; mzpfs; kw;Wk; mzpf; Nfhitfspd; mzpfs; kw;Wk; mzpf; Nfhitfspd; gad;ghLfs;gad;ghLfs;gad;ghLfs;gad;ghLfs; 4 12 10 26

2 ntf;lH ,aw;fzpjk;ntf;lH ,aw;fzpjk;ntf;lH ,aw;fzpjk;ntf;lH ,aw;fzpjk; 6 12 20 38

3 fyg;ngz;fßfyg;ngz;fßfyg;ngz;fßfyg;ngz;fß 4 12 10 26

4 gFKiw tbtfzpjk;gFKiw tbtfzpjk;gFKiw tbtfzpjk;gFKiw tbtfzpjk; 4 6 30 40

5 tif Ez;;fzpjtif Ez;;fzpjtif Ez;;fzpjtif Ez;;fzpjk; gad;ghLfs; k; gad;ghLfs; k; gad;ghLfs; k; gad;ghLfs; ---- I 4 12 20 36

6 tif Ez;;fzpjk; gad;ghLfs;tif Ez;;fzpjk; gad;ghLfs;tif Ez;;fzpjk; gad;ghLfs;tif Ez;;fzpjk; gad;ghLfs;---- II 2 6 10 18

7 njif Ez;;fzpjk; : gad;ghLfs;njif Ez;;fzpjk; : gad;ghLfs;njif Ez;;fzpjk; : gad;ghLfs;njif Ez;;fzpjk; : gad;ghLfs; 4 6 20 30

8 tif nfOr; rkd;ghLfstif nfOr; rkd;ghLfstif nfOr; rkd;ghLfstif nfOr; rkd;ghLfs 4 6 20 30

9 jdp epiyf; fzf;fpay;jdp epiyf; fzf;fpay;jdp epiyf; fzf;fpay;jdp epiyf; fzf;fpay; 4 12 10 26

10 epfo;jfTg; guty;fs;epfo;jfTg; guty;fs;epfo;jfTg; guty;fs;epfo;jfTg; guty;fs; 4 12 10 26

Total 40 96 160 296

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xU kjpg;ngΩ tpdhf;fß (30xU kjpg;ngΩ tpdhf;fß (30xU kjpg;ngΩ tpdhf;fß (30xU kjpg;ngΩ tpdhf;fß (30----kjpg;ngΩfß)kjpg;ngΩfß)kjpg;ngΩfß)kjpg;ngΩfß)

njhFjp njhFjp njhFjp njhFjp ----

1. mzpfs; kw;Wk; mzpf; Nfhitfspd; gad;ghLfs1. mzpfs; kw;Wk; mzpf; Nfhitfspd; gad;ghLfs1. mzpfs; kw;Wk; mzpf; Nfhitfspd; gad;ghLfs1. mzpfs; kw;Wk; mzpf; Nfhitfspd; gad;ghLfs;;(3;(3;(3;(3----tpdhf;fß)tpdhf;fß)tpdhf;fß)tpdhf;fß)

1) A vd;w mzpapd; thpir 3 vdpy; det(KA) vd;gJ

(1) k3 det (A) (2) k2 det (A) (3) k det (A) (4) det (A)

2)

−

−

−

λλ

λ

01

10

01

vd;w mzpapd; juk; 2 vdpy;> λ tpd; kjpg;G fhz;f?

(1) 1 (2) 2 (3) 3 (4) vnjDk; xh; nka;vz;

3) myF mzp I ,d; thpir n, k≠ 0 xU khwpyp vdpy; adj(kI)=

(1) kn (adj I) (2) k (adj I) (3) k2 (adj (I) ) (4) kn -1 (adj I)

4) A= vdpy;, AAT ,d; juk; fhz;f. (1) 1 (2) 2 (3) 3 (4) 0

5)

−

−

541

31

231

k vd;w mzpf;F Neh;khW cz;L vdpy; k-,d; kjpg;Gfs;

(1) k VNjDk; xU nka;naz; (2) k= -4 (3) k ≠ -4 (4) k ≠ 4

6) A,B vd;w VNjDk; ,U mzpfSf;F AB = 0 vd;W ,Ue;J NkYk; A xU G+r;rpakw;w Nfhit mzp vzpy;

(1) (2) xU G+r;rpaf; Nfhit mzp

(3) xU G+r;rpakw;w Nfhit mzp (4)

7) Mfpa rkd;ghLfspd; njhFg;ghdJ xU

ntspg;gilaw;w jPh;it ngw;wpUg;gpd; a−1

1+

b−1

1+

c−1

1 =

(1) 1 (2) 2 (3) -1 (4) 0

8) A=

3

2

1

vdpy; AAT , d; juk; fhz;f.

(1) 3 (2) 0 (3) 1 (4) 2

9)

−

−

0

4

0

2

1

vd;w %iy tpl;l mzpapd; juk; fhz;f?

(1) 0 (2) 2 (3) 3 (4) 5

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10) A=

50

00vdpy;> A12 vd;gJ

(1)

600

00 (2)

1250

00 (3)

00

00 (4)

10

01

11) xU jpirapyp mzpapd; thpir 3> jpirapyp k ≠ o vdpy;> A-1 vd;gJ

(1) Ik2

1 (2) I

k 3

1 (3) I

k

1 (4) KI

12) vd;gj∂ Neh;khW

(1)

−

−

35

12 (2)

−

−

31

52 3)

−−

−

35

13 4)

−

−

21

53

13)

001

010

100

vd;w mzpapd; Neh;khW

(1)

100

010

001

(2)

− 001

010

100

(3)

001

010

100

(4)

−

−

100

010

001

14) kjpg;gpl Ntz;ba %d;W khwpfspy; mike;j %d;W Nehpa mrkgbj;jhdrkd;ghl;Lj; njhFg;gpy; ∆ = 0 kw;Wk; = 0, ≠ 0, = 0 vdpy;> njhFg;Gf;fhdj; jPh;T.

(1) xNu xu jPH;T (2) ,uz;L jPh;Tfs;

3) vz;zpf;iff;ifaw;w jPh;Tfs; (4) jPh;T ,y;yhik 15) kw;Wk;

∆1= qp

ba; ∆ 2=

qd

bc; ∆ 3=

dp

ca; vdpy; ,d; kjpg;G

(1)

∆

∆

∆

∆

1,

1

32 (2)

∆

∆

∆

∆

1

3

1

2 log,log (3)

∆

∆

∆

∆

2

1

3

1 log,log (4)

∆

∆

∆

∆

3

1

2

1 log,log

16)

−

−

−

844

422

211

vd;w mzpapd; juk; fhz;f?

(1) 1 (2) 2 (3) 3 (4) 4

17) xU rJu mzp A,d; thpir n vdpy; | adj A| vd;gJ

(1)| A|2 (2) | A|n (3)| A|n -1 (4) | A|

18) –2x + y + z = 1; x --2y + z = m; x + y -- 2z = n vd;w rkd;ghLfs; l + m + n = 0 vDkhW mikAkhapd; mj;njhFg;gpd; jPh;T

(1) xNu xU G+r;rpakw;w jPH;T (2) ntspg;gilahdj; jPh;T

(3) vz;zpf;iff;ifaw;w jPh;T (4) jPh;T ,y;yhik ngw;W ,Uf;Fk;

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19) A=

43

12 vd;w mzpf;F (adj A)A =

(1)

5

10

05

1

(2)

10

01 (3)

− 50

05 (4)

50

05

2. ntf;lH ,aw;fzpjk;(52. ntf;lH ,aw;fzpjk;(52. ntf;lH ,aw;fzpjk;(52. ntf;lH ,aw;fzpjk;(5----tpdhf;fß)tpdhf;fß)tpdhf;fß)tpdhf;fß)

20) vd;w Nfhsj;jpd; ikak; kw;Wk; Muk;

21) kw;Wk;

vd;w NfhLfs; ntl;bf; nfhs;Sk; Gs;sp

22) , vdpy;>

xU xuyF ntf;lh;

23) f;F ,izahfTk; (1>3>5) Gs;sp topahfTk; nry;yf;$ba Nfhl;bd;

ntf;lh; rkd;ghL

24) , vdpy; f;Fk; f;Fk; ,ilg;gl;l Nfhzk;

(1) (2) (3) (4)

25) (2> 1> -1) vd;w Gs;sp topahfTk; jsq;fs

ntl;bf; nfhs;Sk; Nfhl;il cs;slf;fpaJkhd jsj;jpd; rkd;ghL

26) 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; ,d; kjpg;G

27) kw;Wk; vd;w ,iz NfhLfSf;fpilNa cs;s kpf Fiwe;j

njhiyT

28) xU G+r;rpakw;w ntf;luhfTk; m xU G+r;rpakw;w jpirapypahfTk; ,Ug;gpd; m MdJ xuyF ntf;lh; vdpy;>

(1) (2) l l (3) ||

1

m (4)

29) vd;w ntf;liu xU %iy tpl;lkhfTk; [ xU gf;fkhfTk; nfhΩl ,izfuj;jpd; gug;G

30) kw;Wk; vd;w NfhLfs; ntl;bf; nfhs;Sk; Gs;sp

(1) (0> 0> -4) (2) (1> 0> 0) (3) (0> 2> 0) (4) (1> 2> 0)

31) ,d; kjpg;G

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(1) 0 (2) 1 (3) 2 (4) 4

32) (2>10>1) vd;w Gs;spf;Fk vd;w jsj;jpw;Fk; ,ilg;gl;l kpff; Fiwe;j

Juk; (1) (2) (3) (4)

33) , vdpy;>

-k; -k; ,izahFk;

(4) my;yJ my;yJ -k; -k; ,izahFk;

34) kw;Wk; vd;w NfhLfSf;fpilNa cs;s kpff; Fiwe;j njhiyT

35) kw;Wk; ,uz;L xuyF ntf;lh; kw;Wk; θ vd;gJ mtw;wpw;F ,ilg;gl;l Nfhzk; MdJ xuyF ntf;luhapd;>

36) , Mfpa ntf;lh;fs; nrq;Fj;J ntf;lh;fshapd;>

37) vd;w rkd;ghL Fwpg;gJ

(1) kw;Wk; Gs;spfis ,izf;Fk; Neh;f;NfhL (2) xoy jsk;

(3) yoz jsk; (4) zox jsk; 38) kw;Wk; Mfpait vz;zsT λ nfhz;l ntf;lh;fshapd; | | MdJ

39) vdpy;> ehw;fuk; PQRS ,d; gug;G

(1) (2) (3) (4)

40) kw;Wk; Mfpa ,UNfhLfSk;

(1) ,iz (2) ntl;bf; nfhs;git

(3) xU jsk; mikahjit (4) nrq;Fj;J

41) vdpy; ,d; kjpg;G

42) vDk; tpir vDk; Gs;sptopNar; nray;gLfpwJ vDk; Gs;spiag;

nghWj;J mjd; jpUg;Gj; jpwdpd; msT vdpy; a ,d; kjpg;G

43) vd;gd xU jsk; mikah ntf;lh;fs; NkYk;

vdpy; ,d; kjpg;G

44) xU NfhL x kw;Wk; y mr;RfSld; kpif jpirapy; 45 0 ,60 0 Nfhzq;fis Vw;gLj;JfpwJ

vdpy; z mr;Rld; mJ cz;lhf;Fk; Nfhzk;.

(1) 30 0 (2) 90 0 (3) 45 0 (4) 60 0

45) vd;w NfhLk; vd;w jsKk; ntl;bf;nfhs;Sk;

Gs;sp (1) (8> 6> 22) (2) (-8> -6> -22) (3) (4> 3> 11) (4) (-4> -3> -11)

46) vdpy; ,d; kjpg;G

(1) 32 (2) 8 (3) 128 (4) 0

47) vd;gJ

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(1) , , kw;Wk; f;F nrq;Fj;J

(2) kw;Wk; vd;w ntf;lh;fSf;F ,iz

(3) [ nfhz;l jsKk; [ nfhz;l jsKk; ntl;bf; nfhs;Sk; Nfhl;bw;F ,iz

(4) [ nfhz;l jsKk; [ nfhz;l jsKk; ntl;bf; nfhs;Sk; Nfhl;bw;Fr;nrq;Fj;J

48) vd;w tpir xU Jfis A(3> 3> 3) vDk; epiyapypUe;J B(4> 4> 4)

vDk; epiyf;F efh;j;jpdhy; mt;tpir nra;Ak; NtiyasT (1) 2 myFfs; (2) 3 myFfs; (3) 4 myFfs; (4) 7 myFfs

49) kw;Wk; vdpy; f;Fk; f;Fk; nrq;Fj;jhf cs;s xU xuyF ntf;lh;

50) f;Fk; f;Fk; ,ilg;gl;l Nfhzk; 120 0 NkYk; mtw;wpd; vz;zsTfs; KiwNa

vdpy; MdJ

51) ,d; kPJ ,d; tPoy; kw;Wk; ,d;kPJ ,d; tPoYk; rkkhapd; kw;Wk; f;F ,ilg; gl;l Nfhzk;

52) vd;w xU jskw;w ntf;lh;fSf;F vdpy;

(1) MdJ -f;F ,iz (2) MdJ -f;F ,iz

(3) MdJ -f;F ,iz (4)

53) vd;w myF ntf;lh; kPjhd ,d; tPoyhdJ OPRQ vd;w ,izfuj;jpd;

gug;ig Nghd;W Kk;klq;fhapd; POQ MdJ

54) vdpy;

(1) - k; - k; ,izahFk; (2) - k; - k; nrq;Fj;jhFk;

(3) | | = | | (4) kw;Wk; xuyF ntf;lh;

3. fyg;ngz;fß(33. fyg;ngz;fß(33. fyg;ngz;fß(33. fyg;ngz;fß(3----tpdhf;fß)tpdhf;fß)tpdhf;fß)tpdhf;fß)

55) vd;w rkd;ghl;bd; xU %yk; vdpy; k ,d; kjpg;G

56) ω vd;gJ 1 ,d; Kg;gb %yk; vdpy;(1-ω )(1-ω 2 )(1-ω 4 )(1-ω 8 ) ,d;; kjpg;G

57) vdpy; ,d; kjpg;G 1)

3)

58) a= 3 + i kw;Wk; z = 2 – 3i vdpy; cs;s az, 3az kw;Wk; – az vd;gd xU Mh;fd; jsj;jpy;

(1) nrq;Nfhz Kf;Nfhzj;jpd; Kidg;Gs;spfs; (2) rkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs; (3) ,U rkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs; (4) xNu Nfhliktd

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59) x 2 + y 2 = 1 vd;y; iyx

iyx

−+++

1

1 -d; kjpg;G

60) xU fyg;ngz;zpd; tPr;R 2

π vdpy; me;j vz;

(1) Kw;wpYk; fw;gid vz; (2) Kw;wpYk; nka; vz;

(3) 0 (4) nka;Aky;y fw;gidAky;y

61) a = cosα -- i sin α , b = cos β -- i sin β , c = cos γ -- i sin γ vdpy; vd;gJ

(1) cos2(α - β +γ )+ i sin2(α - β +γ ) (2) –2cos((α - β +γ )

(3) - 2i sin (α - β +γ ) (4) 2cos(α - β +γ )

62) [ ]34/3 πie − vd;w fyg;ngz;zpd; kl;L tPr;R KiwNa

63) vdpy; z 1 z 2 ….. z 6 vd;gJ

64) 2 + i 3 vd;w fyg;ngz;zpd; kl;L

(1) 3 (2) 13 (3) 7 (4) 7

65) vd;gJ ax 2 -bx + c = 0 vd;w rkd;ghl;bd; xU %yk; vdpy; kw;nwhU jPh;T

(1) – i –2 (2) i - 2 (2) 2 + i (2) 2 i + i

66) 4-3i kw;Wk; 4+3i vd;w %yq;fisf; nfhz;l rkd;ghL

2 2 2 2

67) ,d; ,iz fyg;ngz; (1) 1 (2) -1 (3) 0 (4) – i

68) ax 2 +bx+1 =0 vd;w rkd;ghl;bd; xU jPh;T a Ak; b Ak; nka; vdpy; (a,b) vd;gJ

69) - z %d;whk; fhy;gFjpapy; mike;jhy; z mikAk; fhy; gFjp

(1) Kjy; fhy;gFjp (2) ,uz;lhk; fhy;gFjp

(3) %d;whk; fhy;gFjp (4) ehd;fhk;; fhy;gFjp

70) ± i 7 vd;w jPh;Tfisf; nfhz;l ,Ugbr; rkd;ghL

(1) x 2 +7=0 (2) x 2 -7=0 (3) x 2 +x+7=0 (4) x 2 -x-7=0

71) ω vd;gJ 1 ,d; Kg;gb %yk; vdpy;(1-ω +ω 2 ) 4 +(1+ω -- ω 2 ) 4 ,d; kjpg;G (1) 0 (2) 32 (3) -16 (4)-32 72) ,d; kjpg;G vd;gJ 1) i 2) –i 3) 1 4) –1

73)

+−2

31 i 100

+

−−2

31 i 100

,d; kjpg;G

(1) 2 (2) 0 (3) -1 (4) 1 74) vd;w fyngz;zpd; ,iz vz; vdpy; vd;gJ

(1)

−− 82

1 (2)

− 8,2

1 (3)

−8,2

1 (4)

8,

2

1

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75) fy;gngz; ,d; Nghyhh; tbtk;

(1) cos2

π+ i sin

2

π (2) cosπ + i sinπ (3) cosπ -- i sinπ (4) cos

2

π -- i sin

2

π

76) (1) cosθ + i sin θ (2) cosθ - i sin θ (3) sinθ - i cosθ (4) sinθ + i

cosθ

77) p MdJ fyg;G vz; khwp z-If; Fwpf;fpd;wJ| 2z --1|= 2| z|vdpy; P ,d; epakg;ghij

(1) x =4

1 vd;w Neh;NfhL (2) y =

4

1 vd;w Neh;NfhL

(3) z =2

1 vd;w Neh;NfhL (4) x 2 +y 2 -- 4x--1= 0 vd;w tl;lk;

78) fyg;ngz; jsj;jp;y; 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

(1) z 1 + z 4 = z 2 + z 3 (2) z 1 + z 3 = z 2 + z 4

(2) z 1 + z 2 = z 3 + z 4 (4) z 1 – z 2 = z 3 – z 4

79) z xU fyg;ngz;izf; Fwpg;gnjdpy; arg (z) + arg ( z ) vd;gJ

80) x = cosθ + i sinθ vdpy; x n +nx

1 ,d; kjpg;G

(1) 2 cos nθ (2) 2 i sin nθ (3) 2 sin nθ (4) 2i cos nθ

81) z 1 = 4 + 5 i , z 2 = --3 + 2 i vdpy;> vd;gJ

82) iz vd;w fyg;ngz;iz Mjpiag; nghWj;J 2

π Nfhzj;jpy; fbfhu vjph;jpirapy;

Row;Wk;NghJ me;j vz;zpd; Gjpa epiy

(1) iz (2) – iz (3) –z (4) z

83) ω vd;gJ 1 ,d; n Mk; gb %yk; vdpy;

(1) 1+ω 2 +ω 4 +….=ω +ω 3 +ω 5 + ….. (2) ω n =0

(3) ω n =1 (4) ω = ω 1−n

4. gFKiw tbtfzpjk;(34. gFKiw tbtfzpjk;(34. gFKiw tbtfzpjk;(34. gFKiw tbtfzpjk;(3----tpdhf;fß)tpdhf;fß)tpdhf;fß)tpdhf;fß) 84) 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 (1) 36 (2) 18 (3) 72 (4) 144 85) vd;w Neh;NfhL vd;w ePs;tl;lj;jpd; njhLNfhL vdpy;

c d; kjp;g;G

(1) ± 2 3 (2) ± 6 (3) 36 (4) ± 4 86) xy = 32 vd;w nrt;tf mjpgutisaj;jpd; nrt;tfj;jpd; ePsk;

(1) 8 2 (2) 32 (3) 8 (4) 16

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87) 12

2

2

2

=−b

y

a

x vd;w mjpgutisaj;jpd; kPJs;s vNjDk; xU Gs;spapypUe;J

Ftpaj;jpw;F ,ilNaAs;s njhiyTfspy; tpj;jpahrk; 24 kw;Wk; ikaj; njhiyTj;jfT 2 vdpy; mjpgutisaj;jpd; rkd;ghL

(1) 1432144

22

=−yx

(2) 1144432

22

=−yx

(3) 131212

22

=−yx

(4) 112312

22

=−yx

88) vd;w mjpgutisaj;jpd; ikaj; njhiyT jfT

(1) 4 (2) 3 (3) 2 (4) 6

89) 9x 2 +16y 2 =144 vd;w $k;G tistpd; ,af;F tl;lj;jpd; Muk;

(1) 7 (2) 4 (3) 3 (4) 5

90) (8, 0) vd;w Gs;spapypUe;J 13664

22

=−yx

vd;w mjpgutisaj;jpd; njhiyj;

njhLNfhLfSf;F tiuag;gLk; nrq;Fj;J Juq;fspd; ngUf;fy; gyd; (1) 25/576 (2) 576/25 (3) 6/25 (4) 25/6

91) 16x 2 + 25y 2 = 400 vd;w tistiuapd; Ftpaj;jpypUe;J xU njhLNfhl;Lf;F tiuag;gLk; nrq;Fj;Jf; NfhLfspd; mbapd; epakg;ghij

(1) x 2 +y 2 =4 (2) x 2 +y 2 =25 (3) x 2 +y 2 =16 (4) x 2 +y 2 =9

92) y 2 -4x+4y+8=0 vd;w gutisaj;jpd; nrt;tfyj;jpd; ePsk; (1) 8 2) 6 3) 4 4) 2 93) xy = 18 vd;w nrt;tf mjpgutisaj;jpd; xU Ftpak; (1) (6, 6) (2) (3, 3) (3) (4, 4) (4) (5, 5)

94) 1916

2

=−yx

vd;w mjpgutisaj;jpw;F (2>1) vd;w Gs;spapypUe;J tiuag;gLk;

njhLNfhLfspy; njhLehz; (1) 9x -- 8y –72 = 0 (2) 9x + 8y + 72 = 0 3) 8x --9y –72 = 0 4) 8x + 9y – 72 = 0

95) (2, -3) vd;w Kid> x = 4 vd;w ,af;Ftiuiaf; nfhz;l gutisaj;jpd; nrt;tfy ePsk; (1) 2 (2) 4 (3) 6 (4) 8

96) xy = 9 vd;w nrt;tf mjpgutisaj;jpd; kPJs;s vd;w Gs;spapypUe;J tiuag;gLk;

nrq;Fj;J tistiuia kPz;Lk; re;jpf;Fk; Gs;sp

(1)

97) vd;w gutisaj;jpd; mr;R

(1) y = --1 (2) x = --3 (3) x = 3 (4) y =1

98) xy = c2 vd;w nrt;tf mjpgutisaj;jpd; njhiyj;njhLNfhLfs; (1) x = c, y = c (2) x = 0, y = c (3) x = c, y = 0 (4) x = 0, y =0

99) 2x + 3y + 9 = 0 vd;w NfhL y 2 =8x vd;w gutisaj;ijj; njhLk;Gs;sp

(1) (0, --3) (2) (2, 4) 3) (--6,2

9) 4) (

2

9,--6)

100) y 2 = x + 4 vd;w gutisaj;jpd; ,af;Ftiuapd; rkd;ghL

(1) x =4

15 (2) x =

4

15− (3) x =

4

17− (4) x =

4

17

101) xy = 8 vd;w nrt;tf gutisaj;jpd; miu FWf;fr;rpd; ePsk; (1) 2 (2) 4 (3) 16 (4) 8

102) 1916

22

=−yx

vd;w mjpgutisaj;jpd; nrq;Fj;Jj; njhLf;NfhLfspd; ntl;Lk; Gs;spapd;

epakg;ghij (1) x2 + y2 = 25 (2) x2 + y2 = 4 3) x2 + y2 = 3 4) x2 + y2 = 7

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103) 144

2x+

169

2y =1 vd;w ePs;tl;lj;jpd; miu-nel;lr;R kw;Wk; miu-Fw;wr;R ePsq;fs;

(1) 26,12 (2) 13,24 (3) 12,26 (4) 13,12

104) y 2 = 12x vd;w gutisaj;jpd; Ftpehzpd; ,Wjpg;Gs;spfspy; tiuag;gLk; njhLNfhLfs; re;jpf;Fk; Gs;sp mikAk; NfhL

105) xy = 16 vd;w nrt;tf mjpgutisaj;jpd; Kidapd; Maj; njhiyTfs;

106) vd;w mjpgutisaj;jpd; njhiynjhLNfhLfSf;FfpilNaAs;s Nfhzk;

107) 9x 2 + 5y 2 =180 vd;w ePs;tl;lj;jpd; Ftpaq;fSf;fpilNa cs;s njhiyT (1) 4 (2) 6 (3) 8 (4) 2

108) vd;w gutisaj;jpd; Kid

(1)

− 0,8

1

(2)

0,

8

1 (3)

8

1,0 (4)

−8

1,0

109) x + 2y – 5 = 0, 2x – y + 5 = 0 vd;w njhiyj; njhLNfhLfisf; nfhz;l mjpgutisaj;jpd; ikaj; njhiyj; jfT

(1) 3 (2) 2 (3) 3 (4) 2

110) vd;w ePs;tl;lj;jpd; kPJs;s VnjDk; xU Gs;spapypUe;J kw;Wk;

vd;w Gs;spfSf;fpilNa cs;s njhiyTfspd; $Ljy; (1) 4 (2) 8 (3) 6 (4) 18 111) x

2 vd;w mjpgutisaj;jpd; njhiy njhLNfhLfs;

(1) y= ±5

6x (2) y= ±

6

5x (3) y=±

25

36x (4) y= ± x

36

25

112) x 2 =16y vd;w gutisaj;jpd; Ftpak; (1) (4, 0) (2) (0, 4) (3) (--4, 0) (4) (0, --4)

113) 4x 2 -- y 2 =36 f;F 5x -- 2y + 4k = 0 vd;w NfhL xU njhLNfhL vdpy; k ,d; kjpg;G (1) 4/9 (2) 2/3 (3) 9/4 (4) 81/16

114) vd;w $k;Gtistpd; ikaj; njhiyj;jfT e ,d; kjpg;G

(1) 3

1 (2)

3

2 (3)

9

4 (4)

5

2

115) 4 x + 2y = c vd;w NfhL y 2 =16x vd;w gutisaj;jpd; njhLNfhL vdpy; c ,d; kjpg;G (1) -1 (2) -2 (3) 4 (4) - 4 116) XU ePs;tl;lj;jpd; nel;lr;R kw;Wk; mjd; miuFw;wr;Rfspd; ePsq;fs; KiwNa mjd; rkd;ghLfs; kw;Wk; vdpy; ePs;tl;lj;jpd; rkd;ghL

(1) ( )

4

42+x

+( )

16

62−y

=1 (2) ( )

16

42+x

+( )

4

62−y

=1

(3) ( )

16

42+x

-( )

4

62−y

=1 (4) ( )

4

42+x

-( )

16

62−y

=1

117) nrt;tfyj;jpd; ePsk;> Jizar;rpd; ePsj;jpy; ghjp vdj; nfhz;Ls;s mjpgutisaj;jpd; ikaj; njhiyT jfT

(1) 2

3 (2)

3

5 (3)

2

3 (4)

2

5

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118) 16x 2 -- 3y 2 --32x --12y – 44 = 0 vd;gJ

(1) xh; ePs;tl;lkó (2) xh; tl;lk; (3) xh; gutisak; (4) xh; mjpgutisak

119) vd;w Gs;spapypUe;J y 2 =16x f;F tiuag;gLk; ,Uj;njhLNfhLfSf;F ,ilNaAs;s

Nfhzk; (1) 45 0 (2) 30 0 (3) 60 0 (4) 90 0 120) vd;w mjpgutisaj;jpd; ,af;Ftiu

121) vd;w gutisaj;jpd; t1 = t kw;Wk; t 2 = 3t vd;w Gs;spfspy; tiuag;gl;l

njhLNfhLfs; ntl;bf;nfhs;Sk; Gs;sp

(1) (6t 2 , 8t) (2) (8t, 6t 2 ) (3) (t 2 , 4t) (4) (4t, t 2 )

njhFjp njhFjp njhFjp njhFjp –

5. tif Ez;;fzpjk;5. tif Ez;;fzpjk;5. tif Ez;;fzpjk;5. tif Ez;;fzpjk;---- gad;ghLfs; gad;ghLfs; gad;ghLfs; gad;ghLfs; ---- I (3(3(3(3----tpdhf;fß)tpdhf;fß)tpdhf;fß)tpdhf;fß)

122). 0=a , 1=b vdf; nfhz;L 12)( 2 −+= xxxf vd;w rhu;gpw;F nyf;uhQ;rpd;

,ilkjpg;Gj; Njhw;wj;jpd;gbas;s ‘c’ ,d; kjpg;G

(1) -1 (2) 1 (3) 0 (4) 2

1

123) 100kP2 gug;G nfhz;Ls;s nrt;tfj;jpd; kPr;rpW Rw;wsT (1) 10 (2) 20 (3) 40 (4) 60

124) 54)( 2 +−= xxxf vd;w rhHG [0,3] ,y; nfhz;Ls;s kPg;ngU ngUk kjpg;G

(1) 2 (2) 3 (3) 4 (4) 5 125) xxy −= tan vd;w rhh;G

(1) ,y; VWk; rhh;G (2) ,y; ,wq;Fk; rhh;G

(3) ,y; VWk; ,y; ,wq;Fk; (4) ,y; ,wq;Fk; ,y; VWk;.

126) x

x e

x 2

lim∞→

- d; kjpg;G

(1) 2 (2) 0 (3) ∞ (4) 1 127) xU tistiuapd; nrq;NfhL x mr;rpd; kpif jpirapy; θ vd;Dk; Nfhzj;ij Vw;gLj;fpwJ. mr;nrq;NfhL tiuag;gl;l Gs;spapy; tistiuapd; rha;T (1)- cot θ (2) tan θ (3) - tan θ (4) cot θ

128) r Muk; nfhz;l xU tl;lj;jpd; gug;G A ,y; Vw;gLk; khWk; tpjk;>>>>

(1) 2π r (2) 2π rdt

dr (3) π r

2

dt

dr (4) π

dt

dr

129) 1=a kw;Wk; 4=b vdf; nfhz;L xxf =)( vd;w rhu;gpw;F nyf;uhQ;rpd; ,ilkjpg;Gj;

Njhw;wj;jpd;gb mikAk; ‘c’ ,d; kjpg;G

(1) 4

9 (2)

2

3 (3)

2

1 (4)

4

1

130) xU rJuj;jpd; %iy tpl;lj;jpd; ePsk; mjpfupf;Fk; tPjk; 0.1 nr.kP / tpdhb vdpy; gf;f

msT 2

15 nr.kP Mf ,Uf;Fk; NghJ mjd; gug;gsT mjpfupf;Fk; tPjk;

(1) 1.5 nrkP2/ tpdhb (2) 3 nrkP2/ tpdhb

(3) 3 2 nrkP2/ tpdhb (4) 0.15 nrkP2/ tpdhb

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131) 45)( 2 +−= xxxf vd;w rhh;G VWk; ,ilntsp

(1) ( )1,∞− (2) ( )4,1 (3) ( )∞,4 (4) vy;yh Gs;spfsplj;Jk;

132) y = 3x2 vd;w tistiuf;F x ,d; Maj;njhiyT 2 vdf; nfhz;Ls;s Gs;spapy; nrq;Nfhl;bd; rha;thdJ

(1) 13

1 (2)

14

1 (3)

12

1− (4)

12

1

133) - d; kjpg;G

(1) ∞ (2) 0 (3) (4)

134) t

1=θ vDk; tistiuf;F Gs;sp (-3, --1/ 13) vd;w Gs;spapy; nrq;Nfhl;;bd; rkd;ghL

(1) 3θ =27 -80 (2) 5θ =27 -80 (3) 3θ =27 +80 (4)

135) xey 3= kw;Wk; 3

3

−= ea

y vd;Dk; tistiufs; nrq;Fj;jhf ntl;bf; nfhs;fpd;wd vdpy;

‘a’ kjpg;G

(1)-1 (2) 1 (3) 3

1 (4) 3

136) vdpy; ,d;kjpg;G;

(1) 5 (2) -5 (3) 3 (4) -3 137) y = 2x2 -6x-4 vd;w tistiuapy; x-mr;Rf;F ,izahfTs;s njhLNfhl;;bd; njhL Gs;sp

(1)

−2

17,

2

5 (2)

−−2

17,

2

5 (3)

−2

17,

2

5 (4)

−2

17,

2

3

138) 323232 ayx =+ vd;;w tistiuapd; nrq;NfhL x - mr;Rld; vd;Dk;

Nfhzk;Vw;gLj;Jnkdpy;mr; nrq;Nfhl;b∂rha;T

139) x = 2 ,y; y = -2x3 + 3x+ 5 vd;w tistiuapd; rha;T (1) -20 (2) 27 (3) -16 (4) -21 140) gpd;tUtdtw;Ws; vJ (0> ∞ ) ,y; VWk; rhh;G?

(1) ex (2) x

1 (3) -x2 (4) x-2

141) 1925

22

=+yx

kw;Wk; 188

22

=−yx

vDk; tistiufSf;F ,ilgl;l Nfhzk;

(1) 4

π (2)

3

π (3)

6

π (4)

2

π

142) y = x4 vd;w tistiuapd; tisT khw;Wg;Gs;sp (1) x = 0 (2) X = 3 (3) X = 12 (4) vq;Fkpy;iy

143) 2248 xxy −+= vd;w tistiu y-mr;ir ntl;Lk; Gs;spapy; mikAk; njhLf;Nfhl;bd;

rha;T (1) 8 (2) 4 (3) 0 (4) -4

144) y = 5

3xvDk; tistiuf;F (-1, - 1/5) vd;w Gs;spapy; njhLNfhl;bd; rkd;ghL

(1) 5y+3x=2 (2) 5y-3x=2 (3) 3x-5y=2 (4) 3x + 3y = 2

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145) xU Nfhsj;jpd; fd msT kw;Wk; Muj;jpy; Vw;gLk; khWtPjq;fs; vz;zstpy; rkkhf ,Uf;Fk;NghJ Nfhsj;jpd; tisgug;G

(1) 1 (2) π21 (3) 4π (4)

3

4π

146) y2 = x kw;Wk; x2 = y vd;w gutisaq;fSf;fpilNa Mjpapy; mikAk; Nfhzk;

(1) 2 tan-1

4

3 (2) tan-1

3

4 (3)

2

π (4)

4

π

147) xU cUisapd; Muk; 2 nr.kP / tpdhB vd;w tPjj;jpy; mjpfupf;fpd;wJ. mjd; cauk; 3 nr.kP / tpdhb vd;w tPjj;jpy; Fiwfpd;wJ. Muk; 3 nr.kP kw;Wk; cauk; 5 nr.kP Mf ,Uf;Fk; NghJ mjd; fd mstpd; khW tPjk; (1) 23π (2) 33π (3) 43π (4) 53π

148) 2)( xxf = vd;w rhh;G ,wq;Fk; ,ilntsp

(1) ( )∞∞− , (2) ( )0,∞− (3) ( )∞,0 (4) ( )∞− ,2

149) y =3x2 + 3sin x vd;w tistiuf;F x=0 tpy; njhL Nfhl;;bd; rha;T

(1) 3 (2) 2 (3) 1 (4) -1

150) gpd;tUk; tistiufSs; vJ fPo;Nehf;fP FopT ngw;Ws;sJ (1) y = -x2 (2) y = x2 (3) y = ex (4) y = x2+2x-3

151) 74 23 +−= tts vdpy; KLf;fk; Gr;rpakhFk; NghJs;s jpirNtfk;

(1) 3

32m/sec (2)

3

16−m/sec (3)

3

16m/sec (4)

3

32−m/sec

152) 2

cos)(x

xf = vd;w rhu;gpw;F [ ]ππ 3, ,y; Nuhy; Njw;wj;jpd;gb mike;j C ,d; kjpg;G

(1) 0 (2) 2π (3) 2

π (4)

2

3π

153) mxey = kw;Wk; mxey −= , m > 1 vd;Dk; tistiufSf;F ,ilgl;l Nfhzk;

154) tex = cos t ; tey = sin t vd;w tistiuapd; njhLNfhL x - mr;Rf;F ,izahfTs;sJ

vdpy; t ,d; kjpg;G

(1) - 4

π (2)

4

π (3) 0 (4)

2

π

155) 36 xxy −= NkYk; x MdJ tpdhbf;F 5 myFfs; tPjj;jpy; mjpfupf;fpd;wJ.

vDk; NghJ mjd; rha;tpd; khW tPjk; (1) -90 myFfs; / tpdhb (2) 90 myFfs; / tpdhb (3) 180 myFfs; / tpdhb (4) -180 myFfs; / tpdhb 156) nfhLf;fg;gl;Ls;s miu tl;lj;jpd; tpl;lk; 4nr.kP mjDs; tiuag;gLk; nrt;tfj;jpd; ngUk gug;G (1) 2 (2) 4 (3) 8 (4) 16

157) xey −−= vd;w tistiu

(1) X > 0 tpw;F Nky;Nehf;fpf; FopT (2) X > 0 tpw;F fPo;Nehf;ff;p FopT

(3) vg;NghJk; Nky;Nehf;fpf; FopT (4) vg;NghJk; fPo;Nehf;fpf; FopT 158) xU Neu;f;Nfhl;by; efUk; Gs;spapd; jpiNtfkhdJ mf;Nfhl;by; xU epiyg;Gs;spapypUe;J efUk; Gs;spf;F ,ilapy; cs;s njhiytpd; tu;f;fj;jpw;F Neu; tpfpjkhf mike;Js;sJ. vdpy; mjd; KLf;fk; gpd;tUk; xd;wpDf;F tpfpjkhf mike;Js;sJ. (1) s (2) s2 (3) s3 (4) s4

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159) y = x2 vd;w rhu;gpw;F [-2 2 ] ,y; Nuhypd; khwpyp

(1) 3

32 (2) 0 (3) 2 (4) -2

160) XU cUFk; gdpf;fl;bg; Nfhsj;jpd; fd msT 1 nr.kP3./ epkplk; vdf; Fiwfpd;wJ. mjd; tpl;lk; 10 nr.kP vd ,Uf;Fk;NghJ tpl;lk; FiwAk; Ntfk; MdJ

(1) π50

1− nrkP / epkplk; (2)

π50

1 nrkP / epkplk;

(3) π75

11− nrkP / epkplk; (4)

π75

2− nrkP / epkplk;

161) dcxbxaxy ++= 23 vd;w tistiuf;F x = 1 ,y; xU tisT khw;Wg;Gs;sp cz;nldpy;

(1) a + b = 0 (2) a + 3b = 0 (3) 3a + b= 0 (4) 3a + b =1 162) MjpapypUe;J XU Neh;Nfhl;by; x njhiytpy; efUk; Gs;spapy; jpirNtfk; x njhiytpy; efUk; Gs;spapd; jpirNtfk; v vdTk; a + bv2; = x2 vdTk; nfhLf;fg;gl;Ls;sJ. ,q;F a kw;Wk; b khwpypfs;. mjd; KLf;fk; MdJ.

(1) x

b (2)

x

a (3)

b

x (4)

a

x

163) 832 23 ++− xxx mjpfupf;Fk; tPjkhdJ x mjpfupf;Fk; tPjj;ijg; Nghy; ,U klq;F

vdpy; x ,d; kjpg;Gfs;

164) 323232 ayx =+ vDk; tistiuapd; Jiz myFr; rkd;ghLfs;

165) xU fdr; rJuj;jpd; fd msT 4 nr.kP3 / tpdhb vd;w tPjj;jpy;mjpfupf;fpd;wJ. mf;fdr; rJuj;jpd; fd msT 8 f.nr.kP Mf ,Uf;Fk;NghJ mjd; Gwg;gug;gsT mjpfupf;Fk; tPjk; (1) 8 nrkP 2 / tpdhb (2) 16 nrkP 2 / tpdhb (3) 2 nrkP 2/ tpdhb (4) 4 nrkP 2 / tpdhb

6. tif Ez;;fzpjk; gad;ghLfs;6. tif Ez;;fzpjk; gad;ghLfs;6. tif Ez;;fzpjk; gad;ghLfs;6. tif Ez;;fzpjk; gad;ghLfs;---- II. 1111----tpdh tpdh tpdh tpdh

166) 22

1

yxu

+= vdpy; =

∂∂

+∂∂

y

uy

x

ux

(1) u2

1 (2) u (3) u

2

3 (4) - u

167) )1()2( 22 xxxy +=− vd;w tistiuf;Fr;

(1) x mr;Rf;F ,izahd xU njhiyj; njhLNfhL cz;L

(2) y mr;Rf;F ,izahd xU njhiyj; njhLNfhL cz;L (3) ,U mr;RfSf;Fk; ,izahd njhiyj; njhLNfhLfß cz;L (4) njhiyj; njhLNfhLfs; ,y;iy.

168)

=x

yfu vdpy;

y

uy

x

ux

∂∂

+∂∂

,d; kjpg;G

(1) 0 (2) 1 (3) 2 u (4) u

169) y2 (a+x)=x2(3a-x) vd;w tistiu gpd;tUtdtw;Ws; ve;jg; gFjpapy; mikahJ? (1) x >0 (2) 0 < x< 3a (3) x ≤--a kw;Wk; x > 3a (4) –a < x < 3a

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170) θθ sin,cos ryrx == vdpy; =∂∂x

r

(1) sec θ (2) sin θ (3) cos θ (4) cosec θ

171) y2 (a+2x) = x2 (3a-x) vd;w tistiuapd; njhiyj; njhLNfhL

(1) x =3a (2) x = - 2

a (3) x =

2

a (4) x = 0

172)

+=

xy

yxu

22

log vdpy; y

uy

x

ux

∂∂

+∂∂

vd;gJ

(1) 0 (2) u (3) 2u (4) u-1 173) 28 ,d; 11 Mk; gb %y rjtpfpjg; gpio Njhuhakhf 28,d; rjtpfpj gpioiag; Nghy; ........................................................................................ klq;fhFk;

(1) 28

1 (2)

11

1 (3) 11 (4) 28

174) a 2 y 2 = x 2 (a 2 – x 2) vd;w tistiu

(1) x=0 kw;Wk; x=a f;F ,ilapy; xU fz;zp kl;LNk nfhz;Ls;sJ

(2) x=0 kw;Wk; x=a f;F ,ilapy; ,U fz;zpfs; nfhz;Ls;sJ

(3) x= -a kw;Wk; x=a f;F ,ilapy; ,U fz;zpfs; nfhz;Ls;sJ

(4) fz;zp VJkpy;iy

175)

+

+= −

22

441sin

yx

yxu kw;Wk; f = sin u vdpy; rkgbj;jhd rhHG f ,d; gb

(1) 0 (2) 1 (3) 2 (4) 4 176) gpd;tUtdtw;Ws; rupahd $w;Wfs;

(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 ,Umr;Rf;fisg; nghWj;J rkr;rPH ngw;wpUg;gpd;mJ Mjpiag; nghWj;Jk; rkr;rPh ngw;wpUf;Fk

(iii) f (x,y)=0 vd;w tistiu 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;.

177) u = y sin x vdpy; =∂∂

∂yx

u2

(1) cos x (2) cos y (3) sin x (4) 0

178) ay2 = x2 (3a-x) vd;w tistiu y mr;ir ntl;Lk; Gs;spfs; (1) x = -3a, x = 0 (2) x = 0, x = 3a (3) x = 0, x = a (4) x = 0

179) yxu = vdpy; x

u

∂∂

f;Fr; rkkhdJ

(2) U log x (3) u log y (4)

180) 9y2 = x2 (4-x2) vd;w tistiu vjw;F rkr;rPH? (1) y mr;R (2) x mr;R (3) y = x (4) ,U mr;Rfs;

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7. njif Ez;;fzpjk; : gad;ghLfs;7. njif Ez;;fzpjk; : gad;ghLfs;7. njif Ez;;fzpjk; : gad;ghLfs;7. njif Ez;;fzpjk; : gad;ghLfs;----(3(3(3(3----tpdhf;fß)tpdhf;fß)tpdhf;fß)tpdhf;fß)

181) NfhLfs; y = x, y = 1 kw;Wk; x = 0 Mfpait Vw;gLj;Jk; gug;G y mr;ir nghWj;J Row;wg;gLk; jplg;nghUspd; fd msT

182) 43232 =+ yx vd;w tistiuapd; tpy;ypd; ePsk;

(1) 48 (2) 24 (3) 12 (4) 96

183) 12

2

2

2

=+b

y

a

x vd;w ePs;tl;lj;jpd; gug;ig nel;lr;R Fw;wr;R ,tw;iw nghWj;J Row;wg;gLk;

jplg;nghUspd; fd msT tpfpjk; (1) b 2 : a 2 (2) a 2 : b 2 (3) a : b (4) b : a

184) 23 xy += vd;w tistiu x = 0 tpypUe;J x = 4 tiu x mr;ir mrrhf

itj;J Row;wg;gLk; jplg;nghUspd; fd msT

(1) 100 π (2) π9

100 (3) π

3

100 (4)

3

100

185) ∫−

+

2

2cos2

sinπ

π

dxx

x ,d; kjpg;G

(1) 0 (2) 2 (3) log 2 (4) log 4

186) ∫ +−2

0cossin1

cossinπ

dxxx

xx ,d; kjpg;G.

(1) 2

π (2) 0 (3)

4

π (4) π

187) gutis y2 = x f;Fk; mjd; nrt;tfyj;jpw;f;Fk; ,ilg;gl;l gug;G

(1) 3

4 (2)

6

1 (3)

3

2 (4)

3

8

188) ∫π

0

32sin xdxxcox ,d; kjpg;G

(1) π (2) π /2 (3) π /4 (4) 0

189) 12

2

2

2

=+b

y

a

x vd;w ePs; tl;lj;jpw;f;Fk; mjd; Jiz tl;lj;jpw;f;Fk; ,ilg;gl;l gug;G

(1) ( )bab −π (2) 2 ( )baa −π (3) ( )baa −π (4) 2 ( )bab −π

190) x = 0 ,ypUe;J 4

π=x tiuapyhd y = sin x kw;Wk; y = cos x vd;w tistiufspd;

,ilg;gl;l gug;G

(1) 12 + (2) 12 − (3) 222 − (4) 222 +

191) Muk; 5 cs;s Nfhsj;ij jsq;fs; ikaj;jpypUe;J 2 kw;Wk 4 Jhuj;jpy; ntl;Lk; ,U

,izahd jsq;fSf;F ,ilg;gl;l gFjpapd; tisg;gug;G

(1) 20 π (2) 40 π (3) 10 π (4) 30 π

192) ∫ +

2

0

3535

35

sincos

cosπ

dxxx

x ,d; kjpg;G

(1) 2

π (2)

4

π (3) 0 (4) π

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17

193) 1169

22

=+yx

vd;w tistiuia Fw;wr;ir nghWj;J Row;wg;gLk; jplg;nghUspd; fd msT

(1) 48π (2) 64π (3) 32π (4) 128 π 194) y = 2x , x = 0 kw;Wk; x = 2 ,tw;wpf;F ,ilNa Vw;gLk; gug;G x mr;irg; nghWj;J Row;wg;gLk; jplg;nghUspd; tisg;gug;G

(1) π58 (2) π52 (3) π5 (4) π54

195) y = x vd;w Nfhl;bw;f;Fk; x mr;R>NfhLfs; x = 1kw;Wk; x = 2 Mfpatw;wpf;Fk; ,ilg;gl;l gug;G

(1) 2

3 (2)

2

5 (3)

2

1 (4)

2

7

196) (0,0) (3,0) kw;Wk; (3>3) Mfpatw;iw Kidg;Gs;spfshf nfhz;l Kf;Nfhzj;jpd; gug;G x mr;ir nghWj;J Row;wg;gLk; jplg;nghUspd; fd msT (1) 18 π (2) 2 π (3) 36 π (4) 9 π

197) dxx∫4

0

3 2cos

π

,d; kjpg;G .

(1) 3

2 (2)

3

1 (3) 0 (4)

3

2π

198) ∫ −1

0

4)1( dxxx ,d; kjpg;G

(1) 12

1 (2)

30

1 (3)

24

1 (4)

20

1

199) ∫π

0

4sin xdx ,d; kjpg;G

8. tif nfOr; rkd;ghLfs;.(38. tif nfOr; rkd;ghLfs;.(38. tif nfOr; rkd;ghLfs;.(38. tif nfOr; rkd;ghLfs;.(3----tpdhf;fß)tpdhf;fß)tpdhf;fß)tpdhf;fß)

200) tiff;nfO rkd;ghL Qpydx

dy=+ tpd; njhiff; fhuzp

(1) (2) (3) (4) 201) Mjpg;Gs;spia ikakhf nfhz;l tl;lq;fspd; njhFg;gpd; tiff;nfO rkd;ghL (1) x dy + y dx = 0 (2) x dy – y dx = 0

(3) x dx + y dy = 0 (4) x dx – y dy = 0 202) 2 2x ,d; epug;Gr; rhh;G

(1) (Ax + B)ex (2) A cos x + B sin x

(3) (Ax + B)e2x (4) ( Ax + B) e -x

203) 0)(2 =++ dxyxdyx vd;w rkg;gbj;jhd tiff;nfO rkd;ghl;by; y = vx

vdg; gpujpapL nra;Ak; NghJ fpilg;gJ

(1) 0)2( 2 =++ dxvvxdv (2) 0)2( 2 =++ dvxxvdx

(2) 0)( 22 =+− dvxxdxv (3) 0)2( 2 =++ dxxxvdv

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204) yx

yx

dx

dy

+−

= vdpy;

(1) 2 x y + + = c (2) x 2 + y 2 – x + y = c

(3) x 2 + y 2 – 2 x y = c (4) x 2 – y 2 – 2 x y = c

205) 2 ,d; rpwg;G jPHT

(1) xex 2

2

2 (2) xe2x (3) xe-2x (4) xe

x 2

2

−

206)

32

3

3

3

1

dx

yd

dx

dy

c

+

= tiff;nfO rkd;ghl;bd; gb ,q;F c xU khwpyp

(1) 1 (2) 3 (3) -2 (4) 2

207) 2

2.

log

1

xy

xxdx

dy=+ ,d; njhiff;fhuzp

(1) e x (2) log x (3) x

1 (4) e-x

208) xxf =)(' kw;Wk; 2)1( =f vdpy; )(xf vd;gJ

(1) (2) (3) (4)

( ))23

2+xx

209) 3x –3x vd;w rkd;ghl;by; a iaAk; b iaAk; ePf;fpf; fpilf;Fk; tiff;nfO rkd;ghL

(1) 02

2

=+ aydx

yd (2) 09

2

2

=− Ydx

yd

(3) 092

2

=−dx

dy

dx

Yd (4) 09

2

2

=+ xdx

yd

210) QPydx

dy=+ vd;w tiff;nfOr; rkd;ghl;bd; njhiff;fhuzp cos x vdpy; P d; kjpg;G

(1) – cot x (2) cot x (3) tan x (4) – tan x

211) 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;)

(1) p

k

dt

dp= (2) kt

dt

dp= (3) kp

dt

dp= (4) kt

dt

dp−=

212) 2

231

1dx

yd

dx

dy=

+ tiff;nfOr; rkd;ghl;bd; gb

(1) 1 (2) 2 (3) 3 (4) 6

213) vd;w tiff;nfOr; rkd;ghl;bd; njhiff;fhuzp

(1) log x (2) x2 (3) ex (4) x

214) xkey λ= vdpy; mjd; tiff;nfO rkd;ghL

(1) ydx

dyλ= (2) ky

dx

dy= (3) 0=+ ky

dx

dy (4) xe

dx

dy λ=

215) y = cx – c2 vd;gjidg; nghJj; jPHthfg; ngw;w tiff;nfO rkd;ghL (1) (y’)2 – xy’ + y = 0 (2) y” = 0 (3) y’ = c (4) (y’)2+ x y’+ y = 0

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19

216) x y jsj;jpYs;s vy;yh Neh;f; NfhLfspd; njhFg;gpd; tiff;nfO rkd;ghL

(1) =dx

dy xU khwpyp (2) 0

2

2

=dx

yd

(3) 0=+dx

dyy (4) 0

2

2

=+ ydx

yd

217) ydyexdydx y 2sec−=+ ,d;; njhiff;fhuzp

(1) ex (2) e –x (3) ey (4) e –y 218) xU jsj;jpy; cs;s mr;Rf;F nrq;Fj;jy;yhj NfhLfspd; tiff;nfO rkd;ghL

(1) 0=dx

dy (2)

2

2

dx

yd = 0 (3) m

dx

dy= (4) m

dx

yd=

2

2

219) m < 0 Mf ,Ug;gpd; 0=+ mxdy

dx ,d; jPHT

(1)x = cemy (2) x = ce –my (3) x = my + c (4) x = c

220) xydy

dx=+

31

2

5 vd;w tiff;nfOtpd;

(1) thpir 2 kw;Wk; gb 1 (2) thpir 1 kw;Wk; gb 2 (3) thpir 1 kw;Wk; gb 6 (4) thpir 1 kw;Wk; gb 3 221) (3D2 + D – 14 ) y = 13 e 2x ,d; rpwg;Gj; jPHT

(1) 26x e 2x (2) 13x e 2x (3) x e 2x (4) ( x 2/ 2 ) e 2x

222) x vd;w njhlh;gpy; A iaAk; B iaAk; ePf;fp ngwg;gLk; tiff;nfO

rkd;ghL

223) vd;w tiff;nfO rkd;ghl;bd; njhiff;fhuzp

224) 0)(),()()( ≠−= agDgaDDf vdpy; tiff;nfO rkd;ghL axeyDf =)( rpwg;Gj; jPHT

(1) meax (2) )(ag

eax

(3) g (a)eax (4) )(ag

xeax

225) y = mx vd;w NeHNfhLfspd; njhFg;gpd; tiff;nfO rkd;ghL

(1) mdx

dy= (2) 0=− xdyydx (3) 0

2

2

=dx

yd (4) y 0=+ xdydx

9. jdp epiyf; fzf;fpay; (39. jdp epiyf; fzf;fpay; (39. jdp epiyf; fzf;fpay; (39. jdp epiyf; fzf;fpay; (3----tpdhf;fß); tpdhf;fß); tpdhf;fß); tpdhf;fß);

226) ),( 99 +Z ,y; [7] ,d; thpir

(1) 9 (2) 6 (3) 3 (4) 1 227) ngUf;fiyg; nghWj;J Fykhfpa xd;wpd; Kg;gb %yq;fspy; ω 2 ,d; thpir (1) 4 (2) 3 (3) 2 (4) 1 228) xU $l;Lf; $w;W %d;W jdpf;$w;Wfisf; nfhz;ljhf ,Ug;gpd; nka;al;ltizapYs;s

epiufspd; vz;zpf;if

(1) 8 (2) 6 (3) 4 (4) 2

229) )](~[~ qp ∧ d; nka; ml;ltizapy; epiufspd; vz;zpf;if

(1) 2 (2) 4 (3) 6 (4) 2

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20

230) KOf;fspy; * vd;w <UWg;Gr; nrayp a * b = a + b –ab vd tiuaWf;fg;gLfpwJ vdpy; 3*(4*5) ,∂kjpg;G (1) 25 (2) 15 (3) 10 (4) 5 231) rkdpAila miuFf;Fyk; Fykhtjw;F G+Hj;jp nra;a Ntz;ba tpjpahtJ (1) milg;G tpjp (2) NrHg;G tpjp (3) rkdp tpjp (4) vjpHkiw tpjp 232) KOf;fspy;;; * vd;w <UWg;Gr; nrayp a * b = a + b – 1 vd tiuaWf;fg;gLfpwJ vdpy; rkdp cWg;G (1) 0 (2) 1 (3) a (4) b 233) epge;jidf; $w;W qp → f;Fr; rkkhdJ

(1) pvq (2) qpv ~ (3) pvq~ (4) qp^

234) fPo;fz;ltw;Ws; vJ rhp ? (1) xU Fyj;jpd; xU cWg;gpw;F xd;wpw;F Nkw;gl;l vjpHkiw cz;L (2) Fyj;jpd; xt;nthU cWg;Gk; mjd; vjpHkiwahf ,Uf;Fnkdpy; mf;Fyk; xU vgPypad; FykhFk; (3) nka;naz;fis cWg;Gfshff; nfhz;l vy;yh 2 X 2 mzpf;NfhitfSk; ngUf;fy; tpjpapy; FykhFk; (4) vy;yh Gba ∈, f;Fk; (a * b)-1 = a-1 * b -1

235) fPo;fzltw;wpy; vJ R ,y; <UWg;Gr; nrayp my;y?

236) fPof;;fzltw;Ws; vJ Fyk; my;y.

(1) ( )nnZ +, (2) ( )+,Z (3) ( ),.Z (4) ( )+,R

237) p apd;nka; kjpg;G T kwWk; q ,d; nka; kjpg;G F vdpy; gpd;tUtdtw;wpy; vit nka; kjpg;G T vd ,Uf;Fk; (i) pvq (ii) ~ pvq (iii) qpv ~ (iv) qp ~^

(1) (i), (ii), (iii) (2) (i), (ii), (iv) (3) (i), (iii), (iv). (4) (ii), (iii), (iv)

238) ngUf;fiyg; nghWj;J Fykhfpa xd;wpd; nMk; gb %yq;fspd; ωk ,d; vjpHkiw ( )nk <

(1) ω1/k (2) 1−ω (3) kn−ω (4) kn /ω

239) [ ] [ ] [ ]( )653 1111 ++ ,d; kjpg;G

(1) [ ]0 (2) [ ]1 (3) [ ]2 (4) [ ]3

240) qp ↔ f;Fr; rkhdkhdJ.

(1) qp → (2) pq → (3) )()( pqvqp →→ (4) )()^( pqqp →→

241) ngUf;fy; tpjpiag; nghWj;J Fykhfpa xd;wpd; ehyhk; %yq;fspy;; - i d; thpir (1) 4 (2) 3 (3) 2 (4) 1 242) gpd;tUtdtw;Ws; vJ Kuz;ghlhFk;

(1) pvq (2) qp^ (3) ppv ~ (4) pp ~^

243) nka;naz;fspd; fzk; R ,y; * vd;w <UWg;G nrayp vd

tiuaWf;fg;gLfpwJ vdpy; (3 * 4) * 5 ,∂ kjpg;G

(1) 5 (2) 25 (3) 25 (4) 50 244) fPo;fzltw;Ws; vit $w;Wfs;

(i) flTs; cd;id MrPHtjpf;fl;Lk; (ii) Nuhrh xU G+

(iii) ghypd; epwk; ntz;ik (iv) 1 xU gfh vz;

245) gpd;tUtdtw;Ws; vJ nka;ikahFk;

(1) pvq (2) qP ∧ (3) ppv ~ (4) pp ~^

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10. epfo;jfTg; guty;fs;10. epfo;jfTg; guty;fs;10. epfo;jfTg; guty;fs;10. epfo;jfTg; guty;fs;----(3(3(3(3----tpdhf;fß)tpdhf;fß)tpdhf;fß)tpdhf;fß)

246) xU rktha;g;G khwp X, ,ay;epiyg; guty; 25

)100(2/1 2

)(

−−

=x

cexf [

gpd;gw;WfpwJ vdpy; c ,d; kjpg;G

(1) π2 (2) π21 (3) π25 (4)

π25

1

247) gha;]hd; gutypd; gzgsit 25.0=λ vdpy; ,uz;lhtJ tpyf;fg; ngUf;fj; njhif (1) 0.25 (2) 0.3125 (3) 0.0625 (4) 0.025 248) 400khztHfs; vOjpa fzpjj; NjHtpd; kjpg;ngz;fs; ,ay;epiyg; gutiy xj;jpapUf;fpwJ. ,jd; ruhrhp 65. NkYk; 120 khztHfs; 85 kjpg;ngz;fSf;F Nky; ngw;wpUg;gpd; kjpg;ngz;fs; 45 ,ypUe;J 65f;Fs; ngWk; khztHfspd; vz;zpf;if. (1) 120 (2) 20 (3) 80 (4) 160 249) xU gha;]hd; gutypy; P(X = 0) = K vdpy; gutw;gbapd; kjpg;G.

(1) k

1log (2) klog (3) λe (4)

k

1

250) X vd;w rktha;g;G khwpapd; 3>4 kw;Wk; 12 Mfpa kjpg;gfs; KiwNa kw;Wk;

Mfpa epfo;jfTfisf; nfhs;Snkdpy; ,d; kjpg;G (1) 5 (2) 7 (3) 6 (4) 3 251) xU ,ay;epiyg; gutypd; epfo;jfT mlHj;jpr; rhHG )(xf ,d;

ruhrhp µ vdpy; ∫∞

∞−

dxxf )( ,d; kjpg;G

(1) 1 (2) 0.5 (3) 0 (4) 0.25 252) xU ngl;bapy; 6 rptg;G kw;Wk; 4 nts;isg; ge;Jfs; cs;sd. mtw;wpypUe;J 3 ge;Jfs; rktha;g;G Kiwapy; vLf;fg;gl;lhy; 2 nts;isg; ge;Jfs; fpilf;f epfo;jfT

(1) 20

1 (2)

125

18 (3)

25

4 (4)

10

3

253) X vd;w rktha;g;G khwpapd; epfo;jfT epiwr;rhHG guty; gpd;tUkhW λ tpd; kjpg;G (1) 1 (2) 2 (3) 3 (4) 4

254) xU <UWg;Gg; gutypd; ruhrhp 12 kw;Wk; jpl;ltpyf;fk; 2 vdpy; gz;gsit p ,d; kjpg;G

(1) 2

1 (2)

3

1 (3)

3

2 (4)

4

1

255)

=<< 30

2

0)(xkx

xf kw;nwq;fpYk; vd;gJ epfo;jfT mlHj;jpr; rhHG vdpy; K ,d; kjpg;G

(1) 3

1 (2)

6

1 (3)

9

1 (4)

12

1

256) xU ,ay;epiy khwp X ,d; epfo;jfT mlHj;jpr; rhHG )(xf kw;Wk; ),(~ 2σµ X

vdpy; ∫∞−

µ

dxxf )( ,d; kjpg;G

(1) tiuaWf;f KbahJ (2) 1 (3) 0.5 (4) -.5

X -2 3 1

P(X=x)

6

λ

4

λ

12

λ

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257) ∞<<∞−+

= xx

Axf

'216

1)(

π vd;GJ X vd;w njhlH rktha;g;G khwpapd; xU epfo;jfT

mlHj;jpr; rhHG (p.d.f) vdpy; A ,d; kjpg;G (1) 16 (2) 8 (3) 4 (4) 1 258) kw;Wk; vdpy; c ,d; kjpg;G (1) -2 (2) 4 (3) -4 (4) 2 259) X vd;w xU jdp epiy rktha;g;G khwp 0> 1> 2 vd;w kjpg;Gfisf;

nfhs;fpwJ. NkYk; ( ) ,169

1)1(,

169

1440 ==== XPXP vdpy; P(X = 2) ,d; kjpg;G

(1) 169

145 (2)

169

24 (3)

169

2 (4)

169

143

260) xU gfilia 5 Kiw tPRk; NghJ 1 my;yJ 2 fpilg;gJ ntw;wpnadf; fUjg;gLfpwJ vdpy; ntw;wpapd; ruhrhpapd; kjpg;G.

(1) 3

5 (2)

5

3 (3)

9

5 (4)

5

9

261) xU rktha;g;G khwp X gha;]hd; gutiyg; gpd;gw;WfpwJ. NkYk; E(X 2 ) = 30 vdpy; gutypd; gutw;gb. (1) 6 (2) 5 (3) 30 (4) 25 262) ed;F fiyf;fg;gl;l 52 rPl;Lfs; nfhz;l rPl;Lfl;lbypUe;J 2 rPl;Lfs; vLf;fg;gLfpd;wd. ,uz;Lk; xNu epwj;jpy; ,Uf;f epfo;jfT.

(1) 2

1 (2)

51

26 (3)

51

25 (4)

102

25

263) rktha;g;G khwp X ,d; guty; rhHG F(x) xU (1) ,wq;Fk; rhHG (2) VWk; rhu;G (3) khwpypr; rhHG (4) Kjypy; VWk; rhHG gpd;dH ,wq;Fk; rhHG 264) X vd;w rktha;g;G khwpapd; gutw;gb 4 NkYk; ruhrhp 2 vdpy; ,d; ruhrhpapd; E(X2) kjpg;G (1) 2 (2) 4 (3) 6 (4) 8

265) xU jdp epiy rktha;g;G khwp X f;F 202 =µ NkYk; 2761

2 =µ vdpy; X ,d;

ruhrhpapd; kjpg;G (1) 16 (2) 5 (3) 2 (4) 1 266) X vd;w rktha;g;G khwpapd; epfo;jfTg; guty; gpd;tUkhW

X 0 1 2 3 4 5

P(X=x)

2a 3a 4a 5a

( )41 ≤≤ xP ,d; kjpg;G

(1) 21

10 (2)

7

2 (3)

14

1 (4)

2

1

267) xU rktha;g;G khwp X ,d; epfo;jfT epiwr; rhHG (p,d,f) gpd;tUkhW

X 0 1 2 3 4 5 6 7

P(X=x) 0 K 2 k 2 k 3 k k2 2 k2 7 k2+k

K ,d; kjpg;G

(1) 8

1 (2)

10

1 (3) 0 (4) –1 or

10

1

268) xU gfilia 16 Kiwfs; tPRk; NghJ ,ul;ilg;gil vz; fpilg;gJ ntw;wpahFk; vdpy; ntw;wpapd; gutw;gb (1) 4 (2) 6 (3) 2 (4) 256

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269) xU <UWg;Gg; gutypd; ruhrhp 5 NkYk; jpl;ltpyf;fk; 2 vdpy; n kw;Wk; p ,d; kjpg;Gfs;

(1)

25,

5

4 (2)

5

4,25 (3) 25,

5

1 (4)

5

1,25

270) Var (4X+3) ,d; kjpg;G

(1) 7 (2) 16 Var (x) (3) 19 (4) 0

271) xU gha;]hd; gutypy; P(X=2) = P(X=3) vdpy; gz;gsit λ ,d; kjpg;G

(1) 6 (2) 2 (3) 3 (4) 0

Time once lost, is lost forever....

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%%%%%%%%

: : : : ……………….. .. .. ..

1111 31313131 61616161 91919191 1111 31313131 61616161 91919191 121121121121

2222 32323232 62626262 92929292 2222 32323232 62626262 92929292 122122122122

3333 33333333 63636363 93939393 3333 33333333 63636363 93939393 123123123123

4444 34343434 64646464 94949494 4444 34343434 64646464 94949494 124124124124

5555 35353535 65656565 95959595 5555 35353535 65656565 95959595 125125125125

6666 36363636 66666666 96969696 6666 36363636 66666666 96969696 126126126126

7777 37373737 67676767 97979797 7777 37373737 67676767 97979797 127127127127

8888 38383838 68686868 98989898 8888 38383838 68686868 98989898 128128128128

9999 39393939 69696969 99999999 9999 39393939 69696969 99999999 129129129129

10101010 40404040 70707070 100100100100 10101010 40404040 70707070 100100100100 130130130130

11111111 41414141 71717171 101101101101 11111111 41414141 71717171 101101101101 131131131131

12121212 42424242 72727272 102102102102 12121212 42424242 72727272 102102102102 132132132132

13131313 43434343 73737373 103103103103 13131313 43434343 73737373 103103103103 133133133133

14141414 44444444 74747474 104104104104 14141414 44444444 74747474 104104104104 134134134134

15151515 45454545 75757575 105105105105 15151515 45454545 75757575 105105105105 135135135135

16161616 46464646 76767676 106106106106 16161616 46464646 76767676 106106106106 136136136136

17171717 47474747 77777777 107107107107 17171717 47474747 77777777 107107107107 137137137137

18181818 48484848 78787878 108108108108 18181818 48484848 78787878 108108108108 138138138138

19191919 49494949 79797979 109109109109 19191919 49494949 79797979 109109109109 139139139139

20202020 50505050 80808080 110110110110 20202020 50505050 80808080 110110110110 140140140140

21212121 51515151 81818181 111111111111 21212121 51515151 81818181 111111111111 141141141141

22222222 52525252 82828282 112112112112 22222222 52525252 82828282 112112112112 142142142142

23232323 53535353 83838383 113113113113 23232323 53535353 83838383 113113113113 143143143143

24242424 54545454 84848484 114114114114 24242424 54545454 84848484 114114114114 144144144144

25252525 55555555 85858585 115115115115 25252525 55555555 85858585 115115115115 145145145145

26262626 56565656 86868686 116116116116 26262626 56565656 86868686 116116116116 141414146666

27272727 57575757 87878787 117117117117 27272727 57575757 87878787 117117117117 147147147147

28282828 58585858 88888888 118118118118 28282828 58585858 88888888 118118118118 148148148148

29292929 59595959 89898989 119119119119 29292929 59595959 89898989 119119119119 149149149149

30303030 60606060 90909090 120120120120 30303030 60606060 90909090 120120120120 150150150150

121121121121

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(gFjp (gFjp (gFjp (gFjp – M M M M – tpdh tpilfß)tpdh tpilfß)tpdh tpilfß)tpdh tpilfß) 1.mzpfs1.mzpfs1.mzpfs1.mzpfs; kw;Wk; mzpf;Nfhitfsp∂ ga∂ghLfs; kw;Wk; mzpf;Nfhitfsp∂ ga∂ghLfs; kw;Wk; mzpf;Nfhitfsp∂ ga∂ghLfs; kw;Wk; mzpf;Nfhitfsp∂ ga∂ghLfs ;; ;;

( 2 ( 2 ( 2 ( 2 ---- tpdhf;fs; tpdhf;fs; tpdhf;fs; tpdhf;fs; ---- 12 12 12 12 ---- kjpg;ngΩfskjpg;ngΩfskjpg;ngΩfskjpg;ngΩfs;)

vd;w mzpapd; Nru;g;G mzpiaf; fhz;fvd;w mzpapd; Nru;g;G mzpiaf; fhz;fvd;w mzpapd; Nru;g;G mzpiaf; fhz;fvd;w mzpapd; Nru;g;G mzpiaf; fhz;f vd;gijr;vd;gijr;vd;gijr;vd;gijr;

rupghurupghurupghurupghu; jPh;T

vdNt

gapw;rpf; fzf;F vdpy; 2 vd;gijr; rupghu;

kw;Wk;kw;Wk;kw;Wk;kw;Wk; vdpy;vdpy;vdpy;vdpy; vd;gijr; rupghu;vd;gijr; rupghu;vd;gijr; rupghu;vd;gijr; rupghu;

jPh;T ;

=1

=

vdNt

gapw;rpf; fzf;F: kw;Wk; vdpy; vd;gijr; rupghu;

)

2) vdpy; vd;gijr; rupghu;

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26

kw;Wk;kw;Wk;kw;Wk;kw;Wk; vdpy;vdpy;vdpy;vdpy; vd;gijr; rupghuvd;gijr; rupghuvd;gijr; rupghuvd;gijr; rupghu;

jPh;T : kw;Wk;

= =

vdNt

- ,,,,d; Nru;g;G mzp d; Nru;g;G mzp d; Nru;g;G mzp d; Nru;g;G mzp vd epWTf. vd epWTf. vd epWTf. vd epWTf.

jPh;T

Neu;khwzp fhz;fNeu;khwzp fhz;fNeu;khwzp fhz;fNeu;khwzp fhz;f

jPh;T

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27

gapw;rpf; fzf;Ffs : Neu;khwzp fhz;f

vdpy; vd;gijr; rupghu;.

mzpj;juk; fhz;f.mzpj;juk; fhz;f.mzpj;juk; fhz;f.mzpj;juk; fhz;f.

jPh;T

A =

2 2 1 3 3 1

filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ.

,jpy; ,uz;L ©r;rpakw;w epiufs; cs;sjhy;

mzpj;juk; fhz;f.mzpj;juk; fhz;f.mzpj;juk; fhz;f.mzpj;juk; fhz;f.

jPh;T vd;f

∼

∼

filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ ,jpy; 3 ©r;rpakw;w epiufs; cs;sjhy;

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gapw;rpf; fzf;Ffs; : mzpj;juk; fhz;f

mzpKiwapy; Neupa Kiwapy; rkd;ghl;il jPu;f;fTkmzpKiwapy; Neupa Kiwapy; rkd;ghl;il jPu;f;fTkmzpKiwapy; Neupa Kiwapy; rkd;ghl;il jPu;f;fTkmzpKiwapy; Neupa Kiwapy; rkd;ghl;il jPu;f;fTk

jPh;T jug; gl;l rkd;ghLfspd; mzpr; rkd;ghL

and

jPh;Tfs;: ∗∗∗∗ gapw;rpf; fzf;Ffs; mzpKiwapy; Neupa Kiwapy; rkd;ghl;il jPu;f;fTk;

mzpNfhit Kiwapy; jPu;f;fmzpNfhit Kiwapy; jPu;f;fmzpNfhit Kiwapy; jPu;f;fmzpNfhit Kiwapy; jPu;f;f

jPh;T

kw;Wk; vd ,Ug;gjhy; vz;zpf;ifaw;w jPu;Tfisg; ngw;wpUf;Fk;.

Nkw;fz;l njhFg;G vd;w xU jdpr; rkd;ghlhf khWk;.

,jid jPh;f;f vdf; nfhs;f

jPh;T

gapw;rpf; fzf;F mzpf;Nfhit Kiwapy; jPu;f;f.

9) mzpf;Nfhit Kiwapy; gp∂tUk; mrkgbj;jhd rk∂ghl;L;j; njhFg;Gfisj; jPHf;f.9) mzpf;Nfhit Kiwapy; gp∂tUk; mrkgbj;jhd rk∂ghl;L;j; njhFg;Gfisj; jPHf;f.9) mzpf;Nfhit Kiwapy; gp∂tUk; mrkgbj;jhd rk∂ghl;L;j; njhFg;Gfisj; jPHf;f.9) mzpf;Nfhit Kiwapy; gp∂tUk; mrkgbj;jhd rk∂ghl;L;j; njhFg;Gfisj; jPHf;f.

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jPh;T

kw;Wk; njhFg;GxUq;fikT mw;wJ vdNt jPh;Tfs; fpilahJ

10) mzpf;Nfhit Kiwapy; gp∂tUk; mrkgbj;jhd rk∂ghl;L;j; njhFg;Gfisj; jPHf;f.10) mzpf;Nfhit Kiwapy; gp∂tUk; mrkgbj;jhd rk∂ghl;L;j; njhFg;Gfisj; jPHf;f.10) mzpf;Nfhit Kiwapy; gp∂tUk; mrkgbj;jhd rk∂ghl;L;j; njhFg;Gfisj; jPHf;f.10) mzpf;Nfhit Kiwapy; gp∂tUk; mrkgbj;jhd rk∂ghl;L;j; njhFg;Gfisj; jPHf;f.

jPh;T

kw;Wk; ,∂ vy;yh 2 x 2 rpw;wzpf; Nfhitfspd;

kjpg;Gfs; ©r;rpaq;;fs; MFk;. Mdhy; rpy; xU rpw;wzpf;Nfhit ©r;rpakw;wjhAs;sjhy; njhFg;G xUq;fikT mw;wJ. vdNt jPh;Tfs; fpilahJ.

gp∂tUk; rk∂ghLfsp∂ njhFg;G xUq;fikT cilajh vd;gij ju Kiwapy; Muha;f.gp∂tUk; rk∂ghLfsp∂ njhFg;G xUq;fikT cilajh vd;gij ju Kiwapy; Muha;f.gp∂tUk; rk∂ghLfsp∂ njhFg;G xUq;fikT cilajh vd;gij ju Kiwapy; Muha;f.gp∂tUk; rk∂ghLfsp∂ njhFg;G xUq;fikT cilajh vd;gij ju Kiwapy; Muha;f.

jPh;T

∼

,J VWgb tbtpy; cs;sJ. mJ %∂W ©r;rpa kw;w epiufisg; ngw;Ws;sjhy; NkYk;

∴ njhFg;G xUq;fikT mw;wJ vdNt jPh;Tfs; fpilahJ.

gp∂tUk; rk∂ghLfsp∂ njhFg;G xUq;fikT cilajh vd;gij ju Kigp∂tUk; rk∂ghLfsp∂ njhFg;G xUq;fikT cilajh vd;gij ju Kigp∂tUk; rk∂ghLfsp∂ njhFg;G xUq;fikT cilajh vd;gij ju Kigp∂tUk; rk∂ghLfsp∂ njhFg;G xUq;fikT cilajh vd;gij ju Kiwapy; Muha;f.wapy; Muha;f.wapy; Muha;f.wapy; Muha;f.

jPh;T

;

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30

,J VWgb tbtpy; cs;sJ. mJ %∂W ©r;rpa kw;w epiufisg; ngw;Ws;sjhy; NkYk;

∴ njhFg;G xUq;fikT mw;wJ vdNt jPh;Tfs; fpilahJ. gapw;rpf; fzf;F : gp∂tUk; rk∂ghLfsp∂ njhFg;G xUq;fikT cilajh vd;gij ju Kiwapy; Muha;f.

3.3.3.3. fyg;ngz;fs;fyg;ngz;fs;fyg;ngz;fs;fyg;ngz;fs; (gFjp(gFjp(gFjp(gFjp----MMMM----2222----tpdhf;fs;tpdhf;fs;tpdhf;fs;tpdhf;fs;----12121212----kjpg;ngz;fs;)kjpg;ngz;fs;)kjpg;ngz;fs;)kjpg;ngz;fs;)

1) 1) 1) 1) a+iba+iba+iba+ib v∂w jpl;ltbtp;y; vOJf.v∂w jpl;ltbtp;y; vOJf.v∂w jpl;ltbtp;y; vOJf.v∂w jpl;ltbtp;y; vOJf.

jPu;T : =

=

=

=

=

gapw;rpf; fzf;Ffs; a+ib v∂w jpl;ltbtp;y; vOJf.

,∂ tHf;f %yk; fhΩf,∂ tHf;f %yk; fhΩf,∂ tHf;f %yk; fhΩf,∂ tHf;f %yk; fhΩf

jPu;T

tHf;fg;gLj;j

------ (1) ---- (2)

=

------- (3)

~

vdpy; . vdpy;;;; .

.

khw;WKiw khw;WKiw khw;WKiw khw;WKiw ::::

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31

3) ,∂ tHf;f %yk; fhΩf,∂ tHf;f %yk; fhΩf,∂ tHf;f %yk; fhΩf,∂ tHf;f %yk; fhΩf.

jPu;T -

MHf∂jsj;jpy;fyg;ngΩfs;MHf∂jsj;jpy;fyg;ngΩfs;MHf∂jsj;jpy;fyg;ngΩfs;MHf∂jsj;jpy;fyg;ngΩfs; kw;Wk; kw;Wk; kw;Wk; kw;Wk; mikf;Fk; mikf;Fk; mikf;Fk; mikf;Fk; Kf;Nfhzk;Kf;Nfhzk;Kf;Nfhzk;Kf;Nfhzk; xU nrq;Nfhz Kf;NfxU nrq;Nfhz Kf;NfxU nrq;Nfhz Kf;NfxU nrq;Nfhz Kf;Nfhzk; vd epWTfhzk; vd epWTfhzk; vd epWTfhzk; vd epWTf

jPu;T: vDk; Gs;spfs; KiwNa kw;Wk; vDk; fyg;ngΩfis MHf∂jsj;jpy; Fwpf;fl;Lk;

gapw;rpf; fzf;Ffs; : 1) vDk; fyg;ngΩfs; xU rkgf;f Kff;Nfhzj;ij MHf∂ jsj;jpy; cUthf;Fk; v∂W fhl;Lf. 2) kw;Wk; vDk; fyg;ngΩfs; xU nrt;tfj;ij MHf∂ jsj;jpy; xU nrt;tfjij cUthf;Fk; v∂W fhl;Lf. 3) fyg;ngΩfs; vDk; fyg;ngΩfs; MHf∂ jsj;jpy;; xU nrq;Nfhz Kf;Nfhzj;ij mikf;Fk vd epWTf. . . . 4) kw;Wk; vDk; fyg;ngΩfs; xU ,izfuj;ij mikf;Fk; vd epWTf....mikf;Fk

I xUI xUI xUI xU jPh;thff; nfhΩljPh;thff; nfhΩljPh;thff; nfhΩljPh;thff; nfhΩl vDk; rk∂ ghl;b∂ vDk; rk∂ ghl;b∂ vDk; rk∂ ghl;b∂ vDk; rk∂ ghl;b∂

jPh;Tfisf; fhΩfjPh;Tfisf; fhΩfjPh;Tfisf; fhΩfjPh;Tfisf; fhΩf. . . .

jPu;T :

xU jPh;T vdNt kw;;nwhU %yk;

%yk;q;fsp∂ $Ljy;

%yk;q;fsp∂ ngUf;fk;

vdgJ XH fhuzpahfpJ

∴ ≡

x nfOit xg;gpl

kw;;nwhU fhzpahfpwJ

jPHf;f vdNt %yk;q;fs;

gapw;rpf; fzf;Ffs; : I xU jPh;thff; nfhΩl vDk; rk∂ ghl;b∂ jPh;Tfisf; fhΩf.

I xU jPh;thff; nfhΩl vDk; rk∂ ghl;b∂ jPh;Tfisf; fhΩf.



v∂wv∂wv∂wv∂w fyg;ngΩzp∂fyg;ngΩzp∂fyg;ngΩzp∂fyg;ngΩzp∂ kl;L tPr;R fhΩf.kl;L tPr;R fhΩf.kl;L tPr;R fhΩf.kl;L tPr;R fhΩf.

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jPh;T : vd;f.

kl;L kw;Wk; tPr;R

gapw;rpf; fzf;Ffs; : kl;L tPr;R fhΩf.

vdpy;vdpy;vdpy;vdpy; epWTf. epWTf. epWTf. epWTf.

jPu;T :

(i) Given

| | =

| | | | | | . . . | | = | A + i B |

tHf;fg;gLj;j

nghJthf

,∂tPr;R,∂tPr;R,∂tPr;R,∂tPr;R kw;Wkkw;Wkkw;Wkkw;Wk; ,∂tPr;R,∂tPr;R,∂tPr;R,∂tPr;R vdpyvdpyvdpyvdpy; vd epWTfvd epWTfvd epWTfvd epWTf

jPu;T

⇒

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33

gapw;rpf; fzf;Ffs; : vDk; Gs;sp fyg;nΩ khwp Z - If; Fwpj;jhy; P-,∂ epakghijiaf; fhΩf.

vd;wvd;wvd;wvd;w ,U fyg;ngΩfSf;F,U fyg;ngΩfSf;F,U fyg;ngΩfSf;F,U fyg;ngΩfSf;F

(i) vd epWTf. vd epWTf. vd epWTf. vd epWTf.

jPu;T : 1 1 1 2 2 2

z 1 kw;Wk; z 2

gapw;rpf; fzf;Ffs; : vd;w ,U fyg;ngΩfSf;F

vd epWTf.

v∂gJ kpif KO vΩ vdpyv∂gJ kpif KO vΩ vdpyv∂gJ kpif KO vΩ vdpyv∂gJ kpif KO vΩ vdpy ;; ;; vd epWTf. vd epWTf. vd epWTf. vd epWTf.

jPu;T

=

.

gapw;rpf; fzf;Ffs; : n v∂gJ kpif KO vΩ vdpy; .

.

+

vd epWTf.vd epWTf.vd epWTf.vd epWTf.

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34

vdpy;vdpy;vdpy;vdpy; vd vd vd vd

ep&gpep&gpep&gpep&gp.

Kf;Nfhzr; rkdpypia vOjp epWTf. Kf;Nfhzr; rkdpypia vOjp epWTf. Kf;Nfhzr; rkdpypia vOjp epWTf. Kf;Nfhzr; rkdpypia vOjp epWTf. ,U fyg;ngΩfsp∂ $Ljyp∂ kl;L mt;tpU vΩfsp∂ kl;Lfsp∂ $LjYf;Ff; FiwthfNth my;yJ rkkhfNth ,Uf;Fk; mjhtJ

jPu;T : MHf∂ jsj;jpy; kw;Wk; v∂w ,U fyg;ngΩfis A kw;Wk; B Gs;spfshy; Fwpf;f.

OACB v∂w ,izfuj;ij epiwT nra;f. ,q;F C v∂gJ v∂w fyg;ngΩiz Fwpf;fpwJ

kw;Wk;

xU Kf;Nfzj;jpy; VNjDk; ,U gf;f ePsq;fsp∂ $Ljy; %∂whtJ gf;f ePsj;ij tpl nghpaJ

apypUe;J

my;yJ

NkYk; Gs;spfs; xU Nfhliktd vdpy;

,ypUe;J

RUf;FfRUf;FfRUf;FfRUf;Ff

jPu;T :

gapw;rpf; fzf;Ffß : RUf;Ff

vdpy; gp∂ tUgdtw;iw epWTf vdpy; gp∂ tUgdtw;iw epWTf vdpy; gp∂ tUgdtw;iw epWTf vdpy; gp∂ tUgdtw;iw epWTf :

jPu;T :

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35

v∂f

nka; kw;Wk; fw;gidg; gFjpfis xg;gpl>

gapw;rpf; fzf;Ffß : vdpy; gp∂ tUgdw;iw epWTf :

6666----tifEΩfzpjk;:tifEΩfzpjk;:tifEΩfzpjk;:tifEΩfzpjk;: ga∂ghLfs;ga∂ghLfs;ga∂ghLfs;ga∂ghLfs;----IIIIIIII(gFjp(gFjp(gFjp(gFjp----MMMM----1111----tpdhtpdhtpdhtpdh----6666----kjpg;ngz;fs;)kjpg;ngz;fs;)kjpg;ngz;fs;)kjpg;ngz;fs;)

vdpy;vdpy;vdpy;vdpy; vd vd vd vd ep&gpep&gpep&gpep&gp.

,

jPu;T :

$l;l,

vdpy; vdf; fhl;Lf.

jPu;T :

$l;l

rq;fpyp tpjpiag; ga∂gLj;jp rq;fpyp tpjpiag; ga∂gLj;jp rq;fpyp tpjpiag; ga∂gLj;jp rq;fpyp tpjpiag; ga∂gLj;jp fhΩf. fhΩf. fhΩf. fhΩf. ,q;F ,q;F ,q;F ,q;F t – t

jPu;T

,q;F ,q;F ,q;F ,q;F vdpy; rq;fpyp tpjpia vdpy; rq;fpyp tpjpia vdpy; rq;fpyp tpjpia vdpy; rq;fpyp tpjpia

ga∂gLj;jp ga∂gLj;jp ga∂gLj;jp ga∂gLj;jp fzf;fpLf. fzf;fpLf. fzf;fpLf. fzf;fpLf.

jPu;T :

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36

gapw;rpf; fzf;Ffs; : ,q;F vdpy; rq;fpyp tpjpia ga∂gLj;jp

fzf;fpLf. . . . – 1 ,q;F vdpy; rq;fpyp tpjpia ga∂gLj;jp , fzf;fpLf.

2 2 ,q;F,q;F,q;F,q;F vdpy; rq;fpyp tpjpia ga∂gLj;jp vdpy; rq;fpyp tpjpia ga∂gLj;jp vdpy; rq;fpyp tpjpia ga∂gLj;jp vdpy; rq;fpyp tpjpia ga∂gLj;jp

kw;Wkkw;Wkkw;Wkkw;Wk; fzf;fpLf.fzf;fpLf.fzf;fpLf.fzf;fpLf.

jPu;T :

gapw;rpf; fzf;F : – 1 ,q;F vdpy; rq;fpyp tpjpia

ga∂gLj;jp kw;Wk; fzf;fpLf.

2) vd;;W ,Uf;FkhW tiuaWf;fg;ggLfpwJ vdpy;

kw;Wk; If; fhΩf.

kw;Wk; MdJ ,y; Mk; gb rkgbj;jhd rhh;ghap∂

vd epWTf.

jPu;T

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37

xU jdp Cryp∂ ePsk; kw;Wk; KO miyT Neuk; vdpy; ( v∂gJ khwpyp) Jdp Cryp∂ ePsk; nr.kP. ,Ue;J nr.kP. f;F khWk; NghJ Neuj;jpy; Vw;gLk; rj tPjg; gpioiaf; fzf;fpLf

jPu;T :

miyT Neuj;jpy; Vw;gLk; rj tPjg; gpio : FiwT MFk;;.

7 – njhifnjhifnjhifnjhif EΩfzpjk;EΩfzpjk;EΩfzpjk;EΩfzpjk; ga∂ghLfs;.(gFjpga∂ghLfs;.(gFjpga∂ghLfs;.(gFjpga∂ghLfs;.(gFjp ---- MMMM----1111----tpdhtpdhtpdhtpdh :::: 6666----kjpg;ngΩfs;)kjpg;ngΩfs;)kjpg;ngΩfs;)kjpg;ngΩfs;)

kjpg;gpLfkjpg;gpLfkjpg;gpLfkjpg;gpLf ::::

jPu;T :

------(1)

------(2)

2) kjpg;gpLf :kjpg;gpLf :kjpg;gpLf :kjpg;gpLf :

jPu;T :

------(1)

------(2)

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gapw;rpf; fzf;F kjpg;gpLf :

3) kjpg;gpLfkjpg;gpLfkjpg;gpLfkjpg;gpLf : : : :

jPu;T

gapw;rpf; fzf;F kjpg;gpLf :

kjpg;gpLfkjpg;gpLfkjpg;gpLfkjpg;gpLf : : : :

gapw;rpf; fzf;F kjpg;gpLf

kjpg;gpLf : kjpg;gpLf : kjpg;gpLf : kjpg;gpLf :

jPu;T kw;Wk;

ngh;Ndhyp #j;jpuj;ijg; ga∂gLj;j

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kjpg;gpLf kjpg;gpLf kjpg;gpLf kjpg;gpLf : : : :

jPu;T

gapw;rpf; fzf;Ffs; kjpg;gpLf :

v∂w gutisaj;jpw;Fk; mj∂ nrt;tfyj;jpw;Fk; ,ilg;gl;l gug;gpidf; fhΩf.v∂w gutisaj;jpw;Fk; mj∂ nrt;tfyj;jpw;Fk; ,ilg;gl;l gug;gpidf; fhΩf.v∂w gutisaj;jpw;Fk; mj∂ nrt;tfyj;jpw;Fk; ,ilg;gl;l gug;gpidf; fhΩf.v∂w gutisaj;jpw;Fk; mj∂ nrt;tfyj;jpw;Fk; ,ilg;gl;l gug;gpidf; fhΩf.

jPu;T : Njitahdgug;ghdJ kw;Wk; mr;rpdhy; #og;gl;l gug;gp∂ ,U klq;fhFk;

Njitahdgug;G rJu myFfß

gapw;rpf; fzf;F : v∂w NfhL v∂w NfhLfß kw;Wk; mr;R Mfpatw;why; milgLk; muq;fj;jp∂ gug;gpidf; fhΩf

v∂w ePßtl;;lk;v∂w ePßtl;;lk;v∂w ePßtl;;lk;v∂w ePßtl;;lk; Vw;gLj;;Jk; gug;gpid Fw;wr;rpidg; nghWj;Jr; Vw;gLj;;Jk; gug;gpid Fw;wr;rpidg; nghWj;Jr; Vw;gLj;;Jk; gug;gpid Fw;wr;rpidg; nghWj;Jr; Vw;gLj;;Jk; gug;gpid Fw;wr;rpidg; nghWj;Jr;

Row;wpdhy; Vw;gLk; jplg;nghUsp∂ fd msT fhΩfRow;wpdhy; Vw;gLk; jplg;nghUsp∂ fd msT fhΩfRow;wpdhy; Vw;gLk; jplg;nghUsp∂ fd msT fhΩfRow;wpdhy; Vw;gLk; jplg;nghUsp∂ fd msT fhΩf....

jPu;T vdpy;

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fd msT fd myFfß.

9.jdpepiyf;9.jdpepiyf;9.jdpepiyf;9.jdpepiyf; fzf;fpayfzf;fpayfzf;fpayfzf;fpay ;; ;;(gFjp(gFjp(gFjp(gFjp----MMMM :::: 2222----tpdhf;fs;tpdhf;fs;tpdhf;fs;tpdhf;fs;----12121212----kjpgngΩfs;)kjpgngΩfs;)kjpgngΩfs;)kjpgngΩfs;)

tpjpfs;: tpjpfs;: tpjpfs;: tpjpfs;:

: : : : kw;Wk; kw;Wk; kw;Wk; kw;Wk; ,uΩLN,uΩLN,uΩLN,uΩLNk nka;kjpg;G k nka;kjpg;G k nka;kjpg;G k nka;kjpg;G ngw;wpUg;gp∂ ngw;wpUg;gp∂ ngw;wpUg;gp∂ ngw;wpUg;gp∂ ∂ kjpg;Gk; ∂ kjpg;Gk; ∂ kjpg;Gk; ∂ kjpg;Gk; MFk;.MFk;.MFk;.MFk;.

: : : : my;yJ my;yJ my;yJ my;yJ my;yJ ,uΩLNk nka;kjpg;G my;yJ ,uΩLNk nka;kjpg;G my;yJ ,uΩLNk nka;kjpg;G my;yJ ,uΩLNk nka;kjpg;G ngw;wpUg;gp∂ ngw;wpUg;gp∂ ngw;wpUg;gp∂ ngw;wpUg;gp∂ ∂ kjpg;Gk; ∂ kjpg;Gk; ∂ kjpg;Gk; ∂ kjpg;Gk; MFk;.MFk;.MFk;.MFk;.

: : : : kw;Wk; kw;Wk; kw;Wk; kw;Wk; ,uΩLNk nka;kjpg;G ,uΩLNk nka;kjpg;G ,uΩLNk nka;kjpg;G ,uΩLNk nka;kjpg;G ngw;wpUg;gp∂ ngw;wpUg;gp∂ ngw;wpUg;gp∂ ngw;wpUg;gp∂ ∂ kjpg;Gk; ∂ kjpg;Gk; ∂ kjpg;Gk; ∂ kjpg;Gk; MFk;.MFk;.MFk;.MFk;.

: : : : my;yJ my;yJ my;yJ my;yJ my;yJ ,uΩLNk nka;kjpg;G my;yJ ,uΩLNk nka;kjpg;G my;yJ ,uΩLNk nka;kjpg;G my;yJ ,uΩLNk nka;kjpg;G ngw;wpUg;gp∂ ngw;wpUg;gp∂ ngw;wpUg;gp∂ ngw;wpUg;gp∂ ∂ kjpg;Gk; ∂ kjpg;Gk; ∂ kjpg;Gk; ∂ kjpg;Gk; MFk;.MFk;.MFk;.MFk;.

: : : : nka;kjpg;G nka;kjpg;G nka;kjpg;G nka;kjpg;G vdpy; vdpy; vdpy; vdpy; ∂ kjpg;Gk; ∂ kjpg;Gk; ∂ kjpg;Gk; ∂ kjpg;Gk; MFk;. MFk;. MFk;. MFk;.

nka;kjpg;G nka;kjpg;G nka;kjpg;G nka;kjpg;G vdpy; vdpy; vdpy; vdpy; ∂ kjpg;Gk; ∂ kjpg;Gk; ∂ kjpg;Gk; ∂ kjpg;Gk; MFk;.MFk;.MFk;.MFk;.

: : : : nka;kjpg;G nka;kjpg;G nka;kjpg;G nka;kjpg;G ,Ue;J ,Ue;J ,Ue;J ,Ue;J nka;kjpg;G nka;kjpg;G nka;kjpg;G nka;kjpg;G Mf ,Ug;gp∂ Mf ,Ug;gp∂ Mf ,Ug;gp∂ Mf ,Ug;gp∂ ∂ kjpg;G ∂ kjpg;G ∂ kjpg;G ∂ kjpg;G MFk;.MFk;.MFk;.MFk;.

: : : : kw;Wk; kw;Wk; kw;Wk; kw;Wk; ,uΩLNk xNu nka;kjpg;G ngw;wpUg;gp∂ ,uΩLNk xNu nka;kjpg;G ngw;wpUg;gp∂ ,uΩLNk xNu nka;kjpg;G ngw;wpUg;gp∂ ,uΩLNk xNu nka;kjpg;G ngw;wpUg;gp∂ ∂ kjpg;G ∂ kjpg;G ∂ kjpg;G ∂ kjpg;G MFk; MFk; MFk; MFk;

,∂ n,∂ n,∂ n,∂ nka; ml;ltiz mikf;f.ka; ml;ltiz mikf;f.ka; ml;ltiz mikf;f.ka; ml;ltiz mikf;f.

jPh;T:

p q r p∨∨∨∨q ( p ∨∨∨∨ q ) ∨ r

T T T T T

T T F T T

T F T T T

T F F T T

F T T T T

F T F T T

F F T F T

F F F F F 2) nka; ml;ltiz mikf;f. 2) nka; ml;ltiz mikf;f. 2) nka; ml;ltiz mikf;f. 2) nka; ml;ltiz mikf;f.

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jPh;T:

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

3) nka; ml;ltiz mikf;f. 3) nka; ml;ltiz mikf;f. 3) nka; ml;ltiz mikf;f. 3) nka; ml;ltiz mikf;f. jPh;T:

p q r ∼ r p ∧ q (p ∧ q) ∨ ∼ r

T T T F T T

T T F T T T

T F T F F F

T F F T F T

F T T F F F

F T F T F T

F F T F F F

F F F T F T

gapw;rpf; fzf;Ffs;: nka; ml;ltiz mikf;f.

v∂gJ xU nka;ik vdf; fhl;Lf.v∂gJ xU nka;ik vdf; fhl;Lf.v∂gJ xU nka;ik vdf; fhl;Lf.v∂gJ xU nka;ik vdf; fhl;Lf.

jPh;T ,∂ nka; ml;ltiz

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; Mjyhy; xU nka;ik MFk;.

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gapw;rpf; fzf;Ffs; : v∂gJ xU nka;ik vdf; fhl;Lf. (Oct-2009) v∂gJ xU nka;ik vdf; fhl;Lf.

xU KuΩghL vdf;xU KuΩghL vdf;xU KuΩghL vdf;xU KuΩghL vdf;fhl;Lffhl;Lffhl;Lffhl;Lf


filrp epuy; KOtJk; Mjyhy; xU KuΩghL MFk; gapw;rpf; fzf;F : 1) xU KuΩghL vdf; fhl;Lf .

v∂gJ xU nka;ikah vd;gjid nka;

ml;ltiziaf; nfhΩL jPh;khdpf;f.


filrp epuy; KOtJk; Mjyhy; xU nka;ik MFk;. gapw;rpf; fzf;Ffs; : v∂gJ xU nka;ikah vd;gjid nka;

ml;ltiziaf; nfhΩL jPh;.

vdf; fhl;Lf. vdf; fhl;Lf. vdf; fhl;Lf. vdf; fhl;Lf.

jPh;T: ,∂ nka; ml;ltiz ,∂ nka; ml;ltiz

filrp epuy;fs; xNukhjphpahdit ∴

pppp qqqq ∼∼∼∼ pppp ∼∼∼∼ qqqq ((((∼∼∼∼ p ) p ) p ) p ) ∧∧∧∧ ( ( ( ( ∼∼∼∼ q )q )q )q ) ∧∧∧∧

pppp

T T F F F F

T F F T F F

F T T F F F

F F T T T F

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

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vdf; vdf; vdf; vdf; fhl;Lf. fhl;Lf. fhl;Lf. fhl;Lf.

jPh;T: ,∂ nka; ml;ltiz

,∂ nka; ml;ltiz

∴

gapw;rpf; fzf;Ffs; vdf; fhl;Lf.

vdf; fhl;Lf.

vdf; fhl;Lf.

vdf; fhl;Lf. kw;Wk; rkhdkw;wit vdf; fhl;Lf.

, , , MMMMfpa eh∂F mzpfSk; mlq;;fpa fzk; fpa eh∂F mzpfSk; mlq;;fpa fzk; fpa eh∂F mzpfSk; mlq;;fpa fzk; fpa eh∂F mzpfSk; mlq;;fpa fzk;

mzp;g; ngUf;fyp∂ fPo; mzp;g; ngUf;fyp∂ fPo; mzp;g; ngUf;fyp∂ fPo; mzp;g; ngUf;fyp∂ fPo; xU vgpypad; Fyk; mikf;Fk; vd xU vgpypad; Fyk; mikf;Fk; vd xU vgpypad; Fyk; mikf;Fk; vd xU vgpypad; Fyk; mikf;Fk; vd fhl;Lf.fhl;Lf.fhl;Lf.fhl;Lf.

jPh;T vd;f

vd;f mzpg; ngUf;fy;.

• I A B C

I I A B C

A A I C B

B B C I A

C C B A I

ml;ltizapypUe;J

1) milg;G tpjp: ml;ltizapd; vy;yh cWg;GfSk; y; cs;sd. / milg;G tpjp cz;ikahfpwJ.

2) Nrh;g;G tpjp : ngUf;fy; Nrh;g;G tpjp cilaJ. 3) rkdp cWg;G MFk;.

4) vjph;kiw tpjp: d; vjph;kiw

pppp qqqq ∼∼∼∼ pppp ∼∼∼∼ qqqq ∼∼∼∼ p p p p ∨∨∨∨ qqqq ∼∼∼∼q q q q ∨∨∨∨ pppp ((((∼∼∼∼ p p p p ∨∨∨∨ q ) q ) q ) q ) ∧∧∧∧ ((((∼∼∼∼q q q q ∨∨∨∨ p )p )p )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

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d; vjph;kiw d; vjph;kiw d; vjph;kiw

5) ml;ltizapypUe;J ghpkhw;W tpjp cz;ikahfpwJ

vdNt xU vgpypad; FykhFk;.

gapw;rpf; fzf;F : xU Fyk; vd fhl;Lf. v∂w Fyj;jpy; cßs xt;nthU cWg;Gf;Fk; thpiriaf; fhΩfv∂w Fyj;jpy; cßs xt;nthU cWg;Gf;Fk; thpiriaf; fhΩfv∂w Fyj;jpy; cßs xt;nthU cWg;Gf;Fk; thpiriaf; fhΩfv∂w Fyj;jpy; cßs xt;nthU cWg;Gf;Fk; thpiriaf; fhΩf

jPh;T ,J ,∂ $l;ly; kl;Lf;F xU mgpypa∂ FykhFk;.

rkdp cWg;G kw;Wk;

O ( ) = 1

O I Fiwe;j gl;rk; 6 jlitfß $l;bdhy; fpilf;Fk;;

O I Fiwe;j gl;rk; 3 jlitfß $l;bdhy; fpilf;Fk;




xU Fyj;jp∂ ePf;fy; tpjpfis vOjp epWTf. xU Fyj;jp∂ ePf;fy; tpjpfis vOjp epWTf. xU Fyj;jp∂ ePf;fy; tpjpfis vOjp epWTf. xU Fyj;jp∂ ePf;fy; tpjpfis vOjp epWTf.

jPh;T G xU Fyk; v∂f. v∂f (,lJ ePf;fy; tpjp) (tyJ ePf;fy; tpjp) ep&gzk;;: –1 –1

–1 –1

–1 –1

–1 –1

XH Fyk; v∂f. XH Fyk; v∂f. XH Fyk; v∂f. XH Fyk; v∂f. a,b a,b a,b a,b mt;;thuhhap∂ mt;;thuhhap∂ mt;;thuhhap∂ mt;;thuhhap∂ vd epp&gpf;fvd epp&gpf;fvd epp&gpf;fvd epp&gpf;f.

jPh;T : MdJ ,∂ vjph;kiw vdf; fl;bdhy; NghJkhdJ

vdNt ,∂ vjph;kiw

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gFjpgFjpgFjpgFjp----,,,,----tpdhtpdhtpdhtpdh----tpilfß(80tpilfß(80tpilfß(80tpilfß(80---- kjpg;ngz;fs;kjpg;ngz;fs;kjpg;ngz;fs;kjpg;ngz;fs;))))

ntf;lu;,aw;fzpjk;;ntf;lu;,aw;fzpjk;;ntf;lu;,aw;fzpjk;;ntf;lu;,aw;fzpjk;;----2222----tpdhf;fs;tpdhf;fs;tpdhf;fs;tpdhf;fs;----20202020----kjpg;ngz;fs;)kjpg;ngz;fs;)kjpg;ngz;fs;)kjpg;ngz;fs;)

xU Kf;Nfhzj;jp∂ Fj;Jf;NfhLfß xNu Gßspapy; re;jpf;Fk; v∂gjidxU Kf;Nfhzj;jp∂ Fj;Jf;NfhLfß xNu Gßspapy; re;jpf;Fk; v∂gjidxU Kf;Nfhzj;jp∂ Fj;Jf;NfhLfß xNu Gßspapy; re;jpf;Fk; v∂gjidxU Kf;Nfhzj;jp∂ Fj;Jf;NfhLfß xNu Gßspapy; re;jpf;Fk; v∂gjid ntf;lh; Kiwapy; epWTf.ntf;lh; Kiwapy; epWTf.ntf;lh; Kiwapy; epWTf.ntf;lh; Kiwapy; epWTf.

jPh;T -,y; Fj;Jf;;;;NfhLfß Ak;;; -,y; re;jpf;fp∂wd. Fj;Jf;;;;NfhLfß xNu Gßsp topNar; nry;Yk; vd;gij epWt MdJ f;F nrq;Fj;jhf ,Uf;Fk; vdf; fhl;bdhy;

NghJkhdJ. I Mjpahff; nfhßf. ,∂ epiy ntf;lh;fß KiwNa vd;f.

kw;Wk;;

•

⇒ •

⇒ • ( — ) = 0

⇒ • – • = 0 → (1)

⊥ ⇒ •

⇒ •

⇒ • ( – ) = 0

⇒ • – • = 0 → (2)

⇒ • – • = 0

⇒ ( – ) • = 0

⇒ •

⇒ OC ⊥ AB

vdNt %∂W Fj;Jf; NfhLfSk; xNu Gßspapy; re;jpf;Fk; NfhLfshFk;.

vd vd vd vd ntf;lh; Kiwapy; ntf;lh; Kiwapy; ntf;lh; Kiwapy; ntf;lh; Kiwapy; epWTf.epWTf.epWTf.epWTf. jPh;T I ikakhff; nfhΩl myF tl;lj;jp∂ kPJ v∂w ,U Gßspfis vLj;Jf; nfhßf.

kw;Wk; Mdit mr;Rl∂ Vw;gLj;Jk; Nfhzq;fß KiwNa v∂f.

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,∂ Maj; njhiyTfß KiwNa kw;Wk;

v∂w myF ntf;lh;fis mr;R jpirfspy; vLj;Jf; nfhßf.

kw;Wk;

= ( )٠( )

tiuaiuap∂gb

vdNt *

jPh;T I ikakhff; nfhΩl myF tl;lj;jp∂ kPJ v∂w ,U Gßspfis vLj;Jf; nfhßf. kw;Wk; Mdit mr;Rl∂ Vw;gLj;Jk; Nfhzq;fß KiwNa v∂f.



kw;Wk;

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tiuaiuap∂gb

vdNt

jPh;T I ikakhff; nfhΩl myF tl;lj;jp∂ kPJ v∂w ,U Gßspfis vLj;Jf; nfhßf.




kw;Wk;

=

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------- (1)

tiuaiuap∂gb

--------------- (2)

ypUe;J *

jPh;T I ikakhff; nfhΩl myF tl;lj;jp∂ kPJ v∂w ,U Gßspfis vLj;Jf; nfhßf.




kw;Wk;

=

tiuaiuap∂gb

~ . *

, , , vdpy; vdpy; vdpy; vdpy;

( vd;gijr; rupghu;f;f.vd;gijr; rupghu;f;f.vd;gijr; rupghu;f;f.vd;gijr; rupghu;f;f.

jPh;T = =

= =

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( =

= = – 2

= = 1

∴ = .

vdNt (

vdpy; vdpy; vdpy; vdpy;

vd;gijr; rupghu;f;f.vd;gijr; rupghu;f;f.vd;gijr; rupghu;f;f.vd;gijr; rupghu;f;f.

jPh;T = =

= =

( =

= = – 2

= = 1

∴ = .

vdNt ( *

8888 kw;Wk; kw;Wk; kw;Wk; kw;Wk; vd;w NfhLfs; ntl;bf; nfhs;Sk; vdf; vd;w NfhLfs; ntl;bf; nfhs;Sk; vdf; vd;w NfhLfs; ntl;bf; nfhs;Sk; vdf; vd;w NfhLfs; ntl;bf; nfhs;Sk; vdf;

fhl;Lf. NkYk; mit ntl;Lk; Gs;spiaf; fhz;f.fhl;Lf. NkYk; mit ntl;Lk; Gs;spiaf; fhz;f.fhl;Lf. NkYk; mit ntl;Lk; Gs;spiaf; fhz;f.fhl;Lf. NkYk; mit ntl;Lk; Gs;spiaf; fhz;f.

jPu;T

NfhLfs; ntl;bf; nfhs;s epge;jid

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) =

NkYk; , ,izaw;wit. vdNt NfhLfs; ntl;bf;nfhs;Sk;.

vd;w Nfhl;by; cs;s VNjDk; xU Gs;spapd;

mikg;G MFk;.

vd;w Nfhl;by; cs;s VNjDk; xU Gs;spapd;

mikg;G

NfhLfs; ntl;Lk; Gs;spapy;

jPh;f;f

ntl;Lk; Gs;sp MFk;.

kw;Wk; kw;Wk; kw;Wk; kw;Wk; vvvvd;w NfhLfs; ntl;bf; nfhs;Sk; vdf; d;w NfhLfs; ntl;bf; nfhs;Sk; vdf; d;w NfhLfs; ntl;bf; nfhs;Sk; vdf; d;w NfhLfs; ntl;bf; nfhs;Sk; vdf;

fhl;Lf. NkYk; mit ntl;Lk; Gs;spiaf; fhz;f. fhl;Lf. NkYk; mit ntl;Lk; Gs;spiaf; fhz;f. fhl;Lf. NkYk; mit ntl;Lk; Gs;spiaf; fhz;f. fhl;Lf. NkYk; mit ntl;Lk; Gs;spiaf; fhz;f.

jPu;T :

NfhLfs; ntl;bf; nfhs;s epge;jid

=

NkYk; , ,izaw;wit. vdNt NfhLfs; ntl;bf;nfhs;Sk;.

vd;w Nfhl;by; cs;s

VNjDk; xU Gs;spapd; mikg;G MFk;.

vd;w Nfhl;by; cs;s

VNjDk; xU Gs;spapd; mikg;G MFk;.

NfhLfs; ntl;Lk; Gs;spapy;

kw;Wk;

kw;Wk;

ntl;Lk; Gs;sp MFk;.

v∂w Nfhl;ilcßslf;fpaJk; v∂w Nfhl;ilcßslf;fpaJk; v∂w Nfhl;ilcßslf;fpaJk; v∂w Nfhl;ilcßslf;fpaJk; v∂w Nfhl;bw;Fv∂w Nfhl;bw;Fv∂w Nfhl;bw;Fv∂w Nfhl;bw;F

,izahdJkhd,izahdJkhd,izahdJkhd,izahdJkhd jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;;

fhΩf.fhΩf.fhΩf.fhΩf.

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jPu;T: Njitahd jskhdJ v∂w Nfhl;ilcßslf;fpajhy;; jskhdJ

v∂w GßspiaAk; v∂w ,iz ntf;liuAk; cßslf;fpaJ.

NkYk; v∂w ntf;lUf;F ,izahdJ.

jsj;jp∂ ntf;lh; rk∂ghL

= ( ) + s ( ) + t ( )

fhh;Brpa∂ rk∂ghL

v∂w Gßsp topr; nry;tJk; v∂w Gßsp topr; nry;tJk; v∂w Gßsp topr; nry;tJk; v∂w Gßsp topr; nry;tJk; v∂w v∂w v∂w v∂w kw;Wkkw;Wkkw;Wkkw;Wk; v∂w v∂w v∂w v∂w

NfhLfSf;;FNfhLfSf;;FNfhLfSf;;FNfhLfSf;;F ,izahdJkhd,izahdJkhd,izahdJkhd,izahdJkhd jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; fhΩf.jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; fhΩf.jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; fhΩf.jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; fhΩf.

jPu;T : : kw;Wk;

ntf;lh;;;; rk∂ghL

= ( ) + s ( ) + t ( )


12) 12) 12) 12) topNar; nry;yf; $baJk; topNar; nry;yf; $baJk; topNar; nry;yf; $baJk; topNar; nry;yf; $baJk; kw;Wk; kw;Wk; kw;Wk; kw;Wk; Mfpa Mfpa Mfpa Mfpa

NfhLfSf;;F ,izahf cßsJkhdNfhLfSf;;F ,izahf cßsJkhdNfhLfSf;;F ,izahf cßsJkhdNfhLfSf;;F ,izahf cßsJkhd jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂gjsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂gjsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂gjsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; fhΩf.hLfisf;; fhΩf.hLfisf;; fhΩf.hLfisf;; fhΩf.

jPh;T :

kw;Wk;

ntf;lh;;;; rk∂ghL

= ( 2 ) + s (3 ) + t ( )


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v∂w Gßsp topr; nry;tJk; v∂w Gßsp topr; nry;tJk; v∂w Gßsp topr; nry;tJk; v∂w Gßsp topr; nry;tJk; kw;Wkkw;Wkkw;Wkkw;Wk; MfpaMfpaMfpaMfpa

jsq;fSf;;Fr; nrq;Fj;jhdJkhd mike;j jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; jsq;fSf;;Fr; nrq;Fj;jhdJkhd mike;j jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; jsq;fSf;;Fr; nrq;Fj;jhdJkhd mike;j jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; jsq;fSf;;Fr; nrq;Fj;jhdJkhd mike;j jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; fhΩffhΩffhΩffhΩf....

jPu;T : = kw;Wk;

jsj;jp∂ ntf;lh; rk∂ghL

= ( ) + s ( ) + t ( 3i + j + 2k )


⇒

gapw;rpf; fzf;F : v∂w Gßsp topr; nry;tJk; kw;Wk;

2 Mfpa jsq;fSf;;Fr; nrq;Fj;jhdJkhd mike;j jsj;jp∂ ntf;lh; kw;Wk;

fhh;Brpa∂ rk∂ghLfisf;; fhΩf.

kw;Wk; kw;Wk; kw;Wk; kw;Wk; v∂w Gßspfß topNar; nry;yf; $baJk; v∂w Gßspfß topNar; nry;yf; $baJk; v∂w Gßspfß topNar; nry;yf; $baJk; v∂w Gßspfß topNar; nry;yf; $baJk;

v∂w Nfhl;ilbw;F ,izahdJkhdv∂w Nfhl;ilbw;F ,izahdJkhdv∂w Nfhl;ilbw;F ,izahdJkhdv∂w Nfhl;ilbw;F ,izahdJkhd jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;; jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf;;

fhΩf.fhΩf.fhΩf.fhΩf.

jPu;T : kw;Wk;

ntf;lh; rk∂ghL

= (1—s) ( ) + s ( ) + t ( )


⇒

15) (1, 2, 3) 15) (1, 2, 3) 15) (1, 2, 3) 15) (1, 2, 3) kw;Wk; kw;Wk; kw;Wk; kw;Wk; (2, 3, 1) (2, 3, 1) (2, 3, 1) (2, 3, 1) v∂w Gßspfß topNar; nry;yf; $baJk; v∂w Gßspfß topNar; nry;yf; $baJk; v∂w Gßspfß topNar; nry;yf; $baJk; v∂w Gßspfß topNar; nry;yf; $baJk; v∂w v∂w v∂w v∂w

jsj;jpw;F nrq;Fj;jhfTk; mike;j jsj;jp∂ ntf;ljsj;jpw;F nrq;Fj;jhfTk; mike;j jsj;jp∂ ntf;ljsj;jpw;F nrq;Fj;jhfTk; mike;j jsj;jp∂ ntf;ljsj;jpw;F nrq;Fj;jhfTk; mike;j jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisf; fhΩf.h; kw;Wk; fhh;Brpa∂ rk∂ghLfisf; fhΩf.h; kw;Wk; fhh;Brpa∂ rk∂ghLfisf; fhΩf.h; kw;Wk; fhh;Brpa∂ rk∂ghLfisf; fhΩf.

( March – 2006, Sep. – 2006, Oct. – 2007, June – 2008, )

jPh;T v∂w jsj;jp∂ nrq;Fj;J ntf;lh;

,e;j ntf;lh; Njitad jsj;jpw;F ,izahf cßsJ.

; ;

ntf;lh; rk∂ghL

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53


⇒ 16) 16) 16) 16) kw;Wk; kw;Wk; kw;Wk; kw;Wk; Mfpa Gßspfß topNar; nry;yf; $baJk; Mfpa Gßspfß topNar; nry;yf; $baJk; Mfpa Gßspfß topNar; nry;yf; $baJk; Mfpa Gßspfß topNar; nry;yf; $baJk;

v∂w jsj;jpw;F nv∂w jsj;jpw;F nv∂w jsj;jpw;F nv∂w jsj;jpw;F nrq;Fj;jhf miktJkhd jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rq;Fj;jhf miktJkhd jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rq;Fj;jhf miktJkhd jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rq;Fj;jhf miktJkhd jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂

rk∂ghLfisff; fhΩf. rk∂ghLfisff; fhΩf. rk∂ghLfisff; fhΩf. rk∂ghLfisff; fhΩf.

jPh;T ; ;

ntf;lh;;;; rk∂ghL


⇒

17) 17) 17) 17) v∂w Nfhl;ilcßslf;fpaJk; v∂w Nfhl;ilcßslf;fpaJk; v∂w Nfhl;ilcßslf;fpaJk; v∂w Nfhl;ilcßslf;fpaJk; v∂w Gßsp topNar; nry;yf; v∂w Gßsp topNar; nry;yf; v∂w Gßsp topNar; nry;yf; v∂w Gßsp topNar; nry;yf;

$baJkhd jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisff; $baJkhd jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisff; $baJkhd jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisff; $baJkhd jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisff; fhΩf. fhΩf. fhΩf. fhΩf.

jPh;T : ; ;

ntf;lh;;;; rk∂ghL


⇒ 8

kw;Wk; kw;Wk; kw;Wk; kw;Wk; Mfpa Gßspfß topNar; nry;yf; $ba jsj;jp∂ ntf;lh; Mfpa Gßspfß topNar; nry;yf; $ba jsj;jp∂ ntf;lh; Mfpa Gßspfß topNar; nry;yf; $ba jsj;jp∂ ntf;lh; Mfpa Gßspfß topNar; nry;yf; $ba jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghLfisff; fhΩf.kw;Wk; fhh;Brpa∂ rk∂ghLfisff; fhΩf.kw;Wk; fhh;Brpa∂ rk∂ghLfisff; fhΩf.kw;Wk; fhh;Brpa∂ rk∂ghLfisff; fhΩf.

jPh;T ; ;

xNu Nfhl;likahj nfhLf;fg;gl;l %∂W Gßspfß topNar; nry;Yk; jsj;jp∂

ntf;lh; rk∂ghL


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⇒

kw;Wk; kw;Wk; kw;Wk; kw;Wk; Mfpatw;iw epiy ntf;lhf;fshff; nfhΩl Mfpatw;iw epiy ntf;lhf;fshff; nfhΩl Mfpatw;iw epiy ntf;lhf;fshff; nfhΩl Mfpatw;iw epiy ntf;lhf;fshff; nfhΩl Gßspfß topNar; nry;Yk; jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghl;ilf; fhΩf.Gßspfß topNar; nry;Yk; jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghl;ilf; fhΩf.Gßspfß topNar; nry;Yk; jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghl;ilf; fhΩf.Gßspfß topNar; nry;Yk; jsj;jp∂ ntf;lh; kw;Wk; fhh;Brpa∂ rk∂ghl;ilf; fhΩf.

jPh;T :


⇒ 6

ntl;Lj; JΩL tbtpy; xU jsj;ntl;Lj; JΩL tbtpy; xU jsj;ntl;Lj; JΩL tbtpy; xU jsj;ntl;Lj; JΩL tbtpy; xU jsj;jp∂ rk∂ghl;ilj; jUtpf;f. jp∂ rk∂ghl;ilj; jUtpf;f. jp∂ rk∂ghl;ilj; jUtpf;f. jp∂ rk∂ghl;ilj; jUtpf;f. jPh;T kw;Wk;;;; v∂gd KiwNa kw;Wk; ∂ ntl;L JΩLfß v∂f.

/ jskhdJ kw;Wk; v∂w Gßspfß topNar; nry;YfpwJ.


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55

⇒ x bc – abc + y ac + z ab = 0

÷ abc ,

ntf;lh; rk∂ghL :

⇒ .

Hard work never fails you .

3.3.3.3. fyg;ngz;fs;fyg;ngz;fs;fyg;ngz;fs;fyg;ngz;fs;----(gFjp(gFjp(gFjp(gFjp----,,,,----1111----tpdhtpdhtpdhtpdh----10101010----kjpg;ngz;fs;)kjpg;ngz;fs;)kjpg;ngz;fs;)kjpg;ngz;fs;) vDk; Gs;sp fyg;nΩ khwp vDk; Gs;sp fyg;nΩ khwp vDk; Gs;sp fyg;nΩ khwp vDk; Gs;sp fyg;nΩ khwp Z Z Z Z ---- If; Fwpj;jhy; If; Fwpj;jhy; If; Fwpj;jhy; If; Fwpj;jhy; PPPP----,∂ epakghijiaf; fhΩf.,∂ epakghijiaf; fhΩf.,∂ epakghijiaf; fhΩf.,∂ epakghijiaf; fhΩf.

.

jPh;T: vd;f.

⇒

2 2 2 2

∴ P-,∂ epakghij

gapw;rpf; fzf;F

vDk; Gs;sp fyg;nΩ khwp Z - If; Fwpj;jhy; P-,∂ epakghijiaf; fhΩf.

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56

vDk; Gs;sp fyg;nΩ khwp Z - If; Fwpj;jhy; P-,∂ epakghijiaf; fhΩf.

vDk; Gs;sp fyg;nΩ khwp Z - If; Fwpj;jhy; P-,∂ epakghijiaf; fhΩf....

vDk; Gs;sp fyg;nΩ khwp vDk; Gs;sp fyg;nΩ khwp vDk; Gs;sp fyg;nΩ khwp vDk; Gs;sp fyg;nΩ khwp Z Z Z Z ---- If; Fwpj;jhy; If; Fwpj;jhy; If; Fwpj;jhy; If; Fwpj;jhy; PPPP----,∂ epakghijiaf; fhΩf. ,∂ epakghijiaf; fhΩf. ,∂ epakghijiaf; fhΩf. ,∂ epakghijiaf; fhΩf.

jPh;T:

=

⇒

2 2

⇒ +

gapw;rpf; fzf;F: vDk; Gs;sp fyg;nΩ khwp Z - If; Fwpj;jhy; P-,∂ epakghijiaf;

fhΩf.

2 2 2 v∂w rk∂ghl;ghb∂ %yq;fs; v∂w rk∂ghl;ghb∂ %yq;fs; v∂w rk∂ghl;ghb∂ %yq;fs; v∂w rk∂ghl;ghb∂ %yq;fs; kw;Wkkw;Wkkw;Wkkw;Wk; vdpy; vdpy; vdpy; vdpy;

vd epWTf. vd epWTf. vd epWTf. vd epWTf.

jPh;T: 2 2 2 jPh;Tfs ; .

kw;Wk; vd;f

= q

=

=

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57

⇒

gapw;rpf; fzf;F : α, β v∂git 2 ,∂ %yq;fs; kw;Wk;

vdpy; vdf; fhl;Lf.

2 ,∂ %yq;fs; ,∂ %yq;fs; ,∂ %yq;fs; ,∂ %yq;fs; α, kw;Wk; kw;Wk; kw;Wk; kw;Wk; β vdpy;vdpy;vdpy;vdpy; n n n+1 mjpypUe;JmjpypUe;JmjpypUe;JmjpypUe;J

9 9 ∂ kjpg;ig ngWf.∂ kjpg;ig ngWf.∂ kjpg;ig ngWf.∂ kjpg;ig ngWf.

jPh;T 2 jPh;Tfs;

α = β = vd;f.

kw;Wk;

.

vdpy;vdpy;vdpy;vdpy;

vdf; fhl;Lf.vdf; fhl;Lf.vdf; fhl;Lf.vdf; fhl;Lf.

jPu;T :

2

vd;f

m n

gapw;rpf; fzf;F : vdpy;

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58

i) ii) vd ep&gp.

∂ vy;yh kjpg;GfisAk; fhΩf kw;Wk; mj∂ kjpg;Gfsp∂ ngUf;fw;gy∂ 1 ∂ vy;yh kjpg;GfisAk; fhΩf kw;Wk; mj∂ kjpg;Gfsp∂ ngUf;fw;gy∂ 1 ∂ vy;yh kjpg;GfisAk; fhΩf kw;Wk; mj∂ kjpg;Gfsp∂ ngUf;fw;gy∂ 1 ∂ vy;yh kjpg;GfisAk; fhΩf kw;Wk; mj∂ kjpg;Gfsp∂ ngUf;fw;gy∂ 1

vdTk; fhl;Lf. vdTk; fhl;Lf. vdTk; fhl;Lf. vdTk; fhl;Lf. ( Oct. – 2007, March--2008 )

jPu;T : vd;f.

= (cos ( ) + i sin ( ))

=

1/4

1/4

,q;F k

kjpg;Gfs ; :

ngUf;fw;gy∂

∂ vy;yh kjpg;GfisAk; fhΩf∂ vy;yh kjpg;GfisAk; fhΩf∂ vy;yh kjpg;GfisAk; fhΩf∂ vy;yh kjpg;GfisAk; fhΩf

jPu;T : vd;f.

2

,q;F

kjpg;Gfs; :

gapw;rpf; fzf;F ∂ vy;yh kjpg;GfisAk; fhΩf ∂ vy;yh kjpg;GfisAk; fhΩf ∂ vy;yh kjpg;GfisAk; fhΩf ∂ vy;yh kjpg;GfisAk; fhΩf

jPh;f;fjPh;f;fjPh;f;fjPh;f;f 4 3 2

jPu;T : 4 3 2

= 0.

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59

.

kjpg;Gfs ; : ( I ePq;fyhf)

9 5 4 v∂w rk∂ghl;ilj; jPh;f;f.v∂w rk∂ghl;ilj; jPh;f;f.v∂w rk∂ghl;ilj; jPh;f;f.v∂w rk∂ghl;ilj; jPh;f;f.

jPu;T : 9 5 4

5 4 4

⇒

⇒ ;

=

,q;F

%yq;fshtd

,q;F

%yq;fshtd ,t;thW 9 %yq;fs; ngwg;gLfp∂wd.

gapw;rpf; fzf;F7 4 3 . v∂w rk∂ghl;ilj; jPh;f;f.

4.4.4.4. gFKiwgFKiwgFKiwgFKiw tbtf;fzpjk;tbtf;fzpjk;tbtf;fzpjk;tbtf;fzpjk;----gFjpgFjpgFjpgFjp----,,,,----tpdhtpdhtpdhtpdh----tptptptpilfßilfßilfßilfß----2222---- tpdhf;fßtpdhf;fßtpdhf;fßtpdhf;fß----20202020----kjpg;ngΩfßkjpg;ngΩfßkjpg;ngΩfßkjpg;ngΩfß

v∂w Neh;NfhLv∂w Neh;NfhLv∂w Neh;NfhLv∂w Neh;NfhL mjpgutisakmjpgutisakmjpgutisakmjpgutisak; njhLfpwJ vd ep&gpf;f.njhLfpwJ vd ep&gpf;f.njhLfpwJ vd ep&gpf;f.njhLfpwJ vd ep&gpf;f.

NkYk; njhLk; GßspiaAk; fhΩf. NkYk; njhLk; GßspiaAk; fhΩf. NkYk; njhLk; GßspiaAk; fhΩf. NkYk; njhLk; GßspiaAk; fhΩf.

jPh;T : v∂w NfhL mjpgutisak; f;F njhLNfhlhf

mika epge;jid .

kw;Wk;

2 2 ⇒ ,

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60

NkYk;

v∂w Neh;NfhL mjpgutisaj;ij njhLfpwJ

,J mjpgutisaj;ij njhLk; Gßsp

; njhLk; Gßsp

v∂w Neh;NfhL ePßtl;lk;v∂w Neh;NfhL ePßtl;lk;v∂w Neh;NfhL ePßtl;lk;v∂w Neh;NfhL ePßtl;lk; f;F njhLNfhlhf cßsJf;F njhLNfhlhf cßsJf;F njhLNfhlhf cßsJf;F njhLNfhlhf cßsJ vdvdvdvd ep&gpf;f. NkYk; njhLk; GßspiaAk; fhΩf.ep&gpf;f. NkYk; njhLk; GßspiaAk; fhΩf.ep&gpf;f. NkYk; njhLk; GßspiaAk; fhΩf.ep&gpf;f. NkYk; njhLk; GßspiaAk; fhΩf.

jPh;T : v∂w NfhL ePßtl;lk; f;F njhLNfhlhf

mika epge;jid .

kw;Wk;

2 2

NkYk; 2 2 2

v∂w NfhL ePßtl;lj;ij njhLfpwJ.

,J ePßtl;lj;ij njhLk; Gßsp

mjpgutisaj;jpd; ikak; mjpgutisaj;jpd; ikak; mjpgutisaj;jpd; ikak; mjpgutisaj;jpd; ikak; . NkYk; . NkYk; . NkYk; . NkYk; topNa nry;fpwJ. ,j∂ njhiyj;njhL topNa nry;fpwJ. ,j∂ njhiyj;njhL topNa nry;fpwJ. ,j∂ njhiyj;njhL topNa nry;fpwJ. ,j∂ njhiyj;njhL

NfhLfs; NfhLfs; NfhLfs; NfhLfs; kw;Wk; kw;Wk; kw;Wk; kw;Wk; Mfpatw;wpw;F ,izahf ,Uf;fMfpatw;wpw;F ,izahf ,Uf;fMfpatw;wpw;F ,izahf ,Uf;fMfpatw;wpw;F ,izahf ,Uf;fp∂wd. p∂wd. p∂wd. p∂wd.

mjpgutisaj;jpd;mjpgutisaj;jpd;mjpgutisaj;jpd;mjpgutisaj;jpd; rk∂ghL fhΩf.rk∂ghL fhΩf.rk∂ghL fhΩf.rk∂ghL fhΩf.

jPh;T : njhiyj;njhL NfhLfsp∂ rk∂ghLfsp∂ ,izf; NfhLfs;

kw;Wk; / njhiyj;njhL NfhLfsp∂ rk∂ghLfsp∂tbtk;

kw;Wk;

,J mjpgutisaj;jpd; ikak; topNa nry;fpwJ.

vdNt kw;Wk; .

njhiyj; njhL NfhLfsp∂ rk∂ghLfs; kw;Wk; .

vdNt mjpgutisaj;jpd; rk∂ghl;b∂ tbtk;

,J topNa nry;fpwJ.

vdNt mjpgutisaj;jpd; rk∂ghL

x + 2y – 5 = 0 xU njhiyj; njhL NfhlhfTk;> (6>0) kw;xU njhiyj; njhL NfhlhfTk;> (6>0) kw;xU njhiyj; njhL NfhlhfTk;> (6>0) kw;xU njhiyj; njhL NfhlhfTk;> (6>0) kw;Wk; (Wk; (Wk; (Wk; (----3>0) vd;w3>0) vd;w3>0) vd;w3>0) vd;w Gs;spfs; Gs;spfs; Gs;spfs; Gs;spfs;

topNa nry;yf;$baJkhd nrt;tf mjpgutisaj;jpd; rkd;ghL fhz;f.topNa nry;yf;$baJkhd nrt;tf mjpgutisaj;jpd; rkd;ghL fhz;f.topNa nry;yf;$baJkhd nrt;tf mjpgutisaj;jpd; rkd;ghL fhz;f.topNa nry;yf;$baJkhd nrt;tf mjpgutisaj;jpd; rkd;ghL fhz;f.

jPu;T: nrt;tf mjpgutisaj;jpd; njhiyj;njhL NfhLfs; xd;Wf;nfhd;W nrq;Fj;J MFk;.

x + 2y – 5 = 0 tpd; nrq;Fj;J rkd;ghL 2x – y + = 0.

nrt;tf mjpgutisaj;jpd; Nru;g;G rkd;ghL> (x + 2y – 5)(2x – y + ) = 0

vdNt nrt;tf mjpgutisaj;jpd; rkd;ghL>

(x + 2y – 5)(2x – y + ) = -------------- 1

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61

rkd;ghL (6>0) kw;Wk; (-3>0) vd;w Gs;spfs; topNa nry;fpwJ vdpy;>

2 + 3 ⇒ 9 = 36 ⇒ = 4 kw;Wk; = 16

∴∴∴∴ nrt;tf mjpgutisaj;jpd; rkd;ghL> (x + 2y – 5)(2x – y + 4) = 16

xU njhq;F ghyj;jpd; fk;gp tlk; gutisa tbtpypYs;sJ. mjd; ePsk; 40 kPl;lh; MFk;. xU njhq;F ghyj;jpd; fk;gp tlk; gutisa tbtpypYs;sJ. mjd; ePsk; 40 kPl;lh; MFk;. xU njhq;F ghyj;jpd; fk;gp tlk; gutisa tbtpypYs;sJ. mjd; ePsk; 40 kPl;lh; MFk;. 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. topg; ghijahdJ fk;gp tlj;jpd; fPo;kl;lg; Gs;spapypUe;J 5 kPl;lh; fPNo cs;sJ. topg; ghijahdJ fk;gp tlj;jpd; fPo;kl;lg; Gs;spapypUe;J 5 kPl;lh; fPNo cs;sJ. topg; ghijahdJ fk;gp tlj;jpd; fPo;kl;lg; Gs;spapypUe;J 5 kPl;lh; fPNo cs;sJ. fk;gp fk;gp fk;gp fk;gp

tlj;ijj; jhq;Fk; Jhz;fspd; cauq;fs; 55 kPl;lh; vdpy;> 30 kPl;lh; cauj;jpy; fk;gp tlj;ijj; jhq;Fk; Jhz;fspd; cauq;fs; 55 kPl;lh; vdpy;> 30 kPl;lh; cauj;jpy; fk;gp tlj;ijj; jhq;Fk; Jhz;fspd; cauq;fs; 55 kPl;lh; vdpy;> 30 kPl;lh; cauj;jpy; fk;gp tlj;ijj; jhq;Fk; Jhz;fspd; cauq;fs; 55 kPl;lh; vdpy;> 30 kPl;lh; cauj;jpy; fk;gp

tlj;jpw;F xU Jiz jhq;fp $Ljyhff; nfhLf;fg;gl;lhy; mj;Jizj;jhq;fpapd; tlj;jpw;F xU Jiz jhq;fp $Ljyhff; nfhLf;fg;gl;lhy; mj;Jizj;jhq;fpapd; tlj;jpw;F xU Jiz jhq;fp $Ljyhff; nfhLf;fg;gl;lhy; mj;Jizj;jhq;fpapd; tlj;jpw;F xU Jiz jhq;fp $Ljyhff; nfhLf;fg;gl;lhy; mj;Jizj;jhq;fpapd;

ePsj;ijf; fhz;f.ePsj;ijf; fhz;f.ePsj;ijf; fhz;f.ePsj;ijf; fhz;f.

jPh;T: njhq;F ghyj;jpd; fk;gptlk; Nkw;Gwk; jpwg;Gila gutisa mikg;ig ngw;Ws;sJ vdf.

mjhtJ .

nfhLf;fg;gl;l tptuq;fspypUe;J tlj;jpd; Kid topg;ghijapy; ,Ue;J 5kP Nky; mike;Js;sJ.

njhq;F ghyj;jpd; tpl;lk; 40 kPl;lh;. Gs;sp gutisaj;jpd; kPJ mike;J cs;sJ.

Njitahd rkd;ghL

vd;w Gs;sp gutisaj;jpd; kPJ mike;Js;sJ

jhq;fpapd; ePsk; kPl;lh; MFk;.

6666) xU uapy;Nt ghyj;jpd; Nky; tisT gutisaj;jpd; mikgigf; nfhz;ΩLs;sJ me;j ) xU uapy;Nt ghyj;jpd; Nky; tisT gutisaj;jpd; mikgigf; nfhz;ΩLs;sJ me;j ) xU uapy;Nt ghyj;jpd; Nky; tisT gutisaj;jpd; mikgigf; nfhz;ΩLs;sJ me;j ) xU uapy;Nt ghyj;jpd; Nky; tisT gutisaj;jpd; mikgigf; nfhz;ΩLs;sJ me;j tistp∂ mfyk; 100 mbahfTk; mt;tistp∂ cr;rpg;Gs;sap∂ cauk; ghyj;jpypUe;J 10tistp∂ mfyk; 100 mbahfTk; mt;tistp∂ cr;rpg;Gs;sap∂ cauk; ghyj;jpypUe;J 10tistp∂ mfyk; 100 mbahfTk; mt;tistp∂ cr;rpg;Gs;sap∂ cauk; ghyj;jpypUe;J 10tistp∂ mfyk; 100 mbahfTk; mt;tistp∂ cr;rpg;Gs;sap∂ cauk; ghyj;jpypUe;J 10 mbahfTk; cs;sJ .vdpy; ghyj;jpd; kj;jpapypUe;J ,lg;Gwk; my;yJ tyg;Gwk;10 mb Jhuj;jpy;mbahfTk; cs;sJ .vdpy; ghyj;jpd; kj;jpapypUe;J ,lg;Gwk; my;yJ tyg;Gwk;10 mb Jhuj;jpy;mbahfTk; cs;sJ .vdpy; ghyj;jpd; kj;jpapypUe;J ,lg;Gwk; my;yJ tyg;Gwk;10 mb Jhuj;jpy;mbahfTk; cs;sJ .vdpy; ghyj;jpd; kj;jpapypUe;J ,lg;Gwk; my;yJ tyg;Gwk;10 mb Jhuj;jpy; ghyj;jpd; ghyj;jpd; ghyj;jpd; ghyj;jpd; Nky; tisT vt;tsT cauj;jpy; ,Uf;Fk; vdf; fhΩf.Nky; tisT vt;tsT cauj;jpy; ,Uf;Fk; vdf; fhΩf.Nky; tisT vt;tsT cauj;jpy; ,Uf;Fk; vdf; fhΩf.Nky; tisT vt;tsT cauj;jpy; ,Uf;Fk; vdf; fhΩf. jPh;T :

,q;F gutisak; fPo;Nehf;fp jpwg;Gilajhf vLj;Jf; nfhs;Nthk.;

,J topahfnry;fpwJ

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my;yJ

gutisaj;jp∂ Nky ; cs;sJ v∂f.

v∂gJ ghyj;jp∂ ikaj;jpypUe;J tyg;Gwk;10 mb Jhuj;jpy; ghyj;jpd; caukhFk;

Njitahd,lj;jpy; ghyj;jpd; cauk; mb.

7777) xU uhf;nfl; ntbahdJ nfhSj;Jk; NghJ mJ xU gutisag; ghifapy; nry;fpwJ. ) xU uhf;nfl; ntbahdJ nfhSj;Jk; NghJ mJ xU gutisag; ghifapy; nry;fpwJ. ) xU uhf;nfl; ntbahdJ nfhSj;Jk; NghJ mJ xU gutisag; ghifapy; nry;fpwJ. ) xU uhf;nfl; ntbahdJ nfhSj;Jk; NghJ mJ xU gutisag; ghifapy; nry;fpwJ.

mjd; cr;r cauk; 4 nrkP I vl;Lk; NghJ mJ nfhSj;jg;gl;l ,lj;jpypUe;J fpilkl;lJhuk; mjd; cr;r cauk; 4 nrkP I vl;Lk; NghJ mJ nfhSj;jg;gl;l ,lj;jpypUe;J fpilkl;lJhuk; mjd; cr;r cauk; 4 nrkP I vl;Lk; NghJ mJ nfhSj;jg;gl;l ,lj;jpypUe;J fpilkl;lJhuk; mjd; cr;r cauk; 4 nrkP I vl;Lk; NghJ mJ nfhSj;jg;gl;l ,lj;jpypUe;J fpilkl;lJhuk;

6 kP njhiytpYs;sJ. ,Wjpahf fpilkl;l6 kP njhiytpYs;sJ. ,Wjpahf fpilkl;l6 kP njhiytpYs;sJ. ,Wjpahf fpilkl;l6 kP njhiytpYs;sJ. ,Wjpahf fpilkl;lkhf 12 kP njhiytpy; jiuia te;jilfpwJ khf 12 kP njhiytpy; jiuia te;jilfpwJ khf 12 kP njhiytpy; jiuia te;jilfpwJ khf 12 kP njhiytpy; jiuia te;jilfpwJ

vdpy; Gwg;gl;l ,lj;jpy; jiuAld; Vw;gLj;jg;gLk; vwpNfhzk; fhz;f.vdpy; Gwg;gl;l ,lj;jpy; jiuAld; Vw;gLj;jg;gLk; vwpNfhzk; fhz;f.vdpy; Gwg;gl;l ,lj;jpy; jiuAld; Vw;gLj;jg;gLk; vwpNfhzk; fhz;f.vdpy; Gwg;gl;l ,lj;jpy; jiuAld; Vw;gLj;jg;gLk; vwpNfhzk; fhz;f.

gutisaj;jpd; rkd;ghL vd;w mikg;ig cilaJ. (Kidia Mjpahff; nfhs;f)

,J topr; nry;fpwJ.

rkd;ghL :

,g;nghOJ vd;w Gs;spapy; rha;itf; fzf;fpl Ntz;Lk;.

my;yJ tiff;nfO

,y;

Njitahd vwp Nfhzk;

jiukl;lj;jpypUe;J 7.5nrkP cauj;jpy; jiuf;F ,izahf nghUj;jg;gl;l xU FohapypUe;J jiukl;lj;jpypUe;J 7.5nrkP cauj;jpy; jiuf;F ,izahf nghUj;jg;gl;l xU FohapypUe;J jiukl;lj;jpypUe;J 7.5nrkP cauj;jpy; jiuf;F ,izahf nghUj;jg;gl;l xU FohapypUe;J jiukl;lj;jpypUe;J 7.5nrkP cauj;jpy; jiuf;F ,izahf nghUj;jg;gl;l xU FohapypUe;J ntspNaWk; ePh; jiuiaj; njhLk; ghij xU gutisaj;ij Vw;gLj;JfpwJ. ntspNaWk; ePh; jiuiaj; njhLk; ghij xU gutisaj;ij Vw;gLj;JfpwJ. ntspNaWk; ePh; jiuiaj; njhLk; ghij xU gutisaj;ij Vw;gLj;JfpwJ. ntspNaWk; ePh; jiuiaj; njhLk; ghij xU gutisaj;ij Vw;gLj;JfpwJ.

NkYk; ,e;j gutisag; gNkYk; ,e;j gutisag; gNkYk; ,e;j gutisag; gNkYk; ,e;j gutisag; ghijapd; Kid Fohapd; thapy; mikfpwJ. Foha; kl;lj;jpw;F hijapd; Kid Fohapd; thapy; mikfpwJ. Foha; kl;lj;jpw;F hijapd; Kid Fohapd; thapy; mikfpwJ. Foha; kl;lj;jpw;F hijapd; Kid Fohapd; thapy; mikfpwJ. Foha; kl;lj;jpw;F

2.5 kP. fPNo ePhpd; gha;thdJ Fohapd; Kid topahfr; nry;Yk; epiy Fj;Jf; Nfhl;bw;F2.5 kP. fPNo ePhpd; gha;thdJ Fohapd; Kid topahfr; nry;Yk; epiy Fj;Jf; Nfhl;bw;F2.5 kP. fPNo ePhpd; gha;thdJ Fohapd; Kid topahfr; nry;Yk; epiy Fj;Jf; Nfhl;bw;F2.5 kP. fPNo ePhpd; gha;thdJ Fohapd; Kid topahfr; nry;Yk; epiy Fj;Jf; Nfhl;bw;F

3 kP. Jhuj;jpy; cs;sJ vdpy; Fj;Jf; Nfhl;bypUe;J vt;tsT Jhuj;jpw;F mg;ghy; ePuhdJ 3 kP. Jhuj;jpy; cs;sJ vdpy; Fj;Jf; Nfhl;bypUe;J vt;tsT Jhuj;jpw;F mg;ghy; ePuhdJ 3 kP. Jhuj;jpy; cs;sJ vdpy; Fj;Jf; Nfhl;bypUe;J vt;tsT Jhuj;jpw;F mg;ghy; ePuhdJ 3 kP. Jhuj;jpy; cs;sJ vdpy; Fj;Jf; Nfhl;bypUe;J vt;tsT Jhuj;jpw;F mg;ghy; ePuhdJ

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jiuapy; tpOk; vd;gijf; fhz;fjiuapy; tpOk; vd;gijf; fhz;fjiuapy; tpOk; vd;gijf; fhz;fjiuapy; tpOk; vd;gijf; fhz;f.

jPh;T:

nfhLf;fg;gl;l tptuq;fspd;gb gutisak; fPo;Nehf;fp jpwg;Gilajhf mikfpwJ.

mjhtJ

vd;w Gs;sp gutisag; ghijapy; Foha; kl;lj;jpw;F 2.5kP fPNoAk; Fohapd; Kid topNa

nry;Yk; epiy Fj;Jf; Nfhl;bw;F 3kP mg;ghYk; cs;sJ.

vd;gJ . vdNt

/ gutisaj;jpd; rkd;ghL

Fj;Jf;Nfhl;bd; mbapypUe;J Jhuj;Jf;F mg;ghy; ePuhdJ jiuapy; tpOtjhff; nfhs;f.

Mdhy; FohahdJ jiukl;lj;jpypUe;J 7.5 kP cauj;jpy; mike;Js;sJ.

vd;w Gs;sp gutisaj;jpYs;sJ.

/ vdNt jz;zPh; jiuiaf; njhLk; ,lj;Jf;Fk; Fohapd; KidapypUe;J tiuag;gLk;

Fj;Jf;Nfhl;bw;Fk; ,ilg;gl;l Jhuk; kP MFk;.

9999) xU thy; tpz;kPd; MdJ r+hpaidr; Rw;wp gutisiag; ghijapy; nry;fpwJ. Kw;Wk; R+hpad; ) xU thy; tpz;kPd; MdJ r+hpaidr; Rw;wp gutisiag; ghijapy; nry;fpwJ. Kw;Wk; R+hpad; ) xU thy; tpz;kPd; MdJ r+hpaidr; Rw;wp gutisiag; ghijapy; nry;fpwJ. Kw;Wk; R+hpad; ) xU thy; tpz;kPd; MdJ r+hpaidr; Rw;wp gutisiag; ghijapy; nry;fpwJ. Kw;Wk; R+hpad;

gutisaj;jpd; Ftpaj;jpy; mikfpwJ. thy; tpz;kPd; r+hpadpypUe;J 80 kpy;ypad; fp.kP gutisaj;jpd; Ftpaj;jpy; mikfpwJ. thy; tpz;kPd; r+hpadpypUe;J 80 kpy;ypad; fp.kP gutisaj;jpd; Ftpaj;jpy; mikfpwJ. thy; tpz;kPd; r+hpadpypUe;J 80 kpy;ypad; fp.kP gutisaj;jpd; Ftpaj;jpy; mikfpwJ. thy; tpz;kPd; r+hpadpypUe;J 80 kpy;ypad; fp.kP

njhiytpy; mike;J ,Uf;Fk; NghJ thy; tpz;kPidAk; R+hpaidnjhiytpy; mike;J ,Uf;Fk; NghJ thy; tpz;kPidAk; R+hpaidnjhiytpy; mike;J ,Uf;Fk; NghJ thy; tpz;kPidAk; R+hpaidnjhiytpy; mike;J ,Uf;Fk; NghJ thy; tpz;kPidAk; R+hpaidAk; ,izf;Fk; NfhL Ak; ,izf;Fk; NfhL Ak; ,izf;Fk; NfhL Ak; ,izf;Fk; NfhL

ghijapd; mr;Rld; ghijapd; mr;Rld; ghijapd; mr;Rld; ghijapd; mr;Rld; Nfhzj;jpid Vw;gLj;Jkhdhy; Nfhzj;jpid Vw;gLj;Jkhdhy; Nfhzj;jpid Vw;gLj;Jkhdhy; Nfhzj;jpid Vw;gLj;Jkhdhy; thy; tpz; kPdpd; ghijapy; thy; tpz; kPdpd; ghijapy; thy; tpz; kPdpd; ghijapy; thy; tpz; kPdpd; ghijapy;

rkd;ghl;ilf; fhz;f. rkd;ghl;ilf; fhz;f. rkd;ghl;ilf; fhz;f. rkd;ghl;ilf; fhz;f. thy; tpz;kPd; r+hpaDf;F vt;tsT mUfpy; tuKbAk; vd;gijAk; thy; tpz;kPd; r+hpaDf;F vt;tsT mUfpy; tuKbAk; vd;gijAk; thy; tpz;kPd; r+hpaDf;F vt;tsT mUfpy; tuKbAk; vd;gijAk; thy; tpz;kPd; r+hpaDf;F vt;tsT mUfpy; tuKbAk; vd;gijAk;

fhz;f. (ghij tyJ Gwk; jpwg;Gilajhf nffhz;f. (ghij tyJ Gwk; jpwg;Gilajhf nffhz;f. (ghij tyJ Gwk; jpwg;Gilajhf nffhz;f. (ghij tyJ Gwk; jpwg;Gilajhf nfhs;f.)hs;f.)hs;f.)hs;f.)

jPh;T :

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64

gutisaj;jpd; ghij tyJ gf;fk; jpwg;GilaJ. NkYk; Kidg;Gs;sp MjpapYs;sJ.

thy; tpz;kPdpd; epiy vdf; nfhΩlhy; kpy;ypad; fp.kP MFk;

ypUe;J gutisaj;jpd; mR;Rf;F vd;w nrq;Fj;J tiuf.

,q;F vd;f.

Kf;Nfhzk; ypUe;J :

vd;gjhy;) vd;gJ

vd;gJ gutisak; Nky; ,Ug;gjhy;

( Vw;Gilajy;y.)

/ ghijapd; rkd;ghL

R+hpaDf;Fk; thy; tpz;kPDf;Fk; ,ilNaAs;s> kpff; Fiwe;j Jhuk; mjhtJ>

/ kpff; Fiwe;j Jhuk; 20 kpy;ypad; fp.kP MFk;.

xU njhq;F ghyj;jpd; fk;gp tlk; gutisa tbtpypYs;sJ. Mj∂ ghuk; fpilkl;lkhf xU njhq;F ghyj;jpd; fk;gp tlk; gutisa tbtpypYs;sJ. Mj∂ ghuk; fpilkl;lkhf xU njhq;F ghyj;jpd; fk;gp tlk; gutisa tbtpypYs;sJ. Mj∂ ghuk; fpilkl;lkhf xU njhq;F ghyj;jpd; fk;gp tlk; gutisa tbtpypYs;sJ. Mj∂ ghuk; fpilkl;lkhf

rPPuhf gutpAs;sJ.mijj; jhq;Fk; ,U Jhz;fSf;F ,ilNa cs;s Jhuk; 1500 mb. rPPuhf gutpAs;sJ.mijj; jhq;Fk; ,U Jhz;fSf;F ,ilNa cs;s Jhuk; 1500 mb. rPPuhf gutpAs;sJ.mijj; jhq;Fk; ,U Jhz;fSf;F ,ilNa cs;s Jhuk; 1500 mb. rPPuhf gutpAs;sJ.mijj; jhq;Fk; ,U Jhz;fSf;F ,ilNa cs;s Jhuk; 1500 mb.

fk;gp tlj;ijj; jhq;Fk; Gs;spfs; Jhzpy; jiuapypUe;J fk;gp tlj;ijj; jhq;Fk; Gs;spfs; Jhzpy; jiuapypUe;J fk;gp tlj;ijj; jhq;Fk; Gs;spfs; Jhzpy; jiuapypUe;J fk;gp tlj;ijj; jhq;Fk; Gs;spfs; Jhzpy; jiuapypUe;J 200 mb cauj;jpy; mike;Js;sd. 200 mb cauj;jpy; mike;Js;sd. 200 mb cauj;jpy; mike;Js;sd. 200 mb cauj;jpy; mike;Js;sd.

NkYk;NkYk;NkYk;NkYk; jiuapypUe;J fk;gp tlj;jp∂ jho;thd Gs;spap∂ cauk; 70 mb. fk;gptlk; 122 mb jiuapypUe;J fk;gp tlj;jp∂ jho;thd Gs;spap∂ cauk; 70 mb. fk;gptlk; 122 mb jiuapypUe;J fk;gp tlj;jp∂ jho;thd Gs;spap∂ cauk; 70 mb. fk;gptlk; 122 mb jiuapypUe;J fk;gp tlj;jp∂ jho;thd Gs;spap∂ cauk; 70 mb. fk;gptlk; 122 mb

cauj;jpy; jhq;Fk; fk;gj;jpw;F ,ilNa cs;s nrq;;Fj;J ePsk; (jiuf;F ,izahf) fhΩf. cauj;jpy; jhq;Fk; fk;gj;jpw;F ,ilNa cs;s nrq;;Fj;J ePsk; (jiuf;F ,izahf) fhΩf. cauj;jpy; jhq;Fk; fk;gj;jpw;F ,ilNa cs;s nrq;;Fj;J ePsk; (jiuf;F ,izahf) fhΩf. cauj;jpy; jhq;Fk; fk;gj;jpw;F ,ilNa cs;s nrq;;Fj;J ePsk; (jiuf;F ,izahf) fhΩf.

jPh;T :

fk;gp tlj;jp∂ kPJ kpfj;jho;thd Gs;sp KidahFk;. ,jid Mjpahff; nfhs;f. kwWk;

jhq;Fk; JhΩfs; MFk.; ,U JhΩfSf;F ,ilNaAs;s njhiyT 1500 mb

v∂gjhy; mb mb. mb

vdNt v∂gJ .

gutisaj;jpd; rkd;ghL

vd;w Gs;sp gutisaj;jpd; kPJ mike;Js;sJ.

rkd;ghL MFk;.

fk;gk; tpypUe;J fk;gp tlj;jpw;Fr; nrq;Fj;jhd ePsk; v∂f.

.

v∂gJ

gutisaj;jpd; kPJ cs;sJ

mb

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xU NfhxU NfhxU NfhxU Nfh----Nfh tpisahl;L tPuh; tpisahl;Lg; gapw;rpapd; NghJ mtUf;Fk; NfhNfh tpisahl;L tPuh; tpisahl;Lg; gapw;rpapd; NghJ mtUf;Fk; NfhNfh tpisahl;L tPuh; tpisahl;Lg; gapw;rpapd; NghJ mtUf;Fk; NfhNfh tpisahl;L tPuh; tpisahl;Lg; gapw;rpapd; NghJ mtUf;Fk; Nfh----Nfh Nfh Nfh Nfh

Fr;rpfSf;Fk; ,ilNaAs;s Jhuk; vg;NghOJk; 8kP Mf ,Uf;FkhW czh;fpwhh;. mt;tpU Fr;rpfSf;Fk; ,ilNaAs;s Jhuk; vg;NghOJk; 8kP Mf ,Uf;FkhW czh;fpwhh;. mt;tpU Fr;rpfSf;Fk; ,ilNaAs;s Jhuk; vg;NghOJk; 8kP Mf ,Uf;FkhW czh;fpwhh;. mt;tpU Fr;rpfSf;Fk; ,ilNaAs;s Jhuk; vg;NghOJk; 8kP Mf ,Uf;FkhW czh;fpwhh;. mt;tpU

Fr;rpfSf;F ,ilg;gl;l Jhuk; 6kP vdpy; mth; xLk; ghijapd; rkd;gFr;rpfSf;F ,ilg;gl;l Jhuk; 6kP vdpy; mth; xLk; ghijapd; rkd;gFr;rpfSf;F ,ilg;gl;l Jhuk; 6kP vdpy; mth; xLk; ghijapd; rkd;gFr;rpfSf;F ,ilg;gl;l Jhuk; 6kP vdpy; mth; xLk; ghijapd; rkd;ghl;ilf; fhz;f.hl;ilf; fhz;f.hl;ilf; fhz;f.hl;ilf; fhz;f.

jPh;T : Nfh-Nfh Fr;rpfs; ,uz;Lk; F1F1F1F1 kw;Wk; F2F2F2F2 ,y; mike;Js;sd vdf; nfhs;f....

P(x, y)P(x, y)P(x, y)P(x, y) vd;w Gs;spahdJ tpisahl;L tPuhpd; epiy vdf; nfhs;f.

NkYk;

/ ghijapd; rkd;ghL .

12121212) ) ) ) xU ePs; tl;lg; ghijap∂ Ftpaj;jpy; ©kp ,Uf;FkhW xU Jizf;Nfhs; xU ePs; tl;lg; ghijap∂ Ftpaj;jpy; ©kp ,Uf;FkhW xU Jizf;Nfhs; xU ePs; tl;lg; ghijap∂ Ftpaj;jpy; ©kp ,Uf;FkhW xU Jizf;Nfhs; xU ePs; tl;lg; ghijap∂ Ftpaj;jpy; ©kp ,Uf;FkhW xU Jizf;Nfhs; Rw;wp tUfpwJ. Rw;wp tUfpwJ. Rw;wp tUfpwJ. Rw;wp tUfpwJ.

,j∂ ikaj; njhiyj;jfT ,j∂ ikaj; njhiyj;jfT ,j∂ ikaj; njhiyj;jfT ,j∂ ikaj; njhiyj;jfT MfTk; ©kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l kPr;rpW MfTk; ©kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l kPr;rpW MfTk; ©kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l kPr;rpW MfTk; ©kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l kPr;rpW

Jhuk; 400 fpNyh kPl;lu;fs; MfTk; ,Uf;Fkhdhy; ©kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l Jhuk; 400 fpNyh kPl;lu;fs; MfTk; ,Uf;Fkhdhy; ©kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l Jhuk; 400 fpNyh kPl;lu;fs; MfTk; ,Uf;Fkhdhy; ©kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l Jhuk; 400 fpNyh kPl;lu;fs; MfTk; ,Uf;Fkhdhy; ©kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l

mjpfggl;r Jhuk; v∂d ?mjpfggl;r Jhuk; v∂d ?mjpfggl;r Jhuk; v∂d ?mjpfggl;r Jhuk; v∂d ? jPh;T :

glj;;jpypUe;J ©kpap∂ epiy v∂f.

©kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l kPr;rpW Jhuk; fp.kP.

©kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l mjpfggl;r Jhuk;

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©kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l mjpfggl;r Jhuk;

fp.kP.

13131313) #hpa∂ Ftpaj;jpypUf;FkhW nkh;Fhp fpufkhdJ #hpaid xU ePs; tl;lg; ghijap) #hpa∂ Ftpaj;jpypUf;FkhW nkh;Fhp fpufkhdJ #hpaid xU ePs; tl;lg; ghijap) #hpa∂ Ftpaj;jpypUf;FkhW nkh;Fhp fpufkhdJ #hpaid xU ePs; tl;lg; ghijap) #hpa∂ Ftpaj;jpypUf;FkhW nkh;Fhp fpufkhdJ #hpaid xU ePs; tl;lg; ghijapy; Rw;wp y; Rw;wp y; Rw;wp y; Rw;wp

tUfpwJ. mj∂ miu nel;llr;rp∂ ePsk; tUfpwJ. mj∂ miu nel;llr;rp∂ ePsk; tUfpwJ. mj∂ miu nel;llr;rp∂ ePsk; tUfpwJ. mj∂ miu nel;llr;rp∂ ePsk; kpy;ypa∂ iky;fs; MfTk; ikaj; kpy;ypa∂ iky;fs; MfTk; ikaj; kpy;ypa∂ iky;fs; MfTk; ikaj; kpy;ypa∂ iky;fs; MfTk; ikaj;

njhiyj;jfT njhiyj;jfT njhiyj;jfT njhiyj;jfT MfTk; ,Uf;Fkhap∂ MfTk; ,Uf;Fkhap∂ MfTk; ,Uf;Fkhap∂ MfTk; ,Uf;Fkhap∂ nkh;Fhp fpufkhdJ #hpaDf;F kpf nkh;Fhp fpufkhdJ #hpaDf;F kpf nkh;Fhp fpufkhdJ #hpaDf;F kpf nkh;Fhp fpufkhdJ #hpaDf;F kpf

mUfhikapy; tUk; NghJ cs;s Jhuk; mUfhikapy; tUk; NghJ cs;s Jhuk; mUfhikapy; tUk; NghJ cs;s Jhuk; mUfhikapy; tUk; NghJ cs;s Jhuk; nkh;Fhp fpufkhdJ #hpaDf;nkh;Fhp fpufkhdJ #hpaDf;nkh;Fhp fpufkhdJ #hpaDf;nkh;Fhp fpufkhdJ #hpaDf;F kpfj; F kpfj; F kpfj; F kpfj;

njhiytpy; tUk; NghJ cs;s Jhuk; Mfpatw;iwf; fhΩf. njhiytpy; tUk; NghJ cs;s Jhuk; Mfpatw;iwf; fhΩf. njhiytpy; tUk; NghJ cs;s Jhuk; Mfpatw;iwf; fhΩf. njhiytpy; tUk; NghJ cs;s Jhuk; Mfpatw;iwf; fhΩf. jPh;T :

glj;;jpypUe;J #hpadp∂ epiy v∂f.

kpy;ypa∂ iky;fs;;;; nkh;Fhp fpufkhdJ #hpaDf;F kpf mUfhikapy;> kpfj; njhiytpy; ; ; ;

,Uf;;Fk; epiyfs; ; ; ; kw;Wk; v∂f..

#hpaDf;F kpf mUfhikapy; Jhuk;

kpy;ypa∂ iky;fs;;;;

#hpaDf;F kpfj; njhiytpy; Jhuk;

kpy;ypa∂ iky;fs;.

14141414)))) xU ghyj;jpd; tisthdJ miu ePs;tl;ljjp∂ tbtpy; cs;sJ. fpilkl;ljpy; mj∂ xU ghyj;jpd; tisthdJ miu ePs;tl;ljjp∂ tbtpy; cs;sJ. fpilkl;ljpy; mj∂ xU ghyj;jpd; tisthdJ miu ePs;tl;ljjp∂ tbtpy; cs;sJ. fpilkl;ljpy; mj∂ xU ghyj;jpd; tisthdJ miu ePs;tl;ljjp∂ tbtpy; cs;sJ. fpilkl;ljpy; mj∂ mfyk; 40 mbahfTk; ikaj;jpypUe;J mj∂ cauk; 16 mbahfTk; cs;sJ vdpy; mfyk; 40 mbahfTk; ikaj;jpypUe;J mj∂ cauk; 16 mbahfTk; cs;sJ vdpy; mfyk; 40 mbahfTk; ikaj;jpypUe;J mj∂ cauk; 16 mbahfTk; cs;sJ vdpy; mfyk; 40 mbahfTk; ikaj;jpypUe;J mj∂ cauk; 16 mbahfTk; cs;sJ vdpy; ikaj;jpypUe;J tyJ my;yJ ,lg;Gwj;jpy; 9 mb Jhujj;jpy; cs;s jiug; Gs;spapypUe;J ikaj;jpypUe;J tyJ my;yJ ,lg;Gwj;jpy; 9 mb Jhujj;jpy; cs;s jiug; Gs;spapypUe;J ikaj;jpypUe;J tyJ my;yJ ,lg;Gwj;jpy; 9 mb Jhujj;jpy; cs;s jiug; Gs;spapypUe;J ikaj;jpypUe;J tyJ my;yJ ,lg;Gwj;jpy; 9 mb Jhujj;jpy; cs;s jiug; Gs;spapypUe;J

ghyj;jpd; cauk; v∂d ?ghyj;jpd; cauk; v∂d ?ghyj;jpd; cauk; v∂d ?ghyj;jpd; cauk; v∂d ? jPh;T :

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ghyj;jpd; ; ; ; eLg;Gs;spia v∂f. fpil kl;lk; 40 mb.

vdNt Kidfs; kw;Wk; v∂gd kw;Wk;

,q;F

vdNt rkd;ghL

ikaj;jpypypUe;J tyJ Gwj;jpy; 9 mb Jhujj;jpy; ghyj;jpd; cauk; v∂f.

jiug; Gs;spapypUe;J ghyj;jpd; cauk;

15151515) xU tisT miu ePs;tl;l tbtj;jpy; cs;sJ. mjd; mfyk; 48 mb) xU tisT miu ePs;tl;l tbtj;jpy; cs;sJ. mjd; mfyk; 48 mb) xU tisT miu ePs;tl;l tbtj;jpy; cs;sJ. mjd; mfyk; 48 mb) xU tisT miu ePs;tl;l tbtj;jpy; cs;sJ. mjd; mfyk; 48 mb. cauk; . cauk; . cauk; . cauk;

20 mb. jiuapypUe;J 10 mb cauj;jpy;; tistp∂ mfyk;; vd;d?20 mb. jiuapypUe;J 10 mb cauj;jpy;; tistp∂ mfyk;; vd;d?20 mb. jiuapypUe;J 10 mb cauj;jpy;; tistp∂ mfyk;; vd;d?20 mb. jiuapypUe;J 10 mb cauj;jpy;; tistp∂ mfyk;; vd;d? )

jiuap∂ eLg;Gs;spia ikak; Mff; nfhs;f. jiuap∂ mfyk; 48 mb v∂gjy; Kidfs; Mdit MFk;. kw;Wk;

ePßtl;lj;jp∂ rk∂ghL

10 kP cauKs;s Jhzpw;fFk; ikaj;jpw;Fk; ,ilNaAs;s Jhuk; v∂f. vdNt v∂w Gs;sp rk∂ghL (1) I epiwT nra;Ak.

jiuapypUe;J 10 mb cauj;jpy;; tistp∂ mfyk; mb. 16161616) xU EioT thapypd; Nkw;$iuahdJ miu ePs;tl;l tbt) xU EioT thapypd; Nkw;$iuahdJ miu ePs;tl;l tbt) xU EioT thapypd; Nkw;$iuahdJ miu ePs;tl;l tbt) xU EioT thapypd; Nkw;$iuahdJ miu ePs;tl;l tbtj;jpy; cs;sJ. ,jd; mfyk; j;jpy; cs;sJ. ,jd; mfyk; j;jpy; cs;sJ. ,jd; mfyk; j;jpy; cs;sJ. ,jd; mfyk; 20mb ikaj;jpypUe;J mjd; cauk; 18 mb kw;Wk; gf;fk; Rth;fspd; cauk; 12 mb 20mb ikaj;jpypUe;J mjd; cauk; 18 mb kw;Wk; gf;fk; Rth;fspd; cauk; 12 mb 20mb ikaj;jpypUe;J mjd; cauk; 18 mb kw;Wk; gf;fk; Rth;fspd; cauk; 12 mb 20mb ikaj;jpypUe;J mjd; cauk; 18 mb kw;Wk; gf;fk; Rth;fspd; cauk; 12 mb vdpy; VNjDk; xU gf;fr; RthpypUe;J 4 mb Jhuj;jpy; Nkw;$iuapd; cauk; vd;dthf vdpy; VNjDk; xU gf;fr; RthpypUe;J 4 mb Jhuj;jpy; Nkw;$iuapd; cauk; vd;dthf vdpy; VNjDk; xU gf;fr; RthpypUe;J 4 mb Jhuj;jpy; Nkw;$iuapd; cauk; vd;dthf vdpy; VNjDk; xU gf;fr; RthpypUe;J 4 mb Jhuj;jpy; Nkw;$iuapd; cauk; vd;dthf ,Uf;Fk;? ,Uf;Fk;? ,Uf;Fk;? ,Uf;Fk;? (

jPh;T: gf;f RthpypUe;J 4 mb Jhuj;jpy; cs;s Nkw;$iuapd; cauk; vd;f.

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glj;jpd; %yk; mb

d; ePsj;ijf; fhz;Nghk;.

mfyk; 20 mbahf ,Ug;gjhy; Kidfs;

,d; Maj; njhiyfs; vdf; nfhs;f.

glj;jpd; %yk; kw;Wk;

vd;gJ vdpy; ,d; Maj; njhiyfs;

ePs; tl;lj;jpd; kPJ miktjhy;.

//// Njitahd Nkw;$iuapd; cauk; 16.8 mbahFk;.

17171717) xU rkjsj;jpd; Nky; nrq;Fj;jhf mike;Js;s Rthpd; kPJ 15kP ePsKs;s xU VzpahdJ ) xU rkjsj;jpd; Nky; nrq;Fj;jhf mike;Js;s Rthpd; kPJ 15kP ePsKs;s xU VzpahdJ ) xU rkjsj;jpd; Nky; nrq;Fj;jhf mike;Js;s Rthpd; kPJ 15kP ePsKs;s xU VzpahdJ ) xU rkjsj;jpd; Nky; nrq;Fj;jhf mike;Js;s Rthpd; kPJ 15kP ePsKs;s xU VzpahdJ jsj;jpidAk; Rtw;wpjsj;jpidAk; Rtw;wpjsj;jpidAk; Rtw;wpjsj;jpidAk; Rtw;wpidAk; njhLkhW efh;fpwJ nfhz;L ,Uf;fpwJ vdpy;> Vzpapd; fPo;kl;l idAk; njhLkhW efh;fpwJ nfhz;L ,Uf;fpwJ vdpy;> Vzpapd; fPo;kl;l idAk; njhLkhW efh;fpwJ nfhz;L ,Uf;fpwJ vdpy;> Vzpapd; fPo;kl;l idAk; njhLkhW efh;fpwJ nfhz;L ,Uf;fpwJ vdpy;> Vzpapd; fPo;kl;l

KidapypUe;J 6 kP Jhuj;jpy; Vzpapy; mike;Js;sKidapypUe;J 6 kP Jhuj;jpy; Vzpapy; mike;Js;sKidapypUe;J 6 kP Jhuj;jpy; Vzpapy; mike;Js;sKidapypUe;J 6 kP Jhuj;jpy; Vzpapy; mike;Js;s vd;w Gs;spapd; epag;ghijiaf; fhz;f.vd;w Gs;spapd; epag;ghijiaf; fhz;f.vd;w Gs;spapd; epag;ghijiaf; fhz;f.vd;w Gs;spapd; epag;ghijiaf; fhz;f.

jPh;T : vd;gJ Vzp vd;f. Vzpapd; kPJ vd;w Gs;sp kP

,Uf;FkhW vLj;Jf; nfhs;f.

mr;Rf;F nrq;Fj;jhf -Ak; mr;Rf;F

nrq;Fj;jhf ----Ak; tif.

kw;Wk tbnthj;jit.

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6.6.6.6. tiftiftiftif Ez;fzpjk;Ez;fzpjk;Ez;fzpjk;Ez;fzpjk; gad;ghLfs;gad;ghLfs;gad;ghLfs;gad;ghLfs; ---- IIIIIIII gFjpgFjpgFjpgFjp----,,,,----tpdhtpdhtpdhtpdh----tpilfßtpilfßtpilfßtpilfß----1111---- tpdhtpdhtpdhtpdh----10101010----kjpg;ngΩfßkjpg;ngΩfßkjpg;ngΩfßkjpg;ngΩfß

v∂w rhh;gpy;v∂w rhh;gpy;v∂w rhh;gpy;v∂w rhh;gpy; kw;Wkkw;Wkkw;Wkkw;Wk; vDkhW ,Ug;gp∂vDkhW ,Ug;gp∂vDkhW ,Ug;gp∂vDkhW ,Ug;gp∂ kw;Wkkw;Wkkw;Wkkw;Wk;

fhΩffhΩffhΩffhΩf....

jPu;T : kw;Wk;

:

( ,q;F kw;Wk;

vdpy; vdpy; vdpy; vdpy; Âyup∂ Njw;wj;ijg; ga∂gLj;jpÂyup∂ Njw;wj;ijg; ga∂gLj;jpÂyup∂ Njw;wj;ijg; ga∂gLj;jpÂyup∂ Njw;wj;ijg; ga∂gLj;jp

vdf; fhl;Lf. vdf; fhl;Lf. vdf; fhl;Lf. vdf; fhl;Lf.

jPu;T :

RHS rkgbj;jhdrhh;G my;y. vdNt

. vd tiwaWf;fTk;.

,g;NghJ v∂gJ gb cila rkgbj;jhd rhh;G.

Âyhp∂ Njw;wg; gb

3 Âyup∂ Njw;wj;ijg; ga∂gLj;jp Âyup∂ Njw;wj;ijg; ga∂gLj;jp Âyup∂ Njw;wj;ijg; ga∂gLj;jp Âyup∂ Njw;wj;ijg; ga∂gLj;jp vdpy; vdpy; vdpy; vdpy;

vdvdvdvd ep&gpf;f. ep&gpf;f. ep&gpf;f. ep&gpf;f.

jPu;T :

RHS rkgbj;jhdrhh;G my;y. vdNt

vd tiwaWf;fTk;.

,g;NghJ v∂gJ gb 2 cila rkgbj;jhd rhh;G.

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Âyhp∂ Njw;wg;gb

∗∗∗∗

4 f;Ff;Ff;Ff;F Âyup∂ Njw;wj;Âyup∂ Njw;wj;Âyup∂ Njw;wj;Âyup∂ Njw;wj;ij rhpghh;f;fij rhpghh;f;fij rhpghh;f;fij rhpghh;f;f

jPu;T : –1

v∂gJ gb -1 cila rkgbj;jhd rhh;G.

Âyhp∂ Njw;wg; gb

rhpghh;j;jy; : =

=

Âyhp∂ Njw;wk; rhpghh;f;fg; gl;lJ

vdpy; vdpy; vdpy; vdpy; v∂gij rhpghh;.v∂gij rhpghh;.v∂gij rhpghh;.v∂gij rhpghh;.

jPu;T : x

u

∂∂

=

y

u

∂∂

=∂∂

∂yx

u2

=∂∂

∂xy

u2

⇒ =∂∂

∂yx

u2

xy

u

∂∂∂ 2

v∂gijv∂gijv∂gijv∂gij rhpghh;f;f.rhpghh;f;f.rhpghh;f;f.rhpghh;f;f.

7) tifaPLfisg; ga∂gLj;jp gp∂tUtdtw;wp∂ Njuha kjpg;gpid fzf;fpLf.7) tifaPLfisg; ga∂gLj;jp gp∂tUtdtw;wp∂ Njuha kjpg;gpid fzf;fpLf.7) tifaPLfisg; ga∂gLj;jp gp∂tUtdtw;wp∂ Njuha kjpg;gpid fzf;fpLf.7) tifaPLfisg; ga∂gLj;jp gp∂tUtdtw;wp∂ Njuha kjpg;gpid fzf;fpLf.

jPh;T

vd;f.

vd;f

.

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NkYk; vd;f.

vd;f

8) 8) 8) 8) vd;fpw tis tiuia tiufvd;fpw tis tiuia tiufvd;fpw tis tiuia tiufvd;fpw tis tiuia tiuf.

jPh;T: 1. rhh;gfk;> ePl;bg;G> ntl;Lj;Jz;Lfs; kw;Wk; Mjp.

i) rhh;gfk; : (- ii) fpilkl;l ePl;bg;G iii) epiyf;Fj;J ePl;bg;G : (- iv) ntl;Lj;Jz;L : y- ntl;Lj;Jz;L 1. x- ntl;Lj;Jz;L -1 v) Mjp tistiu Mjp top nry;yhJ. 2. rkr;rPh; Nrhjid ,t;tistiuf;F rkr;rPh; gz;G ,y;iy. 3. njhiyj;njhL NfhLfs; njhiyj;njhL NfhLfs; ,y;iy. 4. Xhpay;G jd;ik tistiu KOtJkhf VWKfkhr; nry;Yk;. 5. rpwg;Gg; Gs;spfs; y; tistiu fPo;Nehf;fp FopthfTk;> y; tistiu Nky; Nehf;fpf; FopthfTk; ,Uf;Fk;. vd;gJ tisT khw;wg;Gs;sp.

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

-1

0

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

9)9)9)9) vd;fpw tistiuia tiufvd;fpw tistiuia tiufvd;fpw tistiuia tiufvd;fpw tistiuia tiuf.

jPh;T:

1. rhh;gfk;> ePl;bg;G> ntl;Lj;Jz;Lfs; kw;Wk; Mjp. i) rhh;gfk; : ii) ePl;bg;G fpilkl;l ePl;bg;G

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epiyf;Fj;J ePl;bg;G iii) ntl;Lj;JΩL ntl;Lj;JΩL ntl;Lj;Jz;L vi) Mjp : tistiu Mjp top nry;fpwJ.

2. rkr;rPh; Nrhjid Mjpiag; nghWj;J rkr;rPuhdJ. 3. njhiyj;njhL NfhLfs; njhiyj;njhL NfhLfs; ,y;iy. 4. Xhpay;G jd;ik tistiu KOtJkhf VWKfkhr; nry;Yk;. 5. rpwg;Gg; Gs;spfs; y; tistiu fPo;Nehf;fp FopthfTk;> tistiu y; Nky; Nehf;fpf; FopthfTk; ,Uf;Fk;. vd;gJ tisT khw;wg;Gs;sp.

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

10) 10) 10) 10) vd;fpw tis tiuia tiuf. vd;fpw tis tiuia tiuf. vd;fpw tis tiuia tiuf. vd;fpw tis tiuia tiuf. jPh;T:

1. rhh;gfk;> ePl;bg;G> ntl;Lj;Jz;Lfs; kw;Wk; Mjp. rhh;gfk; : fpilkl;l ePl;bg;G : epiyf;Fj;J ePl;bg;G

ntl;Lj;Jz;L : ntl;Lj;Jz;L ntl;Lj;Jz;L Mjp : tistiu Mjp top nry;fpwJ

2. rkr;rPh; Nrhjid : ,t;tistiu nghWj;J rkr;rPuhdJ. 3. njhiyj;njhL NfhLfs;: njhiyj;njhL NfhLfs; ,y;iy.

4. Xhpay;G jd;ik : f;F tistiu VWKfkh ,Uf;Fk;.

f;F tistiu ,wq;FKfkhf ,Uf;Fk;. 5. rpwg;Gg; Gs;spfs; : vd;gJ tisT khw;wg; Gs;sp my;y.

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-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

8.8.8.8. tifnfOrk∂ghLfßtifnfOrk∂ghLfßtifnfOrk∂ghLfßtifnfOrk∂ghLfß gFjpgFjpgFjpgFjp----,,,,----1111----tpdhf;fßtpdhf;fßtpdhf;fßtpdhf;fß----10101010----kjpg;ngΩfßkjpg;ngΩfßkjpg;ngΩfßkjpg;ngΩfß 1) Nubak; rpijAk; khW tPjkhdJ, mjpy; fhzg;gLk; mstpw;f;F tpfpjkhf 1) Nubak; rpijAk; khW tPjkhdJ, mjpy; fhzg;gLk; mstpw;f;F tpfpjkhf 1) Nubak; rpijAk; khW tPjkhdJ, mjpy; fhzg;gLk; mstpw;f;F tpfpjkhf 1) Nubak; rpijAk; khW tPjkhdJ, mjpy; fhzg;gLk; mstpw;f;F tpfpjkhf mike;Js;sJ.mike;Js;sJ.mike;Js;sJ.mike;Js;sJ. 50 ttttUlq;fspy; Muk;g mstpypUe;J 5 rjtPjk; rpije;jpUf;fpuJ vdpy; Ulq;fspy; Muk;g mstpypUe;J 5 rjtPjk; rpije;jpUf;fpuJ vdpy; Ulq;fspy; Muk;g mstpypUe;J 5 rjtPjk; rpije;jpUf;fpuJ vdpy; Ulq;fspy; Muk;g mstpypUe;J 5 rjtPjk; rpije;jpUf;fpuJ vdpy; 100 tUl Kbtpy; kPjpapUf;Fk; msT vd;d 100 tUl Kbtpy; kPjpapUf;Fk; msT vd;d 100 tUl Kbtpy; kPjpapUf;Fk; msT vd;d 100 tUl Kbtpy; kPjpapUf;Fk; msT vd;d ???? [ A0 I Muk;g msT vdf;nfhs;f] jPu;T : t Neuj;jpy; Nubaj;jpd; msT A vd;f.

⇒ ⇒

∴

50 tUlq;fspy; Nubak; Muk;g epiyapypUe;J 5% rpijTWfpwJ. t = 50 vdpy; A = 0.95A0. ∴

vdpy;

100 tUl Kbtpy; kPjpapUf;Fk; Nubaj;jpd; msT

2) 2) 2) 2) gapw;rpf; fzf;F gapw;rpf; fzf;F gapw;rpf; fzf;F gapw;rpf; fzf;F : xUxUxUxU ,urhad,urhad,urhad,urhad tpistpy;tpistpy;tpistpy;tpistpy; xU nghUßxU nghUßxU nghUßxU nghUß khw;wk;khw;wk;khw;wk;khw;wk; milAk; khWmilAk; khWmilAk; khWmilAk; khW tPjkhdJ tPjkhdJ tPjkhdJ tPjkhdJ NNNNeuj;jpy; khw;wkilahj mg;nghUsp∂ mstpw;F tpfpjkhf cßsJ. xU kzp Neu Kbtpy; euj;jpy; khw;wkilahj mg;nghUsp∂ mstpw;F tpfpjkhf cßsJ. xU kzp Neu Kbtpy; euj;jpy; khw;wkilahj mg;nghUsp∂ mstpw;F tpfpjkhf cßsJ. xU kzp Neu Kbtpy; euj;jpy; khw;wkilahj mg;nghUsp∂ mstpw;F tpfpjkhf cßsJ. xU kzp Neu Kbtpy; 60 fpuhKk; kw;Wk; 4 kzp Neu Kbtpy; 21 f60 fpuhKk; kw;Wk; 4 kzp Neu Kbtpy; 21 f60 fpuhKk; kw;Wk; 4 kzp Neu Kbtpy; 21 f60 fpuhKk; kw;Wk; 4 kzp Neu Kbtpy; 21 fpuhKk; kPjkpUe;jhy;> Muk;g epiyapy;> mg; puhKk; kPjkpUe;jhy;> Muk;g epiyapy;> mg; puhKk; kPjkpUe;jhy;> Muk;g epiyapy;> mg; puhKk; kPjkpUe;jhy;> Muk;g epiyapy;> mg; nghUsp∂ nghUsp∂ nghUsp∂ nghUsp∂ vilapiidf;fhΩfvilapiidf;fhΩfvilapiidf;fhΩfvilapiidf;fhΩf. 3333) & 1000 vd;w njhiff;F njhlur;;rp $l;L tl;b fzf;fplg;gLfpwJ tl;b tpfpjk; ) & 1000 vd;w njhiff;F njhlur;;rp $l;L tl;b fzf;fplg;gLfpwJ tl;b tpfpjk; ) & 1000 vd;w njhiff;F njhlur;;rp $l;L tl;b fzf;fplg;gLfpwJ tl;b tpfpjk; ) & 1000 vd;w njhiff;F njhlur;;rp $l;L tl;b fzf;fplg;gLfpwJ tl;b tpfpjk; Mz;nlhz;wpf;F 4 rjtPjkhf ,Ug;gpd;, mj;njhif vj;jid Mz;Lfspy; Muk;g Mz;nlhz;wpf;F 4 rjtPjkhf ,Ug;gpd;, mj;njhif vj;jid Mz;Lfspy; Muk;g Mz;nlhz;wpf;F 4 rjtPjkhf ,Ug;gpd;, mj;njhif vj;jid Mz;Lfspy; Muk;g Mz;nlhz;wpf;F 4 rjtPjkhf ,Ug;gpd;, mj;njhif vj;jid Mz;Lfspy; Muk;g njhifiag; njhifiag; njhifiag; njhifiag; Nghy; ,Uklq;fhFk;. Nghy; ,Uklq;fhFk;. Nghy; ,Uklq;fhFk;. Nghy; ,Uklq;fhFk;. ( log e

2 = 0.6931)

jPu;T : t vd;w Neuj;jpy; mry; A vd;f.

⇒

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vdpy;

∴

⇒

tUlq;fs; tUlq;fs; 4 khjq;fs; (Njhuhakhf)

4) gapw;rpf; fzf;F gapw;rpf; fzf;F gapw;rpf; fzf;F gapw;rpf; fzf;F : xUxUxUxU tq;fpahdJtq;fpahdJtq;fpahdJtq;fpahdJ njhlu; $l;L tl;bnjhlu; $l;L tl;bnjhlu; $l;L tl;bnjhlu; $l;L tl;b Kiwapy;Kiwapy;Kiwapy;Kiwapy; tl;b tl;b tl;b tl;b fzf;fplg;gLfpwJfzf;fplg;gLfpwJfzf;fplg;gLfpwJfzf;fplg;gLfpwJ.... mjhtJmjhtJmjhtJmjhtJ tl;btl;btl;btl;b tPjjj;ijtPjjj;ijtPjjj;ijtPjjj;ij me;je;jme;je;jme;je;jme;je;j Neuj;jpy;Neuj;jpy;Neuj;jpy;Neuj;jpy; mryp∂mryp∂mryp∂mryp∂ khWkhWkhWkhW tPjj;jpy;tPjj;jpy;tPjj;jpy;tPjj;jpy; fzf;fplg;gLfpwJ.fzf;fplg;gLfpwJ.fzf;fplg;gLfpwJ.fzf;fplg;gLfpwJ. xUtuJxUtuJxUtuJxUtuJ tq;fptq;fptq;fptq;fp ,Ug;gpy;,Ug;gpy;,Ug;gpy;,Ug;gpy; njhlur;;rpahd $l;L tl;bnjhlur;;rpahd $l;L tl;bnjhlur;;rpahd $l;L tl;bnjhlur;;rpahd $l;L tl;b %yk;%yk;%yk;%yk; MΩnlh∂Wf;F MΩnlh∂Wf;F MΩnlh∂Wf;F MΩnlh∂Wf;F tl;btl;btl;btl;b ngWFfpwJ mtuJ tq;fp ,Ug;gp∂ xU tUl fhy mjpfhpg;gp∂ngWFfpwJ mtuJ tq;fp ,Ug;gp∂ xU tUl fhy mjpfhpg;gp∂ngWFfpwJ mtuJ tq;fp ,Ug;gp∂ xU tUl fhy mjpfhpg;gp∂ngWFfpwJ mtuJ tq;fp ,Ug;gp∂ xU tUl fhy mjpfhpg;gp∂ rjtPjj;ijrjtPjj;ijrjtPjj;ijrjtPjj;ijf;f;f;f; fzf;fppLf.fzf;fppLf.fzf;fppLf.fzf;fppLf. ( )

5555) ntg;gepiy 15) ntg;gepiy 15) ntg;gepiy 15) ntg;gepiy 15o c cs;s xU miwapy; itf;fg;cs;s xU miwapy; itf;fg;cs;s xU miwapy; itf;fg;cs;s xU miwapy; itf;fg;gl;Ls;s NjePupd; ntg;ggl;Ls;s NjePupd; ntg;ggl;Ls;s NjePupd; ntg;ggl;Ls;s NjePupd; ntg;g epiy epiy epiy epiy 1000 c MFk;.MFk;.MFk;.MFk;. mJ 5 epkplq;fspy; mJ 5 epkplq;fspy; mJ 5 epkplq;fspy; mJ 5 epkplq;fspy; 600

c Mf Fiwe;J tpLfpwJ. NkYk; 5 epkplk; fopj;J Mf Fiwe;J tpLfpwJ. NkYk; 5 epkplk; fopj;J Mf Fiwe;J tpLfpwJ. NkYk; 5 epkplk; fopj;J Mf Fiwe;J tpLfpwJ. NkYk; 5 epkplk; fopj;J NjePupd; ntg;gNjePupd; ntg;gNjePupd; ntg;gNjePupd; ntg;g epiyiaf; fhz;f. epiyiaf; fhz;f. epiyiaf; fhz;f. epiyiaf; fhz;f. (Oct-2009) jPu;T : t vd;w neuj;jpy; NjePupd; ntg;gepiyia T vd;f.

epA+l;ldpd; Fspu;r;rp tpjpg;gb,

vdpy;

vdpy;

.

6666) xU efuj;jpy; cs;s kf;fs; njhifapd; tsu;rp tPjk; me;Neuj;jpy; cs;s kf;fs; ) xU efuj;jpy; cs;s kf;fs; njhifapd; tsu;rp tPjk; me;Neuj;jpy; cs;s kf;fs; ) xU efuj;jpy; cs;s kf;fs; njhifapd; tsu;rp tPjk; me;Neuj;jpy; cs;s kf;fs; ) xU efuj;jpy; cs;s kf;fs; njhifapd; tsu;rp tPjk; me;Neuj;jpy; cs;s kf;fs; njhiff;F tpfpjkhf mike;Js;sJ. 1960 Mk; Mz;by; kf;fs; njhif 1njhiff;F tpfpjkhf mike;Js;sJ. 1960 Mk; Mz;by; kf;fs; njhif 1njhiff;F tpfpjkhf mike;Js;sJ. 1960 Mk; Mz;by; kf;fs; njhif 1njhiff;F tpfpjkhf mike;Js;sJ. 1960 Mk; Mz;by; kf;fs; njhif 1,30303030,000 000 000 000 vdTk; 1990vdTk; 1990vdTk; 1990vdTk; 1990 y; kf;fsy; kf;fsy; kf;fsy; kf;fs;;;; njhif 1njhif 1njhif 1njhif 1,60606060,000 MfTk; ,Ug;gpd; 2020 Mk; Mz;by; kf;fs; 000 MfTk; ,Ug;gpd; 2020 Mk; Mz;by; kf;fs; 000 MfTk; ,Ug;gpd; 2020 Mk; Mz;by; kf;fs; 000 MfTk; ,Ug;gpd; 2020 Mk; Mz;by; kf;fs; njhif vt;tsthf ,Uf;Fk;.njhif vt;tsthf ,Uf;Fk;.njhif vt;tsthf ,Uf;Fk;.njhif vt;tsthf ,Uf;Fk;. [log e 16 = .2070, e

.42 = 1.52 ]

jPu;T : t vd;w Neuj;jpy; kf;fs; njhif A vd;f.

1960 k; Mz;L kf;fs; njhifapid njhlf;f kf;fs; njhlf;f kf;fs; njhifahf nfhs;f. vdpy;

1990 Mk; Mz;by; vdpy;

2020 Mk; Mz;by; A If; fhz ( t = 2020 - 1960 = 60)

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vdpy;

2020 y; kf;fs; njhif Njhuhakhf 7777) ) ) ) xU fjphpaf;fg; nghUs; rpijAk; khWtPjkhdJ> mjd; vilf;F tpfpjkhf xU fjphpaf;fg; nghUs; rpijAk; khWtPjkhdJ> mjd; vilf;F tpfpjkhf xU fjphpaf;fg; nghUs; rpijAk; khWtPjkhdJ> mjd; vilf;F tpfpjkhf xU fjphpaf;fg; nghUs; rpijAk; khWtPjkhdJ> mjd; vilf;F tpfpjkhf mike;Js;sJ. mike;Js;sJ. mike;Js;sJ. mike;Js;sJ. mjd; vil 10 kp. fpuhk; Mf ,Uf;Fk; NghJ rpijAk; khWtPjk; mjd; vil 10 kp. fpuhk; Mf ,Uf;Fk; NghJ rpijAk; khWtPjk; mjd; vil 10 kp. fpuhk; Mf ,Uf;Fk; NghJ rpijAk; khWtPjk; mjd; vil 10 kp. fpuhk; Mf ,Uf;Fk; NghJ rpijAk; khWtPjk; ehnshd;Wf;F 0.051 kp fpuhk;ehnshd;Wf;F 0.051 kp fpuhk;ehnshd;Wf;F 0.051 kp fpuhk;ehnshd;Wf;F 0.051 kp fpuhk; vdpy; mjd; vil 10 kp. fpuhkpypUe;J 5 kp. vdpy; mjd; vil 10 kp. fpuhkpypUe;J 5 kp. vdpy; mjd; vil 10 kp. fpuhkpypUe;J 5 kp. vdpy; mjd; vil 10 kp. fpuhkpypUe;J 5 kp. fpuhKf;Ff; Fiwa vLj;Jf; nfpuhKf;Ff; Fiwa vLj;Jf; nfpuhKf;Ff; Fiwa vLj;Jf; nfpuhKf;Ff; Fiwa vLj;Jf; nfhs;Sk; fhy msitf; fhz;ffhs;Sk; fhy msitf; fhz;ffhs;Sk; fhy msitf; fhz;ffhs;Sk; fhy msitf; fhz;f. . . . [ / log2 e = 0.6931]

jPu;T : t Neuj;jpy; fjphpaf;fg; nghUspd; vil x vd;f.

⇒ = _________ (1)

⇒

x = 10 vd ,Uf;Fk; NghJ vdf; nfhLf;fg;gl;Ls;sJ

(rpijTWtjhy;)

x = 5 vDk;NghJ t I fhz

ehl;fs;.

gapw;rpf; fzf;Ffß gapw;rpf; fzf;Ffß gapw;rpf; fzf;Ffß gapw;rpf; fzf;Ffß : 8) xU ,we;jtHxU ,we;jtHxU ,we;jtHxU ,we;jtH cliycliycliycliy kUj;JtHkUj;JtHkUj;JtHkUj;JtH ghpNrhjpf;Fk;ghpNrhjpf;Fk;ghpNrhjpf;Fk;ghpNrhjpf;Fk; NghJ ,we;j NNghJ ,we;j NNghJ ,we;j NNghJ ,we;j Neuj;ijj;euj;ijj;euj;ijj;euj;ijj; NjhuhakhfNjhuhakhfNjhuhakhfNjhuhakhf fzf;fplfzf;fplfzf;fplfzf;fpl NtΩbAßsJ.NtΩbAßsJ.NtΩbAßsJ.NtΩbAßsJ. ,we;jthp∂,we;jthp∂,we;jthp∂,we;jthp∂ clyp∂clyp∂clyp∂clyp∂ ntg;gntg;gntg;gntg;g epiyepiyepiyepiy fhiy 10.00fhiy 10.00fhiy 10.00fhiy 10.00 kzpastpy; kzpastpy; kzpastpy; kzpastpy; vdvdvdvdf; f; f; f; Fwpj;Jf; nfhßfpwhH. NkYk;2kzpNeuk;fopj;J ntg;gepiy msit Fwpj;Jf; nfhßfpwhH. NkYk;2kzpNeuk;fopj;J ntg;gepiy msit Fwpj;Jf; nfhßfpwhH. NkYk;2kzpNeuk;fopj;J ntg;gepiy msit Fwpj;Jf; nfhßfpwhH. NkYk;2kzpNeuk;fopj;J ntg;gepiy msit vdf;vdf;vdf;vdf; fhΩfpwhH.fhΩfpwhH.fhΩfpwhH.fhΩfpwhH. miwap∂ ntg;gmiwap∂ ntg;gmiwap∂ ntg;gmiwap∂ ntg;g epiyepiyepiyepiy msTmsTmsTmsT (epiyahdJ) (epiyahdJ) (epiyahdJ) (epiyahdJ) vdpyvdpyvdpyvdpy; ,we;j; ,we;j; ,we;j; ,we;j Neuj;;ijf;Neuj;;ijf;Neuj;;ijf;Neuj;;ijf; fzzf;fpLf.fzzf;fpLf.fzzf;fpLf.fzzf;fpLf.

9) 9) 9) 9) xUxUxUxU Nehahspap∂ rpWePhpypUe;J Ntjpg;nghUßNehahspap∂ rpWePhpypUe;J Ntjpg;nghUßNehahspap∂ rpWePhpypUe;J Ntjpg;nghUßNehahspap∂ rpWePhpypUe;J Ntjpg;nghUß ntspNaWk;ntspNaWk;ntspNaWk;ntspNaWk; mstpidmstpidmstpidmstpid njhlHr;rpahfnjhlHr;rpahfnjhlHr;rpahfnjhlHr;rpahf Nfj;NjlHNfj;NjlHNfj;NjlHNfj;NjlH v∂wv∂wv∂wv∂w fUtpap∂fUtpap∂fUtpap∂fUtpap∂ %yk;%yk;%yk;%yk; fΩfhzpf;fg;gLfpwJ. fΩfhzpf;fg;gLfpwJ. fΩfhzpf;fg;gLfpwJ. fΩfhzpf;fg;gLfpwJ. v∂wv∂wv∂wv∂w Neuj;jpy;Neuj;jpy;Neuj;jpy;Neuj;jpy; Nehahspf;FNehahspf;FNehahspf;FNehahspf;F 10101010 kp.fpuhk;. Ntjpg;kp.fpuhk;. Ntjpg;kp.fpuhk;. Ntjpg;kp.fpuhk;. Ntjpg; nghUßnghUßnghUßnghUß nfhLf;fgnfhLf;fgnfhLf;fgnfhLf;fg;gLfpwJ. ;gLfpwJ. ;gLfpwJ. ;gLfpwJ.

,J ,J ,J ,J kp.fpuhk;kp.fpuhk;kp.fpuhk;kp.fpuhk; kzpkzpkzpkzp v∂Dk;v∂Dk;v∂Dk;v∂Dk; tPjj;jpy;tPjj;jpy;tPjj;jpy;tPjj;jpy; ntspNaWfpwJ.ntspNaWfpwJ.ntspNaWfpwJ.ntspNaWfpwJ. vdpy;vdpy;vdpy;vdpy; Neuk; Neuk; Neuk; Neuk; vDk;NghJ> Nehahspap∂vDk;NghJ> Nehahspap∂vDk;NghJ> Nehahspap∂vDk;NghJ> Nehahspap∂ clypYßs Ntjpg;nghUsp∂ msit clypYßs Ntjpg;nghUsp∂ msit clypYßs Ntjpg;nghUsp∂ msit clypYßs Ntjpg;nghUsp∂ msit fhZk; nghJr; rk∂ghL v∂d?fhZk; nghJr; rk∂ghL v∂d?fhZk; nghJr; rk∂ghL v∂d?fhZk; nghJr; rk∂ghL v∂d? KOikahfKOikahfKOikahfKOikahf Ntjjpg;;Ntjjpg;;Ntjjpg;;Ntjjpg;; nghUßnghUßnghUßnghUß ntspNawntspNawntspNawntspNaw vLj;Jf;vLj;Jf;vLj;Jf;vLj;Jf; nfhßSk;nfhßSk;nfhßSk;nfhßSk; Fiwe;jFiwe;jFiwe;jFiwe;j gl;rgl;rgl;rgl;r fhyfhyfhyfhy msTmsTmsTmsT v∂d?v∂d?v∂d?v∂d? 10) EΩZapHfsp∂ ngUf;fj;jpy; ghf;Bhpahtp∂ ngUf;ftPjkhdJ mjpy;fhzg;gLk; ; 10) EΩZapHfsp∂ ngUf;fj;jpy; ghf;Bhpahtp∂ ngUf;ftPjkhdJ mjpy;fhzg;gLk; ; 10) EΩZapHfsp∂ ngUf;fj;jpy; ghf;Bhpahtp∂ ngUf;ftPjkhdJ mjpy;fhzg;gLk; ; 10) EΩZapHfsp∂ ngUf;fj;jpy; ghf;Bhpahtp∂ ngUf;ftPjkhdJ mjpy;fhzg;gLk; ; ghf;Bhpahtp∂ vΩzpf;iff;F tpfpjkhf mike;JßsJ. ,g;ngUf;fj;jhy;; ghf;Bhpahtp∂ghf;Bhpahtp∂ vΩzpf;iff;F tpfpjkhf mike;JßsJ. ,g;ngUf;fj;jhy;; ghf;Bhpahtp∂ghf;Bhpahtp∂ vΩzpf;iff;F tpfpjkhf mike;JßsJ. ,g;ngUf;fj;jhy;; ghf;Bhpahtp∂ghf;Bhpahtp∂ vΩzpf;iff;F tpfpjkhf mike;JßsJ. ,g;ngUf;fj;jhy;; ghf;Bhpahtp∂

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vΩzpf;if 1 kzp Neuj;jpy; Kk;klq;fhfpwJ vdpy; Ie;J kzp NvΩzpf;if 1 kzp Neuj;jpy; Kk;klq;fhfpwJ vdpy; Ie;J kzp NvΩzpf;if 1 kzp Neuj;jpy; Kk;klq;fhfpwJ vdpy; Ie;J kzp NvΩzpf;if 1 kzp Neuj;jpy; Kk;klq;fhfpwJ vdpy; Ie;J kzp Neu Kbtpy;eu Kbtpy;eu Kbtpy;eu Kbtpy; ghf;Bhpahtp∂ vΩzpf;if Muk;g epiyiaf; fhl;bYk; klq;fhFk; vdf;fhl;Lf.ghf;Bhpahtp∂ vΩzpf;if Muk;g epiyiaf; fhl;bYk; klq;fhFk; vdf;fhl;Lf.ghf;Bhpahtp∂ vΩzpf;if Muk;g epiyiaf; fhl;bYk; klq;fhFk; vdf;fhl;Lf.ghf;Bhpahtp∂ vΩzpf;if Muk;g epiyiaf; fhl;bYk; klq;fhFk; vdf;fhl;Lf.

9.jdpepiyf;9.jdpepiyf;9.jdpepiyf;9.jdpepiyf; fzf;fpayfzf;fpayfzf;fpayfzf;fpay;;;; (gFjp(gFjp(gFjp(gFjp----, : 1, : 1, : 1, : 1----tpdhtpdhtpdhtpdh----10101010----kjpgngΩfs;)kjpgngΩfs;)kjpgngΩfs;)kjpgngΩfs;)

) xU Kbtw;w vgpypad; Fyk; vdf;fhl;Lf. ,q;FxU Kbtw;w vgpypad; Fyk; vdf;fhl;Lf. ,q;FxU Kbtw;w vgpypad; Fyk; vdf;fhl;Lf. ,q;FxU Kbtw;w vgpypad; Fyk; vdf;fhl;Lf. ,q;F MdJMdJMdJMdJ vDkhW vDkhW vDkhW vDkhW

tiuaWf;fg;gl;Ls;stiuaWf;fg;gl;Ls;stiuaWf;fg;gl;Ls;stiuaWf;fg;gl;Ls;sJ.J.J.J.

jPh;T: 1..milg;G tpjp: kw;Wk; 2 KO vz;fs; Mjyhy; xU KO vz; 2.Nrh;g;G tpjp: vd;f.

Nrh;g;G tpjp cz;ik.

3.rkdp tpjp: e rkdp cWg;G vd;f.

rkdp tpjp cz;ikahFk;. 4.vjph;kiw tpjp: vd;f. a- d; vjph;kiw vdpy; –1

–1

–1 vjph;kiw tpjp cz;ikahFk;. ) xU Fyk;. 5.ghpkhw;Wg; gz;G: vd;f. / ghpkhw;W tpjp cz;ikahFk;. xU vgpypad; Fyk;. NkYk;. xU Kbtw;w fzkhjyhy; ,J Kbtw;w vgpypad; FykhFk;.

vd;w mikg;gpy; cs;s mzpfs; ahTk; mlq;fpa fzk; vd;w mikg;gpy; cs;s mzpfs; ahTk; mlq;fpa fzk; vd;w mikg;gpy; cs;s mzpfs; ahTk; mlq;fpa fzk; vd;w mikg;gpy; cs;s mzpfs; ahTk; mlq;fpa fzk; GGGG MdJ MdJ MdJ MdJ

mzp ngUf;fypd; fPo; xU Fyk; vdf;fhl;Lf.mzp ngUf;fypd; fPo; xU Fyk; vdf;fhl;Lf.mzp ngUf;fypd; fPo; xU Fyk; vdf;fhl;Lf.mzp ngUf;fypd; fPo; xU Fyk; vdf;fhl;Lf.

jPh;T: vd;f. mzpg; ngUf;fy;.

1. milg;G tpjp: A = , B = ε G

milg;G tpjp cz;ik.

2. Nrh;g;G tpjp: mzpg; ngUf;fy; vg;NghJk; Nrh;g;G tpjpf;Fl;gl;lJ.

3. rkdpg; gz;G: vd;gJ rkdp cWg;G vd;f.

⇒

⇒

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vd;gJ rkdp cWg;G. / rkdp tpjp cz;ik.

4. vjph;kiw tpjp:

vd;gJ vDkhW cs;sJ vd;f.

⇒ vDkhW cs;sJ.

⇒

d; vjph;kiw MFk;.

vdNt xU FykhFk;.

3)3)3)3) ,,,, mikg;gpy; cs;s vy;yh mzpfSk; mlq;fpa fzk; mzpg; ngUf;fypd; mikg;gpy; cs;s vy;yh mzpfSk; mlq;fpa fzk; mzpg; ngUf;fypd; mikg;gpy; cs;s vy;yh mzpfSk; mlq;fpa fzk; mzpg; ngUf;fypd; mikg;gpy; cs;s vy;yh mzpfSk; mlq;fpa fzk; mzpg; ngUf;fypd;

fPo; xU vgpypad; Fyj;ij mikf;Fk; vdf;fhl;Lf.fPo; xU vgpypad; Fyj;ij mikf;Fk; vdf;fhl;Lf.fPo; xU vgpypad; Fyj;ij mikf;Fk; vdf;fhl;Lf.fPo; xU vgpypad; Fyj;ij mikf;Fk; vdf;fhl;Lf.

jPh;T: mzpg; ngUf;fy;;;;

1. milg;G tpjp : vdpy;

milg;G tpjp cz;ikahFk;. 2. Nrh;g;G tpjp: mzpg; ngUf;fy; vg;NghJk; Nrh;g;G tpjpf;Fl;gl;lJ.

3. rkdpg; gz;G: vd;gJ rkdp cWg;G vd;f..

⇒

vd;gJ rkdp cWg;G.

/rkdp tpjp cz;ik.

4. vjph;kiw tpjp: vd;gJ vDkhW cs;sJ vd;f.

⇒

d; vjph;kiw MFk;.

= ghpkhw;W tpjp cz;ik

vdNt xU vgpypad; FykhFk;. Ij; jtpu kw;w vy;yh tpfpjKW vz;fSk; mIj; jtpu kw;w vy;yh tpfpjKW vz;fSk; mIj; jtpu kw;w vy;yh tpfpjKW vz;fSk; mIj; jtpu kw;w vy;yh tpfpjKW vz;fSk; mlq;fpa fzk; lq;fpa fzk; lq;fpa fzk; lq;fpa fzk; GGGG vd;f. vd;f. vd;f. vd;f. GGGG y; y; y; y; IIII

vDkhW tiuaWg;Nghk;. vDkhW tiuaWg;Nghk;. vDkhW tiuaWg;Nghk;. vDkhW tiuaWg;Nghk;. xU KbTw;w vgpypad; Fyk; xU KbTw;w vgpypad; Fyk; xU KbTw;w vgpypad; Fyk; xU KbTw;w vgpypad; Fyk;

vdf;fhl;Lf. vdf;fhl;Lf. vdf;fhl;Lf. vdf;fhl;Lf.

jPh;T: kw;Wk; 1. milg;G tpjp :

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vdpy;> tpfpjKW vz;fs; kw;Wk;

milg;G tpjp cz;ikahFk;. 2. Nrh;g;G tpjp:

∴ / Nrh;g;G tpjp cz;ikahFk;.

3. rkdp tpjp: vd;gJ rkdp cWg;G vd;f.

/rkdp tpjp cz;ikahFk;. 4. vjph;kiw tpjp: d; vjph;kiw vd;f.

/ vjph;kiw tpjp cz;ikahFk;. 5. ghpkhw;W tpjp: f;F / ghpkhw;W tpjp cz;ik. vdNt xU vgpypad; FykhFk;. NkYk; G Kbtw;wjhjyhy;> KbTw;w vgpypad; Fyk;. 5) 5) 5) 5) ----1 I jtpu kw;w vy;yh tpfpjKW vz;fSk; cs;slf;fpa fzk; 1 I jtpu kw;w vy;yh tpfpjKW vz;fSk; cs;slf;fpa fzk; 1 I jtpu kw;w vy;yh tpfpjKW vz;fSk; cs;slf;fpa fzk; 1 I jtpu kw;w vy;yh tpfpjKW vz;fSk; cs;slf;fpa fzk; GGGG MdJ MdJ MdJ MdJ

vDkhW tiuaWf;fg;gl;l nraypd; vDkhW tiuaWf;fg;gl;l nraypd; vDkhW tiuaWf;fg;gl;l nraypd; vDkhW tiuaWf;fg;gl;l nraypd; d; fPo; xU vgpypad; Fyj;ij d; fPo; xU vgpypad; Fyj;ij d; fPo; xU vgpypad; Fyj;ij d; fPo; xU vgpypad; Fyj;ij

mikf;Fk; vdf; fhl;Lf. mikf;Fk; vdf; fhl;Lf. mikf;Fk; vdf; fhl;Lf. mikf;Fk; vdf; fhl;Lf.

jPh;T: kw;Wk; 1. milg;G tpjp :

vdpy;> tpfpjKW vz;fs; kw;Wk;

milg;G tpjp cz;ikahFk;. 2. Nrh;g;G tpjp:

∴ / Nrh;g;G tpjp cz;ikahFk;. 3. rkdp tpjp: vd;gJ rkdp cWg;G vd;f. /rkdp tpjp cz;ikahFk;. 4. vjph;kiw tpjp: d; vjph;kiw vd;f.

/ vjph;kiw tpjp cz;ikahFk;. 5. ghpkhw;W tpjp: f;F / ghpkhw;W tpjp cz;ik. vdNt xU vgpypad; FykhFk;. NkYk; G Kbtw;wjhjyhy;> KbTw;w vgpypad; Fyk;.

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6) 6) 6) 6) GGGG vd;gJ kpif tpfpjKW vz;fspd; fzk; vd;f. vd;gJ kpif tpfpjKW vz;fspd; fzk; vd;f. vd;gJ kpif tpfpjKW vz;fspd; fzk; vd;f. vd;gJ kpif tpfpjKW vz;fspd; fzk; vd;f. vDkhW tiuaWf;fgl;l nrayp vDkhW tiuaWf;fgl;l nrayp vDkhW tiuaWf;fgl;l nrayp vDkhW tiuaWf;fgl;l nrayp

d; fPo; xU Fyj;ij mikf;Fk; vdf;fhl;Lf. d; fPo; xU Fyj;ij mikf;Fk; vdf;fhl;Lf. d; fPo; xU Fyj;ij mikf;Fk; vdf;fhl;Lf. d; fPo; xU Fyj;ij mikf;Fk; vdf;fhl;Lf.

jPh;T: kpif tpfpj KW vz;fs;

1. milg;G tpjp: vd;f. vd;gd kpif tpfpj KW vz;fshjyhy;

kpif tpfpjKW vz; MFk;.

/ milT tpjp cz;ik.

2. Nrh;g;G tpjp: vdpy;>

/ Nrh;g;G tpjp cz;ik.

3. rkdp tpjp: vd;gJ rkdp cWg;G vd;f.

/ rkdp tpjp cz;ikahFk;. 4. vjph;kiw tpjp: d; vjph;kiw vd;f.

–1

–1 vjph;kiw tpjp cz;ikahFk;.

vdNt xU FykhFk;.

7) 7) 7) 7) v∂gJ $l;liyg; nghWj;;J XU Kbtw;;w vgpypad; Fyj;ij v∂gJ $l;liyg; nghWj;;J XU Kbtw;;w vgpypad; Fyj;ij v∂gJ $l;liyg; nghWj;;J XU Kbtw;;w vgpypad; Fyj;ij v∂gJ $l;liyg; nghWj;;J XU Kbtw;;w vgpypad; Fyj;ij mikf;Fk; vdf;fhl;Lf. mikf;Fk; vdf;fhl;Lf. mikf;Fk; vdf;fhl;Lf. mikf;Fk; vdf;fhl;Lf.

jPh;T

1. milg;G tpjp: x vd;f. ,

milT tpjp cz;ik. 2. Nrh;g;G tpjp : $l;ly; Nrh;g;G tpjp ciaJ.

3. rkdp tpjp: MdJ vy;yh f;Fk;

kw;Wk; vDkhW cs;sJ .

/ rkdp tpjp cz;ikahFk;

4. vjph;kiw tpjp : xt;nthU -f;Fk; MdJ

vDkhW cΩL.

NkYk; MFk; vjph;kiw tpjp cz;ikahFk;.

vdNt xU FykhFk;.

5. ghpkhw;W tpjp : nka;naΩfs; khw;W tpjpf;Fl;gl;lJ. / xU vgpypad; FykhFk;.

NkYk; G Kbtw;wjhjyhy;> Kbtw;w vgpypad; Fyk;. 8) 118) 118) 118) 11----d; kl;Lf;F fhzg; ngw;w ngUf;fypd; fPo; d; kl;Lf;F fhzg; ngw;w ngUf;fypd; fPo; d; kl;Lf;F fhzg; ngw;w ngUf;fypd; fPo; d; kl;Lf;F fhzg; ngw;w ngUf;fypd; fPo; vd;w fzk; XU vd;w fzk; XU vd;w fzk; XU vd;w fzk; XU vgpypad; Fyj;ij mikf;Fk; vdf;fhl;Lf. jPh;T. vgpypad; Fyj;ij mikf;Fk; vdf;fhl;Lf. jPh;T. vgpypad; Fyj;ij mikf;Fk; vdf;fhl;Lf. jPh;T. vgpypad; Fyj;ij mikf;Fk; vdf;fhl;Lf. jPh;T. jPh;T

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ml;ltizapypUe;J 1. milg;G tpjp: ml;ltizapd; vy;yh cWg;GfSk; y; cs;sd. / milg;G tpjp cz;ikahfpwJ. 2. Nrh;g;G tpjp : 11 d; kl;Lf;fhd ngUf;fy; Nrh;g;G tpjp cilaJ. 3. rkdp tpjp : vd;gJ rkdp cWg;G MFk;. 4. vjph;kiw tpjp: d; vjph;kiw d; vjph;kiw

d; vjph;kiw

d; vjph;kiw d; vjph;kiw vdNt vjph;kiw tpjp cz;ikahfpwJ. 5. ghpkhw;W tpjp: 11d; kl;Lf;fhd ngUf;fy; ghpkhw;W tpjpf;Fl;gl;lJ. vdNt xU vgpypad; FykhFk;. 9) 9) 9) 9) xU Fyj;ij mikf;Fk; vdf;fhl;Lf. xU Fyj;ij mikf;Fk; vdf;fhl;Lf. xU Fyj;ij mikf;Fk; vdf;fhl;Lf. xU Fyj;ij mikf;Fk; vdf;fhl;Lf.

ml;ltizapypUe;J

1. milg;G tpjp: ml;ltizapd; vy;yh cWg;GfSk; y; cs;sd. / milg;G tpjp cz;ikahfpwJ. 2. Nrh;g;G tpjp : 9 d; kl;Lf;fhd ngUf;fy; Nrh;g;G tpjp cilaJ. 3. rkdp tpjp : vd;gJ rkdp cWg;G MFk;. 4. vjph;kiw tpjp: d; vjph;kiw d; vjph;kiw

d; vjph;kiw d; vjph;kiw

d; vjph;kiw

d; vjph;kiw vdNt vjph;kiw tpjp cz;ikahfpwJ.

vdNt xU FykhFk 10) g+[;akw;w 10) g+[;akw;w 10) g+[;akw;w 10) g+[;akw;w fyg;ngz;fspd; fyg;ngz;fspd; fyg;ngz;fspd; fyg;ngz;fspd; fzkhd fzkhd fzkhd fzkhd ,y; ,y; ,y; ,y; tiuaWf;fg;gl;l tiuaWf;fg;gl;l tiuaWf;fg;gl;l tiuaWf;fg;gl;l

vd;w rhh;Gfs; ahTk; mlq;fpa fzk; vd;w rhh;Gfs; ahTk; mlq;fpa fzk; vd;w rhh;Gfs; ahTk; mlq;fpa fzk; vd;w rhh;Gfs; ahTk; mlq;fpa fzk;

MdJ rhh;Gfspd; Nrh;g;gpd; fPo; xU vgpypad; Fyk; mikf;Fk; MdJ rhh;Gfspd; Nrh;g;gpd; fPo; xU vgpypad; Fyk; mikf;Fk; MdJ rhh;Gfspd; Nrh;g;gpd; fPo; xU vgpypad; Fyk; mikf;Fk; MdJ rhh;Gfspd; Nrh;g;gpd; fPo; xU vgpypad; Fyk; mikf;Fk; vd epWTf. vd epWTf. vd epWTf. vd epWTf. jPh;T rhh;Gfspd; Nrh;g;G.

*

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ngUf;fy; ml;ltiz:

ml;ltizapypUe;J

1. milg;G tpjp: ml;ltizapd; vy;yh cWg;GfSk; G - y; cWg;Gfshjyhy;> milg;G tpjp cz;ikahfpwJ.

2. Nrh;g;G tpjp : rhh;Gfspd; Nrh;g;G> Nrh;g;G tpjp cilaJ. 3. rkdp tpjp : MdJ rkdp cWg;G MFk;.

4. vjph;kiw tpjp: d; vjph;kiw

d; vjph;kiw d; vjph;kiw d; vjph;kiw vdNt vjph;kiw tpjp cz;ikahfpwJ. vdNt xU FykhFk;.

5. ghpkhw;W tpjp: ml;ltizapypUe;J ghpkhw;W tpjp cz;iknadj;njhpfpwJ.

xU vgpypad; FykhFk;.

11) 11) 11) 11) v∂fpw fzkv∂fpw fzkv∂fpw fzkv∂fpw fzk;; mzpg; mzpg; mzpg; mzpg;

ngUf;fypd; fPo; xU Fyj;ij mikf;Fk; vdf;fhl;Lf. ngUf;fypd; fPo; xU Fyj;ij mikf;Fk; vdf;fhl;Lf. ngUf;fypd; fPo; xU Fyj;ij mikf;Fk; vdf;fhl;Lf. ngUf;fypd; fPo; xU Fyj;ij mikf;Fk; vdf;fhl;Lf. jPh;T

vd;f.

vd;f.

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

ml;ltizapypUe;J

1. milg;G tpjp: ml;ltizapd; vy;yh cWg;GfSk; G - y; cWg;Gfshjyhy;> milg;G tpjp cz;ikahfpwJ.

2. Nrh;g;G tpjp : mzpg; ngUf;fy; Nrh;g;G tpjp cilaJ.

3. rkdp tpjp : MdJ rkdp cWg;G MFk;.

4. vjph;kiw tpjp: d; vjph;kiw

d; vjph;kiw d; vjph;kiw d; vjph;kiw d; vjph;kiw

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d; vjph;kiw vdNt vjph;kiw tpjp cz;ikahfpwJ. vdNt mzpg; ngUf;fypd; fPo; ; ; ; xU FykhFk;.

vDkhW cs;s fyg;ngΩfs; ahTk; mlq;fpa fzk; vDkhW cs;s fyg;ngΩfs; ahTk; mlq;fpa fzk; vDkhW cs;s fyg;ngΩfs; ahTk; mlq;fpa fzk; vDkhW cs;s fyg;ngΩfs; ahTk; mlq;fpa fzk; MdJ fyg;ngΩfsp∂ MdJ fyg;ngΩfsp∂ MdJ fyg;ngΩfsp∂ MdJ fyg;ngΩfsp∂

ngUf;;fyp∂ fPo; ngUf;;fyp∂ fPo; ngUf;;fyp∂ fPo; ngUf;;fyp∂ fPo; xU Fyj;ij mikf;FkxU Fyj;ij mikf;FkxU Fyj;ij mikf;FkxU Fyj;ij mikf;Fk; vdf;fhl;Lf.; vdf;fhl;Lf.; vdf;fhl;Lf.; vdf;fhl;Lf.

v∂w v∂w v∂w v∂w fzkhdJ ngUf;;fyp∂ fPo; fzkhdJ ngUf;;fyp∂ fPo; fzkhdJ ngUf;;fyp∂ fPo; fzkhdJ ngUf;;fyp∂ fPo; xU vgpypad; Fyj;ij mikf;Fk; xU vgpypad; Fyj;ij mikf;Fk; xU vgpypad; Fyj;ij mikf;Fk; xU vgpypad; Fyj;ij mikf;Fk; vdf;fhl;Lf.vdf;fhl;Lf.vdf;fhl;Lf.vdf;fhl;Lf.

xU Fyk; vdf;fhl;Lf.xU Fyk; vdf;fhl;Lf.xU Fyk; vdf;fhl;Lf.xU Fyk; vdf;fhl;Lf.

15) tof;fkhd ngUf;fyp∂ fPo; 115) tof;fkhd ngUf;fyp∂ fPo; 115) tof;fkhd ngUf;fyp∂ fPo; 115) tof;fkhd ngUf;fyp∂ fPo; 1----∂ ∂ ∂ ∂ Mk; gb %yq;;fs; Kbthd FyjMk; gb %yq;;fs; Kbthd FyjMk; gb %yq;;fs; Kbthd FyjMk; gb %yq;;fs; Kbthd Fyj;ij mikf;Fk; ;ij mikf;Fk; ;ij mikf;Fk; ;ij mikf;Fk; vdf;fhl;Lf.vdf;fhl;Lf.vdf;fhl;Lf.vdf;fhl;Lf.

Kf;fpa Fwpg;G :: Kf;fpa Fwpg;G :: Kf;fpa Fwpg;G :: Kf;fpa Fwpg;G :: Gj;jfj;jpy; nfhLf;fg;gl;Ls;s xU kjpg;ngz; tpdhf;fspd; thpir vz;fSk; Gj;jfj;jpy; nfhLf;fg;gl;Ls;s xU kjpg;ngz; tpdhf;fspd; thpir vz;fSk; Gj;jfj;jpy; nfhLf;fg;gl;Ls;s xU kjpg;ngz; tpdhf;fspd; thpir vz;fSk; Gj;jfj;jpy; nfhLf;fg;gl;Ls;s xU kjpg;ngz; tpdhf;fspd; thpir vz;fSk; fl;lfj;jpy; nfhLf;fg;gl;Ls;s xU kjpg;ngz; tpdhf;fspd; thpir vz;fSk; fl;lfj;jpy; nfhLf;fg;gl;Ls;s xU kjpg;ngz; tpdhf;fspd; thpir vz;fSk; fl;lfj;jpy; nfhLf;fg;gl;Ls;s xU kjpg;ngz; tpdhf;fspd; thpir vz;fSk; fl;lfj;jpy; nfhLf;fg;gl;Ls;s xU kjpg;ngz; tpdhf;fspd; thpir vz;fSk; ntt;Ntwhdit.ntt;Ntwhdit.ntt;Ntwhdit.ntt;Ntwhdit. fPo;fz;l ml;ltizapy; cs;s tpfPo;fz;l ml;ltizapy; cs;s tpfPo;fz;l ml;ltizapy; cs;s tpfPo;fz;l ml;ltizapy; cs;s tpilfs; fl;lfj;jpy; ilfs; fl;lfj;jpy; ilfs; fl;lfj;jpy; ilfs; fl;lfj;jpy; cs;s xU kjpg;ngz; tpdhf;fSf;fhd tpilfshFk;.cs;s xU kjpg;ngz; tpdhf;fSf;fhd tpilfshFk;.cs;s xU kjpg;ngz; tpdhf;fSf;fhd tpilfshFk;.cs;s xU kjpg;ngz; tpdhf;fSf;fhd tpilfshFk;.

1111 1 31313131 3 61616161 3 91919191 2 1111 4 31313131 2 61616161 1 91919191 4 121121121121 4

2222 1 32323232 3 62626262 4 92929292 3 2222 3 32323232 1 62626262 4 92929292 2 122122122122 2

3333 4 33333333 4 63636363 2 93939393 1 3333 4 33333333 1 63636363 3 93939393 1 123123123123 4

4444 1 34343434 2 64646464 3 94949494 1 4444 1 34343434 1 64646464 1 94949494 1 124124124124 3

5555 3 35353535 4 65656565 3 95959595 4 5555 2 35353535 2 65656565 2 95959595 2 125125125125 4

6666 1 36363636 3 66666666 3 96969696 3 6666 1 36363636 4 66666666 2 96969696 3 126126126126 2

7777 1 37373737 2 67676767 3 97979797 4 7777 2 37373737 3 67676767 4 97979797 2 127127127127 3

8888 3 38383838 2 68686868 4 98989898 4 8888 1 38383838 2 68686868 3 98989898 2 128128128128 1

9999 3 39393939 3 69696969 4 99999999 4 9999 1 39393939 2 69696969 2 99999999 2 129129129129 1

10101010 2 40404040 3 70707070 1 100100100100 3 10101010 3 40404040 3 70707070 1 100100100100 3 130130130130 4

11111111 3 41414141 1 71717171 3 101101101101 2 11111111 3 41414141 3 71717171 2 101101101101 3 131131131131 2

12121212 1 42424242 2 72727272 1 102102102102 4 12121212 4 42424242 4 72727272 2 102102102102 2 132132132132 2

13131313 3 43434343 1 73737373 3 103103103103 4 13131313 3 43434343 2 73737373 1 103103103103 4 133133133133 3

14141414 4 44444444 4 74747474 1 104104104104 2 14141414 2 44444444 1 74747474 1 104104104104 2 134134134134 3

15151515 3 45454545 2 75757575 4 105105105105 1 15151515 1 45454545 4 75757575 4 105105105105 1 135135135135 3

16161616 1 46464646 2 76767676 2 106106106106 3 16161616 4 46464646 2 76767676 2 106106106106 2 136136136136 3

17171717 3 47474747 3 77777777 1 107107107107 3 17171717 2 47474747 1 77777777 2 107107107107 1 137137137137 1

18181818 3 48484848 2 78787878 2 108108108108 3 18181818 4 48484848 3 78787878 4 108108108108 2 138138138138 2

19191919 4 49494949 4 77779999 3 109109109109 2 19191919 1 49494949 3 79797979 4 109109109109 1 139139139139 1

20202020 3 50505050 2 80808080 1 110110110110 3 20202020 4 50505050 2 80808080 3 110110110110 4 140140140140 2

21212121 3 51515151 1 81818181 3 111111111111 2 21212121 4 51515151 1 81818181 2 111111111111 2 141141141141 3

22222222 3 52525252 3 82828282 3 112112112112 2 22222222 2 52525252 2 82828282 1 112112112112 3 142142142142 2

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23232323 4 53535353 1 83838383 3 113113113113 3 23232323 2 53535353 3 83838383 4 113113113113 2 143143143143 4

24242424 4 54545454 2 84848484 4 114114114114 2 22224444 1 54545454 3 84848484 1 114114114114 3 144144144144 1

25252525 2 55555555 4 85858585 2 115115115115 4 25252525 3 55555555 1 85858585 2 115115115115 3 145145145145 4

26262626 4 56565656 1 86868686 4 116116116116 2 26262626 2 56565656 1 86868686 2 116116116116 3 146146146146 2

27272727 1 57575757 3 87878787 1 117117117117 4 27272727 2 57575757 4 87878787 3 117117117117 3 147147147147 1

28282828 3 58585858 4 88888888 3 118118118118 4 28282828 1 58585858 1 88888888 2 118118118118 4 148148148148 4

29292929 4 59595959 4 89898989 4 119119119119 4 29292929 1 59595959 4 89898989 4 119119119119 4 149149149149 2

30303030 1 60606060 1 90909090 2 120120120120 2 30303030 2 60606060 3 90909090 3 120120120120 1 150150150150 3

121121121121 1

khztHfspd; ftdj;jpw;FkhztHfspd; ftdj;jpw;FkhztHfspd; ftdj;jpw;FkhztHfspd; ftdj;jpw;F 1.1.1.1. khztHfs; fzpjj;jpy; 100 tpOf;fhL NjHr;rp ngWk; tifapy; khztHfs; fzpjj;jpy; 100 tpOf;fhL NjHr;rp ngWk; tifapy; khztHfs; fzpjj;jpy; 100 tpOf;fhL NjHr;rp ngWk; tifapy; khztHfs; fzpjj;jpy; 100 tpOf;fhL NjHr;rp ngWk; tifapy;

,g;gapw;rpf; fl;lfk; jahhpf;fg;gl;Ls;sJ. ,f;fl;lfj;jpy; 158 kjpg;,g;gapw;rpf; fl;lfk; jahhpf;fg;gl;Ls;sJ. ,f;fl;lfj;jpy; 158 kjpg;,g;gapw;rpf; fl;lfk; jahhpf;fg;gl;Ls;sJ. ,f;fl;lfj;jpy; 158 kjpg;,g;gapw;rpf; fl;lfk; jahhpf;fg;gl;Ls;sJ. ,f;fl;lfj;jpy; 158 kjpg;ngz;fSf;fhd tpdhngz;fSf;fhd tpdhngz;fSf;fhd tpdhngz;fSf;fhd tpdh----

tpilfs; %d;W gFjpfshf gphpf;fg;gl;Ls;sd.tpilfs; %d;W gFjpfshf gphpf;fg;gl;Ls;sd.tpilfs; %d;W gFjpfshf gphpf;fg;gl;Ls;sd.tpilfs; %d;W gFjpfshf gphpf;fg;gl;Ls;sd.

Kjy; gFjpapy; gphpT mKjy; gFjpapy; gphpT mKjy; gFjpapy; gphpT mKjy; gFjpapy; gphpT m----f;fhd 271 tpdhf;fhd 271 tpdhf;fhd 271 tpdhf;fhd 271 tpdh----tpilfs; nfhLf;fg;gl;Ls;sd. ,jpypUe;J 30 tpilfs; nfhLf;fg;gl;Ls;sd. ,jpypUe;J 30 tpilfs; nfhLf;fg;gl;Ls;sd. ,jpypUe;J 30 tpilfs; nfhLf;fg;gl;Ls;sd. ,jpypUe;J 30

tpdhf;fs; Nfl;fg;gLk;. 30 kjpg;ngz;fs; ngwyhk;. ,uz;lhk; gFjpapy; gphpT Mtpdhf;fs; Nfl;fg;gLk;. 30 kjpg;ngz;fs; ngwyhk;. ,uz;lhk; gFjpapy; gphpT Mtpdhf;fs; Nfl;fg;gLk;. 30 kjpg;ngz;fs; ngwyhk;. ,uz;lhk; gFjpapy; gphpT Mtpdhf;fs; Nfl;fg;gLk;. 30 kjpg;ngz;fs; ngwyhk;. ,uz;lhk; gFjpapy; gphpT M----f;fhd f;fhd f;fhd f;fhd

tpdhtpdhtpdhtpdh----tpilfs; nfhLf;fg;gl;Ls;sd. ,jpypUetpilfs; nfhLf;fg;gl;Ls;sd. ,jpypUetpilfs; nfhLf;fg;gl;Ls;sd. ,jpypUetpilfs; nfhLf;fg;gl;Ls;sd. ,jpypUe;J muR NjHtpy; 8 tpdhf;fs; Nfl;fg;gLk;. ;J muR NjHtpy; 8 tpdhf;fs; Nfl;fg;gLk;. ;J muR NjHtpy; 8 tpdhf;fs; Nfl;fg;gLk;. ;J muR NjHtpy; 8 tpdhf;fs; Nfl;fg;gLk;.

48 kjpg;ngz;fs; ngwyhk;. %d;whk; gFjpapy; gphpT ,48 kjpg;ngz;fs; ngwyhk;. %d;whk; gFjpapy; gphpT ,48 kjpg;ngz;fs; ngwyhk;. %d;whk; gFjpapy; gphpT ,48 kjpg;ngz;fs; ngwyhk;. %d;whk; gFjpapy; gphpT ,----f;fhd tpdhf;fhd tpdhf;fhd tpdhf;fhd tpdh----tpilfs; tpilfs; tpilfs; tpilfs;

nfhLf;fg;gl;Ls;sd. ,jpypUe;J muR NjHtpy; 8 tpdhf;fs; Nfl;fg;gLk;. 80 kjpg;ngz;fs; nfhLf;fg;gl;Ls;sd. ,jpypUe;J muR NjHtpy; 8 tpdhf;fs; Nfl;fg;gLk;. 80 kjpg;ngz;fs; nfhLf;fg;gl;Ls;sd. ,jpypUe;J muR NjHtpy; 8 tpdhf;fs; Nfl;fg;gLk;. 80 kjpg;ngz;fs; nfhLf;fg;gl;Ls;sd. ,jpypUe;J muR NjHtpy; 8 tpdhf;fs; Nfl;fg;gLk;. 80 kjpg;ngz;fs;

ngwyhk;. ngwyhk;. ngwyhk;. ngwyhk;.

2.2.2.2. xNu khjphpahd tpdhf;fspy; xU tpdhTf;F tpil nfhLf;fg;gl;xNu khjphpahd tpdhf;fspy; xU tpdhTf;F tpil nfhLf;fg;gl;xNu khjphpahd tpdhf;fspy; xU tpdhTf;F tpil nfhLf;fg;gl;xNu khjphpahd tpdhf;fspy; xU tpdhTf;F tpil nfhLf;fg;gl;Ls;sJ kw;wit Ls;sJ kw;wit Ls;sJ kw;wit Ls;sJ kw;wit

gapw;rp tpdhf;fshf jug;gl;Ls;sd. khztHfs; mtw;wpw;F tpil fhz Ntz;Lk;. gapw;rp tpdhf;fshf jug;gl;Ls;sd. khztHfs; mtw;wpw;F tpil fhz Ntz;Lk;. gapw;rp tpdhf;fshf jug;gl;Ls;sd. khztHfs; mtw;wpw;F tpil fhz Ntz;Lk;. gapw;rp tpdhf;fshf jug;gl;Ls;sd. khztHfs; mtw;wpw;F tpil fhz Ntz;Lk;.

fzf;Ffis mbf;fb Nghl;L gof Ntz;Lk;.fzf;Ffis mbf;fb Nghl;L gof Ntz;Lk;.fzf;Ffis mbf;fb Nghl;L gof Ntz;Lk;.fzf;Ffis mbf;fb Nghl;L gof Ntz;Lk;.

3.3.3.3. NjHT neUq;Fk;NghJ mtrukhf fzf;Ffis kdg;ghlk; nra;tij tpLj;J Muk;gk; NjHT neUq;Fk;NghJ mtrukhf fzf;Ffis kdg;ghlk; nra;tij tpLj;J Muk;gk; NjHT neUq;Fk;NghJ mtrukhf fzf;Ffis kdg;ghlk; nra;tij tpLj;J Muk;gk; NjHT neUq;Fk;NghJ mtrukhf fzf;Ffis kdg;ghlk; nra;tij tpLj;J Muk;gk;

KjNy Nghl;L ghHf;f Ntz;Lk;.KjNy Nghl;L ghHf;f Ntz;Lk;.KjNy Nghl;L ghHf;f Ntz;Lk;.KjNy Nghl;L ghHf;f Ntz;Lk;.

4.4.4.4. NjHT vOJk;NghJ Fog;gkpy;yhky; vOJjy; mtrpNjHT vOJk;NghJ Fog;gkpy;yhky; vOJjy; mtrpNjHT vOJk;NghJ Fog;gkpy;yhky; vOJjy; mtrpNjHT vOJk;NghJ Fog;gkpy;yhky; vOJjy; mtrpak;. ak;. ak;. ak;.

5.5.5.5. NjHtpy; gFjp m> M kw;Wk; ,NjHtpy; gFjp m> M kw;Wk; ,NjHtpy; gFjp m> M kw;Wk; ,NjHtpy; gFjp m> M kw;Wk; ,----f;fSf;fhd tpilfis khw;wp khw;wp vOJtij f;fSf;fhd tpilfis khw;wp khw;wp vOJtij f;fSf;fhd tpilfis khw;wp khw;wp vOJtij f;fSf;fhd tpilfis khw;wp khw;wp vOJtij

jtpHf;fTk;. jtpHf;fTk;. jtpHf;fTk;. jtpHf;fTk;.

6.6.6.6. mjpf gf;fq;fs; vOjpdhy; mjpf kjpg;ngz; fpilf;Fk; vd;W epidg;gJ jtW.mjpf gf;fq;fs; vOjpdhy; mjpf kjpg;ngz; fpilf;Fk; vd;W epidg;gJ jtW.mjpf gf;fq;fs; vOjpdhy; mjpf kjpg;ngz; fpilf;Fk; vd;W epidg;gJ jtW.mjpf gf;fq;fs; vOjpdhy; mjpf kjpg;ngz; fpilf;Fk; vd;W epidg;gJ jtW.

7.7.7.7. Njitahd ,lq;fspy; glq;fs; tiua Ntz;Lk;. glj;jpw;F kjpg;ngz;fs; cz;L.Njitahd ,lq;fspy; glq;fs; tiua Ntz;Lk;. glj;jpw;F kjpg;ngz;fs; cz;L.Njitahd ,lq;fspy; glq;fs; tiua Ntz;Lk;. glj;jpw;F kjpg;ngz;fs; cz;L.Njitahd ,lq;fspy; glq;fs; tiua Ntz;Lk;. glj;jpw;F kjpg;ngz;fs; cz;L.

8.8.8.8. Xt;nthU tpilf;Fk; jPHT fhz cjTk; RXt;nthU tpilf;Fk; jPHT fhz cjTk; RXt;nthU tpilf;Fk; jPHT fhz cjTk; RXt;nthU tpilf;Fk; jPHT fhz cjTk; R+j;jpuq;fis mtrpak; vOj Ntz;Lk;.+j;jpuq;fis mtrpak; vOj Ntz;Lk;.+j;jpuq;fis mtrpak; vOj Ntz;Lk;.+j;jpuq;fis mtrpak; vOj Ntz;Lk;.

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9.9.9.9. fl;lfj;jpy; cs;s tpilfs; RUf;fkhfTk; KO kjpg;ngz;fs; ngWk; tifapYk; fl;lfj;jpy; cs;s tpilfs; RUf;fkhfTk; KO kjpg;ngz;fs; ngWk; tifapYk; fl;lfj;jpy; cs;s tpilfs; RUf;fkhfTk; KO kjpg;ngz;fs; ngWk; tifapYk; fl;lfj;jpy; cs;s tpilfs; RUf;fkhfTk; KO kjpg;ngz;fs; ngWk; tifapYk;

nfhLf;fg;gl;Ls;sd. NkYk; ,tw;iw RUf;fp vOJjy; $lhJ.nfhLf;fg;gl;Ls;sd. NkYk; ,tw;iw RUf;fp vOJjy; $lhJ.nfhLf;fg;gl;Ls;sd. NkYk; ,tw;iw RUf;fp vOJjy; $lhJ.nfhLf;fg;gl;Ls;sd. NkYk; ,tw;iw RUf;fp vOJjy; $lhJ.

10.10.10.10. Xt;nthU tpdhtpw;Fk; tpilia jdpNa vLj;J vOjTk;.Xt;nthU tpdhtpw;Fk; tpilia jdpNa vLj;J vOjTk;.Xt;nthU tpdhtpw;Fk; tpilia jdpNa vLj;J vOjTk;.Xt;nthU tpdhtpw;Fk; tpilia jdpNa vLj;J vOjTk;.

“ nja;tj;jhy;; MfhnjdpDk; Kaw;rpj; jd;nja;tj;jhy;; MfhnjdpDk; Kaw;rpj; jd;nja;tj;jhy;; MfhnjdpDk; Kaw;rpj; jd;nja;tj;jhy;; MfhnjdpDk; Kaw;rpj; jd;

nka; tUj;jfnka; tUj;jfnka; tUj;jfnka; tUj;jf; $yp jUk; ; $yp jUk; ; $yp jUk; ; $yp jUk; ”

“ ahidapd; gyk; Jk;gpf;ifapy; ahidapd; gyk; Jk;gpf;ifapy; ahidapd; gyk; Jk;gpf;ifapy; ahidapd; gyk; Jk;gpf;ifapy; ---- kdpjdpd; gyk; ek;gpf;ifapy; kdpjdpd; gyk; ek;gpf;ifapy; kdpjdpd; gyk; ek;gpf;ifapy; kdpjdpd; gyk; ek;gpf;ifapy; ”

vdNt ek;gpf;ifNahL NjHT vOJq;fs;.vdNt ek;gpf;ifNahL NjHT vOJq;fs;.vdNt ek;gpf;ifNahL NjHT vOJq;fs;.vdNt ek;gpf;ifNahL NjHT vOJq;fs;. 100 tpOf;fhL NjHr;rp ngw;W khtl;lj;jpw;Fk;> 100 tpOf;fhL NjHr;rp ngw;W khtl;lj;jpw;Fk;> 100 tpOf;fhL NjHr;rp ngw;W khtl;lj;jpw;Fk;> 100 tpOf;fhL NjHr;rp ngw;W khtl;lj;jpw;Fk;>

khepyj;jpw;Fk; ngUik NrHj;jpl tho;j;Jf;fs;!khepyj;jpw;Fk; ngUik NrHj;jpl tho;j;Jf;fs;!khepyj;jpw;Fk; ngUik NrHj;jpl tho;j;Jf;fs;!khepyj;jpw;Fk; ngUik NrHj;jpl tho;j;Jf;fs;!

,tz;,tz;,tz;,tz;

fzpj fl;lf jahhpg;Gf; FOfzpj fl;lf jahhpg;Gf; FOfzpj fl;lf jahhpg;Gf; FOfzpj fl;lf jahhpg;Gf; FO

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+2 (HSC) Maths Tamil Medium Study Materials 84 Pages