md;ghh;e;j ghlrhiy FOtpdh;fSf;F€¦ · 12/03/2018 · 30. ( ,d; vy;yh kjpg;GfisAk; fhz;f?

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md;ghh;e;j ghlrhiy FOtpdh;fSf;F !!!!!!!!!!!!!!!!!!!!!!!!!!!

+2 fzpj ghlj;jpy; cWjpahf Njh;r;rp ngWtjw;fhd Kf;fpakhd tpdhf;fs; vd;w

jiyg;gpy; 63 gj;J kjpg;ngz; tpdhf;fs; nfhLj;jpUe;Njd;.(mf;Nlhgh; 2017) mjpy;

cWjpahf 5 my;yJ 6 tpdhf;fs; tUnkd;W $wpapUe;Njd;. mjd; mbg;gilapy; New;W

eilg;ngw;w (12.03.2018) + 2 fzpjj;Njh;tpy; cWjpahf 6 tpdhf;fs; ,lk;ngw;wpUe;jJ.

,e;j ghlrhiy ,izajsj;ij ghh;j;j 1000j;jpw;Fk; Nkw;gl;Nlhh; Nghd; %ykhfTk;>

thl;];mg; %ykhfTk; vq;fis Njh;r;rp ngw itj;jjw;F kdkhh;e;j ed;wpapid

njhptpj;jhh;fs;. ,e;j Gfo; midj;Jk; ghlrhiy ,iza jsj;jpidNa NrUk;

vd;Dila 63 tpdhf;fisAk; jkpofk; kw;Wk; ghz;br;NrhpapYs;s midj;J khzt

khztpaUf;Fk; nfhz;L Nrh;j;j ngUik ghlrhiy FOtpdh;fSf;F NrUk; vd;gij

,e;j Neuj;jpy; kpf;f kfpo;r;rpAld; njuptpj;Jf;nfhs;fpNwd;.

63 gj;J kjpg;ngz; tpdhf;fspy; ve;nje;j tpdh vd;W Highlight tz;z

vOj;Jf;fspy; ,q;Nf Fwpg;gpl;Ls;Nsd;

vd;Wk; ed;wpAld;

P.A. godpag;gd; M.Sc.M.Phil.,B.Ed

KJfiy fzpj gl;ljhup Mrpupah;

muR Mz;fs; Nky;epiyg;gs;sp

gl;Lf;Nfhl;il



12Mk; tFg;G fzpjk; - khHr; 2018 nghJj;NjHT tpdhj;jhs;

tpdhf;fs;

1. ep&gpf;f : cos ( A+B ) = cos A cos B – sin A sin B

2. Sin ( A- B ) = sin A cos B – cos A sin B vd ntf;lh; Kiwapy; ep&gp

3. + 3 vdpy;

x ( . ( . ) vd rhpghh;f;f.

4. = + + , = + , + + , = + +2 f;F

( d vd;gijr;

rhpghh;f;f.

5.

kw;Wk;

=

=

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

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

6.

vd;w Nfhl;il cs;slf;fpaJk;

vd;w

Nfhl;bw;F ,izahdJkhd jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad;

rkd;ghLfisf; fhz;f.

7. (1,3,2 ) vd;w Gs;sp topr; nry;tJk;

kw;Wk;

vd;w NfhLfSf;F ,izahdJkhd jsj;jpd;

ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f.

8. (-1,3,2) vd;w Gs;sp topr;nry;tJk; x + 2y + 2z= 5 kw;Wk; 3x + y + 2z = 8

Mfpa jsq;fSf;Fr; nrq;Fj;jhdJkhd jsj;jpd; ntf;lh; kw;Wk;

fhh;Brpad; rkd;ghLfisf; fhz;f.

9. A (1,-2,3) kw;Wk; B(-1,2,-1) vd;w Gs;spfs; topNar; nry;yf;$baJk;

vd;w Nfhl;bw;F ,izahdJkhd jsj;jpd; ntf;lh;

kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f.



10. (1,2,3) kw;Wk; (2,3, 1) vd;w Gs;spfs; topNar; nry;yf; $baJk;

3x -2y +4z -5 = 0 vd;w jsj;jpw;Fr; nrq;Fj;jhfTk; mike;j jsj;jpd;


11.

vd;w Nfhl;il cs;slf;fpaJk; ( -1, 1, -1) vd;w Gs;sp

topNar; nry;yf; $baJkhd ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf;

fhz;f.

12. 3 + 4 + 2 , 2 -2 - kw;Wk; 7 + Mfpatw;iw epiy

ntf;lh;fshff; nfhz;l Gs;spfs; topNar; nry;Yk; jsj;jpd; ntf;lh;

kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f.

13. ntl;Lj;Jz;L tbtpy; xU jsj;jpd; rkd;ghl;ilj; jUtpf;f.

14. xU Kf;Nfhzj;jpd; Fj;Jf;NfhLfs; xNu Gs;spapy; re;jpf;Fk; vd;gjid

ntf;lh; Kiwapy; epWTf.

15. cos (A – B ) = cos A cos B + sin A sin B vd epWTf.

16. sin ( A + B ) = sin A cosB + cos A sin B vd epWTf.

17.

kw;Wk;

vd;w NfhLfs; ntl;Lk; vdf;

fhl;b mit ntl;Lk; Gs;spiaf; fhz;f.

18. ( 2, -1, -3 ) topNar; nry;yf;$baJk;

kw;Wk;

Mfpa NfhLfSf;F ,izahf cs;sJkhd jsj;jpd;


19. ( -1, 1,1 ) kw;Wk; ( 1, -1, 1 ) Mfpa Gs;spfs; topNar; nry;yf; $baJk;

x+ 2y + 2z = 5 vd;w jsj;jpw;F nrq;Fj;jhf miktJkhd jsj;jpd;

ntf;lh; kw;Wk; fhh;Brpad; rkd;ghl;ilf; fhz;f.

20. (2, 2, -1) (3, 4, 2 ) kw;Wk;; ( 7, 0, 6) Mfpa Gs;spfs; topNar; nry;yf;$ba

jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghl;ilf; fhz;f.

21. P vDk; Gs;sp fyg;ngz; khwp z If; Fwpj;jhy; P -,d;

epakg;ghijia gpd;tUtdtw;wpw;f fhz;f.



i) Re

= 1



i) Re

= 1



i) Im

= -2



i) arg

=

25. P vDk; Gs;sp fyg;ngz; khwp z If; Fwpj;jhy; P -,d; epakg;ghijia gpd;tUtdtw;wpw;f fhz;f.

i) arg

=

26. x2 – 2 px + ( p2 + q2 ) = 0 vd;w rkd;ghl;bd; %yq;fs; kw;Wk;

tan

vdpy;

=qn-1

vd epWTf.

27. x2 – 2x + 4 = 0 ,d; %yq;fs; kw;Wk; vdpy;

n - n = i2n+1 sin

mjpypUe;J 9 - 9 d; kjpg;ig ngWf.

28. x +

= 2 cos , y +

= 2 cos vdpy;

(i)

+

= 2 cos ( m - n )

(ii)

-

= 2i sin ( m - n ) vdf; fhl;Lf.

29. a = cos 2 + i sin 2 kw;Wk;

c = cos 2 vdpy; i)

= 2 cos( + )

i)

= 2 cos 2 ( – ) vd ep&gp



30. (

,d; vy;yh kjpg;GfisAk; fhz;f?

31. (

,d; vy;yh kjpg;GfisAk; fhz;f?

32. jPu;f;f : x4 – x3 + x2- x + 1 = 0

33. (

–

,d; vy;yh kjpg;GfisAk fhz;f kw;Wk; mjd; kjpg;Gfspd;

ngUf;fw;gyd; 1 vdTk; fhl;Lf?

34. vd;git x2 – 2x + 2 = 0 ,d; %yq;fs; kw;Wk;

cot vdpy;

=

vdf; fhl;Lf?

35. x9 + x5 – x4 – 1 = 0 vd;w rkd;ghl;ilj; jPh;f;f.

36. x7 + x4 + x3 + 1=0 vd;w rkd;ghl;ilj; jPh;f;f.

37. 5x + 12y = 9 vd;w Neh;f;NfhL mjpgutisak; x2 – 9y2 = 9 – Ij;

njhLfpwJ vd ep&gpf;f . NkYk; njhLk; Gs;spiaAk; fhz;f.

38. x – y + 4 = 0 vd;w Neh;f;NfhL ePs;tl;lk; x2 + 3y2 = 12 f;F njhLNfhlhf

cs;sJ vd ep&gpf;f. NkYk; njhLk; Gs;spiaAk; fhz;f.

39. gpd;tUk; mjpgutisaj;jpd; rkd;ghL fhz;f.

(i) mjpgutisaj;jpd; ikak; ( 2 ,4 ) NkYk; ( 2, 0) topNa nry;fpwJ.

,jd; njhiyj; njhLNfhLfs; x + 2y – 12 = 0 kw;Wk; x – 2y + 8 = 0

Mfpatw;wpw;F ,izahf ,Uf;fpd;wd.

40. x – 2y – 5 = 0 I xU njhiyj; njhLNfhlhfTk; (6,0) kw;Wk; (-3, 0) vd;w Gs;spfs; topNa nry;yf;$baJkhd nrt;tf mjpgutisaj;jpd; rkd;ghL fhz;f.

41. 4y2 = x3 vd;w tistiuapy; x = 0 ,ypUe;J x = 1 tiuAs;s tpy;ypd;

ePsj;ijf; fhz;f.

42.

+

= 1vd;w tistiuapd; ePsj;ijf; fhz;f.



43. y = sin x x = 1 vd;w tistiu x = 0 , x = kw;Wk; x –mr;R Mfpatw;why;

Vw;gLk; gug;gpid x mr;rpidg; nghWj;J Row;Wk; Nghj fpilf;Fk;

jplg;nghUspd; tisgug;G

2

44. x = a(t + sin t), y = a(1 + cos t) vd;w tl;l cUs; tis (cycloid) mjd;

mbg;gf;fj;ijg; (x – mr;R) nghWj;J Row;Wtjhy; Vw;gLk; jplg;nghUspd;

tisgug;igf; fhz;f.

45. Muk; a cila tl;lj;jpd; Rw;wsitf; fhz;f.

46. x = a(t – sin t) , y = a(1 – cos t) vd;w tistiuapd; ePsj;jpid t = 0 Kjy;

t = tiu fzf;fpLf.

47. y2 = 4 vd;w gutisaj;jpd; mjd; nrt;tfyk; tiuapyhd gug;gpid x –

mr;rpd; kPJ Row;Wk;NghJ fpilf;Fk; jplg; nghUspd; tisgug;igf;

fhz;f. .

48. Muk; r myFfs; cs;s Nfhsj;jpd; ikaj;jpypUe;J a kw;Wk; b myFfs; njhiytpy; mike;j ,U ,izahd jsq;fs; Nfhsj;ij ntl;Lk;NghJ ,ilg;gLk; gFjpapd; tisgug;G 2r (b – a) vd epWTf. ,jpypUe;J Nfhsj;jpd; tisgug;ig tUtp (b > a)

49. ( Z, * ) xU Kbtw;w vgPypad; Fyk; vdf; fhl;Lf ,q;F * MdJ a * b= a + b + 2 vDkhW tiuaWf;fg;gl;Ls;sJ.

50.

; x R – {0}, vd;w mikg;gpy; cs;s mzpfs; ahTk; mlq;fpa

fzk; G MdJ mzpg;ngUf;fypd; fPo; xU Fyk; vdf; fhl;Lf.

51. G = {a + b / a, b Q} vd;gJ $l;liyg; nghWj;j xU Kbtw;w vgPypad; Fyk; vdf; fhl;Lf.

52. 1 Ij; jtpu kw;w vy;yh tpfpjKW vz;fSk; mlq;fpa fzk; G vd;f. G

,y; * I a * b = a + b – ab , a, b, G vDkhW tiuaWg;Nghk; ,

( G , * ) xU Kbtw;w vgPypad; Fyk; vdf; fhl;Lf?



53. G+r;rpakw;w fyg;ngz;fspd; fzkhd c- { 0 } ,y; tiuaWf;fg;gl;l

f1(z) = z , f2(z) = -z , f3 (z) =

, f4(z) =

z c-{0} vd;w rhu;Gfs; ahTk;

mlq;fpa fzk; { f1, f2, f3,f4} MdJ rhu;Gfspd; Nrh;g;gpd; fPo; xU vgPypad; Fyk; mikf;Fk; vd epWTf?

54. (Zn , +n) xU Fyk; vdf; fhl;Lf.

55. (z7 – { [ 0 ] } , .7 ) xU Fyj;ij mikf;Fk; vdf; fhl;Lf?

56. tof;fkhd ngUf;fypd; fPo; 1- ,d; n- Mk; gb %yq;fs; Kbthd Fyj;ij mikf;Fk; vdf; fhl;Lf?

57. G vd;gJ kpif tpfpjKW vz; fzk; vd;f. a * b =

vDkhW

tiuaWf;fg;gl;l nrayp * - ,d; fPo; G xU Fyj;ij mikf;Fk;

vdf;fhl;Lf. a, b G

58.

vd;fpw fzk;

mzpg;ngUf;fypd; fPo; xU Fyj;ij mikf;Fk; vdf; fhl;Lf. ( = 1 )

59. | z | = 1 vDkhW cs;s fyg;ngz;fs; ahTk; mlq;fpa fzk; M Mdj fyg;ngz;fspd; ngUf;fypd; fPo; xU Fyj;ij mikf;Fk; vdf; fhl;Lf.

60. -1 I jtpu kw;w vy;yh tpfpjKW vz;fSk; cs;slf;fpa fzk; G MdJ a * b = a + b + ab vDkhW tiuaWf;fg;gl;l nrayp * - ,d; fPo; xU vgPypad; mikf;Fk; vdf; fhl;Lf ?

61. 11- ,d; kl;Lf;F fhzg;ngw;w ngUf;fypd; fPo; { [ 1 ] , [ 3 ] , [ 4 ] , [5 ] ,

[ 9 ] } vd;w fzk; xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf?

62.

a R –{0} mikg;gpy; cs;s vy;yh mzpfSk; mlq;fpa fzk;

mzpg;ngUf;fypd; fPo; xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf?

63. G = { 2n / n Z} vd;w fzkhdJ ngUf;fypd; fPo; xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf.

md;ghh;e;j ghlrhiy FOtpdh;fSf;F€¦ · 12/03/2018 · 30. ( ,d; vy;yh kjpg;GfisAk; fhz;f?

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