Maths-2 Marks- Important Sums-Tamil Medium · X – MATHS QUESTION BANK Page 3 28. 7 + 3 kw;Wk . 7 − 3 Mfpatw;iw %yqfshff nfhzl ,Ugbr rkdghl;bid mikf;f. 29. RUfFf . 6 2+ 9 3 2−

X – MATHS QUESTION BANK Page 1

‘ fzf;Ffs; nra;tjpd; %yk; fzpjj;ijf; fw;Nghk; "

gj;jhk; tFg;G

fzpjk;

tpdh - tq;fp jahhpg;G :

rp.rf;jpNty; , M.Sc.,B.Ed.,

gl;ljhhp Mrphpah; ,

muR Nky; epiyg; gs;sp,

,uhkpak;gl;b, – 635 302

ghg;gpnul;bg;gl;b, ( tl;lk; )

jh;kGhp . ( khtl;lk; )

njhiyNgrp vz; : 94439 50993

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Kf;fpa ,uz;L kjpg;ngz; tpdhf;fs;

ghlk; : 1. fzq;fSk; rhh;GfSk;

1. 𝐴 = −10 , 0 , 1 , 9 , 2 , 4 , 5 , 𝐵 = −1 ,−2 , 5 , 6 , 2 , 3 vd;w fzq;fSf;F gpd;tUtdtw;iw rhpghh;.

i ) fzq;fspd; Nrh;g;G ghpkhw;Wg;; gz;G cilaJ. ii ) fzq;fspd; ntl;L ghpkhw;Wg;; gz;G cilaJ.

2. A = 1 , 2 , 3 , 4 , 5 , B = 3 , 4 , 5 , 6 , C = { 5 , 6 , 7 , 8 } vdpy; A ∪ B ∪ C = A ∪ B ∪ C vd;gijr; rhpghh;.

3. A = { a , b , c , d } B = { a , c , e } kw;Wk; C = { a , e } vdpy; A ∩ B ∩ C = A ∩ B ∩ C vd;gijr; rhpghh;.

4. A ⊂ B vdpy; A ∪ B = B vdf; fhl;Lf. ( ntz;glj;ij gad;gLj;jTk; )

5. A ⊂ B vdpy; A ∩ B kw;Wk; A ⃥ B Mfpatw;iwf; fhz;f. ( ntz;glj;ij gad;gLj;jTk; )

6. ntz;glq;fis gad;gLj;jp A ∩ B ∪ A \ B = A vd;gijr; rhpghh;.

7. A , B kw;Wk; C Mfpa %d;W fzq;fSf;F gpd;tUtdtw;iw tpsf;Fk; ntz;glq;fs; tiuf.

i A ∩ B ∩ C ii A kw;Wk; 𝐵 vd;gd C - d; cl;fzq;fs;.NkYk; mit ntl;lhf; fzq;fs;.

iii A ∩ B \ C iv B ∪ C \ A v A ∪ B ∩ C vi C ∩ B ∪ A

8. nfhLf;fg;gl;Ls;s f = 1 , 3 , 2 , 5 , 4 , 7 , 5 , 9 , 3 , 1 vDk; rhh;gpw;F kjpg;gfk; kw;Wk; tPr;rfk; Mfpatw;iwf; fhz;f.

9. 𝑓 = 12 , 2 , 13 , 3 , 15 , 3 , 14 , 2 , 17 , 17 vd;w rhh;gpy; 2 kw;Wk; 3 Mfpatw;wpd; Kd; cUf;fisf; fhz;f.

10. 𝑅 = 𝑎 , −2 , −5 , 𝑏 , 8 , 𝑐 , 𝑑 , −1 vd;gJ rkdpr; rhh;ig Fwpf;fpwJ vdpy; a , b , c kw;Wk; d Mfpatw;wpd; kjpg;igf;

fhz;f.

ghlk; : 2. nka;naz;fspd; njhlh;thpirfSk; njhlh;fSk;

11. xU $l;Lj;njhlh;thpirapd; Kjy; cWg;G 6 kw;Wk; nghJ tpj;jpahrk; 5 vdpy; mj;njhlh;thpirAk;

mjd; nghJ cWg;igAk; fhz;f.

12. 125 , 120 , 115 , 110 , .......... vd;w $l;Lj; njhlh;thpirapd; nghJ tpj;jpahrj;ijAk; 15 tJ cWg;igAk; fhz;f.

13. 2 , 3 3 , 5 3 vd;w $l;Lj;njhlh; thpirapd; 12 tJ cWg;G ahJ?

14. 4 , 9 , 14 , .......... vd;w $l;Lj;njhlh; thpirapd; 17 tJ cWg;G ahJ?

15.13 My; tFgLk; <hpyf;f kpif KO vz;fspd; vz;zpf;ifiaf; fhz;f.

16. 5 + 11 + 17 + .......... + 95 vd;w $l;Lj;njhlhpd; $Ljy; fhz;f.

17. 1 + 2 + 3 + ............ + 45 d; $Ljy; fhz;f.

18. 2 + 4 + 6 + .......... + 100 d; $Ljy; fhz;f.

19. 7 + 14 + 21 + ........... + 490 d; $Ljy; fhz;f.

20. 12 + 2 2 + ……… . . + 25 2 d; $Ljy; fhz;f.

21. 13 + 2 3 + ……… . . + 20 3 d; $Ljy; fhz;f.

ghlk; : 3.,aw;fzpjk;

22. njhFKiw tFj;jiy gad;gLj;jp gpd;tUtdtw;wpd; <T , kPjp fhz;f.

i ) ( 𝑥 3 + 𝑥 2 − 3𝑥 + 5 ) ÷ ( 𝑥 − 1 ) ii ) ( 3𝑥 3 − 2𝑥 2 + 7𝑥 − 5 ) ÷ ( 𝑥 + 1 ) ) 𝑖𝑖𝑖 ) ( 𝑥 3 + 𝑥 2 − 7𝑥 − 3 ) ÷ ( 𝑥 − 3 )

23. 𝑥 3 − 6𝑥 2 + 11𝑥 − 6 vd;w gy;YWg;Gf; Nfhitf;F ( 𝑥 − 1 ) xU fhuzpah vd Muha;f.

24. 𝑥 3 + 6𝑥 2 + 11 + 6 vd;w gy;YWg;Gf; Nfhitf;F ( 𝑥 + 1 ) xU fhuzpah vd Muha;f.

25. RUf;Ff .6𝑥2− 54

𝑥 2 + 7𝑥 + 12

26. RUf;Ff. 𝑥 3− 27

𝑥 2 − 9

27. RUf;Ff 𝑥

𝑥 + 1 ÷

𝑥 2

𝑥 2− 1

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28. 7 + 3 kw;Wk; . 7 − 3 Mfpatw;iw %yq;fshff; nfhz;l ,Ugbr; rkd;ghl;bid mikf;f.

29. RUf;Ff .6𝑥2+ 9𝑥

3𝑥 2− 12𝑥

30. xU ,Ugb gy;YWg;Gf; Nfhitapd; G+r;rpaq;fspd; $Ljy ; − 4 kw;Wk; mjd; ngUf;fw;gyd; 3 vdpy; mf;Nfhitiaf; fhz;f.

31. xU ,Ugb gy;YWg;Gf; Nfhitapd; G+r;rpaq;fspd; $Ljy; 1

3 kw;Wk; mjd; ngUf;fw;gyd; −

1 3

vdpy; mf;Nfhitiaf; fhz;f.

32. gpd;tUk; rkd;ghLfspd; %yq;fspd; jd;ikiaf; fhz;f.

𝑖 ) 𝑥 2 − 8𝑥 + 12 = 0 𝑖𝑖 ) 2𝑥 2 − 3𝑥 + 4 = 0 , 𝑖𝑖𝑖 ) 9𝑥 2 + 12𝑥 + 4 = 0 𝑖𝑣 ) 𝑥 2 − 11𝑥 − 10 = 0 , 𝑣 ) 2𝑥 2 + 5𝑥 + 5 = 0

ghlk; : 4.mzpfs;

33. 𝐴 = 8 51 −3

24

vdpy; 𝐴 𝑇 kw;Wk; 𝐴 𝑇 𝑇

Mfpatw;iwf; fhz;f.

34. 𝐴 = 2 34 1 5 0

vdpy; A – d; epiu epuy; khw;W mzpiaf; fhz;f.

35. 𝑥 5 45 9 1

= 3 5 𝑧5 𝑦 1

vdpy; x , y kw;Wk; z Mfpatw;wpd; kjpg;G fhz;f.

36. A = 5 6 −21 0 4

32

kw;Wk; B = 3 −1 42 8 2

73

vdpy; A + B If; fhz;f.

37. A = 2 3

−9 5 −

1 57 −1

vdpy; A – d; $l;ly; Neh;khW mzpiaf; fhz;f.

ghlk; : 5.Maj;njhiy tbtpay;

38. ( - 6 , 4 ) kw;Wk; ( 2 , 6 ) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj; Jd;bd; eLg;Gs;sp fhz;f.

39. ( 3 , 0 ) kw;Wk; ( - 1 , 4 ) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj ;Jd;bd; eLg;Gs;sp fhz;f.

40. ( 1 , - 1 ) kw;Wk; ( - 5 , 3 ) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj; Jd;bd; eLg;Gs;sp fhz;f.

41. ( 0 , 0 ) kw;Wk; ( 0 , 4 ) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj; Jd;bd; eLg;Gs;sp fhz;f.

42. xU tl;lj;jpd; ikak; ( - 6 , 4 ) mt;tl;lj;jpd; xU tpl;lj;jpd; xU Kid Mjpg;Gs;sp vdpy; kw;nwhU Kidiaf; fhz;f.

43. A ( 4 , - 6 ) , B ( 3 , - 2 ) kw;Wk; C ( 5 , 2 ) Mfpatw;iw cr;rpfshff;nfhz;l Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; fhz;f.

44. ( 1 , 3 ) , ( 2 , 7 ) kw;Wk; ( 12 , -16 ) Mfpatw;iw cr;rpfshff;nfhz;l Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; fhz;f.

45. ( 3 , 5 ) , ( - 7 , 4 ) kw;Wk; ( 10 , - 2 ) Mfpatw;iw cr;rpfshff; nfhz;l Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; fhz;f.

46. ( 3 , 5 ) , ( 8 , 10 ) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj; Jz;il cl;Gwkhf 2 : 3 vd;w tpfpjj;jpy; gphpf;Fk; Gs;spiaf; fhz;f.

47. ( 3 , 4 ) , ( - 6 , 2 ) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj;Jz;il ntspg;Gwkhf 3 : 2 vd;w tpfpjj;jpy; gphpf;Fk; Gs;spiaf;

fhz;f.

fPo;f;fz;l Gs;spfis Kidfshff; nfhz;l Kf;Nfhzq;fspd; gug;G fhz;f.

48. 1 , 2 , −3 , 4 kw;Wk;;; −5 , −6

49. 0 , 0 , 3 , 0 kw;Wk; 0 , 2

50. 5 , 2 , 3 , −5 kw;Wk; ( −5 , −1 )

51. −4 ,− 5 , 4 , 5 kw;Wk; ( −1 , −6 )

thpirapy; mike;j Kf;Nfhzj;jpd; KidfSk; mit mikf;Fk; Kf;Nfhzj;jpd;

gug;gsTfSk; fPNo nfhLf;fg;gl;Ls;sd. xt;nthd;wpYk; a – tpd; kjpg;G fhz;f.

t.vz; Kidfs; gug;G

52 6 , 7 , −4 , 1 kw;Wk; ( 𝑎 , −𝑎 ) 68 r.m

53 0 , 0 , 4 , 𝑎 kw;Wk; ( 6 , 4 ) 17 r.m

54 𝑎 , 𝑎 , 4 , 5 kw;Wk; ( 6 , −1 ) 9 r.m

55 9 , −3 , 3 , 𝑎 kw;Wk; ( −1 , 5 ) 12 r.m

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gpd;tUk; Gs;spfs; xNu Nfhl;by; mikAk; Gs;spfsh vd Muha;f.

56. 𝐴 2 , 3 , 𝐵 4 , 0 kw;Wk; 𝐶 6 , −3

57. 4 , 3 , 1 , 2 kw;Wk; ( −2 , 1 )

58. −2 , − 2 , −6 , −2 kw;Wk; ( −2 , 2 )

fpNo nfhLf;fg;gl;l Gs;spfs; xU Nfhliktd vdpy; k – d; kjpg;G fhz;f.

59. 𝑘 , −1 , 2 , 1 kw;Wk; ( 4 , 5 )

60. 2 , −5 , 3 , −4 kw;Wk; 9 , 𝑘

61. 𝑘 , 𝑘 , 2 , 3 kw;Wk; ( 4 , −1 )

55. rha;Tf; Nfhzk; 45 0 kw;Wk; y – ntl;L

2

5 vdf;nfhz;l Neh;f;Nfhl;bd; rkd;ghL fhz;f.

62. rha;Tf; Nfhzk; 60 0 kw;Wk; y – ntl;L 3 vdf;nfhz;l Neh;f;Nfhl;bd; rkd;ghL fhz;f.

63. rha;T - 3 kw;Wk; y – ntl;L 4 vdf;nfhz;l Neh;f;Nfhl;bd; rkd;ghL fhz;f.

64. gpd;tUk; tptuq;fSf;F Neh;NfhLfspd; rkd;ghL fhz;f.

i ) rha;T – 4 , ( 1 , 2 ) vd;w Gs;sptopr; nry;fpwJ. ii ) rha;T

2

3 , ( 5 , - 4 ) vd;w Gs;sp topr; nra;fpwJ.

65. gpd;tUk; Neh;f; NfhLfspd; rha;T kw;Wk; y ntl;Lj;Jz;L Mfpadtw;iwf; fhz;f.

𝑖 ) 𝑦 = 𝑥 + 1 𝑖𝑖 ) 5𝑥 = 3𝑦 𝑖𝑖𝑖 ) 4𝑥 − 2𝑦 + 1 = 0 𝑖𝑣 ) 10𝑥 + 15𝑦 + 6 = 0

66. gpd;tUk; Gs;spfs; topr;nry;Yk; Neh;f;Nfhl;bd; rkd;ghL fhz;f.

𝑖 ) −2 , 5 , 3 , 6 𝑖𝑖 ) 0 , − 6 , ( −8 , 2 )

66. x – mr;rpypUe;J 5 myFfs;J}uj;jpy; cs;sJk; x – mr;rpf;F ,izahdJkhd Neh;f;Nfhl;bd; rkd;ghL fhz;f.

67. ( - 5 , - 2 ) vd;w Gs;sp topr;nry;tJk; Ma mr;RfSf;F ,izahdJkhd Neh;f;Nfhl;bd; rkd;ghL fhz;f.

68.gpd;tUk; Neh;f;NfhLfspd; rha;Tfisf; fhz;f

i ) 3x + 4y – 6 = 0 ii ) y = 7x + 6 iii ) 4x = 5y + 3

69. 3x + 2y – 12 = 0 , 6x + 4y =8 = 0 Mfpa Neh;f;NfhLfs; xd;Wf;nfhd;W ,iz vd epWTf.

70. x + 2y + 11 = 0 , 3x + 6y + 2 = 0 Mfpa Neh;f;NfhLfs xd;Wf;nfhd;W ,iz vd epWTf.

71. x + 2y + 1 = 0 , 2x – y + 5 = 0 Mfpa Neh;f;NfhLfs; xd;Wf;nfhd;W nrq;Fj;J vd epWTf.

72. 5x – 2y – 9 = 0 , ay + 2x – 11 = 0 Mfpa Neh;f;NfhLfs; xd;Wf;nfhd;W nrq;Fj;J vdpy; a – tpd; kjpg;G fhz;f.

ghlk; : 6.tbtpay;

73. ∆ 𝐴𝐵𝐶 y; 𝐷𝐸 ‖ 𝐵𝐶 kw;Wk;

𝐴𝐷

𝐷𝐵=

2

3 AE = 3.7 nr.kP vdpy; EC - If; fhz;f.

74. D kw;Wk; E Mfpa Gs;spfs; KiwNa gf;fq;fs; AB kw;Wk; AC fspy; 𝐷𝐸 ‖ 𝐵𝐶 vd;wpUf;FkhW mike;Js;sd.

i ) AD = 6 nr.kP , DB = 9 nr.kP kw;Wk; AE = 8 nr.kP vdpy; AC – fhz;f.

ii ) AD = 8 nr.kP , AB = 9 nr.kP kw;Wk; AE = 12 nr.kP vdpy; CE – fhz;f.

75. AD vd;gJ ∆ 𝐴𝐵𝐶 − y; ∠ 𝐴 d; cl;Gw Nfhz ,Urkntl;b.mJ BC I D - y; re;jpf;fpwJ.

i ) 𝐵𝐷 = 2 nr.kP 𝐴𝐵 = 5 nr.kP 𝐷𝐶 = 3 nr.kP vdpy; AC – fhz;f.

ii ) AB = 5.6 nr.kP AC = 6 nr.kP DC = 3 nr.kP vdpy; BC – fhz;f.

76. gpd;tUtdtw;Ws; AD vd;gJ ∆𝐴𝐵𝐶 y; ∠ 𝐴 d; Nfhz ,Urkntl;b MFkh vdr; Nrhjpf;f.

i ) AB = 4 nr.kP AC = 6 nr.kP BD = 1.6 nr.kP kw;Wk; CD = 2.4 nr.kP

ii ) AB = 6 nr.kP AC = 8 nr.kP BD = 1.5 nr.kP kw;Wk; CD = 3 nr.kP

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77.fPo;f;fhZk; xt;nthU glj;jpYk; x – d; kjpg;G fhz;f.

78. xU tl;lj;jpy; AB , CD vd;Dk; ,U ehz;fs; xd;iwnahd;W cl;Gwkhf P – y; ntl;bf;nfhs;fpwJ vdpy;

i ) CP = 4 nr.kP AP = 8 nr.kP PB = 2 nr.kP vdpy; PD fhz;f.

ii ) AP = 12 nr.kP AB = 15 nr.kP NkYk; CP = PD vdpy; CD – iaf; fhz;f.

79. xU tl;lj;jpy; AB , CD vd;Dk; ,U ehz;fs; xd;iwnahd;W ntspg;Gwkhf P – y; ntl;bf;nfhs;fpwJ vdpy;

i ) AB = 4 nr.kP BP = 5 nr.kP kw;Wk; PD = 3 nr.kP vdpy; CD – iaf; fhz;f.

ii ) BP = 3 nr.kP CP = 6 nr.kP kw;Wk; CD = 2 nr.kP vdpy; CD – iaf; fhz;f.

80. ∆ PQR – d; gf;fq;fs; PQ kw;Wk; PR – fspd; kPJ mike;j Gs;spfs; S kw;Wk; T vd;f. NkYk; 𝑆𝑇 ‖ 𝑄𝑅 , PR = 5.6 nr.kP kw;Wk;

𝑃𝑆

𝑃𝑇=

3

2 vdpy; PT fhz;f.

ghlk; : 7.Kf;Nfhztpay;

80. 𝑆𝑖𝑛 𝜃

𝐶𝑜𝑠𝑒𝑐 𝜃 +

𝐶𝑜𝑠 𝜃

𝑆𝑒𝑐 𝜃= 1 vd;w Kw;nwhUikia epWTf.

81. 1 − 𝐶𝑜𝑠 𝜃1 + 𝐶𝑜𝑠 𝜃

= 𝐶𝑜𝑠𝑒𝑐 𝜃−𝐶𝑜𝑡 𝜃 vd;w Kw;nwhUikia epWTf.

82. 1 − 𝑆𝑖𝑛 𝜃

1 + 𝑆𝑖𝑛 𝜃 = 𝑆𝑒𝑐 𝜃 − 𝑡𝑎𝑛 𝜃 vd;w Kw;nwhUikia epWTf.

83. 𝐶𝑜𝑠 2 𝜃 + 𝑆𝑒𝑐 2 𝜃 = 2 + 𝑆𝑖𝑛 𝜃 vd;w Kw;nwhUikia epWTf.

84. 𝐶𝑜𝑡 2 𝜃 + 𝐶𝑜𝑠 𝜃 = 𝑆𝑖𝑛 2 𝜃 vd;w Kw;nwhUikia epWTf.

85. 200 kP ePsKs;s E}ypdhy; xU fhw;whb fl;lg;gl;L gwe;Jnfhz;bUf;fpwJ. me;j E}y; jiukl;lj;Jld; 30 0 Nfhzj;ij

Vw;gLj;jpdhy; fhw;whb jiukl;lj;jpypUe;J vt;tsT cauj;jpy; gwf;fpwJ vdf; fhz;f.

( ,q;F E}y; xU Neh;f;Nfhl;by; cs;sjhf fUJf )

86. Rthpy; rha;j;J itf;fg;gl;l xU VzpahdJ jiuAld; 60 0 Nfhzj;ij Vw;gLj;JfpwJ. Vzpapd; mb Rtw;wpypUe;J 3.5 kP

J}uj;jpy; cs;sJ vdpy; Vzpapd; ePsj;ij fhz;f.

87. 30 kP ePsKs;s xU fk;gj;jpd; epoypd; ePsk; 10 3 kP vdpy; #hpadpd; Vw;wf; Nfhzj;ij ( jiu kl;lj;jpypUe;J Vw;wf; Nfhzk; )

mstpidf; fhz;f.

88. cauk; 150 nr.kP cs;s xU rpWkp xU tpsf;Ff; fk;gj;jpd; Kd;epd;wthW 150 3 nr.kP ePsKs;s epoiy Vw;gLj;Jfpwhs; vdpy;

tpsf;Ff; fk;gj;jpd; cr;rpapd; Vw;wf;Nfhzj;ijf; fhz;f.

ghlk; : 8.mstpay;

89. kuj;jpdhyhd xU jpd;kf; $k;gpd; mbr;Rw;wsT 44 kP kw;Wk; mjd; cauk; 12 kP vdpy; mj;jpd;kf; $k;gpd; fdmsitf; fhz;f.

90. xU jpd;k Neh;tl;l cUisapd; Muk; 14 nr.kP kw;wk; cauk; 8 nr.kP vdpy; tisgug;G kw;Wk; nkhj;j gug;G Mfpatw;iwf; fhz;f.

91. 98.56 r.nr.kP Gwgug;Gf; nfhz;l xU jpd;kf; Nfhsj;jpd; Muj;ijf; fhz;f.

92. xU ,ilf;fz;l tbtpyhd thspapd; Nkw;Gw kw;Wk; mbg;Gw Muq;fs; KiwNa 15 nr.kP kw;Wk; 8 nr.kP .NkYk; mjd;

Mok; 63 nr.kP vdpy; mjd; nfhs;ssit ypl;lhpy; fhz;f. π = 22

7

93. xU Neh;tl;l cUisapd; tisgug;G 704 r.nr.kP kw;Wk; mjd; cauk; 8 nr.kP vdpy; mt;TUisapd; fd msitf; fhz;f.

94. xU Nfhsj;jpd; fdmsT mjd; Gwg;gug;Gf;Fr; rkk; vdpy; mjd; tpl;lj;ijf; fhz;f.

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95. xU jpd;kf; $k;gpd; Muk; kw;Wk; rhAauk; KiwNa 20 nr.kP kw;Wk; 29 nr.kP vdpy; mj;jpd;kf; $k;gpd; fd msitf; fhz;f.

96. cUisapd; Muk; 14 nr.kP cauk; 8 nr.kP vdpy; cUisapd; fd msitf; fhz;f.

97. xU jpz;k miuf;Nfhsj;jpd; nkhj;jGwgug;G 675 𝜋 r.nr.kP vdpy; mjd; tisgug;G fhz;f.

98. xU jpd;k cUisapd; Muk; 14 nr.kP kw;Wk; cauk; 30 nr.kP vdpy; cUisapd; fd msitf; fhz;f.

99. 448 𝜋 f.nr.kP fdmsT nfhz;l xU jpz;k Neh;tl;l cUisapd; cauk; 7 nr.kP mt;TUisapd; tpl;lj;ijf; fhz;f.

100. xU jpz;k Neh;tl;l cUisapd; Muk; 7 nr.kP kw;Wk; cauk; 20 nr.kP vdpy;

i ) cUisapd; tisgug;G ii ) cUisapd; nkhj;jg; gug;G fhz;f.

101. 8.4 nr.kP tpl;lk; nfhz;l xU Nfhstbt jpz;k cNyhf vwpFz;bd; fd msitf; fhz;f.

102. 62.37 f.nr.kP fd msTf; nfhz;l xU jpz;k Neh;tl;l cUisapd; cauk; 4.5 nr.kP vdpy; mt;TUisapd; Muj;ij fhz;f.

103. xU jpz;k Neh;tl;l cUisapd; fd msT 98 𝜋 f.nr.kP kw;Wk; mjd; cauk; 8 nr.kP vdpy; mt;TUisapd; tisgug;ig fhz;f.

104. xU jpz;k miuf;Nfhsj;jpd; tisgug;G 2772 r.nr.kP vdpy; mjd; nkhj;jgug;igf; fhz;f.

105. xU Neh;tl;l jpz;kf; $k;gpd; cauq;fspd; tpfpjk; 1 : 3 kw;Wk; mtw;wpd; Muq;fspd; tpfpjk; 3 : 1 vdpy; mtw;wpd; fd

msTfspd; tpfpjj;ijf; fhz;f.

101. xU jpz;k Neh;tl;l cUisapd; nkhj;j Gwgug;G 660 r.nr.kP mjd; tpl;lk; 14 nr.kP vdpy; cUisapd; cauj;ijAk; ,

tisgug;igAk; fhz;f.

106. ,uz;L jpz;k miuf;Nfhsq;fspd; Muq;fs; 3 : 5 vd;w tpfpjj;jpy; cs;sd vdpy; mf;Nfhsq;fspd; tisgug;Gfspd; tpfpjk;

fhz;f.

107. xU jpz;k Neh;tl;l $k;gpd; fdmsT 4928 f.nr.kP mjd; cauk; 24 nr.kP vdpy; $k;gpd; Muk; fhz;f.

108. xU Neh;tl;l $k;gpd; fdmsT 216 𝜋 f.nr.kP kw;Wk; mf;$k;gpd; Muk; 9 nr.kP vdpy; mjd; cauk; fhz;f.

ghlk; : 11.Gs;spapay;

109. Kjy; 13 ,ay; vz;fspd; jpl;l tpyf;fj;ijf; fhz;f.

110. xU Gs;sp tpuuj;jpd; khWghl;Lf; nfO 57 kw;Wk; jpl;ltpyf;fk; 6.84 vdpy; mjd; $l;Lr;ruhrhpiaf; fhz;f.

111. 43 , 24 , 38 , 56 , 22 , 39 , 45 Mfpa Gs;sptptuq;fspd; tPr;R kw;Wk; tPr;Rf; nfO Mfpatw;iwf; fhz;f.

112. fz;lwpe;j Gs;sp tptuj; njhFg;gpYs;s 20 kjpg;Gfspd; jpl;ltpyf;fk; 5 vd;f. Gs;sptptuj;jpd; xt;nthU kjpg;igAk; 2 My;

ngUf;fpdhy; fpilf;Fk; Gjpa Gs;sp tptuq;fspd; jpl;ltpyf;fk; kw;Wk; tpyf;f th;f;fr; ruhrhp Mfpatw;iwf; fhz;f.

113. ,U tptuj;njhlh;fspd; khWghl;Lf; nfOf;fs; 58 kw;wk; 69 vd;f.mtw;wpd; jpl;ltpyf;fk; KiwNa 21.2 kw;Wk; 15.6 vdpy;

mtw;wpd; $l;Lr;ruhrhpiaf; fhz;f.

114. 50 msTfspy; kpfg;nghpa kjpg;G 3.84 fp.fp mjd; tPr;R 0.46 fp.fp vdpy; mitfspd ; kPr;rpW kjpg;igf; fhz;f.

115. xU Gs;sp tptuj;jpd; kPr;rpW kjpg;G 12.mjd; tPr;R 59 vdpy; mg;Gs;sp tptuj;jpd; kPg;ngU kjpg;igf; fhz;f.

116. xU Gs;sp tptuj; njhFg;gpd; kPg;ngU kjpg;G 7.44 kw;Wk; mjd; tPr;R 2.26 vdpy; mj;njhFg;gpd; kPr;rpW kjpg;igf; fhz;f.

117. xU FOtpy; 100 Ngh; cs;sdh;.mth;fspd; cauq;fspd; $l;Lr; ruhrhp 163.8 nr.kP kw;Wk; khWghl;L nfO 3.2 vdpy;

mth;fSila cauq;fspd; jpl;ltpyf;fk; fhz;f.

118. 59 , 46 , 30 , 23 , 27 , 40 , 52 , 35 , 29 vd;w Gs;sp tptuj;jpd; tPr;R kw;Wk; tPr;Rf; nfO fhz;f.

ghlk; : 12.epfo;jfT

119. ,U ehzaq;fs; xNu Neuj;jpy; Rz;Lk;NghJ mjpfgl;rkhf xU jiy fpilg;gjw;fhd epfo;jfT fhz;f.

120. A kw;Wk; B vd;w ,U epfo;r;rpfspy; 𝑃 𝐴 = 1

4 , 𝑃 𝐵 =

2

5 kw;Wk; 𝑃 𝐴 ∪ 𝐵 =

1

2 vdpy; 𝑃 𝐴 ∩ 𝐵 fhz;f.

121. Kjy; 20 ,ay; vz;fspypUe;J xU KO vz; rktha;G Kiwapy; Njh;e;njLf;fg;gLfpwJ , me;j vz; xU gfh vz;zhf

,Ug;gjw;fhd epfo;jfT fhz;f.

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122. 12 ey;y Kl;ilfSld; 3 mOfpa Kl;ilfs; fye;Js;sd.rktha;g;G Kiwapy; Njh;e;njLf;fg;gLk; xU Kl;il , mOfpajhf

,Ug;gjw;fhz epfo;jfT fhz;f.

123. xU Fwpg;gpl;l ehspy; kio tUtjw;fhd epfo;jfT 0.76 vdpy; mNj ehspy; kio tuhky; ,Ug;gjw;fhz epfo;jfT fhz;f.

124. ,U gfilfs; xNu Neuj;jpy; xU Nru cUl;lg;gLk;NghJ fpilf;Fk; Kf vz;fspd; $Ljy; 3 My; kw;Wk; 4 My; tFglhky;


125. 3 , 5 , 7 vd;w vz;fisf; nfhz;l xU ,uz;L ,yf;f vz; mikf;fg;gLfpwJ. mt;ntz; 57 I tpl nghpajhf ,Ug;gjw;fhz

epfo;jfT fhz;f.

126. %d;W gfilfs; xNu Neuj;jpy; cUl;lg;gLtjpy; , %d;W gfilfspYk; xNu vz;zhf tUtjw;fhd epfo;jfT fhz;f.

127. xU ngl;bapy; 4 gr;ir , 5 ePyk; kw;Wk; 3 rptg;G epwg; ge;Jfs; cs;sd. rktha;g;G Kiwapy; xU ge;ij Njh;e;njLf;f mJ

gr;ir epwg; ge;jhf ,y;yhky; ,Ug;gjw;fhd epfo;jfT fhz;f.

128. A kw;Wk; B vd;gd xd;iwnahd;W tpyf;Fk; epfo;r;rpfs;.NkYk;; 𝑃 𝐴 = 3

5 , 𝑃 𝐵 =

1

5 kw;Wk; 𝑃 𝐴 ∪ 𝐵 fhz;f.

129. 1 ypUe;J 100 tiuapyhd KO vz;fspypUe;J rktha;g;G Kiwapy; Njh;e;njLf;fg;gLk; xU vz; xU KOth;f;fkhf


130. %d;W ehzaq;fs; xNu Neuj;jpy; Rz;lg;gLfpd;wd.,U G+f;fs; kl;Lk; fpilg;gjw;fhd epfo;jfT fhz;f

131. xU rPuhd gfil ,uz;L Kiw cUl;lg;gLfpwJ. Kf vz;fspd; $Ljy; 9 fpilf;fg; ngWtjw;fhd epfo;jfT fhz;f.

Kf;fpa Ie;J kjpg;ngz; tpdhf;fs;

ghlk; : 1. fzq;fSk; rhh;GfSk;

1. P = a , b , c , d , e , Q = a , e , i , o , u kw;Wk; R = a , c , e , g Mfpa fzq;fspd; ntl;L Nrh;g;G gz;G cilaJ vdr;

rhpghh;.

2. A = { 0 , 1 , 2 , 3 , 4 } , B = { 1 , − 2 , 3 , 4 , 5 , 6 } kw;Wk; C = { 2 , 4 , 6 , 7 } vdpy; A ∪ B ∩ C = A ∪ B ∩ A ∪ C vd;gijr; rhpghh;.

3. A = { −3 , −1 , 0 , 4 , 6 , 8 , 10 } , B = { −1 , −2 , 3 , 4 , 5 , 6 } kw;Wk; C = { −1 , 2 , 3 , 4 , 5 , 7 } vdpy;

( i ) A ∪ B ∩ C = A ∪ B ∩ A ∪ C ( ii ) A ∩ B ∪ C = A ∩ B ∪ A ∩ C Mfpatw;iwr; rhpghh;.

4. A = 10 , 15 , 20 , 25 , 30 , 35 , 40 , 45 , 50 , B = 1 , 5 , 10 , 15 , 20 , 30 kw;Wk; C = 7 , 8 , 15 , 20 , 35 , 45 , 48 Mfpa fzq;fSf;F

A \ B ∩ C = A ⃥ B ∪ A ⃥ C vd;gijr; rhpghh;.

5. A = a , b , c , d , e , f , g , x , y , z , B = 1 , 2 , c , d , e kw;Wk; C = d , e , f , g , 2 , y vdpy;

𝐴 \ 𝐵 ∪ 𝐶 = 𝐴 ⃥ 𝐵 ∩ 𝐴 ⃥ 𝐶 vd;gijr; rhpghh;.

6. gpd;tUk; fzq;fisf; nfhz;L B – khh;fdpd; fz tpj;jpahr tpjpfisr; rhpghh;.

A = 1 , 3 , 5 , 7 , 9 , 11 , 13 , 15 , B = 1 , 2 , 5 , 7 kw;Wk; C = 3 , 9 , 10 , 12 , 13

7. U = −2 ,−1 , 0 , 1 , 2 , 3 , … . , 10 , A = − 2 , 2 , 3 , 4 , 5 kw;Wk; B = 1 , 3 , 5 , 8 , 9 vdpy; B – khh;fdpd; fz epug;gp tpjpfisr; rhpghh;.

8. U = a , b , c , d , e , f , g , h , A = a , b , f , g kw;Wk; B = a , b , c vdpy; B – khh;fdpd; fz epug;gp tpjpfisr; rhpghh;.

9. ntz;glq;fis gad;gLj;jp gpd;tUtdtw;iw rhpah vdr; Nrhjpj;J ghh;f;fTk;.

𝑖 A ∪ B ∩ C = A ∪ B ∩ A ∪ C ii A ∩ B ∪ C = A ∩ B ∪ A ∩ C ( iii ) A ∪ B ′ = A ′ ∩ B ′

𝑖𝑣 A ∩ B ′ = A ′ ∪ B ′ 𝑣 A \ B ∪ C = A ⃥ B ∩ A ⃥ C 𝑣 𝑖 A \ B ∩ C = A ⃥ B ∪ A ⃥ C

10. 𝐴 = 0 , 1 , 2 , 3 kw;Wk; 𝐵 = 1 , 3 , 5 , 7 , 9 vd;gd ,U fzq;fs; vd;f.𝑓 ∶ 𝐴 → 𝐵 vDk; rhh;G 𝑓 𝑥 = 2𝑥 + 1 vdf;

nfhLf;fg;gl;Ls;sJ.,r;rhh;gpid

i ) thpirNrhbfspd; fzk; ii ) ml;ltiz iii ) mk;Gf;Fwpglk; iv ) tiuglk; Mfpatw;why; Fwpf;f.

11. 𝐴 = 6 , 9 , 15 , 18 , 21 , kw;Wk; 𝐵 = 1 , 2 , 4 , 5 , 6 , 𝑓 ∶ 𝐴 → 𝐵 vd;gJ 𝑓 𝑥 = 𝑥−3

3 vd tiuaWf;fg;gl;bUg;gpd; rhh;G 𝑓 − I


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12. 𝐴 = 4 , 6 , 8 , 10 , kw;Wk; 𝐵 = 3 , 4 , 5 , 6 , 7 𝑓 ∶ 𝐴 → 𝐵 vd;gJ 𝑓 𝑥 = 1

2𝑥 + 1 vd tiuaWf;fg;gl;bUg;gpd; rhh;G 𝑓 − I


ghlk; : 2. nka;naz;fspd; njhlh;thpirfSk; njhlh;fSk;

13. xU $l;Lj;njhlh;thpirapd; 9 MtJ cWg;G G+r;rpak; vdpy; 19 tJ vWg;gpd; ,U klq;F 29 MtJ cWg;G vd epWTf.

14. xU $l;Lj;njhlh;thpirapd; 10 kw;Wk; 18 MtJ cWg;Gfs; KiwNa 41 kw;Wk; 73 vdpy; 27 tJ cWg;igf; fhz;f.

15. xU ngUf;Fj;njhlh; thpirapd; ehd;fhtJ cWg;G

2

3 kw;Wk; mjd; VohtJ cWg;G

16

81 vdpy; mg;ngUf;Fj;njhlh; thpiriaf;

fhz;f.

16. 8 – My; tFgLk; midj;J %d;wpyf;f vz;fspd; $Ljy; fhz;f.

17. 9 – My; tFgLk; midj;J %d;wpyf;f vz;fspd; $Ljy; fhz;f.

18. 300 f;Fk; 500 f;Fk; ,ilNa 11 My; tFgLk; midj;J ,ay; vz;fspd; $Ljy; fhz;f.

19. 6 + 66 + 666 + …………….. vDk; njhlhpy; Kjy; n – cWg;Gfspd; $Ljy; fhz;f.

20. 7 + 77 + 777 + …………….. vDk; njhlhpy; Kjy; n – cWg;Gfspd; $Ljy; fhz;f.

21. $Ljy; fhz;f. 12 2 + 13 2 + 14 2 + …… . . + 35 2

22. $Ljy; fhz;f. 16 2 + 17 2 + 18 2 + …… . . + 25 2

23. $Ljy; fhz;f. 16 3 + 17 3 + 18 3 + ……… + 35 3

24. $Ljy; fhz;f. 11 3 + 12 3 + 13 3 + ……… + 28 3

25. 11 nr.kP , 12 nr.kP , 13 nr.kP ........... 24 nr.kP Mfpadtw;iw gf;f msTfshff; nfhz;l 14 rJuq;fspd; nkhj;j gug;G fhz;f.

26. 12 nr.kP , 13 nr.kP , 14 nr.kP ........... 23 nr.kP Mfpadtw;iw gf;f msTfshff; nfhz;l 14 rJuq;fspd; nkhj;j gug;G fhz;f.

27. 16 nr.kP , 17 nr.kP , 18 nr.kP ........... 30 nr.kP Mfpadtw;iw gf;f msTfshff; nfhz;l 15 rJuq;fspd; fd msTfspd;

$Ljy; fhz;f.

ghlk; : 3.,aw;fzpjk;

gpd;tUtdtw;iw fhuzpg;gLj;Jf.

28) 2𝑥3 − 3𝑥2 − 3𝑥 + 2 29 ) 𝑥3 − 3𝑥2 − 10𝑥 + 24 30 ) 𝑥3 − 2𝑥2 − 5𝑥 + 6 31 ) 4𝑥3 − 5𝑥2 + 7𝑥 − 6

32 ) 2𝑥3 − 9𝑥2 + 7𝑥 + 6 33 ) 2𝑥3 + 11𝑥2 − 7𝑥 − 6 34 ) 𝑥3 − 23𝑥2 + 142𝑥 − 120 35 ) 𝑥3 + 13𝑥2 + 32𝑥 + 20

36 ) 𝑥3 − 10𝑥2 − 𝑥 + 10 37 ) 𝑥3 − 5𝑥2 − 2𝑥 + 24

gpd;tUtdtw;wpd; th;f;f %yk; fhz;f.

38 ) 𝑥 4 − 10 𝑥 3 + 37𝑥 2 − 60𝑥 + 36

39 ) 𝑥 4 − 6 𝑥 3 + 19𝑥 2 − 30𝑥 + 25

40 ) 𝑥 4 − 4 𝑥 3 + 10𝑥 2 − 12𝑥 + 9

41) 4𝑥 4 + 8 𝑥 3 + 8𝑥 2 + 4𝑥 + 1

42 ) 9𝑥 4 − 6𝑥 3 + 7𝑥 2 − 2𝑥 + 1

43 ) 4 + 25𝑥 2 − 12𝑥 − 24 𝑥 3 + 16𝑥 4

44 ) 4𝑥 4 − 12 𝑥 3 + 37𝑥 2 + 𝑎𝑥 + 𝑏 MdJ xU KO th;f;fk; vdpy; a , b – d; kjpg;G fhz;f.

45) 𝑥 4 − 4 𝑥 3 + 10𝑥 2 − 𝑎𝑥 + 𝑏 MdJ xU KO th;f;fk; vdpy; a , b – d; kjpg;G fhz;f.

46 ) 9𝑥 4 + 𝑏 𝑥 3 + 109𝑥 2 − 60𝑥 + 36 MdJ xU KO th;f;fk; vdpy; a , b – d; kjpg;G fhz;f.

47) 9𝑥 4 − 𝑏 𝑥 3 + 40𝑥 2 + 24𝑥 + 36 MdJ xU KO th;f;fk; vdpy; a , b – d; kjpg;G fhz;f.

48 ) 𝑚 − 𝑛𝑥 + 28𝑥 2 + 12 𝑥 3 + 9 𝑥 4 MdJ xU KO th;f;fk; vdpy; m , n – d; kjpg;G fhz;f.

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ghlk; : 4.mzpfs;

49. jPh;T fhz;f. 𝑦

3𝑥 =

6 − 2𝑥 31 + 4𝑦

50. 5𝑥 + 2 𝑦 − 4

0 4𝑧 + 6 =

12 −80 2

vdpy; x , y kw;Wk; z - fspd; kjpg;Gfisf; fhz;f.

51. 2𝑥 + 𝑦𝑥 − 3𝑦

= 5

13 vdpy; x kw;Wk; y - fspd; kjpg;Gfisf; fhz;f.

52. 𝐴 = 3 25 1

, 𝐵 = 1 −22 3

kw;Wk; 𝑂 = 0 00 0

vdpy; gpd;tUtdtw;iwr; rhpghh;f;f.

𝑖 ) 𝐴 + 𝐵 = 𝐵 + 𝐴 𝑖𝑖 ) 𝐴 + −𝐴 = 0 = −𝐴 + 𝐴

53. 𝐴 = 4 1 21 −2 30 3 2

𝐵 = 2 0 46 2 8 2 4 6

, 𝐶 = 1 2 −35 0 21 −1 1

vdpy; 𝐴 + 𝐵 + 𝐶 = 𝐴 + 𝐵 + 𝐶 vd;gijr; rhpghh;.

54. 𝐴 = 3 2−1 4

, 𝐵 = −2 5 6 7

kw;Wk; 𝐶 = 1 1−5 3

vdpy; 𝐴 𝐵 + 𝐶 = 𝐴𝐵 + 𝐴𝐶 vd;gijr; rhpghh;.

55. 𝐴 = −2 4 5

kw;Wk; 𝐵 = 1 3 −6 vd;w mzpfSf;F 𝐴𝐵 𝑇 = 𝐵 𝑇 𝐴 𝑇 vd;gijr; rhpghh;.

56. 𝐴 = 1 −1 2 3

vdpy; 𝐴 2 – 4𝐴 + 5𝐼 2 = 0 vd epWTf.

57. 𝐴 = −1 2 1 1 2 3

, 𝐵 = 012

vdpy; 𝐶 = 2 1 vdpy; 𝐴 𝐵 𝐶 = 𝐴 𝐵𝐶 vd;gijr; rhpghh;.

58. 𝐴 = 5 2 7 3

, 𝐵 = 2 −1 −1 1

vdpy; 𝐴𝐵 𝑇 = 𝐵 𝑇 𝐴 𝑇 vd epWTf.

59. 𝐴 = 1 −4−2 3

kw;Wk; 𝐵 = −1 6 3 −2

vdpy; 𝐴 + 𝐵 2 ≠ 𝐴 2 + 2𝐴𝐵 + 𝐵 2 vd;gijr; rhpghh;.

60. 𝐴 = 3 37 6

, 𝐵 = 8 70 9

kw;Wk; 𝐶 = 2 −34 6

vdpy; ( A + B ) C kw;Wk; AC + BC vd;w mzpfisf; fhz;f.

NkYk; ( A + B ) C = AC + BC vd;gJ nka;ahFkh?

ghlk; : 5.Maj;njhiy tbtpay;

gpd;tUtdtw;iw Kidfshff; nfhz;l ehw;fuq;fspd; gug;G fhz;f.

61. 6 , 9 , 7 , 4 , 4 , 2 kw;Wk; ( 3 , 7 )

62. −3 , 4 , −5 ,− 6 , ( 4 , −1 ) kw;Wk; ( 1 , 2 )

63. – 4 , 5 , 0 , 7 , 5 , −5 kw;Wk; −4 , −2

67. ( 7 , 3 ) , ( 6 , 1 ) , ( 8 , 2 ) kw;Wk; ( P, 4 ) vd;gd Xh; ,izfuj;jpd; thpirg;gb mike;j cr;rpfs; vdpy; P - d; kjpg;igf; fhz;f.

ghlk; : 6.tbtpay;

65. Njy;]; Njw;wk; my;yJ mbg;gil tpfpj rk Njw;wj;ij vOjp epWTf.

66. Nfhz ,Urkntl;b Njw;wj;ij vOjp epWTf.

67. gpj;jhfu]; Njw;wj;ij vOjp epWTf.

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ghlk; : 7.Kf;Nfhztpay;

68. Neh;Fj;jhd xU kuj;jpd; Nky; ghfk; fhw;wpdhy; Kwpe;J , mk;Kwpe;j gFjp fPNo tpOe;Jtplhky; kuj;jpd; cr;rp

jiuAld; 30 0 Nfhzj;ij Vw;gLj;JfpwJ.kuj;jpd; cr;rp mjd; mbapypUe;J 30 kP njhiytpy; jiuiaj; njhLfpwJ vdpy;

kuj;jpd; KO cauj;ijf; fhz;f.

69. flw;fiuapy; cs;s nrq;Fj;jhd ghiw xd;wpd; kPJ fl;lg;gl;Ls;s xU fyq;fiu tpsf;fj;jpy; epd;Wnfhd;bUf;Fk; xU rpWkp

fpof;F jpirapy; ,U glFfisg; ghh;fpwhs;.mg;glFfspd; ,wf;ff;Nfhzq;fs; 30 0 , 60 0 KiwNa kw;Wk; ,U glFfSf;F

,ilNa cs;s J}uk; 300 kP vdpy; fly; kl;lj;jpypUe;J fyq;fiu tpsf;ffj;jpd; cr;rpapd; J}uj;ijf; fhz;f.

[ glFfSk; fyq;fiu tpsf;fKk; xNu Neh;f;Nfhl;by; cs;sd ]

70. 60 kP cauKs;s xU NfhGuj;jpypUe;J xU fl;llj;jpd; cr;rp , mb Mfpatw;wpd; ,wf;ff;Nfhzq;fs; KiwNa 30 0 kw;Wk; 60 0

vdpy; fl;llj;jpd; cauj;ijf; fhz;f.

71. xU NfhGuj;jpd; mbapypUe;J 30 3 njhiytpy; epw;Fk; xU ghh;itahsh; , mf;NfhGuj;jpd; cr;rpapid 30 0 Vw;wf;Nfhzj;jpy;

fhz;fpwhh;. jiukl;lj;jpypUe;J mtUila fpilepiy ghh;it Nfhl;bw;F cs;s J}uk; 1.5 kP vdpy; NfhGuj;jpd; cauj;ijf; fhz;f.

72. xU NfhGuj;jpd; mbapypUe;J xU Fd;wpd; cr;rpapd; Vw;wf;Nfhzk; 60 0 vd;f. Fd;wpd; mbapypUe;J NfhGuj;jpd; cr;rpapd;

Vw;wf;Nfhzk; 30 0 kw;Wk; NfhGuj;jpd; cauk; 50 kP vdpy; Fd;wpd; cauj;ijf; fhz;f.

73. 40 nr.kP ePsKs;s xU CryhdJ xU KO miytpd;NghJ mjd; cr;rpapy; 60 0 Nfhzj;ij Vw;gLj;JfpwJ.me;j miytpy;

Cry; Fz;bd; Jtf;f epiyf;Fk; , ,Wjp epiyf;Fk; ,ilNa cs;s kpff; Fiwe;j J}uj;ijf; fhz;f.

ghlk; : 8.mstpay;

74. xU Neh;tl;l cUisapd; tisgug;G 704 r.nr.kP kw;Wk; mjd; cauk; 8 nr.kP vdpy; mt;TUisapd; fd msit ypl;lhpy;

fhz;f.

75. xU jpd;kf; Nfhsj;jpd; fdmsT 7241 1

7 f.nr.kP vdpy; , mjd; Muj;ijf; fhz;f.

76. xU rh;f;f]; $lhukhdJ cUisapd; kPJ $k;G ,ize;j tbtpy; mike;Js;sJ. $lhuj;jpd; nkhj;j cauk; 49 kP .mjd;

mbg;ghfj;jpd; tpl;lk; 42 kP .cUisg;ghfj;jpd; cauk; 21 kP NkYk; 1 r.kP fpj;jhd; Jzpapd; tpiy &.12.50 vdpy; , $lhuk;

mikf;fj; Njitahd fpj;jhd; Jzpapd; tpiyiaf; fhz;f.

77. xU jpd;k kug;nghk;ikahdJ miuf;Nfhsj;jpd; Nky; $k;G mike;j tbtpy; cs;sJ.miuf;Nfhsk; kw;Wk; $k;G Mfpatw;wpd;

Muk; 3.5 nr.kP.NkYk; nghk;ikApd; nkhj;j cauk; 17.5 nr.kP vdpy; mg;nghk;ik jahhpf;fg; gad;gLj;jg;gl;l kuj;jpd; fd

msitf; fhz;f.

78. 7 nr.kP tpl;lKs;s cUistbt Kfitapy; rpwpjsT jz;zPh; cs;sJ.mjpy; xt;nthd;Wk; 1.4 nr.kp tpl;lKs;srpy Nfhstbt

gspq;Ff; fw;fs; Nghlg;gLfpwJ.cUisapYs;s ePhpd; kl;lk; 5.6 nr.kP cau vj;jid gspq;F fw;fis Kfitapy; NghlNtz;Lk;.

79. 8 nr.kP tpl;lKk; 12 nr.kP cauKk; nfhz;l xU Neh;tl;l jpd;k ,Uk;Gf; $k;ghdJ cUf;fg;gl;L 4 kp.kP MuKs;s jpd;k Nfhs

tbt Fz;Lfshf thh;f;fg;gl;lhy; fpilf;Fk; Nfhstbt Fz;Lfspd; vz;zpf;ifiaf; fhz;f.

80. xU cs;sPlw;w Nfhsj;jpd; ntsp kw;Wk; cs; Muq;fs; KiwNa 8 nr.kP kw;Wk; 4 nr.kP . ,f;NfhskhdJ cUf;fg;gl;L 8 nr.kP

tpl;lKs;s Neh;tl;l jpz;kf; $k;ghf khw;wg;gl;lhy; $k;gpd; cauj;ijf; fhz;f.

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ghlk; : 9.nra;Kiw tbtpay;

ikaj;ij gad;gLj;jp tl;lj;jpw;F njhLNfhL tiujy;

81. 3.2 nr.kP MuKs;s tl;lk; tiuf.tl;lj;jpd; Nky; P vd;w Gs;spia Fwpj;J mg;Gs;sptopNa xU njhLNfhL tiuf.

82. 4.2 nr.kP MuKs;s tl;lk; tiue;J mt;tl;lj;jpd; Nky; VNjDk; xU Gs;spiaf; Fwpf;f.tl;lj;jpd; ikaj;ij gad;gLj;jp

mg;Gs;sptopNa njhLNfhL tiuf.

njhLNfhL - ehz; Njw;wj;ij gad;gLj;jp tl;lj;jpd; Nky; mike;j Gs;spapy;

njhLNfhL tiujy;

83. 3.2 nr.kP MuKs;s tl;lk; tiuf. tl;lj;jpd; Nky; P vd;w Gs;spia Fwpj;J mg;Gs;spapy; njhLNfhL - ehz; Njw;wj;ij

gad;gLj;jp njhLNfhL tiuf.

84. 4.2 nr.kP MuKs;s tl;lk; tiue;J mt;tl;lj;jpd; Nky; VNjDk; xU Gs;spiaf; Fwpf;J. mg;Gs;spapy; njhLNfhL - ehz;

Njw;wj;ij gad;gLj;jp njhLNfhL tiuf.

tl;lj;jpw;F ntspNa mike;j Gs;spapypUe;J tl;lj;jpw;F njhLNfhl;L

Nrhbfs; tiujy;

85. 3 nr.kP MuKs;s tl;lk; tiuf.tl;lj;jpd; ikaj;jpypUe;J 7 nr.kP njhiytpy; xU Gs;spiaf; Fwpj;J , mg;Gs;spapypUe;J

tl;lj;jpw;F njhLNfhLfs; tiuf.NkYk; njhLNfhLfspd; ePsq;fis mse;J vOJf.

86. 10 nr.kP tpl;lKs;s tl;lk; tiuf.tl;lj;jpd; ikaj;jpypUe;J 13 nr.kP njhiytpy; P vd;w Gs;spia Fwpj;J mg;Gs;spapypUe;J

tl;lj;jpw;F PA kw;Wk; PB vd;w njhLNfhLfs; tiue;J mjd; ePsq;fis fzf;fpLf.

87. 6 nr.kP MuKs;s xU tl;lk; tiue;J mjd; ikaj;jpypUe;J 10 nr.kP njhiytpYs;s xU Gs;spiaf; Fwpf;f.mg;Gs;spapypUe;J

tl;lj;jpw;F njhLNfhLfs; tiue;J mjd; ePsq;fis fzf;fpLf.

88. 3 nr.kP MuKs;s xU tl;lk; tiue;J mjd; ikaj;jpypUe;J 9 nr.kP njhiytpYs;s xU Gs;spiaf; Fwpf;f.mg;Gs;spapypUe;J

tl;lj;jpw;F njhLNfhLfs; tiue;J mjd; ePsq;fis fzf;fpLf.

tl;l ehw;fuk; tiujy ;

tif : I - %d;W gf;fq;fs; kw;Wk; xU %iytpl;lk; nfhLf;fg;gLk;NghJ tl;l

ehw;fuk; tiujy;

89. 𝐴𝐵 = 6 nr.kP 𝐴𝐶 = 7 nr.kP 𝐵𝐶 = 6 nr.kP kw;Wk; 𝐴𝐷 = 4.2 nr.kP msTfs; nfhz;l ABCD vd;w tl;lehw;fuk; tiuf.

90. 𝑃𝑄 = 6 .5 nr.kP 𝑄𝑅 = 5.5 nr.kP 𝑃𝑅 = 7 nr.kP kw;Wk; 𝑃𝑆 = 4.5 nr.kP msTfs; nfhz;l PQRS vd;w tl;lehw;fuk; tiuf.

91. 𝐴𝐵 = 6 nr.kP 𝐴𝐷 = 4.8 nr.kP 𝐵𝐷 = 8 nr.kP kw;Wk; 𝐶𝐷 = 5.5 nr.kP msTfs; nfhz;l ABCD vd;w tl;lehw;fuk; tiuf.

tif : II - ,uz;L gf;fq;fs; kw;Wk; ,uz;L %iytpl;lk; nfhLf;fg;gLk;NghJ tl;l

ehw;fuk; tiujy ;

92. 𝑃𝑄 = 4 nr.kP 𝑄𝑅 = 6 nr.kP 𝑃𝑅 = 7.5 nr.kP kw;Wk; 𝑄𝑆 = 7 nr.kP msTfs; nfhz;l PQRS vd;w tl;lehw;fuk; tiuf.

93. 𝐾𝐿 = 5.5 nr.kP 𝐾𝑀 = 5 nr.kP 𝐿𝑀 = 4.2 nr.kP kw;Wk; 𝐿𝑁 = 5.3 nr.kP msTfs; nfhz;l KLMN vd;w tl;lehw;fuk; tiuf.

94. 𝐸𝐹 = 7 nr.kP 𝐸𝐻 = 4.8 nr.kP 𝐹𝐻 = 6.5 nr.kP kw;Wk; 𝐸𝐺 = 6.6 nr.kP msTfs; nfhz;l EFGH vd;w tl;lehw;fuk; tiuf.

tif : III - %d;W gf;fq;fs; kw;Wk; xU Nfhzk; nfhLf;fg;gLk;NghJ

tl;l ehw;fuk; tiujy;.

95. 𝐴𝐵 = 6 nr.kP 𝐵𝐶 = 5.5 nr.kP ∠ 𝐴𝐵𝐶 = 80 0 kw;Wk; 𝐴𝐷 = 4.5 nr.kP msTfSf;F ABCD vd;w tl;lehw;fuk; tiuf.

96. 𝑃𝑄 = 5.5 nr.kP 𝑄𝑅 = 4.5 nr.kP ∠ 𝑄𝑃𝑅 = 45 0 kw;Wk; 𝑃𝑆 = 3 nr.kP msTfSf;F PQRS vd;w tl;lehw;fuk; tiuf.

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97. 𝐴𝐵 = 7 nr.kP ∠ 𝐴 = 80 0 , 𝐴𝐷 = 4.5 nr.kP kw;Wk; 𝐵𝐶 = 5 nr.kP msTfSf;F ABCD vd;w tl;lehw;fuk; tiuf

tif : IV - ,uz;L gf;fq;fs; kw;Wk; ,uz;L Nfhzq;fs; nfhLf;fg;gLk;NghJ


98. 𝐸𝐹 = 5.2 nr.kP ∠ 𝐺𝐸𝐹 = 50 0 , 𝐹𝐺 = 6 nr.kP kw;Wk; ∠ 𝐸𝐺𝐻 = 40 0 vd;w msTfSf;F EFGH vd;w tl;lehw;fuk; tiuf.

99. 𝐴𝐵 = 6 nr.kP ∠𝐴𝐵𝐶 = 70 0 , 𝐵𝐶 = 5 nr.kP kw;Wk; ∠ 𝐴𝐶𝐷 = 30 0 vd;w msTfs; nfhz;l ABCD vd;w tl;lehw;fuk; tiuf.

100. 𝑃𝑄 = 5 nr.kP ∠ 𝑄𝑃𝑅 = 35 0 kw;Wk; ∠ 𝑃𝑅𝑆 = 70 0 , 𝑄𝑅 = 4 nr.kP vd;w msTfs; nfhz;l PQRS vd;w tl;lehw;fuk;

tiuf.

tif : V xU gf;fk; kw;Wk; %d;W Nfhzq;fs; nfhLf;fg;gLk;NghJ


101. 𝑃𝑄 = 4 nr.kP ∠ 𝑃 = 100 0 kw;Wk; ∠ 𝑃𝑄𝑆 = 40 0 kW;Wk; ∠ 𝑆𝑄𝑅 = 70 0 vd;w msTfSf;F PQRS vd;w tl;lehw;fuk;

tiuf.

102. 𝐴𝐵 = 5.5 nr.kP ∠ 𝐴𝐵𝐶 = 50 0 kw;Wk; ∠𝐵𝐴𝐶 = 60 0 kW;Wk; ∠ 𝐴𝐶𝐷 = 30 0 vd;w msTfSf;F ABCD vd;w tl;lehw;fuk;

tiuf.

tif : VI ,uz;L gf;fq;fs; xU Nfhzk; kw;Wk; xU ,izNfhL

nfhLf;fg;gLk;NghJ tl;l ehw;fuk; tiujy ;

103. 𝐴𝐵 = 5.8 nr.kP ∠ 𝐴𝐵𝐷 = 35 0 kw;Wk; 𝐴𝐷 = 4.2 nr.kP 𝐴𝐵 ǁ 𝐶𝐷 vd;w msTfs; nfhz;l ABCD vd;w tl;lehw;fuk; tiuf.

104. 𝐴𝐵 = 6.5 nr.kP ∠ 𝐴𝐵𝐶 = 110 0 kw;Wk; 𝐵𝐶 = 5.5 nr.kP 𝐴𝐵 ǁ 𝐶𝐷 vd;w msTfs; nfhz;l ABCD vd;w tl;lehw;fuk; tiuf.

ghlk; : 9.tiuglq;fs;

𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 vDk; tbtpy; mike;j ,Ugbr;rkd;ghl;bd; tiuglk; tiujy ;

fPo;f;fz;l rkd;ghLfSf;F tiuglk; tiuf.

105. 𝑦 = 2𝑥2 106. 𝑦 = 3𝑥2 107. 𝑦 = −4𝑥2 108. 𝑦 = −3𝑥2 109. 𝑦 = 𝑥 + 2 𝑥 + 4 110. 𝑦 = 2𝑥2 − 𝑥 + 3

𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 vDk; tbtpy; mike;j ,Ugbr;rkd;ghl;il tiuglk; %yk; jPh;j;jy;

fPo;f;fz;ltw;iw tiuglk; %yk; jPh;f;f.

111. 𝑥2 − 2𝑥 − 3 = 0 112. 𝑥2 − 3𝑥 − 10 = 0 113. 𝑥2 − 4 = 0

114. 2𝑥2 + 𝑥 − 6 = 0 115. 𝑥 − 5 𝑥 − 1 = 0 116. 2𝑥 + 1 𝑥 − 3 = 0

117. 𝑦 = 2𝑥2 d; tiuglk; tiue;J mjpypUe;J 2𝑥2 + 𝑥 − 6 = 0 vd;w rkd;ghl;il jPh;f;f.

118. 𝑦 = 𝑥2 d; tiuglk; tiue;J mjpypUe;J 𝑥2 − 4 𝑥 − 5 = 0 vd;w rkd;ghl;il jPh;f;f.

119. 𝑦 = 𝑥2 + 3𝑥 + 2 d; tiuglk; tiue;J mjpypUe;J 𝑥2 + 2𝑥 + 4 = 0 vd;w rkd;ghl;il jPh;f;f.

120. 𝑦 = 𝑥2 + 2𝑥 − 3 d; tiuglk; tiue;J mjpypUe;J 𝑥2 − 𝑥 − 6 = 0 vd;w rkd;ghl;il jPh;f;f.

121. 𝑦 = 2𝑥2 + 𝑥 − 6 d; tiuglk; tiue;J mjpypUe;J 2𝑥2 + 𝑥 − 10 = 0 vd;w rkd;ghl;il jPh;f;f.

122. 𝑦 = 𝑥2 − 𝑥 − 8 d; tiuglk; tiue;J mjpypUe;J 𝑥2 − 2 𝑥 − 15 = 0 vd;w rkd;ghl;il jPh;f;f.

123. 𝑦 = 𝑥2 + 𝑥 − 12 d; tiuglk; tiue;J mjpypUe;J 𝑥2 + 2 𝑥 + 2 = 0 vd;w rkd;ghl;il jPh;f;f.

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rpy rpwg;G tiuglq;fs;

124. fPo;f;fhZk; ml;ltizf;Fj; jFe;j tiuglk; tiue;J khwpfspd; khWghl;Lj; jd;ikia fhz;f. mk;khWghl;bd; khwpypiak;

fhz;f.

x 2 3 5 8 10

y 8 12 20 32 40

NkYk; x = 4 vdpy; y d; kjpg;G fhz;f.

125. xU kpjptz;b Xl;Lgth; A vd;w ,lj;jpypUe;J B vd;w ,lj;jpw;F xU rPuhd Ntfj;jpy; xNu topapy; ntt;NtW ehl;fspy; gazk;

nra;fpwhh;.mth; gazk; nra;j Ntfk; , mj;J}uj;jpidf; flf;f vLj;Jf;nfhz;l Neuk; Mfpatw;iwg; gw;wpa tptuq;fs;

( Ntf – fhy ) gpd;tUk; ml;ltizapy; nfhLf;fg;gl;Ls;sd.

Ntfk; ( f.kP / kzp ) 2 4 6 10 12

Neuk; ( kzpapy; ) 60 30 20 12 10

Ntf - fhy tiuglk; tiue;J mjpypUe;J .

i ) mth; kzpf;F 5 fP.kP Ntfj;jpy; nrd;why; J}uj;ijf; flf;f MFk; gaz Neuk;.

ii ) mth; ,f;Fwpg;gpl;l J}uj;ij 40 kzpNeuj;jpy; flf;f ve;j Ntfj;jpy;gazpf;f Ntz;Lk; - Mfpatw;iwf; fhz;f.

126. xU tq;fp %j;j Fbkf;fspd; itg;Gj; njhiff;F 10 % jdptl;b jUfpwJ. itg;Gj;njhiff;Fk; mjw;F Xh; Mz;Lf;Ff;

fpilf;Fk; tl;bf;Fk; ,ilNaahd njhlh;gpidf; fhl;l xU tiuglk; tiuf.mjd; %yk;

i ) &.650 itg;gj;njhiff;Ff; fpilf;Fk; tl;b kw;Wk;

ii ) &.45 tl;bahf fpilf;f tq;fpapy; nrYj;jg;glNtz;ba itg;Gj;njhif Mfpatw;iwf; fhz;f.

127. xU NgUe;J kzpf;F 40 fp.kP Ntfj;jpy; nry;fpwJ. ,jw;Fhpa J}u – fhy njhlh;gpw;fhd tiuglk; tiuf. ,jidg; gad;gLj;jp 3

kzp Neuj;jpy; ,g;NgUe;J gazpj;j J}uj;ijf; fhz;f.

128. thq;fg;gl;l Nehl;L Gj;jfq;fspd; vz;zpf;if kw;Wk; mjw;fhd tpiy tpguk; gpd;tUk; ml;ltidapy; jug;gl;Ls;sJ.

Nehl;L Gj;jfq;fspd; vz;zpf;if x 2 4 6 8 10 12

tpiy ( &.y ) 30 60 90 120 150 180

,jw;F tiuglk; tiue;J mjd; %yk;

i ) VO Nehl;Lg; Gj;jfq;fspd; tpiyiaf; fhz;f. ii ) &.165 – f;F thq;fg;gLk; Nehl;Lg; Gj;jfq;fspd; vz;zpf;ifiaf; fhz;f.

129.

Nkw;fz;l ml;ltizapy; cs;s tptuj;jpw;F tiuglk; tiue;J , mjd; %yk;

i ) x = 4 vdpy; y d; kjpg;G fhz;f. ii ) y = 12 vdpy; x d; kjpg;G fhz;f.

130. xU ypl;lh; ghypd; tpiy &.15 vd;f. ghypd; msTf;Fk; tpiyf;Fk; cs;sj; njhlh;gpidf; fhl;Lk; tiuglk; tiuf.mjidg;

gad;gLj;jp i ) tpfpjrk khwpypiaf; fhz;f. ii ) 3 ypl;lh; ghypd; tpiyiaf; fhz;f.

131. 𝑥𝑦 = 20 , 𝑥 , 𝑦 > 0 vd;gjd; tiuglk; tiuf. mjidg; gad;gLj;jp 𝑥 = 5 vdpy; 𝑦 − d; kjpg;igAk;

𝑦 = 10 vdpy; 𝑥 d; kjpg;igAk; fhz;f.

132

Ntiyahl;fspd; vz;zpf;if x 3 4 6 8 9 16

ehl;fspd; vz;zpf;if y 96 72 48 36 32 18

ml;ltizapy; nfhLf;fg;gl;Ls;s tptuj;jpw;fhd tiuglk; tiuf.mjd; %yk; 12 Ntiyahl;fs; mt;Ntiyia KOtJkhf

nra;J Kbf;f MFk; ehl;fspd; vz;zpf;ifia fhz;f.

x 1 3 5 7 8

y 2 6 10 14 16

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ghlk; : 11.Gs;spapay;

fPo;f;fd;l Gs;sp tptuq;fSf;F jpl;ltpyf;fk; fhz;f.

133 ) 2 , 5 , 8 , 11 , 14 , 6 , 12 , 10

134 ) 65 , 58 , 53 , 50 , 50 , 63 , 52 , 55

135 ) 10 , 20 , 15 , 8 , 3 , 4

136 ) 38 , 40 , 34 , 31 , 28 , 26 , 34

137 ) 18 , 20 , 15 , 12 , 25

138 ) 20 , 18 , 32 , 24 , 26

ghlk; : 12.epfo;jfT

139. xU rPuhd gfil xU Kiw cUl;lg;gLfpwJ.gpd;tUk; epfo;r;rpfSf;fhd epfo;jfT fhz;f.

i ) vz; 4 fpilj;jy; ii ) xU ,ul;ilg;gil vz; fpilj;jy; iii ) 6 – d; gfhf; fhuzp fpilj;jy;

iv ) 4 – I tpl nghpa vz; fpilj;jy;.

140. xU rPuhd ehzak; ,uz;L Kiw Rz;lg;gLfpwJ. fPo;f;fhZk; epfo;r;rpfSf;fhd epfo;jftpid fhz;f.

i ) ,U jiyfs; fpilj;jy; ii ) Fiwe;jJ xU jiy fpilj;jy; iii ) xU G+ kl;Lk; fpilj;jy;

141. ,U rPuhd gfilfs; xU Kiw cUl;lg;gLfpd;wd. fPo;f;fhZk; epfo;r;rpfSf;fhd epfo;jftpid fhz;f.

i ) Kf vz;fspd; $Ljy; 8 Mf ,Uj;jy; ii ) Kf vz;fs; xNu vz;fshf ,Uj;jy;

iii ) Kf vz;fspd; $Ljy; 8 – I tpl mjpfkhf ,Uj;jy;.

142. ed;F fiyj;J itf;fg;gl;l 52 rPl;Lfisf; nfhz;l rPl;Lf;fl;bypUe;J rktha;g;Gr; Nrhjid Kiwapy; xU rPl;L

vLf;fg;gLfpwJ. me;j rPl;L gpd;tUgtdhf ,Uf;f epfo;jfTfisf; fhz;f.

i ) ,uhrh ii ) fUg;G ,uhrh iii ) ];NgL iv ) lakz;l; 10

143. ed;F fiyj;J itf;fg;gl;l 52 rPl;Lfisf; nfhz;l rPl;Lf;fl;bypUe;J rktha;g;Gr; Nrhjid Kiwapy; xU rPl;L

vLf;fg;gLfpwJ. me;j rPl;L gpd;tUgtdhf ,Uf;f epfo;jfTfisf; fhz;f.

i ) vLj;j rPl;L lakz;l Mf ,Uf;f ii ) vLj;j rPl;L lakz;l ,y;yhky; ,Uf;f

iii ) vLj;j rPl;L V]; rPl;lhf ,y;yhky; ,Uf;f.

144. xU ngl;bapy; 4 gr;ir , 5 epyk; kw;Wk; 3 rptg;G epwg; ge;Jfs; cs;sd. rktha;g;G Kiwapy; xU ge;ij Njh;e;njLf;f mJ

i ) rptg;G epwg; ge;jhf ,Uf;f ii ) gr;ir epwg; ge;jhf ,y;yhky; ,Uf;f Mfpatw;wpd; epfo;jfT fhz;f.

145. %d;W ehzaq;fs; xNu Neuj;jpy; Rz;lg;gLfpd;wd. epfo;jftpd; $l;ly; Njw;wj;ij gad;gLj;jp , rhpahf

,U G+f;fs; my;yJ Fiwe;jgl;rk; xU jiyahtJ fpilf;Fk; epfo;r;rpapd; epfo;jftpdif; fhz;f.

146. xU gfil ,UKiw cUl;lg;gLfpwJ.Fiwe;jJ xU cUl;lypyhtJ vz; 5 fpilg;gjw;fhd epfo;jftpidf; fhz;f.

( $l;ly; Njw;wj;ij gad;gLj;Jf )

147. 52 rPl;Lfisf; nfhz;l xU rPl;Lf;fl;bypUe;J rktha;g;G Kiwapy; xU rPl;L vLf;fg;gLk; NghJ , mr;rPl;L xU ,uhrh ( King )

my;yJ xU rptg;G epwr; rPl;lhf fpilg;gjw;fhd epfo;jftpidf; fhz;f.

148. xU igapy; 10 nts;is , 5 fUg;G , 3 gr;ir kw;Wk; 2 rptg;G epwg; ge;Jfs; cs;sd.rktha;g;G Kiwapy; Njh;e;njLf;fg;gLk;

xU ge;J , nts;is my;yJ fUg;G my;yJ gr;ir epwkhf ,Ug;gjw;fhd epfo;jftpidf; fhz;f.

Prepareb by

S.Sakthivel ., M.Sc.,B.Ed .,

B.T , Asst , Govt higher secondary school,

Ramiyampatti,

Pappireddipatti – T.K

Dharmapuri – D.T

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Maths-2 Marks- Important Sums-Tamil Medium · X – MATHS QUESTION BANK Page 3 28. 7 + 3 kw;Wk . 7 − 3 Mfpatw;iw %yqfshff nfhzl ,Ugbr rkdghl;bid mikf;f. 29. RUfFf . 6 2+ 9 3 2−