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+2 fzpj ghlj;jpy; cWjpahf Njh;r;rp ngWtjw;fhd

Kf;fpa tpdhf;fs;

fPNo nfhLf;fg;gl;Ls;s tpdhf;fis kl;Lk; gapw;rp nra;J

ghh;j;jhy; NghJkhdJ fz;bg;ghf 100 kjpg;ngz;fSf;F Nky;

ngwyhk; vd;gJ cz;ik 64 gj;J kjpg;ngz; tpdhf;fs;

nfhLf;fg;gl;Ls;sJ ,jpypUe;J fz;bg;ghf 5 my;yJ 6

tpdhf;fs; tUk;

6 kjpg;ngz; tpdhf;fspy; ghlk; 9-y; jdpepiy fzf;fpay;

gFjpapy; nka;ml;ltiz (Truth Table) apy; fz;bg;ghf xU tpdh

tUk; kw;nwhU tpdh vLj;Jf;fhl;Lfs; 9.12, 9.13, 9.14, 9.15,

9.16, 9.17, 9.19, 9.20 kw;Wk; Fyj;jpd; ePf;fy; tpjpfs;, gpd;

jpUg;Gif tpjpfs ; ,tw;wpy; ,Ue;J tUk;

XU kjpg;ngz; tpdhf;fspy; Book back question Kjy;

njhFjpapy; 121 tpdhf;fSk; ,uz;lhk; njhFjpapy; 150

tpdhf;fSk; Mf nkhj;jk; 271 tpdhf;fs; ,tw;wpy; ,Ue;J

mg;gbNa ve;j khw;wKk; nra;ahky; 30 tpdhf;fs; tUk;



Mf nkhj;jk;

,tw;iw kl;Lk; gbj;J ed;whf gapw;rp nra;J ghh;j;jhy;

fzpjj;jpy; KOikahd ntw;wpg;ngwyhk.; Mrphpa ez;gh;fs;

kPj;jpwd; Fiwthd khzth;fSf;F ,ijkl;Lk; gaw;rp nra;J

ek;Kila ghlj;jpy; 100% rjtPj Njh;r;rpia cWjpnra;J

nfhs;syhk;.

vd;Wk;

10 kjpg;ngz;fs; tpdhf;fs; 6 = 60

6 kjpg;ngz;fs; tpdhf;fs; 2 = 12

1 kjpg;ngz; tpdhf;fs; 30 = 30

nkhj;jk; = 102

xU ehSf;F ,uz;L tpdhf;fs; tPjk; rupahd gapw;rp

nra;jhy; ,uz;L khjj;jpy; KOikahd ntw;wpia

mile;Jtplyhk;



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.

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

miyNgrp vz; : 9443407917

kpf Kf;fpa tpdhf;fs;



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 + 8 = 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.

ENGLISH MEDIUM

1. Prove that cos (A+B) =cosA cosB –sin A sin B

2. Prove that sin (A-B) = sin A cos B – cos A sin B.

3. If = + 3 - , = -2 + 5 , = -3

verify that x ( x )=( . ) -( . )

4. Verify ( x ) x ( x ) = [ ] - [ ] for , and in

= + + , = 2 + , = 2 + + , = + +2

5. Show that the lines

=

=

and

=

=

intersect and find their point of

intersection

6. Altitudes of a triangle are concurrent – prove by vector method

7. Prove that cos (A-B) = cos A cos B + sin A sin B

8. Prove that sin (A +B ) = sin A cos B + cos A sinB

9. Show that the lines

=

=

and

=

=

intersect and find their point of

intersection

10. Find the vector and cartesian equations of the plane passing through the points ( 2,2,-1) ,

(3,4,2) and ( 7 , 0, 6)



11. Find the vector and cartesian equation of the plane containing the line

=

=

and

parallel to the line

=

=

12. Find the vector and cartesian equation of the plane through the point ( 1, 3, 2) and

parallel to the lines

=

=

and

=

=

13. Find the vector and cartesian equation to the plane through the point (-1 , 3, 2 ) and

perpendicular to the planes x+2y ++2z = 5 and

3x+y+2z = 8

14. Find the vector and cartesian equation of the plane passing through the points A(1,-2,3) and

B(-1,2,-1) and is parallel to the line

=

=

15. Find the vector and cartesian of the plane through the points ( 1, 2,3) and ( 2, 3,1)

perpendicular to the plane 3x -2y + 4z-5 = 0

16. Find the vector and cartesian equation of the plane containing the line

=

=

and passing through the point (-1, 1, -1 )

17. Find the vector and cartesian equation of the plane passing through the points with position

vectors

3 +4 + 2 , 2 - 2 - and 7 +

18. Derive the equations of the plane in the intercept form

19. Find the vector and cartesian equations of the plane passing through the points (-1,1,1) and

( 1, -1, 1) and perpendicular to the plane x + 2y + 2z = 5

20. Find the vector and cartesian equations of the plane passing through the points (-1,1,1) and

(1,-1,1) and perpendicular to the plane x + 2y + 2z = 5

21. P represents the variable complex number z . Find the locus of P. if

Re

= 1


Im

= -2


arg

=



24. If are the roots of the equation x2 – 2 px + ( p2 + q2 ) = 0 and tan

=qn-1

25. If and are the roots of x2 – 2x + 4 = 0 Prove that

n - n = i2n+1 sin

and deduct 9 - 9

26. If x +

= 2 cos , y +

= 2 cos show that

(iii)

+

= 2 cos ( m - n )

(iv)

-

= 2i sin ( m - n ) .

27. If a = cos 2 + i sin 2 and

c = cos 2 Prove that

i)

= 2 cos( + )

ii)

= 2 cos 2 ( – )

28. Find all the values of the following : (

29. Solve : x4 – x3 + x2- x + 1 = 0

30. Find all the values of (

–

and hence prove that the product of the values is 1


Re

= 1


arg

=

33. If are the roots of x2 – 2x + 2 = 0 and cot show that

=

34. Solve the equation : x9 + x5 – x4 – 1 = 0

35. Solve the equation : x7 + x4 + x3 + 1=0



36. Find all the equation : (

37. Prove that the line 5x + 12y = 9 touches the hyperbola x2 – 9y2 = 9 and find the point of

content

38. Show that the line x – y + 4 = 0 is a tangent to the ellipse x2 + 3y2 = 12. Find the co –

ordinates of the point of contact.

39. Find the equation of the hyperbola if

(i) its asymptotes are parallel to x + 2y – 12 = 0 and x – 2y + 8 = 0

(2,4) is the centre of the hyperbola and it passes through (2, 0)

40. Find the equation of the rectangular hyperbola which has for one of its asymptotes the line

x + 2y – 5 = 0 and passes through the points (6, 0) and (-3, 0)

41. Find the length of the curve 4y2 = x3 between x = 0 and x = 1

42. Find the length of the curve

+

= 1

43. Show that the surface area of the solid obtained by revolving the arc of the curve y = sin x

from x = 0 to x = about x – axis is 2

44. Find the surface area of the solid generated by revolving the cycloid x = a(t + sin t), y = a(1 +

cos t) about its base (x – axis).

45. Find the perimeter of the circle with radius a.

46. Find the length of the curve x = a(t – sin t) , y = a(1 – cos t) between t = 0 and

47. Find the surface area of the solid generated by revolving the arc of the parabola y2 = 4ax,

bounded by its latus rectum about x – axis.

48. Prove that the curved surface area of a sphere of radius r intercepted between two parallel

planes at a distance a and b from the centre of the sphere is 2r (b – a) and hence deduct

the surface area of the sphere. (b > a)

49. Show that (z, *) is an infinite abelian group where * is defined as a * b = a + b + 2

50. Show that the set G of all matrices of the form

, where x R – {0}, is a group

under matrix multiplication.



51. Show that the set G = {a + b / a, b Q} is an infinite abelian group with respect to

addition.

52. Let G be the set of all rational numbers except 1 and * be defined on G by a * b = a + b – ab

for all a, b G. Show that (G, *) is an infinite abelian group.

53. Prove that the set of four functions f1, f2, f3, f4 on the set of non – zero complex numbers C –

{0} defined by

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

and f4(z) = -

z C – {0} forms an abelian group with respect

to the composition of functions.

54. Show that (Zn , +n) forms group

55. Show that (Z7 – {[0]}, .7) forms a group.

56. Show that the nth roots of unity form an abelian group of finite order with usual

multiplication.

57. Show that set G of all positive rationals forms a group under the composition * defined by a

* b =

for all a, b G

58. Show that

where = 1,

1 form a group with respect to matrix multiplication.

59. Show that the set M of complex numbers z with the condition |z|=1 forms a group with

respect to the operation of multiplication of complex numbers.

60. Show that the set G of all rational numbers except – 1 forms an abelian group with respect

to the operation * given by a * b = a + b + ab for a, b G

61. Show that the set {[1], [3], [4], [5], [9]} forms an abelian group under multiplication modulo

11.

62. Show that the set of all matrices of the form

, a R – {0} forms an abelian group

under matrix multiplication.

63. Show that the set G = {2n / n Z} is an abelian group under multiplication.



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

miyNgrp vz; : 9443407917

2 fzpj ghlj;jpy; cwjpahf njh;r;rp ngwtjw;fhd kf;fpa tpdhf;fs; · pdf file +2 fzpj...

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