12th Maths

7/24/2019 12th Maths

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LRm

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1. mzpfs; kw;Wk; mzpf;nfhitfspd; gad;ghLfs;mzpfs; kw;Wk; mzpf;nfhitfspd; gad;ghLfs;mzpfs; kw;Wk; mzpf;nfhitfspd; gad;ghLfs;mzpfs; kw;Wk; mzpf;nfhitfspd; gad;ghLfs;mzpfs; kw;Wk; mzpf;nfhitfspd; gad;ghLfs;

3 kjpg;bgz; tpdhf;fs;.3 kjpg;bgz; tpdhf;fs;.3 kjpg;bgz; tpdhf;fs;.3 kjpg;bgz; tpdhf;fs;.3 kjpg;bgz; tpdhf;fs;.

1. gpd;tUk; mzpapd; neh;khW mzpiaf;fhz;f :

41

21.

2. A xU g{r;rpa kw;w nfhit mzp vdpy;(AT)-1 = (A-1)T vd epWt[f.


1.

2

5-

1-

7-51

12-1

5-13

vd;w mzpapd; juk; fhz;.

2.

6

3-

4

721-

1-42-

32-1

vd;w mzpapd; juk; fhz;.

3.

0

1-

1

1-11

03-2

210

vd;w mzpapd; juk;fhz;.

4. A =

53

21vd;w mzpapd;neh;g;igf; fz;LA(adj A) A=lAl I vd;gijr; rhpghh;f;f.

5. A =

65

32vdpy; (A-1)T = (AT)-1 vd;gijr; rhpghh;.

6. A =

11

21kw;Wk; B =

21

10 vdpy; (AB)-1= B-1A-1 vd epWt[f.

7. A =

344

101334

d; nrh;g;g[mzpA vd epWt[f.

8. A =

544

434

221

f;FA = A-1 vd epWt[f.

9. neh;khW mzpfhzy; Kiwapy; jPh;f;f :x + y = 3, 2x + 3y = 8.

10. gpd;tUk; mrkgoj;jhd rkd;ghl;Lj; bjhFg;gpidj; jPh; 2x + 3y = 8,4x + 6y = 16.

11. gpd;tUk; mrkgoj;jhd rkd;ghl;Lj; bjhFg;gpidj; jPh;2x + 2y + z = 5 ,x - y + z = 1, 3x + y + 2z = 4



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(mzpf;nfhit Kiwia gad;gLj;Jf).

12. juKiwapid gad;gLj;jp gpd;tUk; rkd;ghLfspd; bjhFg;g[ xUfikt[ cilajh

vdg;gij Muha;f. x + y + z = 7, x + 2y + 3z = 18, y+2z = 6.


1. xU igapy; %. 1 kw;Wk; % 2 kw;Wk; % 5 ehzafs; cs;sd. %gha; 100 kjpg;gpw;F bkhj;jk; 30 ehzafs;

cs;sd. mt;thwhapd; xt;bthU tifapYk; cs;s ehzafspd; vz;zpf;ifiaf; fhz;.

2. xU rpwpa fUj;juF miwapy; 100 ehw;fhypfs;itg;gjw;F nghJkhd ,lKs;sJ. _d;W epwfspy;

ehw;fhypfs; cs;sd. (rptg;g[/ ePyk; kw;Wk; gr;ir) rptg;g[tz;z ehw;fhypapd; tpiy %. 240. ePy tz;z

ehw;fhypapd; tpiy %. 260. gr;ir tz;z ehw;fhypapd; tpiy %. 300. bkhj;jk; %. 25/000 kjpg;g[s;s

ehw ;fhypfs; thfg;gl;lJ. mt;thwhapd; xt;bthU tz;zj;jpYk; thfj;jf;f ehw;fhypfspd;

vz;zpf;iff;F Fiwe;jgl;rk; _d;W jPh;t[fisf; fhz;f.

3. juKiwapidg; gad;gLj;jp x + y + z = 6, x + 2y + 3z = 14, x + 4y + 7z = 30 Mfpa rkd;ghLfs;

xUfiktd vdf; fhl;o mtw;iwj;jPh;f;ft[k;.

4. jug;gl;l rkd;ghl;Lj; bjhFg;g[ xUfike;jjh vd juKiwapidg; gad;gLj;jp rhpghh;j;J mt;thW

xUfike;jjhapd; mjidj;jPh;f;f :2x + 5y + 7z = 52, x + y + z = 9, 2x + y - z = 0.

5. gpd;tUk; rkd;ghLj; bjhFg;g[ xUfikt[ cilajh vd;gij Muha;f. mt;thW xUfikt[

cilajhapd; mjid jPh;f;f : (juKiwia gad;gLj;jt[k;) x + y - z = 1, 2x + 2y - 3z = 13,

2x - 3y + 3z = -32.

6. gpd;tUk; rkd;ghLfspd; xUfikt[j; jd;ikia juKiwapidg; gad;gLj;jp Muha;f

2x - 3y + 7z = 5, 3x + y - 3z =13, 2x + 19y - 47z = 32.

7. mzpf;nfhit Kiwia gad;gLj;jp x + 2y + z = 7, 2x - y + 2z = 4,x + y - 2z = -1 vd;w rkd;ghL.

8. mzpf;nfhit Kiwapy; jPh; 0z

2

y

2

x

35,

z

1

y

4

x

21,

z

1

y

2

x

1==++=+ .

9. neh;khW mzpfhzy; Kiwapy; jPh;f;f : 2x - y + 3z = 9, x + y + z = 6, x - y + z = 2.

10. d; vk;kjpg;gpw;F x + y + 3z = 0, 4x + 3y + z = 0, 2x + y + 2z = 0 vd;w bjhFg;gpw;F(1) btspg;gilj;jPh;t[ (2) btspg;gilaw;w jP;h;t[ fpilf;Fk; (juKiwia gad;gLj;jt[k;)

11. K d; vk;kjpg;g [fSf;F gpd;tUk; rkd;ghl;j; bjhFg;g[ kx + y + z = 1, x + Ky + z = 1,

x + y + kz = 1 (1) xnu xU jPh;t[ (2) xd;Wf;F nkw;gl;l jPh;t[ (3) jPh ;t[ ,y;yhik bgWk;.

(juKiwia gad;gLj;jt[k;)

12. , ,d; vk ;kjpg; g[fSf;F x + y + z = 6, x + 2y +3 z = 10, x + 2y + z = vd;w

rkd;ghLfs; (1) ahbjhU jPh;t[k ; bgw;wpuhJ (2) xnu xU jPh;it bgw;wpUf;Fk; (3)

vz;zpf;ifaw;w jPh;t[fisg; bgw;wpUf;Fk;vd;gjid Muha;f.

13. juKiwapidg; gad;gLj;jp d; vy;yh kjpg;g [fSf;Fk; gpd;tUk; rkd;ghl;Lj; bjhFg;gpd;



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jPh;t[fis Muha;f.x + y + z = 2, 2x + y - 2z = 2, x + y + 4z = 2.

******

2.btf;lh; ,aw;fzpjk;btf;lh; ,aw;fzpjk;btf;lh; ,aw;fzpjk;btf;lh; ,aw;fzpjk;btf;lh; ,aw;fzpjk;

6 kjpg;bgz; tpdhf;fs;:-6 kjpg;bgz; tpdhf;fs;:-6 kjpg;bgz; tpdhf;fs;:-6 kjpg;bgz; tpdhf;fs;:-6 kjpg;bgz; tpdhf;fs;:-

1. xU rha; rJuj;jpd; _iy tpl;lfs; xd;iw xd;W brFj;jhf btl;of; bfhs;Sk; vd;gjid

btf;lh; Kiwapy; epWt[f.

2. vy;yh btf;lh;fs; c,b,a f;Fk; 2cbaaxc,cxb,bxa = . vd epWt[f.

3. tHf;fkhd FwpaPLfSld;sinC

c

sinB

b

sinA

a== vd epWt[f.

4. 0ac,cb,ba = vd epWt[f.

5. (5/5/3) vd;w g[s;sp tHpr; bry;tJk; (1/2/3) ikakhft[k; mika nfhsj;jpd; btf;lh; kw;Wk;

fhh;Orpad; rkd;ghLfisf; fhz;f.

6. k3j4i2-k7j6i2 ++ Wk;kw; vDk; epiy btf; ;lh;fisa[k; g [s;spfs; Kiwna A / B MFk;.

,tw;iw ,izf;Fk; nfhl ;il tpl;lkhff; bfhz;l nfhsj;jpd; btf;lh ; kw ;Wk; fhh;Or pad;rkd;ghLfisf;fhz;f nkYk; ikak;/ Muk; fhz;f.

7. k3ji2 + vDk; epiy btf;liu ikakhft[k; 4 myFfis Mukhft[k; bfhz;l nfhsj;jpd;btf;lh; rkd;ghLfisf; fhz;f kw;Wk; fhh;Orpad; rkd;ghLfisj; jUf.

8. (1, -1, 1)I ikakhft[k; ( )

5k2jir =++ vd;w nfhsj;jpd; Muj;jpw;F rkkhd kjpg;ig

Muj;ijf; bfhz;l nfhsj;jpd; btf;lh; kw;Wk; fhh;Orpad; rkd;ghLfisj; jUf.

9. ( ) 0.50k10j6i8.rr2 =+ vd;w btf;lh; rkd;ghl;ila[ila nfhsj;jpd; ikaj;ija[k; Muj;ija[k; fhz;f.

10 kjpg;bgz; tpdhf;fs; :-10 kjpg;bgz; tpdhf;fs; :-10 kjpg;bgz; tpdhf;fs; :-10 kjpg;bgz; tpdhf;fs; :-10 kjpg;bgz; tpdhf;fs; :-

1. sin (A + B) = sin A cos B + cosA sin B vd btf;lh; Kiwapy; ep%gp.

2. sin (A - B) = sin A cosB -cos A Sin B vd btf;;lh;Kiwapy; ep%gp.

3. cos(A + B) = cos A cos B - sinA sin B vd btf;lh; Kiwapy; ep%gp.

4. cos (A - B) = cos A cos B + sin A sin B vd btf;lh;Kiwapy;ep%gp.

5. xU Kf;nfhzj;jpd; Fj;Jf;nfhLfs; xnu g[s;spapy; re;jpf;Fk; vd;gjid btf;lh; Kiwapy;epWt[f.

6. k3jc,k5i2b,k-j3i2a =+== vdpy; ( ) ( ) ( )cb.abc.acxbxa = vd rhpghh;f;f.

7. k2jid,kji2c,ki2b,kjia ++=++=+=++= vdpy; ) ) dcbacdbadxcxbxa = vd;gijr; rhpghh;f;f.

8. (2/ -1/ -3) vd;w g[s;sp tHpr; bry;tJk;2

2z

3

1y

2

1x

4

3z

2

1y

3

2-x =

+=

=

= Wk;kw; vd;w nfhLfSf;F

,izahdJkhd jsj;jpd; btf;lh;kw;Wk; fhh;Orpad; rkd;ghLfisf; fhz;f.



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9. (-1,-2,1) vd;w g[s;sp tHpr; bry;tJk; x + 2y + 4z + 5 = 0 kw;Wk;2x - y + 3z + 3 = 0 Mfpa jsfSf;Fr;

brFj;jhft[k;cs;s jsj;jpd; btf;lh; kw;Wk; fhh;Orpad; rkd;ghLfis fhz;f.

10. (-1/ 1/ 1) kw;Wk;(1/ -1/ 1) Mfpa g[s;spfs; tHpnar; bry;yf; ToaJk;x + 2y + 2z + 5 = 0 vd;w jsj;jpw;F

brFj;jhf miktJkhd jsj;jpd; btf;lh; kw;Wk; fhh;Orpad; rkd;ghl;ilf; fhz;f.

11. (1/ 2/ 3) kw;Wk; (2/ 3/ 1) vd;w g[s;spfs; tHpnar; bry;yf; ToaJk; 3x - 2y + 4z - 5 = 0 vd;w jsj;jpw;Fr;

brFj;jhft[k; mike;j jsj;jpd; btf;lh; kw;Wk; fhh;Orpad; rkd;ghLfisf; fhz;f.

12. btl;Lj;Jz;L totpy; xU jsj;jpd; rkd;ghl;il btf;lh; KiwapYk; fhh;Orpad; KiwapYk; jUtpf;f.

13.2

1

3

2

2

2-x

=

=

zy vd;w nfhl;il cs;slf;fpaJk; (-1/ 1/ -1) vd;w g[s;sp tHpnar; bry;y ToaJkhd

jsj;jpd; btf;lh; kw;Wk; fhh;Orpad; rkd;ghLfisf; fhz;f.

14. (2/ 2/ -1)/ (3/ 4/ 2) kw;Wk; (7/ 0/ 6) Mfpa g[s;spfs; tHpnar; bry;yf;Toa jsj;jpd; btf;lh;kw;Wk;fhh;Orpad;

rkd;ghl;ilr; fhz;f.

15. ki7kj2i2,k2j4i3 +++ Wk;kw; Mfpatw;iw epiy btf;lh;fshff; bfhz;l g[s;spfs; tHpnar; bry;Yk;


16.3

1

3

2

2

2-x =

=

zy vd;w nfhl;il cs;slf;fpaJk;

1

1

2

1

3

1x +=

=

+ zy vd;w nfhl;ow;F ,izahdJkhd


17. 11z

2

1y

1

2x

3

z

1

1y

1

1-x =

=

=

+= Wk;kw; vd;w nfhLfs; btl;of; bfhs;Sk; vdf; fhl;Lf. nkYk; mit

btl;Lk; g[s;spiaf; fhz;f.

18.3

1z

0

y

2

4x

0

1z

1

1y

3

1-x +==

+=

= Wk;kw; vd;w nfhLfs; btl;Lk;vdf; fhl;o mit btl;Lk; g[s;spiaf;fhz;f.

* * * * * * * * ** * * * * * * * ** * * * * * * * ** * * * * * * * ** * * * * * * * *

3. fyg;bgz;fs;3. fyg;bgz;fs;3. fyg;bgz;fs;3. fyg;bgz;fs;3. fyg;bgz;fs;


1. fyg;bgz;fs; (7 + 9i), (-3 + 7i), (3 + 3i) vDk; fyg;bgz;fs; Mh;fd; jsj;jpy; xU brnfhzKf;nfhzj;ijf;mikf;Fk; vd epWt[f.

2. (-8 - 6i) d;th;f;f _yk; fhz;f.

3. (-7 + 24i) d; th;f;f _yk;fhz;f.

4. fyg;bgz;fspd; Kf;nfhz rkdpypia vGjp ep%gp.

5. (3+i) xU jPh;thff; bfhz;l x4 - 8x3 + 24x2 - 32x + 20 = 0 vDk; rkd;ghl;od; gpw jPh;t[fisf; fhz;.

6. 1+2i I xU _ykhff;bfhz;l x4 - 4x3 + 11x2 - 14x + 10 = 0 vDk;rkd;ghl;od;jPh;t[fisf; fhz;f.



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7. n vd;gJ kpif KG vz; vdpy;(1 + i)n+ (1 - i)n=4

cos2 22

nn+

vd ep%gp.

8. x = cos + i sin , kw;Wk;y = cos + i sin vdpy;xmyn+ nmyx1

= 2 cos (m+ n) vd epWt[f.

9. n N vdpy;(1 + i3)n+ (1 - i3)n=3

cos2 1 nn+

vd epWt[f.


1.4

3

23

2

1

i d; vy;yh kjpg;g[fisa[k; fhz;f kw;Wk; mjd; kjpg;g[fspd; bgUf;fw;gyd; 1 vdt[k;

fhl;Lf.

2. (3 + i)2/3,d;vy;yh kjpg;g[fisa[k; fhz;f.

3. x2-2x+4=0,d; _yfs; kw;Wk; vdpy;n - n= i 2n+1sin3

nmjpypUe;J9 -9 d; kjpg;igf; fhz;f.

4. x2 - 2px + (p2 + q2) = 0 vd;w rkd;ghl;od; _yfs; , kw;Wk; tan =

py

q

+

vdpy;

( )

n

n

nn

qy

sin

sin

-

)(y 1n

=++

vd epWt[f.

5. P vd;Dk; g[s;sp fyg;bgz; khwp z If; Fwpj;jhy; p d; epak ghijia Im .21iz

12z=

+

+f;F fhz;f.

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

7. jPh;f;f : x4 - x3 + x2 - x + 1 = 0.

8. jPh;f;f : x7 + x4 + x3 + 1 = 0.

9. , vd;git x2 - 2x + 2 = 0 _yfs; kw;Wk;cot = y + 1, vdpy;

n

nny

sin

sin

-

)()(y n=

++vdf; fhl;Lf.

* * * * ** * * * ** * * * ** * * * ** * * * *

4. gFKiw totf; fzpjk;4. gFKiw totf; fzpjk;4. gFKiw totf; fzpjk;4. gFKiw totf; fzpjk;4. gFKiw totf; fzpjk;


1. xU ,U rf;fu thfdj;jpd; Kfg;g[ tpsf;fpy; cs;s gpujpgypg;ghd xU gutisa mikg;gpy; cs;sJ.

mjd; tpl;lk; 12 brkP MHk; 4 br.kP vdpy; mjd; mr;rpy;vt;tplj;jpy; gy;gpid bghUj;jpdhy; Kfg;g[

tpsf;F kpfr; rpwe;j Kiwapy; xspiaj; juKoa[k; vdf; fzf;fpLf.

2. bjhiyj; bjhLnfhLfs; 2x + 3y - 8 = 0 kw;Wk; 3x - 2y + 1 = 0 nkYk; (5,3) vd;w g[s;sp tHpahf

mjpgutisak; bry;fpwJ vdpy; mjpgutisaj;jpd; rkd;ghl;oid fhz;f.

3. (1,2)ypUe;J 2x2 - 3y2 = 6 vd;w mjpgutisaj;jpw;F tiuaj;jf;f ,U bjhLnfhLfspd; rkd;ghLfisf;

fhz;f.



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4. brt;tf mjpgutisaj;jpd; VnjDk; xU g[s;spaplj;J tiuag;gLk; bjhLnfhL/ bjhiyj;

bjhLnfhLfSld; mikf;Fk; Kf;nfhzj;jpd; gug;g[xU khwpyp vd epWt[f.


1. y2 + 8x - 6y + 1 = 0 vd;w gutisaj;jpw;F mr;R Kid/ Ftpak; ,af;Ftiu rkd;ghL brt;tyfj;jpd;

rkd;ghL brt;tfyj;jpd; ePsk; Mfpatw;iwf; fhz;f. tiuglk; tiuf.

2. xU bjhF ghyj;jpd; fk;gp tlk; gutisa totpYs;sJ. mjd; ePyk; 40 kPl;lh; Mfk;. tHpg;ghijahdJ

fk;gp tlj;jpd; fPH;kl;lg; g[s;spapypUe;J 5 kPl;lh; fPnH cs;sJ. fk;gp tlj;ijj; jhFk; J]z;fspd;

caufs; 55 kPl;lh; vdpy; 30 kPl;lh; cauj;jpy; fk;gp tlj;jpw;F xU J]iz jhfp TLjyhf

bfhLf;fg;gl;lhy; mj;Jizj; jhfpapd; ePsj;ijf; fhz;.

3. xU uhf;bfl; btoahdJ bfhSj;Jk; nghJ mJ xU gutisag;ghijapy;bry;fpwJ. mjd; cr;r cauk;4 kP I vl;Lk;nghJ mJ bfhSj;jg;gl;l ,lj;jpypUe;J fpil kl;l J]uk;6 kP bjhiytpy;cs;sJ ,Wjpahf

fpilkl;lkhf 12 kP bjhiytpy; jiuia te;jilfpwJ vdpy; g[wg;gl;l ,lj;jpy; jiua[ld;Vw;gLj;jg;gLk;

vwpnfhzk; fhz;f.

4. jiukl;lj;jpypUe;J 7.5 kP cauj;jpy;jiuf;F ,izahf bghUj;jg;gl;l xU FHhapypUe;J btspnaWk;

ePh; jiuiaj;bjhLk; ghij xU gutisaj;ij Vw;gLj;JfpwJ. nkYk; ,e;j gutisag; gijapd;Kid

FHhapd;thapy; mikfpwJ. FHha; kl;lj;jpw;F 2.5 kP fPnH ePhpd; gha;thdJ FHhapd; Kid tHpahfr;

bry;Yk; epiy Fj;Jf;nfhl;ow;F 3 kPl;lh; J]uj;jpy; cs;sJ vdpy; Fj;Jf; nfhl;oypUe;J vt;tst[

J]uj;jpw;F mg;ghy; ePuhdJ jiuapy;tpGk; vd;gijf; fhz;f.

5. xU uapy;;nt ghyj;jpd; nky;tist[ gutisaj;jpd; mikg;igf; bfhz;Ls;sJ. mjd; tistpd; mfyk;

100 moahft[k; mt;tistpd; cr;rpg; g[s;spapd; cauk; ghyj;jpypUe;J 10 moahft[k; cs;sJ vdpy;

ghyj;jpd; kj;jpapypUe;J ,lg;g[wk;my;yJ tyg;g[wk; 10 mo J]uj;jpy; ghyj;jpd;nky; tist[ vt;tst[

cauj;jpy; ,Uf;Fk; vdf; fhz;f.

6. xU rkjsj;jpd; nky;brFj;jhf mike;Js;s Rthpd; kPJ 15 kP ePsKs;s xU VzpahdJ jsj;jpida[k;

Rtw;wpida[k; bjhLkhW efh;e;J bfhz;L ,Uf;fpwJ vdpy; Vzpapd; fPH;kl;l KidapypUe;J 6 kP

J]uj;jpy; Vzpapy; mike;Js;s P vd;w g[s;spapd; epakg;ghijf; fhz;f.

7. xU nfh-nfh tpisahl;L tPuh; tpisahl;Lg; gapw;rpapd; nghJ mtUf;Fk; nfh-nfh Fr;rpfSf;Fk;

,ilnaa[s;s J]uk; vg;bghGJk; 8 kP Mf ,Uf;FkhW czh;fpwhh;. mt;tpU Fr;rpfSf;F ,ilg;gl;l

J]uk; 6 kP vdpy;mth; XLk; ghijapd; rkd;ghl;il fhz;f.

8. Nhpad; Ftpaj;jpypUf;FkhW bkh;Fhp fpufkhdJ Nhpaid xU ePs;tl;lg; ghijapy; Rw;wp tUfpwJ.

mjd; miu bel;lr;rpd; ePsk; 36 kpy;ypad; iky;fy; Mft[k; ikaj; bjhiyj; jft[ 0.206 Mft[k;

,Uf;Fkhapd; (1) bkh;Fhp fpufkhdJ NhpaDf;F kpf mUfhikapy; tUk;nghJ cs;s J]uk;(2) bkh;f;Fhp

fpufkhdJ NhpaDf;F kpfj; bjhiytpy; ,Uf;Fk;nghJ cs;s J[uk;Mfpatw;iwf; fhz;f. (glk;tiuf)



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9

9. xU EiHt[ thapypd; nkw;TiuahdJ miu ePs;tl;l totj;jpy; cs;sJ. ,jd; mfyk; 20 mo

ikaj;jpypUe;J mjd; cauk; 18 mo kw;Wk; gf;f Rth;fspy; cauk; 12 mo vdpy; VnjDk; xU gf;f

RthpypUe;J 4 mo J]uj;jpy;nkw;Tiuapd;cauk; vd;dthf ,Uf;Fk;.

10. x + 2y - 5 = 0 I xU bjhLnfhlhft[k; (6/0) kw;Wk; (-3/0) vd;w g[s;spfs; tHpna bry;yf;ToaJkhd

brt;tf mjpgutisaj;jpd; rkd;ghL fhz;f.

11. xU ePs;tl;lg; ghijapd; Ftpaj;jpy; g{kp ,Uf;FkhW xU Jizf;nfhs;Rw;wp tUfpwJ. ,jd;ikaj;

bjhiyt[j;jft[ Mft[k; g{kpf;Fk; Jizf; nfhSf;Fk; ,ilg;gl;l kPr;rpW J]uk; 400 fpnyh kPl;lh;

Mft[k; ,Uf;Fkhdhy; g{kpf;Fk; Jizf; nfhSf;Fk; ,ilg;gl;l mjpf gl;r J]uk; vd;d?

12. xU tist[miu ePs;tl;l totj;jpy; cs;sJ. mjd; mfyk; 48 mo cauk;20 mo jiuapypUe;J 10 mo

cauj;jpy;tistpd; mfyk; vd;d?

13. xU ghyj;jpd; tisthdJ miuePs; tl;lj;jpd; totpy; cs;sJ. fpil kl;lj;jpy; mjd; mfyk; 40

moahft[k; ikaj;jpypUe;J mjd; cauk; 16 moahft[k; cs;sJ vdpy; ikaj;jpypUe;J tyJ my;yJ

,lg;g[wj;jpy; 9 mo J]uj;jpy; cs;s jiug;g[s;spapypUe;J ghyj;jpd; cauk; vd;d?

14. xU thy; tpz;kPd; (comet) MdJ Nhpaidr; (sun) Rw;wp gutisa ghijapy; bry;fpwJ. kw;Wk; Nhpad;

gutisaj;jpd; Ftpaj;jpy; mikfpwJ. thy; tpz;kPd; NhpadpypUe;J 80 kpy;ypad; fp.kP bjhiytpy;

mike;J ,Uf;Fk; nghJ thy; tpz;kPida[k; Nhpaida[k; ,izf;Fk; nfhL ghijapd; mr;Rld;3

nfhzj;jpid Vw;gLj;Jkhdhy;(1) thy; tpz;kPdpd; ghijapd;rkd;ghl;ilf; fhz;f. (2) thy;tpz;kPd;

NhpaDf;F vt;tst[mUfpy; tuKoa[k;vd;gija[k;fhz;f. (ghij tyJg[wk; jpwg;g[ilajhf bfhs;f.)

15. mjpgutisaj;jpd; ikak; (2/4) nkYk; (2/0) tHpna bry;fpwJ ,jd; bjhiyj; bjhLnfhLfs;

x + 2y - 12 = 0 kw;Wk;x - 2y + 8 = 0, Mfpatw;wpw;F ,izahf ,Uf;fpd;wd vdpy; mjpgutisaj;jpd;

rkd;ghL fhz;f.

16. x2 - 6x - 12y - 3 = 0 vd;w gutisaj;jpd; mr;R/ Kid/ Ftpak;/ ,af;Ftiuapd;rkd;ghL brt;tfyj;jpd;

rkd;ghL/ brt;tfyj;jpd; ePsk; Mfpatw;iwf; fhz;f. nkYk; mjd; tistiuia tiuf.

* * * ** * * ** * * ** * * ** * * *

5. tifEz;fzpjk; - gad;ghLfs; -5. tifEz;fzpjk; - gad;ghLfs; -5. tifEz;fzpjk; - gad;ghLfs; -5. tifEz;fzpjk; - gad;ghLfs; -5. tifEz;fzpjk; - gad;ghLfs; - I

6 kjpg;bgz} tpdhf;fs; :kjpg;bgz} tpdhf;fs; :kjpg;bgz} tpdhf;fs; :kjpg;bgz} tpdhf;fs; :kjpg;bgz} tpdhf;fs; :

1. loge(1 + x) vd;w rhh;gpw;F bkf;yhhpd; tphpt[fhz;f.

2. tan x, -/2 < x < /2 vd;w rhh;gpw;F bkf;yhhpd; tphpt[ fhz;f.

3. gpd;tUtdtw;wpw;F nuhypd; njw;wj;ijr; rupghu;f;f : f(x) = exsinx o

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7. kjpg;g[ fhz;f : ( ) x1

0xcosecxlim

.

8. f(x) = 2x3+ x2 - 20x vd;w rhu;gpd;VWk; kw;Wk; ,wFk; ,ilbtspfisf; fhz;f.

9. ( )20,x tpy; sinx

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11. y = 12x2 - 2x3 - x4 vd;w rhu;g[ve;j ,ilbtspfspy; FHpt[ milfpd;wd vd;gija[k; kw;Wk; tist[

khw;Wg; g[s;spfisa[k; fhz;f.

* * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * *

6. tif Ez;fzpjk; gad;ghLfs; -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


1. tifaPLfisg; gad;gLj;jp njhuha kjpg;g[fis fzf;fpLf. 1.36

2. w = exy, x = t2, y=t3vd;w rhh;gpw;F r';fpyp tpjpiag; gad;gLj;jp =dt

dw- I fzf;fpLf.

3. x = r cos , y = r sin vd;W ,Uf;FkhW w = log (x2+y2)vd tiuaWf;fg;gLfpwJ. vdpy;

r

w

kw;Wk; If;

w

fhz;f.

4. x = u + v , y = u - v vd;W ,Uf;FkhW w = sin-1(xy)vd tiuaWf;fg;gLfpwJ vdpy;/u

w

kw;Wk;

v

w

If;

fhz;f.

5. u vd;gJ x, y,y; -nMk; go rkg;goj;jhd rhh;ghapd;

x

un

yx

uy

x

ux

=

+

)1(

2

2

2 vd epWt[f.

6. u = log (tan x + tan y + tan z) vdpy;sin2xx

u

= 2 vd ep%gp.


1. = u2evvd;w rhh;gpy; u = x/y,v = y log x vdpy;

.

y

w

x

w

Wk;kw;

fhz;f.

2. (i) u =

yx

yx1

sin vdpy;a{yhpd;njw;wj;ijg; gad;gLj;jp .tan21

uy

u

yx

u

x =

+

vdf; fhl;Lf.

(ii) u = tan-1

+

yx

yx33

vdpy; a{yhpd; njw;wj;ijg; gad;gLj;jp uy

uy

x

ux 2sin=

+

vd epWt[f.

3. f(x,y) =

22

1

yx +

f;F a{yhpd;njw;wj;ij rhpghh;f;f.

4. u = sin 3x cos 4y vd;w rhh;gpw;Fxy

y

yx

y

=

22vd;gij rhpghh;f;f.

5. tifaPLfisg; gad;gLj;jp y = 43 02.102.1 + d;njhuha kjpg;gpidf; fzf;fpLf.



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6. tistiuia tiuf.(i) y = x3, (ii) y2=2x3(iii) y = x3+1.

* * * * ** * * * ** * * * ** * * * ** * * * *

. 7. bjhif Ez;fzpjk; gad;ghLfs;7. bjhif Ez;fzpjk; gad;ghLfs;7. bjhif Ez;fzpjk; gad;ghLfs;7. bjhif Ez;fzpjk; gad;ghLfs;7. bjhif Ez;fzpjk; gad;ghLfs;

6 kjpg;bgz; tpdhf;fs}:-6 kjpg;bgz; tpdhf;fs}:-6 kjpg;bgz; tpdhf;fs}:-6 kjpg;bgz; tpdhf;fs}:-6 kjpg;bgz; tpdhf;fs}:-

1. kjpg;gpLf : 2

0

)log(tan

dxx .

2. kjpg;gpLf :(i) +3

6tan1

x

dx(ii) +

3

6cot1

x

dx.

3. ,uz;lhk; mog;gil j; njw;wj;ijg; gad;gLj;jp kjpg;gpLf. xdxscox 2sin2

0

4. kjpg;gpLf. dxxx n)1(

1

0

5. kjpg;gpLf. dxx)(1x10

1

0

6. kjpg;gpLf.

1

0

4 dxxe x

7. 3x - 5y -15=0 vd;w nfhL kw;Wk; x = 1, x = 4 vDk; nfhLfs; kw;Wk; x mr;R Mfpatw;why; Vw;gLk;

mufj;jpd; gug;igf; fhz;f.

8. y = 2x + 4 vd;w nfhL y = 1, y = 3 vd;w nfhLfs;kw;Wk; y - mr;R Mfpatw;why; milgLk; mufj;jpd;

gug;gpidf; fhz;f.

9. y = x vd;w tistiuapy;y = 2, x = 0 Mfpa nfhLfSf;F ,ilg;gl;l gug;gpid y-mr;irg; bghWj;Jr;

RHw;wpdhy; cUthFk; jplg;bghUspd;fd mstpidf;fhz;f.

10. a vd;w MuKs;s tl;lj;jpd; gug;gpidf;fhz;f.

11. y = 1 + x2, x = 1, x = 2, y = 0, x- mr;irg; bghWj;J tistiufs; kw;Wk; nfhLfshy;NHg;gl;l gug;g[r;

RHw;rpapdhy; Vw;gLj;Jk; jplg;bghUspd; fdmsitf; fhz;f.

12. 2ay2= x (x - a)2, a>0, x-mr;irg;bghWj;J tistiufs; kw;Wk; nfhLfshy;NHg;gl;l gug;g[r; RHw;rpapdhy;

Vw;gLj;Jk; jplg;bghUspd; fdmsitf; fhz;f.


1. y = x + 1 vd;w nfhl;ow;Fk; y = x2 - 1 vd;w tistiuf;Fk; ,ilna cs;s mufj;jpd; gug;igf; fhz;f.



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2. 12

2

2

2

=+b

y

a

xvd;w ePs; tl;lj;jpdhy; cUthFk; mufj;jpd; gug;igf;fhz;f.

3. 3ay2 = x (x-a)2 vd;w tistiuapd; fz;zpapd; gug;igf; fhz;f.

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

5. Muk; r myFfs;cs;s nfhsj;jpd; ikaj;jpypUe;J a kw;Wk; b myFfs; bjhiytpy; mike;j ,U

,izahd jsfs;nfhsj;ij btl;Lk;nghJ ,ilg;gLk; gFjpapd; tisgug;g[2 r (b-a) vd epWt[f.

,jpypUe;J nfhsj;jpd; tisgug;ig tUtp (b>a).

6. .13

23

2

=

+

a

y

a

x vd;w tistiuapd; tpy;ypd; ePsk; fhz;f.

7. Muk;r Fj;Jauk; h cila Tk;gpd; fd msitf;fhz;f.

****

8. tiff;bfGr; rkd;ghLfs;.8. tiff;bfGr; rkd;ghLfs;.8. tiff;bfGr; rkd;ghLfs;.8. tiff;bfGr; rkd;ghLfs;.8. tiff;bfGr; rkd;ghLfs;.


1. jPu;f;f : xyyxdx

dy+++=1

2. jPu;tpd;nghJ fpilf;Fk; tistiuahdJ Mjp tHpahfr; bry;fpwJ vdpy;

(x2-y) dx + (y2- x)dy = 0 itj; jPu;f;f.

3. )( yxSindx

dy += iaj;jPu;f;f.

4. jPu;tpd; tistiuahdJ y -mr;rpid btl;of; bfhs;Skhdhy; ydx+xdy=e-xydxIj; jPu;f;f

5. jPu;f;f xdy - ydx = dxyx 22 +

6. (x, y) vd;w g[s;spapy; rha;t[x

y+1 vdf; bfhz;L (1/ 0) vd;w g[s;sp tHpahfr;bry;yf; Toa tistiuapd;

rkd;ghl;ilf; fhz;f.

7. jPu;f;f : 2adxdy

xy = )( .

8. jPu;f;f : (D2 + 9) y = Sin3x

9. jPu;f;f : (D2 - 3D+2) y = x

10. (D2 + 1)y = 0,,F x = 0 vdpy; y = 2,2

=x vdpy; y = - 2itj; jPu;f;f.

11. jPu;f;f : (D2 - 13D + 12) y = e-2x + 5ex

12.jPh;f;f :

x

y

x

y

dx

dytan+= .



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13. jPh;f;f : .cos2cot xxydx

dy=+


1. jPh;f;f : (i) (D2 + 7D + 12)y = e2x(ii) (D2 - 1)y = cos2x - 2sin 2x.

2. nuoak; (Radium) rpija[k; khWtPkhdJ mjpy; fhzg;gLk; mstpw;F tpfpjkhf mike;Js;sJ. 50

tUlfspy; Muk;g mstpypUe;J 5 rjtPjk; rpije;jpUf;fpwJ. vdpy; 100 tUl Kotpy; kPjpapUf;Fk;

mst[ vd;d? (A0I Muk;g mst[ vdf; bfhs;f).

3. xU efuj;jpy; cs;s kf;fs; bjhifapd; tsh;r;rp tPjk; me;neuj;jpy; cs;s kf;fs; bjhiff;F

tpfpjkhf mike;Js;sJ. 1960 Mk; Mz;oy; kf;fs; bjhif 1/30/000 vdt[k; 1990y; kf;fs; bjhif

1/60/000 Mft[k; ,Ug;gpd; 2020 Mk;Mz;oy; kf;fs; bjhif vt;tsthf ,Uf;Fk; ?

(log ( )1316 = 0.2070. e0.42 = 1.52).

4. %. 1000 vd;w bjhiff;F bjhlh; Tl;L tl;o fzf;fplg;gLfpwJ. tl;o tPjk; Mz;blhd;Wf;F 4

rjtPjkhf ,Ug;gpd; mj;bjhif vj;jid Mz;Lfspy; Muk;gj; bjhifapid nghy; ,U klfhFk;.

5. btg;gepiy 15c cs;s xU miwapy; itf;fg;gl;Ls;s njePhpd; btg;gepiy 100c MFk;. mJ 5

epkplfspy; 60CMf Fiwe;J tpLfpwJ. nkYk; 5 epkplk; fHpj;J njePhpd; btg;g epiyapidf; fhz;f.

6. jPu;;f;f : (x + y)2 1=dx

dy.

7. x = 0 Mf ,Uf;Fk;nghJ y = 1 vd ,Uf;Fkhdhy; (1+ex/y) dx + ex/y (1-x/y) dy = 0 vd;w rkd;ghl;od;

jPu;t[ fhz;f.

8. jPu;f;f : (x2 +y2) dx + 3xydy = 0

9. dx + xdy = e-y sec2ydy ia jPu;f;f.

* * * * ** * * * ** * * * ** * * * ** * * * *

9. jdp epiyf; fzf;fpay;9. jdp epiyf; fzf;fpay;9. jdp epiyf; fzf;fpay;9. jdp epiyf; fzf;fpay;9. jdp epiyf; fzf;fpay;6 kjpg;bgz; tpdhf;fs;:-6 kjpg;bgz; tpdhf;fs;:-6 kjpg;bgz; tpdhf;fs;:-6 kjpg;bgz; tpdhf;fs;:-6 kjpg;bgz; tpdhf;fs;:-

1. ((~p) v(~q)) v p bka;ik vdf; fhl;Lf.

2. p q kw;Wk;q p Mfpait rkhd kw;wit vdf; fhl;Lf.

3. Fyj;jpd;ePf;fy;tpjpfis vGjp ep%gp.

4. gpd;tUk; Tw;Wf;F bka; ml;ltizia mikf;f

i) (pvq) n(~q) ii) (p^q) v [~(p^q)] iii) ~(pvq) (~p)^(-q)] vdf; fhl;Lf.

iv) [(~p)vq) v (p^(-q) xU bka;ikah vd nrhjpf;f.



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5. ~(p^q) d[(~p) v (~q)] vdf; fhl;Lf.

6. (z, +) xU Kotw;w vgPypad;Fyk; vd epWt[f.

7. 1 ,d; 3Mk; go _yfs; xU Kothd vgPypad; Fyj;ijg; bgUf;fypd; fPH; mikf;Fk; vdf; fhl;Lf.

8. (C, +) MdJ xU Kotw;w vgPypad;Fyk; vd epWt[f.

9. (R- {0}.) Kot[w;w vgPypad;Fyk; vdf; fhl;Lf.


1. G vd;gJ kpif tpfpjKW vz; fzk; vd;f. a * b3

ab= vDkhW tiuaWf;fg;gl;l brayp * d; fPH; G

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

2. -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 brayp * ,d; fPH; xU vgPypad;Fyj;ij mikf;Fk; vdf;fhl;Lf.

3. g{r;rpakw;w fyg;bgz;fspd; fzkhd C-{0} y; tiuaWf;fg;gl;l f1(z) = z, f

2(z) = -z, f

3(z) =

Z1 ,

f4(z) = -1/z v z c - { 0 } vd;w rhh;g[fs; aht[k; mlfpa fzk; {f1, f2, f3, f4} MdJ rhh;g[fspd; nrh;g;gpd;

fPH; xU vgPypad; Fyk; mikf;Fk; epWt[f.

4. (Z7- {[0]}, .7) xU Fyj;ij mikf;Fk;vdf; fhl;Lf.

5. 11-,d; kl;Lf;F fhzg;bgw;w bgUf;fypd; fPH; {[1], [3], [4], [5], [9]} vd;w fzk; xU vgPypad;

Fyj;ij mikf;Fk; vdf; fhl;Lf.

6.

xx

xx, x R - { 0 } vd;w mikg;gpy; cs;s mzpfs; aht[k; mlfpa fzk; G MdJ mzp bgUf;fypd;

fPH; xU Fyk; vdf; fhl;Lf.

7. G = {a+b2 / a, b, q) vd;gJ Tl;liyg; bghWj;J xU Kotw;w vgPypad; Fyk; vdf; fhl;Lf.

8. |z|=1 vDkhW cs;s fyg;bgz;fs; aht[k; mlfpa fzk; M MdJ mzpg; bgUf;fypd; fPH; xU Fyk;

vdf; fhl;Lf.

9. G = {2n/nz} vd;w fdkhdJ bgUf;fypd; fPH; xU vgPypad;Fyj;ijf; mikf;Fk; vdf; fhl;Lf.

* * * * ** * * * ** * * * ** * * * ** * * * *

10. epfH;jft[g; guty;fy;10. epfH;jft[g; guty;fy;10. epfH;jft[g; guty;fy;10. epfH;jft[g; guty;fy;10. epfH;jft[g; guty;fy;

6 kjpg;bgz; tpdhf;fs;:-kjpg;bgz; tpdhf;fs;:-kjpg;bgz; tpdhf;fs;:-kjpg;bgz; tpdhf;fs;:-kjpg;bgz; tpdhf;fs;:-

1. bjhlh;r;rpahd rktha;g;g[ khwp x d; epfH;j;jft[mlh;j;jpr; rhhh;g[ f(x)

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2. xU gha;rhd; gutypy;P(x = 2) = P(x = 3) vdpy;P(x=5) I fhz;f. (e-3= 0.050)

3. ,U gfilfs; tPRk;nghJ 3fs; fpilg ;gjw;fhd epfH;jft[ epiwr ; rhu ;g [ kw;Wk; nru ;g;g[g ; guty;

rhu;g[ fhz;f.

4. xU bjhlu; rktha;g;g[ khwp x d; epfH;jft[ mlu;j;jp rhu;g[f(x) = =

fpYk;bwkw;0)10,3

2

xx

i) p(xa) ii) p(x>6)=0.05 vdpy; a, b iaf; fhz;f.

5. _d;W gfilfis xU Kiw tPRk;bghGJ 6fs; fpilg;gjw;fhd epfH;jft[g; gutiyf; fhz;f.

6. ,U rPuhd gfilfs; cUl;lg;gLfpd;wd. gfilfspd; nky; cs;s vz;fspd; Tl;Lj; bjhifapd;

vjpu;g;ghu;g;gpidf; fhz;f.


1. xU efuj;jpy; thlif tz;o Xl;Ldh;fshy; Vw;gLk; tpgj;Jf;fspd; vz;zpf;if gha;!hd; gutiy

xj;jpUf;fpwJ. ,jd; gz;gsit 3 vdpy; 1000 Xl;Leh;fspy; (1) xU tUlj;jpy; xU tpgj;Jk; Vw;glhky;

(2) xU tUlj;jpy; 3 tpgj;JfSf;F nky; Vw;glhky; ,Uf;Fk;goahd Xl;Ldh;fspd; vz;zpf;ifiaf;

fhz;f. (e-3= 0.0498).

2. ,ay;epiyg; gutypd; efH;jft[ mlu;j;jpr; rhu;g[ f(x) = Ke-2x2+4x, - < x

=

525

1

5,0

2 x

x

x

vdpy; 5 taJila eha; i) 10 Mz;LfSf;Ff; Fiwthf

ii) 8 Mz;LfSf;Ff; nkyhf iii) 12ypUe;J 15 Mz;Lfs; tiu capu; thH;tjw;fhd epfH;jft[fhz;f.

****


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12th Maths