1

Ts°d Lp®j Õû\

10 Bm YÏl×

fzpjk;jkpo; top

gapw;rpf; fl;lfk;

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

fzpjk;fzpjk;fzpjk;fzpjk;fzpjk;ghlk; 1 : 2 kjpg;ngz;fs;ghlk; 1 : 2 kjpg;ngz;fs;ghlk; 1 : 2 kjpg;ngz;fs;ghlk; 1 : 2 kjpg;ngz;fs;ghlk; 1 : 2 kjpg;ngz;fs;

1. A = {4,6,7,8,9}, B = {2,4,6} kw;Wk; C = {1,2,3,4,5,6} vdpy; A∪(B∩C) fhz;f.

2. A, B vd;gd ,U fzq;fs; kw;Wk; U midj;Jf; fzk;. NkYk; n(u) = 700, n(A) = 200, n(B) = 300kw;Wk; n(A∩B) = 100 vdpy; n(Α1∩Β1) If; fhz;f.

3. n(A) = 285, n(B) = 195, n(u) = 500, kw;Wk; n(A∪B) = 410 vdpy; n(Α1∪Β1) If; fhz;f.

4. fPNo nfhLf;fg;gl;Ls;s mk;Gf;Fwpg; glq;fs; xt;nthd;Wk; xU rhh;gpidf; Fwpf;fpd;wdth vd

Muha;f.

5. x = {1,2,3,4} vd;f. gpd;tUk; xt;nthU cwTk;> x ypUe;J x f;F xU rhh;ghFkh vd Muha;f.

f = {(2,3)(1,4)(2,1)(3,2)(4,4)}6. A = {1,2,3,4,5}, B = N kw;Wk; f : A →B MdJ f(x) = x2 vd tiuaWf;fg;gl;Ls;sJ. f d; tPr;rfj;ij

fhz;f. NkYk; rhh;gpd; tifiaf; fhz;f.

7. f(x) = 2x - 1 vdpy; 2f(-1) - f(3) kjpg;G fhz;f.

5 kjpg;ngz;fs;5 kjpg;ngz;fs;5 kjpg;ngz;fs;5 kjpg;ngz;fs;5 kjpg;ngz;fs;

1. ntd;glq;fisg; gad;gLj;jp gpd;tUtdtw;iwr; rhpah vdr; Nrhjpj;Jg; ghHf;fTk;.

(i) A∪(B∩C)= (A∪Β)∩(Α∪∪∪∪∪C), (ii) A∩(B∪C)=(A∩B)∪(A∩C), (iii) (A∪B)1= A1∩B1,

(iv) A\(B∪C) = (A\B)∩ (A\C)2. A = {0,1,2,3} kw;Wk; B = {1,3,5,7,9} vd;gd ,U fzq;fs; vd;f. f : A→B vd;Dk; rhh;G f(x)= 2x+1

vdf; nfhLf;fg;gl;Ls;sJ. ,r;rhh;gpid (i) thpirr; Nrhbfspd; fzk;> (ii) ml;ltiz> (iii)mk;Gf;Fwpg; glk;> (iv) tiuglk; Mfpatw;iwf; Fwpf;f.

3. rhh;G f : (1, 6) → R MdJ gpd;tUkhW tiuaWf;fg;gl;Ls;sJ.

f(x) = ⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

<≤<≤<≤

−+

6x4x22x1

10- 3x

xx1

2 412 [1,6] = {x < R : 1 < x < 6 } (a) f(5), f(3),f(1) (b) f(2)-f(4) (c)2f (5)-3f(1)

4. rhh;G f : [3,7]→R MdJ gpd;tUkhW tiuaWf;fg;gl;Ls;sJ. f(x) =⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

<≤<≤−<≤

−−

6x44x22x3-

3-2x23x14x2

gpd;tUtdtw;iwf; fhz;f. : (i) f(5) + f(6), (ii) f(1)-f(3), (iii) f(-2)-f(-4) (iv)f(1)2f(6)

1)f(f(3)−−+

.

----------

a

b

c

d

x

y

z

26

2. vz;zpay;vz;zpay;vz;zpay;vz;zpay;vz;zpay;2 kjpg;ngz; tpdhf;fs;.2 kjpg;ngz; tpdhf;fs;.2 kjpg;ngz; tpdhf;fs;.2 kjpg;ngz; tpdhf;fs;.2 kjpg;ngz; tpdhf;fs;.

1. n tJ cWg;G an =

,2n

a 1n

+−

n > 1 ∀

Nn∈

.

2. 20,19¼,18½....vd;w $l;Lj;njhlh; thpirapy; cWg;G tn xU Fiw vz;zhf mika n d; kpfr;

rpwpa kpif KO kjpg;G ahJ?.

3. 125,120,115,110......... vd;w $l;Lj; njhlh; thpirapd; nghJ tpj;jpahrj;ijAk;> cWg;igAk; fhz;f.

4. 24, 23¼, 22½, 21¾ vd;w $l;Lnjhlh; thpirapy; 3 vd;gJ vj;jidahtJ cWg;G MFk;?

5. 4,9,14...... vd;w $l;Lnjhlh; thpirapd; 17tJ cWg;igf; fhz;f.

6. 2/5,6/25,18/125........ vd;w ngUf;Fj; njhlh; thpirapd; nghJ tpfpjk; kw;Wk; nghJ cWg;G fhz;f.

7. 4,-2,-1,-½.... vd;w ngUf;Fj; njhlh; thpirapd; nghJ tpfpjk; fhz;f.

8. 1/4,-1/2,1,-2.... vd;w ngUf;Fj; njhlh; thpirapy; 10tJ cWg;G kw;Wk; nghJ tpfpjk; fhz;f.

9. 1,2,4,8.... vd;w ngUf;Fj; njhlhpy; 1024 vd;w cWg;G vj;jidahtJ cWg;G vdf; fhz;f.

10. a = 5, n = 30, l = 121, vd;w ngUf;Fj; njhlhpy; $Ljy; Sn fhz;f.

11. a = 5, r = 3, n = 12 vd;w ngUf;Fj; njhlhpy; $Ljy; Sn fhz;f.

12. 1 + 3 + 5 + ..... + 25 cWg;Gfs; tiu $Ljy; fhz;f.

13. 12 + 22 + 32 + ..... + 252 $Ljy; fhz;f.

14. 1 + 2 + 3 + ....... + 45 $Ljy; fhz;f.

15. 1 + 2 + 3 + ..... + P = 171, vdpy; 13+23+33+....p3 = ?

Ie;J kjpg;ngz; tpdhf;fs;.Ie;J kjpg;ngz; tpdhf;fs;.Ie;J kjpg;ngz; tpdhf;fs;.Ie;J kjpg;ngz; tpdhf;fs;.Ie;J kjpg;ngz; tpdhf;fs;.

1. 6 + 66 + 666 + ..... vd;w njhlh; thpirapd; n cWg;Gfs; tiu $Ljy; fhz;f.

2. xU $l;Lj; njhlh; thpirapy; mLj;jLj;j %d;W cWg;Gfspd; $Ljy; 6 kw;Wk; mtw;wpd; ngUf;Fj;

njhif -120 vdpy;> mk;%d;W vz;fisf; fhz;f.

3. 300f;Fk; 500f;Fk; ,ilNa 11 My; tFgLk; midj;J ,ay; vz;fspd; $Ljy; fhz;f.

4. G.P. y; 4 tJ cWg;G kw;Wk; 7 tJ cWg;Gfs; KiwNa 54 kw;Wk; 1458 vdpy; mj;njhlh; thpiriaf;

fhz;f.

5. 12 - 22 + 32 - 42 + ...... vd;w njhlhpd; Kjy; 2n cWg;Gfspd; $Ljy; fhz;f.

6. xU $l;Lj; njhlh; thpirapy; 10 kw;Wk; 18tJ cWg;Gfs; KiwNa 41 kw;Wk; 73 vdpy; 27tJ

cWg;igf; fhz;f.

7. 113+123+133+....+283 = ?

8. 12. nr.kP> 13 nr.kP .... 23 nr.kP gf;f msTfSs;s rJuq;fspd; gug;Gfspd; $Ljy; fhz;f.

9. 8 My; tFgLk; midj;J %d;wpyf;f ,ay; vz;fspd; $Ljy; fhz;f.

10. a,b,c,d, vd;gd xU ngUf;Fj; njhlh; thpirapy; cs;sd vdpy; (b-c)2 + (c-a)2 + (d-b)2 = (a-d)2 vd

epWTf.

- - - - - - - - - -- - - - - - - - - -- - - - - - - - - -- - - - - - - - - -- - - - - - - - - -

ghlk; 3 - ,aw;fzpjk;ghlk; 3 - ,aw;fzpjk;ghlk; 3 - ,aw;fzpjk;ghlk; 3 - ,aw;fzpjk;ghlk; 3 - ,aw;fzpjk;I gpd;tUk; gy;YWg;Gf; Nfhitfis fhuzpg;gLj;Jf:-gpd;tUk; gy;YWg;Gf; Nfhitfis fhuzpg;gLj;Jf:-gpd;tUk; gy;YWg;Gf; Nfhitfis fhuzpg;gLj;Jf:-gpd;tUk; gy;YWg;Gf; Nfhitfis fhuzpg;gLj;Jf:-gpd;tUk; gy;YWg;Gf; Nfhitfis fhuzpg;gLj;Jf:-

1. x3-2x2-5x+6 vDk; gy;YWg;Gf; Nfhitia xUgbf; fhuzpfshf fhuzpg;gLj;Jf.

2. 4x3- 7x + 3 vd;Dk; gy;YWg;Gf; Nfhitia xU gbf; fhuzpfshf fhuzpg;gLj;Jf.

27

3. x3 - 23x2 + 142x - 120 vd;Dk; gy;YWg;Gf; Nfhitia xU gbf; fhuzpfshf fhuzpg;gLj;Jf.

4. 4x3 - 5x2 + 7x - 6 vd;Dk; gy;YWg;Gf; Nfhitia xU gbf; fhuzpfshf fhuzpg;gLj;Jf.

5. x3 - 7x + 6 vd;Dk; gy;YWg;Gf; Nfhitia xU gbf; fhuzpfshf fhuzpg;gLj;Jf.

6. x3 + 13x2 + 32x + 20 vd;Dk; gy;YWg;Gf; Nfhitia xU gbf; fhuzpfshf fhuzpg;gLj;Jf.

7. 2x3 - 9x2 + 7x + 6 vd;Dk; gy;YWg;Gf; Nfhitia xU gbf; fhuzpfshf fhuzpg;gLj;Jf.

8. x3 - 5x + 4 vd;Dk; gy;YWg;Gf; Nfhitia xU gbf; fhuzpfshf fhuzpg;gLj;Jf.

9. x3 - 10x2 - x + 10 vd;Dk; gy;YWg;Gf; Nfhitia xU gbf; fhuzpfshf fhuzpg;gLj;Jf.

10. 2x3 + 11x2 -7x - 6 vd;Dk; gy;YWg;Gf; Nfhitia xU gbf; fhuzpfshf fhuzpg;gLj;Jf.

11. x3 + x2 + x - 14 vd;Dk; gy;YWg;Gf; Nfhitia xU gbf; fhuzpfshf fhuzpg;gLj;Jf.

12. x3 - 5x2 - 2x + 24 vd;Dk; gy;YWg;Gf; Nfhitia xU gbf; fhuzpfshf fhuzpg;gLj;Jf.

II gpd;tUk; gy;YWg;Gf; Nfhitfspd; th;f;f %yj;ij tFj;jy; Kiw%yk; fhz;f.gpd;tUk; gy;YWg;Gf; Nfhitfspd; th;f;f %yj;ij tFj;jy; Kiw%yk; fhz;f.gpd;tUk; gy;YWg;Gf; Nfhitfspd; th;f;f %yj;ij tFj;jy; Kiw%yk; fhz;f.gpd;tUk; gy;YWg;Gf; Nfhitfspd; th;f;f %yj;ij tFj;jy; Kiw%yk; fhz;f.gpd;tUk; gy;YWg;Gf; Nfhitfspd; th;f;f %yj;ij tFj;jy; Kiw%yk; fhz;f.

1. x4 - 4x3 + 10x2 - 12x + 92. 4x4 + 8x3 + 8x2 + 4x + 13. 9x4 - 6x3 + 7x2 + 14. 4 + 25x2 -12x - 24x3 +16x4

III gpd;tUk; gy;YWg;Gf; Nfhitfs; KO th;f;fq;fs; vdpy; gpd;tUk; gy;YWg;Gf; Nfhitfs; KO th;f;fq;fs; vdpy; gpd;tUk; gy;YWg;Gf; Nfhitfs; KO th;f;fq;fs; vdpy; gpd;tUk; gy;YWg;Gf; Nfhitfs; KO th;f;fq;fs; vdpy; gpd;tUk; gy;YWg;Gf; Nfhitfs; KO th;f;fq;fs; vdpy; a, b Mfptw;wpd; kjpg;Gfisf;Mfptw;wpd; kjpg;Gfisf;Mfptw;wpd; kjpg;Gfisf;Mfptw;wpd; kjpg;Gfisf;Mfptw;wpd; kjpg;Gfisf;

fhz;f.fhz;f.fhz;f.fhz;f.fhz;f.

1. 4x4 - 12x3 + 37x2 + ax+b2. x4 - 4x3 + 10x2 - ax + b3. ax4 + bx3 +109x2 - 60x +364. ax4 - bx3 + 40x2 +24x + 36

IV1. 3x2 - 5x + 2 = 0 vd;w rkd;ghl;bd; %yq;fs; α, β vdpy; gpd;tUtdtw;wpd; kjpg;Gfisf; fhz;f.

(i)

αβ

βα+

(ii) α² - β² (ii) 22

αβ

βα

+

2. 3x2 - 6x + 1 = 0 vd;w rkd;ghl;bd; %yq;fs; a, b vdpy;> fPo;fhZk; %yq;fisf; nfhz;l

rkd;ghLfis mikf;f.

(a) βα1,1 (b) α-β (c) 2α + β, 2β + a

3. 5x2 - Px +1 = 0 vd;w rkd;ghl;bd; %yq;fs; α kw;Wk; β vd;f. NkYk; α - β = 1 vdpy; P d; kjpg;igf;

fhz;f.

2 kjpg;ngz;fs;:2 kjpg;ngz;fs;:2 kjpg;ngz;fs;:2 kjpg;ngz;fs;:2 kjpg;ngz;fs;:

1. x3 + x2 - 7x - 3 vd;gij x - 3 My; tFf;Fk; NghJ fpilf;Fk; <T kw;Wk; kPjp fhz;f.

2. x3 + x2 - 3x + 5 vd;gij x - 1 My; tFf;Fk; NghJ fpilf;Fk; <T kw;Wk; kPjp fhz;f.

3. 3x3 - 2x2 + 7x - 5 vd;gij x + 3 My; tFf;Fk; NghJ fpilf;Fk; <T kw;Wk; kPjp fhz;f.

4. x3 -6x2 + 11x - 6 vd;gij x - 1 My; tFf;Fk; NghJ fpilf;Fk; <T kw;Wk; kPjp fhz;f.

5. x3 + 6x2 + 11x +6 vd;gij x + 1 My; tFf;Fk; NghJ fpilf;Fk; <T kw;Wk; kPjp fhz;f.

6. RUf;Ff : 287205

++

xx .

7. x2 - 8x - 12 = 0 vd;w rkd;ghl;bd; %yq;fspd; jd;ikia Muha;f.

8. 2x2 - 3x + 4 = 0 vd;w rkd;ghl;bd; %yq;fspd; jd;ikia Muha;f.

28

9. 9x2 + 12x - 4 = 0 vd;w rkd;ghl;bd; %yq;fspd; jd;ikia Muha;f.

10. 2x² - 10x + k = 0 vd;w rkd;ghl;bd; %yq;fspd; nka;naz;fs; kw;Wk; rkkhdit vdpy; k ,d;

kjpg;Gfisf; fz;Lgpb.

11. 12x² - 4kx + 3 = 0 vd;w rkd;ghl;bd; %yq;fspd; nka;naz;fs; kw;Wk; rkkhdit vdpy; k ,d;

kjpg;Gfisf; fz;Lgpb.

12. 7 + √3 kw;Wk; 7 - √3 Mfpatw;iw %yq;fshff; nfhz;l ,Ugbr; rkd;ghL xd;wpid mikf;f.

13. 3 kw;Wk; 4 Mfpatw;iw %yq;fshff; nfhz;l ,Ugbr; rkd;ghL xd;wpid mikf;f.

14. 3 + √7 kw;Wk; 3 - √ 7 Mfpatw;iw %yq;fshff; nfhz;l ,Ugbr; rkd;ghL xd;wpid mikf;f.

15. x² - 6x + 5 = 0 vd;w rkd;ghl;bd; %yq;fspd; $ljy; kw;Wk; ngUf;fw;gyd; Mfpatw;iwf; fhz;f.

16. 3x² - 5x = 0 vd;w rkd;ghl;bd; %yq;fspd; $ljy; kw;Wk; ngUf;fw;gyd; Mfpatw;iwf; fhz;f.

17. 8x² - 25 = 0 vd;w rkd;ghl;bd; %yq;fspd; $Ljy; kw;Wk; ngUf;fw;gyd; Mfpatw;iwf; fhz;f.

V. gpd;tUtdtw;wpd; kP.ngh.t fhz;f.gpd;tUtdtw;wpd; kP.ngh.t fhz;f.gpd;tUtdtw;wpd; kP.ngh.t fhz;f.gpd;tUtdtw;wpd; kP.ngh.t fhz;f.gpd;tUtdtw;wpd; kP.ngh.t fhz;f.

1. 7x² yz4, 21x² y5z³2. 25 bc4 d3, 35b2c5, 45c3 d3. 35 x5y3z4, 49x2yz3, 14 xy2z2

4. c² - d², c(c - d)5. m2-3m-18, m2+5m+6

VI gpd;tUtdtw;wpw;F kP.bgh.k fhz;fgpd;tUtdtw;wpw;F kP.bgh.k fhz;fgpd;tUtdtw;wpw;F kP.bgh.k fhz;fgpd;tUtdtw;wpw;F kP.bgh.k fhz;fgpd;tUtdtw;wpw;F kP.bgh.k fhz;f

i. a2bc, b2ca, c2abii.66 a4b2c3, 44a3b4 c2, 24 a2b3 c4

iii. am+1, am+2, am+3.iv. x²y + xy², x² + xy.

*****

4. mzpmzpmzpmzpmzp2 kjpg;ngz;fs;:-2 kjpg ;ngz;fs;:-2 kjpg ;ngz;fs;:-2 kjpg ;ngz;fs;:-2 kjpg ;ngz;fs;:-

1. aij = jiji

+−

vd;w cWg;Gfisf; nfhz;l 2 x 2 thpirAila A = [aij] iaf; fhz;f.

2. aij =

232 ji −

vd;w cWg;Gfisf; nfhz;l 3 x 2 thpirAila mzp A = [aij] apidf; fhz;f.

3. A = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−

842

906745311

vdpy; i. mzpapd; thpiriaf; fhz;f.

ii. a24 kw;Wk; a32 Mfpa cWg;Gfis vOJf. iii. cWg;G 7 mike;Js;s epiu kw;Wk; epuiyf; fhz;f.

4. A =⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

653542321

, vdpy; (AT) T= A. vd;gjidr; rhpghh;f;f.

5. A = 013

542

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

, vdpy; A apd; epiu epuy; khw;W mzpiaf; fhz;f.

29

6. A = ⎥⎦

⎤⎢⎣

⎡−

−⎥⎦

⎤⎢⎣

⎡− 17

515932

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

7. A = ⎥⎦

⎤⎢⎣

⎡1523

kw;Wk; B = ⎥⎦

⎤⎢⎣

⎡ −3418

vdpy; C = 2A + B vd;w mzpiaf; fhz;f.

8. A = ⎥⎦

⎤⎢⎣

⎡−−

9524

kw;Wk; B = ⎥⎦

⎤⎢⎣

⎡−− 3128 vdpy; 6A - 3B vd;w mzpiaf; fhz;f.

9. ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−+⎥

⎦

⎤⎢⎣

⎡5

1011

32

ba vdpy; a kw;Wk; b Mfpatw;wpd; kjpg;Gfisf; fhz;f.

10. ⎥⎦

⎤⎢⎣

⎡2153

kw;Wk; ⎥⎦

⎤⎢⎣

⎡−

−3152 Mfpad mzpg;ngUf;fiyg; nghUj;J xd;Wf;nfhd;W Neh;khW mzp vd

epWTf.

11. ⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡+−+

20812

640425

zyx

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

12. ⎥⎦

⎤⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡090

00

3321 x

yx

vdpy; x kw;Wk; y d; kjpg;G fhz;f.

5 kjpg;ngz; tpdhf;fs;.5 kjpg;ngz; tpdhf;fs;.5 kjpg;ngz; tpdhf;fs;.5 kjpg;ngz; tpdhf;fs;.5 kjpg;ngz; tpdhf;fs;.

1. A = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

54

2

kw;Wk; [1 3 -6], vd;w mzpfSf;F (AB)T=BTAT vd;gij rhpghh;f;f.

2. A = ⎥⎦

⎤⎢⎣

⎡3725

kw;Wk; B = ⎥⎦

⎤⎢⎣

⎡−

−1112

vdpy; (AB)T BTAT vd;gijr; rhpghh;f;fTk;.

3. A = ⎥⎦

⎤⎢⎣

⎡ −3211

vdpy; A2 - 4A + 5I2 = 0 vd epWTf.

4. A = ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡− 76

524123

B kw;Wk; C = ⎥⎦

⎤⎢⎣

⎡−− 3511

vdpy; A(B + C) = AB + AC vd;gijr; rhpghh;f;fTk;.

5. A = ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡9078

6733

B kw;Wk; C = ⎥⎦

⎤⎢⎣

⎡ −6432

vdpy; (A + B)C = AC + BC vd;gJ rhpahFkh?

6. A = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

210

31

2121-

B kw;Wk; C = [2 1] vdpy; (AB) C = A(BC) vd;gijr; rhpghh;f;f.

7. A = ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡yx

x5735

kw;Wk; C = ⎥⎦

⎤⎢⎣

⎡−−115

kw;Wk; AX = C vdpy; x kw;Wk; y d; kjpg;G fhz;f.

8. jPh;f;f : ⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡492

32

2

yx

yx .

9. jPh;T fhz;f : ⎥⎦

⎤⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡138

5423

yx

10. A = ⎥⎦

⎤⎢⎣

⎡−

−3241

kw;Wk; B = ⎥⎦

⎤⎢⎣

⎡−

−23

61, vdpy; (A + B)2 ≠ A2 + 2AB + B2 vd epWTf.

******

30

5. Maj;njhiy tbtpay;5. Maj;njhiy tbtpay;5. Maj;njhiy tbtpay;5. Maj;njhiy tbtpay;5. Maj;njhiy tbtpay;Kf;fpakhd #j;jpuq;fs;:Kf;fpakhd #j;jpuq;fs;:Kf;fpakhd #j;jpuq;fs;:Kf;fpakhd #j;jpuq;fs;:Kf;fpakhd #j;jpuq;fs;:

1. ,U Gs;spfSf;F ,ilNa cs;s njhiyT

212

212 )()( yyxx −+−

.

2. ,U Gs;spfis ,izf;Fk; Neh;f;Nfhl;Lj; Jz;bid cl;Gwkhf P(x,y) ⎟⎠⎞

⎜⎝⎛

++

++

mlmyly

mlmxlx 1212 , .

3. ntspg;Gwkhf P(x,y) ⎟⎠⎞

⎜⎝⎛

−−

−−

mlmyly

mlmxlx 1212 , .

4. eLg;Gs;sp m(x,y) = ⎟⎠⎞

⎜⎝⎛ ++

2,

22121 yyxx

5. eLf;Nfhl;L ikak; G(x,y)= ⎟⎠⎞

⎜⎝⎛ ++++

33321321 yyy,xxx .

6. Kf;Nfhzj;jpd; gug;gsT :

A = ½ ⎟⎟⎠

⎞⎜⎜⎝

⎛

13

1

21

21

yyxx

yyxx 3

= ½ ( ) ( )( )3231213322 yxyxyxyxyxyx x1 ++−++

7. rha;T :

m = tan θ (Nfhzk;)

m =

12

12

xxyy

−−

(,U Gs;spfs;)

m = nfOd; nfOd;

yx−( rkd;ghL)

8. ,izf;NfhLfs; vdpy; rha;T m1 = m2.9. nrq;Fj;J NfhLfs; vdpy; rha;T m1m2 = -1.Neh; Nfhl;bd; rkd;ghL

10. x mr;R y = 0y mr;R x = 0x mr;Rf;F ,iz y = ky mr;Rf;F ,iz x = kMjpg; Gs;sp topahf y = mxrha;T m ntl;Lj; Jz;L c y = mx + cxU Gs;sp #j;jpuk; y-y1=m(x - x1)

,U Gs;sp #j;jpuk;12

1

yyyy

−−

12

1

xxxx

−−

x ntl;Lj;Jz;L a

y ntl;Lj;Jz;L b

1=−−

bayx

2 kjpg ;ngz;fs;:-2 kjpg ;ngz;fs;:-2 kjpg ;ngz;fs;:-2 kjpg ;ngz;fs;:-2 kjpg ;ngz;fs;:-

31

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

tpil.(1, 2)2. gpd;tUk; Gs;spfis Kidfshff; nfhz;l Kf;Nfhzq;fis eLf;nfhl;L ikaq;fisf; fhz;f.

(3,-5) (-7,4) kw;Wk; (10,-2). tpil : (2, -1)3. xU tl;lj;jpd; ikak; (-6, 4). xU tl;lj;jpd; xU Kid> Mjpg;Gs;sp vdpy; kw;nwhU Kidiaf;

fhz;f. tpil : (-12,-8)4. gphpT R+j;jpuj;ijg; gad;gLj;jp A(1, 0) B(5, 3) C (2, 7) kw;Wk; D(-2, 4) vd;w thpirapy; vLj;Jf;

nfhs;sg;gl;l Gs;spfs; xU ,izfuj;jpd; cr;rpfshFk; vd epWTf..5. (-3,5) kw;Wk; (4,-9) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj; Jz;bid cl;Gwkhf 1:6 vd;w tpfpjj;jpy;

gphpf;Fk; Gs;spapd; mr;Rj; njhiyTfisf; fhz;f. tpil : ( -2,3)6. (3,4) kw;Wk; (-6,2) Mfpa Gs;spfs; ,izf;Fk; Nfhl;Lj;Jz;bid ntspg;Gwkhf 3:2 vd;w tpfpjj;jpy;

gphpf;Fk; Gs;spapd; mr;Rj; njhiyTfisf; fhz;f. tpil : (-24, -2)7. (-4, 0) kw;Wk; B(0,6) vd;w Gs;spfis ,izf;Fk; Nfhl;Lj; Jz;bid ehd;F rkghfq;fshfg;

gphpf;Fk; Gs;spfisf; fhz;f. tpil: (-3,3/2), (-2,3), (-1,9/2).8. gpd;tUk; rha;Tfisf; nfhz;l Neh;f;NfhLfspd; rha;Tf; Nfhzq;fisf; fhz;f.

3

tpil: 60º

9. gpd;tUk; rha;Tf; Nfhzq;fisf; nfhz;L Neh;f;NfhLfspd; rha;Tfisf; fhz;f. 30º tpil : 3

1

10. nfhLf;fg;gl;l Gs;spfs; (3,-2), (7,2). tpil: 111. A(1,2), B(-4,5) kw;Wk; C(0, 1) Mfpad ΔABC d; Kidfs; ,k;Kf;Nfhzj;jpd; xt;nthU

KidapypUe;J ,jd; vjph;g;gf;fj;jpw;F tiuag;gLk; Fj;Jf;NfhLfspd; (altitudes) rha;Tfisf;

fhz;f. tpil: AD=1, BE-d; rha;T -1, CF- d; rha;T 5/3.12. rha;Tf; Nfhzk; 45º kw;Wk; y - ntl;Lj;Jz;L 2/5 Mfpatw;iw nfhz;l Neh;f;Nfhl;bd; rkd;ghl;il

fhz;f. tpil : 5x-5y+2=0.13. (-2, 3) vd;w Gs;sp topr; nry;tJk;> rha;T 1/3 cilaJkhd Neh;f;Nfhl;bd; rkd;ghl;il fhz;f.

tpil : x-3y+11=014. (-1, 1), (2,-4 ) Mfpa Gs;spfspd; topr; nry;Yk; Neh;f;Nfhl;bd; rkd;ghl;ilf; fhz;f.

tpil : 5x+3y+2=015. gpd;tUK; Neh;f;NfhLfspd; rha;Tfisf; fhz;f. 1) y=7x+6. tpil : 716. x+2y+1=0 3x+6y+2=0 Mfpa Neh;f;NfhLfs; ,iz vd epWTf.

17. 3x-5y+7=0,15x+9y+4=0 Mfpa Neh;f;NfhLfs; xd;Wf;F xd;W nrq;Fj;J vd epWTf.18. gpd;tUk; Neh;f;NfhLfspd; rha;Tfisf; fhz;f. 3x+4y-6=0 . tpil : -3/4

5 kjpg;ngz;fs; kjpg;ngz;fs; kjpg;ngz;fs; kjpg;ngz;fs; kjpg;ngz;fs; :1. (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;ig fhz;f. tpil : P=92. (1,2), (-3,4) kw;Wk; (-5,6) Mfpatw;iw Kidfshff; nfhz;l Kf;Nfhzj;jpd; gug;igf; fhz;f.

tpil : 22 sq. units3. gpd;tUtdtw;iw Kidfshff; nfhz;l ehw;fuq;fspd; gug;gsTfisf; fhz;f.(-4,5), (0,7),

(5,-5) kw;Wk; (-4,-2). tpil: 60.5 r.myFfs;

4. A(-2, 3), B(9, 5) Mfpa Gs;spfis ,izf;Fk; Neh;f;NfhL kw;Wk; C(0, 5), D(-2,1) Mfpa Gs;spfis

,izNfhLfs; vdpy;> ‘a’ d; kjpg;ig fhz;f. tpil : a = -1.

32

5. A(2,1), B(-2,3), C(4,5) vd;gd ΔABC d; cr;rpfs;. cr;rp A apypUe;J tiuag;gLk; eLf;Nfhl;bd;(median)

rkd;ghl;ilf; fhz;f. tpil: 3x+y-7=06. A(-6,-5), B(-6,4) vd;gd ,U Gs;spfs; vd;f. Nfhl;Lj; Jz;L AB apd; Nky; AP=

92 AB vd;wthW

mike;Js;s Gs;sp P iaf; fhz;f. tpil:(-6,-3)7. xNu Neh;f;Nfhl;by; mikAk; Gs;spfsh vd Muha;f. (4,3) (1,2) kw;Wk; (-2,1). tpil: xU Nfhliktd.

- - - - - - - - -- - - - - - - - -- - - - - - - - -- - - - - - - - -- - - - - - - - -

6. tbtpay;tbtpay;tbtpay;tbtpay;tbtpay;,U kjpg;ngz;fs;,U kjpg;ngz;fs;,U kjpg;ngz;fs;,U kjpg;ngz;fs;,U kjpg;ngz;fs;:

1. v.fh.1. Δ ABC y; DE || BC kw;Wk; 32

=DBAB . AE = 3.7 nr.kP vdpy; EC If; fhz;f. tpil : 5.55 nr.kP

2. v.fh. 6.5 ΔABC y; A∠ vd;w Nfhzj;jpd; cl;Gw ,U rkntl;b AD MdJ gf;fk; BC I D y;

re;jpf;fpwJ. BD = 2.5 nr.kP., AB = 5 nr.kP. kw;Wk; AC = 4.2 nr.kP. vdpy; DC If; fhz;f.

tpil: 2.1 nr.kP.

3. v.fh. 6.6 ΔABC y;,

A∠

d; ntspg;Gw ,U rkntl;b MdJ BC d; ePl;rpapid E y; re;jpf;fpwJ.

AB = 10 nr.kP., AC = 6 nr.kP. kw;Wk; BC = 12 nr.kP., CE If; fhz;f. tpil: CE =18 nr.kP.

4. gapw;rp 6.1 1(i) D kw;Wk; E Mfpa Gs;spfs; KiwNa ΔABC d; 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 If; fhz;f. tpil: 20 nr.kP.

5. gapw;rp 6.1 (ii) AD=8 nr.kP., AB=12 nr.kP. kw;Wk; AE=12 nr.kP. vdpy; CEIf; fhz;f. tpil: 6 nr.kP.

6. gapw;rp 6.1 (iii) AD=4x-3, BD= 3x-1, AE= 8x-7 kw;Wk; EC=5x-3, vdpy; x d; kjpg;igf; fhz;f. tpil:17. gapw;rp 6.1 AD vd;gJ ΔABC, ∠A d; cl;Gw Nfhz ,U rkntl;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. tpil : 7.5 nr.kP.

8. gapw;rp 6.1 9.(ii) AB=5.6 nr.kP., AC=6 nr.kP. kw;Wk; DC=3 nr.kP. vdpy; BC fhz;f. tpil: 5.8 nr.kP.

9. gapw;rp 6.1 9.(iii) AB= x, AC= x-2, BD= x+2 kw;Wk; DC= x-1 vdpy;> x d; kjpg;igf; fhz;f. tpil: 410. gapw;rp 6.12 xU tl;lj;jpd; Gs;sp A y; tiuag;gLk; njhLNfhL PQ vd;f. AB vd;gJ tl;lj;jpd;

ehz; vd;f. NkYk;

BAC∠

=54º kw;Wk;

BAQ∠

=62º vd;W mikAkhW tl;lj;jpd; Nky;

cs;s Gs;sp C vdpy;

ABC∠

If; fhz;f. tpil: 64º11. v.fh. 6.13 fPo;fhZk; xt;nthU glj;jpYk; x d; kjpg;igf; fhz;f.

i)

tpil : x = 6 tpil : x = 16

12. gapw;rp 6.14 glj;jpy; PA, PB vd;gd O it ikakhf tl;lj;jpw;F ntsp Gs;sp

P apypUe;J tiuag;gl;l njhLNfhLfSf;Fk; CD vd;gJ

tl;lj;jpw;F E vd;Dk; Gs;spapy; tiuag;gl;l njhLNfhL AP=15 nr.kP. vdpy; ΔPCD apd; Rw;wsitf; fz;Lgpb.

tpil : 30 cm

33

13. gapw;rp 6.3 (1) glj;jpy; TP xU njhLNfhL. A, B vd;gd tl;lj;jpd; kPJs;s

Gs;spfs;>

BTP∠

= 72º kw;Wk; ∠ATB = 43° vdpy;

ABT∠

If; fhz;f.

tpil : 65º

14. gapw;rp 6.3 2. xU tl;lj;jpy; AB, CD vd;Dk; ,U ehz;fs; xd;iwnahd;W cl;Gwkhf P apy; ntl;bf;

nfhs;fpd;wd.

(i) CP= 4 nr.kP., AP= 8 nr.kP., PB= 2 nr.kP. vdpy;, PD If; fhz;f. tpil: 4 nr.kP.

15. gapw;rp 6.3 (ii) AP= 12 nr.kP, AB= 15 nr.kP.,CP= PD, vdpy; CD If; fhz;f. tpil: 12 nr.kP.

16. gapw;rp 6.3 AB kw;Wk; CD vd;w ,U ehz;fs; tl;lj;jpw;F ntspNa P vd;Dk; Gs;spapy; ntl;bf;

nfhs;fpd;wd.

(i) AB=4 nr.kP., BP=5 nr.kP. kw;Wk; PD=3 nr.kP. vdpy;, CD If; fhz;f. tpil: 12 nr.kP.

17. gapw;rp 6.3 (ii) BP= 3 nr.kP., CP= 6 nr.kP. kw;Wk; CD= 2 nr.kP. vdpy; AB If; fhz;f. tpil : 5 nr.kP.

Ie;J kjpg;ngz;fs;Ie;J kjpg;ngz;fs;Ie;J kjpg;ngz;fs;Ie;J kjpg;ngz;fs;Ie;J kjpg;ngz;fs;:1. mbg;gil tpfpjrkj; Njw;wk; (my;yJ) Njy;]; Njw;wk;.

2. Nfhz ,U rkntl;bj; Njw;wk;.3. gpjhfu]; Njw;wk;.

*****

7.Kf;Nfhztpay;Kf;Nfhztpay;Kf;Nfhztpay;Kf;Nfhztpay;Kf;Nfhztpay;2 kjpg;ngz; tpdhf;fs;:-2 kjpg;ngz; tpdhf;fs;:-2 kjpg;ngz; tpdhf;fs;:-2 kjpg;ngz; tpdhf;fs;:-2 kjpg;ngz; tpdhf;fs;:-

1. gpd;tUk; Kw;nwhUikia epWTf.

θθ 22 eccossec +

= tan θ + cot θ.

2. 30 kP ePsKs;s xU fk;gj;jpd; epoypd; ePsk; 310 vdpy; #hpadpd; Vw;Wf; Nfhzj;jpd;(jiukl;lj;jpy; ,Ue;J Vw;Wf; Nfhzk;;) mstpidf; fhz;f.

tpil : jiukl;lj;jpy; ,Ue;J #hpadpd; Vw;Wf; Nfhzk; = 60º

3. 1=+θθ

θθ

sincos

eccossin vd;w Kw;nwhUikia epWTf.

4. [ cosec (90º- θ) - sin (90º- θ)] [cosec θ - sin θ ] [tan θ + cot θ ] = 1 vd;w Kw;nwhUikia epWTf.5. (sin 6 θ+ cos6 θ) =1-3 sin2 θ cos2 θ vd;w Kw;nwhUikia epWTf.

6. θθθθθ tan

coscossinsin

=−

−3

3

22

vd;w Kw;nwhUikia epWTf.

7. Rthpy; rha;j;J itf;fg;gl;l xU VzpahdJ jiuAld; 60º Nfhzj;ij Vw;gLj;JfpwJ. Vzpapd;mb Rtw;wpypUe;J 3.5 kP J}uj;jpy; cs;sJ vdpy; Vzpapd; ePsj;ijf; fhz;f.

8. θθθθ coteccos

cos)cos(

−=+−2

2

11 vd;w Kw;nwhUikia epWTf.

5 kjpg;ngz; tpdhf;fs;:-5 kjpg;ngz; tpdhf;fs;:-5 kjpg;ngz; tpdhf;fs;:-5 kjpg;ngz; tpdhf;fs;:-5 kjpg;ngz; tpdhf;fs;:-1. (sinθ+cosec θ)2+ (cosθ+secθ)2= 7+ tan2θ+ cot2θ vd;w Kw;nwhUikia epWTf.

2.θθθθ

tansectansec

+−

= 1- 2sec θ + 2 tan2 θ vd;w Kw;nwhUikia epWTf.

34

3.

θθ

θθ

cossin

secsec

−=

−1

1 2

vd epWTf.

4. Neh;Fj;jhd xU kuj;jpd; Nky;ghfk; fhw;wpdhy; Kwpe;J> mk; Kwpe;j gFjp fPNo tpOe;J tplhky;

kuj;jpd; cr;rp jiuAld; 30º 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.

5. xU NfhGuj;jpd; mbapypUe;J xU Fd;wpd; cr;rpapd; Vw;wf;Nfhzk; 60º vd;f. Fd;wpd; mbapy;

,Ue;J NfhGuj;jpd; cr;rpapd; Vw;wf;Nfhzk; 30º kw;Wk; NfhGuj;jpd; cauk; 50 kP vdpy;> Fd;wpd;

cauj;ijf; fhz;f.

~~~~~~~~

8. mstpay;mstpay;mstpay;mstpay;mstpay;5 kjpg;ngz;fs;.5 kjpg;ngz;fs;.5 kjpg;ngz;fs;.5 kjpg;ngz;fs;.5 kjpg;ngz;fs;.

1. 120 nr.kP ePsKk;> 84 nr.kP tpl;lKk; nfhz;l xU rhiyia rkg;gLj;Jk; cUisiaf; (road roller)nfhz;L xU tpisahl;Lj;jply; rkg;gLj;jg;gLfpwJ. tpisahl;Lj; jpliy rkg;gLj;j ,t;TUis

500 KOr;Rw;Wfs; Roy Ntz;Lk;. tpisahl;Lj; jpliy rkg;gLj;j MFk;. nryitf; fhz;f.

[π = 722

]. tpil : `1188 [v.L.8.4] - gf;fk; 2222. xU jpz;k Neh;tl;lf; $k;gpd; Muk; kw;Wk; rhAauk; KiwNa 35 nr.kP kw;Wk; 37 nr.kP vdpy;

$k;gpd; tisgug;G kw;Wk; nkhj;jg; Gwg;gug;igf; fhz;f. [ π = 722

vd;f.] (v.L. 8.4 gf;fk; 233)

tpil. TSA 7920 r.nr.kP

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

nr.kP. NkYk; Mok; 63 nr.kP vdpy; mjd; nfhs;sit ypl;lhpy; fhz;f. [π = 722

vd;f.]tpil : 26.994 yp.

(v.L. 8.16 gf;fk; 246)

4. fsp kz;izg; gad;gLj;jp xU khztd; 48 nr.kP cauKk; 12 nr.kP MuKk; nfhz;l Neh; tl;l

jpz;kf; $k;igr; nra;jhh;. mf;$k;ig kw;nwhU khzth; XU jpz;kf; Nfhskhf khw;wpdhh;. mt;thW

khw;wg;gl;l Gjpa Nfhsj;jpd; Muj;ijf; fhz;f. tpil : 12 nr.kP

[gapw;rp : 8.3> vz; 5> gf; : 255]5. 14 kP tpl;lKk; kw;Wk; 20 kP MoKs;s xU fpzW cUis tbtpy; ntl;lg;gl;lJ. mt;thW ntl;Lk;

NghJ Njhz;bnaLf;fg;gl;l kz; rPuhf gug;gg;gl;L 20kP x 14 kP msTfspy; mbg;gf;fkhff; nfhz;l

xU Nkilahf mikf;fg;gl;lhy; mk;Nkilapd; cauk; fhz;f.

tpil : 11kP [ gapw;rp : 8.3> vz; : 17] gf;fk; : 256.

6. 12 nr.kP tpl;lKk;> 15 nr.kP cauKk; nfhz;l Neh;tl;l cUis KOtJk; gdpf; $opdhy; (icecream) epug;gg;gl;Ls;sJ. ,g;gdpf; #ohdJ 6nr.kP tpl;lKk;> 12 nr.kP cauKk; nfhz;l Nkw;Gwk;

miuf;Nfhsk; ,ize;j tbtpyike;j $k;gpy; epug;gg;gLfpwJ. vj;jid $k;Gfspy; gdpf; $opid

KOtJkhf epug;gyhk; vdf; fhz;f.

tpil : 10 Nfhz; [ gapw;rp : 8.3> vz; 14] gf; : 256

7. 18 nr.kP MuKs;s jpz;k cNyhff; NfhskhdJ cUf;fg;gl;L %d;W rpwpa ntt;NtW msTs;s

Nfhsq;fshf thh;f;fg;gl;LfpwJ. mt;thW thh;f;fg;gl;l ,uz;L jpz;kf; Nfhsq;fspd; Muq;fs;

KiwNa 2 nr.kP kw;Wk; 12 nr.kP vdpy; %d;whtJ Nfhsj;jpd; Muj;ijf; fhz;f.

tpil : 16 nr.kP [gapw;rp 8.3> vz; 11] gf; : 255

35

8. 8 nr.kP tpl;lKk; 12 nr.kP cauKk; nfhz;l xU Neh; tl;l jpz;k ,Uk;Gf; $k;ghdJ cUf;fg;gl;L

4. kp.kP MuKs;s jpz;kf; Nfhs tbtf; Fz;Lfshf thh;f;fg;gl;lhy; fpilf;Fk; Nfhs tbtf;

Fz;Lfspd; vz;zpf;ifiaf; fhz;f.

tpil : 750 Nfhs tbtf; Fz;Lfs; [gapw;rp : 8.3> vz; 13> gf; 256]9. 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/m2. vdpy; $lhuk; mikf;fj; Njitahd fpj;jhd; Jzpapd;

tpiyiaf; fhz;f. [π = 722

] tpil : ` 6325, [ v.L. 8.23> gf; : 252]10. xU Neh;tl;lf; $k;gpd; ,ilf; fz;lj;jpd; ,UGwKk; mike;j tl;l tpspk;Gfspd; Rw;wsTfs;

KiwNa 44 nr.kP kw;Wk; 8.4π nr.kP vd;f. mjd; cauk; 14 nr.kP vdpy; mt;tpilf; fz;lj;jpd;

fd msitf; fhz;f. tpil : 1408.6 nr.kP3.[gapw;rp : 8.2> vz; 12> gf; 250]11. 21 nr.kP MuKs;s xU tl;lj;jpypUe;J 120º ikaf;Nfhzk; nfhz;l xU tl;l Nfhzg; gFjpia

ntl;bnaLj;J> mjd; Muq;fis xd;wpizj;J xU $k;ghf;fpdhy; fpilf;Fk; $k;gpd;

tisgug;igf; fhz;f. [ π = 722

] tpil : 462 r.nr.kP [v.L. 8.8> gf; : 237]12. xU miuf;Nfhsj;jpd; fd msT 1152π f. nr.kP vdpy; mjd; tisgug;G fhz;f.

tpil : 2881π r.nr.kP²(gapw;rp : 8.2 vz; : 18> gf; : 250)

13. xU jpz;k kuk; nghk;ikahdJ miuf; Nfhsj;jpd; Nky; $k;G ,ize;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. [ π = 722

]tpil : 269.5 f.nr.kP [vL. 8.21 gf; : 251]

2 kjpg ;ngz;fs;:-2 kjpg ;ngz;fs;:-2 kjpg ;ngz;fs;:-2 kjpg ;ngz;fs;:-2 kjpg ;ngz;fs;:-

1. xU jpz;k Neh;tl;l cUisapd; (Solid right Circular Cylinder) Muk; 7 nr.kP kw;Wk; cauk; 20 nr.kP

vdpy; mjd; (i) tisgug;G> (ii) nkhj;j Gwg;gug;G Mfpatw;iwf; fhz;f. [ π = 722

vd;f.].tpil : 1188 r.nr.kP [v.L. 8.1] gf;. -221

2. xU jpz;k miuf; Nfhsj;jpd; nkhj;j Gwg;gug;G 675 π r.nr.kP vdpy; mjd; tisgug;igf; fhz;f.

tpil : 450 π r.nr.kP [v.L. 8.10] gf; - 228.3. xU jpz;k Neh;tl;l $k;gpd; cr;rpf; Nfhzk; kw;Wk; Muk; KiwNa 60º kw;Wk; 15 nr.kP vdpy;

mjd; cauk; kw;Wk; rhAauj;ijf; fhz;f.

tpil : h = 15 3 nr.kP l = 30 nr.kP. [gapw;rp : 8.1 vz; 11] gf; : 241

4. ,uz;L Neh; tl;l cUisfspd; Muq;fspd; tpfpjk; 3 : 2 vd;f. NkYk; mtw;wpd; cauq;fspd;

tpfpjk; 5 : 3 vdpy;> mtw;wpd; tisgug;Gfspd; tpfpjj;ijf; fhz;f.

tpil : C1 : C2 = 5 : 2, [ gapw;rp : 8.1, vz; : 7 ] gf;fk; : 2405. xU jpz;k Neh;tl;lf; $k;gpd; Muk; kw;Wk; cauk; KiwNa 7. nr.kP kw;Wk; 24 nr.kP vdpy; mjd;

tisgug;G kw;Wk; nkhj;g; gug;igf; fhz;f.

tpil : CSA : 550 nr.kP2, TSA : 704 nr.kP2. [ gapw;rp : 8.1 vz; : 10] gf; - 2306. 8.4 nr.kP tpl;lk; nfhz;l xU Nfhs tbt jpz;k cNyhf vwpFz;bd; fd msitf; fhz;f.

[ π = 722

] tpil : 310.464 f. nr.kP (v.L. 8.17 gf; : 248)

7. xU jpz;k cUisapd; Muk; 14. nr.kP> mjd; cauk; 30 nr.kP> mt;TUisapd; fd msitf;

fhz;f. tpil : 1840 nr.kP3[ vapw;rp : 8.2, vz; : 1] gf; - 249

36

8. ,uz;L Neh;tl;l cUisfspd; Muq;fspd; tpfpjk; 2 : 3 NkYk; cauq;fspd; tpfpjk; 5 : 3

vdpy; mtw;wpd; fd msTfspd; tpfpjj;ijf; fhz;f.

tpil : V1 : V2 = 20 : 27 [gapw;rp: 8.2 vz; : 5] gf; - 2389. kuj;jpyhd xU jpz;kf; $k;gpd; mbr;Rw;wsT 44 nr.kP kw;Wk; mjd; cauk; 12 nr.kP vdpy;

mj;jpz;kf; $k;gpd; fd msitf; fhz;f. tpil : 616 nr.kP³ [gapw;rp : 8.2, vz; : 10] gf; - 23910. xU cs;sPlw;w Nfhsj;jpd; ntsp kw;Wk; cs; Muq;fs; KiwNa 12 nr.kP kw;Wk; 10 nr.kP vdpy;

mf;Nfhsj;jpd; fd msitf; fhz;f.

tpil : 3050 2/3 nr.kP3 (gapw;rp : 8.2> vz; 17> gf; : 250)

******

11. Gs;spapay;11. Gs;spapay;11. Gs;spapay;11. Gs;spapay;11. Gs;spapay;2 kjpg;ngz; tpdhf;fs;:-2 kjpg;ngz; tpdhf;fs;:-2 kjpg;ngz; tpdhf;fs;:-2 kjpg;ngz; tpdhf;fs;:-2 kjpg;ngz; tpdhf;fs;:-

1. 43,24,38,56,22,39,45 Mfpa Gs;sp tptuq;fspd; tPr;R kw;Wk; tPr;Rf;nfO fhz;f.

(Eg : 11.1) tpil : 0.4362. xU Gs;sp tptuj; njhFg;gpd; kPg;ngU kjpg;G 7.44 kw;Wk; mjd; 2.26 vdpy;> mj;njhFg;gpd; kPr;rpW

kjpg;Gf; fhz;f. (Eg : 11.3) tpil : 5.183. Kjy; 10 ,ay; vz;fspd; jpl;l tpyf;fk; fhz;f. (Eg : 11.11) tpil : 2.874. 59,46,30,23,27,40,52,35,29 Mfpa Gs;sp tptuq;fspd; tPr;R kw;Wk; tPr;Rf;nfO fhz;f.

(Ex : 11.1) tpil : 36.5. 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. (Ex : 11.1 (2) tpil : 3.746. Kjy; 13 ,ay; vz;fspd; jpl;ltpyf;fj;ijf; fhz;f. (Ex : 11.1 (5) tpil : 257. n = 10, x = 12 kw;Wk; Σx2 = 1530 vdpy;> khWghl;Lf; nfOitf; fzf;fpLf.

(Ex: 11.1 (15) tpil : 3.748. 50 msTfspy; kpfg;nghpa kjpg;G 3.84 fp.fp vdpy; mjd; tPr;R 0.46 fp.fp. mitfspd; kPr;rpW

kjpg;igf; fhz;f. (Ex:11.1(3)) tpil : 3.38 kg.

5 kjpg;ngz; tpdhf;fs;:5 kjpg;ngz; tpdhf;fs;:5 kjpg;ngz; tpdhf;fs;:5 kjpg;ngz; tpdhf;fs;:5 kjpg;ngz; tpdhf;fs;:

1. 2,4,8,11,14,6,12,10 ,g;Gs;sp tptuj;jpd; jpl;ltpyf;fj;ijf; fhz;f. (Eg : 11.4) tpil : 3.742. 18,20,15,12,25 vd;w tptuq;fSf;F khWghl;Lf; nfOitf; fhz;f. (Eg: 11.17) tpil : 24.63. 10,20,15,8,3,4 vd;w Gs;sp tptuj;jpd; jpl;l tpyf;fj;ijf; fhz;f. (Ex : 11.1(7) tpil : 5.974. ml;ltizapy; nfhLf;fg;gl;Ls;s Gs;sp tptuj;jpd; jpl;l tpyf;fj;ijf; fzf;fpLf.

x 3 8 13 18 23f 7 10 15 10 8

Ex : (11.1 (7) tpil : 31.615. khWghl;Lf; nfOitf; fhz;f. 20,18,32,24,26 (Ex : 11.1(16) tpil : 20.4

6. 62,58,53,50,63,52,55 .,g;Gs;sp tptuj;jpd; jpl;ltpyf;fk; fhz;f. (Eg : 11.6) tpil : 4.64

********12. epfo;jfTepfo;jfTepfo;jfTepfo;jfTepfo;jfT

2 kjpg;ngz; tpdhf;fs;.2 kjpg;ngz; tpdhf;fs;.2 kjpg;ngz; tpdhf;fs;.2 kjpg;ngz; tpdhf;fs;.2 kjpg;ngz; tpdhf;fs;.

37

1. Kjy; ,UgJ ,ay; vz;fspypUe;J xU KO vz; rktha;g;G Kiwapy; Njh;e;njLf;fg;gLfpwJ.

me;j vz; xU gfh vz;zhf ,Ug;gjw;fhd epfo;jftpid fhz;f.

v.fh. : 12.3 tpil : 2/52. xU Fwpg;gpl;l ehspy; kio tUtjw;fhd epfo;jfT 0.76 mf;Fwpg;gpl;l ehspy; kio ehspy;

,Ug;gjw;fhd epfo;jfT ahJ? v.fh. : 12.8 tpil : 0.243. xU rPuhd gfil ,uz;L Kiw cUl;lg;gLfpwJ. Kf vz;fspd; $Ljy; 9 fpilf;fg; ngWtjw;fhd

epfo;jfT fhz;f. v.fh. : 12.1 tpil : 0.244. 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;fhd epfo;jfT vd;d? Ex. 12.1. tpil : 1/5.

5. ,U ehzaq;fis xNu rkaj;jpy; Rz;Lk; NghJ> mjpfgl;rkhf xU jiy fpilg;gjw;fhd

epfo;jftpidf; fhz;f. Ex. 12.1 tpil : 3/46. xU ngl;bapy; 4 gr;ir> 5 ePyk; kw;Wk; 3 rptg;G epwg; ge;Jfs; cs;sd. rktha;g;G Kiwapy; xU

ge;ijj; Njh;e;njLf;f mJ i) rptg;G epwg; ge;jhf ,Uf;f> ii) gr;ir epwg; ge;jhf ,y;yhkypUf;f

Mfpadtw;wpd; epfo;jfTfisf; fhz;f. Ex : 12.1 tpil : ¼7. %d;W gfilfs; xNu Neuj;jpy; cUl;lg;gLk; NghJ> %d;W gfilfspYk; xNu vz; fpilg;gjw;fhd

epfo;r;rpapd; epfo;jftpidf; fhz;f. Ex : 12.1 tpil : 1/368. “ ENTERTAINMENT” vd;w nrhy;ypYs;s vOj;JfspypUe;J rktha;g;G Kiwapy; xU vOj;ijj;

Njh;T nra;a> mt;ntOj;J Mq;fpy capnuOj;jhfNth my;yJ vOj;J T MfNth ,Ug;gjw;fhd

epfo;jftpidf; fhz;f. (vOj;Jfs; jpUk;gj; jpUk;g tuyhk;) v.fh. : 12.15 tpil : 8/139. A kw;wk; B vd;gd xd;iwnahd;W tpyf;Fk; epfo;r;rpfs;;> NkYk; P(A) = 3/5, P(B) = 1/5, vdpy;

P(A∪B) If; fhz;f. Ex : 12.2 tpil : 4/5.10. A kw;Wk; B vd;w ,uz;L epfo;r;rpfspy; P(A) = ¼ P(B) = 2/5, P(A∪B) = ½ vdpy; P(A∩B) If;

fhz;f. Ex : 12.2 tpil : 3/20.

5 kjpg;ngz; tpdhf;fs;:-5 kjpg;ngz; tpdhf;fs;:-5 kjpg;ngz; tpdhf;fs;:-5 kjpg;ngz; tpdhf;fs;:-5 kjpg;ngz; tpdhf;fs;:-

1. xU rPuhd ehzak; ,uz;L Kiw Rz;lgLfpwJ. fPo;f;fhZk; epfo;r;rpfSf;fhd epfo;jftpidf;

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

v.fh.: 12.2 Ans: ¼,¾,½2. ,U rPuhd gfilfs; xU Kiw cUl;lg;gLfpd;wd. fPo;f;fhZk; epfo;r;rpfSf;fhd epfo;jftpidf;

fhz;f. i) Kf vz;fspd; $Ljy; 8 Mf ,Uj;jy; ii) Kf vz;fs; xNu vz;zfshf ,Uj;jy; iii) Kf

vz;fspd; $Ljy; 8I tpl mjpfkhf ,Uj;jy;. v.fh.: 12.5 Ans: 5/36,1/6, 5/183. A kw;Wk; B vd;w ,uz;L epfo;r;rpfspy; P(A) = ½, P(B) = 7/10, kw;Wk; P(A∪B) = 1 vdpy; i) P(A∩B)

ii) P(A’ ∪B’) Ex : 12.2 Ans : 1/5,4/54. 1 Kjy; 50 tiuapyhd KOf;fspypUe;J rktha;g;G Kiwapy; Xh; vz; Njh;e;njLf;fg;gLk; NghJ

mt;ntz; 4 my;yJ 6 My; tFgLtjw;fhd epfo;jfT fhz;f. Ex : 12.2 Ans: 8/255. xU igapy; 10 nts;is> 6 rptg;G kw;Wk; 10 fUg;G epwg; ge;Jfs; cs;sd. rkthag;G Kiwapy; xU

ge;jpid vLf;Fk; NghJ mJ nts;is my;yJ rptg;G epwg; ge;jhf ,Ug;gjw;fhd epfo;jftpidf;

fhz;f. Ex: 12.2 Ans: 8/13

38

6. %d;W ehzaq;fs; xNu Neuj;jpy; Rz;lg;gLfpd;wd. gpd;tUk; epfo;r;rpfSf;F epfo;jftpidf;

fhz;f. i. Fiwe;jJ xU jiy fpilg;gJ> ii. ,U G+f;fs; kl;Lk; fpilg;gJ> iii. Fiwe;jJ ,U jiyfs;

fpilg;gJ. Ex : 12.1, Ans 7/8, 3/8, 1/2

**********

nra;Kiw tbtpay;nra;Kiw tbtpay;nra;Kiw tbtpay;nra;Kiw tbtpay;nra;Kiw tbtpay;

1. 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; ePsj;ij

mse;J vOJf.

2. 4.2 kP MuKs;s xU tl;lk; tiue;J mt;tl;lj;jpd; Nky; VNjDk; xU Gs;spiaf; Fwpf;f. tl;lj;jpd;

ikaj;ijg; gad;gLj;jp mg;Gs;sp topNa njhLNfhL tiuf.

3. 10 nr.kP tpl;lKs;s xU tl;lk; tiuf. tl;lj;jpd; ikaj;jpypUe;J 13 nr.kP njhiytpy; P vd;w

Gs;spiaf; Fwpj;J mg;Gs;spapypUe;J tl;lj;jpw;F PA kw;Wk; PB vd;w njhLNfhLfs; tiue;J

mjd; ePsq;fis fzf;fpLf.

4. 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.

5. 3 nr.kP MuKs;s tl;lj;jpd; ikaj;jpypUe;J 9 nr.kP njhiytpy; xU Gs;spiaf; Fwpf;f.

mg;Gs;spapypUe;J tl;lj;jpw;F njhLNfhLfs; tiue;J> mjd; ePsq;fis fzf;fpLf.

- - - - -- - - - -- - - - -- - - - -- - - - -

1. PQ = 6.5 nr.kP> QR = 5.5 nr.kP> PR = 7 nr.kP kw;Wk; PS = 4.5 nr.kP vd;w msTfs; nfhz;l tl;l

ehw;fuk; PQRS tiuf.

2. AB = 6 nr.kP> AD = 4.8 nr.kP> BD = 8 nr.kP kw;Wk; CD = 5.5. nr.kP vd;w msTfs; nfhz;l tl;l

ehw;fuk; ABCD tiuf.

3. PQ = 5.5 nr.kP> QR = 4.5 nr.kP> ∠QPR = 45°, kw;Wk; PS = 3 nr.kP Mfpa msTfs; nfhz;l tl;l

ehw;fuk; PQRS tiuf.

4. AB = 7 nr.kP> ∠A = 80°, AD = 4.5 nr.kP> kw;Wk; BC = 5 nr.kP vd;w msTfs; nfhz;l tl;l ehw;fuk;

ABCD tiuf.

5. KL = 5.5 nr.kP> KM = 5 nr.kP> LM = 4.2 nr.kP> kw;Wk; LN = 5.3 nr.kP Mfpa msTfs; nfhz;l tl;l

ehw;fuk; KLMN tiuf.

6. EF = 7 nr.kP> EH = 4.8 nr.kP> FH = 6.5 nr.kP kw;Wk; EG = 6.6 nr.kP msTfs; nfhz;l tl;l ehw;fuk;

EFGH tiuf.

7. AB = 6 nr.kP> ∠ABC = 70° BC = 5 nr.kP kw;Wk; ∠ACD = 30° Mfpa msTfs; nfhz;l tl;l ehw;fuk;

ABCD tiuf.

8. PQ = 5 nr.kP> QR = 4 nr.kP> ∠QPR = 35° kw;Wk; ∠PRS = 70° Mfpa msTfs; nfhz;l tl;l

ehw;fuk; PQRS tiuf.

- - - - - - - - - - -- - - - - - - - - - -- - - - - - - - - - -- - - - - - - - - - -- - - - - - - - - - -

39

tl;l ehw;fuk; tiujy;tl;l ehw;fuk; tiujy;tl;l ehw;fuk; tiujy;tl;l ehw;fuk; tiujy;tl;l ehw;fuk; tiujy;Nfs;tpfs;:-Nfs;tpfs;:-Nfs;tpfs;:-Nfs;tpfs;:-Nfs;tpfs;:-

1. AB = 6 nr.kP.> AC = 7 nr.kP> BC = 6 nr.kP kw;Wk; AD = 4.2 nr.kP msTfs; nfhz;l tl;l ehw;fuk;

ABCD tiuf.

2. PQ = 4 nr.kP.> QR = 6 nr.kP> PR = 7.5 nr.kP kw;Wk; QS = 7 nr.kP msTfs; nfhz;l tl;l ehw;fuk;

tiuf.

3. AB = 6 nrkP.> BC = 5.5 nr.kP.> ∠ABC = 80° kw;Wk; AD = 4.5 nr.kP msTfs; nfhz;l tl;l ehw;fuk;

tiuf.

4. EF = 5.2 nr.kP> ∠GEF = 50°, FG = 6 nr.kP kw;Wk; ∠EGH = 40°, vd;w msTfs; nfhz;l tl;l

ehw;fuk; EFGH tiuf.

5. PQ = 4 nr.kP. ∠P = 100°, ∠PQS = 40°, kw;Wk; ∠SQR = 70°, vDk;gb tl;l ehw;fuk; PQRS tiuf.

6. AB = 5.8 nr.kP> ∠ABD = 35°, AD = 4.2 nr.kP kw;Wk; AB || CD vd;w msTfs; nfhz;l tl;l ehw;fuk;

ABCD tiuf.

7. AB = 5.5 nr.kP.> ∠ABC = 50°, ÐBAC = 60° kw;Wk; ÐACD = 30° Mfpa msTfs; nfhz;l tl;l

ehw;fuk; ABCD tiuf.

8. AB = 6.5 nr.kP> ∠ABC = 110°, BC = 5.5 nr.kP kw;Wk; AB || CD vd;wthW mikAk; tl;l ehw;fuk;

ABCD tiuf.

- - - - - - - - - - - -- - - - - - - - - - - -- - - - - - - - - - - -- - - - - - - - - - - -- - - - - - - - - - - -

1. ml;ltiz nfhLf;fg;gl;Ls;sit1. ml;ltiz nfhLf;fg;gl;Ls;sit1. ml;ltiz nfhLf;fg;gl;Ls;sit1. ml;ltiz nfhLf;fg;gl;Ls;sit1. ml;ltiz nfhLf;fg;gl;Ls;sit

1. fPo;f;fhZk; ml;ltizf;Fj; jFe;j tiuglk; tiue;J khwpfspd; khWghl;Lj; jd;ikiaf; fhz;.

mk;khWghl;bd; khwpypiaAk; (Constant of proportionality) fhz;f.

x 2 3 5 8 10

y 8 12 20 32 40

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

2. xU kpjptz;b Xl;Lgth; A vd;w ,lj;jpypUe;J B vd;w ,lj;jpypUe;J 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; Mfpadtw;iwg; gw;wpa tptuq;fs; (Ntf fhy) gpd;tUk;

ml;ltizapy; nfhLf;fg;gl;Ls;sd.

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

Neuk; (kzpapy;) y 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 kzp Neuj;jpy; flf;f ve;j Ntfj;jpy;

gazpf;f Ntz;Lk; Mfpadtw;iwf; fhz;f.

3. thq;fg;gl;l Nehl;Lg; Gj;jfq;fspd; vz;zpf;if kw;Wk; mjw;fhd tpiy tptuk; gpd;tUk;

ml;ltizapy; jug;gl;Ls;sJ.

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

tpiy & y 30 60 90 120 150 180

40

,jw;fhd 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.

4. x 1 3 5 7 8

y 2 6 10 14 16

Nkw;fz;l ml;ltizapy; cs;s tptuj;jpw;F tiuglk; tiue;J. mjd; %yk; (i) x = 4, vdpy; y d;

kjpg;igf; fhz;f. (ii) y = 12 vdpy; x d; kjpg;igf; fhz;f.

5. 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;lttizapy; nfhLf;fg;gl;Ls;s tptuj;jpw;fhd tiuglk; tiuf. mjd; %yk; 12 Ntiyahl;fs;

mt;Ntiyia KOtJkhf Kbf;f MFk; ehl;fspd; vz;zpf;ifiaf; fhz;f.

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ml;ltizia jahhpf;f Ntz;baitml;ltizia jahhpf;f Ntz;baitml;ltizia jahhpf;f Ntz;baitml;ltizia jahhpf;f Ntz;baitml;ltizia jahhpf;f Ntz;bait

1. xU tq;fp> %j;jf; Fbkfdpd; itg;Gj; njhiff;F 10% jdptl;b jUfpwJ. itg;Gj; njhiff;Fk;

mjw;F Xh; Mz;Lf;F 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;bahff; fpilf;f

tq;fpapy; nrYj;jg;gl Ntz;ba itg;Gj; njhif Mfpadtw;iwf; fhz;f.

2. xU NgUe;J kzpf;F 40 fp.kP Ntfj;jpy; nry;fpwJ. ,jw;Fhpa J}u fhy njhlh;gpw;fhd tiuglk;

tiuf. ,ijg; gad;gLj;jp 3 kzp Neuj;jpy; ,g;NgUe;J gazpj;jj; J}uj;ijf; fz;Lgpb.

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.

3. xy = 20, x, y > 0 vd;gjd; tiuglk; tiuf. mjidg; gad;gLj;jp x = 5 vdpy; y d; kjpg;igAk; y = 10vdpy;> x d; kjpg;igAk; fhz;f.

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