€¦ · tisgu¥ò L bkh¤j¥ òw¥gu¥ò – 2 Hmo¥g¡f¤Â‹ gu¥ò 660 2 22 7 660 2 2 22 660 2 7 7 7 660 308 352 =−×πr vdnt, 2 352 22 27 352 7 352 7 2227 8 rh h h

g¤jh« tF¥ò

fz¡F

Mark

க.உமாநேடஸ் ் M.Sc.,B.Ed., g£ljhçMÁça® (fâj«)

muR ca®ãiy¥gŸë, [pQ;rk;gl;b,-635104

Ma¤bjhiy toéaš

16

Email: [email protected]

www.Padasalai.Net www.TrbTnpsc.com

http://www.trbtnpsc.com/2013/09/10th-study-materials.html

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g¤jh« tF¥ò

fz¡F

Mark

mséaš

15






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ÏU kÂ¥bg© édh¡fŸ

1. xU cŸÇl‰w cUisæ‹ cŸ k‰W« btë Mu§fŸ Kiwna 12 br.Û. k‰W« 18 br.Û. v‹f.

nkY« mj‹ cau« 14 br.Û våš m›ÎUisæ‹ tisgu¥ò k‰W« bkh¤j¥òw¥gu¥ig¡ fh©f.

( 227

π = v‹f.) (vfh: 8.5)

Ô®Î:

cUisæ‹ cŸMu« 12 br.Û

cUisæ‹ btëMu« 18 br.Û

cUisæ‹ cau« 14 br.Û

tisgu¥ò 2

222 14 (18 12)7222 14 307

2 22 2 30

= × × × +

= × × ×

= × × ×

2640 r.br.Û

bkh¤j¥ òw¥gu¥ò 2

222 (18 12)(18 12 14)7222 30 207

= × × + − +

= × × ×

26400

7= r.br.Û

2. xU Â©k ne® t£l¡ T«Ã‹ Mu« k‰W« rhÍau« Kiwna 35 br.Û k‰W« 37 br.Û. våš

T«Ã‹ tisgu¥ò k‰W« bkh¤j¥ òw¥gu¥ig¡ fh©f. ( 227

π = v‹f.) (vfh: 8.6)

Ô®Î:

T«Ã‹ Mu« 35 br.Û

T«Ã‹ rhÍau« 37 br.Û

18 br Û

12 br Û

14 b

r Û

35 br.Û

37 br.Û

mséaš

க.உமாநேடஸ் M.Sc.,B.Ed., g£ljhçMÁça® (fâj«)

muR ca®ãiy¥gŸë, [pQ;rk;gl;b, -635 104 www.Padasalai.Net www.TrbTnpsc.com


www.Padasalai.Net

tisgu¥ò

22 35 377

22 5 37

= × ×

= × ×

4070 r.br.Û

bkh¤j¥ òw¥gu¥ò

22 35 [37 35]722 35 727

22 5 72

= × × +

= × ×

= × ×

7920 r.br.Û

3. 8.4 br.Û é£l« bfh©l xU nfhstot Â©k cnyhf v¿F©o‹ fd msit¡ fh©f.

( 227

π = v‹f.) (vfh: 8.17)

Ô®Î:

nfhs¤Â‹ Mu« 4.2 br.Û

nfhs¤Â‹ fdmsÎ, 34

3rπ=

4 22 4.2 4.2 4.23 7

224 1.4 4.2 4.27

4 22 0.2 4.2 4.2310.464

= × × × ×

= × × × ×

= × × × ×=

vdnt, cnyhf v¿F©o‹ fd msÎ 310.464 f.br.Û.

4. xU Â©k ne® t£l cUisæ‹ bkh¤j¥ òw¥gu¥ò 660 r. br.Û. mj‹ é£l« 14 br.Û. våš,

m›ÎUisæ‹ cau¤ijÍ«, tisgu¥igÍ« fh©f. (g: 8.1, f: 2)

Ô®Î:

cUisæ‹ Mu« 7 br.Û

cUisæ‹ bkh¤j¥ òw¥gu¥ò 660 r. br.Û

8.4 br Û


muR ca®ãiy¥gŸë, [pQ;rk;gl;b, -635 104



www.Padasalai.Net

tisgu¥ò bkh¤j¥ òw¥gu¥ò – 2 mo¥g¡f¤Â‹ gu¥ò

2660 222660 2 7 77

660 2 22 7660 308352

rπ= − ×

= − × × ×

= − × ×= −=

vdnt,

2 352222 7 3527

352 7 2 22 7

8

rh

h

h

π =

× × × =

×=

× ×=

cUisæ‹ tisgu¥ò 352 r. br.Û


5. xU Â©k ne®t£l¡ T«Ã‹ Mu« k‰W« cau« Kiwna 7 br.Û k‰W« 24 br.Û. våš, mj‹

tisgu¥ò k‰W« bkh¤j¥ òw¥gu¥ig¡ fh©f. (g: 8.1, f: 10)

Ô®Î:


T«Ã‹ cau« 24 br.Û

T«Ã‹ rhÍau« 2 2l r h= +

2 27 24

49 576

625

= +

= +

=

25 = br.Û

tisgu¥ò rlπ=

22 7 257

22 25

= × ×

= ×

550 = r.br.Û

7 br.Û

24

br

.Û

bkh¤j¥ òw¥gu¥ò

660 r. br.Û

7 br Û





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bkh¤j¥

6. xU Â

m¡T«Ã

Ô®Î:

T«Ã‹ r

T«Ã‹ m

tisgu¥

7. xU Â

fh©f.

Ô®Î:

miu¡nf

miu¡nf

8. xU Â©

fh©f.

Ô®Î:

cUisæ

cUisæ

òw¥gu¥ò π=

2

2

2

=

=

=

7=

©k ne® t£

Ã‹ tisgu¥

rhÍau« 1l =

mo¢R‰wsÎ

¥ò rlπ=

118 1= ×

1416 = r

Â©k miu

(g: 8.1,

fhs¤Â‹ ti

fhs¤Â‹ bk

©k cUis

(g: 8.2,

æ‹ Mu«

æ‹ cau«

[ ]r l rπ +

22 7 [25722 7 327

22 32

× ×

× ×

×

704 r.br.Û

£l¡ T«Ã‹

¥ig¡ fh©f

12 br.Û

Î 2 236rπ =

2rπ⇒ =

2

r.br.Û

¡nfhs¤Â‹

f: 17)

isgu¥ò 2π

π⇒

kh¤j¥ òw¥gu

æ‹ Mu«

, f: 1)

br.Û

br.Û

7]+

‹ mo¢R‰ws

f.

6 br.Û

236 1182

=

‹ tisgu¥ò

2 2772r = r

2 27722

r = =

u¥ò

3 1= ×

4158=

br.Û. mj‹

sÎ 236 br.

(g: 8.1, f: 12)

ò 2772 r.br

r.br.Û

1386=

386

8 r.br.Û

‹ cau« b

.Û . k‰W« m

)

r.Û våš,

br.Û våš, m

t

mj‹ rhÍau

mj‹ bkh¤

m›ÎUisæ

2

tisgu¥ò

2772= r.b

r b

u« 12 br.Û v

¤j¥ òw¥gu¥

æ‹ fdms

236 br.Û

12 br

br.Û

br Û

våš,

¥ig¡

it¡

r.Û





www.Padasalai.Net

cUisæ‹ fdmsÎ,

22 14 14 307

22 2 14 30

= × × ×

= × × ×

18480 f.br.Û

9. xU Â©k¡ T«Ã‹ Mu« k‰W« rhÍau« Kiwna 20 br.Û k‰W« 29 br.Û. våš m¤Â©k¡

T«Ã‹ fdmsit¡ fh©f. (g: 8.2, f: 9)

Ô®Î:


T«Ã‹ rhÍau« 29 br.Û

T«Ã‹ cau« 2 2l r= −

2 229 20

841 400

441

= −

= −

=

21 br.Û

T«Ã‹ fdmsÎ 21

3r hπ=

1 22 20 20 213 722 20 20

= × × × ×

= × ×

8800 f.br.Û

10. Kiwna 3 br.Û k‰W« 4 br.Û Mu§fshf¡ bfh©l ÏU nfhs§fë‹ fdmsÎfë‹

é»j¤Âid¡ fh©f.

Ô®Î:

ÏU nfhs§fë‹ Mu§fë‹ é»j« 3 4

ÏU nfhs§fë‹ fdmsÎfë‹ é»j« 3 34 4:

3 3r Rπ π

3 3

3 3

:3 : 427 : 64

r R=

==

20 br.Û

29 br.Û

30 br.Û

14 br Û





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IªJ kÂ¥bg© édh¡fŸ

1. xU ne®t£l cUisæ‹ tisgu¥ò 704 r.br.Û k‰W« mj‹ cau« 8 br.Û våš,

m›ÎUisæ‹ fdmsit è£lçš fh©f. ( 227

π = v‹f.) (vfh: 8.1 2)

Ô®Î:

cUisæ‹ Mu« br.Û


tisgu¥ò 704 r.br.Û

2 704

222 8 7047

704 7 2 22 8

r

r

× × × =

×=

× ×

14 br.Û

cUisæ‹ fdmsÎ,

22 14 14 87

22 2 14 8

= × × ×

= × × ×

4928 f.br.Û

cUisæ‹ fd msÎ4928 4.9281000

= = è£l®. (1000 f.br.Û = 1 è£l®)

2. xU Ïil¡f©l toéyhd thëæ‹ nk‰òw k‰W« mo¥òw Mu§fŸ Kiwna 15 br.Û k‰W« 8

br.Û. nkY«, MH« 63 br.Û våš, mj‹ bfhŸssit è£lçš fh©f. ( 227

π = v‹f.)

(vfh: 8.1 6)

Ô®Î:

thëæ‹ nk‰òw Mu« 15 br.Û

thëæ‹ mo¥òw Mu« 8 br.Û

thëæ‹ cau« 63 br.Û

15 br.Û

8 br.Û

63 b

r.Û

704 r.br.Û

8 b

r.Û




www.Padasalai.Net

thëæ‹ fdmsÎ 2 21 ( )

3h R r Rrπ= + +

2 21 22 63 (15 8 15 8)3 71 22 63 (225 64 120)3 71 22 63 4093 722 3 409

= × × × + + ×

= × × × + +

= × × ×

= × ×

26994 f.br.Û

thëæ‹ fd msÎ26994 26.9941000

= = è£l®. (1000 f.br.Û = 1 è£l®)

3. xU ne®t£l¡ T«Ã‹ Ïil¡f©l¤Â‹ ÏUòwK« mikªj t£l éë«òfë‹ R‰wsÎfŸ

Kiwna 44 br.Û k‰W« 8.4 br.Û v‹f. mj‹ cau« 14 br.Û våš,m›éil¡f©l¤Â‹

fdmsit¡ fh©f. (g: 8.2, f: 12)

Ô®Î:

Ïil¡f©l¤Â‹ nk‰òw R‰wsÎ 2 44 br.Û

222 447

44 7 22 2

7

R

R

⇒ × × =

×=

×=

Ïil¡f©l¤Â‹ mo¥òw R‰wsÎ 2 8.4 br.Û

8.42

4.2

r ππ

⇒ =

=

Ïil¡f©l¤Â‹ nk‰òw Mu« 7 br.Û

Ïil¡f©l¤Â‹ mo¥òw Mu« 4.2 br.Û

Ïil¡f©l¤Â‹ cau« 14 br.Û

44 br.Û

8.4 br.Û 14

br

.Û




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Ïil¡f©l¤Â‹ fdmsÎ 2 21 ( )

3h R r Rrπ= + +

2 21 22 14 (7 4.2 7 4.2)3 71 22 14 (49 17.64 29.4)3 71 22 14 96.043 722 2 96.04

34225.76

31408.586

= × × × + + ×

= × × × + +

= × × ×

× ×=

=

=

1408.6 f.br.Û

4. xU TlhukhdJ cUisæ‹ ÛJ T«ò Ïizªj toéš cŸsJ. Tlhu¤Â‹ bkh¤j cau«

13.5 Û k‰W« é£l« 28 Û. nkY« cUis¥ ghf¤Â‹ cau« 3 Û våš, Tlhu¤Â‹ bkh¤j

òw¥gu¥ig¡ fh©f. (g: 8.3, f: 4)

Ô®Î:

T«Ã‹ Mu« 14 Û

T«Ã‹ cau« 13.5 3 10.5 Û

T«Ã‹ rhÍau« 2 2r h= +

2 214 10.5

196 110.25

306.25

= +

= +

=

17.5 Û

T«Ã‹ tisgu¥ò

22 14 17.57

22 2 17.5

= × ×

= × ×

770 r.Û

cUisæ‹ Mu« 14 Û

cUisæ‹ cau« 3 Û

28 .Û

13.5 .Û

3 .Û




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cUisæ‹ tisgu¥ò 2

222 14 37

2 22 2 3

= × × ×

= × × ×

264 r.Û

Tlhu¤Â‹ bkh¤j òw¥gu¥ò T«Ã‹ tisgu¥ò cUisæ‹ tisgu¥ò

770 264

1034 r.Û

5. fëk©iz¥ ga‹gL¤Â xU khzt‹ 48 br.Û cauK« 12 br.Û MuK« bfh©l ne®

t£lÂ©k¡ T«ig¢ brŒjh®. m¡T«ig k‰bwhU khzt® xU Â©k¡ nfhskhf kh‰¿dh®.

m›thW kh‰w¥g£l òÂa nfhs¤Â‹ Mu¤ij¡ fh©f. (g: 8.3, f: 5)

Ô®Î:


T«Ã‹ cau« 48 br.Û

T«Ã‹ fdmsÎ 21

3r hπ=

1 12 12 483

π= × × × × f.br.Û

nfhs¤Â‹ Mu« br.Û

nfhs¤Â‹ fdmsÎ 34

3rπ=

f.br.Û

nfhs¤Â‹ fdmsÎ T«Ã‹ fdmsÎ

34 1 12 12 483 3

rπ π= × × × ×

3

3

12 12 48 33 4

12 12 12 12

r ππ

× × × ×=

× ×= × ×

=

12r⇒ =

nfhs¤Â‹ Mu« 12 br.Û

12 br.Û

48

br

.Û

r br.Û




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6. 8 br.Û é£lK« 12 br.Û cauK« bfh©l xU ne® t£l Â©k ÏU«ò¡ T«ghdJ cU¡f¥g£L

4 ä.Û MuKŸs Â©k¡ nfhs tot F©Lfshf th®¡f¥g£lhš »il¡F« nfhs tot

F©Lfë‹ v©â¡ifia¡ fh©f. (g: 8.3, f: 13)

Ô®Î:

T«Ã‹ Mu« 4 br.Û 40 ä.Û

T«Ã‹ cau« 12 br.Û 120 ä.Û

T«Ã‹ fdmsÎ 21

3r hπ=

1 40 40 1203

π= × × × × f. ä.Û

nfhs¤Â‹ Mu« 4 ä.Û

nfhs¤Â‹ fdmsÎ 34

3rπ=

4 4 4 43

π= × × × × f. ä.Û

F©Lfë‹ v©â¡if =T«Ã‹ fdmsÎ

nfhs¤Â‹ fdmsÎ

1 40 40 1203

4 4 4 43

3 40 40 1203 4 4 4 4

750

π

π

ππ

× × × ×=

× × × ×

× × × ×=

× × × × ×=

nfhs tot F©Lfë‹ v©â¡if 750

4 br.Û

12 b

r.Û

4 ä.Û




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8. 14 Û é£lK« k‰W« 20 Û MHKŸs xU »zW cUis toéš bt£l¥gL»wJ. m›thW

bt£L«nghJ njh©obaL¡f¥g£l k© Óuhf gu¥g¥g£L 20 Û 14 Û msÎfëš mo¥g¡fkhf¡

bfh©l xU nkilahf mik¡f¥g£lhš, m«nkilæ‹ cau« fh©f. (g: 8.3, f: 17)

Ô®Î:

cUis tot¡ »z‰¿‹ Mu« 7 Û

cUis tot¡ »z‰¿‹ cau« 20 Û

cUis tot¡ »z‰¿‹ fdmsÎ,

22 7 7 207

= × × ×

22 7 20 f.Û

nkilæ‹ Ús« 20 Û

nkilæ‹ mfy« 14 Û

nkilæ‹ cau« Û

nkilæ‹ fdmsÎ,

20 14 f.Û

nkilæ‹ fdmsÎ cUis tot¡ »z‰¿‹ fdmsÎ

20 14 22 7 20

22 7 2020 14

11

h × ×=

×=

nkilæ‹ cau« 11 Û.

14 Û

20 Û

20 Û

14 Û




www.Padasalai.Net



g¤jh« tF¥ò

fz¡F

Mark

K¡nfhz§fŸ tiujš

bra;Kiw totpay; 10

(bjhLBfhL, ;)




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bjhLnfhL tiujš

vL¤J¡fh£L 9.3

3 br.Û MuKŸs t£l« tiuf. t£l¤Â‹ ika¤ÂèUªJ 7 br.Û. bjhiyéš xU òŸëia¡

F¿¤J, m¥òŸëæèUªJ t£l¤Â‰F bjhLnfhLfŸ tiuf. nkY« bjhLnfhLfë‹ Ús¤ij

msªJ vGJf.

cjé¥gl«

A

O P

B

OA = 3 br.Û

OP = 7 br.Û

A O M P B rçgh®¤jš :

3 br.Û

7 br.Û

7 br.Û

3 br.Û 6.3 br.Û

2 2

2 2

PA = OP - OA

= 7 - 3

= 49 - 9

= 40 = 6.3





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gæ‰Á 9.1

3) 10 br.Û é£lKŸs xU t£l« tiuf. t£l¤Â‹ ika¤ÂèUªJ 13 br.Û. bjhiyéš P v‹w

òŸëia¡ F¿¤J m¥òŸëæèUªJ t£l¤Â‰F PA k‰W« PB v‹w bjhLnfhLfŸ tiuªJ

mj‹ Ús§fis fz¡»Lf.

cjé¥gl«

A

O P

B

OA = 5 br.Û

OP = 13 br.Û

rçgh®¤jš :

5 br.Û

13 br.Û

13 br.Û

5 br.Û 12 br.Û

B

O M

P

A

2 2

2 2

PA = OP - OA

= 13 - 5

= 169 - 25

= 144 = 12





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2 2

2 2

PA = OP - OA

= 10 - 6

= 100 - 36

= 64 = 8

4) 6 br.Û MuKŸs xU t£l« tiuªJ mj‹ ika¤ÂèUªJ 10 br.Û bjhiyéYŸs xU

òŸëia¡ F¿¡f. m¥òŸëæèUªJ t£l¤Â‰F bjhLnfhLfŸ tiuªJ mj‹ Ús§fis

fz¡»Lf.

cjé¥gl«

A

O P

B

OA = 6 br.Û

OP = 10 br.Û

rçgh®¤jš :

6 br.Û

10 br.Û

O

P

B

O M

A

10 br.Û

6 br.Û 8 br.Û





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5) 3 br.Û MuKŸs t£l¤Â‹ ika¤ÂèUªJ 9 br.Û bjhiyéš xU òŸëia¡ F¿¡f.

m¥òŸëæèUªJ t£l¤Â‰F bjhLnfhLfŸ tiuªJ, mj‹ Ús§fis fz¡»Lf.

cjé¥gl«

A

O P

B

OA = 3 br.Û

OP = 9 br.Û

rçgh®¤jš :

2 2

2 2

PA = OP - OA

= 9 - 3

= 81 - 9

= 72 = 8.5

3 br.Û

9 br.Û

P

B

O M

A

9 br.Û

3 br.Û 8.5 br.Û





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K¡nfhz§fŸ tiujš

vL¤J¡fh£L 9.4

AB = 6 br.Û. ∠C = 40° k‰W« c¢Á C-æèUªJ AB-¡F tiua¥g£l F¤J¡nfh£o‹ Ús«

4.2 br.Û. bfh©l ∆ABC tiuf.

cjé¥gl«

6 br.Û

40°

4 br.Û

A B

C

D

AB = 6 br.Û

∠C = 40°

CD = 4.2 br.Û

40°

40°

6 br.Û

4.2 br.Û

A B

C H

O

M





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vL¤J¡fh£L 9.5

mo¥g¡f« BC = 5.5 br.Û., ∠A = 60° k‰W« c¢Á A-æèUªJ tiua¥g£l eL¡nfhL AM-‹

Ús« = 4.5 br.Û bfh©l ∆ABC tiuf.

cjé¥gl«

5.5 br.Û

60°

4.5 br.Û

B C

A

M

BC = 5.5 br.Û

∠Α = 60°

AM = 4.5 br.Û

60°

5.5 br.Û

4.5 br.Û

B C

A

O

M60°





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vL¤J¡fh£L 9.6

BC = 4.5 br.Û, ∠A=40° k‰W« c¢Á A-æèUªJ BC ¡F tiua¥g£l eL¡nfh£o‹ Ús« AM = 4.7

br.Û. vd ÏU¡F« go ∆ABC tiuf. nkY« A-æèUªJ BC-¡F tiua¥g£l F¤J¡nfh£o‹

Ús« fh©f.

cjé¥gl«

4.5 br.Û

40°

4.7 br.Û

B C

A

M

BC = 4.5 br.Û

∠Α = 40°

AM = 4.7 br.Û

40°

4.5 br.Û

4.7 br.Û

B C

A

O

M40°





www.Padasalai.Net

Punith

Line

gæ‰Á 9.2

2) ∆PQR-š mo¥g¡f« PQ = 6 br.Û., ∠R = 60° k‰W« c¢ÁR-èUªJ PQ-¡F tiua¥g£l

F¤J¡nfh£o‹ Ús« 4 br.Û vd ÏU¡FkhW DPQR tiuf.

cjé¥gl«

6 br.Û

60°

4 br.Û

P Q

R

M

PQ = 6 br.Û

∠Ρ = 60°

RM = 4 br.Û

60°

6 br.Û

4 br.Û

P Q

R

O

M60°

H





www.Padasalai.Net

3) PQ = 4 br.Û., ∠R = 25° k‰W« c¢Á R-èUªJ PQ -¡F tiua¥g£l F¤J¡nfh£o‹ Ús« 4.5

br.Û. v‹w msÎfŸ bfh©l ∆PQR tiuf.

cjé¥gl«

4 br.Û

25°

4.5 br.Û

P Q

R

M

PQ = 4 br.Û

∠Ρ = 25°

RM = 4.5 br.Û

R

O

H

25°

4.5 br.Û

Q25° 4 br.Û

P M





www.Padasalai.Net

4) ∆ABC-š, BC = 5 br.Û., ∠A = 45° k‰W« c¢Á A-èUªJ BC-¡F tiua¥g£l eL¡nfh£o‹

Ús« 4 br.Û vd ÏU¡F« go ∆ABC tiuf.

cjé¥gl«

5 br.Û

45°

4 br.Û

B C

A

M

BC = 5 br.Û

∠Α = 45°

AM = 4 br.Û

45°

5 br.Û

4 br.Û

B C

A O

M45°





www.Padasalai.Net

5) BC = 5 br.Û., ∠BAC = 40° k‰W« c¢Á A-èUªJ BC-¡F tiua¥g£l eL¡nfh£o‹ Ús« 6

br.Û. v‹w msÎfŸ bfh©l ∆ABC tiuf. nkY« c¢Á A-èUªJ tiua¥g£l F¤J¡nfh£o‹

Ús« fh©f.

cjé¥gl«

5 br.Û

40°

6 br.Û

B C

A

M

BC = 5 br.Û

∠BAC = 40°

AM = 6 br.Û

40°

5 br.Û

4.7 br.Û

B C

A

O

M40°





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g¤jh« tF¥ò

fz¡F



Áw¥òtif tiugl§fŸ

10

Marks




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Áw¥òtif tiugl§fŸ

vL¤J¡fh£L 10.7

Ñœ¡fhQ« m£ltiz¡F¤ jFªj tiugl« tiuªJ kh¿fë‹ khWgh£L¤ j‹ikia¡ fh©.

m«khWgh£o‹ kh¿èiaÍ« (constant of proportionality) fh©f.

x 2 3 5 8 10 y 8 12 20 32 40

nkY« x = 4 våš y-‹ kÂ¥ig¡ fh©f.

m£ltiz :

Ô®Î :

x = 4 våš, y = 16

vL¤J¡fh£L 10.8

xU äÂt©o X£Lgt® A v‹w Ïl¤ÂèUªJ B v‹w Ïl¤Â‰F xU Óuhd ntf¤Âš xnu têæš

bt›ntW eh£fëš gaz« brŒ»wh®. mt® gaz« brŒj ntf«, m¤öu¤Âid¡ fl¡f

vL¤J¡ bfh©l neu« M»adt‰iw¥ g‰¿a étu§fŸ (ntf-fhy) Ã‹tU« m£ltizæš

bfhL¡f¥g£LŸsd.

ntf« (».Û. / kâ)

x 2 4 6 10 12

neu« (kâæš)

y 60 30 20 12 10

ntf - fhy tiugl« tiuªJ mÂèUªJ (i) mt® kâ¡F 5 ».Û ntf¤Âš br‹whš öu¤ij¡

fl¡f MF« gaz neu« (ii) mt® Ï¡F¿¥Ã£l öu¤ij 40 kâneu¤Âš fl¡f vªj ntf¤Âš

gaâ¡f nt©L« M»adt‰iw¡ fh©f.

m£ltiz :

Ô®Î :

(i) kâ¡F 5 ».Û ntf¤Âš br‹whš gaz neu« 24 kâ MF«.

(ii) F¿¥Ã£l öu¤ij 40 kâ neu¤Âš fl¡f, mtUila ntf« 3 ».Û / kâ MF«.

x 2 3 5 8 10 y 8 12 20 32 40

ntf« (».Û. / kâ)

x 2 4 6 10 12

neu« (kâæš)

y 60 30 20 12 10



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vL¤J¡fh£L 10.9

xU t§», _¤j¡Fokfå‹ it¥ò¤ bjhif¡F 10% jåt£o jU»wJ. it¥ò¤ bjhif¡F«

mj‰F X® M©L¡F¡ »il¡F« t£o¡F« Ïilnaahd bjhl®Ãid¡ fh£l xU tiugl«

tiuf. mj‹ _y«,

(i) 650% it¥ò¤ bjhif¡F¡ »il¡F« t£o k‰W«

(ii) 45% t£oahf¡ »il¡f t§»æš brY¤j¥gl nt©oa it¥ò¤ bjhif M»adt‰iw¡

fh©f.

m£ltiz :

Ô®Î :

(i) `650 it¥ò¤ bjhif¡F cça t£o¤ bjhif 65%

(ii) `45 t£oahf¡ »il¡f njitahdit¥ò¤ bjhif 450 %

gæ‰Á 10.2

1. xU ngUªJ kâ¡F 40 ».Û. ntf¤Âš brš»wJ. Ïj‰Fça öu-fhy bjhl®Ã‰fhd tiugl«

tiuf. Ïij¥ ga‹gL¤Â 3 kâneu¤Âš Ï¥ngUªJ gaâ¤j¤ öu¤ij¡ f©LÃo.

m£ltiz :

öu« x 40 80 120 160 200 240 neu« y 1 2 3 4 5 6

Ô®Î :

3 kâneu¤Âš Ï¥ngUªJ gaâ¤j¤ öu« 120».Û.

2. th§f¥g£l neh£L¥ ò¤jf§fë‹ v©â¡if k‰W« mj‰fhd éiy étu« Ã‹tU«

m£ltizæš ju¥g£LŸsJ. neh£L¥ò¤jf§fë‹ v©â¡if

neh£L¥ò¤jf§fë‹

v©â¡if x 2 4 6 8 10 12

éiy

y 30 60 90 120 150 180

Ïj‰fhd tiugl« tiuªJ mj‹ _y«

it¥ò¤ bjhif

x 100 200 300 400 500 600 700

jåt£o y 10 20 30 40 50 60 70



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(i) VG neh£L¥ ò¤jf§fë‹ éiyia¡ fh©f.

(ii) 165%-¡F th§f¥gL« neh£L¥ ò¤jf§fë‹ v©â¡ifia¡ fh©f.

m£ltiz :

neh£L¥ò¤jf§fë‹

v©â¡if x 2 4 6 8 10 12

éiy

y 30 60 90 120 150 180

Ô®Î :

(i) VG neh£L¥ ò¤jf§fë‹ éiy 105%

(ii) 165%-¡F th§f¥gL« neh£L¥ ò¤jf§fë‹ v©â¡if 11

3.

x 1 3 5 7 8 y 2 6 10 14 16

nk‰f©l m£ltizæš cŸs étu¤Â‰F tiugl« tiuªJ, mj‹ _y«

(i) x = 4 våš y-‹ kÂ¥ig¡ fh©f.

(ii) y = 12 våš x-‹ kÂ¥ig¡ fh©f.

m£ltiz :

x 1 3 5 7 8 y 2 6 10 14 16

Ô®Î :

(i) x = 4 våš y = 8

(ii) y = 12 våš x = 6

4. xU è£l® ghè‹ éiy 15% v‹f. ghè‹ msÎ¡F« éiy¡F« cŸs¤ bjhl®Ãid¡ fh£L«

tiugl« tiuf. mjid¥ ga‹gL¤Â,

(i) é»jrk kh¿èia¡ fh©f.

(ii) 3 è£l® ghè‹ éiyia¡ fh©f.



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m£ltiz :

ghè‹ msÎ x 1 2 3 4 5 6

éiy y 15 30 45 60 75 90

Ô®Î :

(i) é»jrk kh¿è = 15

(ii) 3 è£l® ghè‹ éiy 45%

5. xy = 20, x, y > 0 v‹gj‹ tiugl« tiuf. mjid¥ ga‹gL¤Â x = 5 våš, y-‹ kÂ¥igÍ«, y

= 10 våš, x-‹ kÂ¥igÍ« fh©f.

m£ltiz :

x 1 2 4 5 10 20

y 20 10 5 4 2 1

Ô®Î :

(i) x = 5 våš y = 4

(ii) y = 10 våš x = 2

6.

ntiyah£fë‹

v©â¡if x 3 4 6 8 9 16

eh£fë‹

v©â¡if y 96 72 48 36 32 18

m£ltizæš bfhL¡f¥g£LŸs étu¤Â‰fhd tiugl« tiuf. mj‹ _y« 12 ntiyah£fŸ

m›ntiyia KGtJkhf brŒJ Ko¡f MF« eh£fë‹ v©â¡ifia¡ fh©f.

m£ltiz :

ntiyah£fë‹

v©â¡if x 3 4 6 8 9 16

eh£fë‹

v©â¡if y 96 72 48 36 32 18

Ô®Î :

12 ntiyah£fŸ m›ntiyia KGtJkhf brŒJ Ko¡f MF« eh£fë‹ v©â¡if

24 MF«.



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vL¤J¡fh£L 10.7

X’ X

Y

Y’

1 2 3 4 5 6 7 8 9 10

5

0

10

15

20

25

30

35

40

45

50

X

X

X

X

X

(2, 8)

(3, 12)

(5, 20)

(8, 32)

(10, 40)

msÎ¤Â£l« :

X m¢Áš 1 br.Û = 1 myF Y m¢Áš 1 br.Û = 5 myFfŸ



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vL¤J¡fh£L 10.8

X’ X

Y

Y’

X

X

X

X

(12, 10) (10, 12)

(6, 20)

(4, 30)

(2, 60)

X

neu«

ntf«

msÎ¤Â£l« :


2 4 6 8 10 12 14 16 18 20

5

0

10

15

20

25

30

35

40

45

50

55

60

65

70



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vL¤J¡fh£L 10.9

X’ X

Y

Y’

100 200 300 400 500 600 700 800 900 1000

10

0

20

30

40

50

60

70

80

90

100

t£o

it¥ò¤ bjhif

X

X

X

X

(100, 10)

(300, 30)

(400, 40)

(500, 50)

X (200, 20)

X (600, 60)

X (700, 70)

msÎ¤Â£l« :

X m¢Áš 1 br.Û = 100 myFfŸ Y m¢Áš 1 br.Û = 10 myFfŸ



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1

X’ X

Y

Y’

40 80 120 160 200 240 280 320 360 400

1

0

2

3

4

5

6

7

8

9

10

X

X

X

X

(40, 1)

(120, 3)

(160, 4)

(200, 5)

X (80, 2)

X (240, 6)

msÎ¤Â£l« :

X m¢Áš 1 br.Û = 40 myFfŸ Y m¢Áš 1 br.Û = 1 myF

neu«

öu«

öu-fhy bjhl®Ã‰fhd tiugl«



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2

X’ X

Y

Y’

2 4 6 8 10 12 14 16 18 20

30

0

60

90

120

150

180

210

240

270

300

msÎ¤Â£l« :


éiy

neh£L¥ ò¤jf§fë‹

v©â¡if

X

X

X

X

(2, 30)

(6, 90)

(8, 120)

(10, 150)

X (4, 60)

X (12, 180)



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3

X’ X

Y

Y’

1 2 3 4 5 6 7 8 9 10

2

0

4

6

8

10

12

14

16

18

20

msÎ¤Â£l« :


X

X

X

X

X

(1, 2)

(3, 6)

(5, 10)

(7, 14)

(8, 16)



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4

X’ X

Y

Y’

1 2 3 4 5 6 7 8 9 10

15

0

30

45

60

75

90

105

120

135

150

msÎ¤Â£l« :


éiy

ghè‹ msÎ

X

X

X

X

X

(1, 15)

(3, 45)

(5, 75)

(6, 90)

(4, 60)

X (2, 30)

ghè‹ msÎ¡F« - éiy¡F« cŸs¤ bjhl®Ã‰fhd tiugl«



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5

X’ X

Y

Y’

2 4 6 8 10 12 14 16 18 20

2

0

4

6

8

10

12

14

16

18

20

22

24

26

28

msÎ¤Â£l« :


X

X

X

X X

(20, 1) (10, 2)

(5, 4) (4, 5)

(2, 10)

X (1, 20)



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6

X’ X

Y

Y’

2 4 6 8 10 12 14 16 18 20

10

0

20

30

40

50

60

70

80

90

100

110

120

msÎ¤Â£l« :


ntiyah£fë‹

v©â¡if

eh£fë‹

v©â¡if

X

X

X X

X

(16, 18)

(9, 32) (8, 36)

(6, 48)

(4, 72)

X (3, 96)



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9443540473

g¤jh« tF¥ò

fz¡F

Mark



Â£léy¡f«

8



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Â£léy¡f«


1. 43, 24, 38, 56, 22, 39, 45 M»a òŸë étu§fë‹ Å¢R k‰W« Å¢R¡bfG fh©f. (vfh: 11.1 )

Ô®Î :

Û¥bgU kÂ¥ò L = 56

Û¢ÁW kÂ¥ò S = 22

vdnt, Å¢R 56 22 34L S= − = − =

Mfnt, Å¢R¡ bfG 56 22 34 0.43656 22 78

L SL S

− −= = = =

+ +

2. xU òŸë étu¤ bjhF¥Ã‹ Û¥bgU kÂ¥ò 7.44 k‰W« mj‹ Å¢R 2.26 våš, m¤bjhF¥Ã‹

Û¢ÁW kÂ¥ig¡ fh©f. (vfh: 11.3)

Ô®Î :

Û¥bgU kÂ¥ò L = 7.44

Û¢ÁW kÂ¥ò S Å¢R = 2.26

vdnt, Å¢R L S= −

2.26 = 7.44 - S S = 7.44 – 2.26 = 5.18 3. Kjš 10 Ïaš v©fë‹ Â£l éy¡f« fh©f. (vfh: 11.11 )

Ô®Î :

Kjš n Ïaš v©fë‹ Â£l éy¡f« 2 112

n −=

Mfnt, Kjš 10 Ïaš v©fë‹ Â£l éy¡f« 210 1 100 1 99 33 8.25 2.8712 12 12 4

− −= = = = = =

4. 59, 46, 30, 23, 27, 40, 52, 35, 29 kÂ¥òfS¡F Å¢R k‰W« Å¢R¡ bfG fh©f. (g: 11.1, f: 1 )

Ô®Î :

Û¥bgU kÂ¥ò L = 59


vdnt, Å¢R 59 23 36L S= − = − =

Mfnt, Å¢R¡ bfG 59 23 36 0.43959 23 82

L SL S

− −= = = =

+ +




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5. xU òŸë étu¤Â‹ Û¢ÁW kÂ¥ò 12. mj‹ Å¢R 59 våš m¥òŸë étu¤Â‹ Û¥bgU

kÂ¥ig¡ fh©f. (g: 11.1, f: 2 )

Ô®Î :

Û¥bgU kÂ¥ò L


Å¢R = 59


59 = L - 12 L = 59 + 12 = 71 6. 50 msÎfëš äf¥bgça kÂ¥ò 3.84 ».». mj‹ Å¢R 0.46 ».» våš, mitfë‹ Û¢ÁW

kÂ¥ig¡ fh©f. (g: 11.1, f: 3 )

Ô®Î :

Û¥bgU kÂ¥ò L = 3.84

Û¢ÁW kÂ¥ò S Å¢R = 0.46


0.46 = 3.84 - S S = 3.84 – 0.46 = 3.38 Û¢ÁW kÂ¥ò 3.38 ».»

7. f©l¿ªj òŸë étu¤ bjhF¥ÃYŸs 20 kÂ¥òfë‹ Â£l éy¡f« 5 v‹f. òŸë

étu¤Â‹ x›bthU kÂ¥igÍ« 2 Mš bgU¡»dhš »il¡F« òÂa òŸë étu§fë‹ Â£l

éy¡f« k‰W« éy¡f t®¡f¢ ruhrç fh©f. (g: 11.1, f: 4 )

Ô®Î :

bjhF¥ÃYŸs 20 kÂ¥òfë‹ Â£l éy¡f« = 5 x›bthU kÂ¥igÍ« 2 Mš bgU¡»dhš »il¡F« òÂa òŸë étu§fë‹ Â£l éy¡f« =

2 5

òÂa òŸë étu§fë‹ éy¡f t®¡f¢ ruhrç = ( )22 5 20=

8. Kjš 13 Ïaš v©fë‹ Â£l éy¡f¤ij¡ fz¡»Lf. (g: 11.1, f: 5 )

Ô®Î :

Kjš n Ïaš v©fë‹ Â£l éy¡f« 2 112

n −=




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Mfnt, Kjš 13 Ïaš v©fë‹ Â£l éy¡f« 213 1 169 1 168 14 3.7412 12 12

− −= = = = =

9. n = 10, x = 12 k‰W« 2x∑ = 1530 våš, khWgh£L¡ bfGit¡ fz¡»Lf. (g: 11.1, f: 15 )

Ô®Î :

n = 10, x = 12, 2x∑ = 1530

Â£l éy¡f« ( )22

21530 12 153 144 7 2.64610

x xn n

∑ ∑⎛ ⎞= − = − = − = =⎜ ⎟⎝ ⎠

khWgh£L¡ bfG 2.646 264.6. x 100 x 100 22.05

12 12C V

xσ

= = = =

10. xU òŸë étu¤Â‹ khWgh£L¡ bfG 57 k‰W« Â£l éy¡f« 6.84 våš, mj‹ T£L¢

ruhrçia¡ fh©f. (g: 11.1, f: 17 )

Ô®Î :

. x 100 x 100.

6.84 684 x 100 1257 57

C V xC Vx

σ σ= ⇒ =

= = =


1. xU khj¤Âš 8 khzt®fŸ go¤j ò¤jf§fë‹ v©â¡if Ã‹tUkhW. 2, 5, 8, 11, 14, 6, 12, 10.

Ï¥òŸë étu¤Â‹ Â£l éy¡f¤ij¡ fz¡»Lf.

(vfh: 11.4 )

Ô®Î :

Ï§F, n = 8

Â£l éy¡f«,

x d = x - 8 d2 2 -6 36 5 -3 9 8 0 0

11 3 9 14 6 36 6 -2 4

12 4 16 10 2 4

4 114

( )2 22

2114 4 14.25 0.5 14.25 0.25 14 3.748 8

d dn n

σ Σ Σ⎛ ⎞ ⎛ ⎞= − = − = − = − = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠




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2. xU tF¥Ã‰F el¤j¥g£l bghJ m¿Î¤nj®éš bkh¤j kÂ¥bg©fŸ 40-¡F, 6 khzt®fŸ

bg‰w kÂ¥bg©fŸ 20, 14, 16, 30, 21 k‰W« 25. Ï¥òŸë étu¤Â‹ Â£l éy¡f« fh©f.

(vfh: 11.5 )

Ô®Î :

Ï§F, n = 6

Â£l éy¡f«,

3. 62, 58, 53, 50, 63, 52, 55 M»a v©fS¡F Â£l éy¡f« fh©f. (vfh: 11.6 )

Ô®Î :

Ï§F, n = 7

Â£l éy¡f«,

x d = x - 20 d2 20 0 0 14 -6 36 16 -4 16 30 10 100 21 1 1 25 5 25

6 178

x d = x - 55 d2 62 7 49 58 3 9 53 -2 4 50 -5 25 63 8 64 52 -3 9 55 0 0

8 160

( )2 22

2178 6 29.67 1 29.67 1 28.67 5.366 6

d dn n

σ Σ Σ⎛ ⎞ ⎛ ⎞= − = − = − = − = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 22 160 8 1120 64 1056 21.55 4.647 7 49 49 49

d dn n

σ Σ Σ⎛ ⎞ ⎛ ⎞= − = − = − = = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠




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4. 10 khzt®fŸ fâj¤ nj®éš bg‰w kÂ¥bg©fŸ Ã‹tUkhW, 80, 70, 40, 50, 90, 60, 100, 60,

30, 80. Ï«kÂ¥òfS¡F Â£l éy¡f« fh©f. (vfh: 11.7 )

Ô®Î :

Ï§F, n = 10

Â£l éy¡f«,

5. xU fâj édho édh¥ ngh£oæš 48 khzt®fŸ bg‰w kÂ¥bg©fŸ Ã‹tU«

m£ltizæš ju¥g£LŸsd.

kÂ¥bg©fŸ x 6 7 8 9 10 11 12

ãfHbt©fŸ f 3 6 9 13 8 5 4

Ï›étu¤Â‰fhd Â£l éy¡f¤ij¡ fz¡»Lf. (vfh: 11.12 )

Ô®Î :

Â£l éy¡f«,

x d = x – 70 10 d2

30 -4 16 40 -3 9 50 -2 4 60 -1 1 60 -1 1 70 0 0 80 1 1 80 1 1 90 2 4

100 3 9 -4 46

x f d = x – 9 fd fd2 6 3 -3 -9 27 7 6 -2 -12 24 8 9 -1 -9 9 9 13 0 0 0

10 8 1 8 8 11 5 2 10 20 12 4 3 12 36

48 0 124

22

2

2

x

46 4 x 1010 10

4.6 ( 0.4) x 10

4.6 0.16 x 10

4.44 x 10 2.107 x 10 21.07

d d cn n

σ Σ Σ⎛ ⎞= − ⎜ ⎟⎝ ⎠

−⎛ ⎞= − ⎜ ⎟⎝ ⎠

= − −

= −

===

2 22 124 0 2.58 0 2.58 1.6148 48

fd fdf f

σ⎛ ⎞Σ Σ ⎛ ⎞= − = − = − = =⎜ ⎟ ⎜ ⎟Σ Σ ⎝ ⎠⎝ ⎠




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6. Ã‹tU« òŸë étu¤Â‰fhd Â£l éy¡f« fh©f. (vfh: 11.13 )

x 70 74 78 82 86 90

f 1 3 5 7 8 12

Ô®Î :

Â£l éy¡f«,

7. Ã‹tU« gutè‹ éy¡f t®¡f ruhrçia¡ fh©f. (vfh: 11.14 )

ÃçÎ Ïilbtë 3.5-4.5 4.5-5.5 5.5-6.5 6.5-7.5 7.5-8.5

ãfHbt©fŸ 9 14 22 11 17

Ô®Î :

éy¡f t®¡f ruhrç,

x f d = x – 82 fd fd2 70 1 -12 -12 144 74 3 -8 -24 192 78 5 -4 -20 80 82 7 0 0 0 86 8 4 32 128 90 12 8 96 768

36 72 1312

C.I x f d = x – 6 fd fd2 3.5-4.5 4 9 -2 -18 36 4.5-5.5 5 14 -1 -14 14 5.5-6.5 6 22 0 0 0 6.5-7.5 7 11 1 11 11 7.5-8.5 8 17 2 34 68

73 13 129

2 2221312 72 36.44 (2) 36.44 4 32.44 5.7

36 36fd fdf f

σ⎛ ⎞Σ Σ ⎛ ⎞= − = − = − = − = =⎜ ⎟ ⎜ ⎟Σ Σ ⎝ ⎠⎝ ⎠

2 222 2129 13 1.77 (0.18) 1.77 0.0324 1.7376 1.74

73 73fd fdf f

σ⎛ ⎞Σ Σ ⎛ ⎞= − = − = − = − = =⎜ ⎟ ⎜ ⎟Σ Σ ⎝ ⎠⎝ ⎠




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8. cyf¡ fhšgªJ ngh£ofëš 71 K‹dâ Åu®fŸ mo¤j nfhšfë‹ v©â¡ifæ‹

étu§fŸ Ã‹tU« m£ltizæš bfhL¡f¥g£LŸsJ. Ï›étu¤Â‹ Â£l éy¡f« fh©f.

(vfh: 11.15 )

ÃçÎ Ïilbtë 0-10 10-20 20-30 30-40 40-50 50-60 60-70

ãfHbt©fŸ 8 12 17 14 9 7 4

Ô®Î :

Â£l

éy¡f«,

9. 40 f«Ã¤ J©Lfë‹ Ús§fŸ (br.Û-¡F ÂU¤jkhf) ÑnH bfhL¡f¥g£LŸsd.

Ï›étu¤Â‰fhd éy¡f t®¡f¢ ruhrçia¡ fz¡»Lf. (vfh: 11.16 )

Ús« (br.Û) 1-10 11-20 21-30 31-40 41-50 51-60 61-70

f«Ã¤ J©Lfë‹

v©â¡if 2 3 8 12 9 5 1

Ô®Î :

éy¡f t®¡f ruhrç,

C.I x f d = x – 35 10 fd fd2

0-10 5 8 -3 -24 72 10-20 15 12 -2 -24 48 20-30 25 17 -1 -17 17 30-40 35 14 0 0 0 40-50 45 9 1 9 9 50-60 55 7 2 14 28 60-70 65 4 3 12 36

71 -30 210

C.I x f d = x – 35.5 fd fd2 0.5-10.5 5.5 2 -30 -60 1800

10.5-20.5 15.5 3 -20 -60 1200 20.5-30.5 25.5 8 -10 -80 800 30.5-40.5 35.5 12 0 0 0 40.5-50.5 45.5 9 10 90 900 50.5-60.5 55.5 5 20 100 2000 60.5-70.5 65.5 1 30 30 900

40 20 7600

22

2

2

x

210 30 x 1071 71

2.96 ( 0.4) x 10

2.96 0.16 x 10

2.8 x 10 1.673 x 10 16.73

fd fd cf f

σ⎛ ⎞Σ Σ

= − ⎜ ⎟Σ Σ⎝ ⎠

−⎛ ⎞= − ⎜ ⎟⎝ ⎠

= − −

= −

===

2 222 27600 20 190 (0.5) 190 0.25 189.75

40 40fd fdf f

σ⎛ ⎞Σ Σ ⎛ ⎞= − = − = − = − =⎜ ⎟ ⎜ ⎟Σ Σ ⎝ ⎠⎝ ⎠




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10. 18, 20, 15, 12, 25 v‹w étu§fS¡F khWgh£L¡ bfGit¡ fh©f. (vfh: 11.17 )

Ô®Î :

Ï§F, n = 5

T£L ruhrç,

khWgh£L¡ bfG,

11. 5 »ç¡bf£ éisah£L¥ ngh£ofëš Ïu©L k£il Åu®fŸ vL¤j X£l§fŸ (runs)

Ã‹tUkhW. mt®fëš X£l§fŸ vL¥gÂš ah® mÂf Ó®ik¤ j‹ik cilat®? (vfh: 11.18 )

k£il Åu® A 38 47 34 18 33

k£il Åu® B 37 35 41 27 35

Ô®Î :

k£il Åu® A k£il Åu® B

Ï§F, n = 5 Ï§F, n = 5

T£L ruhrç, T£L ruhrç,

Â£l éy¡f«, Â£l éy¡f«,

x d = x - 18 d2 18 0 0 20 2 4 15 -3 9 12 -6 36 25 7 49 90 0 98

x d = x - 34 d2 18 -16 256 33 -1 1 34 0 0 38 4 16 47 13 169

170 0 442

x d = x - 35 d2 27 -8 64 35 0 0 35 0 0 37 2 4 41 6 36

175 0 104

22

298 0 5 5

19.6 0

19.6 4.427

d dn n

σ Σ Σ⎛ ⎞= − ⎜ ⎟⎝ ⎠

⎛ ⎞= − ⎜ ⎟⎝ ⎠

= −

==

90 5

18

xxn

∑=

=

=

4.427 442.7. x 100 x 100 24.618 18

C Vxσ

= = = =

22

2104 0 5 5

20.8 0

20.8 4.6

d dn n

σ Σ Σ⎛ ⎞= − ⎜ ⎟⎝ ⎠

⎛ ⎞= − ⎜ ⎟⎝ ⎠

= −

==

175 355

xxn

∑= = =

22

2442 0 5 5

88.4 0

88.4 9.4

d dn n

σ Σ Σ⎛ ⎞= − ⎜ ⎟⎝ ⎠

⎛ ⎞= − ⎜ ⎟⎝ ⎠

= −

==

170 345

xxn

∑= = =




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khWgh£L¡ bfG, khWgh£L¡ bfG,

vdnt, B v‹w k£il Åu®mÂf Ó®ik¤ j‹ikia¡ bfh©LŸsh®.

12. Ñœ¡fhQ« òŸë étu§fë‹ Â£l éy¡f¤ij¡ fz¡»Lf. (i) 10, 20, 15, 8, 3, 4. (ii) 38, 40,

34, 31, 28, 26, 34. (g: 11.1, f: 6 )

Ô®Î :

Ï§F, n = 6

Â£l éy¡f«,

Ï§F, n = 7

Â£l éy¡f«,

x d = x - 10 d2 3 -7 49 4 -6 36 8 -2 4

10 0 0 15 5 25 20 10 100

0 214

x d = x - 34 d2 26 -8 64 28 -6 36 31 -3 9 34 0 0 34 0 0 38 4 16 40 6 36

-7 161

. x 100

4.6 x 10035460 35

13.14

C Vxσ

=

=

=

=

. x 100

9.4 x 10034940 34

27.65

C Vxσ

=

=

=

=

22

2214 0 6 6

35.67 0

35.67 5.97

d dn n

σ Σ Σ⎛ ⎞= − ⎜ ⎟⎝ ⎠

⎛ ⎞= − ⎜ ⎟⎝ ⎠

= −

==

( )

22

2

2

161 7 7 7

23 1

23 1

22 4.69

d dn n

σ Σ Σ⎛ ⎞= − ⎜ ⎟⎝ ⎠

−⎛ ⎞= − ⎜ ⎟⎝ ⎠

= − −

= −

==




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13. Ñœf©l m£ltizæš bfhL¡f¥g£LŸs òŸë étu¤Â‹ Â£l éy¡f¤ij¡ fz¡»Lf.

(g: 11.1, f: 6)

x 3 8 13 18 23

f 7 10 15 10 8

Ô®Î :

Â£l éy¡f«,

14. xU gŸëæYŸs 200 khzt®fŸ xU ò¤jf¡ f©fh£Áæš th§»a ò¤jf§fë‹

v©â¡ifia¥ g‰¿a étu« Ñœ¡fhQ« m£ltizæš bfhL¡f¥g£LŸsJ.

ò¤jf§fë‹ v©â¡if 0 1 2 3 4

khzt®fë‹ v©â¡if 35 64 68 18 15

Ï¥òŸë étu¤Â‹ Â£l éy¡f¤ij¡ fz¡»Lf. (g: 11.1, f: 8 )

Ô®Î :

Â£l éy¡f«,

x f d = x – 13 fd fd2 3 7 -10 -70 700 8 10 -5 -50 250

13 15 0 0 0 18 10 5 50 250 23 8 10 80 800

50 10 2000

x f d = x – 2 fd fd2 0 35 -2 -70 140 1 64 -1 -64 64 2 68 0 0 0 3 18 1 18 18 4 15 2 30 60 200 -86 282

2 2222000 10 40 (0.2) 40 0.04 39.96 6.32

50 50fd fdf f

σ⎛ ⎞Σ Σ ⎛ ⎞= − = − = − = − = =⎜ ⎟ ⎜ ⎟Σ Σ ⎝ ⎠⎝ ⎠

2 222282 86 1.41 ( 0.43) 1.41 0.1849 1.2251 1.107

200 200fd fdf f

σ⎛ ⎞Σ Σ −⎛ ⎞= − = − = − − = − = =⎜ ⎟ ⎜ ⎟Σ Σ ⎝ ⎠⎝ ⎠




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15. Ã‹tU« òŸë étu¤Â‹ éy¡f t®¡f¢ ruhrçia¡ fz¡»Lf. (g: 11.1, f: 9 )

x 2 4 6 8 10 12 14 16

f 4 4 5 15 8 5 4 5

Ô®Î :

éy¡f t®¡f¢ ruhrç,

16. xU ghjrhç FW¡F¥ ghijia fl¡f¢ Áy® (pedestrian crossing) vL¤J¡ bfh©l neu étu«

Ñœ¡f©l m£ltizæš bfhL¡f¥g£LŸsJ. (g: 11.1, f: 10 )

neu« (éehoæš) 5-10 10-15 15-20 20-25 25-30

eg®fë‹ v©â¡if 4 8 15 12 11

Ï¥òŸë étu¤Â‰F éy¡f t®¡f¢ ruhrç k‰W« Â£l éy¡f¤ij¡ fz¡»Lf.

Ô®Î :


vdnt, Â£l éy¡f«, 36.76 6.06σ = =

x f d = x – 8 fd fd2 2 4 -6 -24 144 4 4 -4 -16 64 6 5 -2 -10 20 8 15 0 0 0

10 8 2 16 32 12 5 4 20 80 14 4 6 24 144 16 5 8 40 320

50 50 804

C.I x f d = x – 17.5 fd fd2 5-10 7.5 4 -10 -40 400

10-15 12.5 8 -5 -40 200 15-20 17.5 15 0 0 0 20-25 22.5 12 5 60 300 25-30 27.5 11 10 110 1100

50 90 2000

2 222 2804 50 16.08 (1) 16.08 1 15.08

50 50fd fdf f

σ⎛ ⎞Σ Σ ⎛ ⎞= − = − = − = − =⎜ ⎟ ⎜ ⎟Σ Σ ⎝ ⎠⎝ ⎠

2 222 22000 90 40 (1.8) 40 3.24 36.76

50 50fd fdf f

σ⎛ ⎞Σ Σ ⎛ ⎞= − = − = − = − =⎜ ⎟ ⎜ ⎟Σ Σ ⎝ ⎠⎝ ⎠




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17. Å£L cçikahs®fŸ 45 ng® mt®fSila bjUé‹ ‘gRik¢ NHš’ Â£l¤Â‰fhf ãÂ

më¤jd®. tNè¡f¥g£l ãÂ¤ bjhif étu« Ã‹tU« m£ltizæš bfhL¡f¥g£LŸsJ.

(g: 11.1, f: 11 )

bjhif 0-20 20-40 40-60 60-80 80-100

Å£L cçikahs®fë‹

v©â¡if 2 7 12 19 5

Ï›étu¤Â‰F éy¡f t®¡f¢ ruhrç k‰W« Â£léy¡f¤ij¡ fz¡»Lf.

Ô®Î :


vdnt, Â£l éy¡f«, 416 20.39σ = =

18. Ñœ¡fhQ« gutè‹ (distribution) éy¡f t®¡f¢ ruhrç fh©f.

(g: 11.1, f: 12 )

ÃçÎ Ïilbtë 20-24 25-29 30-34 35-39 40-44 45-49

ãfHbt©fŸ 15 25 28 12 12 8

Ô®Î :


C.I x f d = x – 50 fd fd2 0-20 10 2 -40 -80 3200

20-40 30 7 -20 -140 2800 40-60 50 12 0 0 0 60-80 70 19 20 380 7600

80-100 90 5 40 200 8000 45 360 21600

C.I x f d = x – 37 fd fd2 19.5-24.5 22 15 -15 -225 3375 24.5-29.5 27 25 -10 -250 2500 29.5-34.5 32 28 -5 -140 700 34.5-39.5 37 12 0 0 0 39.5-44.5 42 12 5 60 300 44.5-49.5 47 8 10 80 800

100 -475 7675

2 222 221600 360 480 (8) 480 64 416

45 45fd fdf f

σ⎛ ⎞Σ Σ ⎛ ⎞= − = − = − = − =⎜ ⎟ ⎜ ⎟Σ Σ ⎝ ⎠⎝ ⎠

2 222 27675 475 76.75 ( 4.75) 76.75 22.5625 54.1875 54.19

100 100fd fdf f

σ⎛ ⎞Σ Σ −⎛ ⎞= − = − = − − = − = =⎜ ⎟ ⎜ ⎟Σ Σ ⎝ ⎠⎝ ⎠




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19. xU tF¥ÃYŸs A, B v‹w ÏU khzt®fŸ bg‰w kÂ¥bg©fŸ Ã‹tUkhW: (g: 11.1, f: 20 )

A 58 51 60 65 66

B 56 87 88 46 43

Ït®fëš ah® äFªj Ó®ik¤ j‹ikia bfh©LŸsh®?

Ô®Î :

A B

Ï§F, n = 5 Ï§F, n = 5

T£L ruhrç, T£L ruhrç,

Â£l éy¡f«, Â£l éy¡f«,

khWgh£L¡ bfG, khWgh£L¡ bfG,

vdnt, khzt® A äFªj Ó®ik¤ j‹ikia¡ bfh©LŸsh®.

x d = x - 35 d2 43 -13 169 46 -10 100 56 0 0 87 31 961 88 32 1024

320 40 2254

x d = x - 60 d2 58 -2 4 51 -9 81 60 0 0 65 5 25 66 6 36

300 0 146

( )

22

2

2

2254 40 5 5

450.8 8

450.8 64

386.8 19.67

d dn n

σ Σ Σ⎛ ⎞= − ⎜ ⎟⎝ ⎠

⎛ ⎞= − ⎜ ⎟⎝ ⎠

= −

= −

==

320 645

xxn

∑= = =

. x 100

19.67 x 10064

1967 64

30.73

C Vxσ

=

=

=

=

22

2146 0 5 5

29.2 0

29.2 5.4

d dn n

σ Σ Σ⎛ ⎞= − ⎜ ⎟⎝ ⎠

⎛ ⎞= − ⎜ ⎟⎝ ⎠

= −

==

300 605

xxn

∑= = =

. x 100

5.4 x 10060540 60

9

C Vxσ

=

=

=

=




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g¤jh« tF¥ò

fz¡F

Mark



08 ãfœjfÎ




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ãfœjfÎ


1. xU Óuhd ehza« Ïu©L Kiw R©l¥gL»wJ. FiwªjJ xU jiy »il¤jš

ãfœ¢ÁfS¡fhd ãfœjféid¡ fh©f. (vfh: )

Ô®Î:

TWbtë S = { HH, HT, TH, TT }. n(S) = 4. A v‹g J FiwªjJ xU jiy »il¡F« ãfœ¢Á.

A = { HH, HT,TH }. n(A) = 3.

Mfnt, ( ) 3( )( ) 4

n AP An S

= =

2. Kjš ÏUgJ Ïaš v©fëèUªJ xU KG v© rkthŒ¥ò Kiwæš nj®ªbjL¡f¥gL»wJ.

mªj v© xU gfh v©zhf ÏU¥gj‰fhd ãfœjféid¡ fh©f. (vfh: 12.3)

Ô®Î:

TWbtë S = { 1, 2, 3, 4, ….., 20 }. n(S) = 20. A v‹g J xU gfh v©zhf ÏU¥gj‰fhd ãfœ¢Á.

A = { 2, 3, 5, 7, 11, 13, 17, 19 }. n(A) = 8.

Mfnt, ( ) 8 2( )( ) 20 5

n AP An S

= = =

3. 35 bghU£fŸ ml§»a bjhF¥ò x‹¿š 7 bghU£fŸ FiwghLilad. m¤bjhF¥ÃèUªJ xU

bghUŸ rkthŒ¥ò Kiwæš nj®ªbjL¡F« nghJ Mj Fiwghl‰w bghUshf ÏU¥gj‰fhd

ãfœjfÎ ahJ? (vfh: 12.4)

Ô®Î:

bkh¤j¥ bghU£fë‹ v©â¡if n(S) = 35.

FiwghLila bghU£fë‹ v©â¡if = 7.

Fiwghl‰w bghUis¤ nj®ªbjL¡F« ãfœ¢Áia A v‹f.

Fiwghl‰w bghU£fë‹ v©â¡if n(A) = 35 - 7 = 28.

vdnt, Fiwghl‰w bghUis¤ nj®ªbjL¡F« ãfœ¢Áæ‹ ãfœjfÎ

( ) 28 4( )( ) 35 5

n AP An S

= = =





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4. xU F¿¥Ã£l ehëš kiH tUtj‰fhd ãfœjfÎ 0.76. m¡F¿¥Ã£l ehëš kiH tuhkš

ÏU¥gj‰fhd ãfœjfÎ ahJ? (vfh: 12.8)

Ô®Î:

F¿¥Ã£l ehëš kiH tU« v‹w ãfœ¢Áia A vdÎ« mªehëš kiH tuhkš ÏU¡F«

ãfœ¢Áia A vdÎ« bfhŸf.

P(A) = 0.76

( )P A ) = 1 - 0.76 = 0.24

vdnt, F¿¥Ã£l ehëš kiH tuhkš ÏU¥gj‰fhd ãfœjfÎ 0.24 MF«.

5. rkthŒ¥ò Kiwæš nj®ªbjL¡f¥gL« be£lh©oš (leap year) 53 btŸë¡»HikfŸ

ÏU¤jš. (vfh: 12.10)

Ô®Î:

xU be£lh©oš cŸs eh£fë‹ v©â¡if = 366.

mjhtJ 52 thu§ fŸ k‰W« 2 eh£fŸ.

TWbtë S = { (PhæW, Â§fŸ), (Â§fŸ, br›thŒ), (br›thŒ, òj‹), (òj‹, éahH‹),

(éahH‹, btŸë), (btŸë, rå) , (rå, PhæW) }. n(S) = 7. A v‹g J xU btŸë¡»Hik tUtj‰fhd ãfœ¢Á.

A = { (éahH‹, btŸë), (btŸë, rå) }. n(A) = 2.

Mfnt, ( ) 2( )( ) 7

n AP An S

= =

6. rkthŒ¥ò Kiwæš nj®ªbjL¡f¥gL« be£lh©oš (leap year) 52 btŸë¡»HikfŸ

ÏU¤jš. (vfh: 12.10)

Ô®Î:

xU be£lh©oš cŸs eh£fë‹ v©â¡if = 366.

mjhtJ 52 thu§ fŸ k‰W« 2 eh£fŸ.

TWbtë S = { (PhæW, Â§fŸ), (Â§fŸ, br›thŒ), (br›thŒ, òj‹), (òj‹, éahH‹),

(éahH‹, btŸë), (btŸë, rå) , (rå, PhæW) }. n(S) = 7. A v‹g J ÛjKŸs ÏU eh£fëš btŸë¡»Hik ÏšyhkèU¥gj‰fhd ãfœ¢Á.

A = { (PhæW, Â§fŸ), (Â§fŸ, br›thŒ), (br›thŒ, òj‹), (òj‹, éahH‹), (rå, PhæW) }. n(A) = 5.

Mfnt, ( ) 5( )( ) 7

n AP An S

= =




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7. rkthŒ¥ò Kiwæš nj®ªbjL¡f¥gL« rhjhuz tUl¤Âš (Non-leap year) 53

btŸë¡»HikfŸ ÏU¤jš. (vfh: 12.10)

Ô®Î:

rhjhuz tUl¤Â‹ bkh¤j eh£fë‹ v©â¡if = 365.

mjhtJ rhjhuz tUl¤Âš 52 thu§fŸ k‰W« xU ehŸ ÏU¡F«.

TWbtë S = { PhæW, Â§fŸ, br›thŒ, òj‹, éahH‹, btŸë , rå }. n(S) = 7. A v‹g J xU btŸë¡»Hik tUtj‰fhd ãfœ¢Á.

A = { btŸë }. n(A) = 1.

Mfnt, ( ) 1( )( ) 7

n AP An S

= =

8. xU igæš cŸs 1 Kjš 100 tiu v©fshš F¿¡f¥g£l 100 Ó£LfëèUªJ xU Ó£L

vL¡f¥gL»wJ. m›thW vL¡f¥gL« Ó£o‹ v© 10 Mš tFgL« v©zhf ÏU¥gj‰fhd

ãfœjféid¡ fh©f. (g: 12.1, f: 1 )

Ô®Î:

TWbtë S = { 1, 2, 3, 4, ….., 100 }. n(S) = 100. A v‹g J xU 10 Mš tFgL« v©zhf ÏU¥gj‰fhd ãfœ¢Á.

A = { 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 }. n(A) = 10.

Mfnt, ( ) 10 1( )( ) 100 10

n AP An S

= = =

9. xU Óuhd gfil Ïu©L Kiw cU£l¥gL»wJ. Kf v©fë‹ TLjš 9 »il¡f¥

bgWtj‰fhd ãfœjfÎ fh©f? (g: 12.1, f: 2 )

Ô®Î:

TWbtë S =

(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

n(S) = 36. A v‹g J Kf v©fë‹ TLjš 9 Mf ÏU¡F« ãfœ¢Á. A = { }(3,6),(4,5),(5,4),(6,3) n(A) = 4.

Mfnt, ( ) 4 1( )( ) 36 9

n AP An S

= = =




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10. 12 ešy K£ilfSl‹ 3 mG»a K£ilfŸ fyªJŸsd. rkthŒ¥ò Kiwæš

nj®ªbjL¡f¥gL« xU K£il, mG»ajhf ÏU¥gj‰fhd ãfœjfÎ v‹d? (g: 12.1, f: 4 )

Ô®Î:

TWbtë S = { 12 ešy K£ilfŸ, 3 mG»a K£ilfŸ }. n(S) = 12+3 = 15. A v‹g J xU K£il, mG»ajhf ÏU¥gj‰fhd ãfœ¢Á.

A = { 3 mG»a K£ilfŸ }. n(A) = 3.

Mfnt, ( ) 3 1( )( ) 15 5

n AP An S

= = =

11. ÏU ehza§fis xnu rka¤Âš R©L«nghJ, mÂfg£rkhf xU jiy »il¥gj‰fhd


Ô®Î:

TWbtë S = { HH, HT, TH, TT }. n(S) = 4. A v‹g J mÂfg£rkhf xU jiy »il¡F« ãfœ¢Á.

A = { TT, HT,TH }. n(A) = 3.

Mfnt, ( ) 3( )( ) 4

n AP An S

= =

12. 1 Kjš 100 tiuæyhd KG v©fëèUªJ rk thŒ¥ò Kiwæš nj®ªbjL¡f¥gL« xU v©

xU KG t®¡fkhf (perfect square) ÏU¡f ãfœjfÎfis¡ fh©f. (g: 12.1, f: 9 )

Ô®Î:

TWbtë S = { 1, 2, 3, 4, ….., 100 }. n(S) = 100. A v‹g J xU xU KG t®¡fkhf ÏU¥gj‰fhd ãfœ¢Á.

A = { 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 }. n(A) = 10.

Mfnt, ( ) 10 1( )( ) 100 10

n AP An S

= = =

13. 3, 5, 7 M»a v©fis Ïy¡f§fshf¡ bfh©L xU Ïu©oy¡f v© mik¡f¥gL»‹wJ.

m›bt© 57 I él¥ bgçajhf ÏU¥gj‰fhd ãfœjfÎ fh©f. (m›bt©âš xnu Ïy¡f¤ij

Û©L« ga‹gL¤j¡ TlhJ). (g: 12.1, f: 13 )

Ô®Î:

TWbtë S = { 35, 37, 53, 57, 73, 75}. n(S) = 6.

A v‹g J nj®ªbjL¡f¥gL« v© 57 I él¥ bgçajhf ÏU¥gj‰fhd ãfœ¢Á.

A = { 73, 75 }. n(A) = 2.

( ) 2 1( )( ) 6 3

n AP An S

= = =




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14. _‹W gfilfŸ xnu neu¤Âš cU£l¥gL«nghJ, _‹W gfilfëY« xnu v©

»il¥gj‰fhd ãfœ¢Áæ‹ ãfœjféid¡ fh©f. (g: 12.1, f: 14 )

Ô®Î:

TWbtë S = { }(1,1,1),(1,1,2),....(6,6,6) n(S) = 6 x 6 x 6 = 216. A v‹g J , _‹W gfilfëY« xnu v© »il¥gj‰fhd ãfœ¢Á. A = { }(1,1,1),(2,2,2),(3,3,3),(4,4,4),(5,5,5),(6,6,6) n(A) = 6.

Mfnt, ( ) 6 1( )( ) 216 36

n AP An S

= = =

15. A k‰W« B v‹gd x‹iwbah‹W éy¡F« ãfœ¢ÁfŸ. nkY« 3( )5

P A = k‰W« 1( )5

P B =

våš, ( )P A B∪ -I¡ fh©f . (g: 12.2, f: 1 )

Ô®Î:

16. A k‰W« B v‹w Ïu©L ãfœ¢Áfëš 1 2( ) , ( )4 5

P A P B= = , k‰W« 1( )2

P A B∪ = våš,

( )P A B∩ -I¡ fh©f. (g: 12.2, f: 2 )

Ô®Î:

17. A k‰W« B v‹w Ïu©L ãfœ¢Áfëš 1 7( ) , ( )2 10

P A P B= = k‰W« ( ) 1P A B∪ = våš, (i)

( )P A B∩ (ii) 1 1( )P A B∪ M»at‰iw¡ fh©f. (g: 12.2, f: 3 )

Ô®Î:

1 1( ) ( ) 1 ( )1 5 1 4 15 5 5

ii P A B P A B∪ = − ∩−

= − = =

( ) ( ) ( )3 1 4 5 5 5

P A B P A P B∪ = +

= + =

( ) ( ) ( ) ( )1 2 1 5 8 10 3 = 4 5 2 20 20 20 20

P A B P A P B P A B∩ = + − ∪

= + − + − =

( ) ( ) ( ) ( ) ( )1 7 5 7 10 2 1 1 = 2 10 10 10 10 10 5

i P A B P A P B P A B∩ = + − ∪

= + − + − = =




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1. xU Óuhd ehza« Ïu©L Kiw R©l¥gL»wJ. ÑH¡fhQ« ãfœ¢ÁfS¡fhd

ãfœjféid¡ fh©f. (i) ÏU jiyfŸ »il¤jš (ii) FiwªjJ xU jiy »il¤jš (iii) xU ó

k£L« »il¤jš. (vfh: 12.2)

Ô®Î:

TWbtë S = { HH, HT, TH, TT }. n(S) = 4. (i) A v‹g J ÏU jiyfŸ »il¡F« ãfœ¢Á.

A = { HH }. n(A) = 1.

Mfnt, ( ) 1( )( ) 4

n AP An S

= =

(ii) B v‹g J FiwªjJ xU jiy »il¡F« ãfœ¢Á.

B = { HH, HT,TH }. n(B) = 3.

Mfnt, ( ) 3( )( ) 4

n BP Bn S

= =

(iii) C v‹g J xU ó k£L« »il¡F« ãfœ¢Á.

C = { HT,TH }. n(C) = 2.

Mfnt, ( ) 2 1( )( ) 4 2

n CP Cn S

= = =

2. ÏU Óuhd gfilfŸ xU Kiw cU£l¥gL»‹wd. ÑH¡fhQ« ãfœ¢ÁfS¡fhd

ãfœjféid¡ fh©f. (i) Kf v©fë‹ TLjš 8 Mf ÏU¤jš (ii) Kf v©fŸ xnu v©fshf

(doublet) ÏU¤jš (iii) Kf v©fë‹ TLjš 8-I él mÂfkhf ÏU¤jš. (vfh: 12.5)

Ô®Î:

TWbtë S =

(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

n(S) = 36. (i) A v‹g J Kf v©fë‹ TLjš 8 Mf ÏU¡F« ãfœ¢Á.

A = { }(2,6),(3,5),(4,4),(5,3),(6,2) n(A) = 5.

Mfnt, ( ) 5( )( ) 36

n AP An S

= =




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(ii) B v‹g J ÏU gfilfëY« xnu v© (Ïu£ilfŸ) »il¡F« ãfœ¢Á.

B = { }(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)

n(B) = 6.

Mfnt, ( ) 6 1( )( ) 36 6

n BP Bn S

= = =

(iii) C v‹g J Kf v©fë‹ TLjš 8-I él mÂfkhf¡ »il¡F« ãfœ¢Á.

C = { }(3,6),(4,5),(4,6),(5,4),(5,5),(5,6),(6,3),(6,4),(6,5),(6,6) n(C) = 10.

Mfnt, ( ) 10 5( )( ) 36 18

n CP Cn S

= = =

3. e‹F fiy¤J it¡f¥g£l 52 Ó£Lfis¡ bfh©l Ó£L¡ f£oèUªJ rkthŒ¥ò¢ nrhjid

Kiwæš xU Ó£L vL¡f¥gL»wJ. mªj¢ Ó£L Ã‹tUtdthf ÏU¡f ãfœjfÎfis¡ fh©f.

(i) Ïuhrh (ii) fU¥ò Ïuhrh (iii) °ngL (iv) lak©£ 10 (vfh: 12.6)

Ô®Î:

TWbtë S = { 52 Ó£LfŸ}. n(S) = 52. (i) A v‹g J Ïuhrh Ó£L »il¡F« ãfœ¢Á.

A = { 4 Ïuhrh Ó£LfŸ }. n(A) = 4.

Mfnt, ( ) 4 1( )( ) 52 13

n AP An S

= = =

(ii) B v‹g J fU¥ò Ïuhrh Ó£L »il¡F« ãfœ¢Á.

B = { 2 fU¥ò Ïuhrh Ó£LfŸ }. n(B) = 2.

Mfnt, ( ) 2 1( )( ) 52 26

n BP Bn S

= = =

(iii) C v‹g J °ngL Ó£L »il¡F« ãfœ¢Á.

C = { 13 °ngL Ó£LfŸ }. n(C) = 13.

Mfnt, ( ) 13 1( )( ) 52 4

n CP Cn S

= = =

(iv) D v‹g J lak©£ 10 Ó£L »il¡F« ãfœ¢Á.

D = { lak©£ 10}. n(D) = 1.

Mfnt, ( ) 1( )( ) 52

n DP Dn S

= =




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4. e‹F fiy¤J mL¡»a 52 Ó£Lfis¡ bfh©l f£oèUªJ rkthŒ¥ò Kiwæš xU Ó£L

vL¡f¥gL»wJ. Ã‹tUtdt‰¿‰F ãfœjfÎfis¡ fh©f. (i) vL¤j Ó£L lak©£ Mf ÏU¡f

(ii) vL¤j Ó£L lak©£ Ïšyhkš ÏU¡f (iii) vL¤j Ó£L V° Ó£lhf Ïšyhkš ÏU¡f.(g: 12.1, f:6 )

Ô®Î:

TWbtë S = { 52 Ó£LfŸ}. n(S) = 52. (i) A v‹g J vL¤j Ó£L lak©£ Mf ÏU¡F« ãfœ¢Á.

A = { 13 lak©£ Ó£LfŸ }. n(A) = 13.

Mfnt, ( ) 13 1( )( ) 52 4

n AP An S

= = =

(ii) B v‹g J vL¤j Ó£L lak©£ Ïšyhkš ÏU¡F« ãfœ¢Á.

Mfnt, 1 4 1 3( ) 1 ( ) 14 4 4

P B P A −= − = − = =

(iii) C v‹g J vL¤j Ó£L V° Ó£lhf ÏU¡F« ãfœ¢Á.

C = { 4 V° Ó£LfŸ }. n(C) = 4.

( ) 4 1( )( ) 52 13

n CP Cn S

= = =

Mfnt, vL¤j Ó£L V° Ó£lhf Ïšyhkš ÏU¡F« ãfœ¢Áæ‹ ãfœjfÎ

1 13 1 121 ( ) 113 13 13

P C −= − = − = =

5. _‹W ehza§fŸ xnu neu¤Âš R©l¥gL»‹wd. Ã‹tU« ãfœ¢ÁfS¡F ãfœjféid¡

fh©f. (i) FiwªjJ xU jiy »il¥gJ (ii) ÏU ó¡fŸ k£L« »il¥gJ (iii) FiwªjJ ÏU

jiyfŸ »il¥gJ. (g: 12.1, f: 7 )

Ô®Î:

TWbtë S = { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT }. n(S) = 8. (i) A v‹g J FiwªjJ xU jiy »il¡F« ãfœ¢Á.

A = { HHH, HHT, HTH, THH, HTT, THT, TTH }. n(A) = 7.

Mfnt, ( ) 7( )( ) 8

n AP An S

= =

(ii) B v‹g J ÏU ó¡fŸ k£L« »il¡F« ãfœ¢Á.

B = { HTT, THT, TTH }. n(B) = 3.

Mfnt, ( ) 3( )( ) 8

n BP Bn S

= =




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(iii) C v‹g J FiwªjJ ÏU jiyfŸ »il¡F« ãfœ¢Á.

C = { HHH, HHT, HTH, THH }. n(C) = 4.

Mfnt, ( ) 4 1( )( ) 8 2

n CP Cn S

= = =

6. xU igæš 1 Kjš 6 tiu v©fŸ F¿¡f¥g£l 6 btŸis ãw¥ gªJfS« k‰W« 7 Kjš 10 tiu

v©fŸ F¿¡f¥g£l 4 Át¥ò ãw¥ gªJfS« cŸsd. rk thŒ¥ò Kiwæš xU gªJ

vL¡f¥gL»wJ våš, Ã‹tU« ãfœ¢ÁfS¡F ãfœjféid¡ fh©f. (i) vL¡f¥g£l gªJ xU

Ïu£il v© bfh©l gªjhf ÏU¤jš (ii) vL¡f¥g£l gªJ xU btŸis ãw¥ gªjhf ÏU¤jš.

(g: 12.1, f: 8 )

Ô®Î:

TWbtë S = { W1 , W2 , W3 , W4 , W5 , W6 , R1 , R2 , R3 , R4}. n(S) = 10. (i) A v‹g J vL¡f¥g£l gªJ xU Ïu£il v© bfh©l gªjhf »il¡F« ãfœ¢Á.

A = { W2 , W4 , W6 , R2 , R4}. n(A) = 5.

Mfnt, ( ) 5 1( )( ) 10 2

n AP An S

= = =

(ii) B v‹g J vL¡f¥g£l gªJ xU btŸis ãw¥ gªjhf »il¡F« ãfœ¢Á.

B = { W1 , W2 , W3 , W4 , W5 , W6 }. n(B) = 6.

Mfnt, ( ) 6 3( )( ) 10 5

n BP Bn S

= = =

7. 1 Kjš 100 tiuæyhd KG v©fëèUªJ rk thŒ¥ò Kiwæš nj®ªbjL¡f¥gL« xU v© (i) xU KG t®¡fkhf (perfect square) ÏU¡f (ii) KG fdkhf Ïšyhkš (not a cube) ÏU¡f

M»adt‰¿‹ ãfœjfÎfis¡ fh©f. (g: 12.1, f: 9 )

Ô®Î:

TWbtë S = { 1, 2, 3, 4, ….., 100 }. n(S) = 100. (i) A v‹g J xU KG t®¡fkhf (perfect square) ÏU¡F« ãfœ¢Á.

A = { 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 }. n(A) = 10.

Mfnt, ( ) 10 1( )( ) 100 10

n AP An S

= = =

(ii) B v‹g J KG fdkhf (a cube) ÏU¡F« ãfœ¢Á.

B = { 1, 8, 27, 64 }. n(B) = 4.




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( ) ( ) ( )5 3 0

13 135 3138

13

P A P B P A B= + − ∩

= + −

+=

=

Mfnt, ( ) 4 1( )( ) 100 25

n BP Bn S

= = =

vdnt, KG fdkhf Ïšyhkš (not a cube) ÏU¡F« ãfœjfÎ 1 25 1 24125 25 25

−= − = = .

8. “ENTERTAINMENT” v‹w brhšèYŸs vG¤JfëèUªJ rkthŒ¥ò Kiwæš xU vG¤ij¤

nj®Î brŒa, m›btG¤J M§»y cæbuG¤jhfnth mšyJ vG¤J T Mfnth ÏU¥gj‰fhd

ãfœjféid¡ fh©f. (vG¤JfŸ ÂU«g¤ ÂU«g tuyh«). (vfh: 12.15)

Ô®Î:

TWbtë S = { E, N, T, E, R, T, A, I, N, M, E, N, T }. n(S) = 13.

A v‹g J nj®Î brŒa¥g£l xU vG¤J M§»y cæbuG¤jhf ÏU¡F« ãfœ¢Á.

A = { E, E, A, I, E }. n(A) =5.

( ) 5( )( ) 13

n AP An S

= =

B v‹g J nj®Î brŒa¥g£l xU vG¤J T Mf ÏU¡F« ãfœ¢Á.

B = { T, T, T }. n(B) = 3.

( ) 3( )( ) 13

n BP Bn S

= =

A∩B = { }. n(A∩B) = 0.

( ) 0( ) 0

( ) 13n A BP A B

n S∩

∩ = = =

vdnt, P(A mšyJ B)

9. 52 Ó£Lfis¡ bfh©l xU Ó£L¡f£oèUªJ rkthŒ¥ò Kiwæš xU Ó£L vL¡f¥gL« nghJ,

m¢Ó£L xU Ïuhrh (King) mšyJ xU Ah®£ (Heart) mšyJ xU Át¥ò ãw¢ Ó£lhf¡

»il¥gj‰fhd ãfœjféid¡ fh©f. (vfh: 12.17)

Ô®Î:

TWbtë S = { 52 Ó£LfŸ}. n(S) = 52.




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A, B k‰W« C v‹gd Kiwna xU Ïuhrh, xU Ah®£ nkY« xU Át¥ò ãw¢ Ó£L »il¡F«

ãfœ¢Á v‹f.

vdnt, n(A) = 4, n(B) = 13, n(C) = 26, n(A ∩ B) = 1, n(B ∩ C) = 13, n(C ∩ A) = 2, n(A ∩ B ∩ C) = 1.

4( )

52P A =

13( )52

P B = 26( )52

P C = 1( )52

P A B∩ = 13( )52

P B C∩ =

2( )

52P C A∩ =

1( )52

P A B C∩ ∩ =

vdnt, P(A mšyJ B mšyJ C)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )4 13 26 1 13 2 1 52 52 52 52 52 52 5244 16

5228 527

13

P A B C P A P B P C P A B P B C P A C P A B C∪ ∪ = + + − ∩ − ∩ − ∩ + ∩ ∩

= + + − − − +

−=

=

=

10. xU igæš 10 btŸis, 5 fU¥ò, 3 g¢ir k‰W« 2 Át¥ò ãw¥ gªJfŸ cŸsd. rkthŒ¥ò

Kiwæš nj®ªbjL¡f¥gL« xU gªJ, btŸis mšyJ fU¥ò mšyJ g¢ir ãwkhf

ÏU¥gj‰fhd ãfœjféid¡ fh©f. (vfh: 12.18)

Ô®Î:

TWbtë S = { 10 btŸis ãw¥ gªJfŸ, 5 fU¥ò ãw¥ gªJfŸ, 3 g¢ir ãw¥ gªJfŸ ,2 Át¥ò

ãw¥ gªJfŸ }. n(S) = 10+5+3+2 = 20.

W, B k‰W« G M»ad Kiwna xU btŸis, xU fU¥ò k‰W« xU g¢ir ãw¥ gªij¤

nj®ªbjL¡F« ãfœ¢ÁfŸ v‹f.

xU btŸis ãw¥ gªij vL¡F« ãfœ¢Áæ‹ ãfœjfÎ ( ) 10( )( ) 20

n WP Wn S

= =

xU fU¥ò ãw¥ gªij vL¡F« ãfœ¢Áæ‹ ãfœjfÎ ( ) 5( )( ) 20

n BP Bn S

= =

xU g¢ir ãw¥ gªij vL¡F« ãfœ¢Áæ‹ ãfœjfÎ ( ) 3( )( ) 20

n GP Gn S

= =




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( ) ( ) ( )18 5 3 23 3 20 536 36 36 36 36 9

P A P B P A B= + − ∩−

= + − = = =

vdnt, xU gªij nj®ªbjL¡f mJ btŸis mšyJ fU¥ò mšyJ g¢irahf ÏU¡f ãfœjfÎ

( W, B k‰W« G M»ad x‹iwbah‹W

éy¡F« ãfœ¢ÁfshF«.)

11. xU gfil ÏUKiw cU£l¥gL»wJ. Kjyhtjhf cU£l¥gL«nghJ xU Ïu£il¥gil v©

»il¤jš mšyJ m›éU cU£lèš Kf v©fë‹ TLjš 8 Mf ÏU¤jš vD« ãfœ¢Áæ‹


Ô®Î:

TWbtë S =

(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

n(S) = 36.

A v‹g J Kjyhtjhf cU£l¥gL«nghJ xU Ïu£il¥gil v© »il¡F« ãfœ¢Á.

A = (2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

n(A) =18.

( ) 18( )( ) 36

n AP An S

= =

B v‹g J Kf v©fë‹ TLjš 8 Mf ÏU¡F« ãfœ¢Á.

B = { }(2,6),(3,5),(4,4),(5,3),(6,2) n(B) = 5.

( ) 5( )( ) 36

n BP Bn S

= =

A∩B = { }(2,6),(4,4),(6,2)

n(A∩B) = 3.

( ) 3( )

( ) 36n A BP A B

n S∩

∩ = =

vdnt, P(A mšyJ B)

( ) ( ) ( ) ( )10 5 3 20 20 2018 209

10

P W B G P W P B P G∪ ∪ = + +

= + +

=

=




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( ) ( ) ( )12 8 450 50 5020 4

501650825

P A P B P A B= + − ∩

= + −

−=

=

=

12. 1 Kjš 50 tiuæyhd KG¡fëèUªJ rkthŒ¥ò Kiwæš X® v© nj®ªbjL¡f¥gL« nghJ

m›bt© 4 mšyJ 6 Mš tFgLtj‰fhd ãfœjfÎ fh©f. (g: 12.2, f: 5 )

Ô®Î:

TWbtë S = { 1, 2, 3, 4, ….., 50 }. n(S) = 50.

A v‹g J nj®ªbjL¡f¥gL« v© 4 Mš tFgL« ãfœ¢Á.

A = { 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48 }. n(A) =12.

( ) 12( )( ) 50

n AP An S

= =

B v‹g J nj®ªbjL¡f¥gL« v© 6 Mš tFgL« ãfœ¢Á.

B = { 6, 12, 18, 24, 30, 36, 42, 48 }. n(B) = 8.

( ) 8( )( ) 50

n BP Bn S

= =

A∩B = { 12, 24, 36, 48 }. n(A∩B) =4.

( ) 4( )

( ) 50n A BP A B

n S∩

∩ = =

vdnt, P(A mšyJ B)

13. xU igæš 50 kiu MâfS« (bolts), 150 ÂUF kiufS« (nuts) cŸsd. mt‰WŸ ghÂ kiu

MâfS«, ghÂ ÂUF kiufS« JU¥Ão¤jit. rkthŒ¥ò Kiwæš VnjD« x‹iw¤

nj®ªbjL¡F« nghJ mJ JU¥Ão¤jjhf mšyJ xU kiu Mâahf ÏU¥gj‰fhd


Ô®Î:

TWbtë S = { 50 kiu MâfŸ (bolts), 150 ÂUF kiufŸ (nuts) }. n(S) = 50 + 150 = 200.




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( ) ( ) ( )100 50 25200 200 200150 25

20012520058

P A P B P A B= + − ∩

= + −

−=

=

=

A v‹g J xU JU¥Ão¤jjhf ÏU¡F« ãfœ¢Á.

A = { 25 kiu MâfŸ, 75 ÂUF kiufŸ }. n(A) =25 + 75 = 100.

( ) 100( )( ) 200

n AP An S

= =

B v‹g J xU kiu Mâahf ÏU¡F« ãfœ¢Á.

B = { 50 kiu MâfŸ }. n(B) = 50.

( ) 50( )( ) 200

n BP Bn S

= =

A∩B = { 25 JU¥Ão¤j kiu MâfŸ }. n(A∩B) =25.

( ) 25( )

( ) 200n A BP A B

n S∩

∩ = =

vdnt, P(A mšyJ B)

14. ÏU gfilfŸ xnu neu¤Âš nru cU£l¥gL«nghJ »il¡F« Kf v©fë‹ TLjš 3 Mš

k‰W« 4 Mš tFglhkèU¡f ãfœjfÎ fh©f. (g: 12.2, f: 7 )

Ô®Î:

TWbtë S =

(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

n(S) = 36.




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( ) ( ) ( )12 9 136 36 3621 1

36203659

P A P B P A B= + − ∩

= + −

−=

=

=

A v‹g J Kf v©fë‹ TLjš 3 Mš tFgL« ãfœ¢Á.

A = (1,2),(1,5),(2,1),(2,4),(3,3),(3,6)(4,2),(4,5),(5,1),(5,4),(6,3),(6,6)

⎧ ⎫⎨ ⎬⎩ ⎭

n(A) =12.

( ) 12( )( ) 36

n AP An S

= =

B v‹g J Kf v©fë‹ TLjš 4 Mš tFgL« ãfœ¢Á.

B = (1,3),(2,2),(2,6),(3,1),(3,5),(4,4)(5,3),(6,2),(6,6)

⎧ ⎫⎨ ⎬⎩ ⎭

n(B) = 9.

( ) 9( )( ) 36

n BP Bn S

= =

A∩B = { }(6,6)

n(A∩B) = 1.

( ) 1( )

( ) 36n A BP A B

n S∩

∩ = =

P(A mšyJ B)

vdnt, Kf v©fë‹ TLjš 3 Mš k‰W« 4 Mš tFglhkèU¡f ãfœjfÎ = P(A mšyJ B) 5 9 5 419 9 9

−= − = =

15. xU Tilæš 20 M¥ÃŸfS« 10 MuŠR¥ gH§fS« cŸsd. mt‰WŸ 5 M¥ÃŸfŸ k‰W« 3

MuŠRfŸ mG»ait. rkthŒ¥ò Kiwæš xUt® xU gH¤ij vL¤jhš, mJ M¥Ãshfnth

mšyJ ešy gHkhfnth ÏU¥gj‰fhd ãfœjféid¡ fh©f. (g: 12.2, f: 8 )

Ô®Î:

TWbtë S = { 20 M¥ÃŸfŸ , 10 MuŠR¥ gH§fŸ }. n(S) = 20 + 10 = 30.




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( ) ( ) ( )20 22 1530 30 3042 15

3027309

10

P A P B P A B= + − ∩

= + −

−=

=

=

A v‹g J vL¤j xU gH« M¥Ãshf ÏU¡F« ãfœ¢Á.

A = { 20 M¥ÃŸfŸ }. n(A) =20.

( ) 20( )( ) 30

n AP An S

= =

B v‹g J vL¤j xU gH« ešy gHkhf ÏU¡F« ãfœ¢Á.

B = { 15 ešy M¥ÃŸfŸ , 7 ešy MuŠR¥ gH§fŸ }. n(B) = 15 + 7 = 22.

( ) 22( )( ) 30

n BP Bn S

= =

A∩B = { ešy M¥ÃŸfŸ }. n(A∩B) = 15.

( ) 15( )

( ) 30n A BP A B

n S∩

∩ = =

vdnt, P(A mšyJ B)

16. xU tF¥Ãš cŸs khzt®fëš 40% ng® fâj édho édh ãfœ¢ÁæY«, 30% ng®

m¿éaš édho édh ãfœ¢ÁæY«, 10% ng® m›éu©L édho édh ãfœ¢ÁfëY« fyªJ

bfh©ld®. m›tF¥ÃèUªJ rkthŒ¥ò Kiwæš xU khzt‹ nj®ªbjL¡f¥g£lhš, mt®

fâj édho édh ãfœ¢Áænyh mšyJ m¿éaš édho édh ãfœ¢Áænyh mšyJ ÏU

ãfœ¢ÁfëYnkh fyªJ bfh©lj‰fhd ãfœjfÎ fh©f. (g: 12.2, f: 9 )

Ô®Î:

TWbtë S = { 100 khzt®fŸ }. n(S) = 100.

A v‹g J khzt®fŸ fâj édho édhéš fyªJ bfh©lj‰fhd ãfœ¢Á.

A = { 40 khzt®fŸ }. n(A) = 40.

( ) 40( )( ) 100

n AP An S

= =




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( ) ( ) ( )40 30 10

100 100 10070 10

10060

10035

P A P B P A B= + − ∩

= + −

−=

=

=

B v‹g J khzt®fŸ m¿éaš édho édhéš fyªJ bfh©lj‰fhd ãfœ¢Á.

B = { 30 khzt®fŸ }. n(B) = 30.

( ) 30( )( ) 100

n BP Bn S

= =

A∩B = { ÏU ãfœ¢ÁfëYnkh fyªJ bfh©l khzt®fŸ }. n(A∩B) = 10.

( ) 10( )

( ) 100n A BP A B

n S∩

∩ = =

vdnt, P(A mšyJ B)

17. e‹F fiy¤J mL¡» it¡f¥g£l 52 Ó£Lfis¡ bfh©l Ó£L¡ f£oèUªJ rkthŒ¥ò

Kiwæš xU Ó£L vL¡f¥gL»wJ. mªj¢ Ó£L °nglhfnth (Spade) mšyJ Ïuhrhthfnth

(King) ÏU¥gj‰fhd ãfœjféid¡ fh©f. (g: 12.2, f: 10 )

Ô®Î:

TWbtë S = { 52 Ó£LfŸ}. n(S) = 52.

A v‹g J vL¤j Ó£L °ngL (Spade) Mf ÏU¡F« ãfœ¢Á.

A = { 13 °ngL Ó£LfŸ }. n(A) =13.

( ) 13( )( ) 52

n AP An S

= =

B v‹g J vL¤j Ó£L Ïuhrh (King) Mf ÏU¡F« ãfœ¢Á.

B = { 4 Ïuhrh Ó£LfŸ }. n(B) = 4.

( ) 4( )( ) 52

n BP Bn S

= =

A∩B = { °ngL k‰W« Ïuhrh Ó£LfŸ }. n(A∩B) = 1.

( ) 1( )

( ) 52n A BP A B

n S∩

∩ = =




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( ) ( ) ( )13 4 152 52 5217 1

5216524

13

P A P B P A B= + − ∩

= + −

−=

=

=

( ) ( )10 626 2616268

13

P A P B= +

= +

=

=

vdnt, P(A mšyJ B)

18. xU igæš 10 btŸis, 6 Át¥ò k‰W« 10 fU¥ò ãw¥ gªJfŸ cŸsd. rkthŒ¥ò Kiwæš xU

gªÂid vL¡F«nghJ mJ btŸis mšyJ Át¥ò ãw¥ gªjhf ÏU¥gj‰fhd ãfœjféid¡

fh©f. (g: 12.2, f: 11 )

Ô®Î:

TWbtë S = { 10 btŸis ãw¥ gªJfŸ, 6 Át¥ò ãw¥ gªJfŸ, 10 fU¥ò ãw¥ gªJfŸ }. n(S) = 10+6+10 = 26.

A v‹g J btŸis ãw¥ gªjhf ÏU¡F« ãfœ¢Á.

A = { 10 btŸis ãw¥ gªJfŸ }. n(A) =10.

( ) 10( )( ) 26

n AP An S

= =

B v‹g J Át¥ò ãw¥ gªjhf ÏU¡F« ãfœ¢Á.

B = { 6 Át¥ò ãw¥ gªJfŸ }. n(B) = 6.

( ) 6( )( ) 26

n BP Bn S

= =

A∩B = { btŸis k‰W« Át¥ò ãw¥ gªJfŸ }. n(A∩B) = 0. vdnt, P(A mšyJ B)




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( ) ( )3 39 96923

P A P B= +

= +

=

=

19. 2, 5, 9 v‹w v©fis¡ bfh©L, X® Ïu©oy¡f v© mik¡f¥gL»wJ. mªj v© 2 mšyJ

5 Mš tFgLkhW mika ãfœjfÎ fh©f. (mik¡f¥gL« v©âš xnu Ïy¡f« Û©L«

tuyh« ) (g: 12.2, f: 12 )

Ô®Î:

TWbtë S = { 22, 25, 29, 52, 55, 59, 92, 95, 99}. n(S) = 9.

A v‹g J nj®ªbjL¡f¥gL« v© 2 Mš tFgL« ãfœ¢Á.

A = { 22, 52, 92 }. n(A) = 3.

( ) 3( )( ) 9

n AP An S

= =

B v‹g J nj®ªbjL¡f¥gL« v© 5 Mš tFgL« ãfœ¢Á.

B = { 25, 55, 95 }. n(B) = 3.

( ) 3( )( ) 9

n BP Bn S

= =

A∩B = { }. n(A∩B) =0.

vdnt, P(A mšyJ B) ( A, B M»ad x‹iwbah‹W éy¡F«

ãfœ¢ÁfshF«.)

20. “ACCOMMODATION” v‹w brhšè‹ x›bthU vG¤J« jå¤jåna Á¿a fh»j§fëš

vGj¥g£L, mªj 13 Á¿a fh»j§fS« xU Kfitæš it¡f¥g£LŸsd. rkthŒ¥ò Kiwæš

KfitæèUªJ xU fh»j¤ij¤ nj®Î brŒÍ« nghJ, mÂš Ïl« bgW« vG¤J (i) ‘A’ mšyJ

‘O’ Mfnth (ii) ‘M’ mšyJ ‘C’ Mfnth ÏU¥gj‰fhd ãfœjfÎfis¡ fh©f. (g: 12.2, f: 13 )

Ô®Î:

TWbtë S = { A, C, C, O, M, M, O, D, A, T, I , O, N }. n(S) = 13. (i) A v‹g J nj®Î brŒa¥g£l xU vG¤J ‘A’ Mf ÏU¡F« ãfœ¢Á.

A = { A, A }. n(A) = 2.

( ) 2( )( ) 13

n AP An S

= =




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( ) ( )2 3

13 135

13

P A P B= +

= +

=

( ) ( )2 2

13 134

13

P C P D= +

= +

=

B v‹g J nj®Î brŒa¥g£l xU vG¤J ‘O’ Mf ÏU¡F« ãfœ¢Á.

B = { O, O, O }. n(B) = 3.

( ) 3( )( ) 13

n BP Bn S

= =

A∩B = { }. n(A∩B) = 0. vdnt, P(A mšyJ B)

(ii) C v‹g J nj®Î brŒa¥g£l xU vG¤J ‘M’ Mf ÏU¡F« ãfœ¢Á.

C = { M, M }. n(C) = 2.

( ) 2( )( ) 13

n CP Cn S

= =

D v‹g J nj®Î brŒa¥g£l xU vG¤J ‘C’ Mf ÏU¡F« ãfœ¢Á.

D = { C, C }. n(D) = 2.

( ) 2( )( ) 13

n DP Dn S

= =

C∩D = { }. n(C∩D) = 0. vdnt, P(C mšyJ D)

21. xU òÂa k»œÎªJ (car) mjDila totik¥Ã‰fhf éUJ bgW« ãfœjfÎ 0.25 v‹f. Áwªj

Kiwæš vçbghUŸ ga‹gh£o‰fhd éUJ bgW« ãfœjfÎ 0.35 k‰W« ÏU éUJfS«

bgWtj‰fhd ãfœjfÎ 0.15 våš, m«k»œÎªJ (i) FiwªjJ VjhtJ xU éUJ bgWjš (ii) xnu

xU éUJ k£L« bgWjš M»a ãfœ¢ÁfS¡fhd ãfœjfÎfis¡ fh©f. (g: 12.2, f: 14 )

Ô®Î:

A v‹g J k»œÎªJ mjDila totik¥Ã‰fhf éUJ bgW« ãfœ¢Á.

( ) 0.25P A = B v‹g J k»œÎªJ Áwªj Kiwæš vçbghUŸ ga‹gh£o‰fhd éUJ bgW« ãfœ¢Á.

( ) 0.35P B =

( ) 0.15P A B∩ =




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( ) ( ) ( ) ( ) 0.25 0.35 0.15 0.60 0.15 0.45

P A B P A P B P A B∪ = + − ∩= + −= −=

( ) ( ) ( ) ( ) ( ) ( ) 0.25 0.15 0.35 0.15 0.60 0.30 0.3

P A B P A B P A P A B P B P A B∩ + ∩ = − ∩ + − ∩= − + −= −=

(i) FiwªjJ VjhtJ xU éUJ bgWjš ãfœ¢ÁfS¡fhd ãfœjfÎ

(ii) xnu xU éUJ k£L« bgWjš ãfœ¢ÁfS¡fhd ãfœjfÎ

22. A, B, C M»nah® xU édhé‰F¤ Ô®Î fh©gj‰fhd ãfœjfÎfŸ Kiwna 4 2 3

, ,5 3 7

v‹f. A

k‰W« B ÏUtU« nr®ªJ Ô®Î fh©gj‰fhd ãfœjfÎ 8

15 . B k‰W« C ÏUtU« nr®ªJ Ô®Î

fh©gj‰fhd ãfœjfÎ 2

7. A k‰W« C ÏUtU« nr®ªJ Ô®Î fhz ãfœjfÎ

12

35, _tU« nr®ªJ

Ô®Î fhz ãfœjfÎ 8

35våš, ahnuD« xUt® m›édhé‹ Ô®Î fh©gj‰fhd ãfœjféid¡

fh©f. (g: 12.2, f: 15 )

Ô®Î:

A, B, C M»ad Kiwna A, B, C M»nah® xU édhé‰F¤ Ô®Î fh©gj‰fhd ãfœ¢ÁfŸ.

4 2 3 8 2 12 8( ) , ( ) , ( ) ( ) , ( ) , ( ) ( )

5 3 7 15 7 35 35,P A P B P C P A B P B C P A C P A B C= = = ∩ = ∩ = ∩ = ∩ ∩ =

ahnuD« xUt® m›édhé‹ Ô®Î fh©gj‰fhd ãfœjfÎ,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

4 2 3 8 2 12 8 5 3 7 15 7 35 3584 70 45 56 30 36 24

105 105 105 105 105 105 105223 122

105

P A B C P A P B P C P A B P B C P A C P A B C∪ ∪ = + + − ∩ − ∩ − ∩ + ∩ ∩

= + + − − − +

= + + − − − +

−=

101 1057

13

=

=




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€¦ · tisgu¥ò L bkh¤j¥ òw¥gu¥ò – 2 Hmo¥g¡f¤Â‹ gu¥ò 660 2 22 7 660 2 2 22 660 2 7 7 7 660 308 352 =−×πr vdnt, 2 352 22 27 352 7 352 7 2227 8 rh h h