An initiative of NIOS towards Massive Open Online Courses - …mooc.nios.ac.in/mooc/pluginfile.php?file=/13701/course... · 2018. 3. 19. · XXX XL L LX LXX LXXX XC C CC CCC 30 40

≤ÃyÌ!õÑ˛ !ü«˛yÎ˚ !mÓy!°Ï≈Ñ˛ !v˛ Ï≤’yõyDIPLOMA IN ELEMENTARY EDUCATION

(D.El.Ed)

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A-24/25 ≤Ã!ì˛¤˛y!òÑ˛ ~úyÑ˛yñ ˆ¢Ñ˛›˛Ó ˚ÈÙÈ62, òÎ˚v˛y

ˆÜÔì˛õ Ó%Âï òÜÓ˚ñ £zv˛z !˛ôÈÙÈ201309

Á ÏÎ˚Ó ¢y£z›˛ ı www.nios.ac.in

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4 ˆ◊!íÑ˛ Ï«˛Ó Óy£z ÏÓ Ü!íì˛ !ü«˛í~Ñ˛Ñ˛ÈÙÈ3 2

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5 !Óîƒyú ÏÎ Ü!íì˛ ¢!õ!ì˛ Ü‡˛ò~Ñ˛Ñ˛ÈÙÈ4 3

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˛ôÂï!ì˛

¢ÇÖƒy ~ÓÇ ¢ÇÖƒyÓ v z ô ÏÓ!Ó!¶˛ß x˛ôy ÏÓüò

5~Ñ˛Ñ˛ÈÙÈ5 2

xyÑ, !ì˛ Á fiÌyò ¢ÇÑ yhs˘˛ïyÓíy

5

Óy!ü!ÓK˛yòÜì˛ ì˛ ÏÌƒÓ !Ó ÏŸ’°Ïí

~Ñ˛Ñ˛ÈÙÈ6 2

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ì˛Ìƒ ˛ô!Ó˚â˛Î≈y 4~Ñ˛Ñ˛ÈÙÈ8 3

˛ôy!›˛Ü!íˆÏì˛Ó˚ ¢yïyÓ˚íÓ) ô Ó#ãÜ!íì˛

4~Ñ˛Ñ˛ÈÙÈ9 2ÓœÑ˛ÈÙÈ3

Ü!í Ïì˛ !ü«˛yÌ≈# ÏîÓ

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3~Ñ˛Ñ ÈÙÈ10 2

4~Ñ˛Ñ ÈÙÈ11 3

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ˆõy›˛¢Ó≈ Ïõy›˛

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63 27 30

63 + 27 + 30 = 120 hrs.

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!â˛e ≤Ãîü≈ò#Ó˚ ãòƒ Ü!íˆÏì˛Ó˚

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fl % Òˆ Ïú ÎyÁÎ˚yÓ ˚

!Ó°Ïˆ ÏÎ ˚Ó ˚ Ü!íì˛

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~Ñ˛Ñ˛ È7 ˛ô!Óõyí Á ˛ô!Óõy˛ô

~Ñ˛Ñ˛ È8 ì˛Ìƒ ˛ô!Óâ˛Î≈y

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≤Ã!ì˛!›˛ ~Ñ˛ˆÏÑ˛ ¶˛yÜ Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ !Ñ˛ä%È !Ó¶˛yÜ ~ÓÇ v z ô!Ó¶˛yˆÏÜ ≤ÃyÌ!õÑ˛ ˛ôÎ≈yˆÏÎ˚ ~Ñ˛ãò !üü% !Ñ˛ ¶˛yˆÏÓ v zÍ¢y£

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ïyÓ˚íyÜ%!ú xyõÓ˚y !üÖˆÏì˛ ˛ôyÓ˚ˆÏÓy–

~Ñ˛Ñ˛-7

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xyÎ˚ì˛òñ Áãò Á ˛ô!Ó˚õy˛ô ˆÑ˛yˆÏòy ≤ÃÌyÜì˛ ~Ñ˛ˆÏÑ˛Ó˚ ¢y£yÎƒ õy˛ôy £Î˚ñ ~äÈyv ¸yÁ xyõÓ˚y ˛ô!Ó˚!â˛ì˛ £ˆÏÓy ˛ô!Ó˚õy˛ô

Á ˛ô!Ó˚õy Ï ôÓ˚ ¢õ ÏÎ˚Ó˚ ¢ ÏAÜ ã!v ¸ì˛ õƒy!› ∆Ñ˛ ˛ôÂï!ì˛Ó˚ ¢ ÏAÜ–

ÓœÑ˛ ˛ô!Ó˚â˛Î˚˛È

~Ñ˛Ñ˛-8

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ˆÑ˛yˆÏòy ì˛ÌƒˆÏÑ˛ v z ôfiÌy˛ôò Ñ˛Ó˚yñ !ÓˆÏŸ’°Ïí Ñ˛Ó˚y ~ÓÇ ì˛ˆÏÌƒÓ˚ ˜Ó!â˛eƒ ¢¡ôˆÏÑ≈˛ ¢!‡˛Ñ˛ ïyÓ˚íyˆÏÑ˛ fiÌy!˛ôì˛ Ñ˛Ó˚ˆÏÓ–

~Ñ˛Ñ˛-9

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!Ó°ÏÎ¢)â˛#

Ñ !õÑ˛ ¢ÇÖƒy ~Ñ˛ ÏÑ˛Ó òyõ ˛ô,¤˛y ¢ÇÖƒy

1 ~Ñ˛Ñ˛ÈÙÈ5 ı ¢ÇÖƒy ~ÓÇ ¢ÇÖƒyÓ v z ô ÏÓ !Ó!¶˛ß x˛ôy ÏÓüò 2

2 ~Ñ˛Ñ˛ÈÙÈ6 ı xyÑ, !ì˛ Á fiÌyò ¢ÇÑ yhs˘˛ ïyÓ˚íy 41

3 ~Ñ˛Ñ˛ÈÙÈ7 ı ˛ô!Óõyí Á ˛ô!Óõy˛ô 89

4 ~Ñ˛Ñ˛-8 ı ì˛Ìƒ ˛ô!Óâ˛Î≈y 123

5 ~Ñ˛Ñ˛-9 ı ˛ôy!›˛Ü!í Ïì˛Ó ¢yïyÓí Ó) ô Ó#ãÜ!íì˛ 151

ˆòy›˛

1504-Block 2 : ¢ÇÖƒy ~ÓÇ ¢ÇÖƒyÓ˚ v˛z˛ô ÏÓ˚ !Ó!¶˛ß¨ x˛ôy ÏÓ˚üò

¢ÇÖƒy ~ÓÇ ¢ÇÖƒyÓ v z ô ÏÓ !Ó!¶˛ß x˛ôy ÏÓüò

ˆòy›˛

2 D.El.Ed. 504


~Ñ˛Ñ˛ÈÈÈÙÙÙÈ5 ı ¢ÇÖƒy ~ÓÇ ¢ÇÖƒyÓ v zô ÏÓ !Ó!¶˛ß

x˛ôy ÏÓüò

!Ó°ÏÎÓhfl%5.0 – ¶) !õÑ˛y

5.1 – !üÖò v z Ïjüƒ

5.2 – ˛!Ó!¶˛ß ïÓ ÏòÓ ¢ÇÖƒyÓ ˆ¢›˛

5.2.1 – ˛¢ÇÖƒyÓ Üíòy ~ÓÇ ˛ô)í≈ ¢ÇÖƒy

5.2.2 – ˛xÖ[˛ ¢ÇÖƒy

5.3 – ¢ÇÖƒyÓ !Ó!¶˛ß x˛ôy ÏÓü ÏòÓ ˜Ó!üT˛ƒ

5.3.1 – ˛fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒy ~ÓÇ ˛ô)í≈ ¢ÇÖƒyÓ x˛ôy ÏÓüò

5.3.2 – ˛xÖ[˛ ¢ÇÖƒyÓ x˛ôy ÏÓüò

5.3.3 – ˛õ)úî ¢ÇÖƒyÓ x˛ôy ÏÓüò

5.4 – v zÍ˛ôyîÑ˛ Ü%íò#ÎÑ˛ ~ÓÇ Ü%!íì˛Ñ˛

5.4.1 – ˛¢yïyÓí Ü%òò#ÎÑ˛ ~ÓÇ Ü!Ó¤˛ ¢yïyÓí Ü%íò#ÎÑ˛

5.4.2 – ˛¢yïyÓí Ü%!íì˛Ñ˛ ~ÓÇ ú!á¤˛ ¢yïyÓí Ü%!íì˛Ñ˛

5.5 – ˛ôy›˛#Ü!íì˛ ~ÓÇ ≤Ã ÏÎyÜ

5.6 – ˛¢yÓyÇü

5.7 – xy˛ôòyÓ x@˘ÃÜ!ì˛ Îyâ˛y£z ÏÎÓ v z_Ó¢õ)£

5.8 – ˛≤Ãhfl Ïy!Óì˛ ˛ôy‡˛ ~ÓÇ ˆÓö˛y ÏÓ™

5.9 – ˛~Ñ˛Ñ˛ xhs˘˛ xò%ü#úò#

5.0 ¶) !õÑ˛y :

˜îò!®ò ã#Ó Ïò xyõÓy ˆÎ ˆÎ Óhfl% ÓƒÓ£yÓ Ñ˛!Ó Óy ˆî!Ö ˆ¢Ü%!ú ÏÑ˛ ˛ô!ÓõyíÜì˛¶˛y ÏÓ ≤ÃÑ˛yü

Ñ˛Óy ≤Ã ÏÎyãò– ˆÎõòñ ˛ô!ÓÓy ÏÓÓ ¢î¢ƒ ¢ÇÖƒyñ ˆ◊!íÑ˛«˛/!Óîƒyú ÏÎ˚Ó˚ !ü«˛yÌ≈#Ó˚ ¢ÇÖƒyñ Ñ˛y˛ôv ¸

ˆÑ˛òyÓ˚ ›˛yÑ˛yñ ¢Ó!ã Á xòƒyòƒ oˆÏÓƒÓ˚ Áãòñ !üü% ÏîÓ˚ ãòƒ Ó£zñ Óy!v ¸ ˆÌˆÏÑ˛ !ÓîƒyúˆÏÎ˚Ó˚ î)Ó˚cñ

áˆÏÓ˚Ó˚ ˜îá≈ƒñ ≤ÃfiÌ £zì˛ƒy!î– ˛ô!Ó˚õy˛ô Ñ˛Ó˚yÓ˚ ãòƒ ¢ÇÖƒyÓ˚ ≤ÃˆÏÎ˚yãò– Óhfl%˛ˆÏÑ˛ ÜíòyÓ˚ ãòƒñ

!Ó!¶˛ß¨ ˛ô!Ó˚õy˛ôˆÏÑ˛ ≤ÃÑ˛yˆÏüÓ˚ ãòƒñ ˜îá≈ƒñ Áãòñ xyÎ˚ì˛òñ ¢õÎ˚ £zì˛ƒy!îÓ˚ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ

¢ÇÖƒyÓ˚ ¶˛#°Ïí ≤Ã ÏÎ˚yãò– xyõy ÏîÓ˚ ã#Ó ÏòÓ˚ ¢y ÏÌ ¢ÇÖƒy ~õò Áì˛ Ï≤Ãyì˛¶˛y ÏÓ ã!v ¸ì˛ ˆÎñ

ˆòy›˛



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Ñ˛!Ó˚ñ ˆ¢Ü%!ú !Ñ˛hs%˘˛ ¢¶˛ƒì˛yÓ˚ xy!îˆÏì˛ !äÈˆÏúy òy– ¢¶˛ƒì˛yÓ˚ !Ó!¶˛ß¨ Î% ÏÜ !Ó!¶˛ß¨ ïÓ˚ˆÏíÓ˚ ¢ÇÖƒy

ì˛ˆÏ_¥Ó˚ !ÓÑ˛yü £ˆÏÎ˚ˆÏäÈ– xy¢%ò ¢¶˛ƒì˛yÓ˚ !Ó!¶˛ß¨ Î% ÏÜ ˆÎ ˆÎ ¢ÇÖƒyì˛ˆÏ_¥Ó˚ !ÓÑ˛yü £ˆÏÎ˚ˆÏäÈ ˆîˆÏÖ

ˆòÁÎ˚y ÎyÑ˛–

õyÎ˚y ¢¶˛ƒì˛yÓ˚ ≤Ãíyú#

ÓƒÓúò#Î˚ ≤Ãíyú#

ˆÓ˚yõyò ≤Ãíyú#

I II III IV V VI VII VIII IX X1 2 3 4 5 6 7 8 9 10

XI XII XIII XIV XV XVI XVII XVIII XIX XX11 12 13 14 15 16 17 18 19 20

XXX XL L LX LXX LXXX XC C CC CCC30 40 50 60 70 80 90 100 200 300

CD D M400 500 1000

~£z ïÓ˚ˆÏòÓ˚ ¢ÇÖƒy ˛ôÂï!ì˛ˆÏì˛ñ !Ó!¶˛ß¨ ¢ÇÖƒyˆÏÑ˛ õˆÏò Ó˚yÖy Ö%Ó Ñ˛!‡˛ò– ~äÈyv ¸yÁ ˆÎyÜñ !ÓˆÏÎ˚yÜ

£zì˛ƒy!î Ñ˛Ó˚yÓ˚ ¢õÎ˚ ~£z ïÓ˚ˆÏòÓ˚ ¢ÇÖƒy ÓƒÓ£yÓ˚ xì˛ƒhs˘˛ Ñ˛!‡˛ò–

¶˛yÓ˚ Ïì˛Ó˚ xÓîyò ı Óì≈ õyòÑ˛y ÏúÓ˚ îü!õÑ˛ ˛ôÂï!ì˛ xÌ≈yÍ 0, 1, 2, 3, 4, 5, 6, 7, 8 ~ÓÇ

9 ~£z îü!›˛ ¢ÇÖƒyÓ˚ !¶˛!_ˆÏì˛ ˜ì˛Ó˚# ˛ôÂï!ì˛ ¶˛yÓ˚ì˛#Î˚ˆÏîÓ˚ ¢,!T˛ ~ÓÇ xyÓ˚ˆÏÓÓ˚ õyò%°Ï ~Ó˚˛ôÓ˚

ˆ¢!›˛ ì˛yˆÏîÓ˚ ˆîˆÏü ≤Ãâ˛úò ü%Ó% Ñ˛ˆÏÓ˚ ~ÓÇ ~£z¶˛yˆÏÓ ˛ô% ÏÓ˚y ˛ôyÿ˛yì˛ƒ ˆîüÜ%!úˆÏì˛ ~Ó˚ ÓƒÓ£yÓ˚

ˆòy›˛

4 D.El.Ed. 504


ü%Ó% £Î˚– ì˛y£z ~£z ˛ôÂï!ì˛ˆÏÑ˛ !£®%ÈÙÈxyÓ˚Ó#Î˚ ¢ÇÖƒy ˛ôÂï!ì˛ Óúy £Î˚–

~¢ ˛ôÂï!ì˛Ó˚ xòƒì˛õ ¢%!Óïy £ú ˆÎñ ˆÎˆÏÑ˛yˆÏòy Óv ¸ ¢ÇÖƒyˆÏÑ˛Á ~£z îü!›˛ ¢ÇÖƒyÓ˚ õyïƒˆÏõ

≤ÃÑ˛yü Ñ˛Ó˚y ÎyÈÎ˚–

¢ÇÖƒyÓ˚ fiÌyò#Î˚ õyò ı ~£z îü!›˛ ¢ÇÖƒy ~Ñ˛Ñ˛ xÇ ÏÑ˛Ó˚ ¢ÇÖƒyÓ˚ ãß√ ˆîÎ˚– Î!î 9-~Ó˚ ˆÓ!ü

¢ÇÖƒyÓ˚ ≤ÃˆÏÎ˚yãò £Î˚ñ ì˛y£ˆÏú xyõÓ˚y î%£z xÇˆÏÑ˛Ó˚ !Ó!¶˛ß¨ ¢ÇÖƒy ˆÎõò 10, 11, 12, ... 25,

... 59, ...98, ~ÓÇ 99 £zì˛ƒy!îÓ˚ ¢,!T˛ Ñ˛!Ó˚– xy˛ô!ò £Î˚ˆÏì˛y ~£z¶˛yˆÏÓ ¢ÇÖƒy ˜ì˛Ó˚# !Ñ˛¶˛yˆÏÓ

Ñ˛Ó˚ˆÏì˛ £Î˚ ì˛y xyˆÏÜ£z ˆãˆÏòˆÏäÈò– ~£z ïÓ˚ˆÏòÓ˚ ¢ÇÖƒyˆÏÑ˛ î%!›˛ fiÌyò ÌyˆÏÑ˛ñ v˛yò!îˆÏÑ˛Ó˚ fiÌyòˆÏÑ˛

~Ñ˛ ÏÑ˛Ó˚ fiÌyò ~ÓÇ Óyõ!î ÏÑ˛Ó˚ fiÌyò ÏÑ˛ îü ÏÑ˛Ó˚ fiÌyò Óúy £Î˚–

~Ñ˛ˆÏÑ˛Ó˚ fiÌyˆÏòÓ˚ ¢ÇÖƒyÓ˚ ~Ñ˛Ñ˛ õyò xyˆÏäÈ xÌ≈yÍ 26 ¢ÇÖƒy!›˛ˆÏì˛ ~Ñ˛ˆÏÑ˛Ó˚ fiÌyˆÏò xyˆÏäÈ 6

~ÓÇ ~Ó˚ õyòÁ 6. îüˆÏÑ˛Ó˚ fiÌyˆÏò xyˆÏäÈ 2 xÌ≈yÍ 26 ¢ÇÖƒy!›˛Ó˚ îüˆÏÑ˛Ó˚ õyò 20 ÎyÓ˚ ¢yïyÓ˚í

õyò 2. xò%Ó)˛ô¶˛yˆÏÓñ üì˛ˆÏÑ˛Ó˚ fiÌyˆÏò ÌyÑ˛y ˆÑ˛yˆÏòy ¢ÇÖƒyÓ˚ fiÌyò#Î˚ õyò ˆ¢£z ¢ÇÖƒyÓ˚ xAÑ˛

õyˆÏòÓ˚ 100 Ü%ò– xy˛ô!ò ~£z ïÓ˚ˆÏòÓ˚ fiÌyò#Î˚ õyˆÏòÓ˚ ¢yˆÏÌ ˛ô!Ó˚!â˛ì˛–

!Ñ˛hs%˛ñ ˆÎ ¢ÇÖƒyÎ˚ ~Ñ˛!›˛£z xAÑ˛ xyˆÏäÈ ì˛yÓ˚ fiÌyò#Î˚ õyò ˆ¢£z ¢ÇÖƒyÓ˚ õyˆÏòÓ˚ ¢yˆÏÌ ¢õyò–

xy˛ô!ò ü)òƒ (0) ¢ÇÖƒy!›˛ ¢¡ô ÏÑ≈ ãy Ïòò– !£®% ¢ÇÖƒyì˛ Ï_¥ ~£z ü)òƒ ¢ÇÖƒyÓ xÓîyò xòfl ∫#Ñ˛yÎ≈–

ˆÎÖyˆÏò£z Ó¢yˆÏòy ˆ£yÑ˛ òy ˆÑ˛ò ~£z ¢ÇÖƒy!›˛Ó˚ fiÌyò#Î˚ õyò ü)òƒ– ì˛y£ˆÏú ~Ó˚ Ü%Ó˚%c Ñ˛#⁄

~Ñ˛!›˛ !ì˛ò!›˛ xˆÏAÑ˛Ó˚ ¢ÇÖƒy !òòñ ˆÎõò 308. 0 ¢ÇÖƒy!›˛Ó˚ îüˆÏÑ˛Ó˚ fiÌyò#Î˚ õyò ü)òƒ– !Ñ˛hs%˘˛

¶˛yÓ%ò ˆì˛yñ Î!î 0 òy ÌyÑ˛ˆÏì˛y ì˛y£ˆÏú ~£z 308 ¢ÇÖƒy!›˛ ˆÑ˛õò £ˆÏì˛y⁄ ¢ÇÖƒy!›˛ ì˛Öò £ˆÏÎ˚

ˆÎ Ïì˛y 38 ~ÓÇ m® ˜ì˛Ó˚# £ ÏÎ˚ ˆÎ Ïì˛y ì˛Öò– ~Öy Ïò 0 ¢ÇÖƒy!›˛ ~£z fiÌyò!›˛ˆÏÑ˛ !ò!î≈T˛ Ó˚yˆÏÖ

~ÓÇ ¢ÇÖƒy!›˛Ó ¢!‡˛Ñ˛ x!hfl Ïc ÏÑ˛ ÓãyÎ Óy ÏÖ– ¢%ì˛ÓyÇ îü!õÑ˛ ˛ôÂï!ì˛ Ïì˛ 0 ¢ÇÖƒy!›˛ ÏÑ˛ fiÌyòïyÓÑ˛

!£¢y ÏÓ õ Ïò Ñ˛Ó˚y £Î˚–

xyõÓ˚y ˜îò!®ò ã#ÓˆÏò ÓƒÓ£,ì˛ !Ñ˛ä%È ˆõÔ!úÑ˛ ¢ÇÖƒyì˛_¥ !òˆÏÎ˚ xyˆÏúyâ˛òy Ñ˛Ó˚Ó ~ÓÇ ~£z

¢ÇÖƒyì˛ˆÏ_¥Ó˚ v z ôÓ˚ !Ó!¶˛ß¨ ˆõÔ!úÑ˛ x˛ôyˆÏÓ˚üˆÏòÓ˚ ˜Ó!üT˛ƒ !òˆÏÎ˚ xyˆÏúyâ˛òy Ñ˛Ó˚Ó–

~£z ~Ñ˛Ñ˛!›˛ˆÏÑ˛ ¢¡ô)í≈ Ñ˛Ó˚yÓ˚ ãòƒñ xy˛ôòyÓ˚ xhs˘˛ì˛˛ôˆÏ«˛ 10 á^˘›˛y xïƒyÓ¢yÎ˚ ≤ÃˆÏÎ˚yãò–

5.1 !üÖò v z Ïjüƒ

~£z ~Ñ˛Ñ˛!›˛ ˛ôv˛yÓ ˛ô ÏÓ xy˛ô!ò

● îü!õÑ˛ ˛ôÂï!ì˛ Ïì˛ ¢ÇÖƒyÓ˚ Ü%Ó%c Ó%G˛ Ïì˛ ¢«˛õ £ ÏÓò

● !Ó!¶˛ß¨ ïÓ˚ˆÏòÓ˚ ¢ÇÖƒyÓ˚ ˆ¢›˛ ˆÎõò fl˛∫y¶˛y!ÓÑ˛ ¢ÇÖƒyñ xÖ[˛ ¢ÇÖƒy ~ÓÇ õ)úî ¢ÇÖƒy

!â˛ò Ïì˛ ¢«˛õ £ ÏÓ–

● !Ñ˛ä%È ˆõÔ!úÑ˛ x˛ôyˆÏÓ˚üò ˆÎõò ˆÎyÜñ !ÓˆÏÎ˚yÜñ Ü%í Á ¶˛yˆÏÜÓ˚ !Ó!¶˛ß¨ ¢ÇÖƒyÓ˚ v˛z˛ôÓ˚

ˆòy›˛



≤Ã ÏÎ˚yÜ ãyò Ïì˛ ¢«˛õ £ ÏÓò–

● fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ Ü%òò#Î˚Ñ˛ Á Ü%!íì˛Ñ˛ !òí≈ ÏÎ˚ ¢«˛õ £ ÏÓò–

5.2 !Ó!¶˛ß ¢ÇÖƒyÓ ˆ¢›˛

5.2.1 ¢ÇÖƒyÓ˚ Üíòy ~ÓÇ ˛ô)í≈ ¢ÇÖƒy

˛≤Ãyâ˛#òÑ˛yˆÏú õyò%°Ï ~£z ¢ÇÖƒyì˛_¥ ˆâ˛ˆÏÎ˚!äÈˆÏúò õ)úì˛ Óhfl% Ó˚ Üíòy Ñ˛yˆÏÎ≈Ó˚ ãòƒñ ì˛y£z ì˛yÓ˚y

Üíòy Ñ˛Ó˚yÓ˚ !Ó!¶˛ß¨ ¢ÇÖƒy ˆÎõò 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11... £zì˛ƒy!îÓ˚ xy!Ó‹ÒyÓ˚

Ñ˛ ÏÓ˚– ~Ü%!ú ÏÑ˛ Óúy £Î˚ ˛ô)í≈ ¢ÇÖƒy–

ì˛yÓ˚y ¢ÇÖƒyÜ%!úˆÏÑ˛ !Ó!¶˛ß¨ Óhfl%˛Ó˚ ¢yˆÏÌ ¢ÇÎ%=˛ Ñ˛ˆÏÓ˚ò ı

¢%Ó˚ì˛yÇñ ÜíòyÓ˚ ãòƒ ≤Ã!ì˛!›˛ ¢ÇÖƒyˆÏÑ˛ ~Ñ˛!›˛ ¢Ç@˘ÃˆÏ£Ó˚ ¢yˆÏÌ ¢ÇÎ%=˛ Ñ˛Ó˚y £ú– ˛ôÎ≈ˆÏÓ«˛í Ñ˛Ó˚%ò

ˆÎñ

(i) 1 £ú «% oì˛õ ÜíòyÓ˚ ¢ÇÖƒy

(ii) ≤Ã!ì˛!›˛ ¢ÇÖƒyÓ˚ ~Ñ˛!›˛ v˛z_Ó˚¢)Ó˚# xyˆÏäÈ Îy õ)ú ¢ÇÖƒy xˆÏ˛ô«˛y 1 ˆÓ!ü xÌ≈yÍ 4ÈÙÈ~Ó˚

v z_Ó˚¢)# 5 ~ÓÇ 29ÈÙÈ~Ó˚ v z_Ó˚¢)Ó˚# 30.

(iii) ≤Ã!ì˛!›˛ ¢ÇÖƒyÓ˚ S1 Óƒì˛#ì˛V ~Ñ˛!›˛ ˛ô)Ó≈¢)Ó˚# xyˆÏäÈ xÌ≈yÍ 7ÈÙÈ~Ó˚ ˛ô)Ó≈¢)Ó˚# 6 ~ÓÇ 60ÈÙÈ~Ó˚

˛ô)Ó≈¢)Ó˚# 59.

(ii) ˆÌˆÏÑ˛ ˛ôyÁÎ˚y ÎyÎ˚ ˆÎñ ˆÎˆÏÑ˛yˆÏòy ÜíòyÓ˚ ¢ÇÖƒyñ ì˛y Îì˛£z ÓˆÏv˛¸y ˆ£yÑ˛ òy ˆÑ˛òñ ì˛yÓ˚

ˆâ˛ˆÏÎ˚Á ÓˆÏv ¸y ÜíòyÓ˚ ¢ÇÖƒyÓ˚ x!hfl˛Ïc xyˆÏäÈ–

˛ô)í≈ ¢ÇÖƒy ı xy˛ô!ò £Î˚ˆÏì˛y ú«˛ƒ Ñ˛ˆÏÓ˚ ÌyÑ˛ˆÏÓò ˆÎ fl˛∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ ˆ¢ˆÏ›˛ 0 xhs˘˛¶)≈˛=˛ òÎ˚–

Ñ˛yÓ˚í Üíòy Ñ˛Ó˚yÓ˚ ¢õÎ˚ ¢Ó≈îy 1 ˆÌˆÏÑ˛ ü%Ó% Ñ˛Ó˚y £Î˚– !Ñ˛hs%˛ ÎÖò xyõÓ˚y ¢ÇÖƒyˆÏÑ˛ xˆÏAÑ˛Ó˚

õyïƒˆÏõ ≤ÃÑ˛yü Ñ˛!Ó˚ñ ì˛Öò xyõÓ˚y 10, 20, 30, ..., 100 £zì˛ƒy!î £zì˛ƒy!î ¢ÇÖƒyÓ˚ ≤ÃÑ˛y Ïü 0

ÓƒÓ£yÓ˚ Ñ˛!Ó˚– xyÓ˚Á ¢£ã ¢Ó˚ú Ñ˛Ó˚yÓ˚ v z Ïj Ïüƒ fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ ˆ¢ Ï›˛ 0 ˆÑ˛ xhs˘˛¶)≈ =˛ Ñ˛ ÏÓ˚

ˆ¢£z òì% ò ˆ¢›˛ˆÏÑ˛ Ú˛ô)í≈ ¢ÇÖƒyÛ Óúy £Î˚ ~ÓÇ W myÓ˚y ~£z ˆ¢›˛ˆÏÑ˛ !â˛!£´ì˛ Ñ˛Ó˚y £Î˚–

E1. ÜíòyÓ˚ ¢ÇÖƒy 1ÈÙÈ~Ó˚ ˆÑ˛ò ˆÑ˛yˆÏòy ˛ô)Ó≈¢)Ó˚# ˆò£z⁄

E2. «%˛oì˛õ ˛ô)í≈¢ÇÖƒy ˆÑ˛yò‰!›˛⁄

Collection

of objects

Number name one Two three four

Numeral 1 2 3 4

Óhfl% Ó˚ ¢Ç@˘Ã£

¢ÇÖƒyÓ˚ òyõ

xAÑ˛

~Ñ˛ î%£z !ì˛ò â˛yÓ˚

ˆòy›˛

6 D.El.Ed. 504


E3. 8 ¢ÇÖƒy!›˛Ó˚ fiÌyò#Î˚ õyò ~ÓÇ xAÑ˛õyò Ñ˛ì˛ ÎÖò ˆ¢!›˛

(i) ~Ñ˛ ÏÑ˛Ó˚ fiÌy Ïò Ìy ÏÑ˛

(ii) îü ÏÑ˛Ó˚ fiÌy Ïò Ìy ÏÑ˛

(iii) üì˛ˆÏÑ˛Ó˚ fiÌyˆÏò ÌyˆÏÑ˛

5.2.2 xÖ[˛ ¢ÇÖƒy

xyõÓ˚y ˜îò!®ò ã#ÓˆÏò ˆÎ ¢õhfl˛Ï ˛ô!Ó˚!fiÌ!ì˛Ó˚ ¢¡ø%Ö#ò ££zñ xyõÓ˚y !Ó!¶˛ß¨ !Ó˛ôÓ˚#ì˛õïõ≈#

˛ô!Ó˚õy˛ô ˆîÖ Ïì˛ ˛ôy£zñ ˆÎõòÈÙÙÙÈ

úy¶˛È Ù È«˛!ì˛

¢¡ôî ÈÙÈ îyÎ˚

ãõy ÈÙÈ ÖÓ˚â˛

v zïπ≈õ%Ö# ÈÙÈ !ò¡¨õ%Ö#

~£z ¢õhfl˛Ï !Ó˛ôÓ˚#ì˛ïõ≈# ïyÓ˚íyÓ˚ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ¢õÎ˚ xyõÓ˚y !Ñ˛ä%È ¶˛yÓ˚¢yõƒ Ö%§ˆÏã ˛ôy£z ı

!Ó˛ôÓ˚#ì˛ ˜Ó!üˆÏT˛ƒÓ˚ ¢yõƒyÓfiÌy

˛ô!Ó˚õy˛ô

úy¶˛ ÈÙÈ «˛!ì˛ ˆÑ˛y Ïòy úy¶˛ Óy ˆÑ˛y Ïòy «˛!ì˛ ˆò£z

¢¡ôî ÈÙÈ îyÎ˚ ˆÑ˛yˆÏòy ¢¡ôî Óy ˆÑ˛yˆÏòy îyÎ˚ ˆò£z

ãõy ÈÙÈ ÖÓ˚â˛ !Ñ˛ä%È ãõy ˆò£z Óy !Ñ˛ä%È ÖÓ˚â˛ £Î˚!ò

v zïπ≈õ%Ö# ÈÙÈ !ò¡¨õ%Ö# v zïπ≈õ%Ö#Á òÎ˚ xyÓyÓ˚ !ò¡¨õ%Ö#Á òÎ˚–

v z ôˆÏÓ˚Ó˚ ¢yõƒyÓfiÌy ˆîˆÏÖ ˆÓyG˛y ÎyˆÏFäÈ ˆÎ ~Ü%!ú ü)òƒ xÓfiÌyˆÏÑ˛ !òˆÏî≈ü Ñ˛Ó˚ˆÏäÈ–

ì˛y£z õyò%°Ï 1, 2, 3 £zì˛ƒy!îÓ˚ !Ó˛ôÓ˚#ì˛ ¢ÇÖƒyÓ˚Á ¢,!T˛ Ñ˛ˆÏÓ˚ˆÏäÈ– ~ÓÇ ~£z¶˛yˆÏÓ xyõÓ˚y !Ñ˛ä%È

¢ÇÖƒyÓ˚ ˆãyv˛¸ ˛ôy£z ÎyÓ˚y ˛ôÓ˚fl˛ôÓ˚ !Ó˛ôÓ˚#ì˛ïõ≈# ˆÎõò ı

+ 1 ~ÓÇ – 1

+ 2 ~ÓÇ – 2

+ 3 ~ÓÇ – 3 £zì˛ƒy!î

~£z ≤Ã!ì˛!›˛ !Ó˛ôÓ˚#ì˛ ˆãy Ïv ¸Ó˚ õ Ïïƒ ¢yõƒÓãyÎ˚ Ó˚yÖyÓ˚ ¢ÇÖƒy £ú ü)òƒ– xyõÓ˚y ˛ôy£z ı

(+1) + (–1) = 0

(+2) + (–2) = 0

(+3) + (–3) = 0 £zì˛ƒy!î

¢%ì˛Ó˚yÇ ¢ÇÖƒyÓ˚ ˆÎ ˆ◊!í xyõÓ˚y ˆ˛ôúyõ ˆ¢!›˛ £ú ı

ˆòy›˛



....... –4, –3, –2, –1, 0, +1, +2, +3, +4, ... ¢ÇÖƒyÓ˚ ~£z ˆ¢›˛ ÏÑ˛ Óúy £Î˚ xÖ[˛ ¢ÇÖƒy

+1, +2, +3, +4, .... £zì˛ƒy!îˆÏÑ˛ Óúy £Î˚ ïòydÑ˛ xÖ[˛ ¢ÇÖƒy ~ÓÇ –1, –2, –3, –4,...

£zì˛ƒy!î ÏÑ˛ Óúy £Î˚ }òydÑ˛ xÖ[˛ ¢ÇÖƒy–

xÖ[˛ ¢ÇÖƒyÓ˚ ˆ¢›˛ ÏÑ˛ Z x«˛Ó˚ myÓ˚y ¢)!â˛ì˛ Ñ˛Ó˚y £Î˚–

(i) ~õò ˆÑ˛yˆÏòy xÖ[˛ ¢ÇÖƒy ˆò£z ÎyˆÏÑ˛ Ó,£_õ Óúy ÎyˆÏÓ ˆÎˆÏÑ˛yˆÏòy ÓˆÏv ¸y xÖ[˛ ¢ÇÖƒy

¶˛yÓ%òñ ~Ó˚ ˆâ˛ ÏÎ˚Á Ó Ïv ¸y xÖ[˛ ¢ÇÖƒyÓ˚ x!hfl Ïc xy ÏäÈ–

(ii) ~õò ˆÑ˛y Ïòy xÖ[˛ ¢ÇÖƒy ˆò£z Îy ÏÑ˛ «% oì˛õ Óúy Îy ÏÓ– ˆÎ ÏÑ˛y Ïòy «% o xÖ[˛ ¢ÇÖƒyÓ˚

Ñ˛Ìy ¶˛yÓ%òñ ~Ó˚ ˆâ˛ˆÏÎ˚Á ˆäÈyˆÏ›˛y xÖ[˛ ¢ÇÖƒyÓ˚ x!hfl˛Ïc xyˆÏäÈ–

(iii) ˆÎ ÏÑ˛y Ïòy +p xÖ[˛ ¢ÇÖƒyÓ˚ ãòƒñ ˆ◊!í Ïì˛ –p ¢ÇÖƒy!›˛Ó˚ x!hfl˛Ïc xyˆÏäÈ ÎyÓ˚ ãòƒ (+p)

+ (–p) = 0 £ ÏÓ– +p ~ÓÇ –p ˆÑ˛ ˛ôÓ˚fl˛ôÓ˚ !Ó˛ôÓ˚#ì˛ ¢ÇÖƒy Óúy £Î˚–

(iv) ü)òƒ S0V ïòydÑ˛Á òy }íydÑ˛Á òy–

Ñ !õÑ˛ xÖ[˛ ¢ÇÖƒy ~ÓÇ ¢ÇÖƒy ˆÓ˚ÖyÎ˚ ì˛y ÏîÓ˚ ≤ÃÑ˛yü ı

!òˆÏâ˛ ˆÎ xÖ[˛ ¢ÇÖƒyÓ˚ ˆ◊!í!›˛ xyˆÏäÈ ˆ¢!›˛ v˛yò!îˆÏÑ˛ Ñ õÓï≈õyò ~ÓÇ Óyõ!îˆÏÑ˛ Ñ õ £…y¢õyò–

........., –3, –2, –1, 0, +1, +2, +3 ............

ì˛y£ˆÏú ..... < –3 < –2 < –1 <0 < +1 < 2 < +3 < ............

xyõÓ˚y xyÓ˚Á ¶˛yÓˆÏì˛ ˛ôy!Ó˚ ˆÎñ ~Ñ˛!›˛ ¢Ó˚úˆÏÓ˚ÖyÓ˚ v z ôˆÏÓ˚Ó˚ !Ó®%Ü%!ú xÖ[˛ ¢ÇÖƒyˆÏÑ˛ ≤ÃÑ˛yü

Ñ˛ˆÏÓ˚– ~£z ïyÓ˚íy xò%¢Ó˚í Ñ˛ˆÏÓ˚ ˛ôy£z ı

(i) ~Ñ˛!›˛ ¢Ó˚ú ÏÓ˚Öy xAÑ˛ò Ñ˛Ó%ò ~ÓÇ òyõ !îò L S~Öy Ïò L ¢õhfl Ï ˆÓ˚Öy!›˛ ÏÑ˛ ¢)!â˛ì˛ Ñ˛ ÏÓ˚–

~!›˛ Á£z ˆÓ˚ÖyÓ˚ v z˛ôˆÏÓ˚ xÓ!fiÌì˛ ˆÑ˛yˆÏòy !Ó®%Ó˚ òyõ òÎ˚ÈV

(ii) !ò!î≈T˛ î)Ó˚c xhs˘˛Ó˚ !Ó®%Ü%!úˆÏÑ˛ !â˛!£´ì˛ Ñ˛Ó%ò–

(iii) ˆÎˆÏÑ˛yˆÏòy ~Ñ˛!›˛ !Ó®%ˆÏÑ˛ O òyõ !îò ~ÓÇ ~›˛yÓ˚ õyïƒˆÏõ O ˆÑ˛ ≤ÃÑ˛yü Ñ˛Ó˚%ò–

(iv) ~ÓyÓ˚ +1, +2, +3 £zì˛ƒy!î ¢ÇÖƒyˆÏÑ˛ ≤ÃÑ˛yü Ñ˛Ó˚%ò 0 ~Ó˚ ˛ôÓ˚Óì≈˛# Ñ ˛!õÑ˛ !Ó®%Ü%!úÓ˚

õyïƒˆÏõ–

(v) 0ÈÙÈ~Ó˚ Óyõ!î ÏÑ˛Ó˚ Ñ !õÑ˛ !Ó®%Ü%!úÓ˚ õyïƒ Ïõ –1, –2, –3 £zì˛ƒy!îˆÏÑ˛ ≤ÃÑ˛yü Ñ˛Ó˚%ò–

~ÓyÓ˚ ~£z ˆÓ˚ÖyÓ˚ òyõ !îò ¢ÇÖƒy ÏÓ˚Öy–

ˆòy›˛

8 D.El.Ed. 504


5.2.3 õ)úî ¢ÇÖƒy

~ÓyÓ˚ ¢õ@˘Ã Óhfl% Ó˚ !Ó!¶˛ß¨ xÇüˆÏÑ˛ ú«˛ƒ Ñ˛Ó%ò ı

!ò¡¨!ú!Öì˛¶˛y ÏÓ ~Ñ˛!›˛ ¢õ@˘Ã xÇ ÏüÓ˚ ¶˛yÜÜ%!ú ÏÑ˛ ˛≤ÃÑ˛yü Ñ˛Ó˚y ÎyÎ˚ ı

~Ñ˛!›˛ Óhfl% Ó˚ 2!›˛ ¢õyò ¶˛yˆÏÜÓ˚ ~Ñ˛!›˛ ¶˛yÜ ı 12 SxˆÏï≈Ñ˛ñ ~ˆÏÑ˛Ó˚ î%£zV

~Ñ˛!›˛ Óhfl%˛Ó˚ 3!›˛ ¢õyò ¶˛yˆÏÜÓ˚ ~Ñ˛!›˛ ¶˛yÜ ı 13 S~Ñ˛ÈÙÈì,˛ì˛#Î˚yÇüñ ~ˆÏÑ˛Ó˚ !ì˛òV

~Ñ˛!›˛ Óhfl%˛Ó˚ 4!›˛ ¢õyò ¶˛y ÏÜÓ˚ ~Ñ˛!›˛ ¶˛yÜ ı 14 S~Ñ˛ÈÙÈâ˛ì% Ì≈yÇüñ ~ ÏÑ˛Ó˚ â˛yÓ˚V

xò%Ó˚)ˆÏ˛ôñ ~£z ˆÓ˚Öy!AÑ˛ì˛ !â˛eÜ%!úˆÏÑ˛ ˆîÖ%ò ı

34 ˆÑ˛ ≤ÃÑ˛yü Ñ˛ˆÏÓ˚ [3 úÓ ~ÓÇ 4 £Ó]

23 ˆÑ˛ ≤ÃÑ˛yü Ñ˛ˆÏÓ˚ [2 úÓ ~ÓÇ 3 £Ó˚]

46 ˆÑ˛ ≤ÃÑ˛yü Ñ˛ˆÏÓ˚ [4 úÓ ~ÓÇ 6 £Ó]

ˆÎ ¢ÇÖƒyÜ%!ú ˆÑ˛yˆÏòy ¢õ@˘Ã Óhfl% Ó˚ !Ó!¶˛ß¨ xÇüˆÏÑ˛ ≤ÃÑ˛yü Ñ˛ˆÏÓ˚ ì˛yˆÏîÓ˚ ¶˛@¿yÇü ¢ÇÖƒy Óy

¶˛@¿yÇü Óúy £Î– !Ó!¶˛ß¨ ïÓ˚ ÏíÓ˚ ¶˛@¿yÇü xy ÏäÈ ı

(i) ≤ÃÑ, ì˛ ¶˛@¿yÇü ı 2 3 5, ,3 8 7 ~Ü%!ú ≤ÃÑ, ì˛ ¶˛@¿yÇüñ ˆÎÖyˆÏò úÓ < £Ó˚–

~Ñ˛ î%!›˛ ¢õyò¶˛yˆÏÜÓ˚~Ñ˛!›˛ ¶˛yÜ

!ì˛ò!›˛ ¢õyò¶˛yˆÏÜÓ˚~Ñ˛!›˛ ¶˛yÜ

â˛yÓ˚!›˛˛ ¢õyò¶˛yˆÏÜÓ˚~Ñ˛!›˛ ¶˛yÜ

ˆòy›˛



(ii) x≤ÃÑ, ì˛ ¶˛@¿yÇü ı 5 11 28, ,3 7 5 £zì˛ƒy!î x≤ÃÑ, ì˛ ¶˛@¿yÇü ˆÎÖy Ïò úÓ > £Ó˚–

(iii) !õ◊ ¶˛@¿yÇü ı 1 22 ,33 7 £zì˛ƒy!î !õ◊ ¢ÇÖƒy ~ÓÇ ~Ü%!ú x≤ÃÑ,˛ì˛ ¶˛@¿yÇˆÏü Ó˚)˛ôyhs˘˛!Ó˚ì˛

£ˆÏì˛ ˛ôyˆÏÓ˚ ~ÓÇ x≤ÃÑ,˛ì˛ ¶˛@¿yÇü Á !õ◊ ¶˛@¿yÇˆÏü Ó˚)˛ôyhs˘˛!Ó˚ì˛ £ˆÏì˛ ˛ôyˆÏÓ˚– ˆÎõò ı

1 7 2 232 , 33 3 7 7= =

(iv) ~Ñ˛Ñ˛ ¶˛@¿yÇü ı ˆÎ¢Ó ¶˛@¿yÇ ÏüÓ˚ úÓ 1, ˆÎõò 1 1 1, ,2 5 9 ~Ñ˛Ñ˛ ¶˛@¿yÇ ÏüÓ˚ v zîy£Ó˚í–

(v) ¢õì% úƒ ¶˛@¿yÇü ı ÎÖò ˆÑ˛y Ïòy ¶˛@¿yÇ ÏüÓ˚ úÓ Á £Ó˚ ÏÑ˛ ˆÑ˛y Ïòy ïòydÑ˛ ¢ÇÖƒy !î ÏÎ˚ Ü%ò

Ñ˛Ó˚y £Î˚ ì˛Öò ì˛yÓ˚ Ó˚)˛ô ˛ô!Ó˚Ó!ì≈˛ì˛ £Î˚ !Ñ˛hs%˘˛ õyò ~Ñ˛£z ÌyˆÏÑ˛– x!ã≈ì˛ ¶˛@¿yÇü!›˛ xy¢ú

¶˛@¿yÇˆÏüÓ˚ ¢õyò£z ÌyˆÏÑ˛–

~£z õyò!Ó!üT˛ ¶˛@¿yÇü ÏÑ˛ ¢õì% úƒ ¶˛@¿yÇü Ó Ïú–

v zîy£Ó˚í ı

(a) 2 4 8 ,3 6 12= = ~ÖyˆÏò 2 4 8, ,

3 6 12 ¢õì% úƒ ¶˛@¿yÇü

(b) 40 20 10 516 8 4 2

= = = ~ÖyˆÏò 40 20 10 5, , ,16 8 4 2 ¢õì% úƒ ¶˛@¿yÇü

(vi) ¢î,ü ¶˛@¿yÇü ı ~Ñ˛£z £Ó˚!Ó!üT˛ ¶˛@¿yÇüˆÏÑ˛ ¢î,ü ¶˛@¿yÇü ÓˆÏú–

1 2 3 4, , ,15 15 15 15 £zì˛ƒy!î ¢î,ü ¶˛@¿yÇ ÏüÓ˚ v zîy£Ó˚í–

õ)úî ¢ÇÖƒy ı pq xyÑ˛yˆÏÓ˚ ≤ÃÑ˛y!üì˛ ¢ÇÖƒy ˆÎÖyˆÏòñ p ~ÓÇ q xÖ[˛ ¢ÇÖƒy ~ÓÇ q ≠ 0

Óúy £Î˚ õ)úî ¢ÇÖƒyñ Q !â˛£´ myÓ˚y õ)úî ¢ÇÖƒyˆÏÑ˛ ¢)!â˛ì˛ Ñ˛Ó˚y £Î˚– fl˛ôT˛¶˛yˆÏÓ£zñ ¶˛@¿yÇü õ)úî

¢ÇÖƒy–

õ)úî ¢ÇÖƒyÓ˚ v zîy£Ó˚í ı 2 3 0 4, , ,7 4 5 9

−−

ú«˛ƒ Ñ˛Ó%ò ı 3 10 8 01 , 2 , 4 ,08 5 2 9

−= = − = =

ˆòy›˛

10 D.El.Ed. 504


ˆîÖy Îy ÏFäÈ ˆÎ ¢Ó xÖ[˛ ¢ÇÖƒy ÏÑ˛ pq xyÑ˛yˆÏÓ˚ ˆúÖy ÎyˆÏÓñ ˆÎÖyˆÏò p ~ÓÇ q xÖ[˛ ¢ÇÖƒy

~ÓÇ q ≠ 0 ¢%ì˛Ó˚yÇ xyõÓ˚y ˆ˛ôúyõ ˆÎñ ≤Ã!ì˛!›˛ xÖ[˛ ¢ÇÖƒy£z õ)úî ¢ÇÖƒy–

fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒy (N), ˛ô)í≈ ¢ÇÖƒy (W)2 xÖ[˛ ¢ÇÖƒy (Z) ~ÓÇ õ)úî ¢ÇÖƒyÓ˚ (Q) xydı¢¡ôÑ≈˛

!ò Ïâ˛Ó˚ !â˛ ÏeÓ˚ ¢y£y ÏÎƒ ≤ÃÑ˛yü Ñ˛Ó˚y ÎyÎ˚–

QÈÙÈ~Ó˚ õ Ïïƒ ¢õhfl Ï xÖ[˛ ¢ÇÖƒy ~ÓÇ xòƒyòƒ ¢ÇÖƒy ˆÎõò 1 2 3, ,2 7 8

− £zì˛ƒy!î xhs˘˛¶%≈˛=˛–

ZÈÙÈ~Ó˚ õˆÏïƒ ¢õhfl˛Ï ˛ô)í≈ ¢ÇÖƒy ~ÓÇ }íydÑ˛ ¢ÇÖƒy –1, –2, –3, . £zì˛ƒy!î xhs˘˛¶%≈˛=˛–

WÈÙÈ~Ó˚ õˆÏïƒ ¢õhfl˛Ï fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒy ~ÓÇ ü)òƒ xhs˘˛¶%≈˛=˛–

¢ÇÖƒy ÏÓ˚ÖyÓ˚ ¢y£y ÏÎƒ õ)úî ¢ÇÖƒy ı xyõÓ˚y ~Ó˚ xyˆÏÜ ˆîˆÏÖ!äÈ ˆÎ ¢ÇÖƒyˆÏÓ˚ÖyÓ˚ v z ôÓ˚ xÖ[˛

¢ÇÖƒy ÏÑ˛ !Ñ˛¶˛y ÏÓ ˆîÖy Ïòy ÎyÎ˚– ~ÓyÓ˚ ˆîÖÓ ¢ÇÖƒy ÏÓ˚ÖyÓ˚ v z ô ÏÓ˚ õ)úî ¢ÇÖƒy ÏÑ˛ !Ñ˛¶˛y ÏÓ

ˆîÖyˆÏòy ÎyÎ˚–

ˆîÖy ÎyÑ˛ (i) 34 , (ii) 2

3− ~ÓÇ (iii) 22

3 ˆÑ˛ !Ñ˛¶˛y ÏÓ ¢ÇÖƒy ÏÓ˚ÖyÓ˚ ¢y£y ÏÎƒ ˆîÖy Ïòy ÎyÎ˚ı

xyõÓ˚y ãy!ò ˆÎñ 34 £ú 4!›˛ ¢õyò ¶˛yˆÏÜÓ˚ 3 ¶˛yÜ–

¢ÇÖƒy ÏÓ˚ÖyÓ˚ v z ôÓ˚ 0 ˆÌˆÏÑ˛ +1 ÙÈ~Ó˚ õˆÏïƒÓ˚ ãyÎ˚ÜyˆÏÑ˛ â˛yÓ˚!›˛ ¢õyò ¶˛yˆÏÜ ¶˛yxÜ Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ

~ÓÇ A, B, C ~£z !Ó®% myÓ ˚ y 4!›˛ ¶˛yÜˆ ÏÑ˛ ˆîÖyˆ Ïòy £ˆ ÏÎ ˚ˆ Ïä È–

0 ˆÌˆÏÑ˛ C ˛ôÎ≈hs˘˛ 4!›˛ ¶˛yˆÏÜÓ˚ 3 ¶˛yÜ xyˆÏäÈ–

∴ C, 34 ˆÑ˛ ≤ÃÑ˛yü Ñ˛Ó˚ ÏäÈ

25

− ˆÑ˛ ≤ÃÑ˛yü Ñ˛Ó˚yÓ˚ ãòƒÈÙÙÙÈ

3/4

–1 O A B C +1

–2/5

–1 D ABC 0

ˆòy›˛



¢ÇÖƒy ÏÓ˚ÖyÎ˚ B !Ó®%!›˛ 25

− ˆÑ˛ ≤ÃÑ˛yü Ñ˛ˆÏÓ˚–

~ Ï«˛ Ïe B !Ó®%!›˛ È 8 223 3= ˆÑ˛ ≤ÃÑ˛yü Ñ˛ˆÏÓ˚–

õ)úî ¢ÇÖƒy ˆúÖyÓ˚ xyîü≈ Ó) ô ı ÎÖò ˆÑ˛y Ïòy õ)úî ¢ÇÖƒy ÏÑ˛ pq xyÑ˛yˆÏÓ˚ ˆúÖy £ˆÏÓ

ˆ¢ÖyˆÏò p ~ÓÇ q ¢ÇÖƒy î%!ÎÓ˚ ~Ñ˛õye ¢yïyÓ˚í Ü%òò#Î˚Ñ˛ 1 ~ÓÇ q > 0, ì˛Öò ˆ¢£z

¢ÇÖƒy!Î ÏÑ˛ xyîü≈ xyÑ˛yÓ˚ Óy Ó) ô Óúy £Î˚–

˛ôÓ˚Óì≈˛# !Ó¶˛yˆÏÜ ÎyÁÎ˚yÓ˚ xyˆÏÜ xy˛ôòyÓ˚ x@˘ÃÜ!ì˛ Îyâ˛y£z Ñ˛Ó˚%ò ı

E4. !ò Ïâ˛Ó˚ !ÓÑ, !ì˛Ü%!ú Ú¢ì˛ƒÛ Óy Ú!õÌƒyÛ !úÖ%ò ı

(a) ¢Ó fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒy xÖ[˛ ¢ÇÖƒy

(b) ¢Ó xÖ[˛ ¢ÇÖƒy£z ˛ô)í≈ ¢ÇÖƒy

(c) ˛ô)í≈ ¢ÇÖƒy õ)úî ¢ÇÖƒy £ˆÏì˛ ˛ôyˆÏÓ˚ òy

(d) ¢Ó õ)úî ¢ÇÖƒy xÖ[˛ ¢ÇÖƒy òÎ˚–

5.3 ¢ÇÖƒyÓ x˛ôy ÏÓü ÏòÓ ˜Ó!üT˛ƒ ı

xyõÓy â˛yÓ!›˛ ˆõÔ!úÑ˛ x˛ôy ÏÓüò ˆÎyÜñ !Ó ÏÎyÜñ Ü%í Á ¶˛y ÏÜÓ ¢y ÏÌ ˛ô!Ó!â˛ì˛ ~ÓÇ ~Ü%!ú

≤Ã ÏÎyÜ Ñ˛Ó Ïì˛ î«˛– ~£z !Ó¶˛y ÏÜ xy˛ôòyÓy !Ó!¶˛ß ¢ÇÖƒyÓ ˆ¢ Ï›˛Ó v z ôÓ ~£z â˛yÓ!›˛ x˛ôy ÏÓü ÏòÓ

˜Ó!üT˛ƒ ãyò Ïì˛ ˛ôyÓ ÏÓò–

5.3.1 fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒy ~ÓÇ ˛ô)í≈ ¢ÇÖƒyÓ˚ x˛ôy ÏÓ˚üò

xyˆÏÜ£z ˆãˆÏòˆÏäÈò ˆÎñ fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒy ~ÓÇ ˛ô)í≈ ¢ÇÖƒyÓ˚ ˛ôyÌ≈Ñ˛ƒ Óì≈˛õyò ~ÓÇ ~£z ˛ôyÌ≈Ñ˛ƒ

‘0’ ¢ÇÖƒy!›˛ˆÏÑ˛ xhs˘˛¶%≈˛!=˛Ó˚ õyïƒˆÏõ– ì˛y£z ~£z â˛yÓ˚!›˛ x˛ôyˆÏÓ˚üò ~£z î%ïÓ˚ˆÏíÓ˚ ¢ÇÖƒyÓ˚ ˆ¢ˆÏ›˛Ó˚

ãòƒ ≤ÃyÎ˚ xò%Ó) ô ~ÓÇ ˆ¢£zãòƒ ~Ü%!úˆÏÑ˛ ~Ñ˛£z¢ˆÏAÜ xyˆÏúyâ˛òy Ñ˛Ó˚y £ˆÏÓ–

(a) ˆÎyÜ ı ÎÖò î%!›˛ xyúyîy xyúyîy ¢Ç@˘Ã£ Óy ¢õ!T˛ˆÏÑ˛ ~Ñ˛!eì˛ Ñ˛Ó˚y £ˆÏÓ ì˛Öò òì% ò

¢Ç@˘ÃˆÏ£ Ñ˛!›˛ Óhfl%˛ ÌyÑ˛ˆÏÓ⁄ ïÓ˚y ÎyÑ˛ñ 2!›˛ ˆîüúy£z Ñ˛y!‡˛ 5!›˛ Ñ˛y!‡˛Ó˚ ¢yˆÏÌ ˆÎyÜ Ñ˛Ó˚y £ú–

xyõÓ˚y !üü%ˆÏîÓ˚ˆÏÑ˛ ~£z ˆÎyÜ î%Ó˚Ñ˛õ¶˛yˆÏÓ ˆüÖyˆÏì˛ ˛ôy!Ó˚– ≤ÃÌõ!›˛ £úñ xyõÓ˚y ~£z î%!›˛ ¢õ!T˛ˆÏÑ˛

2

–1 0 A+2+1 B +3

23

ˆòy›˛

12 D.El.Ed. 504


~Ñ˛!eì˛ Ñ˛ˆÏÓ˚ !üü%ˆÏîÓ˚ Üíòy Ñ˛Ó˚ˆÏì˛ ÓúÓ– !mì˛#Î˚!›˛ £úñ xyõÓ˚y ~Ñ˛!›˛ ¢õ!T˛ˆÏÑ˛ !ò!î≈T˛ Ó˚yÖÓ

SïÓ˚y ÎyÑ˛ 5!›˛ Ñ˛y!‡˛Ó˚ ¢õ!T˛ˆÏÑ˛V ~ÓÇ xòƒ ¢õ!T˛ ˆÌˆÏÑ˛ ~Ñ˛!›˛ Ñ˛ˆÏÓ˚ Ñ˛y!‡˛ ˆÎyÜ Ñ˛Ó˚Ó– ò#ˆÏâ˛

~£z ˛ôÂï!ì˛ ˆîÖy Ïòy £ú ı

= (5 + 1) + 1

= (6 + 1)

= 7

fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒy Á ˛ô)í≈ ¢ÇÖƒyÓ˚ ˆÎy ÏÜÓ˚ !Ñ˛ä%È ˜Ó!üT˛ƒ

(i) xyÓÂï ¢)e ı î%!›˛ fl ∫y¶˛y!ÓÑ˛/˛ô)í≈¢ÇÖƒyÓ˚ ˆÎyÜö˛úÁ ~Ñ˛!›˛ fl˛∫y¶˛y!ÓÑ˛/˛ô)í≈ ¢ÇÖƒy

(ii) !Ó!òõÎ˚ ¢)e ı p + q = q + p ˆÎÖyˆÏò p Á q î%!›˛ fl˛∫y¶˛y!ÓÑ˛/˛ô)í≈ ¢ÇÖƒy–

(iii) xƒyˆÏ¢y!¢ˆÏÎ˚!›˛¶˛ ¢)e ı (p + q) + r = p + (q + r)

= p + q + r

~£z ¢)e!›˛ !ì˛ò!›˛ Óy ì˛yÓ˚ ˆÓ!ü fl˛∫y¶˛y!ÓÑ˛/˛ô)í≈ ¢ÇÖƒyÓ˚ ˆÎyÜ Ñ˛Ó˚yÓ˚ ˛ôÂï!ì˛ ≤Ãîyò Ñ˛ ÏÓ˚–

(iv) ˛ô)í≈¢ÇÖƒyÓ˚ ˆÎyˆÏÜÓ˚ xy£zˆÏv˛ò!›˛!›˛ ı ˛ô)í≈¢ÇÖƒyÓ˚ ˆ¢ˆÏ›˛ñ 4 + 0 = 0 + 4 = 4. xò%Ó˚)ˆÏ˛ô

p + 0 = 0 + p = p SˆÎÖyˆÏò p ˆÎˆÏÑ˛yˆÏòy ˛ô)í≈¢ÇÖƒyV– ¢%ì˛Ó˚yÇ ˛ô)í≈ ¢ÇÖƒyÓ˚ ˆÎyˆÏÜÓ˚

xy£zˆÏv˛ò!›˛!›˛ v˛z˛ôyîyò £ú 0.

(b) !Ó ÏÎ˚yÜ ı !ÓˆÏÎ˚yˆÏÜÓ˚ xÌ≈ Óyî ˆîÁÎ˚y– !Ñ˛ä%È Óhfl% ˆÌˆÏÑ˛ xyõÓ˚y Ñ˛ˆÏÎ˚Ñ˛!›˛ Óy ¢ÓÑ˛!›˛ˆÏÑ˛ !òˆÏÎ˚

!òˆÏì˛ ˛ôy!Ó˚ Óy Óyî !îˆÏì˛ ˛ôy!Ó˚– ˆÎõòñ 5!›˛ Óhfl%˛ ˆÌˆÏÑ˛ xyõÓ˚y 5ÈÙÈ~Ó˚ Ñ˛õ Óy 5!›˛ Óhfl%˛£z Óyî

!îˆÏì˛ ˛ôy!Ó˚– ÎÖò ˆÑ˛yˆÏòy Óhfl% Ó˚ îú ˆÌˆÏÑ˛ ¢ÓÑ˛!›˛ˆÏÑ˛ Óyî ˆîÁÎ˚y £Î˚ ì˛Öò !Ñ˛ä%È£z xÓ!üT˛

ÌyˆÏÑ˛ òy ~ÓÇ ~£z xÓ!üT˛ !Ñ˛ä%È òy ÌyÑ˛yˆÏÑ˛ ü)òƒ Óúy £Î˚– p – p = 0 SˆÎÖy Ïò p ~Ñ˛!›˛

˛ô)í≈ ¢ÇÖƒyV

(c) Ü%í ı Ü%ˆÏíÓ˚ xÌ≈ £ú ~Ñ˛£z ¢ÇÖƒyÓ˚ ÓyÓ˚ ÓyÓ˚ ˆÎyÜ– v˛zîy£Ó˚ífl˛∫Ó˚) ôñ

3 + 3 ˆÑ˛ Óúy ÎyÎ˚ 3 × 2

3 + 3 + 3 ˆÑ˛ Óúy ÎyÎ˚ 3 × 3

3 + 3 + 3 + 3 ˆÑ˛ Óúy ÎyÎ˚ 3 × 4

ˆòy›˛



Ü% ÏíÓ˚ ˜Ó!üT˛ƒ ı

(1) !Ó!òõÎ˚ ¢)e ı !òˆÏâ˛Ó˚ ˆ›˛!Óú!›˛ ú«˛ƒ Ñ˛Ó˚%ò ı

≤Ã!ì˛!›˛ ˆ◊!í Ïì˛ 5!›˛ Ñ˛ ÏÓ˚ ö% ú xy ÏäÈ–

~Ó˚Ñ˛õ 4!›˛ ˆ◊!í xyˆÏäÈ–

∴ ö% ÏúÓ˚ ˆõy›˛ ¢ÇÖƒy = 5 + 5 + 5 + 5 xÌÓy 5 × 4 = 20

~Öy Ïò 5!›˛ hfl˛ÏΩ˛ xyˆÏäÈ–

≤Ã!ì˛!›˛ hfl˛ÏˆÏΩ˛ 4!›˛ Ñ˛ˆÏÓ˚ ö%˛ú xyˆÏäÈ–

∴ ˆõy›˛ ö%˛ú¢ÇÖƒy = 4 + 4 + 4 + 4 + 4 = 4 × 5 = 20

¢%ì˛Ó˚yÇ î%!›˛ ˆ›˛!Óú ˆÌˆÏÑ˛ ~Ñ˛£z ö˛ú ˛ôyÁÎ˚y ÎyˆÏFäÈ–

xyÓ˚Á ˛ôyÁÎ˚y ˆÜúñ 4 × 5 = 5 × 4

xòƒ¶˛yˆÏÓ Óúy ÎyÎ˚ñ ˆÎˆÏÑ˛yˆÏòy î%!›˛ fl ∫y¶˛y!ÓÑ˛/˛ô)í≈¢ÇÖƒy p Á qÈÙÈ~Ó˚ ˆ«˛ Ïe p × q = q × p

¢%ì˛Ó˚yÇ fl ∫y¶˛y!ÓÑ˛ Á ˛ô)í≈ ¢ÇÖƒyÓ˚ ˆ«˛ˆÏe ÚÜ%íÛ !Ó!òõÎ˚ ¢)e ˆõˆÏò â˛ˆÏú–

(i) xyÓÂï ¢)e ı Î!î p Á q î%!›˛ fl˛∫y¶˛y!ÓÑ˛ Óy ˛ô)í≈ ¢ÇÖƒy £Î˚ ì˛y£ˆÏú p×q Á fl˛∫y¶˛y!ÓÑ˛/˛ô)í≈

¢ÇÖƒy £ ÏÓ– xyõÓ˚y Ó!ú ˆÎñ fl ∫y¶˛y!ÓÑ˛/˛ô)í≈ ¢ÇÖƒy Ü% ÏíÓ˚ ˆ«˛ Ïe ÓÂï–

ˆòy›˛

14 D.El.Ed. 504


(ii) xƒyˆÏ¢y!¢ˆÏÎ˚!›˛¶˛ ¢)e ı (p×q) × r = p × (q × r)

SˆÎÖyˆÏò p, q ~ÓÇ r !ì˛ò!›˛ fl˛∫y¶˛y!ÓÑ˛/ ˛ô)í≈ ¢ÇÖƒyV

(iii) Ü%ˆÏíÓ˚ xy£zˆÏv˛ò!›˛!›˛ ı Ü%ˆÏíÓ˚ ˆ«˛ˆÏe ‘1’ ¢ÇÖƒy!›˛Ó˚ !ÓˆÏü°Ï ¶) !õÑ˛y xyˆÏäÈ–

p×1 = 1 × p = p SˆÎÖyˆÏò p ˆÎˆÏÑ˛yˆÏòy fl˛∫y¶˛y!ÓÑ˛ ¢ÇÖƒy ì˛y£z xyõÓ˚y Ó!ú ı ‘1’ Ü% ÏíÓ˚

xy£zˆÏv˛ò!›˛!›˛–

(iv) ˆÎyˆÏÜÓ˚ v z˛ôˆÏÓ˚ Ü% ÏíÓ˚ !Ó!òõÎ˚ ¢)e ı

p × (q + r) = p×q + p×r

Óúy ÎyÎ˚ñ ˆÎy ÏÜÓ˚ v z ô ÏÓ˚ Ü%í !Ó!òõÎ˚ ¢)e ˆõ Ïò â˛ Ïúñ

v zîy£Ó˚íñ 5×(3+4) = 5×7=35

~ÓÇ 5×3+5×4 = 15 + 20 = 25

∴ 5 × (3+4) = 5×3 + 5×4

(d) ¶˛yÜ ı

p Á q î%!›˛ ¢ÇÖƒy ~ÓÇ p×q = r

Óúy ÎyÎ˚ ı

‘r’, ‘p’ myÓ˚y !Ó¶˛yãƒ ~ÓÇ ‘r’, ‘q’ myÓ˚yÁ !Ó¶˛yãƒ

‘p’ Á ‘q’ v z¶˛Î˚£z ‘r’ÈÙÈ~Ó˚ Ü%òò#Î˚Ñ˛ /v˛zÍ˛ôyîÑ˛–

‘r’ £ú ‘p’ Á ‘q’ÈÙÈ~Ó˚ Ü%!íì˛Ñ˛–

xyõÓ˚y ‘÷’ !â˛£´!›˛ ÓƒÓ£yÓ˚ Ñ˛!Ó˚ ~ÓÇ !ú!Ö

r ÷ p = q ~ÓÇ r ÷ q = p

v˛zîy£Ó˚í ı 3 × 5 = 15, ì˛y£z Óúy ÎyÎ˚

(i)15, 3 ~ÓÇ 5 v z¶˛Î˚ myÓ˚y !Ó¶˛yãƒ

(ii) 3 Á 5 v˛z¶˛Î˚£z 15ÈÙÈ~Ó˚ v zÍ˛ôyîÑ˛

(iii) 15 £ú 3 Á 5 v z¶˛ ÏÎ˚Ó˚£z Ü%!íì˛Ñ˛–

xyõÓ˚y !ú!Öñ 15 ÷ 3 = 5 ~ÓÇ 15 ÷ 5 = 3

ú«˛ƒ Ñ˛Ó˚y ÎyÎ˚ ˆÎñ

(a) 1 × 12 = 12 (b) 2 × 6 = 12 (c) 3 × 4 = 12

¢%ì˛Ó˚yÇ (a), (b) Á (c) ˆÌˆÏÑ˛ ˛ôyÁÎ˚y ÎyÎ˚ 1, 2, 3, 4, 6 ~ÓÇ 12 ≤ÃˆÏì˛ƒˆÏÑ˛ 12ÈÙÈ~Ó˚ v zÍ˛ôyîÑ˛–

1, 2, Á 3 ~£z !ì˛ò!›˛ fl˛∫y¶˛y!ÓÑ˛ ¢ÇÖƒy Ñ˛#⁄

~õò ˆÑ˛yˆÏòy î%!›˛ ˛ô,ÌÑ˛ ¢ÇÖƒy ˆò£z ÎyˆÏîÓ˚ Ü%íö˛ú 1.

¢%ì˛Ó˚yÇ 1~Ó˚ ~Ñ˛!›˛ õye v˛zÍ˛ôyîÑ˛ ~ÓÇ ˆ¢!›˛ £ú 1.

ˆòy›˛



ˆÎˆÏ£ì%˛ 1 × 2 = 2 ~ÓÇ ~£z ïÓ˚ˆÏòÓ˚ xyÓ˚Á ˆÑ˛yˆÏòy ˆãyv ¸y ˆò£z ÎyˆÏîÓ˚ Ü%íö˛ú 2 ¢%ì˛Ó˚yÇ

2ÈÙÈ~Ó˚ ˆÑ˛Óúõye 2!›˛ v˛zÍ˛ôyîÑ˛ ~ÓÇ ˆ¢Ü%!ú £ú 1 Á 2

xò%Ó) Ï˛ôñ ~Ó˚Ñ˛õ xˆÏòÑ˛ fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒy Óì≈˛õyò ÎyˆÏîÓ˚ ˆÑ˛Óúõye 2!›˛ v˛zÍ˛ôyîÑ˛ 1, 2, 3,

5, 7, 11, 13.... £zì˛ƒy!î ~£z ïÓ˚ˆÏòÓ˚ ¢ÇÖƒyÓ˚ v zîy£Ó˚í ÎyˆÏîÓ˚ î%!›˛ õye v zÍ˛ôyîÑ˛ xyˆÏäÈ–

~Ü%!ú ÏÑ˛ Óúy £Î˚ ˆõÔ!úÑ˛ ¢ÇÖƒy–

ˆÎ¢Ó fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ ˆÑ˛Óúõye î%!›˛ v zÍ˛ôyîÑ˛ 1 ~ÓÇ ˆ¢£z ¢ÇÖƒy!›˛ !òˆÏãñ ì˛yˆÏîÓ˚ Óúy

£Î˚ ˆõÔ!úÑ˛ ¢ÇÖƒy–

4, 6, 8, 12, 9, ..., 15 £zì˛ƒy!î fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ 2ÈÙÈ~Ó˚ ˆÓ!ü v zÍ˛ôyîÑ˛ xyˆÏäÈ– ~ˆÏîÓ˚ Óúy

£Î˚ ˆÎÔ!ÜÑ˛ ¢ÇÖƒy

ˆõÔ!úÑ˛ ¢ÇÖƒyˆÏÑ˛ fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ ˆ«˛ˆÏe ¢ÇK˛y!Î˚ì˛ Ñ˛Ó˚y £Î˚– 0, 1, –1, –2, ..., 12 , 1

3

£zì˛ƒy!î ¢ÇÖƒy ˆõÔ!úÑ˛Á òÎ˚ xyÓyÓ˚ ˆÎÔ!ÜÑ˛Á òÎ˚–

ˆÎÔ!ÜÑ˛ ¢ÇÖƒyÓ˚ ˆõÔ!úÑ˛ v zÍ˛ôyî ÏÑ˛ !Ó ÏŸ’°Ïí ı

~Ñ˛!›˛ ˆÎÔ!ÜÑ˛ ¢ÇÖƒyˆÏÑ˛ ì˛yÓ˚ ˆõÔ!úÑ˛ v zÍ˛ôyîˆÏÑ˛Ó˚ Ü%íö˛ú xyÑ˛yˆÏÓ˚ ≤ÃÑ˛yü Ñ˛Ó˚yˆÏÑ˛ ˆõÔ!úÑ˛

v zÍ˛ôyî ÏÑ˛ !Ó ÏŸ’°Ïí Óúy £Î˚– ˆÎõòÈÙÙÙÈ

12 = 2 × 2 × 3 v˛zÍ˛ôyîˆÏÑ˛Ó˚ !Óòƒy¢

Ñ˛õ≈ ôÂï!ì˛ ı

¢%ì˛Ó˚yÇ 12 = 2 × 2 × 3 ∴ 12 = 2 × 2 × 3

˛ôÓ˚fl˛ôÓ˚ ˆõÔ!úÑ˛ ¢ÇÖƒy ı

~!›˛ ¢ÇÖƒyˆÏÑ˛ ˛ôÓ˚fl˛ôÓ˚ ˆõÔ!úÑ˛ Óúy £Î˚ Î!î ì˛yˆÏîÓ˚ ˆÑ˛yˆÏòy ¢yïyÓ˚í v˛zÍ˛ôyîÑ˛ òy ÌyˆÏÑ˛– !òˆÏâ˛

v zîy£Ó˚í ˆîÁÎ˚y £ú ı

(i) 8 Á 27 ˛ôÓ˚fl˛ôÓ˚ ˆõÔ!úÑ˛ SÎ!îÁ ~Ó˚y ≤Ã Ïì˛ƒ ÏÑ˛ ˆÎÔ!ÜÑ˛ ¢ÇÖƒyV

(ii) 17 Á 20 ˛ôÓ˚fl˛ôÓ˚ ˆõÔ!úÑ˛–

12

2 6

2 3

ˆòy›˛

16 D.El.Ed. 504


Îõã ˆõÔ!úÑ˛ ı î%!›˛ ˆõÔ!úÑ˛ ¢ÇÖƒyÓ˚ õ Ïïƒ 2 xhs˘˛Ó˚ ÌyÑ˛ Ïú ì˛y ÏîÓ˚ Îõã ˆõÔ!úÑ˛ Ó Ïú–

3 ~ÓÇ 5ñ 5 ~ÓÇ 7, 11 ~ÓÇ 13, 17 ~ÓÇ 19 £zì˛ƒy!î

ˆãyv ¸ ˆõÔ!úÑ˛ ı 2 ~Ñ˛õye ˆõÔ!úÑ˛ ˆãyv ¸ ¢ÇÖƒy 1 ~!›˛ «% oì˛õ ˆõÔ!úÑ˛ ¢ÇÖƒy–

~Ñ˛!›˛ !ò!î≈T˛ ˆ«˛ˆÏeÓ˚ õˆÏïƒ ˆõÔ!úÑ˛ ¢ÇÖƒyÓ˚ üòy=˛Ñ˛Ó˚í ı

!ò Ïâ˛ 1 ˆÌˆÏÑ˛ 100 ˛ôÎ≈hs˘˛ ¢ÇÖƒyÓ˚ õˆÏïƒ ˆõÔ!úÑ˛ ¢ÇÖƒy !â˛!£´ì˛ Ñ˛Ó˚yÓ˚ ˛ôÂï!ì˛ ˆîÁÎ˚y £ú ı

[@˘Ã#Ñ˛ Ü!íì˛!Óî‰ £zÓ˚y Ï›˛y¢ ÏÎò‰ Ï¢Ó˚ ˛ôÂï!ì˛]

˛ôÂï!ì˛ ı

(i) 2ÈÙÈ~Ó˚ ˆÌˆÏÑ˛ ÓˆÏv ¸y 2ÈÙÈ~Ó˚ Ü%!íì˛Ñ˛ˆÏÑ˛ ˆÑ˛ˆÏ›˛ !îò

(ii) 3ÈÙÈ~Ó˚ ˆÌˆÏÑ˛ ÓˆÏv ¸y 3ÈÙÈ~Ó˚ Ü%!íì˛Ñ˛ˆÏÑ˛ ˆÑ˛ˆÏ›˛ !îò

(iii) 5 ,, ,, ,, 5 ,, ,, ,, ,,

(iv) 7 ,, ,, ,, 7 ,, ,, ,, ,,

1 äÈyv ¸y xÓ!üT˛ ¢Ó ¢ÇÖƒyÜ%!ú ˆõÔ!úÑ˛–

~£z ˛ôÂï!ì˛ ˆÑ˛ò 7 !Ü ÏÎ˚ ˆü°Ï £Î˚⁄

ïÓ˚y ÎyÑ˛ñ 100ÈÙÈ~Ó˚ ÓÜ≈õ)ú 10.

10ÈÙÈ~Ó˚ ˆâ˛ ÏÎ˚ ˆäÈy Ï›˛y ˆõÔ!úÑ˛ ¢ÇÖƒy £ú 7– ì˛y£z ~!›˛ 7ÈÙÈ~Ó˚ !Ü ÏÎ˚ ˆü°Ï £Î˚–

¢%ì˛Ó˚yÇ 1 ˆÌ ÏÑ˛ 100 ˛ôÎ≈hs˘˛ ˆõÔ!úÑ˛ ¢ÇÖƒy ı

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73,

79, 83, 89, 97.

ˆòy›˛



~ÓyÓ˚ xy˛ôòyÓ˚ x@˘ÃÜ!ì˛ Îyâ˛y£z Ñ˛Ó%ò ı

E5. ˆÑ˛yˆÏòy fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒy ˆõÔ!úÑ˛ Óy ˆÎÔ!ÜÑ˛ ˆÑ˛yˆÏòy!›˛£z òÎ˚⁄

E6. ˛ô)í≈¢ÇÖƒyÓ˚ ˆ¢ˆÏ›˛ ˆÎyˆÏÜÓ˚ xy£zˆÏv˛ò!›˛!›˛ Ñ˛#⁄

E7. î%!›˛ ˆõÔ!úÑ˛ ¢ÇÖƒyÓ˚ xhs˘˛Ó˚ !ÓˆÏãyv˛¸– ì˛yˆÏîÓ˚ ˆÎyÜö˛ú 15 £ˆÏúñ ˆ¢£z î%!›˛ ˆõÔ!úÑ˛

¢ÇÖƒy Ñ˛# Ñ˛#⁄

E8. 10 ˆÌ ÏÑ˛ 30ÈÙÈ~Ó˚ õˆÏïƒ Ñ˛ì˛Ü%!ú Îõã ˆãyˆÏv ¸Ó˚ ˆãyv ¸y xyˆÏäÈ⁄

E9. ~Ñ˛!›˛ ¶˛yÜ ≤Ã!Ñ Î˚yÎ˚ ¶˛yãÑ˛ñ ¶˛yÜö˛ú Á ¶˛yÜˆÏü°Ï 8, 12 Á 15 £ Ïúñ ¶˛yãƒ Ñ˛ì˛⁄

5.3.2 xÖ[˛ ¢ÇÖƒyÓ˚ ˆ«˛ Ïe x˛ôy ÏÓ˚üò ı

˛A. ˆÎyÜ ı ˛ô)í≈¢ÇÖƒyÓ˚ ˆÎy ÏÜÓ˚ ¢Ó ˜Ó!üT˛ƒ xÌ≈yÍ (i) xyÓÂï ¢)eñ (ii) !Ó!òõÎ˚ ¢)e (iii)

xƒyˆÏ¢y!¢ˆÏÎ˚!›˛¶˛ ¢)eñ (iv) ˆÎyˆÏÜÓ˚ xy£zˆÏv˛ò!›˛!›˛Ó˚ x!hfl˛Ïc £zì˛ƒy!î ¢Ó!Ñ˛ä%È£z xÖ[˛ ¢ÇÖƒyÓ˚

ˆ«˛ˆÏeÁ ≤ÃˆÏÎyãƒ–

xyÓ˚Á !Ñ˛ä%È ˜Ó!üT˛ƒ ò#ˆÏâ˛ ˆîÁÎ˚y £ú ı

(v) ˆÎyˆÏÜÓ˚ !Ó˛ôÓ˚#ì˛ v˛z˛ôyîyˆÏòÓ˚ x!hfl˛Ïc ı Î!î +p xÖu˛ ¢ÇÖƒy £Î˚ñ ì˛y£ Ïú ‘–p’ xÖ[˛

¢ÇÖƒyÓ˚ x!hfl Ïc ÌyÑ˛ ÏÓ ÎyÓ˚ ö˛ Ïú (+p) + (–p) = 0 £ˆÏÓ–

‘+p’ Á ‘–p’ ˆÑ˛ ~ˆÏÑ˛ x˛ôˆÏÓ˚Ó˚ !Ó˛ôÓ˚#ì˛ (Inverse) Óúy £Î˚– ~ÓyÓ˚ñ xÖ[˛ ¢ÇÖƒyÓ˚ ˆÎy ÏÜÓ˚

!Ñ˛ä%È ˛ôÂï!ì˛ ˆîÖy ÎyÑ˛ ı

(a) ïòydÑ˛ xÖ[˛ ¢ÇÖƒyÓ˚ ˆÎyÜ ı

fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ ˆÎyˆÏÜÓ˚ ¢ˆÏAÜ ~£z ˛ôÂï!ì˛ xò%Ó) ô (+5) + (+3) = +8

(b) ~Ñ˛!›˛ ïòydÑ˛ Á ~Ñ˛!›˛ }íydÑ˛ ¢ÇÖƒyÓ˚ ˆÎyÜ ı

ˆÎõò ı (+5) + (–3)

xyõÓ˚y ãy!ò ˆÎñ

+5 = (+1) + (+1) + (+1) + (+1) + (+1)

xò%Ó˚)ˆÏ˛ôñ –8 = (–1) + (–1) + (–1)

~Öòñ (+5) + (–3) = (+1) + (+1) + (+1) + (+1) + (+1) + (–1) + (–1)

= {(+1) + (–1)} + {(+1) + (–1)} + {(+1) + (–1)} (+1) + (+1)

= 0 + 0 + 0 + (+2)

= 0 + (+2)

= +2

¢Ç!«˛Æ ˛ôÂï!ì˛ ı

(+5) + (–3) = (+2) + (+3) + (–3) [(+5) = (+2) + (+3)]

= (+2) + {(+3) + (–3)}

ˆòy›˛

18 D.El.Ed. 504


= (+2) + 0

= +2

xòƒ v˛zîy£Ó˚íñ

(+4) + (–7) = (+4) + (–4) + (–3) [(–7) = (–4) + (–3)]

= {(+4) + (–4)} + (–3)

= 0 + (–3)

= –3

(c) î%!›˛ }íydÑ˛ xÖ[˛ ¢ÇÖƒyÓ˚ ˆÎyÜ ı

(–2) + (–3) = (–1) + (–1) + (–1) + (–1) + (–1)

= –5

B. !Ó ÏÎ˚yÜ ı

!Ó ÏÎ˚yÜ ~ Ï«˛ Ïe !Ó˛ôÓ˚#ì˛ ¢ÇÖƒyÓ˚ ˆÎyÜ Ñ˛Ó˚yÓ˚ ˛ôÂï!ì˛– ¢%ì˛Ó˚yÇ ‘p’ Á ‘q’ î%!›˛ xÖ[˛ ¢ÇÖƒy

£ˆÏúñ p – q = p + (–q)

v zîy£Ó˚í ı

(i) (+5) – (+8) = (+5) + (–8)

(ii) (+4) – (–3) = (+4) + (+3)

(iii) (–5) – (+2) = (–5) + (–2)

(iv) (–7) – (–3) = (–7) + (+3)

ˆÎy ÏÜÓ˚ ˆ«˛ Ïeñ xyõÓ˚y ~Ü%!úÓ˚ ˛ôÂï!ì˛ !ü ÏÖ!äÈ–

xÖ[˛ ¢ÇÖƒyÓ˚ !Ó ÏÎ˚y ÏÜÓ˚ ~Ñ˛!›˛ !Ó Ïü°Ï ˜Ó!üT˛ƒ ı

fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ ˆ«˛ˆÏe p – q xÌ≈˛ô)í≈ £ˆÏÓ Î!î q < p £Î˚– !Ñ˛hs%˛ xÖ[˛ ¢ÇÖƒyÓ˚ ˆ«˛ˆÏe

p – q xÌ≈˛ô)í≈ £Î˚ ÎÖò q < p, q = p xÌÓy q > p £Î–

¢ÇÖƒy ÏÓ˚ÖyÓ˚ v z ôÓ˚ ˆÎyÜ Á !Ó ÏÎ˚yÜ ≤Ã!Ñ Î˚y ı

ˆÎyÜ Á !ÓˆÏÎ˚yˆÏÜÓ˚ ˆ«˛ˆÏe !ò¡¨Ó) Ï˛ô ¢ÇÖƒyˆÏÓ˚ÖyÓ˚ ÓƒÓ£yÓ˚ £Î˚ ı

(a) ˆÎyÜ ı SˆÎyÜ Ñ˛Ó˚yÓ˚ xÌ≈ £ú v˛yò!î ÏÑ˛ x@˘Ã¢Ó˚ £ÁÎ˚yV

(i) (+5) + (+8) = + 13

(ii) (–3) + (+5) = +2

ˆòy›˛



(iii) (+4) + (–6) = (–6) + (+4) = –2 [!Ó!òõÎ˚ ¢)e]

(b) !Ó ÏÎ˚yÜ ı S!ÓˆÏÎ˚yÜ Ñ˛Ó˚yÓ˚ xÌ≈ Óyõ!îÑ˛ x@˘Ã¢Ó˚ £ÁÎ˚yV

(i) (+6) – (+8) = – 2 [+6 ˆÌˆÏÑ˛ xyõÓ˚y Óyõ!îˆÏÑ˛ 8 áÓ˚ Îy ÏÓy]

(ii) (–5) – (–7) = –5 + 7 = 7 – 5 = +2

(C) Ü%í ı Ü% ÏíÓ˚ xÌ≈ £ú î%!›˛ xÖu˛ ¢ÇÖƒyÓ˚ ˆÎyÜ ˆÎÖy Ïò ~Ñ˛!›˛ xÖ[˛ ¢ÇÖƒy x}íydÑ˛–ˆÎõò ı(i) (+5) + (+5) + (+5) + (+3) = (+5) × 4 ì˛y£ Ïú (+5) × (+4) = + 20(ii) (–3) + (–3) + (–3) + (–3) = (–3) ×5ì˛y£ Ïú (–3) × 5 = (–3) + (–3) + (–3) + (–3) + (–3)

= 15xÖ[˛ ¢ÇÖƒyÓ˚ Ü% ÏíÓ˚ ˆ«˛ Ïe !â˛ Ï£´Ó˚ !òÎ˚õ ı

p q p × q

ïòydÑ˛ ïòydÑ˛ ïòydÑ˛P > 0 q > 0 p × q > 0

}òydÑ˛ }òydÑ˛ ïòydÑ˛P < 0 q < 0 p × q > 0

ïòydÑ˛ }òydÑ˛ }òydÑ˛P > 0 q < 0 p × q < 0

}òydÑ˛ ïòydÑ˛ }òydÑ˛P < 0 q > 0 p × q < 0

ïòydÑ˛ ü)òƒ ü)òƒP > 0 q = 0 p × q = 0

}òydÑ˛ ü)òƒ ü)òƒP < 0 q = 0 p × q = 0

ˆòy›˛

20 D.El.Ed. 504


xÖ[˛ ¢ÇÖƒyÓ˚ Ü% ÏíÓ˚ !òÎ˚õ ı

(i) Ü%ò xyÓÂï !òÎ˚õ ˆõ Ïò â˛ Ïú

(ii) xƒyˆÏ¢y!¢ˆÏÎ˚!›˛¶˛ ¢)e ˆõˆÏò â˛ˆÏú

(iii) Ü%ˆÏíÓ˚ ˆ«˛ˆÏe xy£zˆÏv˛ò!›˛!›˛Ó˚ x!hfl˛Ïc xyˆÏäÈ– ˆ¢!›˛ £ú 1–

(iv) Ü%í ˆÎy ÏÜÓ˚ v z ô ÏÓ˚ !Ó!òõÎ˚ ¢)e ˆõ Ïò â˛ Ïú–

Ñ˛õ≈¢)!â˛ 1 :

˛v z ô!Ó˚v z=˛ Ü% ÏíÓ˚ ˜Ó!üT˛ƒÜ%!úˆÏÑ˛ v zîy£Ó˚í¢£ˆÏÎyˆÏÜ Îyâ˛y£z Ñ˛Ó%ò–

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................

D. ¶˛yÜ ı xyõÓ˚y xyˆÏúyâ˛òy Ñ˛ˆÏÓ˚!äÈ ˆÎ fl˛∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ ˆ«˛ˆÏe Ü% ÏíÓ˚ !Ó˛ôÓ˚#ì˛ ˛ôÂï!ì˛ ¶˛yÜ–

~ Ï«˛ ÏeÁ ~!›˛ ≤Ã ÏÎyãƒ– ‘p’ Á ‘q’ xü)òƒ xÖ[˛ ¢ÇÖƒy ~ÓÇ p×q = r. ì˛y£ˆÏú Óúy ÎyÎ˚ñ

(i) r ÷ p = q

(ii) r ÷ q = p

¢%ì˛Ó˚yÇ

(i) (+5) × (+3) = + 15

∴ (+15) ÷ (+5) = + 3

~ÓÇ (+15) ÷ (+3) = + 5

(ii) (+4) × (–6) = –24

(–24) ÷ (+4) = – 6

~ÓÇ (–24) ÷ (–6) = + 4

(iii) (–3) × (–5) = + 15

∴ (+15) ÷ (–3) = – 5

~ÓÇ (+15) ÷ (– 5) = – 3

ˆòy›˛



xÖ[˛ ¢ÇÖƒyÓ˚ ¶˛yˆÏÜÓ˚ ˆ«˛ˆÏe !â˛ˆÏ£´Ó˚ !òÎ˚õ ı

p q p ÷ q

ïòydÑ˛ ïòydÑ˛ ïòydÑ˛

P > 0 q > 0 (p ÷ q) > 0

ïòydÑ˛ }íydÑ˛ }íydÑ˛

P > 0 q < 0 (p ÷ q) < 0

}íydÑ˛ ïòydÑ˛ }íydÑ˛

p < 0 q > 0 (p ÷ q) < 0

}íydÑ˛ }íydÑ˛ }íydÑ˛

p < 0 q < 0 (p ÷ q) < 0

Ü%Ó%c˛ô)í≈ ì˛Ìƒ ı xÖ[˛ ¢ÇÖƒyˆÏÑ˛ ü)òƒ myÓ˚y ¶˛yÜ xÌ≈£#ò–

xy˛ôòyÓ˚ x@˘ÃÜ!ì˛ Îyâ˛y£z Ñ˛Ó˚%ò ı

E10. xÖ[˛ ¢ÇÖƒyÓ˚ ˆÎyˆÏÜÓ˚ ˆ«˛ˆÏe xy£z Ïv˛ò!›˛!›˛ Ñ˛#⁄

E11. +7-~Ó˚ ˆÎyˆÏÜÓ˚ ˆ«˛ˆÏe !Ó˛ôÓ˚#ì˛ v˛z˛ôyîyò Ñ˛#⁄

E12. ˆÑ˛yò xÖ[˛ ¢ÇÖƒyÓ˚ ¢y ÏÌ ˆ¢£z ¢ÇÖƒy ÏÑ˛ Ü%í Ñ˛Ó˚ Ïú Ü%íö˛ú 1 £ˆÏÓ⁄ ~Ó˚Ñ˛õ

Ñ˛ì˛Ü%!ú xÖ[˛ ¢ÇÖƒy Óì≈ õyò⁄

E13. –8 ˆÌ ÏÑ˛ +8 ˛ôÎ≈hs˘˛ ¢õhfl Ï xÖ[˛ ¢ÇÖƒyÓ˚ ˆÎyÜö˛ú !òí≈Î˚ Ñ˛Ó%ò–

5.3.3 õ)úî ¢ÇÖƒyÓ˚ ˆ«˛ Ïe x˛ôy ÏÓ˚üò ı

A. ˆÎyÜ ı

(i) xyõÓ˚y õ)úî ¢ÇÖƒy ÏÑ˛ ¢î,ü ¶˛@¿yÇ Ïü ˛ô!Ó˚íì˛ Ñ˛ ÏÓ˚ ˆÎyÜ Ñ˛!Ó˚–

3 5 3 10

8 12 24 24+ = + Sv z¶˛Î˚ ¢ÇÖƒy ÏÑ˛ ¢î,ü ¶˛@¿yÇ Ïü ˛ô!Ó˚íì˛ Ñ˛Ó˚y £úV

= 9 10

24

+

= 19

24

ˆòy›˛

22 D.El.Ed. 504


(ii) xyõÓ˚y !ò¡¨!ú!Öì˛ ¢) ÏeÓ˚ ≤Ã ÏÎ˚y ÏÜ ˆÎyÜ Ñ˛!Ó˚ ı

p ps rqrq s qs

++ =

v zîy£Ó˚í ı

2 4 2 5 4 3 10 12 22 713 5 3 5 15 15 15

× + × ++ = = = =×

ú«˛ƒ Ñ˛Ó˚%ò ı (–8) ÷ 4 = –2, 8 ÷ (–4) = –2

¢%ì˛Ó˚yÇ 8 8 84 4 4− = = −−

xì˛~Óñ p p p

q q q− = = −−

ˆÎy ÏÜÓ˚ ˜Ó!üT˛ƒ ı

(i) õ)úî ¢ÇÖƒy ˆÎyˆÏÜÓ˚ ˆ«˛ˆÏe xyÓÂï

(ii) õ)úî ¢ÇÖƒyÓ˚ ˆÎyÜ !òõÎ˚ ¢)e ˆõ Ïò â˛ Ïú

(iii) ˆÎyÜ xƒyˆÏ¢y!¢ˆÏÎ˚!›˛¶˛

(iv) ˆÎyˆÏÜÓ˚ xy£z Ïv˛ò!›˛!›˛Ó˚ x!hfl˛Ïc xyˆÏäÈ– ˆ¢!›˛ £ú 0–

(v) ˆÎyˆÏÜÓ˚ !Ó˛ôÓ˚#ì˛ v˛z˛ôyîyò Óì≈˛õyò–

pq ~ÓÇ

pq

− £ú ~ˆÏÑ˛ x˛ôˆÏÓ˚Ó˚ !Ó˛ôÓ˚#ì˛ v˛z˛ôyîyò–

xì˛~Ó 0p pq q

⎛ ⎞ ⎛ ⎞+ − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Ñ˛õ≈¢)!â˛ 2 :

v z ô!Ó˚v z=˛ ˜Ó!üT˛ƒÜ%!úˆÏÑ˛ õ)úî ¢ÇÖƒyÓ˚ v zîy£Ó˚í ¢£ˆÏÎyˆÏÜ ≤Ãõyí Ñ˛Ó%ò–

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................

ˆòy›˛



B. !Ó ÏÎ˚yÜ ı

(i) ~Ñ˛›˛y õ)úî ¢ÇÖƒy ˆÌˆÏÑ˛ xòƒ!›˛ˆÏÑ˛ !ÓˆÏÎ˚yÜ Ñ˛Ó˚yÓ˚ ¢õÎ˚ ì˛yˆÏîÓ˚ ¢î,ü ¶˛@¿yÇˆÏü ˛ô!Ó˚íì˛

Ñ˛Ó˚y £Î˚–

v zîy£Ó˚í ı

5 7 15 14 15 14 18 12 24 24 24 24

−− = − = =

(ii) ~Ñ˛!›˛ õ)úî ¢ÇÖƒy ˆÌˆÏÑ˛ xyˆÏÓ˚Ñ˛!›˛ˆÏÑ˛ !ÓˆÏÎ˚yÜ Ñ˛Ó˚yÓ˚ ¢õÎ˚ ò#ˆÏâ˛Ó˚ ¢)e!›˛ ÓƒÓ£,ì˛ £Î˚

ı

( )p p ps qrr rq s q s qs

−−− = + =

v˛zîy£Ó˚í ı 3 5 ( 2) 43 2 3 2 15 8 7

4 5 4 5 4 5 20 20× + − ×− −− = + = = =×

C. Ü%í ı

pq ~ÓÇ r

s î%!›˛ õ)úî ¢ÇÖƒyñ

p p rrq s q s

×× = ×

v zîy£Ó˚í ı

(i) 2 4 2 4 83 5 3 5 15

×× = =×

(ii) 3 2 3 2 6 34 7 4 7 28 14− − × − −× = = =×

õ)úî ¢ÇÖƒyÓ˚ Ü% ÏíÓ˚ ˜Ó!üT˛ƒ ı

(i) Ü%í xyÓÂï ¢)e ˆõ Ïò â˛ Ïú–

(ii) Ü%í !Ó!òõÎ˚ ¢)e ˆõˆÏò â˛ˆÏú

(iii) Ü%í xƒyˆÏ¢y!¢ˆÏÎ˚!›˛¶˛

ˆòy›˛

24 D.El.Ed. 504


(iv) Ü% ÏíÓ˚ ˆ«˛ˆÏe xy!ˆÏv˛ò!›˛!›˛Ó˚ x!hfl˛Ïc xyˆÏäÈ– ˆ¢!›˛ £ú 1, S~£z ˜Ó!üT˛ƒ!›˛ v zîy£Ó˚ ÏíÓ˚

¢y£yˆÏÎƒ Îyâ˛y£z Ñ˛Ó˚%òV

(v) Ü% ÏíÓ˚ ˆ«˛ˆÏe xˆÏòyòƒÑ˛ v z ôyîyˆÏòÓ˚ x!hfl˛Ïc xyˆÏäÈ–

pq ~ÓÇ

qp ˛ôÓ˚fl˛ôˆÏÓ˚Ó˚ xˆÏòyòƒÑ˛ ÎyÓ˚ ö˛ˆÏú

1p qq P× =

v zîy£Ó˚í ı 23 ~Ó˚ x ÏòyòƒÑ˛ 3

2 – 5-~Ó˚ xˆÏòyòƒÑ˛ 15

(vi) Ü%í ˆÎyˆÏÜÓ˚ v z ôÓ˚ !Ó!òõÎ˚ ¢)e ˆõˆÏò â˛ˆÏú–

ˆÎõò ( )p p pm k m kq n l q n q l× + = × + ×

v˛zîy£Ó˚í ı ( ) ( )2 4 6 2 4 2 63 5 7 3 5 3 7

− + = × − + ×

xÖ[˛ ¢ÇÖƒyÓ˚ õˆÏì˛y£z Ü% ÏíÓ˚ ˆ«˛ˆÏe Á£z ~Ñ˛£z !â˛ˆÏ£´Ó˚ !òÎ˚õ ≤ÃˆÏÎyãƒ–

Ñ˛õ≈¢)â˛# 3 :

õ)úî ¢ÇÖƒyÓ˚ Ü% ÏíÓ˚ ˆ«˛ Ïe (i) ˆÌˆÏÑ˛ (iv) xÓ!ï ˜Ó!üT˛ƒ v zîy£Ó˚í ¢£ ÏÎy ÏÜ Îyâ˛y£z Ñ˛Ó%ò–

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................

D. ¶˛yÜ ı

~Ñ˛!›˛ õ)úî ¢ÇÖƒyˆÏÑ˛ x˛ôÓ˚ myÓ˚y ¶˛yÜ Ñ˛Ó˚yÓ˚ xÌ≈ £ú ≤ÃÌõ ¢ÇÖƒy!›˛ˆÏÑ˛ x˛ôÓ˚!›˛Ó˚ xˆÏòyòƒÑ˛

myÓ˚y Ü%í Ñ˛Ó˚y–

xÌ≈yÍ p p psr sq s q r qr÷ = × =

ˆòy›˛



ˆÎõòñ ( )2 4 2 7 7 113 7 3 4 6 6

−÷ = × = − = −−

ú«˛ƒ Ñ˛Ó˚%ò ı (i) 0 myÓ˚y ¶˛yÜ Ñ˛Ó˚y xÌ≈£#ò

(ii) ¶˛y ÏÜÓ˚ ˆ«˛ Ïe !â˛ Ï£´Ó˚ !òÎ˚õ xÖ[˛ ¢ÇÖƒyÓ˚ ˆ«˛ Ïe !òÎ˚ ÏõÓ˚ õ Ïì˛y£z–

õ)úì˛ ¢ÇÖƒyÓ˚ ¢õì% úƒ îü!õÑ˛ ¢ÇÖƒy ı ˆÎ¢Ó õ)úî ¢ÇÖƒyÓ˚ £Ó˚ 10 Óy 10-~Ó˚ ¢)â˛Ñ˛ñ

ì˛yˆÏîÓ˚ ≤ÃÑ˛yü Ñ˛Ó˚yÓ˚ xòƒ ˛ôÂï!ì˛ xyˆÏäÈ– !ò¡¨!ú!Öì˛ v zîy£Ó˚íÜ%!ú ú«˛ƒ Ñ˛Ó%ò ı

1 2 70.1, 0.2, 0.710 10 10

= = =

1 2 140.01, 0.02, 0.14100 100 100

= = =

xòƒyòƒ õ)úî ¢ÇÖƒyÜ%!úÓ˚ ˆ«˛ˆÏe ı

1 1 5 5 0.52 2 5 10

×= = =×

3 3 25 75 0.754 4 25 100

×= = =×

5 5 125 625 0.6258 8 125 1000

×= = =×

2 2 2 4 0.45 5 2 10

×= = =×

xì˛~Óñ ˆîÖy ˆÜú ˆÎñ ˆÎ¢Ó õ)úî ¢ÇÖƒyÓ˚ £ ÏÓ˚Ó˚ 2 Óy 5 äÈyv˛¸y ˆÑ˛yˆÏòy v˛zÍ˛ôyîÑ˛ ˆò£zñ

ˆ¢Ü%!ú ÏÑ˛ îü!õÑ˛ ¢ÇÖƒyÓ˚ myÓ˚y ≤ÃÑ˛yü Ñ˛Ó˚y ÎyÎ˚– ~Ü%!ú ÏÑ˛ Óúy £Î˚ ¢¢#õ îü!õÑ˛ ¢ÇÖƒy–

!Ñ˛hs%˘˛ 1 1 2, ,3 6 7 £zì˛ƒy!îÓ˚ ˆ«˛ˆÏe Ñ˛# £ˆÏÓ⁄ ~ˆÏ«˛ˆÏe £Ó˚ˆÏÑ˛ 10 Óy 10-Ó˚ ¢)â˛Ñ˛ !£¢yˆÏÓ

˛ô!Ó˚Óì≈˛ò Ñ˛Ó˚y ¢Ω˛Ó òÎ˚–

ì˛y£z xyõÓ˚y 1 ˆÑ˛ 3 myÓ˚y ¶˛yÜ Ñ˛ˆÏÓ˚ ¢õì%˛úƒ ˆÑ˛yˆÏòy îü!õÑ˛ ˆ˛ôˆÏì˛ â˛y£z–

0.333......3 1.000

010910910

ˆòy›˛

26 D.El.Ed. 504


¶˛yÜö˛ˆÏúÓ˚ ˆü°Ï ÌyÑ˛ˆÏÓ òy ~ÓÇ Îì˛ÓyÓ˚£z ¶˛yÜ Ñ˛Ó˚y ˆ£yÑ˛ òy ˆÑ˛ò xÓ!üT˛ ¢õ¢õÎ˚ ¢õyò

ÌyÑ˛ˆÏÓ– xyõÓ˚y 3 ÓyÓ˚ ~Ó˚Ñ˛õ¶˛y ÏÓ x@˘Ã¢Ó˚ £ ÏÎ˚ ˆ˛ôúyõÈÙÙÙÈ

1 0.3333......3=

~!›˛Ó˚ ˆü°Ï ˆò£z ~ÓÇ ÓyÓ˚ÓyÓ˚ 3 ¢ÇÖƒy!›˛ xy¢ˆÏÓ– ì˛y£z xyõÓ˚y ~ˆÏÑ˛ x¢#õ îü!õÑ˛ ¢ÇÖƒy

Ó Ïú Ìy!Ñ˛– ~Ó˚Ñ˛õ xyÓ˚Á!Ñ˛ä%È v zîy£Ó˚í ı

0.232323.........

2.537373737.......

1.342342342......

~£z ¢ÇÖƒyÜ%!ú xòƒ xyÑ˛y ÏÓ˚ñ xòƒ !â˛ Ï£´Ó˚ ¢y£y ÏÎƒ ˆúÖy ÎyÎ˚ ı

0.3333.... = 0.3

0.232323.... = 0.23

2.5373737... = 2.537

1.342342342... = 1.342

x¢#õ îü!õÑ˛ !Ñ˛hs%˛ xyÓ,_ îü!õÑ˛ òÎ˚ ~Ó˚Ñ˛õÁ ¢ÇÖƒy xy ÏäÈ ı

1.12112111211112.........

3.201001000100001......

~ˆÏ«˛ˆÏe ˆÑ˛yˆÏòy ~Ñ˛!›˛ xAÑ˛ Óy ~Ñ˛y!ïÑ˛ xˆÏAÑ˛Ó˚ ˛ô%òÓ˚yÓ,!_ £ˆÏFäÈ òy–

x¢#õ ~ÓÇ xòyÓ,_ îü!õÑ˛ ¢ÇÖƒy~Ñ˛ õ)úî ¢ÇÖƒyÎ˚ ≤ÃÑ˛yü ı

v zîy£Ó˚í ı õ)úî ¢ÇÖƒyÎ˚ ≤ÃÑ˛yü Ñ˛ ÏÓ˚y ı

(i) 0.4 (ii) 0.23

¢õyïyò ı ïÓ˚y ÎyÑ˛ñ 0.4 ........( )x i=

0.444..... x⇒ =

0.444..... 10 10x⇒ × = × Sv˛z¶˛Î˚˛ô«˛ˆÏÑ˛ 10 myÓ˚y Ü%í Ñ˛ ÏÓ˚V

4.444..... 10 .......(2)x⇒ =

ˆòy›˛



(1) Á (2) ˆÌˆÏÑ˛ ˛ôy£z ı

4.444... – 0.444... = 10x – x

4 9x⇒ =

49

x∴ =

(ii) ïÓ˚y ÎyÑ˛ñ 0.23 = x ........(i)

0.232323..... x⇒ =

0.232323...... 100 100x⇒ × = ×

23.232323..... 100 ......(2)x⇒ =

¢õ#Ñ˛Ó˚í (2) ˆÌ ÏÑ˛ (1) !ÓˆÏÎ˚yÜ Ñ˛ˆÏÓ˚ ˛ôy£zÈÙÙÙÈ

23 = 99x

2399

x∴ =

ú«˛ƒ Ñ˛Ó%ò ı ¢¢#õñ x¢#õ ~ÓÇ xyÓ,_ îü!õÑ˛ ¢ÇÖƒy ÏÑ˛ õ)úî ¢ÇÖƒyÓ˚ myÓ˚y ≤ÃÑ˛yü Ñ˛Ó˚y

ÎyÎ˚– !Ñ˛hs%˛ ~Ñ˛!›˛ x¢#õ Á xòyÓ,_ îü!õÑ˛ ¢ÇÖƒy õ)úî ¢ÇÖƒyˆÏÑ˛ ≤ÃÑ˛yü Ñ˛ˆÏÓ˚ òy–

fiÌyò#Î˚ õyò ˛ôÂï!ì˛Ó˚ !ÓÓï≈ò ı

10000 1000 100 10 1 110

1100

11000

= 104 = 103 = 102 = 101 = 100

ì˛y£ˆÏú ˆîÖy ÎyˆÏFäÈñ ~Ñ˛ˆÏÑ˛Ó˚ fiÌyˆÏòÓ˚ v˛yò!îˆÏÑ˛ îüÑ˛ñ üì˛Ñ˛ñ £yãyÓ˚ £zì˛ƒy!îÓ˚ fiÌyò

ÌyˆÏÑ˛–

¢%ì˛Ó˚yÇ 23.715 = 2 × 10 + 3 ×1 + 7 × 110

+ 1 × 1100

+ 5 × 11000

xy˛ôòyÓ˚ x@˘ÃÜ!ì˛ Îyâ˛y£z Ñ˛Ó˚%ò ı

E14. ˆÎyˆÏÜÓ˚ ˆ«˛ˆÏe !Ó˛ôÓ˚#ì˛ ˆÑ˛yò!›˛⁄

(a) 27 (b) 3

8− (c) 0?

ˆòy›˛

28 D.El.Ed. 504


E15. ¢õì% úƒ îü!õÑ˛ ¢ÇÖƒy!›˛ !òí≈Î˚ Ñ˛Ó%òÈÙÙÙÈ

(i) 1225 (ii) 7

8 (iii) 27

E16. Ü% ÏíÓ˚ ˆ«˛ˆÏe xˆÏòƒyòƒÑ˛ !òí≈Î˚ Ñ˛Ó%òÈÙÙÙÈ

(a) 37 (b) 5

8 (c) 0

E17. 0.51 ˆÑ˛ pq xyÑ˛yˆÏÓ˚ ≤ÃÑ˛yü Ñ˛Ó˚%ò–

5.4 Ü%íò#Î˚Ñ˛ ~ÓÇ Ü%!íì˛Ñ˛ ı

fl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ Ü%í Á ¶˛yˆÏÜÓ˚ ˆ«˛ˆÏe xyˆÏúyâ˛òyÓ˚ ˆ«˛ˆÏe xyõÓ˚y Ü%íò#Î˚Ñ˛ Á Ü%!íì˛ˆÏÑ˛Ó˚

Óƒy˛ôyˆÏÓ˚ ˆãˆÏò!äÈ– ~£z !Ó¶˛yˆÏÜñ xyõÓ˚y ¢yïyÓ˚í Ü%íò#Î˚Ñ˛ Á Ü%!íì˛Ñ˛ !òí≈Î˚ Ñ˛Ó˚yÓ˚ Óƒy˛ôyˆÏÓ˚

~ÓÇ Ü!Ó˚¤˛ ¢yïyÓ˚í Ü%íò#Î˚Ñ˛ SÜ.¢y.Ü%.V ~ÓÇ ú!á¤˛ ¢yïyÓ˚í Ü%!íì˛Ñ˛ Sú.¢y.Ü%.V !òí≈Î˚ ~ÓÇ

Óyhfl ÏÓ ¢õ¢ƒyÓ˚ ¢õyïy ÏòÓ˚ ˆ«˛ Ïe ~£z ïyÓ˚íyÓ˚ ÓƒÓ£y ÏÓ˚Ó˚ Ñ˛Ìy ãyò ÏÓy–

5.4.1 ¢yïyÓ˚í Ü%íò#Î˚Ñ˛ ~ÓÇ Ü!Ó˚¤˛ ¢yïyÓ˚í Ü%íò#Î˚Ñ˛ ı

î%!›˛ ¢ÇÖƒy ˆòÁÎ˚y £ú 12 ~ÓÇ 18.

12-~Ó˚ Ü%íò#Î˚Ñ˛ ı 1, 2, 3, 4, 6 ~ÓÇ 12........(a)

18-~Ó˚ Ü%íò#Î˚Ñ˛ ı 1, 2, 3, 6, 9 ~ÓÇ 18........(b)

(a) Á (b) ˆÌ ÏÑ˛ ˛ôyÁÎ˚y ÎyÎ˚ÈÙÙÙÈ

1, 2, 3 Á 6 ~£z Ü%íò#Î˚Ñ˛Ü%!ú v˛z¶˛Î˚ˆÏ«˛ˆÏe£z xyˆÏäÈ– ¢%ì˛Ó˚yÇ 12 Á 18ÈÙÈÓ˚ ¢yïyÓ˚í

Ü%òò#Î˚Ñ˛Ü%!ú £ú ı 1, 2, 3 ~ÓÇ 6

~£z Ü%íò#Î˚Ñ˛Ü%!úÓ˚ õ Ïïƒ Ó,£_õ £ú 6.

ì˛y£z 6 ˆÑ˛ 12 Á 18ÈÙÈ~Ó˚ Ü.¢y.Ü%. Óúy £Î˚–

Ü.¢y.Ü%. !òí≈Î˚ ı

˛ôÂï!ì˛ (i) ≤Ã!ì˛!›˛ ¢ÇÖƒyÎ˚ Ü%íò#Î˚Ñ˛ !òí≈Î˚ Ñ˛Ó˚yÓ˚ ˛ôˆÏÓ˚ñ Ü.¢y.Ü% !òí≈Î˚ Ñ˛Ó˚y ÎyÎ˚ SˆÎõò 12

Á 18 ~Ó˚ ˆ«˛ Ïe ˆîÖy Ïòy £úV

˛ôÂï!ì˛ (ii) ˆõÔ!úÑ˛ v zÍ˛ôyîÑ˛ ˛ôÂï!ì˛ ı

12 = 2 × 2 × 3 = 22 × 31

ˆòy›˛



18 = 2 × 3 × 3 = 21 × 32

v z¶˛Î˚ ¢ÇÖƒyÓ˚ ˆ«˛ˆÏe ¢yïyÓ˚í ˆõÔ!úÑ˛ v zÍ˛ôyîˆÏÑ˛Ó˚ ¢Ó≈!ò¡¨ ¢)â˛ˆÏÑ˛Ó˚ Ü%íö˛ú £ú Ü.¢y.Ü%

∴ 12 Á 18ÈÙÈ~Ó˚ Ü.¢y.Ü% = 21 × 31 = 6

˛ôÂï!ì˛ (iii) Ñ õyÜì˛ ¶˛yÜ ˛ôÂï!ì˛ ı

Step-1 : «% oì˛õ ¢ÇÖƒy myÓ˚y x˛ôÓ˚ î%!›˛ ÓˆÏv ¸y ¢ÇÖƒyˆÏÑ˛ ¶˛yÜ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ ~ÓÇ ¶˛yÜ Ñ˛Ó˚ˆÏì˛

£ ÏÓ ~ÓÇ ¶˛yÜ Ïü°Ï ÏÑ˛ !òí≈Î˚ Ñ˛Ó˚ Ïì˛ £ ÏÓ–

Step-2 : ˛ôÓ˚Óì≈˛#Ñ˛yˆÏú ˛ôyÁÎ˚y ¶˛yÜˆÏü°Ï !îˆÏÎ˚ xyˆÏÜÓ˚ ¶˛yãÑ˛ˆÏÑ˛ ¶˛yÜ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ–

~£z ˛ôÂï!ì˛ ì˛ì˛«˛í â˛ú ÏÓ Îì˛«˛í òy ¶˛yÜ Ïü°Ï ü)òƒ £ ÏÓ– ˆü°Ï ¶˛y ÏÜÓ˚ ˆ«˛ Ïe ¶˛yãÑ˛ ÏÑ˛

Óúy £ ÏÓ !ò Ïí≈Î˚ Ü.¢y.Ü%.–

ú.¢y.Ü%.ÈÙÈ~Ó˚ ≤ÃˆÏÎ˚yÜ ı

ˆ◊!íÑ˛ Ï«˛ 24 ãò ÓyúÑ˛ Á 30 ãò Óy!úÑ˛y xy ÏäÈ– ≤Ã!ì˛!›˛ úy£z Ïò ~Ñ˛£z ¢ÇÖƒÑ˛ !ü«˛yÌ≈# ÏÑ˛

Ó¢y Ïòy Îy ÏÓ ~õò !Ñ˛ä%È ˛ô,ÌÑ˛ úy£zò ˜ì˛Ó˚# Ñ˛Ó˚ Ïì˛ £ ÏÓ– ¢ ÏÓ≈yFâ˛ Ñ˛ì˛ ¢ÇÖƒÑ˛ ÓyúÑ˛ ÏÑ˛ ~Ñ˛

~Ñ˛!›˛ úy£z Ïò Ó¢y Ïú ¢õhfl Ï Óy!úÑ˛y Á ÓyúÑ˛ ÏÑ˛ Ó¢y Ïòy Îy ÏÓ⁄

~Ñ˛ ~Ñ˛!›˛ úy£z Ïò ¢ˆÏÓ≈yFâ˛ !ü«˛yÌ≈#Ó˚ ¢ÇÖƒy = 24 Á 30 ÈÙÈ~Ó˚ Ü.¢y.Ü%.

24 Á 30 ÈÙÈ~Ó˚ Ü.¢y.Ü%. 6

¢%ì˛Ó˚yÇ ~£z¶˛yˆÏÓ xyõÓ˚y v z_Ó˚ ˆ˛ôˆÏÎ˚ ÎyˆÏÓy–

5.4.2 ¢yïyÓ˚í Ü%íò#Î˚Ñ˛ ~ÓÇ ú!á¤˛ ¢yïyÓ˚í Ü%!íì˛Ñ˛ Sú.¢y.Ü%Vı

î%!›˛ ¢ÇÖƒy ˆòÁÎ˚y £ú 8 ~ÓÇ 12

8ÈÙÈ~Ó˚ Ü%!íì˛Ñ˛Ü%!ú £ú ı 8×1, 8×2, 8×3, 8×4 £zì˛ƒy!î

∴ 8ÈÙÈ~Ó˚ Ü%!íì˛Ñ˛ £ú ı 8, 16, 24, 32, 40, 48, 56, 64, 72 .....

S~ÓÇ ~£z ì˛y!úÑ˛y x¢#õV

xò%Ó˚)ˆÏ˛ô 12ÈÙÈ~Ó˚ Ü%!íì˛Ñ˛ £ú ı 12, 24, 36, 48, 60, 72 ....Sx¢#õ ¢ÇÖƒÑ˛V

8 ~ÓÇ 12 ÈÙÈ~Ó˚ ¢yïyÓ˚í Ü%!íì˛Ñ˛Ü%!ú £ú ı 24, 48, 72, ....

~£z ¢yïyÓ˚í Ü%!íì˛ˆÏÑ˛Ó˚ ì˛y!úÑ˛y!›˛Á x¢#õ–

∴ 8È ~ÓÇ 12ÈÙÈ~Ó˚ ú.¢y.Ü%. £ú 24

ú.¢y.Ü% !òí≈Î˚ ı

˛ôÂï!ì˛ (i) ı v˛z˛ô!Ó˚v˛z=˛ ˛ôÂï!ì˛ xò%ÎyÎ˚# î%!›˛ ¢ÇÖƒyÓ˚ Ü%!íì˛ˆÏÑ˛Ó˚ ì˛y!úÑ˛y ˆÌˆÏÑ˛ ¢yïyÓ˚í

ˆòy›˛

30 D.El.Ed. 504


Ü%!íì˛Ñ˛ Ö§%ˆÏã ÓyÓ˚ Ñ˛ˆÏÓ˚ñ ì˛yˆÏîÓ˚ ú.¢y.Ü% ÓyÓ˚ Ñ˛Ó˚y ÎyˆÏÓ–

˛ôÂï!ì˛ (ii) ı ˆõÔ!úÑ˛ v zÍ˛ôyîÑ˛ ˛ôÂï!ì˛ ı

ïÓ˚y ÎyÑ˛ñ 12 Á 18 ~Ó˚ ú.¢y.Ü% !òí≈Î˚ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ–

12 = 2 × 2 × 3 = 22 × 31

18 = 2 × 3 × 3 = 21 × 32

∴ ú.¢y.Ü% = 22 × 32 = 2 × 2 × 3× 3 = 36

î%!›˛ ¢ÇÖƒyÓ˚ ú.¢y.Ü% Á Ü.¢y.Ü% ~Ó˚ ¢¡ôÑ≈˛ ı

!ò Ïâ˛Ó˚ v zîy£Ó˚íÜ%!ú ú«˛ƒ Ñ˛Ó%ò ı

¢ÇÖƒy ¢ÇÖƒyî%!›˛Ó˚ Ü.¢y.Ü% ú.¢y.Ü% Ü.¢y.Ü%

Ü%íö˛ú Á ú.¢y.Ü%

~Ó˚ Ü%íö˛ú

12 Á 18 216 6 36 216

16 Á 28 448 4 112 448

25 Á 35 875 5 175 875

ì˛y£ˆÏú xyõÓ˚y ˆîÖˆÏì˛ ˆ˛ôúyõ ˆÎñ

î%!›˛ ïòydÑ˛ ¢ÇÖƒyÓ˚ Ü%íö˛ú = ¢ÇÖƒyî%!›˛Ó˚ Ü.¢y.Ü% Á ú.¢y.Ü%ÈÙÈ~Ó˚ Ü%íö˛ú

E18. î%!›˛ ˛ôÓ˚fl˛ôÓ˚ ˆõÔ!úÑ˛ ¢ÇÖƒyÓ˚ Ü.¢y.Ü% Ñ˛ì˛⁄

E19. î%!›˛ ¢ÇÖƒyÓ˚ ú.¢y.Ü% Á Ü.¢y.Ü% ÎÌyÑ ˆÏõ 8 Á 96–

~Ñ˛!›˛ ¢ÇÖƒy 24 £ˆÏúñ xòƒ!›˛ Ñ˛ì˛⁄

5.5 ˛ôy›˛#Ü!íì˛ ~ÓÇ ì˛yÓ˚ ≤Ã ÏÎ˚yÜ ı

A ˙!Ñ˛Ñ˛ ˛ôÂï!ì˛ ı

20!›˛ !üü%ˆÏÑ˛ 5 !›˛ î Ïú ¢õyò¶˛y ÏÓ !Ó¶˛=˛ Ñ˛Ó˚ Ïì˛ £ ÏÓ– ~Öy Ïò xyõÓ˚y ¶˛y ÏÜÓ˚ Ñ˛yã Ñ˛Ó˚Ó–

5!›˛ î ÏúÓ˚ ãòƒ 5!›˛ fiÌyò !â˛!£´ì˛ Ñ˛Ó˚y £ú ~ÓÇ ≤Ã!ì˛!›˛ fiÌyˆÏò ~Ñ˛!›˛ Ñ˛ˆÏÓ˚ !üü%ˆÏÑ˛ î§yv˛¸yˆÏì˛

Óúy £ú–

îú ÙÈB îú ÙÈC îú ÙÈD îú ÙÈEîú ÙÈA

ˆòy›˛



20 ãˆÏòÓ˚ õˆÏïƒ 5 ãò Óyî £ ÏÎ˚ ˆÜ Ïúy–

∴ 20 – 5 = 15 ãò xÓ!üT˛ Ó˚£z Ïúy–

≤Ã!ì˛!›˛ fiÌyˆÏò xyˆÏÓ˚Ñ˛ãò Ñ˛ˆÏÓ˚ !üü% ÏÑ˛ î§yv ¸yˆÏì˛ Óúy £ú–

15 ãˆÏòÓ˚ ˆÌˆÏÑ˛ xyÓ˚Á 5 ãò ÏÑ˛ Óyî ˆîÁÎ˚y £ú–

∴ 15 – 5 = 10 ãò xÓ!üT˛ Ó˚£z Ïúy–

~Ó˚˛ôÓ˚ ì, ì˛#Î˚ !üü%!›˛ˆÏÑ˛ ì˛yˆÏîÓ˚ ˛ôyˆÏü î§yv ¸yˆÏì˛ Óúy £ú–

~Öò 5 ãò â˛ˆÏú ÎyÁÎ˚yÓ˚ ˛ôˆÏÓ˚ 10 – 5 = 5 ãò xÓ!üT˛ Ó˚£zú–

~ÓyÓ˚ñ â˛ì%˛Ì≈ !üü%ˆÏÑ˛ xyˆÏÜÓ˚ fiÌyòÜ%!úˆÏì˛ î§yv˛¸yˆÏì˛ Óúy £ú ı

~Öò 5 – 5 = 0 xÓ!üT˛ ÌyÑ˛ Ïúy–

ì˛y£ Ïú ˆîÖy ˆÜú ˆÎñ ≤Ã!ì˛!›˛ î Ïú 4 ãò Ñ˛ˆÏÓ˚ ÌyÑ˛ˆÏÓ–

∴ 20 ÷ 5 = 4 ˛ôyÁÎ˚y ˆÜú–

¢%ì˛Ó˚yÇ ˆîÖy ˆÜú ≤Ã!ì˛!›˛ î Ïú 4 ãò Ñ˛ ÏÓ˚ !üü% ÌyÑ˛ ÏÓ–

xòƒyòƒ v zîy£Ó˚í ı

(i) Î!î 5!›˛ ˛ôyˆÏeÓ˚ ˆõy›˛ 20l Ñ˛ˆÏÓ˚ î%ï ÌyˆÏÑ˛ñ ì˛y£ˆÏú

1!›˛ ˛ôyˆÏe 20l ÷ 5 = 4l Ñ˛ ÏÓ˚ î%ï ÌyÑ˛ ÏÓÎ

(ii) Î!î 5m !ö˛ì˛yÓ˚ îyõ 20 ›˛yÑ˛y £Î˚ñ ì˛y£ Ïú

1 !õ›˛yÓ˚ !ö˛ì˛yÓ˚ îyõ £ˆÏÓ 20 ÷ 5 = 4 ›˛yÑ˛y

îú ÙÈA îú ÙÈB îú ÙÈC îú ÙÈD îú ÙÈE



ˆòy›˛

32 D.El.Ed. 504


● ~ÓyÓ˚ ˆîÖy ÎyÑ˛ñ ~Ñ˛!›˛ Óhfl% Ó˚ ˆ«˛ˆÏe ì˛Ìƒ ãyòy ÌyÑ˛ˆÏúñ

~Ñ˛y!ïÑ˛ Óhfl%˛Ó˚ ˆ«˛ˆÏe ì˛# Ñ˛# £ˆÏÓ⁄

~Ñ˛!›˛ ˆ˛ôˆÏòÓ˚ îyõ 8 ›˛yÑ˛y

ì˛y£ˆÏú 3!›˛ ˆ˛ôˆÏòÓ˚ îyõ Ñ˛ì˛⁄

fl˛ôT˛¶˛y ÏÓñ 3!›˛ ˆ˛ôˆÏòÓ˚ ˆõy›˛ îyõ = 8 + 8 + 8 ›˛yÑ˛y

!Ñ˛hs%˘˛ xyõÓ˚y ãy!ò ˆÎñ 8 × 3 = 8 + 8 + 8

∴ xyõÓ˚y ÓúˆÏì˛ ˛ôy!Ó˚ ˆÎñ 3!›˛ ˆ˛ôˆÏòÓ˚ îyõ = 8 × 3

ì˛y£ Ïú xyõÓ˚y ˆîÖúyõ ˆÎñ

1!›˛ ˆ˛ôˆÏòÓ˚ îyõ 8 ›˛yÑ˛y £ ÏúÈÙÙÙ

È3!›˛ ˆ˛ôˆÏòÓ˚ îyõ £ˆÏÓ 8 × 3 = 24 ›˛yÑ˛y

xòƒyòƒ v zîy£Ó˚í ı

(iii) 1˛ôƒyˆÏÑ˛›˛ úÓˆÏíÓ˚ Áãò 600 @˘Ãyõ–

ì˛y£ˆÏú 4 ˛ôƒyˆÏÑ˛›˛ úÓˆÏíÓ˚ Áãò 600 × 4 @˘Ãyõ

= 2400 @˘Ãyõ

= 2.4 ˆÑ˛!ã

(iv) ~Ñ˛!›˛ ˆì˛ˆÏúÓ˚ ãyˆÏÓ˚Ó˚ Áãò 12 ˆÑ˛!ã

ì˛y£ Ïú 5!›˛ ãyˆÏÓ˚Ó˚ Áãò = 12 ˆÑ˛!ã × 5

= 60 ˆÑ˛!ã

~ÓyÓ˚ v z ô ÏÓ˚ ≤Ãî_ v zîy£Ó˚íÜ%!ú ÏÑ˛ !Ó ÏŸ’°Ïí Ñ˛Ó˚y ÎyÑ˛–

≤Ã!ì˛!›˛ ≤Ãî_ v zîy£Ó˚ ÏíÓ˚ î%!›˛ Ñ˛ ÏÓ˚ â˛úÓ˚y!ü xy ÏäÈ–

v˛zîy£Ó˚í ≤ÃÌõ â˛úÓ˚y!ü !mì˛#Î˚ â˛úÓ˚y!ü

(i) ˛ôyˆÏeÓ˚ ¢ÇÖƒy ïyÓ˚í «˛õì˛y

(ii) ˜îá≈ƒ îyõ

(iii) ˛ôƒyˆÏÑ˛ˆÏ›˛Ó˚ ¢ÇÖƒy Áãò

(iv) ãyˆÏÓ˚Ó˚ ¢ÇÖƒy Áãò

v z ô ÏÓ˚Ó˚ ≤Ã!ì˛!›˛ v zîy£Ó˚ Ïí Îì˛ÓyÓ˚ ≤ÃÌõ â˛úÓ˚y!üÓ˚ v z ÏÕ‘Ö xy ÏäÈñ ì˛ì˛ÓyÓ˚£z !mì˛#Î˚ â˛úÓ˚y!üÓ˚

v z ÏÕ‘Ö xy ÏäÈ–

ˆòy›˛



ˆÎõòÈÙÙÙÈ

î%ÛÓyÓ˚ ˛ôyˆÏeÓ˚ ¢ÇÖƒy ~ˆÏúñ î%ÓyÓ˚ ì˛yˆÏîÓ˚ ïyÓ˚í «˛õì˛yÓ˚ Ñ˛Ìy xy¢ˆÏÓ–

!ì˛òÓyÓ˚ ˜îˆÏá≈ƒÓ˚ Ñ˛Ìy ~ˆÏúñ !ì˛òÓyÓ˚ ì˛yÓ˚ îyõ xy¢ˆÏÓ–

ì˛y£ˆÏúÈÙÙÙÈ

(i) ÎÖò xˆÏòÑ˛ !Ñ˛ä%ÈÓ˚ K˛yò ˆÌˆÏÑ˛ ~Ñ˛!›˛ ¢¡ôˆÏÑ≈˛ Üíòy Ñ˛Ó˚ˆÏì˛ â˛y£z ì˛Öò xyõÓ˚y ¶˛yÜ

Ñ˛Ó˚ˆÏÓy–

(ii) ÎÖò ~Ñ˛!›˛Ó˚ ì˛Ìƒ ˆÌˆÏÑ˛ xˆÏòÑ˛Ü%!ú ¢¡ôˆÏÑ≈˛ ì˛Ìƒ ≤ÃˆÏÎ˚yãòñ ì˛Öò xyõÓ˚y Ü%í

Ñ˛Ó˚ˆÏÓy–

î%!›˛ â˛úÓ˚y!ü Î!î ~õò £Î˚ ˆÎñ ≤ÃÌõ!›˛ Îì˛ÓyÓ˚ v˛zˆÏÕ‘Ö ÌyÑ˛ˆÏÓñ !mì˛#Î˚!›˛Á ì˛ì˛ÓyÓ˚

ÌyÑ˛ˆÏÓñ ì˛Öò ì˛yˆÏîÓ˚ ÓúˆÏÓy ≤Ãì˛ƒ«˛ ¢¡ôÑ≈˛Î%=˛ â˛úÓ˚y!ü–

Óƒyhfl Ïyò% ôy!ì˛Ñ˛ â˛úÓ˚y!ü ı

!òˆÏâ˛Ó˚ ¢õ¢ƒy!›˛ ú«˛ƒ Ñ˛Ó%í ı

3 ãò ◊!õÑ˛ ~Ñ˛!›˛ Ñ˛yã 5 !î Ïò ¢¡ôß¨ Ñ˛ ÏÓ˚–

Î!î ~Ñ˛ãò ◊!õÑ˛ ~Ñ˛y Ñ˛yã!›˛ Ñ˛ˆÏÓ˚ñ ì˛y£ˆÏú ˆ¢!›˛ Ñ˛ì˛!îˆÏò ¢¡ôß¨ £ˆÏÓ⁄

xyõÓ˚y xyõyˆÏîÓ˚ x!¶˛K˛ì˛y ˆÌˆÏÑ˛ ÓúˆÏì˛ ˛ôy!Ó˚ ˆÎñ ◊!õÑ˛!›˛Ó˚ 5 + 5 + 5 = 5 × 3 !îò ¢õÎ˚

úyÜˆÏÓ–

ì˛y£ Ïú xyõÓ˚y ˆîÖúyõ ˆÎñ

Î!î 3 ãò ◊!õÑ˛ ÏîÓ˚ Ñ˛yã!›˛ Ñ˛Ó˚ Ïì˛ ¢õÎ˚ úy ÏÜ 5 !îò ì˛y£ Ïúñ

1 ãò ◊!õ ÏÑ˛Ó˚ ¢õÎ˚ úyÜ ÏÓ 5 × 3 = 15 !îò–

~Ó˚Ñ˛õ xyÓ˚Á !Ñ˛ä%È v zîy£Ó˚í ˆî ÏÖ ˆòÁÎ˚y ÎyÑ˛ÈÙÙÙÈ

(i) 8 ãò õyò%°Ï !Ñ˛ä%È ˛ô!Ó˚õyò Öyîƒ 5 !îˆÏò ˆÖˆÏì˛ ˛ôyˆÏÓ˚– ì˛y£ˆÏúñ

1 ãò õyò%°Ï Á£z ~Ñ˛£z ˛ô!Ó˚õyí Öyîƒ ÖyˆÏÓ 5 × 8 = 40 !îˆÏò–

(ii) ~Ñ˛ãò õyò%°Ï 24 !îˆÏò ~Ñ˛!›˛ Ñ˛yã Ñ˛ˆÏÓ˚ñ ì˛y£ˆÏú

3 ãò õyò%°Ï Á£z Ñ˛yã!›˛ ˆü°Ï Ñ˛Ó˚ˆÏÓ 24 ÷ 3 = 8 !îˆÏò–

(iii) ~Ñ˛!›˛ !ò!î≈T˛ ˛ô!Ó˚õyí ÖyÓyÓ˚ 1 ãò õyò%°Ï 30 !î Ïò ˆü°Ï Ñ˛ ÏÓ˚–

ì˛y£ Ïú 5 ãò õyò%°Ï Á£z ÖyÓyÓ˚ ÖyˆÏÓ 30 ÷ 5 = 6 !îˆÏò–

v˛z˛ô!Ó˚v˛z=˛ v˛zîy£Ó˚ˆÏíÓ˚ â˛úÓ˚y!üÜ%!ú £ú ı

ˆòy›˛

34 D.El.Ed. 504


õyò%ˆÏ°ÏÓ˚ ¢ÇÖƒy Ñ˛yã Ñ˛Ó˚yÓ˚ ¢õÎ˚

!ò!î≈T˛ ˛ô!Ó˚õyí ÖyÓyÓ˚ ÖyÓyÓ˚ ÖyÁÎ˚yÓ˚ ¢õÎ˚

ˆÖ Ïì˛ ˛ôyÓ˚ ÏÓ ~Ó˚Ñ˛õ

õyò%ˆÏ°ÏÓ˚ ¢ÇÖƒy

xyõÓ˚y ˆîÖ!äÈ ˆÎñ

≤ÃÌõ â˛úÓ˚y!ü !mÜ%ò £ˆÏúñ !mì˛#Î˚ â˛úÓ˚y!ü xˆÏï≈Ñ˛ £ˆÏÎ˚ ÎyÎ˚– ≤ÃÌõ â˛úÓ˚y!ü ~Ñ˛ÈÙÈâ˛ì% Ì≈yÇü

£ˆÏúñ !mì˛#Î˚ â˛úÓ˚y!ü â˛yÓ˚Ü%í £ˆÏÓ– ì˛y£z ~£z â˛úÓ˚y!üÜ%!ú Óƒyhfl˛Ïyò% ôy!ì˛Ñ˛ ¢¡ôÑ≈˛Î%=˛–

Ñ˛yÎ≈ Á ¢õ ÏÎ˚ ˙!Ñ˛Ñ˛ !òÎ˚ ÏõÓ˚ ≤Ã ÏÎ˚yÜ ı

~Ñ˛!›˛ v zîy£Ó˚í ˆîÖy ÎyÑ˛ñ

v zîy£Ó˚í ı A ~Ñ˛!›˛ Ñ˛yã 24 !îˆÏò ~ÓÇ B Á£z ~Ñ˛£z Ñ˛yã 18 !î Ïò ˆü°Ï Ñ˛ ÏÓ˚– A ~ÓÇ B

~Ñ˛¢yˆÏÌ Ñ˛yã!›˛ ü%Ó% Ñ˛Ó˚ˆÏúy–

4 !îò ˛ôˆÏÓ˚ A Ñ˛yã!›˛ ˆäÈˆÏv ¸ !îˆÏúy– ~Ó˚˛ôÓ˚ Ñ˛yã!›˛ ¢¡ôß¨ Ñ˛Ó˚ˆÏì˛ Ñ˛ì˛!îò ¢õÎ˚ úyÜˆÏÓ⁄

¢õyïyò ı

A Ñ˛yã!›˛ Ñ˛ ÏÓ˚ 24 !îˆÏò

∴ A 1 !îˆÏò Ñ˛ˆÏÓ˚ Ñ˛yã!›˛Ó˚ 124 xÇü

B ¢õ@˘Ã Ñ˛yã!›˛ Ñ˛ˆÏÓ˚ 18 !îˆÏò

∴ B 1 !îˆÏò Ñ˛ˆÏÓ˚ 118 xÇü

∴ A Á B ~Ñ˛¢ ÏAÜ 1 !îˆÏò Ñ˛ˆÏÓ˚ 124 + 1

18 = 3 4 772 72+ = xÇü

∴ A Á B, 4 !î Ïò Ñ˛yã!›˛Ó˚ 7 472

× xÇü = 718 xÇü

∴ xÓ!üT˛ ÌyˆÏÑ˛ 7 18 7 11118 18 18

−= − = = xÇü

B, 2 !îˆÏò Ñ˛ˆÏÓ˚ 118 xÇü

ˆòy›˛



∴ B, 1118 xÇü Ñ˛yã Ñ˛Ó˚ Ïì˛ ¢õÎ˚ ˆòÎ˚ = 11 1

18 18÷ !îò–

= 11 1818

× !îò

= 11 !îò

∴ ˆõy›˛ ¢õÎ˚ úyÜˆÏÓ = 4 + 11 = 15 !îò–

˙!Ñ˛Ñ˛ !òÎ˚ˆÏõ ¢õyïyò Ñ˛Ó˚yÓ˚ ¢õÎ˚ñ !ò¡¨!ú!Öì˛ !òÎ˚õ õˆÏò Ó˚yÖˆÏì˛ £ˆÏÓ ı

(i) ~Ñ˛Ñ˛ ¢õ ÏÎ˚ Ñ, ì˛Ñ˛yÎ≈ = Ñ˛, ì˛Ñ˛yÎ≈

Ñ˛yã!›˛ Ñ˛Ó˚ Ïì ˛Îy ¢õÎ˚ úyˆÜÏ

(ii) ¢õÎ˚ = ¢õ@˘ÃÑ˛yÎ≈

~Ñ˛Ñ˛ ¢õˆÎÏ Ñ , ì˛Ñ˛yÎ≈

(B) üì˛Ñ˛Ó˚y !òí≈Î˚ ı

ÚÚüì˛Ñ˛Ó˚yÛÛ ü∑!›˛Ó˚ xÌ≈ ı

üì˛Ñ˛Ó˚yÓ˚ xÌ≈ Ú~Ñ˛ˆÏüyÓ˚ õˆÏïƒÛ– !Ñ˛hs%˛ Ñ˛Öò ~!›˛ ÓƒÓ£yÓ˚ Ñ˛Ó˚Ó⁄

A ~ÓÇ B î%ãò !ü«˛yÌ≈#– A 80 ò¡∫ˆÏÓ˚Ó˚ ~Ñ˛!›˛ ˛ôÓ˚#«˛y !îˆÏÎ˚ 64 ˆ˛ôˆÏúy– B 75 ò¡∫ˆÏÓ˚Ó˚

xòƒ ~Ñ˛!›˛ ˛ôÓ˚#«˛yÎ˚ 63 ˆ˛ôú– Ñ˛yÓ˚ ¢yö˛úƒ ˆÓü#⁄ 80ÈÙÈÓ˚ õˆÏïƒ 64 òy!Ñ˛ 75ÈÙÈ~Ó˚ õˆÏïƒ 68?

Î!î î%!›˛ ˛ôÓ˚#«˛yÓ˚ ˛ô)í≈õyò ¢õyò £ˆÏì˛yñ ì˛y£ˆÏú ~£z¶˛yˆÏÓ ì%˛úòy Ñ˛Ó˚y xˆÏòÑ˛ ¢£ã £ì˛–

ì˛y£ˆÏú ïÓ˚y ÎyÑ˛ñ ˛ôÓ˚#«˛yÓ˚ ˛ô)í≈õyò 100–

A 80 ò¡∫ˆÏÓ˚Ó˚ õˆÏïƒ ˆ˛ôˆÏÎ˚ˆÏäÈ 64

ì˛y£ Ïú 100 ò¡∫ˆÏÓ˚Ó˚ õˆÏïƒ 4 ˆ˛ôˆÏÎ˚ˆÏäÈ 64 100 = 8080

×

ì˛y£ Ïú ~Öò xyõÓ˚y Óú Ïì˛ ˛ôy!Ó˚ ˆÎñ A, 100 ÈÙÈ~Ó˚ õ Ïïƒ 80 ˆ˛ôˆÏÎ˚ˆÏäÈ xÌ≈yÍ xyõÓ˚y

ÓúˆÏì˛ ˛ôy!Ó˚ ˆÎñ A, 100ÈÙÈ~Ó˚ õˆÏïƒ 80 ˆ˛ôˆÏÎ˚ˆÏäÈ xÌ≈yÍ üì˛Ñ˛Ó˚y 80 Óy 80% ˆ˛ô ÏÎ˚ ÏäÈ–

xò%Ó˚)ˆÏ˛ô B, 75 ò¡∫ˆÏÓ˚Ó˚ õˆÏïƒ ˆ˛ôˆÏÎ˚ˆÏäÈ 63

∴ 100 ò¡∫ˆÏÓ˚Ó˚ õˆÏïƒ ˆ˛ôˆÏÎ˚ˆÏäÈ 63 100 = 8475×

ì˛y£ˆÏúñ B 100 ò¡∫ˆÏÓ˚Ó˚ õˆÏïƒ 84 xÌ≈yÍ üì˛Ñ˛Ó˚y 84 Óy 84% ˆ˛ôˆÏÎ˚ˆÏäÈ–

ˆòy›˛

36 D.El.Ed. 504


∴ BÈÙÈ~Ó˚ ¢yö˛úƒ A-~Ó˚ ì% úòyÎ˚ ˆÓü#–

ì˛y£ˆÏú xyõÓ˚y ãyòˆÏì˛ ˛ôyÓ˚úyõ ˆÎñ

üì˛Ñ˛Ó˚y £ú ~Ñ˛!›˛ ¢ÇÖƒyÓ˚ ¢yˆÏÌ xòƒ!›˛Ó˚ ì% úòy–

~£z ì% úòyÓ˚ ¢õÎ˚ !mì˛#Î˚ ¢ÇÖƒy!›˛ ÏÑ˛ 100 ïÓ˚y £Î˚–

p ˆÑ˛ qÈÙÈ~Ó˚ ¢yˆÏÌ ì% úòy Ñ˛Ó˚ˆÏú ˛ôy£zñ p

100%q ×

üì˛Ñ˛Ó˚y ≤Ã ÏÎ˚yÜ ı

!ò¡¨!ú!Öì˛ ˛ô!Ó˚!fiÌ!ì˛ˆÏì˛ üì˛Ñ˛Ó˚yÓ˚ ÓƒÓ£yÓ˚ £Î˚ ı

ÓƒÓ¢yÎ˚ñ úy¶˛ Óy «˛!ì˛ˆÏÑ˛ Ñ Î˚õ) ÏúƒÓ˚ üì˛Ñ˛Ó˚y !£¢yˆÏÓ ≤ÃÑ˛yü Ñ˛Ó˚y £Î˚–

‘12% úyˆÏ¶˛Ó˚ xÌ≈ £ú Ñ Î˚õ) ÏúƒÓ˚ Á˛ôÓ˚ 12% úy¶˛–Û

}ò ˆòÁÎ˚yÓ˚ ¢õÎ˚ñ õ)úïˆÏòÓ˚ v z ôˆÏÓ˚ üì˛Ñ˛Ó˚y £yˆÏÓ˚ ¢%î ˆîÁÎ˚y £Î˚–

Ú¢%ˆÏîÓ˚ £yÓ˚ 10%Û ~Ó˚ xÌ≈ £úñ ~Ñ˛ ÓäÈˆÏÓ˚ õ)úïˆÏòÓ˚ v˛z ôÓ˚ 10% ¢%î !î Ïì˛ £ ÏÓ–

v zÍ˛ôyîˆÏòÓ˚ Ó,!Âï Óy £…y¢ˆÏÑ˛ õ)ú v˛zÍ˛ôyîˆÏòÓ˚ £yˆÏÓ˚Ó˚ ¢y£yˆÏÎƒ ≤ÃÑ˛yü Ñ˛Ó˚y ÎyÎ˚–

(a) úy¶˛ ~ÓÇ «˛!ì˛ ı

ÓƒÓ¢yÓ˚ ˆ«˛ˆÏeÈÙÙÙÈ

úy¶˛ = !ÓÑ Î˚õ)úƒ – Ñ Î˚õ)úƒ

«˛!ì˛ = Ñ ˛Î˚õ)úƒ – !ÓÑ Î˚õ)úƒ

úy¶˛% = Ñ Î˚õ) ÏúƒÓ˚ v z ôÓ˚ üì˛Ñ˛Ó˚y úy¶˛

= 100×˛úy¶Ñ˛ Îõú) ƒ

= 100×˛!ÓÑ Îõ˚ ú) ƒÈÙÈÑ Îõ˚ ú) ƒ

Ñ˛ Îõú) ƒ

«˛!ì˛% = Ñ Î˚õ) ÏúƒÓ˚ v z ôÓ˚ üì˛Ñ˛Ó˚y «˛!ì˛

= 100×«˛ ˛˛!ìÑ˛ Îõ)úƒ

= 100×−Ñ˛ Îõú) ƒ !ÓÑ Îõ˚ ú) ƒ

Ñ˛ Îõú) ƒ

ˆòy›˛



ú«˛ƒ Ñ˛Ó%í ı xyõÓ˚y ˆÎõò Óƒy ÏAÑ˛ ›˛yÑ˛y ãõy Ó˚y!Öñ ˆì˛õ!ò ÓƒyAÑ˛ ˆÌ ÏÑ˛ }íÁ !ò£z–

ÎÖò xyõÓ˚y ›˛yÑ˛y ãõy Ó˚y!Öñ xyõÓ˚y ÓƒyAÑ˛ ˆÌˆÏÑ˛ ¢%î ˛ôy£z–

ÎÖò xyõÓ˚y ÓƒyAÑ˛ ˆÌ ÏÑ˛ }í !ò£zñ xyõÓ˚y ÓƒyAÑ˛ ÏÑ˛ ¢%î !î£z–

~£z ¢%î !Ñ˛ Ñ˛ˆÏÓ˚ !£¢yÓ Ñ˛Ó˚y £Î˚⁄

Óƒy ÏAÑ˛Ó˚ ¢y ÏÌ ˆúò Ïî ÏòÓ˚ ¢õÎ˚ñ ÓƒyAÑ˛ ¢% ÏîÓ˚ £yÓ˚ Ñ˛ì˛ ì˛y ãy!ò ÏÎ˚ ˆîÎ˚–

xyõÓ˚y !¢Âïyhs˘˛ !ò£z ˆÎÈÙÙÙÈ

(i) Ñ˛ì˛ ›˛yÑ˛y }í ˆòˆÏÓy SxÌÓy ãõy ˆîˆÏÓyV

(ii) Ñ˛ì˛ !îˆÏòÓ˚ ãòƒ }í ˆòˆÏÓy SÓy ãõy Ó˚yÖˆÏÓyV

ïÓ˚y ÎyÑ˛ñ ‘p’ ›˛yÑ˛y }í ˆòÁÎ˚y £úñ ~ÓÇ ~!›˛ ‘t’ ÓäÈ ÏÓ˚Ó˚ ãòƒ ˆòÁÎ˚y £ú–

ÓƒyˆÏAÑ˛Ó˚ ¢% ÏîÓ˚ £yÓ˚ r%

}ˆÏíÓ˚ ˆõÎ˚yîÑ˛yú ˆü°Ï £ÁÎ˚yÓ˚ ˛ôÓ˚ Ñ˛ì˛ ›˛yÑ˛y ¢%î !îˆÏì˛ £ˆÏÓ⁄ ÓƒyAÑ˛ˆÏÑ˛ ˆõy›˛ Ñ˛ì˛ ›˛yÑ˛y

ˆö˛Ó˚Í !î Ïì˛ £ ÏÓ⁄

Üíòy ı ¢% ÏîÓ˚ £yÓ˚ r%

˛ ~Ó˚ õy Ïòñ 100 ›˛yÑ˛y õ)úïˆÏòÓ˚ ãòƒ 1 ÓäÈ ÏÓ˚Ó˚ ¢%î r

∴ 1 ›˛yÑ˛yÓ˚ ãòƒ 1 ÓäÈ ÏÓ˚Ó˚ ¢%î = r100

∴ P ›˛yÑ˛yÓ˚ ãòƒ 1 ÓäÈˆÏÓ˚Ó˚ ¢%î r PrP100 100

× =

ì˛y£ˆÏú P ›˛yÑ˛yÓ˚ t ÓäÈ ÏÓ˚Ó˚ ¢%î = Pr Prtt100 100

× =

∴ ¢%î I = Prt100 !òÎ˚õ I

}ˆÏíÓ˚ ˆõÎ˚yîÑ˛yˆÏúÓ˚ ˆüˆÏ°Ï ˆõy›˛ ›˛yÑ˛y ˆö˛Ó˚Í !îˆÏì˛ £ˆÏÓ

(A) = P + I

Óyñ A = PrtP100

+

A = P ( )Prt1100

+ !òÎ˚õ II

ˆòy›˛

38 D.El.Ed. 504


!òÎ˚õ ÈÙÈ IÈÙÈ~Ó˚ ≤ÃˆÏÎ˚yÜ

P, t, r ≤Ãî_ñ I !òí≈Î˚ Ñ˛Ó˚y

P, r, I ≤Ãî_ñ t !òí≈Î˚ Ñ˛Ó˚y

P, I, t ≤Ãî_ñ r !òí≈Î˚ Ñ˛Ó˚y

I, t, r ≤Ãî_ñ p !òí≈Î˚ Ñ˛Ó˚y

x@˘ÃÜ!ì˛ Îyâ˛y£z Ñ˛Ó%í ı

E20. ¢y!£ˆÏì˛ƒÓ˚ ~Ñ˛!›˛ ˛ôy‡˛ƒ˛ô%hfl˛ÏˆÏÑ˛Ó˚ ì%˛úòyÎ˚ ~Ñ˛!›˛ Ü!íˆÏì˛Ó˚ ˛ôy‡˛ƒ˛ô%hfl˛ÏˆÏÑ˛Ó˚ õ)úƒ 2 ›˛yÑ˛y

ˆÓ!ü– Î!î 5!›˛ ¢y!£ˆÏì˛ƒÓ˚ ˛ô%hfl˛ÏˆÏÑ˛Ó˚ îyõ 3!›˛ Ü!íˆÏì˛Ó˚ ˛ô%hfl˛ÏˆÏÑ˛Ó˚ ì%˛úòyÎ˚ 38 ›˛yÑ˛y ˆÓ!ü £Î˚ñ

ì˛y£ˆÏú ¢y!£ˆÏì˛ƒÓ˚ ~Ñ˛!›˛ ˛ô%hfl˛ÏˆÏÑ˛Ó˚ îyõ Ñ˛ì˛⁄

E21. !ì˛òãò !õˆÏú ~Ñ˛!›˛ Ñ˛yˆÏãÓ˚ xˆÏï≈Ñ˛ ¢¡ôß¨ Ñ˛ˆÏÓ˚ 8 !îˆÏò– Î!î ~Ñ˛ãò Ñ˛yã!›˛ ˆäÈˆÏv ¸

ˆîÎ˚ñ ì˛y£ˆÏú Óy!Ñ˛ Ñ˛yã!›˛ ˆü°Ï Ñ˛Ó˚ˆÏì˛ xòƒãˆÏòÓ˚ Ñ˛ì˛!îò ¢õÎ˚ úyÜˆÏÓ⁄

E22. ˆÜy˛ôyú 12% ¢%ˆÏî !Ñ˛ä%È ›˛yÑ˛y ïyÓ˚ !òˆÏúy– Î!î 5 ÓäÈÓ˚ ˛ôˆÏÓ˚ ì˛yˆÏÑ˛ ïyÓ˚ ˆüyï Ñ˛Ó˚yÓ˚

ãòƒ ˆõy›˛ 1280 ›˛yÑ˛y !îˆÏì˛ £Î˚ñ ì˛y£ˆÏú ˆ¢ Ñ˛ì˛ ›˛yÑ˛y ïyÓ˚ !òˆÏÎ˚!äÈˆÏúy⁄

5.6 ¢yÓ˚yÇü ı

≤ÃyÌ!õÑ˛ !ü«˛yhfl˛ÏˆÏÓ˚ Ü!íì˛ ˛ôy‡˛Ñ ˆÏõ â˛yÓ˚ ïÓ˚ˆÏíÓ˚ ¢ÇÖƒyì˛ˆÏ_¥Ó˚ Ñ˛Ìy xyˆÏäÈ–

ÈÙÙÙÈfl ∫y¶˛y!ÓÑ˛ ¢ÇÖƒy (N) : 1, 2, 3, 4 ...............

ÈÙÙÙÈ˛ô)í≈ ¢ÇÖƒy (W) : 0, 1, 2, 3 ...............

ÈÙÙÙÈxÖ[˛ ¢ÇÖƒy (Z) : .......–4, –3, –2, –10, +1, +2, +3 ...............

ÈÙÙÙÈõ)úî ¢ÇÖƒy (Q) : pq xyÑ˛yˆÏÓ˚Ó˚ ¢ÇÖƒyñ ˆ¢ÖyˆÏò p Á q xÖ[˛ ¢ÇÖƒy ~ÓÇ q ⊥ p

!Ó!¶˛ß¨ ¢ÇÖƒyÓ˚ ˆ¢ Ï›˛Ó˚ v z ôÓ˚ ˆÎy ÏÜÓ˚ ˜Ó!üT˛ƒ ı

(i) N, W, Z ~ÓÇ Q ~ ˆÎyÜ xyÓÂï ¢)e ˆõ Ïò â˛ Ïú–

(ii) N, W, Z ~ÓÇ Q ÈÙÈ~ ˆÎyÜ !Ó!òõÎ˚ Á xƒyˆÏ¢y!¢ˆÏÎ˚!›˛¶˛ ¢)e ˆõˆÏò â˛ˆÏú–

(iii) W, Z ~ÓÇ QÈÙÈ~ ˆÎyˆÏÜÓ˚ xy£z Ïv˛ò!›˛!›˛ v˛z ôyîyˆÏòÓ˚ x!hfl˛Ïc xyˆÏäÈ ~ÓÇ ˆ¢!›˛ £ú O–

(iv) Z ~ÓÇ QÈÙÈ~ ˆÎy ÏÜÓ˚ !Ó˛ôÓ˚#ì˛ v z ôyîy ÏòÓ˚ x!hfl Ïc xy ÏäÈ–

â˛yÓ˚ ïÓ˚ˆÏíÓ˚ ¢ÇÖƒyÓ˚ ˆ¢ˆÏ›˛Ó˚ v˛z ôÓ˚ Ü% ÏíÓ˚ ˜Ó!üT˛ƒ ı

(i) N, W, Z ~ÓÇ QÈÙÈ~ Ü%í xyÓÂï ¢)e ˆõ Ïò â˛ Ïú–

(ii) N, W, Z ~ÓÇ Q ÈÙÈ~ Ü%í !Ó!òõÎ˚ ¢)e ˆõ Ïò â˛ Ïú–

(iii) N, W, Z ~ÓÇ Q ÈÙÈ~ Ü%í xƒy Ï¢y!¢ ÏÎ˚!›˛¶˛ ¢)e ˆõ Ïò â˛ Ïú–

!òÎ˚õ ÈÙÈ II ÈÙÈ~Ó˚ ≤Ã ÏÎ˚yÜ

P, t, r ≤Ãî_ñ A !òí≈Î˚ Ñ˛Ó˚y

P, r, A ≤Ãî_ñ t !òí≈Î˚ Ñ˛Ó˚y

P, t, A ≤Ãî_ñ r !òí≈Î˚ Ñ˛Ó˚y

A, t, r ≤Ãî_ñ p !òí≈Î˚ Ñ˛Ó˚y

ˆòy›˛



(iv) N, W, Z~ÓÇ QÈÙÈ~ Ü% ÏíÓ˚ xy£z Ïv˛ò!›˛!›˛ Óì≈˛õyò–

(v) QÈÙÈ~ Ü%ˆÏíÓ˚ !Ó˛ôÓ˚#ì˛ v z˛ôyîyò xyˆÏäÈ–

(vi) N, W, A ~ÓÇ QÈÙÈ~ Ü%í ˆÎy ÏÜÓ˚ v z ôÓ˚ !Ó!òõÎ˚ ¢)e ˆõ Ïò â˛ Ïú–

õ)úî ¢ÇÖƒy ÏÑ˛ ≤ÃÑ˛yü Ñ˛Ó˚yÓ˚ ˛ôÂï!ì˛ÈÙÙÙÈ

(i) £ˆÏÓ˚Ó˚ 2 Á 5 äÈyv ¸y ˆÑ˛yˆÏò v zÍ˛ôyîÑ˛ òy ÌyÑ˛ˆÏú ˆ¢!›˛ £ˆÏÓ ¢¢#õ îü!õÑ˛ ¢ÇÖƒy–

(ii) £ˆÏÓ˚Ó˚ 2 Á 5 äÈyv ¸y xòƒ v zÍ˛ôyîÑ˛ ÌyÑ˛ Ïú ˆ¢!›˛ £ ÏÓ x¢#õ îü!õÑ˛ ¢ÇÖƒy–

î%£z Óy ì˛ˆÏì˛y!ïÑ˛ fl˛∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ Ü.¢.Ü%. !òí≈ˆÏÎ˚Ó˚ ˆ«˛ˆÏe ˆõÔ!úÑ˛ v˛zÍ˛ôyîÑ˛ ÓƒÓ£yÓ˚ Ñ˛Ó˚y

£Î˚– xò%Ó˚)ˆÏ˛ôñ î%£z Óy ì˛ˆÏì˛y!ïÑ˛ fl˛∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ ú.¢y.Ü%. !òí≈ˆÏÎ˚Ó˚ ˆ«˛ˆÏe ¢yïyÓ˚í Ü%!íì˛ˆÏÑ˛Ó˚

ÓƒÓ£yÓ˚ Ñ˛Ó˚y £Î˚–

5.7 xy˛ôòyÓ˚ x@˘ÃÜ!ì˛ Îyâ˛y£z Ñ˛Ó˚yÓ˚ v z_Ó˚¢õ)£ ı

E1. «% oì˛õ ÜíòyÓ˚ ¢ÇÖƒy £ú 1

E2. 0

E3. (i) 0, (ii) 72 (iii) 792

E4. (a) ¢ì˛ƒ (b) !õÌƒy (c) !õÌƒy (d) !õÌƒy (e) ¢ì˛ƒ

E5. 1

E6. 0

E7. 2 ~ÓÇ 13

E8. 2

E9. 101

E10. 0

E11. –7

E12. –1 ~ÓÇ +1

E13. 0

E14. (a) 27

− (b) 38 (c) 0

E15. (i) 0.48 (ii) 0.875 (iii) 0.285714

E16. (a) 73 (b) 8

3− (c) x!hfl˛Ïc ˆò£z

ˆòy›˛

40 D.El.Ed. 504


E17. 5199

E18. 1

E19. 32

E20. 22 ›˛yÑ˛y

E21. 12 !îò

E22. 800 ›˛yÑ˛y

5.8 ≤Ãhfl Ïy!Óì˛ ˛ôy‡˛ ~ÓÇ ˆÓ˚ö˛y ÏÓ˚™ ı

VI, VII Á VIII ~Ó˚ ãòƒ N.C.E.R.T myÓ˚y ≤Ãhfl%˛ì˛ Á ≤ÃÑ˛y!üì˛ Ü!íˆÏì˛Ó˚ ˛ôy‡˛ƒ˛ô%hfl˛ÏÑ˛–

5.9 ~Ñ˛Ñ˛È ˆü Ï°ÏÓ˚ xò%ü#úò# ı

1. –30 ~ÓÇ +30-~Ó˚ õ Ïïƒ 3-~Ó˚ Ñ˛!›˛ Ü%!íì˛Ñ˛ xyˆÏäÈ⁄

2. ¢õ!T˛ !òí≈Î˚ Ñ˛Ó˚ ı

(i) 1 – 2 + 3 – 4 + 5 – 6 + ........+ 45

(ii) 1 + 2 – 3 + 4 + 5 – 6 + 7 + 8 – 9 + ........ – 48

3. (i) !òˆÏâ˛Ó˚ ˆ›˛!Óú ˆÌˆÏÑ˛ ˆÎÔ!ÜÑ˛ ¢ÇÖƒyÜ%!ú Ö§% Ïã ÓyÓ˚ Ñ˛Ó%ò ~ÓÇ 20 Á 69 ÈÙÈ~Ó˚ õˆÏïƒ ÌyÑ˛y

ˆõÔ!úÑ˛ ¢ÇÖƒyÜ%!ú ÏÑ˛ !â˛!£´ì˛ Ñ˛Ó%ò–

(ii) ~ ÏîÓ˚ õ Ïïƒ ÚÑ˛Û ˆãyv ¸y Îõã ˆõÔ!úÑ˛ ¢ÇÖƒy xy ÏäÈ⁄

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

4. A, 18 ›˛yÑ˛y ≤Ã!ì˛ ˆÑ˛!ã îˆÏÓ˚ 200 ˆÑ˛!ã â˛yú !Ñ˛òú ~ÓÇ 22 ›˛yÑ˛y ˆÑ˛!ã îˆÏÓ˚ ì˛yÓ˚ ˆÌˆÏÑ˛

150 ˆÑ˛!ã â˛yú !Ó!Ñ Ñ˛Ó˚ˆÏúy ~ÓÇ Óy!Ñ˛ â˛yú 19 ›˛yÑ˛y ≤Ã!ì˛ ˆÑ˛!ã îˆÏÓ˚ !Ó!Ñ Ñ˛Ó˚ˆÏúy–

B ≤Ã!ì˛ ˆÑ˛!ã 20 ›˛yÑ˛y î ÏÓ˚ 250 ˆÑ˛!ã â˛yú !Ñ˛ò Ïúy ~ÓÇ ˆ¢!›˛ 23 ›˛yÑ˛y ≤Ã!ì˛ ˆÑ˛!ã îˆÏÓ˚

!Ó!Ñ Ñ˛Ó˚ Ïúy– Ñ˛yÓ˚ úy¶˛ ˆÓ!ü £ú !òí≈Î˚ Ñ˛Ó%ò–

5. P, ÓƒyÇÑ˛ ˆÌˆÏÑ˛ 8% ¢Ó˚ú ¢% ÏîÓ˚ £yˆÏÓ˚ 80,000.00 ›˛yÑ˛y }í !ò Ïúy– ì˛yÓ˚ }í ¢¡ô)í≈ ˆüyï

Ñ˛Ó˚yÓ˚ ãòƒ 3 ÓäÈÓ˚ ˛ôˆÏÓ˚ ì˛yˆÏÑ˛ Ñ˛ì˛ ›˛yÑ˛y !îˆÏì˛ £ˆÏÓ⁄

ˆòy›˛

41504-Block2 : xyÑ,˛!ì˛ Á fiÌyò ¢ÇÑ ˛yhs˘˛ ïyÓ˚íy

xyÑ, !ì˛ Á fiÌyò ¢ÇÑ yhs˘ ïyÓíy

~Ñ˛Ñ˛ÈÈÈÙÙÙÈ6 ı xyÑ, !ì˛ Á fiÌyò ¢ÇÑ yhs˘ ïyÓíy

~Ñ˛ ÏÑ˛Ó Ñ˛y‡˛y Ïõy

6.0 – ¶) !õÑ˛y

6.1 – !üÖò v z Ïjüƒ

6.2 – ˛Üy!í!ì˛Ñ˛ xyÑ, !ì˛

6.2.1 – ˛¢ÇK˛y£#ò ü∑

6.2.2 – ˛ˆõÔ!úÑ˛ !â˛e

6.3 – ¢yõì˛!úÑ˛ ÓÂï !â˛e

6.3.1 – ˛!e¶% ã

6.3.2 – â˛ì% ¶≈% ã

6.3.3 – ˛Ó,_

6.3.4 – ˛¢Ó≈¢õì˛y Á ≤Ã!ì˛¢yõƒ

6.3.5 – ˛≤Ã!ì˛ö˛úò Á ≤Ã!ì˛¢yõƒ

6.4 – !eõy!eÑ˛ xyÑ, !ì˛

6.5 – ãƒy!õ!ì˛ Ó Ï:Ó v z ôyîy ÏòÓ ¢y£y ÏÎƒ ãƒy!õ!ì˛Ñ˛ !â˛e xAÑ˛ò

6.6 – ˛!ö˛ ÏÓ˚ ˆîÖy

6.7 – !ò ÏãÓ ≤ÃÜ!ì˛ õy˛ôy

6.8 – ≤Ãhfl Ïy!Óì˛ ˛Ü‡˛ò v z ôÑ˛Óí Á ˆÓö˛y ÏÓ™

6.9 – ~Ñ˛ ÏÑ˛Ó v z ôÓ xò%ü#úò#

6.0 ¶) !õÑ˛y :

xyõÓy â˛yÓ˛ôy Ïü ì˛yÑ˛y Ïú£z !Ó!¶˛ß ïÓ ÏíÓ Óhfl% ˆîÖ Ïì˛ ˛ôy£z– ~£z Óhfl% Ü%!úÓ õ Ïïƒ Ñ˛y ÏÓyÓ õ Ïïƒ

xyÑ, !ì˛Üì˛ !õú Ìy ÏÑ˛ñ ˆÎõò ÜyäÈ ˆÌ ÏÑ˛ G% úhs˘˛ ~Ñ˛!›˛ ˆ˛ôyÎyÓy Óy ˆúÓ% Üy ÏäÈ ö˛ Ïú ÌyÑ˛y ˆúÓ%–

xyõyÓ ˆÑ˛y Ïòy ˆÑ˛y Ïòy Óhfl% Ó õ Ïïƒ ~ÓÑ˛õ ¢yïyÓí !õú Ö§% Ïã ˛ôyÁÎy ÎyÎ òy ˆÎõò ˛ôyÌ ÏÓÓ

¶˛yày ›% Ñ˛ ÏÓy–

ˆòy›˛

42 D.El.Ed. 504


!‰þe 6.1

xyõÓy Î!î ~Ñ˛›˛y £z§›˛ !ò£zñ ˆîÖ Ïì˛ ˛ôyÓ ~!›˛Ó !ì˛ò!›˛ !îÑ˛ xy ÏäÈñ xì˛~Ó ~!›˛ ÏÑ˛ xyõÓy

!eõy!eÑ˛ Óhfl% Óú Ïì˛ ˛ôy!Ó (3D Óhfl%˛V–

~!›˛Ó˚ 6!›˛ ì˛úñ 12!›˛ ≤Ãyhs˘˛ Á 8!›˛ ˆÑ˛yíy xyˆÏäÈ– ~!›˛ ˆîÁÎ˚yÓ˚ñ ˆ›˛!ÓˆÏúÓ˚ v˛z˛ô!Ó˚ì˛ú ~Ñ˛!›˛

¢õì˛ˆÏúÓ˚ v zîy£Ó˚í ˆÑ˛yˆÏòyÁ ˛ôyˆÏe Ó˚yÖy ãˆÏúÓ˚ v˛z ô!Ó˚ì˛ú ~Ñ˛!›˛ xò%¶) !õÑ˛ ì˛ˆÏúÓ˚ v˛zîy£Ó˚í–

ˆîÁÎ˚yú Óy ˆõˆÏG˛Ó˚ v˛z˛ôÓ˚ xyõÓ˚y !Ó!¶˛ß¨ ïÓ˚ˆÏíÓ˚ xyÑ˛yÓ˚ Óy !â˛e x§yÑ˛ˆÏì˛ ˛ôy!Ó˚– ò#ˆÏâ˛ ÎyÓ˚

v zîy£Ó˚í ˆîÁÎ˚y £ú–

!â˛e 6.2 ~Ñ˛!›˛ ì˛ˆÏúÓ˚ v˛z ôÓ˚ x§yÑ˛y !â˛e

~Ñ˛!›˛ !mõy!eÑ˛ ì˛úˆÏÑ˛ Óyõ!îˆÏÑ˛ÈÙÈv˛yò!îˆÏÑ˛ñ v˛z˛ôˆÏÓ˚ÈÙÈò#ˆÏâ˛ Ó!ï≈ì˛ £ˆÏì˛ ˛ôyÓ˚ñ ~£z ì˛ˆÏúÓ˚ v˛z˛ôÓ˚

x§yÑ˛y ¢õhfl˛Ï !â˛e£z !mõy!eÑ˛ £ˆÏÓ– ˆ¢£z Ñ˛yÓ˚ˆÏí£z v˛z ôˆÏÓ˚ x§yÑ˛y !â˛eÜ%!ú !mõy!eÑ˛ (2d)– ~£z

~Ñ˛ˆÏÑ˛ 2D ~ÓÇ 3D xyÑ, !ì˛ !òˆÏÎ˚ !Óhfl˛Ïy!Ó˚ì˛ xyˆÏúyâ˛òy Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ–

~£z ~Ñ˛Ñ˛ ˛ôv˛¸yÓ˚ ãòƒ xhs˘˛ì˛ 10 SîüV ár›˛y ¢õÎ˚ !îˆÏì˛ £ˆÏÓ–

6.1 !üÖò v z Ïjüƒ (Learning objectives)

~£z ~Ñ˛Ñ˛ ˛ôv˛yÓ ˛ôÓ xy˛ô!ò Îy Ñ˛Ó Ïú ¢õÌ≈ £ ÏÓò ì˛y £ú ı

● õ)ú ãƒy!õ!ì˛Ñ˛ !â˛e ˆÎõò !Ó®%ñ ¢Óú ÏÓÖyñ ˆÓÖyÇü Á ˆÑ˛yí !â˛ò Ïì˛ ˛ôyÓ ÏÓò–

● ~Ñ˛!›˛ ì˛ Ïú x§yÑ˛y !Ó!¶˛ß ïÓ ÏíÓ ãƒy!õ!ì˛Ñ˛ !â˛e üòy=˛ Ñ˛Ó Ïì˛ ˛ôyÓ ÏÓò–

● î%!›˛ ¢yõì˛!úÑ˛ !â˛ ÏeÓ õ Ïïƒ ¢Ó≈¢õì˛y Á ≤Ã!ì˛¢y ÏõƒÓ üì≈ Óí≈òy Ñ˛Ó Ïì˛ ˛ôyÓ ÏÓò–

● ˆÎ ÏÑ˛yò ¢yõì˛!úÑ˛ !â˛ ÏeÓ ˜Ó!ÖÑ˛ ≤Ã!Ñ˛¢õì˛y Á á)í≈ò ≤Ã!ì˛¢õì˛y !â˛!£´ì˛ Ñ˛Ó Ïì˛ ˛ôyÓ ÏÓò–

● !eõy!eÑ˛ xyÑ, !ì˛ Á ì˛y ÏîÓ ïõ≈ üòy=˛ Ñ˛Ó Ïì˛ ˛ôyÓ ÏÓò–

B C

D

F

H

E

A

ˆòy›˛



6.2 õ)ú ãƒy!õ!ì˛Ñ˛ xyÑ, !ì˛

6.2.1 ¢ÇK˛y£#ò ü∑

Ü!íˆÏì˛Ó˚ !Ó°ÏÎ˚Óhfl% ˆÓyG˛yÓ˚ ãòƒ !Ñ˛ä%È !Ñ˛ä%È Üy!í!ì˛Ñ˛ ü∑ ãyòy ≤ÃˆÏÎ˚yãò– ~£z ü∑ ¢¡ôˆÏÑ≈˛

fl˛ôT˛ ïyÓ˚íy ˛ôyÁÎ˚yÓ˚ ãòƒ ~Ó˚ ¢ÇK˛y ãyòy îÓ˚Ñ˛yÓ˚– ü Ï∑Ó˚ ¢ÇK˛y ÓƒyÖƒy Ñ˛Ó˚yÓ˚ ãòƒ xyÓ˚Á

!Ñ˛ä%È ü∑ ÓƒÓ£yÓ˚ Ñ˛Ó˚yÓ˚ ≤ÃˆÏÎ˚yãò ˛ôˆÏv ¸– ÎÖò ˆÑ˛yò !Ó°ÏˆÏÎ˚Ó˚ ¢)â˛òy £Î˚ñ ì˛Öò !Ó°ÏÎ˚!›˛ ÓƒyÖƒy

Ñ˛Ó˚yÓ˚ ãˆÏòƒ ˆ¢¶˛yˆÏÓ !ò!î≈T˛ ˆÑ˛yˆÏòy ü∑ ÌyˆÏÑ˛ òy– ö˛ˆÏú !Ó°ÏÎ˚!›˛Ó˚ õ)ú ü∑Ü%!úÓ˚ ¢ÇK˛y

ˆîÁÎ˚y ¢Ω˛Ó £Î˚ òy– ~£z ü∑Ü%!ú ÏÑ˛ Óúy £Î˚ ¢ÇK˛y£#ò ü∑– ãƒy!õ!ì˛Ó˚ ˆ«˛ Ïe ~£z ïÓ˚ ÏíÓ˚

!Ñ˛ä%È x!ò!î≈T˛ ü∑ £úÈÙÙÙÈ!Ó®%ñ ˆÓ˚ÖyÇüñ ì˛ú S~Ü%!ú ÓƒyÖƒy Ñ˛Ó˚yÓ˚ ãòƒ ~Ü%!úÓ˚ ¢y£yÎƒ !ò Ïì˛

£Î˚V–

ˆÎˆÏ£ì% v z ôˆÏÓ˚Ó˚ ü∑Ü%!úÓ˚ ãòƒ ˆÑ˛yˆÏòy ¢ÇK˛y ˆîÁÎ˚y ÎyÎ˚ òyñ ˆÎˆÏ£ì% ~£z ü∑Ü%!ú ÎÌyÎÌ¶˛yˆÏÓ

ÓƒÓ£yˆÏÓ˚Ó˚ ãòƒ ò#ˆÏâ˛Ó˚ v z ô˛ôyîƒÜ%!ú ˆîÁÎ˚y £Î˚ ı

ãƒy!õ!ì˛Ó˚ ˆõÔ!úÑ˛ v z ô˛ôyîƒ

I. ≤Ã!ì˛!›˛ ˆÓ˚Öy S¢Ó˚ú ÏÓ˚ÖyV ~ÓÇ ì˛ú Ñ˛ì˛Ñ˛Ü%!ú !Ó®%Ó˚ ¢õ!T˛–

II. ~Ñ˛!›˛ !Ó®% !î ÏÎ˚ x¢ÇÖƒ ¢Ó˚ú ÏÓ˚Öy ›˛yòy ÎyÎ˚–

III. ˆÎˆÏÑ˛yò î%!›˛ !Ó®% !îˆÏÎ˚ ~Ñ˛!›˛ ~ÓÇ ˆÑ˛Óúõye ~Ñ˛!›˛ ¢Ó˚úˆÏÓ˚Öy£z ›˛yòy ÎyÎ˚–

IV. ˆÎˆÏÑ˛yò ì˛ˆÏúÓ˚ v z ô!fiÌì˛ î%!›˛ !Ó®% !îˆÏÎ˚ ÎyÎ˚ ~õò ¢Ó˚úˆÏÓ˚Öy Á£z ì˛ˆÏú£z xÓ!fiÌì˛ £ˆÏÓ–

V. ~Ñ˛!›˛ ì˛ˆÏú xÓ!fiÌì˛ ˆÎˆÏÑ˛yò !ì˛ò!›˛ !Ó®% !îˆÏÎ˚ ¢Ó˚úˆÏÓ˚Öy ›˛yòy òyÁ ˆÎˆÏì˛ ˛ôyˆÏÓ˚–

VI. Î!î î%!›˛ ì˛ú ˛ôÓ˚fl˛ôÓ˚ˆÏÑ˛ ˆäÈî Ñ˛ˆÏÓ˚ñ ì˛yˆÏîÓ˚ ˆäÈîÑ˛ ~Ñ˛!›˛ ¢Ó˚úˆÏÓ˚Öy £ˆÏÓ–

v z ô ÏÓ˚Ó˚ v z ôy˛ôîƒÜ%!ú !ì˛ò!›˛ ¢ÇK˛y£#ò ˛ü∑Ü%!úÓ˚ õ Ïïƒ ¢¡ôÑ≈ fiÌy˛ôò Ñ˛Ó˚ ÏäÈ– ~£z v z ô˛ôyîƒÜ%!ú

x!ò!î≈T˛ ü∑Ü%!ú S!Ó®%ñ ˆÓ˚ÖyÇü Á ì˛úV ˆÎÖy Ïò ÓƒÓ£yÓ˚ £ ÏÎ˚ ÏäÈ ˆ¢£z ãƒy!õ!ì˛Ñ˛ ¢ì˛ƒÜ%!ú

Ó%G˛ Ïì˛ Á ˆÓyG˛y Ïì˛ ¢y£yÎƒ Ñ˛Ó˚ ÏÓ–

6.2.2 ˆõÔ!úÑ˛ !â˛e

!Ó®% ı Ñ˛yÜˆÏãÓ˚ ›%˛Ñ˛ˆÏÓ˚yÓ˚ v˛z˛ôÓ˚ ˆ˛ôò Óy ˆ˛ôò!¢ˆÏúÓ˚ ¢y£yˆÏÎƒ ~Ñ˛!›˛ ö%˛›˛!Ñ˛ !îˆÏú ì˛y !Ó®%

!£¢y ÏÓ Üíƒ Ñ˛Ó˚y £Î˚– ~Ó˚ õy˛ô ¢¡∫ Ïrï xyõy ÏîÓ˚ õ Ïò ˆÑ˛y Ïòy ïyÓ˚íy ˆò£z–

ˆÓ˚ÖyÇü ı ÎÖò ˆÑ˛y Ïòy ˆÓ˚ÖyÇü Óúy £Î˚ñ ì˛y ¢Ó˚ú ÏÓ˚Öy ˆÓyG˛yÎ˚–

CDsuur

!â˛e ı 6.3

ˆòy›˛

44 D.El.Ed. 504


v˛z˛ôˆÏÓ˚Ó˚ !â˛e!›˛ ~Ñ˛!›˛ ¢Ó˚úˆÏÓ˚ÖyÓ˚ñ ~!›˛ ~Ñ˛!›˛ ˆ¢yãy ≤Ãyhs˘˛ Óy !Ñ˛òyÓ˚yÓ˚ ¢y£yˆÏÎƒ x§yÑ˛y £ˆÏÎ˚ˆÏäÈ–

¢Ó˚úˆÏÓ˚ÖyÓ˚ î%!îˆÏÑ˛Ó˚ ì˛#Ó˚ !â˛ˆÏ£´Ó˚ ¢y£yˆÏÎƒ ˆÓyG˛y ÎyˆÏFäÈ ¢Ó˚úˆÏÓ˚Öy!›˛ˆÏÑ˛ î%£z!îÑ˛ ÓÓ˚yÓÓ˚ Îì˛Ö%!ü

!Óhfl ,Ïì˛ Ñ˛Ó˚y ÎyÎ˚–

ì˛ú ı áˆÏÓ˚Ó˚ ˆõˆÏG˛Ó˚ ì˛úñ ˆîÁÎ˚yˆÏúÓ˚ ˛ô,¤˛ñ Ó£z ÏÎ˚Ó˚ ˛ôyì˛y ¢õì˛úˆÏÑ˛ ˆÓyG˛yÎ˚– ˆÎˆÏÑ˛yò ì˛úˆÏÑ˛ñ

ˆ¢£z ì˛ú ÓÓ˚yÓÓ˚ Îì˛Ö%!ü !Óhfl ,Ïì˛ Ñ˛Ó˚y ÎyÎ˚–

î%!›˛ !Ó®%Ó˚ õïƒÓì≈˛# î)Ó˚c ı Î!î A Á B î%!›˛ !Ó®% £Î˚ñ ì˛y£ˆÏú ì˛yˆÏîÓ˚ xhs˘≈˘Óì≈˛# î)Ó˚c ïòƒydÑ˛

Óyhfl˛ÏÓ ¢ÇÖƒy Á ˆ¢!›˛ AB !â˛ ÏeÓ˚ õyïƒ Ïõ ≤ÃÑ˛yü Ñ˛Ó˚y ÎyÎ˚–

õïƒÓì≈˛# S¢ÇK˛yV ı Î!î A, B Á C !ì˛ò!›˛ ˛ô,ÌÑ˛ !Ó®% £Î˚ñ ÎyÓ˚y ~Ñ˛£z ¢Ó˚úˆÏÓ˚ÖyÎ˚ xÓ!fiÌì˛

~ÓÇ AB + BC = AC, ì˛y£ Ïú Óúy ÎyÎ˚ B !Ó®% A Á C Ó˚ õ Ïïƒ xÓfiÌyò Ñ˛Ó˚ ÏäÈ ~ÓÇ

~!›˛ xyõÓ˚y !úÖˆÏì˛ ˛ôy!Ó˚ A–B–C xÌÓy C–B–A–

!â˛e 6.4

v˛z˛ôˆÏÓ˚Ó˚ !â˛e!›˛ˆÏì˛ !ì˛ò!›˛ !Ó®% A, B Á C ~õò¶˛y ÏÓ xÓ!fiÌì˛ ˆÎÖy Ïò A–B–C SxÌ≈yÍ B

!Ó®%ñ A Á C !Ó®%Ó˚ õˆÏïƒ xÓfiÌyò Ñ˛ˆÏÓ˚ˆÏäÈV–

ˆÓ˚ÖyÇü S¢ÇK˛yV ı î%!›˛ ˛ô,ÌÑ˛ !Ó®% A Á B ~ÓÇ ì˛y ÏîÓ˚ õ Ïïƒ xÓ!fiÌì˛ !Ó®%Ü%!ú ˆÓ˚ÖyÇü

!£¢yˆÏÓ A Á B myÓ˚y !â˛!£´ì˛ ~ÓÇ ~!›˛ˆÏÑ˛ AB ~¶˛y ÏÓ ˆúÖy ÎyÎ˚–

A Á B ÓúˆÏì˛ AB Ó˚ î%!î ÏÑ˛Ó˚ ˆü°Ï î%!›˛ !Ó®% ˆÓyG˛yÎ˚– ˆÓ˚ÖyÇü £ú ¢Ó˚ú ÏÓ˚ÖyÎ˚ ~Ñ˛!›˛

xÇüñ ÎyÓ˚ î%!›˛ !îˆÏÑ˛ î%!›˛ ≤Ãhfl˛Ïy!Óì˛ xyˆÏäÈ– ~Ñ˛!›˛ ˆÓ˚ÖyÇˆÏüÓ˚ î%!›˛ ≤Ãyhs˘˛!Ó®% A Á B ú

ˆÓ˚ÖyÇü!›˛ AB ~ÓÇ ~Ó˚ ˜îá≈ƒ AB–

ˆÓ˚ÖyÇ ÏüÓ˚ ˜îá≈ƒ S¢ÇK˛yV ı ~Ñ˛!›˛ ˆÓ˚ÖyÇˆÏüÓ˚ î%!›˛ ≤Ãyhs˘˛!Ó®%Ó˚ õïƒÓì≈˛# î)Ó˚c£z £ú ˆÓ˚ÖyÇˆÏüÓ˚

˜îá≈ƒñ xÌ≈yÍ AB Ó˚ ˜îá≈ƒ £ú AB–

Ó˚!Ÿ¬ S¢ÇK˛yV ı ¢Ó˚úˆÏÓ˚ÖyÓ˚ xÇü Îy ~Ñ˛!›˛ !Ó®% ˆÌˆÏÑ˛ v˛zÍ˛ôß¨ £ˆÏÎ˚ˆÏäÈ ~ÓÇ ˆÎˆÏÑ˛yò ~Ñ˛!›˛

!îˆÏÑ˛ x¢#õ î)Ó˚c xÓ!ï !Óhfl˛,Ïì˛– !â˛e 6.5 ~ A!Ó®% ˆÌˆÏÑ˛ Ó˚!Ÿ¬ v˛zÍ˛ôß¨ £ˆÏÎ˚ˆÏäÈ ~ÓÇ B !Ó®%Ó˚

!îˆÏÑ˛ ~Ñ˛!›˛ x¢#õ î)Ó˚c xÓ!ï !Óhfl˛,Ïì˛ñ ~!›˛ ABuuur

!£¢y ÏÓ ˆúÖy £ ÏÓ ~ÓÇ ˛ôv ¸ Ïì˛ £ ÏÓ ÚÓ˚!Ÿ¬

ABÛ– î%!›˛ x«˛ˆÏÓ˚Ó˚ v˛z˛ôÓ˚ ì˛#Ó˚ !â˛£´ myÓ˚y ˆÓyG˛y ÎyˆÏFäÈ ~!›˛ ~Ñ˛!›˛ Ó˚!Ÿ¬ñ ~ÓÇ A !Ó®% ˆÌˆÏÑ˛

Ó˚!Ÿ¬!›˛ v zÍ˛ôß¨ £ ÏÎ˚ ÏäÈ–

A B C

ˆòy›˛



!â˛e ı 6.5

A !Ó®% £ú ABuuur

Ó˚ v˛zÍ˛ô!_ Óy ≤Ãyhs˘˛!Ó®%–

!â˛e ı 6.6

!Ó˛ôÓ˚#ì˛ Ó˚!Ÿ¬ ı !â˛e 6.6 ÈÙÈ~ ˆîÖy Îy ÏFäÈ ~Ñ˛!›˛ ˆÓ˚Öy AB ˆÎÖy Ïò OAuuur

Á OBuuur

î%!›˛ Ó˚!Ÿ¬ñ

ÎyÓ˚y v˛z¶˛ˆÏÎ˚£z AB Ó˚ xÇüñ O £ú OAuuur

~ÓÇ OBuuur

Ó˚ õ Ïïƒ ~Ñ˛õye ¢yïyÓ˚í !Ó®%– ~£z ˆ«˛ Ïe

OAuuur

~ÓÇ OBuuur

£ú !Ó˛ôÓ˚#ì˛ Ó˚!Ÿ¬– î%!›˛ !Ó˛ôÓ˚#ì˛ Ó˚!Ÿ¬ !õˆÏú ~Ñ˛!›˛ ˆÓ˚Öy Ü‡˛ò Ñ˛Ó˚ˆÏäÈ–

~Ñ˛ ˆãyv ¸y ¢Ó˚ú ÏÓ˚Öy ı ò#ˆÏâ˛Ó˚ !â˛ˆÏe !ì˛òˆÏãyv ¸y ¢Ó˚úˆÏÓ˚Öy ˆîÖy ÎyÎ˚–

!â˛e 6.7

6.7 ~Ó˚ 1 òÇ !â˛ˆÏe ABsuur

Á CDsuur

~õò¶˛y ÏÓ xÓ!fiÌì˛ Îy Ïì˛ ì˛y ÏîÓ˚ î%!î ÏÑ˛ Îì˛î)Ó˚ Ö%!ü Ó!ï≈ì˛

Ñ˛Ó˚ˆÏúÁ ì˛yÓ˚y Ñ˛ÖòÁ ˛ôÓ˚fl˛ôÓ˚ˆÏÑ˛ fl˛ôü≈ Ñ˛Ó˚ˆÏÓ òy– ~£z ïÓ˚ˆÏíÓ˚ ¢Ó˚úˆÏÓ˚Öy üˆÏ∑Ó˚ õˆÏïƒ ˆÑ˛yˆÏòy

¢yïyÓ˚í ˆäÈî!Ó®% ˆò£z ì˛yÓ˚y ˛ôÓ˚fl˛ôÓ˚ ¢õyhs˘˛Ó˚yú–

!â˛e 6.7 ~Ó˚ II òÇÈÙÈ!â˛ Ïe EFsur

Á GHsuur

~Ó˚ ~Ñ˛!›˛ ¢yïyÓ˚í ˆäÈî!Ó®% £ ÏÓ Î!î Á ˆÑ˛ ÎÌyÑ Ïõ

F Á H ~Ó˚ !îˆÏÑ˛ Ó!ï≈ì˛ Ñ˛Ó˚y £Î˚–

!â˛e 6.7 ~Ó˚ III òÇÈÙÈ!â˛ Ïe JKsur

Á LMsuur

~Ó˚ õˆÏïƒ ~Ñ˛!›˛ ¢yïyÓ˚í !Ó®% O v˛z˛ô!fiÌì˛–

¢%ì˛Ó˚yÇ 6.7 ~Ó˚ II òÇ !â˛ˆÏeÓ˚ EFsur

Á GHsuur

ˆÓ˚ÖymÎ˚ˆÏÑ˛ ~ÓÇ 6.7 ~Ó˚ III òÇ !â˛ˆÏeÓ˚ JKsur

Á

LMsuur

ˆÓ˚ÖymÎ˚ ÏÑ˛ x¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚Öy Óy ˆäÈîÑ˛ ¢Ó˚ú ÏÓ˚ÖymÎ˚ Óúy £Î˚– 6.7 ~Ó˚ III òÇ

!â˛ˆÏe O !Ó®% £ú JKsur

Á LMsuur

¢Ó˚ú ÏÓ˚Öym ÏÎ˚Ó˚ ˆäÈî!Ó®%– EFsur

Á GHsuur

~Ó˚ ˆäÈî!Ó®% ÎÌyÑ ˆÏõ

F Á H ~Ó˚ !îˆÏÑ˛ Ó!ï≈ì˛ Ñ˛Ó˚ˆÏú ˛ôyÁÎ˚y ÎyˆÏÓ–

¢õyhs˘˛Ó˚yú ¢Ó˚úˆÏÓ˚ÖyÓ˚ ≤Ãì˛#Ñ˛!â˛£´ ı ABsuur

Á CDsuur

¢õyhs˘˛Ó˚yú ¢Ó˚úˆÏÓ˚ÖymÎ˚ˆÏÑ˛ xyõÓ˚y

A B

A O B

A

BC

D

E

F

HG

J

KL

M

O

ˆòy›˛

46 D.El.Ed. 504


AB || CDsuur suur

!£¢yˆÏÓ ≤ÃÑ˛yü Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚

~Ñ˛ Ïãyv ¸y ¢Ó˚ú ÏÓ˚ÖyÓ˚ ˆäÈîÑ˛ ı

!â˛e 6.8 (I) ~Ñ˛ Ïãyv ¸y ¢õyhs˘˛Ó˚yú òÎ˚ ~õò ¢Ó˚ú ÏÓ˚Öy ABsuur

Á CDsuur

ñ 6.8(II) !â˛ˆÏe ~Ñ˛ˆÏãyv˛¸y

¢õyhs˘˛Ó˚yú ¢Ó˚úˆÏÓ˚Öy PQsuur

Á RSsuur

– !â˛e 6.8 ~Ó (I)ÈÙÈ~ x˛ôÓ˚ ~Ñ˛!›˛ ¢Ó˚ú ÏÓ˚Öy EFsur

Îy ABsuur

Á ˆÓ˚ÖymÎ˚ˆÏÑ˛ ˆäÈî Ñ˛ˆÏÓ˚ˆÏäÈ– ~Ñ˛£z¶˛yˆÏÓ TVsuur ¢Ó˚úˆÏÓ˚Öy 6.8ÈÙÈ~Ó˚ (II) òÇ !â˛ Ïe PQ

suur Á RS

suur

ˆÓ˚Öyî%!›˛ ÏÑ˛ ˆäÈî Ñ˛ ÏÓ˚ ÏäÈ–

!â˛e 6.8

!â˛e 6.8ÙÈ~Ó˚ (I) ÈÙÈ~ EFsur

~ÓÇ (II) ÈÙÈ~ TVsuur £ú ˆäÈîÑ˛ñ EF

sur x¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚ÖmÎ˚ ÏÑ˛

ˆäÈî Ñ˛Ó˚ˆÏäÈ !Ñ˛hs%˘˛ TVsuur ¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚ÖymÎ˚ ÏÑ˛ ˆäÈî Ñ˛ ÏÓ˚ ÏäÈ–

¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚ÖyÓ˚ ˜Ó!üT˛ƒ ı

AB || CDsuur suur

¢Ó˚ú ÏÓ˚Öym ÏÎ˚Ó˚ !â˛ Ïe !ì˛ò Ï›˛ ˆäÈîÑ˛ xy ÏäÈ ÎyÓ˚ È ABsuur

ÙÈ~Ó˚ ¢yˆÏÌ ¢õˆÏÑ˛yí v˛zÍ˛ôß¨

Ñ˛ ÏÓ˚ ÏäÈ–

!â˛e 6.9

P, Q, R !Ó®% Ïì˛ ˆäÈîÑ˛Ü%!ú ABsuur

ˆÓ˚Öy ÏÑ˛ ˆäÈî Ñ˛ ÏÓ˚ ÏäÈ ~ÓÇ S,T,V !Ó®% Ïì˛ ˆäÈîÑ˛Ü%!ú CDsuur

ˆÓ˚ÖyˆÏÑ˛ ˆäÈî Ñ˛ˆÏÓ˚ˆÏäÈ– ~ÖyˆÏò PS , QT ~ÓÇ RV Ó˚ ˜îá≈ƒ ¢õyò–

PS, QT Á RV ≤Ã Ïì˛ƒÑ˛ ¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚ÖymÎ˚ ABsuur

Á CDsuur

ÈÙÈÓ˚ õ Ïïƒ î)Ó˚c ≤ÃÑ˛yü Ñ˛Ó˚ ÏäÈ–

~!›˛ PS = QT = RV– ~!›˛ ~§ ÏÑ˛Á ≤Ãõyí Ñ˛Ó˚y ÎyÎ˚–

C

E

D

A

FB

T

R

P

S

Q

V

EG

I

C F

HK

D

B

S

T

V

ˆòy›˛



~¶˛y ÏÓ xyõÓ˚y ˆîÖ Ïì˛ ˆ˛ôúyõ î%!›˛ ¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚ÖyÓ˚ õ Ïïƒ î)Ó˚c ¢Ó¢õÎ˚ ¢õyò– ¢%ì˛Ó˚yÇ

Óúy ÎyÎ˚ î%!›˛ ¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚ÖyÓ˚ õ Ïïƒ ¢Ó¢õÎ˚ ¢õî)Ó˚c Óì≈ õyò–

ˆÑ˛yí S¢ÇK˛yV ı Î!î A, B, Á C !ì˛ò!›˛ ¢õˆÏÓ˚Ö !Ó®% òy £Î˚ ì˛y£ˆÏú ABuuur Á AC

uuur ˆÓ˚ÖyÇü

!îˆÏÎ˚ ˆÎ !â˛e Ü‡˛ò £ˆÏÓ ì˛y ˆÑ˛yí BAC òyˆÏõ ˛ô!Ó˚!â˛ì˛ £ˆÏÓ S∠BAC !ú!Öì˛ Ó˚) ôV–

ABuuur Á AC

uuur ∠BAC Ó˚ î%!›˛ Óy£% ~ÓÇ A £ú ü#°Ï≈!Ó®%–

!â˛e ı 6.10

A, B Á C Ó˚ xÓfiÌyòˆÏÑ˛ v z˛ôÓ˚ ˆÑ˛yˆÏíÓ˚ !Ó!¶˛ß¨ ïÓ˚ˆÏíÓ˚ xyÑ˛yÓ˚ £Î˚ SäÈ!ÓˆÏì˛ ˆîÁÎ˚y £úV–

!â˛e ı 6.11

ˆÑ˛y ÏíÓ˚ ˛ô!Ó˚õy˛ô ı ãƒy!õ!ì˛ ÓyˆÏ:Ó˚ â§˛yîy !îˆÏÎ˚ ˆÑ˛yí õy˛ôy ÎyÎ˚ ˆÑ˛yˆÏíÓ˚ õy˛ô /~ÓÇ !v˛!@˘ÃˆÏì˛

≤ÃÑ˛yü Ñ˛Ó˚y £Î˚–

!ò Ïâ˛Ó˚ !â˛ Ïe

∠AOB ÈÙÈÓ˚ õy˛ô 40° ~ÓÇ ∠COB Ó˚ õy˛ô 135°–

!â˛e ı 6.12

∠ABC Ó˚ !v˛@˘Ã# ˆõ Ï ô ˆúÁÖy ÎyÎ˚ m∠ABC–

B

AC

B

A CA

B

C

B

CA

0 180

135

CA

B

90120 60

60 12090

40

0 180

O

40

ˆòy›˛

48 D.El.Ed. 504


Î!î !â˛e 6.13Ó˚ ∠ABC Ó˚ õy˛ô 70° £Î˚ñ ì˛y£ˆÏú xyõÓ˚y !úÖˆÏì˛ ˛ôy!Ó˚ m∠ABC = 70°–

!â˛e ı 6.13

fl ∫ì˛É!¢Âï ı ≤Ã!ì˛!›˛ ˆÑ˛y ÏíÓ˚ ¢y ÏÌ OÓ˚ ˆÌ ÏÑ˛ ˆÓ!ü 180° ˆÌ ÏÑ˛ Ñ˛õ Óyhfl ÏÓ¢ÇÖƒy ã!v ¸ì˛–

~£z ¢ÇÖƒy!›˛ ˆÑ˛yˆÏíÓ˚ õy˛ô S!v˛@˘Ã#ˆÏì˛V–

î%!›˛ ¢Ó˚úˆÏÓ˚Öy ˛ôÓ˚fl˛ôÓ˚ ú¡∫¶˛y ÏÓ xÓ!fiÌì˛ ı

!â˛e 6.14 (I) ÈÙÈ~ m∠AOB = 90°– ~ÖyˆÏò OAuuur Á OB

uuur £ú ˛ôÓ˚fl˛ôÓ˚ ú¡∫ ~ÓÇ xyõÓ˚y

!úÖ Ïì˛ ˛ôy!Ó˚ OA OBuuur uuuv

⊥ Óy–

!â˛e ı 6.14

!â˛e 6.14 (II) ˆì˛ñ m∠CEF = 90° xì˛~Ó xyõÓ˚y !úÖˆÏì˛ ˛ôy!Ó˚ FE CDuuur uuur

⊥ – !â˛e 6.14

(III) ˆì˛ m∠GLT = 90°– ~ÖyˆÏò xyõÓ˚y !úÖˆÏì˛ ˛ôy!Ó˚ JK GHsur suur

⊥ Óy GH JKsuur sur

⊥ –

ˆÑ˛y ÏíÓ˚ ¢õ!mÖ[˛Ñ˛ ı Î!î P !Ó®% ∠BACÈÙÈÓ˚ õˆÏïƒ xÓfiÌyò Ñ˛ˆÏÓ˚ ~ÓÇ m∠BAP =

m∠PAC £Î˚ñ ˆ¢ Ï«˛ Ïe ∠BAC ¢õ!mÖ[˛Ñ˛ S!â˛e 6.15)–

!â˛e 6.15

ˆÑ˛y ÏíÓ˚ ˆ◊!í!Ó¶˛yÜ ı ˆÑ˛yˆÏíÓ˚ õyˆÏòÓ˚ !îÑ˛ !îˆÏÎ˚ ˆÑ˛yíˆÏÑ˛ !ò¡¨!ú!Öì˛ ¶˛yˆÏÜ ¶˛yÜ Ñ˛Ó˚y £Î˚ÈÙÙÙÈ

(i) 90° Ó˚ Ñ˛õ Á 30°Ó˚ ˆÓ!ü ˆÑ˛yíˆÏÑ˛ Óúy £Î˚ ¢)∏ˆÏÑ˛yí–

(ii) 90° Ó˚ õyˆÏ˛ôÓ˚ ˆÑ˛yí £ú ¢õˆÏÑ˛yí–

(iii) 90° Ó˚ ˆÓ!ü 180°Ó˚ Ñ˛õ ˆÑ˛yí £ú fiÌ)ú ÏÑ˛yí–

BC

A

A

OB

C D

E

F

J

K

LG H

A

B

C

P

ˆòy›˛



!â˛e ı 6.16

ˆãyv ¸y ˆÑ˛yí ı

¢!ß¨!£ì˛ ˆÑ˛yí ı Î!î î%!›˛ ˆÑ˛yˆÏíÓ˚ ~Ñ˛!›˛ ¢yïyÓ˚í ü#°Ï≈!Ó®% Á ~Ñ˛!›˛ ¢yïyÓ˚í Óy£% äÈyv˛¸y ˆÑ˛yˆÏòy

¢yïyÓ˚í !Ó®% òy ÌyˆÏÑ˛ñ ì˛Öò ˆ¢£z ˆãyv ¸y ˆÑ˛yíˆÏÑ˛ Óúy £Î˚ ¢!ß¨!£ì˛ ˆÑ˛yí–

!â˛e ı 6.17

v˛z˛ôˆÏÓ˚Ó˚ !ì˛ò!›˛ !â˛ˆÏe£z î%ˆÏãyv˛¸y ˆÑ˛yí ∠BAP Á∠CAP Ó A £ú ¢yïyÓ˚í ü#°Ï≈!Ó®%ñ AP

~Ñ˛!›˛ ¢yïyÓ˚í Óy£%ñ ì˛yäÈyv ¸y ˆÑ˛yˆÏòy ¢yïyÓ˚í !Ó®% ˆò£z–

xÌ~Ó ∠BAP Á∠CAP £ú ~Ñ˛ Ïãyv ¸y ¢!ß¨!£ì˛ ˆÑ˛yí–

˛ô)Ó˚Ñ˛ ˆÑ˛yí ı î%!›˛ ˆÑ˛yíˆÏÑ˛ ˛ô)Ó˚Ñ˛ ˆÑ˛yí Óúy £Î˚ Î!î ì˛yˆÏîÓ˚ ˆõy›˛ ˆÑ˛yˆÏíÓ˚ õy˛ô 90° £Î˚ñ

~ˆÏ«˛ˆÏe î%!›˛ ˆÑ˛yí£z ¢)∏ˆÏÑ˛yí £ˆÏÓ–

!â˛e 6.18

A C

P

B

C

PB

C

P

B

B

A

C E

D

F A

R

Q

P

ˆòy›˛

50 D.El.Ed. 504


!â˛e 6.18 (I)ÈÙÈ~ m∠ABC + m∠DEF = 90°–

ì˛y£ Ïú ∠ABC Á ∠DEF ˛ô)Ó˚Ñ˛ ˆÑ˛yí–

!â˛e 6.18 (II)ÈÙÈ~ m∠PAR + m∠RAQ = 90°–

ì˛y£ Ïú ∠PAR Á ∠RAQ £ú ˛ô)Ó˚Ñ˛ ˆÑ˛yí–

ˆòy›˛ ı ~Ñ˛ˆÏãyv˛¸y ˛ô)Ó˚Ñ˛ ˆÑ˛yí ¢!ß¨!£ì˛ ˆÑ˛yí £ˆÏì˛ ˛ôyˆÏÓ˚ Óy òyÁ £ˆÏì˛ ˛ôyˆÏÓ˚–

¢¡ô)Ó˚Ñ˛ ˆÑ˛yí ı ~Ñ˛ˆÏãyv˛¸y ˆÑ˛yíˆÏÑ˛ ¢¡ô)Ó˚Ñ˛ ˆÑ˛yí Óúy £ˆÏÓñ Î!î ì˛yˆÏîÓ˚ ˆõy›˛ õy˛ô 180°

£Î˚–

!â˛e 6.19

!â˛e 6.19 (I)ÈÙÈ~ m∠ABC + m∠DEF = 180°–

xì˛~Ó ∠ABC Á ∠DEF !õˆÏú ¢¡ô)Ó˚Ñ˛ ˆÑ˛yí–

!â˛e 6.19 (II) ÈÙÈ~ m∠PQS + m∠RQS = 180°–

xì˛~Óñ ∠PQS Á ∠RQS !õ Ïú ~Ñ˛ Ïãyv ¸y ¢!ß¨!£ì˛ ¢¡ô)Ó˚Ñ˛ ˆÑ˛yí Ü‡˛ò Ñ˛ ÏÓ˚ ÏäÈ–

ˆòy›˛ ı ~Ñ˛ˆÏãyv ¸y ¢¡ô)Ó˚Ñ˛ ˆÑ˛yí ¢!ß¨!£ì˛ £ˆÏì˛ ˛ôyˆÏÓ˚ Óy òyÁ £ˆÏì˛ ˛ôyˆÏÓ˚–

!Ó≤Ãì˛#˛ô ˆÑ˛yí ı !â˛e 6.20 ˆì˛ ABsuur

Á CDsuur

˛ôÓ˚fl˛ôÓ˚ˆÏÑ˛ O !Ó®%ˆÏì˛ ˆäÈî Ñ˛ˆÏÓ˚ˆÏäÈ– ö˛ˆÏú â˛yÓ˚!›˛

ˆÑ˛yí ¢,!T˛ £ˆÏÎ˚ˆÏäÈ ~ÓÇ ∠AOD â˛yÓ˚!›˛ ˆÑ˛yˆÏíÓ˚ õˆÏïƒ ~Ñ˛!›˛ OAsuur Á OD

suur ˆÌˆÏÑ˛ v˛zÍ˛ôß¨

£ ÏÎ˚ ÏäÈ– OBsuur Á OC

suur ÎÌyÑ Ïõ OAsuur Á OD

suur Ó˚ !Ó˛ôÓ˚#ì˛ Ó˚!Ÿ¬–

!â˛e ı 6.20

OBsuur Á OC

suur ˆÌˆÏÑ˛ ˆÎ ˆÑ˛yíñ xÌ≈yÍ ∠BOC v˛zÍ˛ôß¨ £ˆÏÎ˚ˆÏäÈ ~ÓÇ OAsuur Á OD

suur ˆÌ ÏÑ˛ ˆÎ

ˆÑ˛yíñ ∠AOD v zÍ˛ôß £ ÏÎ ÏäÈ ì˛yÓy ˛ôÓfl˛ôÓ !Ó˛ôÓ#ì˛ ˆÑ˛yí Óy !Ó≤Ãì˛#˛ô ˆÑ˛yí– ~Ñ˛£z Ñ˛yÓíÓüì˛É

∠BOD Á ∠AOC x˛ôÓ˚ ~Ñ˛ˆÏãyv ¸y !Ó≤Ãì˛#˛ô ˆÑ˛yí–

BC

A

E F

D

QR

S

P

B

AD

O

C

ˆòy›˛



xì˛~Ó ÎÖò î%!›˛ ¢Ó˚úˆÏÓ˚Öy ˛ôÓ˚fl˛ôÓ˚ˆÏÑ˛ ˆäÈî Ñ˛ˆÏÓ˚ ì˛Öò î% Ïãyv ¸y !Ó≤Ãì˛#˛ô ˆÑ˛yí v zÍ˛ôß¨ £Î˚–

¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚Öy Á ˆäÈî ÏÑ˛Ó˚ ö˛ Ïú v zÍ˛ôß¨ ˆÑ˛yí ı

!â˛e 6.21 (I) ˆì˛ ABsuur

Á CDsuur

î%!›˛ x¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚Öy Îy ÏîÓ˚ v z¶˛Î˚ ÏÑ˛ PQsuur

¢Ó˚ú ÏÓ˚Öy

ˆäÈî Ñ˛ ÏÓ˚ ÏäÈ– ˆäÈî!Ó®% Ïì˛ ˆÎ ˆÑ˛yíÜ%!ú v zÍ˛ôß¨ £ ÏÎ˚ ÏäÈ ì˛y ÏîÓ˚ 1, 2, 3, 4, 5, 6, 7, Á 8

!£¢y ÏÓ !â˛!£´ì˛ Ñ˛Ó˚y £ ÏÎ˚ ÏäÈ–

!â˛e ı 6.21 (i) !â˛e ı 6.21 (ii)

!â˛e 6.21 (I) ˆì˛ EFsur

Á GHsuur

î%!›˛ ¢õyhs˘˛Ó˚yú ¢Ó˚úˆÏÓ˚Öy ~ÓÇ KLˆäÈîÑ˛ ì˛yˆÏîÓ˚ ˆäÈî Ñ˛ˆÏÓ˚ˆÏäÈ–

!äÈî!Ó®% Ïì˛ ˆÎ ˆÎ ˆÑ˛yíÜ%!ú v zÍ˛ôß¨ £ˆÏÎ˚ˆÏäÈ ì˛yÓ˚y £ú a, b, c, d, e, f, g, Á h–

â˛e 6.21 (I) ˆì˛ 1 ~ÓÇ 5ñ 2 ~ÓÇ 6, 4 ~ÓÇ 8, 3 ~ÓÇ 7 !â˛!£´ì˛ ˆÑ˛yíÜ%!ú xò%Ó) ô ˆÑ˛yí–

1 ~ÓÇ 7, 2 ~ÓÇ 8 !â˛!£´ì˛ ˆÑ˛yíÜ%!ú £ú Ó!£ÉfiÌ ˆÑ˛yíñ 3 ~ÓÇ 5, 4 ~ÓÇ 6 ˆÑ˛yíÜ%!ú

£ú ˆäÈî ÏÑ˛Ó˚ ~Ñ˛yhs˘˛Ó˚ ˆÑ˛yí–

!â˛e 6.21 (II) ˆì˛ a ~ÓÇ h, b ~ÓÇ e, d ~ÓÇ g, c ~ÓÇ f !â˛!£´ì˛ ˆÑ˛yíÜ%!ú £ú xò%Ó˚) ô

ˆÑ˛yí–

ÎÖò î%!›˛ ¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚Öy ÏÑ˛ ~Ñ˛!›˛ ˆäÈîÑ˛ ˆäÈî Ñ˛Ó˚ ÏÓ ì˛Öò ˆÑ˛yíÜ%!ú õy˛ô Ïú ˆîÖy

ÎyˆÏÓÈÙÙÙÈ

(i) Ó!£ÉfiÌ ˆÑ˛yíÜ%!úÓ˚ õy˛ô ¢õyò Á ~Ñ˛yhs˘˛Ó˚ ˆÑ˛yíÜ%!úÓ˚ õyò ¢õyò–

(ii) xò%Ó) ô ˆÑ˛yíÜ%!úÓ˚ õyò ¢õyò–

!ò ÏãÓ˚ ≤ÃÜ!ì˛ õy˛ôy

E1. !â˛e 6.22ˆì˛ O !Ó®%!›˛ ABsuur

Á ODsuur

¢Ó˚ú ÏÓ˚ÖyÓ˚ ¢yïyÓ˚í !Ó®%– ∠AOD Á ∠DOBÓ

õyò ÎÌyÑ Ïõ (x + 30)° ~ÓÇ (x – 25)°– î%!›˛ ˆÑ˛yˆÏíÓ˚ õyò Ñ˛ì˛ !v˛@˘Ã#⁄

A

C

DQ

P

1

3

6

8

E F

G H

K

L

b

e

d

g

4

2

57

Ba

c

f

h

ˆòy›˛

52 D.El.Ed. 504


!â˛e ı 6.22 !â˛e ı 6.23

E2. !â˛e m∠AOC = 72°, ∠AOD, ∠BOC ~ÓÇ ∠BOD Ó˚ õyò Ñ˛ì˛ !v˛@˘Ã#⁄

E3. 6.24 Ó˚ !â˛e ˆîˆÏÖ ò#ˆÏâ˛Ó˚ ˆÑ˛yíÜ%!úÓ˚ òyõ !úÖ%ò–

(i) ~Ñ˛ˆÏãyv˛¸y xò%Ó˚)˛ô ˆÑ˛yí–

(ii) ~Ñ˛ˆÏãyv ¸y ~Ñ˛yhs˘˛Ó˚ ˆÑ˛yí–

(iii) ~Ñ˛yˆÏãyv ¸y !Ó≤Ãì˛#˛ô ˆÑ˛yí–

!â˛e ı 6.24 !â˛e ı 6.25

E4. !â˛e 6.25ˆì˛ AB || CDsuur suur

~ÓÇ PQsuur ì˛yˆÏîÓ˚ ˆäÈî Ñ˛ˆÏÓ˚ˆÏäÈ– ˆÎ ˆÑ˛yíÜ%!ú v˛zÍ˛ôß¨ £ˆÏÎ˚ˆÏäÈ

ì˛y ÏîÓ˚ õyò a, b, c, d, e, f, g ~ÓÇ h– Î!î g = 35°, ì˛y£ˆÏú a, b, c, d, e, f, g ~ÓÇ

h ~Ó˚ õyò Ñ˛ì˛⁄

6.3 !mõy!eÑ˛ ÓÂï !â˛e

~£z xÇˆÏü xyõÓ˚y !ì˛ò ïÓ˚ˆÏíÓ˚ ÓÂï ¢yõì˛!úÑ˛ !â˛eÈÙÙÙÈ!e¶% ãñ â˛ì% ¶≈% ã Á Ó, Ï_Ó˚ ¶˛yÜ Á ì˛yˆÏîÓ˚

ïõ≈ !ò ÏÎ˚ xy Ïúyâ˛òy Ñ˛Ó˚Ó–

6.3.1 !e¶%˛ã

S¢ÇK˛yV Î!î A, B Á C !ì˛ò!›˛ x¢õˆÏÓ˚Ö !Ó®% £Î˚ñ ì˛y£ˆÏú AB BC Á CA !îˆÏÎ˚ ˆÎ xyÑ,˛!ì˛

˜ì˛!Ó˚ £Î˚ ì˛yˆÏÑ˛ !e¶%˛ã ÓˆÏú– (∆ABC)

A

C B

D

O

A

O

D

x – 25x + 30 B

c

A B

C D

P

Q

a

d

f e

hg

P Q

R S

A

B

D

C

b

ˆòy›˛



A, B Á C !ì˛ò!›˛ !e¶%˛ˆÏãÓ˚ !ì˛ò!›˛ ü#°Ï≈!Ó®%– AB , BC Á CA !e¶% ÏãÓ˚ Óy£% ~ÓÇ ∠ABC,

∠BCA ~ÓÇ ∠BAC S¢Ç!«˛Æ¶˛y ÏÓ ∠B, ∠C Á ∠A ˆúÖy £Î˚V ∆ABCÈÙÈÓ˚ !ì˛ò!›˛ ˆÑ˛yí–

!e¶% ÏãÓ˚ x¶˛ƒhs˘˛Ó˚ Á Ó!£¶≈ yÜ ı ∆ABC myÓ˚y ˆÎ ˆÑ˛w!›˛ xyÓÂï S!â˛e 6.26) ˆ¢!›˛ !e¶%˛ˆÏãÓ˚

x¶˛ƒhs˘˛Ó˚– !e¶%˛ã ABC Á ì˛yÓ˚ !¶˛ì˛ˆÏÓ˚Ó˚ xÇü!›˛ˆÏÑ˛ !e¶%˛ãyÑ˛yÓ˚ˆÏ«˛e ÓˆÏú–

!â˛e ı 6.26

!â˛e ı 6.27 ÈÙÈ~ñ P Á R !Ó®% ∆ABC Ó˚ !¶˛ì˛ ÏÓ˚Ó˚ xÇ Ïü xÓ!fiÌì˛– M Á N !Ó®% !e¶% ÏãÓ˚

v˛z˛ôˆÏÓ˚ ~ÓÇ Q, S Á T !Ó®% ∆ABC Ó˚ Óy£zˆÏÓ˚ xÓ!fiÌì˛– ∆ABC Ó˚ Óy£zˆÏÓ˚ ˛ôyì˛yÓ˚ xòƒ

xÇü SˆÎÖy Ïò Q, S, T !Ó®% xÓ!fiÌì˛V !e¶% ÏãÓ˚ Óy£z ÏÓ˚Ó˚ xÇü Ó Ïú ˛ô!Ó˚!â˛ì˛–

!â˛e ı 6.27

!e¶% ÏãÓ˚ ˆ◊í#!Ó¶˛yÜ ı

(a) ˆÑ˛yí Ï¶˛ Ïî !e¶%˛ ÏãÓ˚ ˆ◊!í!Ó¶˛yÜ ı

(i) ¢)∏ ÏÑ˛yí# !e¶%ã ı Î!î !e¶% ÏãÓ !ì˛ò!›˛ ˆÑ˛yí£z ¢)∏ ÏÑ˛yí £Îñ ì˛Öò !e¶% ã!›˛ ÏÑ˛ ¢)∏ ÏÑ˛yí#

!e¶% ã Ó Ïú– !â˛e 6.28(a)ÈÙÈˆì˛ ~£z ïÓ˚ˆÏíÓ˚ !e¶% ˆÏãÓ˚ äÈ!Ó ˆîÁÎ˚y xyˆÏäÈ–

(ii) ¢õˆÏÑ˛yí# !e¶%˛ã ı ˆÎ !e¶% ˆÏãÓ˚ ~Ñ˛!›˛ ˆÑ˛yí ¢õˆÏÑ˛yíñ ì˛Öò !e¶% ã!›˛ ¢õˆÏÑ˛yí# !e¶% ã

£Î˚– ~£z ïÓ˚ ÏíÓ˚ !e¶% ÏãÓ˚ !â˛e 6.28 (b) ˆì˛ ˆîÁÎ˚y £ ÏÎ˚ ÏäÈ–

(iii) fiÌ)ú ÏÑ˛yí# !e¶% ã ı ˆÎ !e¶% ˆÏãÓ˚ ~Ñ˛!›˛ ˆÑ˛yí fiÌ)úˆÏÑ˛yíñ ì˛Öò ˆ¢£z !e¶% ã!›˛ fiÌ)úˆÏÑ˛yí#

!e¶% ã £Î˚– !â˛e 6.28(c) ˆì˛ ~£z ïÓ˚ˆÏíÓ˚ !e¶% ˆÏãÓ˚ äÈ!Ó xyˆÏäÈ–

A

B C

B

A

C

T

N

S

MP

R

Q

ˆòy›˛

54 D.El.Ed. 504


!â˛e 6.28 (a) !â˛e 6.28 (b) !â˛e 6.28 (c)

(b) Óy£% Ï¶˛ Ïî !e¶% ÏãÓ˚ ˆ◊!í!Ó¶˛yÜ ı

(i) ¢õ!mÓy£% !e¶% ã ı ˆÎ !e¶% ÏãÓ˚ î%!›˛ Óy£% ¢õyò ˆ¢!›˛ ¢õ!mÓy£% !e¶% ã– !â˛e ı 6.29 (I)

(ii) !Ó°ÏõÓy£% !e¶%˛ã ı ˆÎ !e¶% ˆÏãÓ˚ ˆÑ˛yˆÏòy î%!›˛ Óy£%£z ¢õyò òÎ˚ ˆ¢!›˛ !Ó°ÏõÓy£% !e¶% ã–

!â˛e ı 6.29 (II)

(iii) ¢õÓy£% !e¶% ã ı ˆÎ !e¶% ÏãÓ˚ !ì˛ò!›˛ Óy£%£z ¢õyò ˆ¢!›˛ ¢õÓy£% !e¶% ã– !â˛e 6.29(III)

[I] [II] [III]

!â˛e 6.29

6.3.2 â˛ì%˛¶≈%˛ã

ò#ˆÏâ˛Ó˚ !â˛e!›˛ ˆîÖ%ò ı

(i) (ii)

!â˛e ı 6.30

!â˛e ı 6.30(I) ÈÙÈ~ â˛yÓ˚!›˛ !Ó®% xyˆÏäÈ– A, B, C Á D– ÎyÓ˚ õˆÏïƒ ˆÑ˛yˆÏòy !ì˛ò!›˛ !Ó®%£z ~Ñ˛!›˛

¢Ó˚úˆÏÓ˚ÖyÎ˚ ˛ôˆÏv˛¸ òy–

!â˛e 6.30(II) ÈÙÈˆì˛ â˛yÓ˚!›˛ !Ó®% !îˆÏÎ˚ ¢Ó˚úˆÏÓ˚ÖyÇü ˆ›˛ˆÏò AB , BC , CD Á DA x§yÑ˛y

A D

B C E F

G

HK

A

B C

D

E F

G

H I

A

BC

D

A

BC

D

ˆòy›˛



£ú–

AB , BC , CD Á DA ÈÙÈÓ˚ ¢Ó˚úˆÏÓ˚ÖyÇüÜ%!ú ~õò ˆÎ ì˛yÓ˚y ü%ï%õye ≤Ãyhs˘˛!Ó®% äÈyv ¸y xòƒ

ˆÑ˛yÌyÁ !õú ÏäÈ òy–

~£z ˆ«˛ Ïe AB , BC , CD Á DA myÓ˚y ˆÎ !â˛e ˜ì˛!Ó˚ £ˆÏÎ˚ˆÏäÈñ ì˛y â˛ì% ¶≈% ã– ˆÎ â˛ì% ¶≈% ã!›˛

˜ì˛!Ó˚ £ú ì˛yÓ˚ òyõ ABCD–

A, B, C Á D ~Ó˚ ü#°Ï≈!Ó®%– AB , BC , CD Á DA ~Ó˚ â˛yÓ˚!›˛ Óy£%–

ABCD â˛ì% ¶≈% ÏãÓ â˛yÓ!›˛ ˆÑ˛yí xy ÏäÈ ∠ABC, ∠BCD ∠CDA Á ∠DAB– AC Á BD

â˛ì% ¶≈% ˆÏãÓ˚ î%!›˛ Ñ˛í≈–

6.31(a) Ó˚ !â˛e!›˛ ˆî!Öñ A, B, C Á D ~Ñ˛£z ì˛ˆÏú xÓ!fiÌì˛ ~ÓÇ ˆÑ˛yˆÏòy !ì˛ò!›˛ !Ó®%£z

¢õ ÏÓ˚Ö òÎ˚–

(a) (b) (c)

!â˛e ı 6.31

!Ñ˛hs%˘˛ AB Á BC ≤Ãyhs˘˛!Ó®% äÈyv˛¸yÁ xòƒ ~Ñ˛!›˛ !Ó®%ˆÏì˛ !õˆÏúˆÏäÈ– xì˛~Ó AB , BC , CD

Á DA â˛ì%˛¶≈%˛ã Ü‡˛ò Ñ˛Ó˚ˆÏì˛ ˛ôyÓ˚!äÈ òy– !â˛e 6.31(b) ~Ñ˛!›˛ Ñ% ∂ â˛ì% ¶≈% ã– 6.32(c) £ú

xÑ% ∂ â˛ì% ¶≈% ã– ~ÖyˆÏò ü%ï%õye Ñ% ∂ â˛ì% ¶≈% ã !òˆÏÎ˚ xyˆÏúyâ˛òy £ˆÏÓ–

â˛ì% ¶≈% ÏãÓ˚ ˆ◊!í!Ó¶˛yÜ ı

(i) ›˛Δy!˛ô!ãÎ˚õ ı ˆÎ â˛ì%˛¶≈%˛ˆÏãÓ˚ ~Ñ˛ˆÏãyv˛¸y !Ó˛ôÓ˚#ì˛ Óy£% ˛ôÓ˚fl˛ôÓ˚ ¢õyhs˘˛Ó˚yú ì˛yˆÏÑ˛ ›˛Δy!˛ô!ãÎ˚yõ

Ó Ïú– 6.32 (I) ÈÙÈ~Ó˚ ›˛Δy!˛ô!ãÎ˚yˆÏõ AB || DC

(I) (II)

!â˛e ı 6.32

A

B

C

D

A

B C

DE

H

G

F

A B

D C

E

FG

H

FGEH

HGEFAB DC

ˆòy›˛

56 D.El.Ed. 504


(ii) ¢yõhs˘˛!Ó˚Ñ˛ ı ˆÎ â˛ì% ¶≈% ˆÏãÓ˚ !Ó˛ôÓ˚#ì˛ Óy£%Ü%!ú ˛ôÓ˚fl˛ôÓ˚ ¢õyhs˘˛Ó˚yú ì˛yˆÏîÓ˚ ¢yõhs˘˛!Ó˚Ñ˛ ÓˆÏú–

!â˛e 6.32(II)ÈÙÈˆì˛ EF || GH ~ÓÇ EG || EH – ì˛y£ Ïú EFGH ~Ñ˛!›˛ ¢yõhs˘˛!Ó˚Ñ˛–

(iii) xyÎ˚ì˛ Ï«˛e ı ˆÎ â˛ì%˛¶%≈˛ˆÏãÓ˚ â˛yÓ˚!›˛ ˆÑ˛yí£z ¢õˆÏÑ˛yí ì˛y xyÎ˚ì˛ˆÏ«˛e– ~›˛yÁ Óúy ÎyÎ˚ñ ˆÎ

¢yõhs˘˛!Ó˚ˆÏÑ˛Ó˚ ~Ñ˛!›˛ ˆÑ˛yí ¢õˆÏÑ˛yí ì˛y xyÎ˚ì˛yÑ˛yÓ˚–

!â˛e 6.33

!â˛e 6.33 ÈÙÈˆì˛ ∠A, ∠B, ∠C Á ∠D ¢õ ÏÑ˛yí– ì˛y£ Ïú ABCD ~Ñ˛!›˛ xyÎ˚ì˛ Ï«˛e–

(iv) Ó˚¡∫¢ ı ˆÎ â˛ì%˛¶≈%˛ˆÏãÓ˚ â˛yÓ˚!›˛ Óy£%£z ¢õyò ì˛y £ú Ó˚¡∫¢–

(v) ÓÜ≈ Ï«˛e ı ˆÎ â˛ì% ¶≈% ˆÏãÓ˚ â˛yÓ˚!›˛ Óy£%£z ¢õyò ~ÓÇ ≤Ã!ì˛!›˛ ˆÑ˛yí ¢õˆÏÑ˛yí ì˛y £ú ÓÜ≈ˆÏ«˛e–

!â˛e 6.33(III) ÈÙÈ~ AB = BC = CD = DA ~ÓÇ ˛∠A, ∠B, ∠C Á ∠D ≤Ã Ïì˛ƒ ÏÑ˛

¢õ ÏÑ˛yí– xì˛~Ó ABCD ~Ñ˛!›˛ ÓÜ≈ˆÏ«˛e–

!Ó!¶˛ß¨ â˛ì%˛¶≈%˛ˆÏãÓ˚ õˆÏïƒ ¢¡ô ÏÑ≈˛Ó˚ â˛y›≈˛ ı

ˆòy›˛ ı !Ó!¶˛ß¨ ïÓ˚ ÏíÓ˚ â˛ì% ¶≈% ÏãÓ˚ ïõ≈ ˛ôÓ˚#«˛y Ñ˛ ÏÓ˚ ãyòy ÎyÎ˚–

â˛ì%˛¶≈%˛ˆÏãÓ˚ ˛ô!Ó˚¢#õy = â˛yÓ˚!›˛ Óy£%Ó˚ ˜î Ïá≈ƒÓ˚ ˆÎyÜö˛ú–

A

C

D A

A A

A A

A

A

AB

â˛ì% ¶≈% ã

!Ó˛ôÓ˚#ì˛ Óy£% Ïãyv ¸y ¢õyhs˘˛Ó˚yú

›˛Δy!˛ô!ãÎ˚yõ

x˛ôÓ˚ Óy£%ˆÏãyv˛¸yÁ ¢õyhs˘˛Ó˚yú

¢yõhs˘ !ÓÑ˛ ~Ñ˛ˆÏãyv˛¸y ¢!ß¨!£ì˛ Óy£% Ó˚¡∫¢

˜î Ïá≈ƒ ¢õyò

~Ñ˛!›˛ ˆÑ˛yí ¢õ ÏÑ˛yí ~Ñ˛!›˛ ˆÑ˛yí ¢õ ÏÑ˛yí

xyÎ˚ì˛ Ï«˛e ~Ñ˛ Ïãyv ¸y ¢!ß¨!£ì˛ Óy£% ÓÜ≈ Ï«˛e

˜î Ïá≈ƒ ¢õyò

ˆòy›˛



â˛ì%˛¶≈%˛ ÏãÓ˚ ˆ«˛eö˛ú = Ñ˛ˆÏí≈Ó˚ õyïƒˆÏõ ˆÎ !e¶% ˆÏãÓ˚ ˆãyv ¸y ˜ì˛!Ó˚ £Î˚ ì˛yˆÏîÓ˚ ˆõy›˛ ˆ«˛eö˛ˆÏúÓ˚

¢õ!T˛–

› Δy!˛ô!ãÎ˚y ÏõÓ˚ ˛ô!Ó˚¢#õy = â˛yÓ˚!›˛ Óy£%Ó˚ ˜îˆÏá≈ƒÓ˚ ˆÎyÜö˛ú–

ABCD ›˛Δy!˛ô!ãÎ˚yˆÏõÓ˚ ˆ«˛eö˛ú S!â˛e ı 6.34)

= ∆ABCÈÙÈÓ˚ ˆ«˛eö˛ú + ∆BCD ˆ«˛eö˛ú

= 1

2AB × h +

1

2CD × h

= 1

2h(AB + CD)

= 1

2h (¢õyhs˘˛Ó˚yú Óy£%m ÏÎ˚Ó˚ ˜î Ïá≈ƒÓ˚ ¢õ!T˛V

¢yõhs˘˛!Ó˚ ÏÑ˛Ó˚ ˛ô!Ó˚¢#õy = AB + BC + CD + DA S!â˛e ı 6.35)

= AB + BC + AB + BC

= 2AB + 2BC

= 2(AB + BC)

ˆ«˛eö˛ú = ¶) !õ ×v˛zFâ˛ì˛y

S¶)˛!õ – DC Á v˛zFâ˛ì˛y h)

xyÎ˚ì˛ Ï«˛ ÏeÓ˚ ˛ô!Ó˚¢#õy ı AB + BC + CD + DA

= l + b + l + b = 2l + 2b

= 2(l + b) [l = ˜îá≈ƒ AB, b = ≤ÃfiÌ]

xyÎ˚ì˛ Ï«˛ ÏeÓ˚ ˆ«˛eö˛ú = l × b

Ó˚¡∫ˆÏ¢Ó˚ ˛ô!Ó˚¢#õy = AB + BC + CD + DA S!â˛e 6.37)

AB + AB + AB + AB = 4AB = 4 × ~Ñ˛!›˛ Óy£%Ó˚ ˜îá≈ƒ

Ó˚¡∫ Ï¢Ó˚ Ñ˛í≈î%!›˛ ˛ôÓ˚fl˛ôÓ˚ˆÏÑ˛ ¢õˆÏÑ˛yˆÏí ˆäÈî Ñ˛ˆÏÓ˚

∴ ABCD Ó˚¡∫ Ï¢Ó˚ ˆ«˛eö˛ú

= ∆ABC Ó˚ ˆ«˛eö˛ú + ∆ACDÓ˚ ˆ«˛eö˛ú

= 1

2AC ×BD +

1

2AC ×DO

A Bh h

C D

!â˛e ı 6.34

A B

CD

h h

!â˛e ı 6.35

h

h h

h

b

l

!â˛e ı 6.36

D C

BA

!â˛e ı 6.37

ˆòy›˛

58 D.El.Ed. 504


= 1

2AC (BO + DO)

= 1

2 A×BD

= 1

2 Ñ˛í≈î%!›˛Ó˚ ˜î Ïá≈ƒÓ˚ Ü%íö˛ú

ÓÜ≈ Ï«˛ ÏeÓ˚ ˛ô!Ó˚¢#õy ı

= 4 × ~Ñ˛!›˛ Óy£%Ó˚ ˜îá≈ƒ SÓ˚¡∫ˆÏ¢Ó˚ òƒyÎ˚V

ÓÜ≈ Ï«˛ ÏeÓ˚ ˆ«˛eö˛ú ı

= AB × AD SxyÎ˚ì˛ˆÏ«˛ˆÏeÓ˚ òƒyÎ˚V [!â˛e 6.38]

= AB ×AB SˆÎˆÏ£ì%˛ ≤Ã!ì˛!›˛ Óy£%Ó˚ ˜îá≈ƒ ¢õyòV

= AB2 = ~Ñ˛!›˛ Óy£%Ó˚ ˜î Ïá≈ƒÓ˚ ÓÜ≈–

6.3.3 Ó,_

Ó,_ £ú ~Ñ˛!›˛ ¢yõì˛!úÑ˛ !â˛e Îy ÓÑ ˛ˆÏÓ˚Öy myÓ˚y Ü!‡˛ì˛ Á ~Ñ˛!›˛ ¢õì˛ˆÏúÓ˚ !Ñ˛ä%È›˛y xÇü !òˆÏÎ˚

xÓfiÌyò Ñ˛ ÏÓ˚–

¢ÇK˛y ı Ó,_ £ú ~Ñ˛!›˛ ì˛ˆÏú xˆÏòÑ˛ !Ó®%Ó˚ ¢õ!T˛ñ ~õò¶˛yˆÏÓ !Ó®%Ü%!ú xÓ!fiÌì˛ Îyì˛ ì˛yÓ˚y

ˆ¢£z ì˛ˆÏúÓ˚ ~Ñ˛!›˛ !ò!î≈T˛ !Ó®% ˆÌˆÏÑ˛ ¢õî)Ó˚ˆÏc xÓfiÌyò Ñ˛ˆÏÓ˚– !ò!î≈T˛ !Ó®%›˛y £ú Ó, Ï_Ó˚ ˆÑ˛wñ

!ò!î≈T˛ î)Ó˚c!›˛ £ú Ó, Ï_Ó˚ Óƒy¢yï≈–

!â˛e ı 6.39

Ó,ˆÏ_Ó˚ ˆÎˆÏÑ˛yò î%!›˛ !Ó®% ˆÎyÜ Ñ˛Ó˚ˆÏú Ó, Ï_Ó˚ ãƒy ˛ôyÁÎ˚y ÎyÎ˚– Ó, Ï_Ó˚ ˆÑ˛w !îˆÏÎ˚ ˆÎ ãƒy ÎyÎ˚

ì˛y £ú Ó, Ï_Ó˚ Óƒy¢– AB SxÌ≈yÍ ABÓ˚ ˜îá≈ƒV Ó, Ï_Ó˚ Óƒy¢ Sd !î ÏÎ˚ !â˛!£´ì˛ Ñ˛Ó˚y £Î˚V– OA

Á OB ≤Ã Ïì˛ƒ ÏÑ˛ ~Ñ˛!›˛ Óƒy¢yï≈ñ xì˛~Ó d = 2r– !â˛e 6.40(a)ˆì˛ ABC Ó,ˆÏ_Ó˚ î%!›˛ !Ó®%

£ú A Á C– ~£z !Ó®% î%!›˛ Ó,_ˆÏÑ˛ î%!›˛ ¶˛yˆÏÜ ¶˛yÜ Ñ˛ˆÏÓ˚ˆÏäÈñ ≤Ã!ì˛!›˛ ¶˛yÜˆÏÑ˛ Ó, Ï_Ó˚ â˛y˛ô Óúy

!â˛e ı 6.37

D C

BA

B AO

r

C

ˆòy›˛



£Î˚– ÎyÓ˚ ≤Ãyhs˘˛!Ó®% £ú A Á C– AC £ú Ó,ˆÏ_Ó˚ ãƒy– P Á Q !Ó®% £ú x˛ôÓ˚ î%!›˛ !Ó®%–

P Á O !Ó®% AC ãƒy!›˛Ó˚ î%!›˛ !Ó˛ôÓ˚#ì˛ !îˆÏÑ˛ xÓfiÌyò Ñ˛Ó˚ˆÏäÈ– !Ñ˛hs%˛ Q Á O !Ó®% AC ãƒyÓ˚

~Ñ˛£z !î ÏÑ˛ xÓfiÌyò Ñ˛Ó˚ ÏäÈ–

!â˛e ı 6.40 (a) 6.40 (b)

ˆÎ Ó,_â˛yˆÏ˛ô P !Ó®% xÓfiÌyò Ñ˛Ó˚ ÏäÈ ì˛yÓ˚ òyõ APC Ó,_â˛y˛ô ~ÓÇ Q !Ó®%!fiÌì˛ Ó,_â˛yˆÏ˛ôÓ˚

òyõ AQC– APC £ú ˆäÈyˆÏ›˛y v˛z˛ôâ˛y˛ô Á AQC £ú Ó Ïv ¸y x!ïâ˛y˛ô– APC Á AQC ˆÑ˛

!Ó˛ôÓ˚#ì˛ Ó,_â˛y˛ô ÓˆÏú–

!â˛e 6.40(b) ˆì˛ î%!›˛ Óƒy¢yï≈ AO ~ÓÇ BO ˆäÈyˆÏ›˛y v˛z˛ôâ˛y˛ô ADBÓ˚ ≤Ãyhs˘˛!Ó®% A Á BˆÏì˛

x§yÑ˛y £ˆÏÎ˚ˆÏäÈ– ö˛ˆÏú ˆÑ˛ˆÏw ∠AOB v zÍ˛ôß¨ £ˆÏÎ˚ˆÏäÈ– ∠AOB £ú ˆÑ˛ Ïw ADBÓ˚ ¢¡ø%Ö!fiÌì˛

ˆÑ˛yí–

∠AOBÓ˚ õyò £ú ADBÓ˚ !v˛@˘Ã#Ó˚ õy˛ô– SÓ,_â˛y˛ô Îì˛ ÓˆÏv˛¸y £ˆÏÓñ ì˛ì˛ ì˛yÓ˚ !v˛@˘Ã#Ó˚ õy˛ô ÓˆÏv˛¸y

£ˆÏÓ–

x!ïâ˛yˆÏ˛ôÓ˚ !v˛@˘Ã#Ó˚ õy˛ô £Î˚ = 360° — ~Ó˚ !Ó˛ôÓ˚#ì˛ Ó,_â˛yˆÏ˛ôÓ˚ !v˛@˘Ã#Ó˚ õy˛ô–

v˛zîy£Ó˚ífl˛∫Ó˚)˛ôñ mACB S!â˛e 6.40(b) = 360° – mADB–

!â˛e 6.41 ÈÙÈˆì˛ AC £ú APC Ó,ˆÏ_Ó˚ Óƒy¢ñ ~ÓÇ î%!›˛ Ó,_â˛y˛ô£ú APC Á AQC, ÎyÓ˚y

≤ÃˆÏì˛ƒˆÏÑ˛ xï≈Ó,_– xï≈Ó,ˆÏ_Ó˚ !v˛@˘Ã# õy˛ô £ú 90°–

!â˛e ı 6.41

!â˛e 6.42(I) ÈÙÈ~ P ~ÓÇ Q v˛z˛ôâ˛y˛ô ABÓ˚ xhs˘˛≈Üì˛ î%!›˛ !Ó®%– ∠APB ~ÓÇ ∠AQB £ú

A C

Q

O

A B

C

D

P

O

A

P

Q

OA

ˆòy›˛

60 D.El.Ed. 504


APB Ó,_â˛yˆÏ˛ôÓ˚ xhs˘˛Ü≈ì˛ ˆÑ˛yí– ~Ñ˛£z¶˛yˆÏÓ ∠ARB ~ÓÇ ∠ASB £ú ARB Ó,_â˛yˆÏ˛ôÓ˚

xhs˘˛Ü≈ì˛ ˆÑ˛yí–

!â˛e ı 6.42

!â˛e ı 6.42(II) ˆì˛ AB £ú ~Ñ˛!›˛ Ó,ˆÏ_Ó˚ Óƒy¢– M Á N xï,Ó,_î%!›˛Ó˚ î%!›˛ !Ó®%ñ ˆÎÖyˆÏò

A ÁB ≤Ãyhs˘˛!Ó®%mÎ˚– ∠AMB Á ∠ANB xï≈Ó, Ï_Ó˚ !ÓÎ%=˛ ˆÑ˛yí–

Ó,_fiÌ â˛ì% ¶≈% ã ı ˆÎ â˛ì% ¶≈% ˆÏãÓ˚ ü#°Ï≈!Ó®%Ü%!ú Ó, Ï_ xÓfiÌyò Ñ˛ˆÏÓ˚ S!â˛e 6.43) ì˛y Ó,_fiÌ

â˛ì% ¶≈% ã– xì˛~Óñ ABCD £ú Ó,_fiÌ â˛ì% ¶≈% ã–

!â˛e ı 6.43

Ó, Ï_Ó˚ Ó!£fiÌ Á xhs˘˛fiÌ !Ó®% ı

!â˛e 6.44ÈÙÈ~ KLM Ó, Ï_Ó˚ ˆÑ˛w £ú O, ¢%ì˛Ó˚yÇ K, L Á M Ó, Ï_Ó˚ !Ó®%–

!â˛e ı 6.44

P Á Q !Ó®% î%!›˛ KLM Ó,ˆÏ_Ó˚ ì˛ˆÏú ~õò¶˛yˆÏÓ xÓ!fiÌì˛ ÎyˆÏì˛ PO < r (r £ú Ó,ˆÏ_Ó˚

Óƒy¢yï≈≈V QO > r, P £ú KLM Ó, Ï_Ó˚ xhs˘˛fiÌ !Ó®% ~ÓÇ Q £ú KLM Ó,_ Ó,ˆÏ_Ó˚ Ó!£fiÌ

!Ó®%–

A B

PQ

R

S

A B

M

N

O

A

BC

D

M

QO

P

K

ˆòy›˛



Ó,ˆÏ_Ó˚ v˛z˛ôˆÏÓ˚ xÓ!fiÌì˛ ≤Ã!ì˛!›˛ !Ó®% Á Ó,ˆÏ_Ó˚ xhs˘˛fiÌ ≤Ã!ì˛!›˛ !Ó®%ˆÏÑ˛ ~Ñ˛ˆÏe Ó,_yÑ˛yÓ˚ xM˛ú

ÓˆÏú–

!â˛e 6.45 ÈÙÈ~ AB £ú ABC Ó,ˆÏ_Ó˚ ãƒyñ AB Ó,_yÑ˛yÓ˚ xM˛úˆÏÑ˛ î%¶˛yˆÏÜ ¶˛yÜ Ñ˛ˆÏÓ˚ˆÏäÈ– ≤Ã!ì˛

¶˛yÜˆÏÑ˛ Ó,_yÇü ÓˆÏú– ˆÎ Ó,_yÇü O ˆÑ˛w ˆò£z ˆ¢£z ˆäÈyˆÏ›˛y xÇüˆÏÑ˛ v˛z ôÓ,_yÇü ÓˆÏú ~ÓÇ

ˆÎ xÇˆÏü ˆÑ˛w O xÓ!fiÌì˛ ˆ¢£z ÓˆÏv ¸y xÇüˆÏÑ˛ x!ïÓ,_yÇü ÓˆÏú–

!â˛e ı 6.45

Ó, Ï_Ó˚ Ñ˛ ÏÎ˚Ñ˛!›˛ ïõ≈ ı

1. ¢õÓ,_â˛yˆÏ˛ô xÓ!fiÌì˛ !ÓÎ%=˛ ˆÑ˛yíÜ%!ú SxÌ≈yÍñ ¢õ Ó,_yÇˆÏüÓ˚

ˆÑ˛yíV ¢õyò– m∠ABD= m ∠ACD, m ∠BAC, m ∠BDC–

2. xï≈Ó, Ï_Ó ˆÑ˛yí ~Ñ˛!›˛ ¢õ ÏÑ˛yí– AB £ú Óƒy¢– APB £ú xï≈Ó,_–

m ∠APB = 90°–

3. Ó,_fiÌ â˛ì% ¶≈% ˆÏãÓ˚ !Ó˛ôÓ˚#ì˛ ˆÑ˛yíÜ%!ú ¢¡ô)Ó˚Ñ˛ ˆÑ˛yí– ABCD £ú

Ó,_fiÌ â˛ì% ¶≈% ã Ñ˛yÓ˚í ~Ó˚ ü#°Ï≈!Ó®%Ü%!ú Ó, Ï_ xÓfiÌyò £ ÏÎ˚ ÏäÈ– m <

A + m < = m < B + m < D = 180°–

4. ~Ñ˛!›˛ Ó,_â˛yˆÏ˛ôÓ˚ !v˛@˘Ã#õy˛ô £ú !Ó˛ôÓ˚#ì˛ Ó,_â˛yˆÏ˛ôÓ˚ !ÓÎ%Ó ˆÑ˛yˆÏíÓ˚

õyˆÏ˛ôÓ˚ !mÜ%ò–

APB 2 ADBm m= ∠ = 2m = ∠ACB

BA

C

B

A D

C

A

P

B

!â˛e ı 6.46

A

B C

D

!â˛e ı 6.47

!â˛e ı 6.48

B

CO

P

!â˛e ı 6.49

ˆòy›˛

62 D.El.Ed. 504


!ò Ïã ÏÑ˛ ≤ÃÜ!ì˛ õy˛ôy ı

E5. ü)òƒfiÌyò ˛ô)Ó˚í ı

(a) ¢yõhs˘˛!Ó˚ˆÏÑ˛Ó˚ î%!›˛ ¢!ß¨!£ì˛ Óy£%Ó˚ ˜îá≈ƒ ¢õò £ˆÏú ˆ¢!›˛ _________–

(b) Ó, Ï_Ó˚ Óƒy¢yï≈ £ú ˆÓ˚ÖyÇü Îy Ó, Ï_Ó˚ ~Ñ˛!›˛ !Ó®% ˆÌˆÏÑ˛ _____________ Î%=˛ £ˆÏÓ–

(c) Î!î ~Ñ˛!›˛ Ó,_â˛yˆÏ˛ôÓ˚ õyò 64° £Î˚ñ ì˛y£ˆÏú !Ó˛ôÓ˚#ì˛ Ó,_â˛yˆÏ˛ôÓ˚ !ÓÎ%=˛ ˆÑ˛yˆÏíÓ˚ õyò £ˆÏÓ

–––––––––––––––

(d) Ó,_fiÌ â˛ì%˛¶≈%˛ˆÏãÓ˚ !Ó˛ôÓ˚#ì˛ ˆÑ˛yíî%!›˛Ó˚ ˆÎyÜ˛ôú £ú ______________ !v˛@˘Ã#–

6.3.4 ¢Ó≈¢õì˛y Á ≤Ã!ì˛¢yõƒ

¢Ó≈¢õì˛y ı

î%!›˛ ¢yõì˛!úÑ˛ !â˛eˆÏÑ˛ ¢Ó≈¢õ Óúy ÎyˆÏÓñ Î!î ~Ñ˛!›˛Ó˚ äÈy˛ô xòƒ!›˛Ó˚ ¢ˆÏAÜ ¢ˆÏÓ≈yì˛¶˛yˆÏÓ !õˆÏú

ÎyÎ˚– ¢Ó≈¢õì˛y ¢¡∫ Ïrï Îy Óúy £ú ì˛y £yˆÏì˛Ñ˛úˆÏõ Ñ˛Ó˚ˆÏú ì˛ˆÏÓ£z ˆîÖy ÎyˆÏÓ– ~!›˛ˆÏÑ˛ Î%!=˛

!îˆÏÎ˚ ≤Ã!ì˛¤˛y Ñ˛Ó˚ˆÏì˛ ˆÜˆÏú !Ñ˛ä%È ¢ÇK˛y Á üì≈˛ !îˆÏì˛ £ˆÏÓ ÎyˆÏì˛ !â˛eÜ%!ú S!e¶% ãV ¢Ó≈¢õ !Ñ˛òy

ì˛y ˆÓyG˛y ÎyÎ˚– ˆÎ !â˛ˆÏ£´Ó˚ ¢y£yˆÏÎƒ ¢Ó≈¢õ ˆÓyG˛yÎ˚ ì˛y £ú –

(i) ˆÓ˚ÖyÇ ÏüÓ˚ ¢Ó≈¢õì˛y ı Î!î î%!›˛ ˆÓ˚ÖyÇˆÏüÓ˚ ˜îá≈ƒ ¢õyò

£Î˚ñ ì˛y£ Ïú ˆÓ˚ÖyÇü î%!›˛ ÏÑ˛ ¢Ó≈¢õ Óúy ÎyÎ˚– xì˛~Ó ¢Ó≈¢õ

Ó˚Öy ÏîÓ˚ ˜îá≈ƒ ¢õyò

AB PQ AB PQ= ⇒ ≅

~ÓÇ AB PQ AB PQ≅ ⇒ =

(ii) ˆÑ˛y ÏíÓ˚ ˛¢Ó≈¢õì˛y ı

î%!›˛ ˆÑ˛yí ÏÑ˛ ¢Ó≈¢õ Óúy ÎyÎ˚ Î!î ì˛y ÏîÓ˚ õyò ¢õyò £Î˚– xì˛~Ó ¢Ó≈¢õ ˆÑ˛y ÏíÓ˚ õyò ¢õyò

£ˆÏÓ–

m∠ABC = m∠PQR ⇒ABC ≅ PQR

~ÓÇ ∠ABC ≅ ∠PQR ⇒ mA∠BC = m∠PQR

!â˛e ı 6.51

P

Q

A B

!â˛e ı 6.50

A

B C

P

Q R

ˆòy›˛



!e¶%˛ˆÏãÓ˚ ¢Ó≈¢õì˛y ı

!e¶% ÏãÓ˚ ¢Ó≈¢õì˛yÓ˚ üì≈ Ü%!ú £ú ı

(i) î%!›˛ !e¶% ã ¢Ó≈¢õ £Î˚ Î!î ~Ñ˛!›˛ !e¶% ˆÏãÓ˚ î%!›˛ Óy£%Ó˚ ˜îá≈ƒ Á ì˛yˆÏîÓ˚ xhs˘˛¶≈% =˛ ˆÑ˛yˆÏíÓ˚

˛ô!Ó˚õy˛ô x˛ôÓ˚ ~Ñ˛!›˛ !e¶%˛ˆÏãÓ˚ xò%Ó˚)˛ô î%!›˛ Óy£%Ó˚ ˜îá≈ƒ Á ì˛yˆÏîÓ˚ xhs˘˛¶≈%˛=˛ ˆÑ˛yˆÏíÓ˚ ˛ô!Ó˚õyˆÏ˛ôÓ˚

¢õyò £Î˚– ~!›˛ˆÏÑ˛ xyõÓ˚y S–A–S üì˛≈ ÓúÓ–

ˆòy›˛ ı üì≈ £ú Îy ÏÑ˛ ≤Ãõyí Ñ˛Ó˚y ÎyÎ˚ òy ˆõ Ïò !ò Ïì˛ £Î˚–

ABC Á PQR !e¶%˛ˆÏã Î!î AB PQ≅ ñ BC QR≅

~ÓÇ ABC PQR∠ ≅ ∠

ì˛y£ˆÏúñ ABC PQRΔ ≅ Δ

!â˛e ı 6.52

(ii) ~Ñ˛!›˛ !e¶%˛ˆÏãÓ˚ !ì˛ò!›˛ Óy£%Ó˚ ˜îá≈ƒ x˛ôÓ˚ !e¶%˛ˆÏãÓ˚ xò%Ó˚)˛ô !ì˛ò!›˛ Óy£%Ó˚ ˜îˆÏá≈ƒÓ˚ ¢õyò

£ Ïú ì˛y ¢Ó≈¢õ£ ÏÓ–

~!›˛ S-S-S ¢Ó≈¢õì˛yÓ˚ üì≈ –

∆ABC Á ∆PQR, Î!î AB PQ≅

BC QR≅ , CA RP≅ ì˛y£ˆÏú ABC PQRΔ ≅ Δ

!â˛e ı 6.53

(iii) ~Ñ˛!›˛ !e¶%˛ˆÏãÓ˚ î%!›˛ ˆÑ˛yˆÏíÓ˚ ˛ô!Ó˚õy˛ô Á ~Ñ˛!›˛ Óy£%Ó˚ ˜îá≈ƒ x˛ôÓ˚ ~Ñ˛!›˛ !e¶%˛ˆÏãÓ˚ î%!›˛

ˆÑ˛yˆÏíÓ˚ ˛ô!Ó˚õy˛ô ~ÓÇ xò%Ó) ô Óy£%Ó˚ ˜îˆÏá≈ƒÓ˚ ¢õyò £ˆÏú ì˛yˆÏÑ˛ ¢Ó≈¢õ !e¶% ã ÓˆÏú–

∆ABC Á ∆PQR

A

B C

P

Q R

B

A

C Q

P

R

ˆòy›˛

64 D.El.Ed. 504


Î!î B Q∠ ≅ ∠ , C R∠ ≅∠ , BC QR≅

ì˛y£ Ïú ABC PQRΔ ≅ Δ

!â˛e ı 6.54

~!›˛ A-A-S üì≈ –

(iv) î%!›˛ ¢õ ÏÑ˛yí# !e¶% ã ¢Ó≈¢õ £ ÏÓ Î!î ì˛y ÏîÓ˚ x!ì˛¶) ÏãÓ˚ ˜îá≈ƒ Á x˛ôÓ˚ ~Ñ˛!›˛ Óy£%Ó˚ ˜îá≈ƒ

¢õyò £Î˚– ì˛yˆÏÑ˛ xyõÓ˚y R-H-S üì≈ ÓúÓ–

!â˛e ı 6.55

∆ABC ˆì˛ñ ∠B £ú ¢õ ÏÑ˛yí ~ÓÇ ∆PQR ÈÙÈ~ ∠Q ¢õ ÏÑ˛yí– Î!î x!ì˛¶) ã AC PR≅

~ÓÇ Óy£% AB PQ≅ £Î˚ñ ì˛ˆÏÓ ì˛y£ˆÏúñ ABC PQRΔ ≅ Δ –

ˆòy›˛ ı ¢Ó≈¢õ !e¶% ÏãÓ˚ ˆ«˛eö˛ú ¢õyò £ ÏÓ–

¢Ó≈¢õì˛yÓ˚ ≤Ã ÏÎ˚yÜ ı ÎÖò !e¶% ˆÏãÓ˚ î%!›˛ Óy£% Óy î%!›˛ ˆÑ˛yíˆÏÑ˛ ¢Ó≈¢õ !£¢yˆÏÓ ≤Ãõyí Ñ˛Ó˚yÓ˚

îÓ˚Ñ˛yÓ˚ ˛ô Ïv ¸ ì˛Öò xyõÓ˚y !e¶% ã î%!›˛ ÏÑ˛ ¢Ó≈¢õ !£¢y ÏÓ ≤Ãõyí Ñ˛ ÏÓ˚ì˛ â˛y£z–

ö˛ˆÏú !e¶% ˆÏãÓ˚ ¢Ó≈¢õì˛yÓ˚ ¢)e ïˆÏÓ˚ xyõÓ˚y !Ó!¶˛ß¨ ãƒy!õ!ì˛Ñ˛ ¢õ¢ƒy ¢õyïyò Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚–

ò#ˆÏâ˛ ~Ñ˛!›˛ v zîy£Ó˚í ˆîÁÎ˚y £ú–

v zîy ı ˛ôyˆÏüÓ˚ äÈ!Ó AB CD≅

ABC BCA∠ ≅∠ , ≤Ãõyí Ñ˛Ó˚ Ïì˛ £ ÏÓ AC BD≅

≤Ãõyíñ ∆ABC Á ∆BCD ˆì˛

AB CD≅ SˆîÁÎ˚y xyˆÏäÈV

A

B C

P

R R

A

B C

P

Q R

A

B C

D

!â˛e ı 6.56

ˆòy›˛



BC £ú î%!›˛ !e¶% ÏãÓ˚ ¢yïyÓ˚í Óy£%

ABC BCD∠ ≅ ∠ SˆîÁÎ˚y xy ÏäÈV

∴ ABC BCDΔ ≅ Δ (S-A-S ¢Ó≈¢õì˛yÓ˚ üì≈ V

AC BD⇒ ≅ [¢Ó≈¢õ !e¶%˛ˆÏãÓ˚ xò%Ó˚)˛ô Óy£%]

!e¶%˛ˆÏãÓ˚ ≤Ã!ì˛¢yõƒ

¢yõì˛!úÑ˛ !â˛ˆÏeÓ˚ î%!›˛ !Ó°ÏÎ˚ ÌyˆÏÑ˛ñ xyÑ˛yÓ˚ Á õy˛ô– Î!î î%!›˛ ¢yõì˛!úÑ˛ !â˛ˆÏeÓ˚ xyÑ˛yÓ˚ Á õy˛ô

¢õyò £Î˚ñ ì˛y£ˆÏú ì˛yÓ˚y ¢Ó≈¢õñ xÌ≈yÍ ~Ñ˛!›˛Ó˚ äÈ!Ó ~§ˆÏÑ˛ x˛ôÓ˚!›˛Ó˚ v˛z˛ôÓ˚ Ó¢yˆÏú ˛ô%ˆÏÓ˚y !õˆÏú

ÌyˆÏÑ˛–

xyõÓ˚yî%!›˛ ¢yõì˛!úÑ˛ !â˛e !òúyõ Îy ÏîÓ˚ xyÑ˛yÓ˚ ~Ñ˛£zÓ˚Ñ˛õ– !Ñ˛ä%È v zîy£Ó˚í ˆîÁÎ˚y £úÈÙÙÙÈ

i) ~Ñ˛ãò Óƒ!=˛Ó˚ ö˛ˆÏ›˛y ì%˛ˆÏú ì˛yÓ˚ ~Ñ˛£z ˆòˆÏÜ!›˛¶˛ ˆÌˆÏÑ˛ !≤Ãr›˛ ˆòÁÎ˚y £úñ !Ñ˛hs%˘˛ õy˛ô

˛ôyˆÏŒ›˛ ˆîÁÎ˚y £ú– ö˛ˆÏ›˛y î%!›˛ ~Ñ˛£zÓ˚Ñ˛õ ˆîÖˆÏì˛ £ˆÏÓ !Ñ˛hs%˛ ì˛yˆÏîÓ˚ õy˛ô !¶˛ß¨ £ˆÏÓ–

ii) ¶˛yÓ˚ˆÏì˛Ó˚ õƒy˛ôÈÙÙÙÈ~Ñ˛!›˛ Ó£zˆÏÎ˚Ó˚ ˛ôyì˛yÎ˚ x§yÑ˛yñ xòƒ!›˛ ˆîÁÎ˚y â˛yˆÏ›≈˛ ˆG˛yúyˆÏòy– õƒy˛ô

î%!›˛Ó˚ xyÑ˛yÓ˚ ~Ñ˛£z !Ñ˛hs%˘˛ õy˛ô xyúyîyñ

(iii)

(a) (b) (c)

!â˛e 6.57

î%!›˛ Ó,_ ~Ñ˛£z Ó˚Ñ˛õ ˆîÖˆÏì˛ ì˛y£z ì˛yÓ˚y ¢î,üñ ~Ñ˛£z¶˛yˆÏÓ â˛ì% ¶%≈˛ã ¢õÓy£% !e¶% ãÁ ¢î,ü

ˆîÖˆÏì˛–

ˆòy›˛ ı ¢Ó≈¢õ !â˛eÓ˚y ¢î,ü £ ÏÓ– !Ñ˛hs%˛ ¢î,ü !â˛e õye£z ¢Ó≈¢õ òÎ˚–

6.3.5 ≤Ã!ì˛ö˛úò Á ≤Ã!ì˛¢yõƒ

xyõÓ˚y ãy!ò xyÎ˚òyÎ˚ ≤Ã!ì˛ö˛úò £Î˚ ~ÓÇ ì˛yÓ˚ ö˛ˆÏú ≤Ã!ì˛!Ó¡∫ Ü!‡˛ì˛ £Î˚– ãƒy!õ!ì˛ˆÏì˛

≤Ã!ì˛¢yõƒÁ xyÎ˚òyÓ˚ ≤Ã!ì˛ö˛ú ÏòÓ˚ õì˛–

(a) ¢Ó˚ú ÏÓ˚ÖyÎ˚ ≤Ã!ì˛ö˛úò ı

(i) ¢Ó˚ú ÏÓ˚ÖyÎ˚ ~Ñ˛!›˛ !Ó®%Ó˚ ≤Ã!ì˛ö˛úò ı !â˛e 6.58 (a)-ˆì˛ ‘L’ ~!›˛ ¢Ó˚ú ÏÓ˚Öy SÑ˛ÖòÁ

Ñ˛ÖòÁ ¢Ó˚ú ÏÓ˚Öy ÏÑ˛ ~Ñ˛!›˛ x«˛Ó˚ !î ÏÎ˚ ≤ÃÑ˛yü Ñ˛Ó˚y £Î˚V ~ÓÇ P ~Ñ˛!›˛ !Ó®%–

ˆòy›˛

66 D.El.Ed. 504


!â˛e 6.58 (a) !â˛e 6.58 (b)

P !Ó®% L ¢Ó˚úˆÏÓ˚ÖyÎ˚ ≤Ã!ì˛ö˛!úì˛ £ˆÏÎ˚ ~Ñ˛!›˛ ≤Ã!ì˛!Óß∫ Ü‡˛ò Ñ˛ˆÏÓ˚– ~ÖyˆÏò ≤Ã!ì˛!Ó¡∫

ˆÑ˛yí!›˛ Á ì˛y ˆÑ˛yÌyÎ˚ ≤Ã!ì˛ö˛!úì˛ £ˆÏÓ⁄

≤Ã!ì˛!Ó¡∫ Ü‡˛ ÏòÓ˚ !òÎ˚õ ı

!â˛e 6.58 (b)- Ïì˛ P !Ó®% ˆÌˆÏÑ˛ñ PM ˆÓ˚Öy L-~Ó˚ ¢ ÏAÜ ú¡∫¶˛y ÏÓ ›˛yòy £ú– PM ÈÙÈ~Ó˚

v˛z˛ôÓ˚ñ ~Ñ˛!›˛ !Ó®% P′ ~õò¶˛yˆÏÓ ˆòÁÎ˚y £ú ÎyˆÏì˛ P-M-P′ ~ÓÇ PM = MP′, ~£z¶˛yˆÏÓ

xyõÓ˚y P !Ó®%Ó˚ P′ ≤Ã!ì˛!Ó¡∫ ˆ˛ôúyõ L ¢Ó˚ú ÏÓ˚ÖyÎ˚ ≤Ã!ì˛ö˛!úì˛ Ñ˛ ÏÓ˚–

L ¢Ó˚ú ÏÓ˚Öy ÏÑ˛ xyÎ˚òy ¢Ó˚ú ÏÓ˚Öy Ó Ïú–

!â˛e 6.65-ˆÏì˛ P1′, L ¢Ó˚ú ÏÓ˚Öy Ïì˛ P

1-~Ó˚ ≤Ã!ì˛!Ó¡∫–

P2′, L ¢Ó˚úˆÏÓ˚ÖyˆÏì˛ P2-Ó˚ ≤Ã!ì˛!Ó¡∫–

P3′, L ¢Ó˚ú ÏÓ˚Öy Ïì˛P3-Ó˚ ≤Ã!ì˛!Ó¡∫–

!â˛e 6.59

L ¢Ó˚úˆÏÓ˚ÖyÓ˚ Îì˛ !òÑ˛ˆÏ›˛ !Ó®% xÓ!fiÌì˛ £ˆÏÓñ ≤Ã!ì˛!Ó¡∫Á L ¢Ó˚ú ÏÓ˚ÖyÎ˚ ì˛ì˛ !òÑ˛ Ï›˛

Ü!‡˛ì˛ £ ÏÓ–

P4 !Ó®% L ¢Ó˚úˆÏÓ˚ÖyÎ˚ xÓ!fiÌì˛– ì˛y£ˆÏú P4-~Ó˚ ≤Ã!ì˛!Ó¡∫ L ¢Ó˚úˆÏÓ˚ÖyˆÏì˛ ˆÑ˛yÌyÎ˚

Ü!‡˛ì˛ £ ÏÓ⁄

P4 ˆÌˆÏÑ˛ L ~Ó˚ î)Ó˚c ü)òƒñ ˆÎˆÏ£ì% P

4, L ¢Ó˚ú ÏÓ˚ÖyÎ˚ xÓ!fiÌì˛–

∴ P4 ~Ó˚ ≤Ã!ì˛!Ó¡∫ L ¢Ó˚ú ÏÓ˚Öy ˆÌ ÏÑ˛ ü)òƒ î)Ó˚ Ïc ÌyÑ˛ ÏÓ–

xì˛~Óñ P4 ~Ó˚ ≤Ã!ì˛!Ó¡∫ L ¢Ó˚ú ÏÓ˚ÖyÎ˚ xÓ!fiÌì˛ £ ÏÓ–

L

LP

M

P�L�

P1

M

P2

P3P4

P�1

P�3

P�2

ˆòy›˛



ì˛y£ Ïú P4 !ò Ïã£z ì˛yÓ˚ ≤Ã!ì˛!Ó¡∫–

xì˛~Ó xyõÓ˚y ÓúˆÏì˛ ˛ôy!Ó˚ÈÙÙÙÈ

xyÎ˚òyˆÏÓ˚ÖyÎ˚ xÓ!fiÌì˛ ˆÑ˛yˆÏòy !Ó®%Ó˚ ≤Ã!ì˛!Ó¡∫ ì˛yÓ˚ !òãfl˛∫ ≤Ã!ì˛!Ó¡∫–

(ii) ¢Ó˚úˆÏÓ˚ÖyÇˆÏüÓ˚ ≤Ã!ì˛!Ó¡∫

L £ú xyÎ˚òyˆÏÓ˚Öy ˆÎÖyˆÏò AB ≤Ã!ì˛ö˛!úì˛ £ˆÏÓ S!â˛e 6.60) AB ˆÓ˚ÖyÓ˚ñ L xyÎ˚òyˆÏÓ˚ÖyÎ˚

≤Ã!ì˛!Ó¡∫ £ú A B′ ′ ñ ˆÎÖyˆÏò A′ Á B′ £ú ÎÌyÑ Ïõ A Á B-~Ó˚ L ¢Ó˚úˆÏÓ˚ÖyÎ˚ Ü!‡˛ì˛

≤Ã!ì˛!Ó¡∫–

!â˛e 6.60 !â˛e 6.61

(iii) !e¶% ÏãÓ ≤Ã!ì˛ö˛úò ı !â˛e 6.61-~ ∆ABC-~Ó ≤Ã!ì˛!Ó¡∫ L xyÎòy ÏÓÖyÎ ∆A′B′C′,

ˆÎÖy Ïò A′, B′ Á C′ £ú L xyÎ˚òyˆÏÓ˚ÖyÓ˚ ÎÌyÑ ˆÏõ A, B Á C-~Ó˚ ≤Ã!ì˛!Ó¡∫–

≤Ã!ì˛¢yõƒ ı !Ñ˛ä%È !Ñ˛ä%È Óhfl%˛Ó˚ xyÑ,˛!ì˛ Óy !v˛ãy£zò xyõyˆÏîÓ˚ xyÑ,˛T˛ Ñ˛ˆÏÓ˚– xyõÓ˚y ÓˆÏú Ìy!Ñ˛

Óhfl%˛!›˛ Óy !â˛e!›˛ Ö%Ó ¢%®Ó˚– ò#ˆÏâ˛ ì˛yÓ˚ !Ñ˛ä%È !â˛e ˆîÁÎ˚y £ú ı

S˛ôyò ˛ôyì˛yV SòÑ˛üyV S≤Ããy˛ô!ì˛V

!â˛e 6.62

v˛z˛ôˆÏÓ˚Ó˚ ≤Ã!ì˛!›˛ !â˛ˆÏe Î!î xyõÓ˚y ÎÌyÎÌ¶˛yˆÏÓ úy£zò ›˛yòˆÏì˛ ˛ôy!Ó˚ñ ~ÓÇ !â˛eÜ%!úÓ˚ äÈ!Ó

~§ˆÏÑ˛ Ñ˛yÜˆÏã ˆÑ˛ˆÏ›˛ !òˆÏÎ˚ xyõÓ˚y Î!î îyÜ›˛y

v z ô ÏÓ˚Ó˚ !â˛eÜ%!úÓ˚ ≤Ã!ì˛¢õ ˆÓ˚Öy ò# Ïâ˛Ó˚ äÈ!Ó Ïì˛ ˆîÁÎ˚y £ú–

A BL

A�

A�

A

B

C

A�

C�

B�

ˆòy›˛

68 D.El.Ed. 504


!â˛e 6.63

!Ñ˛ä%È Ó Ïí≈Ó˚ v zîy£Ó˚í ÎyÓ˚y ˜Ó˚!ÖÑ˛ ≤Ã!ì˛¢õ–

A B D E

xòƒ Óí≈Ü%!úÓ˚ ≤Ã!ì˛¢õ ˆÓ˚Öy ˆÖ§yãyÓ˚ ˆâ˛T˛y Ñ˛Ó%ò–

ãƒy!õ!ì˛Ñ˛ !â˛ˆÏeÓ˚ ˜Ó˚!ÖÑ˛ ≤Ã!ì˛¢õì˛y

ABC ~Ñ˛!›˛ ¢õ!mÓy£% !e¶% ã ~ÓÇ PQuuuur

£ú ~Ó˚ ≤Ã!ì˛¢õ ˆÓ˚Öyñ PQuuur

¢Ó˚ú ÏÓ˚Öy !e¶% ã!›˛Ó˚

A ü#°Ï≈!Ó®% ~ÓÇ ¶)˛!õ BCuuur

ÈÙÈÓ˚ õïƒ!Ó®% Q !îˆÏÎ˚ ÎyˆÏÓ S!â˛e 6.64)–

!â˛e 6.64 !â˛e 6.65

ABCD, ~Ñ˛!›˛ xyÎ˚ì˛ˆÏ«˛e ~ÓÇ ~!›˛Ó˚ 2!›˛ ˜Ó˚!ÖÑ˛ ≤Ã!ì˛¢õ ˆÓ˚Öy xy ÏäÈ– ˆ¢Ü%!ú £ú

PQsuur

~ÓÇ RSsuur ñ ˆÎÖy Ïò P, Q– R ~ÓÇ S £ú ÎÌyÑ Ïõ Óy£% Á ÈÙÈÓ˚ õïƒ!Ó®%– S!â˛e 6.65)

DABC ~Ñ˛!›˛ ¢õÓy£% !e¶% ã– ~!›˛Ó˚ 3!›˛ ˜Ó˚!ÖÑ˛ ≤Ã!ì˛¢õ ˆÓ˚Öy xy ÏäÈñ ~ÓÇ – ~Ó˚y ≤Ã Ïì˛ƒ ÏÑ˛

!e¶%˛ˆÏãÓ˚ ~Ñ˛!›˛ Ñ˛ˆÏÓ˚ ü#°Ï≈!Ó®% Á !Ó˛ôÓ˚#ì˛ Óy£%Ó˚ õïƒ!Ó®% ÏÑ˛ ˆÎyÜ Ñ˛Ó˚ˆÏäÈ– S!â˛e 6.66)

!â˛e 6.66 !â˛e 6.67

P

A

B CQ

B C

A DP

Q

R S

P

Q

R

S

B

V

TC

AA

B CQ

S

DP

R

ˆòy›˛



ABCD ~Ñ˛!›˛ ÓÜ≈ˆÏ«˛e S!â˛e 6.67)ñ ~!›˛Ó˚ â˛yÓ˚!›˛ ˜Ó˚!ÖÑ˛ ≤Ã!ì˛¢õ ˆÓ˚Öy Óì≈˛õyòñ ACsuur

ñ

BDsuur ñ PQ,

suur RS,suur

ˆÎÖy Ïòñ PQ, R Á S £ú ÎÌyÑ ˛ˆÏõ Óy£% AD ñ BC, AB ~ÓÇ CD ÈÙÈÓ˚

õïƒ!Ó®%–

Ó, Ï_Ó˚ ˆÑ˛w !î ÏÎ˚ ˆÎ ¢Ó˚ú ÏÓ˚Öy ›˛yòy ÎyÎ˚ñ ˆ¢!›˛£z £ú Ó, Ï_Ó˚ ˜Ó˚!ÖÑ˛ ≤Ã!ì˛¢õ ˆÓ˚Öy–

ì˛y£ˆÏú xyõÓ˚y ˆîÖúyõ !Ó!¶˛ß¨ ãƒy!õ!ì˛Ñ˛ !â˛ˆÏeÓ˚ ~Ñ˛!›˛ Óy ~Ñ˛y!ïÑ˛ ˜Ó˚!ÖÑ˛ ≤Ã!ì˛¢õ

ˆÓ˚Öy Ìy ÏÑ˛–

á)í≈ò ≤Ã!ì˛¢yõƒ ı

Ñ˛yÜã !îˆÏÎ˚ ÓyòyˆÏòy ˆÖúòy £yÁÎ˚yÈÙÈÑ˛ú Óyì˛yˆÏ¢ ïÓ˚ˆÏú á%Ó˚ˆÏì˛ ÌyˆÏÑ˛–

!â˛e 6.68

£yÁÎ˚y Ñ˛ˆÏúÓ˚ â˛yÓ˚!›˛ ˆÓœ Ïv˛Ó˚ òyõ A, B, C Á D– Î!î ~!›˛ˆÏÑ˛ ¢õˆÏÑ˛yˆÏí ˆáyÓ˚yˆÏòy £Î˚ Sá!v˛¸Ó˚

Ñ§˛y›˛y ˆÎ!îˆÏÑ˛ ˆáyˆÏÓ˚ ì˛yÓ˚ !Ó˛ôÓ˚#ì˛ !îˆÏÑ˛Vñ ˆîÖy ÎyˆÏÓ A, Óì≈˛õyˆÏò B ˆÎ fiÌyˆÏò xyˆÏäÈ ì˛y

îÖú Ñ˛Ó˚ ÏÓñ B, c Á D ÎÌyÑ ˛ˆÏõ C, D Á A-Ó˚ fiÌyò îÖú Ñ˛Ó˚ ÏÓ–

ì˛y£ˆÏúñ £yÁÎ˚y Ñ˛ú!›˛ˆÏì˛ ¢Ñ˛ˆÏúÓ˚ xÓfiÌyò ≤ÃyÌ!õÑ˛ xÓfiÌyˆÏòÓ˚ õˆÏì˛y £ˆÏÎ˚ ÎyˆÏÓ– ì˛y£ˆÏú

~Ñ˛ÓyÓ˚ ˛ô% ÏÓ˚y ˆáyÓ˚y £ˆÏúñ 4!›˛ xÓfiÌy Ïò ˆÓœv˛Ü%!ú ÌyÑ˛ Ïú ≤ÃyÌ!õÑ˛ xÓfiÌy ÏòÓ˚ õ Ïì˛y ˆîÖ Ïì˛

úyÜˆÏÓ– ì˛y£ˆÏú Óúy ÎyÎ˚ !â˛e 6.68ÈÙÈ~Ó˚ ˆÖúòy £yÁÎ˚y Ñ˛ú!›˛Ó˚ á)í≈ò ˛≤Ã!ì˛¢yˆÏõƒÓ˚ õyey 4–

ˆÎ !Ó®% ˆÑ˛w Ñ˛ˆÏÓ˚ £yÁÎ˚y Ñ˛ú!›˛ á%Ó˚ˆÏäÈ ˆ¢£z !Ó®%!›˛ £ú á)í≈ò ≤Ã!ì˛¢yˆÏõƒÓ˚ ˆÑ˛w–

ò#ˆÏâ˛Ó˚ !â˛eÜ%!úÓ˚ á)í≈ò ≤Ã!ì˛¢yõƒ xyˆÏäÈ !Ñ˛òy ˆîÖ%òñ Î!î ÌyˆÏÑ˛ ì˛y£ˆÏú á)í≈ò ≤Ã!ì˛¢yˆÏõƒÓ˚

õyey !òí≈Î˚ Ñ˛Ó%ò–

H N(a) (b) (c) (d) (e)

!â˛e 6.69

ˆòy›˛

70 D.El.Ed. 504


!ò ÏãÓ˚ ≤ÃÜ!ì˛ õy˛ôy

E6 ü)òƒfiÌyò ˛ô)Ó˚í ı

a) î%!›˛ ˆÓ˚ÖyÇü ¢Ó≈ m¢õ Î!î ì˛yˆÏîÓ ________ ¢õyò £Î˚–

b) î%!›˛ ˆÑ˛yí ¢Ó≈¢õ Î!î ì˛y ÏîÓ˚ ________ ¢õyò £Î˚–

c) Î!î î%!›˛ !e¶% ã ¢Ó≈¢õ £Î˚ ì˛y£ Ïú ì˛y ÏîÓ˚ ________ ˆÑ˛y ÏíÓ˚ õyò ¢õyò–

d) î%!›˛ ≤Ã!ì˛¢õ !e¶% ˆÏãÓ ________ ¢õyò–

E7 ∆ABC Á ∆PQR-~ñ AB = 4 cm, BC = 7 cm, PQ = 6 cm, QR = 10.5 cm

~ÓÇ m < B = m < Q.

i) PQ ~Ó˚ õyò !òí≈Î˚ Ñ˛Ó˚%òñ ˆÎÖyˆÏò AC = 8 cm

ii) ∆ABC Á ∆PQR ~Ó˚ ˆ«˛eö˛ˆÏúÓ˚ xò%˛ôyì˛ !òí≈Î˚ Ñ˛Ó˚%ò–

6.4 !eõy!eÑ˛ Óhfl%

xyõÓ˚y ˛ô)ˆÏÓ≈£z !eõy!eÑ˛ Óhfl% Ó˚ Ñ˛Ìy xyˆÏúyâ˛òy Ñ˛ˆÏÓ˚!äÈ– xyõÓ˚y ˆîˆÏÖ!äÈ ˆÎ ¢õhfl˛Ï Óhfl%

!ì˛ò!îˆÏÑ˛ !Óhfl˛,Ïì˛ ˆÎõò Óyõ!îÑ˛ ˆÌˆÏÑ˛ v˛yò!îÑ˛ Ñ˛yäÈ ˆÌˆÏÑ˛ î)ˆÏÓ˚ ~ÓÇ v˛z˛ôÓ˚ ˆÌˆÏÑ˛ ò#ˆÏâ˛ñ

ˆ¢Ü%!ú£z !eõy!eÑ˛ Óhfl% –

ˆÎ ¢õhfl˛Ï Óhfl% ˛ôÓ˚fl˛ôÓ˚ ¢õˆÏÑ˛yí ÓÓ˚yÓÓ˚ !ì˛ò!îˆÏÑ˛ !Óhfl ,Ïì˛ ˆ¢Ü%!ú 3D Óhfl%˛–

!Ó!¶˛ß¨ ïÓ˚ ÏòÓ˚ !eõy!eÑ˛ Óhfl%˛/áòÓhfl%

a) xyÎ˚ì˛áò ı

~Ñ˛!›˛ Óy:ñ £zˆÏ›˛Ó˚ ›%˛Ñ˛ˆÏÓ˚y Óy ~Ü%!úÓ˚ xyÑ,˛!ì˛Ó˚ òƒyÎ˚ Óhfl%˛ xyÎ˚ì˛áò òyˆÏõ ˛ô!Ó˚!â˛ì˛–

!â˛e 6.70ÈÙÈˆì˛ ~Ñ˛!›˛ xyÎ˚ì˛áò ˆîÖy Ïòy £ ÏÎ˚ ÏäÈñ ÎyÓ˚ 8!›˛ ü#°Ï≈!Ó®% (A, B, C, D, E,

F, G, H), 12!›˛ ≤Ãyhs˘˛ ( AB, BC, CD, DA, EF, FG, GH, HE, AH, BE, CF ~ÓÇ

DG V ~ÓÇ 6!›˛ ì˛ú (ABCD, EFGH, ABEH, BEFC, CFGH ~ÓÇ AHGD)

BC, EF, HG ~ÓÇ AD ≤ÃˆÏì˛ƒˆÏÑ˛ xyÎ˚ì˛áˆÏòÓ˚ ˜îá≈ƒ (l) ˆÓyG˛yÎ˚– AB, EH ~ÓÇ FG

~!›˛Ó˚ ≤ÃfiÌ (b) ˆÓyG˛yÎ˚ ~ÓÇ AH, BE, CF ~ÓÇ DG v˛zFâ˛ì˛y (h) ˆÓyG˛yÎ˚–

!â˛e 6.70

A

E

B

F

G

D

xyÎ˚ì˛áò

ˆòy›˛



xyÎ˚ì˛áˆÏòÓ˚ ì˛ˆÏúÓ˚ ˆõy›˛ ˆ«˛eö˛ú = v z ôÓ˚ ì˛ ÏúÓ˚ ò# Ïâ˛Ó˚ ì˛ ÏúÓ˚ ˆ«˛eö˛ú + Óyõ!îˆÏÑ˛Ó˚

Á v˛yò!î ÏÑ˛Ó˚ ì˛ ÏúÓ˚ ˆ«˛eö˛ú + ¢yõ ÏòÓ˚ Á !˛ôäÈ ÏòÓ˚ ì˛ ÏúÓ˚ ˆ«˛eö˛ú = 2 l × b + 2b ×

h + 2l × h = 2 ( l b + bh + l h) ÓÜ≈ ~Ñ˛Ñ˛–

xyÎ˚ì˛áˆÏòÓ˚ xyÎ˚ì˛ò £ú ~!›˛ ˆÎ fiÌyò x!ïÑ˛yÓ˚ Ñ˛ˆÏÓ˚ ÌyˆÏÑ˛ñ xyÎ˚ì˛áˆÏòÓ˚ ˜îá≈ƒñ ≤ÃfiÌ

Á v zFâ˛ì˛yÓ˚ Ü%íö˛ú–

xì˛~Óñ V = l × b × h áò ~Ñ˛Ñ˛

(b) áòÑ˛ ı

ˆÎ xyÎ˚ì˛á ÏòÓ˚ ˜îá≈ƒñ ≤ÃfiÌ Á v zFâ˛ì˛y ¢õyò ì˛y £ú áòÑ˛–

xì˛~Óñ ~Ó˚ ≤Ã!ì˛!›˛ ≤Ãyhs˘˛ ¢õyò ~ÓÇ ≤Ã!ì˛!›˛ ì˛ú ÓÜ≈yÑ˛yÓ˚–

xyÎ˚ì˛ò õy˛ôyÓ˚ ~Ñ˛Ñ˛ ı

!eõy!eÑ˛ Óhfl%˛Ó˚ xyÎ˚ì˛ò õy˛ôyÓ˚ ~Ñ˛Ñ˛ £úñ ~Ñ˛ áò ~Ñ˛ˆÏÑ˛Ó˚ xyÎ˚ì˛òñ xÌ≈yÍ ˆÎ áòˆÏÑ˛Ó˚

≤Ã!ì˛!›˛ ïy Ïâ˛Ó˚ ˜îá≈ƒ ~Ñ˛ ~Ñ˛Ñ˛–

Î!î áòÈÙÈ~Ñ˛ ÏÑ˛Ó˚ ≤Ã!ì˛!›˛ ïy ÏÓ˚Ó˚ ˜îá≈ƒ 1cm £Î˚ñ ì˛y£ Ïú áò ~Ñ˛ ÏÑ˛Ó˚ xyÎ˚ì˛ò 1cm3–

!â˛e ı 6.72

~Ñ˛ áòÈÙÈ~Ñ˛Ñ˛ ÎyÓ˚ ≤Ã!ì˛ Óy£%Ó˚ ˜îá≈ƒ 1cm, ì˛y £ú áòÈÙÈˆ¢!õ– ~Ñ˛ áòÈÙÈ~Ñ˛Ñ˛ ÎyÓ˚ ≤Ã!ì˛

Óy£%Ó˚ ˜îá≈ƒ 1m, ì˛y £ú áòÈÙÈ!õ›˛yÓ˚– áòÈÙÈ!õ›˛yÓ˚ ~Ó˚ !â˛ˆÏe Üyv˛¸ Ó˚ˆÏàÓ˚ xÇü!›˛ ~Ñ˛!›˛ áòÈÙÈˆ¢!õñ

xÌ≈yÍ ~Ó˚ ≤Ã!ì˛ Óy£%Ó˚ ˜îá≈ƒ 1 ˆ¢!õ– xÌ≈yÍ áòÈÙÈ!õ›˛yˆÏÓ˚Ó˚ áòÑ˛!›˛ˆÏÑ˛ Î!î xyõÓ˚y áòÈÙÈˆ¢!õÓ˚

áòˆÏÑ˛ ¶˛y!à ì˛y£ˆÏú xyõÓ˚y 100!›˛ áòÑ˛ ˛ôyÓ AB ≤ÃyˆÏhs˘˛ñ 100!›˛ áòÑ˛ ˛ôyì˛ AC ≤ÃyˆÏhs˘˛

~ÓÇ 100!›˛ áòÑ˛ ˛ôyÓ AD ≤Ãy Ïhs˘˛– ¢%ì˛Ó˚yÇ ˆõy›˛ ˆ¢!õ3 ˛ôyÁÎ˚y Îy ÏÓ 100 × 100 × 100

= 10.00,000– ì˛y£ Ïú 1m3 = 10,00,000 cm3 = 106cm3– 1cm3, 1m3 ~Ü%!ú xyÎ˚ì˛ ÏòÓ˚

áò ~Ñ˛Ñ˛–

!â˛e ı 6.72

1cm

1cm

[Unit -cube

Volume = 1 cm3]

1cm

1 m

A

D

B

1 m

1 m

ˆòy›˛

72 D.El.Ed. 504


ˆòy›˛ ı

(i) ÓÜ≈ Ï¢!õ Á ÓÜ≈!õ›˛y ÏÓ˚Ó˚ ˆ«˛eö˛ú !òí≈Î˚ Ñ˛Ó˚y ÎyÎ˚ òy– ~Ü%!ú ÎÌyÑ Ïõ 1cm2 Á 1m2

!£¢yˆÏÓ ÓƒyÖƒy Ñ˛Ó˚y £Î˚–

(ii) ~Ñ˛!›˛ áòÈÙÈˆ¢!õ Á ~Ñ˛!›˛ áòÈÙÈ!õ›˛y ÏÓ˚Ó˚ ÈÙÈÈÙÈÈÙÈ ÎÌyÑ Ïõ 1cm3 ~ÓÇ 1m3 !£¢y ÏÓ ≤ÃÑ˛yü

Ñ˛Ó˚y £Î˚–

(iii) 1cm2 £ú ˆ«˛eö˛ˆÏúÓ˚ ~Ñ˛Ñ˛ñ !Ñ˛hs%˛ áòÈÙÈˆ¢!õ ≤Ã!ì˛ Óy£%Ó˚ ˜îá≈ƒ £ú 1 ˆ¢!õ–

¢%ì˛Ó˚yÇ 1cm2 ~ÓÇ áòÈÙÈˆ¢!õ î%!›˛ xyúyîy !Ó°ÏÎ˚ ˆÓyG˛yÎ˚–

áòÈÙÈˆ¢!õ ˆ«˛eö˛ú = 1cm2

~Ñ˛£z¶˛yˆÏÓ 1m3 ~ÓÇ áòÈÙÈ!õ›˛yÓ˚ xyúyîy–

~Ñ˛ áòÈÙÈ!õ›˛yˆÏÓ˚Ó˚ xyÎ˚ì˛ò = 1m3–

(b) !≤Ããõ ı

!eõy!eÑ˛ !â˛ ÏeÓ˚ (6.73(i)) ˜Ó!üT˛ƒ

!â˛e ı 6.73

~!›˛Ó˚ 2!›˛ !e¶%˛ãyÑ,˛!ì˛ ì˛ú xyˆÏäÈ– S~Ñ˛!›˛ v˛z˛ôˆÏÓ˚ Á x˛ôÓ˚!›˛ ò#ˆÏâ˛V– ¢yïyÓ˚íì˛ xyõÓ˚y

~Ü%!úˆÏÑ˛ ¶) !õ ÓˆÏú Ìy!Ñ˛–

~!›˛Ó˚ 3!›˛ xyÎ˚ì˛yÑ˛yÓ˚ ì˛ú xyˆÏäÈñ ˆÎÜ%!ú !e¶% ãyÑ, !ì˛ ì˛ˆÏúÓ˚ ¢ˆÏAÜ ú¡∫¶˛yˆÏÓ xÓfiÌyò Ñ˛ˆÏÓ˚ñ

~£z ì˛úÜ%!ú ÏÑ˛ ˛ôyŸª≈#Î˚ì˛ú Ó Ïú–

î%!›˛ !e¶% ãyÑ, !ì˛ ì˛ˆÏúÓ˚ õˆÏïƒÓ˚ î)Ó˚cˆÏÑ˛ !≤ÃãˆÏõÓ˚ v zFâ˛ì˛y ÓˆÏú– S!≤ÃãõˆÏÑ˛ xò%¶) !õÑ˛ xÓfiÌyˆÏò

Ó˚yÖˆÏú ~!›˛ˆÏÑ˛ ˜îá≈ƒÁ Óúy £ˆÏÎ˚ ÌyˆÏÑ˛V–

!â˛e 6.73(II) ÈÙÈ~Ó˚ !≤Ããõ!›˛Ó˚ ¶) !õ î%!›˛ £ú °Ïv˛¸ ÏÑ˛yí#–

!≤Ãã ÏõÓ˚ ˛ô,T˛ì˛ ÏúÓ˚ ˆ«˛eö˛ú ı

A

B C

D

E F

ˆòy›˛

73504-Block2: xyÑ, !ì˛ Á fiÌyò ¢ÇÑ yhs˘˛ ïyÓ˚íy


˛ôyŸª≈#Î˚ì˛ ÏúÓ˚ ˆ«˛eö˛ú = AB × h × BC + h + CA × h áò ~Ñ˛Ñ˛ S~Öy Ïò h Óú Ïì˛ v zFâ˛ì˛y

ˆÓyG˛yˆÏFäÈV

= h (AB + BC + CA) ÓÜ≈~Ñ˛Ñ˛

= h × ¶) !õÓ˚ ˛ô!Ó˚¢#õy

≤Ãyhs˘˛î%!›˛Ó˚ ˆ«˛eö˛ú = 2 × ≤ÃyˆÏhs˘˛Ó˚ ˆ«˛eö˛ú

!≤ÃãˆÏõÓ˚ ˆõy›˛ ˆ«˛eö˛ú = ˛ôyŸª≈#Î˚ ì˛ˆÏúÓ˚ ˆ«˛eö˛ú + ≤Ãyhs˘˛mˆÏÎ˚Ó˚ ˆ«˛eö˛ú

!≤ÃãˆÏõÓ˚ xyÎ˚ì˛ò = ¶)˛!õÓ˚ ˆ«˛eö˛ú × v˛zFâ˛ì˛y

ˆòy›˛ ı ~£z !òÎ˚ Ïõ Ó£%¶% ã ¶) !õ Ïì˛ xy ÏäÈ SˆÎ ˆÑ˛y Ïòy ¢ÇÖƒÑ˛V ~õò !≤Ãã ÏõÓ˚ ˆ«˛ Ïe ˛≤Ã ÏÎyãƒ–

(c) ˆâ˛yà ı ˆâ˛yˆÏàÓ˚ õì˛ ãƒy!õ!ì˛Ñ˛ xyÑ˛yÓ˚ xyõÓ˚y ÜyˆÏäÈÓ˚ Ü%§!v˛¸ˆÏì˛ ˆîÖˆÏì˛ ˛ôy£z–

!â˛e ı 6.74

ˆâ˛yˆÏàÓ˚ ¶)˛!õ î%!›˛ˆÏì˛ î%!›˛ Ó,_ ÌyˆÏÑ˛– ¶)˛!õ äÈyv˛¸yÁ ~Ó˚ ˛ô,¤˛ì˛ú ÓÑ –

î%!›˛ ¶)˛!õÓ˚ õˆÏïƒÓ˚ î)Ó˚c £ú ˆâ˛yˆÏàÓ˚ v zFâ˛ì˛y SÓy ˜îá≈ƒVñ Îy h !£¢y ÏÓ ≤ÃÑ˛yü Ñ˛Ó˚y £Î˚–

ˆâ˛y ÏàÓ˚ ˛ô,¤˛ì˛ ÏúÓ˚ ˆ«˛eö˛ú ı

ÓÑ ˛ ˛ô,¤˛ì˛ˆÏúÓ˚ ˆ«˛eö˛ú ˆâ˛yˆÏàÓ˚ ˛ô,¤˛ì˛ˆÏúÓ˚ ˆ«˛eö˛ú ~Ñ˛!›˛ xyÎ˚ì˛yÑ˛yÓ˚ Ñ˛yÜˆÏãÓ˚ ˆ«˛eö˛ˆÏúÓ˚

¢õyòñ ˆÎ!›˛ ˆâ˛yà!›˛ˆÏÑ˛ ˛ô%ˆÏÓ˚y !áˆÏÓ˚ Ó˚yÖˆÏì˛ ˛ôyˆÏÓ˚– !â˛e 6.75 (b) ˆì˛ ~Ó˚Ñ˛õ£z ~Ñ˛!›˛ Ñ˛yÜ ÏãÓ˚

!â˛e ˆîÖyˆÏòy xyˆÏäÈ–

!â˛e ı 6.75

xyÎ˚ì˛yÑ˛yÓ˚ Ñ˛yÜˆÏãÓ˚ ˜îá≈ƒ (l)

= ¶)˛!õÓ˚ Ó,ˆÏ_Ó˚ ˛ô!Ó˚¢#õy = 2πr ~ÓÇ Ñ˛yÜ ÏãÓ˚ ≤ÃfiÌ (b) ˆâ˛yˆÏàÓ˚ v˛zFâ˛ì˛y = h

A B

D C

ˆòy›˛

74 D.El.Ed. 504


Ñ˛yÜ ÏãÓ˚ ˆ«˛eö˛ú = l × b = 2πr × h = 2πr ÓÜ≈~Ñ˛Ñ˛–

ÓÑ ˛ô,¤˛ì˛ ÏúÓ˚ ˆ«˛eö˛ú = 2πrh ÓÜ≈~Ñ˛Ñ˛

ˆâ˛yˆÏàÓ˚ xyÎ˚ì˛ò = ¶) !õÓ˚ ˆ«˛eö˛ú × v zFâ˛ì˛y = πr2h áò~Ñ˛Ñ˛–

ˆòy›˛ ı ˆâ˛yà £ú Ó,_yÑ˛yÓ˚ ¶) !õÎ%=˛ !≤Ããõ–

(d) !˛ôÓ˚y!õv˛/â˛ì%˛hfl˛ÏÓÑ˛ ı

!õüˆÏÓ˚Ó˚ ö˛yÓ˚yÁˆÏîÓ˚ ¢õy!ï £ú !˛ôÓ˚y!õv˛– !˛ôÓ˚y!õˆÏv˛Ó˚ ¶) !õ £ú !e¶) ãyÑ˛yÓ˚ Óy Ó£%¶% ã

~ÓÇ v˛z˛ôˆÏÓ˚ ~!›˛ ~Ñ˛!›˛ !Ó®%ˆÏì˛ !õ!úì˛ £Î˚– ò#ˆÏâ˛ ~Ó˚Ñ˛õ !Ñ˛ä%È xyÑ˛yÓ˚ ˆîÖyˆÏòy £ú–

!â˛e ı 6.76

!e¶% ãyÑ˛yÓ˚ ¶) !õÎ%=˛ !˛ôÓ˚y!õˆÏv˛Ó˚ 4!›˛ ü#°Ï≈!Ó®%ñ 6!› ≤Ãyhs˘˛ñ ~ÓÇ 4!›˛ ì˛ú ÌyˆÏÑ˛ñ ˆÎÜ%!ú

≤Ã!ì˛!›˛ !e¶%˛ãyÑ,˛!ì˛–

â˛ì%˛¶≈%˛ãyÑ˛yÓ˚ ¶)˛!õÎ%=˛ !˛ôÓ˚y!õˆÏv˛Ó˚ 5!›˛ ü#°Ï≈!Ó®%ñ 8!›˛ ≤Ãyhs˘˛ ~ÓÇ 5!›˛ ì˛ú ÌyˆÏÑ˛ñ ÎyÓ˚ õˆÏïƒ

â˛yú% ì˛úÜ%!ú !e¶% ãyÑ, !ì˛ ~ÓÇ ¶) !õ!›˛ â˛ì% ¶≈% ãyÑ˛yÓ˚–

!˛ôÓ˚y!õˆÏv˛Ó˚ ì˛ˆÏúÓ˚ ˆ«˛eö˛ú = â˛yú% ì˛ˆÏúÓ˚ ˆ«˛eö˛ú + ¶) !õÓ˚ ˆ«˛eö˛ú

!˛ôÓ˚y!õˆÏv˛Ó˚ xyÎ˚ì˛ò = 1

×3

¶) !õÓ˚ ˆ«˛eö˛ú × v zFâ˛ì˛y–

(e) üAÑ% ı

ˆãyÑ˛yˆÏÓ˚Ó˚ õyÌyÓ˚ ›%˛!˛ôñ ö˛yˆÏòˆÏúÓ˚ õ%Ö üAÑ%˛ xyÑ,˛!ì˛Ó˚ £ˆÏÎ˚ ÌyˆÏÑ˛– üAÑ%˛Ó˚ ~Ñ˛!›˛ ü#°Ï≈!Ó®%ñ

~Ñ˛!›˛ Ó,_yÑ˛yÓ˚ ïyÓ˚ñ î%!›˛ ì˛ú xyˆÏäÈ ÎyÓ˚ ~Ñ˛!›˛ ÓÑ ˛yÑ˛yÓ˚ x˛ôÓ˚!›˛ Ó,_yÑ˛yÓ˚ Á â˛ƒy≤Wzy–

!â˛e ı 6.77

[Cone]

hl

r

[Inverted Cone]

r

hl

ˆòy›˛



ˆõy›˛ ˆ«˛eö˛ú = ÓÑ ì˛ ÏúÓ˚ ˆ«˛eö˛ú + Ó,_yÑ˛yÓ˚ ¶) !õÓ˚ ˆ«˛eö˛ú

2 2 2 2 2( )r h r r r h r r= π + + π = π + +

[ˆÎÖyˆÏò r £ú ¶) !õÓ˚ Ó, Ï_Ó˚ Óƒy¢yï≈ñ h £ú üAÑ%˛Ó˚ v˛zFâ˛ì˛y]

â˛y ÏúÓ˚ v zFâ˛ì˛y (l) 2 2h r= +

(f) ˆÜyúÑ˛ ı

ö%˛›˛ÓˆÏúÓ˚ xyÑ,˛!ì˛ ˆÜyúˆÏÑ˛Ó˚ òƒyÎ˚ñ ~Ó˚ ˆÑ˛yˆÏòy ü#°Ï≈!Ó®% Óy ≤Ãyhs˘˛ !Ó®%

ˆò£zñ !Ñ˛hs%˘˛ ~Ñ˛!›˛ ÓÑ ˛ì˛ú xyˆÏäÈ–

ˆÜyúÑ˛ˆÏÓ˚ ˆ«˛eö˛ú = 4πr2

ˆÜyúˆÏÑ˛Ó˚ xyÎ˚ì˛ò 34

3r= π (r £ú ˆÜyú ÏÑ˛Ó˚ Óƒy¢yï≈ ~ÓÇ ÈπÙÈ~Ó˚

õyò £ú 22

7 Óy 3.14)

ˆÖyúy xyÑ˛yÓ˚ Sˆò›˛V ˆÌ ÏÑ˛ áòÓhfl% /!eõy!eÑ˛ Óhfl%˛ Óyòy Ïòy

ˆò›˛ £ú !mõy!eÑ˛ ì˛ˆÏúÓ˚ ~Ñ˛!›˛ Ó˚)˛ôˆÏÓ˚Öyñ Îy ¶§˛yã Ñ˛Ó˚ˆÏú !eõy!eÑ˛ xyÑ,˛!ì˛ ˜ì˛!Ó˚ £Î˚– ˆõy›˛y

Ñ˛yÜˆÏã Sxy›≈˛ ˆ˛ô˛ôyÓ˚V ~Ñ˛›˛y úy£zò ˆ›˛ˆÏò ~Ñ˛›˛y Ó˚)˛ôˆÏÓ˚Öy Óy!òˆÏÎ ì˛y ÓÓ˚yÓÓ˚ ˆÑ˛ˆÏ›˛ ÎÌyÎÌ¶˛yˆÏÓ

¶§˛yã Ñ˛ˆÏÓ˚ !eõy!eÑ˛ xyÑ,˛!ì˛ ÓyòyˆÏòy ÎyÎ˚–

(i) 5 ˆ¢!õ áòÑ˛ Óyòy Ïòy ı

v˛z˛ôˆÏÓ˚Ó˚ ˆò›˛!›˛ˆÏì˛ ≤Ã!ì˛!›˛ ˆÑ˛yí£z ¢õˆÏÑ˛yò xyÓ˚ ≤Ãyhs˘˛ Óy ïyˆÏÓ˚Ó˚ õy˛ô ˆîÁÎ˚y xyˆÏäÈ–

(ii) 3.5 ˆ¢!õ Óƒy¢yˆÏï≈Ó˚ 310 ˆ¢!õ v zFâ˛ì˛y ¢¡ôß¨ ˆâ˛yà Óyòy Ïòy ı

!â˛e ı 6.77

5

5

5

55

55

10

5 5

5

5

5

10

10

5

7

222 ×

a a× 3.5 = 22 cm

10 cm

ˆòy›˛

76 D.El.Ed. 504


a Á b î%!›˛ Ó,_ ÎyˆÏîÓ˚ Óƒy¢yï≈ 3.5 cm, Óy!Ñ˛ xÇü xyÎ˚ì˛ˆÏ«˛e ÎyÓ˚ õy˛ô äÈ!ÓˆÏì˛ ˆîÁÎ˚y

xyˆÏäÈ–

!ò ÏãÓ˚ ≤ÃÜ!ì˛ õy˛ôy ı

E9. ~Ñ˛›˛y Ñ˛yˆÏ‡˛Ó˚ xyÎ˚ì˛áòñ 175ˆ¢!õ ˜îá≈ƒ!Ó!üT˛ 105 ˆ¢!õ ≤ÃfiÌ!Ó!üT˛ ~ÓÇ 63 ˆ¢!õ

v˛zFâ˛ì˛y !Ó!üT˛ ˆÌˆÏÑ˛ ˆÑ˛ˆÏ›˛ ¢ÓˆÏâ˛ˆÏÎ˚ ÓˆÏv˛¸y áòÑ˛ ÓyÓ˚ Ñ˛Ó˚ˆÏì˛ ˆÜˆÏú ì˛yÓ˚ ≤Ã!ì˛ ïyˆÏÓ˚Ó˚ Óy ≤ÃyˆÏhs˘˛Ó˚

õy˛ô Ñ˛ì˛ £ ÏÓ⁄

E10. ~Ñ˛!›˛ 33 ˆ¢!õ ˜îá≈ƒ!Ó!üT˛ Á 22ˆ¢!õ ≤ÃfiÌ !Ó!üT˛ xyÎ˚ì˛áòyÑ˛yÓ˚ Ñ˛yÜ ÏãÓ˚ ›% Ñ˛ ÏÓ˚y

ˆÌˆÏÑ˛ î%ˆÏ›˛y ˆâ˛yà ÓyòyˆÏòy ì˛yˆÏîÓ˚ xyÎ˚ì˛ˆÏòÓ˚ xò%˛ôyì˛ Ñ˛# £ˆÏÓ⁄

6.5 ãƒy!õ!ì˛ Óy Ï:Ó v z ôyîyò ÓƒÓ£yÓ Ñ˛ ÏÓ ãƒy!õ!ì˛Ñ˛ !â˛e xAÑ˛ò

ãƒy!õ!ì˛ Óy Ï:Ó v z ôyîyò £ú ˆflÒúñ !v˛¶˛y£zv˛yÓñ Ñ˛¡ôy¢ñ ˆ¢›˛ˆÏflÒyÎ˚yÓ˚ S~Ñ˛!›˛ 45°– 45°

ˆ¢›˛ˆÏflÒyÎ˚yÓ˚ñ x˛ôÓ˚!›˛ 60°– 30° ˆ¢›˛ ÏflÒyÎ˚yÎ˚V– ÎÖò xyõÓ˚y ˆflÒ ÏúÓ˚ ïyÓ˚ ü%ï%õye ¢Ó˚ú ÏÓ˚Öy

›˛yòyÓ˚ Ñ˛yˆÏã ÓƒÓ£yÓ˚ Ñ˛Ó˚Ó ì˛Öò ˆ¢!›˛ Ó˚%úyÓ˚– Ó˚%úyÓ˚ £ú ¢Ó˚úˆÏÓ˚ÖyÎ%=˛ xyÎ˚ì˛yÑ˛yÓ˚ ›% Ñ˛ˆÏÓ˚y–

Ó%úyÓ˚ Á Ñ˛¡ôy Ï¢Ó˚ ÓƒÓ£yÓ˚ ı

~Ñ˛¢õÎ˚ Ü!íì˛K˛Ó˚y ˆ¶˛ˆÏÓ!äÈˆÏúò Ó%úyÓ˚ Á Ñ˛¡ôyˆÏ¢Ó˚ ¢y£yˆÏÎƒ !Ó!¶˛ß¨ ˆõÔ!úÑ˛ ãƒy!õ!ì˛Ñ˛ !â˛e

ÓyòyˆÏòy ¢Ω˛Ó– !ò¡¨!ú!Öì˛ ãƒy!õ!ì˛Ñ˛ !â˛eÜ%!ú Ó%úyÓ˚ Á Ñ˛¡ôyˆÏ¢Ó˚ ¢y£yˆÏÎƒ x§yÑ˛y ÎyÎ˚–

(i) !ò!î≈T˛ ˜î Ïá≈ƒÓ˚ ¢Ó˚ú ÏÓ˚ÖyÇü xAÑ˛ò

(ii) !ò!î≈T˛ ˜î Ïá≈ƒÓ˚ ¢Ó˚ú ÏÓ˚ÖyÇ ÏüÓ˚ ú¡∫ÈÙÈ¢õ!mÖ[˛Ñ˛ xAÑ˛ò

(iii) ~Ñ˛!›˛ ¢Ó˚úˆÏÓ˚ÖyÓ˚ ¢õyhs˘˛Ó˚yú/ú¡∫ ¢Ó˚ú ÏÓ˚Öy xAÑ˛òñ

(a) ¢Ó˚úˆÏÓ˚ÖyÓ˚ ˆÑ˛yˆÏòy !ò!î≈T˛ !Ó®% Ïì˛

(b) ¢Ó˚úˆÏÓ˚ÖyÓ˚ Óy£z ÏÓ˚Ó˚ ˆÑ˛yˆÏòy !ò!î≈T˛ !Ó®% ˆÌˆÏÑ˛

(iv) !ò!î≈T˛ ˆÑ˛yˆÏíÓ˚ õyˆÏ˛ôÓ˚ ¢õyò ˆÑ˛yí xAÑ˛ò

(v) !ò!î≈T˛ ˆÑ˛y ÏíÓ˚ ¢õ!mÖ[˛Ñ˛ xAÑ˛ò

(vi) 60° ˆÑ˛yí ì˛yÓ˚ Ü%!íì˛Ñ˛ Á v˛z˛ôÜ%!íì˛Ñ˛ S30°, 45°, 15° ≤Ã¶,˛!ì˛ ˆÑ˛yí xAÑ˛òV

(vii) ≤Ãî_ ˆÓ˚ÖyÇüˆÏÑ˛ ¢õ xÇˆÏü ˆäÈyˆÏ›˛y ˆäÈyˆÏ›˛y ÖˆÏ[˛ !Ó¶˛=˛ Ñ˛Ó˚y–

(viii) !ò!î≈T˛ õyˆÏ˛ôÓ˚ !e¶%˛ãñ â˛ì%˛¶≈%˛ã Á Ó,_ xAÑ˛ò–

xyõÓ˚y ¢Ñ˛ˆÏú fl%ÒˆÏú ˛ôv˛¸yÓ˚ ¢õÎ˚ ~£z xAÑ˛òÜ%!ú Ñ˛ˆÏÓ˚!äÈúyõ– ~Ü%!ú ˛ô%òÓ˚yÎ˚ Óy!òˆÏÎ˚ ˆòÁÎ˚yÓ˚

ãòƒ ~Öy Ïò ¢Ç!«˛Æ¶˛y ÏÓ ˛ôÂï!ì˛Ü%!ú ˆîÁÎ˚y £ú–

ˆòy›˛



(i) !ò!î≈T˛ ˜î Ïá≈ƒÓ˚ ¢Ó˚ú ÏÓ˚ÖyÇü xAÑ˛ò ı

(a) ˆflÒ ÏúÓ˚ ˆ¢yãy ˛≤Ãyhs˘˛ ÓÓ˚yÓÓ˚ ~Ñ˛!›˛ ¢Ó˚ú ÏÓ˚Öy ›˛yòy £ú [!â˛e ı 6.79a]

(b) Ñ˛¡ôy Ï¢ !ò!î≈T˛ ˜îá≈ƒ Óƒy¢yï≈ !£¢y ÏÓ ˆõ Ï ô !ò ÏÎ˚ A !Ó®% ÏÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ñ ¢Ó˚úˆÏÓ˚Öy!›˛Ó˚

v˛z˛ôÓ˚ ~Ñ˛!›˛ Ó,_â˛y˛ô ˆÑ˛ˆÏ›˛ ˆòÁÎ˚y £úñ Îy ¢Ó˚úˆÏÓ˚Öy!›˛ˆÏÑ˛ B !Ó®%ˆÏì˛ ˆäÈî Ñ˛Ó˚ú–

[!â˛e ı 6.79(b)]

(a) (b)

!â˛e ı 6.79

(c) AB £ú !ò!î≈T˛ ˜î Ïá≈ƒÓ˚ ¢Ó˚ú ÏÓ˚ÖyÇü–

(ii) AB ¢Ó˚ú ÏÓ˚ÖyÇ ÏüÓ˚ ú¡∫ÈÙÈ¢õ!mÖ[˛Ñ˛ xAÑ˛ò–

ïy˛ô 1 ı Ñ˛¡ôyˆÏ¢ ¢Ó˚úˆÏÓ˚ÖyÇˆÏüÓ˚ xˆÏï≈ˆÏÑ˛Ó˚ ˆÌˆÏÑ˛ ˆÓ!ü õyˆÏ˛ôÓ˚ Óƒy¢yï≈ ˆòÁÎ˚y £ú– AB

¢Ó˚úˆÏÓ˚ÖyÇˆÏ˘üÓ˚ A !Ó®%ˆÏÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ ¢Ó˚úˆÏÓ˚ÖyÓ˚ î%˛ôyˆÏü x§yÑ˛y £ú– [!â˛e ı

6.80]

ïy˛ô 2 ı B !Ó®%ˆÏÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ ¢õÓƒy¢yï≈ !òˆÏÎ˚ñ ≤ÃÌõ ïyˆÏ˛ôÓ˚ òƒyÎ˚ xyÓyÓ˚ î%!›˛ Ó,_â˛y˛ô x§yÑ˛y

£úñ ÎyÓ˚y xyˆÏÜÓ˚ Ó,_â˛y˛ô î%!›˛ˆÏÑ˛ ˆäÈî Ñ˛Ó˚ˆÏÓ–

!â˛e ı 6.80

ïy˛ô 3 ı !mì˛#Î˚ ïyˆÏ˛ô ˛ôyÁÎ˚y Ó,_â˛yˆÏ˛ôÓ˚ ˆäÈî!Ó®% î%!›˛ˆÏÑ˛ ˆÎyÜ Ñ˛ˆÏÓ˚ ~Ñ˛!›˛ ¢Ó˚úˆÏÓ˚Öy ›˛yòy

£ú– ¢Ó˚úˆÏÓ˚Öy!›˛ˆÏÑ˛ PQsuur

òyõ ˆîÁÎ˚y £ú– PQsuur

£ú AB ¢Ó˚úˆÏÓ˚ÖyÎ˚

ú¡∫ÈÙÈ¢õ!mÖ[˛Ñ˛ O !Ó®%!›˛ £ú PQsuur

Á AB Ó˚ ˆäÈî!Ó®% ~ÓÇ AB ¢Ó˚ú ÏÓ˚ÖyÓ˚

õïƒ!Ó®%–

˛ôÓ˚#«˛y Ñ˛ˆÏÓ˚ ˆîÖ%ò Î!î Ñ˛¡ôyˆÏ¢ ≤Ãî_ ¢Ó˚úˆÏÓ˚ÖyÓ˚ xˆÏï≈Ñ˛ Óy xˆÏï≈ˆÏÑ˛Ó˚ Ñ˛õ Óƒy¢yï≈ ˆòÁÎ˚y

£Î˚ ì˛y£ˆÏú Ñ˛# £ˆÏì˛ ˛ôyˆÏÓ˚–

A B

A B

O

Q

P

Q

ˆòy›˛

78 D.El.Ed. 504


(iii) (a) ~Ñ˛!›˛ ¢Ó˚ú ÏÓ˚ÖyÓ˚ !ò!î≈T˛ !Ó®% Ïì˛ ú¡∫ ¢Ó˚ú ÏÓ˚Öy xAÑ˛ò ı

AB Ó˚ P !Ó®% Ïì˛ ú¡∫ ¢Ó˚úˆÏÓ˚Öy xAÑ˛ò ı

ïy˛ô 1 ı P ˆÑ˛w Ñ˛ˆÏÓ˚ ~Ñ˛!›˛ Óƒy¢yï≈ !òˆÏÎ˚ ~õò Ó,_â˛y˛ô x§yÑ˛y £ú Îy AB ˆÑ˛ 2!›˛ !Ó®%ˆÏì˛

ˆäÈî Ñ˛ ÏÓ˚– !Ó®% î%!›˛Ó˚ òyõ ˆîÁÎ˚y £ú M Á N–

!â˛e ı 6.81

ïy˛ô 2 ı ≤ÃÌõ ïyˆÏ˛ô ˆòÁÎ˚y Óƒy¢yˆÏï≈Ó˚ ˆÌˆÏÑ˛ ˆÓ!ü Óƒy¢yï≈ !òˆÏÎ˚ MÁ N !Ó®% Ïì˛ Ó!¢ ÏÎ˚

~õò î%!›˛ Ó,_â˛y˛ô x§yÑ˛y £ú ÎyˆÏì˛ ì˛yÓ˚y AB Ó˚ ~Ñ˛£zïyˆÏÓ˚ ˛ôÓ˚fl˛ôÓ˚ˆÏÑ˛ ˆäÈî Ñ˛ˆÏÓ˚–

!Ó®%!›˛Ó˚ òyõ ˆîÁÎ˚y £ú K–

ïy˛ô 3 ı PKsuur

¢Ó˚ú ÏÓ˚Öy ›˛yòy £ú– PKsuur

£ú ˆ¢£z ¢Ó˚ú ÏÓ˚Öy ˆÎÖy Ïò–

(iii) (b) ~Ñ˛!›˛ ¢Ó˚ú ÏÓ˚ÖyÓ˚ Óy£z ÏÓ˚Ó˚ ˆÑ˛y Ïòy !Ó®% ˆÌ ÏÑ˛ ¢Ó˚ú ÏÓ˚ÖyÓ˚ v z ôÓ˚ ú¡∫ xAÑ˛ò ı

ABsuur Ó˚ ¢ˆÏAÜ ~Ñ˛!›˛ ú¡∫ AB

suur Ó˚ Óy£z ÏÓ˚ xÓ!fiÌì˛ P !Ó®% ˆÌ ÏÑ˛ ›˛yò Ïì˛ £ ÏÓ–

ABsuur £ú ≤Ãî_ ¢Ó˚ú ÏÓ˚Öy ~ÓÇ P !Ó®% AB

suur Ó˚ Óy£z ÏÓ˚ xÓ!fiÌì˛–

ïy˛ô 1 ı P ˆÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ ~Ñ˛!›˛ Óƒy¢yï≈ !òˆÏÎ˚ Ó,_â˛y˛ô ›˛yòy £úñ Îy ABsuur ˆÑ˛ î%!›˛ !Ó®% Ïì˛

ˆäÈî Ñ˛Ó˚ú– !Ó®%î%!›˛ ÏÑ˛ òyõ ˆîÁÎ˚y £ú M Á N–

!â˛e ı 6.82

ïy˛ô 2 ı MN Ó˚ xˆÏï≈ˆÏÑ˛Ó˚ ˆÓ!ü Óƒy¢yï≈ !òˆÏÎ˚ M Á N ˆÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ Ó,_â˛y˛ô x§yÑ˛y £ú

Îy ABsuur Ó˚ ~Ñ˛ ˛ôyˆÏü SP !Ó®% ˆÎ!îˆÏÑ˛ xÓfiÌyò Ñ˛Ó˚ˆÏäÈ ì˛yÓ˚ !Ó˛ôÓ˚#ì˛ !îˆÏÑ˛V

PA

K

M N

M N

P

K

ˆòy›˛



˛ôÓ˚fl˛ôÓ˚ˆÏÑ˛ ˆäÈî Ñ˛Ó˚ú– ˆäÈî!Ó®%!›˛Ó˚ òyõ ˆîÁÎ˚y £ú K1 P ˆÌˆÏÑ˛ k xÓ!ï

¢Ó˚ú ÏÓ˚Öy PKsuur

›˛yòy £ú–

PKsuur

£ú ABsuur Ó˚ v z˛ôÓ˚ ú¡∫ Îy P !Ó®% !îˆÏÎ˚ ÎyˆÏFäÈ–

(iii) (c) ~Ñ˛!›˛ ¢Óú ÏÓÖyÓ Óy£z ÏÓÓ ˆÑ˛y Ïòy !Ó®% ˆÌ ÏÑ˛ ˙ ¢Óú ÏÓÖyÓ ¢õyhs˘ Óyú ¢Óú ÏÓÖy

xAÑ˛ò ı

ABsuur ¢Ó˚ú ÏÓ˚ÖyÓ˚ ¢y ÏÌ P !Ó®% ˆÌ ÏÑ˛ ¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚Öy xy¢ Ïú AB

suur Ó˚ ¢ ÏAÜ

~Ñ˛!›˛ ¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚Öy RSsuur

›˛yòˆÏì˛ £ˆÏÓ–

MNsuuur

~Ñ˛!›˛ ¢Ó˚ú ÏÓ˚Öy P !Ó®% ˆÌˆÏÑ˛ ~õò¶˛yˆÏÓ ›˛yòy £ú ÎyˆÏì˛ ì˛y ABsuur

¢Ó˚ú ÏÓ˚Öy ÏÑ˛

T !Ó®% Ïì˛ ˆäÈî Ñ˛ˆÏÓ˚–

~Ñ˛yhs˘˛Ó˚ ˆÑ˛yímÎ˚ m SPT = m ATP

!â˛e ı 6.83 (a) !â˛e ı 6.83 (b)

!â˛e ı 6.83 (c)

ïy˛ô 1 ı MNsuuur

¢Ó˚ú ÏÓ˚Öy P !Ó®% ˆÌˆÏÑ˛ ~õò¶˛yˆÏÓ ›˛yòy £ú ÎyˆÏì˛ ì˛y ABsuur

ˆÑ˛ ˆÑ˛yò

!Ó®% Ïì˛ ˆäÈî Ñ˛ˆÏÓ˚– ˆäÈî!Ó®%!›˛Ó˚ òyõ ˆîÁÎ˚y £ú T–

ïy˛ô 2 ı ∠BTN ~Ó˚ ¢ˆÏAÜ ¢õyò õyˆÏ˛ôÓ˚ ~Ñ˛!›˛ ˆÑ˛yí PRsuur Ó˚ v˛z˛ôÓ˚ P !Ó®%ˆÏì˛ x§yÑ˛y £ú–

~£z x§yÑ˛yÓ˚ ãòƒ (iii) òÇ xAÑ˛ ÏòÓ˚ ˛ôÂï!ì˛ xÓú¡∫ò Ñ˛Ó˚ Ïì˛ £ ÏÓ–

B

R SP

T

N

M

A A B

M

T

P

N

R P Q

A T

N

B

ˆòy›˛

80 D.El.Ed. 504


xì˛~Óñ xyõÓ˚y P !Ó®% ˆÌˆÏÑ˛ ABsuur

Ó˚ ¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚Öy ˆ˛ôúyõ– RSsuur

òyõÑ˛ ¢Ó˚ú ÏÓ˚Öy!›˛£z

x§yÑ˛ Ïì˛ Óúy £ ÏÎ˚!äÈú–

~Ñ˛!›˛ ¢Ó˚ú ÏÓ˚ÖyÓ˚ ¢ ÏAÜ !ò!î≈T˛ î)Ó˚ Ïc ¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚Öy xAÑ˛ò–

ABsuur £ú ≤Ãî_ ¢Ó˚ú ÏÓ˚Öy– AB

suur Ó˚ ¢ ÏAÜ ¢õyhs˘˛Ó˚yú ¢Ó˚ú ÏÓ˚Öy ›˛yò Ïì˛ £ ÏÓ Îy ABsuur

Ó˚ ˆÌˆÏÑ˛ !ò!î≈T˛ S5 ˆ¢!õV î)Ó˚ Ïc xÓfiÌyò Ñ˛Ó˚ ÏÓ–

ïy˛ô 1 ı ABsuur Ó˚ v˛z˛ôÓ˚ ~Ñ˛!›˛ !Ó®% !òˆÏÎ˚ ì˛yÓ˚ òyõ ˆîÁÎ˚y £ú P– P !Ó®% ˆÌˆÏÑ˛ AB

suur Ó

¢ˆÏAÜ ~Ñ˛!›˛ ¢õyhs˘˛Ó˚yú ›˛yòy £ú Sv˛z˛ôˆÏÓ˚ Ó!í≈ì˛ ˛ôÂï!ì˛ xÓú¡∫ˆÏòV– ~!›˛Ó˚ òyõ

ˆîÁÎ˚y £ú PQsuur

–

(a) (b)

!â˛e ı 6.84

ïy˛ô 2 ı 5 ˆ¢!õ Óƒy¢yï≈ !òˆÏÎ˚ P ˆÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ ~Ñ˛!›˛ Ó,_â˛y˛ô ›˛yòy £ú ÎyˆÏÑ˛ PQsuur

ˆÑ˛ R

!Ó®% Ïì˛ ˆäÈî Ñ˛Ó˚ú–

ïy˛ô 3 ı ∠RPB Ó˚ ¢ ÏAÜ ¢õyòÑ˛ ÏÓ˚ñ R !Ó®% Ïì˛ RPsuur Ó˚ v˛z˛ôˆÏÓ˚

(iii) ˆì˛ Ó!í≈ì˛ ˛ôÂï!ì˛ xÓú¡∫ò Ñ˛ˆÏÓ˚ ~Ñ˛!›˛ ˆÑ˛yí ~õò¶˛yˆÏÓ x§yÑ˛y £úñ ÎyˆÏì˛ ì˛y ∠RPB Ó˚

!Ó˛ôÓ˚#ˆÏì˛ £Î˚– ˆÑ˛yˆÏíÓ˚ Óy£%Ü%!úˆÏÑ˛ Ó!ï≈ì˛ Ñ˛Ó˚y £ú– ~Ó˚ ö˛ˆÏú È AB ÙÈÓ˚ ¢y ÏÌ ~Ñ˛!›˛ ¢õyhs˘˛Ó˚yú

¢Ó˚úˆÏÓ˚Öy !ò!î≈T˛ î)Ó˚ˆÏc S5 ˆ¢!õV ˛ôyÁÎ˚y ˆÜú–

(iv) ˛≤Ãî_ ¢Ó˚ú ÏÓ˚ÖyÓ˚ v z ôÓ˚ !ò!î≈T˛ !Ó®% Ïì˛ ≤Ãî_ ˆÑ˛y ÏíÓ˚ ¢õyò õy Ï ôÓ˚ ˆÑ˛yí xAÑ˛ò–

∠ABC £ú ≤Ãî_ ˆÑ˛yí ~ÓÇ PQ £ú ≤Ãî_ ¢Ó˚ú ÏÓ˚Öy–

PA B

Q

R

P

Q

A B

R

B CP QRPQR

A

M

NS

!â˛e ı 6.85

ˆòy›˛



PQsuur

Ó˚ v˛z˛ôÓ˚ P !Ó®% Ïì˛ ∠ABC Ó˚ ¢õyò õyˆÏ˛ô ˆÑ˛yí x§yÑ˛ˆÏì˛ £ˆÏÓ–

ïy˛ô 1 ı Ñ˛¡ôyˆÏ¢ !Ñ˛ä%È›˛y Óƒy¢yï≈ !òˆÏÎ˚ñ B ˆÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ ~Ñ˛!›˛ Ó,_â˛y˛ô x§yÑ˛y £úñ ˆÎ›˛y

BAsuur Á BC

suur ˆÑ˛ ˆäÈî Ñ˛Ó˚ú– ˆäÈî !Ó®%Ü%!ú £ú ÎÌyÑ ˆÏõ M Á N S!â˛e 6.85a)

¢õ Óƒy¢yï≈ !òˆÏÎ˚ PˆÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ ~Ñ˛!›˛ Ó,_â˛y˛ô x§yÑ˛y £úñ Îy PQsuur

ˆÑ˛ ˆäÈî

Ñ˛Ó˚ ÏäÈ– ˆäÈî!Ó®%!›˛Ó˚ òyõ ˆîÁÎ˚y £ú R S!â˛e ı 685a)

ïy˛ô 2 ı M Á N ~Ó˚ õˆÏïƒÓ˚ î)Ó˚cˆÏÑ˛ Óƒy¢yï≈ !£¢yˆÏÓ !òˆÏÎ˚ RˆÑ˛ ˆÑ˛w Ñ˛ ÏÓ˚ ~Ñ˛!›˛ Ó,_â˛y˛ô

x§yÑ˛ˆÏì˛ £ˆÏÓ– Îy xyˆÏÜÓ˚ Ó,_â˛y˛ôˆÏÑ˛ ˆäÈî Ñ˛ˆÏÓ˚– ˆäÈî!Ó®%Ó˚ òyõ ˆîÁÎ˚y £ú S S!â˛e

ı 6.85c)

ïy˛ô 3 ı PSsur

ˆÓ˚Öy ›˛yòy £ú– ∠SPR £ú ≤Ã ÏÎ˚yãò#Î˚ ˆÑ˛yí Îy ∠ABC Ó˚ ¢ ÏAÜ ¢õyò–

(v) ≤Ãî_ ˆÑ˛yí ÏÑ˛ ¢õ!mÖ[˛Ñ˛ xAÑ˛ò ı

∠ABC £ú ≤Ãî_ ˆÑ˛yí– ~ ÏÑ˛ ¢õ!mÖ[˛Ñ˛ Ó Ïú–

!â˛e ı 6.86

ïy˛ô 1 ı B ˆÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ ~Ñ˛!›˛ Ó,_â˛y˛ô ~õò¶˛yˆÏÓ x§yÑ˛y £ú ÎyˆÏì˛ ì˛y ∠ABC Ó˚ BAsuur

Á BCsuur

ˆäÈî Ñ˛ ÏÓ˚– ˆäÈî!Ó®% î%!›˛Ó˚ òyõ £ú P Á Q–

ïy˛ô 2 ı P Á Q ˆÑ˛ ˆÑ˛w Ñ˛ ÏÓ˚ PQÓ˚ xˆÏï≈ˆÏÑ˛Ó˚ ˆÓ!ü Óƒy¢yï≈ !òˆÏÎ˚ Ó,_â˛y˛ô x§yÑ˛y £úñ

ÎyˆÏì˛ Ó,_â˛y˛ô î%!›˛ ˛ôÓ˚fl˛ôÓ˚ ˆäÈî Ñ˛ˆÏÓ˚– ˆäÈî!Ó®%Ó˚ òyõ £ú R–

ïy˛ô 3 ı BRsuur ¢Ó˚ú ÏÓ˚Öy ›˛yòy £ú– BR

suur £ú ∠ABC Ó˚ ¢õ!mÖ[˛Ñ˛– xÌ~Óñ m∠ABR

= m∠CBR

(vi) !ò!î≈T˛ õy Ï ôÓ˚ ˆÑ˛yí xAÑ˛ò (60° ˆÑ˛yíñ ì˛yÓ˚ Ü%!íì˛Ñ˛ Á v z ôÜ%!íì˛Ñ˛ ˆÑ˛yí xAÑ˛òV

(a) 60° ˆÑ˛yí xAÑ˛ò

ABsuur ~Ñ˛!›˛ ≤Ãî_ ¢Ó˚ú ÏÓ˚Öy– AB

suur Ó˚ v˛z˛ôÓ˚ A !Ó®%ˆÏì˛ 60° ˆÑ˛yí x§yÑ˛ˆÏì˛ £ˆÏÓ–

ïy˛ô 1 ı AˆÏÑ˛ ˆw Ñ˛ˆÏÓ˚ ~õò Óƒy¢yï≈ !òˆÏÎ˚ ~Ñ˛!›˛ Ó,_â˛y˛ô x§yÑ˛ˆÏì˛ £ˆÏÓñ ÎyˆÏì˛ ì˛y AB ˆÓ˚

ˆäÈî Ñ˛ˆÏÓ˚– ˆäÈî!Ó®%Ó˚ òyõ ˆîÁÎ˚y £ú P–

A

B C

P

Q

A

B C

P

Q

R

A

B C

P

Q

R

ˆòy›˛

82 D.El.Ed. 504


!â˛e ı 6.87

ïy˛ô 2 ıP ˆÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ ¢õ Óƒy¢yï≈ !òˆÏÎ˚ ~õò¶˛yˆÏÓ ~Ñ˛!›˛ Ó,_â˛y˛ô x§yÑ˛ˆÏì˛ £ˆÏÓ ÎyˆÏì˛ ì˛y

ïy˛ô 1 ˆì˛ x§yÑ˛y Ó,_â˛y˛ôˆÏÑ˛ ˆäÈî Ñ˛Ó˚ˆÏì˛ ˛ôyˆÏÓ˚– ˆäÈî!Ó®%Ó˚ òyõ ˆîÁÎ˚y £ú Q–

ïy˛ô 3 ı AQsuur

¢Ó˚ú ÏÓ˚Öy ›˛yòy £ú– ∠QAB £ú ≤ÃˆÏÎ˚yãò#Î˚ ˆÑ˛yí ÎyÓ˚ õyò 60°–

(b) 120° ˆÑ˛yí xAÑ˛ò

ïy˛ô 1 ı~Ñ˛!›˛ ¢Ó˚ú ÏÓ˚Öy ABsuur ›˛yòy £ú– A ˆÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ !ò!î≈T˛ Óƒy¢yï≈ !òˆÏÎ˚ ~Ñ˛!›˛

Ó,_â˛y˛ô x§yÑ˛y £ú Îy ABˆÑ˛ P !Ó®%ˆÏì˛ ˆäÈî Ñ˛ˆÏÓ˚– ~Ñ˛!›˛ Ó,_â˛y˛ô ~õò¶˛yˆÏÓ x§yÑ˛y

£ú ÎyˆÏì˛ ~!›˛ ABsuur ˆÑ˛ ˆäÈî Ñ˛ ÏÓ˚ ~ÓÇ ˆäÈî!Ó®%Ó˚ òyõ P–

!â˛e ı 6.88

ïy˛ô 2 ı¢õ Óƒy¢y Ïï≈Ó˚ ~Ñ˛!›˛ Ó,_â˛y˛ô PˆÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ ~õò¶˛yˆÏÓ x§yÑ˛y £ú ÎyˆÏì˛ ì˛y ≤ÃÌõ

Ó,_â˛y˛ô ÏÑ˛ ˆäÈî Ñ˛ ÏÓ˚– ˆäÈî!Ó®%Ó˚ òyõ Q– Q ˆÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ xyÓ˚ ~Ñ˛!›˛ Ó,_â˛y˛ô x§yÑ˛y

£Î˚ ÎyˆÏì˛ ì˛y ïy˛ô 1 ~ x§yÑ˛y Ó,_â˛y˛ôˆÏÑ˛ ˆäÈî Ñ˛ˆÏÓ˚– ˆäÈî!Ó®%Ó˚ òyõ R–

ïy˛ô 3 ı ARsuur ˆÓ˚Öy ›˛yòy £ú– ∠RAB £ú ≤Ã ÏÎ˚yãò#Î˚ ˆÑ˛yí ÎyÓ˚ õy˛ô £ú 120°–

(b) 90° ˆÑ˛yí xAÑ˛ò ı

ïy˛ô 1 ı!â˛e 6.89 ÈÙÈ~Ó˚ õ Ïì˛y xyõÓ˚y ì˛Öò Ñ˛Ó˚úyõ–

BA P BA P

Q

BPA

Q

A PB

QR Q

BPA

R Q

A P B

Q

BA P

S

R

P

!â˛e ı 6.89

ˆòy›˛



ïy˛ô 2 ı∠RAQ ÈÙÈÓ˚ ¢õ!mÖ[˛Ñ˛ x§yÑ˛y £ú– ¢õ!mÖ[˛ ÏÑ˛Ó˚ òyõ £ú ASsuur

– ∠SAB £ú

˛≤Ã ÏÎ˚yãò#Î˚ ˆÑ˛yí ÎyÓ˚ õy˛ô 90°– 45° ˆÑ˛yí ˆ˛ôˆÏì˛ ˆÜˆÏú 90° ˆÑ˛ ú¡∫ ¢õ!mÖ[˛ ÏÑ˛

¶˛yÜ Ñ˛Ó˚ Ïì˛ £ ÏÓ–

122

2

° ˆÑ˛yí x§yÑ˛ Ïì˛ ˆÜ Ïú Ñ˛¡ôy Ï¢Ó˚ 45° ˆÑ˛yí ~§ ÏÑ˛ ì˛y ÏÑ˛ ¢õ!mÖ[˛Ñ˛ Ñ˛Ó˚ Ïì˛ £ ÏÓ–

E11. Óú%ò xy˛ô!ò Ñ˛#¶˛y ÏÓ ~£z ˆÑ˛yíÜ%!ú x§yÑ˛ ÏÓòÈÙÙÙÈ(a) 30° (b) 15° (c) 105°

(vii) ~Ñ˛!›˛ ¢Óú ÏÓÖy ÏÑ˛ ¢õ xÇ Ïü ˆäÈy Ï›˛y ˆäÈy Ï›˛y Ö[˛ !Ó¶˛=˛ Ñ˛Óy ı AB £ú ~Ñ˛!›˛ ¢Óú ÏÓÖyÇü

ÎyˆÏÑ˛ 3!›˛ ¶˛yˆÏÜ ¶˛yÜ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ– ≤Ã!ì˛!›˛ ¶˛yˆÏÜÓ˚ ˜îá≈ƒ x £ ÏÓ–

ïy˛ô 1 ı ˆÓ˚Öy!›˛Ó˚ ~Ñ˛≤Ãyhs˘˛ Sï!Ó˚ AV ÓÓyÓÓ xyõÓy ~õò ~Ñ˛!›˛ ˆÓÖy ›˛yòúyõ Îy ¢Óú ÏÓÖy!›˛Ó

¢ ÏAÜ !õú ÏÓ òy– ~¶˛y ÏÓ AXsuur ›˛yòy £ú–

!â˛e ı 6.90

ïy˛ô 2 ı BYsuur ˆÓ˚Öy AX

suur ~Ó˚ ¢õyhs˘˛Ó˚yú¶˛y ÏÓ ›˛yòy £ú ∠BAXÈÙÈÓ˚ ¢õyò ∠ABY ˆÑ˛ Ï›˛

!òˆÏÎ˚–

ïy˛ô 3 ı A ˆÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ ~Ñ˛!›˛ Óƒy¢yï≈ !òˆÏÎ˚ AXsuur ˆÓ˚ÖyˆÏÑ˛ Ó,_â˛y˛ô !îˆÏÎ˚ ˆäÈî Ñ˛Ó˚yˆÏòy £ú–

ˆäÈî!Ó®%!›˛Ó˚ òyõ ˆîÁÎ˚y £ú P– xyÓyÓ˚ PÈÙÈˆÑ˛ ˆÑ˛w Ñ˛ˆÏÓ˚ ¢õyò Óƒy¢yï≈ !òˆÏÎ˚ AXsuur

ˆÓ˚ÖyˆÏÑ˛ Ó,_â˛y˛ô !îˆÏÎ˚ ˆäÈî Ñ˛Ó˚yˆÏòy £ú– ˆäÈî!Ó®%!›˛Ó˚ òyõ ˆîÁÎ˚y £ú Q– ì˛y£ Ïú

î%!›˛ ¢õyò xÇü AP Á PQ (AB ˆÓ˚ÖyˆÏÑ˛ Îì˛¶˛yˆÏÜ ¶˛yÜ Ñ˛Ó˚ˆÏì˛ Óúy £ˆÏÎ˚ˆÏäÈ ì˛yÓ˚

ˆÌ ÏÑ˛ 1¶˛yÜ Ñ˛õV ˛ôyÁÎ˚y ˆÜú– ˙ ¢õyò Óƒy¢yï≈!›˛ !òˆÏÎ˚ BYsuur ˆÓ˚ÖyˆÏÑ˛ ~Ñ˛£z¶˛yˆÏÓ

P Á Q !Ó®%ˆÏì˛ ˆäÈî Ñ˛Ó˚y £ú S!â˛e 6.90 (b))–

A B

X

A

BP

Q

B

P′

YΘ′

ˆòy›˛

84 D.El.Ed. 504


ïy˛ô 4 ı QRsuur

Á PQsuur

, ABsuur ˆÓ˚Öy ÏÑ˛ ˆÎ ˆÎ !Ó®% Ïì˛ ˆäÈî Ñ˛Ó˚ú ì˛y ÏîÓ˚ ÎÌyÑ Ïõ C ~ÓÇ D

òyõ ˆîÁÎ˚y £ú– [!â˛e 6.90(c)]

xì˛~Óñ ABsuur

ˆÓ˚Öy ÏÑ˛ C Á D !Ó®% Ïì˛ 3 !›˛ ¢õyò ¶˛y ÏÓ ¶˛yÜ Ñ˛Ó˚y £ú–

!â˛e ı 6.90 (c)

!òˆÏãÓ˚ ≤ÃÜ!ì˛ õy˛ôy ı

E12. ~Ñ˛!›˛ ¢Ó˚ú ÏÓ˚ÖyÓ˚ v z ôÓ˚ ú¡∫ÈÙÈ¢õ!mÖ[˛Ñ˛ ~§ ÏÑ˛ ¢Ó˚ú ÏÓ˚Öy ÏÑ˛ Ñ˛!›˛ ¢õyò ¶˛y ÏÜ ¶˛yÜ Ñ˛Ó˚y

ÎyˆÏÓ⁄

E13. ú¡∫ÈÙÈ¢õ!mÖ[˛Ñ˛ ~§ˆÏÑ˛ ~Ñ˛!›˛ ¢Ó˚úˆÏÓ˚ÖyˆÏÑ˛ Ñ˛# ò#ˆÏâ˛Ó˚ ¶˛yˆÏÜ ¶˛yÜ Ñ˛Ó˚y ¢Ω˛Ó⁄

(a) ¢õyò 4 ¶˛y ÏÜ⁄

(b) ¢õyò 8 ¶˛y ÏÜ⁄

(c) ¢õyò 12 ¶˛y ÏÜ⁄

(viii) (a) !e¶% ã xAÑ˛ò ı ¢Ó˚úˆÏÓ˚Öy Á !Ó!¶˛ß¨ õyˆÏ˛ôÓ˚ ˆÑ˛yí x§yÑ˛ˆÏì˛ !üˆÏÖ ˆÜˆÏú Ó%úyÓ˚ Á

Ñ˛¡ôy¢ ÓƒÓ£yÓ˚ Ñ˛ ÏÓ˚ !e¶% ã x§yÑ˛y ¢Ω˛Ó–

!e¶% ã x§yÑ˛yÓ˚ ãòƒ ò)òƒì˛õ ≤Ã ÏÎ˚yãò#Î˚ üì≈ ı 3!›˛ Óy£% 3!›˛ Ñ˛yˆÏíÓ˚ õy˛ô äÈyv˛¸yÁñ !òˆÏ¡¨Ó˚ üì≈˛Ü%!ú

v zˆÏÕ‘Ö ÌyÑ˛ˆÏú !e¶%˛ã x§yÑ˛y ÎyÎ˚ÈÙÙÙÈ

(a) !ì˛ò!›˛ Óy£%Ó˚ ˜îá≈ƒ (S-S-S)

(b) î%!›˛ Óy£%Ó˚ ˜îá≈ƒ Á ì˛yˆÏîÓ˚ xhs˘˛¶≈%˛=˛ ˆÑ˛yˆÏíÓ˚ ˜îá≈ƒ (S-A-S)

(c) ˆÎ ˆÑ˛yò î%!›˛ Óy£%Ó˚ ˜îá≈ƒ ~ÓÇ ˆÎˆÏÑ˛yò ~Ñ˛!›˛ ˆÑ˛yˆÏíÓ˚ õy˛ô (S-S-A)

(d) ˆÎˆÏÑ˛yò î%!›˛ ˆÑ˛yˆÏíÓ˚ õy˛ô ~ÓÇ ~Ñ˛!›˛ Óy£%Ó˚ õy˛ô (A-S-A) Óy (A-A-S)

xyõÓ˚y ˆîÖˆÏì˛ ˆ˛ôúyõñ ˛˛≤Ã!ì˛ ˆ«˛ˆÏe !e¶%˛ã x§yÑ˛yÓ˚ ãòƒ xhs˘˛ì˛ !ì˛ò!›˛ Óhfl%˛Ó˚ õyò ãyòˆÏì˛ £Î˚–

~£z x§yÑ˛yÜ%!ú Ñ˛yˆÏã úy!ÜˆÏÎ˚ !Ó!¶˛ß¨ !e¶% ã x§yÑ˛y ¢Ω˛Ó–

A D C B

P1

Q1

P

Q

X

Y

ˆòy›˛



£y Ïì˛ Ñ˛ú Ïõ Ñ˛yã

î%!›˛ Óy£%Ó˚ õy˛ô ˆîÁÎ˚y ÌyÑ˛ˆÏú ~Ñ˛!›˛ ¢õˆÏÑ˛yí# !e¶% ã Ñ˛#¶˛yˆÏÓ x§yÑ˛y ÎyÎ˚ ì˛yÓ˚ ˛ôÂï!ì˛ Óú%ò–

(b) â˛ì% ¶≈% ã xAÑ˛ò ı !e¶% ã ˛ôÂï!ì˛ ãyò Ïúñ â˛ì% ¶≈% ã x§yÑ˛yÁ ¢£ã £ ÏÎ ÎyÎ– ì˛yÓÑ˛yÓí ¢õhfl Ï

â˛ì% ¶≈% ã ÏÑ˛£z î%!›˛ Ñ˛í≈ ÓÓyÓÓ ¶˛yÜ Ñ˛Ó Ïú î%!›˛ Ñ˛ ÏÓ !e¶% ã ˛ôyÁÎy ÎyÎ– xì˛~Ó â˛ì% ¶≈% ã x§yÑ˛yÓ

ãòƒ ~Ó õ Ïïƒ v z ô!fiÌì˛ !e¶% ã ~§ ÏÑ˛ ì˛yÓ ˆÌ ÏÑ˛ â˛ì% ¶≈% ã x§yÑ˛ Ïì˛ £ ÏÓ–

!â˛e ı 6.91

v zîy£Ó˚ífl ∫Ó) ô Óúy ÎyÎ˚ Î!î ABCD â˛ì%˛¶≈%˛ˆÏãÓ˚ â˛yÓ˚!›˛ Óy£%Ó˚ õy˛ô ~ÓÇ ACÈÙÈÓ˚ ˜îá≈ƒ xÌ≈yÍ

ˆÎˆÏÑ˛yò ~Ñ˛!›˛ Ñ˛ˆÏí≈Ó˚ ˜îá≈ƒ ÓˆÏú ˆîÁÎ˚y £Î˚ñ ì˛y£ˆÏú ~Ó˚ õˆÏïƒÓ˚ ˆÎˆÏÑ˛yò !e¶% ã ïÓ˚y ÎyÑ˛

ABC !e¶% ã x§yÑ˛y ¢Ω˛Óñ ˆ¢Öyò ˆÌ ÏÑ˛ x˛ôÓ˚ !e¶% ã ADC x§yÑ˛y ÎyÎ˚– ~£z¶˛yˆÏÓ v˛z!Õ‘!Öì˛

â˛ì%˛¶≈%˛ã!›˛ x§yÑ˛y ¢Ω˛Ó–

~£z¶˛yˆÏÓ !Ó!¶˛ß¨ ïÓ˚ˆÏíÓ˚ â˛ì%˛¶≈%˛ã x§yÑ˛y ¢Ω˛Ó–

E14. ò#ˆÏâ˛Ó˚ ˆÑ˛yò üì˛≈Ü%!ú v z ÏÕ‘Ö ÌyÑ˛ˆÏúñ â˛ì% ¶≈% ã xAÑ˛ò ¢Ω˛Ó–

(a) â˛y˛ô!›˛ Óy£%Ó˚ õy˛ô–

(b) Ó˚¡∫ˆÏ¢Ó˚ î%!›˛ Ñ˛ˆÏí≈Ó˚ õy˛ô

(c) î%!›˛ ˆÑ˛yˆÏíÓ˚ õy˛ô Á 3!›˛ Óy£%Ó˚ õy˛ô

(d) ¢yõhs˘˛!Ó˚ˆÏÑ˛Ó˚ î%!›˛ ¢!ß¨!£ì˛ Óy£%Ó˚ ˜îá≈ƒ Á ˆÎˆÏÑ˛yò ~Ñ˛!›˛ ˆÑ˛yˆÏíÓ˚ õy˛ô–

6.6 !ö˛ ÏÓ ˆîÖy

¢yõì˛!úÑ˛ ãƒy!õˆÏì˛ˆÏì˛ !Ó®%ñ ˆÓ˚Öy Á ì˛ú ~£z !ì˛ò!›˛ £ú ¢ÇK˛y£#ò ü∑–

ˆÓ˚ÖyÇüñ Ó˚!Ÿ¬ñ ˆÑ˛yíñ !e¶% ãñ â˛ì% ¶≈% ã ≤Ã¶, !ì˛ ˆõÔ!úÑ˛ ü∑Ü%!ú ˙ ¢ÇK˛y£#ò ü∑ !î ÏÎ˚ ÓƒyÖƒy

Ñ˛Ó˚y ÎyÎ˚–

¢!ß¨!£ì˛ ˆÑ˛yíñ ¢)â˛Ñ˛ ˆÑ˛yíñ ¢¡ô)Ó˚Ñ˛ ˆÑ˛yíñ !Ó≤Ãì˛#˛ô ˆÑ˛yí ~Ü%!ú £ú ~Ñ˛ˆÏãyv˛¸y ˆÑ˛yˆÏíÓ˚

v˛zîy£Ó˚í–

A D C B

P1

Q1

P

Q

X

Y

ˆòy›˛

86 D.El.Ed. 504


!e¶% ÏãÓ˚ ˆ◊!í!Ó¶˛yÜ £Î˚– a) Óy£%Ó˚ ˜îá≈ƒ !îˆÏÎ˚ ¢õ!mÖ[˛Ñ˛ ¢õÓy£% Á !Ó°ÏõÓy£% !e¶% ãñ b)

ˆÑ˛yˆÏíÓ˚ õy˛ô !îˆÏÎ˚ ¢)∏ˆÏÑ˛yí ¢õˆÏÑ˛yí# Á fiÌ)úˆÏÑ˛yí# !e¶% ã– !e¶% ˆÏãÓ˚ !ì˛ò!›˛ ˆÑ˛yˆÏíÓ˚ ¢õ!T˛

180°–

!e¶% ÏãÓ˚ ˆ«˛ Ïe a) ˛ô!Ó˚¢#õy = !ì˛ò!›˛ Óy£%Ó˚ ˜îˆÏá≈ƒÓ˚ ˆÎyÜö˛úñ ~ÓÇ b) ˆ«˛eö˛ú = 1/

2 ¶)˛!õ × v˛zFâ˛ì˛y–

›˛Δy!˛ô!ãÎ˚yõñ ¢yõhs˘˛!Ó˚Ñ˛ñ xyÎ˚ì˛ˆÏ«˛eñ Ó˚¡∫¢ Á ÓÜ≈ˆÏ«˛e £ú !Ó!¶˛ß¨ ïÓ˚ˆÏòÓ˚ â˛ì% ¶%≈˛ˆÏãÓ˚ v˛zîy£Ó˚í–

â˛ì%˛¶%≈˛ˆÏãÓ˚ â˛yÓ˚!›˛ ˆÑ˛yˆÏíÓ˚ ¢õ!T˛ 360°– Ó,_ £ú ~Ñ˛!›˛ ì˛ˆÏú xˆÏòÑ˛ !Ó®%Ó˚ ¢õ!T˛ñ !Ó®%Ü%!ú

~õò¶˛yˆÏÓ xÓ!fiÌì˛ ÎyˆÏì˛ ì˛yÓ˚y ˆ¢£z ì˛ˆÏúÓ˚ ~Ñ˛!›˛ !ò!î≈T˛ !Ó®% ˆÌˆÏÑ˛ ¢õî)Ó˚ˆÏc xÓfiÌyò

Ñ˛ˆÏÓ˚ñ !ò!î≈T˛ !Ó®%!›˛ £ú ˆÑ˛w ~ÓÇ ˆÑ˛w ˆÌˆÏÑ˛ Ó, Ï_Ó˚ v˛z ôˆÏÓ˚Ó˚ ˆÎ ˆÑ˛yò !Ó®%Ó˚ î)Ó˚c £ú

Ó,ˆÏ_Ó˚ Óƒy¢yï≈–

Ó,ˆÏ_Ó˚ ˆÎˆÏÑ˛yò î%!›˛ !Ó®% ˆÎyÜ Ñ˛Ó˚ˆÏú Ó,ˆÏ_Ó˚ ãƒy ˛ôyÁÎ˚y ÎyÎ˚– ãƒy !îˆÏÎ˚ Ó,_ˆÏÑ˛ î%!›˛ ¶˛yˆÏÜ

¶˛yÜ Ñ˛Ó˚ˆÏú ~Ñ˛ ~Ñ˛!›˛ ¶˛yÜˆÏÑ˛ Ó, Ï_Ó˚ â˛y˛ô Óúy £Î˚–

~Ñ˛!›˛ Ó,_â˛yˆÏ˛ôÓ˚ õˆÏïƒ xÓ!fiÌì˛ ˆÑ˛yíÜ%!ú ≤ÃˆÏì˛ƒÑ˛!›˛ ¢õyò õyˆÏòÓ˚– xï≈Ó, Ï_Ó˚ ˆÑ˛yí £ú ~Ñ˛

¢õ ÏÑ˛yí–

ˆäÈyˆÏ›˛y v˛z˛ôâ˛yˆÏ˛ôÓ˚ ≤Ãyhs˘˛ !Ó®% î%!›˛ˆÏì˛ Óƒy¢yï≈ ›˛yòˆÏú ˆÑ˛wˆÏì˛ ˆÎ ˆÑ˛yí ˜ì˛!Ó˚ £ˆÏÓ ì˛y !ÓÎ%=˛

ˆÑ˛yí– v z ôâ˛y Ï ôÓ˚ !v˛@˘Ã# õy˛ô £ú ˆÑ˛ Ïw !ÓÎ%=˛ ˆÑ˛y ÏíÓ˚ õy˛ô–

Ó,_fiÌ â˛ì% ¶%≈ ÏãÓ˚ !Ó˛ôÓ˚#ì˛ ˆÑ˛y ÏíÓ˚ ¢õ!T˛ £ú 180°–

¢õ Ïî Ïá≈ƒÓ˚ ˆÓ˚ÖyÇü ¢Ó≈¢õ ~ÓÇ ¢õ õy ÏòÓ˚ ˆÑ˛yí ¢Ó≈¢õ– !Ó!¶˛ß¨ ü Ïì≈ Ó˚ !¶˛!_ Ïì˛ SˆÎõò S-

A-S, S-S-S, A-S-A Á RHS) !e¶% ã ¢Ó≈¢õ £Î˚–

¢Ó≈¢õ !â˛eÓ˚y ¢î,ü £Î˚ !Ñ˛hs%˛ ¢î,ü !â˛e õye£z ¢Ó≈¢õ òÎ˚– î%!›˛ ¢î,ü !e¶% ˆÏãÓ˚ ˆ«˛eö˛ú ì˛yˆÏîÓ˚

Óy£%Ó˚ Ó ÏÜ≈Ó˚ ¢ ÏAÜ ¢õyò% ôy!ì˛Ñ˛–

ãƒy!õ!ì˛Ó˚ ≤Ã!ì˛ö˛úˆÏòÓ˚ ¢ˆÏAÜ xyÎ˚òyÎ˚ ≤Ã!ì˛ö˛úˆÏòÓ˚ ïyÓ˚íyÓ˚ ¢yî,üƒ xyˆÏäÈ– ≤Ã!ì˛ö˛úˆÏòÓ˚

ö˛ Ïú !â˛ ÏeÓ˚ õ Ïïƒ ≤Ã!ì˛¢yõƒ ¢,!T˛ £Î˚–

¢yõì˛!úÑ˛ !â˛ˆÏeÓ˚ ˆ«˛eö˛ˆÏúÓ˚ ¢)e ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ !Ó!¶˛ß¨ !eõy!eÑ˛ Óhfl% ˆÎõò xyÎ˚ì˛áòñ áòÑ˛ñ

ˆâ˛yàñ üAÑ˛ñ !≤Ããõñ !˛ôÓ˚y!õ Ïv˛Ó˚ ˛ô,¤˛ì˛ ÏúÓ˚ ˆ«˛eö˛ú !òí≈Î˚ Ñ˛Ó˚y ÎyÎ˚–

ü%ï%õye Ó%úyÓ˚ Á Ñ˛¡ôyˆÏ¢Ó˚ ¢y£yˆÏÎƒñ !ò¡¨!ú!Öì˛ õ)ú ãƒy!õ!ì˛Ñ˛ !â˛eÜ%!ú xAÑ˛ò Ñ˛Ó˚y ÎyÎ˚ ı

i) !ò!î≈T˛ ˜î Ïá≈ƒÓ˚ ˆÓ˚ÖyÇü xAÑ˛ò

ii) ≤Ãî_ ˆÓ˚ÖyÇ ÏüÓ˚ ú¡∫ÈÙÈ¢õ!mÖ[˛Ñ˛

iii) !ò!î≈T˛ Óƒy¢yˆÏï≈Ó˚ Ó,_ xAÑ˛ò

iv) !ò!î≈T˛ ˆÓ˚ÖyÓ˚ v˛z˛ôÓ˚ !ò!î≈T˛ !Ó®%ˆÏì˛ ~Ñ˛!›˛ ˆÑ˛yˆÏíÓ˚ õyˆÏòÓ˚ ¢õyò õyˆÏòÓ˚ ˆÑ˛yí xAÑ˛ò

v) ~Ñ˛!›˛ ¢Ó˚ú ÏÓ˚ÖyÓ˚ ¢õyhs˘˛Ó˚yú / ú¡∫ ¢Ó˚ú ÏÓ˚Öy xAÑ˛ò

vi) ≤Ãî_ ˆÑ˛y ÏíÓ˚ ¢õ!mÖ[˛Ñ˛ xAÑ˛ò

vii) 60° ˆÑ˛yíñ ì˛yÓ˚ Ü%!íì˛Ñ˛ Á v˛z ôÜ%!íì˛Ñ˛ (30°, 45°, 15° ≤Ã¶, !ì˛V

ˆòy›˛



viii) ÓÂï ¢yõì˛!úÑ˛ !â˛e ˆÎõòñ !e¶% ãñ â˛ì% ¶%≈ ãñ Ó,_ xAÑ˛ò

6.7 !ò ÏãÓ˚ ≤ÃÜ!ì˛ õy˛ôyE1. 921/2°

E2. 108°, 108°, 72°

E3. i) ∠ACQ, ∠CDS, ii) ∠PCD, ∠CDR, iii) ∠QCD, ∠CDR,

E4. a = c = f = h = 145°, e = d = b = 35°

E5. a) Ó˚¡∫¢ñ b) ˆÑ˛wñ c) 32°, d) 180°

E6. a) ˜îá≈ƒñ b) ˛ô!Ó˚õy˛ôñ c) ¢!ß¨!£ì˛ñ d) xyÑ,˛!ì˛

E7. PR = 12 cm

E8. ∆ABC : ∆PQR = 4 : 9

E9. 63 cm

E10.xyÎ˚ì˛ˆÏòÓ˚ xò%˛ôyì˛ £ˆÏÓ 3 : 2

E11. a) 60° ˆÑ˛y ÏíÓ˚ ¢õ!mÖ[˛Ñ˛ ~§ ÏÑ˛

b) 30° ˆÑ˛y ÏíÓ˚ ¢õ!mÖ[˛Ñ˛ ~§ ÏÑ˛

c) ∠ABC = 90°, ∠CBD = 120° ~§ˆÏÑ˛ñ ì˛yÓ˚˛ôÓ˚ BE ˆÓ˚Öy ˆ›˛ˆÏòñ ∠ABD-Ó

¢õ!mÖ[˛Ñ˛ñ ∠EBC Ó˚ õyò = 105°

E12.2!›˛ ¢õyò ¶˛yÜ

E13.a) £§ƒyñ b) £§ƒyñ c) òy

E14.a) òyñ b) £§ƒyñ c) òyñ d) £§ƒy

6.8 ≤Ãhfl Ïy!Óì˛ ˛ô‡˛ò v z ôÑ˛Ó˚í Á ˆÓ˚ö˛y ÏÓ˚™

NCERT-Ó˚ ˛ôM˛õ ˆÌ ÏÑ˛ xT˛õ ˆ◊!íÓ˚ Ü!í Ïì˛Ó˚ ˛ôy‡˛ƒÓ£z

6.9 ~Ñ˛Ñ˛ ˆü Ï°ÏÓ˚ xò%ü#úò#

1. Ó˚%úyÓ˚ Á Ñ˛¡ôyˆÏ¢Ó˚ ¢y£yˆÏÎƒñ ≤Ãî_ !â˛e xò%ÎyÎ˚# !òˆÏ¡¨Ó˚ õyòÜ !òˆÏÎ˚

!â˛e xAÑ˛ò–

AP = 3cm, BP = 4 cm, BC = 7 cm, PQ = 5 cm, QD = 2

cm, BP AD⊥ ~ÓÇ CQ AD⊥!â˛e!›˛ ≤Ã!ì˛¢õ !â˛e ~§ˆÏÑ˛ ˆü°Ï Ñ˛Ó%ò ˆÎÖyˆÏò AD £ú ˜Ó˚!ÖÑ˛ ≤Ã!ì˛¢õ

ˆÓ˚Öy–

A

B

C

D

Q

P

!â˛e 6.92

ˆòy›˛

88 D.El.Ed. 504


2. ≤Ãî_ !â˛ Ïe

AC = BD = PF = DE = 8 cm

∠ABC, ∠BCD, ∠CDA, ∠BDE, ~ÓÇ

∠BPC ≤Ã Ïì˛ƒ ÏÑ˛ ~Ñ˛ ¢õ ÏÑ˛yí–

ˆ¢›˛ ÏflÒyÎ˚yÓ˚ Á ˆflÒú ÓƒÓ£yÓ˚ Ñ˛ ÏÓ˚ ≤Ãî_ õy˛ô xò%ÎyÎ˚#

!â˛e!›˛ xAÑ˛ò Ñ˛Ó%ò– ∆APDÈÙÈÓ˚ myÓ˚y xyúyîy Ñ˛ˆÏÓ˚ ˆîÁÎ˚y

xÇˆÏüÓ˚ ¢ˆÏAÜ Óy!Ñ˛ Üyë˛¸ xÇˆÏüÓ˚ ˆ«˛eö˛ú !òí≈Î˚ Ñ˛ˆÏÓ˚ î%!›˛

xÇ ÏüÓ˚ õ Ïïƒ ì% úòy Ñ˛Ó%ò–

3. !òˆÏ¡¨Ó˚ ˆ›˛!Óú!›˛ ˛ô)Ó˚í Ñ˛Ó˚%ò–

xyÑ,˛!ì˛ á)í≈ò ˆÑ˛w ≤Ã!ì˛¢yˆÏõƒÓ˚ õyey

Ó˚¡∫¢

¢õÓy£% !e¶%˛ã

¢%°Ïõ °Ïv˛¸¶%˛ã

ÓÜ≈ Ï«˛e

4. 20cm ˜îˆÏá≈ƒÓ˚ ˆÑ˛!›˛ ì˛yÓ˚ !òò– ˆ›˛!ÓˆÏú ˆîÁÎ˚y !â˛e!Ó˚ õy˛ô xò%ÎyÎ˚# ì˛yÓ˚!›˛ˆÏÑ˛ Ó§y!Ñ˛ˆÏÎ˚

â˛ì%˛¶%≈˛ã Óyòyò– ≤Ã!ì˛ ˆ«˛ˆÏe ˆ«˛eö˛ú Ñ˛ì˛ £ˆÏÓ ì˛y !òí≈Î˚ Ñ˛Ó˚%ò–

â˛ì% ¶%≈ ÏãÓ˚ ˜îá≈ƒ ≤ÃfiÌ ˆ«˛eö˛ú

8 cm

7 cm

6 cm

5 cm

5. !òˆÏ¡¨Ó˚ ˆ¢ˆÏ›˛ 3!›˛ Ñ˛ˆÏÓ˚ õy˛ô ˆîÁÎ˚y xyˆÏäÈ– ~Ó˚ õˆÏïƒ ˆÑ˛yò õy˛ôÜ%!ú !òˆÏÎ˚ !e¶% ã

ÓyòyˆÏòy ÎyˆÏÓ–

a) 5.5 cm, 6.8 cm ~ÓÇ 7.2 cm

b) 4.7 cm, 5.3 cm ~ÓÇ 7.2 cm

c) 8.0 cm, 7.5 cm ~ÓÇ 9.2 cm

d) 5.8 cm, 12.2 cm ~ÓÇ 6.0 cm

6. Ó˚%úyÓ˚ Á Ñ˛¡ôyˆÏ¢Ó˚ ¢y£yˆÏÎƒ ~Ñ˛!›˛ 52.5° õyˆÏòÓ˚ ˆÑ˛yí ÓyòyˆÏì˛ ˆÜˆÏúñ xõò Ñ˛Ó˚yÓ˚

ãòƒ Ñ˛# Ñ˛# ¶˛yˆÏÜ ˆÑ˛yí!›˛ ¶˛yàˆÏì˛ £ˆÏÓ⁄

B

C D

E

F

!â˛e 6.93

ˆòy›˛

89504-Block2 : ˛ô!Ó˚õyí Á ˛ô!Ó˚õy˛ô

˛ô!Óõyí Á ˛ô!Óõy˛ô

~Ñ˛Ñ˛ÈÈÈÙÙÙÈ7 ı ˛ô!Ó˚õyí Á ˛ô!Ó˚õy˛ô˛ô!Ó˚Ñ˛y‡˛yˆÏõy

7.0 ¶) !õÑ˛y


7.2 ˛ô!Ó˚õyí Á ˛ô!Ó˚õyˆÏ˛ôÓ˚ ïyÓ˚íy

7.3 ˛˛˛Non standard/≤Ãâ˛!úì˛ ~ÓÇ ≤Ãõyí Ó˚y!ü

7.3.1 ˜îˆÏá≈ƒÓ˚ ˛ô!Ó˚õy˛ô

7.3.2 ˆ«˛eö˛ ÏúÓ˚ ˛ô!Ó˚õy˛ô

7.3.3 xyÎ˚ì˛ˆÏòÓ˚ ˛ô!Ó˚õy˛ô

7.3.4 ¶˛ˆÏÓ˚Ó˚ ˛ô!Ó˚õy˛ô

7.4 ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛ˆÏì˛ ˛ô!Ó˚õy˛ô

7.5 ¢õˆÏÎ˚Ó˚ ˛ô!Ó˚õy˛ô

7.6 !ö˛ ÏÓ˚ ˆîÖy

7.7 ˛≤ÃÜ!ì˛ õy˛ôyÓ˚ ãòƒ !Ñ˛ä%È õˆÏv˛ú v z_Ó˚

7.8 ≤Ãhfl˛Ïy!Óì˛ ˛ôy‡˛ Á ˆÓ˚ö˛yˆÏÓ˚™

7.9 ˛ôy‡˛ ˆüˆÏ°ÏÓ˚ xò%ü#úò#

7.0 ¶) !õÑ˛y

xyˆÏÜÓ˚ ~Ñ˛Ñ˛Ü%!úˆÏì˛ xyõÓ˚y xyˆÏúyâ˛òy Ñ˛ˆÏÓ˚!äÈ ˆÎ xyõyˆÏîÓ˚ â˛yÓ˚˛ôyˆÏü ˆÎ ¢õhfl˛Ï ≤ÃyÑ, !ì˛Ñ˛

Ó˚y!ü Óy !ã!ò¢ ˆî!Ö ì˛yÓ˚ î%!›˛ ˜Ó!üT˛ƒÈÙÙÙÈxyÑ˛yÓ˚ Óy xyÑ, !ì˛ ~ÓÇ xyÎ˚ì˛ò– xyÑ, !ì˛ ÓúˆÏì˛

xyõÓ˚y Ó%!G˛ !ã!ò¢!›˛ Óy Ó˚y!ü!›˛ Ñ˛#Ó˚Ñ˛õ ˆîÖˆÏì˛– xyõÓ˚y Ó!úñ ÚÚ~!›˛ ~Ñ˛!›˛ Ó,ˆÏ_Ó˚ äÈ!ÓÛÛ ÚÓ,_Û

ü∑!›˛ ~ÖyˆÏò xyÑ,˛!ì˛ Óy xyÓyÓ˚ ˆÓyG˛yÎ˚– ~Ñ˛£zÓ˚Ñ˛õ¶˛yˆÏÓ xyõÓ˚y ÎÖò ˆÑ˛yˆÏòy õ!®ˆÏÓ˚Ó˚

¢yõˆÏò ˛ôyÌˆÏÓ˚Ó˚ ˜ì˛!Ó˚ !¢Ç£ õ)!ì≈˛ ˆî!Ö xyõÓ˚y Ó!úñ ÚÚ~!›˛ ˛ôyÌˆÏÓ˚Ó˚ ˜ì˛!Ó˚ !¢Ç£ÛÛ– Ú!¢Ç£Û

ü∑!›˛Ó˚ õyïƒˆÏõ ì˛yÓ˚ xyÑ˛yÓ˚ Óy xyÑ,˛!ì˛ ≤ÃÑ˛yü ˛ôyÎ˚–

!Ñ˛hs%˘˛ ÎÖò xyõÓ˚y Ó!ú ˛ôyì˛yÎ˚ x§yÑ˛y ~£z ÓÜ≈!›˛ ÓˆÏv˛¸yñ xyõÓ˚y ~£z ÚÓˆÏv˛¸yÛ ü∑!›˛Ó˚ õˆÏïƒ ì˛yÓ˚

xyÎ˚ì˛ò Ó%!G˛– ¢õ%o ˜¢Ñ˛ˆÏì˛ ˜ì˛!Ó˚ úy£z›˛ £yv˛z¢ Óy Óy!ì˛áÓ˚!›˛ ˆÓü v˛z§â%˛– ~ÖyˆÏò ÚˆÓü v˛z§â%˛Û

Ñ˛ÌyÜ%!ú Óy!ì˛áˆÏÓ˚Ó˚ xyÎ˚ì˛ò!›˛ Ó%G˛ˆÏì˛ ¢y£yÎƒ Ñ˛ˆÏÓ˚–

ò#ˆÏâ˛Ó˚ ÓyÑ˛ƒ!›˛ ˆîˆÏÖyÊ

ÚÚ¢yÑ ≈y Ï¢ ˆÎ ä%È!›˛ ˆãyÑ˛yÓ˚ ˆÖúy ˆîÖy ÏFäÈ ì˛y ÏîÓ˚ õ Ïïƒ ~Ñ˛ãò ˆÓü ˆõy›˛y ~ÓÇ x˛ôÓ˚ãò Ö%Ó

ˆòy›˛

90 D.El.Ed. 504


ˆÓ§ˆÏ›˛– xyõÓ˚y Ñ˛# ˆãyÑ˛yˆÏÓ˚Ó˚ «%˛oì˛yÓ˚ ˛ô!Ó˚õy˛ô Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚⁄ xyõÓ˚y Ñ˛# ˆãyÑ˛yˆÏÓ˚Ó˚ fiÌ)úì˛yÓ˚

˛ô!Ó˚õy˛ô Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚⁄ xyõÓ˚y ¢!‡˛Ñ˛¶˛yˆÏÓ ÓúˆÏì˛ ˛ôy!Ó˚òy–

˛ÎyÓ˚ ¢y£yˆÏÎƒ xyõÓ˚y ÓúˆÏì˛ ˛ôy!Ó˚ Óhfl%˛!›˛ ÓˆÏv˛¸y òy ˆäÈyˆÏ›˛y ì˛yˆÏÑ˛ ˛ô!Ó˚õy˛ô ÓˆÏú–

!Ó!¶˛ß¨ Óhfl%˛Ó˚ ~Ñ˛£zÓ˚Ñ˛õ ˜Ó!üˆÏT˛ƒÓ˚ ~Ñ˛!›˛ !ò!î≈T˛ ˛ô!Ó˚õy˛ô xyˆÏäÈ xyÓyÓ˚ xòƒyòƒ ˜Ó!üˆÏT˛ƒÓ˚

!¶˛ß¨ ˛ô!Ó˚õy˛ô xyˆÏäÈ– ˆÎõòñ ~Ñ˛!›˛ !â˛e ¢õì˛ˆÏú Ñ˛ì˛›˛y ãyÎ˚Üy îÖú Ñ˛ˆÏÓ˚ xyˆÏäÈ ì˛y ˆ¢£z

!â˛ˆÏeÓ˚ ˆ«˛eö˛ˆÏúÓ˚ ˛ô!Ó˚õy˛ô xyÓyÓ˚ ~Ñ˛!›˛ Ñ˛!‡˛ò Óhfl% ˆÏÑ˛ ˛ôyˆÏeÓ˚ ãˆÏú ˆv˛yÓyˆÏú Ñ˛ì˛›˛y ãú

x˛ô¢y!Ó˚ì˛ £ˆÏFäÈ ì˛y ˆ¢£z Óhfl%˛Ó˚ xyÎ˚ì˛ˆÏòÓ˚ ˛ô!Ó˚õy˛ô–

~£z ~Ñ˛ˆÏÑ˛ xyõÓ˚y !Ó!¶˛ß¨ ïÓ˚ˆÏíÓ˚ ˛ô!Ó˚õy˛ôñ ~Ñ˛Ñ˛ñ ˆflÒú Á ˛ô!Ó˚õyˆÏ˛ôÓ˚ ˛ôÂï!ì˛ ~ÓÇ !Ñ˛ä%È

¢yïyÓ˚í !Ó°ÏÎ˚ !òˆÏÎ˚ xyˆÏúyâ˛òy Ñ˛Ó˚ˆÏÓy–

~£z ~Ñ˛Ñ˛!›˛ ˛ôv˛¸ˆÏì˛ xyò%õy!òÑ˛ 7 á^˘›˛y ¢õÎ˚ úyÜˆÏÓ–

7.1 !üÖò v z Ïjüƒ ı

~Ñ˛Ñ˛!›˛ ˛˛ô‡˛ˆÏòÓ˚ ˛ôÓ˚ ˆÎ ˆÎ ¢yõÌ≈ƒ x!ã≈ì˛ £ˆÏÓÈÙÙÙÈ

!Ó!¶˛ß¨ Óhfl%˛ Á ˛ôÂï!ì˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ !Ó!¶˛ß¨ ~Ñ˛Ñ˛ ¢¡∫ˆÏrï ãyòy

≤Ã!ì˛!îˆÏòÓ˚ !Ó!¶˛ß¨ Ñ˛yˆÏÎ≈ !Ó!¶˛ß¨ ~Ñ˛ˆÏÑ˛Ó˚ ÓƒÓ£yÓ˚ Ñ˛Ó˚ˆÏì˛ ˛ôyÓ˚y

˜îá≈ƒñ ˆ«˛eö˛úñ xyÎ˚ì˛òñ ïyÓ˚í «˛õì˛yñ ¶˛Ó˚ Á ¢õÎ˚ ¢¡ô!Ñ≈˛ì˛ Üíòy Ñ˛Ó˚ˆÏì˛ ˛ôyÓ˚y–

7.2 ˛ô!Ó˚õyí Á ˛ô!Ó˚õy Ï ôÓ˚ ïyÓ˚íy

xyõÓ˚y ¢Óy£z ˛ô!Ó˚õy˛ô ¢¡∫ Ïrï xÓÜì˛– xyõyˆÏîÓ˚ ≤Ã!ì˛!îò£z !Ñ˛ä%È òy !Ñ˛ä%È õy˛ôˆÏì˛ £Î˚– ˆÎõòñ

~Ñ˛!›˛ üy›≈˛ ˜ì˛!Ó˚ Ñ˛Ó˚ˆÏì˛ Ñ˛ì˛ ˛ô!Ó˚õyí Ñ˛y˛ôv˛¸ ≤ÃˆÏÎ˚yãòñ ÓyãyÓ˚ ˆÌˆÏÑ˛ ˆÑ˛òyÓ˚ ãòƒ ¢Ó!ã Óy

õ%!îÖyòyÓ˚ !ã!ò¢˛ôˆÏeÓ˚ ˛ô!Ó˚õyíñ ≤Ã!ì˛!îò ˛ôyò Ñ˛Ó˚y ãˆÏúÓ˚ ˛ô!Ó˚õyíñ ˆ◊!íÑ˛ˆÏ«˛ !üü%Ó˚ v zFâ˛ì˛y

Á ÁãˆÏòÓ˚ ˛ô!Ó˚õy˛ôñ ˆ◊!íÑ˛«˛ Óy üÎ˚òÑ˛ˆÏ«˛Ó˚ xyÎ˚ì˛òñ !ÓîƒyúˆÏÎ˚Ó˚ ÓyÜyˆÏòÓ˚ ˆ«˛eö˛úñ Óy!v ¸

ˆÌˆÏÑ˛ !ÓîƒyúˆÏÎ˚ ˆÎˆÏì˛ ¢õˆÏÎ˚Ó˚ ˛ô!Ó˚õy˛ôñ xyõyˆÏîÓ˚ üÓ˚#ˆÏÓ˚Ó˚ ì˛y˛ôõyey ≤Ã¶, !ì˛– ≤Ã!ì˛!›˛ ˆ«˛ˆÏe£z

xyõyˆÏîÓ˚ !Ñ˛ä%È õy˛ôˆÏì˛ £Î˚– v˛z˛ôˆÏÓ˚y=˛ ¢õhfl˛Ï ˆ«˛ˆÏe õyò Á ì%˛úò#Î˚ ~Ñ˛ˆÏÑ˛Ó˚ õyïƒˆÏõ ˛ô!Ó˚õy˛ô

Óƒ=˛ Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ– xÌ≈yÍ ˛ô!Ó˚õy˛ô !òí≈Î˚ Ñ˛Ó˚ˆÏì˛ ˆÎ ˆÑ˛yˆÏòy Óhfl%˛Ó˚ xyÎ˚ì˛ò Óy xyÎ˚ì˛ˆÏòÓ˚ !ò!î≈T˛

!Ó°ÏÎ˚ ˆÎõò ˜îá≈ƒñ ˆ«˛eö˛ú ¢ÇÖƒyÎ˚ Óƒ=˛ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ– ˛ô!Ó˚õyˆÏ˛ô ¢ÇÖƒyÓ˚ ÓƒÓ£yÓ˚ !üü% ÏîÓ˚ˆÏÑ˛

˜îò!®ò ã#ÓˆÏò Ü!íˆÏì˛Ó˚ Ü%Ó%c Ó%G˛ˆÏì˛ ~ÓÇ !Ó!¶˛ß¨ Üy!í!ì˛Ñ˛ ïyÓ˚íy Á î«˛ì˛y xã≈ò Ñ˛Ó˚ˆÏì˛

¢y£yÎƒ Ñ˛ ÏÓ˚–

≤ÃyÌ!õÑ˛ !ÓîƒyúˆÏÎ˚ ˛ôy‡˛Ó˚ì˛ !üü%ˆÏîÓ˚ !Ó!¶˛ß¨ Óhfl%˛ õy˛ôˆÏì˛ ˆüÖy ≤ÃˆÏÎ˚yãò– v˛zFâ˛ ≤ÃyÌ!õÑ˛

ˆòy›˛



!ÓîƒyúÎ˚Ü%!úˆÏì˛Á ˛ôy‡˛Ñ ˛ˆÏõ ˜îá≈ƒñ ˆ«˛eö˛úñ xyÎ˚ì˛òñ ¢õÎ˚ ˛ô!Ó˚õy˛ô Ñ˛Ó˚ˆÏì˛ £Î˚– !Ñ˛hs%˛ ~!›˛ 7

ˆÌ ÏÑ˛ 9 ÓäÈÓ ÓÎ¢# !üü%Ó Ñ˛y ÏäÈ !Ó°ÏÎ!›˛ !Óõ)ì≈ – ˆ¢£z Ñ˛yÓ Ïí ~Ñ˛ãò !ü«˛Ñ˛ !£ Ï¢ ÏÓ xyõy ÏîÓ ÏÑ˛

~£z ¢Ó ïyÓ˚íyÓ˚ ¢ ÏAÜ !üü% ÏîÓ˚ ˛ô!Ó˚!â˛ì˛ Ñ˛Ó˚y Ïì˛ £ ÏÓ–

Ó!£≈ãÜ Ïì˛Ó˚ ¢ ÏAÜ ˛ô!Ó˚!â˛ì˛ £ÁÎ˚yÓ˚ ¢ ÏAÜ ¢ ÏAÜ ~Ñ˛!›˛ !üü% !eõy!eÑ˛ (3-D) Ñ˛!‡˛ò Óhfl%˛ ˆÎõò

ˆÖúòyñ ö˛úñ Ó˚!àò ÓœÑ˛ñ áòÑ˛ñ xyÎ˚ì˛áò Óhfl% ñ ˆîüúy£zñ Óy:ñ òúyÑ˛yÓ˚ Óhfl% ˆÎõò â˛Ñ˛ñ ˆÓ˚yúyÓ˚

£zì˛ƒy!îÓ˚ ¢ˆÏAÜ ˛ô!Ó˚!â˛ì˛ £ˆÏì˛ £Î˚– !eõy!eÑ˛ Óhfl%˛ òyv˛¸yâ˛yv˛¸y Ñ˛ˆÏÓ˚£z ~Ñ˛!›˛ !üü%Ó˚ !mõy!eÑ˛ Óhfl%˛Ó˚

(2D) äÈ!Ó ¢¡∫ˆÏrï ïyÓ˚íy £Î˚– ~Ñ˛!›˛ !üü% áòÑ˛ Óy ˆîüúy£z Óy: !òˆÏÎ˚ ˆÖúˆÏì˛ ˆÖúˆÏì˛ xÌÓy

~£z Óhfl%˛Ó˚ Ñ˛yÜˆÏã äÈ!Ó x§yÑ˛ˆÏì˛ x§yÑ˛ˆÏì˛ Ó%G˛ˆÏì˛ ˛ôyˆÏÓ˚ Ñ˛#¶˛yˆÏÓ Ñ˛!‡˛ò Óhfl%˛Ü%!úˆÏÑ˛ Ñ˛yÜˆÏã Óy

ˆ›˛!ÓˆÏúÓ˚ v˛z˛ôˆÏÓ˚ ¢õì˛ú xyÑ,˛!ì˛ˆÏì˛ x§yÑ˛ˆÏì˛ £Î˚– !eõy!eÑ˛ Óhfl% Ó˚ ì˛úˆÏîü ˛ôÎ≈ˆÏÓ«˛í Ñ˛ˆÏÓ˚ Óy

ˆ¢Ü%!úÓ˚ !mõy!eÑ˛ ≤Ã!ì˛Ó)˛ô ˆîˆÏÖ SÑ˛yÜˆÏã Óy ˆîÁÎ˚yˆÏú xÌÓy ˆõˆÏG˛ˆÏì˛ !eõy!eÑ˛ Óhfl% Ó˚ äÈ!Ó

~§ˆÏÑ˛V !üü% !mõy!eÑ˛ Óhfl%˛Ó˚ ˜Ó!üˆÏT˛ƒÓ˚ ¢ˆÏAÜ ˛ô!Ó˚!â˛ì˛ £Î˚– !eõy!eÑ˛ Óhfl% Ó˚ ïyÓ˚íy !mõy!eÑ˛

Óhfl%˛Ó˚ Óy£% ~ÓÇ !Ñ˛ä%È ¢yïyÓ˚í Óhfl%˛ ˆÎõò ¢Ó˚% ¢%ˆÏì˛yñ ì˛yÓ˚ ˆÌˆÏÑ˛ ~Ñ˛õy!eÑ˛ Óhfl%˛Ó˚ ïyÓ˚íy ˜ì˛!Ó˚

£Î˚–

!Ó!¶˛ß¨ Óhfl%˛ õy˛ôyÓ˚ ïyÓ˚íy Ü‡˛ˆÏòÓ˚ ãòƒ !üü% ÏÑ˛ !Ó!¶˛ß¨ £yˆÏì˛Ñ˛úˆÏõ Ñ˛yã Ñ˛Ó˚ˆÏì˛ !îˆÏì˛ £ˆÏÓ

ˆÎõò !Ó!¶˛ß¨ Óhfl%˛Ó˚ xyÑ,˛!ì˛Ó˚ v˛z˛ôÓ˚ !ò¶≈˛Ó˚ Ñ˛ˆÏÓ˚ ˆ¢Ü%!ú !Ó¶˛=˛ Ñ˛Ó˚yñ !mõy!eÑ˛ Á !eõy!eÑ˛ Óhfl%˛Ó˚

äÈ!Ó x§yÑ˛y– ~£z¢Ó £yˆÏì˛Ñ˛úˆÏõ Ñ˛yˆÏãÓ˚ õyïƒˆÏõ !üü% Ó%G˛ˆÏì˛ ˛ôyÓ˚ˆÏÓ ˆÎ ~Ñ˛õy!eÑ˛ Óhfl% ü%ï%

˜îá≈ƒ Ó˚ ÏÎ˚ ÏäÈ– !mõy!eÑ˛ Óhfl% Ó˚ ˜îá≈ƒ Á ≤ÃfiÌ Ó˚ ÏÎ˚ ÏäÈ Á !eõy!eÑ˛ Óhfl% Ó˚ ˜îá≈ƒñ ≤ÃfiÌ Á v zFâ˛ì˛y

Ó˚ˆÏÎ˚ˆÏäÈ– !Ó!¶˛ß¨ xyÑ,˛!ì˛ Á xyÎ˚ì˛ˆÏòÓ˚ Óhfl%˛ !òˆÏÎ˚ £yˆÏì˛Ñ˛úˆÏõ Ñ˛yˆÏãÓ˚ õyïƒˆÏõ ˜îá≈ƒñ ˆ«˛eö˛úñ

xyÎ˚ì˛ˆÏòÓ˚ ˛ô!Ó˚õy˛ô ¢¡∫ˆÏrï– !üü%ˆÏîÓ˚ ïyÓ˚í Ü‡˛ò £ˆÏÓ– ~Ñ˛!›˛ !üü% !ÓîƒyúˆÏÎ˚ Óy ˜îò!®ò

ã#ÓˆÏòÓ˚ !Ñ ˛Î˚yÑ˛úyˆÏ˛ôÓ˚ õyïƒˆÏõ !òˆÏ¡ÿ=˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ !Ó°ÏÎ˚Ü%!ú Ó%G˛ˆÏì˛ ˛ôyÓ˚ˆÏÓ– ˛ôÓ˚Óì≈˛# !Ó¶˛yˆÏÜ

xyõÓ˚y ~£z !Ó°ÏÎ˚ !òˆÏÎ˚ !Óhfl˛Ïy!Ó˚ì˛ xyˆÏúyâ˛òy Ñ˛Ó˚Ó–

S1V î)Ó˚c ˛ô!Ó˚õy˛ô ı î)Ó˚c õy˛ôˆÏì˛ ˜îá≈ƒñ ≤ÃfiÌñ v zFâ˛ì˛y Á Óƒy¢yï≈ õy˛ôˆÏì˛ £Î˚– ≤ÃˆÏì˛ƒÑ˛!›˛ Ó˚y!ü

î%!›˛ !Ó®%Ó˚ õyˆÏG˛Ó˚ î)Ó˚c !òÓ˚)˛ôí Ñ˛ˆÏÓ˚–

!â˛e 7.1

!â˛e 7.1 ÈÙÈ~ xyõÓ˚y !ì˛ò!›˛ Ñ˛yˆÏ‡˛Ó˚ ›%˛Ñ˛ˆÏÓ˚y Óy ÓœÑ˛ ˆîÖˆÏì˛ ˛ôy!FäÈ– ≤Ã!ì˛!›˛Ó˚ î%!›˛ ≤Ãyhs˘˛¶˛yÜ xyˆÏäÈ

(L, M). ~£z î%!›˛ ≤Ãyhs˘˛¶˛yˆÏÜÓ˚ õˆÏïƒ ~Ñ˛›˛y î)Ó˚c xyˆÏäÈ– !â˛eˆÏì˛ ˆîÖy ÎyˆÏFäÈ ˆÎ (b)ÙÈ~Ó˚ ˆ«˛ Ïe

ˆòy›˛

92 D.El.Ed. 504


x˛ôÓ˚î%!›˛Ó˚ (a, c) ˆÌˆÏÑ˛ ~£z î)Ó˚c!›˛ ˆÓ!ü– xì˛~Ó Óœ ÏÑ˛Ó˚ î%£z ≤Ãyhs˘˛¶˛yˆÏÜÓ˚ õyˆÏG˛Ó˚ î)Ó˚c!›˛

Óœ ÏÑ˛Ó˚ ~Ñ˛!›˛ ˜Ó!üT˛ƒ– ~!›˛ ÏÑ˛ Ó Ïú ˜îá≈ƒ–

S2V ˆ«˛eö˛ú ˛ô!Ó˚õy˛ô ı ≤Ã!ì˛!›˛ !mõy!eÑ˛ xyÑ,˛!ì˛ !Ñ˛ä%È›˛y ãyÎ˚Üy ˛ô!Ó˚ˆÏÓT˛ò Ñ˛ˆÏÓ˚ ÌyˆÏÑ˛– ~£z

˛ô!Ó˚ ÏÓ!T˛ì˛ xÇü!›˛ ÏÑ˛ Ó Ïú ˆ«˛eö˛ú–

!â˛e 7.2

≤Ã!ì˛!›˛ xyÑ,˛!ì˛ Ñ˛yÜˆÏãÓ˚ ~Ñ˛!›˛ !ò!î≈T˛ xÇü !áˆÏÓ˚ Ó˚ˆÏÎ˚ˆÏäÈ– ì˛y£z !mõy!eÑ˛ xyÑ˛yˆÏÓ˚Ó˚ ˆ«˛eö˛ú

Ó˚ˆÏÎ˚ˆÏäÈ–

S3V xyÎ˚ì˛ò ˛ô!Ó˚õy˛ô ı !eõy!eÑ˛ Óhfl% !Ñ˛ä%È›˛y ãyÎ˚Üy îÖú Ñ˛ˆÏÓ˚ ÌyˆÏÑ˛– ~£z ãyÎ˚Üy!›˛ˆÏÑ˛ ÓˆÏú

ì˛yÓ˚ xyÎ˚ì˛ò– ~Ñ˛!›˛ !eõy!eÑ˛ Óhfl% ˆÏÑ˛ SˆÎ!›˛ ãˆÏú oÓí#Î˚ òÎ˚V ãˆÏú ˆv˛yÓyˆÏú !Ñ˛ä%È ˛ô!Ó˚õyí

ãú x˛ô¢y!Ó˚ì˛ £Î˚– ~£z x˛ô¢y!Ó˚ì˛ ãˆÏúÓ˚ ˛ô!Ó˚õy˛ô £ú Óhfl% !›˛Ó˚ xyÎ˚ì˛ò–

S4V ¶˛Ó˚ ˛ô!Ó˚õy˛ô ı õy!›˛ ˆÌˆÏÑ˛ ˆÑ˛yˆÏòy Óhfl%˛ˆÏÑ˛ Á˛ôˆÏÓ˚ ì%˛úˆÏú ˆÑ˛yˆÏòy ˆÑ˛yˆÏòy Óhfl%˛Ó˚ ˆ«˛ˆÏe

xyõyˆÏîÓ˚ ì%˛úˆÏì˛ Ñ˛T˛ £Î˚ xyÓyÓ˚ ˆÑ˛yˆÏòy Óhfl%˛ xyõÓ˚y ¢£ˆÏã£z ì%˛úˆÏì˛ ˛ôy!Ó˚– !eõy!eÑ˛ Óhfl%˛

xyÓyÓ˚ ò#ˆÏâ˛Ó˚ !îˆÏÑ˛ xÌ≈yÍ ˛ô,!ÌÓ#Ó˚ !îˆÏÑ˛ ›˛yò xò%¶˛Ó Ñ˛ˆÏÓ˚– ~£z ›˛yˆÏòÓ˚ ˆÓ!ü Ñ˛ˆÏõÓ˚ v˛z˛ôÓ˚

Óhfl%˛Ó˚ ¶˛Ó˚ !ò¶≈˛Ó˚ Ñ˛ˆÏÓ˚– ÚÓhfl% Ó˚ ¶˛y!Ó˚ £ÁÎ˚yÛ ~£z ˜Ó!üT˛ƒ!›˛ ¶˛Ó˚ ˆÓyG˛yÎ˚–

S5V ¢õÎ˚ ˛ô!Ó˚õy˛ô ı á›˛òy!›˛ !îˆÏòÓ˚ ˆÓ!ü ¢õÎ˚ áˆÏ›˛ˆÏäÈ⁄ xyõyˆÏîÓ˚ Ñ˛yã!›˛ Ñ˛Ó˚ˆÏì˛ Ñ˛ì˛ ¢õÎ˚

úyÜ ÏÓ⁄ ~£z ≤Ã ÏŸ¿Ó˚ v z_Ó˚ !î Ïì˛ xyõy ÏîÓ˚ ¢õÎ˚ õy˛ô Ïì˛ £ ÏÓ–

î%!›˛ ¢î,ü Óhfl% Ó˚ ì% úòyÓ˚ õyïƒ Ïõ ˛ô!Ó˚õy˛ô ı

˛òˆÏÓ˚ü ì˛yÓ˚ °Ï¤˛ ˆ◊í#Ó˚ äÈyeäÈye#ˆÏîÓ˚ !ÓîƒyúˆÏÎ˚Ó˚ ÓyÜyˆÏò ~Ñ˛!›˛ ã!õ ˜ì˛!Ó˚ Ñ˛Ó˚ˆÏì˛ ¢y£yÎƒ

Ñ˛Ó˚!äÈú– ã!õÓ˚ ¢#õyòy !ò!î≈T˛ Ñ˛Ó˚yÓ˚ ¢õÎ˚ îú!›˛ !‡˛Ñ˛ Ñ˛Ó˚ú ˆÎ !Óîƒyú ÏÎ˚Ó˚ ã!õ ˆÌ ÏÑ˛ ~Ñ˛!›˛

xyÎ˚ì˛yÑ˛yÓ˚ ã!õ ì˛yÓ˚y ÓyÓ˚ Ñ˛Ó˚ˆÏÓ ÎyÓ˚ ˜îá≈ƒ ≤ÃˆÏfiÌÓ˚ !mÜ%í– ÎÖò xòƒyòƒ äÈyeäÈye#Ó˚y õy˛ôyÓ˚

Îs˛f Ö%§ãˆÏì˛ Óƒhfl˛Ï ì˛Öò !ò!ì˛ò ~Ñ˛!›˛ î%£yì˛ ú¡∫y úy!‡˛ !òˆÏÎ˚ ã!õÓ˚ ~Ñ˛!îÑ˛ ˆõˆÏ˛ôˆÏäÈ î%ÓyÓ˚

úy!‡˛!›˛ Ó!¢ˆÏÎ˚ Á xòƒ!îˆÏÑ˛ ˆõˆÏ˛ôˆÏäÈ â˛yÓ˚ÓyÓ˚ úy!‡˛!›˛ Ó!¢ˆÏÎ˚–

˛ˆ¢!îò ˆ¢yõÓyÓ˚ Á !Óîƒyú ÏÎ˚Ó˚ ˛ôM˛õ ˆ◊!íÓ˚ äÈyeäÈye# ÏîÓ˚ ˛ôyúy !äÈú Ñ˛y ÏäÈÓ˚ òúÑ) ˛ô

ˆÌˆÏÑ˛ ˛ôyò#Î˚ ãú ~ˆÏò ãˆÏúÓ˚ ˛ôye ¶˛ˆÏÓ˚ Ó˚yÖyÓ˚– ì˛yˆÏîÓ˚ˆÏÑ˛ ~Ñ˛!›˛ ˆäÈyˆÏ›˛y Óyú!ì˛ ˆîÁÎ˚y

£ ÏÎ˚!äÈú ãú xyòyÓ˚ ãòƒ– ì˛yÓ˚y ˆîÖú !Óîƒyú ÏÎ˚Ó˚ ã ÏúÓ˚ ˛ôye!›˛ ¶˛Ó˚ Ïì˛ ì˛y ÏîÓ˚ 18 Óyú!ì˛

ãú úy ÏÜ–

ˆòy›˛



v˛z˛ôˆÏÓ˚y=˛ î%!›˛ v zîy£Ó˚í ïÓ˚y ÎyÑ˛– ≤ÃÌõ v zîy£Ó˚ˆÏíÓ˚ ˆ«˛ˆÏe ã!õÓ˚ ˜îá≈ƒ õy˛ôyÓ˚ ãòƒ ~Ñ˛!›˛ úy!‡˛

ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚!äÈú– xÌ≈yÍ úy!‡˛Ó˚ ˜îˆÏá≈ƒÓ˚ ¢ˆÏAÜ ì% úòy Ñ˛ˆÏÓ˚ ã!õÓ˚ ˜îá≈ƒ !òÓ˚) ôí Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ–

~ˆÏ«˛ˆÏe ã!õÓ˚ ˜îá≈ƒ!›˛ ≤ÃˆÏfiÌÓ˚ !mÜ%í ~£z üì≈˛!›˛ ˛ô)Ó˚í Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ– !mì˛#Î˚ v˛zîy£Ó˚í!›˛ˆÏì˛

!ÓîƒyúˆÏÎ˚Ó˚ ãú Ó˚yÖyÓ˚ ˛ôyˆÏeÓ˚ xyÎ˚ì˛ò ˆÓyG˛yÓ˚ ãòƒ Óyú!ì˛Ó˚ ¢y£yˆÏÎƒ ˆòÁÎ˚y £ˆÏÎ˚ˆÏäÈ–

xyõÓ˚y !Ñ˛ ã!õÓ˚ ˜îá≈ƒ õy˛ôyÓ˚ ãòƒ Óyú!ì˛ Á ãú ˛ôyˆÏeÓ˚ xyÎ˚ì˛ò õy˛ôyÓ˚ ãòƒ úy!‡˛Ó˚ ¢y£yÎƒ

!ò Ïì˛ ˛ôy!Ó˚⁄

~£z v˛zîy£Ó˚ˆÏíÓ˚ õyïƒˆÏõ xyõÓ˚y Ó%G˛ˆÏì˛ ˛ôyÓ˚úyõ ˆÎ ˛ô!Ó˚õy˛ô ~Ñ˛!›˛ ˛ôÂï!ì˛ ÎyÓ˚ õyïƒˆÏõ î%!›˛

¢î,ü Óhfl% Ó˚ ì% úòy Ñ˛Ó˚y £Î˚– ã!õÓ˚ ˜îá≈ƒ Á ≤ÃfiÌ õy˛ôyÓ˚ ãòƒ ~Ñ˛!›˛ !ò!î≈T˛ ˜îˆÏá≈ƒÓ˚ !õ›˛yÓ˚ ˆflÒú

Óy ~Ñ˛!›˛ !ò!î≈T˛ õyˆÏ˛ôÓ˚ úy!‡˛ ÓƒÓ£yÓ˚ Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚– ~Ñ˛£z¶˛yˆÏÓ ãˆÏúÓ˚ ˛ôyˆÏeÓ˚ xyÎ˚ì˛ò

õy˛ôyÓ˚ ãòƒ ~Ñ˛!›˛ !ò!î≈T˛ xyÎ˚ì˛ˆÏòÓ˚ ˛ôye ÓƒÓ£,ì˛ £ˆÏÎ˚ˆÏäÈ ˆÎõò ~Ñ˛ !ú›˛yÓ˚ Ñ˛ƒyò xÌÓy

ˆÓyì˛ú Óy ~Ñ˛!›˛ !ò!î≈T˛ xyÎ˚ì˛ˆÏòÓ˚ Óyú!ì˛– ˆÎˆÏ£ì%˛ !õ›˛yÓ˚ ˆflÒú ~ÓÇ !ú›˛yÓ˚ Ñ˛ƒyò ~Ñ˛£z

˛ô!Ó˚õyˆÏ˛ôÓ˚ Óhfl%˛ òÎ˚–

î%!›˛ ¢î,ü Óhfl%˛Ó˚ ì%˛úòyÓ˚ ãòƒ !òˆÏ¡ÿ=˛ ˛ô§yâ˛!›˛ ˛ôÂï!ì˛Ó˚ ~Ñ˛!›˛ Óy ~ˆÏÑ˛Ó˚ ˆÓ!ü ˛ôÂï!ì˛ ÓƒÓ£yÓ˚

Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚– ~Ü%!ú £úÈÙÙÙÈ

Ñ˛. ˛ôÎ≈ˆÏÓ«˛ˆÏíÓ˚ õyïƒˆÏõ

Ö. ~Ñ˛!›˛ Óhfl%˛Ó˚ v˛z˛ôÓ˚ xyˆÏÓ˚Ñ˛!›˛ xyˆÏÓ˚y˛ô Ñ˛ˆÏÓ˚

Ü. ˛ôˆÏÓ˚y«˛ ˛ôÂï!ì˛Ó˚ õyïƒˆÏõ

á. õyòƒ òÎ˚ ~õò ~Ñ˛Ñ˛ Óy ≤Ãâ˛!úì˛ ~Ñ˛Ñ˛ ÓƒÓ£yˆÏÓ˚Ó˚ õyïƒˆÏõ

à. ≤Ãõyí ~Ñ˛Ñ˛ ÓƒÓ£y ÏÓ˚Ó˚ õyïƒ Ïõ

˛ôÓ˚Óì≈˛# ~Ñ˛ˆÏÑ˛ ˆüˆÏ°ÏÓ˚ î%!›˛ ˛ôÂï!ì˛ !òˆÏÎ˚ xyˆÏúyâ˛òy Ñ˛Ó˚y £ˆÏÓ– xyõÓ˚y ~Öò ˆîÖˆÏÓy Ñ˛#¶˛yˆÏÓ

î%!›˛ ¢î,ü Óhfl%˛Ó˚ ¢y£yˆÏÎƒ ˛ô!Ó˚õy˛ô Ñ˛Ó˚y ÎyÎ˚–

~£z ˛ôÂï!ì˛!›˛ ≤ÃÌõ Á !mì˛#Î˚ ˆ◊!íÓ˚ !üü%ˆÏîÓ˚ ˆ«˛ˆÏe ÓƒÓ£yÓ˚ Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚– ì˛yˆÏîÓ˚ îü!›˛

Ó˚!àò Ñ˛y!‡˛ !îˆÏÎ˚ ì˛yˆÏîÓ˚ ˆäÈyˆÏ›˛y ˆÌˆÏÑ˛ ÓˆÏv˛¸y ¢yãyˆÏì˛ ÓúˆÏì˛ £ˆÏÓ– ˆîÖy ÎyˆÏÓ !üü%Ó˚y Ñ˛y!‡˛Ü%!ú

~ˆÏÑ˛ x˛ôˆÏÓ˚Ó˚ ¢yˆÏÌ ì%˛úòy Ñ˛ˆÏÓ˚ ¢yãyˆÏFäÈ– ~Ñ˛£z¶˛yˆÏÓ ì˛yˆÏîÓ˚ ˛ô§yâ˛!›˛ ˛ôyÌÓ˚ !îˆÏÎ˚ ¶˛y!Ó˚ £yúÑ˛y

xò%ÎyÎ˚# ¢yãyˆÏì˛ Óúy ˆÎˆÏì˛ ˛ôyˆÏÓ˚ÈÙÙÙÈ¢Ó ˆÌˆÏÑ˛ ¶˛y!Ó˚ ˛ôyÌÓ˚!›˛ ≤ÃÌˆÏõ Á £yúÑ˛y ˛ôyÌÓ˚!›˛ ¢Ó

ˆüˆÏ°Ï– Ö%Ó ¢£ˆÏã Á ¢!‡˛Ñ˛¶˛yˆÏÓ ì˛yÓ˚y Ñ˛yã!›˛Ó˚ ¢õyïyò Ñ˛Ó˚ˆÏì˛ ˛ôyÓ˚ˆÏÓ– ~£z v˛zîy£Ó˚íÜ%!ú

¢Ó£z ˛ôÎ≈ˆÏÓ«˛ˆÏíÓ˚ õyïƒˆÏõ £ˆÏÎ˚ˆÏäÈ ˆ◊!íÑ˛ˆÏ«˛ !ü«˛Ñ˛/!ü!«˛Ñ˛yˆÏÑ˛ î%!›˛ Ó˚!àò !ö˛ˆÏì˛ !òˆÏì˛

£ˆÏÓÈÙÙÙÈúyú!›˛ v˛yò £yˆÏì˛ Á ¢Ó%ã!›˛ Ó§y £yˆÏì˛– !ü«˛yÌ≈#ˆÏîÓ˚ !ãK˛y¢y Ñ˛Ó˚ˆÏì˛ £ˆÏÓ ˆÑ˛yò !ö˛ˆÏì˛!›˛

ÓˆÏv˛¸y– ˆÑ˛v˛z ÓúˆÏÓ úyú!›˛ ÓˆÏv ¸y xyÓyÓ˚ ˆÑ˛v z ¢Ó%ã!›˛ ˆîÖyˆÏÓ– ü%ï%õye ˛ôÎ≈ˆÏÓ«˛í Ñ˛ˆÏÓ˚ ¢!‡˛Ñ˛

˛ô!Ó˚õy˛ô Ñ˛Ó˚y ÎyˆÏÓ òy– !ü«˛Ñ˛ !ãK˛y¢y Ñ˛Ó˚ˆÏÓò ÚÚÑ˛#¶˛yˆÏÓ ˆì˛yõÓ˚y ˛ôÓ˚#«˛y Ñ˛Ó˚ˆÏÓ ˆÑ˛yò

ˆòy›˛

94 D.El.Ed. 504


!ö˛ˆÏì˛!›˛ ú¡∫y⁄ÛÛ ~Ñ˛ãò !ü«˛yÌ≈# ~Ñ˛!›˛ !ö˛ˆÏì˛Ó˚ v˛z˛ôÓ˚ xyÓ˚ ~Ñ˛!›˛ !ö˛ˆÏì˛ ˆÓ˚ˆÏÖ ˆîÖú ¢Ó%ã

!ö˛ˆÏì˛!›˛ úyˆÏúÓ˚ ˆÌˆÏÑ˛ ˆÓ!ü ú¡∫y– ~õòÁ xˆÏòÑ˛ ˛ô!Ó˚!fiÌ!ì˛ Ó˚ˆÏÎ˚ˆÏäÈ ˆÎÖyˆÏò ~Ñ˛!›˛ !ã!òˆÏ¢Ó˚

v˛z˛ôÓ˚ xyÓ˚ ~Ñ˛!›˛ ˛Óhfl%˛ ˆÓ˚ˆÏÖ ì%˛úòy Ñ˛Ó˚y ¢Ω˛Ó £Î˚ òy– v˛zîy£Ó˚ífl˛∫Ó˚)˛ô ~Ñ˛!›˛ ¢Ó˚% @’y¢ Á ~Ñ˛!›˛

â˛Áv ¸y @’y¢ !òˆÏÎ˚ ì˛yˆÏîÓ˚ xyÎ˚ì˛ò ¢õˆÏrï ≤ÃŸ¿ Ñ˛Ó˚ˆÏì˛ £ˆÏÓÈÙÙÙÈˆÑ˛yò!›˛Ó˚ xyÎ˚ì˛ò ˆÓ!ü⁄ ~Ñ˛!›˛

˛ôÂï!ì˛ £ˆÏì˛ ˛ôyˆÏÓ˚ ~Ñ˛!›˛ ˛@’yˆÏ¢ ãú ˆë˛ˆÏú ˆ¢£z ãú!›˛ xòƒ @’yˆÏ¢ ë˛yúy– Î!î ≤ÃÌõ @’yˆÏ¢Ó˚

¢¡ô)í≈ ãú !mì˛#Î˚ @’yˆÏ¢ ¶˛Ó˚y ¢ˆÏ_¥Á !mì˛#Î˚ @’y¢!›˛ Öy!ú ÌyˆÏÑ˛ xyõÓ˚y ÓúˆÏì˛ ˛ôy!Ó˚ ˆÎ

!mì˛#Î˚ @’y¢!›˛Ó˚ xyÎ˚ì˛ò ≤ÃÌõ!›˛Ó˚ ˆÌˆÏÑ˛ ˆÓ!ü–

7.3 Non standard/˛≤Ãâ˛!úì˛ ~ÓÇ ≤Ãõyí ~Ñ˛Ñ˛

≤Ã!ì˛!›˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ¢ˆÏAÜ ~Ñ˛!›˛ õyò Óy ¢ÇÖƒy Î%=˛ ÌyˆÏÑ˛ Ñ˛yÓ˚í ˆ¢£z õyò Óy ¢ÇÖƒyÓ˚ õyïƒˆÏõ

xyõÓ˚y ÓˆÏv˛¸y Óy ˆäÈyˆÏ›˛y Ó%G˛ˆÏì˛ ˛ôy!Ó˚– ~£z ¢ÇÖƒyÓ˚ õyïƒˆÏõ Óhfl%˛Ó˚ ì%˛úòy Ñ˛Ó˚y £Î˚– v˛zîy£Ó˚ífl˛∫Ó˚)˛ô

!â˛e 6.1(a)ÈÙÈˆì˛ ÓœÑ˛!›˛Ó˚ ˜îá≈ƒ õy˛ôyÓ˚ ãòƒ ~Ñ˛!›˛ ¢ÇÖƒyÓ˚ ≤ÃˆÏÎ˚yãò ˆ¢£z ¢ÇÖƒyÓ˚ ¢yˆÏÌ xyõÓ˚y

xòƒ ÓœˆÏÑ˛Ó˚ ˜îˆÏá≈ƒÓ˚ ¢ÇÖƒy!›˛ ì%˛úòy Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚– xì˛~Ó Óhfl%˛!›˛Ó˚ ˜îá≈ƒ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ ~Ñ˛!›˛

!ò!î≈T˛ ˜îá≈ƒ ≤ÃˆÏÎ˚yãò ÎyÓ˚ ¢yˆÏÌ ~!›˛ ì% úòy Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚– ~£z !ò!î≈T˛ ˜îá≈ƒˆÏÑ˛ ~Ñ˛Ñ˛ ˜îá≈ƒ

ÓˆÏú– Óhfl%˛!›˛Ó˚ ˜îá≈ƒ Á ~Ñ˛Ñ˛ ˜îˆÏá≈ƒÓ˚ xò% ôyˆÏì˛Ó˚ õyïƒˆÏõ xyõÓ˚y ÓœÑ˛!›˛Ó˚ ˜îá≈ƒ ˛ô!Ó˚õy˛ô Ñ˛Ó˚ˆÏì˛

˛ôyÓ˚Ó–

~Ñ˛£z¶˛y ÏÓ ˆ«˛eö˛ ÏúÓ ~Ñ˛Ñ˛ñ xyÎì˛ ÏòÓ ~Ñ˛Ñ˛ ~ÓÇ ¶˛ ÏÓÓ ~Ñ˛ ÏÑ˛Ó ¢y£y ÏÎƒ ˆ«˛eö˛úñ xyÎì˛ò

Á ¶˛Ó˚ õy˛ôy ˆÎˆÏì˛ ˛ôyˆÏÓ˚– ≤Ã!ì˛!›˛ Ó˚y!üÓ˚ !ò!î≈T˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ~Ñ˛Ñ˛ Ó˚ˆÏÎ˚ˆÏäÈ– ≤ÃˆÏÎ˚yãò Á xÓfiÌy

xò%¢y ÏÓ˚ ~£z ~Ñ˛Ñ˛Ü%!ú !¶˛ß¨ £ Ïì˛ ˛ôy ÏÓ˚– v zîy£Ó˚ífl ∫Ó) ôÈÙÙÙÈ

ãyòúyÓ˚ ˛ôî≈y ›˛yàyˆÏòyÓ˚ Ó˚v˛!›˛ ˆ¶˛ˆÏà ˆÜˆÏäÈ ì˛y£z ÓyãyÓ˚ ˆÌˆÏÑ˛ xyÓ˚ ~Ñ˛!›˛ !Ñ˛ˆÏò xyòˆÏì˛ £ˆÏÓ–

¢!‡˛Ñ˛ õyˆÏ˛ôÓ˚ Ó˚v˛ !òˆÏÎ˚ xy¢yÓ˚ ãòƒ ˆÎ ˆÎ ˛ôÂï!ì˛ ˆòÁÎ˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚ ˆ¢Ü%!ú £úÈÙÙÙÈ

1– ¶˛yày Ó˚v˛!›˛ ˆîyÑ˛yˆÏò !òˆÏÎ˚ ˆ¢£z ~Ñ˛£z õyˆÏ˛ôÓ˚ Ó˚v˛ !Ñ˛ˆÏò xyòy–

2– ~Ñ˛!›˛ úy!‡˛ !îˆÏÎ˚ Ó˚v˛!›˛ ˆõˆÏ˛ô ˆ¢£z ~Ñ˛£z õyˆÏ˛ôÓ˚ úy!‡˛ !òˆÏÎ˚ xòƒ ~Ñ˛!›˛ Ó˚v˛ !Ñ˛ˆÏò xyòy–

3– Ó˚ˆÏv˛Ó˚ ˜îá≈ƒ!›˛ ˛ôy !îˆÏÎ˚ ˆõˆÏ˛ô Ñ˛ì˛ Ú˛ôyÛ Ó˚v˛ ˆúˆÏÜˆÏäÈ ˆ¢£z xò%ÎyÎ˚# Ó˚v˛ ˆÑ˛òy–

4– ¢%ˆÏì˛y !îˆÏÎ˚ Ó˚v˛ ˆõˆÏ˛ô ¢% Ïì˛yÓ˚ õy˛ô xò%ÎyÎ˚# Ó˚v˛ ˆÑ˛òy–

5– !õ›˛yÓ˚ ˆflÒú !î ÏÎ˚ Ó˚ Ïv˛Ó˚ ˜îá≈ƒ õy˛ôy Á Ó˚v˛ ˆÑ˛òy–

v z ôˆÏÓ˚y=˛ 2ñ 3ñ Á 4 òÇ ¢õyïyòÜ%!ú Óƒ!=˛ xò%ÎyÎ˚# ÓîˆÏú ˆÎˆÏì˛ ˛ôyˆÏÓ˚– ~£z v zîy£Ó˚íÜ%!ú

~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ ˛ô!Ó˚õy˛ô ~ÓÇ ~Ó˚ !Ó!¶˛ß¨ ïõ≈Ü%!ú xyõÓ˚y ˛ôÓ˚Óì≈˛# !Ó¶˛yˆÏÜ xyˆÏúyâ˛òy

Ñ˛Ó˚Ó– xòƒ !îˆÏÎ˚ ¢yÓ˚y ˛ô,!ÌÓ#ˆÏì˛ !õ›˛yÓ˚ ˆflÒˆÏúÓ˚ ÓƒÓ£yÓ˚ ˜îá≈ƒ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ~Ñ˛!›˛ ≤Ãõyí ~Ñ˛Ñ˛

ˆòy›˛



!£ˆÏ¢ˆÏÓ Üíƒ– ~Ñ˛ !õ›˛yˆÏÓ˚Ó˚ ˜îá≈ƒ fiÌyòñ Ñ˛yú Á ˛ôye !ò!Ó≈ˆÏüˆÏ°Ï ~Ñ˛– xì˛~Ó ˜îá≈ƒ ˛ô!Ó˚õyˆÏ˛ôÓ˚

~!›˛ ~Ñ˛!›˛ ≤Ãõyí ~Ñ˛Ñ˛–

≤Ãõyí ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ ì% úòyõ)úÑ˛¶˛y ÏÓ ¢£ã Á ¢yÓ˚y ˛ô,!ÌÓ#Ó˚ õyò%°Ï ÏîÓ˚ Ñ˛y ÏäÈ ¢£ã ÏÓyïƒ–

Î!îÁ ~£z ≤Ãõyí ~Ñ˛ˆÏÑ˛Ó˚ ˆÑ˛yˆÏòy î,ë ¸ ˆÎÔ!=˛Ñ˛ì˛y ˆò£zñ ì˛Ó% Ó£% Î%Ü ïˆÏÓ˚ !ÓK˛yò#Ó˚y ÜˆÏÓ°Ïíy

Ñ˛ˆÏÓ˚ ~!›˛ˆÏÑ˛ @˘Ã£íˆÏÎyÜƒ Ñ˛ˆÏÓ˚ ì%˛ˆÏúˆÏäÈ– ì˛y£z ¢!‡˛Ñ˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ ~£z ~Ñ˛Ñ˛Ü%!ú !ò¶%≈˛ú

Ñ˛Ó˚yÓ˚ ˆâ˛T˛y Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ– v z˛ôÜ%!íì˛Ñ˛ ~Ñ˛Ñ˛ SˆÎõò ˆ¢!r›˛!õ›˛yÓ˚ Á !õ!ú!õ›˛yÓ˚ ¢Ó !õ›˛yˆÏÓ˚Ó˚

v˛z˛ô~Ñ˛Ñ˛V ~ÓÇ Ü%!íì˛Ñ˛ ~Ñ˛Ñ˛ SˆÎõò !Ñ˛ˆÏúy!õ›˛yÓ˚ñ !õ›˛yˆÏÓ˚Ó˚ ÓˆÏv˛¸y ~Ñ˛Ñ˛V £ú ≤Ãõyí ~Ñ˛Ñ˛

~ÓÇ ~Ü%!ú ˆÌˆÏÑ˛ !¶˛ß¨–

≤Ãõyí ~Ñ˛Ñ˛ xy!Ó‹ÒyˆÏÓ˚ xyˆÏÜ !Ñ˛ä%È ¢õ¢ƒy Óy â˛y!£îy ˛ô)Ó˚ˆÏíÓ˚ ãòƒ õyòƒ òÎ˚ ~õò ~Ñ˛Ñ˛

ÓƒÓ£yÓ˚ Ñ˛Ó˚y £ì˛– ˛ôyˆÏÎ˚¢ ˜ì˛!Ó˚ Ñ˛Ó˚yÓ˚ ¢õÎ˚ ¢!‡˛Ñ˛¶˛yˆÏÓ â˛yúñ î%ïñ !â˛!òÓ˚ õy˛ô !òï≈yÓ˚í Ñ˛!Ó˚òy–

xyõÓ˚y x!¶˛K˛ì˛yÓ˚ !¶˛!_ˆÏì˛ ~Ñ˛õ% Ï‡˛y â˛yˆÏúÓ˚ ¢ˆÏAÜ ˛ô§yâ˛ â˛yõâ˛ !â˛!ò Á î%@’y¢ î%ï !õ!üˆÏÎ˚

˛ôyˆÏÎ˚¢ ˜ì˛!Ó˚ Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚ Îy !ò!î≈T˛ ˛ô!Ó˚õy˛ô Ñ˛ˆÏÓ˚ ˜ì˛!Ó˚ Ñ˛Ó˚yÓ˚ ˛ôyˆÏÎ˚ˆÏ¢Ó˚ õˆÏì˛y£z ¢õyò

¢%fl ∫yî%– ~£z ˛ôÂï!ì˛ ~Ñ˛ãˆÏòÓ˚ ãòƒ ¢%!ÓïyãòÑ˛ £ˆÏúÁ xòƒãˆÏòÓ˚ ˆ«˛ˆÏe ~!›˛ v z ôˆÏÎyÜ# òyÁ

£ˆÏì˛ ˛ôyˆÏÓ˚ Á ì˛yÓ˚ â˛y!£îy !¶˛ß¨ £ˆÏì˛ ˛ôyˆÏÓ˚–

ˆÑ˛yˆÏòy ~Ñ˛!›˛ ~úyÑ˛y Óy ¢¡±îyˆÏÎ˚Ó˚ õyò%ˆÏ°ÏÓ˚y Ó£%Ñ˛yú ïˆÏÓ˚ ˜îá≈ƒñ ¶˛Ó˚ñ ˆ«˛eö˛ú ~ÓÇ xyÎ˚ì˛ò

õy˛ôyÓ˚ ˆÎ ¢õhfl Ï ≤Ãâ˛!úì˛ ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛ ÏÓ˚ xy¢ ÏäÈ ˆ¢£z ~úyÑ˛y Óy ¢¡±îy ÏÎ˚ Á£z ~Ñ˛Ñ˛Ü%!ú

≤Ãõyí ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ ÓƒÓ£,ì˛ £ˆÏÎ˚ ÌyˆÏÑ˛– xyõÓ˚y ≤ÃˆÏì˛ƒÑ˛ ¢Çfl,Ò!ì˛ˆÏì˛£z ~£z ~Ñ˛Ñ˛ ˆîÖˆÏì˛ ˛ôyÓ

!Ñ˛hs%˛ ˆ¢£z ~Ñ˛Ñ˛Ü%!ú ü%ï%õye ˆ¢£z ¢¡±îyÎ˚ Á ¢Çfl,Ò!ì˛Ó˚ ãòƒ£z ≤Ã ÏÎyãƒ–

!Ñ Î˚yÑ˛úy˛ô ÈÙÈ 1

ˆì˛yõyÓ˚ ~úyÑ˛yÎ˚ ˆÎ ¢õhfl˛Ï ˛ô!Ó˚õyˆÏ˛ôÓ˚ ~Ñ˛Ñ˛ ≤Ãâ˛!úì˛ ì˛yÓ˚ ~Ñ˛!›˛ ì˛y!úÑ˛y Ñ˛ˆÏÓ˚y– ~£z

xyhs˘˛ã≈y!ì˛Ñ˛¶˛yˆÏÓ ≤Ãõy!íì˛ òÎ˚ ~£z ~Ñ˛Ñ˛Ü%!úÓ˚ ¢yˆÏÌ ≤Ãõyí ~Ñ˛ˆÏÑ˛Ó˚ ¢õyò ì˛y !òÓ) ôí Ñ˛ˆÏÓ˚y

~ÓÇ ˆì˛yõyÓ˚ ~úyÑ˛yÓy¢# ÏîÓ˚ ~£z ~Ñ˛Ñ˛Ü%!ú ÓƒÓ£y ÏÓ˚ ¢%!Óïy Á x¢%!Óïy ˆú ÏÖy–

ÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈ

ÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈ


˛≤ÃÜ!ì˛ õy˛ôy

E1 !ÓˆÏŸª ≤Ãõyí ~Ñ˛Ñ˛ Á xyhs˘˛ã≈y!ì˛Ñ˛¶˛yˆÏÓ ≤Ãõy!íì˛ òÎ˚ ~õò ~Ñ˛Ñ˛Ü%!úÓ˚ õˆÏïƒ !ì˛ò!›˛ ˛ôyÌ≈Ñ˛ƒ

ˆúˆÏÖy–

E2 ˛ô!Ó˚õy Ï ôÓ˚ ãòƒ !Ó ÏŸª ≤Ãõyí ~Ñ˛ ÏÑ˛Ó˚ ≤Ã ÏÎ˚yãò#Î˚ì˛y Ñ˛# Ñ˛#⁄

ˆòy›˛

96 D.El.Ed. 504



~Ñ˛!›˛ ¢%ˆÏì˛y/î!v ¸ !î ÏÎ˚ !ò ÏãÓ˚ Ñ˛!∂ Á ÜúyÓ˚ ˜îá≈ƒ õy˛ô– ˆì˛yõyÓ˚ ÜúyÓ˚ ˜îá≈ƒ Ñ˛!∂Ó˚ ˜î Ïá≈ƒÓ˚

Ñ˛ì˛ Ü%í ì˛y !òÓ)˛ôí Ñ˛ˆÏÓ˚y– !òˆÏãÓ˚ v˛z_ˆÏÓ˚Ó˚ ¢ˆÏAÜ ¢£˛ôy‡˛#ˆÏîÓ˚ v˛z_Ó˚ ˆõúyÁ–





@˘Ãyö˛ ˆ˛ô˛ôyˆÏÓ˚ !Ó!¶˛ß¨ xyÑ˛yˆÏÓ˚Ó˚ xyÎ˚ì˛yÑ˛yÓ˚ !â˛e x§yÑ˛– xyÎ˚ì˛yÑ˛yˆÏÓ˚Ó˚ ˜îá≈ƒ Á ≤ÃfiÌ ˆîÁÎ˚y

ÌyÑ˛ Ïú ˆ«˛eö˛ú !òí≈Î˚ Ñ˛Ó˚yÓ˚ !òÎ˚õ ÓyÓ˚ Ñ˛ ÏÓ˚y–




≤Ãõyí ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ ¢%!ÓïyãòÑ˛ £ ÏúÁ !üÖ ÏòÓ˚ ≤ÃÌõ ïy Ï ô xyhs˘˛ã≈y!ì˛Ñ˛¶˛y ÏÓ ≤Ãõy!íì˛ òÎ˚

~õò ~Ñ˛Ñ˛Ü%!ú ÓƒÓ£yÓ˚ Ñ˛Ó˚y ≤Ã ÏÎ˚yãò– !ü«˛yÌ≈# ÏîÓ˚ Ñ˛y ÏäÈ ~£z¢Ó ~Ñ˛Ñ˛Ü%!ú ˆÓ!ü ˛ô!Ó˚!â˛ì˛–

ö˛úì˛ !Ó!¶˛ß¨¶˛yˆÏÓ ì%˛úòy Ñ˛Ó˚yÑ˛yú#ò ì˛yÓ˚y ï#ˆÏÓ˚ ï#ˆÏÓ˚ ≤Ãõyí ~Ñ˛ˆÏÑ˛Ó˚ ≤ÃˆÏÎ˚yãò#Î˚ì˛y Ó%G˛ˆÏì˛

˛ôyÓ˚ˆÏÓ–

≤Ãïyòì˛É î%!›˛ ïÓ˚ˆÏòÓ˚ non standard ~Ñ˛Ñ˛ xyˆÏäÈ– ~£z ïÓ˚ˆÏòÓ˚ ˛ô!Ó˚õyˆÏ˛ô !Óáì˛˛ Óy Ú£yì˛Û Óy

Ú˛ôyˆÏÎ˚Ó˚ ˛ôyì˛yÛ £zì˛ƒy!îˆÏÑ˛ ˜îˆÏá≈ƒÓ˚ ~Ñ˛Ñ˛ ïˆÏÓ˚ ˜îá≈ƒ ˛ô!Ó˚õy˛ô Ñ˛Ó˚ˆÏì˛ !ÜˆÏÎ˚ ˆîÖy ÎyÎ˚ ~ˆÏÑ˛Ñ˛

ãˆÏòÓ˚ ˆ«˛ˆÏe ~ˆÏÑ˛Ñ˛ ˛ô!Ó˚õy˛ô ˛ôyÁÎ˚y ÎyÎ˚– ˆ◊!íÑ˛ˆÏ«˛ !Ó!¶˛ß¨ Óhfl% ˆÎõò ˆ›˛!ÓˆÏúÓ˚ ˜îá≈ƒ

õy˛ôyÓ˚ ¢õÎ˚ xyõÓ˚y ~ ïÓ˚ˆÏòÓ˚ ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚– !Óáì˛ Óy £yˆÏì˛Ó˚ ÓƒÓ£yÓ˚ Ñ˛Ó˚yÓ˚

!Ó!¶˛ß¨ !ü«˛yÌ≈#Ó˚ ˆ«˛ˆÏe ì˛y£z !¶˛ß¨ ˛ô!Ó˚õy˛ô ˛ôyÁÎ˚y ÎyˆÏÓ Îy xyõÓ˚y ~Ñ˛!›˛ äÈˆÏÑ˛Ó˚ ì˛y!úÑ˛y¶%˛=˛

Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚–

≤ÃyE˛úò Á ˛ô!Ó˚õy˛ô ı

!ü«˛yÌ≈#ˆÏîÓ˚ £yì˛ Óy úy!‡˛ !îˆÏÎ˚ ˆ›˛!ÓˆÏúÓ˚ ïyˆÏÓ˚Ó˚ ˜îá≈ƒ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ÓƒÓ£yˆÏÓ˚Ó˚ ˛ô)ˆÏÓ≈ ˆ¢£z

˜îá≈ƒ!›˛ xò%õyò Ñ˛ˆÏÓ˚ ÓúˆÏì˛ Óú%ò– ì˛yˆÏîÓ˚ xò%õyò Á ¢!‡˛Ñ˛¶˛yˆÏÓ !ÓˆÏÓâ˛òy Ñ˛Ó˚yÓ˚ ¢% ÏÎyÜ Ñ˛ˆÏÓ˚

!îò– !ü«˛yÌ≈#ˆÏîÓ˚ ~Ñ˛£z¶˛yˆÏÓ ÜyˆÏäÈÓ˚ ˜îá≈ƒ õˆÏÜÓ˚ xyÎ˚ì˛ˆÏòÓ˚ !¶˛!_ˆÏì˛ Óyú!ì˛Ó˚ xyÎ˚ì˛ò xò%õyò

Ñ˛Ó˚ˆÏì˛ Óú%ò– ≤ÃyE˛úò Óy !ÓˆÏÓâ˛òy Ñ˛#¶˛yˆÏÓ !ü«˛yÌ≈#ˆÏîÓ˚ ˛ôÓ˚Óì≈˛# ˆ«˛ˆÏe ¢y£yÎƒ Ñ˛Ó˚ˆÏÓ⁄

˛ô!Ó˚õyˆÏ˛ôÓ˚ ˆ«˛ˆÏe ≤ÃyE˛úò ~Ñ˛!›˛ Ü%Ó˚%c˛ô)í≈ !Ó°ÏÎ˚– ~Ó˚ ¢y£yˆÏÎƒ !ü«˛yÌ≈# xò%¢rïyò Ñ˛Ó˚ˆÏì˛

£zFä%ÈÑ˛ £Î˚ Á ¶%˛ú Ñ˛Ó˚y £…y¢ ˛ôyÎ˚– ˆÑ˛yˆÏòy Ó˚y!üˆÏÑ˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ˛ô)ˆÏÓ≈ ì˛y xò%õyˆÏòÓ˚ õyïƒˆÏõ

ˆòy›˛



!ü«˛yÌ≈# ¢!‡˛Ñ˛ ~Ñ˛Ñ˛ ÓyäÈy£z Ñ˛Ó˚ˆÏì˛ ¢«˛õ £Î˚– ö˛úì˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ˛ôÂï!ì˛!›˛ ì˛yÓ˚ Ñ˛yˆÏäÈ xyˆÏÓ˚y

Î%!=˛˛ô)í≈ £ˆÏÎ˚ ÁˆÏ‡˛–

ˆÑ˛yˆÏòy ≤ÃyÑ,˛!ì˛Ñ˛ Ó˚y!ü ˛ô!Ó˚õyˆÏ˛ôÓ˚ ˛ô)ˆÏÓ≈ Î!î !ü«˛yÌ≈#Ó˚y ì˛y xò%õyò Ñ˛ˆÏÓ˚ ì˛ˆÏÓ ì˛yˆÏîÓ˚ Ñ˛yˆÏäÈ

ˆ¢£z Ó˚y!üÓ˚ xò%õ!ì˛ õyò/¢ÇÖƒy!›˛ Ü%Ó%c˛ô)í≈ £ˆÏÎ˚ ÁˆÏ‡˛– ˆÎõò ~Ñ˛!›˛ !ü«˛yÌ≈# ì˛yÓ˚ ˆÓˆÏM˛Ó˚

˜î Ïá≈ƒÓ˚ ˛ô!Ó˚õy˛ô ˆ˛ô ÏÎ˚ ÏäÈ 63 ˆ¢!õ SÎ!îÁ ì˛yÓ˚ xy¢ú õyò £ú 37 ˆ¢!õV Ñ˛yÓ˚í ˆ¢ !õ›˛yÓ˚

ˆflÒú!›˛ v˛zˆÏŒ›˛y !îÑ˛ ˆÌˆÏÑ˛ ˆõˆÏ˛ôˆÏäÈ– Î!î !ü«˛yÌ≈#!›˛ xyˆÏúy ˆÌˆÏÑ˛ ˆ¢!›˛ ¢!‡˛Ñ˛ xò%õyò Ñ˛Ó˚ˆÏì˛

˛ôyÓ˚ì˛ SïÓ˚y ÎyÑ˛ 40 ˆ¢!õV ì˛y£ˆÏú ˆ¢ ì˛yÓ˚ ~£z ¶%˛ú!›˛ ¢£ˆÏã£z ïÓ˚ˆÏì˛ ˛ôyÓ˚Ó–

˛˛≤ÃÜ!ì˛ õy˛ôy

E3. !üÖ ÏòÓ˚ ≤ÃÌõ ïy Ï ô non standard ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛Ó˚yÓ˚ ˆÎˆÏÑ˛yˆÏòy î%!›˛ ¢%!Óïy v˛zˆÏÕ‘Ö

Ñ˛ˆÏÓ˚y–

E4. ˆÑ˛yˆÏòy ≤ÃyÑ,˛!ì˛Ñ˛ Ó˚y!üÓ˚ ˛ô!Ó˚õyˆÏ˛ôÓ˚ õyò xò%õyò Ñ˛Ó˚yÓ˚ î%!›˛ ≤ÃˆÏÎ˚yãò#Î˚ì˛yÓ˚ v˛zîy£Ó˚í

îyÁ–

7.3.1 ˜îá≈ƒ ˛ô!Ó˚õyí

xyõÓ˚y xyˆÏÜ£z ˆãˆÏò ~ˆÏ¢!äÈ ˆÎ Ó˚y!üÓ˚ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ˆ«˛ˆÏe ≤Ãâ˛!úì˛ non standard ~Ñ˛Ñ˛Ü%!ú

≤ÃÌˆÏõ ÓƒÓ£yÓ˚ Ñ˛Ó˚y £Î˚– !ü«˛yÌ≈#ˆÏîÓ˚ !îˆÏÎ˚ ≤ÃyÑ, !ì˛Ñ˛ Ó˚y!ü õy˛ôyÓ˚ xyˆÏÜ ì˛y xò%õyò Ñ˛Ó˚ˆÏì˛

˛ôyÓ˚yÓ˚ î«˛ì˛yˆÏÑ˛ v zÍ¢y!£ì˛ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ ÎyÓ˚ ö˛ˆÏú ì˛yÓ˚y ¢!‡˛Ñ˛ ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛Ó˚ˆÏì˛ !üÖˆÏÓ–

!ü«˛yÌ≈#Ó˚ ˛ôy!Ó˚˛ôy!Ÿª≈Ñ˛ ãÜÍ ˆÌˆÏÑ˛ !Ó!¶˛ß¨ !ã!ò¢ ~£z ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ ÓƒÓ£yÓ˚ Ñ˛Ó˚ ˆÎˆÏì˛

˛ôyˆÏÓ˚ ˆÎõò úy!‡˛ñ ì˛yÓ˚ñ ¢%ˆÏì˛yñ ˛ôyì˛yñ Ñ˛yÜãñ xyÑ˛°Ï≈ ≤Ã¶,˛!ì˛– ˆÓ!üÓ˚¶˛yÜ ˆ«˛ˆÏe£z ¢yÓ˚y ˛ô,!ÌÓ#

ã% Ïv ¸ üÓ˚# ÏÓ˚Ó˚ !Ó!¶˛ß¨ xÇü ÏÑ˛ nonstandard ~Ñ˛Ñ˛ !£ Ï¢ ÏÓ ÓƒÓ£yÓ˚ Ñ˛Ó˚yÓ˚ ≤Ãâ˛úò xy ÏäÈ–

~£z¢Ó ~Ñ˛Ñ˛Ü%!ú !Ó!¶˛ß¨ ¶˛y°ÏyÎ˚ !Ó!¶˛ß¨ òy Ïõ ˛ô!Ó˚!â˛ì˛–

˜îá≈ƒ ˛ô!Ó˚õy Ï ôÓ˚ ˆ«˛ Ïe üÓ˚# ÏÓ˚Ó˚ !Ó!¶˛ß¨ xÇ ÏüÓ˚ ~Ñ˛Ñ˛ !£ Ï¢ ÏÓ ÓƒÓ£yÓ˚ÈÙÙÙÈ

xyà%úÈÙÙÙÈ ≤ÃÌõ xAÜ%ú#Ó˚ !Óhfl ÏyÓ˚ / â˛Áv ¸y

!Óáì˛ ÈÙÙÙÈ xAÜ%ú#¢£ £yˆÏì˛Ó˚ ˛ôyì˛yÓ˚ !Óhfl˛ÏyÓ˚

£yì˛ ÈÙÙÙÈ Ñ˛ò%£z ˆÌˆÏÑ˛ õïƒõy xÓ!ï £yˆÏì˛Ó˚ ˜îá≈ƒ

˛ôîˆÏ«˛˛ô ÈÙÙÙÈ â˛úyÓ˚ ˆ«˛ˆÏe ~Ñ˛ ˛ôîˆÏ«˛ˆÏ˛ô ˛ô!Ó˚õy˛ô

Ó§yÁ ÈÙÙÙÈ Ó§y Á v˛yò £yì˛ î%!îˆÏÑ˛ ≤Ã¢y!Ó˚ì˛ Ñ˛Ó˚ˆÏú Ó§y £yˆÏì˛Ó˚ xAÜ%ú# ˆÌˆÏÑ˛ v˛yò £yˆÏì˛Ó˚

xAÜ%ú#Ó˚ î)Ó˚c

£z!M˛ ÈÙÙÙÈ Ó,ÂïyAÜ% Ï¤˛Ó˚ x@˘Ã¶˛yÜ ˆÌˆÏÑ˛ ì˛yÓ˚ Ü§y›˛ Óy xAÜ%ú# ˛ôˆÏÓ≈Ó˚ î)Ó˚c

ˆòy›˛

98 D.El.Ed. 504


!â˛e 7.3

ÓÎ˚¢ Á !ü«˛y !ò!Ó≈ Ïü Ï°Ï ˜îá≈ƒ ˛ô!Ó˚õy Ï ôÓ˚ ãòƒ ≤Ãâ˛!úì˛ non standard ~Ñ˛ˆÏÑ˛Ó˚ ÓƒÓ£yÓ˚

ˆ«˛eö˛ú Á xyÎ˚ì˛ò õy˛ôyÓ˚ ˆÌˆÏÑ˛ ì% úòyõ)úÑ˛¶˛yˆÏÓ ˆÓ!ü ≤Ãâ˛!úì˛– ≤ÃyÌ!õÑ˛¶˛yˆÏÓ Non stan-

dard ~Ñ˛ˆÏÑ˛Ó˚ ÓƒÓ£yÓ˚ !Ó!¶˛ß¨ Ñ˛yÓ˚ˆÏí Ü%Ó%c˛ô)í≈ ı

SÑ˛V ˜îá≈ƒ ˛ô!Ó˚õyˆÏ˛ôÓ˚ î«˛ì˛y xã≈ˆÏòÓ˚ ˆ«˛ˆÏe !ü«˛yÌ≈#Ó˚y ì˛yˆÏîÓ˚ ˛ô!Ó˚!â˛ì˛ ãÜˆÏì˛Ó˚ !Ó!¶˛ß¨ Óhfl%

≤Ãâ˛!úì˛ /non standard ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ ÌyˆÏÑ˛ ö˛ˆÏú òì%˛ò òyõ Óy ü∑ ì˛yˆÏîÓ˚

xyÎ˚_ Ñ˛Ó˚ˆÏì˛ £Î˚òy–

SÖV ≤ÃyÌ!õÑ˛¶˛y ÏÓ ˜îá≈ƒ ˛ô!Ó˚õy Ï ô ≤Ãâ˛!úì˛/ non standard ~Ñ˛ ÏÑ˛Ó˚ ÓƒÓ£yÓ˚ ˆÓ!ü v z ôÎ%=˛–

!òˆÏãÓ˚ Órï%Ó˚ v˛zFâ˛ì˛y ˛ô!Ó˚õy˛ô xÌÓy ˆ›˛!Óú Óy ˆÓˆÏM˛Ó˚ ïyÓ˚ õy˛ôyÓ˚ ãòƒ ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ

ˆ¢!r›˛!õ›˛yÓ˚ ÓƒÓ£yÓ˚ Ñ˛Ó˚ Ïú Úõyò/¢ÇÖƒyÛ!›˛ x ÏòÑ˛ ˆÓ!ü £ ÏÓ–

SÜV Non-standard ~Ñ˛Ñ˛ ÓƒÓ£yˆÏÓ˚ !ü«˛yÌ≈#ˆÏîÓ˚ v˛zÍ¢y!£ì˛ Ñ˛Ó˚ˆÏú ì˛yˆÏîÓ˚ õˆÏò ˆÑ˛yˆÏòy ~Ñ˛!›˛

!ò!î≈T˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ ≤Ãõyí ~Ñ˛Ñ˛ xy!Ó‹ÒyˆÏÓ˚Ó˚ ì˛y ÓƒÓ£yˆÏÓ˚Ó˚ ïyÓ˚íy ˜ì˛!Ó˚ £ˆÏÓ–

SáV Non-standard ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛Ó˚ Ïì˛ Ñ˛Ó˚ Ïì˛ !üü% õò ≤Ãõyí ~Ñ˛ ÏÑ˛Ó˚ ≤Ã ÏÎ˚yãò#Î˚ì˛y

xò%¶˛Ó Ñ˛Ó˚ˆÏÓ– ÎÖò ì˛yÓ˚y ˆîÖˆÏÓ ˆ›˛!ÓˆÏúÓ˚ ïyˆÏÓ˚Ó˚ ˜îˆÏá≈ƒÓ˚ ˛ô!Ó˚õy˛ô ì˛yˆÏîÓ˚ £yì˛ xò%ÎyÎ˚# 2

£yì˛ xyÓyÓ˚ ì˛yˆÏîÓ˚ !ü«˛Ñ˛/!ü!«˛Ñ˛yÓ˚ £yì˛ xò%ÎyÎ˚# ì˛y Ñ˛ˆÏõ 11/2 £yì˛ £Î˚– ì˛yÓ˚y ï# ÏÓ˚ ï# ÏÓ˚

Ó%G˛ˆÏì˛ ˛ôyÓ˚ˆÏÓ ~Ñ˛£z Óhfl%˛Ó˚ ˜îá≈ƒÓ˚ ˛ô!Ó˚õy˛ô !¶˛ß¨ !¶˛ß¨ !ü«˛yÌ≈#Ó˚ ˆ«˛ˆÏe !¶˛ß¨ £ˆÏFäÈ ì˛Öò ì˛yÓ˚y

≤Ãõyí ~Ñ˛ ÏÑ˛Ó˚ ≤Ã ÏÎ˚yãò#Î˚ì˛y xò%¶˛Ó Ñ˛Ó˚ ÏÓ–


ˆì˛yõyÓ˚ ˛ôy!Ó˚˛ôy!Ÿª≈Ñ˛ ãÜˆÏì˛ ˆÎ ¢õhfl˛Ï Óhfl% ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ ÓƒÓ£yÓ˚ Ñ˛Ó˚y ÎyÎ˚

ì˛yÓ˚ ~Ñ˛!›˛ ì˛y!úÑ˛y ÓyòyÁ




Hand

Thumb

First finger

Hand span

Ó%ˆÏv˛¸y xyAÜ%ú

ì˛ã≈!ò

!Óáì˛

£yì˛

ˆòy›˛



˜îá≈ƒÓ˚ xyhs˘˛ã≈y!ì˛Ñ˛ ≤Ãõyí ~Ñ˛Ñ˛ ı v zFâ˛ ≤ÃyÌ!õÑ˛ hfl Ï ÏÓ˚ ˜îá≈ƒÓ˚ ≤Ãõyí ~Ñ˛Ñ˛ !£ Ï¢ ÏÓ ˆ¢!r›˛!õ›˛yÓ˚ñ

!õ›˛yÓ˚ñ !õ!ú!õ›˛yÓ˚ Á !Ñ˛ˆÏúy!õ›˛yˆÏÓ˚Ó˚ ÓƒÓ£yÓ˚ £Î˚– ≤ÃyÌ!õÑ˛¶˛yˆÏÓ !üü%Ó˚ Ñ˛yˆÏäÈ !Ñ˛ˆÏúy!õ›˛yˆÏÓ˚Ó˚

ïyÓ˚íy Ö%Ó fl˛ôT˛¶˛yˆÏÓ £Î˚ òy– ¢%ì˛Ó˚yÇ ~£z ˛ôÎ≈yÎ˚ !ü«˛yÌ≈#ˆÏîÓ˚ ≤Ãõyí ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ !õ›˛yÓ˚ ~ÓÇ

v˛z˛ôÜ%!íì˛Ñ˛ ~Ñ˛Ñ˛ ˆ¢!r›˛!õ›˛yÓ˚ Á !õ!ú!õ›˛yˆÏÓ˚Ó˚ ÓƒÓ£yˆÏÓ˚Ó˚ ¢ˆÏAÜ ˛ô!Ó˚!â˛ì˛ Ñ˛Ó˚yˆÏì˛ £ˆÏÓ–

!ü«˛yÌ≈#ˆÏîÓ˚ Ñ˛yˆÏäÈ Ñ˛Öò Á Ñ˛#¶˛yˆÏÓ ˜îá≈ƒ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ≤Ãõyí ~Ñ˛ˆÏÑ˛Ó˚ ≤ÃÓì≈˛ò/v˛z˛ôfiÌy!˛ôì˛

Ñ˛Ó˚ˆÏì˛ £ˆÏÓ⁄

!ü«˛yÌ≈#Ó˚y ÎÖò ˆÑ˛yˆÏòy Ó˚y!ü ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ ≤Ãõyí ˆflÒˆÏúÓ˚ ≤ÃˆÏÎ˚yãò#Î˚ì˛y xò%¶˛Ó Ñ˛Ó˚ˆÏì˛

ì˛Öò£z ≤Ãõyí ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ !õ›˛yÓ˚ Á ˆ¢!r›˛!õ›˛yˆÏÓ˚Ó˚ v z ôfiÌy˛ôò Ñ˛Ó˚ˆÏì˛ £ˆÏÓ– ≤ÃyÌ!õÑ˛¶˛yˆÏÓ

!ü«˛yÌ≈# ÏîÓ˚ non standard ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ ˆÎ ˆÑ˛yˆÏòy Óhfl%˛Ó˚ ˜îá≈ƒ ˛ô!Ó˚õyˆÏ˛ôÓ˚ î«˛ì˛y

xã≈ò Ñ˛Ó˚ Ïì˛ £ ÏÓ ~ÓÇ ≤ÃyÆ õyò Á ~Ñ˛Ñ˛ ~Ñ˛¢y ÏÌ ≤ÃÑ˛yü Ñ˛Ó˚ Ïì˛ £ ÏÓ ˆÎõò 3 £yì˛ñ 2

!Óáì˛ 5 ˛ôîˆÏ«˛˛ô ≤Ã¶,˛!ì˛– !mì˛#Î˚yˆÏï≈ ÎÖò ì˛yÓ˚y v˛z˛ôˆÏÓ˚y=˛ î«˛ì˛y xã≈ò Ñ˛Ó˚ˆÏì˛ ˛ôyÓ˚ˆÏÓ ~ÓÇ

¢!‡˛Ñ˛ ˛ô!Ó˚õy˛ô Ñ˛# £ ÏÓ ˆ¶˛ ÏÓ ì˛y ÏîÓ˚ õ Ïò ≤ÃŸ¿ ãyÜ Ïì˛ ˆ¢£z ¢õÎ˚ Ñ˛y!‡˛/úy!‡˛ ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ

ÓƒÓ£yÓ˚ Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚– ˛ôˆÏÓ˚ ÎÖò ì˛yÓ˚y ~£z Ñ˛y!‡˛ Óy úy!‡˛ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ ˜îá≈ƒ ˛ô!Ó˚õyˆÏ˛ô

¢«˛õ £ˆÏÓ ì˛Öò ≤Ãõyí ˆflÒú xÌ≈yÍ !õ›˛yÓ˚ ˆflÒú v˛z˛ôfiÌy!˛ôì˛ Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚ ~ÓÇ ì˛y

!ü«˛yÌ≈#ˆÏîÓ˚ ˛ôˆÏ«˛ v z˛ôˆÏÎyÜ# £ˆÏÓ–

!õ›˛yÓ˚ ˆflÒˆÏúÓ˚ ¢y£yˆÏÎƒ ˛ô!Ó˚õy˛ô ı !ü«˛yÌ≈#ˆÏîÓ˚ úy!‡˛Ó˚ ¢y£yˆÏÎƒ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ¢õÎ˚ !Ñ˛ä%È

!Ó°ÏÎ˚ÈÙÙÙÈòãÓ˚ !î Ïì˛ £ ÏÓ ˆÎõò Óhfl% Ó˚ ~Ñ˛!î ÏÑ˛Ó˚ ≤Ãyhs˘˛¶˛y ÏÜÓ˚ ¢y ÏÌ úy!‡˛Ó˚ ≤Ãyhs˘˛¶˛yÜ ~Ñ˛£z

!Ó®% Ïì˛ !õ!úì˛ £ˆÏÓ– ˆÓˆÏM˛Ó˚ ˜îá≈ƒ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ úy!‡˛Ó˚ ~Ñ˛!›˛ ≤Ãyhs˘˛¶˛yÜ ˆÓˆÏM˛Ó˚ ~Ñ˛!îˆÏÑ˛Ó˚

≤Ãyhs˘˛¶˛yˆÏÜÓ˚ ¢ˆÏAÜ ~Ñ˛£z !Ó®%ˆÏì˛ !õ!úì˛ £ˆÏÓ– úy!‡˛Ó˚ x˛ôÓ˚ ≤Ãyhs˘˛¶˛yÜ!›˛ ˆÎÖyˆÏò Ó˚ˆÏÎ˚ˆÏäÈ

ˆÓ ÏM˛ ˆ¢£z ãyÎ˚Üy!›˛ !â˛!£´ì˛ Ñ˛Ó˚ Ïì˛ £ ÏÓ– ~Ó˚ ôÓ˚ úy!‡˛›˛y ì% Ïú !ò ÏÎ˚ ˆÓ ÏM˛Ó˚ !â˛!£´ì˛ ãyÎ˚Üy!›˛ Ïì˛

úy!‡˛Ó˚ ≤Ãyhs˘˛¶˛yÜ!›˛ xyÓyÓ˚ !Ó®%Ó˚ ¢ˆÏAÜ !õ!úˆÏÎ˚ Ó˚yÖˆÏì˛ £ˆÏÓ Á úy!‡˛Ó˚ ˆü°Ï ≤Ãyhs˘˛¶˛yÜ!›˛ ˆÓˆÏM˛

!â˛!£´ì˛ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ– ~£z¶˛yˆÏÓ !ü«˛yÌ≈#Ó˚y Ó%G˛ˆÏì˛ ˛ôyÓ˚ˆÏÓ ˆÓˆÏM˛Ó˚ ˜îá≈ƒ!›˛ ˛ô!Ó˚õy˛ô Ñ˛Ó˚ˆÏì˛ Ñ˛ˆÏì˛y

ÓyÓ˚ úy!‡˛›˛y Ó¢yˆÏì˛ £ˆÏFäÈ xÌ≈yÍ ˆÓˆÏM˛Ó˚ ˜îá≈ƒ!›˛ úy!‡˛Ó˚ ˜îá≈ƒÓ˚ Ñ˛ì˛Ü%í ˆÓ!ü– ~£zÓyÓ˚ ì˛yÓ˚

!õ›˛yÓ˚ ˆflÒú ÓƒÓ£yˆÏÓ˚Ó˚ ãòƒ ≤Ãhfl%˛ì˛ £ˆÏÓ–

!â˛e 7.4

ˆòy›˛

100 D.El.Ed. 504


v z˛ôˆÏÓ˚y=˛ ˛ôÂï!ì˛ˆÏ£z !õ›˛yÓ˚ ˆflÒú ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ ˜îá≈ƒ õy˛ôˆÏì˛ £Î˚– !õ›˛yÓ˚ ˆflÒú ÓƒÓ£yÓ˚ Ñ˛Ó˚yÓ˚

¢õÎ˚ !ü«˛yÌ≈#ˆÏîÓ˚ !Ñ˛ä%È !Ó°ÏˆÏÎ˚ !ÓˆÏü°Ï òãÓ˚ !îˆÏì˛ £ˆÏÓ ÎyˆÏì˛ ì˛yÓ˚y ¢!‡˛Ñ˛¶˛yˆÏÓ ˜îá≈ƒ ˛ô!Ó˚õy˛ô

Ñ˛Ó˚ˆÏì˛ ˛ôyˆÏÓ˚Ê

ˆflÒ ÏúÓ˚ v z ô~Ñ˛ ÏÑ˛Ó˚ ¢ ÏAÜ !ü«˛yÌ≈#Ó˚ ˛ô!Ó˚!â˛!ì˛ ı

ì˛yÓ˚y Ó%G˛ˆÏì˛ ˛ôyÓ˚ˆÏÓ ˆÎ ˆflÒˆÏúÓ˚ ≤Ã!ì˛!›˛ xÇü ¢õõyˆÏòÓ˚– ì˛yÓ˚y !õ›˛yÓ˚ ˆflÒˆÏú ˆ¢!r›˛!õ›˛yˆÏÓ˚Ó˚

¶˛yÜÜ%!ú !â˛!£´ì˛ Ñ˛Ó˚ˆÏì˛ ˛ôyÓ˚ˆÏÓ– ≤ÃÌˆÏõ ì˛yÓ˚y ˜îá≈ƒ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ ˆ¢!r›˛!õ›˛yÓ˚ ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ

ÓƒÓ£yÓ˚ Ñ˛Ó˚ˆÏÓ Á ˛ôˆÏÓ˚ !õ›˛yÓ˚ Á !õ!ú!õ›˛yˆÏÓ˚Ó˚ ¢ˆÏAÜ ˛ô!Ó˚!â˛ì˛ £ˆÏÓ–

!â˛e 7.5 !õ›˛yÓ˚ ˆflÒú

Óhfl%˛Ó˚ v˛z˛ôÓ˚ ¢!‡˛Ñ˛¶˛yˆÏÓ ˆflÒú!›˛ Ó!¢ˆÏÎ˚ ˛ô!Ó˚õy˛ô Ñ˛Ó˚y ~ÓÇ ¢ÇÖƒy˛ ˛ô!Ó˚õy˛ô ˆòÁÎ˚y Ö%Ó

Ü%Ó%c˛ô)í≈– ~£z ˛˛ôÂï!ì˛!›˛ !ü«˛yÌ≈#ˆÏîÓ˚ Ñ˛ˆÏÓ˚ ˆîÖyˆÏì˛ £ˆÏÓ– !ü«˛yÌ≈#ˆÏîÓ˚ ¢!‡˛Ñ˛¶˛yˆÏÓ ˆflÒú!›˛ Óhfl% Ó˚

v˛z˛ôÓ˚ Ó˚yÖˆÏì˛ £ˆÏÓ ÎyˆÏì˛ ˆflÒˆÏúÓ˚ ‘0’ !â˛£´!›˛ Óhfl% Ó˚ ~Ñ˛!îˆÏÑ˛Ó˚ ≤Ãyhs˘˛¶˛yˆÏÜÓ˚ ¢ˆÏAÜ ~Ñ˛!›˛ !Ó®% Ïì˛

!õ!úì˛ £Î˚–

!â˛e 7.6

!ü«˛yÌ≈#ˆÏîÓ˚ ˆflÒú!›˛ ~õò¶˛yˆÏÓ Óhfl% Ó˚ ~Ñ˛!îˆÏÑ˛ ≤Ãyhs˘˛¶˛yˆÏÜ Ó¢yˆÏì˛ ÓúˆÏì˛ £ˆÏÓ ÎyˆÏì˛ ˆ¢!›˛

ˆòy›˛



ˆflÒ ÏúÓ˚ 1/2/3 !â˛ Ï£´Ó˚ ¢ ÏAÜ !õ!úì˛ £Î˚ S‘0’ äÈyv ¸yV– ~Ó˚˛ôÓ˚ î%!›˛ õyˆÏòÓ˚ !ÓˆÏÎ˚yÜö˛ú ˆÌˆÏÑ˛

Óhfl%˛Ó˚ ˜îá≈ƒÓ˚ ˛ô!Ó˚õy˛ô!›˛ ˆÓ!Ó˚ˆÏÎ˚ xy¢ˆÏÓ–

!â˛e 7.7

!òí≈Î˚ Ñ˛Ó˚yÓ˚ xyˆÏÜ ˜îá≈ƒ!›˛ xò%õyò Ñ˛Ó˚y– ˛ô) ÏÓ≈ xyˆÏúyâ˛òy Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ ˆÎ !ü«˛yÌ≈#ˆÏîÓ˚

≤Ãõyí ˆflÒú !îˆÏÎ˚ Óhfl%˛Ó˚ ˜îá≈ƒ õy˛ôyÓ˚ ˛ô)ˆÏÓ≈ ˆ¢!›˛ xò%õyò Ñ˛Ó˚yÓ˚ ãòƒ v˛zÍ¢y!£ì˛ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ–

¢!‡˛Ñ˛¶˛y ÏÓ ˜îá≈ƒ !òí≈Î˚ Ñ˛Ó˚y– ≤ÃyÑ, !ì˛Ñ˛ Ó˚y!üÓ˚ ˜îá≈ƒ õy˛ôyÓ˚ ãòƒ !ü«˛yÌ≈# ÏîÓ˚ ÓyÓ˚ ÓyÓ˚

!Ó!¶˛ß¨ Óhfl% Ó˚ ˆflÒú !î ÏÎ˚ ˜îá≈ƒ õy˛ôyÓ˚ x¶˛ƒy¢ Ñ˛Ó˚ Ïì˛ £ ÏÓ–

≤ÃÌˆÏõ !ü«˛yÌ≈# ˆflÒˆÏúÓ˚ ‘0’ !Ó®%Ó˚ ¢ˆÏAÜ Óhfl% Ó˚ ~Ñ˛!îˆÏÑ˛ ≤Ãyhs˘˛¶˛yÜ!›˛ !õ!úˆÏÎ˚ Ó¢yˆÏÓ–

Óhfl% Ó˚ xòƒ ≤Ãyhs˘˛¶˛yÜ!›˛ ˆflÒˆÏú !â˛!£´ì˛ ˆÎ ¢ÇÖƒy!›˛ Óy !Ó˚!v˛Ç ˆîÖyˆÏÓ ˆ¢!›˛£z £ú Óhfl% Ó˚ ˜îá≈ƒ–

Î!î !ü«˛yÌ≈# ˆflÒˆÏúÓ˚ ‘0’ !Ó®%Ó˚ ˛ô!Ó˚ÓˆÏì≈˛ xòƒ ˆÑ˛yˆÏòy !Ó®%ˆÏì˛ SÎÌy 1, 2 Óy 3) Ó¢yÎ˚

ì˛y£ˆÏú ˆflÒˆÏúÓ˚ î%£z ≤Ãyhs˘˛ !Ó®%Ó˚ !ÓˆÏÎ˚yÜö˛ú£z £ˆÏÓ Óhfl% Ó˚ ˜îá≈ƒ–

˜îˆÏá≈ƒÓ˚ ~Ñ˛Ñ˛ !òˆÏÎ˚ Ñ˛yã Ñ˛Ó˚yÓ˚ ¢õÎ˚ !ü«˛yÌ≈#Ó˚y ~õò !ã!ò¢ !òˆÏÎ˚ ˜îá≈ƒ õy˛ôˆÏÓ ÎyˆÏì˛

˜îˆÏá≈ƒÓ˚ õyò ˛ô)í≈ ¢ÇÖƒy £Î˚ ˆÎõò !ü«˛yÌ≈#Ó˚y Î!î ì˛yÓ˚ñ Ó˚v˛ñ úy!‡˛ Ñ˛yÜˆÏãÓ˚ !fi›˛˛ô !òˆÏÎ˚ Ñ˛yã

Ñ˛ˆÏÓ˚ ì˛ˆÏÓ ì˛yˆÏîÓ˚ ˛ô!Ó˚õy˛ô ˛ô)í≈ ~Ñ˛ˆÏÑ˛ ˆÓÓ˚ˆÏÓ SxÌ≈yÍ 3 ˆ¢!õñ 5 ˆ¢!õñ 10 ˆ¢!õV– ˛ôˆÏÓ˚

ì˛yˆÏîÓ˚ v z ôÜ%!íì˛Ñ˛ ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛Ó˚ˆÏì˛ !îˆÏì˛ £ˆÏÓ– SxÌ≈yÍ !õ!ú!õ›˛yÓ˚V–

˛ô!Ó˚õyˆÏ˛ôÓ˚ î«˛ì˛y xã≈ò ÈÙÙÙÈ ≤Ãõyí ˆflÒú ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ ˜îá≈ƒ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ˆ«˛ˆÏe !Ñ˛ä%È !Ó°ÏÎ˚

õyÌyÎ˚ Ó˚yÖˆÏì˛ £ˆÏÓ ˆÎõò ¢!‡˛Ñ˛¶˛yˆÏÓ ˆflÒú!›˛ Óhfl% Ó˚ Á˛ôÓ˚ Ó¢yˆÏòyñ ¢!‡˛Ñ˛ !Ó˚!v˛Ç ˆòÁÎ˚yñ xÌÓy

!Ó˚!v˛Ç ˆÌˆÏÑ˛ ˜îá≈ƒ !ÓˆÏÎ˚yÜ Ñ˛Ó˚y– ~Ó˚ õˆÏïƒ ˆÑ˛yˆÏòy›˛y£z !ü«˛yÌ≈#ˆÏîÓ˚ ˛ôˆÏ«˛ xã≈ò Ñ˛Ó˚y Ñ˛!‡˛ò òÎ˚

!Ñ˛hs%˘˛ x¢yÓïyòì˛y Óüì˛É ì˛yÓ˚y õyˆÏG˛ õyˆÏG˛ ¶%˛ú Ñ˛Ó˚ˆÏì˛ ˛ôyˆÏÓ˚– ≤ÃÌõ ˆÌˆÏÑ˛£z Î!î ˆ¢!îˆÏÑ˛ òãÓ˚

ˆîÁÎ˚y ÎyÎ˚ ì˛y£ Ïú ì˛yÓ˚y ¢£ Ïã£z î«˛ì˛y xã≈ò Ñ˛Ó˚ Ïì˛ ˛ôyÓ˚ ÏÓ–

v˛z˛ôÎ%=˛ ˆflÒú ÓyäÈy£z ÓyãyˆÏÓ˚ !Ó!¶˛ß¨ ïÓ˚ˆÏòÓ˚ Á õyˆÏ˛ôÓ˚ ˆflÒú ˛ôyÁÎ˚y ÎyÎ˚– !ÓîƒyúˆÏÎ˚ ˆÎ

¢õhfl Ï ˆflÒú ÓƒÓ£yÓ˚ Ñ˛Ó˚y £Î˚ ˆ¢Ü%!ú £úÈÙÙÙÈ1 !õ›˛yÓ˚ñ 30 ˆ¢!r›˛!õ›˛yÓ˚ Á 15 ˆ¢!r›˛!õ›˛yÓ˚–

ˆòy›˛

102 D.El.Ed. 504


~äÈyv˛¸yÁ Ñ˛y‡˛ !õ!fl˛f Óy Ó˚yã!õ!fl˛f ÓƒÓ£,ì˛ õy˛ôyÓ˚ !ö˛ˆÏì˛ Á Ñ˛y˛ôˆÏv˛¸Ó˚ ˆîyÑ˛yˆÏò ÓƒÓ£,ì˛ Ó˚ˆÏv˛Ó˚

¢ ÏAÜÁ ì˛y ÏîÓ˚ ˛ô!Ó˚!â˛!ì˛ á›˛y Ïì˛ £ ÏÓ– ≤ÃÌõ!î ÏÑ˛ ˆ◊!íÑ˛ Ï«˛ ≤Ãõyí ~Ñ˛ ÏÑ˛Ó˚ ¢y£y ÏÎƒ ˜îá≈ƒ

˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ 30 ˆ¢!õ ˜îˆÏá≈ƒÓ˚ ˆflÒú ÓƒÓ£yÓ˚ Ñ˛Ó˚y ÓyN˛ò#Î˚ Ñ˛yÓ˚í ˆÎ ¢õhfl˛Ï Óhfl% !ü«˛yÌ≈#Ó˚y

˛ô!Ó˚õy˛ô Ñ˛Ó˚ˆÏì˛ ˆ¢Ü%!ú ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ ~£z ˆflÒú!›˛ v˛z˛ôˆÏÎyÜ#– ú¡∫y Óhfl%˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ

ÓˆÏv˛¸y ˆflÒú Óy õy˛ôyÓ˚ !ö˛ˆÏì˛ ÓƒÓ£yÓ˚ Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚– !ÓîƒyúˆÏÎ˚Ó˚ ˆ◊!íÑ˛ˆÏ«˛ Óy ÓyÓ˚y®y

õy˛ôyÓ˚ ãòƒ ÓˆÏv˛¸y !õ›˛yÓ˚ ˆflÒú ÓƒÓ£yÓ˚ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ xyÓyÓ˚ ÎÖò Öyì˛yÎ˚ x§yÑ˛y ˆÓ˚Öy Óy ˆäÈyˆÏ›˛y

Óhfl% õy˛ôyÓ˚ ãòƒ 15 ˆ¢!õ ú¡∫y ˆflÒú ÓƒÓ£yÓ˚ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ–

Óhfl%˛Ó˚ÈÙÈ ˜îá≈ƒ õy˛ôy !ü«˛yÌ≈#ˆÏîÓ˚ ãòƒ ¢£ã– ~Ó˚ãòƒ !ü«˛yÌ≈#ˆÏîÓ˚ ˆ◊!íÑ˛ˆÏ«˛Ó˚ !¶˛ì˛ˆÏÓ˚ Á

Óy£zˆÏÓ˚ !Ó!¶˛ß¨ !Ñ ˛Î˚yÑ˛úyˆÏ˛ô v˛zÍ¢y!£ì˛ ˆîÁÎ˚y £úÈÙÙÙÈ

Non-standard ˆflÒú ˜ì˛!Ó˚ ˆÎõò úy!‡˛Ó˚ ¢õxÇü !â˛!£´ì˛ Ñ˛ˆÏÓ˚ ì˛y Ó˚à Ñ˛Ó˚y

ˆ¢!õ ˆflÒú ˜ì˛!Ó˚ Ñ˛Ó˚y ˆÎõò 10 ˆ¢!õñ 15 ˆ¢!õñ 20 ˆ¢!õ £zì˛ƒy!î– ~Ñ˛ ~Ñ˛!›˛ ˆflÒˆÏú 1

ˆ¢!õ !â˛!£´ì˛ Ñ˛Ó˚ˆÏì˛ !¶˛ß¨ Ó˚à ÓƒÓ£yÓ˚ Ñ˛Ó˚y–

!Ó!¶˛ß¨ ˆÖúy ˆÎõò Ñ˛yÓy!v˛ñ ˆÖy ˆÖyñ Óƒyv˛!õr›˛ò ˆÖúyÓ˚ ãòƒ ˆÖúyÓ˚ õy‡˛ ˆÓ˚Öy!AÑ˛ì˛

Ñ˛Ó˚y–

úà ãy¡ôñ £y£z ãy Ï¡ô xÇü@˘Ã£í Ñ˛Ó˚y Á ≤Ã!ì˛ Ï«˛ Ïe ˜îá≈ƒ õy˛ôy

!Ó!¶˛ß¨ ˜îˆÏá≈ƒÓ˚ ˆÓ˚ÖyÇü !îˆÏÎ˚ òÑ˛üy x§yÑ˛y–

7.3.2 ˆ«˛eö˛ú ˛ô!Ó˚õy˛ô

ˆÑ˛yˆÏòy !ã!òˆÏ¢Ó˚ ˆ«˛eö˛ú õy˛ôyÓ˚ xyˆÏÜ xyõyˆÏîÓ˚ xyˆÏÜ ãyòˆÏì˛ £ˆÏÓ ˆ«˛eö˛ú Ñ˛yˆÏÑ˛ ÓˆÏú–

xyõÓ˚y xyˆÏÜ ˆîˆÏÖ!äÈ ˆÓ˚ÖyÇüˆÏÑ˛ ˆflÒˆÏúÓ˚ ¢y£yˆÏÎƒ õy˛ôˆÏì˛ £Î˚– xyõÓ˚y ~Ñ˛õy!eÑ˛ Óhfl% Ó˚ ˜îá≈ƒ

˛ô!Ó˚õy˛ô Ñ˛!Ó˚ ˆÎõò ˆ›˛!ÓˆÏúÓ˚ ïyÓ˚ñ ì˛yÓ˚ñ õƒyˆÏ˛ôÓ˚ ˆÓ˚Öyñ ˆ˛ôò!¢ú ≤Ã¶,˛!ì˛– ~ÓyÓ˚ ò#ˆÏâ˛Ó˚

!â˛e!›˛ ˆî ÏÖy ÈÙÙÙÈ

(a) (b) (c) (d)

!â˛e 7.8

xyõyˆÏîÓ˚ ˜îá≈ƒ õy˛ôyÓ˚ ˛ô)Ó≈Óì≈˛# K˛yò !òˆÏÎ˚ xyõÓ˚y !â˛ˆÏeÓ˚ ˆÑ˛yò ˆÑ˛yò xÇüÜ%!ú õy˛ôˆÏì˛ ˛ôyÓ˚Ó⁄

ÚÚxyÑ˛yÓ˚Ü%!úÓ˚ ¢#õyòy ˆÎ ˆÎ ˆÓ˚ÖyÇü !îˆÏÎ˚ ˜ì˛!Ó˚ ì˛yÓ˚ ˜îá≈ƒÛÛ

ÚÚ!Ñ˛hs%˘˛ ü%ï% ˜îˆÏá≈ƒÓ˚ õyïƒˆÏõ Ñ˛# !â˛ˆÏeÓ˚ xyÑ˛yÓ˚ Óy xyÎ˚ì˛ò Óí≈òy Ñ˛Ó˚y ÎyÎ˚⁄ÛÛ

ˆòy›˛



!â˛ˆÏeÓ˚ ¢#õyòy Á ˜îá≈ƒ ˆÌˆÏÑ˛ ü%ï%õye xyÑ˛yÓ˚ ¢¡∫ˆÏrï ïyÓ˚íy ˛ôyÁÎ˚y ÎyÎ˚ !Ñ˛hs%˘˛ ì˛yÓ˚ xyÎ˚ì˛ò

ˆÓyG˛y ÎyÎ˚ òy– v˛zîy£Ó˚ífl˛∫Ó˚)˛ôñ ò#ˆÏâ˛Ó˚ äÈ!ÓÓ˚ ˆãyv˛¸yÜ%!ú ì%˛úòy Ñ˛ˆÏÓ˚y ı

(a) (b) (c)

!â˛e 7.9

≤Ã!ì˛!›˛ ˆãyv˛¸yÓ˚ õˆÏïƒ xˆÏòÑ˛ ¢yî,üƒ xyˆÏäÈ– ÚÚ!Ñ˛hs%˛ ≤Ã!ì˛ ˆãyv˛¸yÓ˚ õˆÏîƒ Ñ˛# Ñ˛# ˛ôyÌ≈Ñ˛ƒ Ó˚ˆÏÎ˚ˆÏäÈ⁄ÛÛ

≤Ã!ì˛ ˆãyv˛¸yÎ˚ î%!›˛ äÈ!ÓÓ˚ xyÑ˛yÓ˚ ~Ñ˛ !Ñ˛hs%˘˛ xyÎ˚ì˛ò xyúyîy– Î!î !ü«˛yÌ≈#ˆÏîÓ˚ î%!›˛ ˆãyv˛¸yÓ˚

õˆÏïƒ ˆÑ˛yò!›˛ ÓˆÏv˛¸y !â˛!£´ì˛ Ñ˛Ó˚ˆÏì˛ Óúy £Î˚ ì˛y£ˆÏú ì˛yÓ˚y ì˛y ¢!‡˛Ñ˛¶˛yˆÏÓ !â˛!£´ì˛ Ñ˛Ó˚ˆÏì˛

˛ôyÓ˚ˆÏÓ–

ÎÖò !ü«˛yÌ≈#ˆÏîÓ˚ !ãK˛y¢y Ñ˛Ó˚y £ú ˆÑ˛ò ì˛yÓ˚y ÓúˆÏäÈ ~Ñ˛!›˛ äÈ!Ó x˛ôÓ˚!›˛Ó˚ ˆÌˆÏÑ˛ ÓˆÏv˛¸y v˛z_ˆÏÓ˚

ì˛yÓ˚y Ó Ïú ÏäÈ ı

ÚÚ~Ñ˛!›˛ xòƒ!›˛Ó˚ ˆÌˆÏÑ˛ ÓˆÏv˛¸y–ÛÛ

ÚÚ~Ñ˛!›˛ xyÑ˛yˆÏÓ˚Ó˚ Á˛ôÓ˚ xyÓ˚ ~Ñ˛!›˛ Ó˚yÖˆÏú ˆÓyG˛y ÎyˆÏFäÈ ˆÎ ˆÑ˛yò!›˛ ÓˆÏv˛¸y–ÛÛ

ÚÚˆÎ xyÑ˛yÓ˚!›˛ Ñ˛yÜˆÏãÓ˚ ˆÓ!ü xÇü !òˆÏÎ˚ˆÏäÈ ˆ¢!›˛£z ÓˆÏv˛¸y–ÛÛ

ÚÚˆ›˛!ÓˆÏú Ó˚yÖˆÏú ˆÎ äÈ!Ó›˛y ÓˆÏv˛¸y xÇü äÈ!v˛¸ˆÏÎ˚ Ó˚ˆÏÎ˚ˆÏäÈ ˆ¢!›˛ ÓˆÏv˛¸y–ÛÛ

v˛z_Ó˚Ü%!ú ˆÌˆÏÑ˛ ~›˛y fl˛ôT˛ ˆÎ !ü«˛yÌ≈#ˆÏîÓ˚ ˆÑ˛yˆÏòy äÈ!ÓÓ˚ ≤Ãühfl˛Ïì˛y Óy ¢ÇÑ˛#í≈ì˛y ¢¡∫ Ïrï ïyÓ˚íy

xyˆÏäÈ ~ÓÇ ì˛yÓ˚y xyÎ˚ì˛ˆÏòÓ˚ ïyÓ˚íyÓ˚ Ö%Ó Ñ˛yäÈyÑ˛y!äÈ ˆ˛ô§ÔˆÏäÈ ˆÜˆÏäÈ– ¢õì˛ú xyÑ, !ì˛Ó˚ xòƒì˛õ

˜Ó!üT˛ƒ £ú xyÎ˚ì˛ò– ~Ñ˛!›˛ xyÑ,˛!ì˛ Ñ˛yÜˆÏã Óy ˆ›˛!ÓˆÏú Óy Ñ§˛yˆÏâ˛Ó˚ ˆ≤’ˆÏ›˛ Ñ˛ì˛›˛y xÇü îÖú

Ñ˛ˆÏÓ˚ Ó˚ˆÏÎ˚ˆÏäÈ ì˛yÓ˚ v z˛ôÓ˚ !ò¶≈˛Ó˚ü#ú–

!Ñ˛hs%˘˛ !ü«˛yÌ≈#ˆÏîÓ˚ ÎÖò ò#ˆÏâ˛Ó˚ î%!›˛ ~Ñ˛£zÓ˚Ñ˛õ !â˛ˆÏeÓ˚ ˛ô!Ó˚ÓˆÏì≈˛ î%!›˛ !¶˛ß¨ äÈ!ÓÓ˚ ì%˛úòy Ñ˛Ó˚ˆÏì˛

ˆîÁÎ˚y £Î˚ ì˛Öò ì˛yÓ˚y ¢õ¢ƒyÓ˚ ¢¡ø%Ö#ò £Î˚–

(a) (b)

!â˛e 7.10

v˛z˛ôˆÏÓ˚y=˛ î%!›˛ äÈ!ÓÓ˚ õˆÏïƒ ˆÑ˛yò xyÑ˛yÓ˚!›˛ ˆÓ!ü ãyÎ˚Üy îÖú Ñ˛ˆÏÓ˚ˆÏäÈ xÌÓy ˆÑ˛yò xyÑ˛yÓ˚!›˛Ó˚

xyÎ˚ì˛ò ˆÓ!ü⁄ xyˆÏÜÓ˚ v˛zîy£Ó˚ˆÏí xyõÓ˚y ~Ñ˛!›˛ xyÑ˛yˆÏÓ˚Ó˚ v˛z˛ôÓ˚ xyÓ˚ ~Ñ˛!›˛ xyÑ˛yÓ˚ Ó!¢ˆÏÎ˚

ˆòy›˛

104 D.El.Ed. 504


î%!›˛Ó˚ xyÎ˚ì˛ˆÏòÓ˚ ì%˛úòy Ñ˛ˆÏÓ˚!äÈ– ~ˆÏ«˛ˆÏe xyÎ˚ì˛ò ˛ô!Ó˚õyˆÏ˛ôÓ˚ ~Ñ˛!›˛ ~Ñ˛Ñ˛ ïyÎ≈ Ñ˛ˆÏÓ˚ xyõÓ˚y

≤Ã!ì˛!›˛ xyÑ˛yˆÏÓ˚Ó˚ xyÎ˚ì˛ò !òí≈Î˚ Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚–

â˛ˆÏúy xyõÓ˚y !Ñ˛ä%È ˆîüúy£z Óy: ˆãyÜyv˛¸ Ñ˛!Ó˚ ÎyˆÏîÓ˚ xyÎ˚ì˛ò ~Ñ˛ ~ÓÇ ≤Ã!ì˛!›˛ ~Ñ˛£z ˆÑ˛y¡ôyò#Ó˚

S~Ó˚ ö˛ˆÏú ≤Ã!ì˛!›˛ ˆîüúy£z ÓyˆÏ:Ó˚ ò#ˆÏâ˛Ó˚ ì˛ˆÏúÓ˚ ˆ«˛eö˛ú ~Ñ˛ £ˆÏÓV– ~Ó˚˛ôÓ˚ ≤Ã!ì˛!›˛ !â˛ˆÏe

xyõÓ˚y ˆîüúy£z Óy:Ü%!ú ÜyˆÏÎ˚ ÜyˆÏÎ˚ úy!ÜˆÏÎ˚ ˛ôÓ˚ ˛ôÓ˚ ¢yãy£z ÎyˆÏì˛ î%!›˛ ˆîüúy£z ÓyˆÏ:Ó˚

õˆÏïƒ ˆÑ˛yˆÏòy ö§˛yÑ˛y òy ÌyˆÏÑ˛– ò#ˆÏâ˛ !â˛e!›˛ ˆîÁÎ˚y £ú ı

(a) (b)

!â˛e 7.11

xyõÓ˚y ˆîÖˆÏì˛ ˛ôy!FäÈ !â˛e 7.11ÈÙÈ~Ó˚ SÑ˛V äÈ!Ó!›˛ 16!›˛ ˆîüúy£z Óy: !îˆÏÎ˚ Ü!‡˛ì˛ xyÓ˚ SÖV

!â˛e!›˛ 14!›˛ ˆîüúy£z Óy: !îˆÏÎ˚ Ü!‡˛ì˛– ¢%ì˛Ó˚yÇ xyõÓ˚y ÓúˆÏì˛ ˛ôy!Ó˚ !â˛e 7.11ÈÙÈÓ˚ SÑ˛V ~Ó˚

ˆ«˛eö˛ú !â˛e 7.11ÈÙÈÓ˚ SÖV ~Ó˚ ˆ«˛eö˛ ÏúÓ˚ ˆÌ ÏÑ˛ ˆÓ!ü– ì˛yäÈyv ¸yÁ ˆîüúy£z Óy: ÏÑ˛ ~Ñ˛Ñ˛

!£ˆÏ¢ˆÏÓ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ xyõÓ˚y î%!›˛ !â˛ˆÏeÓ˚ ˆ«˛eö˛ˆÏúÓ˚ !òí≈Î˚Á Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚–

ˆì˛yõÓ˚y Ñ˛# ò# Ïâ˛ äÈ!ÓÜ%!úÓ˚ ˆ«˛eö˛ú !òí≈Î˚ Ñ˛Ó˚ Ïì˛ ˛ôyÓ˚⁄

(a) (b) (c) (d) (e)

!â˛e 7.12

xyõÓ˚y v˛z˛ôˆÏÓ˚y=˛ äÈ!ÓÜ%!úÓ˚ ˆ«˛eö˛ú !òí≈Î˚ Ñ˛Ó˚ˆÏì˛ ˛ôyÓ˚ˆÏÓy òy Ñ˛yÓ˚í ≤Ã!ì˛!›˛ ˆÑ˛yˆÏòy ¢õì˛ú

!á ÏÓ˚ ˆÑ˛y Ïòy ãyÎ˚Üy îÖú Ñ˛Ó˚ ÏäÈ òy– !e¶% ã Óy â˛ì% ¶%≈ ã £ú ÓÂï xyÑ˛y ÏÓ˚Ó˚ !â˛ ÏeÓ˚ v zîy£Ó˚í

!Ñ˛hs%˘˛ v˛z˛ôˆÏÓ˚Ó˚ äÈ!ÓÜ%!úÓ˚ ¢Ó õ%=˛ xyÑ˛yˆÏÓ˚Ó˚ !â˛ˆÏeÓ˚ v˛zîy£Ó˚í–

≤ÃÌõ!îˆÏÑ˛ ˆ«˛eö˛ú ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ ~Ñ˛!›˛ !ò!î≈T˛ ˆ◊!íÓ˚ ˆîüúy£z Óy: ~Ñ˛£z xyÎ˚ì˛ˆÏòÓ˚

≤’y!fi›˛Ñ˛ ÓÜ≈ Óy ~Ñ˛£z xyÎ˚ì˛ˆÏòÓ˚ ÓÜ≈yÑ˛yÓ˚ Ó˚!àò Ñ˛yÜãñ Ó£zñ Öyì˛y £zì˛ƒy!î ÓƒÓ£yÓ˚ Ñ˛Ó˚ˆÏì˛

˛ôy!Ó˚–

S

ˆòy›˛



ˆ◊!íÑ˛ˆÏ«˛ x¶˛ƒyˆÏ¢Ó˚ ãòƒ !Ñ˛ä%È !Ñ Î˚yÑ˛úy˛ôˆÏÓ˚ v zîy£Ó˚í ˆîÁÎ˚y £ú ı

~Ñ˛£z xyÑ˛yˆÏÓ˚Ó˚ Ó£z Óy Öyì˛y !îˆÏÎ˚ !ü«˛ˆÏÑ˛Ó˚ ˆ›˛!Óú!›˛ õy˛ôy Á Ñ˛ˆÏì˛yÜ%ˆÏúy Ó£z Óy Öyì˛y

ˆú ÏÜ ÏäÈ ì˛y !òí≈Î˚ Ñ˛Ó˚y–

~Ñ˛£z xyÑ˛yÓ˚/xyÎ˚ì˛ˆÏòÓ˚ Ó£z Óy Öyì˛y !îˆÏÎ˚ !ü«˛yÌ≈#Ó˚ ˆÓˆÏM˛Ó˚ v˛z ô!Ó˚ì˛ú õy˛ô !òí≈Î˚ Ñ˛Ó˚y

~Ñ˛£z xyÑ˛yÓ˚ Á ˆ◊!íÓ˚ ˆîüúy£z Óy: !îˆÏÎ˚ Ó£z Óy Öyì˛yÓ˚ v z ô!Ó˚ì˛ˆÏúÓ˚ õy˛ô !òí≈Î˚ Ñ˛Ó˚y–

S~ˆÏ«˛ˆÏe !ü«˛yÌ≈#Ó˚ Ñ˛!›˛ ˆîüúy£z Óy: ˆúˆÏÜˆÏäÈ ì˛y !òí≈Î˚ Ñ˛Ó˚ˆÏì˛ £ˆÏÓV

ˆ◊!íÑ˛ˆÏ«˛Ó˚ ˆõˆÏG˛ Óy ˆîÁÎ˚yˆÏú Ó˚!àò Ñ˛yÜã xy›˛Ñ˛yˆÏòy ˆäÈyˆÏ›˛y xyÑ˛yˆÏÓ˚Ó˚ !â˛ˆÏeÓ˚ ãòƒ

ˆäÈy Ï›˛y ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛Ó˚y ˆÎ Ïì˛ ˛ôy ÏÓ˚ ˆÎõò ˆîüúy£z Óy:–

~ˆÏ«˛ˆÏe ˆîüúy£z Óy: ˛ôyüy˛ôy!ü Ó!¢ˆÏÎ˚ xyõÓ˚y !â˛e!›˛ˆÏÑ˛ ˆë˛ˆÏÑ˛ ˆîˆÏÓy– !Ñ˛hs%˘˛ Î!î ˆîÖy ÎyÎ˚

!â˛ˆÏeÓ˚ !Ñ˛ä%È›˛y xÇü ˆîüúy£z Óy: !îˆÏÎ˚ ¶˛Ó˚y ÎyÎ˚!ò– ~ˆÏ«˛ˆÏe Î!î ˆ¢£z xÇü!›˛ ~Ñ˛ ~Ñ˛ˆÏÑ˛Ó˚

xˆÏÂï≈ˆÏÑ˛Ó˚ SxÌ≈yÍ 1/2 ~Ñ˛ˆÏÑ˛V ˆÌˆÏÑ˛ Ñ˛õ £Î˚ ì˛y£ˆÏú xyõÓ˚y ˆ¢!›˛ ÜíòyÓ˚ xhs˘˛¶%≈˛=˛ Ñ˛Ó˚ˆÏÓy òy

!Ñ˛hs%˘˛ Î!î ˆ¢£z xÇü!›˛ ~Ñ˛ ~Ñ˛ˆÏÑ˛Ó˚ xˆÏÂï≈ˆÏÑ˛Ó˚ SxÌ≈yÍ 1/2 ~Ñ˛ ÏÑ˛Ó˚V ˆÌ ÏÑ˛ ˆÓ!ü £Î˚ ì˛y£ Ïú

xyõÓ˚y ˆ¢!›˛ ˛ô% ÏÓ˚y˛ô%!Ó˚ ~Ñ˛ ~Ñ˛Ñ˛ ïÓ˚ˆÏÓy–

v z ôˆÏÓ˚Ó˚ xyˆÏúyâ˛òy ˆÌˆÏÑ˛ xyõÓ˚y ÓúˆÏì˛ ˛ôy!Ó˚ ˆÎ !mõy!eÑ˛ xyÑ˛yˆÏÓ˚Ó˚ ¢ˆÏAÜ ˆ«˛eö˛ú ¢¡ô!Ñ≈˛ì˛

~Ñ˛Ñ˛ ÓÂï xyÑ˛yˆÏÓ˚Ó˚ !â˛ˆÏeÓ˚ ˆ«˛eö˛ú !òí≈ˆÏÎ˚Ó˚ ãòƒ xyõÓ˚y ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ ˆîüúy£z Óy:

ÓƒÓ£yÓ˚ Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚– ì˛yäÈyv ¸y ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ ˆîüúy£z Óy: äÈyv ¸yÁ xyõÓ˚y xòƒyòƒ ¢õ xyÑ˛yˆÏÓ˚Ó˚

Óhfl% !îˆÏÎ˚ ˆ«˛eö˛ú !òí≈Î˚ Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚– ˆÑ˛yˆÏòy !â˛ˆÏeÓ˚ ˆ«˛eö˛ú !òí≈Î˚ Ñ˛Ó˚yÓ˚ ¢õÎ˚ Î!î xˆÏÂï≈Ñ˛

~Ñ˛ˆÏÑ˛Ó˚ Ñ˛õ xÇü Óyî ˛ôˆÏÓ˚ ÎyÎ˚ ì˛ˆÏÓ xyõÓ˚y ˆ¢!›˛ xhs˘˛¶%≈˛=˛ Ñ˛Ó˚ÏÓ òy !Ñ˛hs%˘˛ Î!î xˆÏÂï≈Ñ˛ ~Ñ˛ˆÏÑ˛Ó˚

ˆÓ!ü xÇü Óyî ˛ôˆÏÓ˚ ÎyÎ˚ ì˛ˆÏÓ ˆ¢!›˛ˆÏÑ˛ ˛ô)í≈ ~Ñ˛ ~Ñ˛Ñ˛ ïˆÏÓ˚ ˆ«˛eö˛ú Üíòy Ñ˛Ó˚ˆÏÓy–

ˆ«˛eö˛ú ˛ô!Ó˚õy Ï ôÓ˚ ≤Ãõyí ~Ñ˛Ñ˛ ı

¢yïyÓ˚íì˛ı ~Ñ˛!›˛ ÓÜ≈ˆÏ«˛ˆÏeÓ˚ ˆ«˛eö˛úˆÏÑ˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ ≤Ãõyí

~Ñ˛Ñ˛ !£ Ï¢ ÏÓ Üíƒ Ñ˛Ó˚y £Î˚– v zîy£Ó˚ífl ∫Ó) ô ˛ôy ÏüÓ˚ !â˛e 7.13 ˆì˛

~Ñ˛!›˛ 1 !õı Óy£%!Ó!üT˛ ~Ñ˛!›˛ ˛ÓÜ≈ Ï«˛e ÏÑ˛ ˆîÖy Ïòy £ ÏÎ ÏäÈ ÓÜ≈ Ï«˛e!›˛Ó

ˆ«˛eö˛ú £ú 1 ÓÜ≈ Ï¢!õı !â˛e 7.13 Óy 1m2.

ˆòy›˛ ı 1 ÓÜ≈ !õı ˆ«˛eö˛ú ÏÑ˛ 1m2 !£ Ï¢ ÏÓ ˆúÖy £ ÏÎ ÏäÈ– ˆ◊!íÑ˛ Ï«˛Ó

ˆ«˛eö˛úñ !Óîƒyú ÏÎ˚Ó˚ ˆîÁÎ˚y ÏúÓ˚ ˆ«˛eö˛ú xÌÓy ÓyÜy ÏòÓ˚ ˆ«˛eö˛ú ≤Ã¶, !ì˛ ÓÂï xyÑ˛y ÏÓ˚Ó˚

ãyÎ˚ÜyÎ˚ ˆ«˛eö˛ú ˛ô!Ó˚õy Ï ôÓ˚ ãòƒ 1 ÓÜ≈ !õÉ v˛z˛ôÎ%=˛ ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ Üíƒ £Î˚– !Ñ˛hs%˘˛ ˆäÈyˆÏ›˛y

!â˛ˆÏeÓ˚ Óy Óhfl%˛Ó˚ ˆ«˛eö˛ú ˆÎõò ˆòy›˛ Ó£zˆÏÎ˚Ó˚ ˛ôyì˛yÎ˚ x§yÑ˛y äÈ!Óñ ˆòy›˛Ó£z ˆõyv˛¸yˆÏòyÓ˚ ãòƒ

õúyˆÏ›˛Ó˚ ˛ô!Ó˚õy˛ôñ Óy Ó%õyú ≤Ã¶, !ì˛ ãòƒ 1 ÓÜ≈ ˆ¢!õ Óy 1cm2 xˆÏòÑ˛ ˆÓ!ü v˛z˛ôÎ%=˛ ~Ñ˛Ñ˛

!£ Ï¢ ÏÓ Üíƒ £ ÏÓ–

1m

1m

!â˛e 7.13

ˆòy›˛

106 D.El.Ed. 504


Ñ˛#¶˛y ÏÓ ≤Ãõyí ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛ ÏÓ˚ ãƒy!õ!ì˛Ñ˛ !â˛ ÏeÓ˚ ˆ«˛eö˛ú Üíòy Ñ˛Ó˚ Ïì˛ £Î˚⁄

ò#ˆÏâ˛Ó˚ xyÎ˚ì˛yÑ˛yÓ˚ ABCDÈÙÈÓ˚ äÈ!Ó!›˛ ˆîÖy ÎyÑ˛ ˜îá≈ƒ 5 ˆ¢!õ Á ≤ÃfiÌ 3 ˆ¢!õ– ˆÎ Ï£ì% Óy£%Ó˚

õy˛ô ˆ¢!õ ˆflÒˆÏú ˆòÁÎ˚y £ˆÏÎ˚ˆÏäÈ ì˛y£z ˆ«˛eö˛ú ˛ô!Ó˚õyˆÏ˛ôÓ˚ ˆ«˛ˆÏeÁ 1 ÓÜ≈ ˆ¢!õ Óy 1cm2

~Ñ˛Ñ˛ !£ Ï¢ ÏÓ ïÓ˚y £ ÏÓ–

!â˛e 7.14

â˛ˆÏúy v z˛ôˆÏÓ˚Ó˚ äÈ!Ó!›˛ˆÏì˛ 1 ÓÜ≈ ˆ¢!õ ÓÜ≈yÑ˛yÓ˚ Ñ˛yÜã !îˆÏÎ˚ ¶˛!Ó˚ ÎyˆÏì˛ ~Ñ˛!›˛ ÓÜ≈ xòƒ ÓÜ≈ˆÏÑ˛

ë˛yÑ˛y òy ˆîÎ˚ xyÓyÓ˚ ≤Ã!ì˛!›˛ ÓˆÏÜ≈Ó˚ õˆÏïƒ ˆÑ˛yò ö§˛yÑ˛ òy ˆÌˆÏÑ˛ ÎyÎ˚– xyõÓ˚y ˆîÖˆÏì˛ ˛ôy!FäÈ 3!›˛

¢y!Ó˚ £ˆÏÎ˚ˆÏäÈ Á ≤Ã!ì˛!›˛ ¢y!Ó˚ˆÏì˛ 5!›˛ Ñ˛ˆÏÓ˚ ÓÜ≈yÑ˛yÓ˚ Ñ˛yÜã Ó˚ˆÏÎ˚ˆÏäÈ– ¢%ì˛Ó˚Ç ABCD xyÎ˚ì˛Ñ˛

!â˛e!›˛ 15!›˛ ˆäÈyˆÏ›˛y ÓÜ≈ !îˆÏÎ˚ ¶˛Ó˚y £ˆÏÎ˚ˆÏäÈ– xyõÓ˚y ÓúˆÏì˛ ˛ôy!Ó˚ ABCD xyÎ˚ì˛Ñ˛ ÏÓ˚Ó˚ ˆ«˛eö˛ú

1 ÓÜ≈ ˆ¢!õÓ˚ 15 Ü%í xÌ≈yÍ 15 ÓÜ≈ ˆ¢!õ Ó 15cm2.

1cm2 ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ xyÎ˚ì˛Ñ˛yÓ˚ !â˛e!›˛Ó˚ ˆ«˛eö˛ú !òí≈Î˚ Ñ˛Ó˚yÓ˚ ˛ôÂï!ì˛ £ú ı

ABCD xyÎ˚ì˛Ñ˛yÓ˚!›˛ S!â˛e 7.14) AB Á AD ˆÓ˚ÖyÇüÓ˚ ¢ˆÏAÜ ¢õyhs˘˛Ó˚yú Ñ˛ˆÏÓ˚ ˆÓ˚Öy x§yÑ˛y

£ú ÎyÓ˚ ö˛ Ïú xyÎ˚ì˛Ñ˛Ó˚!›˛ ÓÜ≈ ˆ¢!õ Ïì˛ !Ó¶˛=˛ £ú–

~Ñ˛!›˛ ¢y!Ó˚ˆÏì˛ ˆÎ ¢ÇÖƒÑ˛ ÓÜ≈ Ó˚ˆÏÎ˚ˆÏäÈ £ú 5 ~ÓÇñ 3!›˛ ¢y!Ó˚ˆÏì˛ ÓˆÏÜ≈Ó˚ ¢ÇÖƒ = 5 × 3 = 15

∴ xyÎ˚ì˛Ñ˛ ÏÓ˚Ó˚ ˆ«˛eö˛ú = ˜îá≈ƒ xò%ÎyÎ˚# ~Ñ˛ˆÏÑ˛Ó˚ ¢ÇÖƒy × ≤ÃfiÌ xò%ÎyÎ˚# ~Ñ˛ˆÏÑ˛Ó˚ ¢ÇÖƒy

xÌ≈yÍ xyÎ˚ì˛Ñ˛ˆÏÓ˚Ó˚ ˆ«˛eö˛ú = S˜îá≈ƒ × ≤ÃfiÌV cm2

= l × b cm2

ˆÎÖyˆÏò l = Ïîá≈ƒ ~Ñ˛Ñ˛ Á b = ≤ÃfiÌ ~Ñ˛Ñ˛–

v˛z˛ôˆÏÓ˚Ó˚ !â˛e 7.14ÈÙÈ~ l = 5 S5 ˆ¢!õ òÎ˚V ~ÓÇ b = 3 S3 ˆ¢!õ òÎ˚V

xÌ≈yÍ ABCD xyÎ˚ì˛!›˛Ó˚ ˆ«˛eö˛ú = (l × b) cm2

= (5 × 3)cm2 = 15cm2

~Ñ˛£z¶˛y ÏÓ Ó ÏÜ≈Ó˚ ˆ«˛eö˛ú = (l × l)cm2


E5. ò#ˆÏâ˛ ˆîÁÎ˚y äÈ!ÓÜ%!úÓ˚ ˆ«˛eö˛ú !òí≈Î˚ Ñ˛ˆÏÓ˚y ˆÎÖyˆÏò ≤Ã!ì˛!›˛ ˆäÈyˆÏ›˛y ÓˆÏÜ≈Ó˚ ˆ«˛eö˛ú

1cm2.

A D5cm

3cm

B C

ˆòy›˛



(a) (b) (c)

xòƒyòƒ ãƒy!õ!ì˛Ñ˛ xyÑ, !ì˛ ˆÎõò !e¶% ãñ â˛ì% ¶%≈˛ã Ó£%¶) ˆÏãÓ˚ ˆ«˛eö˛úÁ xyÎ˚ì˛ ˆ«˛eö˛ˆÏúÓ˚

ÓyÓ˚ Ñ˛Ó˚yÓ˚ !òÎ˚õ ˆõˆÏò Üíòy Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚–

x!òÎ˚!õì˛ xyÑ˛y ÏÓ˚Ó˚ ˆ«˛eö˛ú ı x!òÎ˚!õì˛ !â˛ ÏeÓ˚ ˆ«˛eö˛ú !òí≈ ÏÎ˚Ó˚ ãòƒ ˆ¢!õ @˘Ãyö˛ Óy

ˆ¢!õ ˆÌ v˛ @˘Ãyö˛ ÓƒÓ£yÓ˚ Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚– î%ˆÏ›˛y !ã!ò¢£z ~Ñ˛£z ò#!ì˛ˆÏì˛ Ñ˛yã Ñ˛ˆÏÓ˚– ~Ñ˛!›˛

x!òÎ˚!õì˛ xyÑ˛yÓ˚ ˆÎõò ~Ñ˛!›˛ ˛ôyì˛y !òˆÏÎ˚ ˆ¢!›˛ @˘Ãyö˛ Ñ˛yÜˆÏãÓ˚ v˛z˛ôÓ˚ ˆÓ˚ˆÏÖ ì˛yÓ˚ ¢#õyòy

!â˛!£´ì˛ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ– S!â˛e 7.15) SxÌÓy ˆÌ v˛ @˘ÃyˆÏö˛ !â˛e!›˛ Ó˚yÖˆÏì˛ £ˆÏÓV– ˛ôyì˛yÓ˚ ˆ«˛eö˛ú

!òí≈ˆÏÎ˚Ó˚ ãòƒ ˛ôyì˛yÓ˚ ¢#õyòy ïˆÏÓ˚ ÓÜ≈Ü%!ú Ü%íˆÏì˛ £ˆÏÓ– Î!î ¢#õyòy!›˛ ÓˆÏÜ≈Ó˚ xˆÏï≈Ñ˛ Óy

xˆÏï≈ˆÏÑ˛Ó˚ ˆÓ!ü xÇü îÖú Ñ˛ˆÏÓ˚ ì˛ˆÏÓ ˆ¢!›˛ˆÏÑ˛ ~Ñ˛!›˛ ˛ô)í≈ ÓÜ≈ !£ˆÏ¢ˆÏÓ Üíòy Ñ˛Ó˚ˆÏì˛ £ˆÏÓ– Î!î

~Ñ˛ÓˆÏÜ≈Ó˚ ˆÌˆÏÑ˛ Ñ˛õ xÇü £Î˚ ì˛ˆÏÓ ˆ¢!›˛ˆÏÑ˛ x@˘Ãy£ƒ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ– @˘Ãyö˛ Ñ˛yÜˆÏã ˆõy›˛ ˆÎ

¢ÇÖƒÑ˛ ÓÜ≈ Üíòy Ñ˛Ó˚y £ ÏÎ˚ ÏäÈ ˆ¢!›˛ ~£z !â˛ ÏeÓ˚ xyò%õy!òÑ˛ ˆ«˛eö˛ú (cm2 ~Ñ˛ˆÏÑ˛V– !Ñ˛hs%˘˛ ~!›˛

Ñ˛yäÈyÑ˛y!äÈ ~Ñ˛!›˛ Üíòy– ~Ñ˛!›˛ x!òÎ˚!õì˛ ÓÂï xyÑ˛yˆÏÓ˚Ó˚ !â˛ˆÏeÓ˚ ˆ«˛eö˛ú !òï≈yÓ˚í Ñ˛Ó˚ˆÏì˛ £ˆÏú

ˆÜy›˛y !â˛e!›˛ v˛zÕ‘¡∫ Á xò%¶)˛!õÑ˛ xyÎ˚ì˛Ñ˛yˆÏÓ˚ !Ó¶˛=˛ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ Á ¢õhfl˛Ï ˆÎyÜ Ñ˛ˆÏÓ˚ ˆõy›˛

ˆ«˛eö˛ú!›˛ !òí≈Î˚ Ñ˛Ó˚ Ïì˛ £ ÏÓ–

!â˛e 7.15

v˛zFâ˛ì˛Ó˚ Á !ò¡¨ì˛yÓ˚ ˆ«˛eö˛ˆÏúÓ˚ ~Ñ˛Ñ˛ ı ABCD ~Ñ˛!›˛ !õ›˛yÓ˚ ÓÜ≈ S!â˛e 7.16V. AB Á AD

ˆòy›˛

108 D.El.Ed. 504


ˆÓ˚ÖyÇ ÏüÓ˚ ¢õyhs˘˛Ó˚yú ˆÓ˚ÖyÇü ~§ ÏÑ˛ ABCD ÓÜ≈!›˛ ˆ¢!r›˛!õ›˛yÓ˚ ÓˆÏÜ≈ !Ó¶˛=˛ Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ–

xÌ≈yÍ ABCD ÓˆÏÜ≈Ó˚ AB Á AD Óy£%Ó˚ ïyˆÏÓ˚ 100!›˛ Ñ˛ˆÏÓ˚ ˆäÈyˆÏ›˛y ÓÜ≈ xyˆÏäÈ–

∴ ABCDÈÙÈˆì˛ ˆõy›˛ Ó ÏÜ≈Ó˚ ¢ÇÖƒy = 100 × 100

= 10,000

≤Ã!ì˛!›˛ ˆäÈyˆÏ›˛y ÓˆÏÜ≈Ó˚ ˆ«˛eö˛ú 1cm2

¢%ì˛Ó˚yÇñ 1m2 = 10,000 cm2.

~Ñ˛£z¶˛yˆÏÓ Óúy ˆÎˆÏì˛ ˛ôyˆÏÓ˚ñ 1cm2 = (10 × 10) mm2

= 100 mm2

!â˛e 7.16

¶) !õ ˛ô!Ó˚õy˛ô ı ≤Ãyâ˛#òÑ˛yú ˆÌ ÏÑ˛£z ¶) !õÓ˚ ˆ«˛eö˛ú ˛ô!Ó˚õy Ï ôÓ˚ ãòƒ Ú~Ñ˛Ó˚Û ÏÑ˛ ≤Ãõyí ~Ñ˛Ñ˛

!£ Ï¢ ÏÓ ÓƒÓ£yÓ˚ Ñ˛Ó˚y £ ÏFäÈ–

ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛ˆÏì˛ Úˆ£QÓ˚Û Á ÚxyÓ˚Û ˆ«˛eö˛ú õy˛ôyÓ˚ ≤Ãõyí ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ ÓƒÓ£,ì˛ £Î˚–

1 ˆ£QÓ˚ = 10,000m2 ~ÓÇ 1 xyÓ˚ = 100m2

∴ 1 ˆ£QÓ˚ = 100 xyÓ˚ ~ÓÇ 100 ˆ£QÓ˚ = 1km2

xyÓyÓ˚ñ 1 ˆ£QÓ˚ = 2.471 ~Ñ˛Ó˚

7.3.3 xyÎ˚ì˛ ÏòÓ˚ ˛ô!Ó˚õy˛ô

xyõÓ˚y ~ì˛«˛í ~Ñ˛õy!eÑ˛ Á !mõy!eÑ˛ Óhfl% Ó˚ ˜îá≈ƒ Á ˆ«˛eö˛ú ˛ô!Ó˚õy˛ô !òˆÏÎ˚ xyˆÏúyâ˛òy

Ñ˛ˆÏÓ˚!äÈ– ~Öò xyõÓ˚y !eõy!eÑ˛ Óhfl% ˛ô!Ó˚õy˛ô !òˆÏÎ˚ xyˆÏúyâ˛òy Ñ˛Ó˚ˆÏÓy– xyõyˆÏîÓ˚ â˛y!Ó˚˛ôyˆÏü ˆÎ

¢õhfl˛Ï Óhfl%˛ Ó˚ˆÏÎ˚ˆÏäÈ ì˛yˆÏîÓ˚ !ì˛ò!›˛ õyey Ó˚ˆÏÎ˚ˆÏäÈÈÙÙÙÈ˜îá≈ƒñ ≤ÃfiÌ Á v˛zFâ˛ì˛y Óy â˛Áv˛¸y– xyõyˆÏîÓ˚

â˛y!Ó˚ ôy Ïü xyõÓ˚y ˆÎ ¢õhfl Ï !ã!ò¢ ˆîÖ Ïì˛ ˛ôy!FäÈ ˆÎõòÈÙÙÙÈˆâ˛Î˚yÓ˚ñ ˆ›˛!Óúñ ˆÓM˛ñ Ó£zñ Óúñ

Óƒy›˛ñ ˆ˛ôò!¢úñ â˛Ñ˛ ~Ü%!ú ¢Ó£z !eõy!eÑ˛ Óhfl% Îy ~Ñ˛›˛y ãyÎ˚Üy îÖú Ñ˛ˆÏÓ˚ xyˆÏäÈ– ~Ñ˛!›˛ Óhfl%

ˆÎ ˛ô!Ó˚õyí ãyÎ˚Üy îÖú Ñ˛ˆÏÓ˚ ˆ¢!›˛ ˆ¢£z Óhfl% Ó˚ xyÎ˚ì˛ò–

B C

A D

QP

M

K

N

1m

1m

ˆòy›˛



Óhfl% ˆÎ ãyÎ˚Üy îÖú Ñ˛ˆÏÓ˚ xyˆÏäÈ ì˛y xyõÓ˚y Ñ˛#¶˛yˆÏÓ !òí≈Î˚ Ñ˛Ó˚ˆÏÓy⁄ ~£z ≤ÃˆÏŸ¿Ó˚ v z_Ó˚ ˆîÁÎ˚yÓ˚

xyˆÏÜ xyõyˆÏîÓ˚ ˆîÖˆÏì˛ £ˆÏÓ !ü«˛yÌ≈#ˆÏîÓ˚ xyÎ˚ì˛ˆÏòÓ˚ ïyÓ˚íy fl˛ôT˛ £ˆÏÎ˚ˆÏäÈ !Ñ˛òy–

ÚÚ~£z ˆ˛ôò!¢ú!›˛ Ñ˛# Ó%úyÓ˚!›˛Ó˚ ˆÌˆÏÑ˛ ˆÓ!ü ãyÎ˚Üy îÖú Ñ˛ˆÏÓ˚ˆÏäÈ⁄ÛÛ

ÚÚ!Ñ ˆÏÑ˛›˛ Óú Ñ˛# ö% ›˛ÓˆÏúÓ˚ ˆÌˆÏÑ˛ ˆÓ!ü ãyÎ˚Üy ˆòÎ˚⁄ÛÛ

ÚÚ~Ñ˛!›˛ â˛Ñ˛ Ñ˛# v˛yfi›˛yˆÏÓ˚Ó˚ ˆÌˆÏÑ˛ ˆÓ!ü ãyÎ˚Üy îÖú Ñ˛ˆÏÓ˚⁄ÛÛ

ÚÚ~Ñ˛!›˛ xyõ Ñ˛# ˛ôy!ì˛ˆÏúÓ%Ó˚ ˆÌˆÏÑ˛ ˆÓ!ü ãyÎ˚Üy ˆòÎ˚⁄ÛÛ

ÚÚÜÓ% Ñ˛# ˆõyˆÏ°ÏÓ˚ ˆÌˆÏÑ˛ ˆÓ!ü ãyÎ˚Üy îÖú Ñ˛ˆÏÓ˚⁄ÛÛ

xyˆÏÜ xyõyˆÏîÓ˚ !ò!ÿ˛ì˛ £ˆÏì˛ £ˆÏÓ ˆÎ !ü«˛yÌ≈#Ó˚y !Ó!¶˛ß¨ Óhfl%˛ ì%˛úòy Óyv˛¸ˆÏì˛ ˛ôyˆÏÓ˚ ì˛yÓ˚˛ôÓ˚ !Ñ˛ä%È

Ñ˛yˆÏ‡˛Ó˚ ÓœÑ˛ ˆî!ÖˆÏÎ˚ ì˛yˆÏîÓ˚ xyÎ˚ì˛ˆÏòÓ˚ ì%˛úòy Ñ˛Ó˚ˆÏì˛ ÓúˆÏì˛ £ˆÏÓ– S!â˛e 7.17V

(a) (b) (c)

!â˛e 7.17

~ÖˆÏò !ü«˛yÌ≈#ˆÏîÓ˚ ÓœÑ˛Ü%!úÓ˚ xyÎ˚ì˛ò !òí≈Î˚ Ñ˛Ó˚ˆÏì˛ ¢õ¢ƒyÓ˚ ¢¡ø%Ö#ò £ˆÏì˛ £ˆÏÓ– ì˛y£z ì˛yˆÏîÓ˚

xyÎ˚ì˛ò !òí≈ˆÏÎ˚Ó˚ ˛ôÂï!ì˛!›˛ ãyòˆÏì˛ £ˆÏÓ–

˜îá≈ƒ Á ˆ«˛eö˛ˆÏúÓ˚ õˆÏì˛y xyÎ˚ì˛ò ˛ô!Ó˚õyˆÏ˛ôÓ˚ ≤Ãõyí ~Ñ˛Ñ˛ v˛z ôfiÌy!˛ôì˛ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ– ~Ñ˛!›˛

áòˆÏÑ˛Ó˚ ˛~Ñ˛Ñ˛ £ú 1 ˆ¢!õ × 1 ˆ¢!õ × 1 ˆ¢!õ Îy xyÎ˚ì˛ò ˛ô!Ó˚õy Ï ôÓ˚ ≤Ãõyí ~Ñ˛Ñ˛ !£ Ï¢ ÏÓ

ÓƒÓ£,ì˛ £Î˚–

(a) (b)

!â˛e 7.18

~£z 1 ˆ¢!õ Óy£% !Ó!üT˛ áòÑ˛!›˛ áò ˆ¢!r›˛!õ›˛yÓ˚ S!â˛e 7.18 SÑ˛V ÓˆÏú– ~Ó˚ xyÎ˚ì˛ò £ú ~Ñ˛

áò ˆ¢!õ Óy 1cm3.

4cm

5cm

3cm

ˆòy›˛

110 D.El.Ed. 504


ÓˆÏv˛¸y Óhfl%˛Ó˚ xyÎ˚ì˛ò õy˛ôˆÏì˛ ÓˆÏv˛¸y ~Ñ˛Ñ˛ ˆòÁÎ˚y £Î˚ ˆÎõò ~Ñ˛ áò !õı Óy 1m3.

â˛ˆÏúy xyõÓ˚y ãy!ò Ñ˛#¶˛yˆÏÓ ~Ñ˛!›˛ xyÎ˚ì˛ áò!›˛Ó˚ xyÎ˚ì˛ò õy˛ôˆÏì˛ £Î˚ S!â˛e 7.18SÖVV äÈ!Ó

xò%ÎyÎ˚# áòÑ˛!›˛ 3 ˆ¢!õ × 4 ˆ¢!õ × 5 ˆ¢!õ â˛Áv ¸y 5!›˛ fl ‘ƒyÓ !îˆÏÎ˚ ˜ì˛!Ó˚ ~ÓÇ ~Ñ˛ ~Ñ˛!›˛

fl˛‘ƒyˆÏÓ 3!›˛ ¢y!Ó˚ Á ~Ñ˛ ~Ñ˛!›˛ ¢y!Ó˚ˆÏì˛ 4 áòÑ˛ Ó˚ˆÏÎ˚ˆÏäÈ– ¢%ì˛Ó˚yÇ ~Ñ˛!›˛ fl˛‘ƒyˆÏÓ (4 × 3) = 12

cm3 ~ÓÇ 5!›˛ fl˛‘ƒyˆÏÓ ¢Ó≈ˆÏõy›˛ 60cm3 xyˆÏäÈ– xì˛~Óñ ~£z xyÎ˚ì˛ áò!›˛Ó˚ xyÎ˚ì˛ò £ú 60

áò ˆ¢!õ Óy 60cm3.

ì˛y£ Ïú xyõÓ˚y Óú Ïì˛ ˛ôy!Ó˚ñ

xyÎ˚ì˛áò xyÎ˚ì˛ò = (l × b × h) áò ~Ñ˛Ñ˛ ˆÎÖyˆÏò

l = ˜îá≈ƒ b = ≤ÃfiÌ h = v˛zFâ˛ì˛y

xyõÓ˚ ãy!ò ~Ñ˛!›˛ áò ÏÑ˛ñ l = b = h

¢%ì˛Ó˚yÇ áòˆÏÑ˛Ó˚ xyÎ˚ì˛ò = l3 áò ~Ñ˛Ñ˛

xyõyˆÏîÓ˚ Ñ˛yˆÏäÈ Î!î xyÎ˚ì˛áò Óhfl%˛ Óy !eõy!eÑ˛ Óhfl%˛ ÌyˆÏÑ˛ ì˛y£ˆÏú xyõÓ˚y ~£z ö˛õ%≈úy!›˛

xò%ÎyÎ˚# xyÎ˚ì˛áòÓ˚ xyÎ˚ì˛ò !òí≈Î˚ Ñ˛Ó˚ Ïì˛ ˛ôyÓ˚Ó–

Ñ˛!‡˛ò Óhfl%˛Ó˚ xyÎ˚ì˛ò ˛ô!Ó˚õyˆÏ˛ôÓ˚ ~Ñ˛!›˛ ˛ôÂï!ì˛ xyˆÏäÈ– Î!î xyõÓ˚y Ñ˛!‡˛ò Óhfl%˛!›˛ ãˆÏú v%˛!ÓˆÏÎ˚

Ó˚y!Ö ì˛ˆÏÓ ˆÎ ˛ô!Ó˚õyò ãú v˛z ôˆÏâ˛ ˛ôv˛¸ ÏÓ ˆ¢!›˛ £ú ˆ¢£z Óhfl% !›˛Ó˚ xyÎ˚ì˛ò–


~Ñ˛!›˛ Ñ§˛yˆÏâ˛Ó˚ ˛ôye xÌÓy fl˛∫FäÈ ≤’y!fi›˛ˆÏÑ˛Ó˚ Óyú!ì˛ˆÏì˛ ãú ¶˛ˆÏÓ˚y– ãˆÏúÓ˚ v˛zFâ˛ì˛y !â˛!£´ì˛ Ñ˛ˆÏÓ˚y–

~Ó˚˛ôÓ˚ ~Ñ˛!›˛ ˛ôyÌˆÏÓ˚ î!v˛¸ ˆÓ§ˆÏï ˆ¢!›˛ˆÏÑ˛ ãˆÏú ˆv˛yÓyÁ– ~ÓyˆÏÓ˚ ãˆÏúÓ˚ v˛zFâ˛ì˛y õyˆÏ˛ôy Á ˆ¢!›˛

w1 !£ˆÏ¢ˆÏÓ !â˛!£´ì˛ Ñ˛ˆÏÓ˚y– ì˛yÓ˚˛ôÓ˚ xyÓ˚ ~Ñ˛›˛y xyˆÏÓ˚y ÓˆÏv˛¸y ˛ôyÌÓ˚ òyÁ Á xyˆÏÜÓ˚ ÓyˆÏÓ˚Ó˚

õˆÏì˛y î!v ¸ ˆÓ§ˆÏï ˛ôyÌÓ˚!›˛ ãˆÏú ˆv˛yÓyÁ xyÓ˚ ãˆÏúÓ˚ ~ÖòÑ˛yÓ˚ v zFâ˛ì˛yˆÏÑ˛ w2 ÓˆÏú !â˛!£´ì˛

Ñ˛ˆÏÓ˚y– ãˆÏúÓ˚ v zFâ˛ì˛y (W1 Á W

2) ÈÙÈÓ˚ õˆÏïƒ ˆÑ˛yˆÏòy ˛ôyÌ≈Ñ˛ƒ ú«˛ƒ Ñ˛Ó˚äÈ Ñ˛#⁄

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!â˛e 7.19

W1

W2

ˆòy›˛



~£z ˜Ó!üT˛ƒˆÏÑ˛ Ñ˛yˆÏã úy!ÜˆÏÎ˚È xyÎ˚ì˛ò õy˛ô!ò ˆâ˛yˆÏàÓ˚ ¢y£yˆÏÎƒ Ñ˛!‡˛ò Óhfl%˛Ó˚ xyÎ˚ì˛ò ˛ô!Ó˚õy˛ô

Ñ˛Ó˚y ÎyÎ˚ Á xyÎ˚ì˛ˆÏòÓ˚ ~Ñ˛Ñ˛ˆÏÑ˛ xyõÓ˚y áò ˆ¢!r›˛!õ›˛yÓ˚ (c.c)ÈÙÈ~Ó˚ õyïƒ Ïõ ≤ÃÑ˛yü Ñ˛!Ó˚– ~£z

˛ôÂï!ì˛Ó˚ õyïƒˆÏõ xyÎ˚ì˛ò õy˛ô!ò ˆâ˛yà!›˛ ≤ÃÌˆÏõ ì˛Ó˚ú ˛ôîyÌ≈ xÌÓy ãú !îˆÏÎ˚ ¶˛!ì≈˛ Ñ˛ˆÏÓ˚ ˆ¢!›˛

ˆâ˛yˆÏàÓ˚ ˆîÎ˚yˆÏúÓ˚ ˆflÒˆÏú ˆÎ îyÜ fl˛ôü≈ Ñ˛Ó˚ˆÏäÈ ì˛yÓ˚ !Ó˚!v˛Ç!›˛ ˆòÁÎ˚y £Î˚– ~Ó˚˛ôÓ˚ ˆÎ Ñ˛!‡˛ò Óhfl% Ó˚

xyÎ˚ì˛ò ˛ô!Ó˚õy˛ô Ñ˛Ó˚ˆÏì˛ £ˆÏÓ ˆ¢!›˛ Á£z ì˛Ó˚ú/ã Ïú ¢¡ô)í≈ v% !Ó ÏÎ˚ ~ÓyÓ˚ ãú ˆâ˛y ÏàÓ˚ ˆîÎ˚y ÏúÓ˚

ˆflÒˆÏú ˆÎ îyÜ fl˛ôü≈ Ñ˛Ó˚ˆÏäÈ ì˛yÓ˚ !Ó˚!v˛Ç ˆòÁÎ˚y £Î˚– î%!›˛ !Ó˚!v˛Ç~Ó˚ !ÓˆÏÎ˚yÜö˛ú £ú Óhfl% Ó˚

xyÎ˚ì˛ò–

xòƒ ~Ñ˛!›˛ ˛ôÂï!ì˛ˆÏì˛ ˆÎ ˆÑ˛yˆÏòy ~Ñ˛!›˛ ˛ôye Ñ˛yòyÎ˚ Ñ˛yòyÎ˚ ãú Óy xòƒ ì˛Ó˚ú !îˆÏÎ˚ ˛ô)í≈

Ñ˛Ó˚ˆÏì˛ £ˆÏÓ ÎyˆÏì˛ x!ì˛!Ó˚=˛ ~Ñ˛ˆÏö˛y›˛y ãú ˛ôv˛¸ˆÏú ì˛y ˛ôye ˆÌˆÏÑ˛ v˛z˛ôˆÏâ˛ ˛ôv˛¸ˆÏÓ– ~Ó˚˛ôÓ˚ Óhfl%˛Ó˚

xyÎ˚ì˛ò ˛ô!Ó˚õy˛ô Ñ˛Ó˚ˆÏì˛ £ˆÏÓ ì˛y Á£z ˛ôyˆÏeÓ˚ ãˆÏú ˆv˛yÓyˆÏì˛ £ˆÏÓ ~ÓÇ ˆÎ ˛ô!Ó˚õyí ãú/

ì˛Ó˚ú ˛ôye!›˛ ˆÌˆÏÑ˛ v z˛ôˆÏâ˛ ˛ôv ¸ˆÏÓ ì˛y ¢Ç@˘Ã£ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ– ~£z ¢ÇÜ,£#ì˛ ãú£z Óhfl% Ó˚ xyÎ˚ì˛ò–

ïyÓ˚í «˛õì˛y ~ÓÇ xyÎ˚ì˛ò ı xyõÓ˚y ~ì˛«˛í Ñ˛!‡˛ò Óhfl% Ó˚ xyÎ˚ì˛ò ˛ô!Ó˚õyˆÏ˛ôÓ˚ ˆÎ ˛ôÂï!ì˛ Óí≈òy

Ñ˛ˆÏÓ˚!äÈ ˆ¢£z ˛ôÂï!ì˛ ì˛Ó˚ú ˛ôîyˆÏÌ≈Ó˚ xyÎ˚ì˛ò ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ v˛z ôÎ%=˛ òÎ˚– ì˛yäÈyv˛¸yÁñ ì˛Ó˚ú

˛ôîyˆÏÌ≈Ó˚ ˆÑ˛yˆÏòy !ò!î≈T˛ xyÑ˛yÓ˚ ˆò£z ì˛yÓ˚y ˛ôyˆÏeÓ˚ xyÑ˛yÓ˚ ïyÓ˚í Ñ˛ˆÏÓ˚– ˛ôyˆÏeÓ˚ ïyÓ˚í «˛õì˛y£z

ˆ¢£z ì˛Ó˚ú ˛ôîyÌ≈ Óy Óy!ú Óy Á£z ¢õ ÏÜye#Î˚ !ã!ò Ï¢Ó˚ xyÎ˚ì˛ò Ó Ïú Üíƒ Ñ˛Ó˚y £Î˚– Î!î

~Ñ˛!›˛ ãˆÏúÓ˚ ˛ôye ˛ô)í≈ Ñ˛Ó˚ˆÏì˛ 20 ˆÓyì˛ú S~Ñ˛£z xyÑ˛y ÏÓ˚Ó˚V ã ÏúÓ˚ ≤Ã ÏÎ˚yãò ì˛y£ Ïú Á£z

ãˆÏúÓ˚ ˛ôyeÓ˚ ïyÓ˚í «˛õì˛y 20 ˆÓyì˛ú ãú xyÓ˚ Î!î ~Ñ˛!›˛ ˆÓyì˛ ÏúÓ˚ ïyÓ˚í «˛õì˛y 1 !ú›˛yÓ˚

ì˛ˆÏÓ Á ãˆÏúÓ˚ ˛ôyeÓ˚ ïyÓ˚í «˛õì˛y 20 !ú›˛yÓ˚–

!üÖˆÏòÓ˚ ≤ÃÌõ ïy˛ôÜ%!úˆÏì˛ !ü«˛yÌ≈#ˆÏîÓ˚ xˆÏòÑ˛ ¢% ÏÎyÜ Ñ˛ˆÏÓ˚ !îˆÏì˛ £ˆÏÓ ÎyˆÏì˛ ì˛yÓ˚y !Ó!¶˛ß¨

˛ôyˆÏeÓ˚ ïyÓ˚í «˛õì˛y ˛ô!Ó˚õy˛ô Ñ˛Ó˚ˆÏì˛ ˛ôyˆÏÓ˚– !ÓîƒyúˆÏÎ˚ xÌÓy !ÓîƒyúˆÏÎ˚Ó˚ Óy£z ÏÓ˚ ˆÎ ¢õhfl˛Ï

!Ñ ˛Î˚yÑ˛úy˛ô Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyÓ˚ ò#ˆÏâ˛ ì˛yÓ˚ ~Ñ˛!›˛ ì˛y!úÑ˛y ˆîÁÎ˚y £úÈÙÙÙÈ

â˛y ÏÎ˚Ó˚ ˛ôye/ˆÑ˛›˛!ú â˛yˆÏÎ˚Ó˚ Ñ˛yˆÏ˛ôÓ˚ ¢y£yˆÏÎƒ ãú !îˆÏÎ˚ ˛ô)í≈ Ñ˛Ó˚y

~Ñ˛!›˛ !ò!î≈T˛ ˆÓyì˛ˆÏúÓ˚ ¢y£yˆÏÎƒ ~Ñ˛!›˛ ãˆÏúÓ˚ ˛ôye ˛ô)í≈ Ñ˛Ó˚y

ãˆÏúÓ˚ Óyú!ì˛Ó˚ ¢y£yˆÏÎƒ !ÓîƒyúˆÏÎ˚Ó˚ ãˆÏúÓ˚ ˛ôye ˛ô)í≈ Ñ˛Ó˚y

ˆäÈyˆÏ›˛y Ñ˛y˛ô !îˆÏÎ˚ ˛ôyˆÏe !â˛!ò ¶˛Ó˚y

!›˛ˆÏòÓ˚ ˛ôye !îˆÏÎ˚ Óy!ú ˛ô!Ó˚õy˛ô

ˆäÈyˆÏ›˛y Ñ˛y˛ô/≤’y!fi›˛Ñ˛ ˛ôyˆÏeÓ˚ ¢y£yˆÏÎƒ â˛yú/Üõ/xy›˛y/!Ó!¶˛ß¨ v˛yú ˛ô!Ó˚õy˛ô

â˛yõâ˛ñ Ñ˛y˛ôñ ãÜñ !›˛ˆÏòÓ˚ ˛ôyeñ ≤’y!fi›˛Ñ˛ õÜ !îˆÏÎ˚ ˆÑ˛yˆÏòy !ã!ò¢ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ¢õÎ˚ õyÌyÎ˚

Ó˚yÖˆÏì˛ £ˆÏÓ ˆÎ ˆÎˆÏÑ˛yˆÏòy ~Ñ˛!›˛ !ã!ò¢ !îˆÏÎ˚£z xyõyˆÏîÓ˚ ¢¡ô)í≈ â˛yú/xy›˛y/Üõ/ãú ˛ô!Ó˚õy˛ô

Ñ˛Ó˚ Ïì˛ £ ÏÓ–

ˆòy›˛

112 D.El.Ed. 504


xyõÓ˚y â˛yú ˛ô!Ó˚õyˆÏ˛ôÓ˚ ¢õÎ˚ !¶˛ß¨ !¶˛ß¨ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ˛ôye ÓƒÓ£yÓ˚ Ñ˛Ó˚ˆÏì˛£z ˛ôy!Ó˚ !Ñ˛hs%˘˛ xyõyˆÏîÓ˚

ˆ¢£z â˛yúˆÏÑ˛ xyÓyÓ˚ ~Ñ˛!›˛ !ò!î≈T˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ˛ôye !îˆÏÎ˚ ˆõˆÏ˛ô ˛ô!Ó˚õyˆÏ˛ôÓ˚ ˛ôyÌ≈Ñ˛ƒ!›˛ ú«˛ƒ

Ñ˛Ó˚ Ïì˛ £ ÏÓ–

!ü«˛yÌ≈#Ó˚y !Ó!¶˛ß¨ nonstandard ~Ñ˛Ñ˛ !îˆÏÎ˚ ˛ô!Ó˚õyˆÏ˛ôÓ˚ î«˛ì˛y xã≈ò Ñ˛Ó˚yÓ˚ ˛ôÓ˚£z ≤Ãõyí ~Ñ˛Ñ˛

ÎÌy áò ˆ¢!õ (cc) ~ÓÇ !ú›˛yÓ˚ (l)ÈÙ~Ó˚ v˛z ôfiÌy˛ôò Ñ˛Ó˚ˆÏì˛ £ˆÏÓ– £Î˚ì˛ ˆÑ˛yˆÏòy õ%!îÖyòyÓ˚

ˆîyÑ˛yˆÏò !ü«˛yÌ≈#Ó˚y ˆì˛ú ˛ô!Ó˚õyˆÏ˛ôÓ˚ !›˛ˆÏòÓ˚ ˛ôye!›˛ ˆîˆÏÖ ÌyÑ˛ˆÏÓ–

ì˛Ó˚ú ˛ôîyÌ≈ ˛ô!Ó˚õy Ï ôÓ˚ ≤Ãõyí ~Ñ˛Ñ˛ £ú !ú›˛yÓ˚–

1 !ú›˛yÓ˚ = 1000 áò ˆ¢!õ Óy 1000cm3.

E6. ãˆÏúÓ˚ x˛ô¢Ó˚í á!›˛ˆÏÎ˚ Ñ˛!‡˛ò Óhfl% Ó˚ xyÎ˚ì˛ò õy˛ôyÓ˚ ˛ôÂï!ì˛ˆÏì˛ ≤Ãõyí ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ Ñ˛#

ÓƒÓ£yÓ˚ÈÙÙÙÈ Ñ˛Ó˚y £Î˚⁄

E7. ~Ñ˛!›˛ ã ÏúÓ˚ ˆâ˛ÔÓyFâ˛yÓ˚ ˜îá≈ƒ 3 !õ›˛yÓ˚ñ ≤ÃfiÌ 2 !õı Á v˛zFâ˛ì˛y 1 !õÉ ˆâ˛ÔÓyFâ˛y!›˛ˆÏì˛ Ñ˛ì˛

!ú›˛yÓ˚ ãú ïÓ˚ ÏÓ⁄

7.3.4 Áãò ˛ô!Ó˚õy˛ô

ˆäÈyˆÏ›˛y ÓÎ˚¢ ˆÌˆÏÑ˛£z !ü«˛yÌ≈#ˆÏîÓ˚ Óhfl% Ó˚ ¶˛Ó˚ !òˆÏÎ˚ ÓyãyˆÏÓ˚ !Ó!¶˛ß¨ x!¶˛K˛ì˛y £Î˚– £Î˚ì˛ ì˛yˆÏîÓ˚

î§y!v ¸ ôyÕ‘y ÓƒÓ£yÓ˚ Ñ˛ ÏÓ˚ â˛yúñ ¢Ó!ãñ õ%!îÖyòyÓ˚ !ã!ò¢ Á xòƒyòƒ ÖyÓyÓ˚ !ã!ò¢ Áã ÏòÓ˚

x!¶˛K˛ì˛y £ˆÏÎ˚ˆÏäÈ– !Ñ˛hs%˘˛ ≤ÃÌõ!îˆÏÑ˛ !ü«˛yÌ≈#îÓ˚ !Ó!¶˛ß¨ nonstandard ~Ñ˛Ñ˛ ˆÎõò ˛ôyÌˆÏÓ˚Ó˚ Óy

£z§ˆÏ›˛Ó˚ ›%˛Ñ˛ˆÏÓ˚yñ Ñ˛y‡˛ñ ˆúy£y Óy xòƒyòƒ ïyì%˛Ó˚ !ã!ò¢ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ Áãò/¶˛Ó˚ õy˛ôyÓ˚ î«˛ì˛y

xã≈ˆÏòÓ˚ ¢%ˆÏÎyÜ Ñ˛ˆÏÓ˚ !îˆÏì˛ £ˆÏÓ– ì˛yÓ˚y !òˆÏ¡ÿ=˛ !Ñ ˛Î˚yÑ˛úy˛ô Ü%!ú Ñ˛Ó˚ˆÏì˛ ˛ôyˆÏÓ˚ ı

î§y!v˛¸˛ôyÕ‘yÓ˚ xyîˆÏú ~Ñ˛!›˛ úy!‡˛Ó˚ î%ïyˆÏÓ˚ î%!›˛ ˛ôye G%˛!úˆÏÎ˚ î§y!v˛¸˛ôyÕ‘y ˜ì˛!Ó˚ ı ~Ñ˛!›˛ îˆÏu˛Ó˚

!‡˛Ñ˛ õyG˛ÖyˆÏò ~Ñ˛!›˛ î!v˛¸ xy›˛Ñ˛yˆÏì˛ £ˆÏÓ– î%!›˛ ¢õõyˆÏòÓ˚ Áãò î%!îˆÏÑ˛Ó˚ ˛ôyˆÏe ˆÓ˚ˆÏÖ õyˆÏG˛Ó˚

î!v ¸ ïˆÏÓ˚ Á˛ôˆÏÓ˚ ì% úˆÏú î[˛!›˛ õy!›˛Ó˚ ¢ˆÏAÜ ¢õyhs˘˛Ó˚yú ÌyÑ˛ˆÏÓ–

non standard ~Ñ˛Ñ˛ !î ÏÎ˚ ¶˛Ó˚ õy˛ôy ı !ü«˛yÌ≈#Ó˚y ì˛yˆÏîÓ˚ ˜ì˛!Ó˚ î§y!v˛¸˛ôyÕ‘y !îˆÏÎ˚ !Ó!¶˛ß¨

!ã!òˆÏ¢Ó˚ ¶˛Ó˚ õy˛ôˆÏì˛ v˛zÍ˛¢y!£ì˛ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ ˆÎõò Óy!úñ ˛ôyì˛yñ Ó#ã £zì˛ƒy!î– ~£z Ñ˛yˆÏãÓ˚

õyïƒˆÏõ ì˛yÓ˚y î%¶˛yˆÏÓ î§y!v˛¸˛ôyÕ‘y ÓƒÓ£yÓ˚ Ñ˛Ó˚yÓ˚ î«˛ì˛y xã≈ò Ñ˛Ó˚ˆÏÓ– ≤ÃÌõì˛ñ !ü«˛yÌ≈#Ó˚y

î§y!v ¸˛ôyÕ‘yÓ˚ ÓƒÓ£yÓ˚ñ î[˛ ïˆÏÓ˚ ˆì˛yúyñ ˛ôyÕ‘yÎ˚ Áãò ˆîÁÎ˚yñ ¢!‡˛Ñ˛ ˛ôyÕ‘yÎ˚ õy˛ôyÓ˚ Ó˚v˛ ˆîÁÎ˚y

≤Ã¶,˛!ì˛Ó˚ î«˛ì˛y xã≈ò Ñ˛Ó˚ˆÏÓ– !mì˛#Î˚ì˛ñ !ü«˛yÌ≈#Ó˚y î§y!v˛¸˛ôyÕ‘yÎ˚ Áãò Ñ˛Ó˚yÓ˚ Óhfl%˛ˆÏÑ˛ î%!ì˛òˆÏ›˛

¢õ˛ô!Ó˚õyí ¶˛y ÏÜ ¶˛yÜ Ñ˛Ó˚ Ïì˛ !üÖ ÏÓ–

~£z ˛ôÂï!ì˛ˆÏì˛ x¶˛ƒhfl˛Ï £ÁÎ˚yÓ˚ ˛ôÓ˚ !ü«˛yÌ≈#Ó˚y ¢!‡˛Ñ˛ î§y!v˛¸ ôyÕ‘y Á ≤Ãõyí Áãò ÓƒÓ£yÓ˚ Ñ˛Ó˚yÓ˚

≤ÃˆÏÎ˚yãò#Î˚ì˛y xò%¶˛Ó Ñ˛Ó˚ˆÏÓ– ï#ˆÏÓ˚ ï#ˆÏÓ˚ ì˛yÓ˚y !Ó!¶˛ß¨ Áãò ÎˆÏs˛fÓ˚ ¢ˆÏAÜ ˛ô!Ó˚!â˛ì˛ £ˆÏÓ ˆÎõò

ˆòy›˛



ì%˛úy Îs˛fñ £zˆÏúQ…!òÑ˛ Áãò õy˛ôyÓ˚ Îs˛fñ Ö%Ó Ñ˛õ Óy Ö%Ó ˆÓ!ü ˛ô!Ó˚õyí Áãò õy˛ôyÓ˚ Îs˛f–

¢yïyÓ˚íì˛ Óhfl%˛Ó˚ ¶˛Ó˚ õy˛ôˆÏì˛ @˘Ãyõ Á !Ñ˛ˆÏúy@˘Ãyõ ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ ÓƒÓ£,ì˛ £Î˚– !ü«˛yÌ≈#Ó˚y ˆîˆÏÖˆÏäÈ

¢Ó!ã Óy õ%!îÖyòyÓ˚ !ã!ò¢ õy˛ôyÓ˚ ãòƒ !Ñ˛ Ïúy@˘Ãyõñ 500 @˘Ãyõñ 250 @˘ÃyˆÏõÓ˚ Óy›˛ÖyÓ˚y ÓƒÓ£,ì˛

£Î˚–

7.8 ˆõ!› ΔÑ˛ ˛ôÂï!ì˛ Ïì˛ ˛ô!Ó˚õy˛ô

!Ó!¶˛ß¨ ˆîˆÏü ˛ô!Ó˚õyˆÏ˛ôÓ˚ ~Ñ˛Ñ˛ ÓƒÓ£yˆÏÓ˚Ó˚ î%!›˛ ≤Ãïyò ˛ôÂï!ì˛ xyˆÏäÈ– ˆ¢ î%!›˛ £úÈÙÙÙÈˆõ!›˛ΔÑ˛

˛ôÂï!ì˛ ~ÓÇ !Ó !›˛ü ˛ôÂï!ì˛– ≤Ã!ì˛!›˛ ˛ôÂï!ì˛ˆÏì˛£z î%!›˛ ïÓ˚ˆÏòÓ˚ ~Ñ˛Ñ˛ xyˆÏäÈ ˆõÔ!úÑ˛/≤ÃyÌ!õÑ˛

~Ñ˛Ñ˛ ~ÓÇ úkï ~Ñ˛Ñ˛– ˜îá≈ƒñ Áãò/¶˛Ó˚ñ ¢õÎ˚ S~Ó˚ ¢ ÏAÜ ì˛y˛ôõyeyñ ì˛!v ¸Í≤ÃÓy£ñ ˛ôîy ÏÌ≈Ó˚

˛ô!Ó˚õyíV ≤Ã¶, !ì˛ õy˛ôyÓ˚ ãòƒ ˆõÔ!úÑ˛ ~Ñ˛Ñ˛ ÓƒÓ£,ì˛ £Î˚ ~ÓÇ xòƒyòƒ ˆ«˛eÜ%!úˆÏì˛ñ ˆÎõò

ˆ«˛eö˛úñ xyÎ˚ì˛òñ ïyÓ˚í«˛õì˛yñ ˆÓÜ ≤Ã¶, !ì˛ úkï ~Ñ˛Ñ˛ ÓƒÓ£,ì˛ £Î˚ Îy ˆõÔ!úÑ˛ ~Ñ˛ˆÏÑ˛Ó˚ v z ôÓ˚

!¶˛!_ Ñ˛ ÏÓ˚ Üíòy Ñ˛Ó˚y £Î˚– ˆÎõò ˆ«˛eö˛ ÏúÓ˚ ~Ñ˛Ñ˛ cm2ñ xyÎ˚ì˛ ÏòÓ˚ ~Ñ˛Ñ˛ cm3 xÌÓy

ˆÓÜÈÙÈ~Ó˚ ~Ñ˛Ñ˛ km/hour. ~£z ˆõÔ!úÑ˛ ~Ñ˛ˆÏÑ˛Ó˚ v˛z˛ôÓ˚ !¶˛!_ Ñ˛ˆÏÓ˚£z ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛ˆÏÑ˛ C – G

– S Sˆ¢!õÈÙÈ@˘ÃyõÈÙÈˆ¢ˆÏÑ˛[˛V ˛ôÂï!ì˛ Óy m–k–s S!õ›˛yÓ˚ÈÙÈ!Ñ˛ Ïúy@˘ÃyõÈÙÈˆ¢ ÏÑ˛[˛V ˛ôÂï!ì˛ Óúy £Î˚–

~Ñ˛£z¶˛yˆÏÓ !Ó !›˛ü ˛ôÂï!ì˛ˆÏÑ˛ f-p-s Sö%˛›˛ÈÙÈ˛ôyv˛z[˛ÈÙÈˆ¢ˆÏÑ˛u˛V ˛ôÂï!ì˛ ÓˆÏú–

ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛ˆÏÑ˛ xyhs˘˛ã≈y!ì˛Ñ˛ v˛yˆÏÑ˛ fl˛∫#Ñ,˛!ì˛ ˆîÁÎ˚y £ˆÏÎ˚ˆÏäÈ ì˛y£z ~!›˛ˆÏÑ˛ ÚÚxyhs˘˛ã≈y!ì˛Ñ˛ ~Ñ˛Ñ˛

˛ôÂï!ì˛ Óy xyhs˘˛ã≈y!ì˛Ñ˛ ≤Ãõyí ~Ñ˛Ñ˛ (S.I.) Ó Ïú SSIÈÙÈÓy ˆö ÏM˛ Ó Ïú Systeme international

d’unites’sV– SI ˛ôÂï!ì˛ˆÏì˛ î%!›˛ ˆ◊!íÓ˚ ~Ñ˛Ñ˛ xyˆÏäÈ– ≤ÃÌõ ˆ◊!íÓ˚ xhs˘˛¶%≈˛=˛ ˆõÔ!úÑ˛ ~Ñ˛Ñ˛

ˆÎõò ˜îá≈ƒñ Áãòñ ¢õÎ˚ñ ì˛y˛ôõyeyñ ì˛!v ¸Í≤ÃÓy£ ≤Ã¶, !ì˛Ó˚ ~Ñ˛Ñ˛– !mì˛#Î˚ ˆ◊!íÓ˚ xhs˘˛¶%≈˛=˛ £ú

úkï ~Ñ˛Ñ˛ Îy ˆõÔ!úÑ˛ ~Ñ˛ ÏÑ˛Ó˚ v z ôÓ˚ !¶˛!_ Ñ˛ ÏÓ˚ ˜ì˛!Ó˚ ˆÎõò Ñ˛yã/Ñ˛yÎ≈ñ Óúñ ü!=˛ £zì˛ƒy!î–

~£z !Ó¶˛y ÏÜ xyõÓ˚y ˜îá≈ƒñ ïyÓ˚í «˛õì˛y Á ¶˛Ó˚/Áã ÏòÓ˚ SI ~Ñ˛Ñ˛ !òˆÏÎ˚ xyˆÏúyâ˛òy Ñ˛Ó˚ˆÏÓy–

xyõÓ˚y ãy!ò ˆ«˛eö˛ú Á xyÎ˚ì˛ ÏòÓ˚ ~Ñ˛Ñ˛ÈÙÈ~Ó˚ ¢ ÏAÜ ˜î Ïá≈ƒÓ˚ ~Ñ˛Ñ˛ Áì˛ Ï≤Ãyì˛¶˛y ÏÓ ã!v ¸ì˛–

ˆÑ˛yˆÏòy ˆÑ˛yˆÏòy ¢õÎ˚ xyõyˆÏîÓ˚ ˆîˆÏü ¢yïyÓ˚í õyò%°Ï myÓ˚y ÓƒÓ£,ì˛ ≤Ãâ˛!úì˛ ~Ñ˛Ñ˛ xÌÓy !Ó !›˛ü

˛ôÂï!ì˛ˆÏì˛ ÓƒÓ£,ì˛ ~Ñ˛ˆÏÑ˛Ó˚ ¢ˆÏAÜ SI ˛ôÂï!ì˛Ó˚ ~Ñ˛ˆÏÑ˛Ó˚ ì% úòy Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ–

ˆõ!› ΔÑ˛ ˛ôÂï!ì˛Ó˚ £z!ì˛£y¢

ˆ°Ïyv˛¸ü Á ¢Æîü üì˛ˆÏÑ˛ ö ˛yˆÏ™ ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛Ó˚ ≤ÃÓì≈˛ò Ñ˛Ó˚y £Î˚ ~ÓÇ Üƒy!Ó Î˚yú õyv˛z›˛òñ ˆ¢r›˛

˛ôú â˛yˆÏâ≈˛Ó˚ !v˛Ñ˛yÓ˚ñ !úÎ˚òñ ö ˛y™ ~£z ˛ôÂï!ì˛Ó˚ ≤ÃÓì≈˛Ñ˛– 1670 Ö #É !ì˛!ò îü!õÑ˛ ˛ôÂï!ì˛ˆÏì˛

˛ô!Ó˚õy˛ô Ñ˛Ó˚yÓ˚ ≤ÃÓì≈ ò Ñ˛ ÏÓ˚ò–

ö˛Ó˚y¢# !Ó≤’ˆÏÓÓ˚ ¢õÎ˚ñ 1790 Ö #ıñ ö ˛yˆÏ™Ó˚ òƒyüòyú ~ˆÏ¢¡∫ú# ˆö ˛M˛ ~Ñ˛yˆÏv˛!õÓ˚ !ÓK˛yò

ˆòy›˛

114 D.El.Ed. 504


üyÖyˆÏÑ˛ xò%ˆÏÓ˚yï Ñ˛ˆÏÓ˚òñ ‘‘deduce an invariable standard for all the measures and

all the weights’’ Ñ˛!õüò myÓ˚y !òˆÏÎ˚y!ãì˛ ~Ñ˛yˆÏv˛!õ ~Ñ˛!›˛ ¢£ã ~ÓÇ ˜ÓK˛y!òÑ˛ ˛ôÂï!ì˛

˜ì˛!Ó˚ Ñ˛ˆÏÓ˚ò–

˛ô,!ÌÓ#Ó˚ ˛ô!Ó˚!ïÓ˚ ~Ñ˛ xÇü £ú ˜îˆÏá≈ƒÓ˚ ~Ñ˛Ñ˛–

xyÎ˚ì˛ò Á ¶˛Ó˚ÈÙÈ~Ó˚ ~Ñ˛Ñ˛ úkï ~Ñ˛Ñ˛ ~ÓÇ ~£z î%!›˛ ˜îˆÏá≈ƒÓ˚ ˆõÔ!úÑ˛ ~Ñ˛Ñ˛ ˆÌˆÏÑ˛ ˜ì˛!Ó˚

£ ÏÎ˚ ÏäÈ– ~Ó˚ õyïƒ Ïõ ˆõÔ!úÑ˛ ~Ñ˛Ñ˛Ü%!ú ~ ÏÑ˛ x˛ô ÏÓ˚Ó˚ ¢y ÏÌ Á ≤ÃÑ, !ì˛Ó˚ ¢ ÏAÜÁ ¢¡ô!Ñ≈ ì˛–

≤Ã!ì˛!›˛ ~Ñ˛ˆÏÑ˛Ó˚ ÓˆÏv˛¸y Á ˆäÈyˆÏ›˛y ~Ñ˛Ñ˛Ü%!ú ≤Ãõyí ~Ñ˛Ñ˛ˆÏÑ˛ 10 Óy ì˛yÓ˚ power 10 !î ÏÎ˚

Ü%ò Óy ¶˛yÜ Ñ˛ ÏÓ˚ ˛ôyÁÎ˚y ÎyÎ˚– ˜îá≈ƒÓ˚ ~Ñ˛Ñ˛ !£ Ï¢ ÏÓ Ñ˛!õüò ‘metre’ S!õ›˛yÓ˚V ü∑!›˛ ÓƒÓ£yÓ˚

Ñ˛ˆÏÓ˚ˆÏäÈ– ~£z ü∑!›˛ @˘Ã#Ñ˛ ü∑ ‘metron’ ˆÌˆÏÑ˛ ˆòÁÎ˚y £ˆÏÎ˚ˆÏäÈ ÎyÓ˚ õyˆÏò Ú˛ô!Ó˚õy˛ôÛ– Ú˜îá≈ƒÛ

≤ÃyÑ,˛!ì˛Ñ˛ Ó˚y!ü!›˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ !õ›˛yÓ˚ ÓƒÓ£,ì˛ £Î˚ Îy v˛z_Ó˚ ˆõÓ˚% ˆÌˆÏÑ˛ !Ó°Ï%ÓˆÏÓ˚Öy xÓ!ï

î)Ó˚ˆÏcÓ˚ ~Ñ˛~Ó˚ îü !õ!úÎ˚ò ~ÓÇ ~£z ˆõ!Ó˚!v˛Î˚ò (meridian) ö ˛yˆÏ™Ó˚ £zÇ!úü â˛ƒyˆÏòˆÏúÓ˚

v%˛ò!Ñ˛Ñ≈˛ Á ˆfl˛ôˆÏòÓ˚ ÓyˆÏ¢≈ˆÏúyòyÓ˚ v˛z ôÓ˚ !îˆÏÎ˚ ˆÜˆÏäÈ– !˛ôÎ˚yÓ˚#ÈÙÈö yˆÏAÑ˛y£z¢ ~ˆÏ[˛ !õˆÏò£zò ~ÓÇ

!ãÈòÈÙÈÓƒy˛ô!›˛fi›˛ ˆãy Ï¢ö˛ !v˛ ÏúõÓy ÏÜÓ˚ (Pierre-Fnancois-Andre Mechain and Jean-

Baptist Joseph Delembre) xhs˘˛Ü≈ì˛ ~Ñ˛!›˛ ˛ô!Ó˚îü≈òÑ˛yÓ˚# îú äÈÓäÈÓ˚ ïˆÏÓ˚ ö ˛yˆÏ™Ó˚ £zÇ!úü

â˛ƒyˆÏòˆÏúÓ˚ v˛yò!Ñ˛Ñ≈˛ Á ˆfl˛ôˆÏòÓ˚ ÓyˆÏ¢≈ˆÏúyòyÓ˚ myÓ˚y ˛ô,!ÌÓ#Ó˚ ˆÎ â˛y˛ô (arc)!›˛ ˜ì˛Ó˚# £ˆÏÎ˚ˆÏäÈ ì˛y

˛ô!Ó˚õyˆÏ˛ôÓ˚ ˆâ˛T˛y Ñ˛ˆÏÓ˚ˆÏäÈò– ~£z Ñ˛yÎ≈ â˛úyÑ˛yú#ò ˛ô!Ó˚îü≈òÑ˛yÓ˚#ˆÏîÓ˚ !Ó!¶˛ß¨ ïÓ˚ˆÏòÓ˚ x¢%!ÓïyÓ˚

¢¡ø%Ö#ò £ˆÏì˛ £ˆÏÎ˚ˆÏäÈ ~õò!Ñ˛ Ñ˛yÓ˚yÜyˆÏÓ˚ Ó®# £ˆÏì˛Á £ˆÏÎ˚ˆÏäÈ– ˛ôˆÏÓ˚ ˆîÖy !ÜˆÏÎ˚ˆÏäÈ ˆÎ ˛ô!Ó˚õyˆÏ˛ôÓ˚

¢õÎ˚ Delambre ~ÓÇ Mechain ˛ô,!ÌÓ#Ó˚ ~£z Ñ˛õúyˆÏúÓ%Ó˚ õˆÏì˛y ˆÜyúyÑ, !ì˛Ó˚ !ÓˆÏÓâ˛òy Ñ˛ˆÏÓ˚

ˆîˆÏÖò!ò– Îy£z Ï£yÑ˛ !õ›˛yÓ˚ xyãÁ ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛Ó˚ xyhs˘˛ã≈y!ì˛Ñ˛¶˛yˆÏÓ ≤Ãõyí ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ

ÓƒÓ£,ì˛ £Î˚– Î!îÁ Ú!õ›˛yÓ˚Û ü∑!›˛Ó˚ ÓƒyÖƒy Óy ¢ÇK˛yÎ˚ ÎÌyÌ≈ì˛y xyòyÓ˚ ãòƒ ˛ô!Ó˚Ó!ì≈˛ì˛ £ˆÏÎ˚ˆÏäÈ

!Ñ˛hs%˘˛ ~Ó˚ ˜îˆÏá≈ƒÓ˚ ˆÑ˛yˆÏòy ˛ô!Ó˚Óì≈˛ò £Î˚!ò–

xòƒ!îˆÏÑ˛ ˜ÓK˛y!òÑ˛Ó˚y Ú!õ›˛yÓ˚ÛÈÙÈ~Ó˚ v˛z ôÓ˚ !¶˛!_ Ñ˛ˆÏÓ˚ xòƒyòƒ ~Ñ˛Ñ˛Ü%!ú !òí≈Î˚ Ñ˛Ó˚yÓ˚ ˆâ˛T˛y

Ñ˛ ÏÓ ÏäÈò–

ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛ˆÏì˛ ÚÓ˚ÛÈÙÈ~Ó˚ ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ Ú@˘ÃyõÛÈÙÈˆÑ˛ ïyÎ≈ Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ Îy– áò ˆ¢!õ xyÎ˚ì˛ˆÏòÓ˚

~Ñ˛!›˛ áò ÏÑ˛Ó˚ ¶˛Ó˚ Óy ˆ¢£z ~Ñ˛£z ˛ô!Ó˚õyí ã ÏúÓ˚ S4°c ì˛y˛ôõyeyÓ˚V x˛ô¢Ó˚í á›˛yÎ˚–

ïyÓ˚í «˛õì˛yÓ˚ ãòƒ Ú!ú›˛yÓ˚Û ü∑!›˛ ~Ñ˛Ñ˛ !£ˆÏ¢ˆÏÓ ÓƒÓ£yÓ˚ Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ Îy 1 áò ˆv˛!¢!õ›˛yÓ˚

xyÎ˚ì˛ò ˆÓyG˛yÎ˚–

ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛!›˛ !òï≈y!Ó˚ì˛ £ÁÎ˚yÓ˚ ˛ôÓ˚ ö ˛yˆÏ™ ~£z ˛ôÂï!ì˛!›˛ fl˛∫˛ôˆÏ«˛ Á !Ó˛ôˆÏ«˛ xˆÏòÑ˛ xyˆÏúyâ˛òy

£ˆÏÎ˚ˆÏäÈ– ˛ôˆÏÓ˚ 7£z ~!≤Ãú 1795 ¢yˆÏú ö yˆÏ™Ó˚ ¢Ó˚Ñ˛yÓ˚ ~!›˛ ¢Ó˚Ñ˛yÓ˚#¶˛yˆÏÓ @˘Ã£í Ñ˛ˆÏÓ˚ò– 1798

ˆòy›˛



ˆÌ ÏÑ˛ 1799ÈÙÈ~ ˆòîyÓ˚úƒyu˛ ¢%£zãyÓ˚úƒyu˛ñ ˆv˛òõyÑ≈˛ñ ˆfl˛ôò Á £zì˛y!ú ˆÌˆÏÑ˛ ≤Ã!ì˛!ò!ï !òˆÏÎ˚

~Ñ˛!›˛ ˜ÓK˛y!òÑ˛ xyˆÏúyâ˛òy ¢¶˛y £Î˚ ˆÎÖyˆÏò ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛Ó˚ ˜Óïƒì˛y !ÓˆÏÓ!â˛ì˛ £Î˚ Á xyÓ˚Á

òõ%òy ≤Ãõyí ~Ñ˛ˆÏÑ˛Ó˚ òÑ˛üy Ñ˛Ó˚y £Î˚– ≤’y!›˛òyõ !îˆÏÎ˚ ˜ì˛!Ó˚ !õ›˛yÓ˚ Á !Ñ˛ˆÏúy@˘ÃyˆÏõÓ˚ ~Ñ˛Ñ˛ ˜ì˛!Ó˚

Ñ˛Ó˚y £Î˚– Îy xy£z ÏòÓ˚ õyïƒ Ïõ 10£z !v˛ˆÏ¢¡∫Ó˚ 1799 ¢yˆÏú ¢Ó˚Ñ˛yÓ˚# ¶˛yˆÏÓ ö ˛yˆÏ™ â˛yú% Ñ˛Ó˚y £Î˚–

Î!îÁ ≤ÃÌõ!îˆÏÑ˛ ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛!›˛ v zÍ¢yˆÏ£Ó˚ ¢ˆÏAÜ Ü,£#ì˛ £Î˚!ò ì˛Ó%Á 1840 ¢yú ö ˛yˆÏ™ ~!›˛

Ü,£#ì˛ £ÁÎ˚yÓ˚ ˛ôÓ˚ xòƒyòƒ ˆîüÜ%!úˆÏì˛ ï#ˆÏÓ˚ ï#ˆÏÓ˚ ~Ü%!úÓ˚ ÓƒÓ£yÓ˚ Óyïƒì˛yõ)úÑ˛ Ñ˛Ó˚y £Î˚

1950 Á 1960 ¢yˆÏú ˆÓ!üÓ˚¶˛yÜ ˆîü£z ~£z ˛ôÂï!ì˛ @˘Ã£í Ñ˛ˆÏÓ˚– !Ñ˛hs%˛ xyˆÏõ!Ó˚Ñ˛yÓ˚ õˆÏì˛y

Ñ˛ˆÏÎ˚Ñ˛!›˛ ˆîü xyãÁ ~£z ˛ôÂï!ì˛ @˘Ã£í Ñ˛ˆÏÓ˚!ò– 1956 ¢yˆÏúÓ˚ !v˛ˆÏ¢¡∫Ó˚ õyˆÏ¢ ¶˛yÓ˚ˆÏì˛Ó˚ ¢Ç¢î

Standard of weights and Measures xy£zò xò%¢yˆÏÓ˚ ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛!›˛ @˘Ã£í Ñ˛ˆÏÓ˚ ~ÓÇ ì˛y

˛ôÎ˚úy x ÏQyÓÓ˚ 1958 ¢yú ˆÌ ÏÑ˛ â˛yú% £Î˚–

ˆõ!› ΔÑ˛ ˛ôÂï!ì˛Ó˚ ~Ñ˛Ñ˛ ı ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛ˆÏì˛ v˛z ôÜ%!íì˛Ñ˛ ~Ñ˛Ñ˛Ü%!ú SˆäÈyˆÏ›˛y ~Ñ˛Ñ˛V ˆõÔ!úÑ˛

~Ñ˛ˆÏÑ˛Ó˚ 1 1

,10 100

≤Ã¶, !ì˛ xÇü ~ÓÇ Ü%!íì˛Ñ˛ ~Ñ˛Ñ˛Ü%!ú ˆõÔ!úÑ˛ ~Ñ˛ˆÏÑ˛Ó˚ 10, 100 £zì˛ƒy!î

Ü%í ˆÓ!ü–

!Ñ˛ˆÏúy ~Ñ˛Ñ˛ = 1000 ~Ñ˛Ñ˛

Ü%!íì˛Ñ˛ Ó!Âï≈ì˛ ˆ£Qy ~Ñ˛Ñ˛ = 100 ~Ñ˛Ñ˛

~Ñ˛Ñ˛ ~Ñ˛Ñ˛ ˆv˛Ñ˛y ~Ñ˛Ñ˛ = 10 ~Ñ˛Ñ˛

~Ñ˛Ñ˛

v˛z˛ôÜ%!íì˛Ñ˛ ~Ñ˛Ñ˛ ˆv˛!¢ ~Ñ˛Ñ˛ = 1

10 ~Ñ˛Ñ˛

˛ ˆ¢!r›˛ ~Ñ˛Ñ˛ = 1

100 ~Ñ˛Ñ˛

!õ!ú ~Ñ˛Ñ˛ = 1

1000 ~Ñ˛Ñ˛

â˛ˆÏúy ˆî!Ö ˆõ!›˛ΔÑ˛ ~Ñ˛ˆÏÑ˛Ó˚ ~£z ˛ô!Ó˚Ñ˛y‡˛yˆÏõy ˛ô!Ó˚õyˆÏ˛ôÓ˚ xòƒyòƒ ~Ñ˛Ñ˛ ˆÎõò ˜îá≈ƒñ ïyÓ˚í

«˛õì˛y Á ¶˛Ó˚ SÁãòV ˛ô!Ó˚õy Ï ô Á ¢ÇK˛y ˜ì˛!Ó˚ Ïì˛ Ñ˛#¶˛y ÏÓ ¢y£yÎƒ Ñ˛ ÏÓ˚–

ˆòy›˛

116 D.El.Ed. 504


˜îá≈ƒ ˛ô!Ó˚õy Ï ôÓ˚ ~Ñ˛Ñ˛ ı ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛ˆÏì˛ ˜îá≈ƒ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ~Ñ˛Ñ˛ £ú !õ›˛yÓ˚–

!Ñ˛ˆÏúy!õ›˛yÓ˚ (km) = 1000 !õ›˛yÓ˚ (m)

Ü%!íì˛Ñ˛ ˆ£Qy !õ›˛yÓ˚ (hm) = 100 !õ›˛yÓ˚ (m)

~Ñ˛Ñ˛ ˆv˛Ñ˛y !õ›˛yÓ˚ (dam) = 10 !õ›˛yÓ˚ (m)

~Ñ˛Ñ˛ ı !õ›˛yÓ˚ (m)

v˛z˛ôÜ%!íì˛Ñ˛ ~Ñ˛Ñ˛ ˆv˛!¢ !õ›˛yÓ˚ (dm) = 1

10m

ˆ¢!r›˛ !õ›˛yÓ˚ (cm) = 1

100m

!õ!ú !õ›˛yÓ˚ (mm) = 1

1000m

v z ô ÏÓ˚Ó˚ ~Ñ˛Ñ˛Ü%!úÓ˚ õ Ïïƒ S¢yïyÓ˚íì˛V km, m Á cm ~Ñ˛Ñ˛Ü%!ú î)Ó˚c õy˛ôyÓ˚ ãòƒ ÓƒÓ£,ì˛

£Î˚–

Î!îÁ xyõy ÏîÓ˚ ˆî Ïü 1958 ¢yú ˆÌ ÏÑ˛ ˆõ!› ΔÑ˛ ˛ôÂï!ì˛!›˛ ≤Ãâ˛!úì˛ ì˛Ó%Á ã!õñ Ñ˛y˛ôv˛ ˛ô!Ó˚õy Ï ôÓ˚

ãòƒ xyõÓ˚y ~Ö Ïòy £z!M˛ (= 2.54cm), ö% ›˛ (= 12 £z!M˛) Á Üã S= 3 ö% ›˛V ÓƒÓ£yÓ˚ Ñ˛ ÏÓ˚

Ìy!Ñ˛

ïyÓ˚í «˛õì˛y / xyÎ˚ì˛ò ˛ô!Ó˚õy Ï ôÓ˚ ~Ñ˛Ñ˛ ı xyõÓ˚y ãy!ò ì˛Ó˚ú ˛ôîyˆÏÌ≈Ó˚ ˆÑ˛yˆÏòy !ò!î≈T˛

xyÑ˛yÓ˚ ˆò£z ˆ¢!›˛ ˆÎ ˛ôyˆÏe Ó˚yÖy £Î˚ ì˛yÓ˚ xyÑ˛yÓ˚ @˘Ã£í Ñ˛ˆÏÓ˚– ¢%ì˛Ó˚yÇ ì˛Ó˚ˆÏúÓ˚ ïyÓ˚í«˛õì˛yÓ˚

~Ñ˛Ñ˛ !£¢yˆÏÓ xyõÓ˚y Áãò Óy xyÎ˚ì˛ò õy˛ôyÓ˚ ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ Ìy!Ñ˛–

xyÎ˚ì˛ ÏòÓ˚ ~Ñ˛Ñ˛ £ú 1 !ú›˛yÓ˚ S= 1000 cm3V ì˛Ó˚ˆÏúÓ˚ ïyÓ˚í«˛õì˛y ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ !Ó!¶˛ß¨

xyÑ˛yˆÏÓ˚Ó˚ ˛ôye ÓƒÓ£,ì˛ £Î˚–

Ü%!íì˛Ñ˛ !Ñ˛ Ïúy !ú›˛yÓ˚ (kl) = 1000l

~Ñ˛Ñ˛ ˆ£Qy !ú›˛yÓ˚ (hl) = 100l

ˆv˛Ñ˛y !ú›˛yÓ˚ (dal) = 10l

~Ñ˛Ñ˛ ı !ú›˛yÓ (l)

v˛z˛ôÜ%!íì˛Ñ˛ ~Ñ˛Ñ˛ ˆv˛!¢ !ú›˛yÓ˚ (dl)˛ = 1

10l˛

˛ ˆ¢!r›˛ !ú›˛yÓ˚ (cl) = 1

100l

!õ!ú !ú›˛yÓ˚ (ml)˛ = 1

1000l

ˆòy›˛



î%ïñ ì˛úñ ˆì˛ú ãyì˛#Î˚ ì˛Ó˚ú ˛ô!Ó˚õyˆÏ˛ôÓ˚ ãòƒ Ú!ú›˛yÓ˚ÛÈÙÈ~Ó˚ ÓƒÓ£yÓ˚ Ó£%ú ≤Ãâ˛!úì˛–

¶˛Ó˚ SÁãòV ˛ô!Ó˚õy Ï ôÓ˚ ~Ñ˛Ñ˛ ı ˆõÔ!úÑ˛ ~Ñ˛Ñ˛ £ú Ú@˘ÃyõÛ–

!Ñ˛ˆÏúy @˘Ãyõ (kg) = 1000g

Ü%!íì˛Ñ˛ ˆ£Qy @˘Ãyõ (hg) = 100g

~Ñ˛Ñ˛ ˆv˛Ñ˛y @˘Ãyõ (dag) = 10g

~Ñ˛Ñ˛ ı @˘Ãyõ (g)

v˛z˛ôÜ%!íì˛Ñ˛ ~Ñ˛Ñ˛ ˆv˛!¢ @˘Ãyõ (dg) = 1

10g

ˆ¢!r›˛ @˘Ãyõ (cg) = 1

100g

!õ!ú @˘Ãyõ (mg) = 1

1000g

~Ó˚ õˆÏïƒ !Ñ˛ˆÏúy@˘Ãyõñ @˘Ãyõ Á !õ!ú@˘Ãyõ ˆÓ!ü ÓƒÓ£,ì˛ £Î˚– ~äÈyv˛¸yÁ Ñ% £zr›˛yúñ ˆõ!›˛ΔÑ˛ ›˛ò ¶˛yÓ˚#

!ã!ò¢ Áã ÏòÓ˚ ãòƒ ÓƒÓ£yÓ˚ Ñ˛Ó˚y £Î˚–

1 Ñ%˛£zr›˛yú = 100 kg

1 ›˛ò = 1000 kg = 10 Ñ%˛£zr›˛yú


E8. !Ó !›˛ü ˛ôÂï!ì˛Ó˚ ì%˛úòyÎ˚ ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛ˆÏì˛ Ñ˛# Ñ˛# ¢%!Óïy Ó˚ˆÏÎ˚ˆÏäÈ ˆúˆÏÖy–

E9. 1 ˆÑ˛!ã â˛y ÏúÓ˚ îyõ r 25 £ Ïúñ 5 Ñ% £zr›˛yú â˛yˆÏúÓ˚ îyõ Ñ˛ì˛⁄ â˛yˆÏúÓ˚ ãòƒ 20 ˆÑ˛!ã

ÁãˆÏòÓ˚ ˛ôƒyˆÏÑ˛›˛ ˜ì˛!Ó˚ £ˆÏúñ ˛ô%ˆÏÓ˚y â˛yú›˛y ¶˛Ó˚ˆÏì˛ Ñ˛ˆÏì˛y ˛ôƒyˆÏÑ˛›˛ úyÜˆÏÓ⁄

7.5 ¢õ ÏÎ˚Ó˚ ˛ô!Ó˚õy˛ô

¢) ÏÎ≈Ó˚ â˛y!Ó˚!îˆÏÑ˛ ˛ô,!ÌÓ#Ó˚ ˛ô!Ó˚Ñ õyÓ˚ Sxy!£´Ñ˛ Á Óy!°Ï≈Ñ˛ Ü!ì˛V ö˛ˆÏú ¢õˆÏÎ˚Ó˚ ˛ô!Ó˚Óì≈˛ò £Î˚–

î%!›˛ ¢)ˆÏÎ≈yîˆÏÎ˚Ó˚ õyˆÏG˛Ó˚ ¢õÎ˚ˆÏÑ˛ ~Ñ˛!›˛ !îò ÓˆÏú– !Ñ˛hs%˘˛ ˜ÓK˛y!òÑ˛Ó˚y î%!›˛ ˛ôÓ˚ õïƒÓ˚yˆÏì˛Ó˚ õyˆÏG˛Ó˚

¢õÎ˚!›˛ˆÏÑ˛ ~Ñ˛!îò ïˆÏÓ˚– ¢%ì˛Ó˚yÇ !îˆÏòÓ˚ ü%Ó% £Î˚ÈÙÙÙÈõïƒÓ˚yì˛ ˆÌˆÏÑ˛ Á ˆü°Ï £Î˚ õïƒÓ˚yˆÏì˛ !ÜˆÏÎ˚–

~£z ¢õÎ˚ ÏÑ˛ ˆ¢ÔÓ˚!îò Ó Ïú–

˛ô,!ÌÓ# ì˛yÓ˚ xy!£´Ñ˛ Ü!ì˛ ¢¡ô)í≈ Ñ˛Ó˚ˆÏì˛ ˆÎ ¢õÎ˚ ˆòÎ˚ ì˛yˆÏÑ˛Ï ~Ñ˛ ˆ¢ÔÓ˚!îò ÓˆÏú–

xy!£´Ñ˛ Ü!ì˛ ¢¡ô)í≈ Ñ˛Ó˚ˆÏì˛ ˛ô,!ÌÓ#Ó˚ !òˆÏãÓ˚ xˆÏ«˛ 360° á%Ó˚ Ïì˛ £Î˚– ¢õÎ˚ ˛ô!Ó˚õy˛ô Á !v˛@˘Ã#

˛ô!Ó˚õyˆÏ˛ô ¢õì˛y xyòyÓ˚ ãòƒ 360ÈÙÈ~Ó˚ v z ôÜ%!íì˛Ñ˛ !£ Ï¢ ÏÓ ~Ñ˛Ñ˛ ¶˛yÜ Ñ˛Ó˚y £ ÏÎ˚ ÏäÈ– ö˛úì˛É

~Ñ˛!›˛ ˆ¢ÔÓ˚!îòˆÏÑ˛ 24 ¢õ¶˛y ÏÜ ¶˛yÜ Ñ˛Ó˚y £ ÏÎ˚ ÏäÈ ˆÎÖy Ïò 1 ¶˛yÜ 1 á^˘›˛yÓ˚ ¢õyò

ˆòy›˛

118 D.El.Ed. 504


1 á^˘›˛y = 60 !õ!ò›˛ 1 ˆ¢ÔÓ˚!îò = 24 á^˘›˛y

1 !õ!ò›˛ = 60 ˆ¢ ÏÑ˛u˛

˛ô,!ÌÓ#Ó˚ ¢) ÏÎ≈Ó˚ â˛y!Ó˚!îˆÏÑ˛ á%Ó˚ˆÏì˛ 1 ˆ¢ÔÓ˚ ÓäÈÓ˚ úyˆÏÜ –

ˆ¢ÔÓ˚ ÓäÈÓ˚ Á ˆ¢ÔÓ˚!î ÏòÓ˚ ¢¡ôÑ≈ ı

1 ˆ¢ÔÓ˚ ÓäÈÓ˚ = 365 !îò 5 á^˘›˛y 48 !õ!ò›˛ 47 ˆ¢ ÏÑ˛[˛ xÌ≈yÍ 1 ˆ¢ÔÓ˚ÓäÈÓ˚ = 3651

14!îò

Ñ˛ƒyˆÏúu˛yÓ˚ xò%ÎyÎ˚# 1 ÓäÈˆÏÓ˚ 365 !îò ÌyˆÏÑ˛–

~£z¶˛yˆÏÓ ≤Ã!ì˛ ÓäÈÓ˚ xyõÓ˚y 1/4 !îò ÏÑ˛ ï!Ó˚ òy–

ì˛y£z ≤ÃˆÏì˛ƒÑ˛ 4 ÓäÈÓ˚ xyõÓ˚y ~£z 1 1 1 1

4 4 4 4+ + + !îò = 1 !îò ïˆÏÓ˚ ˆ¢£z ÓäÈˆÏÓ˚ ¢Ó≈ˆÏõy›˛

366 !îò ï!Ó˚–

ˆ¢£z ¢yú!›˛ 4 !îˆÏÎ˚ ¶˛yÜ Ñ˛Ó˚y ÎyÎ˚ ì˛yˆÏÑ˛ !ú˛ô £zÎ˚yÓ˚ ÓˆÏú ~ÓÇ ˆ¢£z ÓäÈˆÏÓ˚ ˆö˛Ó %Î˚yÓ˚# õyˆÏ¢

24 !îˆÏòÓ˚ ÓîˆÏú 27 !îò ïÓ˚y £Î˚– Ñ˛ƒyˆÏúu˛yÓ˚ xò%ÎyÎ˚# ãyò%Î˚yÓ˚#ñ õyâ≈˛ñ ˆõñ ã%úy£zñ xyÜfi›˛ñ

x ÏQyÓÓ˚ Á !v˛ Ï¢¡∫Ó˚ õy Ï¢ 31 !îò ÌyˆÏÑ˛– ~ÓÇ ~!≤Ãúñ ã%òñ ˆ¢ˆÏ≤Wz¡∫Ó˚ Á òˆÏ¶˛¡∫Ó˚ õyˆÏ¢ 30

!îò ÌyˆÏÑ˛ ~ÓÇ ˆö˛Ó %Î˚yÓ˚#ˆÏì˛ 28 !îò Á !ú˛ô £zÎ˚yÓ˚ 29 !îò ÌyˆÏÑ˛– ò#ˆÏâ˛Ó˚ Üyò!›˛ !ü«˛yÌ≈#ˆÏîÓ˚

~£z !£ Ï¢Ó õ Ïò Ó˚yÖ Ïì˛ ¢y£yÎƒ Ñ˛Ó˚ ÏÓ–

A Song of the Months

Thirty days hath September

April June and November

All the rest have thirty one

Except for February alone

And that has twenty eight days clear

And twenty nine in each leap year

Óƒ!ì˛Ñ ˛õ# ı

ˆÎ ¢õhfl˛Ï ÓˆÏ°Ï≈Ó˚ ~Ñ˛Ñ˛ Á îüˆÏÑ˛Ó˚ fiÌyˆÏò ü)òƒ xyˆÏäÈ ˆ¢£z ¢õhfl˛Ï Ó°Ï≈Ü%!ú 4 !î ÏÎ˚ ¶˛yÜ £ ÏúÁ

ˆ¢Ü%!ú !ú˛ô £zÎ˚yÓ˚ ÓˆÏú Üíƒ £ˆÏÓ òy– Á£z Ó°Ï≈Ü%!úˆÏÑ˛ Î!î 400 !îˆÏÎ˚ ¶˛yÜ Ñ˛Ó˚y ÎyÎ˚ ì˛ˆÏÓ£z

ˆ¢£z Ó°Ï≈Ü%!ú !ú˛ô £zÎ˚yÓ˚ £ˆÏÓ–

ˆÎõò 2000 ¢yú !ú˛ô £zÎ˚yÓ˚ £ˆÏúÁ 1900, 1800, 2100, 2200, 2300 !ú˛ô £zÎ˚yÓ˚ òÎ˚–

á!v ¸Ó˚ ¢õÎ˚ ı î%ïÓ˚ˆÏòÓ˚ á!v˛¸ xyˆÏäÈ 12 á^˘›˛yÓ˚ á!v˛¸ Á 24 á^˘›˛yÓ˚ á!v ¸–

xyõÓ˚y ¢yïyÓ˚íì˛ 12 á^˘›˛yÓ˚ á!v˛¸ ÓƒÓ£yÓ˚ Ñ˛!Ó˚– ~£z á!v˛¸ˆÏì˛ 1 ˆÌˆÏÑ˛ 12 á^˘›˛y ¶˛yÜ Ñ˛Ó˚y

ˆòy›˛



xyˆÏäÈ– á^˘›˛yÓ˚ Ñ§˛y›˛y á!v˛¸ˆÏì˛ 12 á^˘›˛yÎ˚ ~Ñ˛ÓyÓ˚ ˆáyˆÏÓ˚ ~ÓÇ !õ!òˆÏ›˛Ó˚ Ñ§˛y›˛y 1 á^˘›˛yÎ˚ ~Ñ˛ÓyÓ˚

ˆáyˆÏÓ˚–

õïƒy£´ Á õïƒÓ˚yì˛ 12 !îˆÏÎ˚ ˆÓyG˛y ÎyÎ˚– xyõÓ˚y Ó!ú î%˛ô%Ó˚ 12›˛y Á Ó˚yì˛ 12›˛yñ Ó˚yì˛ 12›˛y

ˆÌˆÏÑ˛ î%˛ô%Ó˚ 12›˛y am !£ˆÏ¢ˆÏÓ !ÓˆÏÓ!â˛ì˛ ˆÎõò xyõÓ˚y ˆ¶˛yÓ˚ ˛ô§yâ˛›˛yÓ˚ Ó!ú 5am, ˆì˛õ!ò Ó!ú

6.30 am, 8 am ~ÓÇ î%˛ô%Ó˚ 12 ›˛y ˆÌˆÏÑ˛ Ó˚yì˛ 12›˛y pm !£ˆÏ¢ˆÏÓ !ÓˆÏÓ!â˛ì˛ ˆÎõò !ÓˆÏÑ˛ˆÏú

4pm ¢rïƒyÎ˚ 7pm.

ˆÓ˚ú˛ôÌ Á xyÑ˛yü˛ôˆÏÌ 24 á^˘›˛yÓ˚ á!v˛¸ ÓƒÓ£,ì˛ £Î˚–

õïƒÓ˚yˆÏì˛ õyˆÏò 24 á^˘›˛y ˆÓyG˛yÎ˚ ~ÓÇ ì˛yÓ˚˛ôÓ˚ ˆÌˆÏÑ˛ 1 á^˘›˛yñ 2 á^˘›˛y £zì˛ƒy!î– ~ÖyˆÏò am

Á pm ÈÙÈ~Ó˚ ÓƒÓ£yÓ˚ ˆò›˛– á!v ¸ Ïì˛ 1 ˆÌˆÏÑ˛ 24 á^˘›˛y ˆîÖyÎ˚ Á á^˘›˛yÓ˚ Ñ§˛y›˛y á!v˛¸ˆÏì˛ 24 á^˘›˛yÎ˚

~Ñ˛ÓyÓ˚ ˆáy ÏÓ˚–

¢õÎ˚ K˛yò ı á!v ¸ ˆîÖ Ïì˛ ˆüÖyÓ˚ xy ÏÜ£z !ü«˛yÌ≈# ÏîÓ˚ ¢õÎ˚ñ xì˛#ì˛ñ Óì≈ õyòñ ¶˛!Ó°ÏƒÍ ¢¡∫ Ïrï

~Ñ˛!›˛ ˛ô)Ó≈ ïyÓ˚íy ÌyˆÏÑ˛– !üü% ÎÖò 6 ÓäÈÓ˚ ÓÎ˚ Ï¢ ≤ÃÌõ !Óîƒyú ÏÎ˚ xy Ï¢ ì˛Öò ì˛yÓ˚ ~£z ¢õÎ˚

!Ñ˛ä%È›˛y £ˆÏúÁ ˜ì˛!Ó˚ £Î˚– Üì˛Ñ˛yú Óy Üì˛ ¢ÆyˆÏ£ Ñ˛# áˆÏ›˛ˆÏäÈñ ~Öò ì˛yÓ˚y Ñ˛# Ñ˛Ó˚ˆÏäÈ Á Ñ˛yú

ì˛yÓ˚y Ñ˛# Ñ˛Ó˚ˆÏÓ ˆ¢ ¢¡∫rï ì˛yÓ˚y ÓúˆÏì˛ ˛ôyˆÏÓ˚– Üì˛Ñ˛yúñ xyã Á xyÜyõ#Ñ˛yˆÏúÓ˚ ïyÓ˚íy !îˆÏÎ˚

ü%Ó˚% Ñ˛ˆÏÓ˚ !üü%ˆÏîÓ˚ Üì˛ õy¢ñ Üì˛ ÓäÈÓ˚ Óy !Ñ˛ä%È ÓäÈÓ˚ xyˆÏÜÓ˚ !Ñ˛ä%È ì˛y!Ó˚Ö !îˆÏÎ˚ á›˛òy ˛ôÎ≈yÎ˚Ñ ˆÏõ

¢yãy Ïì˛ ˆîÁÎ˚y ˆÎ Ïì˛ ˛ôy ÏÓ˚– ~Ñ˛£z¶˛y ÏÓ xyÜyõ#Ñ˛yúñ xyÜyõ# ¢Æy£ Óy õy¢ Óy ÓäÈ ÏÓ˚Ó˚Á

ïyÓ˚íy ˆîÁÎ˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚– !üü% ÏîÓ˚ ¢ˆÏAÜ ¢õÎ˚ ¢¡∫ Ïrï !Ó!¶˛ß¨ xyˆÏúyâ˛òy Á ≤ÃˆÏŸ¿y_ˆÏÓ˚Ó˚

õyïƒˆÏõ xì˛#ì˛ñ Óì≈˛õyò Á ¶˛!Ó°ÏƒˆÏì˛Ó˚ ïyÓ˚íyÓ˚ xÓì˛yÓ˚íy Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚– v zîy£Ó˚ífl ∫Ó) ô

ò# Ïâ˛ !Ñ˛ä%È ≤ÃŸ¿ ˆîÁÎ˚y £ú ı

xyã Ñ˛# ÓyÓ˚⁄

Üì˛Ñ˛yú Ñ˛# ÓyÓ˚ !äÈú Óy xyÜyõ#Ñ˛yú Ñ˛# ÓyÓ˚ £ ÏÓ⁄

xyã ˆÑ˛ ˆÑ˛ ˆ◊!íÑ˛ˆÏ«˛ xò% ô!fiÌì˛⁄

Üì˛Ñ˛yú Ñ˛yÓ˚y xò%˛ô!fiÌì˛ !äÈú⁄

xyÜyõ#Ñ˛yú ˆÑ˛yò !˛ô!Ó˚Î˚ˆÏv˛ xˆÏAÑ˛Ó˚ Ñœ y¢ Ó˚ˆÏÎ˚ˆÏäÈ⁄

ˆÑ˛ ÓÎ˚ˆÏ¢ ÓˆÏv˛¸yñ ì%˛!õ òy ˆì˛yõyÓ˚ Órï%⁄

~£z ¢õhfl˛Ï xyˆÏúyâ˛òyÓ˚ õyïƒˆÏõ !üü% ÏîÓ˚ ¢õÎ˚ K˛yò ¢%ì˛#«¯˛ £ˆÏÓ– ~£z xyˆÏúyâ˛òyÓ˚ ¢ˆÏAÜ ¢ˆÏAÜ

!üü%ˆÏîÓ˚ xì˛#ì˛ Á ¢%î)Ó˚ xì˛#ì˛ ˆÌˆÏÑ˛ ü%Ó˚% Ñ˛ˆÏÓ˚ Óì≈˛õyò Á ¶˛!Ó°ÏƒˆÏì˛Ó˚ á›˛òy ˛ôÎ≈yÎ˚Ñ ˛ˆÏõ

¢yãyˆÏì˛ ˆîÁÎ˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚– ¢õÎ˚yò%¢yˆÏÓ˚ Î!î ì˛yÓ˚y á›˛òy ¢yãyˆÏì˛ ¢«˛õ £Î˚ ì˛ˆÏÓ£z á!v˛¸ Á

¢õÎ˚ Üíòy !ò ÏÎ˚ Ñ˛yã ü%Ó% Ñ˛Ó˚ Ïì˛ £ ÏÓ–

¢õˆÏÎ˚Ó˚ Óƒy!Æ

ˆòy›˛

120 D.El.Ed. 504


á›˛òyÓ˚ ¢õˆÏÎ˚Ó˚ v˛z˛ôÓ˚ !ò¶≈˛Ó˚ Ñ˛ˆÏÓ˚ ¢õÎ˚ ò!Ì¶%˛=˛ Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚– v˛zîy£Ó˚ífl˛∫Ó˚)˛ô ò#ˆÏâ˛Ó˚ !ì˛ò!›˛

á›˛òy ˛ôy‡˛ Ñ˛ˆÏÓ˚y–

¶˛yÓ˚ì˛Ó°Ï≈ Ñ˛ì˛ ÓäÈÓ˚ xyˆÏÜ fl ∫yï#òì˛y úy¶˛ Ñ˛ˆÏÓ˚ˆÏäÈ⁄

ˆì˛yõyÓ˚ î%£z Órï% ¢#õy Á ˆ¢£òyÓ˚ õˆÏïƒ Ñ˛ì˛ ÓäÈˆÏÓ˚Ó˚ ì˛ö˛yÍ⁄

ˆ◊!íˆÏì˛ xˆÏAÑ˛Ó˚ ˛ôÓ˚#«˛yÓ˚ v˛z_Ó˚ Ñ˛Ó˚ˆÏì˛ ˆÓ˚y!£ˆÏì˛Ó˚ Ñ˛ì˛ ¢õÎ˚ ˆúˆÏÜˆÏäÈ⁄

≤ÃÌõ ≤ÃˆÏŸ¿Ó˚ ˆ«˛ˆÏe xyõyˆÏîÓ˚ ãyòˆÏì˛ £ˆÏÓ ˆÑ˛yò ¢yˆÏú ¶˛yÓ˚ì˛Ó°Ï≈ fl ∫yï#òì˛y úy¶˛ Ñ˛ˆÏÓ˚!äÈú

(1947 A.D.). ~ÓÇ Óì≈ õyò ÓäÈÓ˚!›˛ SˆÎõò 2018 A.D)– ì˛yÓ˚ ôÓ˚ î%!›˛ ¢yú !Ó ÏÎ˚yÜ Ñ˛ ÏÓ˚

v zì˛Ó˚ ˛ôyÁÎ˚y Îy ÏÓ–

!mì˛#Î˚ ≤Ã ÏŸ¿Ó˚ ˆ«˛ Ïe xyõy ÏîÓ˚ î%£z Órï%Ó˚ ãß√ ì˛y!Ó˚Öñ õy¢ Á ¢yú¢£ ãyò Ïì˛ £ ÏÓ– ~!›˛

xyˆÏÜÓ˚!›˛Ó˚ ˆÌˆÏÑ˛ ~Ñ˛›%˛ Ñ˛!‡˛ò–˚

ïÓ˚y ÎyÑ˛ î%£z Órï%Ó˚ ãß√ ì˛y!Ó˚Ö ÎÌyÑ Ïõ 18£z x ÏQyÓÓ˚ 1999 S¢#õyV Á 12£z ˆ¢ˆÏ≤Wz¡∫Ó˚

2000 Sˆ¢£òyV–

î%¶˛yˆÏÓ ì˛yˆÏîÓ˚ ÓÎ˚ˆÏ¢Ó˚ ˛ôyÌ≈Ñ˛ƒ !òí≈Î˚ Ñ˛Ó˚y ÎyÎ˚ÈÙÙÙÈ

(i) î%£z Órï%Ó˚ Óì≈˛õyò ÓÎ˚¢ ÓyÓ˚ Ñ˛ˆÏÓ˚ ì˛yÓ˚˛ôÓ˚ ˆ¢£z ÓÎ˚¢ !ÓˆÏÎ˚yÜ Ñ˛ˆÏÓ˚ ÓÎ˚ˆÏ¢Ó˚ ˛ôyÌ≈Ñ˛ƒ ÓyÓ˚

Ñ˛ˆÏÓ˚y–

(ii) î%£z Órï%Ó˚ ãß√ ì˛y!Ó˚Ö õy¢ Á ¢y ÏúÓ˚ !Ó ÏÎ˚yÜ ö˛ú !òí≈Î˚ Ñ˛Ó˚y–

xyõÓ˚y ~ÖyˆÏò !mì˛#Î˚ ˛ôÂï!ì˛!›˛ ˆî!ÖˆÏÎ˚!äÈ– ≤ÃÌˆÏõ ãß√ì˛y!Ó˚Öñ ÓäÈÓ˚ õy¢ Á !îò xò%ÎyÎ˚# ˛ôÓ˚

˛ôÓ˚ ¢y!ã ÏÎ˚ !úÖ Ïì˛ £ ÏÓÈÙÙÙÈ

ÓäÈÓ˚ õy¢ !îò

ˆ¢£òy ı 2000 09 12

¢#õy ı – 1999 10 18

00 10 24

≤ÃÌ Ïõ Ú!îòÛ ïÓ˚ Ïú xyõÓ˚y ˆîÖ Ïì˛ ˛ôy!FäÈ 12 !îò ˆÌˆÏÑ˛ 18 !îò !ÓˆÏÎ˚yÜ Ñ˛Ó˚yÓ˚ ÎyÎ˚ òy ì˛y£z

xyõy ÏîÓ˚ Úõy¢Û ˆÌ ÏÑ˛ 1 õy¢ (30 !îòV ïyÓ˚ Ñ˛Ó˚ˆÏú xyõÓ˚y !îò ˛ôyÓ (30 + 12) = 42 !îò

~ÓÇ 42 !îò ˆÌˆÏÑ˛ 18 !îò Óyî !îˆÏú xyõÓ˚y ˛ôy£z 24 !îò– ~ÓyÓ˚ õy¢ ˛ôˆÏv˛¸ Ó˚ˆÏÎ˚ˆÏäÈ 8 ÎyÓ˚

ˆÌ ÏÑ˛ 10 !ÓˆÏÎ˚yÜ Ñ˛Ó˚y ÎyÎ˚ òy– ì˛y£z xyÓyÓ˚ ÓäÈÓ˚ ˆÌˆÏÑ˛ 1 ÓäÈÓ˚ (=12 õy¢V ïyÓ˚ Ñ˛ˆÏÓ˚ ˆ¢!›˛

8 õy Ï¢Ó˚ ¢ ÏAÜ ˆÎyÜ Ñ˛Ó˚ Ïú xyõÓ˚y ˛ôy£z (8 + 12 = 20) õy¢– 20 õy¢ ˆÌ ÏÑ˛ 10 õy¢ !òˆÏÎ˚

!òˆÏú ˛ôˆÏv˛¸ ÌyˆÏÑ˛ 10 õy¢–

¢%ì˛Ó˚yÇ ¢#õy ˆ¢£òyÓ˚ ˆÌ ÏÑ˛ 10 õy¢ 24 !î ÏòÓ˚ Ó Ïv ¸y–

ˆòy›˛



!ü«˛yÌ≈# ÏîÓ ¢õÎ ¢¡∫ Ïrï ïyÓíy ˆîÁÎyÓ ãòƒ xyõy ÏîÓ !Ó!¶˛ßß ïÓ ÏòÓ !Ñ ÎyÑ˛úy Ï ôÓ ˛ô!ÓÑ˛“òy

Ñ˛Ó˚ˆÏì˛ £ˆÏÓ ˆÎõò á!v˛¸ ˆîÖyñ am Á pm ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ ¢õÎ˚ Óúyñ î%!›˛ á›˛òyÓ˚ ¢õˆÏÎ˚Ó˚ ˛ôyÌ≈Ñ˛ƒ

!òí≈Î˚ ≤Ã¶, !ì˛–

≤ÃÜ!ì˛ õy˛ôy

E10. Ó˚yõ 9.11.2011 ì˛y!Ó˚ˆÏÖ ¢Ñ˛yˆÏú ì˛yÓ˚ Ñ˛ˆÏúã £ˆÏfi›˛ˆÏúÓ˚ v˛z ÏjˆÏüƒ Îyey Ñ˛ˆÏÓ˚!äÈú ~ÓÇ

12.12.2012 ì˛y!Ó˚ˆÏÖ Ó˚yˆÏe ì˛yÓ˚ Óy!v˛¸ ˆö˛Ó˚yÓ˚ Ñ˛Ìy– Ñ˛ì˛ !îˆÏòÓ˚ ãòƒ ˆ¢ Óy!v˛¸Ó˚

Óy£z ÏÓ˚ ÌyÑ˛ ÏÓ⁄

E11. ò#ˆÏâ˛ ˆîÁÎ˚y ¢yúÜ%!ú ˆÌˆÏÑ˛ ˆÑ˛yòÜ%!ú !ú˛ô £zÎ˚yÓ˚ ì˛y !â˛!£´ì˛ Ñ˛ˆÏÓ˚y⁄

E12. 2008 ¢y ÏúÓ˚ 10£z ãyò%Î˚yÓ˚# !ÓîƒyúˆÏÎ˚Ó˚ ˆõÓ˚yõ!ì˛Ó˚ Ñ˛yã ü%Ó˚% £ˆÏÎ˚ˆÏäÈ ~ÓÇ ì˛y 65

!îò xÓ!ï â˛úˆÏÓ– Ñ˛ì˛ ì˛y!Ó˚ˆÏÖ Ñ˛yã!›˛ ¢õyÆ £ˆÏÓ⁄

7.6 !ö˛ ÏÓ˚ ˆîÖy

ˆÑ˛yˆÏòy Óhfl%˛Ó˚ ~Ñ˛!›˛ !ÓˆÏü°Ï ≤ÃyÑ,˛!ì˛Ñ˛ Ó˚y!üÓ˚ ¢ˆÏAÜ ~Ñ˛£zÓ˚Ñ˛õ xyÓ˚ ~Ñ˛!›˛ Óhfl%˛Ó˚ !ò!î≈T˛

˛ô!Ó˚õyˆÏ˛ôÓ˚ ~Ñ˛ˆÏÑ˛Ó˚ ì%˛úòy Á ì˛y !òí≈ˆÏÎ˚Ó˚ ˛ôÂï!ì˛ˆÏÑ˛˛ ˛ô!Ó˚õy˛ô ÓˆÏú–

˜îá≈ƒ ~Ñ˛õy!eÑ˛ x§yÑ˛ˆÏì˛ ˆ«˛eö˛ú Á xyÎ˚ì˛ò !mõy!eÑ˛ Á !eõy!eÑ˛ xyÑ˛yˆÏÓ˚Ó˚ v˛zîy£Ó˚í–

˛ôÎ≈ˆÏÓ«˛íñ ~Ñ˛!›˛ Óhfl%˛Ó˚ Á˛ôÓ˚ xyÓ˚ ~Ñ˛!›˛ Óhfl%˛ ˆÓ˚ˆÏÖ ì%˛úòy ~ÓÇ ˛ôˆÏÓ˚y«˛ ˛ôÂï!ì˛Ó˚ õyïƒˆÏõ

≤Ãâ˛!úì˛ ~ÓÇ ≤Ãõyí ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛ ÏÓ˚ Óhfl% Ó˚ ˛ô!Ó˚õy˛ô Ñ˛Ó˚y–

≤Ãõyí ~Ñ˛ ÏÑ˛Ó˚ õ Ïì˛y ≤Ãâ˛!úì˛ / non-standard ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ ˛ô!Ó˚õy˛ô ÎÌyÌ≈¶˛yˆÏÓ

Ñ˛Ó˚y ÎyÎ˚ òy !Ñ˛hs%˛ !ü«˛yÌ≈#Ó˚y ≤ÃÌõ!î ÏÑ˛ ~£z¢Ó non-standard ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛ ÏÓ˚£z ˛ô!Ó˚õy Ï ôÓ˚

ïyÓ˚íy úy¶˛ Ñ˛ ÏÓ˚–

≤Ãõyí ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ Óhfl%˛Ó˚ ˜îá≈ƒñ ˆ«˛eö˛úñ xyÎ˚ì˛òñ ¶˛Ó˚ñ ïyÓ˚í«˛õì˛y õy˛ôyÓ˚ ˛ô) ÏÓ≈

!ü«˛yÌ≈#ˆÏîÓ˚ ˛ô!Ó˚õyˆÏ˛ôÓ˚ !Ó!¶˛ß¨ ˜Ó!üT˛ƒÓ˚ ¢yˆÏÌ ˛ô!Ó˚!â˛ì˛ £ˆÏì˛ £ˆÏÓ– ≤Ãõyí ~Ñ˛Ñ˛ ÓƒÓ£yˆÏÓ˚Ó˚

ˆ«˛ˆÏeÁ !ü«˛yÌ≈#ˆÏîÓ˚ non standard ~Ñ˛Ñ˛ ÓƒÓ£y ÏÓ˚Ó˚ x!¶˛K˛ì˛y Ñ˛y Ïã úyÜy ÏòyÓ˚ ãòƒ v zÍ¢y£

!îˆÏì˛ £ˆÏÓ–

≤ÃyÑ,˛!ì˛Ñ˛ Ó˚y!ü ˆõÔ!úÑ˛ñ Ü%!íì˛Ñ˛ Á v˛z˛ôÜ%!íì˛Ñ˛ ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛ˆÏì˛ Óhfl%˛Ó˚

˜îá≈ƒñ ˆ«˛eö˛úñ xyÎ˚ì˛òñ ïyÓ˚í«˛õì˛y Á ¶˛ˆÏÓ˚Ó˚ ˛ô!Ó˚õyˆÏ˛ôÓ˚ î«˛ì˛y xã≈ò !ü«˛yÌ≈#Ó˚y v˛zFâ˛

≤ÃyÌ!õÑ˛ !Óîƒyú ÏÎ˚ Ñ˛Ó˚ ÏÓ–

ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛ xyhs˘˛ã≈y!ì˛Ñ˛ ≤Ãõyí (SI) ˛ôÂï!ì˛ ÓˆÏú fl˛∫#Ñ,˛!ì˛ úy¶˛ Ñ˛ˆÏÓ˚ˆÏäÈ ~ÓÇ ~£z

˛ôÂï!ì˛ˆÏì˛ ˜îá≈ƒñ ¶˛Ó˚ñ ¢õÎ˚ñ ì˛y˛ôõyeyñ ì˛!v˛¸Í≤ÃÓy£ñ ˛ô!Ó˚õyíñ xyˆÏúyÓ˚ ì˛#«¯˛ì˛y ~£z ¢yì˛!›˛

ˆõÔ!úÑ˛ ~Ñ˛Ñ˛Á xyhs˘˛ã≈y!ì˛Ñ˛¶˛yˆÏÓ fl ∫#Ñ,˛!ì˛ ˆ˛ôˆÏÎ˚ˆÏäÈ–

ˆòy›˛

122 D.El.Ed. 504


ˆ¢ÔÓ˚ ÓäÈÓ˚ñ ˆ¢ÔÓ˚ õy¢ñ !îòñ á^˘›˛yñ !õ!ò›˛ Á ˆ¢ˆÏÑ˛ˆÏ[˛Ó˚ õyïƒˆÏõ ¢õÎ˚ ˛ô!Ó˚õy˛ô Ñ˛Ó˚y £Î˚–

á›˛òyÓ˚ ¢õÎ˚ ò!Ì¶%˛=˛ Ñ˛Ó˚ˆÏì˛ Á á›˛òyÓ˚ ¢õÎ˚ !òí≈Î˚ Ñ˛Ó˚ˆÏì˛ !ü«˛yÌ≈#ˆÏîÓ˚ î«˛ì˛y xã≈ò Ñ˛Ó˚yÓ˚

¢% ÏÎyÜ Ñ˛ ÏÓ˚ !î Ïì˛ £ ÏÓ–

7.7 ≤ÃÜ!ì˛ õy˛ôyÓ˚ ãòƒ õ Ïv˛ú v z_Ó˚

E1. ˆÎˆÏÑ˛yˆÏòy !ì˛ò!›˛ ˛ôyÌ≈Ñ˛ƒ ˆúˆÏÖy

E2. ~£zÜ%ˆÏúy ¢yÓ˚y ˛ô,!ÌÓ# ã% Ïv˛¸ ÎÌyÌ≈ Á ~Ñ˛ ~ÓÇ ˜ÓK˛y!òÑ˛¶˛yˆÏÓ ˛ô!Ó˚õyˆÏ˛ô ÓƒÓ£,ì˛

E3. v zîy£í îyÁ

E4. ≤Ãõyí ˆflÒú/~Ñ˛Ñ˛ ÓƒÓ£yˆÏÓ˚Ó˚ ˛ô) ÏÓ≈ !ü«˛yÌ≈#Ó˚y non standard/˛≤Ãâ˛!úì˛ ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚

Ñ˛Ó˚ˆÏÓ Ñ˛yÓ˚í !ü«˛yÌ≈#Ó˚y ~Ü%!úÓ˚ ¢yˆÏÌ ˆÓ!ü ˛ô!Ó˚!â˛ì˛

E5. (A) 8cm2 (B) 9cm2 (C) 24cm2

E6. áò ˆ¢!r›˛!õ›˛yÓ˚ S!ú›˛yÓ˚ òÎ˚ Ñ˛yÓ˚í ˆ¢!›˛ ì˛Ó˚ú ˛ôîyˆÏÌ≈Ó˚ xyÎ˚ì˛ò ˛ô!Ó˚õyˆÏ˛ô ÓƒÓ£,ì˛ £Î˚V

E7. 6000 !ú›˛yÓ˚

E8. ˆõ!›˛ΔÑ˛ ˛ôÂï!ì˛ˆÏì˛ Ü%!íì˛Ñ˛ Á v˛z ôÜ%!íì˛Ñ˛ ~Ñ˛Ñ˛Ü%!ú ˆõÔ!úÑ˛ ~Ñ˛ˆÏÑ˛Ó˚ 10 Ü%í Óy 10

¶˛yˆÏÜÓ˚ 1 ¶˛yÜ

E9. r 12500.00, 25 ˛ôƒyˆÏÑ˛›˛

E10. 398 !îò

E11. 1536, 1600, 1820, 2000, 2012

E12. 14£z õyâ≈˛ñ 2008

7.8 ≤Ãhfl Ïy!Óì˛ ˛ôy‡˛ Á ˆÓ˚ö˛y ÏÓ˚™

IGNOU (2008). Teaching of primary school mathematics(vol.5): Measurement.

New Delhi: IGNOU.

7.9 ˛ôy‡˛ ˆü Ï°ÏÓ˚ xò%ü#úò#

S1V !Ó!¶˛ß¨ Óhfl%˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ˆ«˛ˆÏe ì%˛úòy Ñ˛Ó˚yÓ˚ ˛ôÂï!ì˛!›˛ Óí≈òy Ñ˛ˆÏÓ˚y

S2V v zîy£Ó˚í ¢£ˆÏÎyˆÏÜ non standard/≤Ãâ˛!úì˛ ~Ñ˛Ñ˛ ~ÓÇ ≤Ãõyí ~Ñ˛Ñ˛ ÓƒÓ£yÓ˚ Á ˜Ó!ü ÏT˛ƒÓ˚

ì% úòy Ñ˛Ó˚y

S3V ˆÑ˛ò ˆõ!› ΔÑ˛ ˛ôÂï!ì˛ Ïì˛ ÓƒÓ£,ì˛ ˜îá≈ƒñ ¶˛Ó˚ ~ÓÇ ¢õ ÏÎ˚Ó˚ ~Ñ˛Ñ˛ SIÈÙÈ~Ó˚ ˆõÔ!úÑ˛ ~Ñ˛Ñ˛

!£ˆÏ¢ˆÏÓ !ÓˆÏÓ!â˛ì˛ ~ÓÇ ˆ«˛eö˛úñ xyÎ˚ì˛ò Á ïyÓ˚í«˛õì˛yñ ˆÓÜ ~Ü%!ú úkï ~Ñ˛Ñ˛

!£ˆÏ¢ˆÏÓ !ÓˆÏÓ!â˛ì˛⁄

ˆòy›˛

123504-Block2 : ì˛Ìƒ ˛ô!Ó˚â˛Î≈y

ì˛Ìƒ ˛ô!Óâ˛Î≈y

~Ñ˛Ñ˛È ÙÈ 8 ˛˛ì˛Ìƒ ˛ô!Ó˚â˛Î≈y

˛Ü‡˛ò

8.0 ¶) !õÑ˛y

8.1 v z Ïjüƒ !üÖò

8.2 ì˛Ìƒ ¢Ç@˘Ã£ Á v z ôfiÌy˛ôò

8.2.1 ì˛Ìƒ ¢Ç@˘Ã£

8.2.2 ì˛ ÏÌƒÓ˚ ¢yÓ˚í# Órï v z ôfiÌy˛ôò

8.3 ˛ì˛ˆÏÌƒÓ˚ !â˛ey!Î˚ì˛

8.3.1 !â˛e ¢¡∫rï#Î˚

8.3.2 î[˛ !â˛e

8.3.3 xyÎ˚ì˛ ÏúÖ

8.3.4 ˛ôy£z!â˛e

8.4 ì˛Ìƒ !ÓˆÏŸ’°Ïí

8.4.1 ˆÑ˛w#Î˚ ≤ÃÓíì˛yÓ˚ ˛ô!Ó˚õy˛ô

8.4.1.1 Üy!í!ì˛Ñ Üv ¸

8.4.1.2 õïƒõy

8.4.1.3 ¶) !Î˚T˛Ñ˛

8.5 ¢Ç!«˛ÆÑ˛Ó˚í

8.6 xyîü≈ v z_Ó˚õyúy

8.7 ˛@˘ÃrÌ˛ôO#

8.8 ˛≤ÃŸ¿yÓú#

8.0 ¶) !õÑ˛y

ˆì˛yõÓ˚y !Ó!¶˛ß¨ Ü‡˛òõ)úÑ˛ Ñ˛yˆÏã ã!õñ Ó˚yãfl ∫ñ Ñ, !°Ïã˛ôíƒñ õò%°Ïƒ «˛õì˛y £zì˛ƒy!î ¢ÇÑ yhs˘˛ !Ó!¶˛ß¨

ì˛Ìƒ ~Ñ˛›˛y !ò!î≈T˛ ¢õÎ˚ xhs˘˛Ó˚ ¢Ç@˘Ã£ Ñ˛Ó˚y £Î˚ ~ÓÇ ¢ÇÓ˚«˛í Ñ˛Ó˚y £Î˚– xyõyˆÏîÓ˚ ˜îò!®ò

ã#Ó Ïò ¢ÇÖƒyñ äÈ!Óñ òyõ £zì˛ƒy!î ¢ÇÑ yhs˘˛ ì˛Ìƒ ¢ÇÜ,!£ì˛ Á ÓƒÓ£,ì˛ £Î˚– ì˛Ìƒ ¢Ç@˘Ã£ ¢!Iì˛Ñ˛Ó˚í

~ÓÇ ~£z ì˛Ìƒ ˆÌˆÏÑ˛ ˆÑ˛yò !¢Âïyhs˘˛ v z˛ô!õì˛ £ÁÎ˚y ˆÎˆÏÑ˛yò Ü‡˛òõ)úÑ˛ Ñ˛yˆÏãÓ˚ õ)ú !Ó°ÏÎ˚Óhfl% –

!ÓîƒyúÎ˚ !ü«˛yÓ˚ ˆ«˛ˆÏeÁ Á£z ~Ñ˛£z Ñ˛Ìy ≤ÃˆÏÎyãƒ !üü% ÏîÓ˚ ì˛Ìƒ ˛ô!Ó˚â˛Î≈yÎ˚ K˛yò ÌyÑ˛y îÓ˚Ñ˛yÓ˚–

ì˛yÓ˚y !Ó!¶˛ß¨ ˆÖúyï)úy Á Ñ˛y ÏãÓ˚ ¢õÎ˚ òyòyò ïÓ˚ ÏòÓ˚ ì˛ ÏÌƒÓ˚ ¢¡ø%Ö#ò £Î˚– ˆ¢£zãòƒ ì˛Ìƒ

ˆòy›˛

124 D.El.Ed. 504


˛ô!Ó˚â˛Î≈y Ñ˛Ó˚yÓ˚ !Ñ˛ä%È ˆÑ˛Ôüú !üü% ÏîÓ˚ ãyòy îÓ˚Ñ˛yÓ˚–

~£z ~Ñ˛ˆÏÑ˛ñ ≤ÃyÌ!õÑ˛ ˆ◊í#ˆÏì˛ ì˛Ìƒ ¢ÇÑ yhs˘˛ !Ñ˛ä%È xyˆÏúyâ˛òy Ñ˛Ó˚y £ˆÏÓ–

~£z ~Ñ˛Ñ˛!›˛ ¢õyÆ Ñ˛Ó˚yÓ˚ ãòƒ xyò%õy!òÑ˛ 7 S¢yì˛V á^˘›˛y ¢õ ÏÎ˚Ó˚ ≤Ã ÏÎ˚yãò–


~£z ~Ñ˛Ñ˛!›˛ ˛ôy‡˛ Ñ˛ˆÏÓ˚ !ü«˛yÌ≈#Ó˚yÈÙÙÙÈ

ˆÑ˛õò Ñ˛ ÏÓ˚ ì˛Ìƒ ¢Ç@˘Ã£ Á !Óòƒy¢Ñ˛Ó˚í Ñ˛Ó˚ Ïì˛ £Î˚ !üÖ ÏÓ

ì˛ÌƒˆÏÑ˛ ¢yÓ˚í# äÈˆÏÑ˛Ó˚ õyïƒˆÏõ v˛z ôfiÌy˛ôò Ñ˛Ó˚ˆÏì˛ ˛ôyÓ˚ˆÏÓ–

!˛ôˆÏQy@˘Ãyö˛ñ î[˛!â˛eñ xyÎ˚ì˛ˆÏúÖñ ˛ôy£z!â˛eñ ˜ì˛Ó˚# Ñ˛Ó˚ˆÏì˛ ˛ôyÓ˚ˆÏÓ

ì˛Ìƒ ˆÌˆÏÑ˛ !¢Âïyhs˘˛ v˛z˛ô!òì˛ £ˆÏì˛ ˛ôyÓ˚ˆÏÓ–

8.2 ì˛Ìƒ ¢Ç@˘Ã£ Á v z ôfiÌy˛ôò

˜îò!®ò ã#Óˆò xyõy ÏîÓ˚ òyòyÏò ≤Ã ÏÎ˚yãò ˆõ›˛y ÏòyÓ˚ ãòƒ òyòyÓ˚Ñ˛ ÏõÓ˚ ì˛Ìƒ ¢Ç@˘Ã Ï£Ó˚ ≤ÃÎ˚yãò

£Î˚– ˆÎõòñ õy!¢Ñ˛ õy!£òy ˛ôyÁÎ˚yÓ˚ ˛ôÓ˚ xyõyˆÏîÓ˚ ÖÓ˚ˆÏâ˛Ó˚ ~Ñ˛!›˛ ˛ô!Ó˚Ñ˛“òy Ñ˛Ó˚ˆÏì˛ £Î˚– !Ñ˛

!Ñ˛ !ã!ò¢ !Ñ˛òˆÏì˛ £ˆÏÓñ !Óî%ƒˆÏì˛Ó˚ !Óúñ ãyõyÑ˛y˛ôv˛¸ ˆÑ˛òyñ ÓyFâ˛yˆÏîÓ˚ fl%ÒˆÏúÓ˚ !ö˛ñ !â˛!Ñ˛Í¢y

ÖÓ˚â˛ñ Îyì˛yÎ˚yì˛ ÖÓ˚â˛ £zì˛ƒy!î ¢ÇÑ ˛yhs˘˛ ~Ñ˛!›˛ ì˛y!úÑ˛y xyõÓ˚y Ñ˛ˆÏÓ˚ Ìy!Ñ˛– ˛ô)Ó≈Óì≈˛# õy¢Ü%!úÓ˚

ÖÓ˚ˆÏâ˛Ó˚ ˛ô!Ó˚ˆÏ≤Ã!«˛ˆÏì˛ ~Ñ˛!›˛ xyò%õy!òÑ˛ !£¢yÓ xyõÓ˚y Ñ˛ˆÏÓ˚ Ìy!Ñ˛– xÌ≈yÍ Óúy ÎyÎ˚ ˛ô)Ó≈Óì≈˛#

õy¢Ü%!ú ÖÓ˚â˛ ¢¡ô ÏÑ≈ xyõÓ˚y ì˛Ìƒ ¢Ç@˘Ã£ Ñ˛ ÏÓ˚ Ìy!Ñ˛–

Ñ˛yã ÈÙÈ 1

ˆì˛yõÓ˚y fl%ÒˆÏú ˜îò!®ò Á Óy!°Ï≈Ñ˛ ˆÎ ¢õhfl˛Ï ì˛Ìƒ ¢Ç@˘Ã£ Ñ˛ˆÏÓ˚y ì˛yÓ˚ ~Ñ˛!›˛ ì˛y!úÑ˛y ˜ì˛Ó˚# Ñ˛Ó˚–




!ü«˛Ñ˛ !£¢yˆÏÓ xy˛ô!ò !ü«˛yÌ≈#ˆÏîÓ˚ ˜îò!®òñ ¢yÆy!£Ñ˛ñ õy!¢Ñ˛ñ Óy!°Ï≈Ñ˛ v˛z˛ô!fiÌ!ì˛ñ ≤Ã!ì˛

ˆ◊í# Ïì˛ ˆäÈ Ïú ˆõ ÏÎ˚Ó˚ ¢ÇÖƒyñ SC, ST ÓyFâ˛yˆÏîÓ˚ ¢ÇÖƒyñ ˆÑ˛yˆÏòy õyˆÏ¢ Ñ˛yˆÏãÓ˚ !îˆÏòÓ˚ ¢ÇÖƒy

õy!¢Ñ˛ õy!£òy £zì˛ƒy!î ¢¡ôˆÏÑ≈˛ òyòy ì˛Ìƒ ¢Ç@˘Ã£ Ñ˛ˆÏÓ˚ ÌyˆÏÑ˛ò– ~£z ¢õhfl˛Ï ì˛Ìƒ£z ¢ÇÖƒyÓ˚

!Óâ˛yˆÏÓ˚ Ñ˛Ó˚y £Î˚– ~£z ïÓ˚ˆÏòÓ˚ ì˛ˆÏÌƒÓ˚ ¢yÇ!ÖƒÑ˛ !ÓÓÓ˚íˆÏÑ˛£z ì˛Ìƒ Óy data Ó Ïú–

ì˛Ìƒ ≤Ãì˛ƒ«˛ Á˛ ˛ôˆÏÓ˚y«˛ !Ó!¶˛ß¨¶˛yˆÏÓ ¢Ç@˘Ã£ Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚– ~£z ¢ÇÜ,£#ì˛ ì˛Ìƒ ˛ôÓ˚Óì≈˛#Ñ˛yˆÏú

ÓƒÓ£y ÏÓ˚Ó˚ ãòƒ ¢%¤% ¶˛y ÏÓ ¢ÇÓ˚!«˛ì˛ Ñ˛Ó˚y ≤Ã ÏÎ˚yãò– !Ñ˛¶˛y ÏÓ ì˛Ìƒ ¢ÇÜ,£#ì˛ Ñ˛Ó˚ Ïì˛ £Î˚ñ ¢yÓ˚í#ÓÂï

ˆòy›˛



Ñ˛Ó˚ˆÏì˛ £Î˚ ~ÓÇ äÈ!ÓÓ˚ õyïƒˆÏõ ˆîÖyˆÏì˛ £Î˚ ì˛y xyõyˆÏîÓ˚ ˆüÖyÓ˚ îÓ˚Ñ˛yÓ˚– ¢ÇÜ,£#ì˛ñ ¢ÇÓ˚!«˛ì˛

~ÓÇ v˛z˛ôfiÌy!˛ôì˛ ì˛Ìƒ xyõyˆÏîÓ˚ x!¶˛K˛ì˛y ~ÓÇ ˆÑ˛yˆÏòy !¢ÂïyˆÏhs˘˛ v˛z˛ô!òì˛ £ˆÏì˛ ¢y£yÎƒ Ñ˛ˆÏÓ˚–

8.2.1 ì˛Ìƒ ¢Ç@˘Ã£

ˆÑ˛yˆÏòy ~Ñ˛!›˛ !¢ÂïyˆÏhs˘˛ v z ô!òì˛ £ÁÎ˚yÓ˚ ãòƒ ˆ¢£z !Ó°ÏÎ˚ ¢¡ôˆÏÑ≈˛ ˆÎ ¢õhfl˛Ï ˆÖyã ÖÓÓ˚ ˆòÁÎ˚y

£Î˚ ì˛y£z £ú ì˛Ìƒ– ì˛Ìƒ ¢Ç@˘ÃˆÏ£Ó˚ !Ó!¶˛ß¨ v zÍ¢ ¢¡ôˆÏÑ≈˛ xyˆÏúyâ˛òy Ñ˛Ó˚y ÎyÑ˛–

ïÓ˚y ÎyÑ˛ÈÙÙÙÈ

v˛zîyÈÙÈ1 ı ¢Æõ ˆ◊í#Ó˚ äÈye/äÈye#Ó˚y ~Ñ˛!›˛ !˛ôÑ‰ !òÑ‰ ~Ó˚ xyˆÏÎ˚yãò Ñ˛Ó˚ˆÏäÈ– !ü«˛Ñ˛ õ£yüÎ˚

äÈyeäÈye#ˆÏîÓ˚ Ñ˛úyñ xyˆÏ˛ôúñ ˆúÓ%ñ ˆ˛ôÎ˚yÓ˚y ˆÌˆÏÑ˛ ~Ñ˛!›˛ ö˛ú !òÎ≈y¢ Ñ˛Ó˚ˆÏì˛ ÓúˆÏúò– !ü«˛Ñ˛

õ£yüÎ˚ ≤ÃˆÏì˛ƒˆÏÑ˛Ó˚ !òÓ≈yâ˛ò !ú!˛ôÓÂï Ñ˛Ó˚ˆÏúò– ~ˆÏ«˛ˆÏe ì˛Ìƒ ¢Ó˚y¢!Ó˚ !ü«˛yÌ≈#ˆÏîÓ˚ Ñ˛yäÈ ˆÌˆÏÑ˛

¢ÇÜ,£#ì˛ £ú– ~!›˛ õ%Öƒ v zÍ Ï¢Ó˚ v zîy£Ó˚í–

v˛zîyÈÙÈ2 ı ïÓ˚y ÎyÑ˛ñ ˆÑ˛yˆÏòy ~Ñ˛!›˛ ü£Ó˚/@˘ÃyˆÏõÓ˚ !Ó!¶˛ß¨ xyˆÏÎ˚Ó˚ õyò% Ï°ÏÓ˚ ¢ÇÖƒy xyõÓ˚y ãyòˆÏì˛

â˛y£z–

~ˆÏ«˛ˆÏe xyõyˆÏîÓ˚ !õv˛z!ò!¢˛ôƒy!ú!›˛/˛ôM˛yˆÏÎ˚ì˛ x!ö˛ˆÏ¢Ó˚ Ó˚!«˛ì˛ ˆÓ˚Ñ˛v≈˛ Sˆ¢ò¢y¢≈ !Ó˚ˆÏ˛ôy›≈˛V ~Ó˚

v˛z˛ôÓ˚ !ò¶≈˛Ó˚ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ– xÌ≈yÍ ¢Ó˚y¢!Ó˚ ì˛Ìƒ ¢Ç@˘Ã£ òy Ñ˛ˆÏÓ˚ ˆÑ˛yò ˆÓ˚Ñ˛ˆÏv≈˛Ó˚ !˛ôò xyõÓ˚y

!ò¶≈˛Ó˚ Ñ˛!Ó˚– ~£z ïÓ˚ˆÏòÓ˚ ˛ôˆÏÓ˚y«˛ v zÍ¢ˆÏÑ˛ Óúy £Î˚ ˆÜÔí v˛zÍ¢–

¢ÇÜ,£#ì˛ xˆÏÜyäÈyˆÏúy ì˛ÌƒˆÏÑ˛ Óúy £Î˚ Ñ§˛yâ˛y ì˛Ìƒ–

v˛zîyÙÈ3 ı ~Ñ˛!›˛ ˆ›˛fi›˛ !¢!Ó˚ˆÏãÓ˚ !Ó!¶˛ß¨ £z!òÇˆÏ¢ ˆì˛u%˛úÑ˛ˆÏÓ˚Ó˚ Ó˚yòÜ%!ú £ú 16, 56, 25, 8,

3, 33, 23, 107

~ˆÏ«˛ˆÏe Ó˚yòÜ%!ú !‡˛Ñ˛õì˛ ¢yãyˆÏòy ˆò£z– ~›˛y ~Ñ˛!›˛ Ñ§˛yâ˛y ì˛ˆÏÌƒÓ˚ v˛zîy£Ó˚í–

ì˛ÌƒÜ%!úˆÏÑ˛ ˆÎõò ˆÑ˛yˆÏòy ~Ñ˛!›˛ ˛ô!Ó˚ˆÏ≤Ã!«˛ˆÏì˛ îúÓÂï Ñ˛Ó˚y £Î˚ ~õò ì˛yˆÏÑ˛ Óúy £Î˚ !Ó¶˛yãò–

≤ÃyÆ ò¡∫Ó˚ñ v zFâ˛ì˛yñ ÓÎ˚¢ñ xyÎ˚ £zì˛ƒy!î ¢ÇÑ yhs˘˛ ˆÎ ¢ÇÖƒyÜ%!ú ˛ôyÁÎ˚y ÎyÎ˚ ˆ¢Ü%!ú £ú ˆflÒyÓ˚–

8.2.2 ì˛ ÏÌƒÓ˚ ˆ◊í#ÓÂï v z ôfiÌy˛ôò ı

x“ ¢ÇÖƒÑ˛ ì˛Ìƒ ˆÎõò Ñ õÓï≈õyò Óy Ñ õ £…y¢õyò v z ôyˆÏÎ˚ ¢yãyˆÏòy £Î˚ ì˛Öò ì˛yˆÏÑ˛ Óúy £ÈÎ˚

ˆ◊í#ÓÂï ì˛Ìƒ–

v˛zîyÈÙÈ4 ı Ñ˛§yâ˛y ì˛Ìƒ ı 25 ò¡∫ˆÏÓ˚Ó˚ ~Ñ˛!›˛ £zv˛z!ò›˛ ˆÌˆÏÑ˛ 12 ãò !ü«˛yÌ≈#Ó˚ ≤ÃyÆ ò¡∫Ó˚ ˆîÁÎ˚y

£úÈÙÙÙÈ S˛ô)í≈õyò ÈÙ 25V16, 7, 23, 10, 18, 9, 21, 20, 12, 17, 16, 21

ˆ◊í#ÓÂï ì˛Ìƒ 7, 9,10, 12, 16, 16, 17, 18, 20, 21, 21, 23 SÑ ˛õÓï≈õyò v˛z˛ôyˆÏÎ˚ ¢yãyòV

ˆòy›˛

126 D.El.Ed. 504


Ñ˛yã ÈÙÈ 2(1) Ñ ˛õ£…y¢õyò v˛z˛ôyˆÏÎ˚ v˛z˛ôˆÏÓ˚Ó˚ ì˛ÌƒÜ%!ú ¢yãyò–(2) 88, 25, 16, 43, 7, 10, 16, 34, 61, 52, 97

Ñ ˛õÓï≈õyò ~ÓÇ Ñ ˛õ£…y¢õyò v˛z ôyˆÏÎ˚ ¢yãyò–ÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈÈÙÈ

ˆÓü# ¢ÇÖƒyÓ˚ ì˛Ìƒ £ Ïú ì˛y ÏîÓ˚ ¢yÓ˚í#ÓÂï Ñ˛Ó˚y x¢Ω˛Ó– ì˛Öò ì˛y ÏîÓ˚ !ÓÑ˛“ v z ôy ÏÎ˚ ¢!Iì˛

Ñ˛Ó˚y £Î˚ ÎyˆÏì˛ ˆÑ˛yò !¢Âïyhs˘˛ ¢£ˆÏã£z ˆòÁÎ˚y ÎyÎ˚–

v˛zîy ÈÙÈ 5 : ~°Ïõy ì˛yÓ˚ ˆ◊í#Ó˚ ¢Ó äÈye#Ó˚ ã% Ïì˛yÓ˚ õy˛ô ¢Ç@˘Ã£ Ñ˛ˆÏÓ˚!äÈú– !òˆÏâ˛ !Ó!¶˛ß¨ õyˆÏ˛ôÓ˚

ã% Ïì˛yÓ˚ !ÓÓÓ˚í ˆîÁÎ˚y £ Ïúy–

5 4 7 5 6 7 6 5 6 6

4 5 6 8 7 4 6 5 6 4

5 7 6 7 5 6 6 4 8 7

~Öò ~£z ˆ›˛!Óú ˆÌˆÏÑ˛ ˆÑ˛yò !¢Âïyhs˘˛ xy¢y Ö%Ó£z ¢£ã–

ˆ›˛!Óú ÈÙÈ 8.1

ì%˛!õ !Ñ˛ ãyò⁄

ˆÑ˛yò ˆflÒyÓ˚ ÓyÓ˚ÓyÓ˚ xyõyÓ˚ ãòƒ ˆÎ !â˛£´ ˆÑ˛yò SîyÜV ÓƒÓ£yÓ˚ Ñ˛Ó˚y £ú ˆ¢!›˛ £ú ›˛ƒy!ú õyÑ≈˛–

~Ñ˛£z ˆflÒyÓ˚ Ñ˛ì˛ÓyÓ˚ xy¢ˆÏäÈ ì˛yˆÏÑ˛ Óúy £Î˚ ˛ô!Ó˚¢ÇÖƒy (f).

v˛z˛ôˆÏÓ˚Ó˚ v˛zîy£Ó˚í!›˛ ~Ñ˛!›˛ x!Óòƒhfl Ï ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yã ÏòÓ˚ v˛zîy£Ó˚í–

ˆ›˛!Óú ÈÙÈ 8.1 ã%ˆÏì˛yÓ˚ õy˛ô

ã%ˆÏì˛yÓ˚ õy˛ô ›˛ƒy!ú õyÑ≈˛ äÈye#Ó˚ ¢ÇÖƒy ( f )

4 5

5 7

6 10

7 6

8 2

ˆõy›˛ 30

ˆòy›˛



Ñ˛yã ÈÙÈ 2

!ò¡¨!ú!Öì˛ ì˛Ìƒ ¢Ç@˘Ã£ Ñ˛Ó%ò ~ÓÇ ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yãˆÏòÓ˚ ˆ›˛!Óú ≤Ãhfl% ì˛ Ñ˛Ó%ò–

(a) ~Ñ˛!›˛ ˆ◊í# Ïì˛ äÈy ÏeÓ˚ ÓÎ˚¢

(b ) ~Ñ˛!›˛ ˆ◊í#ˆÏì˛ äÈyˆÏeÓ˚ v zFâ˛ì˛y–

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v zîy£Ó˚í ÈÙÈ 6 °Ï¤˛ ˆ◊í#Ó˚ Ü!íì˛ ˛ôÓ˚#«˛yÎ˚ 40 ãò äÈyˆÏeÓ˚ ≤ÃyÆ ò¡∫Ó˚ ı

8, 48, 55, 52, 78, 42, 93, 85, 7, 37, 94, 66, 72, 73, 66, 91, 52, 78, 85, 9, 68,

81, 64, 60, 75, 84, 78, 10, 63, 21, 14, 30, 19, 25, 93, 33, 15, 29, 25, 13

Ñ õÓï≈õyò v z ôy ÏÎ˚ ¢yãyòñ ˆflÒy ÏÓ˚Ó˚ !Ó¶˛yãòÜ%!ú £ú S!Óòƒy¢ xò%ÎyÎ˚#V

7, 8, 9, 10, 13, 14, 15, 19, 21, 25, 25, 29, 30, 33, 37, 42, 48, 52, 52, 55, 60, 63, 64,66, 66, 68, 72, 73, 75, 78, 78, 78, 81, 84, 85, 85, 91, 93, 93, 94.

!Óüî ¢ÇÖƒÑ˛ ˆflÒyÓ˚ ~£z¶˛yˆÏÓ ¢yãyˆÏú xyõÓ˚y ˆÑ˛yò !Ñ˛ä%È£z !‡˛Ñ˛õì˛ ÓúˆÏì˛ ˛ôy!Ó˚ òy– v˛z ôÓ˚hs%˛

~£z x!Óòƒhfl˛Ï !Óòƒy¢ Ö%Ó x“ ˆ«˛ˆÏe£z ÓƒÓ£,ì˛ £Î˚–

xyõÓ˚y ~£z ˆflÒyÓ˚Ü%!úˆÏÑ˛ xòƒ ˆÑ˛yò v˛z˛ôyˆÏÎ˚ ¢yãyˆÏòyÓ˚ ˆâ˛T˛y Ñ˛!Ó˚ÈÙÙÙÈ

~ˆÏ«˛ˆÏe xyõÓ˚y ¢Ó ˆflÒyÓ˚Ü%!úˆÏÑ˛ xyúyîy xyúyîy òy Ñ˛ˆÏÓ˚ !Ñ˛ä%È ˆflÒyÓ˚ !òˆÏÎ˚ ~Ñ˛!›˛ îú ˜ì˛Ó˚#

Ñ˛Ó˚y £ú– ~£z îúÜ%!ú ÏÑ˛ Óúy £Î˚ ˆ◊í# !Ó¶˛yÜ (C.I) ~Ñ˛!›˛ ˆ◊í#ˆÏì˛ Îì˛Ü%!ú ˆflÒyÓ˚ ÌyÑ˛ˆÏì˛

˛ôyÓ˚ ÏÓ ˆ¢!›˛ £ú ˆ◊í# ˜îá≈ƒ–

10-19 ~£z ˆ◊í#!Ó¶˛y ÏÜ 10, 13, 14, 15, 19 ~£z ˛ô§yâ˛!›˛ ˆflÒyÓ˚ xyˆÏäÈ– ~£z ˛ô§yâ˛ ¢ÇÖƒy!›˛

£ú ~£z ˆ◊í# !Ó¶˛y ÏÜÓ˚ ˛ô!Ó˚¢ÇÖƒy–

!ò¡¨!ú!Öì˛ v z ôyˆÏÎ˚ !Ó¶˛yãò !Óòƒy¢ Ñ˛Ó˚í Ñ˛Ó˚y £Î˚ÈÙÙÙÈ

~£z !Ó¶˛yã Ïò ¢Ó≈!ò¡¨ ˆflÒyÓ˚ 7 ~ÓÇ ¢ˆÏÓ≈yFâ˛ ˆflÒyÓ˚ 94.

˛ô!Ó˚¢Ó = ¢ ÏÓ≈yFâ˛ ˆflÒyÓ˚ – ¢Ó≈!ò¡¨ ˆflÒyÓ˚

= 94 – 7 = 87

˛ô!Ó˚¢Ó˚ xyõyˆÏîÓ˚ ˆ◊í#!Ó¶˛yˆÏÜÓ˚ ¢ÇÖƒy Îy ˆ◊í#˜Ïîá≈ƒ !òï≈yÓ˚í Ñ˛Ó˚ˆÏì˛ ¢£yÎ˚ì˛y Ñ˛ˆÏÓ˚–

ˆòy›˛

128 D.El.Ed. 504


~ˆÏ«˛ˆÏe Î!î xyõÓ˚y ˆ◊í#˜Ïîá≈ƒ 10 !‡˛Ñ˛ Ñ˛!Ó˚ ì˛ˆÏÓ ˆ◊í#!Ó¶˛yˆÏÜÓ˚ ¢ÇÖƒy £ˆÏÓ 10.

ˆ›˛!Óú ÈÙÈ 8.2 îúÜì˛ ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yãò

~!›˛ ~Ñ˛!›˛ îúÜì˛ ˛ô!Ó¢ÇÖƒy !Ó¶˛yãò– ˆÎ ˆÑ˛yòy ˆ◊í#Ó ˆ◊í# Ïîá≈ƒ !òí≈Î Ñ˛Óy ÎyÎ– ~£z¶˛y ÏÓÈÙÙÙÈ

ˆÎõò 60-69 ~£z ˆ◊í#Ó˚ !ò¡¨¢#õy 60 ~Ñ˛£z v˛zFâ˛¢#õy 69.

∴ ˆ◊í# Ïîá≈ƒ = 69 – 60 + 1 = 10

~ˆÏ«˛ˆÏe ¢Ó ˆ◊í#Ó˚ ˆ◊í#˜Ïîá≈ƒ ¢õyò– ˆÑ˛yò ˆÑ˛yò ˆ«˛ˆÏe ˆ◊í#!Ó¶˛yÜÜ%!ú !ò¡¨ ˛v z ôyˆÏÎ˚ Ñ˛Ó˚y

£Î˚ÈÙÙÙÈ

0 - 9 ~Ó˚ ˛ô!Ó˚Ó Ïì≈ 0 - 10

10 - 10 ~Ó˚ ˛ô!Ó˚Ó Ïì≈ 10 - 20

20 - 29 ~Ó˚ ˛ô!Ó˚Ó Ïì≈ 20 - 30

£zì˛ƒy!î

~£z ïÓ˚ˆÏíÓ˚ !ÓòƒyˆÏ¢ ˆÑ˛yò ˆ◊í#Ó˚ v zFâ˛¢#õy ˙ ˆ◊í#Ó˚ xhs˘˛¶≈% =˛ ˆflÒyÓ˚ òÎ˚–

~ˆÏ«˛ˆÏe v zFâ˛¢#õy Á !ò¡¨¢#õyÓ˚ ˛ôyÌ≈Ñ˛ƒ£z £ú ˆ◊í# ˜îá≈ƒ– ˆÎõò 30 – 40 ~£z ˆ◊í#Ó˚

ˆ◊í# Ïîá≈ƒ £ú 40 – 30 = 10–

ˆ◊í#!Ó¶˛yÜ ›˛ƒy!úõyÑ≈ ˛ô!Ó˚¢ÇÖƒy (f)

0-9 03

10-19 05

20-29 04

30-39 03

40-49 02

50-59 03

60-69 06

70-79 06

80-89 04

90-99 04

ˆõy›˛ (N) 40

ˆòy›˛



Ñ˛yã ÈÙÙÙÈ 3

¢Æõ ˆ◊í#Ó˚ !ü«˛yÌ≈# ÏîÓ˚ Áãò ˛ô!Ó˚õy˛ô Ñ˛Ó%ò– ì˛y ÏîÓ˚ Áã ÏòÓ˚ îúÜì˛ ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yãò

äÈÑ˛ ˜ì˛Ó˚# Ñ˛Ó%ò–

.....................................................................................................................

.....................................................................................................................

.....................................................................................................................

8.3 ì˛ ÏÌƒÓ˚ !â˛ey!Î˚ì˛ Ó) ô

!â˛e xyõy ÏîÓ˚ ˆâ˛yÖ ÏÑ˛ xyÑ˛°Ï≈í Ñ˛ ÏÓ˚ñ õ Ïò î#á≈fiÌyÎ˚# £Î˚ ~ÓÇ Ó%G˛ Ïì˛Á ¢%!Óïy £Î˚ñ ˆ¢£z

Ñ˛yÓ˚ˆÏí ì˛ÌƒˆÏÑ˛ v˛z˛ôfiÌy˛ôò Ñ˛Ó˚yÓ˚ ãòƒ !Ó!¶˛ß¨ !â˛ey!Î˚ì˛ Ó˚)˛ô Üv˛¸ v˛zˆÏ‡˛ˆÏäÈ– ~£z xò%ˆÏFäÈˆÏî!˛ô ÏQy@˘Ãyö˛ñ î[˛!â˛eñ xyÎ˚ì˛ ÏúÖñ ˛ôÎ≈î!â˛e ~£z Ü%!ú ¢¡ô ÏÑ≈ xy Ïúyâ˛òy Ñ˛Ó˚y £ú–

8.3.1 !˛ô ÏQy@˘Ãyö˛ ı

xyõÓ˚y Üy!í!ì˛Ñ˛ ì˛Ìƒ v z˛ôfiÌy˛ôò Ñ˛Ó˚yÓ˚ ãòƒ äÈ!Ó !â˛£´ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ Ìy!ÏÑ˛– äÈ!ÓÓ˚ ~Ñ˛!›˛

ì˛yÍ«˛!íÑ˛ xò%¶)˛!ì˛ xyˆÏäÈ– ò#ˆÏâ˛Ó˚ Üy!í!ì˛Ñ˛ ì˛Ìƒ !˛ôˆÏQy@˘ÃyˆÏö˛ v˛z ôfiÌy˛ôò Ñ˛Ó˚y £ú–

v˛zîyı ÈÙÙÙÈ 7 : ~Ñ˛!›˛ !Óîƒyú ÏÎ˚ !Ó!¶˛ß¨ ˆ◊í# Ïì˛ ˛ôv ¸y äÈye#Ó˚ ¢ÇÖƒy ˆîÁÎ˚y £ú ıˆ›˛!Óú ÈÙÈ 8.3 !Óîƒyú ÏÎ˚ äÈye#¢ÇÖƒy

ˆ◊í# I II III IV V

äÈye#¢ÇÖƒy 25 20 30 10 15

v z ôˆÏÓ˚Ó˚ ì˛Ìƒ!˛ôˆÏQy@˘ÃyˆÏö˛ ˆîÖyˆÏòy £ú ı

ˆ◊í# äÈye# ¢ÇÖƒy

V

IV

III

II

I

˛~£z !â˛£´ˆÏÑ˛ ˛ô§yâ˛ãò äÈye#ˆÏÑ˛ !òˆÏî≈ü Ñ˛ˆÏÓ˚

ˆòy›˛

130 D.El.Ed. 504


!˛ô ÏQy@˘Ãyö˛ xAÑ˛ò ı

ˆ◊í# äÈye# ¢ÇÖƒy

1 ã ÏòÓ˚ ãòƒ !â˛£´ ÓƒÓ£,ì˛ £Î˚–






(a) ÖÓˆÏÓ˚Ó˚ Ñ˛yÜã Óy õƒyÜy!ãò ˆÌˆÏÑ˛ 2/3 !›˛ !˛ô ÏQy@˘Ãyö˛ ¢Ç@˘Ã£ Ñ˛ ÏÓ˚ !ü“# ÏîÓ˚ ˆîÖyò–

(b) xy˛ôòyÓ˚ !ÓîƒyúˆÏÎ˚Ó˚ !Ó!¶˛ß¨ ˆ◊í#Ó˚ xò% ô!fiÌ!ì˛ äÈ!ÓÓ˚ õyïƒˆÏõ ˆîÖyò–

...............................................................................................................

...............................................................................................................

...............................................................................................................

8.3.2 î[˛!â˛e ı

!˛ôˆÏQy@˘Ãyö˛ xAÑ˛ò Ñ˛Ó˚y ˆÑ˛yò ˆÑ˛yòÁ ¢õÎ˚ x¢%!ÓïyãòÑ˛ ~ÓÇ ¢õÎ˚ ¢yˆÏ˛ôˆÏ«˛– ì˛ÌƒˆÏÑ˛

xòƒ v z ôyˆÏÎ˚Á v z ôfiÌy˛ôò Ñ˛Ó˚y £Î˚– v zîy£Ó˚í - 7 ~Ó˚ ì˛ÌƒˆÏÑ˛ î[˛!â˛e ~Ó˚ õyïƒˆÏõ v z ôfiÌy˛ôò

Ñ˛Ó˚y £ú–

!â˛e ı 8.6 ˆ◊í# xò%ÎyÎ˚# äÈye#Ó˚ ¢ÇÖƒy

v˛z˛ôˆÏÓ˚Ó˚ ÓyÓ˚!â˛e!›˛ ˛ôÎ≈ˆÏÓ«˛í Ñ˛ˆÏÓ˚ ˆîÖy ÎyˆÏÓ–

ˆ◊í# I ˆ◊í# II ˆ◊í# III ˆ◊í# IV ˆ◊í# V

ˆòy›˛



¢õyò ≤ÃfiÌ ÏFäÈ ÏîÓ˚ Îì˛Ü%!ú î[˛ v zÕ‘¡∫Óy ÏÓ˚ x§yÑ˛y £ ÏÎ˚ ÏäÈ–î%!›˛ îˆÏ[˛Ó˚ õˆÏïƒ ¢õyò ö§˛yÑ˛ xyˆÏäÈ– ≤ÃˆÏì˛ƒÑ˛!›˛ îˆÏ[˛Ó˚ v˛zFâ˛ì˛y ˛ô!Ó˚¢ÇÖƒyˆÏÑ˛ !òˆÏî≈ü Ñ˛ˆÏÓ˚–~£z ïÓ˚ ÏíÓ˚ !â˛e ÏÑ˛ Óúy £Î˚ î[˛!â˛e– î[˛!â˛e xò%¶) !õÑ˛ ¶˛y ÏÓÁ x§yÑ˛y ÎyÎ˚– î[˛Ü%!ú ÏÑ˛

~Ñ˛£z Ó˚Ç Óy ˆüv˛ Ñ˛Ó˚y ÎyÎ˚–ì%˛!õ !Ñ˛ ãyˆÏòy⁄x«˛ ˛ô!Ó˚Óì≈ ò Ñ˛ ÏÓ˚Á xò%¶) !õÑ˛¶˛y ÏÓ î[˛ xAÑ˛ò Ñ˛Ó˚y ÎyÎ˚–

¢Ó î[˛ ~Ñ˛£z Ó˚Ç £ˆÏì˛ £ˆÏÓ–

î[˛!â˛e xAÑ˛ò ≤Ãíyú# ı

ïy˛ô 1 ı xò%¶) !õÑ˛ Á v zÕ‘¡∫ ˆÓ˚Öy xAÑ˛ò

ïy˛ô II ı xò%¶) !õÑ˛ ˆÓ˚ÖyÓ˚ ˆflÒyÓ˚/˛ôî Ó¢yò

ïy˛ô III ı v zÕ‘¡∫ ˆÓ˚ÖyÎ˚ ˛ô!Ó˚¢ÇÖƒyò Ó¢yò

ïy˛ô IV ı ≤ÃˆÏì˛ƒÑ˛!›˛ ˛ôˆÏîÓ˚ v z˛ôÓ˚ ÎÖò ≤ÃfiÌˆÏFäÈˆÏîÓ˚ ¢yÓ˚# ˜ì˛Ó˚# Ñ˛Ó˚%ò–

ïy˛ô V ı ¢y!Ó˚Ü%!úÓ˚ Ó˚Ç Óy ˆüv˛ Ñ˛Ó%ò


~Ñ˛ ¢Æy Ï£Ó˚ !ü«˛yÌ≈# ÏîÓ˚ xò% ô!fiÌ!ì˛ ˆîÁÎ˚y £ú– î[˛!â˛e ˜ì˛Ó˚# Ñ˛Ó%ò–

!îò Ó˚!Ó ˆ¢yõ õAÜú Ó%ï Ó,£ ü%Ñ ü!òv˛z˛ô!fiÌì˛ 25 7 16 11 9 18 20

¢ÇÖƒy

............................................................................................................................

............................................................................................................................

............................................................................................................................

ˆÎÔ!ÜÑ˛ î[˛!â˛e ı

î%£z Óy ì˛yÓ˚ ˆÓü# ïÓ˚ˆÏòÓ˚ ì˛ˆÏÌƒÓ˚ Ó) ô ˆÎÔ!ÜÑ˛ î[˛!â˛e ÓƒÓ£,ì˛ £Î˚– ˆÎõò ¢Æõ ˆ◊í#Ó˚ ˆÑ˛yò

!ü«˛yÌ≈#Ó˚ xï≈Óy!°Ï≈Ñ˛ Á Óy!°Ï≈Ñ˛ ˛ôÓ˚#«˛yÎ˚ !Ó!¶˛ß¨ !Ó°ÏˆÏÎ˚ ≤ÃyÆ ò¡∫Ó˚ ˆîÁÎ˚y £ú– ~£z ïÓ˚ˆÏòÓ˚

ì˛ÌƒˆÏÑ˛ ˆÎÔ!ÜÑ˛ î[˛!â˛ˆÏe v˛z˛ôfiÌy!˛ôì˛ Ñ˛Ó˚y ÎyÎ˚– ~£z !â˛e ˆÌˆÏÑ˛ ì˛yÓ˚ ˛ôyÓ˚î!ü≈ì˛yÓ˚ ˛ôyÌ≈Ñ˛ƒ Ñ˛Ó˚y

ÎyÎ˚–

v zîy£Ó˚í ı 8

~õ.xy£z.~ú £zÇ!úü xAÑ˛ !ÓK˛yò ¢yõy!ãÑ˛ !ü«˛y

xï≈Óy!°Ï≈Ñ˛ 61 65 90 70 58

Óy!°Ï≈Ñ˛ 68 72 80 57 76

!Ó°ÏÎ˚˛ôÓ˚#«˛y

ˆòy›˛

132 D.El.Ed. 504


~ˆÏ«˛ˆÏe ≤ÃˆÏì˛ƒÑ˛ !Ó°ÏˆÏÎ˚Ó˚ ãòƒ ˛ôyüy˛ôy!ü î%!›˛ î[˛!â˛e ÓƒÓ£,ì˛ £Î˚– î[˛!â˛e î%!›˛ ˛ôyÌ≈Ñ˛ƒ Ñ˛Ó˚yÓ˚

!Ó!¶˛ß¨ Ó˚Ç Óy !Ó!¶˛ß¨ ˆüv˛ Ñ˛Ó˚y £Î˚–

!â˛e ı 8.7

8.3.3 xyÎ˚ì˛ˆÏúÖ ı

x!Ó!FäÈß¨ îúÜì˛ ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yãˆÏòÓ˚ ˆ«˛ˆÏe xyÎ˚ì˛ˆÏúÖ ÓƒÓ£yÓ˚ Ñ˛Ó˚y £Î˚– Ñ˛yÓ˚í

~ˆÏ«˛ˆÏe î%!›˛ ˆ◊í#Ó˚ õˆÏïƒ ö§˛yÑ˛y Ó˚yÖyÓ˚ ˆÑ˛yò xÓÑ˛yü ˆò£z–

v zîy£Ó˚í - 9 ˛ôM˛õ ˆ◊í#Ó˚ ~Ñ˛!›˛ £zv˛z!ò›˛ ˆ›˛ˆÏfi›˛Ó˚ ≤ÃyÆ ò¡∫ˆÏÓ˚Ó˚ xyÎ˚ì˛ˆÏúÖ xAÑ˛ò ı

ˆ›˛!Óú ÈÙÈ 8.5

ˆ◊í# !Ó¶˛yÜ 0-5 5-10 10-15 15-20 20-25

!ü«˛yÌ≈# 5 7 10 12 6

!â˛e ı 8.8 Ü!íì˛ ˛˛ôÓ˚#«˛yÓ˚ õyòÈÙÈ~Ó˚ xyÎ˚ì˛ ÏúÖ

~õ.xy£z.~ú £zÇ!úü xAÑ˛ !ÓK˛yò ¢yõy!ãÑ˛

!ü«˛yxï≈Óy!°Ï≈Ñ˛ Óy!°Ï≈Ñ˛

õyò

äÈye

¢ÇÖƒ

y

˛ôM˛õ ˆ◊í#Ó˚ Ü!íì˛ ˛ôÓ˚#«˛yÓ˚ õyò

ˆòy›˛



8.3.4 ˛ôy£z!â˛e ı

îú£#ò ì˛ˆÏÌƒÓ˚ ˆ«˛ˆÏe ˛ôy‡˛!â˛e ÓƒÓ£,ì˛ £Î˚– ˛ôy‡˛!â˛ˆÏe ãyì˛#Î˚ Ó,_yÑ˛yÓ˚ ˆ«˛e ¢õhfl˛Ï ïÓ˚ˆÏòÓ˚

ì˛ÌƒˆÏÑ˛ !òˆÏî≈ü Ñ˛ˆÏÓ˚– ˆÑ˛yò ~Ñ˛›˛ ì˛Ìƒ Ó,_yÑ˛yÓ˚ ˆ«˛ˆÏeÓ˚ ~Ñ˛!›˛ xÇü ≤ÃˆÏì˛ƒÑ˛ ì˛ˆÏÌƒÓ˚ ãòƒ

ˆÑ˛wfiÌ ˆÑ˛yò !ò¡¨¶˛y ÏÓ ˆÓÓ˚ Ñ˛Ó˚y £Î˚ÈÙÙÙÈ

ˆÎÖyˆÏò f = ˙ ì˛ ÏÌƒÓ˚ ˛ô!Ó˚¢ÇÖƒy

N = ˆõy›˛ ˛ô!Ó˚¢ÇÖƒy

v˛zîy£Ó˚í ı 10 ò#ˆÏâ˛Ó˚ v zîy£Ó˚í!›˛Ó˚ ˛ôy‡˛ !â˛e xAÑ˛ò–

ˆ›˛!Óú 8.6 õy!¢Ñ˛ ÖÓ˚â˛

!Ó!¶˛ß¨ ÖÓ˚â˛ ÖÓ˚â˛

Óy!v˛¸¶˛yv˛¸y 21

£zˆÏú!Q…!¢!›˛ 03

!ü«˛y 36

˛ô!Ó˚Ó£ò 06

Öyîƒ 42

!Ó!Óï 12

ˆõy›˛ 120

ˆ›˛!Óú 8.7 õy!¢Ñ˛ ÖÓ˚â˛

ÖÓ˚ Ïâ˛Ó˚ v˛z˛ôÑ˛Ó˚í ÖÓ˚â˛ ˆ¢r›˛Δyú ~ƒ ÏAÜú θθθθθ

Óy!v˛¸¶˛yv˛¸y 21

£zˆÏúÑ˛!›˛Δ!¢!›˛ 03

!ü«˛y 36

˛ô!Ó˚Ó£ò 06

ˆòy›˛

134 D.El.Ed. 504


Öyîƒ 42

!Ó!Óï 12

!â˛e 8.9

ïy˛ôÈÙÈI : ≤Ã!ì˛!›˛ ˆ¢QÓ˚~Ó˚ ˆÑ˛w#Î˚ ˆÑ˛yí θ !òí≈Î˚ Ñ˛Ó˚%ò–ïy˛ôÈÙÈII : ˆÎˆÏÑ˛yò Óƒy¢yˆÏï≈Ó˚ ~Ñ˛!›˛ Ó,_ xAÑ˛ò Ñ˛Ó˚%ò–ïy˛ôÈÙÈIII : Óƒy¢yˆÏï≈Ó˚ myÓ˚y ≤Ã!ì˛!›˛ ˆ¢QˆÏÓ˚Ó˚ ˆÑ˛yòÜ%!ú ˛ôÓ˚˛ôÓ˚ xAÑ˛ò Ñ˛Ó˚%ò–ïy˛ôÈÙÈIV : ≤Ã!ì˛!›˛ ˆ¢QÓ˚ !Ó!¶˛ß¨ Ó˚Ç Óy ˆüv˛ Ñ˛Ó˚%ò–

E1. 5!›˛ @˘ÃyˆÏõ ˆõy›˛ ≤Ãyí# ¢ÇÖƒy ˆîÁÎ˚y £ú ı

òÓ˚£!Ó˚˛ô%Ó˚ ˛ô%Ó˚òyˆÏÑ˛y›˛ ˛ôy›˛úy ¢yòyõ%[˛y Ñ˛y!hs˘˛õy!ú

80 100 60 120 50

E2. 1¢ÆyˆÏ£ Ó£z !ÓˆÏÑ ì˛yÓ˚ !Ó!Ñ ì˛ Ó£z ÏÎ˚Ó˚ ¢ÇÖƒy ˆîÁÎ˚y £ú ı

!îò Ó˚!Ó ˆ¢yõ õAÜú Ó%ï Ó,£ ü%Ñ ü!ò

Ó£zˆÏÎ˚Ó˚ ¢ÇÖƒy 60 40 35 50 25 70 30

E3. v z!v ¸°ÏƒyÓ˚ !Ó!¶˛ß¨ ÓÎ˚ Ï¢Ó˚ ˆúy ÏÑ˛Ó˚ ¢ÇÖƒy ˆîÁÎ˚y £ú ı

ÓÎ˚¢ 1-15 15-30 30-45 45-60 60-75 75ÈÙÈ~Ó˚ ˆÓü#

ˆúyÑ˛¢ÇÖƒy 24 30 42 36 18 12

E4. !Ó!¶˛ß¨ ÎyòÓy£ˆÏò Ñ˛ˆÏÓ˚ xy¢y äÈyˆÏeÓ˚ ¢ÇÖƒy ı

Îyì˛yÈÎ˚yì˛ fl%Òú Óy¢ ¢y£zˆÏÑ˛ú ˆõy›˛Ó˚ ¢y£zˆÏÑ˛ú Üy!v˛¸ xòƒyòƒ

/fl%Ò›˛yÓ˚

äÈyeÈ/äÈye#Ó˚ ¢ÇÖƒy 160 80 60 20 40

˛ôy£z ¢yÓ˚í# õy!¢Ñ˛ ÖÓ˚â˛ ˆîÖyÎ˚

Óy!v˛¸ ¶˛yv˛¸y£z ÏúÑ˛!› Δ!¢!›˛!ü«˛y˛ô!Ó˚Ó£òÖyîƒ!Ó!Óï

ˆòy›˛



8.4 ì˛Ìƒ !Ó ÏŸ’°Ïí ı

¢ÇÜ,£#ì˛ ì˛ÌƒˆÏÑ˛ !Ñ˛¶˛yˆÏÓ äÈ!Ó ~ÓÇ @˘ÃyˆÏö˛Ó˚ õyïƒˆÏõ v˛z˛ôfiÌy!˛ôì˛ Ñ˛Ó˚y ÎyÎ˚ ì˛y xyõÓ˚y

!üˆÏÖ!äÈ– ~äÈyv ¸y xòƒ ˆÑ˛yò v z ôyˆÏÎ˚ ì˛ÌƒˆÏÑ˛ !ÓˆÏŸ’°Ïí Ñ˛Ó˚y ÎyÎ˚⁄

~ ˛ôÎ≈hs˘˛ ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yãò !òˆÏÎ˚ ˆÎ xyˆÏúyâ˛òy £ˆÏÎ˚ˆÏäÈñ ì˛y ˆÌˆÏÑ˛ î%!›˛ !Ó°ÏÎ˚ ˆîÖy ÎyÎ˚ı

ˆflÒyˆÏÓ˚Ó˚ õyòÜ%!ú ~Ñ˛!›˛ ˆÑ˛w#Î˚õyˆÏòÓ˚ !îˆÏÑ˛ ãˆÏv ¸y £ˆÏì˛ ÌyˆÏÑ˛– ~£z ïÓ˚ˆÏòÓ˚ ˜Ó!üT˛ƒˆÏÑ˛

Óúy £Î˚ ì˛ ÏÌƒÓ˚ ˆÑ˛w#Î˚ ≤ÃÓíì˛y– ˙ ˆÑ˛w#Î˚ õyò!›˛ ÏÑ˛ ~Ñ˛!›˛ ≤Ã!ì˛!ò!ï !£¢y ÏÓ Üíƒ Ñ˛Ó˚y £Î˚–

ü%ï%õye ˆÑ˛w#Î˚ ≤ÃÓíì˛yÓ˚ ˛ô!Ó˚õyí myÓ˚y !Ó¶˛yãò#Î˚ ˜Ó!üT˛ƒ ¢¡ô)í≈Ó) Ï˛ô Ó%G˛y ÎyÎ˚ òy–

ˆÑ˛w#Î˚ õyˆÏòÓ˚ ˛ôyüy˛ôy!üñ ˆflÒyÓ˚Ü%!ú ˙ ˆÑ˛w#Î˚ õyˆÏòÓ˚ v z_Ó˚!îˆÏÑ˛ Ñ˛ì˛î)Ó˚ !Óhfl ,Ïì˛ ì˛yÁ ãyòyÓ˚

≤Ã ÏÎ˚yãò– ò# Ïâ˛Ó˚ !Ó¶˛yãò î%!›˛ ˆîÖ Ïú ˆÓyG˛y Îy ÏÓ–

(a) 10, 11, 12, 13, 14, 15, 16

(b) 5, 9, 10, 13, 19, 18

v˛z˛ôˆÏÓ˚Ó˚ î%!›˛ !Ó¶˛yãˆÏòÓ˚ ˆÑ˛w#Î˚ õyò 13. !Ñ˛hs%˛ ≤ÃÌõ !Ó¶˛yãò!›˛Ó˚ ˆflÒyÓ˚Ü%!ú 13ÈÙÈ~Ó˚ Ö%Ó

Ñ˛yäÈyÑ˛y!äÈ xyˆÏäÈ– ~£z ¢Ó˚ú v zîy£Ó˚í!›˛ ˆÌˆÏÑ˛ ¢£ˆÏã£z ˆflÒyÓ˚Ü%!ú !Óhfl˛,Ï!ì˛ Óy !Ó°Ïõì˛y ¢¡ôˆÏÑ≈˛

ãyòyÓ˚ ≤ÃˆÏÎ˚yãò– ˆflÒyÓ˚Ü%!úÓ˚ ~£z ïÓ˚ˆÏòÓ˚ ˛ô!Ó˚Óì≈˛òü#úyˆÏÑ˛ Óúy £ÈÎ˚ !Óõ°Ïì˛y– ~£z !Ó°Ïõì˛yÓ˚

˛ô!Ó˚õy˛ô ˆÌˆÏÑ˛ xyõÓ˚y ¢£ˆÏã£z Ó%G˛ˆÏì˛ ˛ôy!Ó˚ ˆflÒyÓ˚Ü%!ú Ñ˛ì˛î)Ó˚ ˛ôÎ≈hs˘˛ !Óhfl˛,Ïì˛–

~£z xò%ˆÏFäÈˆÏî ˆÑ˛w#Î˚ ≤ÃÓíì˛yÓ˚ ˛ô!Ó˚õy˛ô ~ÓÇ !Ó°Ïõì˛yÓ˚ ˛ô!Ó˚õy˛ô ¢¡ôˆÏÑ≈˛ xyˆÏúyâ˛òy

Ñ˛Ó˚y £ˆÏÓ–

8.4.1 ˆÑ˛w#Î˚ ≤ÃÓíì˛yÓ˚ ˛ô!Ó˚õy˛ô¢õ)£ ı

ˆÑ˛w#Î˚ õyò!›˛ ¢¡ô)í≈ !Ó¶˛yãò!›˛Ó˚ ~Ñ˛!›˛ ≤Ã!ì˛!ò!ï äÈye#Ó˚ õyò– xy«˛!Ó˚Ñ˛ xˆÏÌ≈ ˆÑ˛w#Î˚

≤ÃÓíì˛y Óú Ïì˛ ¢õhfl Ï ˆflÒyÓ˚Ü%!ú ~Ñ˛ ˆÑ˛w#Î˚ õy ÏòÓ˚ !î ÏÑ˛ x@˘Ã¢Ó˚ £Î˚– xyõÓ˚y !ì˛ò ïÓ˚ ÏòÓ˚

ˆÑ˛w#Î˚ ≤ÃÓíì˛yÓ˚ ˛ô!Ó˚õy˛ô !ò ÏÎ˚ xy Ïúyâ˛òy Ñ˛Ó˚Ó– ˆ¢Ü%!ú £úÈÙÙÙÈ

(i) Üy!í!ì˛Ñ˛ Üv ¸

(ii) õïƒõy

(iii) ¶) !õT˛Ñ˛

8.4.1.1 Üy!í!ì˛Ñ˛ Üv ¸ ı

Üy!í!ì˛Ñ˛ Üv ¸ £ Ïúy ¢Ó ÏÌ ÏÑ˛ ˆÓü# õyeyÎ˚ ≤Ã ÏÎyãƒ ˆÑ˛w#Î˚ ≤ÃÓíì˛yÓ˚ ˛ô!Ó˚õy˛ô– £z£y Üv ¸

òy ÏõÁ ˛ô!Ó˚!â˛ì˛–

ˆòy›˛

136 D.El.Ed. 504


SÑ˛V x!ÓòƒfiÌ ì˛ ÏÌƒÓ˚ Üv ¸ ı Ñ˛ì˛Ñ˛Ü%!ú ˆîÁÎ˚y ì˛ ÏÌƒÓ˚ (x1, x

2, x

3 .....x

n) Üv ¸ ÏÑ˛ ¢ÇK˛y!Î˚ì˛

Ñ˛Ó˚y £Î˚ÈÙÙÙÈ

(Σ ~£z !â˛£´!›˛ ˆÎy ÏÜÓ˚ ãòƒ ÓƒÓ£,ì˛V

v˛zîy ı 11 : ~Ñ˛!îˆÏòÓ˚ !Ñ ˛ˆÏÑ˛›˛ !¢!Ó˚ˆÏã ˆïyò# 33, 17, 60, 25 ~ÓÇ 45Ó˚yò Ñ˛ˆÏÓ˚ˆÏäÈ– ì˛yÓ˚

Ó˚y ÏòÓ˚ Üv ¸ Ñ˛ì˛⁄

ˆõy›˛ Ó˚yò = 33 + 17 + 60 + 25 + 45 = 180

ˆõy›˛ õƒyâ˛ ~Ó˚ ¢ÇÖƒy = 5

∴ ≤ÃˆÏì˛ƒÑ˛ õƒyˆÏâ˛Ó˚ Üv˛¸ Ó˚yò = 36

Ñ˛yã ÈÙÈ 6 ı

ˆÑ˛yò ~Ñ˛!›˛ ¢ÆyˆÏ£ xy˛ôòyÓ˚ ˛ôv˛¸yÓ˚ ¢õˆÏÎ˚Ó˚ Üv˛¸ !òí≈Î Ñ˛ˆÏÓ˚y–

~Ñ˛ ¢Æy Ï£ xy˛ôòyÓ˚ á%õy ÏòyÓ˚ ¢õ ÏÎ˚Ó˚Ó˚ Üv ¸ !òí≈Î˚ Ñ˛ ÏÓ˚y–

__________________________________________

__________________________________________

__________________________________________

SÖV x!ÓòƒfiÌ ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yã ÏòÓ˚ Üv ¸ ı ò# Ïâ˛Ó˚ v zîy£Ó˚í!›˛ ˆòÁÎ˚y ÎyÑ˛–

v˛zîy ÈÙÈ 12 : Ó˚!OˆÏì˛Ó˚ ˆÑ˛yò ~Ñ˛!›˛ ö˛yˆÏõ≈Ó˚ ˜î!òÑ˛ ˛ôy!Ó˚◊!õÑ˛ !òˆÏ¡¨Ó˚ ¢yÓ˚!íˆÏì˛ ˆîÁÎ˚y £ú–

ì˛yÓ˚ ˜î!òÑ˛ ˛ôy!Ó˚◊!õˆÏÑ˛Ó˚ Üv ¸ !òí≈Î˚ Ñ˛Ó%í ı

˜î!òÑ˛ ˛ôy!Ó˚◊!õÑ˛ 120 130 140 145 150

!îò ¢ÇÖƒy 5 4 7 6 8

SÖVÜv ¸ ˜î!òÑ˛ õã%Ó˚# !òí≈Î˚ Ñ˛Ó˚yÓ˚ ãòƒ xyõÓ˚y ò# Ïâ˛Ó˚ ¢yÓ˚í#!›˛ ˜ì˛!Ó˚ Ñ˛Ó˚úyõ–

ˆòy›˛



¢yÓ˚í# 8.8 : ~Ñ˛ õy Ï¢Ó˚ ˜î!òÑ˛ ˛ôy!Ó˚◊!õÑ˛

˜î!òÑ˛ ˛ôy!Ó˚◊!õÑ˛ !îò ¢ÇÖƒy f × x

S˛ô!Ó˚¢ÇÖƒy = f)

120 5 600

130 4 520

140 7 980

145 6 870

150 8 1200

Σf = 30 Σfx = 4170

Ó˚!O Ïì˛Ó˚ Üv ¸ ˜î!òÑ˛ õã%Ó˚# = 139.00 ›˛yÑ˛y

Ñ˛yã ÈÙÙÙÈ 7 : ò# Ïâ˛Ó˚ ≤ÃyÆ ò¡∫ ÏÓ˚Ó˚ Üv ¸ !òí≈Î˚ Ñ˛Ó%ò ı

≤ÃyÆ òÇ 15 17 20 22 24 25

!ü«˛yÌ≈#Ó˚ ¢ÇÖƒy 3 5 9 4 7 2

SÜV ˆ◊í#ÓÂï ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yã ÏòÓ˚ Üv ¸ ı ˆ◊í#ÓÂï ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yãˆÏòÓ˚ ˆ«˛ˆÏe ≤ÃˆÏì˛ƒÑ˛!›˛

ˆ◊í#Ó˚ ˛ô!Ó˚ÓˆÏì≈˛ ~Ñ˛!›˛ ≤Ã!ì˛!ò!ï fiÌyò#Î˚ õyòˆÏÑ˛ ïˆÏÓ˚ !òˆÏÎ˚ x@˘Ã¢Ó˚ £ˆÏì˛ £Î˚– ~£z ≤Ã!ì˛!ò!ï

fiÌyò#Î˚ õyò!›˛ £ú ˙ ˆ◊í#Ó˚ õïƒõyò–

¢%ì˛Ó˚yÇñ ˆÑ˛yò ~Ñ˛!›˛ ˆ◊í#!Ó¶˛yãˆÏòÓ˚ õïƒõyò ñ ˆÎÖy Ïò l1 ~ÓÇ l

2 ˙ ˆ◊í#Ó˚ !ò¡¨

Á v zFâ˛¢#õy–

v zîy£Ó˚í ı 13 !òˆÏ¡¨Ó˚ !Ó¶˛yãò!›˛Ó˚ Üv˛¸ !òí≈Î˚ Ñ˛Ó˚%ò

¢yÓ˚í# 8.9 ˆ◊í#ÓÂï ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yã ÏòÓ˚ Üv ¸

ˆ›˛!Óú 8.9 ˆ◊í#ÓÂï ˛ô!Ó˚¢ÇÖƒy Ó^˘›˛ò

ˆflÒyÓ˚ ˛ô!Ó˚¢ÇÖƒy õïƒõy f × x(C.I.) (f) (x)

0-10 5 5 25

10-20 6 15 90

20-30 12 25 300

30-40 8 35 280

ú

ˆòy›˛

138 D.El.Ed. 504


40-50 10 45 450

50-60 6 55 330

60-70 6 65 390

50=∑ f

¢%ì˛Ó˚yÇ Üv ¸

!Ó.o. ı õïƒõyˆÏòÓ˚ ˛ô!Ó˚ˆÏ≤Ã!«˛ˆÏì˛ ˆ◊í#Ü%!ú ˛ô!Ó˚Ó!ì≈˛ì˛ Ñ˛Ó˚yÎ˚ ˆ◊í#ÓÂï !ÓK˛y˛ôò x!Óòƒhfl˛Ï

!Ó¶˛yã Ïò ˛ô!Ó˚Ó!ì≈ ì˛ £Î˚ ~ÓÇ ~£z õïƒõyòÜ%!ú ˆflÒyÓ˚ !£¢y ÏÓ ˛ô!Ó˚!â˛ì˛ £Î˚–

Ñ˛yãÈÙÈ8

Üv ¸ !òí≈Î˚ Ñ˛Ó˚

ˆ◊í# !Ó¶˛yãò 0-10 10-20 20-30 30-40 40-50 50-60

˛ô!Ó˚¢ÇÖƒy 8 12 15 9 11 5

E-5 : ≤ÃÌõ 10 !›˛ fl˛∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ Üv˛¸ Ñ˛ì˛⁄

ì% !õ !Ñ˛ ãyò⁄

¢yïyÓ˚í !Ó¶˛yã Ïò Üv ¸ £ú ¢Ó ÏÌ ÏÑ˛ !ÓŸªhfl Ï ˆÑ˛w#Î˚ ≤ÃÓíì˛yÓ˚ ˛ô!Ó˚õy˛ô

ˆflÒyÓ˚ÈÙÈ~Ó˚ xyÑ˛yˆÏÓ˚Ó˚ Á˛ôÓ˚ Üv˛¸ !ò¶≈˛Ó˚ Ñ˛ˆÏÓ˚– Î!î ˆÑ˛yò ˆflÒyÓ˚ C ~Ñ˛Ñ˛ ˛ô!Ó˚õyí Óyv˛¸yˆÏòy

SÓy Ñ˛õyˆÏòyV ÎyÎ˚ñ ì˛ˆÏÓ Üv˛¸Á ~Ñ˛Ñ˛ ˛ô!Ó˚õyò ÓyˆÏv ¸ SÓy Ñ˛ˆÏõV–

Î!î ≤ÃˆÏì˛ƒÑ˛ ˆflÒyÓ˚ÈÈÙÈ~Ó˚ ¢yˆÏÌ c ~Ñ˛Ñ˛ ˆÎyÜ Ñ˛Ó˚y ÎyÎ˚ ì˛ˆÏÓ Üv ¸Á C ~Ñ˛Ñ˛ ˛ô!Ó˚õyò Óy Ïv ¸–

xÌ≈yÍ Î!î Üv ¸ £Î˚ ì˛ ÏÓ òì% ò Üv ¸ £ ÏÓ c+

8.4.1.2 õïƒõy

ˆflÒyÓ˚Ü%ˆÏúyˆÏÑ˛ Ñ ˛õyß∫ˆÏÎ˚ SÓ!ï≈°÷Ï$ Óy «˛!Î˚°÷Ï$V xyÑ˛yˆÏÓ˚ ¢yãyˆÏòy ÎyÎ˚ ì˛ˆÏÓ !‡˛Ñ˛ õyˆÏG˛Ó˚ fiÌyò!›˛

£ú õïƒõy–

¢%ì˛Ó˚yÇ õïƒõy £ú ~õò ~Ñ˛›˛y ˆflÒyÓ˚ ˆÎ!›˛ ¢¡ôß¨ ˆflÒyÓ˚Ü%ˆÏúyˆÏÑ˛ î%!›˛ xˆÏï≈ !Ó¶˛=˛ Ñ˛ˆÏÓ˚– xÌ≈yÍ

õïƒõy õyˆÏòÓ˚ v˛z˛ôˆÏÓ˚ xˆÏï≈Ñ˛ ˆflÒyÓ˚ ~ÓÇ ò#ˆÏâ˛ xˆÏï≈Ñ˛ ˆflÒyÓ˚ ÌyÑ˛ˆÏÓ–

ˆòy›˛



SÑ˛V x!Óòƒhfl Ï ì˛ ÏÌƒÓ˚ õïƒõy ı

ˆflÒyÓ˚Ü%ˆÏúyˆÏÑ˛ Ñ ˛õÓï≈õyò Óy Ñ ˛õ£…y¢ õyò ¶˛yˆÏÓ ¢yãyˆÏòyÓ˚ ˛ôÓ˚ !‡˛Ñ˛ õyˆÏG˛Ó˚ ˆflÒyÓ˚!›˛ £ú õïƒõy–

x!Óòƒhfl Ï ì˛ ÏÌƒÓ˚ õïƒõy !òí≈Î˚ ˛ôÂï!ì˛ ı ≤ÃÌ Ïõ ˆflÒyÓ˚Ü% Ïúy ÏÑ˛ Ñ õÓï≈õyò Óy Ñ õ£…y¢õyò¶˛y ÏÓ

¢yãyˆÏì˛ £ˆÏÓ– ïÓ˚y ÎyÑ˛ ˆõy›˛ ì˛ˆÏÌƒÓ˚ ¢ÇÖƒy n

ˆÑ˛¢ÈÙÙÙÈ(I) n ÎÖò !Ó Ïãyv ¸ ¢ÇÖƒy

õïƒõy ì˛õ ˆflÒyˆÏÓ˚Ó˚ õyò–

ˆÑ˛¢ÈÙÙÙÈ(II) n ÎÖò ˆãyv ¸ ¢ÇÖƒy

õïƒõy

ò# Ïâ˛Ó˚ v zîy£Ó˚í!›˛ ˆòÁÎ˚y ÎyÑ˛

v zîy£Ó˚í ı 14 ˆÑ˛yò ~Ñ˛!›˛ £zfl%ÒˆÏúÓ˚ 10 ãò !ü«˛ ÏÑ˛Ó˚ ÓÎ˚¢ SÓäÈÓ˚VÜ%!ú £ú 37, 34, 52, 45,

50, 41, 31, 40, 36 ~ÓÇ 55 ì˛yˆÏîÓ˚ õïƒõy ÓÎ˚¢ Ñ˛ì˛⁄

ÓÎ˚¢Ü%!ú ÏÑ˛ Ñ õÓï≈õy Ïò ¢y!ã ÏÎ˚ ˛ôyÁÎ˚y ÎyÎ˚ ı 31, 34, 36, 37, 40, 41, 45, 50, 52

~ÓÇ 55

~ Ï«˛ ÏeÏ n = 10, ˆ¢!›˛ ˆãyv˛¸ ¢ÇÖƒy

¢%ì˛Ó˚yÇ ~£z !Ó¶˛yãˆÏòÓ˚ !‡˛Ñ˛ õïƒ ˆflÒyÓ˚ £ú î%!›˛ 40 ~ÓÇ 41

∴ õïƒõy = 40 ~ÓÇ 41 ~Ó˚ Üv ¸

(40 + 41)

81×

= 40.5

¢%ì˛Ó˚yÇ !ü«˛Ñ˛ ÏîÓ˚ õïƒõy ÓÎ˚¢ 40.5 ÓäÈÓ˚–

v zîy£Ó˚í 15 : ~Ñ˛!îˆÏòÓ˚ ˆÖúyÎ˚ ¶˛yÓ˚ì˛#Î˚ !Ñ ˆÏÑ˛›˛ îˆÏúÓ˚ Ó˚yòÜ%!ú £ú 95, 40, 2, 55, 10,

38, 33, 22, 0, 18, 8, õïƒõy ˆflÒyÓ˚!›˛ !òí≈Î˚ Ñ˛Ó%ò–

ˆflÒyÓ˚Ü%!úˆÏÑ˛ Ñ õ£…y¢õyò ¶˛yˆÏÓ ¢yãyˆÏú ˛ôyÁÎ˚y ÎyÎ˚ÈÙÙÙÈ

95, 55, 40, 38, 33, 22, 18, 10, 8, 2, 0

~ˆÏ«˛ˆÏe n = 11 ˆ¢!›˛ !ÓˆÏãyv˛¸ ¢ÇÖƒy

ì˛õ ˆflÒyÓ˚ì˛õ ˆflÒyÓ˚

ˆòy›˛

140 D.El.Ed. 504


∴ õïƒõyÓ˚ ˆflÒyÓ˚ £ú (11 1)× + ì˛õ ˆflÒyÓ˚

12× ì˛õ ˆflÒyÓ˚

= 6 ì˛õ ˆflÒyÓ˚

= 22

¢%ì˛Ó˚yÇ ~ˆÏ«˛ˆÏe õïƒõy ˆflÒyÓ˚!›˛ £ú 22

SÖV x!Óòƒhfl Ï ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yã ÏòÓ˚ õïƒõy ı

x!Óòƒhfl˛Ï ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yãˆÏòÓ˚ ˆ«˛ˆÏe õïƒõy !òí≈ˆÏÎ˚Ó˚ ãòƒ ò#ˆÏâ˛Ó˚ ïy˛ôÜ%!ú xò%¢Ó˚í Ñ˛Ó˚y

£Î˚ÈÙÙÙÈ

ïy˛ôÈÙÈ1 ı ˆflÒyÓ˚Ü%!úˆÏÑ˛ Ñ õÓï≈õyò Óy Ñ õ£…y¢õyò v˛z ôyˆÏÎ˚ ¢yãyˆÏì˛ £ˆÏÓ–

ïy˛ôÈÙÈ2 : Ñ õ ÏÎÔ!ÜÑ˛ ˛ô!Ó˚¢ÇÖƒy !òí≈Î˚ Ñ˛Ó˚ Ïì˛ £ ÏÓ–

ïy˛ôÈÙÈ3 ı ˆõy›˛ ˛ô!Ó˚¢ÇÖƒy N !òí≈Î˚ Ñ˛Ó˚ Ïì˛ £ ÏÓ–

Î!î N !Ó Ïãyv ¸ ¢ÇÖƒy £Î˚ ì˛ ÏÓ õïƒõy = ì˛õ ˆflÒyÓ˚

ò# Ïâ˛Ó˚ v zîy£Ó˚í!›˛ ˆòÁÎ˚y ÎyÑ˛

v zîy£Ó˚í 16 : ò# Ïâ˛Ó˚ !Ó¶˛yãò!›˛Ó˚ õïƒõy !òí≈Î˚ Ñ˛Ó%ò

≤ÃyÆ ò¡∫Ó˚ 17 20 15 22 18 25

äÈye/äÈye#Ó˚ ¢ÇÖƒy (f) 5 8 6 7 10 3

~Öò Ñ õ ÏÎÔ!ÜÑ˛ ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yãò ¢yÓ˚í# ˜ì˛Ó˚# Ñ˛Ó˚y £ Ïúy

ˆ›˛!Óú 8.10 : Ñ õ ÏÎÔ!ÜÑ˛ ˛ô!Ó˚¢ÇÖƒy !Óì˛Ó˚í

≤ÃyÆ ò¡∫Ó˚ (x) äÈye/äÈye#Ó˚ ¢ÇÖƒy (f) Ñ õ ÏÎÔ!ÜÑ˛ ˛ô!Ó˚¢ÇÖƒy (cf)

15 6 617 5 6 + 5 = 11

18 10 11 + 10 = 21*

20 8 21 + 8 = 29

22 7 29 + 7 = 36

25 3 36 + 3 = 39

N = 39f =∑

20 ì˛õ fiÌyò

ˆòy›˛



¢%ì˛Ó˚yÇ õïƒõyÓ˚ fiÌyò £ú ı ì˛õ fiÌyò Óy ì˛õ fiÌyò Óy 20ì˛õ fiÌyò

~Ó˚ xÌ≈ 20ì˛õ fiÌyˆÏò õïƒõy!›˛ xÓ!fiÌì˛– ~£z v˛zîy£Ó˚ˆÏí 20ì˛õ fiÌyˆÏò 18 ˆflÒyÓ˚!›˛ !Óîƒõyò–

SÜV ˆ◊í#ÓÂï !Ó¶˛yã ÏòÓ˚ õïƒõy ı

¢Æõ ˆ◊í#Ó˚ !ü«˛yÌ≈# Ü!í Ïì˛ ≤ÃyÆ ò¡∫Ó˚ ò# Ïâ˛Ó˚ ˆ◊í# !Ó¶˛yã Ïò ˆîÁÎ˚y £úÈÙÙÙÈ

ˆ›˛!Óú 8.11 : ˆ◊í#ÓÂï !Ó¶˛yã ÏòÓ˚ õïƒõy

≤ÃyÆ ò¡∫Ó˚ (x) äÈye/äÈye#Ó˚ ¢ÇÖƒy (f) Ñ õ ÏÎÔ!ÜÑ˛ ˛ô!Ó˚¢ÇÖƒy (cf)

30-39 1 140-49 3 450-59 8 1260-69 15 2770-79 10 3780-89 9 46

90-99 4 50

N = 50

ˆ◊í#ÓÂï !Ó¶˛yã ÏòÓ˚ õïƒõy !òí≈Î˚ Ñ˛Ó˚yÓ˚ ãòƒ ò# Ïâ˛Ó˚ ¢)e!›˛ ÓƒÓ£yÓ˚ Ñ˛Ó˚y £ÈÎ˚

õïƒõy =

ˆÎÖyˆÏò Lm = õïƒõy ˆ◊í#Ó˚ ≤ÃÑ, ì˛ !ò¡¨¢#õy

N = ˆõy›˛ ˆflÒyÓ˚

F = õïƒõy ˆ◊í#Ó˚ ò#ˆÏâ˛Ó˚ ˆ◊í# ˛ôÎ≈hs˘˛ ˆõy›˛ ˛ô!Ó˚¢ÇÖƒy

fm = õïƒõy ˆ◊í#Ó˚ ˛ô!Ó˚¢ÇÖƒy

i = õïƒõy ˆ◊í#Ó˚ ˜îá≈ƒ–

v˛z˛ôˆÏÓ˚Ó˚ v˛zîy£Ó˚í!›˛ˆÏì˛ N = 50–

õïƒõy 25ì˛õ ~ÓÇ 26ì˛õ fiÌy ÏòÓ˚ õ Ïïƒ xÓfiÌyò Ñ˛ ÏÓ˚ xÌ≈yÍ 60-69 ~£z ˆ◊í#Ó˚ õ Ïïƒ

xÓfiÌyò Ñ˛ ÏÓ˚–

ˆòy›˛

142 D.El.Ed. 504


~ Ï«˛ Ïeñ Lm = 60-69 ˆ◊í#Ó˚ ≤ÃÑ, ì˛ !ò¡¨¢#õy = 59.5

F = 12, fm = 15 ~ÓÇ i = 10

∴ õïƒõy = 59.5 + = 59.5+8.67= 68.17

õïƒõy !òí≈ˆÏÎ˚Ó˚ ïy˛ôÜ%!ú ı

1. Ñ õ ÏÎÔ!ÜÑ˛ ˛ô!Ó˚¢ÇÖƒy !òí≈Î˚ Ö[˛ò ~ÓÇ ˆÎ ˆ◊í# Ïì˛ õïƒõy xy ÏäÈ ˆ¢!›˛ ¢òy=˛ Ñ˛Ó%ò–

2. õïƒõy ˆ◊í#Ó˚ ≤ÃÑ, ì˛ !ò¡¨¢#õy !òï≈yÓ˚í Ñ˛Ó˚%ò–

3. Ñ õˆÏÎÔ!ÜÑ˛ ˛ô!Ó˚¢ÇÖƒy ˆÌˆÏÑ˛ õïƒõy ˆ◊í#Ó˚ !ò¡¨ Ï◊í# ˛ôÎ≈hs˘˛ Ñ õˆÏÎÔ!ÜÑ˛ ˛ô!Ó˚¢ÇÖƒy !òí≈Î˚

Ñ˛Ó%ò–

4. õïƒõy ˆ◊í#Ó˚ ˛ô!Ó˚¢ÇÖƒy !òï≈yÓ˚í Ñ˛Ó%ò–

5. v z ô!Ó˚v z=˛ õyòÜ%!ú ¢) Ïe Ó!¢ˆÏÎ˚ õïƒõyÓ˚ õyò !òí≈Î˚ Ñ˛Ó%ò–

E6. ≤ÃÌõ xy›˛!›˛ !ÓˆÏãyv˛¸ fl˛∫y¶˛y!ÓÑ˛ ¢ÇÖƒyÓ˚ õïƒõy !òí≈Î˚ Ñ˛Ó˚%ò–

E7. ò#ˆÏâ˛Ó˚ ¢yÓ˚í# ˆÌˆÏÑ˛ õïƒõy Áãò !òí≈Î˚ Ñ˛Ó%ò–

Áãò SˆÑ˛!ãV 40 42 45 46 48 50 52

äÈye/äÈye#Ó˚ ¢ÇÖƒy 8 5 6 9 7 4 2

ì% !õ !Ñ˛ ãyò ı

● õïƒõy £ú ¢¡ô)í≈ !Ó¶˛yãò!›˛Ó˚ !‡˛Ñ˛ õyG˛ÖyˆÏò ˆÎ ˆflÒyÓ˚!›˛ xyˆÏäÈ ˆ¢!›˛–

● õïƒõy £ú xÓfiÌyò Üì˛ ˆflÒyÓ˚ ~ÓÇ ˆ¢£z Ñ˛yÓ˚ Ïí ¢#õyfiÌ ˆflÒy ÏÓ˚Ó˚ Á˛ôÓ˚ £z£y !ò¶≈ Ó˚ Ñ˛ ÏÓ˚

òy–

● Ö%Ó «% o ¢ÇÖƒÑ˛ Óy Ö%Ó Ó,£Í ˆflÒy ÏÓ˚Ó˚ ˆ«˛ Ïe õïƒõy ÓƒÓ£yÓ˚ Ñ˛Ó˚y £Î˚–

● Ö%Ó ì˛yv ¸yì˛y!v ¸ ˆÑ˛w#Î˚ ≤ÃÓíì˛yÓ˚ ˛ô!Ó˚õy˛ô ãyòyÓ˚ ãòƒ õïƒõy ÓƒÓ£,ì˛ £Î˚–

8.4.1.3 ¶) !°Ï‹ÒÑ˛

Üv ¸ ~ÓÇ õïƒõy äÈyv ¸y ˆÑ˛w#Î˚ ≤ÃÓíì˛y ˛ô!Ó˚õyˆÏ˛ôÓ˚ xyˆÏÓ˚Ñ˛!›˛ ˛ôÂï!ì˛ £ú ¶) !°Ï¤˛Ñ˛–

ˆòy›˛



v˛zîy£Ó˚í 17 : õy!¢Ñ˛ â˛y!£îy !òí≈Î˚ Ñ˛Ó˚yÓ˚ ãòƒ ~Ñ˛ãò ˆîyÑ˛yòîyÓ˚ !Ó!¶˛ß¨ õyˆÏ˛ôÓ˚ ã% Ïì˛yÓ˚

!£¢yÓ ˆÓ˚ ÏÖ ÏäÈò– ~Ñ˛ õy Ï¢Ó˚ ì˛Ìƒ ˆîÁÎ˚y £ ÏúyÈÙÙÙÈ

õy˛ô 5 6 7 8 9 10 ˆõy›˛

!Ó!Ñ ˛ì˛ ã%ˆÏì˛y 20 51 70 35 10 6 192

ã%ˆÏì˛y !Ó!Ñ ˛Ó˚ ˛ôÓ˚ = 32

~Ñ˛ãò ã%ˆÏì˛y !ÓˆÏÑ ˛ì˛y !Ñ˛ ≤Ã!ì˛!›˛ õyˆÏ˛ôÓ˚ 32 ˆãyv ¸y Ñ˛ˆÏÓ˚ ã% Ïì˛y Ñ˛# Ó˚yÖˆÏÓò⁄ Î!î !ì˛!ò ì˛y

Ñ˛ˆÏÓ˚ò ì˛ˆÏÓ Ñ˛# ˆÓ!üÓ˚¶˛yÜ ˆÑ ì˛yÓ˚ â˛y!£îy ˛ô)Ó˚í Ñ˛Ó˚ˆÏì˛ ˛ôyÓ˚ˆÏÓò⁄ ì˛Ìƒ ˆÌˆÏÑ˛ ˆîÖy ÎyÎ˚ 6,

7 ~ÓÇ 8 õyˆÏ˛ôÓ˚ ã%ˆÏì˛yÓ˚ â˛y!£îy ˆÓ!ü– ~ÓÇ ì˛yÓ˚ õˆÏïƒ 7 òÇ ã% Ïì˛yÓ˚ â˛y!£îy ¢Ó ˆÌˆÏÑ˛ ˆÓ!ü–

ˆîyÑ˛yòîyÓ˚ ˆ¢ ã%ˆÏì˛y ¢Ó ˆÌˆÏÑ˛ ˆÓ!ü !Ó!Ñ ˛ì˛ £Î˚ñ ˆ¢!›˛Ó˚ Á˛ôˆÏÓ˚ !ì˛!ò ˆÓ!ü Ü%Ó˚%c xyˆÏÓ˚y˛ô

Ñ˛ˆÏÓ˚ò– ~£z ¢ÇÖƒy!›˛ £ú xyˆÏÓ˚Ñ˛!›˛ ≤Ã!ì˛!ò!ï fiÌyò#Î˚ õyò– ~£z ≤Ã!ì˛!ò!ï fiÌyò#Î˚ õyò!›˛ õ)ú

¶)˛!°Ï‹Ò–

¢%ì˛Ó˚yÇ ~Ñ˛!›˛ !Ó¶˛yãˆÏò ˆÎ ¢ÇÖƒy!›˛ ¢Ó ˆÌˆÏÑ˛ ˆÓ!üÓyÓ˚ áˆÏ›˛ ÌyˆÏÑ˛ ˆ¢£z ¢ÇÖƒy!›˛ £ú ¶) !°Ï‹Ò–

v zîy£Ó˚í 18 : ¶˛yÓ˚ì˛#Î˚ !Ñ ˛ˆÏÑ˛›˛ ˆÖˆÏúyÎ˚yˆÏv˛¸Ó˚ ÓÎ˚¢ ˆîÁÎ˚y £ú– ì˛yˆÏîÓ˚ ¶) !°Ï‹Ò ÓÎ˚¢ !òí≈Î˚

Ñ˛Ó%ò⁄ 29, 38, 19, 24, 34, 29, 24, 38, 23, 28, 24, 25, 21, 26, 24

ÓÎ˚¢Ü%!úˆÏÑ˛ Ñ õyò%¢yˆÏÓ˚ ¢y!ãˆÏÎ˚ ˛ôyÁÎ˚y ÎyÎ˚ 19, 21, 23, 24, 24, 24, 24, 25, 26, 28,

29, 34, 38, 38 ~ˆÏ«˛ˆÏe ¶)˛!°Ï‹Ò!›˛ £ú 24 Ñ˛yÓ˚í ~£z ˆflÒyÓ˚!›˛ xòƒ ˆflÒyÓ˚Ü%!úÓ˚ ì%˛úòyÎ˚

ˆÓü#ÓyÓ˚ xyÓ!ì≈ ì˛ £ ÏÎ˚ ÏäÈ–

v zîy£Ó˚í 19 : ò#ˆÏâ˛Ó˚ ì˛ˆÏÌƒÓ˚ ¶)˛!°Ï‹Ò !òí≈Î˚ Ñ˛Ó˚%ò

≤ÃyÆ òÇ 10 12 13 15 18 19 20

äÈye/äÈye#Ó˚ ¢ÇÖƒy 3 4 5 2 5 2 1

~ˆÏ«˛ˆÏe ¶) !°ÏT˛Ñ˛ £ú 13 ~ÓÇ 18 ˆÑ˛òòy ¢ÓˆÏÌˆÏÑ˛ ˆÓü# ˛ô!Ó˚¢ÇÖƒy (5) ~£z î%!› ˆflÒyˆÏÓ˚Ó˚

ãòƒ£z ~ Ï«˛ Ïe ˆîÖy ÎyÎ˚–

!Ó.o ı ˆÑ˛yò !Ó¶˛yãˆÏòÓ˚ 1~Ó˚ ˆÓü# xÌ≈yÍ Ó£% ¶)˛!°Ï‹Ò ÌyÑ˛ˆÏì˛ ˛ôyˆÏÓ˚–

● ˛ôÎ≈ ÏÓ«˛í ¢ÇÖƒy Ö%Ó ˆÓü# £ Ïú xyõÓ˚y x!Óòƒhfl Ï ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yãò äÈÑ˛ ˜ì˛Ó˚# Ñ˛Ó˚Ó

ì˛yÓ˚˛ôÓ˚ Üv˛¸. õïƒõy Á ¶)˛!°Ï‹Ò !òí≈Î˚ Ñ˛Ó˚Ó–

ˆòy›˛

144 D.El.Ed. 504


ì% !õ !Ñ˛ ãyò⁄

● ˛¶)˛!°Ï‹ÒÑ˛ !òˆÏ¡¨Ó˚ ¢)ˆÏeÓ˚ ¢y£yˆÏÎƒÁ !òí≈Î˚ Ñ˛Ó˚y ÎyÎ˚– ¶)˛!°Ï‹ÒÑ˛ = 3 × õïƒõy – 2 × Üv˛

● Óƒ!ì˛Ñ ˛õ# ˆ«˛ˆÏe ¶)˛!°Ï‹ÒÑ˛ ÓƒÓ£yÓ˚ Ñ˛Ó˚y £Î˚–

● ˛¶) !°Ï‹ÒÑ˛ ¢Ó ÏÌ ÏÑ˛ x!fiÌÓ˚ ˆÑ˛w ≤ÃÓíì˛yÓ˚ ˛ô!Ó˚õy˛ô

● ¶)˛!°Ï‹ÒÑ˛ ˆÑ˛yò ˛ô!Ó˚¢ÇÖƒy òÎ˚ £z£y ü%ï%õye ~Ñ˛!›˛ ˆflÒyÓ˚

8.4.2 !Ó°Ïõì˛yÓ˚ ˛ô!Ó˚õy˛ô

î%£z ïÓ˚ˆÏòÓ˚ !Ó°Ïõì˛yÓ˚ ˛ô!Ó˚õy˛ô xyˆÏäÈÈÙÙÙÈî)Ó˚c ˛ô!Ó˚õy˛ô ~ÓÇ Üv˛¸ â%˛ƒ!ì˛Ó˚ ˛ô!Ó˚õy˛ô

î)Ó˚c ˛ô!Ó˚õy˛ô ı î)Ó˚c ˛ô!Ó˚õy˛ô £ú !Ñ˛ä%È !òÓ≈y!â˛ì˛ ˛ô!Ó˚õyˆÏ˛ôÓ˚ î)Ó˚c– î%£z ïÓ˚ˆÏòÓ˚ î)Ó˚c ˛ô!Ó˚õy˛ô

xy ÏäÈÈÙÙÙÈ≤Ã¢yÓ˚ ~ÓÇ xyhs˘˛â˛ì% Ì≈Ñ˛ ≤Ã¢yÓ˚–

≤Ã¢yÓ˚ ı ≤Ã¢yÓ˚ £ Ïúy ¢Ó ˆÌ ÏÑ˛ ¢£ã !Ó°Ïõì˛yÓ˚ ˛ô!Ó˚õy˛ô– £z£y ¢Ó≈!ò¡¨ Á ¢ ÏÓ≈yFâ˛ ˛ôÎ≈ ÏÓ«˛ ÏíÓ˚

˛ô!Ó˚õy˛ô– £z£y ¢Ó≈!ò¡¨ Á ¢ˆÏÓ≈yFâ˛ ˛ôÎ≈ˆÏÓ«˛ˆÏíÓ˚ ˛ôyÌ≈Ñ˛ƒ– v˛zîy£Ó˚í fl˛∫Ó˚) ô 2, 5, 6, 4, 12, 10,

9 ~ÓÇ 8 ~£z !Ó¶˛yãò!›˛Ó˚ ≤Ã¢yÓ˚ £ú 12 – 2 = 10

~Ñ˛£z ¶˛yˆÏÓñ –2, 0, 3, 7, ~ÓÇ 9 ~ ÏîÓ˚ ≤Ã¢yÓ˚ £ú 9 – (–2) = 11

~£z ïÓ˚ˆÏòÓ˚ ≤Ã¢yÓ˚ˆÏÑ˛ Óúy £Î˚ Óƒy!ì˛ˆÏÓ˚Ñ˛ ≤Ã¢yÓ˚ ~ÓÇ ÎÖò ≤ÃÑ,˛ì˛ v˛zFâ˛¢#õy ~ÓÇ ≤ÃÑ,˛ì˛

!ò¡¨¢#õyÓ˚ õ Ïïƒ ˛ôyÌ≈Ñ˛ƒ Ñ˛Ó˚y £Î˚ ì˛Öò ì˛y ÏÑ˛ Ó Ïú xyhs˘˛ ÏÓ˚Ö ≤Ã¢yÓ˚–

v z ôˆÏÓ˚Ó˚ v zîy£Ó˚í!›˛ˆÏì˛ 12 ˆflÒyÓ˚!›˛Ó˚ ¢ˆÏÓ≈yFâ˛ ≤ÃÑ, ì˛ ¢#õy 12.5 ~ÓÇ 2 ˆflÒyÓ˚!›˛Ó˚ ≤ÃÑ,˛ì˛ ¢Ó≈!ò¡¨

¢#õy 1.5

¢%ì˛Ó˚yÇ xyhs˘˛ˆÏÓ˚Ö ≤Ã¢yÓ˚ £ú 12.5 – 1.5 = 11

Î!îÁ ~ ÏîÓ˚ Óƒy!ì˛ ÏÓ˚Ñ˛ ≤Ã¢yÓ˚ £ú 10

E8. xyhs˘˛!Ó˚Ñ˛ ≤Ã¢yÓ˚ !òí≈Î˚ Ñ˛Ó˚ 1, 2, 5, –5, 4, 6 ~ÓÇ 7

E9. Óƒy!ì˛ ÏÓ˚Ö Á xyhs˘˛ ÏÓ˚Ö ˛≤Ã¢y ÏÓ˚Ó˚ õ Ïïƒ ˛ôyÌ≈Ñ˛ƒ

Î!îÁ î%£z ïÓ˚ˆÏíÓ˚ ≤Ã¢yÓ˚ xyˆÏäÈÈÙÙÙÈÓƒy!ì˛ˆÏÓ˚Ñ˛ Á xyhs˘˛ˆÏÓ˚Ñ˛ñ ~!›˛ ¢£ãì˛õ !Ó°Ïõì˛yÓ˚ ˛ô!Ó˚õy˛ô–

äÈv˛¸yˆÏòy ˆflÒyˆÏÓ˚Ó˚ ˆ«˛ˆÏe ~Ó˚ ˆÑ˛yò ≤Ã!ì˛ö˛úò ˆò£z– v˛zîy£Ó˚ífl˛∫Ó˚)˛ô Î!î 100!›˛ ˆflÒyÓ˚ 1-10 ~Ó

õˆÏïƒ !Óhfl ,Ïì˛ £Î˚ ì˛ˆÏÓ ì˛yˆÏîÓ˚ xyhs˘˛ˆÏÓ˚Ö ≤Ã¢yÓ˚ 10. !Ñ˛hs%˘˛ Î!î ~Ñ˛!›˛ ˆflÒyÓ˚ 1 ~ÓÇ xyˆÏÓ˚Ñ˛!›˛

ˆflÒyÓ˚ 10 ~£z î%!›˛ ˆflÒyˆÏÓ˚Ó˚ õˆÏïƒ Î!î 5 ˆflÒyÓ˚!›˛ 98 òÎ˚ ˛ôyÁÎ˚y ÎyÎ˚ ˆ¢ˆÏ«˛ˆÏe ˙ xyhs˘˛!Ó˚Ñ˛

≤Ã¢yÓ˚ 10 Á£z v˛zîy£Ó˚í ˆÌˆÏÑ˛ ˆîÖy ÎyÎ˚ ~£z ≤Ã¢yÓ˚ £ú !Ó°Ïõì˛yÓ˚ ¢ÓˆÏÌˆÏÑ˛ ì˛yv˛¸yì˛y!v˛¸

˛ô!Ó˚õyˆÏ˛ôÓ˚ ~Ñ˛!›˛ v˛z˛ôyÎ˚–

ˆòy›˛



xyhs˘˛â˛ì% Ì≈Ñ˛ ≤Ã¢yÓ˚ ı

xyhs˘˛â˛ì%˛Ì≈Ñ˛ !Óhfl˛ÏyÓ˚ !Ó°Ïõì˛y ˛ô!Ó˚õyˆÏ˛ôÓ˚ xyÓ˚Ñ˛!›˛ î)Ó˚c ≤Ã¢yÓ˚– xyˆÏÜ£z xyõÓ˚y xyˆÏúyâ˛òy

Ñ˛ ÏÓ!äÈñ õïƒõy ˆõy›˛ !Ó¶˛yãò!›˛ ÏÑ˛ î%!›˛ x Ïï≈ !Ó¶˛=˛ Ñ˛ ÏÓ– ~Ñ˛£z¶˛y ÏÓ â˛ì% Ì≈Ñ˛ ¢¡ô)í≈ !Ó¶˛yãò!›˛ ÏÑ˛

4!›˛ ¢õyò xÇˆÏü !Ó¶˛=˛ Ñ˛ˆÏÓ˚– ¢ÇK˛y xò%¢yˆÏÓ˚ ≤ÃÌõ â˛ì% Ì≈Ñ˛ (Q1)– £ú ~Ñ˛!›˛ ˆflÒyÓ˚ ˆÎ!›˛

!Ó¶˛yãò!›˛ ÏÑ˛ ò# Ïâ˛Ó˚ 25% ˆflÒyÓ˚ˆÏÑ˛ xyúyîy Ñ˛ˆÏÓ˚– !mì˛#Î˚ â˛ì% Ì≈ (Q2) £ú ~Ñ˛!›˛ ˆflÒyÓ˚ ÎyÓ˚

ò# Ïâ˛Ó˚ 50% ˆflÒyÓ˚ !Óîƒõyò– ú«˛ƒ Ñ˛Ó%ò !mì˛#Î˚ â˛ì% Ì≈Ñ˛!›˛ £ú õïƒõyÓ˚ ¢yˆÏÌ ¢õyò– xÓˆÏüˆÏ°Ï

ì, ì˛#Î˚ â˛ì% Ì≈Ñ˛ (Q3) £ú ~Ñ˛!›˛ ˆflÒyÓ˚ ÎyÓ˚ ò#ˆÏâ˛ xÇü ¢ÇÖƒÑ ˆflÒyÓ˚ !Óîƒõyò–

¢%ì˛Ó˚yÇ xyhs˘˛ â˛ì% Ì≈Ñ˛ ≤Ã¢yÓ˚ ÏÑ˛ ¢ÇK˛y!Î˚ì˛ Ñ˛Ó˚y ÎyÎ˚ ~£z¶˛y ÏÓÈÙÙÙÈ

xyhs˘˛â˛ì%˛Ì≈Ñ˛ ≤Ã¢yÓ˚ = Q3 – Q1

ÎÖò ~£z xyhs˘˛â˛ì%˛Ì≈Ñ˛ ≤Ã¢yÓ˚ !Ó°Ïõì˛yÓ˚ ˛ô!Ó˚õy˛ô !£¢yˆÏÓ ÓƒÓ£,ì˛ £Î˚ ì˛Öò ~Ñ˛ xï≈ xyhs˘˛â˛ì% Ì≈Ñ˛

≤Ã¢yÓ˚ !£¢yˆÏÓ ˛ô!Ó˚íì˛ Ñ˛Ó˚y £Î˚– ˆÎ›˛y Ö%Ó ¢£ã¶˛yˆÏÓ xyhs˘˛â˛ì% Ì≈Ñ˛ ≤Ã¢yˆÏÓ˚Ó˚ xˆÏï≈Ñ˛– xÌ≈yÍ

â˛ì% Ì≈Ñ˛ â% ƒ!ì˛

v zîy£Ó˚í 20 : !òˆÏ¡¨Ó˚ !Ó¶˛yãò!›˛Ó˚ â˛ì%˛Ì≈Ñ˛ â%˛ƒ!ì˛ !òí≈Î˚ Ñ˛Ó˚%òÈÙÙÙÈ3, 4, 6, 9, 11, 12, 14, 15

~ˆÏ«˛ˆÏe ˆõy›˛ ˆflÒyÓ˚~Ó˚ ¢ÇÖƒy 8– 8 ~Ó˚ 25% £ú 2

~ÓÇ 8 ~Ó˚ 75% £ú 6

ÎyÓ˚ xÌ≈ Q1 £ú ~õò ~Ñ˛!›˛ ˆflÒyÓ˚ ÎyÓ˚ ò#ˆÏâ˛ î%!›˛ ˆflÒyÓ˚ xyˆÏäÈ 3 ~ÓÇ 4

¢%ì˛Ó˚yÇ Q1 £ú 5

~Ñ˛£z¶˛yˆÏÓ Q3 £ú ~õò ~Ñ˛!›˛ ˆflÒyÓ˚ ÎyÓ˚ ò#ˆÏâ˛ 6!›˛ ˆflÒyÓ˚ xy ÏäÈ–

ˆ¢ Ï«˛ Ïe Q3 £ú 14

¢%ì˛Ó˚yÇ â˛ì%˛Ì≈ â%˛ƒ!ì˛ £ú

Üv ¸ â%˛ƒ!ì˛Ó˚ ˛ô!Ó˚õy˛ô ı Üv ¸ ˆÌˆÏÑ˛ ˆflÒyÓ˚Ü%ˆÏúyÓ˚ î)Ó˚c£z £ú â% ƒ!ì˛– ÎÖò£z â% ƒ!ì˛Ü% ÏúyˆÏÑ˛ ÜˆÏv˛¸Ó˚

ãòƒ ÓƒÓ£yÓ˚ Ñ˛Ó˚y £Î˚ ˆ¢ˆÏ«˛ˆÏe !Ó°Ïõì˛yÓ˚ ˛ô!Ó˚õy˛ô!›˛ xyˆÏÓ˚y !ò¶%≈˛ú ~ÓÇ !ÓŸªhfl˛Ï– xyõÓ˚y

~ÖyˆÏò î%ïÓ˚ˆÏíÓ˚ Üv ¸ â% ƒ!ì˛Ó˚ ˛ô!Ó˚õy˛ô !òˆÏÎ˚ xyˆÏúyâ˛òy Ñ˛Ó˚ÓÈÙÙÙÈ

Üv ¸ â% ƒ!ì˛ ~ÓÇ ¢õƒÑ˛ â% ƒ!ì˛–

ÎÖò ˆflÒyÓ˚Ü%!ú ÜˆÏv ¸ Ö%Ó Ñ˛yäÈyÑ˛y!äÈ ÌyˆÏÑ˛ ì˛Öò Üv ¸ â% ƒ!ì˛Ó˚ ˛ô!Ó˚õy˛ô Ö%Ó£z ¢yõyòƒ– ˆ¢ˆÏ«˛ˆÏe

xyõyˆÏîÓ˚ !Ó¶˛yãò!›˛Ó˚ ≤ÃÑ,˛!ì˛ Ó%G˛ˆÏì˛ xˆÏòÑ˛ ¢£yÎ˚ì˛y Ñ˛ˆÏÓ˚– v˛zîy£Ó˚ífl˛∫Ó˚)˛ô ~Ñ˛!›˛ ˛ôÓ˚#«˛yÎ˚

Ü!í Ïì˛Ó˚ Üv ¸ ˆflÒyÓ˚ 65 ~ÓÇ ¢õƒÑ˛ â% ƒ!ì˛ 10 xyÓyÓ˚ ¶˛y°ÏyÓ˚ Üv˛¸ ˆflÒyÓ˚ 60 ~ÓÇ ¢õƒÑ˛ â% ƒ!ì˛

ˆòy›˛

146 D.El.Ed. 504


5 ì˛y£ˆÏú xyõÓ˚y ˆîÖˆÏì˛ ˛ôy£z ˆÓ!üÓ˚¶˛yÜ !ü«˛yÌ≈#Ó˚ ¶˛y°ÏyÓ˚ ˆflÒyÓ˚ 60 ~Ó˚ Ñ˛yäÈyÑ˛y!äÈ– !Ñ˛hs%˘˛

Ü!íˆÏì˛Ó˚ ˆflÒyÓ˚ xˆÏòÑ˛ ˆÓü# î)Ó˚ !Óhfl˛,Ïì˛– ~ˆÏ«˛ˆÏe Ö%Ó x“ ¢ÇÖƒÑ˛ !ü«˛yÌ≈# Ü!íˆÏì˛ x“ ò¡∫Ó˚

ˆ˛ôˆÏÎ˚ˆÏäÈ– Üv˛¸ â%˛ƒ!ì˛Ó˚ ˛ô!Ó˚õy˛ô äÈyv˛¸y ü%ï%õye ˆÑ˛w#Î˚ ≤ÃÓíì˛yÓ˚ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ¢y£yˆÏÎƒ ˆÑ˛yò

¢!‡˛Ñ˛ !¢ÂïyˆÏhs˘˛ xyõÓ˚y v˛z˛ôò#ì˛ £ˆÏì˛ ˛ôy!Ó˚ òy–

Üv ¸ â% ƒ!ì˛ ı â%˛ƒ!ì˛Ó˚ ˆflÒyÓ˚ = ˆflÒyÓ˚ – Üv ¸

= X –

Î!î ˆÑ˛yò ~Ñ˛!›˛ Üv ¸ 50 ì˛ˆÏÓ 55 ˆflÒyÓ˚!›˛Ó˚ â%˛ƒ!ì˛ £ú X – = 55 – 50 = 5

ú«˛ƒ Ñ˛Ó%ò â%˛ƒ!ì˛Ó˚ ˆ«˛ˆÏe î%!›˛ xÇü xyˆÏäÈ ~Ñ˛!›˛ £ú !â˛£´ S‘+’ xÌÓy ‘–’V ~ÓÇ ¢ÇÖƒy– !â˛£´

xyõyˆÏîÓ˚ !òˆÏî≈ü Ñ˛ˆÏÓ˚ ˆÎ ˆflÒyÓ˚Ü%!ú ÜˆÏv˛¸Ó˚ Á˛ôˆÏÓ˚ xyˆÏäÈ òy ò#ˆÏâ˛ xyˆÏäÈ– xyÓ˚ ¢ÇÖƒy xyõyˆÏîÓ˚

!òˆÏî≈ü Ñ˛ˆÏÓ˚ Üv ¸ ˆÌˆÏÑ˛ ≤ÃÑ, ì˛ î)Ó˚c– ¢%ì˛Ó˚yÇ Üv˛¸ â% ƒ!ì˛ !òí≈ˆÏÎ˚Ó˚ ˆ«˛ˆÏe ïòydÑ˛ Óy }íydÑ˛

!â˛£´ˆÏÑ˛ v˛zˆÏ˛ô«˛y Ñ˛ˆÏÓ˚ ü%ï%õye ïòydÑ˛ !â˛ˆÏ£´Ó˚ ¢y£yˆÏÎƒ ¢Ñ˛ú â%˛ƒ!ì˛ˆÏÑ˛ !òˆÏÎ˚ Üv˛¸ â%˛ƒ!ì˛ !òí≈Î˚

Ñ˛Ó˚y £Î˚– 5, 8, 11, 12, 14 ~£z ˆflÒyÓ˚Ü%!úÓ˚ Üv ¸ 10 ~ ÏîÓ˚ Üv ¸ â% ƒ!ì˛ Üíòy Ñ˛Ó˚yÓ˚ ãòƒ xyõÓ˚y

!ò¡¨!ú!Öì˛ ¢yÓ˚í#!›˛ xò%¢yˆÏÓ˚ â˛úÓ–

x!Óòƒhfl Ï ˆflÒy ÏÓ˚Ó˚ Üv ¸â%˛ƒ!ì˛

ˆflÒyÓ˚ (X) â%˛ƒ!ì˛ (X – ) |X – |

5 5-10= -5 5

8 8-10= -2 2

11 11-10=1 1

12 12-10=2 2

14 14-10=4 4

= 14

ˆ›˛!Óú 8.12

Üv ¸ â%˛ƒ!ì˛ (AD) = = = 2.8.

ú«˛ƒ Ñ˛Ó%ò â% ƒ!ì˛Ó˚ ˆÎyÜö˛ú 0, ˆÑ˛ò⁄

!Óòƒhfl Ï Üí¢ÇÖƒy !ò ÏÓüò ˆÌ ÏÑ˛ñ

ˆòy›˛



ˆ◊í#ÓÂï !Ó¶˛yã ÏòÓ˚ Üv ¸

ˆ◊í#Óƒ!Æ Üí¢ÇÖƒy ˆ◊í# õïƒõyò

30-34 3 32 10.38 31.14

35-39 9 37 5.38 48.42

40-44 15 42 0.38 5.70

45-49 8 47 4.62 36.96

50-54 5 52 9.62 48.10

N = 40 170.32

ˆ›˛!Óú 8.13

~ˆÏ«˛ˆÏe n = 40 ~ÓÇ Üv˛¸ ~ÓÇ = 170.32.

Üv ¸ â% ƒ!ì˛ (AD) =

v˛z˛ôˆÏÓ˚Ó˚ v˛zîy£Ó˚í!›˛ˆÏì˛ AD = = 40

32.170= 4.26.

¢õƒÑ˛ â%˛ƒ!ì˛ ı ¢Ó ˆÌˆÏÑ˛ ˆÓ!ü õyeyÎ˚ ÓƒÓ£,ì˛ !Ó°Ïõì˛yÓ˚ ˛ô!Ó˚õy˛ô £ú ¢õƒÑ˛ â% ƒ!ì˛– ÜíòyÓ˚

˛ô!Ó˚ˆÏ≤Ã!«˛ˆÏì˛ ~ˆÏÑ˛ xˆÏòÑ˛ ¢õÎ˚ ÓÜ≈â% ƒ!ì˛Ó˚ ÜˆÏv ¸Ó˚ õ)ú–

~Öyò ˆÌ ÏÑ˛ ¢õƒÑ˛â% ƒ!ì˛Ó˚ !òí≈Î˚ Ñ˛Ó˚yÓ˚ ˆÎ ïy˛ôÜ%!ú îÓ˚Ñ˛yÓ˚ ˆ¢Ü%!ú £úÈÙÙÙÈ

ïy˛ôÈÙÈ1 !Ó¶˛yãò!›˛Ó˚ Üv˛¸ !òí≈Î˚ Ñ˛Ó˚%ò

ïy˛ôÈÙÈ2 Üv ¸ ˆÌˆÏÑ˛ ˆflÒyÓ˚Ü%!úÓ˚ â% ƒ!ì˛ !òí≈Î˚ Ñ˛Ó%ò

ïy˛ôÈÙÈ3 ≤Ã!ì˛!›˛ â%˛ƒ!ì˛Ó˚ ÓÜ≈ !òí≈Î˚ Ñ˛Ó˚%ò

ïy˛ôÈÙÈ4 â%˛ƒ!ì˛ Ü%!úÓ˚ ÓˆÏÜ≈Ó˚ Üv ¸ !òí≈Î˚ Ñ˛Ó%ò– ~£z Üv˛¸ ÏÑ˛ ÓˆÏú Ú¶˛ƒy!Ó˚ˆÏÎ˚™Û–

ïy˛ô-5 ~£z ¶˛ƒy!Ó˚ˆÏÎ˚™ÈÙÈ~Ó˚ }íydÑ˛ ÓÜ≈õ)ú !òí≈Î˚ Ñ˛Ó˚ ~ÓÇ ~£z v z_Ó˚!›˛ £ú ¢õƒÑ˛ â% ƒ!ì˛

!òˆÏ¡¨Ó˚ ˆflÒyÓ˚Ü%!úÓ˚ ¢õƒÑ˛â% ƒ!ì˛ (SD) !òí≈Î˚ Ñ˛Ó%ò ı 5, 6, 8, 10, 11, 14 Üv ¸ £ú 9

ˆòy›˛

148 D.El.Ed. 504


x!Óòƒhfl Ï ì˛ ÏÌƒÓ˚ ¢õƒÑ˛ â%˛ƒ!ì˛

ˆflÒyÓ˚ â%˛ƒì˛ ÓÜ≈â%˛ƒ!ì˛

(x = X – X) (x2)

5 - 16

6 - 9

8 - 1

10 1 1

11 2 4

14 5 25

ˆ›˛!Óú 8.14

ÓÜ≈â%˛ƒ!ì˛Ó˚ ˆÎyÜö˛ú = ∑ 2x = 16 + 9 + 1 + 1 + 4 + 25 = 56

¶˛ƒy!Ó˚ ÏÎ˚™ = ÓÜ≈â%˛ƒ!ì˛Ó˚ Üv˛¸ = N

x∑ 2

= 6

56=9.33

¢õƒÑ˛ â%˛ƒ!ì˛ (SD)=N

x∑ 2

= 33.9 = 3.05

ˆ◊í#ÓÂï!Ó¶˛yã ÏòÓ˚ ¢õƒÑ˛ â% ƒ!ì˛ ı ò# Ïâ˛Ó˚ ¢yÓ˚í# ˆÌ ÏÑ˛ ¢õƒÑ˛ â% ƒ!ì˛ !òí≈Î˚ Ñ˛Ó%ò

ˆ◊í#ÓÂï !Ó¶˛yã ÏòÓ˚ ¢õƒÑ˛ â% ƒ!ì˛

C.I. f Mid-point Deviation fx fx2

of C.I. (X) x X X= −

30-34 3 32 -10.38 -31.14 323.2332

35-39 9 37 -5.38 -48.42 260.4996

40-44 15 42 -0.38 -5.70 2.166

45-49 8 47 4.62 36.96 170.7552

50-54 5 52 9.62 48.10 462.722

∑ f =N=40 ∑ 2fx =1219.3760

ˆ›˛!Óú 8.15

¢õƒÑ˛ â% ƒ!ì˛ !òí≈ˆÏÎ˚Ó˚ ¢)e ı SD = N

fx∑ 2

ˆòy›˛



ˆÎÖy Ïò x £ú ˆ◊í# õïƒõyò

f £ú ˆ◊í#Ó˚ ˛ô!Ó˚¢ÇÖƒy

N £ú ˆõy›˛ ˛ô!Ó˚¢ÇÖƒy

v˛z˛ôˆÏÓ˚Ó˚ ¢yÓ˚í# ˆÌˆÏÑ˛ñ xyõÓ˚y ˛ôy£z

SD = N

fx∑ 2

= 40

3760.1219= 4844.30 = 5.52

!Óhfl˛ÏyÓ˚ ˛ô!Ó˚õy˛ô Óy !Ó!¶˛ß¨ì˛yÓ˚ ÓƒÓ£yÓ˚

● Ö%Ó ì˛yv ¸yì˛y!v ¸ ïyÓ˚íy ˛ôyÁÎ˚yÓ˚ ãòƒ ≤Ã¢yÓ˚ ÓƒÓ£,ì˛ £Î˚–

● ÎÖò !Ó¶˛yãˆÏò !Ñ˛ä%È â) v ¸yhs˘˛ ˆflÒyÓ˚ ÌyˆÏÑ˛ ~ÓÇ ˆÑ˛w#Î˚ ≤ÃÓíì˛yÓ˚ ˛ô!Ó˚õy˛ô !£¢yˆÏÓ õïƒõy

ÓƒÓ£,ì˛ £Î˚ ì˛Öò â˛ì% Ì≈Ñ˛ â% ƒ!ì˛ˆÏÑ˛ x@˘ÃÜíƒì˛y ˆîÁÎ˚y £Î˚–

● ÎÖò ˆflÒyÓ˚Ü%!ú Ö%Ó ˆÓü# õyeyÎ˚ !Óhfl ,Ïì˛ ÌyˆÏÑ˛ ì˛Öò Üv ¸ â% ƒ!ì˛ ÓƒÓ£,ì˛ £Î˚–

● ¢õhfl˛Ï !Ó°Ïõì˛yÓ˚ ˛ô!Ó˚õyˆÏ˛ôÓ˚ õˆÏïƒ ¢õƒÑ˛ â%˛ƒ!ì˛ (SD) ¢Ó ˆÌ ÏÑ˛ ˆÓ!ü õyeyÎ˚ !fiÌÓ˚ ~ÓÇ

!òÖ%ì˛–

E10. ò#ˆÏâ˛Ó˚ ˆflÒyÓ˚Ü%!úÓ˚ SD !òí≈Î˚ Ñ˛Ó%ò 11, 13, 15, 17, 19, 21, 23

8.5 ¢Ç£!ì˛Ñ˛Ó˚í

● !Ñ˛ä%È ì˛Ìƒ ¢Ó˚y¢!Ó˚ ¢Ç@˘Ã£ Ñ˛Ó˚y ÎyÎ˚– !Ñ˛ä%È ì˛Ìƒ ˆÜÔí v˛zÍ¢ ˆÎõò ˆÓ˚Ñ˛v˛≈ñ ãyò≈yúñ

ˆ¢ò¢y¢ £zì˛ƒy!îÓ˚ Á˛ôÓ˚ !ò¶≈˛Ó˚ Ñ˛ˆÏÓ˚ ˛ôyÁÎ˚y ÎyÎ˚–

● ˆflÒyÓ˚Ü%!ú ÏÑ˛ x!Óòƒhfl Ï ~ÓÇ ˆ◊í#ÓÂï ˛ô!Ó˚¢ÇÖƒy !Ó¶˛yã Ïò ¢yãy Ïòy ÎyÎ˚–

● ì˛ÌƒÜ%!úˆÏÑ˛ äÈ!Óñ ˆúÖ!â˛e £zì˛ƒy!îÓ˚ õyïƒˆÏõ ˛v z ôfiÌy˛ôò Ñ˛Ó˚y ÎyÎ˚ ~ÓÇ !â˛ey!Î˚ì˛ Ó) ôñ

î[˛!â˛eñ xyÎ˚ì˛ ÏúÖñ ˛ôy£z!â˛e £ú ¢£ã ˜Ó˚!ÖÑ˛ xyÑ˛yÓ˚

● Üy!í!ì˛Ñ˛ Üv ¸ñ õïƒõy ~ÓÇ ¶) !°Ï¤˛Ñ˛ ~£z !ì˛ò!›˛ £ú ˆÑ˛w#Î˚ ≤ÃÓíì˛yÓ˚ ˛ô!Ó˚õy˛ô–

● !Ó°Ïõì˛yÓ˚ ˛ô!Ó˚õyˆÏ˛ôÓ˚ ˆ«˛ˆÏe ≤Ã¢yÓ˚ ~ÓÇ â˛ì% Ì≈Ñ˛ â% ƒ!ì˛ £ú î)Ó˚ˆÏcÓ˚ ˛ô!Ó˚õy˛ôñ xyÓyÓ˚ Üv˛¸

â%˛ƒ!ì˛ ~ÓÇ ¢õƒÑ˛ â%˛ƒ!ì˛ £ú Üv ¸ â%˛ƒ!ì˛Ó˚ ˛˛ô!Ó˚õy˛ô–

8.6 xyîü≈ v z_Ó˚

E1. y x«˛ ÓÓ˚yÓÓ˚ @˘ÃyˆÏõÓ˚ òyõ ~ÓÇ x x«˛ ÓÓ˚yÓÓ˚ ˛ôü%Ó˚ òyõ !òˆÏÎ˚ !â˛ey!Î˚ì˛ Ó˚) ô xAÑ˛ò

Ñ˛Ó%ò–

ˆòy›˛

150 D.El.Ed. 504


E2. 1 ˆ¢!r›˛!õ›˛yÓ˚ = 10 ~Ñ˛Ñ˛ ïˆÏÓ˚ !òˆÏÎ˚ ~Ñ˛!›˛ î[˛!â˛e xAÑ˛ò Ñ˛Ó˚%ò–

E3. x x«˛ ÓÓ˚yÓÓ˚ ˆ◊í# ˜îá≈ƒ ~ÓÇ y x«˛ ÓÓ˚yÓÓ˚ ˆúyÑ˛ ¢ÇÖƒy ï ÏÓ˚ ~Ñ˛›˛y xyÎ˚ì˛ ÏúÖ

xAÑ˛ò Ñ˛Ó%ò–

E4. 3 ˆ¢!õ Óy 4 ˆ¢!õ Óƒy¢yˆÏï≈Ó˚ Ó,_ xAÑ˛ò Ñ˛ˆÏÓ˚ ˛ôy£z!â˛e xAÑ˛ò Ñ˛Ó˚%ò– ~ˆÏ«˛ˆÏe ˆÑ˛w#Î˚

ˆÑ˛yò

E5. Üv˛¸ = 5.5

E6. õïƒõy = 8

E7. õïƒõyÓ˚ Áãò = 45 ˆÑ˛!ã

E8. 13

E9. 1

E10 4

8.7 ≤ÃŸ¿yÓú#1. !˛ôˆÏQy@˘Ãyö˛ Á xyÎ˚ì˛ˆÏúÖ ˜ì˛Ó˚# Ñ˛Ó˚%ò ı

!Ó°ÏÎ˚ ~õ.xy£z.~ú £zÇÓ˚yã# xAÑ˛ !ÓK˛yò ¢yõy!ãÑ˛!Óîƒy

¢ÇÖƒy !òÓ˚%˛ôí 60 55 80 55 50

2. Üv ¸ õïƒõy ¶) !°Ï‹ÒÑ˛ !òí≈Î˚ Ñ˛Ó%ò ı16, 24, 14, 10, 20, 14, 15, 21, 15, 12, 13, 16, 19, 17

3. Üv˛¸ õïƒõy ~ÓÇ ¶)˛!°Ï¤˛Ñ˛ !òí≈Î˚ Ñ˛Ó˚%ò

ˆflÒyÓ˚ 5 6 7 8 9 10 11 12

˛ô!Ó˚¢ÇÖƒy 8 10 15 20 16 12 9 10

4. ò#ˆÏâ˛Ó˚ ì˛ˆÏÌƒÓ˚ ¢õƒÑ˛ â%˛ƒ!ì˛ !òí≈ÈÎ˚ Ñ˛Ó˚%ò ı

ˆ◊í#Óƒ!Æ 30-39 40-49 50-59 60-69 70-79 80-89 90-99

ˆflÒyÓ˚ 6 9 15 25 13 7 5

8.8 @˘ÃrÌ˛ôO#NCERT Maths Text Book for Classes VI, VII, VIII

Teaching of Maths of Upper Primary Level Vol-I, An IGNOU Publication

äÈye¢ÇÖƒy

ˆòy›˛

151504-Block2 : ˛ôy!›˛Ü!í Ïì˛Ó˚ ¢yïyÓ˚í Ó˚)˛ô Ó#ãÜ!íì˛

˛ôy!›˛Ü!í Ïì˛Ó ¢yïyÓí Ó) ô Ó#ãÜ!íì˛

~Ñ˛Ñ˛ÈÈÈÙÙÙÈ9 ı ˛ôy!›˛Ü!í Ïì˛Ó˚ ¢yïyÓ˚í Ó) ô Ó#ãÜ!íì˛Ü‡˛ò

9.0 ¶) !õÑ˛y


9.2 ¢ÇÖƒyÓ˚ ˛ô!Ó˚Ó Ïì≈ !â˛ Ï£´Ó˚ ÓƒÓ£yÓ˚

9.3 Ó#ãÜy!í!ì˛Ñ˛ ˛ôî ~ÓÇ Ó˚y!ü

9.3.1 Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!ü

9.3.2 â˛úÑ˛ ~ÓÇ ï %ÓÑ˛

9.3.3 Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!ü ˛ôî

9.3.4 ˆ≤Ãyv˛yQ v zÍ˛ôyîÑ˛ ~ÓÇ ¢£Ü

9.3.5 ¢õãyì˛#Î˚ Á !Ó°Ïõãyì˛#Î˚ ˛ôî

9.4 Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üÓ˚ v˛z˛ôÓ˚ !Ó!¶˛ß¨ ≤Ã!Ñ ˛Î˚y

9.4.1 ˆÎyÜ

9.4.2 !ÓˆÏÎ˚yÜ

9.4.3 Ü%í

9.4.4 ¶˛yÜ

9.5 ~Ñ˛áyì˛ Ó#ãÜy!í!ì˛Ñ˛ ¢õ#Ñ˛Ó˚í ~ÓÇ ¢õyïyò

9.5.1 ~Ñ˛áyì˛ Ó#ãÜy!í!ì˛Ñ˛ ¢õ#Ñ˛Ó˚í

9.5.2 ~Ñ˛áyì˛ ¢õ#Ñ˛Ó˚ˆÏíÓ˚ ¢õyïyò

9.6 Ó#ãÜy!í!ì˛Ñ˛ ˛ôÂï!ì˛Ó˚ ÓƒÓ£yÓ˚

9.7 ˛¢Ç£!ì˛Ñ˛Ó˚í

9.8 x@˘ÃÜ!ì˛ Îyâ˛y£z Ñ˛Ó˚%ò

9.9 @˘ÃrÌ˛ôO#

9.10. ~Ñ˛Ñ˛ÈÙÈˆü Ï°Ï ≤ÃŸ¿yÓú#

9.0 ¶) !õÑ˛y

Ü!íˆÏì˛Ó˚ ˛üyÖy !£¢yˆÏÓ ˛ôy!›˛Ü!íì˛ ˛ô!Ó˚!â˛ì˛– £z£y õ)ì≈˛ ¢ÇÖƒy ˆÎõòñ 1, 2, 25, 37, 456

£zì˛ƒy!î ~ÓÇ òyòyò Ó˚Ñ˛ˆÏõÓ˚ ≤Ã!Ñ ˛Î˚y ˆÎõòñ ˆÎyÜñ !ÓˆÏÎ˚yÜñ Ü%íñ ¶˛yÜ !òˆÏÎ˚ â˛y!úì˛– !Ñ˛hs%˛

¢ÇÖƒyÓ˚ ˛ô!Ó˚ÓˆÏì≈˛ !â˛£´ ~ÓÇ !â˛£´ !òˆÏÎ˚ ˛ôy!›˛Üy!í!ì˛Ñ˛ ≤Ã!Ñ Î˚y â˛yúòyÓ˚ õyïƒˆÏõ ˛ôy!›˛Ü!íì˛ˆÏÑ˛

ˆòy›˛

152 D.El.Ed. 504


xyÓ˚Á Óƒy˛ôÑ˛ xˆÏÌ≈ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ Ó#ãÜ!íì˛ ÜˆÏv˛¸ v˛z Ï‡˛ˆÏäÈ–

ì˛y£z Óúy ÎyÎ˚ñ Ó#ãÜ!íì˛ £ú Ü!íˆÏì˛Ó˚ ~Ñ˛!›˛ üyÖy Îy ˛ôy!›˛Ü!íˆÏì˛Ó˚ ò#!ì˛Ü%!ú ¢¡ô)í≈¶˛yˆÏÓ

!â˛ Ï£´Ó˚ v z ôÓ˚ ≤Ã ÏÎ˚yÜ ÏÎyÜƒ– Ó#ãÜ!íì˛ Ö%Ó£z xyÑ˛°Ï≈í#Î˚ ÓƒÓ£yÓ˚ ÏÎyÜƒ ~ÓÇ Óƒy˛ôÑ˛–

Ó#ãÜ!í Ïì˛Ó˚ v zÍ¢ ı

Ó#ãÜ!íì˛ ü∑!›˛ ~ˆÏ¢ˆÏäÈ xyÓ˚Ó#Î˚ ü∑ ÚxyúãyÓyÓ˚Û ˆÌˆÏÑ˛– ÎyÓ˚ õyˆÏò ˛˛ô%ò!õ≈úò– ~!›˛

ÓƒÓ£yÓ˚ Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ Úxyú!Ñ˛ì˛yÓÛ Úxyú õ%=˛yÛ–

!ÓÖƒyì˛ @˘Ã#Ñ˛ Üy!í!ì˛Ñ˛ v˛yˆÏÎ˚yö˛ƒyr›˛y¢ ì, ì˛#Î˚ üì˛y∑#ˆÏì˛ Ó#ãÜ!íˆÏì˛Ó˚ ãòÑ˛ !£ˆÏ¢ˆÏÓ Öƒyì˛

!äÈˆÏúò–


~£z ~Ñ˛Ñ˛!›˛ ˛ôy‡˛ Ñ˛Ó˚yÓ˚ ˛ôÓ˚ xy˛ô!ò

Ó#ãÜy!í!ì˛Ñ˛ ˛ôîñ Ó˚y!ü ÓƒyÖƒy Ñ˛Ó˚ˆÏì˛ ˛ôyÓ˚ˆÏÓò ~ÓÇ Ó˚y!üõyúyÜ%!úˆÏÑ˛ ˆ◊í#ÓÂï Ñ˛Ó˚ˆÏì˛

˛ôyÓ˚ˆÏÓò–

â˛úÑ˛ ~ÓÇ ï %ÓÑ˛ñ ¢õ Á x¢õãyì˛#Î˚ ˛ôˆÏîÓ˚ ˛ôyÌ≈Ñ˛ƒ Ñ˛Ó˚ˆÏì˛ ˛ôyÓ˚ˆÏÓò–

Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üõyúyÓ˚ v z ôÓ˚ !Ó!¶˛ß¨ ≤Ã!Ñ Î˚y â˛yúyˆÏì˛ ˛ôyÓ˚ˆÏÓò–

~Ñ˛ â˛úÑ˛Î%=˛ ~Ñ˛áyì˛ ¢õ#Ñ˛Ó˚ ÏíÓ˚ ¢õyïyò Ñ˛Ó˚ Ïì˛ ˛ôyÓ˚ ÏÓò–

Üy!í!ì˛Ñ˛ ¢õ¢ƒy ¢õyïyò Ñ˛Ó˚yÓ˚ ãòƒ Ó#ãÜy!í!ì˛Ñ˛ ˛ôÂï!ì˛ ≤Ã ÏÎ˚yÜ Ñ˛Ó˚ Ïì˛ ˛ôyÓ˚ ÏÓò–

9.2 ¢ÇÖƒyÓ˚ ˛ô!Ó˚Ó Ïì≈ !â˛ Ï£´Ó˚ ÓƒÓ£yÓ˚

Ó#ãÜ!íˆÏì˛Ó˚ õ)ú !Ó°ÏÎ˚ £ú ˛ôy!›˛Ü!íˆÏì˛Ó˚ !ÓˆÏü°Ï Ó) ôˆÏÑ˛ !â˛£´ˆÏÑ˛ ¢ÇÖƒy !£¢yˆÏÓ ÓƒÓ£yÓ˚ Ñ˛Ó˚yÓ˚

õyïƒˆÏõ ˆÑ˛yò Üy!í!ì˛Ñ˛ ¢õ¢ƒy ¢õyïyò Ñ˛Ó˚y– !â˛£´ ÓƒÓ£yÓ˚ Ñ˛Ó˚yÓ˚ ö˛ˆÏú ˆÎ ˆÑ˛yò ¢)e ~ÓÇ

!òÎ˚õˆÏÑ˛ xyõÓ˚y Óƒy˛ôÑ˛ xˆÏÌ≈ ÓƒÓ£yÓ˚ Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚–

ò# Ïâ˛Ó˚ v zîy£Ó˚íÜ%!ú ˆîÖy ÎyÑ˛–

v zîy£Ó˚íÈÙÈ1. xy ÏÎ˚¢yÓ˚ Ñ˛y ÏäÈ 3!›˛ ˆ˛ôò Á ì˛yÓ˚ ¶˛y£z xÓ˚!ÓˆÏòÓ˚ Ñ˛yˆÏäÈ 2!›˛ ˆ˛ôò xyˆÏäÈ– ¢%ì˛Ó˚yÇ

ì˛y ÏîÓ˚ Ñ˛y ÏäÈ ˆõy›˛ 3 + 2 = 5!›˛ ˆ˛ôò xyˆÏäÈ–

v z ôˆÏÓ˚Ó˚ v zîy£Ó˚ˆÏíñ ¢ÇÖƒyÜ%!ú ÓƒÓ£,ì˛ £ˆÏÎ˚ˆÏäÈ ˛ô!Ó˚õyí ˆÓyG˛yÓyÓ˚ ãˆÏòƒ– ¢õyïyò ¢) Ïeñ

ïÓ˚y ÎyÑ˛ xyˆÏÎ˚¢yÓ˚ Ñ˛yˆÏäÈ x ¢ÇÖƒÑ˛ ˆ˛ôò Á xÓ˚!Ó ÏòÓ˚ Ñ˛y ÏäÈ y ¢ÇÖƒÑ˛ ˆ˛ôò– ì˛y£ˆÏú xyõÓ˚y

!Ñ˛ ˆõy›˛ ˆ˛ôˆÏòÓ˚ ¢ÇÖƒy ÓúˆÏì˛ ˛ôy!Ó˚⁄ £§ƒy xyõÓ˚y ÓúˆÏì˛ ˛ôy!Ó˚ Îy ì˛yˆÏîÓ˚ Ñ˛yˆÏäÈ (x + y)

¢ÇÖƒÑ˛ ˆ˛ôò xyˆÏäÈ– ¢%ì˛Ó˚yÇ x Á y î%!›˛ !ò!î≈T˛ ¢ÇÖƒy ˆÓyG˛yˆÏFäÈ–

ˆòy›˛



v zîy£Ó˚í 2. a) !â˛e 9.1 £ú ~Ñ˛!›˛ ÓÜ≈ Ï«˛e ÎyÓ˚ ≤Ã Ïì˛ƒÑ˛ !î ÏÑ˛Ó˚ ˜îá≈ƒ £ú 2 ˆ¢!õ.–

£z£yÓ˚ ˛ô!Ó˚¢#õy = AB + BC + CD + DA

= (2 + 2 + 2 + 2) ˆ¢!õ

= 2 × 4 ˆ¢!õ. = 4 × 2 ˆ¢!õ = 8 ˆ¢!õ–

b) !â˛e 9.2 ÓÜ≈ˆÏ«˛ˆÏeÓ˚ ˛ô!Ó˚¢#õy !òí≈Î˚ Ñ˛Ó˚–

˛ô!Ó˚¢#õy KLMN = KL + LM + NM + KN

= (5 + 5 + 5 + 5) ˆ¢!õ

= 5 × 4 ˆ¢!õ = 4 × 5 ˆ¢!õ = 20 ˆ¢!õ

~£z¶˛yˆÏÓ xyõÓ˚y ˆÓÓ˚ Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚ ÓÜ≈ˆÏ«˛ˆÏe ˛ô!Ó˚¢#õy ÎÖò ˜îá≈ƒ ˆîÁÎ˚y ÌyˆÏÑ˛–

v˛z˛ôˆÏÓ˚Ó˚ v zîy£Ó˚í ˆÌˆÏÑ˛ xyõÓ˚y Óú!ì˛ ˛ôy!Ó˚ ÓÜ≈ˆÏ«˛ˆÏeÓ˚ ˛ô!Ó˚¢#õy ì˛yÓ˚ ≤ÃˆÏì˛ƒÑ˛ !îˆÏÑ˛Ó˚

˜îá≈ƒÓ˚ 4 Ü%í–

ÓÜ≈ Ï«˛ ÏeÓ˚ ˛ô!Ó˚¢#õy = 4 × Óy£%Ó˚ ˜îá≈ƒ

Î!î ÓÜ≈ Ï«˛ ÏeÓ˚ Óy£%Ó˚ ˜îá≈ƒ ‘a’ £Î˚ñ ì˛y£ Ïú ˛ô!Ó˚¢#õy ‘P’ˆÏÑ˛ ≤ÃÑ˛yü Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚

P = 4 × a

~ÖyˆÏò a £ú !ò!î≈T˛ ÓÜ≈ˆÏ«˛ˆÏeÓ˚ Óy£%Ó˚ ˜îá≈ƒ ~ÓÇ xyõÓ˚y ˆÎ ˆÑ˛yˆÏòy ÓÜ≈ˆÏ«˛ˆÏeÓ˚ ˛ô!Ó˚¢#õy

!òí≈Î˚ Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚ !Ó!¶˛ß¨ ‘a’ÈÙÈ~Ó˚ õy ÏòÓ˚ v z ôÓ˚–

~£z¶˛y ÏÓ xyõÓ˚y !â˛£´ ÏÑ˛ ¢ÇÖƒy !£¢y ÏÓ ≤ÃÑ˛yü Ñ˛Ó˚ Ïì˛ ˛ôy!Ó˚– x«˛Ó˚ a, b, c, .....x, y, zˆÏÑ˛

¢ÇÖƒy !£¢y ÏÓ ≤ÃÑ˛yü Ñ˛Ó˚y £Î˚– ~£z¶˛y ÏÓ x ÏòÑ˛ ¢õ¢ƒy ¢õyïyò Ñ˛Ó˚y ÎyÎ˚–

9.3 Ó#ãÜy!í!ì˛Ñ˛ ˛ôî Á Ó˚y!üõyúy

9.3.1 Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üõyúy ı

˛ôy›˛#Ü!íˆÏì˛ xyõÓ˚y !úˆÏÖ Ìy!Ñ˛ñ (3 × 8) + 2, (16 ÷ 5) + (3 × 20) – 7 £zì˛ƒy!î ~£z

v zîy£Ó˚íÜ%!ú Ïì˛ xyõÓ˚y ˆîÖ Ïì˛ ˛ôy£z

(i) Ó˚y!üõyúyÜ%!ú ¢ÇÖƒy ˆÌˆÏÑ˛ ˜ì˛Ó˚#

(ii) â˛yÓ˚!›˛ õ)ú ≤Ã!Ñ ˛Î˚y ˆÎyÜñ !ÓˆÏÎ˚yÜñ Ü%íñ ¶˛yÜ Óy ì˛yˆÏîÓ˚ !Ñ˛ä%È !òˆÏÎ˚ Ó˚y!üõyúyÜ%!ú

Ü!‡˛ì˛–

xyõÓ˚y â˛úˆÏÑ˛Ó˚ myÓ˚yÁ Ó˚y!üõyúy ˜ì˛Ó˚# Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚–

v zîy£Ó˚í - 3 : ÓyÓú% °Ï¤˛ ˆ◊í#ˆÏì˛ ˛ôˆÏv ¸– ì˛yÓ˚ ˆ◊í#ˆÏì˛ ‘m’ ¢ÇÖƒÑ˛ !ü«˛yÌ≈#– Óyú ÏÑ˛Ó˚

¢ÇÖƒy Óy!úÑ˛yÓ˚ ¢ÇÖƒyÎ˚ ˆÌˆÏÑ˛ 7 Ñ˛õ– ì˛yÓ˚ ˆ◊í#Ó˚ ˆõy›˛ !ü«˛yÌ≈# ¢ÇÖƒy Ñ˛ì˛⁄

Óy!úÑ˛yÓ˚ ¢ÇÖƒy = m

!â˛e 9.1

!â˛e 9.2

ˆòy›˛

154 D.El.Ed. 504


Óyú ÏÑ˛Ó˚ ¢ÇÖƒy = m – 7

ˆõy›˛ !ü«˛yÌ≈# = m + (m – 7) = 2m – 7

~ˆÏ«˛ˆÏe 2m – 7 ~Ñ˛!›˛ Ó˚y!üõyúy ˆÎÖy Ïò m ~Ñ˛!›˛ â˛úÑ˛ñ 2 ~ÓÇ 7 ï %ÓÑ˛– ~ˆÏ«˛ˆÏe

!ÓˆÏÎ˚yÜ Á Ü%í ≤Ã!Ñ Î˚y ÓƒÓ£,ì˛ £ˆÏÎ˚ˆÏäÈ ~!›˛ ~Ñ˛!›˛ Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üõyúyÓ˚ v zîy£Ó˚í–

¢%ì˛Ó˚yÇ Óúy ÎyÎ˚ ˆÑ˛yò Ó˚y!üõyúy Îy !Ñ˛ä%È â˛úÑ˛ ~ÓÇ ¢Ó Óy !Ñ˛ä%È ¢ÇÖƒÑ˛ ≤Ã!Ñ Î˚y +, –

, ×, ÷ myÓ˚y Ü!‡˛ì˛ñ ì˛y£z £ú Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üõyúy–

v zîy£Ó˚ífl ∫Ó) ô ı 7m, 2py + 1, 5,2a − m + n – 2 ~£zÜ%!ú ¢Ó£z Ó#ãÜy!í!ì˛Ñ˛

Ó˚y!üõyúy–

!ò Ïî≈ü Ó˚y!üõyúy

P-~Ó˚ ¢yˆÏÌ 16 ˆÎyÜ P + 16

r ˆÌ ÏÑ˛ 25 Ñ˛õ

(–6) ~Ó˚ ‘P’ Ü%í

xˆÑ˛ 3 !îˆÏÎ˚ ¶˛yÜ

‘m’ ~Ó˚ 3 Ü%í ~ÓÇ

Ü%íö˛ ÏúÓ˚ ˆÌ ÏÑ˛ 8 ˆÓü#

xyÓyÓ˚ñ ÎÖò Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üõyúy ˆîÁÎ˚y ÌyˆÏÑ˛ñ ì˛yÓ˚ ˆÌˆÏÑ˛ xyõÓ˚y ÓúˆÏì˛ ˛ôy!Ó˚ ˆÎ

!Ñ˛¶˛yˆÏÓ Ü!‡˛ì˛ £ˆÏÎ˚ˆÏäÈ– ~ÓyÓ˚ v˛zîy£Ó˚í ˛ôv˛Yò ~ÓÇ ü)òƒfiÌyò ˛ô)Ó˚í Ñ˛Ó˚%ò–

Ó˚y!üõyúy !Ñ˛¶˛y ÏÓ Ü!‡˛ì˛ £ ÏÎ˚ ÏäÈ

S – 1 S ˆÌˆÏÑ˛ 1 Ñ˛õ

T + 25

11a

25b

2n – 4

xyõÓ˚y ˆîÖˆÏì˛ ˛ôy£z Ó˚y!üõyúyÎ˚ ~Ñ˛ Óy ~ˆÏÑ˛Ó˚ ˆÓ!ü ˛ôî ÌyÑ˛ˆÏì˛ ˛ôyˆÏÓ˚– ˛ôˆÏîÓ˚ ¢yˆÏ˛ôˆÏ«˛

Ó˚y!üõyúy !Ó!¶˛ß¨ ¶˛yˆÏÜ ¶˛yÜ Ñ˛Ó˚y ÎyÎ˚–

~Ñ˛˛ôî# Ó˚y!ü ı ˆÎ Ó˚y!ü ˆÑ˛Óúõye ~Ñ˛!›˛ ˛ôî myÓ˚y Ü!‡˛ì˛ ì˛y£z £ú ~Ñ˛˛ôî# Ó˚y!ü–

ˆÎõòÈÙÙÙÈ 7xy, 2x, –4x, –8, 3a2b

ˆòy›˛



!m˛ôî Ó˚y!ü ı ˆÎ Ó˚y!üõyúy î%!›˛ x¢õ ˛ôî myÓ˚y Ü!‡˛ì˛ ì˛y£z £ú !m˛ôî Ó˚y!ü– ˆÎõòÈÙÙÙÈ x

+ y, 2p – 3q, z + 1, 3xy + 2x.

!e˛ôî Ó˚y!ü ı !ì˛ò!›˛ x¢õ ˛ôî myÓ˚y Ü!‡˛ì˛ Ó˚y!üõyúy £ú !e˛ôî Ó˚y!ü– ˆÎõòÈÙÙÙÈ2a – 5b

+ 3c, x + y – 3, pq + p – 2q.

Ó£% ôî Ó˚y!ü ı ˆÎ Ó˚y!üõyúy ~Ñ˛ Óy ì˛yÓ˚ ˆÓü# ˛ôî myÓ˚y Ü!‡˛ì˛ ì˛y£z £ú Ó£% ôî Ó˚y!ü–

ˆÎõòÈÙÙÙÈ 5x, 2a + 3b, m + n – 3.

ì%˛!õ !Ñ˛ ãyò⁄

3xy !m˛ôî Ó˚y!ü òÎ˚– £z£y ~Ñ˛˛ôî# Ó˚y!ü– m + n – 3 !m˛ôî Ó˚y!ü òÎ˚– £z£y !e˛ôî Ó˚y!ü–

3a + 3a !m˛ôî Ó˚y!ü òÎ˚– ~ Ï«˛ Ïe ˛ôîÜ%!ú x¢õ òÎ˚–

~Ñ˛˛ôî#ñ !m˛ôî#ñ !e˛ôî# ¢Ó£z Ó£%˛ôî# Ó˚y!ü–

9.3.2 â˛úÑ˛ ~ÓÇ ï %ÓÑ˛

P = 4a ~£z Üy!í!ì˛Ñ˛ !ÓÓ,!ì˛ ˆÌ ÏÑ˛ ˆîÖy ÎyÎ˚ÈÙÙÙÈÎÖò a = 1, ì˛Öò P = 4 × 1 = 4.

ÎÖò a = 2, ì˛Öò P = 4 × 2 = 8. ÎÖò a = 3, ì˛Öò P = 4 × 3 = 12.

¢%ì˛Ó˚yÇ a ~Ó˚ !Ó!¶˛ß¨ õyˆÏò P-~Ó˚ õyò ˛ô!Ó˚Óì≈ ò £Î˚– xÌ≈yÍ a – x ˛ô!Ó˚Óì≈˛ˆÏòÓ˚ ¢yˆÏÌ

P-~Ó˚ ˛ô!Ó˚Óì≈˛ò £Î˚– ì˛y£z xyõÓ˚y ÓúˆÏì˛ ˛ôy!Ó˚ ‘a’ ~ÓÇ ‘P’ v˛z¶˛ˆÏÎ˚£z ˛ô!Ó˚Óì≈˛òü#ú ì˛y£z

xyõÓ˚y ÓúˆÏì˛ ˛ôy!Ó˚–

~Ñ˛!›˛ ≤Ãì˛#Ñ˛ Óy !â˛£´ ÎyÓ˚ ˆÑ˛yò !ò!î≈T˛ õyò ˆò£zñ !Ñ˛hs%˛ ≤ÃˆÏÎ˚yãò xò%¢yˆÏÓ˚ ˆÎ ˆÑ˛yò õyò

@˘Ã£í Ñ˛Ó˚ Ïì˛ ˛ôy ÏÓ˚ ì˛y£z £ú â˛úÑ˛–

xyÓyÓ˚ñ xyõÓ˚y Î!î Ó!ú !e¶%˛ˆÏãÓ˚ Óy£% ¢ÇÖƒy 3!›˛ äÈyv˛¸y xòƒ !Ñ˛ä%È £ˆÏì˛ ˛ôyˆÏÓ˚ !Ñ˛⁄

xÓüƒ£z òy– ¢%ì˛Ó˚yÇ !e¶% ˆÏãÓ˚ Óy£%Ó˚ ¢ÇÖƒy !ò!î≈T˛ ~ÓÇ ~!›˛ˆÏÑ˛ xyõÓ˚y ÓúˆÏì˛ ˛ôy!Ó˚ ï %ÓÑ˛–

¢%ì˛Ó˚yÇñ

~Ñ˛!›˛ ≤Ãì˛#Ñ˛ Óy !â˛£´ Î!î ~Ñ˛!›˛ !ò!î≈T˛ õyò @˘Ã£í Ñ˛ˆÏÓ˚ñ ì˛Öò ì˛yˆÏÑ˛ ÓˆÏú ï %ÓÑ˛–

P = 4a ~£z !ÓÓ,!ì˛ˆÏì˛ ‘a’ ~ÓÇ ‘p’ â˛úÑ˛ ~ÓÇ ‘4’ ï %ÓÑ˛–

ãyò !Ñ˛⁄

â˛ú ÏÑ˛Ó˚ ˆÑ˛yò !ò!î≈T˛ õyò ˆò£z–

x, y, z, p, z, r £zì˛ƒy!î Óí≈Ü%!ú ¢yïyÓ˚í â˛ú ÏÑ˛Ó˚ ãòƒ ÓƒÓ£,ì˛ £Î˚–

¢õhfl Ï Óyhfl ÏÓ ¢ÇÖƒy£z ï %ÓÑ˛–

Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üõyúy â˛úÑ˛ ~ÓÇ ï %ÓÑ˛ !îˆÏÎ˚ Ü!‡˛ì˛–

v zîy£Ó˚íÈÙÈ5. ˛ôy˛ô%ú% 2!›˛ ~Ñ˛ïÓ˚ˆÏòÓ˚ ˆ˛ôò !Ñ˛ˆÏò!äÈú– ≤ÃˆÏì˛ƒÑ˛›˛y ˆ˛ôˆÏòÓ˚ õ)úƒ 10 ›˛yÑ˛y– Ñ˛ì˛

›˛yÑ˛y ˆ¢ ˆîyÑ˛yòîyÓ˚ ÏÑ˛ ˆî ÏÓ⁄

ˆòy›˛

156 D.El.Ed. 504


fl˛ôT˛ì˛£zñ 2!›˛ ˆ˛ôˆÏòÓ˚ õ)úƒ = Rs. 10 × 2 = Rs. 20, Îy ˛ôy˛ôú%ˆÏÑ˛ !îˆÏì˛ £ˆÏÓ–

ˆõy›˛ õ)úƒ 1!›˛ !ã!òˆÏ¢Ó˚ õ)úƒ × !ã!òˆÏ¢Ó˚ ¢ÇÖƒy

Î!î c = ˆõy›˛ õ)úƒ ~ÓÇ n = !ã!òˆÏ¢Ó˚ ¢ÇÖƒy £Î˚– ì˛y£ˆÏúñ C = 10n

~ÖyˆÏò n ~ÓÇ c v z¶˛ ÏÎ˚£z â˛úÑ˛ ~ÓÇ 10 ï %ÓÑ˛–

v zîy£Ó˚í 6. ~¢õy ~ÓÇ ˆÓ˚üõy î%£z ˆÓyò– ~¢õy ˆÓ˚üõyÓ˚ ˆÌ ÏÑ˛ 4 ÓäÈˆÏÓ˚Ó˚ ÓˆÏv˛¸y– ~Öò

ˆ›˛!Óú›˛y ˛ôv˛Yò ~ÓÇ ö§˛yÑ˛y ãyÎ˚Üy ˛ô)Ó˚í Ñ˛Ó˚%ò–

ˆÓ˚üõyÓ˚ ÓÎ˚¢ ÓäÈ ÏÓ˚ ~¢õyÓ˚ ÓÎ˚¢ ÓÎ˚ Ï¢

7 7 + 4 = 11

9

x

xy˛ôòyÓ˚ v z_Ó˚ñ ˆüˆÏ°ÏÓ˚ ÓyˆÏ: £ˆÏÓ– (x + 4) õyˆÏò ~¢õyÓ˚ ÓÎ˚¢ £ˆÏÓ (x + 4) ÎÖò

ˆÓ˚üõyÓ˚ ‘x’ ÓäÈÓ˚ ÓÎ˚¢– ~Öy Ïò (x + 4) £ Ïúy Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!ü ˆÎÖy Ïò x £ˆÏúy â˛úÑ˛

~ÓÇ 4 £ˆÏúy ï %ÓÑ˛–

~Öò !ò Ï¡¨Ó˚ Ó˚y!üõyúy ˆÌ ÏÑ˛ ï %ÓÑ˛ Á â˛úÑ˛ !úÖ%ò–

Ó˚y!üõyúy â˛úÑ˛ ï %ÓÑ˛

y – 7

32s +

2p + 3

9.3.3 Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üõyúyÓ˚ ˛ôî

v˛z˛ôˆÏÓ˚Ó˚ xyˆÏúyâ˛òy ˆÌˆÏÑ˛ xyõÓ˚y ãyòˆÏì˛ ˛ôy!Ó˚ Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üõyúy ~Ñ˛ Óy ì˛yÓ˚ ˆÓü#

˛ôî myÓ˚y Ü!‡˛ì˛– v zîy£Ó˚ífl ∫Ó) ôÈÙÙÙÈ

v zîy£Ó˚í 7. 2p + 3 ~Ñ˛!›˛ Ó˚y!üõyúy–

~£z Ó˚y!üõyúy Ü‡˛ò Ñ˛Ó˚yÓ˚ ãòƒ xyõÓ˚y ≤ÃÌ Ïõ 2 ~ÓÇ p Ü%í Ñ˛ ÏÓ˚ 2p Ü‡˛ò Ñ˛Ó˚úyõ ~ÓÇ

ì˛yÓ˚˛ôÓ˚ ì˛yÓ˚ ö˛ˆÏú 3 ˆÎyÜ Ñ˛Ó˚úyõ–

v˛zîy£Ó˚í 8. xy + 3z – 5 ~Ñ˛!›˛ Ó˚y!üõyúy–

~£z Ó˚y!üõyúyÓ˚ ãòƒ ≤ÃÌ Ïõ x ~ÓÇ y Ü%í Ñ˛ ÏÓ˚ xy ˜ì˛!Ó˚ Ñ˛Ó˚y £ú– ì˛yÓ˚ ôÓ˚ 3 ~ÓÇ z

Ü%í Ñ˛ ÏÓ˚ 3z Ü‡˛ò Ñ˛Ó˚y £ú– ì˛yÓ˚˛ôÓ˚ xy ~ÓÇ 3z ˆÎyÜ Ñ˛ˆÏÓ˚ ì˛yÓ˚ ˆÌˆÏÑ˛ 5 !ÓˆÏÎ˚yÜ Ñ˛Ó˚y £ú–

v zîy£Ó˚í 7-~ 2p ~ÓÇ 3 £ˆÏúy ˛ôî ~ÓÇ v zîy£Ó˚í 8~ñ xy, 3z ~ÓÇ 5 £ˆÏúy ˛ôî–

ˆòy›˛



ì% !õ !Ñ˛ ãy Ïòy⁄

˛ ˛ôî ÏÑ˛ ˆÎyÜ Ñ˛Ó˚y £Î˚ Ó˚y!üõyúy Ü‡˛ ÏòÓ˚ ãòƒ–

˛ôˆÏîÓ˚ !‡˛Ñ˛ xyˆÏÜÓ˚ !â˛£´›˛y£z ˙ ˛ôî!›˛Ó˚ !â˛£´–

!òˆÏ¡¨Ó˚ Ó˚y!üõyúy ˆÌˆÏÑ˛ ˛ôî ~Ç ˛ôˆÏîÓ˚ ¢ÇÖƒy !òí≈Î˚ Ñ˛Ó˚%ò–

Ó˚y!üõyúy ˛ô ÏîÓ˚ ¢ÇÖƒy ˛ô ÏîÓ˚ òyõ

2ab – 3 2 2ab, –3

13k +

2xyz−

16 – x + 3y2

ˆâ˛T˛y Ñ˛Ó˚%ò ı 4!›˛ ˛ôî !îˆÏÎ˚ 2!›˛ Ó˚y!üõyúy Ü‡˛ò Ñ˛Ó%ò–

9.3.4 Ü%íö˛úñ v zÍ˛ôyîÑ˛ ~ÓÇ ¢£Ü

xyõÓ˚y ãy!ò 2 × 5 = 10.

~ˆÏ«˛ˆÏe 10 £ú Ü%íö˛ú ~ÓÇ 2 Á 5 £ú 10ÈÙÈ~Ó˚ v zÍ˛ôyîÑ˛

ÎÖò î%!›˛ â˛úÑ˛ ÏÑ˛ Ü%í Ñ˛Ó˚y £Î˚ñ ì˛y ÏîÓ˚ Ü%íö˛ú !Ñ˛⁄

3 Á z ~Ó˚ Ü%íö˛ú = 3 × z = 3z

y Á z ~Ó˚ Ü%íö˛ú = y × z = yz

xyõÓ˚y ˆãˆÏò!äÈ ~Ñ˛!›˛ Ó˚y!üõyúy ~Ñ˛ Óy ì˛yÓ˚ ˆÓü# ˛ôî myÓ˚y Ü!‡˛ì˛– ˆÎõòñ 2ab – 3ˆÏì˛

2ab ~ÓÇ (–3) ~£z î%!›˛ ˛ôî xyˆÏäÈ– ~ˆÏ«˛ˆÏe 2ab, 2, a ~ÓÇ b ~Ó˚ Ü%íö˛ú– ¢%ì˛Ó˚yÇ 2ab

~Ó˚ v zÍ˛ôyîÑ˛Ü%!ú £ú 2, a ~ÓÇ b.

ˆâ˛T˛y Ñ˛Ó˚ ı (i) 3xy + 5y

(ii) 7ab – 5a + 2

Ó˚y!üõyúy

˛ôî

v zÍ˛ôyîÑ˛

ˆòy›˛

158 D.El.Ed. 504


ì%˛!õ !Ñ˛ ãyˆÏòy⁄

● ~Ñ˛!›˛ ï %ÓÑ˛ v˛zÍ˛ôyîÑ˛ˆÏÑ˛ ¢yÇ!ÖƒÑ˛ v˛zÍ˛ôyîÑ˛Á ÓˆÏú–

● ~Ñ˛!›˛ â˛úò v zÍ˛ôyîÑ˛ˆÏÑ˛ xy«˛!Ó˚Ñ˛ SÓ#ãÜy!í!ì˛Ñ˛V v˛zÍ˛ôyîÑ˛Á ÓˆÏú–

v˛zîy£Ó˚í 9. Ó˚y!ü°Ïõyúy 3xy – 5y-~Ó˚ î!›˛ ˛ôî xyˆÏäÈ– 3xy ~ÓÇ –5y ˛ôˆÏîÓ˚ ¢yÇ!ÖƒÑ˛

v zÍ˛ôyîÑ˛ £ Ïúy ¢yÇ!ÖƒÑ˛ ¢£ xÌÓy ¢yïyÓ˚í ¢£Ü– 3xy ~Ó˚ ¢£Ü £ˆÏúy 3 ~ÓÇ y-~Ó˚ ¢£Ü

£ Ïúy 5.

ˆâ˛T˛y Ñ˛ ÏÓ˚y ı !ò Ï¡¨Ó˚ Ó˚y!üõyúyÓ˚ ˛ôîÜ%!úÓ˚ ¢£Ü !â˛!£´ì˛ Ñ˛Ó˚ ı

(i) –6ab (ii) 3pq−

9.3.5 ¢õ ~ÓÇ x¢õ ˛ôî

2pq, –pq, 5pq, ( )1 ,2

pq ~£z Ó˚y!üÜ%!úÓ˚ ≤ÃˆÏì˛ƒÑ˛!›˛ˆÏì˛£z pq v˛zÍ˛ôyîÑ˛!›˛ xyˆÏäÈ– !Ñ˛hs%˘˛

2p, 3pq, –5q Ó˚y!üÜ%!úˆÏì˛ !Ó!¶˛ß¨ xy«˛!Ó˚Ñ˛ v˛zÍ˛ôyîÑ˛ xyˆÏäÈ– ¢%ì˛Ó˚yÇ ÓúyÎyÎ˚ˆÏÎ ˛ôîÜ%!úˆÏì˛

~Ñ˛£z Ó#ãÜy!í!ì˛Ñ˛ v zÍ˛ôyîÑ˛ xy ÏäÈ ˆ¢Ü%!ú ¢õãyì˛#Î˚ ˛ôî– xyÓ˚ ˆÎÜ%!ú Ïì˛ ~Ñ˛£z Ó#ãÜy!í!ì˛Ñ˛

v zÍ˛ôyîÑ˛ ˆò£z ˆ¢£z ˛ôîÜ%!ú !Ó°Ïõãyì˛#Î˚ ˛ôî–

v zîy£Ó˚í 1. 2a + 5ab – 3a – b Ó˚y!üõyúyÎ˚ 2a, –3aˆì˛ ~Ñ˛£z Ó#ãÜy!í!ì˛Ñ˛ v˛zÍ˛ôyîÑ˛

a !Óîƒõyò– ¢%ì˛Ó˚yÇ ~Ü%!ú ¢õãyì˛#Î˚ ˛ôî– xòƒ!î ÏÑ˛ 5ab, –b !Ó°Ïõãyì˛#Î˚ ˛ôî–

ˆâ˛T˛y Ñ˛Ó˚ ı

(1) ¢õ˛ô ÏîÓ˚ !Ó¶˛yÜ Ñ˛Ó˚ ı 7x, 7, –8y, –8x, x, –y, 15y

(2) ≤Ã!ì˛!›˛ !Ó¶˛yÜ ˛ô)í≈ Ñ˛Ó˚ ı

˛ôî v zÍ˛ôyîÑ˛ Ó#ãÜy!í!ì˛Ñ˛ ¢õ/!Ó°Ïõ ˛ôî

v˛zÍ˛ôyîÑ˛ ~Ñ˛

~ÓÇ !Ó!¶˛ß¨

15x 15, x !Ó!¶˛ß¨ !Ó°Ïõ

12y 12, y

9z

–13z

6xy

2x

–2ab

5ba

{{{{

ˆòy›˛



x@˘ÃÜ!ì˛ Îyâ˛y£z Ñ˛Ó%ò ı

~ì˛«˛í ˛ôˆÏv˛¸ Îy Ó%G˛ˆÏú ì˛yÓ˚ Á˛ôÓ˚ !¶˛!_ Ñ˛ˆÏÓ˚ !òˆÏ¡¨Ó˚ ≤ÃŸ¿Ü%!úÓ˚ v˛z_Ó˚ ˆîÁÎ˚yÓ˚ ˆâ˛T˛y

Ñ˛ˆÏÓ˚y–

E1. p ~ÓÇ q â˛úÑ˛ˆÏÑ˛ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ 2!›˛ Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üõyúy ˆúÖ–

E2. !ò¡¨!ú!Öì˛ ˆ«˛ˆÏe Ó˚y!üõyúy ˆúÖ–

(i) x ~ÓÇ y ~Ó˚ Ü%íö˛ ÏúÓ˚ 5 Ü% ÏíÓ˚ ˆÌ ÏÑ˛ 3 ˆÓ!ü–

(ii) a ~ÓÇ b ~Ó˚ ˆÎyÜö˛ú ì˛yˆÏîÓ˚ Ü%íö˛ú ˆÌˆÏÑ˛ !ÓˆÏÎ˚yÜ Ñ˛Ó˚–

E3. 2y – 3z + 5 ~ñ ï %ÓÑ˛ ~ÓÇ â˛úÑ˛ ¢òy=˛ Ñ˛Ó˚–

E4. ≤ÃˆÏì˛ƒÑ˛ ˆ«˛ˆÏe x ~Ó˚ ¢£Ü ˆúÖ–

(i) –x

(ii) 2xy + yz + 2

(iii) 223

x y

E5. ¢õ˛ôî ¢òy=˛ Ñ˛Ó˚–

2p, –3pq, 12

pqr , –5, 3p

, 3pqr, 5pq

E6. !ò¡¨!ú!Öì˛ ~Ñ˛˛ôî# Ó˚y!ü ˆÌ ÏÑ˛ v zÍ˛ôyîÑ˛ ˆúÖ–

(i) 5xy (ii) –3 abc

E7. ≤Ã!ì˛!›˛ Ó˚y!üõyúy ˆÌˆÏÑ˛ ˛ôî ~ÓÇ v zÍ˛ôyîÑ˛ ¢òy=˛ Ñ˛Ó˚–

(i) 3xy – 5y (ii) ab + 2a – 3y

9.4 Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üõyúyÓ˚ v z ôÓ˚ ≤Ã!Ñ Î˚y

xyõÓ˚y ˆÎyÜñ !ÓˆÏÎ˚yÜñ Ü%í ¶˛yÜ ~£z ≤Ã!Ñ Î˚yÜ%!úÓ˚ ¢yˆÏÌ ¢ÇÖƒyÓ˚ ˆ«˛ˆÏe ˛ô!Ó˚!â˛ì˛– ~Öò

xyõÓ˚y !‡˛Ñ˛ ~Ñ˛£z¶˛yˆÏÓ ˛ôy!›˛Ü!íì˛ ˆÌˆÏÑ˛ Ó#ãÜ!íˆÏì˛Ó˚ !îˆÏÑ˛ x@˘Ã¢Ó˚ £ˆÏÓy– ˆÎˆÏ£ì% !â˛£´ ˆÑ˛yò

¢õÖƒyˆÏÑ˛£z ≤ÃÑ˛yü Ñ˛ˆÏÓ˚ñ ¢%ì˛Ó˚yÇ ¢ÇÖƒyÓ˚ v z ôÓ˚ ¢õhfl˛Ï ≤Ã!Ñ Î˚y!›˛ !â˛ˆÏ£´Ó˚ v z ôÓ˚ ≤ÃˆÏÎyãƒ–

2!›˛ ¶˛yˆÏÓ xyõÓ˚y ¢õhfl˛Ï ≤Ã!Ñ Î˚y!›˛ xyˆÏúyâ˛òy Ñ˛Ó˚Ó–

(i) !â˛ˆÏ£´Ó˚ ≤Ã!Ñ ˛Î˚y

(ii) Ó˚y!üõyúyÓ˚ ≤Ã!Ñ ˛Î˚y

9.4.1 ˆÎyÜ

Óyhfl Ï ÏÓ ≤Ãâ% Ó˚ ¢õ¢ƒy xy ÏäÈ ˆÎÖy Ïò Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üõyúy ÓƒÓ£yÓ˚ Ñ˛Ó˚y £Î˚ ~ÓÇ ˛ôy›˛#Ü!íì˛

ˆòy›˛

160 D.El.Ed. 504


≤Ã!Ñ Î˚yˆÏì˛ ≤ÃˆÏÎ˚yÜ Ñ˛Ó˚y £Î˚– ˆîÑ˛y ÎyÑ˛ !Ñ˛¶˛yˆÏÓ ~!›˛ ¢¡ôß¨ £Î˚–

(a) ~Ñ˛˛ôî# â˛úÓ˚y!ü/¢ÇÖƒyõyúyÓ˚ ˆÎyÜ

xyõÓ˚y ãy!òñ 2 + 2 + 2 = 2 × 3 = 3 × 2 = 6

~Ñ˛£z¶˛yˆÏÓñ xyõÓ˚y ˛ôy£zñ x + x + x= x × 3 = 3 × x = 3x

~ÓÇ x + x + x + x + x = 5x ~Ó˚ ˆÎyÜö˛ú

= 3x + 5x

= (x + x + x) + (x + x + x + x +x)

xyÓyÓ˚ñ 3x + 5x = (3 × x) + (5 × x)

= (3 + 5) × x S!Ó¶˛yãò xy£zòV

= 8 × x = 8x

!ò Ïâ˛Ó˚ v zîy£Ó˚íÜ%!ú ˆîÖy ÎyÑ˛ÈÙÙÙÈ

v zîy£Ó˚í 10 5ab, 7ab ~ÓÇ ab ~Ó˚ ˆÎyÜ Ñ˛Ó˚–

ˆÎyÜ = 5ab + 7ab + ab

= (5 × ab) + (7 × ab) + (1 × ab)

= (5 + 7 + 1) × ab

= 13 × ab

= 13ab

ì˛y£zñ 5ab + 7ab + ab = 13 ab.

ˆâ˛T˛y Ñ˛ˆÏÓ˚y ı

ˆÎyÜ !òí≈Î˚ Ñ˛Ó˚–

(i) 3p, p ~ÓÇ 7p

(ii) 6xyz ~ÓÇ 12xyz

~£z¶˛yˆÏÓ xyõÓ˚y ãyòˆÏì˛ ˛ôy!Ó˚ î%£z Óy î%£z ÏÎ˚Ó˚ ˆÓü# ¢õ˛ôî ˆÎyÜ Ñ˛Ó˚y ÎyÎ˚– ~ÓyÓ˚ ¶˛y!Ó

!Ó°Ïõ˛ôˆÏîÓ˚ ˆÎyÜ–

â˛úñ 5!›˛ xyõ ~ÓÇ 5!›˛ Ñ˛õúyÓ˚ ˆÎyÜ !òí≈Î˚ Ñ˛!Ó˚–

xyõÓ˚y Ñ˛ÖˆÏòy£z ÓúˆÏì˛ ˛ôy!Ó˚ òy ˆÎñ 8!›˛ xyõ xÌÓy 8!›˛ Ñ˛õúy–

~Ñ˛£z¶˛yˆÏÓñ 5x ~ÓÇ 3y ~Ó˚ ˆÎyÜö˛ú Ñ˛Ö Ïòy£z ~Ñ˛Ñ˛ ˛ôî òÎ˚– xyõÓ˚y !ú!Ö 5x + 3y

ˆÎÖy Ïò v z¶˛Î˚ ˛ôî£z xy ÏäÈ–

(b) Ó#ãÜy!íì˛#Ñ˛ Ó˚y!üõyúyÓ˚ ˆÎyÜ ı

xy¢%ò xyõÓ˚y v˛zîy£Ó˚í !îˆÏÎ˚ xyˆÏúyâ˛òy Ñ˛!Ó˚ ı

ˆòy›˛



v zîy£Ó˚í 11. 5a + 7 ~ÓÇ 2a – 5 ~Ó˚ ˆÎyÜ

= 5a + 7 + 2a – 5

= (5a + 2a) + (7 – 5) S¢õ˛ôîÜ%!ú ÏÑ˛ ~Ñ˛¢y ÏÌ ˆÓ˚ ÏÖV

= 7a + 2

v zîy£Ó˚í 12. 4x + 3y, 8 + 2x ~ÓÇ 2y – 5

= 4x + 3y + 8 + 2x + 2y – 5

= (4x + 2x) + (2y + 3y) + (8 – 5) S~Ñ˛!eì˛ Ñ˛ ÏÓ˚V

= 2x + 5y + 3

ˆâ˛T˛y Ñ˛ ÏÓ˚y ı Ó˚y!üõyúyÜ%!ú ˆÎyÜ Ñ˛Ó˚ ı

(i) mn + 5 ~ÓÇ 2nm – 7

(ii) 2a + 3b – 1, 3a + 7 ~ÓÇ 5b – 3

9.4.2 !Ó ÏÎ˚yÜ

a) ~Ñ˛˛ôî Ó˚y!üÓ˚ !ÓˆÏÎ˚yÜ

5x ˆÌ ÏÑ˛ 2x !Ó ÏÎ˚yÜ Ñ˛Ó%òı

5x – 2x = (x + x + x + x + x) – (x + x)

= x + x + x + x + x – x – x

= 0 + 0 + x + x + x

= 0 + 3x = 3x

xòƒ¶˛y ÏÓ xyõÓ˚y Óú Ïì˛ ˛ôy!Ó˚

5x – 2x = 5 × x – 2 × x

= (5 × 2) × x = 3 × x = 3x

v zîy£Ó˚í 13. 16mn ˆÌˆÏÑ˛ 7mn !Ó ÏÎ˚yÜ Ñ˛Ó˚–

16mn – 7mn = 16 × mn – 7 × mn

= (16 – 7)mn

= 9 × mn

= 9mn

!Ñ˛hs%˘˛ î%!›˛ !Ó°Ïõ òî# ˛ôyÌ≈Ñ˛ƒ ~Ñ˛˛ôîÓ˚y!ü òÎ˚ ÓÓ˚Ç ~›˛y £ú !m˛ôî–

v˛zîy£Ó˚í ı 5x – 3y

ˆâ˛T˛y Ñ˛Ó˚ ı !Ó ÏÎ˚yÜ

(i) 11m ˆÌ ÏÑ˛ 5m

(ii) 10ab ˆÌˆÏÑ˛ 6ab

ˆòy›˛

162 D.El.Ed. 504


(iii) 3xy ˆÌ ÏÑ˛ 5xy

(b) Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üõyúyÓ˚ !Ó ÏÎ˚yÜ ı

(4a + 5b – 2) – (3a + 2b)

= 4a + 5b – 2 – 3a – 2b

= (4a – 3a) + (5b – 2b) – 2

= a + 3b – 2

xòƒ ˛ôÂï!ì˛ xyõÓ˚y ¢õ˛ôî# Ó˚y!üÜ%!úˆÏÑ˛ ~Ñ˛›˛yÓ˚ !òˆÏâ˛ ~Ñ˛›˛y Ñ˛úˆÏõÓ˚ xyÑ˛yˆÏÓ˚ Ó¢yÓ˚ ~ÓÇ

xyúyîy xyúyîy ¶˛y ÏÓ !Ó ÏÎ˚yÜ Ñ˛Ó˚Ó–

4a + 5b – 2

– 3a – 2b——————— a + 3b – 2


(i) 5x – 9 ˆÌ ÏÑ˛ 10x + 7b – 3

(ii) 4pq – 5q + 2 ˆÌˆÏÑ˛ 6pq – 1 + 3q

9.4.3 Ü%í Ñ˛Ó˚ ı

(a) ~Ñ˛ÈÙÈ˛ôî# Ó˚y!üÓ˚ Ü%í

a × a = a2 ÎÖò 2 £Î˚ ~Ñ˛›˛y ¢ÇÖƒy Îy a ~Ó˚ ¢ÇÖƒyˆÏÑ˛ ˛ô!Ó˚ˆÏÓüò Ñ˛ˆÏÓ˚

a2 2 ˆÑ˛ ¢)â˛Ñ˛ Ó Ïú ~ÓÇ 2ˆÑ˛ ¶˛#ì˛ Ó Ïú–

a~Ó˚ ¢ÇÖƒy Ü%í ¶˛#ì˛ ¢)â˛Ñ˛

a×a×a×a a3 a 3

a×a×a×a×a

xy¢%ò !ü!Ö î%ˆÏ›˛y ˛ôˆÏîÓ˚ Üí a ~ÓÇ b ~Ó˚ Ü%íñ a × b ˆÑ˛ ‘ab’ !£¢y ÏÓ ˆúÖy ÎyÎ˚

~Ñ˛£z¶˛yˆÏÓ a × a × b = a2b

a × a × b × b = a2b2 ...

ì˛y£ˆÏú ~›˛y !Ñ˛ (i) x × x × x × y × y

(ii) m × m × m × n × n × n ...........

v zîy£Ó˚í È15 : Ü%í Ñ˛Ó˚ 2x ˆÑ˛ 3y !îˆÏÎ˚

2x + 3y = 2 × x × 3 × y

= 2 × 3 × x × y = 6xy

ˆòy›˛



v˛zîy£Ó˚í 16 : Ü%í Ñ˛Ó˚ 3mn ˆÑ˛ – 5mn. !îˆÏÎ˚

3mn × (–5mn) = 3 × m × n × (–5) × m × n

= 3 × (–5) × (m×m) × (n×n)

= – 15 × m2 × n2 = –15m2n2

v˛zîy£Ó˚í 17 : Ü%í Ñ˛Ó˚ ı – 5pq, 4pq ~ÓÇ 2r

(–5pq) × (4pqr) × 2r = (–5) × p × q × 4 × p × q × r × 2 × r

= (–5) × 4 × 2 × p × p × q × q × r × r

= –40 × p2 × q2 × r2 = –40p2q2r2

ˆâ˛T˛y Ñ˛Ó˚ ı ˆîÖ ˆÎ

(i) 4xy × 2x2

(ii) 5m × 3n × 7mn

(b) Ü%í Ñ˛Ó˚ ÈÙÙÙÈ ~Ñ˛˛ôî# Ó˚y!üˆÏÑ˛ Ó£% ôî# Ó˚y!ü !îˆÏÎ˚

v˛zîy£Ó˚í 18 : Ü%í Ñ˛Ó˚ ı (3x – 5) ¢ ÏAÜ 2x

(3x – 5) × 2x

= 3x × 2x – 5 × 2n

= 3 × 2 × n × n – 5 × 2 × x

= 6x2 – 10x

v zîy£Ó˚í 19 : Ü%í Ñ˛Ó˚ 3a ¢ˆÏAÜ (5a – 2b + 4)

= 3a × (5a – 2b + 4)

= 3a × 5a + 3a × (–2b) + 3a × 4

= 3 × a × 5 × a + 3 × a × (–2) × b + 3 × a × 4

= 15a2 – 6ab + 12a


Ü%í Ñ˛Ó˚ ı

(i) 2x – 3y ~ÓÇ 5xy

(ii) 3mn ~ÓÇ (5m – 7mn + 3n)

(c) Ó£% ôî# Ó˚y!üÓ˚ ¢ ÏAÜ Ó£% ôî# Ó˚y!üÓ˚ Ü%í Ñ˛Ó˚ ı

v zîy£Ó˚í 20 : (a + b) (3a – 5b)

¢õyïyò ı (a + b) (3a – 5b)

= a(3a – 5b) + b (3a – 5b)

= a × 3a – a × 5b + b × 3a + b × (–5b)

= 3 × a × a – 5 × a × b + 3 × a × b – 5 × b × b

ˆòy›˛

164 D.El.Ed. 504


= 3a2 – 5ab + 3ab – 5b2

= 3a2 – 2ab – 5b2

v zîy£Ó˚í 21 : Ü%í Ñ˛Ó˚ ı (2x + 5) ˆÑ˛ (x2 – 3x + 2) myÓ˚y

¢õyïyò ı (2x + 5) × (x2 – 3x + 2)

= 2x × (x2 – 3x + 2) + 5 (x2 – 3x + 2)

= 2x × x2 – 2x × 3x + 2x × 2 + 5 × x2 – 5 × 3x + 5 × 2

= 2x3 – 2 × 3 × x × x + 2 × 2x + 5x2 – 15x + 10

= 2x3 – 6x2 + 5x2 + 4x – 15x + 10

= 2x3 + (–6 + 5) x2 +(4 – 15) x + 10

= 2x3 – x2 – 11x + 10


Ü%í Ñ˛Ó˚ ı

(i) (2m + 3n) ~ÓÇ (m – 2n)

(ii) (pq – 1) ~ÓÇ (2p + 3q – 5)

9.4.4 ¶˛yÜ ı

¢ÇÖƒyÓ˚ ¶˛yÜ ≤Ã!Ñ Î˚y ì% !õ ãyˆÏòy– ~Ñ˛£z¶˛yˆÏÓ xyõÓ˚y ¶˛yÜ Ñ˛Ó˚ˆÏÓy Ó#ãÜy!í!ì˛Ñ˛ Ó˚y!üõyúyÓ˚

ˆ«˛ ÏeÁ–

(a) ~Ñ˛˛ôî#Ó˚y!ü ÏÑ˛ ~Ñ˛˛ôî#Ó˚y!ü !î ÏÎ˚ ¶˛yÜ

(i) ¶˛yãƒ ÏÑ˛ úÓ !£ Ï¢ ÏÓ ~ÓÇ ¶˛yãÑ˛ ÏÑ˛ £Ó˚ !£ Ï¢ ÏÓñ

(ii) úÓ Á £Ó˚ v z¶˛Î˚ˆÏÑ˛£z v˛zÍ˛ôyîˆÏÑ˛Ó˚ Ü%íö˛ú !£ˆÏ¢ˆÏÓ ≤ÃÑ˛yü Ñ˛Ó˚

v zîy£Ó˚í 22 :

(i) 5m myÓ˚y 15mn ¶˛yÜ 15 3 5 35 5mn m n nm m

× × ×= = =×

(ii) 2 2

2 2 3 618 ( 6 ) 18 3

6 6x y x x y y

x y xy xyxy x y

× × × × ×÷ − = = = −− − × ×


Devide (i) 25xy myÓ˚y – 5y (ii) 30a2b2c myÓ˚y 6ab

(b) Ó£% ôî# Ó˚y!üÓ˚ ¶˛yÜ ~Ñ˛˛ôî# Ó˚y!ü !î ÏÎ˚ ı

~ÖyˆÏò ¶˛yãƒ £ˆÏúy Ó£% ôî# Ó˚y!ü ~ÓÇ ¶˛yãÑ˛ £ˆÏúy ~Ñ˛˛ôî# Ó˚y!ü–

!òÎ˚õ ı

¶˛y ÏãƒÓ˚ ≤Ã Ïì˛ƒÑ˛ ˛ôî ¶˛yÜ Ñ˛Ó%ò ¶˛yãÑ˛ myÓ˚y– ≤Ã Ïì˛ƒÑ˛ ¶˛@¿yÇü ¢Ó˚ú#Ñ˛Ó˚í Ñ˛Ó%ò–

ˆòy›˛



v zîy£Ó˚í 23 :

(i) ¶˛yÜ ı 9x2 – 15xy myÓ˚y 3x

~Öò (9x2 – 15xy) ÷ (3x) 2 29 15 159 3 5

3 3 3x xy xyx x y

x x x−= = − = −

(ii) ¶˛yÜ 8a2b – 12ab2 + 20ab myÓ˚y (–4ab)

~Öòñ (8a2b – 12ab2 + 20ab) ÷ (–4ab) = – 2a + 3b – 5

ˆâ˛T˛y Ñ˛Ó%ò ı

¶˛yÜ (i) 6m2n – 9mn2 myÓ˚y 3mn

(ii) 10x3y – 15x2y2 – 5x2y2 myÓ˚y 5xy

(c) Ó£% ôî Ó˚y!ü !î ÏÎ˚ Ó£% ôî Ó˚y!üÓ˚ ¶˛yÜ ÈÙÈ

v zîy£Ó˚í 24 : ¶˛yÜ 11x + 15x2 – 12 myÓ˚y 5x – 3

ïy˛ôÈÙÈ1 «˛õì˛yÓ˚ Ñ õ£…y¢õyò xò%ÎyÎ˚# ¶˛yãÑ˛ ~ÓÇ ¶˛yãƒ ¢y!ã ÏÎ˚ !ò Ïì˛ £ ÏÓ–

5x – 3 15x2 + 11x – 12

ïy˛ôÈÙÈ2 ¶˛yãˆÏÑ˛Ó˚ ≤ÃÌõ ˛ôî !îˆÏÎ˚ ¶˛yˆÏãƒÓ˚ ≤ÃÌõ ˛ôî ¶˛yÜ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ ~ÓÇ ˛ôyÁÎ˚y ÎyˆÏÓ

¶˛yÜö˛ ÏúÓ˚ ≤ÃÌõ ˛ôî

~ÖyˆÏòñ 215 3

5x xx

=

ïy˛ôÈÙÈ3 ¶˛yãˆÏÑ˛Ó˚ ˛≤ÃˆÏì˛ƒÑ˛ ˛ôîˆÏÑ˛ 3x !îˆÏÎ˚ Ü%í Ñ˛Ó˚ˆÏì˛ £ˆÏÓ ~ÓÇ ö˛úyö˛ú›˛y ò#ˆÏâ˛ !úÖˆÏÓò–

ì˛yÓ˚˛ôÓ˚ ¶˛yãƒ ˆÌˆÏÑ˛ !ÓˆÏÎ˚yÜ Ñ˛Ó˚ˆÏì˛ £ˆÏÓ–

ïy˛ôÈÙÈ5 ïy˛ô 2 ˆÌˆÏÑ˛ 3 xÓ!ï ≤Ã!Ñ ˛Î˚y!›˛Ó˚ ˛ô%òÓ˚yÓ,!_ Ñ˛Ó˚ˆÏì˛

£ˆÏÓ ~£z ïˆÏÓ˚ ˆÎ 20x – 12 £ˆÏúy òì%˛ò ¶˛yãƒ–

~Öy Ïò 2Î˚ ˛ôî 20 45

xx

= =

¶˛yÜö˛ ÏúÓ˚ 2Î˚ ˛ôî ~ÓÇ ¶˛yã ÏÑ˛Ó˚ Ü%íö˛ú òì% ò ¶˛yãƒ ˆÌ ÏÑ˛

!Ó ÏÎ˚yÜ Ñ˛Ó˚y £ ÏÓ– ö˛úyö˛ú ‘0’–

~Öy Ïòñ ¶˛yÜö˛ú = 3x + 4

¶˛yÜˆÏü°Ï = 0.

∴ (15x2 + 11x – 12) ÷ (5x – 3)

= 3x + 4

2

2

3 4

5 3 15 11 12

15 9–

20 1220 12

0

x

x x x

x x

xx

+− + −

− ↓+

−−

− +

2

2

3

5 3 15 11 12

15 9–

20 12

x

x x x

x x

x

− + −− ↓+

−

ˆòy›˛

166 D.El.Ed. 504



E8. ¢Ó˚ú#Ñ˛Ó˚í ı

E9. !ÓˆÏÎ˚yÜ Ñ˛Ó˚%ò 3m – 5n + 7

(2m – 3n + 5) ~ÓÇ (8 + 4n) ~Ó˚ ˆÎyÜö˛ú ˆÌ ÏÑ˛

E10. ü)òƒfiÌyò ˛ô)Ó˚í Ñ˛Ó˚%ò ı

(a2 – 5ab – 3a + 7) + (...............) = 3a2 + 2ab – 5b + 2

E11. Ü%í Ñ˛Ó˚ ı

(a) (3x – 2) myÓ˚y (2x + 3)

(b) (p2 + pq + q2) myÓ˚y (p – q)

E12. ¶˛yÜ ı 3a3 + 16a2 + 20 + 21a myÓ˚y (a + 4)


~Ñ˛áyì˛ ¢õ#Ñ˛Ó˚í ~ÓÇ ì˛yÓ˚ ¢õyïyò– ~Ñ˛!›˛ ˛¢õ#Ñ˛Ó˚íˆÏÑ˛ îy!v ¸ ôyÕ‘yÓ˚ ¢yˆÏÌ ì% úòy Ñ˛Ó˚y ÎyÎ˚–

îy!v ¸ ôyÕ‘yÓ˚ î%!›˛ ˛ôye ¢õ#Ñ˛Ó˚ˆÏíÓ˚ î%!›˛ ˛ô«˛ÈÙÙÙÈÓyõ˛ô«˛ Á v˛yò˛ô«˛– ¢õì˛y!â˛£´ !òˆÏî≈ü Ñ˛ˆÏÓ˚ ˆÎ

˛ôyeî%!›˛ ¶˛yÓ˚¢yõƒ ÓãyÎ˚ ˆÓ˚ ÏÖ ÏäÈ– v zîy£Ó˚ífl ∫Ó) ô ı

3x – 5 = 16 ~£z ¢õ#Ñ˛Ó˚ ÏíÓ˚ 3x – 5 £ú Óyõ˛ô«˛ ~ÓÇ 16 £ú v˛yò˛ô«˛– ~ Ï«˛ Ïe

¢õ#Ñ˛Ó˚ˆÏíÓ˚ x £ú ~Ñ˛!›˛ xãyòy Ó˚y!ü–

~£z xò% ÏFäÈ Ïî ~Ñ˛áyì˛ ¢õ#Ñ˛Ó˚í ¢¡ô ÏÑ≈ !Óüî¶˛y ÏÓ xy Ïúyâ˛òy Ñ˛Ó˚y £ ÏÓ–

9.5.1 ~Ñ˛áyì˛ Ó#ãÜy!í!ì˛Ñ˛ ¢õ#Ñ˛Ó˚í ı

~Ñ˛!›˛ ¢£ã v zîy£Ó˚í ˆòÁÎ˚y ÎyÑ˛ÈÙÙÙÈÚÚ~Ñ˛!›˛ ¢ÇÖƒy !òí≈Î˚ Ñ˛Ó˚ ÎyÓ˚ !mÜ% ÏòÓ˚ ¢y ÏÌ 5 ˆÎyÜ

Ñ˛Ó˚ˆÏú 15 ˛ôyÁÎ˚y ÎyÎ˚– ~£z !ÓÓ,!ì˛ˆÏÑ˛ Üy!í!ì˛Ñ˛ !ÓÓ,!ì˛ˆÏì˛ ≤ÃÑ˛yü Ñ˛Ó˚yÓ˚ ãòƒ xyõÓ˚y ~Ñ˛!›˛

xãyòy Ó˚y!ü x !òúyõ– ì˛yÓ˚˛ôÓ˚ !mÜ%ò Ñ˛ˆÏÓ˚ 2x ˆ˛ôúyõ– ì˛yÓ˚˛ôÓ˚ 5 ˆÎyÜ Ñ˛ ÏÓ˚ 2x + 5 ˛ôyÁÎ˚y

ˆÜú– ~Öò ˆîÁÎ˚y xy ÏäÈ ˆÎ £z£y 15 ~Ó˚ ¢õyò– ¢%ì˛Ó˚yÇ xyõÓ˚y !úÖˆÏì˛ ˛ôy!Ó˚ñ

ˆòy›˛



2x + 5 = 15, Îy £ú v z ô ÏÓ˚Ó˚ !ÓÓ,!ì˛Ó˚ ¢õ#Ñ˛Ó˚í–

!Ñ˛ä%È !â˛£´ !îˆÏÎ˚ ˆÎ Ó£%˛ôî Ó˚y!ü Ü‡˛ò Ñ˛Ó˚y ÎyÎ˚ ~ÓÇ xì˛É˛ôÓ˚ xyÓ˚ ~Ñ˛!›˛ Ó˚y!üÓ˚ ¢yˆÏÌ ¢õì˛y

xyòy £Î˚ñ ì˛y£z £ú ¢õ#Ñ˛Ó˚í–

¢%ì˛Ó˚yÇñ 2x + 5 = 15, ~Ñ˛!›˛ ¢Ó˚ú Ó#ãÜy!í!ì˛Ñ˛ ¢õ#Ñ˛Ó˚í–

!ÓÓ,!ì˛ ¢õ#Ñ˛Ó˚í

x ~ÓÇ 7 ~Ó˚ ˆÎyÜö˛ú £ú 16 x + 7 = 16

y ˆÌ ÏÑ˛ 3 !ÓˆÏÎ˚yÜ Ñ˛Ó˚ˆÏú 10 £ÈÎ˚

x ~Ó˚ 9 Ü%í £ˆÏúy 36

m ~Ó˚ 34 £ Ïúy 12

v˛z˛ôˆÏÓ˚Ó˚ ˆ›˛!Óú!›˛ˆÏì˛ ˆÎ ¢õ#Ñ˛Ó˚íÜ%!ú xy˛ôòyÓ˚y !úÖˆÏúò ˆÎÜ%!ú ~Ñ˛!›˛ xãyòy Ó˚y!üÎ%=˛ ~Ñ˛õì˛

¢õ#Ñ˛Ó˚í ~ÓÇ ~£z xãyòy Ó˚y!üÓ˚ áyì˛ 1. ì˛y£z xyõÓ˚y ÓúˆÏì˛ ˛ôy!Ó˚ ~Ñ˛áyì˛ Ó£% ôîÓ˚y!ü ¢¡∫!úì˛

¢õ#Ñ˛Ó˚í £ú ~Ñ˛áyì˛ ¢õ#Ñ˛Ó˚í ~ÓÇ £z£yÓ˚ ¢yïyÓ˚í xyÑ˛yÓ˚ ax + b = 0, ˆÎÖyˆÏò a, b, ï %ÓÑ˛ Ó˚yü

~ÓÇ x xãyòy Ó˚y!ü ~ÓÇ x xãyòy Óy â˛úÑ˛ Ó˚y!ü–

x2 + 7x + 6 = 0 ~Ñ˛áyì˛ ¢õ#Ñ˛Ó˚í òÎ˚ Ñ˛yÓ˚í ~ Ï«˛ Ïe xãyòy Ó˚y!üÓ˚ ¢ ÏÓ≈yFâ˛ áyì˛ 2

x + 2y = 5 ~!›˛ î%!›˛ xãyòy Ó˚y!ü x ~ÓÇ y Î%=˛ ~Ñ˛áyì˛ ¢õ#Ñ˛Ó˚í–

9.5.2 ~Ñ˛áyì˛ ¢õ#Ñ˛Ó˚ ÏíÓ˚ ¢õyïyò

x + 7 = 16 ~£z ¢õ#Ñ˛Ó˚ Ïí x-~Ó˚ ˆÑ˛yò õyˆÏòÓ˚ ãòƒ ¢õì˛y ÓãyÎ˚ ÌyÑ˛ˆÏÓ– xyõÓ˚y x ~Ó˚

õyò 1, 2, 3 £zì˛ƒy!î £zì˛ƒy!î õyò @˘Ã£í Ñ˛!Ó˚ Îì˛«˛í òy ˛ôÎ≈hs˘˛ ~£z ¢õì˛y ÓãyÎ˚ ÌyˆÏÑ˛– ~£z¶˛yˆÏÓ

˛ôÓ˚#«˛y Ñ˛Ó˚ˆÏì˛ Ñ˛Ó˚ˆÏì˛ xyõÓ˚y x = 9 ~Ó˚ ãòƒ ~£z ¢õì˛y ˆ˛ôúyõ–

¢%ì˛Ó˚yÇ x ~Ó˚ ~Ñ˛!›˛õye õyˆÏòÓ˚ ãòƒ£z ~£z ¢õì˛y ˆ˛ôúyõ–

xyÓyÓ˚ 2y = 6 ~£z ¢õ#Ñ˛Ó˚ Ïí y ~Ó˚ õyò 3 £ˆÏú ¢õì˛y ÓãyÎ˚ ÌyˆÏÑ˛– ~Ñ˛£zÓ˚)ˆÏ˛ô m –

3 = 4 ~Ó˚ ˆ«˛ˆÏe m ~Ó˚ õyò 7 £ Ïú ¢õì˛y ÓãyÎ˚ Ìy ÏÑ˛–

ˆÎ õyˆÏòÓ˚ ãòƒ ˆÑ˛yˆÏòy ¢õ#Ñ˛Ó˚í ~Ñ˛!›˛ ¢!ì˛ƒ !ÓÑ, !ì˛ !£¢yˆÏÓ ≤Ã!ì˛˛ôß¨ £Î˚ ˆ¢£z õyò!›˛ £ú

¢õ#Ñ˛Ó˚ ÏíÓ˚ ¢õyïyò Óy Ó#ã–

~Ñ˛áyì˛ ¢õ#Ñ˛Ó˚ ÏíÓ˚ ¢õyïy ÏòÓ˚ ˛ôÂï!ì˛ ı

(a) ˛ôÓ˚#«˛y ˛ôÂï!ì˛ ı x + 7 = 16 ~Ó˚ ¢õyïyò ˛ôyÁÎ˚yÓ˚ ãòƒ ˛ôÓ˚#«˛y ˛ôÂï!ì˛ˆÏì˛ ò#ˆÏâ˛Ó˚

ˆÎ !Óú!›˛ ˛ô)Ó˚í Ñ˛Ó˚y ÎyÑ˛ÈÙÙÙÈ

ˆòy›˛

168 D.El.Ed. 504


x 1 2 3 4 5 6 7 8 9 10

x + 7

~ Ï«˛ Ïe x ~Ó˚ !Ó!¶˛ß¨ õyò !òˆÏÎ˚ ˆ›˛!Óú!›˛ ˛ô)Ó˚í Ñ˛Ó˚y £ˆÏÎ˚ˆÏäÈ– ~£z ≤Ã!Ñ Î˚yˆÏÑ˛ Óúy £Î˚

˛ôÓ˚#«˛y ˛ôÂï!ì˛– !Ñ˛hs%˘˛ ~£z ˛ôÂï!ì˛!›˛ Ö%Ó£z ¢õÎ˚ ¢yˆÏ˛ô«˛ ~ÓÇ Óv˛¸ Óv˛¸ ¢ÇÖƒyÓ˚ ˆ«˛ˆÏe £z£y

xò%˛ôÎ%=˛–


(i) m – 5 = 16

(ii) 2y – 1 = 17

(b) ˆÎyÜÈÙÈ!Ó ÏÎ˚yÜ ˛ôÂï!ì˛ ı

xyõÓ˚y xyˆÏÜ£z ÓˆÏú!äÈ ¢õ#Ñ˛Ó˚ˆÏíÓ˚ î%!›˛ ˛ô«˛ ì%˛úyÎˆÏs˛f î%!›˛ ì%˛úy ˛ôye– xyõÓ˚y ì%˛úyÎs˛f!›˛ˆÏÑ˛

xò%¶)˛!õÑ˛ Ó˚yÖyÓ˚ ãòƒ ì%˛úy˛ôyˆÏe !Ñ˛ä%È !ã!ò¢ Óy ˛ô!Ó˚õy˛ô Ó˚y!Ö Óy ≤ÃˆÏÎ˚yãò v˛z_Ó˚ ì%˛úy˛ôyˆÏe

ˆÌˆÏÑ˛ !Ñ˛ä%È !ã!ò¢ ì%˛ˆÏú ˆò£z– ¢õ#Ñ˛Ó˚ˆÏíÓ˚ ˆ«˛ˆÏeÁ ˆ¢£z ~Ñ˛£z á›˛òy ≤ÃˆÏÎyãƒ– ïÓ˚y ÎyÑ˛ ‘x’

xãyòy ˛ô!Ó˚õyˆÏ˛ôÓ˚ ~Ñ˛!›˛ ˛ôƒyˆÏÑ˛›˛ Îy Óyõ ì%˛úy˛ôyˆÏe xyˆÏäÈ– x˛ôÓ˚!îˆÏÑ˛ ~Ñ˛!›˛ Óy›˛Öyv˛¸y w

v˛yò˛ôyˆÏe ˆÓ˚ˆÏÖ ì%˛úyÎs˛f!›˛ˆÏì˛ xò%¶) !õÑ˛ Ñ˛Ó˚y £ú– ¢%ì˛Ó˚yÇ x = w.

ˆÑ˛¢ÈÙÈI : xyõÓ˚y Î!î î%ˆÏ›˛y ˛ôyˆÏe£z Áãò ‘c’ ˆÎyÜ Ñ˛!Ó˚ ì˛y£ˆÏú Óy›˛ÖyÓ˚y!›˛ xò%¶)˛!õÑ˛

ÌyÑ˛ ÏÓ òy ˆ£ Ïú ÌyÑ˛ ÏÓ⁄

xÓüƒ£z ~›˛y xò%¶) !õÑ˛ ÌyÑ˛ ÏÓ–

¢%ì˛Ó˚yÇ x + c = w + c

ˆÑ˛¢ II : Î!î î% Ï›˛y ˛ôye ˆÌ ÏÑ˛£z Áãò ‘c’ ¢!Ó˚ˆÏÎ˚ ˆòÁÎ˚y £Î˚ñ ì˛y£ˆÏú Óy›˛ÖyÓ˚y!›˛ ˆÑ˛yò

xÓfiÌyÎ˚ ÌyÑ˛ ÏÓ⁄

xÓüƒ£z xò%¶)˛!õÑ˛–

¢%ì˛Ó˚yÇñ x – c = w – c

ˆÑ˛¢ III : ~Ñ˛£z ¶˛yˆÏÓ Î!î xyõÓ˚y î%ˆÏ›˛y ˛ôyˆÏe£z ‘c’Ü%ò Áãò Ó˚y!Öñ ì˛y£ˆÏú Óy›˛ÖyÓ˚y!›˛

xò%¶) !õÑ˛ ÌyÑ˛ ÏÓ–

¢%ì˛Ó˚yÇñ ~!›˛ xyõÓ˚y ≤ÃÑ˛yü Ñ˛Ó˚ˆÏì˛ ˛ôy!Ó˚ñ xc = wc.

ˆÑ˛¢ IV : xyÓyÓ˚ Î!î xyõÓ˚y î%ˆÏ›˛y ˛ôyˆÏe£z 1c

SˆÎÖy Ïò 0c ≠ V Ü%ò Áãò Ó˚y!Ö ì˛y£ Ïú

Óy›˛ÖyÓ˚y!›˛ ¢õì˛yÎ˚ ÌyÑ˛ ÏÓ–

ˆòy›˛



¢%ì˛Ó˚yÇ xyõÓ˚y ˛ôy£z x wc c= SˆÎÖyˆÏò 0c ≠ V

v z ôˆÏÓ˚Ó˚ xyˆÏúyâ˛òy ˆÌˆÏÑ˛ ¢õ#Ñ˛Ó˚í ¢õyïyò Ñ˛Ó˚yÓ˚ ˆ«˛ˆÏe xyõÓ˚y ¢õyïyò Ñ˛Ó˚yÓ˚ !òÎ˚õÜ%!ú

˛ôy£zÈÙÙÙÈ

● ~Ñ˛£z Ó˚y!ü v z¶˛Î˚˛ôˆÏ«˛ ˆÎyÜ Ñ˛Ó˚y ÎyÎ˚–

● ~Ñ˛£z Ó˚y!ü v z¶˛Î˚ ˛ô«˛ ˆÌˆÏÑ˛ !ÓˆÏÎ˚yÜ Ñ˛Ó˚y ÎyÎ˚–

● ~Ñ˛£z Ó˚y!ü !îˆÏÎ˚ v˛z¶˛Î˚˛ôˆÏ«˛ Ü%ò Ñ˛Ó˚y ÎyÎ˚–

● ü)òƒ äÈyv ¸y xòƒ ˆÎˆÏÑ˛yˆÏòy Ó˚y!ü !îˆÏÎ˚ v z¶˛Î˚˛ôˆÏ«˛ ¶˛yÜ Ñ˛Ó˚y ÎyÎ˚–

v zîy£Ó˚í 25 : ¢õyïyò y – 5 = 11

¢õyïyò ı ¢õyïyò Ñ˛Ó˚yÓ˚ xÌ≈ xãyòy Ó˚y!üÓ˚ õyò !òí≈Î˚–

¢%ì˛Ó˚yÇ y – 5 = 11

( 5) 5 11 5y⇒ − + = + Sv z¶˛Î˚!î ÏÑ˛ 5 ˆÎyÜ Ñ˛!Ó˚Î˚yV

( 5 5) 16y⇒ + − + = S¢ÇˆÏÎyÜV

0 16y⇒ + =

16y⇒ =

v˛zîy£Ó˚í 26 : ¢õyïyò Ñ˛Ó%ò z + 4 = 8

¢%ì˛Ó˚yÇ z + 4 = 8

( 4) 4 8 4z⇒ + − = − Sv z¶˛Î˚!îÑ˛ ˆÌ ÏÑ˛ 4 !ÓˆÏÎ˚yÜ Ñ˛!Ó˚Î˚yV

(4 4) 4z⇒ + − = S¢ÇˆÏÎyÜV

0 4z⇒ + =

4z⇒ =

v zîy£Ó˚í 27 : ¢õyïyò Ñ˛Ó˚%ò 123x =

¢õyïyò ı 123x =

3 12 33x⇒ × = × Sv z¶˛Î˚!îÑ˛ ÏÑ˛ 3 myÓ˚y Ü%í Ñ˛!Ó˚Î˚yV

ˆòy›˛

170 D.El.Ed. 504


( )1 3 363

x⇒ × × = S¢ÇˆÏÎyÜV

1 36x⇒ × =

36x⇒ =

v zîy£Ó˚í 28 : ¢õyïyò Ñ˛Ó%ò 5x – 2 = 28

¢õyïyò ı ~Öy Ïò ¢Ó≈≤ÃÌ Ïõ ¢ÇÖƒy Sï %ÓÑ˛V ¢!Ó˚ ÏÎ˚ ˆîÁÎ˚y £ Ïúy–

¢%ì˛Ó˚yÇñ 5x – 2 = 28

(5 2) 2 28 2x⇒ − + = + Sv z¶˛Î˚!î ÏÑ˛ 2 ˆÎyÜ Ñ˛!Ó˚Î˚yV

5 ( 2 2) 30x⇒ + − + = S¢ÇˆÏÎyÜV

5 0 30x⇒ + =

5 30x⇒ =

6 (Ans)x⇒ = Sv z¶˛Î˚!îÑ˛ ÏÑ˛ 5 myÓ˚y ¶˛yÜ Ñ˛!Ó˚Î˚yV

(c) ˛ô«˛yhs˘˛Ó˚ ≤Ã!Ñ Î˚y ı

¢õ#Ñ˛Ó˚í ¢õyïyˆÏòÓ˚ ˆ«˛ˆÏe xyõÓ˚y ˆÑ˛yò Ó˚y!ü Óyõ˛ô«˛ ˆÌˆÏÑ˛ v˛yò˛ôˆÏ«˛ Óy v˛yò˛ô«˛ ˆÌˆÏÑ˛

Óyõ˛ôˆÏ«˛ xy¢ˆÏì˛ ˛ôy!Ó˚– ~£z ≤Ã!Ñ ˛Î˚yÎ˚ ü%ï%õye (+) ˆÌˆÏÑ˛ (–) Óy (–) ˆÌ ÏÑ˛ (+)~ ˛ô!Ó˚Óì≈˛ò

£Î˚–

xÌ≈yÍ

(i) ˆÎyÜ ˛ô!Ó˚Óì≈ ò £Î˚ !Ó ÏÎ˚y ÏÜ

(ii) !ÓˆÏÎ˚yÜ ˛ô!Ó˚Óì≈˛ò £Î˚ ˆÎyˆÏÜ

(iii) Ü%í ˛ô!Ó˚Óì≈˛ò £Î˚ ¶˛yˆÏÜ

(iv) ¶˛yÜ ˛ô!Ó˚Óì≈˛ò £Î˚ Ü%ˆÏí–

~£z !òÎ˚õˆÏÑ˛ Óúy £Î˚ ˛ô«˛yhs˘˛Ó˚ (Transposition) !òÎ˚õ–

v˛zîy£Ó˚í 29. 2x – 7 = 5

¢õyïyò ı 2x – 7 = 5

⇒ 2x = 5 + 7⇒ 2x = 12

ˆòy›˛



⇒ 122

x =

⇒ x = 6

v˛zîy£Ó˚í 30. 5 2

63

y − =

¢õyïyò ı 5 2

63

y − =

⇒ 5y – 2 = 6 × 3

⇒ 5y – 2 = 18

⇒ 5y =18 + 2

⇒ 5y = 20

⇒ 20 45

y = =

4y∴ =


(i) 3p + 2 = 17 (ii) 2 (x + 4) = 12

(d) ÓLÜ%íò ≤Ã!Ñ Î˚y ı

Î!î ¢õ#Ñ˛Ó˚ Ïí ¶˛@¿yÇü Ìy ÏÑ˛ñ ì˛y£ Ïú ¶˛@¿yÇü ÏÑ˛ Óyî !î ÏÎ˚ ~Ñ˛›˛y ¢£ã ˛ôÂï!ì˛ !ü!Ö–

õˆÏò Ñ˛!Ó˚ ¶˛@¿yÇü!›˛ a cb d=

ca bd

⇒ = × S‘b’ ˆÑ˛ v˛yò˛ôˆÏ«˛ ¢!Ó˚ˆÏÎ˚V

c bad×⇒ =

⇒ a × d = c × b S‘c’ ˆÑ˛ Óyõ˛ô Ï«˛ ¢!Ó˚ ÏÎ˚V

¢%ì˛Ó˚yÇñ a c a d c bb d= ⇒ × = ×

xòƒ¶˛yˆÏÓñ a cb d=

a cbd bdb d

⇒ × = × (v z¶˛Î˚ ô Ï«˛ ‘bd’ Ü%í Ñ˛ˆÏÓ˚V

⇒ ad = cb

~!›˛ £ Ïúy ÓLÜ%íò ˛ôÂï!ì˛–

xy¢%ò !Ñ˛ä%È v zîy£Ó˚í xy Ïúyâ˛òy Ñ˛!Ó˚–

ˆòy›˛

172 D.El.Ed. 504


v˛zîy£Ó˚í 31. ≤Ãõyí Ñ˛Ó˚ ı 3 1 72 4

x x+ +=

¢õyïyò ı 3 1 72 4

x x+ +=

⇒ (3x + 1)× 4 = (x + 7) × 2 SÓLÜ%íò ˛ôÂï!ì˛ myÓ˚yV⇒ 12x + 4 = 2x + 14 S!ÓˆÏFäÈî ˛ôÂï!ì˛V⇒ 12x – 2x = 14 – 4 SÓyõ˛ô Ï«˛ xãyòy ˛ôîÜ%!ú ˆÓ˚ ÏÖ ~ÓÇ v˛yò!î ÏÑ˛

ï %ÓÑ˛ ˆÓ˚ˆÏÖV⇒ 10x = 10

⇒ 1010

x = S10ˆÑ˛ v˛yò˛ôˆÏ«˛ ¢!Ó˚ˆÏÎ˚V

⇒ x = 1

v˛zîy£Ó˚í 32. ¢õyïyò ı 3 1 52 3 7

yy− =+

¢õyïyò ı 3 1 52 3 7

yy− =+

⇒ (3y – l) × 7 = (2y + 3) × 5 (ÓLÜ%íò)

⇒ 21y – 7 = 10y +15

⇒ 21y – 10y = 15 + 7

⇒ 11y = 22

⇒ 22 211

y = =


E13 !ò¡¨!ú!Öì˛ ¢õ#Ñ˛Ó˚ ÏíÓ˚ ¢õyïyò Ñ˛Ó%ò–

(i) 7x = 28 (ii) 3y – 2 = 19 (iii) 2x – 3 = 6


(i) 28 = 4 + 3 (t + 5) (ii) 2(2p – 3) = 6


(i)3

4

63

8 =+x

x(ii)

8

5

7x3

3x2 =++

ˆòy›˛



9.6 Óyhfl ÏÓ ¢õ¢ƒyÓ˚ ¢õyïyò / Ó#ãÜy!í!ì˛Ñ˛ ≤Ã!Ñ Î˚yÓ˚ ≤Ã ÏÎ˚yÜ ı

xyõÓ˚y !üˆÏÖ!äÈ Üy!í!ì˛Ñ˛ !ÓÓ,!ì˛ˆÏì˛ ¢yïyÓ˚í ¢õ#Ñ˛Ó˚ˆÏíÓ˚ ˛ô!Ó˚Óì≈˛ò Ñ˛Ó˚ˆÏì˛ £úñ xyõÓ˚y xyÓ˚Á

!ü ÏÖ!äÈ !Ñ˛¶˛y ÏÓ ¢yïyÓ˚í ¢õ#Ñ˛Ó˚í ÏÑ˛ ¢õyïyò Ñ˛Ó˚ Ïì˛ £Î˚– Óyhfl ÏÓ ¢õ¢ƒy ¢õyïyò Ñ˛Ó˚ Ïì˛ ˆÎ ˛ôÂï!ì˛

xÓú¡∫ò Ñ˛Ó˚ Ïì˛ £Î˚ ˆ¢!›˛ £úÈÙÙÙÈ

(i) ¢õ¢ƒy!›˛Ó˚ ˛ô!Ó˚!fiÌ!ì˛ Ó%G˛ˆÏì˛ ˛ôyÓ˚y–

(ii) xãyòy Ó˚y!üÓ˚ ãòƒ v z˛ôÎ%=˛ ≤Ãì˛#Ñ˛ Óy !â˛£´ !‡˛Ñ˛ Ñ˛Ó˚y–

(iii) Á£z ¢õ¢ƒy!›˛Ó˚ ¢õ#Ñ˛Ó˚í ˆúÖy–

(iv) ¢õ#Ñ˛Ó˚í!›˛Ó˚ ¢õyïyò Ñ˛Ó˚y Á xãyòy Ó˚y!üÓ˚ õyò !òí≈Î˚ Ñ˛Ó˚y–

(v) ¢õyïyò!›˛Ó˚ !ò¶%≈˛úì˛y Îyâ˛y£z Ñ˛Ó˚y–

v˛zîy£Ó˚íÜ%!ú xyˆÏúyâ˛òy Ñ˛!Ó˚

v zîy£Ó˚í 33. !õì˛y !â˛hs˘˛y Ñ˛ˆÏÓ˚ ¢ÇÖƒy x– ˆ¢ Î!î ¢ÇÖƒy!›˛ 4 Ü% ÏíÓ˚ ˆÌˆÏÑ˛ 7 ¢!Ó˚ˆÏÎ˚ ˆòÎ˚ñ

ì˛y£ˆÏú ö˛úyö˛ú £ˆÏÓ 17–

¢ÇÖƒy!›˛ Ñ˛ì˛⁄

õˆÏò Ñ˛!Ó˚ !õì˛y !â˛hs˘˛y Ñ˛ˆÏÓ˚ ¢ÇÖƒy x.

¢ÇÖƒy!›˛ 4 Ü%í = 4x

4x ˆÌ ÏÑ˛ 7 ¢!Ó˚ˆÏÎ˚ !òˆÏú, !õì˛yÓ˚ ÌyˆÏÑ˛ 4x - 7.

¢%ì˛Ó˚yÇ ≤ÃŸ¿yò%¢yˆÏÓ˚ñ 4x - 7 = 17

~£z¶˛y ÏÓ x ~Ó˚ õyò ˛ôyÁÎ˚y ÎyÎ˚–

~ÓyÓ˚ ¢õ¢ƒy›˛yÓ˚ ¢õyïyò Ñ˛Ó%ò–

4x - 7 = 17

7174 +=⇒ x

244 =⇒ x

4

24x =⇒

6x =⇒

¢%ì˛Ó˚yÇ ¢ÇÖƒy!›˛ £ú 6

v˛z_Ó˚ Îyâ˛y£z Ñ˛Ó˚%ò ı

Óyõ˛ô«˛ 4x – 7 = 4 × 6 – 7 = 24 – 7 = 17 = v˛yò˛ô«˛ ≤Ã ÏÎ˚yãòyò%ÎyÎ˚#

ˆòy›˛

174 D.El.Ed. 504


v˛zîy£Ó˚í 34 !¶˛!Ñ˛ !Ó˚!Ñ˛Ó˚ ˆÌˆÏÑ˛ 5 ÓäÈˆÏÓ˚Ó˚ ÓˆÏv˛¸y– 15 ÓäÈÓ˚ xy ÏÜ !¶˛!Ñ˛Ó˚ ÓÎ˚¢ !Ó˚!Ñ˛Ó˚ ÓÎ˚ Ï¢Ó˚!mÜ%í !äÈú– ì˛y ÏîÓ˚ Óì≈ õyò ÓÎ˚¢ Ñ˛ì˛⁄

¢õyïyò ı

ì˛yˆÏîÓ˚ õˆÏïƒ !Ó˚!Ñ˛ ˆäÈyˆÏ›˛yñ ì˛y£z ïÓ˚y ÎyÑ˛ !Ó˚!Ñ˛Ó˚ Óì≈˛õyò ÓÎ˚¢ £ú z.

!¶˛!Ñ˛Ó˚ Óì≈˛õyò ÓÎ˚¢ = (z + 5) ÓäÈÓ˚

15 ÓäÈÓ˚ xy ÏÜ !¶˛!Ñ˛Ó˚ ÓÎ˚¢ !äÈú z + 5 – 15 = z – 10 ÓäÈÓ˚

~ÓÇ ì˛Öò !Ó˚!Ñ˛Ó˚ ÓÎ˚¢ !äÈú (z – 15) ÓäÈÓ˚

≤ÃŸ¿yò%¢yˆÏÓ˚ñ

!¶˛!Ñ˛Ó˚ ÓÎ˚¢ £ú !Ó˚!Ñ˛Ó˚ ÓÎ˚ˆÏ¢Ó˚ !mÜ%í–

⇒ z – 10 = 2 (z - 15)

⇒ z – 10 = 2z – 30

⇒ z – 2z = -30 + 10

⇒ z = 20

¢%ì˛Ó˚yÇ !Ó˚!Ñ˛Ó˚ Óì≈˛õyò ÓÎ˚¢ = 20 ÓäÈÓ˚ ~ÓÇ !¶˛!Ñ˛Ó˚ ÓÎ˚¢ = 20 + 5 = 25 ÓäÈÓ˚


E16. î%!›˛ ¢ÇÖƒyÓ˚ ˆÎyÜö˛ú £ú 64. ~Ñ˛!›˛ £ú xòƒ!›˛Ó˚ ˆÌˆÏÑ˛ 14 ˆÓ!ü– ¢ÇÖƒyÜ%!ú !Ñ˛ !Ñ˛⁄

E17. ¢õ¢ƒyÓ˚ ¢õyïyò Ñ˛Ó%ò–

(Ñ˛) ¢!â˛ò ˆïy!òÓ˚ !mÜ%ˆÏíÓ˚Á ˆÓ!ü Ó˚yò Ñ˛ˆÏÓ˚– ì˛yˆÏîÓ˚ ~Ñ˛ˆÏe Ó˚yò ~Ñ˛ˆÏüyÓ˚ ˆÌˆÏÑ˛ ~Ñ˛

Ñ˛õ– ì˛yÓ˚y ≤ÃˆÏì˛ƒˆÏÑ˛ Ñ˛ì˛ Ñ˛ˆÏÓ˚ Ó˚yò Ñ˛ˆÏÓ˚!äÈú⁄

(Ö) òÓ˚£!Ó˚˛ô%ˆÏÓ˚Ó˚ ˆúyˆÏÑ˛Ó˚y 102 !›˛ ÜyäÈ ì˛y ÏîÓ˚ @˘Ãy ÏõÓ˚ ÓyÜy Ïò úy!Ü ÏÎ˚!äÈú– !ò‹≥˛ ÏúyÑ˛

Üy ÏäÈÓ˚ ¢ÇÖƒy ö˛úhs˘˛ Üy ÏäÈÓ˚ ¢ÇÖƒyÓ˚ 3 !ì˛ò Ü%ˆÏíÓ˚ ˆÌˆÏÑ˛ 2 ˆÓ!ü– ö˛úhs˘˛ Üy ÏäÈÓ˚

¢ÇÖƒy Ñ˛ì˛⁄

(Ü) ¢y!òÎ˚yÓ˚ ÓÎ˚¢ ì˛yÓ˚ ÓyÓyÓ˚ ÓÎ˚ˆÏ¢Ó˚ xˆÏï≈Ñ˛– ~ÓÇ ì˛yÓ˚ ÓyÓyÓ˚ ÓÎ˚¢ ì˛yÓ˚ îyî%Ó˚ ÓÎ˚ˆÏ¢Ó˚

xˆÏï≈Ñ˛– 20 ÓäÈÓ˚ ˛ôÓ˚ ¢y!òÎ˚yÓ˚ ÓÎ˚¢ ì˛yÓ˚ ÓyÓyÓ˚ Óì≈ õyò ÓÎ˚ Ï¢Ó˚ ¢õyò £ ÏÓ– ì˛y£ Ïú

¢y!òÎ˚yÓ˚ñ ì˛yÓ˚ ÓyÓyÓ˚ ~ÓÇ ì˛yÓ˚ îyî%Ó˚ ÓÎ˚¢ Ñ˛ì˛⁄

ˆòy›˛



E18. ¢õyïyò Ñ˛Ó˚ ı

2 3 5

3 7 8

x

x

+ =−

9.7 ¢Ç!«˛ÆÑ˛Ó˚í

● Ó#ãÜ!íì˛ £ú ˛ôy›˛#Ü!íˆÏì˛Ó˚ ~Ñ˛!›˛ Óƒy˛ôÑ˛ Ó˚)ˆÏ˛ô ˆÎÖyˆÏò Óí≈ˆÏÑ˛ ¢ÇÖƒyÓ˚ ˛ô!Ó˚ÓˆÏì≈˛ ≤Ãì˛#Ñ˛!£ˆÏ¢ˆÏÓ ÓƒÓ£yÓ˚ Ñ˛Ó˚y £Î˚– ≤ÃˆÏì˛ƒÑ˛ ¢ÇÖƒy£z £ú ï %ÓÑ˛ ~ÓÇ ≤ÃˆÏì˛ƒÑ˛ ≤Ãì˛#Ñ˛ !Ó!¶˛ß¨ ˛ô!Ó˚!fiÌ!ì˛ˆÏì˛!Ó!¶˛ß¨ õyò @˘Ã£í Ñ˛ ÏÓ˚–

● Ó#ã Üy!í!ì˛Ñ˛ Ó˚y!üõyúy ≤Ãì˛#Ñ˛ ~ÓÇ ï %ÓÑ˛ !îˆÏÎ˚ Ü!‡˛ì˛ xyõÓ˚y â˛yÓ˚!›˛ õ)ú ≤Ã!Ñ Î˚y ~£z ≤Ãì˛#Ñ˛~ÓÇ ï %ÓˆÏÑ˛Ó˚ v˛z˛ôÓ˚ ÓƒÓ£yÓ˚ Ñ˛ˆÏÓ˚ Ìy!Ñ˛– ˛ôî £ú Ó˚y!üõyúyÓ˚ ~Ñ˛!›˛ xÇü ÎyÓ˚ ‘+’ xÌÓy ‘_’

!â˛£´Ó˚ myÓ˚y ˛ô,ÌÑ˛#Ñ,˛ì˛– ~£z ˛ôî ï %ÓÑ˛ Óy â˛úÑ˛ v˛z¶˛Î˚£z £ˆÏì˛ ˛ôyˆÏÓ˚–

● ~Ñ˛ Óy ˆÓü# ˛ôî myÓ˚y Ü!‡˛ì˛ Ó˚y!üõyúyˆÏÑ˛ Ó£%˛ôî Ó˚y!ü Óúy £Î˚– ≤ÃÑ,˛ì˛˛ôˆÏ«˛ ~Ñ˛˛ôîÎ%=˛Ó˚y!üˆÏÑ˛ Óúy £Î˚ ~Ñ˛˛ôîÎ%=˛ Ó˚y!üñ î%ˆÏ›˛y ˛ôî Î%=˛ Ó˚y!üˆÏÑ˛ Óúy £Î˚ !m˛ôî Ó˚y!ü ~ÓÇ ˆÑ˛yˆÏòyÓ£% ôî Ó˚y!üÓ˚ xãyòy Ó˚y!üÓ˚ ¢ ÏÓ≈yFâ˛ áyì˛ ÏÑ˛ Óúy £Î˚ Ó£% ôî Ó˚y!üÓ˚ õyey–

● ~Ñ˛!›˛ ¢õ#Ñ˛Ó˚ˆÏí î%!›˛ ˛ô«˛ ~Ñ˛!›˛ ì%˛úy ÎˆÏs˛f î%!›˛ ˛ôyˆÏeÓ˚ ¢ˆÏAÜ ì%˛úòy Ñ˛Ó˚y ÎyÎ˚–

● ~Ñ˛áyì˛ ¢õ#Ñ˛Ó˚í ¢õyïyò â˛yÓ˚!›˛ ˛ôÂï!ì˛ ˆÎ ˆÑ˛yˆÏòy ~Ñ˛!›˛ myÓ˚y Ñ˛Ó˚y ˆÎˆÏì˛ ˛ôyˆÏÓ˚ñ ˛ôÓ˚#«˛í˛ôÂï!ì˛ñ ˆÎyÜñ !ÓˆÏÎ˚yÜ ˛ôÂï!ì˛ñ ˛ô«˛yhs˘˛Ó˚ ˛ôÂï!ì˛ñ ÓLÜ%íò ˛ôÂï!ì˛

9.8 x@˘ÃÜ!ì˛ Îyâ˛y£z Ñ˛Ó%ò

E1. p + q, 2p – q, 3p + 2q – 1 xÌÓy ˆÎˆÏÑ˛yˆÏòy !ì˛ò!›˛ ~Ñ˛Ó˚Ñ˛õ Ó˚y!üõyúy

E2. (i) 5xy + 3 (ii) ab - (a + b)

E3. â˛úÑ˛ = y & z, ï %ÓÑ˛ = 2, 3, 5

E4. (i) -1, (ii) 2y (iii) xy3

2

E5. ⎟⎠⎞

⎜⎝⎛

3

p,p2 , (-3pq, 5pq), ⎟

⎠⎞

⎜⎝⎛

pqr3,pqr2

1

E6. (i) xy ~Ó˚ ¢£Ü = 5 , xÌÓy 5 = xy £zì˛ƒy!î

(ii) 3 ~Ó˚ ¢£Ü £ú = – abc, xÌÓy abc = –3 £zì˛ƒy!î

ˆòy›˛

176 D.El.Ed. 504


E7. (i) ˛ôî = 3xy, 5y ; 3xy ~Ó˚ v zÍ˛ôyîÑ˛ = 3, x, y ; 5y ~Ó˚ v zÍ˛ôyîÑ˛ = 5, y

(ii) ˛ôî = ab, 2a ~ÓÇ 3y

E8. 4x2 – 4xy – 6y2

E9. - m+ 6n + 6

E10. 2a2 + 7ab + 3a - 5b – 5

E11. (a) 6x2 + 5x – 6 (b) p3 – q3

E12. (i) 3a2 + 4a + 5

E13. (i) x = 4 (ii) y = 7 (iii) x = 18

E14. (i) t = 3 (ii) p = 3

E15. (i) x = 2 (ii) x = 11

E16. 25 ~ÓÇ 39

E17. (a) ˆïyò# = 33, ¢!â˛ò = 66 (b) 25 (c) 20 ,40, 80

E18. x = 11

9.9 SUGGESTED READINGS AND REFERENCES

Bansal, R.K. (2007). Middle school mathematics. New Delhi: Selina Publication.

NCTM (1999). Activities for Junior High School and Middle School Mathematics,

Vol.2, The National Council of Teachers of Mathematics(NCTM), INC, USA.

Teaching of Maths at Upper Primary Level, Vol.- II Published by DEP-SSA, IGNOU,

New Delhi

Text Book of Cl- VI, Vii and VIII published by NCERT, New Delhi.

9.10 ~Ñ˛Ñ˛ ˆü Ï°ÏÓ˚ ≤ÃŸ¿Óú#

1. !ò¡¨!ú!Öì˛Ü%!úÓ˚ Ó˚y!üõyúy !úÖ%ò–

ˆòy›˛



(i) p Á q ~Ó˚ ˆÎyÜö˛ˆÏúÓ˚ ~Ñ˛ ì,˛ì˛#Î˚yÇü

(ii) a ~ÓÇ b ˆÎyÜö˛ú ì˛yˆÏîÓ˚ Ü%íö˛ú ˆÌˆÏÑ˛ !ÓˆÏÎ˚yÜ

(iii) x Á y ~Ó˚ Ü%íö˛ˆÏúÓ˚ î%Ü%ˆÏíÓ˚ ¢yˆÏÌ 8 ˆÎyÜ Ñ˛Ó%ò–

2. ¢£Ü !òí≈Î Ñ˛Ó%ò– (i) of xy in – 5xyz, (ii) of m2 in 2 m2n2 (iii) 9 in – 9 pqr.

3. ¢yÇ!ÖƒÑ˛ ¢£Ü ¢òy=˛ Ñ˛Ó%ò– (i) –t (ii) pq3

2 (iii) – 8 x2y2

4. ~Ñ˛!›˛ v˛zîy£Ó˚í !îò (i) ~Ñ˛˛ôî# Ó˚y!ü (ii) !m˛ôî# Ó˚y!ü (iii) !e˛ôî# Ó˚y!ü

5. ¢õ˛ôîÜ%!ú !úÖ%ò–

(i) ab2, – 4ab, 2a2b, ab, – 3ab2, 3

2ab, –5a2b, a2b2

(ii) 2x, –5xy, –x, 2

xy, 3y

6. !òˆÏâ˛Ó˚ Ó˚y!üÜ%!ú ˆÎyÜ Ñ˛Ó%ò–

(i) a + b – 5, b – a + 3 ~ÓÇ a – b + 6

(ii) 4x + 3y – 7xy, 3xy – 2x ~ÓÇ 2xy – y

(iii) m2 – n2 – 1, n2 – 1 – m2 ~ÓÇ 1 – m2 – n2

7. !ÓˆÏÎ˚yÜ Ñ˛Ó˚%ò ı

(i) y (3 – x) ˆÌˆÏÑ˛ x (y – 3)

(ii) 5m – 10 Ì ÏÑ˛ m2 + 10m – 5

(iii) 2ab – 3a2 – 3b2 ˆÌˆÏÑ˛ 5a2 – 7ab + 5b2

8. ¢Ó˚ú Ñ˛Ó%ò ı

(i) 10x2 – 8x + 5 + -5x – 4x2 – 6m – 10

(ii) 20mn – 10n – 17m – 12n + 14m + 2

9 Ü%í Ñ˛Ó%ò ı

(i) (a – b) × (a2 + ab + b2)

(ii) (p + q – 5) × (p - q)

ˆòy›˛

178 D.El.Ed. 504


10. ¶˛yÜ Ñ˛Ó˚%ò ı

(i) (8m2 + 4m – 60) ÷ (2m – 5)

(ii) (6 a2b2 – 7abc – 3b2) ÷ (3ab + c)

11. ¢õyïyò Ñ˛Ó%ò ı

(i) 2y – 5 = 9 (ii)5

13

5

3 =+ x

(iii) 4053=+ zz

(iv) 3

4

36

8 −=+ x

x

12. !ò Ïâ˛Ó˚ ¢õ¢ƒyÜ%!úÓ˚ ¢õyïyò Ñ˛Ó%ò ı

(i) xyÎ˚ì˛yÑ˛yÓ˚ ~Ñ˛!›˛ ã!õÓ˚ ˜îá≈ƒñ ≤Ã ÏfiÌÓ˚ ˆÌ ÏÑ˛ 5m ˆÓ!ü– ã!õ!›˛Ó˚ ˛ô!Ó˚¢#õy 70m–

ã!õ!›˛Ó˚ ˜îá≈ƒ !òí≈Î˚ Ñ˛Ó%ò–

(ii) ~Ñ˛!›˛ ¢õ!mÓy£% !e¶%˛ˆÏãÓ˚ ¶)˛!õfiÌ ˆÑ˛yíÜ%!ú ¢õyò– ü#°≈ˆÏÑ˛yí!›˛ ≤ÃˆÏì˛ƒÑ˛!›˛ ¶)˛!õfiÌ

ˆÑ˛yí ˆÌ ÏÑ˛ 15° ˆÓ!ü– !e¶% ã!›˛Ó˚ ˆÑ˛yíÜ%!ú !òí≈Î˚ Ñ˛Ó%ò–

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