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10. Á¤š¦·„Žr¨³—¸Áš°¦¤r œ· œ´ šr 20 æŠÁ¦¸¥œ­»¦µ¬‘¦›r µœ¸
11. Áª„Á˜°¦r 2 ¨³ 3 ¤˜· · 36 æŠÁ¦¸¥œ¡»œ¡š· ¥µ‡¤
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13. š§¬‘„¸ ¦µ¢ 15 æŠÁ¦¥¸ œ¼¦–³¦µÎ ¨„¹

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16. „µ¦Á¦¥¸ Š­´ Áž¨É¥¸ œÂ¨³„µ¦‹´—®¤¼n 30 æŠÁ¦¥¸ œÁ˜¦¥¸ ¤°—» ¤«„¹ ¬µ£µ‡Ä˜o

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ƒ „µ¦œÎµÁ­œ°…°o ¤¼¨Â¨³‡µn „¨µŠ (12 ‡µ) æŠÁ¦¥¸ œ¼¦–³¦µÎ ¨¹„

ƒ „µ¦„¦³‹µ¥…°Š…o°¤¼¨ (25 ‡µ) æŠÁ¦¥¸ œ­»¦µ¬‘¦›r µœ¸

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1
Á¦°ºÉ Š 7
Ÿœ„µ¦‹—´ „µ¦Á¦¸¥œ¦¼ošÉ¸ 1 Á¦Éº°Š ‹Îµœªœ‹¦Š· 8
Á°„­µ¦Âœ³ÂœªšµŠš¸É 1 9
Á°„­µ¦ „f ®´—šÉ¸ 1 11
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Ÿœ„µ¦‹—´ „µ¦Á¦¥¸ œ¦oš¼ ɸ 2 Á¦ºÉ°Š ­¤˜´ ·…°Š¦³‹µÎ œªœ‹¦Š· 17
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 2.1 18
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 2.2 19
Á°„­µ¦Âœ³ÂœªšµŠš¸É 2.3 20
Á°„­µ¦Âœ³ÂœªšµŠš¸É 2.4 22
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 2.5 24
Á°„­µ¦ „f ®´—š¸É 2 25
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 2.6 27
Á°„­µ¦‡ªµ¤¦¼oš¸É 2 28
Á°„­µ¦ f„®´—Á¡É¤· Á˜·¤ 30
Ÿœ„µ¦‹´—„µ¦Á¦¸¥œ¦š¼o ɸ 3 Á¦ºÉ°Š „µ¦œÎµ­¤´˜…· °Š‹Îµœªœ‹¦·ŠÅžÄÄo œ„µ¦¡­· ‹¼ œr 33
Á°„­µ¦Âœ³ÂœªšµŠš¸É 3 35
Ÿœ„µ¦‹—´ „µ¦Á¦¥¸ œ¦oš¼ ¸É 4 Á¦ºÉ°Š „µ¦¨Â¨³„µ¦®µ¦‹Îµœªœ‹¦·Š 36
Á°„­µ¦Âœ³ÂœªšµŠš¸É 4.1 38
Á°„­µ¦Âœ³ÂœªšµŠš¸É 4.2 40
Ÿœ„µ¦‹´—„µ¦Á¦¥¸ œ¦¼šo ɸ 5 Á¦ºÉ°Š š§¬‘¸šÁ«¬Á®¨º° 41
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 5.1 43
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 5.2 46
Ÿœ„µ¦‹—´ „µ¦Á¦¸¥œ¦oš¼ ɸ 6 Á¦Éº°Š „µ¦Â„o­¤„µ¦˜´ªÂž¦Á—¥¸ ª 47
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 6.1 48
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 6.2 51
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 6.3 56
Ÿœ„µ¦‹—´ „µ¦Á¦¸¥œ¦š¼o ɸ 7 Á¦É°º Š ­¤´˜·„µ¦Å¤nÁšµn 57
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 7.1 58
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 7.2 59
Á°„­µ¦ „f ®´—šÉ¸ 7
Á°„­µ¦‡ªµ¤¦¼ošÉ¸ 7

Ÿœ„µ¦‹—´ „µ¦Á¦¸¥œš¸É 8 Á¦Éº°Š nªŠ 2
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 8
Á°„­µ¦ „f ®—´ šÉ¸ 8.1 60
Á°„­µ¦ f„®´—š¸É 8.2 64
Ÿœ„µ¦‹´—„µ¦Á¦¥¸ œ¦¼šo ¸É 9 Á¦°Éº Š „µ¦Â„°o ­¤„µ¦ 65
Á°„­µ¦ „f ®—´ šÉ¸ 9 66
Ÿœ„µ¦‹—´ „µ¦Á¦¥¸ œ¦šo¼ ɸ 10 Á¦°ºÉ Š „µ¦Â„o°­¤„µ¦(˜n°) 67
Á°„­µ¦Âœ³ÂœªšµŠš¸É 10 73
Á°„­µ¦‡ªµ¤¦¼ošÉ¸ 10 74
Ÿœ„µ¦‹—´ „µ¦Á¦¥¸ œ¦o¼š¸É 11 Á¦ºÉ°Š ‡µn ­¤´ ¦¼ –r 78
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 11.1 80
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 11.2 82
Á°„­µ¦ f„®—´ š¸É 11.1 87
Á°„­µ¦ „f ®´—šÉ¸ 11.2 90
Ÿœ„µ¦‹´—„µ¦Á¦¥¸ œ¦oš¼ ¸É 12 Á¦Éº°Š ‡µn ­¤´ ¦¼ –r(˜°n ) 91
Á°„­µ¦ f„®´—šÉ¸ 12 92
Ÿœ„µ¦‹´—„µ¦Á¦¥¸ œ¦o¼š¸É 13 Á¦º°É Š ­´‹¡‹œr…°Š‡ªµ¤¦· ¼¦–r 93
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 13 97
Á°„­µ¦ „f ®—´ š¸É 13 98
Ÿœ„µ¦‹—´ „µ¦Á¦¸¥œ¦¼šo ɸ 14 Á¦º°É Š „µ¦®µ¦¨Š˜´ª 102
Á°„­µ¦Âœ³ÂœªšµŠš¸É 14 103
Á°„­µ¦ „f ®´—š¸É 14 104
Ÿœ„µ¦‹—´ „µ¦Á¦¥¸ œ¦šo¼ ¸É 15 Á¦°Éº Š ­¤´˜…· °Š‹ÎµœªœÁ˜È¤ (˜°n ) 109
Ÿœ„µ¦‹´—„µ¦Á¦¸¥œ¦¼ošÉ¸ 16 Á¦Éº°Š ˜´ª®µ¦¦nª¤¤µ„(®.¦.¤.) 110
Á°„­µ¦ „f ®´—š¸É 16 111
Ÿœ„µ¦‹—´ „µ¦Á¦¸¥œ¦¼oš¸É 17 Á¦ºÉ°Š ˜ª´ ‡¼–¦nª¤œ°o ¥(‡.¦.œ) 116
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123
125

⌦ 1
⌦

Ÿœ„µ¦‹´—„µ¦Á¦¥¸ œ¦oš¼ ¸É 1

Á¦Éº°Š ‹µÎ œªœ‹¦·Š Êœ´ ¤›´ ¥¤«¹„¬µžešÉ¸ 4
ª·µ ‡–·˜«µ­˜¦r Áª¨µ 2 ´ªÉ äŠ

********************************************************** * * * * * * * * * * * * * * * * * * * * * *
Ÿ¨„µ¦Á¦¥¸ œ¦¼oš‡É¸ µ—®ª´Š

¤¸‡ªµ¤‡·—¦ª¥°—Á„¥¸É ª„´ ¦³‹Îµœªœ‹¦·Š

1. ‹—» ž¦³­Š‡„r µ¦Á¦¸¥œ¦¼o œ´„Á¦¥¸ œ­µ¤µ¦™
1.1 Á…¸¥œÁŽ˜…°Š‹Îµœªœœ´ ŗo
1.2 Á…¸¥œÁŽ˜…°Š‹ÎµœªœÁ˜¤È ŗo
1.3 Á…¸¥œÁŽ˜…°Š‹Îµœªœ˜¦¦„¥³Å—o
1.4 °„Å—ªo µn ‹µÎ œªœ‹¦Š· ėÁžœ} ‹µÎ œªœ˜¦¦„¥³®¦º°°˜¦¦„¥³
1.5 °„Å—ªo nµ‹µÎ œªœšÉ„¸ µÎ ®œ—Ä®Áo žœ} ‹Îµœªœœ—· ė
1.6 °„‡ªµ¤­´¤¡´œ›r…°ŠÁŽ˜…°Š‹Îµœªœ˜nµŠÇ ŗo

2. œª‡ªµ¤‡—· ®¨´„
¦³‹µÎ œªœ‹¦Š· Áž}œÃ‡¦Š­¦oµŠšµŠ‡–·˜«µ­˜¦r ž¦³„°—ªo ¥ÁŽ˜…°Š‹Îµœªœ‹¦·Š „´Ã°Áž°Á¦´Éœ

ª„¨³‡–¼ š¸­É °—‡¨°o Š„´ ­´‹¡‹œr 15 …o°

3. Áœº°Ê ®µ­µ¦³
ÁŽ˜…°Š‹ÎµœªœšÉ¸Ážœ} ­´ ÁŽ˜…°Š‹µÎ œªœ‹¦·Š ¨³‡ªµ¤­´¤¡´œ›r…°Š‹µÎ œªœœ—· ˜µn ŠÇ Ĝ¦¼žÂŸœŸ´Š

4. „¦³ªœ„µ¦‹´—„µ¦Á¦¸¥œ¦o¼
4.1 ššªœ‡ªµ¤Â˜„˜nµŠ…°Š‡ªµ¤®¤µ¥…°Š‹Îµœªœ„´˜´ªÁ¨… ˜µ¤š¸Éœ´„Á¦¸¥œÁ‡¥Á¦¸¥œ¤µÂ¨oªÄœ

Ê´œ¤´›¥¤«¹„¬µ˜°œ˜oœ ¡¦o°¤š´ÊŠÁ¨nµž¦³ª´˜·‡ªµ¤Áž}œ¤µ…°Š˜´ªÁ¨…¨³‹Îµœªœ¡°­´ŠÁ…ž (ץĮo
œ´„Á¦¸¥œš„» ‡œ°nµœ‹µ„®œŠ´ ­°º Á¦¥¸ œ­µ¦³„µ¦Á¦¸¥œ¦oÁ¼ ¡·É¤Á˜·¤ ‡–·˜«µ­˜¦r Á¨n¤ 1 Êœ´ ¤›´ ¥¤«¹„¬µžše ¸É 4 )

4.2 Ä®oœ´„Á¦¸¥œnª¥„´œ¥„˜´ª°¥nµŠ˜´ªÁ¨…šÉ¸œ´„Á¦¸¥œÁ‡¥Á¦¸¥œ˜´ÊŠÂ˜nÊ´œ¤´›¥¤«¹„¬µ˜°œ˜oœ ×¥„µ¦
™µ¤˜° ¨oª‡¦¼Á…¸¥œœ„¦³—µœ—ε ‹œ„¦³š´ÉŠÅ—o˜´ªÁ¨…Äœ¦¼žÂ˜nµŠÇ „´œ ŽÉ¹Šœ´„Á¦¸¥œ‡ª¦˜°Å—o‡¦
š„» ¦ž¼  ‡°º ‹µÎ œªœÁ˜È¤ª„®¦º°‹Îµœªœœ´, ‹ÎµœªœÁ˜È¤¨, «¼œ¥r, Á«¬­nªœ, š«œ·¥¤ŽÊε, š«œ·¥¤Å¤nŽÎʵ,

‹µÎ œªœšÉ˜¸ —· Á‡¦°Éº Š®¤µ¥„¦–”r Ánœ 3, 3, 2 , 0, 0.5, 0. 5, 0.14253948. , 2, 3 5 Ážœ} ˜œo

3

4.3 ‡¦¼Äoª·›¸„µ¦™µ¤˜° Á¡ºÉ°Ä®oœ´„Á¦¸¥œÂ¥„ž¦³Á£š‹Îµœªœš¸Éŗo‹µ„…o° 1 °°„Áž}œ 2 „¨n»¤Ž¹ÉŠ
œ„´ Á¦¸¥œ‡ª¦˜°Å—oªµn

2 ⌫ ⌫  ⌦
 ⌫     ⌫  

⌦ 3
⌦

4 ⌫ ⌫  ⌦
 ⌫     ⌫  

6. 3 0 ¤¸‡nµÁšnµÅ¦ 3
7. 3 u 0 ¤¸‡µn Ášµn Ŧ 0

8. 3 ¤¸‡µn ÁšnµÅ¦ (œ´„Á¦¸¥œ°µ‹˜° 0 ®¦º° 3 ®¦°º °¥µn Š°Éœº ŽÉŠ¹ ‡¦¥¼ ´ŠÅ¤Án Œ¨¥‡µÎ ˜°šÉ¸™„¼ )

0

3 ¤‡¸ µn Ášµn Ŧ 1

9.
3

5 ¤¸‡µn Ášµn Ŧ 1

10.
5

0 ¤¸‡µn ÁšnµÅ¦ (œ„´ Á¦¸¥œ°µ‹˜° 0 ®¦º° 1 ®¦°º °¥µn Š°ºœÉ ŽÉŠ¹ ‡¦¼¥Š´

11.
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8 Áž}œ‹¦·Š®¦°º Áš‹È (‹¦·Š)
4 Ážœ} ‹¦·Š®¦°º Áš‹È (‹¦Š· )

2

8 4u2

Á¡É°º šÉ‹¸ ³®µ‡Îµ˜°…°Š 3 ªnµ¤‡¸ µn Ášµn Ŧ ¨oª‡¼–„´«¼œ¥rŗoŸ¨¨´¡›r 3 Ž¹ÉŠŸ¼oÁ¦¸¥œ‹³

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0

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‹³˜°Å—®o ¨µ¥‡µn Ž¹ÉŠÁžœ} ‡Îµ˜°š¸É™„¼ š»„‡nµ

­—Šªµn 0 Áž}œ¦¼žÂš¸¥É Š´ Ťn„ε®œ— (Indeterminate form)
0

—´ŠœœÊ´ ‹ÎµœªœšÉÁ¸ …¸¥œÄœ¦ž¼ a Á¤ºÉ° a,b Ážœ} ‹ÎµœªœÁ˜È¤ ˜ª´ ®µ¦‡°º b ‹³Áž}œ«œ¼ ¥År ¤Ån —o
b

4.10 ‡¦¼Â‹„Á°„­µ¦Âœ³ÂœªšµŠš¸É 1 Á¦Éº°Š„µ¦Áž¨¸É¥œš«œ·¥¤Ä®oÁž}œÁ«¬­nªœ Ä®oœ´„Á¦¸¥œšÎµÁž}œ

„µ¦oµœ

4.11 ‡¦Â¼ ¨³œ„´ Á¦¥¸ œnª¥„œ´ ÁŒ¨¥Á°„­µ¦Âœ³ÂœªšµŠš¸É 1 ×¥„µ¦™µ¤˜° Á¡Éº°®µ…o°­¦»ž„µ¦šÎµ

š«œ·¥¤ŽÎʵĮoÁž}œÁ«¬­nªœ Áž}œ„µ¦ššªœšÉ¸Á‡¥Á¦¸¥œ¤µÂ¨oªÄœÊ´œ ¤.3 Ž¹ÉŠ­¦»ž Áž}œª·›¸­´ÊœÇ ŗo—´ŠœÊ¸

⌦ 5
⌦

6 ⌫ ⌫  ⌦
 ⌫     ⌫  

5. ®¨nŠ„µ¦Á¦¥¸ œ¦¼o
5.1 Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 1
5.2 Á°„­µ¦ f„®—´ š¸É 1
5.3 ŸœŸŠ´ ­—Š‡ªµ¤­¤´ ¡œ´ ›r…°Š‹Îµœªœœ—· ˜µn ŠÇ
5.5 ®œŠ´ ­º°Á¦¥¸ œ­µ¦³„µ¦Á¦¥¸ œ¦Á¼o ¡É¤· Á˜¤· ‡–˜· «µ­˜¦r Á¨¤n 1 ´Êœ¤´›¥¤«„¹ ¬µžše ¸É 4 ­­ªš.
5.6 š—­°

6. „¦³ªœ„µ¦ª—´ ¨³ž¦³Á¤·œŸ¨ „µ¦ž¦³Á¤œ· Ÿ¨
1. œ„´ Á¦¸¥œ˜°‡µÎ ™µ¤Å—™o ¼„˜°o ŠÁž}œ­ªn œ¤µ„
„µ¦ª´—Ÿ¨ 2. œ„´ Á¦¸¥œÄ®o‡ªµ¤­œÄ‹—¸ ˜´ÊŠÄ‹Á¦¸¥œ
1. ­´ŠÁ„˜‹µ„„µ¦˜°‡Îµ™µ¤ 3. œ„´ Á¦¸¥œšµÎ ŗ™o „¼ ˜o°Šž¦³¤µ– 90 %
2. ­Š´ Á„˜‹µ„‡ªµ¤­œÄ‹ 4. œ„´ Á¦¥¸ œšµÎ ŗ™o ¼„˜o°Šž¦³¤µ– 85 %
3. šÎµÁ°„­µ¦Âœ³ÂœªšµŠšÉ¸ 1 5. œ„´ Á¦¥¸ œšÎµÅ—™o „¼ ˜o°Šž¦³¤µ– 90 %
4. šµÎ ˚¥Ár °„­µ¦ „f ®´—š¸É 1
5. šÎµÂš—­°

7. œ´ š„¹ ®¨´Š­°œ
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8. „·‹„¦¦¤Á­œ°Âœ³
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⌦ 7
⌦

Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 1

‡µÎ ­ŠÉ´ ‹ŠÁž¨É¸¥œ‹Îµœªœ˜n°ÅžœÄʸ ®Áo žœ} Á«¬­ªn œ

1. 0.252525… 0.252525… = x ………( 1 )
ª·›š¸ ε Ä®o 25.252525… = 100 x ……….( 2 )
( 1 ) u100 ;

( 2 )-( 1 ) ; 25 = 99x

x = ………..
œÉœ´ ‡°º 0.252525… = ……….

2. 0.453 26 0.45326326… = x ……….( 1 )
ª·›¸šÎµ Ä®o

( 1 ) u 100 ; …………………….. = …………………( 2 )

( 1 ) u 100000; ………………….. = ………………….( 3 )

( 3 ) – ( 2 ) ………………………… = …………………………

x = …………………………

œœ´É ‡º° 0.45326326… = …………………………

…°o ­´ŠÁ„˜ 0.45326326… = =45326 45 45281

99900 99900

3. 2.354

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4. 4.9999…

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8 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦ „f ®—´ šÉ¸ 1

1. ‹Š¡‹· µ¦–µªµn ‹ÎµœªœÄ—Áž}œ­¤µ·„…°ŠÁŽ˜Ä— ¨oªÁ…¸¥œÁ‡¦º°É Š®¤µ¥ — ¨ŠÄœ°n ŠÁŽ˜œÊ´œ

‹Îµœªœ ‹µÎ œªœœ´ ‹ÎµœªœÁ˜¤È ‹ÎµœªœÁ˜¤È ‹µÎ œªœ ‹Îµœªœ ‹Îµœªœ‹¦·Š
N I ¨ I- ˜¦¦„¥³ Q °˜¦¦„¥³ Q/ R
0
5
6
2

1
3

2

- 9 1

( 4)2

6 16
3 8 2

1.333…

3.999…

22
7

2. …o°‡ªµ¤˜°n ŞœÁʸ žœ} ‹¦Š· ®¦º°Áš‹È
……… 2.1 3 Ťnčn‹µÎ œªœ°˜¦¦„¥³
……… 2.2 3.999…  I
……… 2.3 4  R
……… 2.4 5  Q/
……… 2.5 0.3033033303… Ážœ} ‹µÎ œªœ˜¦¦„¥³
……… 2.6 0.303300330003… Áž}œ‹Îµœªœ°˜¦¦„¥³
……… 2.7 2 , 3, 4 , 5 ¨³ 7 Áž}œ‹Îµœªœ°˜¦¦„¥³šÊ´Š®¤—
……… 2.8 1.1 54 3.667 8 Áž}œ‹µÎ œªœ˜¦¦„¥³
……… 2.9 ™µo a Ážœ} ‹µÎ œªœ°˜¦¦„¥³Â¨oª a2 ‹³Áž}œ‹µÎ œªœ˜¦¦„¥³

……… 2.10 Q ˆ Qc I

⌦ 9
⌦

š—­°

‡Îµ­Š´É ‹ŠÁ¨°º „…°o š¸É™„¼ šÉ¸­»—Á¡¥¸ Š…°o Á—¥¸ ª ¨ªo šµÎ Á‡¦ºÉ°Š®¤µ¥ (u) Ä®o˜¦Š„´…o°šÁɸ ¨°º „
1. ‹ÎµœªœÄœ…°o ėÁž}œ‹Îµœªœ˜¦¦„¥³

„. 4.6394…
…. 1.458458458…
‡. 1.3244235…
Š. 0.112111211112…
2. ‹ÎµœªœÄœ…°o ėŤÄn ‹n 圪œ‹¦Š·
„. 2S
…. 2
‡. ( 3)2
Š. 4
3. …°o ėÁž}œ‹¦·Š
„. ™oµ x Ážœ} ‹µÎ œªœ‹¦Š· ¨oª x>0 ®¦º° x<0
…. ™µo x Áž}œ‹µÎ œªœ˜¦¦„¥³Â¨ªo x Ážœ} ‹µÎ œªœ°˜¦¦„¥³
‡. ™oµ x Áž}œ‹µÎ œªœ‡Ân¼ ¨ªo x2 Ážœ} ‹Îµœªœ‡n¼
Š. ÁŽ˜…°Š‹µÎ œªœ‡Áɸ žœ} ‹Îµœªœ°˜¦¦„¥³
4. …o°Ä—Ť‹n ¦Š·
„. 4.999… Ážœ} ‹µÎ œªœœ´
…. 0 Ážœ} ‹µÎ œªœ°˜¦¦„¥³

S

‡. ¤‹¸ 圪œ˜¦¦„¥³°¥¼¦n ³®ªnµŠ 0.5 „´ 0.6
Š. ‹µÎ œªœÁ˜È¤¨¤­¸ ¤˜´ ·žd—…°Š„µ¦ª„
5. …o°Ä—„¨nµª™¼„˜o°Š
„. ‹ÎµœªœÁ˜¤È š¸É¤µ„š­¸É »— ¨³¤µ„„ªnµ 5 ‡º° 6
…. Ážœ} ‹ÎµœªœÁ˜¤È ª„ ¨³Ážœ} ‹µÎ œªœ˜¦¦„¥³
‡. ‹µÎ œªœ‹¦·Šš¸Éœ°o ¥š­¸É »—¨³œ°o ¥„ªµn 8 ‡º° 7
Š. ‹µÎ œªœÁ˜¤È š¤¸É µ„š­¸É »—š¸œÉ °o ¥„ªµn 11.06 ‡º° 11

10 ⌫ ⌫  ⌦
 ⌫     ⌫  

ŸœŸŠ´ ­—Š‡ªµ¤­¤´ ¡œ´ ›…r °Š‹Îµœªœœ—· ˜nµŠÇ

‹ÎµœªœÁŠ· Ž°o œ

‹µÎ œªœ‹¦·Š ‹µÎ œªœÁŠ· Žo°œš¸ÉŤnčn‹µÎ œªœ‹¦Š·

‹µÎ œªœ°˜¦¦„¥³ ‹Îµœªœ˜¦¦„¥³

‹µÎ œªœ˜¦¦„¥³š¸ÉŤÄn n‹ÎµœªœÁ˜È¤ ‹µÎ œªœÁ˜È¤

‹Îµœª‹œµÎ Áœ˜ªÈ¤œ¨Á˜¤È ¨ ‹ÎµœªœÁ˜È¤«œ¼ ¥r ‹ÎµœªœÁ˜¤È ª„
®¦°º ‹µÎ œªœœ´

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⌦ 11
⌦

Ÿœ„µ¦‹—´ „µ¦Á¦¸¥œ¦o¼šÉ¸ 2

Á¦É°º Š ­¤˜´ …· °Š¦³‹µÎ œªœ‹¦Š· Êœ´ ¤›´ ¥¤«„¹ ¬µžeš¸É 4
ª·µ ‡–˜· «µ­˜¦r Áª¨µ 3 ª´É äŠ

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

Ÿ¨„µ¦Á¦¥¸ œ¦o¼šÉ¸‡µ—®ª´Š
œµÎ ­¤´˜˜· nµŠÇ Á„¸É¥ª„´ ‹Îµœªœ‹¦Š· ¨³„µ¦—εÁœ·œ„µ¦ÅžÄÅo —o

1. ‹—» ž¦³­Š‡„r µ¦Á¦¥¸ œ¦¼o œ´„Á¦¸¥œ­µ¤µ¦™
1.1 °„­¤´˜·„µ¦Ášµn „´œÄœ¦³‹µÎ œªœ‹¦Š· ŗo
1.2 œÎµ­¤˜´ „· µ¦Ášnµ„´œÄœ¦³‹µÎ œªœ‹¦Š· ŞčÅo —o
1.3 °„­¤´˜·…°Š‹µÎ œªœ‹¦·ŠÁ„¥É¸ ª„´ „µ¦ª„Å—o
1.4 °„­¤˜´ ·…°Š‹Îµœªœ‹¦Š· Á„É¥¸ ª„´ „µ¦‡¼–Å—o
1.5 œÎµ­¤˜´ …· °Š‹Îµœªœ‹¦·ŠÁ„¸É¥ª„´ „µ¦ª„¨³„µ¦‡¼–ŞčoĜ„µ¦‡Îµœª–Å—o

2. œª‡ªµ¤‡—· ®¨„´
­¤´˜·¡Êºœ“µœ…°Š¦³‹Îµœªœ‹¦·Š ‡º° ­´‹¡‹œr 11 …o°Â¦„ ŗo„n ­¤´˜·žd—­Îµ®¦´„µ¦

ª„¨³„µ¦‡¼– ­¤´˜·„µ¦­¨´šÉ¸­Îµ®¦´„µ¦ª„¨³„µ¦‡¼– ­¤´˜·„µ¦Áž¨É¸¥œ„¨»n¤­Îµ®¦´„µ¦ª„
¨³„µ¦‡¼– ­¤´˜·„µ¦¤¸Á°„¨´„¬–r­Îµ®¦´„µ¦ª„¨³„µ¦‡¼– ­¤´˜·„µ¦¤¸°·œÁª°¦r­­Îµ®¦´„µ¦ª„
¨³„µ¦‡¼– ¨³­¤´˜„· µ¦Â‹„‹Š

3. Áœº°Ê ®µ­µ¦³
1. „µ¦Ášµn „´œÄœ¦³‹Îµœªœ‹¦Š·
­¤´˜·„µ¦Ášnµ„œ´ Á¤ºÉ° a,b, c  R
1.1 ­¤´˜·„µ¦­³šo°œ a a
1.2 ­¤˜´ „· µ¦­¤¤µ˜¦ ™oµ a b ¨ªo b a
1.3 ­¤˜´ „· µ¦™µn ¥š°— ™µo a b ¨³ b c ¨oª a c
1.4 ­¤˜´ „· µ¦ª„—ªo ¥‹µÎ œªœšÁ¸É šµn „´œ ™oµ a b ¨ªo a c b c
1.5 ­¤˜´ ·„µ¦‡–¼ —oª¥‹ÎµœªœšÁɸ šµn „œ´ ™µo a b ¨ªo ac bc
2. „µ¦ª„¨³„µ¦‡¼–Äœ¦³‹µÎ œªœ‹¦·Š

12 ⌫ ⌫  ⌦
 ⌫     ⌫  

šœ·¥µ¤ 1 Ĝ¦³‹Îµœªœ‹¦·Š Á¦¥¸ „‹µÎ œªœ‹¦Š· š¸É ª„„´‹µÎ œªœ‹¦·ŠÄ—„Șµ¤ ŗŸo ¨¨¡´ ›r
Ážœ} ‹Îµœªœ‹¦·ŠœÊœ´ ªnµ “Á°„¨„´ ¬–r„µ¦ª„” „¨nµª‡°º Ä®o x ƒ ¨³ e ‡°º Á°„¨´„¬–r„µ¦ª„
—Š´ œœ´Ê x + e = x = e + x

Ĝ¦³‹Îµœªœ‹¦Š· ¤¸ 0 Ážœ} Á°„¨„´ ¬–…r °Š„µ¦ª„
0+a = a= a+0

šœ·¥µ¤ 2 Ĝ¦³‹µÎ œªœ‹¦Š· °·œÁª°¦­r „µ¦ª„…°Š‹µÎ œªœ‹¦·Š a šœ—oª¥ –a
Ž¹ÉŠ®¤µ¥™¹Š ‹Îµœªœ‹¦Š· šÉ¸ ª„„´ a ¨ªo ŗo«¼œ¥r „¨µn ª‡°º a+(-a) = 0 = (-a) + a

šœ·¥µ¤ 3 Ĝ¦³‹Îµœªœ‹¦Š· Á¦¥¸ „‹Îµœªœ‹¦·ŠšÉ¸Å¤Án ž}œ«¼œ¥r ŽŠ¹É Á¤ºÉ°œµÎ ¤µ‡–¼ „´ ‹Îµœªœ
‹¦·Š ė¨ªo ŗŸo ¨¨´¡›rÁž}œ‹µÎ œªœ‹¦·ŠœœÊ´ ªµn “Á°„¨´„¬–r„µ¦‡¼–”

™oµ e Ážœ} Á°„¨´„¬–„r µ¦‡–¼ ¨³ e z 0 ¨ªo
ea a ae Á¤°ºÉ a  R

Ĝ¦³‹µÎ œªœ‹¦·Š‹³¤¸ 1 Áž}œÁ°„¨„´ ¬–„r µ¦‡¼–

šœ·¥µ¤ 4 Ĝ¦³‹µÎ œªœ‹¦Š· °œ· Áª°¦r­„µ¦‡–¼ …°Š‹Îµœªœ‹¦Š· a Á¤º°É a z 0 ‹³Á…¥¸ œ
šœ—ªo ¥ a 1 ŽŠÉ¹ ®¤µ¥™¹Š ‹µÎ œªœš‡É¸ ¼–„´ a ¨oªÅ—o 1

—´ŠœÊœ´ aa-1 = 1 = a-1a
™µo a  R ¨³ a z 0 ‹³Å—o a 1 1

a

3. ­¤´˜·…°Š¦³‹Îµœªœ‹¦·Š
¦³‹Îµœªœ‹¦·Šž¦³„°—oª¥ÁŽ˜…°Š‹Îµœªœ‹¦·Š R „´„µ¦ª„¨³„µ¦‡¼–ŽÉ¹Š¤¸­¤´˜·
—´Š˜°n Şœ¸Ê

3.1 ­¤˜´ ·ž—d …°Š„µ¦ª„ Á¤ºÉ° a  R ¨³ b  R ‹³Å—o a b  R
3.2 ­¤˜´ „· µ¦­¨´š…¸É °Š„µ¦ª„ Á¤Éº° a, b  R ‹³Å—o a b b a

⌦ 13
⌦

3.3 ­¤˜´ „· µ¦Áž¨¸¥É œ„¨»n¤…°Š„µ¦ª„ Á¤°Éº a, b, c  R ‹³Å—o
a b c a b c

3.4 Á°„¨„´ ¬–r„µ¦ª„ Ĝ¦³‹Îµœªœ‹¦Š· ¤¸ 0 Ážœ} Á°„¨´„¬–„r µ¦ª„

­Îµ®¦´ ‹µÎ œªœ‹¦Š· a ė Ç ŽŠÉ¹ a + 0 = 0 + a = a

3.5 °œ· Áª°¦r­„µ¦ª„Äœ¦³‹Îµœªœ‹¦·Š ™oµ a  R ‹³¤¸ a  R

Ž¹ŠÉ a + (-a) = 0 = (-a)+a

3.6 ­¤˜´ ·žd—…°Š„µ¦‡–¼ Á¤°ºÉ a  R ¨³ b  R ‹³Å—o a ˜ b  R

3.7 ­¤´˜„· µ¦­¨´ šÉ…¸ °Š„µ¦‡–¼ Á¤Éº° a, b  R ‹³Å—o ab ba

3.8 ­¤˜´ „· µ¦Áž¨É¸¥œ„¨¤n» …°Š„µ¦‡¼– Á¤ºÉ° a, b, c  R ‹³Å—o ab c a bc

3.9 ­¤´˜·„µ¦¤¸Á°„¨„´ ¬–„r µ¦‡¼– ¤¸ 1 Áž}œÁ°„¨„´ ¬–r„µ¦‡¼–

­µÎ ®¦´ ‹Îµœªœ‹¦Š· a Ä—Ç 1 ˜ a a ˜ 1 a

3.10 ­¤´˜„· µ¦¤¸°œ· Áª°¦­r „µ¦‡–¼ ™oµ a  R ¨³ a z 0 ¨ªo ‹³¤¸ a 1  R

ŽÉŠ¹ a ˜ a 1 a 1 ˜ a 1 ™oµ a  R ¨³ a z 0, a 1 1

a

3.11 ­¤´˜„· µ¦Â‹„‹Š Á¤ºÉ° a, b, c  R ‹³Å—o a b c ab ac
b c a ba ca

4. „¦³ªœ„µ¦‹´—„µ¦Á¦¥¸ œ¦o¼

4.1 ‡¦¼„¨nµª™¹Š­´¨´„¬–ršœ„µ¦Ášnµ„´œ®¦º°Áž}œ­É·ŠÁ—¸¥ª„´œÄœšµŠ‡–·˜«µ­˜¦r ‹³Äo “=”

‡¦¼™µ¤œ´„Á¦¸¥œªnµ­´¨´„¬–ršœ„µ¦„¦³šÎµšÉ¸šÎµÄ®oÁ„·—„µ¦Ášnµ„´œ¤¸°³Å¦oµŠ œ´„Á¦¸¥œ‡ª¦˜°Å—oªnµ

Á‡¦°Éº Š®¤µ¥ª„ ¨ ‡¼– ®µ¦Â¨³ Á‡¦°ºÉ Š®¤µ¥„¦–”r

Ánœ 2 3 5

5 1 4
2 u 5 10
20 y 4 5

42
1.9 2
‡¦¼Ä®oœ´„Á¦¸¥œ­´ŠÁ„˜ªnµ‹ÎµœªœÁ—¸¥ª„´œÂšœÅ—o—oª¥®¨µ¥­´¨´„¬–r ˜n­´¨´„¬–rÁ—¸¥ª‹³Âšœ‹Îµœªœ
Á¡¸¥Š‹ÎµœªœÁ—¸¥ª Áœn 4 2, 4 2, 1.9 2

2

4.2 ‡¦¼Â‹„Á°„­µ¦Âœ³ÂœªšµŠš¸É 2.1 Ä®oœ´„Á¦¸¥œ«¹„¬µÂ¨³˜°‡Îµ™µ¤ÄoÁª¨µ 10 œµš¸ ¨oª‡¦¼
¨³œ´„Á¦¸¥œnª¥„œ´ ÁŒ¨¥‡Îµ˜° Á¡É°º Áž}œ„µ¦¥ÎµÊ ­¤´˜·„µ¦Ášµn „œ´ …°Š‹µÎ œªœ‹¦·ŠÄ®oœ„´ Á¦¥¸ œÁ…oµÄ‹¥ŠÉ· …œ¹Ê

14 ⌫ ⌫  ⌦
 ⌫     ⌫  

4.3 nŠœ´„Á¦¸¥œ°°„Áž}œ 8 „¨n»¤ „¨»n¤¨³ž¦³¤µ– 5 ‡œ ‡¦¼Â‹„Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 2.2, 2.3
Ä®oš»„‡œÅ—«o „¹ ¬µ ¨³nª¥„´œ‡·—˜°‡Îµ™µ¤ÄoÁª¨µ 15 œµš¸ ¨oª‡¦¼Äoª·›¸­»n¤˜´ªÂšœ…°ŠÂ˜n¨³„¨»n¤ „¨»n¤
¨³ 1 ‡œ °°„¤µ­¦»žœ·¥µ¤…°Š Á°„¨´„¬–r„µ¦ª„ °·œÁª°¦r­„µ¦ª„ Á°„¨´„¬–r„µ¦‡¼– ¨³°·œÁª°¦r­
„µ¦‡¼– ×¥‡¦‡¼ °¥Ä®o‡ÎµÂœ³œÎµ

4.4 Ä®œo „´ Á¦¥¸ œšÎµÂ „f ®—´ 2.2 „ …o° 1, 2, 3 Ĝ®œ´Š­°º Á¦¸¥œ­µ¦³„µ¦Á¦¥¸ œ¦oÁ¼ ¡É·¤Á˜¤· ‡–·˜«µ­˜¦r
Á¨n¤ 1 ´Êœ¤›´ ¥¤«„¹ ¬µžše ¸É 4

4.5 ‡¦¼ššªœœ·¥µ¤…°ŠÁ°„¨´„¬–r¨³°·œÁª°¦r­…°Š„µ¦ª„¨³„µ¦‡¼–Äœ¦³‹Îµœªœ‹¦·Š
×¥‡¦¼¥„˜´ª°¥nµŠ‹Îµœªœ ¨oªÄ®oœ´„Á¦¸¥œ˜° Ánœ °·œÁª°¦r­„µ¦ª„¨³„µ¦‡¼–…°Š 3 1 ‡º°°³Å¦
œ´„Á¦¸¥œ‡ª¦˜°Å—oªµn

°œ· Áª°¦­r „µ¦ª„…°Š 3 1 ‡º° 3 1
°œ· Áª°¦r­„µ¦‡–¼ …°Š 3 1 ‡°º 1

3 1

4.6 ‡¦¼Â‹„Á°„­µ¦Âœ³ÂœªšµŠš¸É 2.4 ¨³ 2.5 Ä®oœ´„Á¦¸¥œ«¹„¬µ­¤´˜·…°Š‹Îµœªœ‹¦·ŠÁ„ɸ¥ª„´
„µ¦ª„¨³„µ¦‡–¼ ¨oª˜°‡Îµ™µ¤ÄoÁª¨µ 15 œµš¸

4.7 ‡¦¼Â¨³œ„´ Á¦¸¥œnª¥„´œÁŒ¨¥Á°„­µ¦Âœ³ÂœªšµŠš¸É 2.4 ¨³ 2.5 ×¥„µ¦™µ¤˜° ¤¸­nªœÅ®œ
šÉ¸œ´„Á¦¸¥œ‡œÄ—ŤnÁ…oµÄ‹ ‡¦¼‹³°›·µ¥Ã—¥¥„˜´ª°¥nµŠÁ¡·É¤Á˜·¤Â¨³­¦»ž­¤´˜·‹µ„Á°„­µ¦Âœ³ÂœªšµŠšÉ¸
2.4 ¨³ 2.5 °¸„‡¦Š´Ê

4.8 ‡¦¼Â‹„Á°„­µ¦ f„®´—š¸É 2 Ä®oœ´„Á¦¸¥œš»„‡œšÎµ čoÁª¨µ 10 œµš¸ ¨oª‡¦¼ÁŒ¨¥‡Îµ˜°œ
„¦³—µœ—ε

4.9 Ä®œo „´ Á¦¥¸ œšµÎ  f„®´— 2.2 „ …°o 1 Ĝ®œŠ´ ­º°Á¦¥¸ œ­µ¦³„µ¦Á¦¸¥œ¦¼oÁ¡É·¤Á˜·¤
4.10 ‡¦¼ÁŒ¨¥Â f„®´— 2.2 …°o šœÉ¸ „´ Á¦¸¥œ­nªœÄ®šn 執— ®¦º°…°o šÉœ¸ „´ Á¦¥¸ œ­Š­¥´
4.11 ‹„Á°„­µ¦Âœ³ÂœªšµŠš¸É 2.6 Ä®œo ´„Á¦¥¸ œš„» ‡œ«¹„¬µÁ„¥¸É ª„´ „µ¦œµÎ ­¤´˜…· °Š‹Îµœªœ‹¦Š·
Şčo„´„µ¦„¦³šÎµ (Operation) °ÉºœÇ œ°„Á®œº°‹µ„„µ¦ª„¨³„µ¦‡¼– Ánœ
,†, ' ×¥„µ¦„ε®œ—
œ·¥µ¤…°Š„µ¦„¦³šÎµ
,†, ' Ä®¤nčoÁª¨µ 10 œµš¸ ¨oª‡¦¼ÁŒ¨¥‡Îµ˜°Â¨³°›·µ¥Á¡É·¤Á˜·¤­Îµ®¦´
œ„´ Á¦¥¸ œš¥É¸ ´ŠÅ¤Án …µo ċ
4.12 Ä®oœ´„Á¦¸¥œ«¹„¬µÁ°„­µ¦‡ªµ¤¦¼ošÉ¸ 2 Ž¹ÉŠÁž}œ˜´ª°¥nµŠ„µ¦˜¦ª‹­°­¤´˜·…°ŠÃ°Áž°Á¦É´œš¸É
„ε®œ—Ä®o ™µo ¤¸­nªœÅ®œÅ¤nÁ…oµÄ‹ ‡¦¼‹³°›· µ¥Á¡É·¤Á˜¤·
4.13 Ä®oœ´„Á¦¸¥œnª¥„´œ­¦»ž­¤´˜·…°Š‹Îµœªœ‹¦·ŠÁ„ɸ¥ª„´„µ¦ª„ 5 …o° ¨³Á„¸É¥ª„´„µ¦‡¼– 6
…o° ¨³Â‹oŠÄ®oœ´„Á¦¸¥œš¦µªnµ­¤´˜·…°Š‹Îµœªœ‹¦·ŠšÊ´Š 11 …o°œ¸Ê ¤¸‡ªµ¤­Îµ‡´Â¨³‹³ÄoĜ„µ¦¡·­¼‹œr
š§¬‘¸š˜nµŠÇ Á„¸É¥ª„´‹µÎ œªœ‹¦Š· ˜°n Ş Ä®oœ´„Á¦¸¥œ—¼­¤´˜·…°Š‹Îµœªœ‹¦·Šš´ÊŠ 11 …o° ‹µ„®œ´Š­º°Á¦¸¥œ
­µ¦³„µ¦Á¦¸¥œ¦Á¼o ¡É¤· Á˜·¤² ®œµo 58
4.14 Ä®oœ´„Á¦¸¥œšÎµÂ f„®´—š¸É 2.2 „ …o° 4 ®œ´Š­º°Á¦¸¥œ­µ¦³„µ¦Á¦¸¥œ¦¼oÁ¡·É¤Á˜·¤‡–·˜«µ­˜¦r
Á¨n¤ 1 …°Š ­­ªš.

⌦ 15
⌦

5. ®¨Šn „µ¦Á¦¥¸ œ¦¼o
5.1 Á°„­µ¦Âœ³ÂœªšµŠš¸É 2.1 ™¹Š 2.6
5.2 Á°„­µ¦ „f ®—´ šÉ¸ 2
5.3 Á°„­µ¦‡ªµ¤¦šo¼ ¸É 2
5.4 ˜µ¦µŠÂ­—Š­¤´˜…· °Š¦³‹Îµœªœ‹¦·Š 11 …o°
5.5 ®œ´Š­°º Á¦¸¥œ­µ¦³„µ¦Á¦¥¸ œ¦Áo¼ ¡¤É· Á˜¤· ‡–·˜«µ­˜¦r Á¨¤n 1 Ê´œ¤›´ ¥¤«¹„¬µžešÉ¸ 4 …°Š ­­ªš.

6. „¦³ªœ„µ¦ª—´ ¨³ž¦³Á¤œ· Ÿ¨

„µ¦ª´—Ÿ¨ „µ¦ž¦³Á¤·œŸ¨

1. ­´ŠÁ„˜‹µ„„µ¦˜°‡Îµ™µ¤ 1. œ„´ Á¦¸¥œ˜°‡µÎ ™µ¤Å—™o „¼ ˜o°Š—¸¤µ„

2. ­Š´ Á„˜‹µ„„µ¦¦ªn ¤„‹· „¦¦¤ 2. œ„´ Á¦¸¥œ¦ªn ¤„·‹„¦¦¤—¸

3. ­Š´ Á„˜‹µ„‡ªµ¤­œÄ‹ 3. œ´„Á¦¥¸ œÄ®‡o ªµ¤­œÄ‹—¸

4. šÎµÃ‹š¥rÁ°„­µ¦Âœ³ÂœªšµŠšÉ¸ 2.1–2.6 4. œ„´ Á¦¸¥œšµÎ ŗ™o „¼ ˜°o Šž¦³¤µ– 95 %

¨³ šÎµÃ‹š¥rÁ°„­µ¦ f„®—´ š¸É 2

5. šÎµÃ‹š¥rĜ®œ´Š­º°ÂÁ¦¸¥œÂ f„®´—š¸É 5. œ´„Á¦¸¥œšÎµÅ—™o „¼ ˜o°Šž¦³¤µ– 95 %

2.2 „

7. ´œš¹„®¨´Š­°œ
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………

8. „‹· „¦¦¤Á­œ°Âœ³
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………

16 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦Âœ³ÂœªšµŠš¸É 2.1

‡µÎ ­Š´É ‹Š«¹„¬µ­¤´˜„· µ¦Ášµn „´œ…°Š‹µÎ œªœ‹¦Š·

­¤˜´ „· µ¦Ášµn „´œ Á¤ºÉ° a,b,c R
­¤˜´ ·„µ¦­³š°o œ

aa

­¤˜´ „· µ¦­¤¤µ˜¦

™oµ a b ¨ªo b a

­¤˜´ ·„µ¦™µn ¥š°—

™oµ a b ¨³ b c ¨oª a c

­¤˜´ „· µ¦ª„—oª¥‹µÎ œªœšÁɸ šnµ„œ´

™oµ a b ¨ªo a c b c

­¤´˜·„µ¦‡–¼ —ªo ¥‹ÎµœªœšÉÁ¸ šnµ„´œ

™µo a b ¨oª ac bc

‹µ„­¤´˜·„µ¦Ášnµ„´œ…oµŠ˜oœ ‹ŠÁ˜·¤‡Îµ˜°¨ŠÄœn°ŠªnµŠ ¡¦o°¤šÊ´Š°„­¤´˜·…°Š„µ¦Ášnµ„´œšÉ¸

­°—‡¨°o Š„´…°o ‡ªµ¤

1. ™oµ a 4 ¨oª 4 ___________________ ­¤´˜· ___________________________________

2. ™oµ x y ¨oª y ___________________ ­¤´˜· ___________________________________

3. ™oµ a 3 ¨oª a 2 _______________ ­¤´˜· ___________________________________

4. ™oµ b 3.5 ¨ªo b _____ 3.5 8 ­¤˜´ · ___________________________________

5. ™oµ c 8 ¨ªo c u10 8u______________ ­¤´˜· ___________________________________

6. ™oµ x 3 ¨³ y 3 ‹³Å—o x _________ ­¤´˜· ___________________________________

7. ™µo 15 x ¨³ y x ‹³Å—o ______=____ ­¤˜´ ·____________________________________

8. ™µo y 5 ¨ªo 8y __________________ ­¤˜´ · ___________________________________

9. ™µo x 3 8 ¨ªo x 5 ­¤˜´ _· ___________________________________

10. ™µo 5x 20 ¨oª x 4 ­¤´˜·____________________________________

⌦ 17
⌦

Á°„­µ¦Âœ³ÂœªšµŠš¸É 2.2

1. ‹ŠÁ˜¤· ‡Îµ˜°¨ŠÄœn°Šªµn Š
1.1 3 0 _________
1.2 _____ 3 3

1.3 2 0 _________
1.4 _____ 5 0

1.5 3 0 _________
1.6 x 0 _________

®¤µ¥Á®˜» ‹µ„…o° 1.1-1.6 Á¦¸¥„ 0 ªnµÁžœ} Á°„¨„´ ¬–r…°Š„µ¦ª„

2. ‹ŠÁ˜¤· ‡Îµ˜°¨ŠÄœn°ŠªnµŠ
2.1 2 ___ 0

2.2 ___ 3 3

2.3 5 ___ 0
2.4 2 ___ 0
2.5 a ___ 0

®¤µ¥Á®˜» ‹µ„…o° 2.1 Á¦¥¸ „ –2 ªnµÁžœ} °œ· Áª°¦r­„µ¦ª„…°Š 2
2.2 Á¦¸¥„ 3 ªµn Ážœ} ____________________________
2.3 °·œÁª°¦­r „µ¦ª„…°Š 5 ‡°º ______________
2.4 °·œÁª°¦­r „µ¦ª„…°Š 2 ‡°º ____________
2.5 °œ· Áª°¦r­„µ¦ª„…°Š a ‡º° ________________

­¦»žÁž}œšœ¥· µ¤Å——o ´ŠœÊ¸

šœ·¥µ¤ Ĝ¦³‹Îµœªœ‹¦Š· °œ· Áª°¦­r „µ¦ª„…°Š‹µÎ œªœ‹¦Š· a Á…¸¥œÂšœ—ªo ¥ - a
Ž¹ŠÉ ®¤µ¥™Š¹ ‹µÎ œªœ‹¦·ŠšÉ¸ ª„„´ a ¨oªÅ—o 0 „¨µn ª‡°º

a - a 0 ( a) + a

®¦°º „¨nµªªµn a ¨³ - a Ážœ} °œ· Áª°¦­r „µ¦ª„ŽÉŠ¹ „œ´ ¨³„œ´

18 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦Âœ³ÂœªšµŠš¸É 2.3

‡µÎ ­´ŠÉ ‹ŠÁ˜·¤‡Îµ˜°¨ŠÄœn°Šªµn Š

1. 5u___ 5 6. 2 u ___ 1

2. ___u 5 5 7. 3u ___ 1
8.
3. 2 u ___ 2 1 u ___ 1
5
1 u ___ 1
4. 3 3 9. ___u 3 1

5. au___ a,aR 10. 1 u ___ 1,a z 0
a

®¤µ¥Á®˜» ‡Îµ˜°Äœ…°o 1-5 ‡°º 1 Á¡¦µ³ªµn 1 ‡–¼ ‹ÎµœªœÄ—„Șµ¤ ‹³Å—o‹µÎ œªœ‹¦·ŠœœÊ´ Á­¤° ‹³Á¦¥¸ „ 1
ªµn Áž}œÁ°„¨„´ ¬–„r µ¦‡–¼ Ĝ¦³‹Îµœªœ‹¦Š·

šœ¥· µ¤ Ĝ¦³‹Îµœªœ‹¦Š· Á¦¥¸ „‹Îµœªœ‹¦·ŠšÅɸ ¤Án ž}œ 0 ŽÉ¹ŠÁ¤°Éº œÎµÅž‡–¼ „´ ‹Îµœªœ‹¦·Š
ė¨oªÅ—o‹Îµœªœ‹¦Š· œœÊ´ ªµn Á°„¨„´ ¬–r„µ¦‡¼–

®¤µ¥Á®˜» ‡Îµ˜°Äœ…o° 6-10 Ÿ¨‡¼–…°Š­°Š‹ÎµœªœÄ—¤¸‡nµÁž}œ 1 ‹³Á¦¸¥„­°Š‹ÎµœªœœÊ´œªnµÁž}œ

°·œÁª°¦­r „µ¦‡–¼ ŽŠ¹É „´œÂ¨³„œ´

—Š´ œœÊ´ °·œÁª°¦­r „µ¦‡–¼ …°Š 2 ‡°º 1

2

°·œÁª°¦­r „µ¦‡¼–…°Š -3 ‡°º 1

3

°œ· Áª°¦r­„µ¦‡–¼ …°Š 1 ‡°º –5

5

°·œÁª°¦­r „µ¦‡–¼ …°Š 3 ‡º° 1
3

°·œÁª°¦r­„µ¦‡¼–…°Š 1 ‡º° a , a z 0
a

­¦ž» Ážœ} šœ¥· µ¤Å——o Š´ œÊ¸

šœ¥· µ¤ Ĝ¦³‹Îµœªœ‹¦Š· °·œÁª°¦r­„µ¦‡¼–…°Š‹µÎ œªœ‹¦Š· a Á¤º°É a z 0 ‹³Âšœ—ªo ¥
a 1 ®¤µ¥™¹Š‹ÎµœªœšÉ‡¸ ¼–„´ a ¨oªÅ—o Á°„¨„´ ¬–„r µ¦‡–¼

—Š´ œ´Êœ a ˜ a 1 a 1 ˜ a 1
™µo a  R ¨³ a z 0 ¨oª a 1 1

a

⌦ 19
⌦

Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 2.4

‡Îµ­ÉŠ´ ‹Š«„¹ ¬µ…o°‡ªµ¤Äœ˜°œšÉ¸ 1 ¨oª˜°‡Îµ™µ¤

˜°œšÉ¸ 1 ­¤´˜·…°Š‹µÎ œªœ‹¦·ŠÁ„¸¥É ª„´ „µ¦ª„

1. ­¤´˜ž· d—…°Š„µ¦ª„

Á¤ºÉ° a  R ¨³ b  R ‹³Å—o a b  R

Áœn 2  R ¨³ 5  R ‹³Å—o 2 5  R

2. ­¤˜´ „· µ¦­¨´ š¸…É °Š„µ¦ª„

Á¤°Éº a, b  R ‹³Å—o a b b a
2,3  R ‹³Å—o 2 3 __________

3. ­¤˜´ „· µ¦Áž¨É¸¥œ„¨»¤n …°Š„µ¦ª„

Á¤É°º a, b, c  R ‹³Å—o a b c a b c
2 3 4 _______________

4. Á°„¨´„¬–„r µ¦ª„
Ĝ¦³‹µÎ œªœ‹¦Š· ¤¸ ________Ážœ} Á°„¨„´ ¬–„r µ¦ª„ ­µÎ ®¦´‹Îµœªœ‹¦·Š a Ä—Ç ŽÉ¹Š

a _____ _____ a

5. °œ· Áª°¦r­„µ¦ª„

Ĝ¦³‹µÎ œªœ‹¦·Š ™µo a  R ‹³¤¸ a  R

ŽŠÉ¹ a a a a __________

˜°œšÉ¸ 2 ‹Š°„­¤˜´ ·…°Š‹Îµœªœ‹¦·Š‹µ„…o°‡ªµ¤˜n°Åžœ¸Ê

1. 2 S Áž}œ‹µÎ œªœ‹¦Š· ­¤´˜·žd—…°Š„µ¦ª„

2. 12 0 12 __________________________________
__________________________________
3. 3 4 5 4 3 5 __________________________________
4. 1 3 5 5 1 3 __________________________________
5. 2 4 6 2 4 6

6. 0 0 0 __________________________________
__________________________________
7. 5 7 9 5 9 7 __________________________________
8. b b 0

20 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 2.5

˜°œš¸É 1 Ä®oœ„´ Á¦¥¸ œ«„¹ ¬µ…°o ‡ªµ¤˜n°Åžœ¸Ê ¨ªo Á˜·¤‡µÎ ˜°¨ŠÄœ°n Šªµn Š

­¤˜´ ·…°Š¦³‹µÎ œªœ‹¦·ŠÁ„É¥¸ ª„´„µ¦‡–¼

1. ­¤´˜·žd—…°Š„µ¦‡¼–

Á¤°ºÉ a  R ¨³ b  R ‹³Å—o a ˜ b  R

Ánœ 2  R ¨³ 2  R ‹³Å—o _______________

2. ­¤´˜„· µ¦­¨´š…¸É °Š„µ¦‡¼–

Á¤Éº° a, b  R ‹³Å—o ab ba

2,5  R ‹³Å—o _______________

3. ­¤˜´ „· µ¦Áž¨É¥¸ œ„¨¤n» …°Š„µ¦‡¼–

Á¤ºÉ° a, b, c  R ‹³Å—o ab c a bc

2,4,6  R ‹³Å—o _______________

4. ­¤˜´ ·„µ¦¤¸Á°„¨´„¬–„r µ¦‡–¼

Ĝ¦³‹µÎ œªœ‹¦Š· ¤¸ 1 Ážœ} Á°„¨´„¬–„r µ¦‡¼– ­Îµ®¦´ ‹Îµœªœ‹¦·Š a ėÇ

œœÉ´ ‡°º 1 ˜ a a ˜ 1 a

5. ­¤´˜„· µ¦¤°¸ ·œÁª°¦­r „µ¦‡–¼

™µo a  R ¨³ a z 0 ¨oª‹³¤¸ a 1  R ŽÉŠ¹ a ˜ a 1 a 1 ˜ a 1

™µo a  R ¨³ a z 0, a 1 1

a

2 ˜ 2 1 2 ˜ 1 1
2

1 ˜ ____ ____ ____
3

6. ­¤´˜„· µ¦Â‹„‹Š

Á¤°Éº a, b, c  R ‹³Å—o a b c ab ac

b c a ba ca
5 x 2 __________
Á¤Éº° x  R

⌦ 21
⌦

˜°œšÉ¸ 2 ‹Š°„­¤´˜…· °Š‹Îµœªœ‹¦·Šš­¸É °—‡¨o°Š„´ …°o ‡ªµ¤˜n°ÅžœÊ¸

1. 20 u 5 Ážœ} ‹µÎ œªœ‹¦Š· ­¤˜´ · _______________________

2. 2 u 3 u 5 3 u 2 u 5 ­¤´˜· _______________________

3. 2 u 3 u 5 2 u 3 u 5 ­¤´˜· _______________________
­¤´˜· _______________________
4. 1u 3 2 3 2

5. 3 u 4 1 ­¤˜´ · _______________________
­¤˜´ · _______________________
43

6. 2 5 7 2 u 5 2 u 7

7. 77 3 99 2 77 3 99 77 3 2 ­¤˜´ · _______________________

8. 895 1 104 3 (895 104) (1 3) ­¤˜´ · _______________________
44 44

22 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦ „f ®´—š¸É 2

1. ‹Š°„­¤˜´ ·…°Š‹µÎ œªœ‹¦Š· š­É¸ °—‡¨°o Š„´…o°‡ªµ¤˜°n ޜʸ

1.1 x 2 5 Ážœ} ‹µÎ œªœ‹¦Š· __________________________________

1.2 3 x 5 3x 15 __________________________________

1.3 2 2 __________________________________
__________________________________
1.4 3 3 0

1.5 ™oµ x x 2x ¨oª 2x x x __________________________________
__________________________________
1.6 8 4 1 1 8 4 __________________________________
1.7 8 4 1 8 4 1

1.8 ™µo x y 3 ¨³ z 3 ¨ªo x y z __________________________________

1.9 1 u 3 1 __________________________________
__________________________________
3

1.10 ™oµ x y ¨oª 3x 3y

2. ‹ŠÁ˜·¤n°ŠªnµŠÄ®­o ¤¦¼ –r

…°o š¸É ‹µÎ œªœ °œ· Áª°¦r­

1 10 „µ¦ª„ „µ¦‡–¼
2 1

4

3 1

3

45

5 3 2

⌦ 23
⌦

3. ‹ŠÄo­¤´˜·„µ¦­¨´šÉ¸®¦°º „µ¦Áž¨¥¸É œ„¨»¤n …°Š„µ¦ª„®µŸ¨ª„Ĝ˜¨n ³…°o ˜n°ÅžœÊ¸

3.1 1+2+3+4+…+20 = ____________________________________________________

= ____________________________________________________

= ____________________________________________________

3.2 2+4+6+8+…+28 = ____________________________________________________

= ____________________________________________________

3.3 103 1 116 1 = ____________________________________________________
55 = ____________________________________________________
= ____________________________________________________

= ____________________________________________________

4. ‹ŠÄ­o ¤´˜…· °Š„µ¦Â‹„‹Š®µ‡nµ˜°n Şœ¸Ê

4.1 98u10 1 = ____________________________________________________
7

= ____________________________________________________

4.2 27 u 99 = ____________________________________________________
= ____________________________________________________

= ____________________________________________________

= ____________________________________________________

5. ‹ŠÁž¨¥É¸ œ 12345 Ä®Áo žœ} Á¨…“µœ 10 ץčo­¤˜´ „· µ¦Â‹„‹Š
__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

24 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 2.6

‡µÎ Âʸ ‹Š Ä®œo „´ Á¦¸¥œ«„¹ ¬µ…°o ‡ªµ¤…µo Š¨nµŠœ¸Ê ¨³˜°‡Îµ™µ¤šÁɸ ªœo Ūo
¦³‹µÎ œªœ£µ¥Ä˜o„µ¦„¦³šÎµ (Operation) (
)
ÁŽ˜…°Š‹Îµœªœ¤¸„µ¦„¦³šÎµ¡Êºœ“µœ¦³®ªnµŠ‹Îµœªœ 2 ‹Îµœªœ ‡º° , ,u,y Á¦µ°µ‹œ·¥µ¤

„µ¦„¦³šµÎ °ºÉœÇ šÉŽ¸ ´Ž°o œ¤µ„…œÊ¹ ŗo ץŤ¤n ¸„‘Á„–”šr Â¸É œœn °œ Ánœ

˜´ª°¥µn Š „ε®œ— a, b  R Á¤ºÉ° R Áž}œÁŽ˜…°Š‹Îµœªœ‹¦·Š ¨³ a
b a b 2

‹µ„ a
b a b 2

™µo a 2, b 3 ‹³Å—o 2
3 2 3 2 7

™µo a 5, b 1 ‹³Å—o 5
1 _________________________

™oµ a 1,b 1 ‹³Å—o 1
§¨ 1 ¸· _________________________
2 4 2 © 4 ¹

‹ŠÁ˜·¤‡Îµ˜°¨ŠÄœn°ŠªnµŠ Ä®o­°—‡¨o°Š„´„‘Á„–”ršÉ¸„ε®œ—Ä®o Á¤ºÉ° R Áž}œÁŽ˜…°Š‹Îµœªœ
‹¦Š· ¨³ a, b  R

1. a
b a b 2ab

3
7 3 7 2 3 7

2. a † b a b 1

2

5 † 2 _________________________

3. a'b a ab
4'2 _________________________

4. a
b a b 10
3
2 _________________________

3
2
5 _________________________

2
5 ______________________________

3
2
5 _________________________

⌦ 25
⌦

Á°„­µ¦‡ªµ¤¦oš¼ ɸ 2

„µ¦˜¦ª‹­°­¤´˜·¡ºœÊ “µœ…°ŠÁŽ˜ A £µ¥Ä˜o„µ¦„¦³šµÎ

1. ­¤˜´ ·ž—d

™oµ a, b  A ¨ªo a
b  A
­—Šªnµ A ¤­¸ ¤˜´ ž· d—£µ¥Ä˜o

2. ­¤´˜·„µ¦­¨´šÉ¸
™oµ a, b  A ¨oª a
b b
a
­—Šªµn A ¤¸­¤´˜·„µ¦­¨´šÉ£¸ µ¥Ä˜o

3. ­¤´˜·„µ¦Áž¨É¸¥œ„¨¤n»

™oµ a, b, c  A ¨ªo a
b
c a
b
c

­—Šªµn A ¤­¸ ¤´˜·„µ¦Áž¨¥É¸ œ„¨»¤n £µ¥Ä˜o

4. ­¤˜´ ·„µ¦¤¸Á°„¨´„¬–r

™µo a  A ¨³¤¸ e  A ŽÉ¹Š a
e a e
a
Á¦¥¸ „ e ªnµÁ°„¨´„¬–£r µ¥Ä˜„o µ¦„¦³šµÎ

­—Šªnµ A ¤Á¸ °„¨´„¬–r £µ¥Ä˜o

5. ­¤˜´ ·„µ¦¤°¸ œ· Áª°¦r­
™oµ a  A ¨³¤¸ a 1  A ŽÉŠ¹ a
a 1 e a 1
a
Á¦¥¸ „ a 1 ªnµÁž}œ°œ· Áª°¦r­…°Š a £µ¥Ä˜„o µ¦„¦³šµÎ

­—Šªµn A ¤°¸ ·œÁª°¦r­ £µ¥Ä˜o


˜´ª°¥nµŠ „ε®œ—Ä®o a
b a b 4 Á¤É°º a, b  R ‹Š˜¦ª‹­°ªµn ÁŽ˜ R ¤­¸ ¤˜´ ˜· n°Åžœ£Ê¸ µ¥Ä˜o

®¦º°Å¤n
1. ­¤´˜·ž—d
2. ­¤´˜„· µ¦­¨´šÉ¸
3. ­¤˜´ „· µ¦Áž¨É¥¸ œ„¨¤»n ŗo
4. ­¤´˜·„µ¦¤Á¸ °„¨„´ ¬–r
5. ­¤˜´ „· µ¦¤°¸ ·œÁª°¦r­
6. ‹Š®µ°·œÁª°¦­r …°Š 5 £µ¥Ä˜„o µ¦„¦³šµÎ

26 ⌫ ⌫  ⌦
 ⌫     ⌫  

ª·›š¸ µÎ

1. ‹µ„ a
b a b 4 Á¤º°É a, b  R

ÁœÉº°Š‹µ„ a, b  R —Š´ œœÊ´ a b 4  R

? R ¤¸­¤´˜ž· d—£µ¥Ä˜o


2. ‹µ„ a
b a b 4 Á¤É°º a, b  R

4 b
a b a 4
a b 4

?a
b b
a

? R ¤¸­¤´˜„· µ¦­¨´ š¸É£µ¥Ä˜o


3. ‹µ„ a
b a b 4

¡‹· µ¦–µ a
b
c a
b c 4

a b 4 c 4

¡‹· µ¦–µ a b c 8

a
b
c a b
c 4
a b c 4 4

a b c 8

4 a
b b
a

—´ŠœœÊ´ ÁŽ˜ R ¤­¸ ¤´˜·„µ¦Áž¨¥¸É œ„¨»¤n £µ¥Ä˜o


4. Ä®o e Ážœ} Á°„¨´„¬–r­µÎ ®¦´
, a  R a
‹³Å—o a
e a ¨³ e
a a
a e 4 a ¨³ e a 4
e 4 0 ¨³ e 4 0
? e 4 ¨³ e 4
—´ŠœÊœ´ ÁŽ˜ R ¤¸ –4 Áž}œÁ°„¨´„¬–£r µ¥Ä˜o


5. Ä®o a 1 Áž}œ°·œÁª°¦r­…°Š a ­Îµ®¦´
Á¤É°º a, a 1  R

‹³Å—o a 1
a 4 ¨³ a
a 1 4

a 1 a 4 4 ¨³ a a 1 4 4

a 1 8 a ¨³ a 1 8 a

—Š´ œœÊ´ ÁŽ˜ R ¤¸ 8 a Ážœ} °·œÁª°¦­r …°Š a £µ¥Ä˜o


6. 4 °·œÁª°¦r­…°Š a ‡°º a 8
—Š´ œ´Êœ °œ· Áª°¦­r …°Š 5 ‡º° 5 8 13

…\\\\\\…

⌦ 27
⌦

Á°„­µ¦ „f ®´—Á¡É·¤Á˜·¤

1. ‹ŠÁ˜¤· ‡Îµ˜°¨ŠÄœ°n Šªµn Š
„µÎ ®œ— a † b = a+b-ab Á¤º°É a, b Ážœ} ‹µÎ œªœ‹¦Š· ‹Š®µ
1.1 5 † 3 = _________________________
1.2 (2 † 4 ) † 4 = _______________________
1.3 Á°„¨´„¬–r…°Š † ‡°º ________________________
1.4 °œ· Áª°¦­r …°Š 5 ­Îµ®¦´ † ‡°º ________________________

2. Á¤º°É a ,b  R ×¥š¸É a † b = ab ¨³ a * b = ab
…o°Ä—Ážœ} ‹¦·Š…o°Ä—Ážœ} Áš‹È (‹ŠÂ­—Šª›· ‡¸ ·—)
2.1 a † (b * c ) = (a † b) * ( a † c )
2.2 (b*c) † a = (b† a) * (c † a )
2.3 a * (b † c) = (a* b) † (a*c)
2.4 (b † c) * a = (b* a) † (c* a)

ª›· ¸‡·—
___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

28 ⌫ ⌫  ⌦
 ⌫     ⌫  

Ÿœ„µ¦‹´—„µ¦Á¦¥¸ œ¦oš¼ ¸É 3

Á¦Éº°Š „µ¦œÎµ­¤´˜·…°Š‹µÎ œªœ‹¦·ŠÅžÄÄo œ„µ¦¡·­‹¼ œr Êœ´ ¤›´ ¥¤«¹„¬µžeš¸É 4
ª·µ ‡–˜· «µ­˜¦r Áª¨µ 2 ª´É äŠ

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

Ÿ¨„µ¦Á¦¸¥œ¦¼oš‡É¸ µ—®ªŠ´

¡­· ‹¼ œrš§¬‘¸ šÁ„¥¸É ª„´‹Îµœªœ‹¦Š· ŗo

1. ‹»—ž¦³­Š‡„r µ¦Á¦¥¸ œ¦¼o œ´„Á¦¥¸ œ­µ¤µ¦™
1.1 °„­¤´˜·…°Š‹Îµœªœ‹¦·Šš´ŠÊ 11 …o°Å—o
1.2 œµÎ ­¤´˜…· °Š‹µÎ œªœ‹¦Š· š´ÊŠ 11 …°o ¤µÄoĜ„µ¦¡­· ¼‹œšr §¬‘¸šÅ—o

2. œª‡ªµ¤‡·—®¨„´
­¤´˜¡· ʺœ“µœ…°Š¦³‹Îµœªœ‹¦Š· 11 …°o Á¡¥¸ Š¡°š‹¸É ³œµÎ ¤µÄ¡o ­· ¼‹œšr §¬’¸š

3. Áœ°Êº ®µ­µ¦³

­¤˜´ ·…°Š‹Îµœªœ‹¦·Š 11 …°o

­¤´˜…· °Š‹Îµœªœ‹¦·Š…o°š¸É 12, 13 ¨³ 14

…°o š¸É 12 ™µo aR Ž¹ÉŠ a z 0 ¨oª˜o°ŠÁž}œž¦³„µ¦Ä—ž¦³„µ¦®œŠ¹É Ášµn œ´œÊ ‡°º
„. aR ®¦º° …. - aR

…°o š¸É 13 ™oµ a,bR ¨oª a bR
…°o šÉ¸ 14 ™oµ a, bR ¨oª abR

œ°„‹µ„œ¸Ê Ĝ¦³‹Îµœªœ‹¦·Š¥´Š¤¸­¤´˜·šÉ¸­Îµ‡´°¸„ž¦³„µ¦®œÉ¹Š ‡º° ­¤´˜·‡ªµ¤¦·¼¦–r

Ž¹ŠÉ ‹³„¨µn ªÃ—¥¨³Á°¥¸ —£µ¥®¨Š´

4. „¦³ªœ„µ¦‹—´ „µ¦Á¦¥¸ œ¦¼o
4.1 œ„´ Á¦¥¸ œªn ¥„´œ­¦ž» ­¤˜´ ·…°Š‹Îµœªœ‹¦·ŠÁ„ɸ¥ª„´„µ¦ª„¨³„µ¦‡¼– 11 …o° ×¥„µ¦™µ¤

˜°Â¨ªo ‡¦œ¼ µÎ Ÿœ£¼¤­· ¦ž» ­¤˜´ …· °Š‹µÎ œªœ‹¦·ŠÄ®oœ´„Á¦¥¸ œ—°¼ „¸ ‡¦Ê´Š
4.2 ‡¦¼Â‹oŠÄ®oœ´„Á¦¸¥œš¦µªnµœ°„‹µ„­¤´˜·…°Š‹Îµœªœ‹¦·Š 11 …o°˜µ¤šÉ¸„¨nµªÂ¨oª ¦³

‹Îµœªœ‹¦·Š¥´Š¤¸¦³¥n°¥ R ŽÉ¹Š R  R ¨³¤¸­¤´˜·Á¡É·¤Á˜·¤°¸„ 4 ž¦³„µ¦ ‡º°…o°š¸É 12, 13, 14
¨³­¤´˜·‡ªµ¤¦· ¦¼ –r Áž}œ…°o šÉ¸ 15 ŽŠÉ¹ ‹³„¨nµª¨³Á°¥¸ —Äœ£µ¥®¨´Š

4.3 nŠœ´„Á¦¸¥œ°°„Áž}œ 6 „¨n»¤ ‡¦¼Â‹„Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 3 Ä®oš»„‡œ«¹„¬µ ¨³Ä®o
nª¥„´œ¡·­¼‹œrš§¬‘¸šš»„„¨»n¤ „¨»n¤¨³ 1 š ×¥„µ¦‹´Œ¨µ„ Á¤ºÉ°š¦µªnµÅ—ošÅ®œÂ¨oªÄ®ož¦¹„¬µ
„´œÂ¨³nª¥„´œšÎµ ¨³¤µÂ­—Šª·›¸šÎµœ„¦³—µœ—ε čoÁª¨µ 5-7 œµš¸ œ„¦³—µœ—ε‹³¤¸š§¬‘¸ššÉ¸
1-6 Ä®œo ´„Á¦¥¸ œš„» ‡œÄœ®°o Šnª¥„œ´ ˜¦ª‹­°‡ªµ¤™¼„˜o°ŠÃ—¥‡¦¼Ážœ} Ÿ¼™o µ¤Ä®oœ´„Á¦¥¸ œnª¥„´œ˜°

4.4 Ä®oœ´„Á¦¸¥œnª¥„´œ­¦»ž­¤´˜·‹Îµœªœ‹¦·Š šÊ´Š 14 …o° ×¥‡¦¼™µ¤Ä®oœ´„Á¦¸¥œnª¥„´œ˜°Â¨³
‡¦¼Á…¸¥œœ„¦³—µœ—ε

⌦ 29
⌦

­¤˜´ …· °Š¦³‹µÎ œªœ‹¦Š·

™oµ a,b ¨³ c Áž}œ‹Îµœªœ‹¦·ŠÄ—Ç

„µ¦ª„ „µ¦‡–¼

„µ¦ž—d 1. a+bR 6. abR

„µ¦­¨´ šÉ¸ 2. a+b = b+a 7. ab = ba

„µ¦Áž¨¥¸É œ„¨¤»n 3. (a+b)+c = a+(b+c) 8. (ab)c = a(bc)

„µ¦¤Á¸ °„¨„´ ¬–r 4. ¤¸ 0 Ž¹ÉŠ a+0 = a = 0+a 9. ¤¸ 1 Ž¹ŠÉ 1˜a = a = a˜1
10. ˜¨n ³ aR š¸É az0 ‹³¤¸ a 1 R Ž¹ÉŠ
„µ¦¤°¸ ·œÁª°¦­r 5. ˜¨n ³ aR ¤¸ (-a)R ŽÉ¹Š
a a 1 1 a 1 a
a+(-a) = 0 = (-a)+a

„µ¦Â‹„‹Š 11. a(b+c) = ab+ac

12. ­¤´˜·Å˜¦ª·£µ‡ ™oµ a Áž}œ‹Îµœªœ‹¦·ŠÄ—Ç …o°˜n°ÅžœÊ¸ …o°Ä—…o°®œÉ¹ŠÂ¨³Á¡¸¥Š…o°Á—¸¥ª
‹³˜°o ŠÁžœ} ‹¦Š· ‡°º 1. a = 0 ®¦º° 2. aR ®¦°º 3. - aR

13. ™oµ a, bR ¨oª a bR
14. ™µo a, bR ¨ªo abR

5. ®¨nŠ„µ¦Á¦¥¸ œ¦o¼
Á°„­µ¦Âœ³ÂœªšµŠš¸É 3

6. „¦³ªœ„µ¦ª´—¨³ž¦³Á¤·œŸ¨ „µ¦ž¦³Á¤œ· Ÿ¨
1. œ„´ Á¦¥¸ œ¦ªn ¤¤°º „´œšÎµŠµœ—¸
„µ¦ª´—Ÿ¨ 2. š„» ‡œÄ®o‡ªµ¤­œÄ‹—¸
1. ­´ŠÁ„˜„µ¦šÎµŠµœ„¨n»¤ 3. œ„´ Á¦¥¸ œ˜°‡µÎ ™µ¤Å—o™„¼ ˜o°Š—¤¸ µ„
2. ­Š´ Á„˜‡ªµ¤­œÄ‹ 4. œ„´ Á¦¸¥œšµÎ ŗ™o „¼ ˜o°Šž¦³¤µ– 80 %
3. ­Š´ Á„˜„µ¦˜°‡µÎ ™µ¤
4. šÎµÁ°„­µ¦Âœ³ÂœªšµŠšÉ¸ 3

7. œ´ š¹„®¨´Š­°œ
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
8. „·‹„¦¦¤Á­œ°Âœ³
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………

30 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦Âœ³ÂœªšµŠš¸É 3

‡µÎ Âʸ ‹Š Ä®oœ„´ Á¦¥¸ œ«¹„¬µš§¬‘¸ š˜°n Şœ¸Ê ¨oªÁ˜·¤Á®˜»Ÿ¨Á„ɸ¥ª„´ ­¤´˜·…°Š ‹Îµœªœ‹¦·Š®¨´Š…o°‡ªµ¤šÉ¸
¡·­‹¼ œr

š§¬‘¸š 1 („‘„µ¦˜´—°°„­Îµ®¦´ „µ¦ª„)
Á¤É°º a,b,c Áž}œ‹Îµœªœ‹¦Š· ėÇ
1. ™µo a+c = b+c ¨ªo a = b
2. ™µo c+a = c+b ¨ªo a = b

¡­· ‹¼ œr 1. ‹µ„ a+c = b+c
a+c+(-c) = b+c+(-c) „µ¦ª„—ªo ¥‹µÎ œªœšÁ¸É šµn „œ´

a+[c+(-c)] = b+[c+(-c)] _____________________________________________

a+0 = b+0 _____________________________________________

a=b _____________________________________________
2. ‹µ„ c+a = c+b

_____________________________________________________________

_____________________________________________________________

_____________________________________________________________

_____________________________________________________________

š§¬‘¸ š 2 („‘„µ¦˜—´ °°„­Îµ®¦´ „µ¦‡¼–)
Á¤ºÉ° a,b,c Áž}œ‹µÎ œªœ‹¦Š· ėÇ
1. ™oµ ac = bc ¨³ cz0 ¨ªo a = b
2. ™µo ca = cb ¨³ cz0 ¨oª a = b

¡­· ¼‹œr 1. ‹µ„ ac bc „µ¦‡–¼ —ªo ¥‹µÎ œªœšÁɸ šµn „´œ
_____________________________________________
ac c 1 bc c 1 _____________________________________________
_____________________________________________
a c ˜c 1 b c ˜c 1

a ˜1 b ˜1

ab

⌦ 31
⌦

2. ‹µ„ ca cb
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________

š§¬‘¸š 3 Á¤º°É aR

1. a˜0 = 0

2. 0˜a = 0 mm

¡­· ‹¼ œr

1. ‹µ„ 0+0=0 Á°„¨„´ ¬–„r µ¦ª„
__________________________________
a(0 + 0) = a˜0 __________________________________
__________________________________
a˜0 + a˜0 = a˜0 + 0 __________________________________
__________________________________
a˜0 = 0

2. ‹µ„ 0+0=0

(0 + 0)˜a = 0˜a

_____________________________________________________________

_____________________________________________________________

_____________________________________________________________

_____________________________________________________________

š§¬‘¸ š 4 Ä®o aR (-1)a = -a __________________________________
¡­· ‹¼ œr __________________________________
1 + (-1) = 0 __________________________________
(1 + (-1))˜a = 0˜a __________________________________
1˜a + (-1)˜a = 0 __________________________________
a + (-1) ˜a = a + (-a)

(-1)˜a = -a

š§¬‘¸š 5 Á¤É°º aR _________________________________

™oµ ab = 0 ¨oª a = 0 ®¦º° b = 0

¡·­‹¼ œr „¦–š¸ ɸ 1™oµ a = 0 ‹³Å—o 0˜b = 0
„¦–¸šÉ¸ 2™µo a z 0 ‹³¤¸ a 1
‹µ„ ab 0
a 1ab a 1 0

_____________________________________________________________

_____________________________________________________________

32 ⌫ ⌫  ⌦
 ⌫     ⌫  

š§¬‘¸š 6 Á¤°Éº aR

1. a(-b) = -ab

2. (-a)b = -ab

3. (-a)(-b) = ab ‹µ„š§¬‘¸š 4
¡­· ¼‹œr ­¤˜´ „· µ¦Áž¨¸É¥œ„¨n»¤…°Š„µ¦‡¼–
‹µ„š§¬‘¸š 4
1. a(-b) = a(-1)b
= (-1)(ab)
= -ab

2. (-a)b = ___________________________

= ___________________________

= ___________________________

3. (-a)(-b) = ___________________________

= ___________________________

= ___________________________

ª ®œµo œ¸¤Ê ¦¸ µŠª´¨ ©

ª›· š¸ ε˜n°Åžœ¸ÊŸ—· ˜¦ŠÅ®œ

ª·›š¸ µÎ Áœ°ºÉ Š‹µ„ a2 - a2 = (a + a) (a – a)
(a+a)(a–a)
—´ŠœÊœ´ a ( a – a ) = (a + a)
2.a
×¥­¤´˜·„µ¦˜—´ °°„ a= 2

—´Šœ´œÊ 1. a =

×¥­¤´˜„· µ¦˜—´ °°„ 1=

…bbbbbb…

⌦ 33
⌦

Ÿœ„µ¦‹—´ „µ¦Á¦¸¥œ¦¼oš¸É 4

Á¦º°É Š „µ¦¨Â¨³„µ¦®µ¦‹Îµœªœ‹¦Š· œÊ´ ¤´›¥¤«¹„¬µžše ɸ 4
ª· µ ‡–·˜«µ­˜¦r Áª¨µ 1 Éª´ äŠ

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Ÿ¨„µ¦Á¦¸¥œ¦š¼o ¸‡É µ—®ª´Š
œµÎ œ·¥µ¤„µ¦¨Â¨³„µ¦®µ¦‹Îµœªœ‹¦Š· ŞčÄo œ„µ¦¡­· ‹¼ œšr §¬‘¸ šÅ—o

1. ‹—» ž¦³­Š‡„r µ¦Á¦¸¥œ¦¼o œ„´ Á¦¥¸ œ­µ¤µ¦™
1.1 °„œ·¥µ¤„µ¦¨‹Îµœªœ‹¦Š· ŗo
1.2 œµÎ œ¥· µ¤„µ¦¨‹Îµœªœ‹¦·ŠÅžÄo¡·­¼‹œšr §¬‘¸ šÅ—o
1.3 °„œ·¥µ¤„µ¦®µ¦‹Îµœªœ‹¦Š· ŗo
1.4 œµÎ œ·¥µ¤„µ¦®µ¦‹Îµœªœ‹¦·ŠÅžÄo¡­· ¼‹œšr §¬‘¸ šÅ—o

2. œª‡ªµ¤‡—· ®¨´„
„µ¦¨Â¨³„µ¦®µ¦‹Îµœªœ‹¦·Š Áž}œ¦¼žÂ®œ¹ÉŠ…°Š„µ¦ª„¨³„µ¦‡¼–‹Îµœªœ‹¦·Š ŽÉ¹Š‹³¤¸

ž¦³Ã¥œ˜r °n „µ¦œÎµÅžÄ¡o ·­¼‹œšr §¬’¸š „µ¦Â„o­¤„µ¦Â¨³°­¤„µ¦˜n°Åž

3. Áœ°ºÊ ®µ­µ¦³
„µ¦¨Â¨³„µ¦®µ¦‹Îµœªœ‹¦·Š

šœ·¥µ¤ 1 Á¤°ºÉ a ¨³ b Áž}œ‹µÎ œªœ‹¦Š· Ä—Ç a – b = a + (-b)

šœ¥· µ¤ 2 Á¤É°º a ¨³ b Áž}œ‹µÎ œªœ‹¦Š· Ä—Ç Â¨³ bz0 a a ˜ b 1

,

b

4. „¦³ªœ„µ¦‹—´ „µ¦Á¦¸¥œ¦¼o
4.1 ‡¦¼Â‹„Á°„­µ¦Âœ³ÂœªšµŠš¸É 4.1 Ä®œo „´ Á¦¸¥œ«¹„¬µÂ¨³˜°‡Îµ™µ¤ čÁo ª¨µ 5-7 œµš¸
4.2 ‡¦¼ÁŒ¨¥Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 4.1 ×¥„µ¦™µ¤˜° Á¡ºÉ°Ä®oœ´„Á¦¸¥œ­¦»ž‡ªµ¤®¤µ¥…°Š

„µ¦¨‹Îµœªœ‹¦·ŠÄœ¦¼ž°·œÁª°¦r­…°Š„µ¦ª„ ¨³‡ªµ¤®¤µ¥…°Š„µ¦®µ¦‹Îµœªœ‹¦·ŠÄœ¦¼ž…°Š„µ¦
‡¼–—ªo ¥°œ· Áª°¦­r „µ¦‡¼–…°Š˜ª´ ®µ¦ Á¤É°º ˜ª´ ®µ¦Å¤Án ž}œ«œ¼ ¥r

34 ⌫ ⌫  ⌦
 ⌫     ⌫  

4.3 nŠ„¨n»¤œ´„Á¦¸¥œ°°„Áž}œ 5 „¨n»¤ Ä®oš»„„¨n»¤«¹„¬µÁ°„­µ¦Âœ³ÂœªšµŠš¸É 4.2 čo Áª¨µ 15
œµš¸ Á¡Éº°ªn ¥„œ´ ¡­· ¼‹œrš§¬‘¸ š˜µn ŠÇ

4.4 ­n»¤˜´ªÂšœ…°ŠÂ˜n¨³„¨»n¤ ­—Š„µ¦¡·­¼‹œrœ„¦³—µœ „¨n»¤¨³ 1 ‡œ ×¥Á¨º°„ ¡·­¼‹œr…o°
Å®œ„Èŗo ˜n˜o°ŠÅ¤nŽÊε„´œ ¨oªÄ®oš»„‡œÄœ®o°Šnª¥„´œ—¼‡ªµ¤™¼„˜o°Š ™oµœ´„Á¦¸¥œ­nªœÄ®nŤnÁ…oµÄ‹
®¦°º ¤¸…°o ėšÅɸ ¤n¤¸„¨¤»n ėšÎµÅ—o ‡¦¼‹³°›·µ¥Ä®Áo …oµÄ‹ ¨³Â­—Šª·›¡¸ ·­¼‹œrÄ®—o ‹¼ œ„¦³šÉŠ´ Á…oµÄ‹

4.5 Ä®oš»„‡œ„¨´ ޚ嚧¬‘¸š 3 Áž}œ„µ¦oµœ°¸„‡¦Š´Ê Ážœ} „µ¦¥Îʵ‡ªµ¤Á…oµÄ‹
4.6 ‡¦¼­¦»žœ·¥µ¤„µ¦¨Â¨³„µ¦®µ¦ ×¥„µ¦™µ¤˜°°¸„‡¦´ÊŠ Á¡Éº°Ä®oÁ®Èœž¦³Ã¥œrĜ„µ¦
œµÎ Şčo

5. ®¨Šn „µ¦Á¦¸¥œ¦¼o
Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 4.1, 4.2

6. „¦³ªœ„µ¦ª—´ ¨³ž¦³Á¤·œŸ¨ „µ¦ž¦³Á¤œ· Ÿ¨
1. œ´„Á¦¥¸ œ˜°‡µÎ ™µ¤Å—™o ¼„˜°o Š—¤¸ µ„
„µ¦ª´—Ÿ¨ 2. œ„´ Á¦¸¥œ¦nª¤„‹· „¦¦¤—¸
1. ­Š´ Á„˜‹µ„„µ¦˜°‡µÎ ™µ¤ 3. œ´„Á¦¸¥œÄ®o‡ªµ¤­œÄ‹—¸
2. ­´ŠÁ„˜„µ¦¦ªn ¤„‹· „¦¦¤ 4. œ´„Á¦¥¸ œšÎµÅ—™o ¼„˜°o Šž¦³¤µ– 85 %
3. ­Š´ Á„˜‹µ„‡ªµ¤­œÄ‹
4. šÎµÁ°„­µ¦Âœ³ÂœªšµŠšÉ¸ 4.1 , 4.2

7. œ´ š¹„®¨Š´ ­°œ
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
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8. „·‹„¦¦¤Á­œ°Âœ³
....................................................................................................................................................................
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⌦ 35
⌦

Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 4.1

‡Îµ­ÉŠ´ Ä®oœ„´ Á¦¸¥œÁ˜¤· ‡Îµ˜°¨ŠÄœ°n ŠªnµŠ Á¡ºÉ°®µ…°o ­¦»žÁžœ} œ·¥µ¤„µ¦¨Â¨³„µ¦®µ¦‹Îµœªœ‹¦Š·
1. °œ· Áª°¦r­„µ¦ª„…°Š 3 ‡°º ____________________
2. °œ· Áª°¦­r „µ¦ª„…°Š 5 ‡º° ____________________
3. °œ· Áª°¦r­„µ¦ª„…°Š a ‡°º ____________________
4. ‹ŠÁ˜·¤‹µÎ œªœ¨ŠÄœªŠÁ¨È

4.1 10 3 10 ( 3)

4.2 2 5 5 2 5 _____ 4.4 25 5 25 _____

4.3 2 1 2 _____ 4.5 a b a _____
33 3 Á¤É°º a,bR

5. ‹µ„…o° 4. ‹³Á®ÈœÅ—oªnµ‹ÎµœªœÄœªŠÁ¨È—oµœ…ªµÁž}œ°·œÁª°¦r­„µ¦ª„…°Š‹ÎµœªœšÉ¸Áž}œ˜´ª¨Äœ

—µo œŽµo ¥

­¦»žÁž}œšœ·¥µ¤Å—ªo nµ

Á¤Éº° a ¨³ b Ážœ} ‹Îµœªœ‹¦·ŠÄ—Ç a – b ‡°º Ÿ¨ª„…°Š __________ „´ _______________

6. °œ· Áª°¦r­„µ¦‡–¼ …°Š 3 ‡º°____________________

7. °œ· Áª°¦r­„µ¦‡–¼ …°Š 2 ‡°º ____________________

8. 5 5 ˜¨©§ 1 ¸¹· 10 10˜ _____
3 3 2

7 7˜ _____ a y b a a ˜ _____ Á¤°ºÉ bz0
4 b

9. ‹µ„…o° 8 ‹³Á®ÈœÅ—oªnµ‹ÎµœªœÄœªŠÁ¨È—oµœ…ªµÁž}œ____________________…°Š‹Îµœªœš¸ÉÁž}œ

˜´ª____________________Ĝ—oµœŽoµ¥

­¦»žÁž}œšœ·¥µ¤Å—ªo nµ

Á¤Éº° a ¨³ b Áž}œ‹µÎ œªœ‹¦Š· Ä—Ç a ‡°º Ÿ¨‡–¼ …°Š ______ „´ _______…°Š _______

b

®¤µ¥Á®˜» ‹µ„šœ¥· µ¤ Á¤°Éº a,bR ¨³ bz0

a y b a ˜b 1 ¨³ a a ˜ 1 —´ŠœÊ´œ 1 b 1

bb b

…












…

36 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦Âœ³ÂœªšµŠš¸É 4.2

‡µÎ ­É´Š Ä®oœ„´ Á¦¸¥œ«¹„¬µª·›¸„µ¦¡­· ‹¼ œrš§¬‘¸ š˜n°Åžœ¸Ê ¨oªÁ˜¤· …°o ‡ªµ¤ ¨ŠÄœ°n Šªµn ŠšÁɸ ªœo Ūo

š§¬‘¸ š1 Ä®o a,b ¨³ c R

1. a b c ab ac

2. a b c ac bc

3. a b c ab ac

¡­· ‹¼ œr

1. a b c a b c œ¥· µ¤„µ¦¨

ab a c „µ¦Â‹„‹Š

ab ac š§¬‘¸š 6 Á¦ºÉ°Š„µ¦‡–¼ ‹µÎ œªœ‹¦Š·

2. a b c ____________________

____________________

____________________
3. a b c ____________________

____________________

____________________

š§¬‘¸ š 2 ™µo a z 0 ‹³Å—o a 1 z 0
¡­· ‹¼ œr ­¤¤»˜·Ä®o a-1 = 0

—´ŠœÊœ´ a.a-1= a.0

=0
˜n a.a-1 = 1 Á„·—„µ¦…—´ Â¥Šo
—Š´ œÊ´œ a-1 z 0

š§¬‘¸ š 3

1. §¨ a ·¸ a Á¤Éº° b,cz0
© b ¹

c bc

a ac Á¤É°º b,cz0
2.

b bc Á¤º°É b,dz0
3. a c ad bc

b d bd

4. ©¨§ a ¹¸·©§¨ c ·¸¹ ac Á¤Éº° b,dz0
b d bd

⌦ 37
⌦

©§¨ b ·¸¹ 1 c
c b
5. Á¤Éº° b,cz0
Á¤º°É b,cz0
6. a ac
Á¤°Éº b,c,dz0
¨©§ b ¸·¹ b
c

7. ¨©§ a ¹¸· ad
b bc

©§¨ c ·¹¸
d

¡­· ¼‹œr

1. §¨© a ·¹¸ ab 1 (œ¥· µ¤„µ¦®µ¦)
b (œ¥· µ¤„µ¦®µ¦)
(„µ¦Áž¨¥¸É œ„¨¤»n …°Š„µ¦‡¼–)
cc
(œ¥· µ¤„µ¦®µ¦)
ab 1 c 1

a b 1c 1

a bc 1

a

bc
Ä®oœ„´ Á¦¥¸ œ¡­· ¼‹œr…°o 2.-7 Áž}œ„µ¦oµœ

38 ⌫ ⌫  ⌦
 ⌫     ⌫  

Ÿœ„µ¦‹—´ „µ¦Á¦¥¸ œ¦o¼š¸É 5

Á¦°Éº Š š§¬‘¸ šÁ«¬Á®¨º° œÊ´ ¤´›¥¤«„¹ ¬µžeš¸É 4
ª·µ ‡–˜· «µ­˜¦r Áª¨µ 1 ´ÉªÃ¤Š

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

Ÿ¨„µ¦Á¦¥¸ œ¦šo¼ ‡É¸ µ—®ª´Š
®µÁ«¬‹µ„„µ¦®µ¦¡®»œµ¤Ã—¥Äoš§¬’¸ šÁ«¬Á®¨º°Å—o

1. ‹—» ž¦³­Š‡„r µ¦Á¦¥¸ œ¦¼o
1.1 °„š§¬‘¸šÁ«¬Á®¨°º ŗo
1.2 čšo §¬‘¸ šÁ«¬Á®¨°º ®µÁ«¬‹µ„„µ¦®µ¦¡®œ» µ¤—ªo ¥¡®œ» µ¤š„ɸ 宜—Ä®oŗo

2. œª‡ªµ¤‡·—®¨´„
Ĝ„µ¦®µÁ«¬‹µ„„µ¦®µ¦¡®»œµ¤—oª¥¡®»œµ¤šÉ¸„ε®œ—Ä®o ‹³®µÅ—o¦ª—Á¦Èª…ʹœ™oµœÎµš§¬‘¸š

Á«¬Á®¨º°¤µÄo ¨³Ÿ¨š¸˜É µ¤¤µ‹³¤¸ž¦³Ã¥œr°¥µn Š¥ŠÉ· Ĝ„µ¦Â¥„˜ª´ ž¦³„° Á¡ºÉ°Â„­o ¤„µ¦¡®œ» µ¤

3. ÁœÊ°º ®µ­µ¦³

š§¬‘¸ šÁ«¬Á®¨°º

Á¤°ºÉ P(x) ‡º° ¡®œ» µ¤ anxn+an-1xn-1+an-2xn-2+…+a1x+a0×¥š¸É n Ážœ} ‹ÎµœªœÁ˜È¤ª„ an,an-1, …,a1,a0
Ážœ} ‹Îµœªœ‹¦·Š Ž¹ÉŠ anz 0 ™oµ®µ¦¡®œ» µ¤ P(x) —oª¥¡®œ» µ¤ x-c Á¤Éº° c Ážœ} ‹µÎ œªœ‹¦Š· ¨oª Á«¬‹³
Ášnµ„´ P(c)

®¤µ¥Á®˜» ¡®œ» µ¤šÉ¸ x Áž}œ˜ª´ ž¦Á—¥¸ ª °µ‹‹³Âšœ—ªo ¥ P(x) ®¦°º Q(x) ®¦°º R(x)
Ánœ P(x) = 3x2+2x-1

4. „¦³ªœ„µ¦‹—´ „µ¦Á¦¸¥œ¦¼o
4.1 ššªœ„µ¦®µ¦¡®»œµ¤—oª¥¡®»œµ¤ ×¥‡¦¼„ε®œ—Ëš¥r Ánœ 2x4-7x3+x2+5x-2 ®µ¦

—ªo ¥ x- 3 Ä®oœ´„Á¦¥¸ œªn ¥„œ´ ®µŸ¨®µ¦ ¨³ Á«¬ ˜µ¤ª·›¸š¸ÉÁ‡¥Á¦¸¥œ¤µÂ¨oª ŽÉ¹Šœ´„Á¦¸¥œ‡ª¦‹³®µÅ—oªnµ
Ÿ¨®µ¦ = 2x3-x2-2x-1 ¨³Á«¬ = -5

4.2 Ä®oœ´„Á¦¸¥œ‹´‡n¼Á¡Éº°nª¥„´œ«¹„¬µÁ°„­µ¦Âœ³ÂœªšµŠš¸É 5.1 čoÁª¨µ 10 œµš¸ Á­¦È‹Â¨oª‡¦¼

⌦ 39
⌦

ÁŒ¨¥Á°„­µ¦Ã—¥„µ¦™µ¤˜° Á¡ºÉ°­¦»ž…´Êœ˜°œ®¦º°ª·›¸„µ¦Ä®oœ´„Á¦¸¥œš¦µªnµÄœ„µ¦®µÁ«¬‹µ„„µ¦
®µ¦¡®œ» µ¤—ªo ¥¡®œ» µ¤—„¸ ¦¸ 1 Á¦µ‹³Äoš§¬‘¸ šÁ«¬Á®¨°º ªn ¥Äœ„µ¦®µ

4.3 ‹„Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 5.2 Ä®oœ´„Á¦¸¥œnª¥„´œ«¹„¬µ°¸„ 15 œµš¸ ¨oª‡¦¼°›·µ¥¥Êε°¸„
‡¦´ŠÊ ­Îµ®¦´ ˜ª´ š§¬‘¸ šÁ«¬Á®¨°º ¨³˜ª´ °¥nµŠÄœ„µ¦œµÎ š§¬‘¸ šÁ«¬Á®¨º°ÅžÄ®o µÁ«¬

4.4 ¥µÎÊ Ä®oœ„´ Á¦¸¥œÅ—˜o ¦³®œ´„™Š¹ ‡ªµ¤­µÎ ‡´ …°Š„µ¦®µÁ«¬Ã—¥Äoš§¬‘¸šÁ«¬Á®¨º° ×¥ÁŒ¡µ³
°¥µn Š¥·ŠÉ Áª¨µšÄ¸É Äo œ„µ¦®µ‹³Á¦Èª„ªµn ª·›„¸ µ¦˜´ŠÊ ®µ¦ Á¤ºÉ°œ„´ Á¦¸¥œš»„‡œÁ…µo ċ˜¦Š„œ´ ¨ªo Ä®šo µÎ  „f ®—´
šÉ¸ 2.6 …°o 1 ®œµo 72 Ĝ®œŠ´ ­°º Á¦¥¸ œ­µ¦³„µ¦Á¦¥¸ œ¦Áo¼ ¡¤É· Á˜¤· ²

5. ®¨Šn „µ¦Á¦¥¸ œ¦¼o
5.1 ®œ´Š­º°Á¦¸¥œ­µ¦³„µ¦Á¦¸¥œ¦¼oÁ¡É·¤Á˜¤· ‡–˜· «µ­˜¦r Á¨¤n 1 …°Š ­­ªš.
5.2 Á°„­µ¦Âœ³ÂœªšµŠš¸É 5.1 , 5.2

6. „¦³ªœ„µ¦ª´—¨³ž¦³Á¤·œŸ¨ „µ¦ž¦³Á¤·œŸ¨
1. œ„´ Á¦¸¥œ˜°‡µÎ ™µ¤Å—™o „¼ ˜°o ŠÁžœ} ­ªn œ¤µ„
„µ¦ª—´ Ÿ¨ 2. œ´„Á¦¥¸ œ­œÄ‹Â¨³˜ÊŠ´ ċÁ¦¸¥œ
1. ­Š´ Á„˜‹µ„„µ¦˜°‡Îµ™µ¤ 3. œ„´ Á¦¥¸ œšµÎ ŗ™o ¼„˜o°Šž¦³¤µ– 80 %
2. ­Š´ Á„˜‹µ„‡ªµ¤­œÄ‹ 4. œ´„Á¦¥¸ œšÎµÅ—™o ¼„˜o°Šž¦³¤µ– 85 %
3. šÎµÁ°„­µ¦Âœ³ÂœªšµŠš¸É 5.1 , 5.2
4. šÎµÃ‹š¥Âr  „f ®—´ 2.2 …o° 1,2 Ĝ

®œŠ´ ­°º Á¦¸¥œ­µ¦³„µ¦Á¦¥¸ œ¦¼Áo ¡É·¤Á˜·¤²

7. œ´ š„¹ ®¨Š´ ­°œ
…………………………………………………………………………………………………………....
…………………………………………………………………………………………………………....
…………………………………………………………………………………………………………....

8. „‹· „¦¦¤Á­œ°Âœ³
…………………………………………………………………………………………………………....
…………………………………………………………………………………………………………....
…………………………………………………………………………………………………………....

40 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 5.1

‡Îµ­ŠÉ´ Ä®oœ„´ Á¦¸¥œ«„¹ ¬µÁ°„­µ¦Â¨³˜°‡µÎ ™µ¤˜°n Şœ¸Ê°¥nµŠ˜n°ÁœÉ°º Š

1. ¡®œ» µ¤ 2x4- 7x3+x2+7x-3 ®µ¦—ªo ¥ x + 2 (×¥ª·›˜¸ ´ÊŠ®µ¦)
‹³Å—Ÿo ¨®µ¦ = …………………………… Á«¬ = …………………
™µo Ä®o ˜ª´ ˜ŠÊ´ = (˜´ª®µ¦ u Ÿ¨®µ¦) + Á«¬
‹µ„Ëš¥‹r ³Å—o 2x4- 7x3+x2+7x-3 = (x+2) (…………………….) + ………….

2. ™µo Ä®o P(x) šœ¡®»œµ¤šÁɸ žœ} ˜´ª˜´ÊŠ

x-c šœ¡®œ» µ¤šÁɸ ž}œ˜´ª®µ¦

q(x) šœ¡®œ» µ¤šÁɸ žœ} Ÿ¨®µ¦

r(x) šœÁ«¬šÅɸ —o‹µ„„µ¦®µ¦ ×¥š¸É r(x) = 0 ®¦°º r(x) Áž}œ¡®œ» µ¤š¸¤É —¸ „¸ ¦¸

œ°o ¥„ªµn —„¸ ¦¸…°Š x-c œÉœ´ ‡º° ¤—¸ „¸ ¦«¸ ¼œ¥r ®¦°º r(x) Áž}œ˜´ª‡Š˜´ª ™µo Ä®o r(x) = R Á¤É°º R Áž}œ˜ª´ ‡Š˜´ª

—´Šœ´Êœ P(x) = (x-c)(q(x)) + R

™µo Ä®o x = c ‹³Å—o P(c) = (c-c)q(c) +R

=R

3. ‹µ„…o° 1 P(x) = 2x4- 7x3+x2+7x-3

x-c = x+ 2
‹³Å—o c = -2
šœ‡µn x = -2 Ĝ P(x) ‹³Å—o P(-2) = 2(-2)4 –7(-2)3+(-2)2+7(-2)-3

= …………………………………………………………...
‡µn …°Š P(-2) = Á«¬šÉŸ —o‹µ„„µ¦œµÎ (x+2) Ş®µ¦ P(x) ®¦°º Ťn ………………………………...

4. „µÎ ®œ— P(x) = 2x4-5x3+2x2-x+2 Á¤°ºÉ œµÎ (x-3) Ş®µ¦ P(x)
‹³Å—o P(3) = ………………………………………….. = ………………………………………
—´Šœ´œÊ Á«¬šÅ¸É —o‹µ„„µ¦®µ¦ = …………………………………………………………………….

⌦ 41
⌦

Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 5.2

‡µÎ ­É´Š Ä®oœ„´ Á¦¸¥œ«„¹ ¬µš§¬‘¸ šÁ«¬Á®¨°º ¨³˜´ª°¥µn Š ¨ªo ˜°‡µÎ ™µ¤

š§¬‘¸šÁ«¬Á®¨°º (Remainder Theorem)

Á¤É°º P(x) ‡°º ¡®»œµ¤ anxn+an-1xn-1+an-2xn-2+ … +a1x+a0 ×¥š¸É n Ážœ} ‹ÎµœªœÁ˜¤È ª„ an,an-1,
…,a1,a0 Ážœ} ‹Îµœªœ‹¦Š· ŽŠ¹É anz 0 ™oµ®µ¦¡®»œµ¤ P(x) —ªo ¥¡®»œµ¤ x-c Á¤É°º c Áž}œ‹µÎ œªœ‹¦Š·
¨oª Á«¬‹³Ášµn „´ P(c)

®¤µ¥Á®˜» ¡®œ» µ¤š¸É x Áž}œ˜ª´ ž¦Á—¸¥ª °µ‹‹³Âšœ—ªo ¥ P(x) ®¦°º Q(x) ®¦°º R(x)
Áœn P(x) = 3x2+2x-1

™µo x Áž¨¸¥É œÁžœ} a ‹³Å—o
P(a) = 3a2+2a-1

™µo x Áž¨¥¸É œÁž}œ c ‹³Å—o
P(c) = 3c2+2c-1

˜ª´ °¥nµŠ ‹Š®µÁ«¬ Á¤°Éº 2x4-7x3+3x-5 ®µ¦—oª¥ x-3
ª·›š¸ ε Ä®o P(x) = 2x4-7x3+3x-5
¨³ x-c = x- 3

c =3
‹µ„š§¬‘¸šÁ«¬Á®¨°º Á¤Éº°®µ¦ P(x) —ªo ¥ x-c
‹³Å—o Á«¬ = P(c) = P(3)
‹µ„ P(x) = 2x4-7x3+3x-5

P(3) = 2(3)4-7(3)3+3(3)-5

= - 23
? Á«¬‹µ„„µ¦®µ¦ = - 23

‹µ„„µ¦«„¹ ¬µš§¬‘¸šÁ«¬Á®¨°º Ä®œo „´ Á¦¥¸ œÁ˜·¤‡µÎ ˜°Äœ°n Šªµn Š
1. Á¤°ºÉ ®µ¦ 2x3+7x2+3x —oª¥ x + 3

‹µ„š§¬‘¸ šÁ«¬Á®¨º° P(x) = ________________________________________________

x – c = ________________________________________________
—Š´ œ´œÊ c = ________________________________________________

42 ⌫ ⌫  ⌦
 ⌫     ⌫  

⌦ 43
⌦