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Transcript of +2 MATHEMATICS 10MARKS STUDY MATERIAL · +2 mathematics 10marks study material times.com page 1...
+2 MATHEMATICS 10MARKS STUDY MATERIAL
WWW.MATHS TIMES.COM Page 1
ªõ‚ì˜ ÞòŸèí¤î‹
𝟏. ´Õ Ó째¡½ò¾¢ý ÌòÐ째¡Î¸û ´§Ã ÒûǢ¢ø ºó¾¢ìÌõ
±ýÀ¾¨É ¦Åì¼÷ ӨȢø ¿¢Ú׸.O-2006,J-2008
¾£÷×: Ó째¡½ò¾¢ý ¯îº¢¸û 𝐴, 𝐵, 𝐶 ±ý¸.
ÌòÐ째¡Î¸û 𝐴𝐷, 𝐵𝐸 Ũø.
«¨Å¸û ¦ÅðÎõ ÒûÇ¢ 𝑂 ±ý¸.
𝐶𝑂 ¨Åî §º÷òÐ ¿£ð¼ «Ð 𝐴𝐵 ¨Â
𝐹 þø ºó¾¢ì¸¢ÈÐ ±ý¸.
ÌòÐ째¡Î¸û ´§Ã ÒûǢ¢ø ºó¾¢ìÌõ
±ýÀ¾¨É ¿¢ÚÅ, 𝐶𝐹 ⊥ 𝐴𝐵 ±É ¿¢ÚŢɡø §À¡Ðõ.
𝑂 ¨Åô ¦À¡ÚòÐ 𝐴, 𝐵, 𝐶 ¸Ç¢ý ¿¢¨Ä ¦Åì¼÷¸û 𝑎 , 𝑏 , 𝑐 ±ý¸.
𝑂𝐴 = 𝑎 , 𝑂𝐵 = 𝑏 , 𝑂𝐶 = 𝑐
𝐴𝐷 ⊥ 𝐵𝐶 ⇒ 𝑂𝐴 ⊥ 𝐵𝐶
⇒ 𝑂𝐴 ⋅ 𝐵𝐶 =0 ⇒ 𝑂𝐴 ⋅ (𝑂𝐶 − 𝑂𝐵 )=0 ⇒ 𝑎 ⋅ ( 𝑐 − 𝑏 )=0
𝑎 ⋅ 𝑐 − 𝑎 ⋅ 𝑏 =0 (1)
𝐵𝐸 ⊥ 𝐶𝐴 ⇒ 𝑂𝐵 ⊥ 𝐶𝐴
⇒ 𝑂𝐵 ⋅ 𝐶𝐴 =0 ⇒ 𝑂𝐵 ⋅ 𝑂𝐴 − 𝑂𝐶 =0 ⇒ 𝑏 ⋅ ( 𝑎 − 𝑐 )=0
𝑏 ⋅ 𝑎 − 𝑏 ⋅ 𝑐 =0 (2)
(1) ÁüÚõ (2) ¨Âì Üð¼
𝑎 ⋅ 𝑐 − 𝑎 ⋅ 𝑏 + 𝑏 ⋅ 𝑎 − 𝑏 ⋅ 𝑐 =0
⇒ 𝑎 ⋅ 𝑐 − 𝑏 ⋅ 𝑐 =0
⇒ 𝑎 − 𝑏 ⋅ 𝑐 =0 ⇒ 𝑂𝐴 − 𝑂𝐵 ⋅ 𝑂𝐶 =0
⇒ 𝐵𝐴 ⋅ 𝑂𝐶 =0 ⇒ 𝐵𝐴 ⊥ 𝑂𝐶 ⇒ 𝐶𝐹 ⊥ 𝐴𝐵 ±É§Å ãýÚ ÌòÐ째¡Î¸Ùõ ´§Ã ÒûǢ¢ø ºó¾¢ìÌõ §¸¡Î¸Ç¡Ìõ.
𝟐. 𝐜𝐨𝐬 𝑨 − 𝑩 = 𝐜𝐨𝐬𝑨 𝐜𝐨𝐬𝑩 + 𝐬𝐢𝐧𝑨 𝐬𝐢𝐧𝑩±ýÀ¾¨É ¦Åì¼÷ ӨȢø
¿¢Ú׸.
¾£÷×:𝑂 ¨Â ¨ÁÂÁ¡¸×õ ´ÃÄÌ ¬ÃÓõ
¦¸¡ñ¼ ´Õ Åð¼õ Ũø.∠𝑋𝑂𝑃 = 𝐴, ∠𝑋𝑂𝑄 = 𝐵
±ýÈÅ¡Ú «ùÅð¼ò¾¢ý À⾢¢ø 𝑃, 𝑄 ±ýÈ
þÕÒûÇ¢¸¨Ç ÌÈ¢ì¸.
∴ ∠𝑃𝑂𝑄 = ∠𝑃𝑂𝑋 − ∠𝑄𝑂𝑋 = 𝐴 − 𝐵.
𝑃𝑀, 𝑄𝐿 ⊥ 𝑂𝑋 Ũø.𝑃, 𝑄- ý ¬Âò¦¾¡¨Ä¸û
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ӨȧÂ(cos 𝐴, sin 𝐴 ) ÁüÚõ (cos 𝐵, sin 𝐵 ).
𝑥, 𝑦 − «îÍò ¾¢¨º¸Ç¢ø ¦ºÂøÀÎõ «ÄÌ ¦Åì¼÷¸û 𝑖 ÁüÚõ 𝑗 ±ý¸.
𝑂𝑃 = 𝑂𝑀 + 𝑀𝑃 = cos 𝐴 𝑖 + sin 𝐴 𝑗
𝑂𝑄 = 𝑂𝐿 + 𝐿𝑄 = cos 𝐵 𝑖 + sin 𝐵 𝑗 ÒûÇ¢ô¦ÀÕì¸Ä¢ý ŨèÈô ÀÊ
𝑂𝑃 ⋅ 𝑂𝑄 = 𝑂𝑃 𝑂𝑄 cos 𝐴 − 𝐵 = 1 1 cos 𝐴 − 𝐵 = cos 𝐴 − 𝐵 (1) ÒûÇ¢ô¦ÀÕì¸Ä¢ý Á¾¢ôÀ¢ý ÀÊ
𝑂𝑃 ⋅ 𝑂𝑄 = cos 𝐴 𝑖 + sin 𝐴 𝑗 ⋅ cos 𝐵 𝑖 + sin 𝐵 𝑗 = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵 (2)
(1) ÁüÚõ (2) Ä¢ÕóÐ
cos 𝐴 − 𝐵 = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵
𝟑. 𝐜𝐨𝐬 𝑨 + 𝑩 = 𝐜𝐨𝐬𝑨 𝐜𝐨𝐬𝑩 − 𝐬𝐢𝐧𝑨 𝐬𝐢𝐧𝑩±ýÀ¾¨É ¦Åì¼÷ ӨȢø
¿¢Ú׸. M-2006,M-2008
¾£÷×: 𝑂 ¨Â ¨ÁÂÁ¡¸×õ ´ÃÄÌ
¬ÃÓõ ¦¸¡ñ¼ ´Õ Åð¼õ Ũø.
∠𝑥𝑂𝑃 = 𝐴, ∠𝑥𝑂𝑄 = 𝐵 ±ýÈÅ¡Ú «ùÅð¼ò¾¢ý
À⾢¢ø 𝑃, 𝑄 ±ýÈ þÕÒûÇ¢¸¨Ç
ÌÈ¢ì¸.∴ ∠𝑃𝑂𝑄 = ∠𝑃𝑂𝑥 + ∠𝑄𝑂𝑥 = 𝐴 + 𝐵.
𝑃, 𝑄- ý ¬Âò¦¾¡¨Ä¸û
Өȧ (cos 𝐴, sin 𝐴 ) ÁüÚõ(cos 𝐵, −sin 𝐵 ).
𝑥, 𝑦 − «îÍò ¾¢¨º¸Ç¢ø ¦ºÂøÀÎõ
«ÄÌ ¦Åì¼÷¸û 𝑖 ÁüÚõ 𝑗 ±ý¸.𝑃𝑀, 𝑄𝐿 ⊥ 𝑂𝑋 Ũø.
𝑂𝑃 = 𝑂𝑀 + 𝑀𝑃 = cos 𝐴 𝑖 + sin 𝐴 𝑗
𝑂𝑄 = 𝑂𝐿 + 𝐿𝑄 = cos 𝐵 𝑖 − sin 𝐵 𝑗 ÒûÇ¢ô¦ÀÕì¸Ä¢ý ŨèÈô ÀÊ
𝑂𝑃 ⋅ 𝑂𝑄 = 𝑂𝑃 𝑂𝑄 cos 𝐴 + 𝐵 = 1 1 cos 𝐴 + 𝐵 = cos 𝐴 + 𝐵 (1)
ÒûÇ¢ô¦ÀÕì¸Ä¢ý Á¾¢ôÀ¢ý ÀÊ
𝑂𝑃 ⋅ 𝑂𝑄 = cos 𝐴 𝑖 + sin 𝐴 𝑗 ⋅ cos 𝐵 𝑖 − sin 𝐵 𝑗
= cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵 (2)
(1) ÁüÚõ (2) Ä¢ÕóÐ,
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cos 𝐴 + 𝐵 = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵
𝟒. 𝐬𝐢𝐧 𝑨 − 𝑩 = 𝐬𝐢𝐧𝑨 𝐜𝐨𝐬𝑩 − 𝐜𝐨𝐬𝑨𝐬𝐢𝐧𝑩±ýÀ¾¨É
¦Åì¼÷ ӨȢø ¿¢Ú׸. J-2007, O-2007, O-2008,O-2010,M-2012
¾£÷×: 𝑂 ¨Â ¨ÁÂÁ¡¸×õ ´ÃÄÌ ¬ÃÓõ ¦¸¡ñ¼´Õ Åð¼õ
Ũø.∠𝑋𝑂𝑃 = 𝐴, ∠𝑋𝑂𝑄 = 𝐵 ±ýÈÅ¡Ú «ùÅð¼ò¾¢ý À⾢¢ø 𝑃, 𝑄
±ýÈ þÕÒûÇ¢¸¨Ç ÌÈ¢ì¸.
∴ ∠𝑃𝑂𝑄 = ∠𝑃𝑂𝑋 − ∠𝑄𝑂𝑋 = 𝐴 − 𝐵.
𝑃, 𝑄- ý ¬Âò¦¾¡¨Ä¸û ӨȧÂ
(cos 𝐴, sin 𝐴 ) ÁüÚõ (cos 𝐵, sin 𝐵 ).
𝑥, 𝑦 − «îÍò ¾¢¨º¸Ç¢ø ¦ºÂøÀÎõ
«ÄÌ ¦Åì¼÷¸û 𝑖 ÁüÚõ 𝑗 ±ý¸.
𝑃𝑀, 𝑄𝐿 ⊥ 𝑂𝑋 Ũø.
𝑂𝑃 = 𝑂𝑀 + 𝑀𝑃 = cos 𝐴 𝑖 + sin 𝐴 𝑗
𝑂𝑄 = 𝑂𝐿 + 𝐿𝑄 = cos 𝐵 𝑖 + sin 𝐵 𝑗 ÌÚìÌô¦ÀÕì¸Ä¢ý ŨèÈô ÀÊ
𝑂𝑄 × 𝑂𝑃 = 𝑂𝑄 𝑂𝑃 sin 𝐴 − 𝐵 𝑘
= 1 1 sin 𝐴 − 𝐵 𝑘 = sin 𝐴 − 𝐵 𝑘 (1) ÌÚìÌô¦ÀÕì¸Ä¢ý Á¾¢ôÀ¢ý ÀÊ
𝑂𝑄 × 𝑂𝑃 = 𝑖 𝑗 𝑘
cos 𝐵 sin 𝐵 0cos 𝐴 sin 𝐴 0
= 𝑘 (sin𝐴 cos 𝐵 − cos 𝐴 sin 𝐵) (2)
(1) ÁüÚõ (2) Ä¢ÕóÐ, sin 𝐴 − 𝐵 = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵
𝟓. 𝐬𝐢𝐧 𝑨 + 𝑩 = 𝐬𝐢𝐧𝑨 𝐜𝐨𝐬𝑩 + 𝐜𝐨𝐬𝑨 𝐬𝐢𝐧𝑩±ýÀ¾¨É ¦Åì¼÷ ӨȢø
¿¢Ú׸.
¾£÷×: 𝑂 ¨Â ¨ÁÂÁ¡¸×õ ´ÃÄÌ ¬ÃÓõ ¦¸¡ñ¼ ´Õ Åð¼õ Ũø.
∠𝑥𝑂𝑃 = 𝐴, ∠𝑥𝑂𝑄 = 𝐵 ±ýÈÅ¡Ú «ùÅð¼ò¾¢ý À⾢¢ø 𝑃, 𝑄
±ýÈþÕÒûÇ¢¸¨Ç ÌÈ¢ì¸.
∴ ∠𝑃𝑂𝑄 = ∠𝑃𝑂𝑥 + ∠𝑄𝑂𝑥 = 𝐴 + 𝐵.
𝑃, 𝑄- ý ¬Âò¦¾¡¨Ä¸û ӨȧÂ
(cos 𝐴, sin 𝐴 ) ÁüÚõ (cos 𝐵, −sin 𝐵 ).
𝑥, 𝑦 − «îÍò ¾¢¨º¸Ç¢ø ¦ºÂøÀÎõ
«ÄÌ ¦Åì¼÷¸û 𝑖 ÁüÚõ 𝑗 ±ý¸.
𝑃𝑀, 𝑄𝐿 ⊥ 𝑂𝑋 Ũø.
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𝑂𝑃 = 𝑂𝑀 + 𝑀𝑃 = cos 𝐴 𝑖 + sin 𝐴 𝑗
𝑂𝑄 = 𝑂𝐿 + 𝐿𝑄 = cos 𝐵 𝑖 − sin 𝐵 𝑗 ÌÚìÌô¦ÀÕì¸Ä¢ý ŨèÈô ÀÊ
𝑂𝑄 × 𝑂𝑃 = 𝑂𝑄 𝑂𝑃 sin 𝐴 + 𝐵 𝑘
= 1 1 sin 𝐴 + 𝐵 𝑘 = sin 𝐴 + 𝐵 𝑘 (1)
ÌÚìÌô¦ÀÕì¸Ä¢ý Á¾¢ôÀ¢ý ÀÊ
𝑂𝑄 × 𝑂𝑃 = 𝑖 𝑗 𝑘
cos 𝐵 − sin 𝐵 0cos 𝐴 sin 𝐴 0
= 𝑘 (sin𝐴 cos 𝐵 + cos 𝐴 sin 𝐵) (2)
(1) ÁüÚõ (2) Ä¢ÕóÐ, sin 𝐴 + 𝐵 = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵
6. 𝒂 = 𝟐 𝒊 + 𝟑 𝒋 − 𝒌 , 𝒃 = −𝟐 𝒊 + 𝟓 𝒌 , 𝒄 = 𝒋 − 𝟑 𝒌 ±É¢ø
𝒂 × 𝒃 × 𝒄 = 𝒂 ⋅ 𝒄 𝒃 − 𝒂 ⋅ 𝒃 𝒄 ±É ºÃ¢À¡÷ì¸. M-2007, O-2008,O-2009
¾£÷×:
𝑏 × 𝑐 = 𝑖 𝑗 𝑘
−2 0 50 1 −3
= 0 − 5 𝑖 − 6 − 0 𝑗 + −2 − 0 𝑘
= −5 𝑖 − 6 𝑗 − 2 𝑘
𝑎 × 𝑏 × 𝑐 = 𝑖 𝑗 𝑘
2 3 −1−5 −6 −2
= −6 − 6 𝑖 − −4 − 5 𝑗 + (−12 + 15) 𝑘
= −12 𝑖 + 9 𝑗 + 3 𝑘 (1) 𝑎 ⋅ 𝑐 = 2 0 + 3 1 + −1 −3 = 0 + 3 + 3 = 6
𝑎 ⋅ 𝑏 = 2 −2 + 3 0 + −1 5 = −4 + 0 − 5 = −9
𝑎 ⋅ 𝑐 𝑏 − 𝑎 ⋅ 𝑏 𝑐 = (6) 𝑏 − (−9) 𝑐 = 6 𝑏 + 9 𝑐
= 6 −2 𝑖 + 5 𝑘 + 9( 𝑗 − 3 𝑘 )
= −12 𝑖 + 30 𝑘 + 9 𝑗 − 27 𝑘
= −12 𝑖 + 9 𝑗 + 3 𝑘 (2)
(1) ÁüÚõ (2) Ä¢ÕóÐ, 𝑎 × 𝑏 × 𝑐 = 𝑎 ⋅ 𝑐 𝑏 − 𝑎 ⋅ 𝑏 𝑐
7. 𝒂 = 𝒊 + 𝒋 + 𝒌 , 𝒃 = 𝟐 𝒊 + 𝒌 , 𝒄 = 𝟐 𝒊 + 𝒋 + 𝒌
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𝒅 = 𝒊 + 𝒋 + 𝟐 𝒌 ±É¢ø
𝒂 × 𝒃 × 𝒄 × 𝒅 = [ 𝒂 𝒃 𝒅 ] 𝒄 − [ 𝒂 𝒃 𝒄 ] 𝒅 ±ýÀ¨¾î ºÃ¢À¡÷ì¸.
M-2009 ¾£÷×:
𝑎 × 𝑏 = 𝑖 𝑗 𝑘
1 1 12 0 1
= 1 − 0 𝑖 − 1 − 2 𝑗 + 0 − 2 𝑘 = 𝑖 + 𝑗 − 2 𝑘
𝑐 × 𝑑 = 𝑖 𝑗 𝑘
2 1 11 1 2
= 2 − 1 𝑖 − 4 − 1 𝑗 + 2 − 1 𝑘 = 𝑖 − 3 𝑗 + 𝑘
𝑎 × 𝑏 × 𝑐 × 𝑑 = 𝑖 𝑗 𝑘
1 1 −21 −3 1
= 1 − 6 𝑖 − 1 + 2 𝑗 + −3 − 1 𝑘
= −5 𝑖 − 3 𝑗 − 4 𝑘 (1)
𝑎 𝑏 𝑐 = 1 1 12 0 12 1 1
= 1 0 − 1 − 1 2 − 2 + 1(2 − 0)
= 1 −1 − 1 0 + 1 2 = −1 − 0 + 2 = 1
𝑎 𝑏 𝑑 = 1 1 12 0 11 1 2
= 1 0 − 1 − 1 4 − 1 + 1 2 − 0
= 1 −1 − 1 3 + 1 2 = −1 − 3 + 2 = −2
𝑎 𝑏 𝑑 𝑐 − 𝑎 𝑏 𝑐 𝑑 = −2 𝑐 − 1 𝑑 = −2 𝑐 – 𝑑
= −2 2 𝑖 + 𝑗 + 𝑘 − ( 𝑖 + 𝑗 + 2 𝑘 )
= −4 𝑖 − 2 𝑗 − 2 𝑘 − 𝑖 − 𝑗 − 2 𝑘
= −5 𝑖 − 3 𝑗 − 4 𝑘 (2)
(1) ÁüÚõ (2) Ä¢ÕóÐ,
𝒂 × 𝒃 × 𝒄 × 𝒅 = 𝒂 𝒃 𝒅 𝒄 − 𝒂 𝒃 𝒄 𝒅
8. 𝒙−𝟏
𝟑=
𝒚−𝟏
−𝟏=
𝒛+𝟏
𝟎ÁüÚõ
𝒙−𝟒
𝟐=
𝒚
𝟎=
𝒛+𝟏
𝟑±ýÈ §¸¡Î¸û ¦ÅðÎõ ±Éì
¸¡ðÊ «¨Å ¦ÅðÎõ ÒûÇ¢¨Âì ¸¡ñ¸. J-2007,J-2009 ¾£÷×:
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𝐿1:𝑥 − 1
3=
𝑦 − 1
−1=
𝑧 + 1
0
⇒ 𝑎1 = 𝑖 + 𝑗 − 𝑘 §ÁÖõ 𝑢 = 3 𝑖 − 𝑗
𝐿2:𝑥 − 4
2=
𝑦
0=
𝑧 + 1
3
⇒ 𝑎2 = 4 𝑖 − 𝑘 §ÁÖõ 𝑣 = 2 𝑖 + 3 𝑘
𝑎2 − 𝑎1 = 4 𝑖 − 𝑘 − 𝑖 − 𝑗 + 𝑘 = 3 𝑖 − 𝑗
𝑎2 − 𝑎1 𝑢 𝑣 = 3 −1 03 −1 02 0 3
= 0 ∵ 𝑅1 ≡ 𝑅2
þíÌ 𝑢 -õ, 𝑣 -õ þ¨½ÂüȨŸû.
¦¸¡Îì¸ôÀð¼ þÕ §¸¡Î¸Ùõ ¦ÅðÎõ. 𝑥−1
3=
𝑦−1
−1=
𝑧+1
0= 𝜆 ±ý¸.
𝑥 − 1
3= 𝜆 ⇒ 𝑥 − 1 = 3𝜆 ⇒ 𝑥 = 3𝜆 + 1
𝑦 − 1
−1= 𝜆 ⇒ 𝑦 − 1 = −𝜆 ⇒ 𝑦 = −𝜆 + 1
𝑧 + 1
0= 𝜆 ⇒ 𝑧 + 1 = 0 ⇒ 𝑧 = −1
∴ 𝐿1þý Á£ÐûÇ ²§¾Ûõ ´Õ ÒûǢ¢ý «¨ÁôÒ(3𝜆 + 1, −𝜆 + 1, −1) ¬Ìõ.
𝑥−4
2=
𝑦
0=
𝑧+1
3= 𝜇 ±ý¸.
𝑥 − 4
2= 𝜇 ⇒ 𝑥 − 4 = 2𝜇 ⇒ 𝑥 = 2𝜇 + 4
𝑦
0= 𝜇 ⇒ 𝑦 = 0
𝑧 + 1
3= 𝜇 ⇒ 𝑧 + 1 = 3𝜇 ⇒ 𝑧 = 3𝜇 − 1
∴ 𝐿2þý Á£ÐûÇ ²§¾Ûõ ´Õ ÒûǢ¢ý «¨ÁôÒ(2𝜇 + 4,0,3𝜇 − 1)
¬Ìõ.𝐿1, 𝐿2 ±ýÈ §¸¡Î¸û ¦ÅðÊ즸¡ûž¡ø, ²§¾Ûõ 𝜆, 𝜇 ìÌ
3𝜆 + 1, −𝜆 + 1, −1 = 2𝜇 + 4,0,3𝜇 − 1
−𝜆 + 1 = 0 ⇒ −𝜆 = −1 ⇒ 𝜆 = 1
𝜆 = 1 ±É¢ø 3𝜆 + 1 = 3 + 1 = 4, , −𝜆 + 1 = −1 + 1 = 0
∴ ¦ÅðÎõ ÒûÇ¢ = (4,0, −1)
9. 𝒙−𝟏
𝟏=
𝒚+𝟏
−𝟏=
𝒛
𝟑ÁüÚõ
𝒙−𝟐
𝟏=
𝒚−𝟏
𝟐=
−𝒛−𝟏
𝟏±ýÈ §¸¡Î¸û ¦ÅðÎõ ±Éì
¸¡ðÊ «¨Å ¦ÅðÎõ ÒûÇ¢¨Âì ¸¡ñ¸. J-2006,J-2010
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¾£÷×:
𝐿1:𝑥 − 1
1=
𝑦 + 1
−1=
𝑧
3
⇒ 𝑎1 = 𝑖 − 𝑗 §ÁÖõ 𝑢 = 𝑖 − 𝑗 + 3 𝑘
𝐿2:𝑥 − 2
1=
𝑦 − 1
2=
−𝑧 − 1
1⇒
𝑥 − 2
1=
𝑦 − 1
2=
𝑧 + 1
−1
⇒ 𝑎2 = 2 𝑖 + 𝑗 − 𝑘 §ÁÖõ 𝑣 = 𝑖 + 2 𝑗 − 𝑘
𝑎2 − 𝑎1 = 2 𝑖 + 𝑗 − 𝑘 − 𝑖 + 𝑗 = 𝑖 + 2 𝑗 − 𝑘
𝑎2 − 𝑎1 𝑢 𝑣 = 1 2 −11 −1 31 2 −1
= 0 ∵ 𝑅1 ≡ 𝑅3
þíÌ 𝑢 -õ, 𝑣 -õ þ¨½ÂüȨŸû.
¦¸¡Îì¸ôÀð¼ þÕ §¸¡Î¸Ùõ ¦ÅðÎõ. 𝑥−1
1=
𝑦+1
−1=
𝑧
3= 𝜆 ±ý¸.
𝑥 − 1
1= 𝜆 ⇒ 𝑥 − 1 = 𝜆 ⇒ 𝑥 = 𝜆 + 1
𝑦 + 1
−1= 𝜆 ⇒ 𝑦 + 1 = −𝜆 ⇒ 𝑦 = −𝜆 − 1
𝑧
3= 𝜆 ⇒ 𝑧 = 3𝜆
∴ 𝐿1þý Á£ÐûÇ ²§¾Ûõ ´Õ ÒûǢ¢ý «¨ÁôÒ (𝜆 + 1, −𝜆 − 1,3𝜆) ¬Ìõ.
𝑥−2
1=
𝑦−1
2=
−𝑧−1
1= 𝜇 ±ý¸.
𝑥 − 2
1= 𝜇 ⇒ 𝑥 − 2 = 𝜇 ⇒ 𝑥 = 𝜇 + 2
𝑦 − 1
2= 𝜇 ⇒ 𝑦 − 1 = 2𝜇 ⇒ 𝑦 = 2𝜇 + 1
−𝑧 − 1
1= 𝜇 ⇒ −𝑧 − 1 = 𝜇 ⇒ −𝑧 = 𝜇 + 1 ⇒ 𝑧 = −𝜇 − 1
∴ 𝐿2þý Á£ÐûÇ ²§¾Ûõ ´Õ ÒûǢ¢ý «¨ÁôÒ(𝜇 + 2,2𝜇 + 1, −𝜇 − 1)
¬Ìõ.§¸¡Î¸û ¦ÅðÊ즸¡ûž¡ø, ²§¾Ûõ 𝜆, 𝜇 ìÌ
𝜆 + 1, −𝜆 − 1,3𝜆 = 𝜇 + 2,2𝜇 + 1, −𝜇 − 1 𝜆 + 1 = 𝜇 + 2
⇒ 𝜆 − 𝜇 = 1 (1)
−𝜆 − 1 = 2𝜇 + 1 ⇒ −𝜆 − 2𝜇 = 2 (2) 1 + 2 ⇒ 𝜆 − 𝜇 − 𝜆 − 2𝜇 = 1 + 2
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⇒ −3𝜇 = 3 ⇒ 𝜇 = −1
𝜇 = −1 ±É¢ø
𝜇 + 2 = −1 + 2 = 1,
2𝜇 + 1 = −2 + 1 = −1,
−𝜇 − 1 = 1 − 1 = 0 ∴ ¦ÅðÎõ ÒûÇ¢ = (1, −1,0)
10. (𝟐,−𝟏,−𝟑)ÅÆ¢§Âî ¦ºøÄìÜÊÂÐõ 𝒙−𝟐
𝟑=
𝒚−𝟏
𝟐=
𝒛−𝟑
−𝟒ÁüÚõ
𝒙−𝟏
𝟐=
𝒚+𝟏
−𝟑=
𝒛−𝟐
𝟐¬¸¢Â §¸¡Î¸ÙìÌ þ¨½Â¡¸ ¯ûÇÐÁ¡É ¾Çò¾¢ý
¦Åì¼÷ ÁüÚõ ¸¡÷˺¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸. M-2010 ¾£÷×:
¦Åì¼÷ ºÁýÀ¡Î
þíÌ 𝑎 = 2 𝑖 − 𝑗 − 3 𝑘 , 𝑢 = 3 𝑖 + 2 𝑗 − 4 𝑘 , 𝑣 = 2 𝑖 − 3 𝑗 + 2 𝑘
∴§¾¨ÅÂ¡É ¦Åì¼÷ ºÁýÀ¡Î 𝒓 = 𝒂 + 𝒔 𝒖 + 𝒕 𝒗 , 𝒕, 𝒔 ¾¢¨ºÂ¢Ä¢¸û
𝑟 = 2 𝑖 − 𝑗 − 3 𝑘 + 𝑠 3 𝑖 + 2 𝑗 − 4 𝑘 + 𝑡 2 𝑖 − 3 𝑗 + 2 𝑘
¸¡÷˺¢Âý ºÁýÀ¡Î
þíÌ
𝑥1, 𝑦1, 𝑧1 = 2, −1, −3 ; 𝑙1, 𝑚1, 𝑛1 = 3,2, −4 ; 𝑙2, 𝑚2, 𝑛2 = (2, −3,2)
¾Çò¾¢ý ºÁýÀ¡Î
𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑙1 𝑚1 𝑛1
𝑙2 𝑚2 𝑛2
= 0
𝑥 − 2 𝑦 + 1 𝑧 + 3
3 2 −42 −3 2
= 0
𝑥 − 2 4 − 12 − 𝑦 + 1 6 + 8 + 𝑧 + 3 −9 − 4 = 0 𝑥 − 2 (−8) − (𝑦 + 1)(14) + (𝑧 + 3)(−13) = 0
−8𝑥 + 16 − 14𝑦 − 14 − 13𝑧 − 39 = 0 −8𝑥 − 14𝑦 − 13𝑧 − 37 = 0
8𝑥 + 14𝑦 + 13𝑧 + 37 = 0
11. (𝟏,𝟑, 𝟐)±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøÅÐõ 𝒙+𝟏
𝟐=
𝒚+𝟏
−𝟏=
𝒛+𝟑
𝟑ÁüÚõ
𝒙−𝟐
𝟏=
𝒚+𝟏
𝟐=
𝒛+𝟐
𝟐¬¸¢Â §¸¡Î¸ÙìÌ þ¨½Â¡¸ ¯ûÇÐÁ¡É ¾Çò¾¢ý
¦Åì¼÷ ÁüÚõ ¸¡÷˺¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.
¾£÷×:
¦Åì¼÷ ºÁýÀ¡Î
þíÌ
𝑎 = 𝑖 + 3 𝑗 + 2 𝑘 , 𝑢 = 2 𝑖 − 𝑗 + 3 𝑘 , 𝑣 = 𝑖 + 2 𝑗 + 2 𝑘 ∴§¾¨ÅÂ¡É ¦Åì¼÷ ºÁýÀ¡Î 𝒓 = 𝒂 + 𝒔 𝒖 + 𝒕 𝒗 , 𝒕, 𝒔 ¾¢¨ºÂ¢Ä¢¸û
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𝑟 = 𝑖 + 3 𝑗 + 2 𝑘 + 𝑠 2 𝑖 − 𝑗 + 3 𝑘 + 𝑡 𝑖 + 2 𝑗 + 2 𝑘
¸¡÷˺¢Âý ºÁýÀ¡Î
þíÌ 𝑥1, 𝑦1, 𝑧1 = 1,3,2 ; 𝑙1, 𝑚1, 𝑛1 = 2, −1,3 ; 𝑙2, 𝑚2, 𝑛2 = (1,2,2)
¾Çò¾¢ý ºÁýÀ¡Î
𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑙1 𝑚1 𝑛1
𝑙2 𝑚2 𝑛2
= 0
𝑥 − 1 𝑦 − 3 𝑧 − 2
2 −1 31 2 2
= 0
𝑥 − 1 −2 − 6 − 𝑦 − 3 4 − 3 + 𝑧 − 2 4 + 1 = 0
𝑥 − 1 (−8) − (𝑦 − 3)(1) + (𝑧 − 2)(5) = 0
−8𝑥 + 8 − 𝑦 + 3 + 5𝑧 − 10 = 0
−8𝑥 − 𝑦 + 5𝑧 + 1 = 0
8𝑥 + 𝑦 − 5𝑧 − 1 = 0
12. (−𝟏,𝟑, 𝟐)±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøÅÐõ 𝒙 + 𝟐𝒚 + 𝟐𝒛 = 𝟓ÁüÚõ 𝟑𝒙 +
𝒚 + 𝟐𝒛 = 𝟖 ¬¸¢Â ¾Çí¸ÙìÌ ¦ºíÌò¾¡ÉÐÁ¡É ¾Çò¾¢ý ¦Åì¼÷
ÁüÚõ ¸¡÷˺¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.
¾£÷×:
¦Åì¼÷ ºÁýÀ¡Î
þíÌ
𝑎 = − 𝑖 + 3 𝑗 + 2 𝑘 , 𝑢 = 𝑖 + 2 𝑗 + 2 𝑘 , 𝑣 = 3 𝑖 + 𝑗 + 2 𝑘
∴§¾¨ÅÂ¡É ¦Åì¼÷ ºÁýÀ¡Î 𝒓 = 𝒂 + 𝒔 𝒖 + 𝒕 𝒗 , 𝒕, 𝒔 ¾¢¨ºÂ¢Ä¢¸û
𝑟 = − 𝑖 + 3 𝑗 + 2 𝑘 + 𝑠 𝑖 + 2 𝑗 + 2 𝑘 + 𝑡 3 𝑖 + 𝑗 + 2 𝑘
¸¡÷˺¢Âý ºÁýÀ¡Î
þíÌ 𝑥1, 𝑦1, 𝑧1 = −1,3,2 ; 𝑙1, 𝑚1, 𝑛1 = 1,2,2 ; 𝑙2,𝑚2, 𝑛2 = (3,1,2)
¾Çò¾¢ý ºÁýÀ¡Î
𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑙1 𝑚1 𝑛1
𝑙2 𝑚2 𝑛2
= 0
𝑥 + 1 𝑦 − 3 𝑧 − 2
1 2 23 1 2
= 0
𝑥 + 1 4 − 2 − 𝑦 − 3 2 − 6 + 𝑧 − 2 1 − 6 = 0 𝑥 + 1 2 − 𝑦 − 3 −4 + 𝑧 − 2 −5 = 0
2𝑥 + 2 + 4𝑦 − 12 − 5𝑧 + 10 = 0
2𝑥 + 4𝑦 − 5𝑧 = 0
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13. (−𝟏,−𝟐, 𝟏)±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøÅÐõ 𝒙 + 𝟐𝒚 + 𝟒𝒛 + 𝟕 = 𝟎
ÁüÚõ 𝟐𝒙 − 𝒚 + 𝟑𝒛 + 𝟑 = 𝟎 ¬¸¢Â ¾Çí¸ÙìÌ
¦ºíÌò¾¡ÉÐÁ¡É ¾Çò¾¢ý ¦Åì¼÷ ÁüÚõ ¸¡÷˺¢Âý
ºÁýÀ¡Î¸¨Çì ¸¡ñ¸. J-2006,M-2008 ¾£÷×:
¦Åì¼÷ ºÁýÀ¡Î
þíÌ 𝑎 = − 𝑖 − 2 𝑗 + 𝑘 , 𝑢 = 𝑖 + 2 𝑗 + 4 𝑘 , 𝑣 = 2 𝑖 − 𝑗 + 3 𝑘
∴§¾¨ÅÂ¡É ¦Åì¼÷ ºÁýÀ¡Î 𝒓 = 𝒂 + 𝒔 𝒖 + 𝒕 𝒗 , 𝒕, 𝒔 ¾¢¨ºÂ¢Ä¢¸û
𝑟 = − 𝑖 − 2 𝑗 + 𝑘 + 𝑠 𝑖 + 2 𝑗 + 4 𝑘 + 𝑡 2 𝑖 − 𝑗 + 3 𝑘
¸¡÷˺¢Âý ºÁýÀ¡Î
þíÌ 𝑥1, 𝑦1, 𝑧1 = −1, −2,1 ; 𝑙1, 𝑚1, 𝑛1 = 1,2,4 ; 𝑙2, 𝑚2, 𝑛2 = (2, −1,3)
¾Çò¾¢ý ºÁýÀ¡Î
𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑙1 𝑚1 𝑛1
𝑙2 𝑚2 𝑛2
= 0
𝑥 + 1 𝑦 + 2 𝑧 − 1
1 2 42 −1 3
= 0
𝑥 + 1 6 + 4 − 𝑦 + 2 3 − 8 + 𝑧 − 1 −1 − 4 = 0
𝑥 + 1 10 − 𝑦 + 2 −5 + 𝑧 − 1 −5 = 0
10𝑥 + 10 + 5𝑦 + 10 − 5𝑧 + 5 = 0
10𝑥 + 5𝑦 − 5𝑧 + 25 = 0
2𝑥 + 𝑦 − 𝑧 + 5 = 0
14. (𝟏,𝟐, −𝟐)ÅÆ¢§Âî ¦ºøÄìÜÊÂÐõ𝒙+𝟐
𝟑=
𝒚+𝟏
−𝟐=
𝒛−𝟒
−𝟒±ýÈ §¸¡ðÊüÌ
þ¨½Â¡¸×õ, 𝟐𝒙 + 𝟑𝒚 + 𝟑𝒛 = 𝟖±ýÈ ¾Çò¾¢üÌ ¦ºíÌò¾¡¸×õ
¯ûÇ ¾Çò¾¢ý ¦Åì¼÷ ÁüÚõ ¸¡÷˺¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.
O-2010 ¾£÷×:
¦Åì¼÷ ºÁýÀ¡Î
þíÌ 𝑎 = 𝑖 + 2 𝑗 − 2 𝑘 , 𝑢 = 3 𝑖 − 2 𝑗 − 4 𝑘 , 𝑣 = 2 𝑖 + 3 𝑗 + 3 𝑘
∴§¾¨ÅÂ¡É ¦Åì¼÷ ºÁýÀ¡Î 𝒓 = 𝒂 + 𝒔 𝒖 + 𝒕 𝒗 , 𝒕, 𝒔 ¾¢¨ºÂ¢Ä¢¸û
𝑟 = 𝑖 + 2 𝑗 − 2 𝑘 + 𝑠 3 𝑖 − 2 𝑗 − 4 𝑘 + 𝑡 2 𝑖 + 3 𝑗 + 3 𝑘
¸¡÷˺¢Âý ºÁýÀ¡Î
þíÌ 𝑥1, 𝑦1, 𝑧1 = 1,2, −2 ; 𝑙1, 𝑚1, 𝑛1 = 3, −2, −4 ; 𝑙2, 𝑚2, 𝑛2 = (2,3,3)
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¾Çò¾¢ý ºÁýÀ¡Î
𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑙1 𝑚1 𝑛1
𝑙2 𝑚2 𝑛2
= 0
𝑥 − 1 𝑦 − 2 𝑧 + 2
3 −2 −42 3 3
= 0
𝑥 − 1 −6 + 12 − 𝑦 − 2 9 + 8 + 𝑧 + 2 9 + 4 = 0
𝑥 − 1 6 − 𝑦 − 2 17 + 𝑧 + 2 13 = 0
6𝑥 − 6 − 17𝑦 + 34 + 13𝑧 + 26 = 0
6𝑥 − 17𝑦 + 13𝑧 + 54 = 0
15. 𝒙−𝟐
𝟐=
𝒚−𝟐
𝟑=
𝒛−𝟏
𝟑±ýÈ §¸¡ð¨¼ ¯ûǼ츢ÂÐõ,
𝒙+𝟏
𝟑=
𝒚−𝟏
𝟐=
𝒛+𝟏
𝟏±ýÈ
§¸¡ðÊüÌ þ¨½Â¡ÉÐÁ¡É ¾Çò¾¢ý ¦Åì¼÷ ÁüÚõ ¸¡÷˺¢Âý
ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.
¾£÷×:
¦Åì¼÷ ºÁýÀ¡Î
þíÌ 𝑎 = 2 𝑖 + 2 𝑗 + 𝑘 , 𝑢 = 2 𝑖 + 3 𝑗 + 3 𝑘 , 𝑣 = 3 𝑖 + 2 𝑗 + 𝑘
∴§¾¨ÅÂ¡É ¦Åì¼÷ ºÁýÀ¡Î 𝒓 = 𝒂 + 𝒔 𝒖 + 𝒕 𝒗 , 𝒕, 𝒔 ¾¢¨ºÂ¢Ä¢¸û
𝑟 = 2 𝑖 + 2 𝑗 + 𝑘 + 𝑠 2 𝑖 + 3 𝑗 + 3 𝑘 + 𝑡 3 𝑖 + 2 𝑗 + 𝑘
¸¡÷˺¢Âý ºÁýÀ¡Î
þíÌ 𝑥1, 𝑦1, 𝑧1 = 2,2,1 ; 𝑙1,𝑚1, 𝑛1 = 2,3,3 ; 𝑙2, 𝑚2, 𝑛2 = (3,2,1)
¾Çò¾¢ý ºÁýÀ¡Î
𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑙1 𝑚1 𝑛1
𝑙2 𝑚2 𝑛2
= 0
𝑥 − 2 𝑦 − 2 𝑧 − 1
2 3 33 2 1
= 0
𝑥 − 2 3 − 6 − 𝑦 − 2 2 − 9 + 𝑧 − 1 4 − 9 = 0 𝑥 − 2 −3 − 𝑦 − 2 −7 + 𝑧 − 1 −5 = 0
−3𝑥 + 6 + 7𝑦 − 14 − 5𝑧 + 5 = 0 −3𝑥 + 7𝑦 − 5𝑧 − 3 = 0
3𝑥 − 7𝑦 + 5𝑧 + 3 = 0
16. 𝑨(𝟏, −𝟐,𝟑)ÁüÚõ 𝑩(−𝟏,𝟐, −𝟏)±ýÈ ÒûÇ¢¸û ÅÆ¢§Âî
¦ºøÄìÜÊÂÐõ 𝒙−𝟐
𝟐=
𝒚+𝟏
𝟑=
𝒛−𝟏
𝟒±ýÈ §¸¡ðÊüÌ þ¨½Â¡ÉÐÁ¡É
¾Çò¾¢ý ¦Åì¼÷ ÁüÚõ ¸¡÷˺¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.
¾£÷×:
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¦Åì¼÷ ºÁýÀ¡Î
þíÌ 𝑎 = 𝑖 − 2 𝑗 + 3 𝑘 , 𝑏 = − 𝑖 + 2 𝑗 − 𝑘 , 𝑣 = 2 𝑖 + 3 𝑗 + 4 𝑘
∴§¾¨ÅÂ¡É ¦Åì¼÷ ºÁýÀ¡Î
𝒓 = (𝟏 − 𝒕) 𝒂 + 𝒕 𝒃 + 𝒔 𝒗 , 𝒕, 𝒔 ¾¢¨ºÂ¢Ä¢¸û
𝑟 = 1 − 𝑡 𝑖 − 2 𝑗 + 3 𝑘 + 𝑡 − 𝑖 + 2 𝑗 − 𝑘 + 𝑠 2 𝑖 + 3 𝑗 + 4 𝑘
¸¡÷˺¢Âý ºÁýÀ¡Î
þíÌ 𝑥1, 𝑦1, 𝑧1 = 1, −2,3 ; 𝑥2, 𝑦2, 𝑧2 = −1,2, −1 ; 𝑙, 𝑚, 𝑛 = (2,3,4) ¾Çò¾¢ý ºÁýÀ¡Î
𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1
𝑙 𝑚 𝑛 = 0
𝑥 − 1 𝑦 + 2 𝑧 − 3
−1 − 1 2 + 2 −1 − 32 3 4
= 0
𝑥 − 1 𝑦 + 2 𝑧 − 3−2 4 −42 3 4
= 0
𝑥 − 1 16 + 12 − 𝑦 + 2 −8 + 8 + 𝑧 − 3 −6 − 8 = 0 𝑥 − 1 28 − 𝑦 + 2 0 + 𝑧 − 3 −14 = 0
28𝑥 − 28 + 0 − 14𝑧 + 42 = 0
28𝑥 − 14𝑧 + 14 = 0
2𝑥 − 𝑧 + 1 = 0
17. (𝟏, 𝟐, 𝟑)ÁüÚõ (𝟐,𝟑, 𝟏)±ýÈ ÒûÇ¢¸û ÅÆ¢§Âî ¦ºøÄìÜÊÂÐõ
𝟑𝒙 − 𝟐𝒚 + 𝟒𝒛 − 𝟓 = 𝟎 ±ýÈ ¾Çò¾¢üÌ ¦ºíÌò¾¡¸×õ «¨Áó¾
¾Çò¾¢ý ¦Åì¼÷ ÁüÚõ ¸¡÷˺¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.
M-2006, O-2006, O-2007,J-2008,M-2012 ¾£÷×:
¦Åì¼÷ ºÁýÀ¡Î
þíÌ 𝑎 = 𝑖 + 2 𝑗 + 3 𝑘 , 𝑏 = 2 𝑖 + 3 𝑗 + 𝑘 , 𝑣 = 3 𝑖 − 2 𝑗 + 4 𝑘
∴§¾¨ÅÂ¡É ¦Åì¼÷ ºÁýÀ¡Î
𝒓 = (𝟏 − 𝒕) 𝒂 + 𝒕 𝒃 + 𝒔 𝒗 , 𝒕, 𝒔 ¾¢¨ºÂ¢Ä¢¸û
𝑟 = 1 − 𝑡 𝑖 + 2 𝑗 + 3 𝑘 + 𝑡 2 𝑖 + 3 𝑗 + 𝑘 + 𝑠 3 𝑖 − 2 𝑗 + 4 𝑘
¸¡÷˺¢Âý ºÁýÀ¡Î
þíÌ 𝑥1, 𝑦1, 𝑧1 = 1,2,3 ; 𝑥2, 𝑦2, 𝑧2 = 2,3,1 ; 𝑙, 𝑚, 𝑛 = (3, −2,4)
¾Çò¾¢ý ºÁýÀ¡Î 𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1
𝑙 𝑚 𝑛 = 0
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𝑥 − 1 𝑦 − 2 𝑧 − 32 − 1 3 − 2 1 − 3
3 −2 4 = 0
𝑥 − 1 𝑦 + 2 𝑧 − 3
1 1 −23 −2 4
= 0
𝑥 − 1 4 − 4 − 𝑦 − 2 4 + 6 + 𝑧 − 3 −2 − 3 = 0 𝑥 − 1 0 − 𝑦 − 2 10 + 𝑧 − 3 −5 = 0
−10𝑦 + 20 − 5𝑧 + 15 = 0 −10𝑦 − 5𝑧 + 35 = 0
2𝑦 + 𝑧 − 7 = 0
18. (−𝟏, 𝟏, 𝟏)ÁüÚõ (𝟏,−𝟏, 𝟏)±ýÈ ÒûÇ¢¸û ÅÆ¢§Âî ¦ºøÄìÜÊÂÐõ
𝒙 + 𝟐𝒚 + 𝟐𝒛 = 𝟓 ±ýÈ ¾Çò¾¢üÌ ¦ºíÌò¾¡¸×õ «¨Áó¾ ¾Çò¾¢ý
¦Åì¼÷ ÁüÚõ ¸¡÷˺¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.
M-2007,M-2009,J-2010 ¾£÷×:
¦Åì¼÷ ºÁýÀ¡Î
þíÌ 𝑎 = − 𝑖 + 𝑗 + 𝑘 , 𝑏 = 𝑖 − 𝑗 + 𝑘 , 𝑣 = 𝑖 + 2 𝑗 + 2 𝑘
∴§¾¨ÅÂ¡É ¦Åì¼÷ ºÁýÀ¡Î
𝒓 = (𝟏 − 𝒕) 𝒂 + 𝒕 𝒃 + 𝒔 𝒗 , 𝒕,𝒔 ¾¢¨ºÂ¢Ä¢¸û
𝑟 = 1 − 𝑡 − 𝑖 + 𝑗 + 𝑘 + 𝑡 𝑖 − 𝑗 + 𝑘 + 𝑠 𝑖 + 2 𝑗 + 2 𝑘
¸¡÷˺¢Âý ºÁýÀ¡Î
þíÌ 𝑥1, 𝑦1, 𝑧1 = −1,1,1 ; 𝑥2, 𝑦2, 𝑧2 = 1, −1,1 ; 𝑙, 𝑚, 𝑛 = (1,2,2)
¾Çò¾¢ý ºÁýÀ¡Î 𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1
𝑙 𝑚 𝑛 = 0
𝑥 + 1 𝑦 − 1 𝑧 − 11 + 1 −1 − 1 1 − 1
1 2 2 = 0
𝑥 + 1 𝑦 − 1 𝑧 − 1
2 −2 01 2 2
= 0
𝑥 + 1 −4 − 0 − 𝑦 − 1 4 − 0 + 𝑧 − 1 4 + 2 = 0 𝑥 + 1 −4 − 𝑦 − 1 4 + 𝑧 − 1 6 = 0
−4𝑥 − 4 − 4𝑦 + 4 + 6𝑧 − 6 = 0 −4𝑥 − 4𝑦 + 6𝑧 − 6 = 0
2𝑥 + 2𝑦 − 3𝑧 + 3 = 0
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19. 𝒙−𝟐
𝟐=
𝒚−𝟐
𝟑=
𝒛−𝟏
−𝟐±ýÈ §¸¡ð¨¼ ¯ûǼ츢ÂÐõ (−𝟏, 𝟏, −𝟏)±ýÈ
ÒûÇ¢ ÅÆ¢§Âî ¦ºøÄì ÜÊÂÐÁ¡É ¾Çò¾¢ý ¦Åì¼÷ ÁüÚõ
¸¡÷˺¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.
¾£÷×:
¦Åì¼÷ ºÁýÀ¡Î
þíÌ 𝑎 = − 𝑖 + 𝑗 − 𝑘 , 𝑏 = 2 𝑖 + 2 𝑗 + 𝑘 , 𝑣 = 2 𝑖 + 3 𝑗 − 2 𝑘
∴§¾¨ÅÂ¡É ¦Åì¼÷ ºÁýÀ¡Î
𝒓 = 𝟏 − 𝒕 𝒂 + 𝒕 𝒃 + 𝒔 𝒗 , 𝒕, 𝒔 ¾¢¨ºÂ¢Ä¢¸û
𝑟 = 1 − 𝑡 − 𝑖 + 𝑗 − 𝑘 + 𝑡 2 𝑖 + 2 𝑗 + 𝑘 + 𝑠 2 𝑖 + 3 𝑗 − 2 𝑘
¸¡÷˺¢Âý ºÁýÀ¡Î
þíÌ 𝑥1, 𝑦1, 𝑧1 = −1,1, −1 , 𝑥2, 𝑦2, 𝑧2 = 2,2,1 , 𝑙, 𝑚, 𝑛 = (2,3, −2)
¾Çò¾¢ý ºÁýÀ¡Î 𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1
𝑙 𝑚 𝑛 = 0
𝑥 + 1 𝑦 − 1 𝑧 + 12 + 1 2 − 1 1 + 1
2 3 −2 = 0
𝑥 − 1 𝑦 − 1 𝑧 + 1
3 1 22 3 −2
= 0
𝑥 + 1 −2 − 6 − 𝑦 − 1 −6 − 4 + 𝑧 + 1 9 − 2 = 0 𝑥 + 1 −8 − 𝑦 − 1 −10 + 𝑧 + 1 7 = 0
−8𝑥 − 8 + 10𝑦 − 10 + 7𝑧 + 7 = 0 −8𝑥 + 10𝑦 + 7𝑧 − 11 = 0
8𝑥 − 10𝑦 − 7𝑧 + 11 = 0
20. 𝟑 𝒊 + 𝟒 𝒋 + 𝟐 𝒌 , 𝟐 𝒊 − 𝟐 𝒋 − 𝒌 ÁüÚõ 𝟕 𝒊 + 𝒌 ¬¸¢ÂÅü¨È ¿¢¨Ä ¦Åì¼÷¸Ç¡¸ì ¦¸¡ñ¼ ÒûÇ¢¸û ÅÆ¢§Âî
¦ºøÖõ ¾Çò¾¢ý ¦Åì¼÷ ÁüÚõ ¸¡÷˺¢Âý ºÁýÀ¡Î¸¨Çì
¸¡ñ¸. J-2009 ¾£÷×:¦Åì¼÷ ºÁýÀ¡Î
þíÌ 𝑎 = 3 𝑖 + 4 𝑗 + 2 𝑘 , 𝑏 = 2 𝑖 − 2 𝑗 − 𝑘 , 𝑐 = 7 𝑖 + 𝑘
∴§¾¨ÅÂ¡É ¦Åì¼÷ ºÁýÀ¡Î
𝒓 = (𝟏 − 𝒔 − 𝒕) 𝒂 + 𝒔 𝒃 + 𝒕 𝒄 , 𝒕, 𝒔 ¾¢¨ºÂ¢Ä¢¸û
𝑟 = 1 − 𝑠 − 𝑡 3 𝑖 + 4 𝑗 + 2 𝑘 + 𝑠 2 𝑖 − 2 𝑗 − 𝑘 + 𝑡 7 𝑖 + 𝑘
¸¡÷˺¢Âý ºÁýÀ¡Î
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þíÌ 𝑥1, 𝑦1, 𝑧1 = 3,4,2 ; 𝑥2, 𝑦2, 𝑧2 = 2, −2,−1 ; 𝑥3, 𝑦3, 𝑧3 = (7,0,1)
¾Çò¾¢ý ºÁýÀ¡Î
𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1
𝑥3 − 𝑥1 𝑦3 − 𝑦1 𝑧3 − 𝑧1
= 0
𝑥 − 3 𝑦 − 4 𝑧 − 22 − 3 −2 − 4 −1 − 27 − 3 0 − 4 1 − 2
= 0
𝑥 − 3 𝑦 − 4 𝑧 − 2−1 −6 −34 −4 −1
= 0
𝑥 − 3 6 − 12 − 𝑦 − 4 1 + 12 + 𝑧 − 2 4 + 24 = 0 (𝑥 − 3) −6 − 𝑦 − 4 13 + 𝑧 − 2 28 = 0
−6𝑥 + 18 − 13𝑦 + 52 + 28𝑧 − 56 = 0 −6𝑥 − 13𝑦 + 28𝑧 + 14 = 0
6𝑥 + 13𝑦 − 28𝑧 − 14 = 0
21. 𝟐,𝟐, −𝟏 , (𝟑, 𝟒, 𝟐)ÁüÚõ (𝟕, 𝟎,𝟔)¬¸¢Â ÒûÇ¢¸û ÅÆ¢§Âî
¦ºøÄìÜÊ ¾Çò¾¢ý ¦Åì¼÷ ÁüÚõ ¸¡÷˺¢Âý ºÁýÀ¡Î¸¨Çì
¸¡ñ¸. O-2009 ¾£÷×: ¦Åì¼÷ ºÁýÀ¡Î
þíÌ 𝑎 = 2 𝑖 + 2 𝑗 − 𝑘 , 𝑏 = 3 𝑖 + 4 𝑗 + 2 𝑘 , 𝑐 = 7 𝑖 + 6 𝑘
∴§¾¨ÅÂ¡É ¦Åì¼÷ ºÁýÀ¡Î
𝒓 = 𝟏 − 𝒔 − 𝒕 𝒂 + 𝒔 𝒃 + 𝒕 𝒄 , 𝒕, 𝒔 ¾¢¨ºÂ¢Ä¢¸û
𝑟 = 1 − 𝑠 − 𝑡 2 𝑖 + 2 𝑗 − 𝑘 + 𝑠 3 𝑖 + 4 𝑗 + 2 𝑘 + 𝑡 7 𝑖 + 6 𝑘
¸¡÷˺¢Âý ºÁýÀ¡Î
þíÌ 𝑥1, 𝑦1, 𝑧1 = 2,2, −1 ; 𝑥2, 𝑦2, 𝑧2 = 3,4,2 ; 𝑥3, 𝑦3, 𝑧3 = (7,0,6)
¾Çò¾¢ý ºÁýÀ¡Î
𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1
𝑥3 − 𝑥1 𝑦3 − 𝑦1 𝑧3 − 𝑧1
= 0
𝑥 − 2 𝑦 − 2 𝑧 + 13 − 2 4 − 2 2 + 17 − 2 0 − 2 6 + 1
= 0
𝑥 − 2 𝑦 − 2 𝑧 + 1
1 2 35 −2 7
= 𝟎
𝑥 − 2 14 + 6 − 𝑦 − 2 7 − 15 + 𝑧 + 1 −2 − 10 = 0
𝑥 − 2 20 − 𝑦 − 2 −8 + 𝑧 + 1 −12 = 0
20𝑥 − 40 + 8𝑦 − 16 − 12𝑧 − 12 = 0
20𝑥 + 8𝑦 − 12𝑧 − 68 = 0
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5𝑥 + 2𝑦 − 3𝑧 − 17 = 0
22. ¦ÅðÎòÐñÎ ÅÊÅ¢ø ´Õ ¾Çò¾¢ý ºÁýÀ¡ð¨¼ì ¸¡ñ¸.M-2010 ¾£÷×: ¸¡÷˺¢Âý ºÁýÀ¡Î
´Õ ¾Çò¾¢ý, 𝑥 − ¦ÅðÎòÐñÎ 𝑎, 𝑦 − ¦ÅðÎòÐñÎ 𝑏, 𝑧 −
¦ÅðÎòÐñÎ 𝑐 ±ý¸.
∴ ¾ÇÁ¡ÉÐ 𝑎, 0,0 , (0, 𝑏, 0) ÁüÚõ (0,0, 𝑐)¬¸¢Â ÒûÇ¢¸û ÅÆ¢§Âî
¦ºøÖõ.
¦Åì¼÷ ºÁýÀ¡Î
þíÌ 𝑎 = 𝑎 𝑖 , 𝑏 = 𝑏 𝑗 , 𝑐 = 𝑐 𝑘
∴§¾¨ÅÂ¡É ¦Åì¼÷ ºÁýÀ¡Î
𝒓 = (𝟏 − 𝒔 − 𝒕) 𝒂 + 𝒔 𝒃 + 𝒕 𝒄 , 𝒕, 𝒔 ¾¢¨ºÂ¢Ä¢¸û
𝒓 = 𝟏 − 𝒔 − 𝒕 𝒂 𝒊 + 𝒔𝒃 𝒋 + 𝒕𝒄 𝒌
þíÌ 𝑥1, 𝑦1, 𝑧1 = 𝑎, 0,0 , 𝑥2, 𝑦2, 𝑧2 = 0, 𝑏, 0 ,
𝑥3, 𝑦3, 𝑧3 = (0,0, 𝑐)
¾Çò¾¢ý ºÁýÀ¡Î
𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1
𝑥3 − 𝑥1 𝑦3 − 𝑦1 𝑧3 − 𝑧1
= 0
𝑥 − 𝑎 𝑦 − 0 𝑧 − 00 − 𝑎 𝑏 − 0 0 − 00 − 𝑎 0 − 0 𝑐 − 0
= 0 ⇒ 𝑥 − 𝑎 𝑦 𝑧−𝑎 𝑏 0−𝑎 0 𝑐
= 0
𝑥 − 𝑎 𝑏𝑐 − 0 − 𝑦 −𝑎𝑐 − 0 + 𝑧 0 + 𝑎𝑏 = 0 𝑏𝑐𝑥 − 𝑎𝑏𝑐 + 𝑎𝑐𝑦 + 𝑎𝑏𝑧 = 0
𝑎𝑏𝑐 ¬ø ÅÌì¸,
𝑥
𝑎− 1 +
𝑦
𝑏+
𝑧
𝑐= 0 ⇒
𝑥
𝑎+
𝑦
𝑏+
𝑧
𝑐= 1
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fy¥bg§fŸ
1. 𝑃v‹D« òŸë fy¥ò v§ kh¿ 𝑧I¡ F¿¤jhš 𝑃 Ï‹ ãak¥ghijia
arg 𝑧−1
𝑧+1 =
𝜋
3v‹w f£L¥gh£o‰F c£g£L fh§f.
¾£÷×
𝑧 = 𝑥 + 𝑖𝑦 ±ý¸.
𝑧 − 1 = 𝑥 + 𝑖𝑦 − 1 = 𝑥 − 1 + 𝑖𝑦 𝑧 + 1 = 𝑥 + 𝑖𝑦 + 1 = 𝑥 + 1 + 𝑖𝑦 𝑧 − 1
𝑧 + 1=
𝑥 − 1 + 𝑖𝑦
𝑥 + 1 + 𝑖𝑦
= 𝑥 − 1 + 𝑖𝑦
𝑥 + 1 + 𝑖𝑦 ×
𝑥 + 1 − 𝑖𝑦
𝑥 + 1 − 𝑖𝑦
= 𝑥 − 1 𝑥 + 1 − 𝑖 𝑥 − 1 𝑦 + 𝑖𝑦 𝑥 + 1 − 𝑖2𝑦2
(𝑥 + 1)2 + 𝑦2
=𝑥2 − 1 + 𝑦2 − 𝑖 𝑥 − 1 𝑦 − 𝑦 𝑥 + 1
(𝑥 + 1)2 + 𝑦2
=𝑥2 + 𝑦2 − 1 − 𝑖 𝑥𝑦 − 𝑦−𝑥𝑦 − 𝑦
(𝑥 + 1)2 + 𝑦2
=𝑥2 + 𝑦2 − 1
(𝑥 + 1)2 + 𝑦2+ 𝑖
2𝑦
(𝑥 + 1)2 + 𝑦2
¦ÁöÀ̾¢ =𝑥2 + 𝑦2 − 1
(𝑥 + 1)2 + 𝑦2¸üÀ¨É À̾¢=
2y
(x+1)2+y2
¸ðÎôÀ¡Î:
arg 𝑧 − 1
𝑧 + 1 =
𝜋
3⇒ tan−1
¸üÀ¨ÉÀ̾¢
¦ÁöÀ̾¢ =
𝜋
3
⇒¸üÀ¨ÉÀ̾¢
¦ÁöÀ̾¢= tan
𝜋
3= 3 ⇒
2𝑦
(𝑥+1)2+𝑦2
𝑥2+𝑦2−1
(𝑥+1)2+𝑦2
= 3 ⇒2𝑦
𝑥2 + 𝑦2 − 1= 3
⇒ 2𝑦 = 3(𝑥2 + 𝑦2 − 1) ⇒ 3(𝑥2 + 𝑦2) − 2𝑦 − 3 = 0
3(𝑥2 + 𝑦2) − 2𝑦 − 3 = 0v‹gJ njitahd ãak¥ghijahF«.
2. 𝑃v‹D« òŸë fy¥ò v§ kh¿ 𝑧I¡ F¿¤jhš 𝑃 Ï‹ ãak¥ghijia
Re 𝑧−1
𝑧+1 = 1v‹w f£L¥gh£o‰F c£g£L fh§f.
¾£÷×
𝑧 = 𝑥 + 𝑖𝑦 ±ý¸.
𝑧 − 1 = 𝑥 + 𝑖𝑦 − 1 = 𝑥 − 1 + 𝑖𝑦
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𝑧 + 1 = 𝑥 + 𝑖𝑦 + 1 = 𝑥 + 1 + 𝑖𝑦 𝑧 − 1
𝑧 + 1=
𝑥 − 1 + 𝑖𝑦
𝑥 + 1 + 𝑖𝑦
= 𝑥 − 1 + 𝑖𝑦
𝑥 + 1 + 𝑖𝑦 ×
𝑥 + 1 − 𝑖𝑦
𝑥 + 1 − 𝑖𝑦
= 𝑥 − 1 𝑥 + 1 − 𝑖 𝑥 − 1 𝑦 + 𝑖𝑦 𝑥 + 1 − 𝑖2𝑦2
(𝑥 + 1)2 + 𝑦2
=𝑥2 − 1 + 𝑦2 − 𝑖 𝑥 − 1 𝑦 − 𝑦 𝑥 + 1
(𝑥 + 1)2 + 𝑦2
=𝑥2 + 𝑦2 − 1 − 𝑖 𝑥𝑦 − 𝑦−𝑥𝑦 − 𝑦
(𝑥 + 1)2 + 𝑦2
=𝑥2 + 𝑦2 − 1
(𝑥 + 1)2 + 𝑦2 + 𝑖2𝑦
(𝑥 + 1)2 + 𝑦2
¦ÁöÀ̾¢ =𝑥2 + 𝑦2 − 1
(𝑥 + 1)2 + 𝑦2,¸üÀ¨É À̾¢=
2y
(x+1)2+y2
¸ðÎôÀ¡Î:
Re 𝑧 − 1
𝑧 + 𝑖 = 1 ⇒
𝑥 𝑥 − 1 + 𝑦 𝑦 + 1
𝑥2 + (𝑦 + 1)2 = 1
⇒ 𝑥2 − 𝑥 + 𝑦2 + 𝑦 = 𝑥2 + 𝑦2 + 2𝑦 + 1 ⇒ −𝑥 − 𝑦 = 1 ∴ 𝑥 + 𝑦 + 1 = 0v‹gJ njitahd ãak¥ghijahF«.
3. 𝑃v‹D« òŸë fy¥ò v§ kh¿ 𝑧I¡ F¿¤jhš 𝑃 Ï‹ ãak¥ghijia
arg 𝑧−1
𝑧+3 =
𝜋
2v‹w f£L¥gh£o‰F c£g£L fh§f.
¾£÷×
𝑧 = 𝑥 + 𝑖𝑦 ±ý¸.
𝑧 − 1 = 𝑥 + 𝑖𝑦 − 1 = 𝑥 − 1 + 𝑖𝑦 𝑧 + 3 = 𝑥 + 𝑖𝑦 + 3 = 𝑥 + 3 + 𝑖𝑦 𝑧 − 1
𝑧 + 3=
𝑥 − 1 + 𝑖𝑦
𝑥 + 3 + 𝑖𝑦
= 𝑥 − 1 + 𝑖𝑦
𝑥 + 3 + 𝑖𝑦 ×
𝑥 + 3 − 𝑖𝑦
𝑥 + 3 − 𝑖𝑦
= 𝑥 − 1 𝑥 + 3 − 𝑖 𝑥 − 1 𝑦 + 𝑖𝑦 𝑥 + 3 − 𝑖2𝑦2
(𝑥 + 3)2 + 𝑦2
=𝑥2 + 2𝑥 − 3 + 𝑦2 − 𝑖 𝑥 − 1 𝑦 − 𝑦 𝑥 + 3
(𝑥 + 1)2 + 𝑦2
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=𝑥2 + 𝑦2 + 2𝑥 − 3 − 𝑖 𝑥𝑦 − 𝑦−𝑥𝑦 − 3𝑦
(𝑥 + 3)2 + 𝑦2
=𝑥2 + 𝑦2 + 2𝑥 − 3
(𝑥 + 3)2 + 𝑦2− 𝑖
4𝑦
(𝑥 + 3)2 + 𝑦2
¦ÁöÀ̾¢ =𝑥2 + 𝑦2 + 2𝑥 − 3
(𝑥 + 3)2 + 𝑦2¸üÀ¨É À̾¢= −
4𝑦
(𝑥 + 3)2 + 𝑦2
¸ðÎôÀ¡Î:
arg 𝑧 − 1
𝑧 + 3 =
𝜋
2⇒ tan−1
¸üÀ¨ÉÀ̾¢
¦ÁöÀ̾¢ =
𝜋
2
⇒¸üÀ¨ÉÀ̾¢
¦ÁöÀ̾¢= tan
𝜋
2= ∞ ⇒ ¦ÁöÀ̾¢ = 0
⇒𝑥2 + 𝑦2 + 2𝑥 − 3
(𝑥 + 3)2 + 𝑦2 = 0 ⇒ 𝑥2 + 𝑦2 + 2𝑥 − 3 = 0
𝑥2 + 𝑦2 + 2𝑥 − 3 = 0v‹gJ njitahd ãak¥ghijahF«.
4. 𝑃v‹D« òŸë fy¥ò v§ kh¿ 𝑧I¡ F¿¤jhš 𝑃 Ï‹ ãak¥ghijia
Im 2𝑧+1
𝑖𝑧+1 = −2 v‹w f£L¥gh£o‰F c£g£L fh§f.
¾£÷×
𝑧 = 𝑥 + 𝑖𝑦 ±ý¸.
2𝑧 + 1 = 2 𝑥 + 𝑖𝑦 + 1 = 2𝑥 + 1 + 𝑖2𝑦 𝑖𝑧 + 1 = 𝑖 𝑥 + 𝑖𝑦 + 1 = 1 − 𝑦 + 𝑖𝑥 2𝑧 + 1
𝑖𝑧 + 1=
2𝑥 + 1 + 𝑖2𝑦
1 − 𝑦 + 𝑖𝑥×
1 − 𝑦 − 𝑖𝑥
1 − 𝑦 − 𝑖𝑥
= ¦ÁöôÀ̾¢ + 𝑖2𝑦 1 − 𝑦 − 𝑥 2𝑥 + 1
(1 − 𝑦)2 + 𝑥2
¸üÀ¨ÉÀ̾¢ =2𝑦(1 − 𝑦) − 𝑥(2𝑥 + 1)
(1 − 𝑦)2 + 𝑥2
¸ðÎôÀ¡Î:
Im 2𝑧 + 1
𝑖𝑧 + 1 = −2 ⇒
2𝑦 1 − 𝑦 − 𝑥 2𝑥 + 1
(1 − 𝑦)2 + 𝑥2= −2
⇒ 2𝑦 1 − 𝑦 − 𝑥 2𝑥 + 1 = −2 (1 − 𝑦)2 + 𝑥2 ⇒ 2𝑦 − 2𝑦2 − 2𝑥2 − 𝑥 = −2 1 − 2𝑦 + 𝑦2 + 𝑥2 ⇒ 2𝑦−2𝑦2−2𝑥2 − 𝑥 = −2 + 4𝑦−2𝑦2−2𝑥2 ⇒ 2𝑦 − 𝑥 − 4𝑦 + 2 = 0 ⇒ −𝑥 − 2𝑦 + 2 = 0
∴ 𝑥 + 2𝑦 − 2 = 0v‹gJ njitahd ãak¥ghijahF«.
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5. 𝑃v‹D« òŸë fy¥ò v§ kh¿ 𝑧I¡ F¿¤jhš 𝑃 Ï‹ ãak¥ghijia
Re 𝑧+1
𝑧+𝑖 = 1v‹w f£L¥gh£o‰F c£g£L fh§f.
¾£÷×
𝑧 = 𝑥 + 𝑖𝑦 ±ý¸.
𝑧 + 1 = 𝑥 + 𝑖𝑦 − 1 = 𝑥 + 1 + 𝑖𝑦 𝑧 + 𝑖 = 𝑥 + 𝑖𝑦 + 𝑖 = 𝑥 + 𝑖 𝑦 + 1 𝑧 + 1
𝑧 + 𝑖=
𝑥 + 1 + 𝑖𝑦
𝑥 + 𝑖 𝑦 + 1
= 𝑥 + 1 + 𝑖𝑦
𝑥 + 𝑖 𝑦 + 1 ×
𝑥 − 𝑖 𝑦 + 1
𝑥 − 𝑖 𝑦 + 1
= 𝑥 + 1 𝑥 − 𝑖 𝑥 − 1 𝑦 + 1 + 𝑖𝑦𝑥 − 𝑖2𝑦 𝑦 + 1
𝑥2 + (𝑦 + 1)2
= 𝑥 + 1 𝑥 − 𝑖 𝑥 − 1 𝑦 + 1 + 𝑖𝑦𝑥 + 𝑦 𝑦 + 1
𝑥2 + (𝑦 + 1)2
=𝑥2 + 𝑥 + 𝑦2 + 𝑦
𝑥2 + 𝑦2 + 2𝑦 + 1+ 𝑖 ¸üÀ¨ÉÀ̾¢
¦ÁöÀ̾¢ =𝑥2 + 𝑥 + 𝑦2 + 𝑦
𝑥2 + 𝑦2 + 2𝑦 + 1
¸ðÎôÀ¡Î:
Re 𝑧 + 1
𝑧 + 𝑖 = 1
⇒𝑥2 + 𝑥 + 𝑦2 + 𝑦
𝑥2 + 𝑦2 + 2𝑦 + 1= 1
⇒ 𝑥2 + 𝑥 + 𝑦2 + 𝑦 = 𝑥2 + 𝑦2 + 2𝑦 + 1 ⇒ 𝑥 − 𝑦 = 1
∴ 𝑥 − 𝑦 − 1 = 0v‹gJ njitahd ãak¥ghijahF«.
6. 𝑥2 − 2𝑝𝑥 + (𝑝2 + 𝑞2) = 0v‹w rk‹gh£o‹ _y¦fŸ𝛼,𝛽k‰W«
tan𝜃 =𝑞
𝑦+𝑝ßm våš
(𝑦+𝛼)𝑛−(𝑦+𝛽)𝑛
𝛼−𝛽= 𝑞𝑛−1 sin 𝑛𝜃
si n𝑛𝜃 vd ãWÎf.
¾£÷×
𝑥2 − 2𝑝𝑥 + (𝑝2 + 𝑞2) = 0 𝑎 = 1, 𝑏 = −2𝑝, 𝑐 = 𝑝2 + 𝑞2
∴ 𝑥 =−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎=
2𝑝 ± 4𝑝2 − 4 𝑝2 + 𝑞2
2
=2𝑝 ± 4𝑝2 − 4𝑝2 − 4𝑞2
2=
2𝑝 ± −4𝑞2
2=
2𝑝 ± 𝑖2𝑞
2= 𝑝 ± 𝑖𝑞
𝛼 = 𝑝 + 𝑖𝑞; 𝛽 = 𝑝 − 𝑖𝑞 ±ý¸.
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tan𝜃 =𝑞
𝑦 + 𝑝⇒ 𝑦 + 𝑝 =
𝑞
tan𝜃⇒ 𝑦 = 𝑞
cos 𝜃
sin 𝜃− 𝑝
∴ 𝑦 + 𝛼 = 𝑞cos 𝜃
sin 𝜃− 𝑝 + 𝑝 + 𝑖𝑞 = 𝑞
cos𝜃 + 𝑖sin𝜃
sin𝜃
𝑦 + 𝛼 𝑛 = 𝑞 cos𝜃 + 𝑖sin𝜃
sin𝜃
𝑛
= 𝑞𝑛(cos𝜃 + 𝑖sin𝜃)𝑛
sin𝑛𝜃=
𝑞𝑛
sin𝑛𝜃 cos𝑛𝜃 + 𝑖sin𝑛𝜃 ……… … . (1)
þ§¾ §À¡Ä,
(𝑦 + 𝛽)𝑛 =𝑞𝑛
sin𝑛𝜃 cos𝑛𝜃 − 𝑖sin𝑛𝜃 ⋯⋯⋯ (2)
(1)– (2) ⇒
𝑦 + 𝛼 𝑛 − (𝑦 + 𝛽)𝑛 =𝑞𝑛
sin𝑛𝜃 cos𝑛𝜃 + 𝑖sin𝑛𝜃 −
𝑞𝑛
sin𝑛𝜃 cos𝑛𝜃 − 𝑖sin𝑛𝜃
=𝑞𝑛
sin𝑛𝜃 cos𝑛𝜃 + 𝑖sin𝑛𝜃 − cos𝑛𝜃 + 𝑖sin𝑛𝜃
=𝑞𝑛
sin𝑛𝜃 2𝑖sin𝜃
𝛼 − 𝛽 = 𝑝 + 𝑖𝑞 − 𝑝 + 𝑖𝑞 = 2𝑖𝑞
∴(𝑦 + 𝛼)𝑛 − (𝑦 + 𝛽)𝑛
𝛼 − 𝛽=
𝑞𝑛
si n𝑛𝜃 2𝑖sin𝜃
2𝑖𝑞= 𝑞𝑛−1
sin𝑛𝜃
sin𝑛𝜃
7. 𝑥2 − 2𝑥 + 2 = 0v‹w rk‹gh£o‹ _y¦fŸ𝛼,𝛽k‰W« cot 𝜃 = 𝑦 + 1våš
(𝑦+𝛼)𝑛 −(𝑦+𝛽)𝑛
𝛼−𝛽=
sin 𝑛𝜃
si n𝑛𝜃 vd ãWÎf.
¾£÷×
𝑥2 − 2𝑥 + 2 = 0
∴ 𝑥 =−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎=
2 ± 4 − 8
2=
2 ± −4
2=
2 ± 𝑖2
2= 1 ± 𝑖
𝛼 = 1 + 𝑖; 𝛽 = 1 − 𝑖±ý¸.
cot𝜃 = 𝑦 + 1 ⇒ 𝑦 = cot𝜃 − 1
∴ 𝑦 + 𝛼 = cot𝜃 − 1 + 1 + 𝑖 = cot𝜃 + 𝑖 =cos 𝜃
sin 𝜃+ 𝑖 =
cos𝜃 + 𝑖sin𝜃
sin𝜃
∴ 𝑦 + 𝛼 𝑛 = cos𝜃 + 𝑖sin𝜃
sin𝜃
𝑛
=(cos𝜃 + 𝑖sin𝜃)𝑛
sin𝑛𝜃
=cos𝑛𝜃 + 𝑖sin𝑛𝜃
sin𝑛𝜃⋯⋯⋯⋯⋯⋯ (1)
þ§¾ §À¡Ä,
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(𝑦 + 𝛽)𝑛 =cos𝑛𝜃 − 𝑖sin𝑛𝜃
sin𝑛𝜃⋯⋯⋯⋯⋯⋯ (2)
(1)– (2) ⇒
𝑦 + 𝛼 𝑛 − (𝑦 + 𝛽)𝑛 =cos𝑛𝜃 + 𝑖sin𝑛𝜃
sin𝑛𝜃−
cos𝑛𝜃 − 𝑖sin𝑛𝜃
sin𝑛𝜃
=cos𝑛𝜃 + 𝑖sin𝑛𝜃 − cos𝑛𝜃 + 𝑖sin𝑛𝜃
sin𝑛𝜃=
2𝑖sin𝜃
sin𝑛𝜃
𝛼 − 𝛽 = 1 + 𝑖 − 1 + 𝑖 = 2𝑖
∴(𝑦 + 𝛼)𝑛 − (𝑦 + 𝛽)𝑛
𝛼 − 𝛽=
2𝑖sin 𝜃
si n𝑛𝜃
2𝑖=
sin𝑛𝜃
sin𝑛𝜃
8. 𝑥2 − 2𝑥 + 4 = 0 þý ãÄí¸û𝛼ÁüÚõ𝛽 ±É¢ø 𝛼𝑛 − 𝛽𝑛 = 𝑖2𝑛+1sin𝑛𝜋
3
±É ¿¢Ú׸. «¾¢Ä¢ÕóÐ 𝛼9 − 𝛽9 -ý Á¾¢ô¨À ¦ÀÚ¸.
¾£÷×
𝑥2 − 2𝑥 + 4 = 0
𝑥 =−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎=
2 ± 2 − 16
2=
2 ± −12
2
=2 ± 4 × −3
2=
2 ± 𝑖2 3
2= 1 ± 𝑖 3
𝛼 = 1 + 𝑖 3; 𝛽 = 1 − 𝑖 3 ±ý¸.
1 + 𝑖 3 = 𝑥 + 𝑖𝑦
𝑥 = 1, 𝑦 = 3
∴ 𝑟 = 𝑥2 + 𝑦2 = 1 + 3 = 2
𝛼 = tan−1 𝑦
𝑥 = tan−1
3
1 = tan−1( 3) =
𝜋
3
1 + 𝑖 3, I¬ÅÐ ¸¡øÀ̾¢Â¢ø ¯ûÇÐ. ∴ 𝜃 = 𝛼 =𝜋
3
1 + 𝑖 3 = 2 cos𝜋
3+ 𝑖sin
𝜋
3
þ§¾ §À¡Ä, 1 − 𝑖 3 = 2 cos𝜋
3− 𝑖sin
𝜋
3
𝛼𝑛 = 1 + 𝑖 3 𝑛
= 2𝑛 cos𝑛𝜋
3+ 𝑖sin
𝑛𝜋
3
𝛽𝑛 = 1 − 𝑖 3 𝑛
= 2𝑛 cos𝑛𝜋
3− 𝑖sin
𝑛𝜋
3
∴ 𝛼𝑛 − 𝛽𝑛 = 2𝑛 cos𝑛𝜋
3+ 𝑖sin
𝑛𝜋
3 − 2𝑛 cos
𝑛𝜋
3− sin
𝑛𝜋
3
= 2𝑛 cos𝑛𝜋
3+ 𝑖sin
𝑛𝜋
3− cos
𝑛𝜋
3+ 𝑖sin
𝑛𝜋
3 = 2𝑛 2𝑖sin
𝑛𝜋
3
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= 𝑖2𝑛+1sin𝑛𝜋
3
𝑛 = 9 ±É À¢Ã¾¢Â¢¼
𝛼9 − 𝛽9 = 𝑖29+1sin9𝜋
3= 𝑖210sin3𝜋 = 0
9. 𝑥 +1
𝑥= 2cos𝜃, 𝑦 +
1
𝑦= 2cos𝜑 ±É¢ø
(i) 𝑥𝑚
𝑦𝑛+
𝑦𝑛
𝑥𝑚= 2cos 𝑚𝜃 − 𝑛𝜑 (ii)
𝑥𝑚
𝑦𝑛− +
𝑦𝑛
𝑥𝑚= 2𝑖sin 𝑚𝜃 − 𝑛𝜑 ±Éì
¸¡ðθ.
¾£÷×:
𝑥 +1
𝑥= 2cos𝜃 ⇒ 𝑥2 + 1 = 2cos𝜃 𝑥
𝑥2 − 2cos𝜃 𝑥 + 1 = 0 𝑎 = 1, 𝑏 = −2cos𝜃, 𝑐 = 1
𝑥 =−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎=
2cos𝜃 ± 4cos2𝜃 − 4
2
=2cos𝜃 ± 2 cos2𝜃 − 1
2=
2cos𝜃 ± 𝑖2sin𝜃
2= cos𝜃 ± 𝑖sin𝜃
𝑥 = cos𝜃 + 𝑖sin𝜃 ±ý¸.
þ§¾ §À¡Ä, 𝑦 +1
𝑦= 2cos𝜑 ⇒ 𝑦 = cos𝜑 + 𝑖sin𝜑
𝑥𝑚 = cos𝑚𝜃 + 𝑖sin𝑚𝜃, 𝑦𝑛 = cos𝑛𝜑 + 𝑖sin𝑛𝜑
𝑥𝑚
𝑦𝑛=
cos𝑚𝜃 + 𝑖sin𝑚𝜃
cos𝑛𝜑 + 𝑖sin𝑛𝜑= cos 𝑚𝜃 − 𝑛𝜑 + 𝑖sin 𝑚𝜃 − 𝑛𝜑 1
𝑦𝑛
𝑥𝑚=
1
cos(𝑚𝜃 − 𝑛𝜑) + 𝑖sin(𝑚𝜃 − 𝑛𝜑)
= cos 𝑚𝜃 − 𝑛𝜑 − 𝑖sin 𝑚𝜃 − 𝑛𝜑 (2) 1 + 2 ⇒𝑥𝑚
𝑦𝑛+
𝑦𝑛
𝑥𝑚
== 2cos 𝑚𝜃 − 𝑛𝜑
1 − 2 ⇒𝑥𝑚
𝑦𝑛−
𝑦𝑛
𝑥𝑚= 2𝑖sin 𝑚𝜃 − 𝑛𝜑
10. 𝑎 = cos2𝛼 + 𝑖sin2𝛼, 𝑏 = cos2𝛽 + 𝑖sin2𝛽, 𝑐 = cos2𝛾 + 𝑖sin2𝛾±É¢ø
(i) 𝑎𝑏𝑐 +1
𝑎𝑏𝑐= 2cos(𝛼 + 𝛽 + 𝛾)
(ii) 𝑎2𝑏2+𝑐2
𝑎𝑏𝑐= 2cos2(𝛼 + 𝛽 − 𝛾)±Éì ¸¡ðθ.
¾£÷×:
𝑎𝑏𝑐 = cos2𝛼 + 𝑖sin2𝛼 cos2𝛽 + 𝑖sin2𝛽 cos2𝛾 + 𝑖sin2𝛾 = cos𝛼 + 𝑖sin𝛼 2 cos𝛽 + 𝑖sin𝛽 2 cos𝛾 + 𝑖sin𝛾 2
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𝑎𝑏𝑐 = cos𝛼 + 𝑖sin𝛼 cos𝛽 + 𝑖sin𝛽 cos𝛾 + 𝑖sin𝛾 = cos 𝛼 + 𝛽 + 𝛾 + 𝑖sin 𝛼 + 𝛽 + 𝛾
1
𝑎𝑏𝑐=
1
cos 𝛼 + 𝛽 + 𝛾 + 𝑖sin 𝛼 + 𝛽 + 𝛾 = cos 𝛼 + 𝛽 + 𝛾 − 𝑖sin 𝛼 + 𝛽 + 𝛾
∴ 𝑎𝑏𝑐 +1
𝑎𝑏𝑐= 2cos 𝛼 + 𝛽 + 𝛾
𝑎2𝑏2 + 𝑐2
𝑎𝑏𝑐=
𝑎2𝑏2
𝑎𝑏𝑐+
𝑐2
𝑎𝑏𝑐=
𝑎𝑏
𝑐+
𝑐
𝑎𝑏
𝑎𝑏
𝑐=
cos2𝛼 + 𝑖sin2𝛼 cos2𝛽 + 𝑖sin2𝛽
cos2𝛾 + 𝑖sin2𝛾
= cos 2𝛼 + 2𝛽 − 2𝛾 + 𝑖sin 2𝛼 + 2𝛽 − 2𝛾 𝑐
𝑎𝑏=
1𝑎𝑏
𝑐
=1
cos 2𝛼 + 2𝛽 − 2𝛾 + 𝑖sin 2𝛼 + 2𝛽 − 2𝛾
= cos 2𝛼 + 2𝛽 − 2𝛾 − 𝑖sin 2𝛼 + 2𝛽 − 2𝛾
𝑎𝑏
𝑐+
𝑐
𝑎𝑏= 2 cos 2𝛼 + 2𝛽 − 2𝛾
𝑎2𝑏2 + 𝑐2
𝑎𝑏𝑐=
𝑎𝑏
𝑐+
𝑐
𝑎𝑏= 2cos2(𝛼 + 𝛽 − 𝛾)
11. − 3 − 𝑖 2
3 þý ±øÄ¡ Á¾¢ôÒ¸¨ÇÔõ ¸¡ñ¸.
¾£÷×:
− 3 − 𝑖 = 𝑥 + 𝑖𝑦 ±ý¸.
𝑟 = 𝑥2 + 𝑦2 = 3 + 1 = 2
𝛼 = tan−1 𝑦
𝑥 = tan−1
−1
− 3 = tan−1
1
3 =
𝜋
6
− 3 − 𝑖, III¬ÅÐ ¸¡øÀ̾¢Â¢ø ¯ûÇÐ..
∴ 𝜃 = −π + 𝛼 = −𝜋 +𝜋
6= −
5𝜋
6
∴ − 3 − 𝑖 = 2 cos −5𝜋
6 + 𝑖sin −
5𝜋
6
− 3 − 𝑖 2
= 22 cos −5𝜋
6 + 𝑖sin −
5𝜋
6
2
= 22 cos 2 × −5𝜋
6 + 𝑖sin 2 × −
5𝜋
6
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= 22 cos −5𝜋
3 + 𝑖sin
−5𝜋
3
− 3 − 𝑖 2
3 = 22
3 cos −5𝜋
3 + 𝑖sin
−5𝜋
3
1
3
= 22
3 cos 2𝑘𝜋 −5𝜋
3 + 𝑖sin 2𝑘𝜋 −
5𝜋
3
1
3
= 22
3 cos1
3 2𝑘𝜋 −
5𝜋
3 + 𝑖sin
1
3 2𝑘𝜋 −
5𝜋
3 , 𝑘 = 0,1,2
= 22
3 cos 6𝑘 − 5 𝜋
9+ 𝑖sin 6𝑘 − 5
𝜋
9 , 𝑘 = 0,1,2
− 3 − 𝑖 2
3-ý ±øÄ¡ Á¾¢ôÒ¸û:
22
3cis −5𝜋
9 , 2
2
3cis 𝜋
9 , 2
2
3cis 7𝜋
9
12. 3 + 𝑖 2
3 þý ±øÄ¡ Á¾¢ôÒ¸¨ÇÔõ ¸¡ñ¸.
¾£÷×:
3 + 𝑖 = 𝑥 + 𝑖𝑦 ±ý¸.
𝑟 = 𝑥2 + 𝑦2 = 3 + 1 = 2
𝛼 = tan−1 𝑦
𝑥 = tan−1
1
3 = tan−1
1
3 =
𝜋
6
3 + 𝑖, I¬ÅÐ ¸¡øÀ̾¢Â¢ø ¯ûÇÐ..
∴ 𝜃 = 𝛼 =𝜋
6
3 + 𝑖 = 2 cos𝜋
6+ 𝑖sin
𝜋
6
3 + 𝑖 2
3 = 22
3 cos𝜋
6+ 𝑖sin
𝜋
6
2
3
= 22
3 cos 2 ×𝜋
6 + 𝑖sin 2 ×
𝜋
6
1
3= 2
2
3 cos𝜋
3+ 𝑖sin
𝜋
3
1
3
= 22
3 cos 2𝑘𝜋 +𝜋
3 + 𝑖sin 2𝑘𝜋 +
𝜋
3
1
3
= 22
3 cos1
3 2𝑘𝜋 +
𝜋
3 + 𝑖sin
1
3 2𝑘𝜋 +
𝜋
3 , 𝑘 = 0,1,2
= 22
3 cos 6𝑘 + 1 𝜋
9+ 𝑖sin 6𝑘 + 1
𝜋
9 , 𝑘 = 0,1,2
3 + 𝑖 2
3-ý ±øÄ¡ Á¾¢ôÒ¸û:
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22
3cis𝜋
9, 2
2
3cis7𝜋
9, 2
2
3cis13𝜋
9
13. 1
2− 𝑖
3
2
3
4-ý ±øÄ¡ Á¾¢ôÒ¸¨ÇÔõ ¸¡ñ¸. ÁüÚõ «¾ý Á¾¢ôҸǢý
¦ÀÕì¸üÀÄý 1 ±É×õ ¸¡ðθ.
¾£÷×:
1
2− 𝑖
3
2= 𝑥 + 𝑖𝑦 ±ý¸.
∴ 𝑥 =1
2, 𝑦 = −
3
2
∴ r = 𝑥2 + 𝑦2 = 1
4+
3
4= 1
𝛼 = tan−1 𝑦
𝑥 = tan−1
− 3
21
2
= tan−1 3 =𝜋
3
1
2− 𝑖
3
2, 4 ¬ÅÐ ¸¡øÀ̾¢Â¢ø ¯ûÇÐ.
∴ 𝜃 = −α = −𝜋
3
1
2− 𝑖
3
2= cos −
𝜋
3 + 𝑖sin −
𝜋
3
∴ 1
2− 𝑖
3
2
3
4
= cos −𝜋
3 + 𝑖sin −
𝜋
3
3
4
= cos 3 × −𝜋
3 + 𝑖sin 3 × −
𝜋
3
1
4
= cos −𝜋 + 𝑖sin −𝜋 1
4 = cos 2𝑘𝜋 − 𝜋 + 𝑖sin 2𝑘𝜋 − 𝜋 1
4
= cos 2𝑘𝜋 − 𝜋
4 + 𝑖sin
2𝑘𝜋 − 𝜋
4 , 𝑘 = 0,1,2,3
= cos 2𝑘 − 1
4 𝜋 + 𝑖sin
2𝑘 − 1
4 𝜋, 𝑘 = 0,1,2,3
1
2− 𝑖
3
2
3
4
-ý ±øÄ¡ Á¾¢ôÒ¸û:
cis −𝜋
4 , cis
𝜋
4, cis
3𝜋
4, cis
5𝜋
4
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þÅüÈ¢ý ¦ÀÕì¸üÀÄý = cis −𝜋
4 × cis
𝜋
4× cis
3𝜋
4× cis
5𝜋
4
= cis −𝜋
4+
𝜋
4+
3𝜋
4+
5𝜋
4
= cis 8𝜋
4 = cis2𝜋
= cos2𝜋 + isin2𝜋 = 1 14.¾£÷ì¸: 𝑥4 − 𝑥3 + 𝑥2 − 𝑥 + 1 = 0 ¾£÷×:
𝑥4 − 𝑥3 + 𝑥2 − 𝑥 + 1 = 0 þÕ ÒÈÓõ 𝑥 + 1 ¬ø ¦ÀÕì¸,
𝑥 + 1 𝑥4 − 𝑥3 + 𝑥2 − 𝑥 + 1 = 0 ⇒ 𝑥5 − 𝑥4 + 𝑥3 − 𝑥2 + 𝑥 + 𝑥4 − 𝑥3 + 𝑥2 − 𝑥 + 1 = 0 ⇒ 𝑥5 + 1 = 0 ⇒ 𝑥5 = −1
⇒ 𝑥 = −1 1
5 = (cos𝜋 + 𝑖sin𝜋)1
5 = cos 2𝑘𝜋 + 𝜋 + 𝑖sin 2𝑘𝜋 + 𝜋 1
5
= cos 2𝑘𝜋 + 𝜋
5 + 𝑖sin
2𝑘𝜋 + 𝜋
5 , 𝑘 = 0,1,2,3,4
= cos 2𝑘 + 1 𝜋
5+ 𝑖sin 2𝑘 + 1
𝜋
5, 𝑘 = 0,1,2,3,4
𝑥5 + 1 = 0 -ý ¾£÷׸û:
cis𝜋
5, cis
3𝜋
5, cis
5𝜋
5= cis𝜋 = −1, cis
7𝜋
5, cis
9𝜋
5
þ¾¢ø 𝑥 + 1 ¬ø ¦ÀÕì¸ì¸¢¨¼ìÌõ ¾£÷× cis𝜋 = −1 ¨Â ¿£ì¸¢Å¢¼,
𝑥4 − 𝑥3 + 𝑥2 − 𝑥 + 1 = 0 -ý ¾£÷׸û:
cis𝜋
5, cis
3𝜋
5, cis
7𝜋
5, cis
9𝜋
5
15.¾£÷ì¸: 𝑥9 + 𝑥5 − 𝑥4 − 1 = 0. ¾£÷×:
𝑥9 + 𝑥5 − 𝑥4 − 1 = 0 ⇒ 𝑥5 𝑥4 + 1 − 1 𝑥4 + 1 = 0 𝑥4 + 1 𝑥5 − 1 = 0 ⇒ 𝑥5 − 1 = 0; 𝑥4 + 1 = 0
𝑥 = 1 1
5; 𝑥 = −1 1
4
i ⇒ 𝑥 = 1 1
5 = (cos0 + 𝑖sin0)1
5
= cos2𝑘𝜋 + 𝑖sin2𝑘𝜋 1
5
= cos2𝑘𝜋
5+ 𝑖sin
2𝑘𝜋
5, 𝑘 = 0,1,2,3,4
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∴ cis0 = 1, cis2𝜋
5, cis
4𝜋
5, cis
6𝜋
5, cis
8𝜋
5±ýÀÉ 𝑥5 − 1 = 0 -ý ¾£÷׸ǡÌõ.
ii ⇒ 𝑥 = −1 1
4 = (cos𝜋 + 𝑖sin𝜋)1
4
= cos 2𝑟𝜋 + 𝜋 + 𝑖sin 2𝑟𝜋 + 𝜋 1
4
= cos 2𝑟𝜋 + 𝜋
4 + 𝑖sin
2𝑟𝜋 + 𝜋
4 , 𝑟 = 0,1,2,3
= cos 2𝑘 + 1 𝜋
4+ 𝑖sin 2𝑘 + 1
𝜋
4, 𝑟 = 0,1,2,3.
∴ cis𝜋
4, cis
3𝜋
4, cis
5𝜋
4, cis
7𝜋
4±ýÀÉ 𝑥4 + 1 = 0 -ý ¾£÷׸ǡÌõ.
∴ 1, cis2𝜋
5, cis
4𝜋
5, cis
6𝜋
5, cis
8𝜋
5, cis
𝜋
4, cis
3𝜋
4, cis
5𝜋
44, cis
7𝜋
4
±ýÀɧ¾¨Å¡ɾ£÷׸û . 16.¾£÷ì¸: 𝑥7 + 𝑥4 + 𝑥3 + 1 = 0. ¾£÷×:
𝑥7 + 𝑥4 + 𝑥3 + 1 = 0 ⇒ 𝑥4 𝑥3 + 1 + 1 𝑥3 + 1 = 0 𝑥4 + 1 𝑥3 + 1 = 0 ⇒ 𝑥4 + 1 = 0; 𝑥3 + 1 = 0
𝑥 = −1 1
4;𝑥 = −1 1
3
ii ⇒ 𝑥 = −1 1
4 = (cos𝜋 + 𝑖sin𝜋)1
4
= cos 2𝑟𝜋 + 𝜋 + 𝑖sin 2𝑟𝜋 + 𝜋 1
4
= cos 2𝑟𝜋 + 𝜋
4 + 𝑖sin
2𝑟𝜋 + 𝜋
4 , 𝑟 = 0,1,2,3
= cos 2𝑘 + 1 𝜋
4+ 𝑖sin 2𝑘 + 1
𝜋
4, 𝑟 = 0,1,2,3.
∴ cis𝜋
4, cis
3𝜋
4, cis
5𝜋
4, cis
7𝜋
4±ýÀÉ 𝑥4 + 1 = 0 -ý ¾£÷׸ǡÌõ.
ii ⇒ 𝑥 = −1 1
3 = (cos𝜋 + 𝑖sin𝜋)1
3
= cos 2𝑟𝜋 + 𝜋 + 𝑖sin 2𝑟𝜋 + 𝜋 1
3
= cos 2𝑟𝜋 + 𝜋
3 + 𝑖sin
2𝑟𝜋 + 𝜋
3 , 𝑟 = 0,1,23
= cos 2𝑘 + 1 𝜋
3+ 𝑖sin 2𝑘 + 1
𝜋
3, 𝑟 = 0,1,2,3.
∴ cis𝜋
3, cis
3𝜋
3= cis𝜋 = −1, cis
5𝜋
3±ýÀÉ 𝑥3 + 1 = 0 -ý ¾£÷׸ǡÌõ.
∴ cis𝜋
4, cis
3𝜋
4, cis
5𝜋
4, cis
7𝜋
4, cis
𝜋
3, −1, cis
5𝜋
3±ýÀɧ¾¨Å¡ɾ£÷׸û
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ANALYTICAL GEOMETRY
1. 𝒚𝟐 − 𝟖𝒙 + 𝟔𝒚 + 𝟗 = 𝟎±ýÈ ÀÃŨÇÂò¾¢ý «îÍ, Ó¨É, ÌÅ¢Âõ,
þÂìÌŨâý ºÁýÀ¡Î, ¦ºùŸÄò¾¢ý ºÁýÀ¡Î, ¦ºùŸÄò¾¢ý ¿£Çõ
¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý ŨÃô À¼ò¨¾ Ũø. J-2008,O-2010
¾£÷×: 𝑦2 − 8𝑥 + 6𝑦 + 9 = 0 𝑦2 + 6𝑦 = 8𝑥 − 9
𝑦2 + 6𝑦 + 9 = 8𝑥 − 9 + 9 𝑦 + 3 2 = 8𝑥
𝒀𝟐 = 𝟖𝑿 þíÌ 𝑌 = 𝑦 + 3; 𝑋 = 𝑥
𝟒𝒂 = 𝟖 ⇒ 𝒂 = 𝟐 𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
«îÍ 𝑌 = 0 𝑦 + 3 = 0  𝑉(0,0) 𝑉(0,−3)
ÌÅ¢Âõ 𝐹 𝑎, 0 = 𝐹(2,0) 𝐹(2 + 0,0 − 3)
= 𝐹(2,−3)
þÂìÌŨâý ºÁýÀ¡Î 𝑋 = −𝑎 𝑋 = −2
𝑥 = −2
¦ºùŸÄò¾¢ý ºÁýÀ¡Î 𝑋 = 𝑎 𝑋 = 2
𝑥 = 2
¦ºùŸÄò¾¢ý ¿£Çõ 4𝑎 = 8 8
2. 𝒚𝟐 − 𝟖𝒙 − 𝟐𝒚 + 𝟏𝟕 = 𝟎±ýÈ ÀÃŨÇÂò¾¢ý «îÍ, Ó¨É, ÌÅ¢Âõ,
þÂìÌŨâý ºÁýÀ¡Î, ¦ºùŸÄò¾¢ý ºÁýÀ¡Î, ¦ºùŸÄò¾¢ý ¿£Çõ
¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý ŨÃô À¼ò¨¾ Ũø. J-2007
¾£÷×: 𝑦2 − 8𝑥 − 2𝑦 + 17 = 0
𝑦2 − 2𝑦 = 8𝑥 − 17
𝑦2 − 2𝑦 + 1 = 8𝑥 − 17 + 1
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𝑦 − 1 2 = 8𝑥 − 16
𝑦 − 1 2 = 8(𝑥 − 2)
𝒀𝟐 = 𝟖𝑿 þíÌ 𝑌 = 𝑦 − 1; 𝑋 = 𝑥 − 2
𝟒𝒂 = 𝟖 ⇒ 𝒂 = 𝟐
𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
«îÍ 𝑌 = 0 𝑦 − 1 = 0
 𝑉(0,0) 𝑉(2,1)
ÌÅ¢Âõ 𝐹 𝑎, 0 = 𝐹(2,0) 𝐹(2 + 2,0 + 1)
= 𝐹(4,1)
þÂìÌŨâý ºÁýÀ¡Î 𝑋 = −𝑎
𝑋 = −2 𝑥 = 0
¦ºùŸÄò¾¢ý ºÁýÀ¡Î 𝑋 = 𝑎
𝑋 = 2 𝑥 = 4
¦ºùŸÄò¾¢ý ¿£Çõ 4𝑎 = 8 8
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3. 𝒚𝟐 + 𝟖𝒙 − 𝟔𝒚 + 𝟏 = 𝟎±ýÈ ÀÃŨÇÂò¾¢ý «îÍ, Ó¨É, ÌÅ¢Âõ,
þÂìÌŨâý ºÁýÀ¡Î, ¦ºùŸÄò¾¢ý ºÁýÀ¡Î, ¦ºùŸÄò¾¢ý ¿£Çõ
¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý ŨÃôÀ¼ò¨¾ Ũø. O-2006, M-2007
¾£÷×:
𝑦2 + 8𝑥 − 6𝑦 + 1 = 0
𝑦2 − 6𝑦 = −8𝑥 − 1
𝑦2 − 6𝑦 + 𝟗 = −8𝑥 − 1 + 𝟗
𝑦2 − 6𝑦 + 9 = −8𝑥 + 8
𝑦 − 3 2 = −8(𝑥 − 1)
𝒀𝟐 = −𝟖𝑿 þíÌ 𝑌 = 𝑦 − 3; 𝑋 = 𝑥 − 1
𝟒𝒂 = 𝟖 ⇒ 𝒂 = 𝟐
𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
«îÍ 𝑌 = 0 𝑦 − 3 = 0  𝑉(0,0) 𝑉(1,3)
ÌÅ¢Âõ 𝐹 −𝑎, 0 = 𝐹(−2,0) 𝐹(−2 + 1,0 + 3)
= 𝐹(−1,3)
þÂìÌŨâý ºÁýÀ¡Î 𝑋 = 𝑎 𝑋 = 2
𝑥 − 1 = 2 𝑥 − 3 = 0
¦ºùŸÄò¾¢ý ºÁýÀ¡Î 𝑋 = −𝑎 𝑋 = −2
𝑥 − 1 = −2 𝑥 + 1 = 0
¦ºùŸÄò¾¢ý ¿£Çõ 4𝑎 = 8 8
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4. 𝒙𝟐 − 𝟔𝒙 − 𝟏𝟐𝒚 − 𝟑 = 𝟎±ýÈ ÀÃŨÇÂò¾¢ý «îÍ, Ó¨É, ÌÅ¢Âõ,
þÂìÌŨâý ºÁýÀ¡Î, ¦ºùŸÄò¾¢ý ºÁýÀ¡Î, ¦ºùŸÄò¾¢ý ¿£Çõ
¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý ŨÃô À¼ò¨¾ Ũø. M-2010
¾£÷×: 𝑥2 − 6𝑥 − 12𝑦 − 3 = 0
𝑥2 − 6𝑥 = 12𝑦 + 3
𝑥2 − 6𝑥 + 𝟗 = 12𝑦 + 3 + 𝟗
𝑥2 − 6𝑥 + 9 = 12𝑦 + 12
𝑥 − 3 2 = 12(𝑦 + 1)
𝑿𝟐 = 𝟏𝟐𝒀 þíÌ 𝑋 = 𝑥 − 3;𝑌 = 𝑦 + 1
𝟒𝒂 = 𝟏𝟐 ⇒ 𝒂 = 𝟑
𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
«îÍ 𝑋 = 0 𝑥 − 3 = 0  𝑉(0,0) 𝑉(3,−1)
ÌÅ¢Âõ 𝐹 0, 𝑎 = 𝐹(0,3) 𝐹(0 + 3,3 − 1)
= 𝐹(3,2)
þÂìÌŨâý ºÁýÀ¡Î 𝑌 = −𝑎 𝑌 = −3
𝑦 + 1 = −3 𝑦 + 4 = 0
¦ºùŸÄò¾¢ý ºÁýÀ¡Î 𝑌 = 𝑎 𝑌 = 3
𝑦 + 1 = 3 𝑦 − 2 = 0
¦ºùŸÄò¾¢ý ¿£Çõ 4𝑎 = 12 12
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5. 𝒙𝟐 − 𝟐𝒙 + 𝟖𝒚 + 𝟏𝟕 = 𝟎±ýÈ ÀÃŨÇÂò¾¢ý «îÍ, Ó¨É, ÌÅ¢Âõ,
þÂìÌŨâý ºÁýÀ¡Î, ¦ºùŸÄò¾¢ý ºÁýÀ¡Î, ¦ºùŸÄò¾¢ý ¿£Çõ
¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý ŨÃôÀ¼ò¨¾ Ũø.
¾£÷×:
𝑥2 − 2𝑥 + 8𝑦 + 17 = 0
𝑥2 − 2𝑥 = −8𝑦 − 17
𝑥2 − 2𝑥 + 𝟏 = −8𝑦 − 17 + 𝟏
𝑥2 − 2𝑥 + 1 = −8𝑦 − 16
𝑥 − 1 2 = −8 𝑦 + 2
𝑿𝟐 = −𝟖𝒀
þíÌ 𝑋 = 𝑥 − 1; 𝑌 = 𝑦 + 2
𝟒𝒂 = 𝟖 ⇒ 𝒂 = 𝟐
𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
«îÍ 𝑋 = 0 𝑥 − 1 = 0  𝑉(0,0) 𝑉(1,−2)
ÌÅ¢Âõ 𝐹 0, −𝑎 = 𝐹(0, −2) 𝐹(0 + 1, −2 − 2)
= 𝐹(1,−4)
þÂìÌŨâý ºÁýÀ¡Î 𝑌 = 2 𝑌 = 2
𝑦 + 2 = 2 𝑦 = 0
¦ºùŸÄò¾¢ý ºÁýÀ¡Î 𝑌 = −𝑎 𝑌 = −2
𝑦 + 2 = −2 𝑦 + 4 = 0
¦ºùŸÄò¾¢ý ¿£Çõ 4𝑎 = 8 8
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6. 𝒙𝟐 + 𝟒𝒚𝟐 − 𝟖𝒙 − 𝟏𝟔𝒚 − 𝟔𝟖 = 𝟎±ýÈ ¿£ûÅð¼ò ¾¢ý ¨ÁÂò¦¾¡¨Ä× Å¢¸¢¾õ,
¨ÁÂõ, ÌÅ¢Âí¸û, Өɸû ¿£Çõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý
ŨÃôÀ¼ò¨¾ Ũø.
¾£÷×: 𝑥2 + 4𝑦2 − 8𝑥 − 16𝑦 − 68 = 0
𝑥2 − 8𝑥 + 4𝑦2 − 16𝑦 = 68
𝑥2 − 8𝑥 + 4 𝑦2 − 4𝑦 = 68
𝑥2 − 8𝑥 + 𝟏𝟔 − 𝟏𝟔 + 4 𝑦2 − 4𝑦 + 𝟒 − 𝟒 = 68
𝑥2 − 8𝑥 + 16 − 16 + 4 𝑦2 − 4𝑦 + 4 − 16 = 68
𝑥 − 4 2 + 4 𝑦 − 2 2 = 16 + 16 + 68
𝑥 − 4 2 + 4 𝑦 − 2 2 = 100
𝑥 − 4 2
100+
𝑦 − 2 2
25= 1 ⇒
𝑿𝟐
𝟏𝟎𝟎+
𝒀𝟐
𝟐𝟓= 𝟏
þíÌ 𝑋 = 𝑥 − 4; 𝑌 = 𝑦 − 2.
𝑎2 = 100, 𝑏2 = 25 ⟹ 𝑒 = 1 −𝑏2
𝑎2= 1 −
25
100
⟹ 𝑒 = 1 −1
4=
3
4=
3
2
𝑎𝑒 = 10 × 3
2= 5 3
𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
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¨ÁÂõ 𝐶 0,0 𝐶(4,2)
ÌÅ¢Âí¸û 𝐹1 𝑎𝑒, 0 = 𝐹1(5 3, 0) 𝐹1(4 + 5 3, 2
𝐹2 −𝑎𝑒, 0 = 𝐹2(−5 3, 0) 𝐹2(4 − 5 3, 2)
Өɸû 𝐴 𝑎, 0 = 𝐴(10,0) 𝐴(14,2)
𝐴′ −𝑎, 0 = 𝐴′(−10,0) 𝐴′(−6,2)
¨ÁÂò¦¾¡¨Ä× Å¢¸¢¾õ 𝑒 = 3
2 𝑒 =
3
2
7. 𝟏𝟔𝒙𝟐 + 𝟗𝒚𝟐 + 𝟑𝟐𝒙 − 𝟑𝟔𝒚 = 𝟗𝟐±ýÈ ¿£ûÅð¼ò ¾¢ý ¨ÁÂò¦¾¡¨Ä× Å¢¸¢¾õ,
¨ÁÂõ, ÌÅ¢Âí¸û, Өɸû ¿£Çõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý
ŨÃôÀ¼ò¨¾ Ũø. J-2009
¾£÷×: 16𝑥2 + 9𝑦2 + 32𝑥 − 36𝑦 = 92
16𝑥2 + 32𝑥 + 9𝑦2 − 36𝑦 = 92
16 𝑥2 + 2𝑥 + 9 𝑦2 − 4𝑦 = 92
16 𝑥2 + 2𝑥 + 𝟏 − 𝟏 + 9 𝑦2 − 4𝑦 + 𝟒 − 𝟒 = 92
16 𝑥2 + 2𝑥 + 1 − 16 + 9 𝑦2 − 4𝑦 + 4 − 36 = 92
16 𝑥 + 1 2 + 9 𝑦 − 2 2 = 16 + 36 + 92
16 𝑥 + 1 2 + 9 𝑦 − 2 2 = 144
𝑥 + 1 2
9+
𝑦 − 2 2
16= 1
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𝑋2
9+
𝑌2
16= 1
þíÌ 𝑋 = 𝑥 + 1; 𝑌 = 𝑦 − 2.
𝑎2 = 16, 𝑏2 = 9 ⟹ 𝑒 = 1 −𝑏2
𝑎2= 1 −
9
16
⟹ 𝑒 = 16 − 9
16=
7
16=
7
4
𝑎𝑒 = 4 × 7
4= 7
𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
¨ÁÂõ 𝐶 0,0 𝐶(−1,2)
ÌÅ¢Âí¸û 𝐹1 0, 𝑎𝑒 = 𝐹1(0, 7) 𝐹1(−1,2 + 7)
𝐹2 0,−𝑎𝑒 = 𝐹2(0, − 7) 𝐹2(−1,2 − 7)
Өɸû 𝐴 0, 𝑎 = 𝐴(0,4) 𝐴(−1,6)
𝐴′ 0, −𝑎 = 𝐴′(0,−4) 𝐴′(−1, −2) ¨ÁÂò¦¾¡¨Ä× Å¢¸¢¾õ
𝑒 = 7
4 𝑒 =
7
4
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8. 𝟑𝟔𝒙𝟐 + 𝟒𝒚𝟐 − 𝟕𝟐𝒙 + 𝟑𝟐𝒚 − 𝟒𝟒 = 𝟎±ýÈ ¿£û Åð¼ò¾¢ý ¨ÁÂò¦¾¡¨Ä×
Å¢¸¢¾õ, ¨ÁÂõ, ÌÅ¢Âí¸û, Өɸû ¿£Çõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ
«¾ý ŨÃôÀ¼ò¨¾ Ũø. M-2006,J-2006
¾£÷×: 36𝑥2 + 4𝑦2 − 72𝑥 + 32𝑦 − 44 = 0
36𝑥2 − 72𝑥 + 4𝑦2 + 32𝑦 = 44
36 𝑥2 − 2𝑥 + 4 𝑦2 + 8𝑦 = 44
36 𝑥2 − 2𝑥 + 𝟏 − 𝟏 + 4 𝑦2 + 8𝑦 + 𝟏𝟔 − 𝟏𝟔 = 44
36 𝑥2 − 2𝑥 + 1 − 36 + 4 𝑦2 + 8𝑦 + 16 − 64 = 44
36 𝑥 − 1 2 + 4 𝑦 + 4 2 = 36 + 64 + 44
36 𝑥 − 1 2 + 4 𝑦 + 4 2 = 144
𝑥 − 1 2
4+
𝑦 + 4 2
36= 1
𝑋2
4+
𝑌2
36= 1
þíÌ 𝑋 = 𝑥 − 1; 𝑌 = 𝑦 + 4.
𝑎2 = 36, 𝑏2 = 4 ⟹ 𝑒 = 1 −𝑏2
𝑎2= 1 −
4
36
⟹ 𝑒 = 36 − 4
36=
32
36=
8
9=
2 2
3
𝑎𝑒 = 6 ×2 2
3= 4 2
𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
¨ÁÂõ 𝐶 0,0 𝐶(1, −4)
ÌÅ¢Âí¸û 𝐹1 0, 𝑎𝑒 = 𝐹1(0,4 2) 𝐹1(1,−4 + 4 2)
𝐹2 0,−𝑎𝑒 = 𝐹2(0, −4 2) 𝐹2(1, −4 − 4 2)
Өɸû 𝐴 0,𝑎 = 𝐴(0,6) 𝐴(1,2)
𝐴′ 0, −𝑎 = 𝐴′(0,−6) 𝐴′(1, −10)
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¨ÁÂò¦¾¡¨Ä× Å¢¸¢¾õ 𝑒 =2 2
3 𝑒 =
2 2
3
9. 𝒙𝟐 − 𝟒𝒚𝟐 + 𝟔𝒙 + 𝟏𝟔𝒚 − 𝟏𝟏 = 𝟎±ýÈ «¾¢ÀÃŨÇÂò¾¢ý ¨ÁÂò¦¾¡¨Ä×
Å¢¸¢¾õ, ¨ÁÂõ, ÌÅ¢Âí¸û, Өɸû ¿£Çõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ
«¾ý ŨÃôÀ¼ò¨¾ Ũø. M-2010
¾£÷×: 𝑥2 − 4𝑦2 + 6𝑥 + 16𝑦 − 11 = 0
𝑥2 + 6𝑥 + −4𝑦2 + 16𝑦 = 11
𝑥2 + 6𝑥 − 4 𝑦2 − 4𝑦 = 11
𝑥2 + 6𝑥 + 𝟗 − 𝟗 − 4 𝑦2 − 4𝑦 + 𝟒 − 𝟒 = 11
𝑥2 + 6𝑥 + 9 − 9 − 4 𝑦2 − 4𝑦 + 4 + 16 = 11
𝑥 + 3 2 − 4 𝑦 − 2 2 = 9 − 16 + 11
𝑥 + 3 2 − 4 𝑦 − 2 2 = 4
𝑥 + 3 2
4−
𝑦 − 2 2
1= 1
𝑋2
4−
𝑌2
1= 1
þíÌ 𝑋 = 𝑥 + 3; 𝑌 = 𝑦 − 2.
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𝑎2 = 4, 𝑏2 = 1 ⟹ 𝑒 = 1 +𝑏2
𝑎2= 1 +
1
4
⟹ 𝑒 = 4 + 1
4=
5
4=
5
2
𝑎𝑒 = 2 × 5
2= 5
𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
¨ÁÂõ 𝐶 0,0 𝐶(−3,2)
ÌÅ¢Âí¸û 𝐹1 𝑎𝑒, 0 = 𝐹1( 5, 0) 𝐹1(−3 + 5, 2)
𝐹2 −𝑎𝑒, 0 = 𝐹2(− 5, 0) 𝐹2(−3 − 5, 2)
Өɸû 𝐴 𝑎, 0 = 𝐴(2,0) 𝐴(−1,2)
𝐴′ −𝑎, 0 = 𝐴′(−2,0) 𝐴′(−5,2)
¨ÁÂò¦¾¡¨Ä× Å¢¸¢¾õ 𝑒 = 5
2 𝑒 =
5
2
10. 𝟗𝒙𝟐 − 𝟏𝟔𝒚𝟐 − 𝟏𝟖𝒙 − 𝟔𝟒𝒚 − 𝟏𝟗𝟗 = 𝟎±ýÈ «¾¢ÀÃŨÇÂò¾¢ý
¨ÁÂò¦¾¡¨Ä× Å¢ ¢̧¾õ, ¨ÁÂõ, ÌÅ¢Âí¸û, Өɸû ¿£Çõ ¬ ¢̧ÂÅü¨Èì
¸¡ñ¸. §ÁÖõ «¾ý ŨÃôÀ¼ò¨¾ Ũø.
¾£÷×:
9𝑥2 − 16𝑦2 − 18𝑥 − 64𝑦 − 199 = 0
9𝑥2 − 18𝑥 + −16𝑦2 − 64𝑦 = 199
9 𝑥2 − 2𝑥 − 16 𝑦2 + 4𝑦 = 199
9 𝑥2 − 2𝑥 + 𝟏 − 𝟏 − 16 𝑦2 + 4𝑦 + 𝟒 − 𝟒 = 199
9 𝑥2 − 2𝑥 + 1 − 9 − 16 𝑦2 + 4𝑦 + 4 + 64 = 199
9 𝑥 − 1 2 − 16 𝑦 + 2 2 = 9 − 64 + 199
9 𝑥 − 1 2 − 16 𝑦 + 2 2 = 144
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𝑥 − 1 2
16−
𝑦 + 2 2
9= 1
𝑋2
16−
𝑌2
9= 1
þíÌ 𝑋 = 𝑥 − 1; 𝑌 = 𝑦 + 2.
𝑎2 = 16, 𝑏2 = 9 ⟹ 𝑒 = 1 +𝑏2
𝑎2= 1 +
9
16
⟹ 𝑒 = 16 + 9
16=
25
16=
5
4
𝑎𝑒 = 4 ×5
4= 5
𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
¨ÁÂõ 𝐶 0,0 𝐶(1, −2)
ÌÅ¢Âí¸û 𝐹1 𝑎𝑒, 0 = 𝐹1(5,0) 𝐹1(6,−2)
𝐹2 −𝑎𝑒, 0 = 𝐹2(−5,0) 𝐹2(−4, −2)
Өɸû 𝐴 𝑎, 0 = 𝐴(4,0) 𝐴(5, −2)
𝐴′ −𝑎, 0 = 𝐴′(−4,0) 𝐴′(−3, −2)
¨ÁÂò ¦¾¡¨Ä× Å¢¸¢¾õ 𝑒 =5
4 𝑒 =
5
4
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11. 𝒙𝟐 − 𝟑𝒚𝟐 + 𝟔𝒙 + 𝟔𝒚 + 𝟏𝟖 = 𝟎±ýÈ «¾¢ÀÃÅ¨Ç Âò¾¢ý ¨ÁÂò¦¾¡¨Ä×
Å¢¸¢¾õ, ¨ÁÂõ, ÌÅ¢Âí¸û, Өɸû ¿£Çõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ
«¾ý ŨÃôÀ¼ò¨¾ Ũø. M-2008,O-2008,O-2009,J-2010
¾£÷×: 𝑥2 − 3𝑦2 + 6𝑥 + 6𝑦 + 18 = 0
𝑥2 + 6𝑥 + −3𝑦2 + 6𝑦 = −18
𝑥2 + 6𝑥 − 3 𝑦2 − 2𝑦 = −18
𝑥2 + 6𝑥 + 𝟗 − 𝟗 − 3 𝑦2 − 2𝑦 + 𝟏 − 𝟏 = −18
𝑥2 + 6𝑥 + 9 − 9 − 3 𝑦2 − 2𝑦 + 1 + 3 = −18
𝑥 + 3 2 − 3 𝑦 − 1 2 = 9 − 3 − 18
𝑥 + 3 2 − 3 𝑦 − 1 2 = −12
𝑥 + 3 2
12−
𝑦 − 1 2
4= −1
𝑋2
12−
𝑌2
4= −1
þíÌ 𝑋 = 𝑥 + 3; 𝑌 = 𝑦 − 1.
𝑌2
4−
𝑋2
12= 1
𝑎2 = 4, 𝑏2 = 12 ⟹ 𝑒 = 1 +𝑏2
𝑎2= 1 +
12
4
⟹ 𝑒 = 1 + 3 = 4 = 2
𝑎𝑒 = 2 × 2 = 4
𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
¨ÁÂõ 𝐶 0,0 𝐶(−3,1)
ÌÅ¢Âí¸û 𝐹1 0, 𝑎𝑒 = 𝐹1(0,4) 𝐹1(−3,5)
𝐹2 0,−𝑎𝑒 = 𝐹2(0, −4) 𝐹2(−3, −3)
Өɸû 𝐴 0,𝑎 = 𝐴(0,2) 𝐴(−3,3)
𝐴′ 0, −𝑎 = 𝐴′(0, −2) 𝐴′(−3, −1) ¨ÁÂò¦¾¡¨Ä× Å¢¸¢¾õ 𝑒 = 2 𝑒 = 2
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12. 𝟗𝒙𝟐 − 𝟏𝟔𝒚𝟐 + 𝟑𝟔𝒙 + 𝟑𝟐𝒚 + 𝟏𝟔𝟒 = 𝟎±ýÈ «¾¢ÀÃŨÇÂò¾¢ý
¨ÁÂò¦¾¡¨Ä× Å¢ ¢̧¾õ, ¨ÁÂõ, ÌÅ¢Âí¸û, Өɸû ¿£Çõ ¬ ¢̧ÂÅü¨Èì
¸¡ñ¸. §ÁÖõ «¾ý ŨÃôÀ¼ò¨¾ Ũø.
¾£÷×: 9𝑥2 − 16𝑦2 + 36𝑥 + 32𝑦 + 164 = 0
9𝑥2 + 36𝑥 + −16𝑦2 + 32𝑦 = −164
9 𝑥2 + 4𝑥 − 16 𝑦2 − 2𝑦 = −164
9 𝑥2 + 4𝑥 + 𝟒 − 𝟒 − 16 𝑦2 − 2𝑦 + 𝟏 − 𝟏 = −164
9 𝑥2 + 4𝑥 + 4 − 36 − 16 𝑦2 − 2𝑦 + 1 + 16 = −164
9 𝑥 + 2 2 − 16 𝑦 − 1 2 = 36 − 16 − 164
9 𝑥 + 2 2 − 16 𝑦 − 1 2 = −144
𝑥 + 2 2
16−
𝑦 − 1 2
9= −1 ⇒
𝑋2
16−
𝑌2
9= −1
þíÌ 𝑋 = 𝑥 + 2; 𝑌 = 𝑦 − 1.
𝒀𝟐
𝟗−
𝑿𝟐
𝟏𝟔= 𝟏
𝑎2 = 9, 𝑏2 = 16 ⟹ 𝑒 = 1 +𝑏2
𝑎2= 1 +
16
9
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⟹ 𝑒 = 9 + 16
9=
25
9=
5
3
𝑎𝑒 = 3 ×5
3= 5
𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
¨ÁÂõ 𝐶 0,0 𝐶(−2,1)
ÌÅ¢Âí¸û 𝐹1 0, 𝑎𝑒 = 𝐹1(0,5) 𝐹1(−2,6)
𝐹2 0,−𝑎𝑒 = 𝐹2(0, −5) 𝐹2(−2, −4)
Өɸû 𝐴 0,𝑎 = 𝐴(0,3) 𝐴(−2,4)
𝐴′ 0, −𝑎 = 𝐴′(0, −3) 𝐴′(−2, −2)
¨ÁÂò¦¾¡¨Ä× Å¢¸¢¾õ 𝑒 =5
3 𝑒 =
5
3
13. 𝟏𝟐𝒙𝟐 − 𝟒𝒚𝟐 − 𝟐𝟒𝒙 + 𝟑𝟐𝒚 − 𝟏𝟐𝟕 = 𝟎±ýÈ «¾¢ÀÃŨÇÂò¾¢ý
¨ÁÂò¦¾¡¨Ä× Å¢ ¢̧¾õ, ¨ÁÂõ, ÌÅ¢Âí¸û, Өɸû ¿£Çõ ¬ ¢̧ÂÅü¨Èì
¸¡ñ¸. §ÁÖõ «¾ý ŨÃôÀ¼ò¨¾ Ũø. O-2007
¾£÷×: 12𝑥2 − 4𝑦2 − 24𝑥 + 32𝑦 − 127 = 0
12𝑥2 − 24𝑥 + −4𝑦2 + 32𝑦 = 127
12 𝑥2 − 2𝑥 − 4 𝑦2 − 8𝑦 = 127
12 𝑥2 − 2𝑥 + 𝟏 − 𝟏 − 4 𝑦2 − 8𝑦 + 𝟏𝟔 − 𝟏𝟔 = 127
12 𝑥2 − 2𝑥 + 1 − 12 − 4 𝑦2 − 8𝑦 + 16 + 64 = 127
12 𝑥 − 1 2 − 4 𝑦 − 4 2 = 12 − 64 + 127
12 𝑥 − 1 2 − 4 𝑦 − 4 2 = 75
𝑥 − 1 2
75
12
− 𝑦 − 4 2
75
4
= 1 ⇒𝑋2
75
12
−𝑌2
75
4
= 1
þíÌ 𝑋 = 𝑥 − 1; 𝑌 = 𝑦 − 4.
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𝑎2 =75
12, 𝑏2 =
75
4⟹ 𝑒 = 1 +
𝑏2
𝑎2= 1 +
75
475
12
⟹ 𝑒 = 1 +75
4×
12
75= 1 + 3 = 4 = 2
𝑎𝑒 = 75
12× 2 =
25
4× 2 = 5
𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
¨ÁÂõ 𝐶 0,0 𝐶(1,4)
ÌÅ¢Âí¸û 𝐹1 𝑎𝑒, 0 = 𝐹1(5,0) 𝐹1(6,4)
𝐹2 −𝑎𝑒, 0 = 𝐹2(−5,0) 𝐹2(−4,4)
Өɸû
𝐴 𝑎, 0 = 𝐴 5
2, 0 𝐴
7
2, 4
𝐴′ −𝑎, 0 = 𝐴′ −5
2, 0 𝐴′ −
3
2, 4
¨ÁÂò¦¾¡¨Ä× Å¢¸¢¾õ 𝑒 = 2 𝑒 = 2
14. 𝟓𝒙 + 𝟏𝟐𝒚 = 𝟗 ±ýÈ §¿÷째¡Î «¾¢ÀÃŨÇÂõ 𝒙𝟐 − 𝟗𝒚𝟐 = 𝟗 -³ò
¦¾¡Î¸¢ÈÐ ±É ¿¢åÀ¢ì¸. §ÁÖõ ¦¾¡Îõ ÒûÇ¢¨ÂÔõ ¸¡ñ¸. J-
2009
¾£÷×:
§¿÷째¡Î 5𝑥 + 12𝑦 = 9
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12𝑦 = −5𝑥 + 9
𝑦 = −5
12𝑥 +
9
12
𝑦 = 𝑚𝑥 + 𝑐 ¯¼ý ´ôÀ¢¼
𝑚 = −5
12, 𝑐 =
9
12=
3
4
«¾¢ÀÃŨÇÂõ 𝑥2 − 9𝑦2 = 9
𝑥2
9−
𝑦2
1= 1
𝑥 2
𝑎2 −𝑦2
𝑏2 = 1 ¯¼ý ´ôÀ¢¼
𝑎2 = 9, 𝑏2 = 1
𝑦 = 𝑚𝑥 + 𝑐±ýÈ §¿÷째¡Î «¾¢ÀÃŨÇÂõ 𝑥 2
𝑎2 −𝑦2
𝑏2 = 1 -³ò ¦¾¡¼ ¸ðÎôÀ¡Î
𝑐2 = 𝑎2𝑚2 − 𝑏2
𝑐2 = 3
4
2
=9
16
𝑎2𝑚2 − 𝑏2 = 9 −5
12
2
− 1 = 9 25
144 − 1
=225
144− 1 =
225 − 144
144=
81
144=
9
16
∴ 𝑐2 = 𝑎2𝑚2 − 𝑏2
∴ 5𝑥 + 12𝑦 = 9±ýÈ §¿÷째¡Î «¾¢ÀÃŨÇÂõ
𝑥2 − 9𝑦2 = 9 -³ò ¦¾¡Î¸¢ÈÐ.
¦¾¡Îõ ÒûÇ¢ : −𝒂𝟐𝒎
𝒄,−𝒃𝟐
𝒄
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−𝑎2𝑚
𝑐= −9 × −
5
12 ×
4
3= 5
−𝑏2
𝑐= −1 ×
4
3= −
4
3
¦¾¡Îõ ÒûÇ¢ = −𝑎2𝑚
𝑐,−𝑏2
𝑐 = 5, −
4
3
15. 𝒙 − 𝒚 + 𝟒 = 𝟎±ýÈ §¿÷째¡Î ¿£ûÅð¼õ
𝒙𝟐 + 𝟑𝒚𝟐 = 𝟏𝟐 -³ò ¦¾¡Î¸¢ÈÐ ±É ¿¢åÀ¢ì¸. §ÁÖõ ¦¾¡Îõ ÒûÇ¢¨ÂÔõ
¸¡ñ¸.
¾£÷×:
§¿÷째¡Î 𝑥 − 𝑦 + 4 = 0 ⇒ −𝑦 = −𝑥 − 4 ⇒ 𝑦 = 𝑥 + 4
𝑦 = 𝑚𝑥 + 𝑐 ¯¼ý ´ôÀ¢¼
𝑚 = 1, 𝑐 = 4
¿£ûÅð¼õ 𝑥2 + 3𝑦2 = 12
𝑥2
12+
𝑦2
4= 1
𝑥 2
𝑎2 +𝑦2
𝑏2 = 1 ¯¼ý ´ôÀ¢¼
𝑎2 = 12, 𝑏2 = 4
𝑦 = 𝑚𝑥 + 𝑐±ýÈ §¿÷째¡Î ¿£ûÅð¼õ 𝑥2
𝑎2 +𝑦2
𝑏2 = 1 -³ò ¦¾¡¼ ¸ðÎôÀ¡Î
𝑐2 = 𝑎2𝑚2 + 𝑏2
𝑐2 = 42 = 16
𝑎2𝑚2 + 𝑏2 = 12 1 2 + 4 = 12 + 4 = 16
∴ 𝑐2 = 𝑎2𝑚2 + 𝑏2
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𝑥 − 𝑦 + 4 = 0±ýÈ §¿÷째¡Î ¿£ûÅð¼õ
𝑥2 + 3𝑦2 = 12 -³ò ¦¾¡Î¸¢ÈÐ.
¦¾¡Îõ ÒûÇ¢ : −𝒂𝟐𝒎
𝒄,−𝒃𝟐
𝒄
−𝑎2𝑚
𝑐= −12 × 1 ×
1
4= −3
𝑏2
𝑐= 4 ×
1
4= 1
¦¾¡Îõ ÒûÇ¢ = −𝑎2𝑚
𝑐,𝑏2
𝑐 = −3,1
16. 𝒙 + 𝟐𝒚 − 𝟓 = 𝟎-³ ´Õ ¦¾¡¨Äò ¦¾¡Î§¸¡¼¡ ¸×õ(𝟔, 𝟎)ÁüÚõ (−𝟑, 𝟎)±ýÈ
ÒûÇ¢¸û ÅÆ¢§Â ¦ºøÄìÜÊÂÐÁ¡É ¦ºùŸ «¾¢ÀÃŨÇÂò¾¢ý ºÁýÀ¡Î
¸¡ñ¸.O-2006,M-2007,J-2007,M-2008,O-2008,O-2010
¾£÷×:
¦ºùŸ «¾¢ÀÃŨÇÂò¾¢ý ´Õ ¦¾¡¨Äò ¦¾¡Î§¸¡Î
𝑥 + 2𝑦 − 5 = 0
±É§Å, Áü¦È¡Õ ¦¾¡¨Äò ¦¾¡Î§¸¡ðÊý ÅÊÅõ
2𝑥 − 𝑦 + 𝑘 = 0
¦¾¡¨Äò ¦¾¡Î §¸¡Î¸Ç¢ý §º÷ôÒ ºÁýÀ¡ðÊý ÅÊÅõ
(𝑥 + 2𝑦 − 5)(2𝑥 − 𝑦 + 𝑘) = 0
±É§Å,«¾¢ÀÃŨÇÂò¾¢ý ºÁýÀ¡ðÊý ÅÊÅõ
𝑥 + 2𝑦 − 5 2𝑥 − 𝑦 + 𝑘 + 𝑐 = 0
«¾¢ÀÃŨÇÂõ (6,0) ±ýÈ ÒûÇ¢ ÅƢ¡¸î ¦ºøž¡ø,
6 − 0 − 5 12 − 0 + 𝑘 + 𝑐 = 0
1 12 + 𝑘 + 𝑐 = 0
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12 + 𝑘 + 𝑐 = 0
𝑘 + 𝑐 = −12 (1)
«¾¢ÀÃŨÇÂõ (−3,0) ±ýÈ ÒûÇ¢ ÅƢ¡¸î ¦ºøž¡ø,
−3 − 0 − 5 −6 − 0 + 𝑘 + 𝑐 = 0
−8 −6 + 𝑘 + 𝑐 = 0
48 − 8𝑘 + 𝑐 = 0
−8𝑘 + 𝑐 = −48 (2)
1 − 2 ⇒ 𝑘 + 𝑐 + 8𝑘 − 𝑐 = −12 + 48
⇒ 9𝑘 = 36 ⇒ 𝒌 = 𝟒
𝑘 = 4 ±É (1) þø À¢Ã¾¢Â¢¼,
4 + 𝑐 = −12 ⇒ 𝒄 = −𝟏𝟔 «¾¢ÀÃŨÇÂò¾¢ý ºÁýÀ¡Î
𝑥 + 2𝑦 − 5 2𝑥 − 𝑦 + 4 − 16 = 0
17. «¾¢ÀÃŨÇÂò¾¢ý ¨ÁÂõ 𝟐,𝟒 .§ÁÖõ (𝟐, 𝟎) ÅÆ¢§Â ¦ºø ¢̧ÈÐ. þ¾ý
¦¾¡¨Äò ¦¾¡Î§¸¡Î¸û 𝒙 + 𝟐𝒚 − 𝟏𝟐 = 𝟎ÁüÚõ𝒙 − 𝟐𝒚 + 𝟖 =
𝟎¬¸¢ÂÅüÈ¢üÌ þ¨½Â¡¸ þÕ츢ýÈÉ ±É¢ø «¾¢ÀÃŨÇÂò¾¢ý
ºÁýÀ¡Î ¸¡ñ¸. M-2006,J-2006,J-2008,M-2009
¾£÷×: ¦¾¡¨Äò ¦¾¡Î§¸¡Î¸Ç¢ý þ¨½ §¸¡Î¸û
𝑥 + 2𝑦 − 12 = 0 ÁüÚõ 𝑥 − 2𝑦 + 8 = 0
∴ ¦¾¡¨Äò ¦¾¡Î§¸¡Î¸Ç¢ý ºÁýÀ¡Î¸Ç¢ý ÅÊÅõ
𝑥 + 2𝑦 + 𝑙 = 0 ÁüÚõ 𝑥 − 2𝑦 + 𝑚 = 0
þÐ «¾¢ÀÃŨÇÂò¾¢ý ¨ÁÂõ 2,4 ÅƢ¡¸î ¦ºø¸¢ÈÐ. ±É§Å
2 + 8 + 𝑙 = 0 ⇒ 10 + 𝑙 = 0 ⇒ 𝒍 = −𝟏𝟎
2 − 8 + 𝑚 = 0 ⇒ −6 + 𝑚 = 0 ⇒ 𝒎 = 𝟔
∴ ¦¾¡¨Äò ¦¾¡Î§¸¡Î¸Ç¢ý ºÁýÀ¡Î¸û
𝑥 + 2𝑦 − 10 = 0 ÁüÚõ 𝑥 − 2𝑦 + 6 = 0 ¦¾¡¨Äò ¦¾¡Î §¸¡Î¸Ç¢ý §º÷ôÒ ºÁýÀ¡Î
(𝑥 + 2𝑦 − 10)(𝑥 − 2𝑦 + 6) = 0 ±É§Å,«¾¢ÀÃŨÇÂò¾¢ý ºÁýÀ¡ðÊý ÅÊÅõ
𝑥 + 2𝑦 − 10 𝑥 − 2𝑦 + 6 + 𝑘 = 0
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«¾¢ÀÃŨÇÂõ (2,0) ±ýÈ ÒûÇ¢ ÅƢ¡¸î ¦ºøž¡ø,
(2 + 0 − 10)( 2 − 0 + 6 + 𝑘 = 0
⇒ (−8)( 8 + 𝑘 = 0 ⇒ −64 + 𝑘 = 0 ⇒ 𝒌 = 𝟔𝟒
«¾¢ÀÃŨÇÂò¾¢ý ºÁýÀ¡Î 𝑥 + 2𝑦 − 10 𝑥 − 2𝑦 + 6 + 64 = 0
18. ´Õ ¦¾¡íÌ À¡Äò¾¢ý ¸õÀ¢ żõ ÀÃŨÇ ÅÊÅ¢ÖûÇÐ. «¾ý À¡Ãõ
¸¢¨¼Áð¼Á¡¸ º£Ã¡¸ ÀÃÅ¢ÔûÇÐ. «¨¾ò ¾¡íÌõ þÕ àñ¸ÙìÌ
þ¨¼§ÂÔûÇ àÃõ 𝟏𝟓𝟎𝟎 «Ê. ¸õÀ¢ żò¨¾ ¾¡íÌõ ÒûÇ¢¸û རø
¾¨Ã¢ĢÕóÐ 𝟐𝟎𝟎 «Ê ¯ÂÃò¾¢ø «¨ÁóÐûÇÉ. §ÁÖõ ¾¨Ã¢ĢÕóÐ
¸õÀ¢ żò¾¢ý ¾¡úÅ¡É ÒûǢ¢ý ¯ÂÃõ 𝟕𝟎 «Ê, ¸õÀ¢Å¼õ 𝟏𝟐𝟐 «Ê
¯ÂÃò¾¢ø ¾¡íÌõ ¸õÀò¾¢üÌ þ¨¼§Â ¯ûÇ ¦ºíÌòÐ ¿£Çõ
¸¡ñ¸.(¾¨ÃìÌ þ¨½Â¡¸) O-2007
¾£÷×:¦¾¡íÌ À¡Äò¾¢ý ¸õÀ¢ żõ §ÁüÒÈõ ¾¢ÈôÒ¨¼Â ÀÃŨÇ ÅÊÅ¢ÖûÇÐ
±Éì ¦¸¡û¸. ±É§Å, ¸õÀ¢ żò¾¢ý ºÁýÀ¡Î 𝑥2 = 4𝑎𝑦.
¸¡Îì¸ôÀð¼ Å¢Ãí¸Ç¢Ä¢ÕóРżò¾¢ý  ÅÆ¢ôÀ¡¨¾Â¢Ä¢ÕóÐ 70 «Ê
§Áø «¨ÁóÐûÇÐ. ¦¾¡íÌ À¡Äò¾¢ý Å¢ð¼õ 1500 «Ê.
ÒûÇ¢ 𝐵(750,130) ÀÃŨÇÂò¾¢ý Á£Ð «¨ÁóÐûÇÐ.
750 2 = 4𝑎 130 ⇒ 4𝑎 =750 × 750
130=
75 × 750
13
¸õÀ¢ żò¾¢ý ºÁýÀ¡Î 𝑥2 =75×750
13𝑦.
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𝑄(𝑥1, 52) ±ýÈ ÒûÇ¢ ÀÃŨÇÂò¾¢ý Á£Ð «¨ÁóÐûÇÐ.
∴ 𝑥12 =
75 × 750
13× 52 = 75 × 750 × 4
𝑥12 = 75 × 75 × 10 × 4
⇒ 𝑥1 = 75 × 2 10 = 150 10
𝑃𝑄 = 2𝑥1 = 300 10 «Ê
19. ´Õ ¦¾¡íÌ À¡Äò¾¢ý ¸õÀ¢ żõ ÀÃŨÇ ÅÊÅ¢ÖûÇÐ. «¾ý ¿£Çõ
𝟒𝟎Á£ð¼÷ ¬Ìõ. ÅÆ¢ôÀ¡¨¾Â¡ÉÐ ¸õÀ¢ żò¾¢ý £̧úÁð¼ô ÒûǢ¢ĢÕóÐ
𝟓 Á£ð¼÷ £̧§Æ ¯ûÇÐ. ¸õÀ¢ żò¨¾ ¾¡íÌõ àñ¸Ç¢ý ¯ÂÃí¸û 𝟓𝟓
Á£ð¼÷ ±É¢ø 𝟑𝟎 Á£ð¼÷ ¯ÂÃò¾¢ø ¸õÀ¢ żò¾¢üÌ ´Õ Ш½ ¾¡í ¢̧
Üξġ¸ì ¦¸¡Îì¸ôÀð¼¡ø «òШ½ò¾¡í¸¢Â¢ý ¿£Çò¨¾ì ¸¡ñ¸.J-
2006
¾£÷×:
¦¾¡íÌ À¡Äò¾¢ý ¸õÀ¢ żõ §ÁüÒÈõ ¾¢ÈôÒ¨¼Â ÀÃŨÇ ÅÊÅ¢ÖûÇÐ
±Éì ¦¸¡û¸.
±É§Å, ¸õÀ¢ żò¾¢ý ºÁýÀ¡Î 𝑥2 = 4𝑎𝑦.
¦¸¡Îì¸ôÀð¼ Å¢Ãí¸Ç¢Ä¢ÕóРżò¾¢ý  ÅÆ¢ôÀ¡¨¾Â¢Ä¢ÕóÐ 40 Á£ð¼÷
§Áø «¨ÁóÐûÇÐ. ¦¾¡íÌ À¡Äò¾¢ý Å¢ð¼õ 40 Á£ð¼÷.
ÒûÇ¢ 𝐴(20,50) ÀÃŨÇÂò¾¢ý Á£Ð «¨ÁóÐûÇÐ.
20 2 = 4𝑎 50 ⇒ 4𝑎 =20 × 20
50= 8
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¸õÀ¢ żò¾¢ý ºÁýÀ¡Î 𝑥2 = 8𝑦.
𝑄(𝑥1, 25) ±ýÈ ÒûÇ¢ ÀÃŨÇÂò¾¢ý Á£Ð «¨ÁóÐûÇÐ.
∴ 𝑥12 = 8 × 25 = 2 × 4 × 25
⇒ 𝑥1 = 2 × 2 × 5 = 10 2
𝑃𝑄 = 2𝑥1 = 20 2 Á£ð¼÷.
20. ´Õ âø§Å À¡Äò¾¢ý §Áø ŨÇ× ÀÃŨÇÂò¾¢ý «¨Áô¨Àì
¦¸¡ñÎûÇÐ. «ó¾ ŨÇÅ¢ý «¸Äõ 𝟏𝟎𝟎 «Ê¡¸×õ «ùŨÇÅ¢ý
¯îº¢ôÒûǢ¢ý ¯ÂÃõ À¡Äò¾¢Ä¢ÕóÐ 𝟏𝟎 «Ê¡¸ ×õ ¯ûÇÐ ±É¢ø,
À¡Äòò¾¢ý Áò¾¢Â¢Ä¢ ÕóÐ þ¼ôÒÈõ «øÄÐ ÅÄôÒÈõ 𝟏𝟎 «Ê àÃò¾¢ø
À¡Äò¾¢ý §Áø ŨÇ× ±ùÅÇ× ¯ÂÃò¾¢ø þÕìÌõ?M-2009
¾£÷×:
þíÌ ÀÃŨÇÂõ £̧ú§¿¡ì¸¢ò ¾¢üôÒ¨¼Â¾¡¸ ±ÎòÐì ¦¸¡û§Å¡õ.
±É§Å,âø§Å À¡Äò¾¢ý §Áø ŨÇÅ¢ý ºÁýÀ¡Î
𝑥2 = −4𝑎𝑦
þÐ (50, −10) ÅƢ¡¸î ¦ºø ¢̧ÈÐ.
∴ 50 × 50 = −4𝑎(−10) ⇒ 𝒂 =𝟐𝟓𝟎
𝟒
∴ 𝑥2 = −4 250
4 𝑦 ⇒ 𝒙𝟐 = −𝟐𝟓𝟎𝒚
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ÀÃŨÇÂò¾¢ý §Áø ¯ûÇ ÒûÇ¢ 𝐵 10, 𝑦1 ±ý¸.
∴ 100 = −250𝑦1 ⇒ 𝑦1 = −100
250= −
2
5
𝐴𝐵 ±ýÀÐ À¡Äò¾¢ý ¨ÁÂò¾¢Ä¢ÕóÐ ÅÄôÒÈò¾¢ø 10 «Ê ¦¾¡¨ÄÅ¢ø
À¡Äò¾¢ý ¯ÂÃÁ¡Ìõ.
𝐴𝐶 = 10 ÁüÚõ 𝐵𝐶 =2
5
𝐴𝐵 = 10 −2
5=
50−2
5=
48
5= 9
3
5 «Ê
«¾¡ÅÐ, §¾¨ÅôÀð¼ þ¼ò¾¢ø À¡Äò¾¢ý ¯ÂÃõ 93
5 «Ê ¬Ìõ.
21. ´Õ Å¡ø Å¢ñÁ£ý ¬ÉÐ Ýâ¨Éî ÍüÈ¢ ÀÃŨÇÂô À¡¨¾Â¢ø ¦ºø¸¢ÈÐ
ÁüÚõ ÝâÂý ÀÃŨÇÂò¾¢ý ÌÅ¢Âò¾¢ø «¨Á¸¢ÈÐ. Å¡ø Å¢ñÁ£ý
ÝâÂɢĢÕóÐ 𝟖𝟎 Á¢øÄ¢Âý ¢̧.Á£. ¦¾¡¨ÄÅ¢ø «¨ÁóÐ þÕìÌõ §À¡Ð
Å¡ø Å¢ñÁ£¨ÉÔõ Ýâ¨ÉÔõ þ¨½ìÌõ §¸¡Î «îͼý 𝝅
𝟑
§¸¡½ò¾¢¨É ²üÀÎòÐÁ¡É¡ø (i) Å¡ø Å¢ñÁ£É¢ý À¡¨¾Â¢ý
ºÁýÀ¡ð¨¼ì ¸¡ñ¸. (ii) Å¡ø Å¢ñÁ£ý ÝâÂÛìÌ ±ùÅÇ× «Õ¸¢ø
ÅÃÓÊÔõ ±ýÀ¨¾Ôõ ¸¡ñ¸.(À¡¨¾ ÅÄÐÒÈõ ¾¢ÈôÒ¨¼Â¾¡¸ ¦¸¡û¸)
M-2008
¾£÷×: Å¡ø Å¢ñÁ£É¢ý À¡¨¾ 𝑦2 = 4𝑎𝑥 ±ý¸.
Å¡ø Å¢ñÁ£É¢ý ¿¢¨Ä 𝑃 ±ý¸. ÌÅ¢Âõ 𝐹 ±É¢ø 𝐹𝑃 = 80 Á¢øÄ¢Âý ¸¢.Á£.
¬Ìõ.
𝑃𝑄 ⊥ 𝑥 − «îÍ Å¨Ã¸. 𝐹𝑄 = 𝑥1 ±ý¸.
Δ 𝐹𝑄𝑃 þÄ¢ÕóÐ
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sin𝜋
3=
𝑃𝑄
𝐹𝑃⇒ 𝑃𝑄 = 𝐹𝑃 ⋅ sin
𝜋
3= 80 ×
3
2= 40 3
cos𝜋
3=
𝐹𝑄
𝐹𝑃⇒ 𝐹𝑄 = 𝐹𝑃 ⋅ cos
𝜋
3= 80 ×
1
2= 40
∴ 𝑉𝑄 = 𝑉𝐹 + 𝐹𝑄 = 𝑎 + 40.
𝑃 ±ýÀÐ 𝑉𝑄, 𝑃𝑄 = (𝑎 + 40,40 3).
𝑃 ±ýÀÐ ÀÃŨǾ¢ý Á£ÐûǾ¡ø,
40 3 2
= 4𝑎(𝑎 + 40)
1600 × 3 = 4𝑎2 + 160𝑎
4𝑎2 + 160𝑎 − 4800 = 0
𝑎2 + 40𝑎 − 1200 = 0
𝑎 + 60 𝑎 − 20 = 0
𝑎 = −60 «øÄÐ 𝑎 = 20
𝑎 = −60 ²üÒ¨¼Â¾øÄ.
Å¡ø Å¢ñÁ£É¢ý À¡¨¾Â¢ý ºÁýÀ¡Î
𝑦2 = 4 × 20 × 𝑥 ⇒ 𝑦2 = 80𝑥
ÝâÂÛìÌõ Å¡ø Å¢ñÁ£ÛìÌõ þ¨¼§ÂÔûÇ Á¢¸ì ̨Èó¾ àÃõ 𝑉𝐹 = 𝑎 =
20Á¢øÄ¢Âý ¸¢.Á£.
22. ´Õ á즸ð ¦ÅÊ¡ÉÐ ¦¸¡ÙòÐõ §À¡Ð «Ð ´Õ ÀÃŨÇÂô À¡¨¾Â¢ø
¦ºø¸¢ÈÐ. «¾ý ¯îº ¯ÂÃõ 4 Á£ð¼÷ ³ ±ðÎõ§À¡Ð «Ð
¦¸¡Ùò¾ôÀð¼ þ¼ò¾¢Ä¢ÕóÐ ¸¢¨¼Áð¼ àÃõ 6 Á£ð¼÷ ¦¾¡¨ÄÅ¢ÖûÇÐ.
þÚ¾¢Â¡¸ ¸¢¨¼Áð¼Á¡¸ 12Á£ ¦¾¡¨ÄÅ¢ø ¾¨Ã¨Â Å󾨼¸¢ÈÐ ±É¢ø
ÒÈôÀð¼ þ¼ò¾¢ø ¾¨ÃÔ¼ý ²üÀÎòÐõ ±È¢§¸¡½õ ¸¡ñ¸.
M-2006, J-2009,J-2010
¾£÷×:
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´Õ á즸ð ¦ÅÊ¡ÉÐ ¦ºøÖõ ÀÃŨÇÂô À¡¨¾Â¢ý ºÁýÀ¡Î 𝑥2 = −4𝑎𝑦
¬Ìõ. þÐ (6, −4) ÅÆ¢î ¦ºø¸¢ÈÐ.
∴ 62 = −4𝑎 −4
36 = 16𝑎 ⇒ 𝑎 =36
16=
9
4
ÀÃŨÇÂô À¡¨¾Â¢ý ºÁýÀ¡Î
𝑥2 = −4 9
4 𝑦
𝑥2 = −9𝑦
𝑥- ³ ¦À¡ÚòРŨ¸Â¢¼,
2𝑥 = −9𝑑𝑦
𝑑𝑥⇒
𝑑𝑦
𝑑𝑥=
2𝑥
−9= −
2
9𝑥
(−6,−4) þø
𝑑𝑦
𝑑𝑥= −
2
9× −6 =
12
9=
4
3
«¾¡ÅÐ,
tan 𝜃 =4
3⇒ 𝜃 = tan−1
4
3
∴á즸ð ¦ÅÊ¡ÉÐ ÒÈôÀð¼ þ¼ò¾¢ø ¾¨ÃÔ¼ý ²üÀÎòÐõ ±È¢§¸¡½õ
tan−1 4
3
23. ¾¨ÃÁð¼ò¾¢Ä¢ÕóÐ 7.5Á£ ¯ÂÃò¾¢ø ¾¨ÃìÌ þ¨½Â¡¸ ¦À¡Õò¾ôÀð¼
´Õ Ìơ¢ĢÕóÐ ¦ÅÇ¢§ÂÚõ ¿£÷ ¾¨Ã¨Âò ¦¾¡Îõ À¡¨¾ ´Õ
ÀÃŨÇÂò¨¾ ²üÀÎòи¢ÈÐ. §ÁÖõ þó¾ ÀÃŨÇÂô À¡¨¾Â¢ý Ó¨É
Ìơ¢ý š¢ø «¨Á¸¢ÈÐ. ÌÆ¡ö Áð¼ò¾¢üÌ 2.5Á£ £̧§Æ ¿£Ã¢ý
À¡öÅ¡ÉÐ Ìơ¢ý  ÅƢ¡¸î ¦ºøÖõ ¿¢¨Ä ÌòÐ째¡ðÊüÌ 3 Á£ð¼÷ àÃò¾¢ø ¯ûÇÐ ±É¢ø ÌòÐì §¸¡ðÊÄ¢ÕóÐ ±ùÅÇ× àÃò¾¢üÌ
«ôÀ¡ø ¿£Ã¡ÉÐ ¾¨Ã¢ø Å¢Øõ.O-2009 ¾£÷×:¸½ì ¢̧ýÀÊ, Ìơ¢ĢÕóÐ ¦ÅÇ¢§ÂÚõ ¿£÷ À¡¨¾ £̧ú§¿¡ì¸¢ ¾¢ÈôÒ¨¼Â
ÀÃŨÇÂõ ¬Ìõ. ÀÃŨÇÂô À¡¨¾Â¢ý ºÁýÀ¡Î 𝑥2 = −4𝑎𝑦 ¬Ìõ.
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𝑃±ýÈ ÒûÇ¢ ÀÃŨÇÂô À¡¨¾Â¢ø ÌÆ¡ö Áð¼ò¾¢üÌ 2.5 Á£ £̧§ÆÔõ, Ìơ¢ý
 ÅƢ¡¸î ¦ºøÖõ ÌòÐ째¡ðÊüÌ 3 Á£ð¼÷ «ôÀ¡Öõ ¯ûÇÐ
±ýÀ¾¡ø
𝑃 ±ýÀÐ (3,−2 ⋅ 5) ¬Ìõ.
𝑥2 = −4𝑎𝑦 ±ýÀÐ (3,−2.5) ÅÆ¢î ¦ºø¸¢ÈÐ.
∴ 32 = −4𝑎 −2.5
9 = 10𝑎 ⇒ 𝑎 =9
10
ÀÃŨÇÂô À¡¨¾Â¢ý ºÁýÀ¡Î
𝑥2 = −4 9
10 𝑦
ÌòÐì §¸¡ðÊÄ¢ÕóÐ 𝑥1 Á£ð¼÷ àÃò¾¢üÌ «ôÀ¡ø ¿£Ã¡ÉÐ ¾¨Ã¢ø Å¢Øõ
±ý¸. Ìơ¡ÉÐ ¾¨ÃÁð¼ò¾¢Ä¢ÕóÐ 7.5Á£ ¯ÂÃò¾¢ø «¨ÁóÐûǾ¡ø,
𝑥1, −7 ⋅ 5 ±ýÈ ÒûÇ¢Ôõ ÀÃŨÇÂô À¡¨¾Â¢ø «¨ÁóÐ þÕìÌõ. ±É§Å
𝑥12 = −4 ×
9
10× −7 ⋅ 5 = 30 ×
9
10= 9 × 3
𝑥1 = 3 3
±É§Å ÌòÐì §¸¡ðÊÄ¢ÕóÐ 3 3 Á£ð¼÷ àÃò¾¢üÌ «ôÀ¡ø ¿£Ã¡ÉÐ ¾¨Ã¢ø
Å¢Øõ.
24. ´Õ ŨÇ× «¨Ã-¿£ûÅð¼ ÅÊÅ¢ø ¯ûÇÐ. «¾ý «¸Äõ 48 «Ê, ¯ÂÃõ
20 «Ê. ¾¨Ã¢ĢÕóÐ 10 «Ê ¯ÂÃò¾¢ø ŨÇÅ¢ý «¸Äõ ±ýÉ?O-2006
¾£÷×: 2𝑎 = 48 ⇒ 𝑎 = 24
𝑏 = 20
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¿£ûÅð¼¾¢ý ºÁýÀ¡Î
𝑥2
𝑎2+
𝑦2
𝑏2= 1
⇒𝑥2
242+
𝑦2
202= 1
¨ÁÂò¾¢Ä¢ÕóÐ ¾¨Ã¢ĢÕóÐ 10 «Ê ¯ÂÃò¾¢ø ŨÇÅ¢ý «¸Äõ 𝑥1 ±ý¸.
±É§Å (𝑥1, 10) ±ýÈ ÒûÇ¢ ¿£ûÅð¼¾¢ý Á£ÐûÇÐ.
∴𝑥1
2
242+
102
202= 1
𝑥12
242= 1 −
100
400= 1 −
1
4=
3
4
∴ 𝑥12 = 242
3
4
𝑥1 = 24 × 3
2= 12 3
¨ÁÂò¾¢Ä¢ÕóÐ ¾¨Ã¢ĢÕóÐ 10 «Ê ¯ÂÃò¾¢ø ŨÇÅ¢ý «¸Äõ 2𝑥1 = 2 ×
12 3 = 24 3 «Ê.
25. ´Õ À¡Äò¾¢ý ŨÇÅ¡ÉÐ «¨Ã-¿£ûÅð¼ ÅÊÅ¢ø ¯ûÇÐ.
¸¢¨¼Áð¼ò¾¢ø «¾ý «¸Äõ 40 «Ê¡¸×õ, ¨ÁÂò¾¢Ä¢ÕóÐ «¾ý
¯ÂÃõ 16 «Ê¡¸×õ ¯ûÇÐ ±É¢ø ¨ÁÂò¾¢Ä¢ÕóÐ ÅÄÐ «øÄÐ þ¼ô
ÒÈò¾¢ø 9 «Ê àÃò¾¢ø ¯ûÇ ¾¨ÃôÒûǢ¢ĢÕóÐ À¡Äò¾¢ý ¯ÂÃõ
±ýÉ? O-2010
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¾£÷×: 2𝑎 = 40 ⇒ 𝑎 = 20
𝑏 = 16
¿£ûÅð¼¾¢ý ºÁýÀ¡Î
𝑥2
𝑎2+
𝑦2
𝑏2= 1
𝑥2
202+
𝑦2
162= 1 ⇒
𝑥2
400+
𝑦2
256= 1
¨ÁÂò¾¢Ä¢ÕóÐ ÅÄôÒÈò¾¢ø 9 «Ê àÃò¾¢ø ¯ûÇ ¾¨ÃôÒûǢ¢ĢÕóÐ
À¡Äò¾¢ý ¯ÂÃõ 𝑦1 ±ý¸. ±É§Å (9, 𝑦1) ±ýÈ ÒûÇ¢ ¿£ûÅð¼¾¢ý Á£ÐûÇÐ.
∴92
400+
𝑦12
256= 1
𝑦12
256= 1 −
92
400= 1 −
81
400=
400 − 81
400=
319
400
∴ 𝑦12 = 256
319
400
𝑦1 =16
20 319 =
4
5 319
¨ÁÂò¾¢Ä¢ÕóÐ ÅÄÐ «øÄÐ þ¼ô ÒÈò¾¢ø 9 «Ê àÃò¾¢ø ¯ûÇ
¾¨ÃôÒûǢ¢ĢÕóÐ À¡Äò¾¢ý ¯ÂÃõ4
5 319 «Ê.
26. ´Õ ѨÆ× Å¡Â¢Ä¢ý §ÁüܨáÉÐ «¨Ã-¿£ûÅð¼ ÅÊÅ¢ø ¯ûÇÐ.
þ¾ý «¸Äõ 20 «Ê. ¨ÁÂò¾¢Ä¢ÕóÐ «¾ý ¯ÂÃõ 18 «Ê ÁüÚõ Àì¸î
ÍÅâ¸Ç¢ý ¯ÂÃõ 12 «Ê ±É¢ø ²§¾Ûõ ´Õ Àì¸î ÍÅâĢÕóÐ 4 «Ê
àÃò¾¢ø §Áüܨâý ¯ÂÃõ ±ýÉÅ¡¸ þÕìÌõ? M-2007,M-2010
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¾£÷×: Àì¸î ÍÅâĢÕóÐ 4 «Ê àÃò¾¢ø §Áüܨâý ¯ÂÃõ 𝑃𝑄𝑅±ý¸.
À¼ò¾¢ý ãÄõ𝑃𝑄 = 12 «Ê.
À¼ò¾¢ý ãÄõ , Өɸû𝐴(10,0)ÁüÚõ𝐴′ (−10,0).
À¼ò¾¢ý ãÄõ , 𝐴𝐴′ = 2𝑎 = 20 ⇒ 𝑎 = 10
𝑏 = 18 − 12 = 6 ¿£ûÅð¼¾¢ý ºÁýÀ¡Î
𝑥2
𝑎2+
𝑦2
𝑏2= 1 ⇒
𝑥2
102+
𝑦2
62= 1
Àì¸î ÍÅâĢÕóÐ 4 «Ê àÃò¾¢ø §Áüܨâý ¯ÂÃõ 12 + 𝑦1 ±ý¸. ±É§Å
𝑅(6,𝑦1) ±ýÈ ÒûÇ¢ ¿£ûÅð¼¾¢ý Á£ÐûÇÐ.
∴62
100+
𝑦12
36= 1
𝑦12
36= 1 −
36
100=
100 − 36
100=
64
100
∴ 𝑦12 = 36
64
100
𝑦1 = 6 ×8
10=
48
10= 4 ⋅ 8
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Àì¸î ÍÅâĢÕóÐ 4 «Ê àÃò¾¢ø §Áüܨâý ¯ÂÃõ 12 + 𝑦1 = 12 + 4 ⋅ 8 =
16 ⋅ 8 «Ê.
27. ´Õ ¿£ûÅð¼ô À¡¨¾Â¢ý ÌÅ¢Âò¾¢ø âÁ¢ þÕìÌÁ¡Ú ´Õ Ш½ì§¸¡û
ÍüÈ¢ ÅÕ ¢̧ÈÐ. þ¾ý ¨ÁÂò ¦¾¡¨Ä× ¾¸× 𝟏
𝟐 ¬¸×õ âÁ¢ìÌõ Ш½ì
§¸¡ÙìÌõ þ¨¼ôÀð¼ Á£îº¢Ú àÃõ 𝟒𝟎𝟎 ¸¢§Ä¡ Á£ð¼÷¸û ¬¸×õ
þÕìÌÁ¡É¡ø Ш½ì §¸¡ÙìÌõ âÁ¢ìÌõ þ¨¼ôÀð¼ «¾¢¸Àðº
àÃõ ±ýÉ? J-2007,J-2008
¾£÷×: À¼ò¾¢ø âÁ¢Â¢ý ¿¢¨Ä 𝐹1 ±ý¸.
âÁ¢ìÌõ Ш½ì §¸¡ÙìÌõ þ¨¼ôÀð¼ Á£îº¢Ú àÃõ 𝐹1𝐴 = 400 ¸¢§Ä¡
Á£ð¼÷¸û. Ш½ì §¸¡ÙìÌõ âÁ¢ìÌõ þ¨¼ôÀð¼ «¾¢¸Àðº àÃõ 𝐹1𝐴′.
𝐶𝐴′ = 𝐶𝐴 = 𝑎, 𝐶𝐹1 = 𝑎𝑒, 𝐹1𝐴 = 400
𝐶𝐴 = 𝐶𝐹1 + 𝐹1𝐴 ⇒ 𝑎 = 𝑎𝑒 + 400
𝑎 = 𝑎 1
2 + 400
𝑎 −𝑎
2= 400
𝑎
2= 400 ⇒ 𝑎 = 800
𝐹1𝐴′ = 𝐶𝐹1 + 𝐶𝐴′ = 𝑎𝑒 + 𝑎 =
𝑎
2+ 𝑎 =
800
2+ 800 = 400 + 800 = 1200
Ш½ì §¸¡ÙìÌõ âÁ¢ìÌõ þ¨¼ôÀð¼ «¾¢¸Àðº àÃõ àÃõ 1200 ¸¢§Ä¡
Á£ð¼÷¸û.
28. ÝâÂý ÌÅ¢Âò¾¢Ä¢ÕìÌÁ¡Ú ¦Á÷ìÌâ ¸¢Ã¸Á¡ÉÐ Ý̢嬃 ´Õ ¿£ûð¼ô
À¡¨¾Â¢ø ÍüÈ¢ ÅÕ ¢̧ÈÐ. «¾ý «¨Ã ¦¿ð¼îº¢ý ¿£Çõ 𝟑𝟔 Á¢øÄ¢Âý
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¨Áø¸û ¬¸×õ ¨ÁÂò ¦¾¡¨Ä× ¾¸× 𝟎 ⋅ 𝟐𝟎𝟔 ¬¸×õ þÕìÌÁ¡Â¢ý (𝒊) ¦Á÷ìÌâ ¸¢Ã¸Á¡ÉÐ ÝâÂÛìÌ Á¢¸ «Õ¸¡¨Á¢ø ÅÕõ§À¡Ð ¯ûÇ àÃõ
(𝒊𝒊) ¦Á÷ìÌâ ¸¢Ã¸Á¡ÉÐ ÝâÂÛìÌ Á¢¸ò ¦¾¡¨ÄÅ¢ø þÕìÌõ §À¡Ð
¯ûÇ àÃõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸. O-2009,J-2010 ¾£÷×:
À¼ò¾¢ø ÝâÂÉ¢ý ¿¢¨Ä 𝐹1 ±ý¸.
𝐶𝐴 = 𝑎 = 36, 𝑒 = 0 ⋅ 206
𝑎𝑒 = 36 × 0 ⋅ 206 = 7 ⋅ 416
(𝑖)¦Á÷ìÌâ ¢̧øÁ¡ÉÐ ÝâÂÛìÌ Á¢¸ «Õ¸¡¨Á¢ø ÅÕõ§À¡Ð ¯ûÇ àÃõ
𝐹1𝐴 = 𝐶𝐴 − 𝐶𝐹1 = 𝑎 − 𝑎𝑒 = 36 − 7 ⋅ 416 = 28 ⋅ 584 ¦Á÷ìÌâ ¸¢Ã¸Á¡ÉÐ ÝâÂÛìÌ Á¢¸ «Õ¸¡¨Á¢ø ÅÕõ§À¡Ð ¯ûÇ àÃõ
28 ⋅ 584 Á¢øÄ¢Âý ¨Áø¸û
𝑖𝑖 ¦Á÷ìÌâ ¸¢Ã¸Á¡ÉÐ ÝâÂÛìÌ Á¢¸ò ¦¾¡¨ÄÅ¢ø þÕìÌõ §À¡Ð ¯ûÇ
àÃõ
𝐹1𝐴′ = 𝐶𝐴′ + 𝐶𝐹1 = 𝑎 + 𝑎𝑒 = 36 + 7 ⋅ 416 = 43 ⋅ 416
¦Á÷ìÌâ ¢̧øÁ¡ÉÐ ÝâÂÛìÌ Á¢¸ò ¦¾¡¨ÄÅ¢ø þÕìÌõ §À¡Ð ¯ûÇ àÃõ
43 ⋅ 416 Á¢øÄ¢Âý ¨Áø¸û
29. ´Õ §¸¡-§¸¡ Å¢¨Ç¡ðΠţÃ÷ Å¢¨Ç¡ðÎô À¢üº¢Â¢ý §À¡Ð «ÅÕìÌõ
§¸¡-§¸¡ Ìì¸ÙìÌõ þ¨¼§ÂÔûÇ àÃõ ±ô¦À¡ØÐõ
𝟖 Á£ ¬¸ þÕìÌÁ¡Ú ¯½÷¸¢È¡÷. «ùÅ¢Õ Ì¸ÙìÌ þ¨¼ôÀð¼ àÃõ
𝟔 Á£ ±É¢ø «Å÷ µÎõ À¡¨¾Â¢ý ºÁýÀ¡¨¼ì ¸¡ñ¸.
¾£÷×:
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§¸¡-§¸¡ Ì¸û þÃñÎõ 𝐹1 ÁüÚõ 𝐹2 þø «¨ÁóÐûÇÉ ±Éì ¦¸¡û¸.
𝑃(𝑥, 𝑦) ±ýÈ ÒûǢ¡ÉРŢ¨Ç¡ðΠţÃâý ¿¢¨Ä ±Éì ¦¸¡û¸.
∴ 𝐹1𝑃 + 𝐹2𝑃 = 2𝑎 = 8
∴ 𝑎 = 4
𝐹1𝐹2 = 2𝑎𝑒 = 6
𝑎𝑒 = 3
4𝑒 = 3 ⇒ 𝑒 =3
4
§ÁÖõ,
𝑏2 = 𝑎2(1 − 𝑒2) = 16 1 −9
16 = 16 ×
16 − 9
16= 7
∴ À¡¨¾Â¢ý ºÁýÀ¡Î
𝑥2
42+
𝑦2
7= 1 ⇒
𝒙𝟐
𝟏𝟔+
𝒚𝟐
𝟕= 𝟏
30. ´Õ ºÁ¾Çò¾¢ý §Áø ¦ºíÌò¾¡¸ «¨ÁóÐûÇ ÍÅâý Á£Ð 𝟏𝟓Á£ ¿£ÇÓûÇ
²½¢Â¡ÉÐ ¾Çò¾¢¨ÉÔõ ÍÅüÈ¢¨ÉÔõ ¦¾¡ÎÁ¡Ú ¿¸÷óÐ ¦¸¡ñÎ
þÕ츢ÈÐ ±É¢ø ²½¢Â¢ý £̧úÁð¼ ӨɢĢÕóÐ 𝟔Á£ àÃò¾¢ø ²½¢Â¢ø
«¨ÁóÐûÇ 𝑷 ±ýÈ ÒûǢ¢ý ¿¢ÂÁôÀ¡¨¾ì ¸¡ñ¸.O-2007,O-2008
¾£÷×:
𝐴𝐵 ±ýÀÐ ²½¢ ±ý¸. ²½¢Â¢ý
Á£Ð 𝑃(𝑥1, 𝑦1) ±ýÈ ÒûÇ¢ 𝐴𝑃 = 6Á£
±É þÕìÌÁ¡Ú ±ÎòÐì ¦¸¡û¸.
𝑥 − «îÍìÌ ¦ºíÌò¾¡¸ 𝑃𝐷 Ôõ,
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𝑦 − «îÍìÌ ¦ºíÌò¾¡¸ 𝑃𝐶 Ôõ Ũø.
Δ 𝐴𝐷𝑃 ÁüÚõ Δ 𝑃𝐶𝐵 Åʦšò¾¨Å.
∠𝑂𝐴𝑃 = ∠𝐶𝑃𝐵 = 𝜃±ý¸.
⊿𝐴𝐷𝑃¢ø
sin 𝜃 =𝑃𝐷
𝑃𝐴=
𝑦1
6
⊿𝐶𝑃𝐵¢ø
cos 𝜃 =𝑃𝐶
𝑃𝐵=
𝑂𝐷
𝑃𝐵=
𝑥1
9
cos2 𝜃 + sin2 𝜃 = 1 ⇒ 𝑥1
9
2
+ 𝑦1
6
2
= 1
⇒𝑥1
2
81+
𝑦12
36= 1
∴ 𝑃(𝑥1, 𝑦1) þý ¿¢ÂÁôÀ¡¨¾
𝑥2
81+
𝑦2
36= 1.
þÐ ´Õ ¿£ûÅð¼Á¡Ìõ.
31. 𝟗𝒙𝟐 + 𝟐𝟓𝒚𝟐 − 𝟏𝟖𝒙 − 𝟏𝟎𝟎𝒚 − 𝟏𝟏𝟔 = 𝟎±ýÈ ¿£ûÅð¼ò¾¢ý ¨ÁÂò¦¾¡¨Ä×
Å¢¸¢¾õ, ¨ÁÂõ, ÌÅ¢Âí¸û, Өɸû ¿£Çõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ
«¾ý ŨÃôÀ¼ò¨¾ Ũø. M-2009
¾£÷×:
9𝑥2 + 25𝑦2 − 18𝑥 − 100𝑦 − 116 = 0
9𝑥2 − 18𝑥 + 25𝑦2 − 100𝑦 = 116
9 𝑥2 − 2𝑥 + 25 𝑦2 − 4𝑦 = 116
9 𝑥2 − 2𝑥 + 𝟏 − 𝟏 + 25 𝑦2 − 4𝑦 + 𝟒 − 𝟒 = 116
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9 𝑥2 − 2𝑥 + 1 − 9 + 25 𝑦2 − 4𝑦 + 4 − 100 = 116
9 𝑥 − 1 2 + 25 𝑦 − 2 2 = 9 + 100 + 116
9 𝑥 − 1 2 + 25 𝑦 − 2 2 = 225
𝑥 − 1 2
25+
𝑦 − 2 2
9= 1
𝑋2
25+
𝑌2
9= 1
þíÌ 𝑋 = 𝑥 − 1; 𝑌 = 𝑦 − 2.
𝑎2 = 25, 𝑏2 = 9 ⟹ 𝑒 = 1 −𝑏2
𝑎2= 1 −
9
25
⟹ 𝑒 = 25 − 9
25=
16
25=
4
5
𝑎𝑒 = 5 ×4
5= 4
𝑋, 𝑌 ³ô ¦À¡ÚòÐ 𝑥, 𝑦 ³ô ¦À¡ÚòÐ
¨ÁÂõ 𝐶 0,0 𝐶(1,2)
ÌÅ¢Âí¸û 𝐹1 𝑎𝑒, 0 = 𝐹1(4,0) 𝐹1(5,2)
𝐹2 −𝑎𝑒, 0 = 𝐹2(−4,0) 𝐹2(−3,2)
Өɸû 𝐴 𝑎, 0 = 𝐴(5,0) 𝐴(6,2)
𝐴′ −𝑎, 0 = 𝐴′(−5,0) 𝐴′(−4,2)
¨ÁÂò¦¾¡¨Ä× Å¢¸¢¾õ 𝑒 =4
5 𝑒 =
4
5
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tifE©fâj«- ga‹ghLfŸ-II
1. 𝑢 = sin−1 𝑥−𝑦
𝑥+ 𝑦 ±É¢ø äÄâý §¾üÈò¨¾ô ÀÂýÀÎò¾¢ 𝑥
∂𝑢
∂𝑥+ 𝑦
∂𝑢
∂𝑦=
1
2tan𝑢
±Éì ¸¡ðθ.
¾£÷×:
𝑢 = sin−1 𝑥 − 𝑦
𝑥 + 𝑦 𝑓 = sin𝑢 =
𝑥 − 𝑦
𝑥 + 𝑦±ý¸.
𝑓±ýÀÐ ÀÊ 1
2 ¯¨¼Â ºÁôÀÊò¾¡ý º¡÷Ò.
±É§Å, äÄâý §¾üÈôÀÊ,
𝑥∂𝑓
∂𝑥+ 𝑦
∂𝑓
∂𝑦=
1
2𝑓
𝑥∂ sin𝑢
∂𝑥+ 𝑦
∂ sin𝑢
∂𝑦=
1
2sin𝑢
𝑥cos𝑢∂𝑢
∂𝑥+ 𝑦cos𝑢
∂𝑢
∂𝑦=
1
2sin𝑢
sin𝑢 ¬ø ÅÌì¸,
𝑥∂𝑢
∂𝑥+ 𝑦
∂𝑢
∂𝑦=
1
2
sin𝑢
cos𝑢
⇒ 𝑥∂𝑢
∂𝑥+ 𝑦
∂𝑢
∂𝑦=
1
2tan𝑢
2. 𝑢 = tan−1 𝑥 3+𝑦3
𝑥−𝑦 ±É¢ø äÄâý §¾üÈò¨¾ô ÀÂýÀÎò¾¢ 𝑥
∂𝑢
∂𝑥+ 𝑦
∂𝑢
∂𝑦= sin2𝑢
±Éì ¸¡ðθ.
¾£÷×:
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𝑢 = tan−1 𝑥3 + 𝑦3
𝑥 − 𝑦
𝑓 = tan𝑢 =𝑥3 + 𝑦3
𝑥 − 𝑦±ý¸.
𝑓±ýÀÐ ÀÊ 2 ¯¨¼Â ºÁôÀÊò¾¡ý º¡÷Ò.
±É§Å, äÄâý §¾üÈôÀÊ,
𝑥∂𝑓
∂𝑥+ 𝑦
∂𝑓
∂𝑦= 2𝑓
𝑥∂ tan𝑢
∂𝑥+ 𝑦
∂ tan𝑢
∂𝑦= 2tan𝑢
𝑥sec2𝑢∂𝑢
∂𝑥+ 𝑦sec2𝑢
∂𝑢
∂𝑦= 2tan𝑢
sec2𝑢 ¬ø ÅÌì¸,
𝑥∂𝑢
∂𝑥+ 𝑦
∂𝑢
∂𝑦= 2
tan𝑢
sec2𝑢
= 2sin𝑢
cos𝑢× cos2𝑢 = 2sin𝑢cos𝑢 = sin2𝑢
3. 𝑓 =1
𝑥2+𝑦2 ±ýÈ º¡÷À¢üÌ äÄâý §¾üÈò¨¾î ºÃ¢À¡÷ì¸.
¾£÷×:
𝑓 =1
𝑥2 + 𝑦2
𝑓±ýÀÐ ÀÊ −1 ¯¨¼Â ºÁôÀÊò¾¡ý º¡÷Ò.
±É§Å, äÄâý §¾üÈôÀÊ,𝑥∂𝑓
∂𝑥+ 𝑦
∂𝑓
∂𝑦= −𝑓.
ºÃ¢À¡÷ò¾ø:
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𝑓 =1
𝑥2 + 𝑦2= 𝑥2 + 𝑦2 −
1
2
∂𝑓
∂𝑥= −
1
2 𝑥2 + 𝑦2 −
1
2−1 2𝑥 = −𝑥 𝑥2 + 𝑦2 −
3
2
𝑥∂𝑓
∂𝑥= −𝑥2 𝑥2 + 𝑦2 −
3
2
þ§¾ §À¡ýÚ,
𝑦∂𝑓
∂𝑦= −𝑦2 𝑥2 + 𝑦2 −
3
2
Üð¼,
𝑥∂𝑓
∂𝑥+ 𝑦
∂𝑓
∂𝑦= −𝑥2 𝑥2 + 𝑦2 −
3
2 − 𝑦2 𝑥2 + 𝑦2 −3
2
= − 𝑥2 + 𝑦2 −3
2 𝑥2 + 𝑦2 = − 𝑥2 + 𝑦2 −1
2 = −𝑓
∴ äÄâý §¾üÈõ ºÃ¢À¡÷ì¸ôÀð¼Ð.
4. 𝑢 = tan−1 𝑥
𝑦 ±ýÈ º¡÷ÒìÌ
∂2𝑢
∂𝑥 ∂𝑦=
∂2𝑢
∂𝑦 ∂𝑥 ±ýÀ¨¾î ºÃ¢À¡÷ì¸.
¾£÷×:
𝑢 = tan−1 𝑥
𝑦
∂𝑢
∂𝑥=
1
1 + 𝑥
𝑦
2 ×1
𝑦=
𝑦
𝑥2 + 𝑦2
∂𝑢
∂𝑦=
1
1 + 𝑥
𝑦
2 × 𝑥 −1
𝑦2 = −
𝑥
𝑥2 + 𝑦2
∂
∂𝑥 ∂𝑢
∂𝑦 = −
𝑥2 + 𝑦2 × 1 − 𝑥 2𝑥
(𝑥2 + 𝑦2)2 =
𝑥2 − 𝑦2
(𝑥2 + 𝑦2)2
∂
∂𝑦 ∂𝑢
∂𝑥 =
𝑥2 + 𝑦2 × 1 − 𝑦 2𝑦
(𝑥2 + 𝑦2)2=
𝑥2 − 𝑦2
(𝑥2 + 𝑦2)2
⇒∂2𝑢
∂𝑥 ∂𝑦=
∂2𝑢
∂𝑦 ∂𝑥
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𝑢 =𝑥
𝑦2 −𝑦
𝑥 2 ±ýÈ º¡÷ÒìÌ ∂2𝑢
∂𝑥 ∂𝑦=
∂2𝑢
∂𝑦 ∂𝑥 ±ýÀ¨¾î ºÃ¢À¡÷ì¸.
𝑢 = sin3𝑥cos4𝑦 ±ýÈ º¡÷ÒìÌ ∂2𝑢
∂𝑥 ∂𝑦=
∂2𝑢
∂𝑦 ∂𝑥 ±ýÀ¨¾î ºÃ¢À¡÷ì¸.
7. 𝑤 = 𝑢2𝑒𝑣 ±ýÈ º¡÷À¢ø 𝑢 =
𝑥
𝑦 ÁüÚõ 𝑣 = 𝑦log𝑥 ±ÛÁ¡Ú þÕôÀ¢ý
∂𝑤
∂𝑥
ÁüÚõ ∂𝑤
∂𝑦 ¸¡ñ¸.
¾£÷×:
𝑤 = 𝑢2𝑒𝑣
∂𝑤
∂𝑢= 2𝑢𝑒𝑣
∂𝑤
∂𝑣= 𝑢2𝑒𝑣
𝑢 =𝑥
𝑦
∂𝑢
∂𝑥=
1
𝑦
∂𝑢
∂𝑦=
−𝑥
𝑦2
𝑣 = 𝑦log𝑥 ∂𝑣
∂𝑥=
𝑦
𝑥 ;
∂𝑢
∂𝑦= log𝑥
∂𝑤
∂𝑥=
∂𝑤
∂𝑢
∂𝑢
∂𝑥+
∂𝑤
∂𝑣
∂𝑣
∂𝑥
= 2𝑢𝑒𝑣 1
𝑦 + 𝑢2𝑒𝑣
𝑦
𝑥
= 𝑢𝑒𝑣 2
𝑦+
𝑢𝑦
𝑥
=𝑥
𝑦𝑒𝑦log 𝑥
2
𝑦+
𝑥
𝑦⋅𝑦
𝑥
=𝑥
𝑦𝑒 log𝑥𝑦
2
𝑦+ 1
=𝑥
𝑦𝑥𝑦
2
𝑦+ 1
∂𝑤
∂𝑥=
𝑥
𝑦𝑥𝑦
2
𝑦+ 1
∂𝑤
∂𝑦=
∂𝑤
∂𝑢
∂𝑢
∂𝑦+
∂𝑤
∂𝑣
∂𝑣
∂𝑦
= 2𝑢𝑒𝑣 −𝑥2
𝑦 + 𝑢2𝑒𝑣 ⋅ log𝑥
= 𝑢𝑒𝑣 −2𝑥
𝑦2+ 𝑢log𝑥
=𝑥
𝑦𝑒𝑦log 𝑥 −
2𝑥
𝑦2+
𝑥
𝑦log𝑥
=𝑥
𝑦𝑒 log𝑥𝑦
−2𝑥 + 𝑥𝑦log𝑥
𝑦2
=𝑥
𝑦𝑥𝑦
−2𝑥 + 𝑥𝑦log𝑥
𝑦2
=𝑥2
𝑦3𝑥𝑦 −2 + 𝑦log𝑥
∂𝑤
∂𝑦=
𝑥2
𝑦3𝑥𝑦 𝑦log𝑥 − 2
tistiu tiujš
ºÁýÀ¡Î 𝑦 = 𝑥3 + 1 𝑦 = 𝑥3 𝑦2 = 2𝑥3
(1) º¡÷À¸õ, ¿£ðÊôÒ, ¦ÅðÎòÐñθû ÁüÚõ ¬¾¢:
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º¡÷À¸õ (−∞, ∞) (−∞, ∞) [0, ∞)
¿£ðÊôÒ
¸¢¨¼Áð¼
¿£ðÊôÒ −∞ < 𝑥 < ∞ −∞ < 𝑥 < ∞ 0 ≤ 𝑥 < ∞
¿¢¨ÄìÌòÐ
¿£ðÊôÒ −∞ < 𝑦 < ∞ −∞ < 𝑦 < ∞ −∞ < 𝑦 < ∞
¦ÅðÎò
Ðñθû
𝑥 − ¦ÅðÎ
ÐñÎ −1 0 0
𝑦 − ¦ÅðÎ
ÐñÎ 1 0 0
¬¾¢ ŨÇŨà ¬¾¢
ÅÆ¢î ¦ºøÄ¡Ð.
ŨÇŨà ¬¾¢
ÅÆ¢î ¦ºøÖõ.
ŨÇŨà ¬¾¢
ÅÆ¢î ¦ºøÖõ.
2 ºÁ÷ §º¡¾¨É
ŨÇŨáÉÐ
ºÁ÷
¾ý¨Á¨Â
¦ÀÈÅ¢ø¨Ä.
ŨÇŨÃ
¬¾¢¨Âô
¦À¡ÚòÐ
ºÁáÉÐ.
ŨÇŨà 𝑥 −
¦À¡ÚòÐ
ºÁáÉÐ.
3 ¦¾¡¨Äò¦¾¡Î
§¸¡Î¸û
ŨÇŨÃìÌ
±ó¾ ´Õ
¦¾¡¨Äò¦¾¡Î
§¸¡Î¸Ùõ
þø¨Ä.
ŨÇŨÃìÌ
±ó¾ ´Õ
¦¾¡¨Äò¦¾¡Î
§¸¡Î¸Ùõ
þø¨Ä.
ŨÇŨÃìÌ
±ó¾ ´Õ
¦¾¡¨Äò¦¾¡Î
§¸¡Î¸Ùõ
þø¨Ä.
4 ´Ã¢ÂøÒ ¾ý¨Á:
𝑦 ′ ≥ 0 , ∀ 𝑥∈ −∞, ∞ ±ýÀ¾¡ø
(−∞,∞) þø
ÓØÅÐÁ¡¸
²ÚÓ¸Á¡¸î
¦ºøÖõ.
𝑦 ′ ≥ 0 , ∀ 𝑥∈ −∞, ∞ ±ýÀ¾¡ø
(−∞,∞) þø
ÓØÅÐÁ¡¸
²ÚÓ¸Á¡¸î
¦ºøÖõ.
𝑦 = 2𝑥3
2 ±ýÈ
¸¢¨Ç¢ø
ŨÇŨÃ
²ÚÓ¸Á¡¸
þÕìÌõ.
𝑦 = − 2𝑥3
2 ±ýÈ ¢̧¨Ç¢ø
ŨÇŨÃ
þÈíÌ Ó¸Á¡¸
þÕìÌõ.
5 º¢ÈôÒô ÒûÇ¢¸û:
(−∞, 0) ±ýÈ
þ¨¼¦ÅǢ¢ø
£̧ú §¿¡ì¸¢
ÌƢš¸×õ
ÁüÚõ, (0, ∞) ±ýÈ
þ¨¼¦ÅǢ¢ø
§Áø §¿¡ì¸¢
ÌƢš¸×õ
þÕìÌõ.
0,1 ±ýÀÐ
ŨÇ× Á¡üÚô
ÒûÇ¢.
(−∞, 0) ±ýÈ
þ¨¼¦ÅǢ¢ø
£̧ú §¿¡ì¸¢
ÌƢš¸×õ
ÁüÚõ, (0, ∞) ±ýÈ
þ¨¼¦ÅǢ¢ø
§Áø §¿¡ì¸¢
ÌƢš¸×õ
þÕìÌõ.
0,0 ±ýÀÐ
ŨÇ× Á¡üÚô
ÒûÇ¢.
0,0 ±ýÀÐ
ŨÇ× Á¡üÚô
񞂢忀.
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ŨÃôÀ¼õ
õ¬è‚ªè¿„êñ¡ð£´èœ
1. ¾£÷: 𝒙 + 𝒚 𝟐 𝒅𝒚
𝒅𝒙= 𝒂𝟐
. O-2010
¾£÷×:
𝑥 + 𝑦 2 𝑑𝑦
𝑑𝑥= 𝑎2
(1)
𝑥 + 𝑦 = 𝑧 ±ý¸. (2)
𝑥 ³ ¦À¡ÚòРŨ¸ôÀÎò¾,
1 +𝑑𝑦
𝑑𝑥=
𝑑𝑧
𝑑𝑥
⇒𝑑𝑦
𝑑𝑥=
𝑑𝑧
𝑑𝑥− 1 (3)
(3),(2) ³ (1) þø À¢Ã¾¢Â¢¼,
𝑧2 𝑑𝑧
𝑑𝑥− 1 = 𝑎2
⇒𝑑𝑧
𝑑𝑥− 1 =
𝑎2
𝑧2⇒
𝑑𝑧
𝑑𝑥=
𝑎2
𝑧2+ 1
⇒𝑑𝑧
𝑑𝑥=
𝑎2 + 𝑧2
𝑧2
Á¡È¢¸¨Çô À¢Ã¢ôÀ¾¡ø ¢̧¨¼ôÀÐ
𝑧2
𝑧2 + 𝑎2𝑑𝑧 = 𝑑𝑥
¦¾¡¨¸Â¢¼ì ¸¢¨¼ôÀÐ,
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𝑧2
𝑧2 + 𝑎2𝑑𝑧 = 𝑑𝑥 ⇒
𝑧2 + 𝑎2 − 𝑎2
𝑧2 + 𝑎2𝑑𝑧 = 𝑥 + 𝑐
⇒ 𝑧2 + 𝑎2
𝑧2 + 𝑎2𝑑𝑧 −
𝑎2
𝑧2 + 𝑎2𝑑𝑧 = 𝑥 + 𝑐
⇒ 𝑑𝑧 − 𝑎2 1
𝑧2 + 𝑎2𝑑𝑧 = 𝑥 + 𝑐
⇒ 𝑧 − 𝑎2 ⋅1
𝑎tan−1
𝑧
𝑎= 𝑥 + 𝑐
⇒ 𝑥 + 𝑦 − 𝑎 tan−1 𝑥 + 𝑦
𝑎 = 𝑥 + 𝑐
⇒ 𝒚 − 𝒂𝐭𝐚𝐧−𝟏 𝒙 + 𝒚
𝒂 = 𝒄
2. ¾£÷: 𝒙 + 𝒚 𝟐 𝒅𝒚
𝒅𝒙= 𝟏. O-2008
¾£÷×:
𝑥 + 𝑦 2 𝑑𝑦
𝑑𝑥= 1 (1)
𝑥 + 𝑦 = 𝑧 ±ý¸. (2)
𝑥 ³ ¦À¡ÚòРŨ¸ôÀÎò¾,
1 +𝑑𝑦
𝑑𝑥=
𝑑𝑧
𝑑𝑥
⇒𝑑𝑦
𝑑𝑥=
𝑑𝑧
𝑑𝑥− 1 (3)
(3),(2) ³ (1) þø À¢Ã¾¢Â¢¼,
𝑧2 𝑑𝑧
𝑑𝑥− 1 = 1
⇒𝑑𝑧
𝑑𝑥− 1 =
1
𝑧2⇒
𝑑𝑧
𝑑𝑥=
1
𝑧2+ 1
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⇒𝑑𝑧
𝑑𝑥=
1 + 𝑧2
𝑧2
Á¡È¢¸¨Çô À¢Ã¢ôÀ¾¡ø ¢̧¨¼ôÀÐ
𝑧2
𝑧2 + 1𝑑𝑧 = 𝑑𝑥
¦¾¡¨¸Â¢¼ì ¸¢¨¼ôÀÐ,
𝑧2
𝑧2 + 1𝑑𝑧 = 𝑑𝑥 ⇒
𝑧2 + 1 − 1
𝑧2 + 1𝑑𝑧 = 𝑥 + 𝑐
⇒ 𝑧2 + 1
𝑧2 + 1𝑑𝑧 −
1
𝑧2 + 1𝑑𝑧 = 𝑥 + 𝑐
⇒ 𝑑𝑧 − 1
𝑧2 + 1𝑑𝑧 = 𝑥 + 𝑐
⇒ 𝑧 − tan−1 𝑧 = 𝑥 + 𝑐
⇒ 𝑥 + 𝑦 − tan−1(𝑥 + 𝑦) = 𝑥 + 𝑐
⇒ 𝒚 − 𝐭𝐚𝐧−𝟏 𝒙 + 𝒚 = 𝒄
3. ´Õ ÓôÀÊô ÀøÖÚôÒì §¸¡¨Å 𝒙 = −𝟏 ±Ûõ §À¡Ð ¦ÀÕÁ Á¾¢ôÒ 𝟒
¬¸×õ 𝒙 = 𝟏 ±Ûõ §À¡Ð º¢ÚÁ Á¾¢ôÒ 𝟎 ¬¸×õ þÕôÀ¢ý
«ì§¸¡¨Å¨Âì ¸¡ñ¸.
¾£÷×:
𝑥 þø ÓôÀÊô ÀøÖÚôÒì §¸¡¨Å¨Â 𝑦 = 𝑓(𝑥) ±ý¸.
𝑥 = −1 ±Ûõ §À¡Ð ¦ÀÕÁ Á¾¢ô¨ÀÔõ
𝑥 = 1 ±Ûõ §À¡Ð º¢ÚÁ Á¾¢ô¨ÀÔõ ¦ÀÚž¡ø
𝑥 = −1, 𝑥 = 1 ¬¸¢Â Á¾¢ôÒ¸ÙìÌ 𝑑𝑦
𝑑𝑥= 0
𝑑𝑦
𝑑𝑥= 𝑘 𝑥 + 1 𝑥 − 1 = 𝑘(𝑥2 − 1)
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Á¡È¢¸¨Çô À¢Ã¢ôÀ¾¡ø ¢̧¨¼ôÀÐ,
𝑑𝑦 = 𝑘 𝑥2 − 1 𝑑𝑥
𝑑𝑦 = 𝑘 𝑥2 − 1 𝑑𝑥
𝑦 = 𝑘 𝑥3
3− 𝑥 + 𝑐 (1)
𝑥 = −1 ±É¢ø 𝑦 = 4
1 ⇒ 4 = 𝑘 −1
3+ 1 + 𝑐 ⇒ 4 = 𝑘
2
3+ 𝑐
⇒ 𝟏𝟐 = 𝟐𝒌 + 𝟑𝒄 (2)
𝑥 = 1 ±É¢ø 𝑦 = 0
1 ⇒ 0 = 𝑘 1
3− 1 + 𝑐 ⇒ 0 = 𝑘
−2
3+ 𝑐
⇒ 𝟎 = −𝟐𝒌 + 𝟑𝒄 3
2 + 3 ⇒ 12 = 6𝑐 ⇒ 𝑐 = 2
𝑐 = 2 ±É (3) þø À¢Ãò¢Â¢¼,
0 = −2𝑘 + 6 ⇒ 2𝑘 = 6 ⇒ 𝑘 = 3
𝑘 = 3, 𝑐 = 2 ±É (1) þø À¢Ãò¢Â¢¼,
𝑦 = 3 𝑥3
3− 𝑥 + 2
𝒚 = 𝒙𝟑 − 𝟑𝒙 + 𝟐
4. ¾£÷: 𝒙𝟑 + 𝟑𝒙𝒚𝟐 𝒅𝒙 + 𝒚𝟑 + 𝟑𝒙𝟐𝒚 𝒅𝒚 = 𝟎.
¾£÷×:
𝑥3 + 3𝑥𝑦2 𝑑𝑥 + 𝑦3 + 3𝑥2𝑦 𝑑𝑦 = 0
𝑥3𝑑𝑥 + 3𝑥𝑦2𝑑𝑥 + 𝑦3𝑑𝑦 + 3𝑥2𝑦𝑑𝑦 = 0
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𝑥3𝑑𝑥 + 𝑦3𝑑𝑦 = −3𝑥𝑦2𝑑𝑥 − 3𝑥2𝑦𝑑𝑦
𝑥3𝑑𝑥 + 𝑦3𝑑𝑦 = −3𝑥𝑦 𝑦𝑑𝑥 + 𝑥𝑑𝑦
𝑥3𝑑𝑥 + 𝑦3𝑑𝑦 = −3 𝑥𝑦 𝑦𝑑𝑥 + 𝑥𝑑𝑦
𝑥3𝑑𝑥 + 𝑦3𝑑𝑦 = −3 𝑥𝑦 𝑑 𝑥𝑦
𝑥4
4+
𝑦4
4= −3
𝑥𝑦 2
2+ 𝑐
4 ¬ø þÕÒÈÓõ ¦ÀÕì¸
𝑥4 + 𝑦4 + 6𝑥2𝑦2 = 4𝑐
𝒙𝟒 + 𝒚𝟒 + 𝟔𝒙𝟐𝒚𝟐 = 𝒌 þíÌ 𝑘 = 4𝑐
5. ¾£÷: 𝒙𝟐 + 𝒚𝟐 𝒅𝒙 + 𝟑𝒙𝒚𝒅𝒚 = 𝟎. O-2007
¾£÷×:
𝑥2 + 𝑦2 𝑑𝑥 + 3𝑥𝑦𝑑𝑦 = 0
3𝑥𝑦𝑑𝑦 = − 𝑥2 + 𝑦2 𝑑𝑥
𝑑𝑦
𝑑𝑥= −
𝑥2 + 𝑦2
3𝑥𝑦
𝑦 = 𝑣𝑥 ±ý¸.
L. H. S. = 𝑣 + 𝑥𝑑𝑣
𝑑𝑥;
R. H. S. = − 𝑥2 + 𝑣2𝑥2
3𝑥2𝑣
= − 𝑥2 1 + 𝑣2
3𝑥2𝑣
= − 1 + 𝑣2
3𝑣
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𝑣 + 𝑥𝑑𝑣
𝑑𝑥= −
1 + 𝑣2
3𝑣
𝑥𝑑𝑣
𝑑𝑥= −
1 + 𝑣2
3𝑣 − 𝑣
=−1 − 𝑣2 − 3𝑣2
3𝑣
=−1 − 4𝑣2
3𝑣
Á¡È¢¸¨Çô À¢Ã¢ôÀ¾¡ø ¢̧¨¼ôÀÐ,
3𝑣
1 + 4𝑣2𝑑𝑣 = −
𝑑𝑥
𝑥
þÕÒÈÓõ 8 ¬ø ¦ÀÕì¸,
38𝑣
1 + 4𝑣2𝑑𝑣 = −8
𝑑𝑥
𝑥
3 8𝑣
1 + 4𝑣2𝑑𝑣 = −8
𝑑𝑥
𝑥
3 log 1 + 4𝑣2 = −8 log 𝑥 + log 𝑐
3 log(1 + 4𝑣2) + 8 log 𝑥 = log 𝑐
log 1 + 4𝑣2 3 + log 𝑥8 = log 𝑐
1 + 4𝑣2 3𝑥8 = 𝑐
1 + 4𝑦2
𝑥2
3
𝑥8 = 𝑐
𝑥2 + 4𝑦2
𝑥2
3
𝑥8 = 𝑐
𝑥2 + 4𝑦2 3
𝑥6𝑥8 = 𝑐
𝒙𝟐 + 𝟒𝒚𝟐 𝟑𝒙𝟐 = 𝑐
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6. ¾£÷: 𝟐 𝒙𝒚 − 𝒙 𝒅𝒚 + 𝒚𝒅𝒙 = 𝟎.
¾£÷×:
2 𝑥𝑦 − 𝑥 𝑑𝑦 + 𝑦𝑑𝑥 = 0
𝑦𝑑𝑥 = − 2 𝑥𝑦 − 𝑥 𝑑𝑦
𝑑𝑥
𝑑𝑦= −
2 𝑥𝑦 − 𝑥
𝑦
𝑥 = 𝑣𝑦 ±ý¸.
L. H. S. = 𝑣 + 𝑦𝑑𝑣
𝑑𝑦; R. H. S. = −
2 𝑦2𝑣 − 𝑣𝑦
𝑦
𝑣 + 𝑦𝑑𝑣
𝑑𝑦= −
𝑦 2 𝑣 − 𝑣
𝑦= − 2 𝑣 − 𝑣
𝑦𝑑𝑣
𝑑𝑦= − 2 𝑣 − 𝑣 − 𝑣
= −2 𝑣 + 𝑣 − 𝑣 = −2 𝑣
Á¡È¢¸¨Çô À¢Ã¢ôÀ¾¡ø ¢̧¨¼ôÀÐ,
1
𝑣𝑑𝑣 = −2
𝑑𝑦
𝑦
𝑣−1
2𝑑𝑣 = −2 𝑑𝑦
𝑦
𝑣−1
2+1
−1
2+ 1
= −2 log 𝑦 + 2 log 𝑐
𝑣1
2
1
2
= −2 log 𝑦 + 2 log 𝑐
2 𝑣 = −2 log 𝑦 + 2 log 𝑐
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𝑣 = − log 𝑦 + log 𝑐
𝑥
𝑦= loge
𝑐
𝑦
𝑐
𝑦= 𝑒
𝑥
𝑦
𝒚𝒆
𝒙
𝒚 = 𝒄
7. 𝒙 = 𝟎¬¸ þÕìÌõ §À¡Ð 𝒚 = 𝟏 ±É þÕìÌÁ¡É¡ø 𝟏 + 𝒆𝒙
𝒚 𝒅𝒙 +
𝒆𝒙
𝒚 𝟏 −𝒙
𝒚 𝒅𝒚 = 𝟎 ±ýÈ ºÁýÀ¡ðÊý ¾£÷× ¸¡ñ¸:
¾£÷×:
1 + 𝑒𝑥
𝑦 𝑑𝑥 + 𝑒𝑥
𝑦 1 −𝑥
𝑦 𝑑𝑦 = 0
1 + 𝑒𝑥
𝑦 𝑑𝑥 = −𝑒𝑥
𝑦 1 −𝑥
𝑦 𝑑𝑦
𝑑𝑥
𝑑𝑦=
−𝑒𝑥
𝑦 1 −𝑥
𝑦
1 + 𝑒𝑥
𝑦
𝑥 = 𝑣𝑦 ±ý¸.
L. H. S. = 𝑣 + 𝑦𝑑𝑣
𝑑𝑦;
R. H. S. =−𝑒
𝑣𝑦
𝑦 1 −𝑣𝑦
𝑦
1 + 𝑒𝑣𝑦
𝑦 =
−𝑒𝑣 1 − 𝑣
1 + 𝑒𝑣
𝑣 + 𝑦𝑑𝑣
𝑑𝑦= −
−𝑒𝑣 1 − 𝑣
1 + 𝑒𝑣
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𝑦𝑑𝑣
𝑑𝑦= −
−𝑒𝑣 1 − 𝑣
1 + 𝑒𝑣 − 𝑣
=−𝑒𝑣 + 𝑒𝑣𝑣 − 𝑣 − 𝑣𝑒𝑣
1 + 𝑒𝑣
=−𝑒𝑣 − 𝑣
1 + 𝑒𝑣
=− 𝑣 + 𝑒𝑣
1 + 𝑒𝑣
Á¡È¢¸¨Çô À¢Ã¢ôÀ¾¡ø ¢̧¨¼ôÀÐ,
1 + 𝑒𝑣
𝑣 + 𝑒𝑣𝑑𝑣 = −
𝑑𝑦
𝑦
1 + 𝑒𝑣
𝑣 + 𝑒𝑣 𝑑𝑣 = −
𝑑𝑦
𝑦
log 𝑣 + 𝑒𝑣 = − log 𝑦 + log 𝑐
log 𝑣 + 𝑒𝑣 = log𝑐
𝑦
𝑣 + 𝑒𝑣 =𝑐
𝑦
⇒𝑥
𝑦+ 𝑒
𝑥
𝑦 =𝑐
𝑦
𝑦 ¬ø þÕÒÈÓõ ¦ÀÕì¸
⇒ 𝑥 + 𝑦𝑒𝑥
𝑦 = 𝑐
𝑥 = 0¬¸ þÕìÌõ §À¡Ð 𝑦 = 1 ±ýÀ¾¡ø
0 + 1 ⋅ 𝑒0
1 = 𝑐 ⇒ 0 + 1 ⋅ 𝑒0 = 𝑐 ⇒ 0 + 1 ⋅ 1 = 𝑐 ⇒ 𝑐 = 1
∴ 𝒙 + 𝒚𝒆𝒙
𝒚 = 𝟏
8. ¾£÷: 𝟏 − 𝒙𝟐 𝒅𝒚
𝒅𝒙+ 𝟐𝒙𝒚 = 𝒙 𝟏 − 𝒙𝟐 . M-2007
¾£÷×:
1 − 𝑥2 𝑑𝑦
𝑑𝑥+ 2𝑥𝑦 = 𝑥 1 − 𝑥2
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1 − 𝑥2 = 𝑡 ±ý¸
⇒ 1 − 𝑥2 = 𝑡2
⇒ −2𝑥𝑑𝑥 = 2𝑡𝑑𝑡
⇒ 𝑥𝑑𝑥 = −𝑡𝑑𝑡
þÕÒÈÓõ 1 − 𝑥2 ¬ø ÅÌì¸
𝑑𝑦
𝑑𝑥+
2𝑥
1 − 𝑥2 𝑦 =
𝑥
1 − 𝑥2
þÐ 𝑦 þø §¿Ã¢Âî ºÁýÀ¡Î.
þíÌ
𝑃 =2𝑥
1 − 𝑥2; 𝑄 =
𝑥
1 − 𝑥2
𝑃𝑑𝑥 = 2𝑥
1 − 𝑥2𝑑𝑥 = − log 1 − 𝑥2 = log
1
1 − 𝑥2
I. F. = 𝑒 𝑃𝑑𝑥 = 𝑒log
1
1−𝑥2 =1
1 − 𝑥2
§¾¨ÅÂ¡É ¾£÷×
𝑦 I. F. = 𝑄 I. F. 𝑑𝑥 + 𝑐
𝑦 ×1
1 − 𝑥2=
𝑥
1 − 𝑥2×
1
1 − 𝑥2𝑑𝑥 + 𝑐
𝑦
1 − 𝑥2= −
1
𝑡×
1
𝑡2× 𝑡𝑑𝑡 + 𝑐
= − 1
𝑡2𝑑𝑡 + 𝑐
= − 𝑡−2 𝑑𝑡 + 𝑐 = −𝑡−1
−1+ 𝑐
=1
𝑡+ 𝑐 =
1
1 − 𝑥2+ 𝑐
𝒚
𝟏 − 𝒙𝟐=
𝟏
𝟏 − 𝒙𝟐+ 𝒄
9. ¾£÷: 𝟏 + 𝟐𝒙𝟑 𝒅𝒚
𝒅𝒙+ 𝟔𝒙𝟐𝒚 = 𝐜𝐨𝐬𝐞𝐜𝟐 𝒙. J-2010
¾£÷×:
1 + 2𝑥3 𝑑𝑦
𝑑𝑥+ 6𝑥2𝑦 = cosec2 𝑥
þÕÒÈÓõ 1 + 2𝑥3 ¬ø ÅÌì¸
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𝑥2 = 𝑡 ±ý¸
⇒ 2𝑥𝑑𝑥 = 𝑑𝑡
⇒ 𝑥𝑑𝑥 =1
2𝑑𝑡
𝑑𝑦
𝑑𝑥+
6𝑥2
1 + 2𝑥3𝑦 =
cosec2 𝑥
1 + 2𝑥3
þÐ 𝑦 þø §¿Ã¢Âî ºÁýÀ¡Î. þíÌ
𝑃 =6𝑥2
1 + 2𝑥3; 𝑄 =
cosec2 𝑥
1 + 2𝑥3
𝑃𝑑𝑥 = 6𝑥2
1 + 2𝑥3𝑑𝑥 = log 1 + 2𝑥3
I. F. = 𝑒 𝑃𝑑𝑥 = 𝑒 log 1+2𝑥3 = 1 + 2𝑥3
§¾¨ÅÂ¡É ¾£÷×
𝑦 I. F. = 𝑄 I. F. 𝑑𝑥 + 𝑐
𝑦 × 1 + 2𝑥3 = cosec2 𝑥
1 + 2𝑥3× 1 + 2𝑥3 𝑑𝑥 + 𝑐
𝑦 1 + 2𝑥3 = cosec2 𝑥 𝑑𝑥 + 𝑐
= − cot 𝑥 + 𝑐
10. ¾£÷: 𝒅𝒚
𝒅𝒙+
𝒚
𝒙= 𝐬𝐢𝐧(𝒙𝟐).
¾£÷×:
𝑑𝑦
𝑑𝑥+
𝑦
𝑥= sin(𝑥2)
þÐ 𝑦 þø §¿Ã¢Âî ºÁýÀ¡Î.
þíÌ
𝑃 =1
𝑥; 𝑄 = sin(𝑥2)
𝑃𝑑𝑥 = 1
𝑥𝑑𝑥 = log 𝑥
I. F. = 𝑒 𝑃𝑑𝑥 = 𝑒 log 𝑥 = 𝑥
§¾¨ÅÂ¡É ¾£÷×
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𝑑𝑣 = 𝑒−𝑥𝑑𝑥 𝑢 = 𝑥 𝑣 = −𝑒−𝑥
𝑢′ = 1 𝑣1 = 𝑒−𝑥
𝑦 I. F. = 𝑄 I. F. 𝑑𝑥 + 𝑐
𝑦 × 𝑥 = sin(𝑥2) × 𝑥𝑑𝑥 + 𝑐
𝑦𝑥 = sin 𝑡 ×1
2𝑑𝑡 + 𝑐
=1
2 sin 𝑡 𝑑𝑡 + 𝑐 = −
1
2cos 𝑡 + 𝑐
= −1
2cos 𝑥2 + 𝑐
𝟐𝒙𝒚 + 𝐜𝐨𝐬 𝒙𝟐 = 𝒄
11. ±ó¾¦Å¡Õ ÒûǢ¢Öõ º¡ö× 𝒚 + 𝟐𝒙 ±Éì ¦¸¡ñÎ ¬¾¢ÅƢ¡¸î ¦ºøÖõ
ŨÇŨâý ºÁýÀ¡Î 𝒚 = 𝟐(𝒆𝒙 − 𝒙 − 𝟏) ±Éì ¸¡ðθ. M-2010
¾£÷×:
º¡ö× = 𝑦 + 2𝑥
𝑑𝑦
𝑑𝑥= 𝑦 + 2𝑥
𝑑𝑦
𝑑𝑥− 𝑦 = 2𝑥
þÐ 𝑦 þø §¿Ã¢Âî ºÁýÀ¡Î.
þíÌ
𝑃 = −1;𝑄 = 2𝑥
𝑃𝑑𝑥 = − 𝑑𝑥 = − 𝑥
I. F. = 𝑒 𝑃𝑑𝑥 = 𝑒−𝑥
§¾¨ÅÂ¡É ¾£÷×
𝑦 I. F. = 𝑄 I. F. 𝑑𝑥 + 𝑐
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𝑦 × 𝑒−𝑥 = 2𝑥𝑒−𝑥𝑑𝑥 + 𝑐
𝑦𝑒−𝑥 = 2 𝑢𝑑𝑣 + 𝑐
= 2 𝑢𝑣 − 𝑢′𝑣1 + 𝑐
= 2 𝑥 ⋅ −𝑒−𝑥 − 1 ⋅ 𝑒−𝑥 + 𝑐
= 2 −𝑥𝑒−𝑥 − 𝑒−𝑥 + 𝑐
= −2𝑒−𝑥 𝑥 + 1 + 𝑐
𝑦 = −2 𝑥 + 1 + 𝑐𝑒𝑥
ŨÇŨà ¬¾¢ ÅÆ¢î ¦ºøž¡ø , 𝑥 = 0 ±É¢ø 𝑦 = 0
0 = −2 0 + 1 + 𝑐𝑒0
0 = −2 + 𝑐
𝑐 = 2 ŨÇŨâý ºÁýÀ¡Î
𝑦 = −2 𝑥 + 1 + 2𝑒𝑥
= −2𝑥 − 2 + 2𝑒𝑥
𝒚 = 𝟐(𝒆𝒙 − 𝒙 − 𝟏)
12. ¾£÷: 𝟏 + 𝒚𝟐 𝒅𝒙 = 𝐭𝐚𝐧−𝟏 𝒚 − 𝒙 𝒅𝒚. J-2007
¾£÷×:
1 + 𝑦2 𝑑𝑥 = tan−1 𝑦 − 𝑥 𝑑𝑦
𝑑𝑥
𝑑𝑦=
tan−1 𝑦 − 𝑥
1 + 𝑦2=
tan−1 𝑦
1 + 𝑦2−
𝑥
1 + 𝑦2
𝑑𝑥
𝑑𝑦+
𝑥
1 + 𝑦2=
tan−1 𝑦
1 + 𝑦2
þÐ 𝑥 þø §¿Ã¢Âî ºÁýÀ¡Î.
þíÌ
𝑃 =1
1 + 𝑦2; 𝑄 =
tan−1 𝑦
1 + 𝑦2
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tan−1 𝑦 = 𝑡 ±ý¸
⇒1
1 + 𝑦2𝑑𝑦 = 𝑑𝑡
𝑑𝑣 = 𝑒𝑡𝑑𝑡
𝑢 = 𝑡 𝑣 = 𝑒𝑡
𝑢′ = 1 𝑣1 = 𝑒𝑡
tan−1 𝑦 = 𝑡 ±ý¸
⇒1
1 + 𝑦2𝑑𝑦 = 𝑑𝑡
𝑃𝑑𝑦 = 1
1 + 𝑦2𝑑𝑦 = tan−1 𝑦
I. F. = 𝑒 𝑃𝑑𝑦 = 𝑒ta n−1 𝑦
§¾¨ÅÂ¡É ¾£÷×
𝑥 I. F. = 𝑄 I. F. 𝑑𝑦 + 𝑐 𝑥𝑒ta n−1 𝑦 = tan−1 𝑦
1 + 𝑦2× 𝑒ta n−1 𝑦𝑑𝑦 + 𝑐
= 𝑡𝑒𝑡𝑑𝑡 + 𝑐
= 𝑢𝑑𝑣 + 𝑐
= 𝑢𝑣 − 𝑢′𝑣1 + 𝑐
= 𝑡 ⋅ 𝑒𝑡 − 1 ⋅ 𝑒𝑡 + 𝑐
= 𝑡𝑒𝑡 − 𝑒𝑡 + 𝑐
= 𝑒𝑡 𝑡 − 1 + 𝑐
= 𝑒ta n−1 𝑦 tan−1 𝑦 − 1 + 𝑐
𝒙𝒆𝐭𝐚𝐧−𝟏 𝒚 = 𝒆𝐭𝐚𝐧−𝟏 𝒚 𝐭𝐚𝐧−𝟏 𝒚 − 𝟏 + 𝒄
13. ¾£÷:𝒅𝒙
𝒅𝒚+
𝒙
𝟏+𝒚𝟐 =𝐭𝐚𝐧−𝟏 𝒚
𝟏+𝒚𝟐 .
¾£÷×:
𝑑𝑥
𝑑𝑦+
𝑥
1 + 𝑦2=
tan−1 𝑦
1 + 𝑦2
þÐ 𝑥 þø §¿Ã¢Âî ºÁýÀ¡Î.
þíÌ
𝑃 =1
1 + 𝑦2; 𝑄 =
tan−1 𝑦
1 + 𝑦2
𝑃𝑑𝑦 = 1
1 + 𝑦2𝑑𝑦 = tan−1 𝑦
I. F. = 𝑒 𝑃𝑑𝑦 = 𝑒ta n−1 𝑦
§¾¨ÅÂ¡É ¾£÷×
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𝑑𝑣 = 𝑒𝑡𝑑𝑡
𝑢 = 𝑡 𝑣 = 𝑒𝑡
𝑢′ = 1 𝑣1 = 𝑒𝑡
𝑥 I. F. = 𝑄 I. F. 𝑑𝑦 + 𝑐
𝑥𝑒tan−1 𝑦 = tan−1 𝑦
1 + 𝑦2× 𝑒tan−1 𝑦𝑑𝑦 + 𝑐
= 𝑡𝑒𝑡𝑑𝑡 + 𝑐 = 𝑢𝑑𝑣 + 𝑐
= 𝑢𝑣 − 𝑢′𝑣1 + 𝑐
= 𝑡 ⋅ 𝑒𝑡 − 1 ⋅ 𝑒𝑡 + 𝑐
= 𝑡𝑒𝑡 − 𝑒𝑡 + 𝑐
= 𝑒𝑡 𝑡 − 1 + 𝑐
= 𝑒ta n−1 𝑦 tan−1 𝑦 − 1 + 𝑐
𝒙𝒆𝐭𝐚𝐧−𝟏 𝒚 = 𝒆𝐭𝐚𝐧−𝟏 𝒚 𝐭𝐚𝐧−𝟏 𝒚 − 𝟏 + 𝒄
14. ¾£÷: 𝒅𝒙 + 𝒙𝒅𝒚 = 𝒆−𝒚 𝐬𝐞𝐜𝟐 𝒚 𝒅𝒚.
¾£÷×:
𝑑𝑥 + 𝑥𝑑𝑦 = 𝑒−𝑦 sec2 𝑦 𝑑𝑦
𝑑𝑦 ¬ø þÕÒÈÓõ ÅÌì¸
𝑑𝑥
𝑑𝑦+ 𝑥 = 𝑒−𝑦 sec2 𝑦
þÐ 𝑥 þø §¿Ã¢Âî ºÁýÀ¡Î.
þíÌ
𝑃 = 1; 𝑄 = 𝑒−𝑦 sec2 𝑦
𝑃𝑑𝑦 = 𝑑𝑦 = 𝑦
I. F. = 𝑒 𝑃𝑑𝑦 = 𝑒𝑦
§¾¨ÅÂ¡É ¾£÷×
𝑥 I. F. = 𝑄 I. F. 𝑑𝑦 + 𝑐
𝑥𝑒𝑦 = 𝑒−𝑦 sec2 𝑦 × 𝑒𝑦𝑑𝑦 + 𝑐
= sec2 𝑦𝑑𝑦 + 𝑐
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= tan 𝑦 + 𝑐
𝒙𝒆𝒚 = 𝐭𝐚𝐧𝒚 + 𝒄
15. ¾£÷: 𝒅𝟐𝒚
𝒅𝒙𝟐 − 𝟑𝒅𝒚
𝒅𝒙+ 𝟐𝒚 = 𝟐𝒆𝟑𝒙
þíÌ 𝒙 = 𝐥𝐨𝐠𝟐 ±É¢ø 𝒚 = 𝟎 ÁüÚõ 𝒙 = 𝟎
±É¢ø 𝒚 = 𝟎. J-2006,M-2008
¾£÷×: º¢ÈôÒî ºÁýÀ¡Î 𝑝2 − 3𝑝 + 2 = 0.
𝑝 − 2 𝑝 − 1 = 0
⇒ 𝑝 = 2 ÁüÚõ 𝑝 = 1
C. F. ±ýÀÐ 𝐴𝑒2𝑥 + 𝐵𝑒𝑥 .
º¢ÈôÒò¾£÷×
𝑃. 𝐼. =1
𝐷2 − 3𝐷 + 22𝑒3𝑥
= 2 ⋅1
32 − 3 ⋅ 3 + 2𝑒3𝑥
= 2 ⋅1
9 − 9 + 2𝑒3𝑥
= 2 ⋅1
2𝑒3𝑥 = 𝑒3𝑥
±É§Å ¦À¡Ðò¾£÷×
𝑦 = 𝐶. 𝐹. +𝑃. 𝐼.
= 𝐴𝑒2𝑥 + 𝐵𝑒𝑥 + 𝑒3𝑥
𝑥 = log 2 ±É¢ø 𝑦 = 0
0 = 𝐴𝑒2 log 2 + 𝐵𝑒 log 2 + 𝑒3 log 2
0 = 𝐴𝑒 log 22+ 𝐵𝑒 log 2 + 𝑒 log 23
0 = 𝐴𝑒 log 4 + 𝐵𝑒 log 2 + 𝑒 log 8
0 = 4𝐴 + 2𝐵 + 8
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2𝐴 + 𝐵 = −4 (1)
𝑥 = 0 ±É¢ø 𝑦 = 0
0 = 𝐴𝑒0 + 𝐵𝑒0 + 𝑒0
0 = 𝐴 + 𝐵 + 1
𝐴 + 𝐵 = −1 (2)
1 − 2 ⇒ 2𝐴 + 𝐵 − 𝐴 − 𝐵 = −4 + 1 ⇒ 𝐴 = −3
𝑨 = −𝟑
2 ⇒ 𝐴 + 𝐵 = −1 ⇒ −3 + 𝐵 = −1 ⇒ 𝐵 = 2
𝑩 = 𝟐
±É§Å º¢ÈôÒò¾£÷×
𝑦 = −3𝑒2𝑥 + 2𝑒𝑥 + 𝑒3𝑥
𝒚 = 𝒆𝒙 𝟐 − 𝟑𝒆𝒙 + 𝒆𝟐𝒙
16. ¾£÷: 𝑫𝟐 − 𝟏 𝒚 = 𝐜𝐨𝐬𝟐𝒙 − 𝟐𝐬𝐢𝐧𝟐𝒙M-2007,O-2008,J-2009
¾£÷×: º¢ÈôÒî ºÁýÀ¡Î 𝑝2 − 1 = 0.
𝑝 − 1 𝑝 + 1 = 0
⇒ 𝑝 = 1 ÁüÚõ 𝑝 = −1
C. F. ±ýÀÐ 𝐴𝑒𝑥 + 𝐵𝑒−𝑥 .
𝑃𝐼1 =1
𝐷2 − 1cos 2𝑥
=1
−22 − 1cos 2𝑥 =
1
−4 − 1cos 2𝑥
= −1
5cos 2𝑥
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𝑃𝐼2 =1
𝐷2 − 1(−2 sin 2𝑥)
= −2 ⋅1
−22 − 1sin 2𝑥
= −2 ⋅1
−4 − 1sin 2𝑥
=2
5sin 2𝑥
±É§Å ¦À¡Ðò¾£÷×
𝑦 = 𝐶. 𝐹. +𝑃. 𝐼.1+ 𝑃. 𝐼.2
𝒚 = 𝑨𝒆𝒙 + 𝑩𝒆−𝒙 −𝟏
𝟓𝐜𝐨𝐬𝟐𝒙 +
𝟐
𝟓𝐬𝐢𝐧𝟐𝒙
17. ¾£÷: 𝑫𝟐 − 𝟐𝑫 + 𝟐 𝒚 = 𝐬𝐢𝐧𝟐𝒙 + 𝟓. O-2006
¾£÷×: º¢ÈôÒî ºÁýÀ¡Î 𝑝2 − 2𝑝 + 2 = 0.
𝑝 =−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎
=2 ± 4 − 8
2
=2 ± −4
2
=2 ± 2𝑖
2= 1 ± 𝑖
C. F. ±ýÀÐ 𝑒𝑥 𝐴 cos 𝑥 + 𝐵 sin 𝑥 .
𝑃𝐼1 =1
𝐷2 − 2𝐷 + 2sin 2𝑥
=1
−22 − 2𝐷 + 2sin 2𝑥
=1
−4 − 2𝐷 + 2sin 2𝑥 =
1
−2𝐷 − 2sin 2𝑥
= −1
2 𝐷 + 1 sin 2𝑥
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= −1
2⋅
𝐷 − 1
𝐷 + 1 𝐷 − 1 sin 2𝑥
= −1
2⋅ 𝐷 − 1
𝐷2 − 1sin 2𝑥
= −1
2⋅ 𝐷 − 1
−22 − 1sin 2𝑥
= −1
2⋅
1
−4 − 1 𝐷 sin 2𝑥 − 1 sin 2𝑥
=1
10(2 cos 2𝑥 − sin 2𝑥)
𝑃𝐼2 =1
𝐷2 − 2𝐷 + 25
=1
0 − 0 + 25 =
5
2
±É§Å ¦À¡Ðò¾£÷×
𝑦 = 𝐶. 𝐹. +𝑃. 𝐼1 + 𝑃. 𝐼.2
𝒚 = 𝒆𝒙 𝑨𝐜𝐨𝐬𝒙 + 𝑩𝐬𝐢𝐧𝒙 +𝟏
𝟏𝟎(𝟐𝐜𝐨𝐬𝟐𝒙 − 𝐬𝐢𝐧𝟐𝒙) +
𝟓
𝟐
18. ¾£÷: 𝑫𝟐 − 𝟓𝑫 + 𝟔 𝒚 = 𝐬𝐢𝐧𝟐𝒙 + 𝟐𝒆𝟐𝒙. O-2009
¾£÷×: º¢ÈôÒî ºÁýÀ¡Î 𝑝2 − 5𝑝 + 6 = 0.
𝑝 − 3 𝑝 − 2 = 0
𝑝 = 3, 𝑝 = 2
C. F. ±ýÀÐ 𝐴𝑒3𝑥 + 𝐵𝑒2𝑥 .
𝑃𝐼1 =1
𝐷2 − 5𝐷 + 6sin 2𝑥
=1
−22 − 5𝐷 + 6sin 2𝑥
=1
−4 − 5𝐷 + 6sin 2𝑥
=1
−5𝐷 + 2sin 2𝑥
= −1
5𝐷 − 2 sin 2𝑥
= − 5𝐷 + 2
5𝐷 − 2 5𝐷 + 2 sin 2𝑥
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= − 5𝐷 + 2
25𝐷2 − 4sin 2𝑥
= − 5𝐷 + 2
25(−22) − 4sin 2𝑥
= −1
−100 − 4 5𝐷(sin 2𝑥) + 2(sin 2𝑥)
=1
104 10 cos 2𝑥 + 2 sin 2𝑥
=1
52 5 cos 2𝑥 + sin 2𝑥
𝑃𝐼2 =1
𝐷2 − 5𝐷 + 62𝑒2𝑥 =
1
𝐷 − 3 𝐷 − 2 2𝑒2𝑥
=1
2 − 3 𝑥 2𝑒2𝑥 = −2𝑥𝑒2𝑥
±É§Å ¦À¡Ðò¾£÷×
𝑦 = 𝐶. 𝐹. +𝑃. 𝐼1 + 𝑃. 𝐼.2
𝒚 = 𝑨𝒆𝟑𝒙 + 𝑩𝒆𝟐𝒙 +𝟏
𝟓𝟐(𝟓𝐜𝐨𝐬𝟐𝒙 + 𝐬𝐢𝐧𝟐𝒙) − 𝟐𝒙𝒆𝟐𝒙
19. ¾£÷: 𝑫𝟐 − 𝟔𝑫 + 𝟗 𝒚 = 𝒙 + 𝒆𝟐𝒙M-2006,J-2008,M-2009
¾£÷×: º¢ÈôÒî ºÁýÀ¡Î 𝑝2 − 6𝑝 + 9 = 0.
𝑝 − 3 2 = 0
⇒ 𝑝 = 3 ÁüÚõ 𝑝 = 3
C. F. ±ýÀÐ (𝐴 + 𝐵𝑥)𝑒3𝑥 .
𝑃𝐼1 =1
𝐷2−6𝐷+9𝑥 = ℓ𝑥 + 𝑚 ±ý¸. 1
𝑥 = 𝐷2 − 6𝐷 + 9
= 𝐷2 ℓ𝑥 + 𝑚 − 6𝐷 ℓ𝑥 + 𝑚 + 9 ℓ𝑥 + 𝑚
=𝑑2
𝑑𝑥2 ℓ𝑥 + 𝑚 − 6
𝑑
𝑑𝑥 ℓ𝑥 + 𝑚 + 9 ℓ𝑥 + 𝑚
= 0 − 6ℓ + 9ℓ𝑥 + 9𝑚
𝑥 = 9ℓ𝑥 + (9𝑚 − 6ℓ)
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𝑡 𝐴
0 𝐴0
50 0 ⋅ 95 𝐴0
100 ?
þÕÒÈÓõ 𝑥 þý ¦¸Ø ÁüÚõ Á¡È¢Ä¢¸¨Çî ºÁý¦ºöÂ
9ℓ = 1 ⇒ ℓ =1
9
9𝑚 − 6ℓ = 0 ⇒ 9𝑚 = 6ℓ = 6 ×1
9=
2
3⇒ 𝑚 =
2
27
1 ⇒ 𝑃𝐼1 =1
9𝑥 +
2
27
𝑃𝐼2 =1
𝐷2 − 6𝐷 + 9𝑒2𝑥
=1
22 − 6 ⋅ 2 + 9𝑒2𝑥
=1
4 − 12 + 9𝑒2𝑥
=1
13 − 12𝑒2𝑥 = 𝑒2𝑥
±É§Å ¦À¡Ðò¾£÷×
𝑦 = 𝐶. 𝐹. +𝑃. 𝐼.1+ 𝑃. 𝐼.2
𝒚 = 𝑨 + 𝑩𝒙 𝒆𝟑𝒙 + 𝟏
𝟗𝒙 +
𝟐
𝟐𝟕 + 𝒆𝟐𝒙
20. §ÃÊÂõ º¢¨¾Ôõ Á¡ÚÅ£¾Á¡ÉÐ, «¾¢ø ¸¡½ôÀÎõ «ÇÅ¢üÌ Å¢¸¢¾Á¡¸
«¨ÁóÐûÇÐ. 𝟓𝟎 ÅÕ¼í ¸Ç¢ø ¬ÃõÀ «ÇŢĢÕóÐ 𝟓 º¾Å£¾õ
º¢¨¾ó¾¢Õì ¸¢ÈÐ ±É¢ø 𝟏𝟎𝟎 ÅÕ¼ ÓÊÅ¢ø Á£¾¢Â¢ÕìÌõ «Ç× ±ýÉ?
[𝑨𝟎³ ¬ÃõÀ «Ç× ±Éì ¦¸¡û¸.]M-2006,J-2009,M-2010
¾£÷×: 𝑡 ±Ûõ ÅÕ¼ò¾¢ø Á£¾Â¢ÕìÌõ §ÃÊÂò¾¢ý «Ç× 𝐴 ±ý¸. ¸½ì¸¢ý ÀÊ,
𝑑𝐴
𝑑𝑡∝ 𝐴 ⇒
𝑑𝐴
𝑑𝑡= 𝑘𝐴 ⇒ 𝑨 = 𝒄𝒆𝒌𝒕 (𝟏)
𝑡 = 0 ±É¢ø 𝐴 = 𝐴0
1 ⇒ 𝐴0 = 𝑐𝑒0⋅𝑘
⇒ 𝐴0 = 𝑐𝑒0
⇒ 𝐴0 = 𝑐 ⋅ 1
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𝑡 𝐴
1 60
4 21
0 ?
⇒ 𝒄 = 𝑨𝟎
∴ 𝐴 = 𝐴0𝑒𝑘𝑡 (2)
𝑡 = 50 ±É¢ø 𝐴 = 0 ⋅ 95 𝐴0
2 ⇒ 0 ⋅ 95𝐴0 = 𝐴0𝑒50𝑘
𝑒50𝑘 = 0 ⋅ 95
𝑡 = 100 ±É¢ø 𝐴 =?
𝐴 = 𝐴0𝑒100𝑘 = 𝐴0 𝑒
50𝑘 2
= 𝐴0 0 ⋅ 95 2 = 0 ⋅ 9025𝐴0
100 ÅÕ¼ ÓÊÅ¢ø Á£¾¢Â¢ÕìÌõ «Ç× 0 ⋅ 9025𝐴0
21. ´Õ þú¡ÂÉ Å¢¨ÇÅ¢ø, ´Õ ¦À¡Õû Á¡üÈõ «¨¼Ôõ Á¡ÚÅ£¾Á¡ÉÐ 𝒕 §¿Ãò¾¢ø Á¡üÈÁ¨¼Â¡¾ «ô¦À¡ÕÇ¢ý «ÇÅ¢üÌ Å¢¸¢¾Á¡¸ ¯ûÇÐ. ´Õ
Á½¢ §¿Ã ÓÊÅ¢ø 𝟔𝟎 ¸¢Ã¡Óõ ÁüÚõ 𝟒 Á½¢ §¿Ã ÓÊÅ¢ø 𝟐𝟏 ¸¢Ã¡Óõ
Á£¾¢Â¢Õó¾¡ø ¬ÃõÀ ¿¢¨Ä¢ø «ô¦À¡ÕÇ¢ý ±¨¼Â¢¨Éì ¸¡ñ¸.
¾£÷×:𝑡 ±Ûõ §¿Ãò¾¢ø ¦À¡ÕÇ¢ý þÕôÒ 𝐴 ±ý¸.
¸½ì¸¢ý ÀÊ,
𝑑𝐴
𝑑𝑡∝ 𝐴 ⇒
𝑑𝐴
𝑑𝑡= 𝑘𝐴 ⇒ 𝑨 = 𝒄𝒆𝒌𝒕 ⋯ (𝟏)
𝑡 = 1 ±É¢ø 𝐴 = 60
1 ⇒ 60 = 𝑐𝑒𝑘 ⋯⋯⋯⋯⋯⋯ (2)
𝑡 = 4 ±É¢ø 𝐴 = 21
1 ⇒ 21 = 𝑐𝑒4𝑘 ⋯⋯⋯⋯⋯⋯ (3)
2 4 ⇒ 604 = 𝑐4𝑒4𝑘 ⋯⋯⋯⋯⋯⋯ (4)
(4)
(3)⇒ 𝑐3 =
604
21⇒ 𝒄 = 𝟖𝟓 ⋅ 𝟏𝟓
∴ ¬ÃõÀ ¿¢¨Ä¢ø ¦À¡ÕÇ¢ý ±¨¼ 85 ⋅ 15 ¸¢Ã¡õ (¦¾¡Ã¡ÂÁ¡¸)
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𝑡 𝐴
0 1000
? 2000
22. ´Õ Åí¸¢Â¡ÉÐ ¦¾¡¼÷ ÜðÎ ÅðÊ Ó¨È¢ø ÅðʨÂì ¸½ì ¢̧θ¢ÈÐ.
«¾¡ÅÐ ÅðÊ Å£¾¾ò¨¾ «ó¾ó¾ §¿Ãò¾¢ø «ºÄ¢ý Á¡Ú Å£¾¾ò¾¢ø
¸½ì¸¢Î¸¢ÈÐ. ´ÕÅÃÐ Åí¸¢ þÕôÀ¢ø ¦¾¡¼÷ Â¡É ÜðÎ ÅðÊ ãÄõ
¬ñ¦¼¡ýÚìÌ 𝟖% ÅðÊ ¦ÀÕ̸¢ÈÐ ±É¢ø «ÅÃÐ Åí¸¢Â¢ÕôÀ¢ý ´Õ
ÅÕ¼ ¸¡Ä «¾¢¸Ã¢ôÀ¢ý º¾Å£¾ò¨¾ ¸½ì¸¢Î¸.
[𝒆𝟎⋅𝟎𝟖 ≈ 𝟏 ⋅ 𝟎𝟖𝟑𝟑 ±ÎòÐì ¦¸¡û¸] O-2007
¾£÷×:𝑡 ±Ûõ §¿Ãò¾¢ø «ºø 𝐴(𝑡) ±ý¸.
¸½ì¸¢ý ÀÊ,
𝑑𝐴
𝑑𝑡∝ 𝐴 ⇒
𝑑𝐴
𝑑𝑡= 0 ⋅ 08𝐴
⇒ 𝑨(𝒕) = 𝒄𝒆0⋅08𝒕
𝑡 = 0 ±É¢ø 𝐴 0 = 𝑐
𝑡 = 1 ±É¢ø 𝐴 1 = 𝑐𝑒0⋅08
∴ ´Õ ÅÕ¼ «¾¢¸Ã¢ôÒ
𝐴 1 − 𝐴 0 = 𝑐𝑒0⋅08 − 𝑐
= 𝑐 𝑒0⋅08 − 1
= 𝑐 1 ⋅ 0833 − 1 = 0 ⋅ 0833𝑐
∴ ´Õ ÅÕ¼ «¾¢¸Ã¢ôÒº¾Å£¾õ
𝐴 1 − 𝐴 0
𝐴 0 × 100 =
0 ⋅ 0833𝑐
𝑐× 100
= 0 ⋅ 0833 × 100
= 8 ⋅ 33%
23. å. 𝟏𝟎𝟎𝟎 ±ýÈ ¦¾¡¨¸ìÌ ¦¾¡¼÷ ÜðÎ ÅðÊ ¸½ì¸¢¼ôÀθ¢ÈÐ. ÅðÊ
Å£¾õ ¬ñ¦¼¡ýÚìÌ 𝟒 º¾Å£¾Á¡¸ þÕôÀ¢ý «ò¦¾¡¨¸ ±ò¾¨É
¬ñθǢø ¬ÃõÀò ¦¾¡¨¸¨Âô §À¡ø þÕ Á¼í¸¡Ìõ? O-2006, J-
2007,J-2008,O-2010 𝐥𝐨𝐠𝒆 𝟐 = 𝟎 ⋅ 𝟔𝟗𝟑𝟏 .
¾£÷×: 𝑡 ±Ûõ §¿Ãò¾¢ø «ºø 𝐴(𝑡) ±ý¸.
¸½ì¸¢ý ÀÊ,
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¬ñÎ 𝑡 𝐴
1960 0 130000
1990 30 160000
2020 60 ?
𝑑𝐴
𝑑𝑡∝ 𝐴 ⇒
𝑑𝐴
𝑑𝑡= 0 ⋅ 04𝐴
⇒ 𝑨(𝒕) = 𝒄𝒆0⋅04𝒕
𝑡 = 0 ±É¢ø 𝐴 0 = 1000
⇒ 1000 = 𝑐𝑒0 ⇒ 𝑐 = 1000
⇒ 𝑨(𝒕) = 𝟏𝟎𝟎𝟎𝒆0⋅04𝒕
𝐴 = 2000 ±É¢ø 𝑡 =?
2000 = 1000𝒆0⋅04𝒕
𝒆0⋅04𝒕 =2000
1000= 2
0 ⋅ 04𝑡 = log𝑒 2
𝑡 =log𝑒 2
0 ⋅ 04=
0 ⋅ 6931
0 ⋅ 04=
69 ⋅ 31
4= 17
17(§¾¡Ã¡ÂÁ¡¸) ¬ñθǢø ¬ÃõÀò ¦¾¡¨¸¨Âô §À¡ø þÕ Á¼í¸¡Ìõ.
24. ´Õ ¿¸Ãò¾¢ø ¯ûÇ Áì¸û ¦¾¡¨¸Â¢ý ÅÇ÷ Å£¾õ «ó§¿Ãò¾¢ø ¯ûÇ
Áì¸û ¦¾¡¨¸ìÌ Å¢¸¢¾Á¡¸ «¨ÁóÐûÇÐ. 𝟏𝟗𝟔𝟎 ¬õ ¬ñÊø Áì¸û
¦¾¡¨¸ 𝟏, 𝟑𝟎, 𝟎𝟎𝟎 ±É×õ 𝟏, 𝟔𝟎,𝟎𝟎𝟎 ¬¸×õ þÕôÀ¢ý 𝟐𝟎𝟐𝟎 ¬õ ¬ñÊø
Áì¸û ¦¾¡¨¸ ±ùÅÇÅ¡¸ þÕìÌõ? M-2008,J-2010
𝐥𝐨𝐠𝒆 𝟏𝟔
𝟏𝟑 = 𝟎 ⋅ 𝟐𝟎𝟕𝟎; 𝒆𝟎⋅𝟒𝟐 = 𝟏 ⋅ 𝟓𝟐 .
¾£÷×: 𝑡 ±Ûõ §¿Ãò¾¢ø Áì¸û ¦¾¡¨¸ 𝐴(𝑡) ±ý¸.
¸½ì¸¢ý ÀÊ,
𝑑𝐴
𝑑𝑡∝ 𝐴 ⇒
𝑑𝐴
𝑑𝑡= 𝑘𝐴
⇒ 𝑨(𝒕) = 𝒄𝒆𝑘𝒕
1960 ¬õ ¬ñÎ Áì¸û ¦¾¡¨¸Â¢¨É ¦¾¡¼ì¸ Áì¸û ¦¾¡¨¸Â¡¸ì ¦¸¡û¸.
𝑡 = 0 ±É¢ø 𝐴 0 = 130000
⇒ 130000 = 𝑐𝑒0 ⇒ 𝑐 = 130000
⇒ 𝑨(𝒕) = 𝟏𝟑𝟎𝟎𝟎𝟎𝒆𝑘𝒕
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𝑡 𝐴 𝑑𝐴
𝑑𝑡
0 10 −0 ⋅ 051
? 5
𝑡 = 30 ±É¢ø 𝐴(30) = 160000
160000 = 130000𝒆30𝑘
𝑒30𝑘 =160000
130000=
16
13
𝑡 = 60 ±É¢ø 𝐴 60 =?
𝐴 60 = 130000𝑒60𝑘
= 130000 𝑒30𝑘 2
= 130000 16
13
2
= 130000 ×16
13×
16
13= 10000 ×
256
13
= 196923
2020 ¬õ ¬ñÊø Áì¸û ¦¾¡¨¸ §¾¡Ã¡ÂÁ¡¸ 197000 þÕìÌõ.
25. ´Õ ¸¾¢Ã¢Âì¸ô ¦À¡Õû º¢¨¾Ôõ Á¡ÚÅ£¾Á¡ÉÐ «¾ý ±¨¼ìÌ Å¢¸¢¾Á¡¸
«¨ÁóÐûÇÐ. «¾ý ±¨¼ 𝟏𝟎 Á¢. ¢̧áõ ¬¸ þÕìÌõ §À¡Ð º¢¨¾Ôõ
Á¡ÚÅ£¾õ ¿¡¦Ç¡ýÚìÌ 𝟎 ⋅ 𝟎𝟓𝟏 Á¢.¸¢Ã¡õ ±É¢ø «¾ý ±¨¼ 𝟏𝟎
¸¢Ã¡Á¢Ä¢ÕóÐ 𝟓 ¸¢Ã¡Á¡¸ì ̨È ±ÎòÐì ¦¸¡ûÙõ ¸¡Ä «Ç¨Åì
¸¡ñ¸? 𝐥𝐨𝐠𝒆 𝟐 = 𝟎 ⋅ 𝟔𝟗𝟑𝟏 .
¾£÷×: 𝑡 ±Ûõ §¿Ãò¾¢ø ¸¾¢Ã¢Âì¸ô ¦À¡ÕÇ¢ý ±¨¼ 𝐴(𝑡) ±ý¸.
¸½ì¸¢ý ÀÊ,
𝑑𝐴
𝑑𝑡∝ 𝐴 ⇒
𝑑𝐴
𝑑𝑡= 𝑘𝐴
⇒ 𝑨(𝒕) = 𝒄𝒆𝑘𝒕
𝑡 = 0 ±É¢ø 𝐴 0 = 10
⇒ 10 = 𝑐𝑒0 ⇒ 𝑐 = 10
⇒ 𝑨(𝒕) = 𝟏𝟎𝒆𝑘𝒕
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𝑡 𝑇
0 100
5 60
10 ?
𝑑𝐴
𝑑𝑡= 10𝑒𝑘𝑡 × 𝑘 = 𝑘𝐴
𝐴 = 10 ±É¢ø
𝑑𝐴
𝑑𝑡= −0 ⋅ 051
−0 ⋅ 051 = 10𝑘 ⇒ 𝑘 = −0 ⋅ 0051
𝐴 𝑡 = 10𝑒−0⋅0051𝑡
𝐴 = 5 ±É¢ø 5 = 10𝑒−0⋅0051𝑡 ⇒1
2= 𝑒−0⋅0051𝑡
⇒ 2 = 𝑒0⋅0051𝑡
⇒ log𝑒 2 = 0 ⋅ 0051𝑡
⇒ 𝑡 =log𝑒 2
0 ⋅ 0051=
0 ⋅ 6931
0 ⋅ 0051=
6931
51≈ 136
¸¾¢Ã¢Âì¸ô ¦À¡ÕÇ¢ý ±¨¼ 10 ¸¢Ã¡Á¢Ä¢ÕóÐ 5 ¸¢Ã¡Á¡¸ì ̨È ±ÎòÐì
¦¸¡ûÙõ ¸¡Ä «Ç× 136 ¿¡ð¸û.(§¾¡Ã¡ÂÁ¡¸)
26. ¦ÅôÀ ¿¢¨Ä 𝟏𝟓𝟎𝑪 ¯ûÇ ´Õ «¨È¢ø ¨Åì¸ôÀðÎûÇ §¾¿£Ã¢ý ¦ÅôÀ
¿¢¨Ä 𝟏𝟎𝟎𝟎𝑪 ¬Ìõ. «Ð 𝟓 ¿¢Á¢¼í¸Ç¢ø 𝟔𝟎𝟎𝑪 ¬¸ ̨ÈóРŢθ¢ÈÐ.
§ÁÖõ 𝟓 ¿¢Á¢¼õ ¸Æ¢òÐ §¾¿£Ã¢ý ¦ÅôÀ ¿¢¨Ä¢¨É ¸¡ñ¸. O-2009
¾£÷×: 𝑡 ±Ûõ §¿Ãò¾¢ø §¾¿£Ã¢ý ¦ÅôÀ ¿¢¨Ä 𝑇(𝑡) ±ý¸.
¿¢äð¼É¢ý ÌÇ¢÷ Å¢¾¢ôÀÊ,
𝑑𝑇
𝑑𝑡∝ (𝑇 − 𝑆) ⇒
𝑑𝑇
𝑑𝑡= 𝑘(𝑇 − 𝑆)
⇒ 𝑇 − 𝑆 = 𝑐𝑒𝑘𝑡
⇒ 𝑻 = 𝑺 + 𝒄𝒆𝑘𝒕 = 𝟏𝟓 + 𝒄𝒆𝑘𝒕
𝑡 = 0 ±É¢ø 𝑇 = 100
⇒ 100 = 15 + 𝑐𝑒0 ⇒ 𝑐 = 100 − 15 = 85
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⇒ 𝑻 = 𝟏𝟓 + 𝟖𝟓𝒆𝑘𝒕
𝑡 = 5 ±É¢ø 𝑇 = 60
⇒ 60 = 15 + 85𝑒5𝑘 ⇒ 45 = 85𝑒5𝑘
⇒ 𝑒5𝑘 =45
85=
9
17
𝑡 = 10 ±É¢ø 𝑇 =?
𝑇 = 15 + 85𝑒10𝑘
= 15 + 85 𝑒5𝑘 2
= 15 + 85 9
17
2
= 15 + 85 ×81
17 × 17
= 15 + 5 ×81
17
= 15 +405
17
= 15 + 23 ⋅ 82
= 38 ⋅ 82
§ÁÖõ 5 ¿¢Á¢¼õ ¸Æ¢òÐ §¾¿£Ã¢ý ¦ÅôÀ ¿¢¨Ä
38 ⋅ 820𝐶
27. ´Õ þÈó¾Å÷ ¯¼¨Ä ÁÕòÐÅ÷ À⧺¡¾¢ìÌõ §À¡Ð þÈó¾ §¿Ãò¨¾
§¾¡Ã¡ÂÁ¡¸ ¸½ì¸¢¼ §ÅñÊÔûÇÐ. þÈó¾Å÷ ¯¼Ä¢ý ¦ÅôÀ ¿¢¨Ä
¸¡¨Ä 𝟏𝟎.𝟎𝟎 Á½¢ÂÇÅ¢ø 𝟗𝟑. 𝟒𝟎𝑭 ±É ÌÈ¢òÐì ¦¸¡û¸¢È¡÷. §ÁÖõ 𝟐
Á½¢ §¿Ãõ ¸Æ¢òÐ ¦ÅôÀ ¿¢¨Ä «Ç¨Å 𝟗𝟏. 𝟒𝟎𝑭 ±Éì ¸¡ñ ¢̧È¡÷.
«¨È¢ý ¦ÅôÀ ¿¢¨Ä «Ç× (¿¢¨Ä¡ÉÐ) 𝟕𝟐𝟎𝑭 ±É¢ø , þÈó¾ §¿Ãò¨¾ì
¸½ì¸¢Î.(´Õ ÁÉ¢¾ ¯¼Ä¢ý º¡¾¡Ã½ ¦ÅôÀ ¿¢¨Ä «Ç× 𝟗𝟖. 𝟔𝟎𝑭 ±Éì
¦¸¡û¸.)
𝐥𝐨𝐠𝒆
𝟏𝟗. 𝟒
𝟐𝟏. 𝟒= −𝟎. 𝟎𝟒𝟐𝟔 × 𝟐. 𝟑𝟎𝟑 , 𝐥𝐨𝐠𝒆
𝟐𝟔. 𝟔
𝟐𝟏. 𝟒= 𝟎. 𝟎𝟗𝟒𝟓 × 𝟐. 𝟑𝟎𝟑
¾£÷×:𝑡 ±Ûõ §¿Ãò¾¢ø ¯¼Ä¢ý ¦ÅôÀ ¿¢¨Ä𝑇 ±ý¸.
¿¢äð¼É¢ý ÌÇ¢÷ Å¢¾¢ôÀÊ,
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Á½¢ 𝑡 𝑇
10.00 0 93.4
12.00 120 91.4
? ? 98.6
𝑑𝑇
𝑑𝑡∝ (𝑇 − 72)
⇒𝑑𝑇
𝑑𝑡= 𝑘(𝑇 − 72)
⇒ 𝑇 − 72 = 𝑐𝑒𝑘𝑡
⇒ 𝑻 = 𝟕𝟐 + 𝒄𝒆𝑘𝒕
𝑡 = 0 ±É¢ø 𝑇 = 93.4
⇒ 93.4 = 72 + 𝑐𝑒0 ⇒ 𝑐 = 93.4 − 72 = 21.4
⇒ 𝑻 = 𝟕𝟐 + 𝟐𝟏. 𝟒𝒆𝑘𝒕
𝑡 = 120 ±É¢ø 𝑇 = 91.4
⇒ 91.4 = 72 + 21.4𝑒120𝑘
⇒ 19.4 = 21.4𝑒120𝑘
⇒ 𝑒120𝑘 =19.4
21.4
⇒ 120𝑘 = log𝑒 19.4
21.4
⇒ 𝑘 =1
120log𝑒
19.4
21.4 =
1
120 −0.0426 × 2.303
⇒𝟏
𝒌= −𝟏𝟐𝟎
𝟏
𝟎. 𝟎𝟒𝟐𝟔 × 𝟐. 𝟑𝟎𝟑
𝑡1 ±ýÀÐ þÈó¾ §¿Ãò¾¢üÌô À¢ý ¸¡¨Ä 10.00 Á½¢ìÌ ¯ûÇ¡É §¿Ãõ ±ý¸.
𝑡 = 𝑡1 ±Ûõ §À¡Ð 𝑇 = 98 ⋅ 6
⇒ 98 ⋅ 6 = 72 + 21 ⋅ 4𝑒𝑘𝑡1
⇒ 98 ⋅ 6 − 72 = 21 ⋅ 4𝑒𝑘𝑡1
⇒ 𝑒𝑘𝑡1 =26.6
21 ⋅ 4
⇒ 𝑘𝑡1 = log𝑒 26.6
21.4 = 0.0945 × 2.303
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𝑡1 =1
𝑘 0.0945 × 2.303
=−120 × 0.0945 × 2.303
0.0426 × 2.303≈ −𝟐𝟔𝟔
¸¡¨Ä10.00 Á½¢ìÌ ÓýÒ ÍÁ¡÷ 266 ¿¢Á¢¼í¸ÙìÌ ÓýÒ þÈôÒ
§¿÷ó¾¢ÕìÌõ. «¾¡ÅÐ ÍÁ¡÷ 4 Á½¢ 26 ¿¢Á¢¼í¸û.þÈó¾ §¿Ãõ §¾¡Ã¡ÂÁ¡¸
«¾¢¸¡¨Ä 𝟓 ⋅ 𝟑𝟒
28. ´Õ §¿¡Â¡Ç¢Â¢ý º¢Ú¿£Ã¢Ä¢ÕóÐ §Å¾¢ô¦À¡Õû ¦ÅÇ¢§ÂÚõ «ÇÅ¢¨É
¦¾¡¼÷¡¸ §¸ò§¾¼÷ ±ýÈ ¸ÕŢ¢ý ãÄõ ¸ñ¸¡½¢ì¸ôÀθ¢ÈÐ.
𝒕 = 𝟎 ±ýÈ §¿Ãò¾¢ø §¿¡Â¡Ç¢ìÌ 𝟏𝟎 Á¢.¸¢Ã¡õ §Å¾¢ô ¦À¡Õû
¦¸¡Îì¸ôÀθ¢ÈÐ. þÐ −𝟑𝒕𝟏
𝟐 Á¢. ¢̧áõ/ Á½¢ ±ýÛõ Å£¾ò¾¢ø
¦ÅÇ¢§ÂÚ¸¢ÈÐ ±É¢ø,
(𝒊)§¿Ãõ 𝒕 > 0 ±Ûõ §À¡Ð §¿¡Â¡Ç¢Â¢ý ¯¼Ä¢ÖûÇ §Å¾¢ô¦À¡ÕÇ¢ý
«Ç¨Åì ¸¡Ïõ ¦À¡Ðî ºÁýÀ¡Î ±ýÉ?
(𝒊𝒊)ÓبÁ¡¸ §Å¾¢ô¦À¡Õû ¦ÅÇ¢§ÂÈ ±ÎòÐì ¦¸¡ûÙõ ̨Èó¾Àðº
¸¡Ä «Ç× ±ýÉ?
¾£÷×:(𝑖)𝑡 ±Ûõ §¿Ãò¾¢ø §Å¾¢ô¦À¡ÕÇ¢ý ±¨¼ 𝐴 ±ý¸.
§Å¾¢ô¦À¡Õû §ÅÇ¢§ÂÚõ Å£¾õ= −3𝑡1
2
𝑑𝐴
𝑑𝑡= −3𝑡
1
2 ⇒ 𝑑𝐴 = −3𝑡1
2𝑑𝑡
𝑑𝐴 = −3𝑡1
2𝑑𝑡
𝐴 = −3𝑡
1
2+1
1
2+ 1
+ 𝑐
𝐴 = −3𝑡
3
2
3
2
+ 𝑐
𝐴 = −2𝑡3
2 + 𝑐
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𝑡 𝐴
0 𝐴0
1 3𝐴0
5 ?
𝑡 = 0 ±É¢ø 𝐴 = 10 ⇒ 𝑐 = 10
𝐴(𝑡) = 10 − 2𝑡3
2
(𝑖𝑖)𝐴 = 0 ±É¢ø §Å¾¢ô¦À¡ÕûÓبÁ¡¸¦ÅÇ¢§ÂÈ¢ Å¢ð¼Ð ±Éô ¦À¡Õû.
0 = 10 − 2𝑡3
2 ⇒ 10 = 2𝑡3
2 ⇒ 5 = 𝑡3
2
𝑡3 = 52 = 25 ⇒ 𝑡 = 2.9
ÓبÁ¡¸ §Å¾¢ô¦À¡Õû ¦ÅÇ¢§ÂÈ ±ÎòÐì ¦¸¡ûÙõ ̨Èó¾À𺠸¡Ä
«Ç× 2 ⋅ 9 Á½¢.
29. ÑñÏ¢÷¸Ç¢ý ¦ÀÕì¸ò¾¢ø, À¡ìËâ¡Ţý ¦ÀÕì¸Å£¾Á¡ÉÐ «¾¢ø
¸¡½ôÀÎõ À¡ìËâ¡Ţý ±ñ½¢ì¨¸ìÌ Å¢¸¢¾Á¡¸ «¨ÁóÐûÇÐ.
þô¦ÀÕì ¸ò¾¡ø À¡ìËâ¡Ţý ±ñ½¢ì¨¸ 𝟏 Á½¢ §¿Ãò¾¢ø
ÓõÁ¼í¸¡ ¢̧ÈÐ ±É¢ø ³óÐ Á½¢ §¿Ã ÓÊÅ¢ø À¡ìËâ¡Ţý
±ñ½¢ì¨¸ ¬ÃõÀ ¿¢¨Ä¨Âì ¸¡ðÊÖõ 𝟑𝟓 Á¼í¸¡Ìõ ±Éì ¸¡ðθ.
J-2006,M-2009
¾£÷×:𝑡 ±Ûõ §¿Ãò¾¢ø À¡ìËâ¡Ţý ±ñ½¢ì¨¸ 𝐴 ±ý¸.
¸½ì¸¢ý ÀÊ,
𝑑𝐴
𝑑𝑡∝ 𝐴 ⇒
𝑑𝐴
𝑑𝑡= 𝑘𝐴
⇒ 𝑨(𝒕) = 𝒄𝒆𝑘𝒕
¬ÃõÀ ¿¢¨Ä¢ø ¯ûÇ À¡ìËâ¡Ţý ±ñ½¢ì¨¸ 𝐴0 ±ý¸.
𝑡 = 0 ±É¢ø 𝐴0 = 𝑐𝑒0 ⇒ 𝑐 = 𝐴0
⇒ 𝑨 𝒕 = 𝑨𝟎𝒆𝑘𝒕
𝑡 = 1 ±É¢ø 𝐴 1 = 3𝐴0 ⇒ 3𝐴0 = 𝐴0𝑒𝑘 ⇒ 𝑒𝑘 = 3
𝑡 = 5 ±É¢ø 𝐴 5 = 𝐴0𝑒5𝑘 = 𝐴0 𝑒
𝑘 5 = 𝐴0 3 5 = 35 ⋅ 𝐴0
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∴ ³óÐ Á½¢ §¿Ã ÓÊÅ¢ø À¡ìËâ¡Ţý ±ñ½¢ì¨¸ ¬ÃõÀ ¿¢¨Ä¨Âì
¸¡ðÊÖõ 35 Á¼í¸¡Ìõ.
ÌÄí¸û
ŨèÈ:
𝐺 ´Õ ¦ÅüÈüÈ ¸½õ ±ý¸. ∗ ´Õ ®ÕÚôÒî ¦ºÂÄ¢, (𝐺,∗) ¬ÉÐ
ÌÄÁ¡¸Â¢Õì¸ À¢ýÅÕõ ¿¢Àó¾¨É¸û ¯ñ¨Á¡¸ §ÅñÎõ.
(1) «¨¼ôÒ Å¢¾¢ : 𝑎, 𝑏 ∈ 𝐺 ⇒ 𝑎 ∗ 𝑏 ∈ 𝐺
(2) §º÷ôÒ Å¢¾¢ : ∀ 𝑎, 𝑏, 𝑐 ∈ 𝐺,
𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎 ∗ (𝑏 ∗ 𝑐)
(3) ºÁÉ¢ Å¢¾¢ : 𝑒 ∈ 𝐺-³ 𝑎 ∗ 𝑒 = 𝑒 ∗ 𝑎 = 𝑎, ∀ 𝑎 ∈ 𝐺 ±ÛÁ¡Ú
¸¡½Ä¡õ.
(4) ±¾¢÷Á¨È Å¢¾¢ : ´ù¦Å¡Õ 𝑎 ∈ 𝐺-ìÌõ, 𝑎−1 ∈ 𝐺 -³ 𝑎−1 ∗ 𝑎 = 𝑎 ∗ 𝑎−1 =
𝑒 ±ÛÁ¡Ú ¸¡½ÓÊÔõ.
𝑒 ¬ÉÐ 𝐺– þý ºÁÉ¢ ¯ÚôÒ ±ÉôÀÎõ.
𝑎−1 ¬ÉÐ 𝑎 - þý ±¾¢÷Á¨È ±ÉôÀÎõ.
1) (𝒁𝟕 − 𝟎 ,∙𝟕) ´Õ ÌÄõ
¾£÷×: 𝐺 = { 1 , 2 , 3 , 4 , 5 , 6 } ±ý¸.
∙𝟕 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 1 3 5 3 3 6 2 5 1 4 4 4 1 5 2 6 3 5 5 3 1 6 4 2
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6 6 5 4 3 2 1
«¨¼ôÒ Å¢¾¢: «ð¼Å¨½Â¢ý ±øÄ¡ ¯ÚôÒ¸Ùõ 𝐺 -þý ¯ÚôҸǡÌõ.
∴ «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
§º÷ôÒ Å¢¾¢ 7-þý ÁðÎì¸¡É ¦ÀÕì¸ø, ±ô¦À¡ØÐõ §º÷ôÒ
Å¢¾¢ìÌðÀÎõ.
ºÁÉ¢ Å¢¾¢ ºÁÉ¢ÔÚôÒ 1 ∈ 𝐺 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢ ¯ÚôÒ ±¾¢÷Á¨È
[1] [1]
[2] [4]
[3] [5]
[4] [2]
[5] [3]
[6] [6]
±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
∴ 𝑮 = (𝒁𝟕 − 𝟎 ,∙𝟕) ´Õ ÌÄõ
2) 𝟏𝟏 − þý ÁðÎìÌ ¸¡½ô¦ÀüÈ ¦ÀÕì¸Ä¢ý £̧ú { 𝟏 , 𝟑 , 𝟒 , 𝟓 , 𝟗 } ±ýÈ
¸½õ ´Õ ±À£Ä¢Âý ÌÄò¨¾ «¨ÁìÌõ
¾£÷×:
𝐺 = { 1 , 3 , 4 , 5 , 9 } ±ý¸.
∙𝟏𝟏 1 3 4 5 9 1 1 3 4 5 9 3 3 9 1 4 5 4 4 1 5 9 3 5 5 4 9 3 1 9 9 5 3 1 4
«¨¼ôÒ Å¢¾¢: «ð¼Å¨½Â¢ý ±øÄ¡ ¯ÚôÒ¸Ùõ 𝐺 -þý¯ÚôҸǡÌõ.
∴ «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
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§º÷ôÒ Å¢¾¢ 11-þý ÁðÎì¸¡É ¦ÀÕì¸ø, ±ô¦À¡ØÐõ §º÷ôÒ
Å¢¾¢ìÌðÀÎõ.
ºÁÉ¢ Å¢¾¢ ºÁÉ¢ÔÚôÒ 1 ∈ 𝐺 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢ ¯ÚôÒ ±¾¢÷Á¨È
[1] [1]
[3] [4]
[4] [3]
[5] [9]
[9] [5] ±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
∴ (𝑮,∙𝟏𝟏) ´Õ ÌÄõ
ÀâÁ¡üÚ Å¢¾¢ «ð¼Å¨½Â¢Ä¢ÕóÐ ÀâÁ¡üÚ Å¢¾¢Ôõ ¯ñ¨Á.
∴ (𝑮,∙𝟏𝟏) ´Õ ±À£Ä¢Âý ÌÄÁ¡Ìõ.
3) âÂÁüÈ ¸øô¦Àñ¸Ç¢ý ¸½Á¡É 𝑪 − {𝟎} þø ŨÃÂÚì¸ôÀð¼
𝒇𝟏 𝒛 = 𝒛, 𝒇𝟐 𝒛 = −𝒛,𝒇𝟑 𝒛 =𝟏
𝒛, 𝒇𝟒 𝒛 = −
𝟏
𝒛, ∀𝒛 ∈ 𝑪 − {𝟎} ±ýÈ º¡÷Ò¸û
¡×õ «¼í¸¢Â ¸½õ {𝒇𝟏,𝒇𝟐, 𝒇𝟑, 𝒇𝟒} ¬ÉÐ º¡÷ҸǢý §º÷ôÀ¢ý £̧ú ´Õ
±À£Ä¢Âý ÌÄò¨¾ «¨ÁìÌõ.
¾£÷×: 𝐺 = {𝑓1 , 𝑓2, 𝑓3, 𝑓4} ±ý¸.
∘ 𝑓1 𝑓2 𝑓3 𝑓4
𝑓1 𝑓1 𝑓2 𝑓3 𝑓4
𝑓2 𝑓2 𝑓1 𝑓4 𝑓3
𝑓3 𝑓3 𝑓4 𝑓1 𝑓2
𝑓4 𝑓4 𝑓3 𝑓2 𝑓1
«¨¼ôÒ Å¢¾¢: «ð¼Å¨½Â¢ý ±øÄ¡ ¯ÚôÒ¸Ùõ 𝐺 -þý ¯ÚôҸǡÌõ. ∴ «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
§º÷ôÒ Å¢¾¢ º¡÷ҸǢý §º÷ôÒ ±ô¦À¡ØÐõ §º÷ôÒ Å¢¾¢ìÌðÀÎõ.
ºÁÉ¢ Å¢¾¢ ºÁÉ¢ÔÚôÒ 𝑓1 ∈ 𝐺 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢ ¯ÚôÒ ±¾¢÷Á¨È
𝑓1 𝑓1
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𝑓2 𝑓2
𝑓3 𝑓3
𝑓4 𝑓4
±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
(𝑮,∘) ´Õ ÌÄõ
ÀâÁ¡üÚ Å¢¾¢ «ð¼Å¨½Â¢Ä¢ÕóÐ ÀâÁ¡üÚ Å¢¾¢Ôõ ¯ñ¨Á.
∴ (𝑮,∘) ´Õ ±À£Ä¢Âý ÌÄÁ¡Ìõ.
4) 𝟏 𝟎𝟎 𝟏
, 𝝎 𝟎𝟎 𝝎𝟐 , 𝝎
𝟐 𝟎𝟎 𝝎
, 𝟎 𝟏𝟏 𝟎
, 𝟎 𝝎𝟐
𝝎 𝟎 ,
𝟎 𝝎𝝎𝟐 𝟎
±ý¸¢ýÈ ¸½õ
«½¢ô¦ÀÕì¸Ä¢ý £̧ú ´Õ ÌÄò¨¾ «¨ÁìÌõ. (𝝎𝟑 = 𝟏)
¾£÷×: 𝐼 = 1 00 1
, 𝐴 = 𝜔 00 𝜔2 , 𝐵 = 𝜔
2 00 𝜔
, 𝐶 = 0 11 0
, 𝐷 = 0 𝜔2
𝜔 0 ,
𝐸 = 0 𝜔𝜔2 0
±ý¸.𝐺 = {𝐼, 𝐴, 𝐵, 𝐶, 𝐷, 𝐸} ±ý¸.
∙ 𝐼 𝐴 𝐵 𝐶 𝐷 𝐸
𝐼 𝐼 𝐴 𝐵 𝐶 𝐷 𝐸
𝐴 𝐴 𝐵 𝐼 𝐸 𝐶 𝐷
𝐵 𝐵 𝐼 𝐶 𝐷 𝐸 𝐶 𝐶 𝐶 𝐷 𝐸 𝐼 𝐴 𝐵
𝐷 𝐷 𝐸 𝐶 𝐵 𝐼 𝐴
𝐸 𝐸 𝐶 𝐷 𝐴 𝐵 𝐼
«¨¼ôÒ Å¢¾¢: «ð¼Å¨½Â¢ý ±øÄ¡ ¯ÚôÒ¸Ùõ 𝐺 -þý ¯ÚôҸǡÌõ. ∴ «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
§º÷ôÒ Å¢¾¢ «½¢ô¦ÀÕì¸ø ±ô¦À¡ØÐõ §º÷ôÒ Å¢¾¢ìÌðÀÎõ.
ºÁÉ¢ Å¢¾¢ ºÁÉ¢ÔÚôÒ 𝐼 ∈ 𝐺 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢ ¯ÚôÒ ±¾¢÷Á¨È
𝐼 𝐼
𝐴 𝐵
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𝐵 𝐴
𝐶 𝐶
𝐷 𝐷
𝐸 𝐸
±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
(𝑮,∙) ´Õ ÌÄõ
5) (𝒁𝒏, +𝒏)´Õ ÌÄõ.
¾£÷×: 𝑍𝑛 = { 0 , 1 , 2 , 3 , 4 , ⋯ . [𝑛 − 1]}
«¨¼ôÒ Å¢¾¢:
𝑙 , 𝑚 ∈ 𝑍𝑛 ±ý¸. þíÌ 0 ≤ 𝑙, 𝑚 < 𝑛.
𝑙 +𝑛 𝑚 = [𝑙 + 𝑚]
𝑙 + 𝑚 < 𝑛 ±É¢ø 𝑙 +𝑛 𝑚 = [𝑙 + 𝑚] ∈ 𝑍𝑛 .
𝑙 + 𝑚 ≥ 𝑛 ±É¢ø 𝑙 +𝑛 𝑚 = 𝑙 + 𝑚 = 𝑟 ∈ 𝑍𝑛 þíÌ 𝑟 ±ýÀÐ 𝑙 + 𝑚 ³ 𝑛 ¬ø
ÅÌì¸ì ¸¢¨¼ìÌõ Á£¾¢Â¡Ìõ. §ÁÖõ 0 ≤ 𝑟 < 𝑛 ¬Ìõ. ±É§Å «¨¼ôÒ Å¢¾¢
¯ñ¨Á¡Ìõ.
§º÷ôÒ Å¢¾¢:
𝑛 −þý ÁðÎì¸¡É Üð¼ø ±ô¦À¡ØÐõ §º÷ôÒ Å¢¾¢ìÌðÀÎõ.
ºÁÉ¢ Å¢¾¢:
ºÁÉ¢ÔÚôÒ [0] ∈ 𝑍𝑛 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢:
[𝑎] ∈ 𝑍𝑛 þý ±¾¢÷Á¨È ¯ÚôÒ 𝑛 − 𝑎 ∈ 𝑍𝑛
±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
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(𝒁𝒏, +𝒏)´Õ ÌÄõ.
6) ´ýÈ¢ý 𝒏 ¬õ ÀÊ ãÄí¸Ç¢ý ¸½õ ¦ÀÕì¸Ä¢ý £̧ú ÓÊ×Ú ±À£Ä¢Âý
ÌÄõ.
¾£÷×: 𝐺 = 1, 𝜔, 𝜔2, 𝜔3, ⋯ , 𝜔𝑛−1 , 𝜔𝑛 = 1 ±ý¸.
«¨¼ôÒ Å¢¾¢:
𝜔𝑙 , 𝜔𝑚 ∈ 𝐺 ±ý¸. þíÌ 0 ≤ 𝑙, 𝑚 < 𝑛.
𝑙 + 𝑚 < 𝑛 ±É¢ø 𝜔𝑙 ⋅ 𝜔𝑚 = 𝜔𝑙+𝑚 ∈ 𝐺.
𝑙 + 𝑚 ≥ 𝑛 ±É¢ø 𝜔𝑙 ⋅ 𝜔𝑚 = 𝜔𝑙+𝑚 = 𝜔𝑟 ∈ 𝐺 þíÌ 𝑟 ±ýÀÐ 𝑙 + 𝑚 ³ 𝑛 ¬ø
ÅÌì¸ì ¸¢¨¼ìÌõ Á£¾¢Â¡Ìõ. §ÁÖõ 0 ≤ 𝑟 < 𝑛 ¬Ìõ.
𝜔𝑙 , 𝜔𝑚 ∈ 𝐺 ⇒ 𝜔𝑙 ⋅ 𝜔𝑚 ∈ 𝐺
±É§Å «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
§º÷ôÒ Å¢¾¢:
¦ÀÕì¸ø ±ô¦À¡ØÐõ §º÷ôÒ Å¢¾¢ìÌðÀÎõ.
ºÁÉ¢ Å¢¾¢:
ºÁÉ¢ÔÚôÒ 1 ∈ 𝐺 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢:
𝜔𝑙 ∈ 𝐺 þý ±¾¢÷Á¨È ¯ÚôÒ 𝜔𝑛−𝑙 ∈ 𝐺
±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
(𝑮,⋅)´Õ ÌÄõ.
ÀâÁ¡üÚ Å¢¾¢:
¦ÀÕì¸ø ±ô¦À¡ØÐõ ÀâÁ¡üÚ Å¢¾¢ìÌðÀÎõ.
∴ (𝑮,⋅) ´Õ ±À£Ä¢Âý ÌÄÁ¡Ìõ.
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𝐺 þø ÓÊ×Ú ¯ÚôÒ¸û «¾¡ÅÐ 𝑛 ¯ÚôÒ¸û ÁðΧÁ ¯ûǾ¡ø (𝐺,⋅) £̧ú
ÓÊ×Ú ±À£Ä¢Âý ÌÄõ.
7) (ℤ,∗)´Õ ÓÊÅüÈ ±À£Ä¢Âý ÌÄõ. þíÌ 𝒂 ∗ 𝒃 = 𝒂 + 𝒃 + 𝟐
¾£÷×:
«¨¼ôÒ Å¢¾¢:
𝑎, 𝑏 ∈ 𝑍 ±ý¸. À¢ýÉ÷ 𝑎 + 𝑏 + 2 ∈ ℤ.
𝑎, 𝑏 ∈ ℤ ⇒ 𝑎 ∗ 𝑏 ∈ ℤ
±É§Å «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
§º÷ôÒ Å¢¾¢:
𝑎, 𝑏, 𝑐 ∈ ℤ ±ý¸.
𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎 ∗ 𝑏 + 𝑐 + 2
= 𝑎 + 𝑏 + 𝑐 + 2 + 2
= 𝑎 + 𝑏 + 𝑐 + 4 (1)
𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎 + 𝑏 + 2 ∗ 𝑐
= 𝑎 + 𝑏 + 2 + 𝑐 + 2
= 𝑎 + 𝑏 + 𝑐 + 4 (2)
1 , (2) Ä¢ÕóÐ 𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎 ∗ 𝑏 ∗ 𝑐.
±É§Å §º÷ôÒ Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
ºÁÉ¢ Å¢¾¢:
𝑒 ∈ ℤ ±ýÀÐ ºÁÉ¢ ¯ÚôÒ ±ý¸.
ºÁÉ¢ ¯ÚôÀ¢ý ŨèÈôÀÊ, 𝑎 ∗ 𝑒 = 𝑎.
∗ þý ŨèÈôÀÊ, 𝑎 ∗ 𝑒 = 𝑎 + 𝑒 + 2
⇒ 𝑎 + 𝑒 + 2 = 𝑎
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⇒ 𝑒 = −2 ∈ ℤ
ºÁÉ¢ÔÚôÒ −2 ∈ ℤ «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢:
𝑎−1 ∈ ℤ ±ýÀÐ 𝑎 ∈ ℤ þý ±¾¢÷Á¨È ¯ÚôÒ ±ý¸.
±¾¢÷Á¨È ¯ÚôÀ¢ý ŨèÈôÀÊ,
𝑎 ∗ 𝑎−1 = −2.
∗ þý ŨèÈôÀÊ, 𝑎 ∗ 𝑎−1 = 𝑎 + 𝑎−1 + 2
⇒ 𝑎 + 𝑎−1 + 2 = −2
⇒ 𝑎−1 = −𝑎 − 4 ∈ ℤ
𝑎 ∈ ℤ þý ±¾¢÷Á¨È ¯ÚôÒ −𝑎 − 4 ∈ ℤ
±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
(ℤ,∗)´Õ ÌÄõ.
ÀâÁ¡üÚ Å¢¾¢:
𝑎, 𝑏 ∈ ℤ ±ý¸. À¢ýÉ÷
𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 2 = 𝑏 + 𝑎 + 2 = 𝑏 ∗ 𝑎.
±É§Å ÀâÁ¡üÚ â÷ò¾¢Â¡¸¢ÈÐ.
∴ (ℤ,∗) ´Õ ±À£Ä¢Âý ÌÄÁ¡Ìõ.
ℤ þø ÓÊÅüÈ ¯ÚôÒ¸û ¯ûǾ¡ø (ℤ,∗) ´Õ ÓÊÅüÈ ±À£Ä¢Âý ÌÄõ.
8) 𝑮±ýÀÐ Á¢¨¸ Å¢ ¢̧¾ÓÚ ±ñ¸Ç¢ý ¸½õ ±ý¸. (𝑮,∗) ´Õ ÌÄõ. þíÌ
𝒂 ∗ 𝒃 =𝒂𝒃
𝟑
¾£÷×:
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«¨¼ôÒ Å¢¾¢: 𝑎, 𝑏 ∈ 𝐺 ±ý¸. À¢ýÉ÷ 𝑎𝑏 ∈ 𝐺.
⇒𝑎𝑏
3∈ 𝐺
𝑎, 𝑏 ∈ 𝐺 ⇒ 𝑎 ∗ 𝑏 ∈ 𝐺
±É§Å «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
§º÷ôÒ Å¢¾¢: 𝑎, 𝑏, 𝑐 ∈ 𝐺 ±ý¸.
𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎 ∗ 𝑏𝑐
3 =
𝑎 𝑏𝑐
3
3=
𝑎𝑏𝑐
9
𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎𝑏
3 ∗ 𝑐 =
𝑎𝑏
3 𝑐
3=
𝑎𝑏𝑐
9
𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎 ∗ 𝑏 ∗ 𝑐.
±É§Å §º÷ôÒ Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
ºÁÉ¢ Å¢¾¢: 𝑒 ∈ 𝑍 ±ýÀÐ ºÁÉ¢ ¯ÚôÒ ±ý¸.
ºÁÉ¢ ¯ÚôÀ¢ý ŨèÈôÀÊ, 𝑎 ∗ 𝑒 = 𝑎.
∗ þý ŨèÈôÀÊ,
𝑎 ∗ 𝑒 =𝑎𝑒
3⇒
𝑎𝑒
3= 𝑎 ⇒ 𝑒 = 3 ∈ 𝐺
ºÁÉ¢ÔÚôÒ 3 ∈ 𝐺 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢: 𝑎−1 ∈ 𝑍 ±ýÀÐ 𝑎 ∈ 𝑍 þý ±¾¢÷Á¨È ¯ÚôÒ ±ý¸.
±¾¢÷Á¨È ¯ÚôÀ¢ý ŨèÈôÀÊ,
𝑎 ∗ 𝑎−1 = 3.
∗ þý ŨèÈôÀÊ,
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𝑎 ∗ 𝑎−1 =𝑎𝑎−1
3⇒
𝑎𝑎−1
3= 3 ⇒ 𝑎−1 =
9
𝑎∈ 𝐺
±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
(𝑮,∗)´Õ ÌÄõ.
9) 𝑮 = ℚ − {−𝟏}(𝑮,∗) ´Õ ±À£Ä¢Âý ÌÄõ. þíÌ 𝒂 ∗ 𝒃 = 𝒂 + 𝒃 + 𝒂𝒃
¾£÷×: 𝐺 = ℚ − {−1}
«¨¼ôÒ Å¢¾¢:
𝑎, 𝑏 ∈ 𝐺 ±ý¸. À¢ýÉ÷ 𝑎 ≠ −1, 𝑏 ≠ −1..
𝑎 ∗ 𝑏 ∈ 𝐺 ±É ¿¢ÚÅ 𝑎 ∗ 𝑏 ≠ −1 ±É ¿¢ÚŢɡø §À¡ÐÁ¡ÉÐ.
Á¡È¡¸ 𝑎 ∗ 𝑏 = −1 ±Éì ¦¸¡û¸.
𝑎 ∗ 𝑏 = −1
𝑎 + 𝑏 + 𝑎𝑏 = −1
𝑏 + 𝑎𝑏 = −1 − 𝑎
𝑏 1 + 𝑎 = − 1 + 𝑎
𝑏 = −1 + 𝑎
1 + 𝑎= −1 ∵ 𝑎 ≠ −1
𝑏 = −1
þÐ 𝑏 ≠ −1 ìÌ ÓÃñÀ¡Î.
±É§Å 𝑎 ∗ 𝑏 = −1 ±ýÀÐõ ÓÃñÀ¡Î. ±É§Å 𝑎 ∗ 𝑏 ≠ −1
𝑎, 𝑏 ∈ 108𝐺 ⇒ 𝑎 ∗ 𝑏 ∈ 𝐺
±É§Å «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
§º÷ôÒ Å¢¾¢:
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𝑎, 𝑏, 𝑐 ∈ 𝐺 ±ý¸.
𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎 ∗ 𝑏 + 𝑐 + 𝑏𝑐
= 𝑎 + 𝑏 + 𝑐 + 𝑏𝑐 + 𝑎(𝑏 + 𝑐 + 𝑏𝑐)
= 𝑎 + 𝑏 + 𝑐 + 𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎 + 𝑎𝑏𝑐
𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎 + 𝑏 + 𝑎𝑏 ∗ 𝑐
= 𝑎 + 𝑏 + 𝑎𝑏 + 𝑐 + 𝑎 + 𝑏 + 𝑎𝑏 𝑐
= 𝑎 + 𝑏 + 𝑐 + 𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎 + 𝑎𝑏𝑐
⇒ 𝒂 ∗ 𝒃 ∗ 𝒄 = 𝒂 ∗ 𝒃 ∗ 𝒄 .
±É§Å §º÷ôÒ Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
ºÁÉ¢ Å¢¾¢: 𝑒 ∈ 𝑍 ±ýÀÐ ºÁÉ¢ ¯ÚôÒ ±ý¸.
ºÁÉ¢ ¯ÚôÀ¢ý ŨèÈôÀÊ, 𝑎 ∗ 𝑒 = 𝑎.
∗ þý ŨèÈôÀÊ,
𝑎 ∗ 𝑒 = 𝑎 + 𝑒 + 𝑎𝑒
⇒ 𝑎 + 𝑒 + 𝑎𝑒 = 𝑎
⇒ 𝑒 + 𝑎𝑒 = 𝑎 − 𝑎
⇒ 𝑒 1 + 𝑎 = 0
⇒ 𝑒 =0
1 + 𝑎= 0 ∵ 𝑎 ≠ −1
ºÁÉ¢ÔÚôÒ 0 ∈ 𝐺 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢: 𝑎−1 ∈ 𝐺 ±ýÀÐ 𝑎 ∈ 𝐺 þý ±¾¢÷Á¨È ¯ÚôÒ ±ý¸.
±¾¢÷Á¨È ¯ÚôÀ¢ý ŨèÈôÀÊ,
𝑎 ∗ 𝑎−1 = 0.
∗ þý ŨèÈôÀÊ,
𝑎 ∗ 𝑎−1 = 𝑎 + 𝑎−1 + 𝑎𝑎−1
⇒ 𝑎 + 𝑎−1 + 𝑎𝑎−1 = 0
⇒ 𝑎−1 + 𝑎𝑎−1 = −𝑎
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⇒ 𝑎−1 1 + 𝑎 = −𝑎
⇒ 𝑎−1 = −𝑎
1 + 𝑎∈ 𝐺 ∵ 𝑎 ≠ −1
±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
(𝑮,∗)´Õ ÌÄõ.
ÀâÁ¡üÚ Å¢¾¢: 𝑎, 𝑏 ∈ 𝐺 ±ý¸. À¢ýÉ÷
𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 𝑎𝑏 = 𝑏 + 𝑎 + 𝑏𝑎 = 𝑏 ∗ 𝑎.
±É§Å ÀâÁ¡üÚ â÷ò¾¢Â¡¸¢ÈÐ.
∴ (𝐺,∗) ´Õ ±À£Ä¢Âý ÌÄÁ¡Ìõ.
10) 𝑮 = ℚ − {𝟏}(𝑮,∗) ´Õ ±À£Ä¢Âý ÌÄõ. þíÌ 𝒂 ∗ 𝒃 = 𝒂 + 𝒃 − 𝒂𝒃
¾£÷×:
𝐺 = ℚ − {1}
«¨¼ôÒ Å¢¾¢:
𝑎, 𝑏 ∈ 𝐺 ±ý¸. À¢ýÉ÷ 𝑎 ≠ 1, 𝑏 ≠ 1..
𝑎 ∗ 𝑏 ∈ 𝐺 ±É ¿¢ÚÅ 𝑎 ∗ 𝑏 ≠ 1 ±É ¿¢ÚŢɡø §À¡ÐÁ¡ÉÐ.
Á¡È¡¸ 𝑎 ∗ 𝑏 = 1 ±Éì ¦¸¡û¸.
𝑎 ∗ 𝑏 = 1
𝑎 + 𝑏 − 𝑎𝑏 = 1
𝑏 − 𝑎𝑏 = 1 − 𝑎
𝑏 1 − 𝑎 = 1 − 𝑎
𝑏 =1 − 𝑎
1 − 𝑎= −1 ∵ 𝑎 ≠ 1
𝑏 = 1
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þÐ 𝑏 ≠ 1 ìÌ ÓÃñÀ¡Î.
±É§Å 𝑎 ∗ 𝑏 = 1 ±ýÀÐõ ÓÃñÀ¡Î.
±É§Å 𝑎 ∗ 𝑏 ≠ 1
𝑎, 𝑏 ∈ 𝐺 ⇒ 𝑎 ∗ 𝑏 ∈ 𝐺
±É§Å «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
§º÷ôÒ Å¢¾¢:
𝑎, 𝑏, 𝑐 ∈ 𝐺 ±ý¸.
𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎 ∗ 𝑏 + 𝑐 − 𝑏𝑐
= 𝑎 + 𝑏 + 𝑐 − 𝑏𝑐 − 𝑎(𝑏 + 𝑐 − 𝑏𝑐)
= 𝑎 + 𝑏 + 𝑐 − 𝑎𝑏 − 𝑏𝑐 − 𝑐𝑎 + 𝑎𝑏𝑐
𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎 + 𝑏 − 𝑎𝑏 ∗ 𝑐
= 𝑎 + 𝑏 − 𝑎𝑏 + 𝑐 − 𝑎 + 𝑏 − 𝑎𝑏 𝑐
= 𝑎 + 𝑏 + 𝑐 − 𝑎𝑏 − 𝑏𝑐 − 𝑐𝑎 + 𝑎𝑏𝑐
⇒ 𝒂 ∗ 𝒃 ∗ 𝒄 = 𝒂 ∗ 𝒃 ∗ 𝒄 .
±É§Å §º÷ôÒ Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
ºÁÉ¢ Å¢¾¢:
𝑒 ∈ 𝑍 ±ýÀÐ ºÁÉ¢ ¯ÚôÒ ±ý¸.
ºÁÉ¢ ¯ÚôÀ¢ý ŨèÈôÀÊ, 𝑎 ∗ 𝑒 = 𝑎.
∗ þý ŨèÈôÀÊ,
𝑎 ∗ 𝑒 = 𝑎 + 𝑒 − 𝑎𝑒
⇒ 𝑎 + 𝑒 − 𝑎𝑒 = 𝑎
⇒ 𝑒 − 𝑎𝑒 = 𝑎 − 𝑎
⇒ 𝑒 1 − 𝑎 = 0
⇒ 𝑒 =0
1 − 𝑎= 0 ∵ 𝑎 ≠ 1
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ºÁÉ¢ÔÚôÒ 0 ∈ 𝐺 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢:
𝑎−1 ∈ 𝐺 ±ýÀÐ 𝑎 ∈ 𝐺 þý ±¾¢÷Á¨È ¯ÚôÒ ±ý¸.
±¾¢÷Á¨È ¯ÚôÀ¢ý ŨèÈôÀÊ,
𝑎 ∗ 𝑎−1 = 0.
∗ þý ŨèÈôÀÊ,
𝑎 ∗ 𝑎−1 = 𝑎 + 𝑎−1 − 𝑎𝑎−1
⇒ 𝑎 + 𝑎−1 − 𝑎𝑎−1 = 0
⇒ 𝑎−1 − 𝑎𝑎−1 = −𝑎
⇒ 𝑎−1 1 − 𝑎 = −𝑎
⇒ 𝑎−1 = −𝑎
1 − 𝑎∈ 𝐺 ∵ 𝑎 ≠ 1
±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
(𝑮,∗)´Õ ÌÄõ.
ÀâÁ¡üÚ Å¢¾¢: 𝑎, 𝑏 ∈ 𝐺 ±ý¸. À¢ýÉ÷
𝑎 ∗ 𝑏 = 𝑎 + 𝑏 − 𝑎𝑏 = 𝑏 + 𝑎 − 𝑏𝑎 = 𝑏 ∗ 𝑎. ±É§Å ÀâÁ¡üÚ â÷ò¾¢Â¡¸¢ÈÐ.
∴ (𝐺,∗) ´Õ ±À£Ä¢Âý ÌÄÁ¡Ìõ.
11) 𝒙 𝒙𝒙 𝒙
𝒙 ∈ ℝ − {𝟎 } ±ýÈ ¸½õ «½¢ô¦ÀÕì¸Ä¢ý £̧ú ÌÄõ.
¾£÷×:
𝐺 = 𝑥 𝑥𝑥 𝑥
𝑥 ∈ ℝ − {0 } ±ý¸.
«¨¼ôÒ Å¢¾¢: 𝐴 = 𝑥 𝑥𝑥 𝑥
, 𝐵 = 𝑦 𝑦𝑦 𝑦 ∈ 𝐺 ±ý¸.
À¢ýÉ÷ 𝑥, 𝑦 ∈ ℝ − {0}.
𝐴𝐵 = 𝑥 𝑥𝑥 𝑥
𝑦 𝑦𝑦 𝑦
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= 𝑥𝑦 + 𝑥𝑦 𝑥𝑦 + 𝑥𝑦𝑥𝑦 + 𝑥𝑦 𝑥𝑦 + 𝑥𝑦
= 2𝑥𝑦 2𝑥𝑦2𝑥𝑦 2𝑥𝑦
∈ 𝐺 ∵ 𝑥 ≠ 0, 𝑦 ≠ 0 ⇒ 2𝑥𝑦 ≠ 0
𝐴, 𝐵 ∈ 𝐺 ⇒ 𝐴𝐵 ∈ 𝐺
±É§Å «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
§º÷ôÒ Å¢¾¢:
«½¢ ¦ÀÕì¸ø ±ô¦À¡ØÐõ §º÷ôÒ Å¢¾¢ìÌðÀÎõ.
ºÁÉ¢ Å¢¾¢:
𝐸 = 𝑒 𝑒𝑒 𝑒
113±ýÀÐ ºÁÉ¢ ¯ÚôÒ ±ý¸.
ºÁÉ¢ ¯ÚôÀ¢ý ŨèÈôÀÊ,
𝐴𝐸 = 𝐴 = 𝑥 𝑥𝑥 𝑥
.
«½¢ ¦ÀÕì¸Ä¢ý ÀÊ,,𝐴𝐸 = 2𝑥𝑒 2𝑥𝑒2𝑥𝑒 2𝑥𝑒
⇒ 2𝑥𝑒 2𝑥𝑒2𝑥𝑒 2𝑥𝑒
= 𝑥 𝑥𝑥 𝑥
⇒ 2𝑥𝑒 = 𝑥 ⇒ 𝑒 =1
2∈ 𝐺 ∵ 𝑥 ≠ 0
ºÁÉ¢ÔÚôÒ 𝐸 =
1
2
1
21
2
1
2
∈ 𝐺 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢: 𝐵 ∈ 𝐺 ±ýÀÐ 𝐴 ∈ 𝐺 þý ±¾¢÷Á¨È ¯ÚôÒ ±ý¸.
±¾¢÷Á¨È ¯ÚôÀ¢ý ŨèÈôÀÊ,
𝐴𝐵 = 𝐸 =
1
2
1
21
2
1
2
«½¢ ¦ÀÕì¸Ä¢ý ÀÊ,
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𝐴𝐵 = 2𝑥𝑦 2𝑥𝑦2𝑥𝑦 2𝑥𝑦
⇒ 2𝑥𝑦 =1
2⇒ 𝑦 =
1
4𝑥≠ 0 ∵ 𝑥 ≠ 0
𝐴 = 𝑥 𝑥𝑥 𝑥
∈ 𝐺 þý ±¾¢÷Á¨È ¯ÚôÒ
𝐵 =
1
4𝑥
1
4𝑥1
4𝑥
1
4𝑥
∈ 𝐺
±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
𝐺 = 𝑥 𝑥𝑥 𝑥
𝑥 ∈ ℝ − {0 } ±ýȸ½õ«½¢ô¦ÀÕì¸Ä¢ý £̧úÌÄõ.
12) 𝒂 𝟎𝟎 𝟎
𝒂 ∈ ℝ − {𝟎 } ±ýÈ ¸½õ «½¢ô¦ÀÕì¸Ä¢ý £̧ú ÌÄõ.
¾£÷×:
𝐺 = 𝑎 00 0
𝑎 ∈ ℝ − {0 } ±ý¸.
«¨¼ôÒ Å¢¾¢:
𝐴 = 𝑎 00 0
, 𝐵 = 𝑏 00 0
∈ 𝐺 ±ý¸.
À¢ýÉ÷ 𝑎, 𝑏 ∈ ℝ − {0}.
𝐴𝐵 = 𝑎 00 0
𝑏 00 0
= 𝑎𝑏 + 0 0 + 00 + 0 0 + 0
= 𝑎𝑏 00 0
∈ 𝐺 ∵ 𝑎 ≠ 0, 𝑏 ≠ 0 ⇒ 𝑎𝑏 ≠ 0
𝐴, 𝐵 ∈ 𝐺 ⇒ 𝐴𝐵 ∈ 𝐺
±É§Å «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
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§º÷ôÒ Å¢¾¢:
«½¢ ¦ÀÕì¸ø ±ô¦À¡ØÐõ §º÷ôÒ Å¢¾¢ìÌðÀÎõ.
ºÁÉ¢ Å¢¾¢:
𝐸 = 𝑒 00 0
115±ýÀÐ ºÁÉ¢ ¯ÚôÒ ±ý¸.
ºÁÉ¢ ¯ÚôÀ¢ý ŨèÈôÀÊ,
𝐴𝐸 = 𝐴 = 𝑎 00 0
.
«½¢ ¦ÀÕì¸Ä¢ý ÀÊ,
𝐴𝐸 = 𝑎𝑒 00 0
⇒ 𝑎𝑒 00 0
= 𝑎 00 0
⇒ 𝑎𝑒 = 𝑎 ⇒ 𝑒 = 1 ∈ 𝐺 ∵ 𝑎 ≠ 0
ºÁÉ¢ÔÚôÒ 𝐸 = 1 00 0
∈ 𝐺 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢:
𝐵 ∈ 𝐺 ±ýÀÐ 𝐴 ∈ 𝐺 þý ±¾¢÷Á¨È ¯ÚôÒ ±ý¸.
±¾¢÷Á¨È ¯ÚôÀ¢ý ŨèÈôÀÊ,
𝐴𝐵 = 𝐸 = 𝑒 00 0
«½¢ ¦ÀÕì¸Ä¢ý ÀÊ,
𝐴𝐵 = 𝑎𝑏 00 0
⇒ 𝑎𝑏 = 1 ⇒ 𝑏 =1
𝑎≠ 0 ∵ 𝑎 ≠ 0
𝐴 = 𝑎 00 0
∈ 𝐺 þý ±¾¢÷Á¨È ¯ÚôÒ
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𝐵 = 1
𝑎0
0 0
∈ 𝐺
±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
𝐺 = 𝑎 00 0
𝑎 ∈ ℝ − {0 } ±ýȸ½õ«½¢ô¦ÀÕì¸Ä¢ý £̧úÌÄõ.
ÀâÁ¡üÚ Å¢¾¢: 𝐴, 𝐵 ∈ 𝐺 ±ý¸. À¢ýÉ÷
𝐴𝐵 = 𝑎𝑏 00 0
= 𝑏𝑎 00 0
= 𝐵𝐴
±É§Å ÀâÁ¡üÚ â÷ò¾¢Â¡¸¢ÈÐ.
∴ 𝑮 ±ýÈ ¸½õ «½¢ô¦ÀÕì¸Ä¢ý £̧ú ÌÄõ.
13) 𝑮 = 𝒂 + 𝒃 𝟐 𝒂,𝒃 ∈ ℚ ±ýÈ ¸½õ Üð¼Ä¢ý £̧ú ´Õ ÓÊÅüÈ ±À£Ä¢Âý
ÌÄõ.
¾£÷×: 𝑮 = 𝒂 + 𝒃 𝟐 𝒂, 𝒃 ∈ ℚ
«¨¼ôÒ Å¢¾¢: 𝛼 = 𝑎 + 𝑏 2, 𝛽 = 𝑐 + 𝑑 2 ∈ 𝐺 ±ý¸.
À¢ýÉ÷ 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℚ.
𝛼 + 𝛽 = 𝑎 + 𝑏 2 + 𝑐 + 𝑐 2
= 𝑎 + 𝑐 + 𝑏 + 𝑑 2 ∈ 𝐺 ∵ 𝑎 + 𝑐, 𝑏 + 𝑑 ∈ ℚ
𝛼, 𝛽 ∈ 𝐺 ⇒ 𝛼𝛽 ∈ 𝐺
±É§Å «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
§º÷ôÒ Å¢¾¢: Üð¼ø ±ô¦À¡ØÐõ §º÷ôÒ Å¢¾¢ìÌðÀÎõ.
ºÁÉ¢ Å¢¾¢: ºÁÉ¢ÔÚôÒ 0 = 0 + 0 2 ∈ 𝐺 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢: 𝛼 = 𝑎 + 𝑏 2 ∈ 𝐺 þý ±¾¢÷Á¨È ¯ÚôÒ −𝛼 = −𝑎 − 𝑏 2 ∈ 𝐺
±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
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𝐺 ±ýȸ½õÜð¼Ä¢ý £̧úÌÄõ.
ÀâÁ¡üÚ Å¢¾¢: Üð¼ø, ±ô¦À¡ØÐõ ÀâÁ¡üÚ Å¢¾¢ìÌðÀÎõ.
±É§Å ÀâÁ¡üÚ â÷ò¾¢Â¡¸¢ÈÐ.
𝐺 þø ÓÊÅüÈ ¯ÚôÒ¸û ¯ûǾ¡ø (𝐺, +) ´Õ ÓÊÅüÈ ±À£Ä¢Âý ÌÄõ.
14) 𝑮 = {𝟐𝒏 𝒏 ∈ ℤ} ±ýÈ ¸½õ ¦ÀÕì¸Ä¢ý £̧ú ±À£Ä¢Âý ÌÄõ.
¾£÷×:
𝑮 = 𝟐𝒏 𝒏 ∈ ℤ
= {⋯ , 𝟐−𝟑, 𝟐−𝟐, 𝟐−𝟏, 𝟐𝟎 = 𝟏, 𝟐𝟏, 𝟐𝟐, ⋯ }
«¨¼ôÒ Å¢¾¢:
𝛼 = 2𝑙 ,𝛽 = 2𝑚 ∈ 𝐺 ±ý¸.
À¢ýÉ÷ 𝑙, 𝑚 ∈ ℤ.
𝛼𝛽 = 2𝑙 ⋅ 2𝑚 = 2𝑙+𝑚 ∈ 𝐺 ∵ 𝑙 + 𝑚 ∈ ℤ
𝛼, 𝛽 ∈ 𝐺 ⇒ 𝛼𝛽 ∈ 𝐺
±É§Å «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
§º÷ôÒ Å¢¾¢:
¦ÀÕì¸ø ±ô¦À¡ØÐõ §º÷ôÒ Å¢¾¢ìÌðÀÎõ.
ºÁÉ¢ Å¢¾¢:
ºÁÉ¢ÔÚôÒ 1 = 20 ∈ 𝐺 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢:
𝛼 = 2𝑙 ∈ 𝐺 þý ±¾¢÷Á¨È ¯ÚôÒ
1
𝛼=
1
2𝑙= 2−𝑙 ∈ 𝐺
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±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
𝐺 ±ýȸ½õ ¦ÀÕì¸Ä¢ý £̧úÌÄõ.
ÀâÁ¡üÚ Å¢¾¢:
¦ÀÕì¸ø, ±ô¦À¡ØÐõ ÀâÁ¡üÚ Å¢¾¢ìÌðÀÎõ.
±É§Å ÀâÁ¡üÚ â÷ò¾¢Â¡¸¢ÈÐ.
(𝐺,⋅) ´Õ ±À£Ä¢Âý ÌÄõ.
15) 𝑴 = {𝒛 ∈ ℂ 𝒛 = 𝟏} ±ýÈ ¸½õ ¦ÀÕì¸Ä¢ý £̧ú ±À£Ä¢Âý ÌÄõ.
¾£÷×: 𝑴 = {𝒛 ∈ ℂ 𝒛 = 𝟏}
«¨¼ôÒ Å¢¾¢: 𝑧1, 𝑧2 ∈ 𝐺 ±ý¸.
À¢ýÉ÷ 𝑧1 = 𝑧2 = 1. 𝑧1𝑧2 = 𝑧1 𝑧2 = 1 × 1 = 1
𝑧1, 𝑧2 ∈ 𝐺 ⇒ 𝑧1𝑧2 ∈ 𝐺
±É§Å «¨¼ôÒ Å¢¾¢ ¯ñ¨Á¡Ìõ.
§º÷ôÒ Å¢¾¢:
¸Äô¦Àñ¸Ç¢ý ¦ÀÕì¸ø ±ô¦À¡ØÐõ §º÷ôÒ Å¢¾¢ìÌðÀÎõ.
ºÁÉ¢ Å¢¾¢:
ºÁÉ¢ÔÚôÒ 1 ∈ 𝐺 «Ð ºÁÉ¢ Å¢¾¢¨Âô â÷ò¾¢ ¦ºöÔõ.
±¾¢÷Á¨È Å¢¾¢:
𝑧 ∈ 𝐺 þý ±¾¢÷Á¨È ¯ÚôÒ
1
𝑧∈ 𝐺 ∵
1
𝑧 =
1
𝑧 =
1
1= 1
±É§Å ±¾¢÷Á¨È Å¢¾¢ â÷ò¾¢Â¡¸¢ÈÐ.
𝐺 ±ýȸ½õ ¸Äô¦Àñ¸Ç¢ý ¦ÀÕì¸Ä¢ý £̧úÌÄõ.
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epfo;jfT guty;
(1) xU rktha;g;G khwp 𝑿-,d; epfo;jfT epiwr;rhHG guty; gpd;tUkhW cs;sJ :
𝑿 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔
𝑷 𝑿 = 𝒙 𝒌 𝟑𝒌 𝟓𝒌 𝟕𝒌 𝟗𝒌 𝟏𝟏𝒌 𝟏𝟑𝒌
(1) 𝒌-,d; kjpg;G fhz;f.
(2) 𝑷 𝑿 < 4 , 𝑷 𝑿 ≥ 𝟓 , 𝑷(𝟑 < 𝑋 ≤ 6),tw;wpd;
kjpg;G fhz;f.
(3) 𝑷 𝑿 ≤ 𝒙 >𝟏
𝟐Mf ,Uf;f 𝒙,d; kPr;rpW kjpg;G fhz;f.
¾£÷×: (1)𝑃(𝑋 = 𝑥) ´Õ ¿¢¸ú¾¸× ¿¢¨Èî º¡÷Ò ±ýÀ¾¡ø
𝑃(𝑋 = 𝑥)
6
𝑥=0
= 1.
i. e. , 𝑃 𝑋 = 0 + 𝑃 𝑋 = 1 + 𝑃 𝑋 = 2 + 𝑃 𝑋 = 3 + 𝑃 𝑋 = 4 + 𝑃 𝑋 = 5 + 𝑃(𝑋
= 6) = 1
⇒ 𝑘 + 3𝑘 + 5𝑘 + 7𝑘 + 9𝑘 + 11𝑘 + 13𝑘 = 1
⇒ 49𝑘 = 1 ⇒ 𝒌 =𝟏
𝟒𝟗
(2) 𝑃 𝑋 < 4 = 𝑃 𝑋 = 0 + 𝑃 𝑋 = 1 + 𝑃 𝑋 = 2 + 𝑃 𝑋 = 3
= 𝑘 + 3𝑘 + 5𝑘 + 7𝑘 = 16𝑘 =16
49
𝑃 𝑋 ≥ 5 = 𝑃 𝑋 = 5 + 𝑃 𝑋 = 6 = 11𝑘 + 13𝑘 = 24𝑘 =24
49
𝑃 3 < 𝑋 ≤ 6 = 𝑃 𝑋 = 4 + 𝑃 𝑋 = 5 + 𝑃 𝑋 = 6
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= 9𝑘|11𝑘 + 13𝑘 = 33𝑘 =33
49
(3) 𝑥 þý Á£îº¢Ú Á¾¢ô¨À §º¡¾¨É ӨȢø ¸¡½Ä¡õ.
𝑃 𝑋 ≤ 0 = 𝑃 𝑋 = 0 = 𝑘 =1
49<
1
2
𝑃 𝑋 ≤ 1 = 𝑃 𝑋 = 0 + 𝑃 𝑋 = 1 = 4𝑘 =4
49<
1
2
𝑃 𝑋 ≤ 2 = 𝑃 𝑋 = 0 + 𝑃 𝑋 = 1 + 𝑃 𝑋 = 2
= 9𝑘 =9
49<
1
2
𝑃 𝑋 ≤ 3 = 𝑃 𝑋 = 0 + 𝑃 𝑋 = 1 + 𝑃 𝑋 = 2 + 𝑃 𝑋 = 3
= 16𝑘 =16
49<
1
2
𝑃 𝑋 ≤ 4 = 𝑃 𝑋 = 0 + 𝑃 𝑋 = 1 + 𝑃 𝑋 = 2
+𝑃 𝑋 = 3 + 𝑃(𝑋 = 4)
= 25𝑘 =25
49>
1
2
𝑃 𝑋 ≤ 𝑥 >1
2 ¬¸ þÕì¸ 𝑥 þý Á£îº¢Ú Á¾¢ôÒ 4 ¬Ìõ.
(2) xU nfhs;fyj;jpy; 4 nts;is kw;Wk; 3 rptg;Gg; ge;JfSk; cs;sd. 3 ge;Jfis xt;nthd;whf vLf;Fk;NghJ> rptg;G epwg; ge;Jfspd; vz;zpf;ifapd; epfo;jfTg; guty; (epiwr;rhHG) fhz;f.
(i) jpUg;gp itf;Fk; Kiwapy;
(ii) jpUg;gp itf;fh Kiwapy;
¾£÷×: 𝑋±ýÀÐ 3 Ó¨È Àóи¨Ç ±ÎìÌõ §À¡Ð ¸¢¨¼ìÌõ º¢ÅôÒô ÀóиǢý
±ñ½¢ì¨¸ ±ý¸.
∴ 𝑋 = 0,1,2,3
𝑅 ±ýÀÐ º¢ÅôÒ ÀóÐ ±ÎìÌõ ¿¢¸ú ±É×õ, 𝑊 ±ýÀÐ ¦Åû¨Ç ÀóÐ
±ÎìÌõ ¿¢¸ú ±É×õ ¦¸¡û¸.
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(i) ¾¢ÕôÀ¢ ¨ÅìÌõ ӨȢø
121𝑃 𝑅 =3
7, 𝑃 𝑊 =
4
7
𝑃 𝑋 = 0 = 𝑃 𝑊𝑊𝑊 =4
7×
4
7×
4
7=
64
343
𝑃 𝑋 = 1 = 𝑃 𝑅𝑊𝑊 + 𝑃 𝑊𝑅𝑊 + 𝑃 𝑊𝑊𝑅
= 3
7×
4
7×
4
7 +
4
7×
3
7×
4
7 +
4
7×
3
7×
4
7
= 3 ×48
343=
144
343
𝑃 𝑋 = 2 = 𝑃 𝑅𝑅𝑊 + 𝑃 𝑊𝑅𝑅 + 𝑃 𝑅𝑊𝑅
= 3
7×
3
7×
4
7 +
4
7×
3
7×
3
7 +
3
7×
3
7×
4
7
= 3 ×36
343=
108
343
𝑃 𝑋 = 3 = 𝑃 𝑅𝑅𝑅 =3
7×
3
7×
3
7=
27
343
∴ §¾¨ÅÂ¡É ¿¢¸ú¾× ÀÃÅø À¢ýÅÕÁ¡Ú:
𝑋 0 1 2 3 𝑃(𝑋 = 𝑥) 64
343
144
343
108
343
27
343
ÌÈ¢ôÒ:
64
343+
144
343+
108
343+
27
343=
343
343= 1
(ii) ¾¢ÕôÀ¢ ¨Å측 ӨȢø
´ù¦Å¡Õ Ó¨ÈÔõ ¦¸¡û¸Äò¾¢Ä¢ÕóÐ Àó¨¾ ±ÎìÌõ §À¡Ð ÀóÐì¸Ç¢ý
±ñ½¢ì¨¸ ´ýÚ Ì¨ÈÔõ.
𝑃 𝑋 = 0 = 𝑃 𝑊𝑊𝑊 =4
7×
3
6×
2
5=
4
35
𝑃 𝑋 = 1 = 𝑃 𝑅𝑊𝑊 + 𝑃 𝑊𝑅𝑊 + 𝑃 𝑊𝑊𝑅
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𝛽𝑥𝛼 = 𝑡±ý¸.
𝑥 0 ∞
𝑡 0 ∞
𝛼𝛽𝑥𝛼−1𝑑𝑥 = 𝑑𝑡
𝑥𝛼−1𝑑𝑥 =𝑑𝑡
𝛼𝛽
= 3
7×
4
6×
3
5 +
4
7×
3
6×
3
5 +
4
7×
3
6×
3
5
= 3 ×36
210=
18
35
𝑃 𝑋 = 2 = 𝑃 𝑅𝑅𝑊 + 𝑃 𝑊𝑅𝑅 + 𝑃 𝑅𝑊𝑅
= 3
7×
2
6×
4
5 +
4
7×
3
6×
2
5 +
3
7×
4
6×
2
5
= 3 ×24
210=
12
210
𝑃 𝑋 = 3 = 𝑃 𝑅𝑅𝑅 =3
7×
2
6×
1
5=
1
35
∴ §¾¨ÅÂ¡É ¿¢¸ú¾× ÀÃÅø À¢ýÅÕÁ¡Ú:
𝑋 0 1 2 3 𝑃(𝑋 = 𝑥) 4
35
18
35
12
35
1
35
ÌÈ¢ôÒ:
4
35+
18
35+
12
35+
1
35=
35
35= 1
(3) xU rktha;g;G khwp𝑿,d; epfo;jfT mlHj;jpr; rhHG
𝒇 𝒙 = 𝒌𝒙𝜶−𝟏𝒆−𝜷𝒙𝜶
, 𝒙, 𝜶, 𝜷 > 0𝟎, 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞
vdpy;(i)𝒌,d; kjpg;G fhz;f.(ii)𝑷 ( 𝑿 > 10 )fhz;f.
¾£÷×: (i)𝑓 𝑥 ´Õ ¿¢¸ú¾¸× «¼÷ò¾¢î º¡÷Ò ±ýÀ¾¡ø
𝑓(𝑥)
∞
−∞
𝑑𝑥 = 1.
⇒ 𝑘 𝑥𝛼−1𝑒−𝛽𝑥𝛼∞
0𝑑𝑥 = 1
⇒ 𝑘 𝑒−𝑡
∞
0
𝑑𝑡
𝛼𝛽= 1
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𝛽𝑥𝛼 = 𝑡±ý¸.
𝑥 10 ∞
𝑡 𝛽 10 𝛼 ∞
𝛼𝛽𝑥𝛼−1𝑑𝑥 = 𝑑𝑡
𝑥𝛼−1𝑑𝑥 =𝑑𝑡
𝛼𝛽
⇒𝑘
𝛼𝛽 𝑒−𝑡
∞
0
𝑑𝑡 = 1 ⇒𝑘
𝛼𝛽 𝑒−𝑡
−1
0
∞
= 1 ⇒𝑘
𝛼𝛽 0 + 1 = 1
⇒𝑘
𝛼𝛽= 1 ⇒ 𝒌 = 𝜶𝜷
(ii)
𝑃 𝑋 > 10 = 𝑘 𝑥𝛼−1𝑒−𝛽𝑥𝛼
∞
10
𝑑𝑥
= 𝑘 𝑒−𝑡
∞
𝛽 10 𝛼
𝑑𝑡
𝛼𝛽
=𝑘
𝛼𝛽 𝑒−𝑡
∞
𝛽 10 𝛼
𝑑𝑡
=𝑘
𝛼𝛽 𝑒−𝑡
−1 𝛽 10 𝛼
∞
=𝛼𝛽
𝛼𝛽 0 + 𝑒−𝛽 10 𝛼 = 𝑒−𝛽 10 𝛼
(4) Ie;J taJila xU caHe;j tif ehapd; KO Mal;fhyk; xU rktha;g;G khwpahFk;. mjd; guty; rhHG (NrHg;G)
𝑭 𝒙 = 𝟎, 𝒙 ≤ 𝟓
𝟏 −𝟐𝟓
𝒙𝟐, 𝒙 > 5
vdpy; 5 taJila eha;
(i) 10Mz;LfSf;F Nkyhf
(ii) 8Mz;LfSf;Ff; Fiwthf
(iii) 12,ypUe;J15Mz;Lfs; tiu capH
tho;tjw;fhd epfo;jfT fhz;f.
¾£÷×: (i)𝑋±ýÀÐ 5 ÅÂШ¼Â ´Õ ¯Â÷ó¾ Ũ¸ ¿¡Â¢ý ¬Ôð¸¡Äõ ±ý¸.
𝑃 𝑋 > 10 = 1 − 𝑃 𝑋 ≤ 10
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= 1 − 𝐹 10
= 1 − 1 −25
102
= 1 − 1 +25
100
=1
4
¿¡ö 10 ¬ñθÙìÌ §ÁÄ¡¸ ¯Â¢÷ Å¡Æ ¿¢¸ú¾¸× 𝟏
𝟒 ¬Ìõ.
(ii)
𝑃 𝑋 < 8 = 𝐹 8 = 1 −25
82= 1 −
25
64=
64 − 25
64=
39
64
¿¡ö 8 ¬ñθÙìÌ Ì¨ÈÅ¡¸ ¯Â¢÷ Å¡Æ ¿¢¸ú¾¸× 𝟑𝟗
𝟔𝟒 ¬Ìõ.
(iii)
𝑃 12 < 𝑋 < 15 = 𝐹 15 − 𝐹 12
= 1 −25
152 − 1 −
25
122
= 1 −25
225− 1 +
25
144
= 25 1
144−
1
225
= 25 225 − 144
144 × 225
=81
144 × 9
=9
144=
1
16
¿¡ö 12 þÄ¢ÕóÐ 15 ¬ñθû Ũà ¯Â¢÷ Å¡Æ ¿¢¸ú¾¸× 𝟏
𝟏𝟔 ¬Ìõ.
(5) xU NgUe;J epiyaj;jpy;> xU epkplj;jpw;F cs;Ns tUk; NgUe;Jfspd;
vz;zpf;if gha;]hd; gutiyg; ngw;wpUf;fpwJ vdpy;𝝀 = 0 ⋅ 9vdf; nfhz;L
(i) 5 epkpl fhy ,ilntspapy; rhpahf 9 NgUe;Jfs; cs;Ns tu
(ii) 8 epkpl; fhy ,ilntspapy; 10 f;Fk; Fiwthf NgUe;Jfs; cs;Ns tu
(iii) 11 epkpl fhy ,ilntspapy; Fiwe;jgl;rk; 14 NgUe;Jfs; cs;Ns tu epfo;jfT fhz;f.
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¾£÷×: (i)
´Õ ¿¢Á¢¼ò¾¢ø ¯û§Ç ÅÕõ §ÀÕóиÙ측É
𝝀 = 𝟎 ⋅ 𝟗
∴ 𝟓 ¿¢Á¢¼ò¾¢ø ¯û§Ç ÅÕõ §ÀÕóиÙì¸¡É ºÃ¡ºÃ¢
5𝜆 = 5 × 0 ⋅ 9 = 4 ⋅ 5
𝟓 ¿¢Á¢¼ò¾¢ø ºÃ¢Â¡¸ 𝟗 §ÀÕóиû ¯û§Ç Åà ¿¢¸ú¾¸×
125𝑃 𝑋 = 9 =𝑒5𝜆 × 5𝜆 9
9! =
𝑒4⋅5 × 4 ⋅ 5
9!
9
(ii)
∴ 8 ¿¢Á¢¼ þ¨¼¦ÅǢ¢ø ¯û§Ç ÅÕõ §ÀÕóиÙì¸¡É ºÃ¡ºÃ¢ 8𝜆 = 8 × 0 ⋅
9 = 7 ⋅ 2
8 ¿¢Á¢¼ þ¨¼¦ÅǢ¢ø 10 ìÌ Ì¨ÈÅ¡¸ §ÀÕóиû ¯û§Ç Åà ¿¢¸ú¾¸×
125𝑃 𝑋 < 10 = 𝑒8𝜆 × 8𝜆 𝑥
𝑥!
9
𝑥=0
= 𝑒7⋅2 × 7 ⋅ 2 𝑥
𝑥!
9
𝑥=0
(iii)
∴ 11 ¿¢Á¢¼ þ¨¼¦ÅǢ¢ø ¯û§Ç ÅÕõ §ÀÕóиÙì¸¡É ºÃ¡ºÃ¢ 11𝜆 = 11 ×
0 ⋅ 9 = 9 ⋅ 9
111 ¿¢Á¢¼ þ¨¼¦ÅǢ¢ø ̨Èó¾Àðºõ14§ÀÕóиû¯û§Ç Åà ¿¢¸ú¾¸×
𝑃 𝑋 ≥ 14 = 1 − 𝑃 𝑋 < 14
= 1 − 𝑒11𝜆 × 11𝜆 𝑥
𝑥!
13
𝑥=0
= 1 − 𝑒9⋅9 × 9 ⋅ 9 𝑥
𝑥!
13
𝑥=0
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= 1000 × 0 ⋅ 0498
= 49 ⋅ 8 = 50
= 1000 × 0 ⋅ 3526
= 352 ⋅ 6 = 353
(6) xU efuj;jpy; thlif tz;b Xl;LdHfshy; Vw;gLk; tpgj;Jfspd; vz;zpf;if
gha;]hd; gutiy xj;jpUf;fpwJ. ,jd; gz;gsit 3 vdpy;>1000
Xl;LeHfspy;(i)xU tUlj;jpy; xU tpgj;Jk; Vw;glhky; (ii)xU tUlj;jpy; %d;W tpgj;JfSf;F Nky; Vw;glhky; ,Uf;Fk;gbahd Xl;LdHfspd; vz;zpf;ifiaf;
fhz;f.[𝒆−𝟑 = 𝟎 ⋅ 𝟎𝟒𝟗𝟖].
¾£÷×: (i)𝑋±ýÀÐ ´Õ ÅÕ¼ò¾¢ø ²üÀÎõ Å¢ÀòÐì¸Ç¢ý ±ñ½¢ì¨¸ ±ý¸.
´Õ ÅÕ¼ò¾¢ø ²üÀÎõ Å¢ÀòÐì¸Ç¢ý ºÃ¡ºÃ¢ ±ñ½¢ì¨¸ = 3 𝑖. 𝑒. 𝜆 = 3
𝑃 𝑋 = 𝑥 =𝑒−𝜆𝜆𝑥
𝑥!=
𝑒−33𝑥
𝑥!
(i)
𝑃 𝑋 = 0 =𝑒−330
0!= 𝑒−3 = 0 ⋅ 0498
1000´ðÎÉ÷¸Ç¢ø ´Õ Å¢Àòиû
þøÄ¡Áø µðÎõ µðÎÉ÷¸Ç¢ý
±ñ½¢ì¨¸
(ii)
𝑃 𝑋 > 3 = 1 − 𝑃 𝑋 ≤ 3
= 1 − 𝑃 𝑋 = 𝑥
3
𝑥=0
= 1 − 𝑒−3 × 3 𝑥
𝑥!
3
𝑥=0
= 1 − 𝑒−3 30
0!+
31
1!+
32
2!+
33
3!
= 1 − 𝑒−3 1
1+
3
1+
9
2+
27
5
= 1 − 𝑒−3 1 + 3 +9
2+
9
2
= 1 − 𝑒−3 1 + 3 + 9 = 1 − 13𝑒−3
= 1 − 13 × 0 ⋅ 0498 = 1 − 0 ⋅ 6474
= 0 ⋅ 3526
1000´ðÎÉ÷¸Ç¢ø 3 Å¢ÀòиÙìÌ
²üÀÎòÐõ µðÎÉ÷¸Ç¢ý
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= 1000 × 0 ⋅ 5471
= 547 ⋅ 1 = 547
±ñ½¢ì¨¸
(7) xU NjHtpy; 1000 khztHfspd; ruhrhp kjpg;ngz; 34 kw;Wk; jpl;l tpyf;fk;
16 MFk;. kjpg;ngz; ,ay;epiyg; gutiy ngw;wpUg;gpd; (i) 30 ,ypUe;J 60
kjpg;ngz;fSf;fpilNa kjpg;ngz; ngw;w khztHfspd; vz;zpf;if (ii)kj;jpa
70%khztHfs; ngWk; kjpg;ngz;fspd; vy;iyfs; ,tw;iwf; fhz;f.
𝒁 𝟎 ⋅ 𝟐𝟓 𝟏 ⋅ 𝟎𝟒 𝟏 ⋅ 𝟔𝟑 gug;G 𝟎 ⋅ 𝟎𝟗𝟖𝟕 𝟎 ⋅ 𝟑𝟓𝟎𝟎 𝟎 ⋅ 𝟒𝟒𝟖𝟒
¾£÷×:𝑋±ýÀÐ ´Õ §¾÷Å¢ø Á¡½ù÷¸û ¦ÀÚõ Á¾¢ô¦Àñ ±ý¸.
þíÌ 𝜇 = 34, 𝜎 = 16, 𝑛 = 1000
𝑍 =𝑋 − 𝜇
𝜎=
𝑋 − 34
16
(i)
𝑋 = 30 ⇒ 𝑍 =𝑋 − 34
16=
30 − 34
16= −
4
16= −
1
4= −0 ⋅ 25
𝑋 = 60 ⇒ 𝑍 =𝑋 − 34
16=
60 − 34
16=
26
16=
13
8= 1 ⋅ 63
𝑃 30 < 𝑋 < 60 = 𝑃 −0 ⋅ 25 < 𝑍 < 1 ⋅ 63
= 𝜙 𝑧
1⋅63
−0⋅25
𝑑𝑧
= 𝜙 𝑧
0
−0⋅25
𝑑𝑧 + 𝜙 𝑧
1⋅63
0
𝑑𝑧
= 𝜙 𝑧
0⋅25
0
𝑑𝑧 + 𝜙 𝑧
1⋅63
0
𝑑𝑧
= 0 ⋅ 0987 + 0 ⋅ 4484 = 0 ⋅ 5471
30þÄ¢ÕóÐ 60 Á¾¢ô¦Àñ¸Ùì
¸¢¨¼§Â Á¾¢ô¦Àñ ¦ÀüÈ
Á¡½Å÷¸Ç¢ý ±ñ½¢ì¨¸
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= −1 ⋅ 04
𝑍1 =𝑋 − 34
16= −1 ⋅ 04
𝑋1 − 34 = −1 ⋅ 04 × 16
𝑋1 = 34 − 16 ⋅ 64
= 17 ⋅ 36
𝑍2 =𝑋 − 34
16= 1 ⋅ 04
𝑋2 − 34 = 1 ⋅ 04 × 16
𝑋2 = 34 + 16 ⋅ 64
= 50 ⋅ 64
(ii)
ÀÃôÒ «ð¼Å¨½Â¢Ä¢ÕóÐ
0 ⋅ 35 ÀÃôÀ¢ü¸¡É 𝑍1 Á¾¢ôÒ
[ 𝑍 = 0 ìÌ þ¼ôÒÈõ 𝑍1 «¨Áž¡ø ]
þÐ §À¡Ä§Å 𝑍2 = 1 ⋅ 04
∴Áò¾¢Â 70% Á¡½Å÷¸û 17 ⋅ 36 þÄ¢ÕóÐ 50 ⋅ 64 ìÌ þ¨¼ôÀð¼
Á¾¢ô¦Àñ¸¨Çô ¦ÀÚ ¢̧È¡÷¸û.
(8) etPd rpw;We;Jfspy; nghUj;jg;gLk; rf;fuq;fspypUe;J rktha;g;G Kiwapy; NjHe;njLf;fg;gLk; rf;fuj;jpd; fhw;wOj;jk; ,ay;epiyg; gutiy xj;jpUf;fpwJ.
fhw;wOj;j ruhrhp 𝟑𝟏 𝐩𝐬𝐢NkYk; jpl;l tpyf;fk; 𝟎 ⋅ 𝟐 𝐩𝐬𝐢vdpy;
(i) (a)30.5 psif;Fk;31.5 psif;Fk; ,ilg;gl;l fhw;wOj;jk;
(b)30 psif;Fk;32 psif;Fk; ,ilg;gl;l fhw;wOj;jk; vd ,Uf;Fk;gbahf rf;fuj;jpid NjHe;njLf;f epfo;jfT fhz;f.
(ii) rktha;g;G Kiwapy; NjHe;njLf;fg;gLk; rf;fuj;jpd; fhw;wOj;jk; 30.5
psif;F mjpfkhf ,Uf;f epfo;jfT fhz;f.
𝒁 𝟐 ⋅ 𝟓 𝟓 gug;G 𝟎 ⋅ 𝟒𝟗𝟑𝟖 𝟎 ⋅ 𝟓𝟎𝟎𝟎
¾£÷×:
𝑋±ýÀÐ ¿Å£É º¢üÚóиǢø ¦À¡Õò¾ôÀÎõ ºì¸Ãò¾¢ý ¸¡üÈØò¾õ ±ý¸.
þíÌ 𝜇 = 31, 𝜎 = 0 ⋅ 2, 𝑛 = 500
𝑍 =𝑋 − 𝜇
𝜎=
𝑋 − 31
0 ⋅ 2
(i) (a)
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𝑋 = 30 ⋅ 5 ⇒ 𝑍 =𝑋 − 31
0 ⋅ 2=
30 ⋅ 5 − 31
0 ⋅ 2= −
0 ⋅ 5
0 ⋅ 2= −
5
2= −2 ⋅ 5
𝑋 = 31 ⋅ 5 ⇒ 𝑍 =𝑋 − 31
0 ⋅ 2=
31 ⋅ 5 − 31
0 ⋅ 2=
0 ⋅ 5
0 ⋅ 2=
5
2= 2 ⋅ 5
𝑃 30 ⋅ 5 < 𝑋 < 31 ⋅ 5 = 𝑃 −2 ⋅ 5 < 𝑍 < 2 ⋅ 5
= 𝜙 𝑧
2⋅5
−2⋅5
𝑑𝑧
= 2 𝜙 𝑧
2⋅5
0
𝑑𝑧
= 2 × 0 ⋅ 4938 = 0 ⋅ 9876
(i) (b)
𝑋 = 30 ⇒ 𝑍 =𝑋 − 31
0 ⋅ 2=
30 − 31
0 ⋅ 2= −
1
0 ⋅ 2= −
10
2= −5
𝑋 = 32 ⇒ 𝑍 =𝑋 − 31
0 ⋅ 2=
32 − 31
0 ⋅ 2=
1
0 ⋅ 2=
10
2= 5
𝑃 30 < 𝑋 < 32 = 𝑃 −5 < 𝑍 < 5
= 𝜙 𝑧
5
−5
𝑑𝑧
= 2 𝜙 𝑧
5
0
𝑑𝑧
= 2 × 0 ⋅ 5 = 1
(ii)
𝑋 = 30 ⋅ 5 ⇒ 𝑍 =𝑋 − 31
0 ⋅ 2=
30 ⋅ 5 − 31
0 ⋅ 2= −
0 ⋅ 5
0 ⋅ 2= −
5
2= −2 ⋅ 5
𝑃 𝑋 > 30 ⋅ 5 = 𝑃 𝑍 > −2 ⋅ 5
= 𝜙 𝑧
∞
−2⋅5
𝑑𝑧
= 𝜙 𝑧
0
−2⋅50
𝑑𝑧 + 𝜙 𝑧
∞
0
𝑑𝑧
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= 𝜙 𝑧
2⋅50
0
𝑑𝑧 + 𝜙 𝑧
∞
0
𝑑𝑧
= 0 ⋅ 4938 + 0 ⋅ 5000 = 0 ⋅ 9938
(9) xU Fwpg;gpl;l fy;Y}hpapy; 500 khztHfspd; vilfs; xU ,ay;epiyg;
gutiy xj;jpUg;gjhff; nfhs;sg; gLfpwJ. ,jd; ruhrhp 151
gTz;LfshfTk; jpl;l tpyf;fk; 15 gTz;LfshfTk; cs;sd. (i)
120gTz;Lf;Fk;155 gTz;Lf;Fk; ,ilNaAs;s khztHfs;(ii) 185gTz;Lf;F
Nky; epiwAs;s khztHfspd; vz;zpf;if fhz;f.
𝒁 𝟐 ⋅ 𝟎𝟔𝟕 𝟎 ⋅ 𝟐𝟔𝟔𝟕 𝟐 ⋅ 𝟐𝟔𝟔𝟕 gug;G 𝟎 ⋅ 𝟒𝟖𝟎𝟑 𝟎 ⋅ 𝟏𝟎𝟐𝟔 𝟎 ⋅ 𝟒𝟖𝟖𝟏
¾£÷×:
𝑋±ýÀÐ Á¡½Å÷¸Ç¢ý ±¨¼¸û±ý¸.
þíÌ 𝜇 = 151, 𝜎 = 15
𝑍 =𝑋 − 𝜇
𝜎=
𝑋 − 151
15
(i)
𝑋 = 120 ⇒ 𝑍 =𝑋 − 151
15=
120 − 151
15=
−31
15= −2 ⋅ 067
𝑋 = 155 ⇒ 𝑍 =𝑋 − 151
15=
155 − 151
15=
4
15= 0 ⋅ 2667
𝑃 120 < 𝑋 < 155 = 𝑃 −2 ⋅ 067 < 𝑍 < 0 ⋅ 2667
= 𝜙 𝑧
0⋅2667
−2⋅067
𝑑𝑧
= 𝜙 𝑧
0
−2⋅0667
𝑑𝑧 + 𝜙 𝑧
0⋅2667
0
𝑑𝑧
= 𝜙 𝑧
2⋅0667
0
𝑑𝑧 + 𝜙 𝑧
0⋅2667
0
𝑑𝑧
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= 500 × 0 ⋅ 5829
= 291 ⋅ 45 = 291
= 500 × 0 ⋅ 0119
= 5 ⋅ 95 = 6
= 0 ⋅ 4803 + 0 ⋅ 1026 = 0 ⋅ 5829
∴ 500 Á¡½Å÷¸Ç¢ý 120þÄ¢ÕóÐ 150 À×ñÊüÌû
±¨¼ þÕìÌõ
Á¡½Å÷¸Ç¢ý ±ñ½¢ì¨¸
(ii)
𝑋 = 185 ⇒ 𝑍 =𝑋 − 151
15=
185 − 151
10=
34
15= 2 ⋅ 2667
𝑃 𝑋 > 185 = 𝑃 𝑍 > 2 ⋅ 2667
= 𝜙 𝑧
∞
2⋅2667
𝑑𝑧
= 𝜙 𝑧
∞
0
𝑑𝑧 − 𝜙 𝑧
2⋅2667
0
𝑑𝑧
= 0 ⋅ 5000 − 0 ⋅ 4881 = 0 ⋅ 0119
∴ 500 Á¡½Å÷¸Ç¢ý 185 À×ñÊüÌ §Áø ±¨¼
þÕìÌõ
Á¡½Å÷¸Ç¢ý ±ñ½¢ì¨¸
(10) ,ay;epiy khwp𝑿-d; ruhrhp 𝟔 kw;Wk; jpl;l tpyf;fk; 𝟓 MFk;.
(i)𝑷(𝟎 ≤ 𝑿 ≤ 𝟖)
(ii)𝑷( 𝑿 − 𝟔 < 𝟏𝟎)Mfpatw;iwf; fhz;f.
𝒁 𝟏 ⋅ 𝟐 𝟎 ⋅ 𝟒 𝟐 gug;G 𝟎 ⋅ 𝟑𝟖𝟒𝟗 𝟎 ⋅ 𝟏𝟓𝟓𝟒 𝟎 ⋅ 𝟒𝟕𝟕𝟐
¾£÷×:
þíÌ 𝜇 = 6, 𝜎 = 5
𝑍 =𝑋 − 𝜇
𝜎=
𝑋 − 6
5
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(i)
𝑋 = 0 ⇒ 𝑍 =𝑋 − 6
5=
0 − 6
5=
−6
5= −1 ⋅ 2
𝑋 = 8 ⇒ 𝑍 =𝑋 − 6
5=
8 − 6
5=
2
5= 0 ⋅ 4
𝑃 0 < 𝑋 < 8 = 𝑃 −1 ⋅ 2 < 𝑍 < 0 ⋅ 4
= 𝜙 𝑧
0⋅4
−1⋅2
𝑑𝑧
= 𝜙 𝑧
0
−1⋅2
𝑑𝑧 + 𝜙 𝑧
0⋅4
0
𝑑𝑧
= 𝜙 𝑧
1⋅2
0
𝑑𝑧 + 𝜙 𝑧
0⋅4
0
𝑑𝑧
= 0 ⋅ 3849 + 0 ⋅ 1554 = 0 ⋅ 5403
(i)
𝑋 − 6 < 10 ⇒ −10 < 𝑋 − 6 < 10
⇒ −10 + 6 < 𝑋 − 6 + 6 < 10 + 6
⇒ −4 < 𝑋 < 16
𝑋 = −4 ⇒ 𝑍 =𝑋 − 6
5=
−4 − 6
5=
−10
5= −2
𝑋 = 16 ⇒ 𝑍 =𝑋 − 6
5=
16 − 6
5=
10
5= 2
𝑃 𝑋 − 6 < 10 = 𝑃 −4 < 𝑋 < 16
= 𝑃 −2 < 𝑍 < 2
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= 𝜙 𝑧
2
−2
𝑑𝑧
= 2 × 𝜙 𝑧
2
0
𝑑𝑧
= 2 × 0 ⋅ 4772 = 0 ⋅ 9544
(11) xU ,ay;epiyg; gutypd; epfo;jfTg; guty;
𝒇 𝒙 = 𝒄𝒆−𝒙𝟐+𝟑𝒙 , −∞ < 𝑋 < ∞
vdpy; 𝒄, 𝝁, 𝝈𝟐,tw;iwf; fhz;f.
¾£÷×:
−𝑥2 + 3𝑥 = − 𝑥2 − 3𝑥
= − 𝑥2 − 3𝑥 +9
4−
9
4
= − 𝑥2 − 3𝑥 +9
4 +
9
4= − 𝑥 −
3
2
2
+9
4
= −1
2
𝑥 −3
2
2
1
2
+9
4= −
1
2
𝑥 −3
21
2
2
+9
4
𝑓 𝑥 = 𝑐𝑒−𝑥2+3𝑥
= 𝑐𝑒−
1
2
𝑥−32
1
2
2
+9
4
= 𝑐𝑒−
1
2
𝑥−32
1
2
2
𝑒9
4
= 𝑐𝑒9
4𝑒−
1
2
𝑥−32
1
2
2
⋯⋯⋯⋯⋯ (1)
1 ¨Â 𝑓 𝑥 =1
𝜎 2𝜋𝑒−
1
2 𝑥−𝜇
𝜎
2
¯¼ý ´ôÀ¢¼
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𝜇 =3
2 𝜎 =
1
2⇒ 𝜎2 =
1
2
𝑐𝑒9
4 =1
𝜎 2𝜋
𝑐 =1
𝑒9
41
2 2𝜋
=𝑒−
9
4
𝜋
(12) xU ,ay;epiyg; gutypd; epfo;jfTg; guty;
𝒇 𝒙 = 𝒌𝒆−𝟐𝒙𝟐+𝟒𝒙 , −∞ < 𝑋 < ∞
vdpy; 𝒌, 𝝁, 𝝈𝟐,tw;iwf; fhz;f.
¾£÷×:
−2𝑥2 + 4𝑥 = −2 𝑥2 − 2𝑥
= −2 𝑥2 − 2𝑥 + 1 − 1 = −2 𝑥2 − 2𝑥 + 1 + 2
= −2 𝑥 − 1 2 + 2 = −2 ×1
2
𝑥 − 1 2
1
2
+ 2
= − 𝑥 − 1 2
1
2
+ 2 = −1
2
𝑥 − 1 2
1
2×
1
2
+ 2
= −1
2
𝑥 − 11
2
2
+ 2
𝑓 𝑥 = 𝑘𝑒−2𝑥2+4𝑥
= 𝑘𝑒−
1
2
𝑥−112
2
+2
= 𝑘𝑒−
1
2
𝑥−112
2
𝑒2
= 𝑘𝑒2𝑒−
1
2
𝑥−112
2
⋯⋯⋯⋯⋯ (1)
1 ¨Â 𝑓 𝑥 =1
𝜎 2𝜋𝑒
−1
2 𝑥−𝜇
𝜎
2
¯¼ý ´ôÀ¢¼
𝜇 = 1 𝜎 =1
2⇒ 𝜎2 =
1
4 𝑘𝑒2 =
1
𝜎 2𝜋
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𝑘 =1
𝑒2 1
2 2𝜋
= 2𝑒−2
𝜋
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