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+2 MATHEMATICS 10MARKS STUDY MATERIAL

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ªõ‚ì˜ ÞòŸèí¤î‹

𝟏. ´Õ Óì§¸¡½ò¾¢ý ÌòÐì§¸¡Î¸û ´§Ã ÒûÇ¢Â¢ø ºó¾¢ìÌõ

±ýÀ¾¨É ¦Åì¼÷ Ó¨ÈÂ¢ø ¿¢Ú×¸.O-2006,J-2008

¾£÷×: Óì§¸¡½ò¾¢ý ¯îº¢¸û 𝐴, 𝐵, 𝐶 ±ý¸.

ÌòÐì§¸¡Î¸û 𝐴𝐷, 𝐵𝐸 Å¨Ã¸.

«¨Å¸û ¦ÅðÎõ ÒûÇ¢ 𝑂 ±ý¸.

𝐶𝑂 ¨Åî §º÷òÐ ¿£ð¼ «Ð 𝐴𝐵 ¨Â

𝐹 þø ºó¾¢ì¸¢ÈÐ ±ý¸.

ÌòÐì§¸¡Î¸û ´§Ã ÒûÇ¢Â¢ø ºó¾¢ìÌõ

±ýÀ¾¨É ¿¢ÚÅ, 𝐶𝐹 ⊥ 𝐴𝐵 ±É ¿¢ÚÅ¢É¡ø §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 ÌòÐì§¸¡Î¸Ùõ ´§Ã ÒûÇ¢Â¢ø ºó¾¢ìÌõ §¸¡Î¸Ç¡Ìõ.

𝟐. 𝐜𝐨𝐬 𝑨 − 𝑩 = 𝐜𝐨𝐬𝑨 𝐜𝐨𝐬𝑩 + 𝐬𝐢𝐧𝑨 𝐬𝐢𝐧𝑩±ýÀ¾¨É ¦Åì¼÷ Ó¨ÈÂ¢ø

¿¢Ú×¸.

¾£÷×:𝑂 ¨Â ¨ÁÂÁ¡¸×õ ´ÃÄÌ ¬ÃÓõ

¦¸¡ñ¼ ´Õ Åð¼õ Å¨Ã¸.∠𝑋𝑂𝑃 = 𝐴, ∠𝑋𝑂𝑄 = 𝐵

±ýÈÅ¡Ú «ùÅð¼ò¾¢ý ÀÃ¢¾¢Â¢ø 𝑃, 𝑄 ±ýÈ

þÕÒûÇ¢¸¨Ç ÌÈ¢ì¸.

∴ ∠𝑃𝑂𝑄 = ∠𝑃𝑂𝑋 − ∠𝑄𝑂𝑋 = 𝐴 − 𝐵.

𝑃𝑀, 𝑄𝐿 ⊥ 𝑂𝑋 Å¨Ã¸.𝑃, 𝑄- ý ¬Âò¦¾¡¨Ä¸û

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Ó¨È§Â(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 Å¨ÃÂ¨Èô 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÷ Ó¨ÈÂ¢ø

¿¢Ú×¸. M-2006,M-2008

¾£÷×: 𝑂 ¨Â ¨ÁÂÁ¡¸×õ ´ÃÄÌ

¬ÃÓõ ¦¸¡ñ¼ ´Õ Åð¼õ Å¨Ã¸.

∠𝑥𝑂𝑃 = 𝐴, ∠𝑥𝑂𝑄 = 𝐵 ±ýÈÅ¡Ú «ùÅð¼ò¾¢ý

ÀÃ¢¾¢Â¢ø 𝑃, 𝑄 ±ýÈ þÕÒûÇ¢¸¨Ç

ÌÈ¢ì¸.∴ ∠𝑃𝑂𝑄 = ∠𝑃𝑂𝑥 + ∠𝑄𝑂𝑥 = 𝐴 + 𝐵.

𝑃, 𝑄- ý ¬Âò¦¾¡¨Ä¸û

Ó¨È§Â (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 Å¨Ã¸.


𝑂𝑄 = 𝑂𝐿 + 𝐿𝑄 = cos 𝐵 𝑖 − sin 𝐵 𝑗 ÒûÇ¢ô¦ÀÕì¸Ä¢ý Å¨ÃÂ¨Èô 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÷ Ó¨ÈÂ¢ø ¿¢Ú×¸. J-2007, O-2007, O-2008,O-2010,M-2012

¾£÷×: 𝑂 ¨Â ¨ÁÂÁ¡¸×õ ´ÃÄÌ ¬ÃÓõ ¦¸¡ñ¼´Õ Åð¼õ

Å¨Ã¸.∠𝑋𝑂𝑃 = 𝐴, ∠𝑋𝑂𝑄 = 𝐵 ±ýÈÅ¡Ú «ùÅð¼ò¾¢ý ÀÃ¢¾¢Â¢ø 𝑃, 𝑄

±ýÈ þÕÒûÇ¢¸¨Ç ÌÈ¢ì¸.

∴ ∠𝑃𝑂𝑄 = ∠𝑃𝑂𝑋 − ∠𝑄𝑂𝑋 = 𝐴 − 𝐵.

𝑃, 𝑄- ý ¬Âò¦¾¡¨Ä¸û Ó¨È§Â

(cos 𝐴, sin 𝐴 ) ÁüÚõ (cos 𝐵, sin 𝐵 ).


«ÄÌ ¦Åì¼÷¸û 𝑖 ÁüÚõ 𝑗 ±ý¸.

𝑃𝑀, 𝑄𝐿 ⊥ 𝑂𝑋 Å¨Ã¸.


𝑂𝑄 = 𝑂𝐿 + 𝐿𝑄 = cos 𝐵 𝑖 + sin 𝐵 𝑗 ÌÚìÌô¦ÀÕì¸Ä¢ý Å¨ÃÂ¨Èô 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÷ Ó¨ÈÂ¢ø

¿¢Ú×¸.

¾£÷×: 𝑂 ¨Â ¨ÁÂÁ¡¸×õ ´ÃÄÌ ¬ÃÓõ ¦¸¡ñ¼ ´Õ Åð¼õ Å¨Ã¸.

∠𝑥𝑂𝑃 = 𝐴, ∠𝑥𝑂𝑄 = 𝐵 ±ýÈÅ¡Ú «ùÅð¼ò¾¢ý ÀÃ¢¾¢Â¢ø 𝑃, 𝑄

±ýÈþÕÒûÇ¢¸¨Ç ÌÈ¢ì¸.

∴ ∠𝑃𝑂𝑄 = ∠𝑃𝑂𝑥 + ∠𝑄𝑂𝑥 = 𝐴 + 𝐵.

𝑃, 𝑄- ý ¬Âò¦¾¡¨Ä¸û Ó¨È§Â

(cos 𝐴, sin 𝐴 ) ÁüÚõ (cos 𝐵, −sin 𝐵 ).


«ÄÌ ¦Åì¼÷¸û 𝑖 ÁüÚõ 𝑗 ±ý¸.

𝑃𝑀, 𝑄𝐿 ⊥ 𝑂𝑋 Å¨Ã¸.

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𝑂𝑄 = 𝑂𝐿 + 𝐿𝑄 = cos 𝐵 𝑖 − sin 𝐵 𝑗 ÌÚìÌô¦ÀÕì¸Ä¢ý Å¨ÃÂ¨Èô 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¨½ÂüÈ¨Å¸û.

¦¸¡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 ÒûÇ¢Â¢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 ÒûÇ¢Â¢ý «¨ÁôÒ(2𝜇 + 4,0,3𝜇 − 1)

¬Ìõ.𝐿1, 𝐿2 ±ýÈ §¸¡Î¸û ¦ÅðÊì¦¸¡ûÅ¾¡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¨½ÂüÈ¨Å¸û.

¦¸¡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 ÒûÇ¢Â¢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 ÒûÇ¢Â¢ý «¨ÁôÒ(𝜇 + 2,2𝜇 + 1, −𝜇 − 1)

¬Ìõ.§¸¡Î¸û ¦ÅðÊì¦¸¡ûÅ¾¡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 ¸¡÷Ëº¢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

¸¡÷Ëº¢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 ¸¡÷Ëº¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.

¾£÷×:


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


þíÌ 𝑥1, 𝑦1, 𝑧1 = 1,3,2 ; 𝑙1, 𝑚1, 𝑛1 = 2, −1,3 ; 𝑙2, 𝑚2, 𝑛2 = (1,2,2)


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 ¸¡÷Ëº¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.

¾£÷×:


þíÌ

𝑎 = − 𝑖 + 3 𝑗 + 2 𝑘 , 𝑢 = 𝑖 + 2 𝑗 + 2 𝑘 , 𝑣 = 3 𝑖 + 𝑗 + 2 𝑘


𝑟 = − 𝑖 + 3 𝑗 + 2 𝑘 + 𝑠 𝑖 + 2 𝑗 + 2 𝑘 + 𝑡 3 𝑖 + 𝑗 + 2 𝑘


þíÌ 𝑥1, 𝑦1, 𝑧1 = −1,3,2 ; 𝑙1, 𝑚1, 𝑛1 = 1,2,2 ; 𝑙2,𝑚2, 𝑛2 = (3,1,2)


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 ¸¡÷Ëº¢Âý

ºÁýÀ¡Î¸¨Çì ¸¡ñ¸. J-2006,M-2008 ¾£÷×:


þíÌ 𝑎 = − 𝑖 − 2 𝑗 + 𝑘 , 𝑢 = 𝑖 + 2 𝑗 + 4 𝑘 , 𝑣 = 2 𝑖 − 𝑗 + 3 𝑘


𝑟 = − 𝑖 − 2 𝑗 + 𝑘 + 𝑠 𝑖 + 2 𝑗 + 4 𝑘 + 𝑡 2 𝑖 − 𝑗 + 3 𝑘


þíÌ 𝑥1, 𝑦1, 𝑧1 = −1, −2,1 ; 𝑙1, 𝑚1, 𝑛1 = 1,2,4 ; 𝑙2, 𝑚2, 𝑛2 = (2, −1,3)


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 ¸¡÷Ëº¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.

O-2010 ¾£÷×:


þíÌ 𝑎 = 𝑖 + 2 𝑗 − 2 𝑘 , 𝑢 = 3 𝑖 − 2 𝑗 − 4 𝑘 , 𝑣 = 2 𝑖 + 3 𝑗 + 3 𝑘


𝑟 = 𝑖 + 2 𝑗 − 2 𝑘 + 𝑠 3 𝑖 − 2 𝑗 − 4 𝑘 + 𝑡 2 𝑖 + 3 𝑗 + 3 𝑘


þíÌ 𝑥1, 𝑦1, 𝑧1 = 1,2, −2 ; 𝑙1, 𝑚1, 𝑛1 = 3, −2, −4 ; 𝑙2, 𝑚2, 𝑛2 = (2,3,3)

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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 ¯ûÇ¼ì¸¢ÂÐõ,

𝒙+𝟏

𝟑=

𝒚−𝟏

𝟐=

𝒛+𝟏

𝟏±ýÈ

§¸¡ðÊüÌ þ¨½Â¡ÉÐÁ¡É ¾Çò¾¢ý ¦Åì¼÷ ÁüÚõ ¸¡÷Ëº¢Âý

ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.

¾£÷×:


þíÌ 𝑎 = 2 𝑖 + 2 𝑗 + 𝑘 , 𝑢 = 2 𝑖 + 3 𝑗 + 3 𝑘 , 𝑣 = 3 𝑖 + 2 𝑗 + 𝑘


𝑟 = 2 𝑖 + 2 𝑗 + 𝑘 + 𝑠 2 𝑖 + 3 𝑗 + 3 𝑘 + 𝑡 3 𝑖 + 2 𝑗 + 𝑘


þíÌ 𝑥1, 𝑦1, 𝑧1 = 2,2,1 ; 𝑙1,𝑚1, 𝑛1 = 2,3,3 ; 𝑙2, 𝑚2, 𝑛2 = (3,2,1)


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 ¸¡÷Ëº¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.

¾£÷×:

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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


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 ¸¡÷Ëº¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.

M-2006, O-2006, O-2007,J-2008,M-2012 ¾£÷×:


þíÌ 𝑎 = 𝑖 + 2 𝑗 + 3 𝑘 , 𝑏 = 2 𝑖 + 3 𝑗 + 𝑘 , 𝑣 = 3 𝑖 − 2 𝑗 + 4 𝑘


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


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 ¸¡÷Ëº¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.

M-2007,M-2009,J-2010 ¾£÷×:


þíÌ 𝑎 = − 𝑖 + 𝑗 + 𝑘 , 𝑏 = 𝑖 − 𝑗 + 𝑘 , 𝑣 = 𝑖 + 2 𝑗 + 2 𝑘


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


þíÌ 𝑥1, 𝑦1, 𝑧1 = −1,1,1 ; 𝑥2, 𝑦2, 𝑧2 = 1, −1,1 ; 𝑙, 𝑚, 𝑛 = (1,2,2)


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 ¯ûÇ¼ì¸¢ÂÐõ (−𝟏, 𝟏, −𝟏)±ýÈ

ÒûÇ¢ ÅÆ¢§Âî ¦ºøÄì ÜÊÂÐÁ¡É ¾Çò¾¢ý ¦Åì¼÷ ÁüÚõ

¸¡÷Ëº¢Âý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.

¾£÷×:


þíÌ 𝑎 = − 𝑖 + 𝑗 − 𝑘 , 𝑏 = 2 𝑖 + 2 𝑗 + 𝑘 , 𝑣 = 2 𝑖 + 3 𝑗 − 2 𝑘


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


þíÌ 𝑥1, 𝑦1, 𝑧1 = −1,1, −1 , 𝑥2, 𝑦2, 𝑧2 = 2,2,1 , 𝑙, 𝑚, 𝑛 = (2,3, −2)


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 ¸¡÷Ëº¢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


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


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þíÌ 𝑥1, 𝑦1, 𝑧1 = 3,4,2 ; 𝑥2, 𝑦2, 𝑧2 = 2, −2,−1 ; 𝑥3, 𝑦3, 𝑧3 = (7,0,1)


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

¦ºøÄìÜÊÂ ¾Çò¾¢ý ¦Åì¼÷ ÁüÚõ ¸¡÷Ëº¢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


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


þíÌ 𝑥1, 𝑦1, 𝑧1 = 2,2, −1 ; 𝑥2, 𝑦2, 𝑧2 = 3,4,2 ; 𝑥3, 𝑦3, 𝑧3 = (7,0,6)


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£÷×: ¸¡÷Ëº¢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.


þíÌ 𝑎 = 𝑎 𝑖 , 𝑏 = 𝑏 𝑗 , 𝑐 = 𝑐 𝑘


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)


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

¦ÁöÀÌ¾¢ =𝑥2 + 𝑦2 − 1

(𝑥 + 1)2 + 𝑦2¸üÀ¨É ÀÌ¾¢=

2y

(x+1)2+y2

¸ðÎôÀ¡Î:

arg 𝑧 − 1

𝑧 + 1 =

𝜋

3⇒ tan−1

¸üÀ¨ÉÀÌ¾¢

¦ÁöÀÌ¾¢ =

𝜋

3

⇒¸üÀ¨ÉÀÌ¾¢

¦ÁöÀÌ¾¢= 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«.


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

¦ÁöÀÌ¾¢ =𝑥2 + 𝑦2 − 1

(𝑥 + 1)2 + 𝑦2,¸üÀ¨É ÀÌ¾¢=

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«.


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

¦ÁöÀÌ¾¢ =𝑥2 + 𝑦2 + 2𝑥 − 3

(𝑥 + 3)2 + 𝑦2¸üÀ¨É ÀÌ¾¢= −

4𝑦

(𝑥 + 3)2 + 𝑦2

¸ðÎôÀ¡Î:

arg 𝑧 − 1

𝑧 + 3 =

𝜋

2⇒ tan−1

¸üÀ¨ÉÀÌ¾¢

¦ÁöÀÌ¾¢ =

𝜋

2

⇒¸üÀ¨ÉÀÌ¾¢

¦ÁöÀÌ¾¢= tan

𝜋

2= ∞ ⇒ ¦ÁöÀÌ¾¢ = 0

⇒𝑥2 + 𝑦2 + 2𝑥 − 3

(𝑥 + 3)2 + 𝑦2 = 0 ⇒ 𝑥2 + 𝑦2 + 2𝑥 − 3 = 0

𝑥2 + 𝑦2 + 2𝑥 − 3 = 0v‹gJ njitahd ãak¥ghijahF«.


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

= ¦ÁöôÀÌ¾¢ + 𝑖2𝑦 1 − 𝑦 − 𝑥 2𝑥 + 1

(1 − 𝑦)2 + 𝑥2

¸üÀ¨ÉÀÌ¾¢ =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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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¨ÉÀÌ¾¢

¦ÁöÀÌ¾¢ =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 ¸¡øÀÌ¾¢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

¸¡ðÎ¸.

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 ¸¡ðÎ¸.

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 ¸¡øÀÌ¾¢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 ¸¡øÀÌ¾¢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¢ôÒ¸Ç¢ý

¦ÀÕì¸üÀÄý 1 ±É×õ ¸¡ðÎ¸.

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 ¸¡øÀÌ¾¢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£÷×¸Ç¡Ìõ.

ii ⇒ 𝑥 = −1 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£÷×¸Ç¡Ìõ.

∴ 1, cis2𝜋

5, cis

4𝜋

5, cis

6𝜋

5, cis

8𝜋

5, cis

𝜋

4, cis

3𝜋

4, cis

5𝜋

44, cis

7𝜋

4

±ýÀÉ§¾¨ÅÂ¡É¾£÷×¸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


4


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£÷×¸Ç¡Ìõ.

ii ⇒ 𝑥 = −1 1


3


3


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£÷×¸Ç¡Ìõ.

∴ cis𝜋

4, cis

3𝜋

4, cis

5𝜋

4, cis

7𝜋

4, cis

𝜋

3, −1, cis

5𝜋

3±ýÀÉ§¾¨ÅÂ¡É¾£÷×¸û

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ANALYTICAL GEOMETRY

1. 𝒚𝟐 − 𝟖𝒙 + 𝟔𝒚 + 𝟗 = 𝟎±ýÈ ÀÃÅ¨ÇÂò¾¢ý «îÍ, Ó¨É, ÌÅ¢Âõ,

þÂìÌÅ¨ÃÂ¢ý ºÁýÀ¡Î, ¦ºùÅ¸Äò¾¢ý ºÁýÀ¡Î, ¦ºùÅ¸Äò¾¢ý ¿£Çõ

¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý Å¨Ãô À¼ò¨¾ Å¨Ã¸. 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 Ó¨É 𝑉(0,0) 𝑉(0,−3)

ÌÅ¢Âõ 𝐹 𝑎, 0 = 𝐹(2,0) 𝐹(2 + 0,0 − 3)

= 𝐹(2,−3)

þÂìÌÅ¨ÃÂ¢ý ºÁýÀ¡Î 𝑋 = −𝑎 𝑋 = −2

𝑥 = −2

¦ºùÅ¸Äò¾¢ý ºÁýÀ¡Î 𝑋 = 𝑎 𝑋 = 2

𝑥 = 2

¦ºùÅ¸Äò¾¢ý ¿£Çõ 4𝑎 = 8 8

2. 𝒚𝟐 − 𝟖𝒙 − 𝟐𝒚 + 𝟏𝟕 = 𝟎±ýÈ ÀÃÅ¨ÇÂò¾¢ý «îÍ, Ó¨É, ÌÅ¢Âõ,


¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý Å¨Ãô À¼ò¨¾ Å¨Ã¸. 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

Ó¨É 𝑉(0,0) 𝑉(2,1)

ÌÅ¢Âõ 𝐹 𝑎, 0 = 𝐹(2,0) 𝐹(2 + 2,0 + 1)

= 𝐹(4,1)

þÂìÌÅ¨ÃÂ¢ý ºÁýÀ¡Î 𝑋 = −𝑎

𝑋 = −2 𝑥 = 0

¦ºùÅ¸Äò¾¢ý ºÁýÀ¡Î 𝑋 = 𝑎

𝑋 = 2 𝑥 = 4

¦ºùÅ¸Äò¾¢ý ¿£Çõ 4𝑎 = 8 8

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3. 𝒚𝟐 + 𝟖𝒙 − 𝟔𝒚 + 𝟏 = 𝟎±ýÈ ÀÃÅ¨ÇÂò¾¢ý «îÍ, Ó¨É, ÌÅ¢Âõ,


¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý Å¨ÃôÀ¼ò¨¾ Å¨Ã¸. 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


«îÍ 𝑌 = 0 𝑦 − 3 = 0 Ó¨É 𝑉(0,0) 𝑉(1,3)

ÌÅ¢Âõ 𝐹 −𝑎, 0 = 𝐹(−2,0) 𝐹(−2 + 1,0 + 3)

= 𝐹(−1,3)

þÂìÌÅ¨ÃÂ¢ý ºÁýÀ¡Î 𝑋 = 𝑎 𝑋 = 2

𝑥 − 1 = 2 𝑥 − 3 = 0

¦ºùÅ¸Äò¾¢ý ºÁýÀ¡Î 𝑋 = −𝑎 𝑋 = −2

𝑥 − 1 = −2 𝑥 + 1 = 0

¦ºùÅ¸Äò¾¢ý ¿£Çõ 4𝑎 = 8 8

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4. 𝒙𝟐 − 𝟔𝒙 − 𝟏𝟐𝒚 − 𝟑 = 𝟎±ýÈ ÀÃÅ¨ÇÂò¾¢ý «îÍ, Ó¨É, ÌÅ¢Âõ,


¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý Å¨Ãô À¼ò¨¾ Å¨Ã¸. 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


«îÍ 𝑋 = 0 𝑥 − 3 = 0 Ó¨É 𝑉(0,0) 𝑉(3,−1)

ÌÅ¢Âõ 𝐹 0, 𝑎 = 𝐹(0,3) 𝐹(0 + 3,3 − 1)

= 𝐹(3,2)

þÂìÌÅ¨ÃÂ¢ý ºÁýÀ¡Î 𝑌 = −𝑎 𝑌 = −3

𝑦 + 1 = −3 𝑦 + 4 = 0

¦ºùÅ¸Äò¾¢ý ºÁýÀ¡Î 𝑌 = 𝑎 𝑌 = 3

𝑦 + 1 = 3 𝑦 − 2 = 0

¦ºùÅ¸Äò¾¢ý ¿£Çõ 4𝑎 = 12 12

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5. 𝒙𝟐 − 𝟐𝒙 + 𝟖𝒚 + 𝟏𝟕 = 𝟎±ýÈ ÀÃÅ¨ÇÂò¾¢ý «îÍ, Ó¨É, ÌÅ¢Âõ,


¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý Å¨ÃôÀ¼ò¨¾ Å¨Ã¸.

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


«îÍ 𝑋 = 0 𝑥 − 1 = 0 Ó¨É 𝑉(0,0) 𝑉(1,−2)

ÌÅ¢Âõ 𝐹 0, −𝑎 = 𝐹(0, −2) 𝐹(0 + 1, −2 − 2)

= 𝐹(1,−4)

þÂìÌÅ¨ÃÂ¢ý ºÁýÀ¡Î 𝑌 = 2 𝑌 = 2

𝑦 + 2 = 2 𝑦 = 0

¦ºùÅ¸Äò¾¢ý ºÁýÀ¡Î 𝑌 = −𝑎 𝑌 = −2

𝑦 + 2 = −2 𝑦 + 4 = 0

¦ºùÅ¸Äò¾¢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, Ó¨É¸û ¿£Çõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý

Å¨ÃôÀ¼ò¨¾ Å¨Ã¸.

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


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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)

Ó¨É¸û 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, Ó¨É¸û ¿£Çõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ «¾ý

Å¨ÃôÀ¼ò¨¾ Å¨Ã¸. 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


¨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)

Ó¨É¸û 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, Ó¨É¸û ¿£Çõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸. §ÁÖõ

«¾ý Å¨ÃôÀ¼ò¨¾ Å¨Ã¸. 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


¨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)

Ó¨É¸û 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¢ÀÃÅ¨ÇÂò¾¢ý ¨ÁÂò¦¾¡¨Ä×


«¾ý Å¨ÃôÀ¼ò¨¾ Å¨Ã¸. 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


¨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)

Ó¨É¸û 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¢ÀÃÅ¨ÇÂò¾¢ý

¨ÁÂò¦¾¡¨Ä× Å¢ ¢̧¾õ, ¨ÁÂõ, ÌÅ¢Âí¸û, Ó¨É¸û ¿£Çõ ¬ ¢̧ÂÅü¨Èì

¸¡ñ¸. §ÁÖõ «¾ý Å¨ÃôÀ¼ò¨¾ Å¨Ã¸.

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


¨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)

Ó¨É¸û 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×


«¾ý Å¨ÃôÀ¼ò¨¾ Å¨Ã¸. 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


¨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)

Ó¨É¸û 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¢ÀÃÅ¨ÇÂò¾¢ý


¸¡ñ¸. §ÁÖõ «¾ý Å¨ÃôÀ¼ò¨¾ Å¨Ã¸.

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


¨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)

Ó¨É¸û 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¢ÀÃÅ¨ÇÂò¾¢ý


¸¡ñ¸. §ÁÖõ «¾ý Å¨ÃôÀ¼ò¨¾ Å¨Ã¸. 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


¨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)

Ó¨É¸û

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 §¿÷ì§¸¡Î «¾¢ÀÃÅ¨ÇÂõ 𝒙𝟐 − 𝟗𝒚𝟐 = 𝟗 -³ò

¦¾¡Î¸¢ÈÐ ±É ¿¢åÀ¢ì¸. §ÁÖõ ¦¾¡Îõ ÒûÇ¢¨ÂÔõ ¸¡ñ¸. J-

2009

¾£÷×:

§¿÷ì§¸¡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¢ÀÃÅ¨ÇÂõ 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 §¿÷ì§¸¡Î «¾¢ÀÃÅ¨ÇÂõ 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 §¿÷ì§¸¡Î «¾¢ÀÃÅ¨ÇÂõ

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 §¿÷ì§¸¡Î ¿£ûÅð¼õ

𝒙𝟐 + 𝟑𝒚𝟐 = 𝟏𝟐 -³ò ¦¾¡Î¸¢ÈÐ ±É ¿¢åÀ¢ì¸. §ÁÖõ ¦¾¡Îõ ÒûÇ¢¨ÂÔõ

¸¡ñ¸.

¾£÷×:

§¿÷ì§¸¡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 §¿÷ì§¸¡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 §¿÷ì§¸¡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 ¦ºùÅ¸ «¾¢ÀÃÅ¨ÇÂò¾¢ý ºÁýÀ¡Î

¸¡ñ¸.O-2006,M-2007,J-2007,M-2008,O-2008,O-2010

¾£÷×:

¦ºùÅ¸ «¾¢ÀÃÅ¨ÇÂò¾¢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¢ÀÃÅ¨ÇÂò¾¢ý ºÁýÀ¡ðÊý ÅÊÅõ

𝑥 + 2𝑦 − 5 2𝑥 − 𝑦 + 𝑘 + 𝑐 = 0

«¾¢ÀÃÅ¨ÇÂõ (6,0) ±ýÈ ÒûÇ¢ ÅÆ¢Â¡¸î ¦ºøÅ¾¡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¢ÀÃÅ¨ÇÂõ (−3,0) ±ýÈ ÒûÇ¢ ÅÆ¢Â¡¸î ¦ºøÅ¾¡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¢ÀÃÅ¨ÇÂò¾¢ý ºÁýÀ¡Î

𝑥 + 2𝑦 − 5 2𝑥 − 𝑦 + 4 − 16 = 0

17. «¾¢ÀÃÅ¨ÇÂò¾¢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¡¸ þÕì¸¢ýÈÉ ±É¢ø «¾¢ÀÃÅ¨ÇÂò¾¢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¢ÀÃÅ¨ÇÂò¾¢ý ¨ÁÂõ 2,4 ÅÆ¢Â¡¸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¢ÀÃÅ¨ÇÂò¾¢ý ºÁýÀ¡ðÊý ÅÊÅõ

𝑥 + 2𝑦 − 10 𝑥 − 2𝑦 + 6 + 𝑘 = 0

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«¾¢ÀÃÅ¨ÇÂõ (2,0) ±ýÈ ÒûÇ¢ ÅÆ¢Â¡¸î ¦ºøÅ¾¡ø,

(2 + 0 − 10)( 2 − 0 + 6 + 𝑘 = 0

⇒ (−8)( 8 + 𝑘 = 0 ⇒ −64 + 𝑘 = 0 ⇒ 𝒌 = 𝟔𝟒

«¾¢ÀÃÅ¨ÇÂò¾¢ý ºÁýÀ¡Î 𝑥 + 2𝑦 − 10 𝑥 − 2𝑦 + 6 + 64 = 0

18. ´Õ ¦¾¡íÌ À¡Äò¾¢ý ¸õÀ¢ Å¼õ ÀÃÅ¨ÇÂ ÅÊÅ¢ÖûÇÐ. «¾ý À¡Ãõ

¸¢¨¼Áð¼Á¡¸ º£Ã¡¸ ÀÃÅ¢ÔûÇÐ. «¨¾ò ¾¡íÌõ þÕ àñ¸ÙìÌ

þ¨¼§ÂÔûÇ àÃõ 𝟏𝟓𝟎𝟎 «Ê. ¸õÀ¢ Å¼ò¨¾ ¾¡íÌõ ÒûÇ¢¸û à½¢ø

¾¨ÃÂ¢Ä¢ÕóÐ 𝟐𝟎𝟎 «Ê ¯ÂÃò¾¢ø «¨ÁóÐûÇÉ. §ÁÖõ ¾¨ÃÂ¢Ä¢ÕóÐ

¸õÀ¢ Å¼ò¾¢ý ¾¡úÅ¡É ÒûÇ¢Â¢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¢ Å¼õ §ÁüÒÈõ ¾¢ÈôÒ¨¼Â ÀÃÅ¨ÇÂ ÅÊÅ¢ÖûÇÐ

±Éì ¦¸¡û¸. ±É§Å, ¸õÀ¢ Å¼ò¾¢ý ºÁýÀ¡Î 𝑥2 = 4𝑎𝑦.

¸¡Îì¸ôÀð¼ Å¢Ãí¸Ç¢Ä¢ÕóÐ Å¼ò¾¢ý Ó¨É ÅÆ¢ôÀ¡¨¾Â¢Ä¢ÕóÐ 70 «Ê

§Áø «¨ÁóÐûÇÐ. ¦¾¡íÌ À¡Äò¾¢ý Å¢ð¼õ 1500 «Ê.

ÒûÇ¢ 𝐵(750,130) ÀÃÅ¨ÇÂò¾¢ý Á£Ð «¨ÁóÐûÇÐ.

750 2 = 4𝑎 130 ⇒ 4𝑎 =750 × 750

130=

75 × 750

13

¸õÀ¢ Å¼ò¾¢ý ºÁýÀ¡Î 𝑥2 =75×750

13𝑦.

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𝑄(𝑥1, 52) ±ýÈ ÒûÇ¢ ÀÃÅ¨ÇÂò¾¢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¢ Å¼õ ÀÃÅ¨ÇÂ ÅÊÅ¢ÖûÇÐ. «¾ý ¿£Çõ

𝟒𝟎Á£ð¼÷ ¬Ìõ. ÅÆ¢ôÀ¡¨¾Â¡ÉÐ ¸õÀ¢ Å¼ò¾¢ý £̧úÁð¼ô ÒûÇ¢Â¢Ä¢ÕóÐ

𝟓 Á£ð¼÷ £̧§Æ ¯ûÇÐ. ¸õÀ¢ Å¼ò¨¾ ¾¡íÌõ àñ¸Ç¢ý ¯ÂÃí¸û 𝟓𝟓

Á£ð¼÷ ±É¢ø 𝟑𝟎 Á£ð¼÷ ¯ÂÃò¾¢ø ¸õÀ¢ Å¼ò¾¢üÌ ´Õ Ð¨½ ¾¡í ¢̧

ÜÎ¾Ä¡¸ì ¦¸¡Îì¸ôÀð¼¡ø «òÐ¨½ò¾¡í¸¢Â¢ý ¿£Çò¨¾ì ¸¡ñ¸.J-

2006

¾£÷×:

¦¾¡íÌ À¡Äò¾¢ý ¸õÀ¢ Å¼õ §ÁüÒÈõ ¾¢ÈôÒ¨¼Â ÀÃÅ¨ÇÂ ÅÊÅ¢ÖûÇÐ

±Éì ¦¸¡û¸.

±É§Å, ¸õÀ¢ Å¼ò¾¢ý ºÁýÀ¡Î 𝑥2 = 4𝑎𝑦.

¦¸¡Îì¸ôÀð¼ Å¢Ãí¸Ç¢Ä¢ÕóÐ Å¼ò¾¢ý Ó¨É ÅÆ¢ôÀ¡¨¾Â¢Ä¢ÕóÐ 40 Á£ð¼÷

§Áø «¨ÁóÐûÇÐ. ¦¾¡íÌ À¡Äò¾¢ý Å¢ð¼õ 40 Á£ð¼÷.

ÒûÇ¢ 𝐴(20,50) ÀÃÅ¨ÇÂò¾¢ý Á£Ð «¨ÁóÐûÇÐ.

20 2 = 4𝑎 50 ⇒ 4𝑎 =20 × 20

50= 8

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¸õÀ¢ Å¼ò¾¢ý ºÁýÀ¡Î 𝑥2 = 8𝑦.

𝑄(𝑥1, 25) ±ýÈ ÒûÇ¢ ÀÃÅ¨ÇÂò¾¢ý Á£Ð «¨ÁóÐûÇÐ.

∴ 𝑥12 = 8 × 25 = 2 × 4 × 25

⇒ 𝑥1 = 2 × 2 × 5 = 10 2

𝑃𝑄 = 2𝑥1 = 20 2 Á£ð¼÷.

20. ´Õ ÃÂ¢ø§Å À¡Äò¾¢ý §Áø Å¨Ç× ÀÃÅ¨ÇÂò¾¢ý «¨Áô¨Àì

¦¸¡ñÎûÇÐ. «ó¾ Å¨ÇÅ¢ý «¸Äõ 𝟏𝟎𝟎 «ÊÂ¡¸×õ «ùÅ¨ÇÅ¢ý

¯îº¢ôÒûÇ¢Â¢ý ¯ÂÃõ À¡Äò¾¢Ä¢ÕóÐ 𝟏𝟎 «ÊÂ¡¸ ×õ ¯ûÇÐ ±É¢ø,

À¡Äòò¾¢ý Áò¾¢Â¢Ä¢ ÕóÐ þ¼ôÒÈõ «øÄÐ ÅÄôÒÈõ 𝟏𝟎 «Ê àÃò¾¢ø

À¡Äò¾¢ý §Áø Å¨Ç× ±ùÅÇ× ¯ÂÃò¾¢ø þÕìÌõ?M-2009

¾£÷×:

þíÌ ÀÃÅ¨ÇÂõ £̧ú§¿¡ì¸¢ò ¾¢üôÒ¨¼Â¾¡¸ ±ÎòÐì ¦¸¡û§Å¡õ.

±É§Å,ÃÂ¢ø§Å À¡Äò¾¢ý §Áø Å¨ÇÅ¢ý ºÁýÀ¡Î

𝑥2 = −4𝑎𝑦

þÐ (50, −10) ÅÆ¢Â¡¸î ¦ºø ¢̧ÈÐ.

∴ 50 × 50 = −4𝑎(−10) ⇒ 𝒂 =𝟐𝟓𝟎

𝟒

∴ 𝑥2 = −4 250

4 𝑦 ⇒ 𝒙𝟐 = −𝟐𝟓𝟎𝒚

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ÀÃÅ¨ÇÂò¾¢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 ÝÃ¢Â¨Éî ÍüÈ¢ ÀÃÅ¨ÇÂô À¡¨¾Â¢ø ¦ºø¸¢ÈÐ

ÁüÚõ ÝÃ¢Âý ÀÃÅ¨ÇÂò¾¢ý ÌÅ¢Âò¾¢ø «¨Á¸¢ÈÐ. Å¡ø Å¢ñÁ£ý

ÝÃ¢ÂÉ¢Ä¢ÕóÐ 𝟖𝟎 Á¢øÄ¢Âý ¢̧.Á£. ¦¾¡¨ÄÅ¢ø «¨ÁóÐ þÕìÌõ §À¡Ð

Å¡ø Å¢ñÁ£¨ÉÔõ ÝÃ¢Â¨ÉÔõ þ¨½ìÌõ §¸¡Î «îÍ¼ý 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 ÝÃ¢ÂÛìÌ ±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 ÀÃÅ¨ÇÂ¾¢ý Á£ÐûÇ¾¡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

ÝÃ¢ÂÛìÌõ Å¡ø Å¢ñÁ£ÛìÌõ þ¨¼§ÂÔûÇ Á¢¸ì Ì¨Èó¾ àÃõ 𝑉𝐹 = 𝑎 =

20Á¢øÄ¢Âý ¸¢.Á£.

22. ´Õ Ã¡ì¦¸ð ¦ÅÊÂ¡ÉÐ ¦¸¡ÙòÐõ §À¡Ð «Ð ´Õ ÀÃÅ¨ÇÂô 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 Åó¾¨¼¸¢ÈÐ ±É¢ø

ÒÈôÀð¼ þ¼ò¾¢ø ¾¨ÃÔ¼ý ²üÀÎòÐõ ±È¢§¸¡½õ ¸¡ñ¸.

M-2006, J-2009,J-2010

¾£÷×:

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´Õ Ã¡ì¦¸ð ¦ÅÊÂ¡ÉÐ ¦ºøÖõ ÀÃÅ¨ÇÂô 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

ÀÃÅ¨ÇÂô À¡¨¾Â¢ý ºÁýÀ¡Î

𝑥2 = −4 9

4 𝑦

𝑥2 = −9𝑦

𝑥- ³ ¦À¡ÚòÐ Å¨¸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

∴Ã¡ì¦¸ð ¦ÅÊÂ¡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 ÌÆ¡Â¢Ä¢ÕóÐ ¦ÅÇ¢§ÂÚõ ¿£÷ ¾¨Ã¨Âò ¦¾¡Îõ À¡¨¾ ´Õ

ÀÃÅ¨ÇÂò¨¾ ²üÀÎòÐ¸¢ÈÐ. §ÁÖõ þó¾ ÀÃÅ¨ÇÂô À¡¨¾Â¢ý Ó¨É

ÌÆ¡Â¢ý Å¡Â¢ø «¨Á¸¢ÈÐ. ÌÆ¡ö Áð¼ò¾¢üÌ 2.5Á£ £̧§Æ ¿£Ã¢ý

À¡öÅ¡ÉÐ ÌÆ¡Â¢ý Ó¨É ÅÆ¢Â¡¸î ¦ºøÖõ ¿¢¨Ä ÌòÐì§¸¡ðÊüÌ 3 Á£ð¼÷ àÃò¾¢ø ¯ûÇÐ ±É¢ø ÌòÐì §¸¡ðÊÄ¢ÕóÐ ±ùÅÇ× àÃò¾¢üÌ

«ôÀ¡ø ¿£Ã¡ÉÐ ¾¨ÃÂ¢ø Å¢Øõ.O-2009 ¾£÷×:¸½ì ¢̧ýÀÊ, ÌÆ¡Â¢Ä¢ÕóÐ ¦ÅÇ¢§ÂÚõ ¿£÷ À¡¨¾ £̧ú§¿¡ì¸¢ ¾¢ÈôÒ¨¼Â

ÀÃÅ¨ÇÂõ ¬Ìõ. ÀÃÅ¨ÇÂô À¡¨¾Â¢ý ºÁýÀ¡Î 𝑥2 = −4𝑎𝑦 ¬Ìõ.

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𝑃±ýÈ ÒûÇ¢ ÀÃÅ¨ÇÂô À¡¨¾Â¢ø ÌÆ¡ö Áð¼ò¾¢üÌ 2.5 Á£ £̧§ÆÔõ, ÌÆ¡Â¢ý

Ó¨É ÅÆ¢Â¡¸î ¦ºøÖõ ÌòÐì§¸¡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

ÀÃÅ¨ÇÂô À¡¨¾Â¢ý ºÁýÀ¡Î

𝑥2 = −4 9

10 𝑦

ÌòÐì §¸¡ðÊÄ¢ÕóÐ 𝑥1 Á£ð¼÷ àÃò¾¢üÌ «ôÀ¡ø ¿£Ã¡ÉÐ ¾¨ÃÂ¢ø Å¢Øõ

±ý¸. ÌÆ¡Â¡ÉÐ ¾¨ÃÁð¼ò¾¢Ä¢ÕóÐ 7.5Á£ ¯ÂÃò¾¢ø «¨ÁóÐûÇ¾¡ø,

𝑥1, −7 ⋅ 5 ±ýÈ ÒûÇ¢Ôõ ÀÃÅ¨ÇÂô 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¨ÃÂ¢ø

Å¢Øõ.

24. ´Õ Å¨Ç× «¨Ã-¿£ûÅð¼ ÅÊÅ¢ø ¯ûÇÐ. «¾ý «¸Äõ 48 «Ê, ¯ÂÃõ

20 «Ê. ¾¨ÃÂ¢Ä¢ÕóÐ 10 «Ê ¯ÂÃò¾¢ø Å¨ÇÅ¢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¨ÃÂ¢Ä¢ÕóÐ 10 «Ê ¯ÂÃò¾¢ø Å¨ÇÅ¢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¨ÃÂ¢Ä¢ÕóÐ 10 «Ê ¯ÂÃò¾¢ø Å¨ÇÅ¢ý «¸Äõ 2𝑥1 = 2 ×

12 3 = 24 3 «Ê.

25. ´Õ À¡Äò¾¢ý Å¨ÇÅ¡ÉÐ «¨Ã-¿£ûÅð¼ ÅÊÅ¢ø ¯ûÇÐ.

¸¢¨¼Áð¼ò¾¢ø «¾ý «¸Äõ 40 «ÊÂ¡¸×õ, ¨ÁÂò¾¢Ä¢ÕóÐ «¾ý

¯ÂÃõ 16 «ÊÂ¡¸×õ ¯ûÇÐ ±É¢ø ¨ÁÂò¾¢Ä¢ÕóÐ ÅÄÐ «øÄÐ þ¼ô

ÒÈò¾¢ø 9 «Ê àÃò¾¢ø ¯ûÇ ¾¨ÃôÒûÇ¢Â¢Ä¢ÕóÐ 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¨ÃôÒûÇ¢Â¢Ä¢ÕóÐ

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¨ÃôÒûÇ¢Â¢Ä¢ÕóÐ À¡Äò¾¢ý ¯ÂÃõ4

5 319 «Ê.

26. ´Õ Ñ¨Æ× Å¡Â¢Ä¢ý §ÁüÜ¨ÃÂ¡ÉÐ «¨Ã-¿£ûÅð¼ ÅÊÅ¢ø ¯ûÇÐ.

þ¾ý «¸Äõ 20 «Ê. ¨ÁÂò¾¢Ä¢ÕóÐ «¾ý ¯ÂÃõ 18 «Ê ÁüÚõ Àì¸î

ÍÅÃ¢¸Ç¢ý ¯ÂÃõ 12 «Ê ±É¢ø ²§¾Ûõ ´Õ Àì¸î ÍÅÃ¢Ä¢ÕóÐ 4 «Ê

àÃò¾¢ø §ÁüÜ¨ÃÂ¢ý ¯ÂÃõ ±ýÉÅ¡¸ þÕìÌõ? M-2007,M-2010

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¾£÷×: Àì¸î ÍÅÃ¢Ä¢ÕóÐ 4 «Ê àÃò¾¢ø §ÁüÜ¨ÃÂ¢ý ¯ÂÃõ 𝑃𝑄𝑅±ý¸.

À¼ò¾¢ý ãÄõ𝑃𝑄 = 12 «Ê.

À¼ò¾¢ý ãÄõ , Ó¨É¸û𝐴(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 ÍÅÃ¢Ä¢ÕóÐ 4 «Ê àÃò¾¢ø §ÁüÜ¨ÃÂ¢ý ¯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 ÍÅÃ¢Ä¢ÕóÐ 4 «Ê àÃò¾¢ø §ÁüÜ¨ÃÂ¢ý ¯ÂÃõ 12 + 𝑦1 = 12 + 4 ⋅ 8 =

16 ⋅ 8 «Ê.

27. ´Õ ¿£ûÅð¼ô À¡¨¾Â¢ý ÌÅ¢Âò¾¢ø âÁ¢ þÕìÌÁ¡Ú ´Õ Ð¨½ì§¸¡û

ÍüÈ¢ ÅÕ ¢̧ÈÐ. þ¾ý ¨ÁÂò ¦¾¡¨Ä× ¾¸× 𝟏

𝟐 ¬¸×õ âÁ¢ìÌõ Ð¨½ì

§¸¡ÙìÌõ þ¨¼ôÀð¼ Á£îº¢Ú àÃõ 𝟒𝟎𝟎 ¸¢§Ä¡ Á£ð¼÷¸û ¬¸×õ

þÕìÌÁ¡É¡ø Ð¨½ì §¸¡ÙìÌõ âÁ¢ìÌõ þ¨¼ôÀð¼ «¾¢¸Àðº

àÃõ ±ýÉ? J-2007,J-2008

¾£÷×: À¼ò¾¢ø âÁ¢Â¢ý ¿¢¨Ä 𝐹1 ±ý¸.

âÁ¢ìÌõ Ð¨½ì §¸¡ÙìÌõ þ¨¼ôÀð¼ Á£îº¢Ú àÃõ 𝐹1𝐴 = 400 ¸¢§Ä¡

Á£ð¼÷¸û. Ð¨½ì §¸¡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

Ð¨½ì §¸¡ÙìÌõ âÁ¢ìÌõ þ¨¼ôÀð¼ «¾¢¸Àðº àÃõ àÃõ 1200 ¸¢§Ä¡

Á£ð¼÷¸û.

28. ÝÃ¢Âý ÌÅ¢Âò¾¢Ä¢ÕìÌÁ¡Ú ¦Á÷ìÌÃ¢ ¸¢Ã¸Á¡ÉÐ ÝÃ¢Â¨É ´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÷ìÌÃ¢ ¸¢Ã¸Á¡ÉÐ ÝÃ¢ÂÛìÌ Á¢¸ «Õ¸¡¨ÁÂ¢ø ÅÕõ§À¡Ð ¯ûÇ àÃõ

(𝒊𝒊) ¦Á÷ìÌÃ¢ ¸¢Ã¸Á¡ÉÐ ÝÃ¢ÂÛìÌ Á¢¸ò ¦¾¡¨ÄÅ¢ø þÕìÌõ §À¡Ð

¯ûÇ àÃõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸. O-2009,J-2010 ¾£÷×:

À¼ò¾¢ø ÝÃ¢ÂÉ¢ý ¿¢¨Ä 𝐹1 ±ý¸.

𝐶𝐴 = 𝑎 = 36, 𝑒 = 0 ⋅ 206

𝑎𝑒 = 36 × 0 ⋅ 206 = 7 ⋅ 416

(𝑖)¦Á÷ìÌÃ¢ ¢̧Ã¸Á¡ÉÐ ÝÃ¢ÂÛìÌ Á¢¸ «Õ¸¡¨ÁÂ¢ø ÅÕõ§À¡Ð ¯ûÇ àÃõ

𝐹1𝐴 = 𝐶𝐴 − 𝐶𝐹1 = 𝑎 − 𝑎𝑒 = 36 − 7 ⋅ 416 = 28 ⋅ 584 ¦Á÷ìÌÃ¢ ¸¢Ã¸Á¡ÉÐ ÝÃ¢ÂÛìÌ Á¢¸ «Õ¸¡¨ÁÂ¢ø ÅÕõ§À¡Ð ¯ûÇ àÃõ

28 ⋅ 584 Á¢øÄ¢Âý ¨Áø¸û

𝑖𝑖 ¦Á÷ìÌÃ¢ ¸¢Ã¸Á¡ÉÐ ÝÃ¢ÂÛìÌ Á¢¸ò ¦¾¡¨ÄÅ¢ø þÕìÌõ §À¡Ð ¯ûÇ

àÃõ

𝐹1𝐴′ = 𝐶𝐴′ + 𝐶𝐹1 = 𝑎 + 𝑎𝑒 = 36 + 7 ⋅ 416 = 43 ⋅ 416

¦Á÷ìÌÃ¢ ¢̧Ã¸Á¡ÉÐ ÝÃ¢ÂÛìÌ Á¢¸ò ¦¾¡¨ÄÅ¢ø þÕìÌõ §À¡Ð ¯ûÇ àÃõ

43 ⋅ 416 Á¢øÄ¢Âý ¨Áø¸û

29. ´Õ §¸¡-§¸¡ Å¢¨ÇÂ¡ðÎ Å£Ã÷ Å¢¨ÇÂ¡ðÎô ÀÂ¢üº¢Â¢ý §À¡Ð «ÅÕìÌõ

§¸¡-§¸¡ Ìîº¢ì¸ÙìÌõ þ¨¼§ÂÔûÇ àÃõ ±ô¦À¡ØÐõ

𝟖 Á£ ¬¸ þÕìÌÁ¡Ú ¯½÷¸¢È¡÷. «ùÅ¢Õ Ìîº¢¸ÙìÌ þ¨¼ôÀð¼ àÃõ

𝟔 Á£ ±É¢ø «Å÷ µÎõ À¡¨¾Â¢ý ºÁýÀ¡¨¼ì ¸¡ñ¸.

¾£÷×:

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§¸¡-§¸¡ Ìîº¢¸û þÃñÎõ 𝐹1 ÁüÚõ 𝐹2 þø «¨ÁóÐûÇÉ ±Éì ¦¸¡û¸.

𝑃(𝑥, 𝑦) ±ýÈ ÒûÇ¢Â¡ÉÐ Å¢¨ÇÂ¡ðÎ Å£ÃÃ¢ý ¿¢¨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 ÍÅÃ¢ý Á£Ð 𝟏𝟓Á£ ¿£ÇÓûÇ

²½¢Â¡ÉÐ ¾Çò¾¢¨ÉÔõ ÍÅüÈ¢¨ÉÔõ ¦¾¡ÎÁ¡Ú ¿¸÷óÐ ¦¸¡ñÎ

þÕì¸¢ÈÐ ±É¢ø ²½¢Â¢ý £̧úÁð¼ Ó¨ÉÂ¢Ä¢ÕóÐ 𝟔Á£ àÃò¾¢ø ²½¢Â¢ø

«¨ÁóÐûÇ 𝑷 ±ýÈ ÒûÇ¢Â¢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 Å¨Ã¸.

Δ 𝐴𝐷𝑃 ÁüÚõ Δ 𝑃𝐶𝐵 ÅÊ¦Å¡ò¾¨Å.

∠𝑂𝐴𝑃 = ∠𝐶𝑃𝐵 = 𝜃±ý¸.

⊿𝐴𝐷𝑃Â¢ø

sin 𝜃 =𝑃𝐷

𝑃𝐴=

𝑦1

6

⊿𝐶𝑃𝐵Â¢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×


«¾ý Å¨ÃôÀ¼ò¨¾ Å¨Ã¸. 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


¨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)

Ó¨É¸û 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 äÄÃ¢ý §¾üÈò¨¾ô ÀÂýÀÎò¾¢ 𝑥

∂𝑢

∂𝑥+ 𝑦

∂𝑢

∂𝑦=

1

2tan𝑢

±Éì ¸¡ðÎ¸.

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, äÄÃ¢ý §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 äÄÃ¢ý §¾üÈò¨¾ô ÀÂýÀÎò¾¢ 𝑥

∂𝑢

∂𝑥+ 𝑦

∂𝑢

∂𝑦= sin2𝑢

±Éì ¸¡ðÎ¸.

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, äÄÃ¢ý §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 äÄÃ¢ý §¾üÈò¨¾î ºÃ¢À¡÷ì¸.

¾£÷×:

𝑓 =1

𝑥2 + 𝑦2

𝑓±ýÀÐ ÀÊ −1 ¯¨¼Â ºÁôÀÊò¾¡ý º¡÷Ò.

±É§Å, äÄÃ¢ý §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

∴ äÄÃ¢ý §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


𝑢 = sin3𝑥cos4𝑦 ±ýÈ º¡÷ÒìÌ ∂2𝑢

∂𝑥 ∂𝑦=

∂2𝑢


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, ¦ÅðÎòÐñÎ¸û ÁüÚõ ¬¾¢:

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º¡÷À¸õ (−∞, ∞) (−∞, ∞) [0, ∞)

¿£ðÊôÒ

¸¢¨¼Áð¼

¿£ðÊôÒ −∞ < 𝑥 < ∞ −∞ < 𝑥 < ∞ 0 ≤ 𝑥 < ∞

¿¢¨ÄìÌòÐ

¿£ðÊôÒ −∞ < 𝑦 < ∞ −∞ < 𝑦 < ∞ −∞ < 𝑦 < ∞

¦ÅðÎò

ÐñÎ¸û

𝑥 − ¦ÅðÎ

ÐñÎ −1 0 0

𝑦 − ¦ÅðÎ

ÐñÎ 1 0 0

¬¾¢ Å¨ÇÅ¨Ã ¬¾¢

ÅÆ¢î ¦ºøÄ¡Ð.

Å¨ÇÅ¨Ã ¬¾¢

ÅÆ¢î ¦ºøÖõ.

Å¨ÇÅ¨Ã ¬¾¢

ÅÆ¢î ¦ºøÖõ.

2 ºÁîº£÷ §º¡¾¨É

Å¨ÇÅ¨ÃÂ¡ÉÐ

ºÁîº£÷

¾ý¨Á¨Â

¦ÀÈÅ¢ø¨Ä.

Å¨ÇÅ¨Ã

¬¾¢¨Âô

¦À¡ÚòÐ

ºÁîº£Ã¡ÉÐ.

Å¨ÇÅ¨Ã 𝑥 −

¦À¡ÚòÐ

ºÁîº£Ã¡ÉÐ.

3 ¦¾¡¨Äò¦¾¡Î

§¸¡Î¸û

Å¨ÇÅ¨ÃìÌ

±ó¾ ´Õ

¦¾¡¨Äò¦¾¡Î

§¸¡Î¸Ùõ

þø¨Ä.

Å¨ÇÅ¨ÃìÌ

±ó¾ ´Õ

¦¾¡¨Äò¦¾¡Î

§¸¡Î¸Ùõ

þø¨Ä.

Å¨ÇÅ¨ÃìÌ

±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

¸¢¨ÇÂ¢ø

Å¨ÇÅ¨Ã

²ÚÓ¸Á¡¸

þÕìÌõ.

𝑦 = − 2𝑥3

2 ±ýÈ ¢̧¨ÇÂ¢ø

Å¨ÇÅ¨Ã

þÈíÌ Ó¸Á¡¸

þÕìÌõ.

5 º¢ÈôÒô ÒûÇ¢¸û:

(−∞, 0) ±ýÈ

þ¨¼¦ÅÇ¢Â¢ø

£̧ú §¿¡ì¸¢

ÌÆ¢Å¡¸×õ

ÁüÚõ, (0, ∞) ±ýÈ


§Áø §¿¡ì¸¢

ÌÆ¢Å¡¸×õ

þÕìÌõ.

0,1 ±ýÀÐ

Å¨Ç× Á¡üÚô

ÒûÇ¢.

(−∞, 0) ±ýÈ


£̧ú §¿¡ì¸¢

ÌÆ¢Å¡¸×õ

ÁüÚõ, (0, ∞) ±ýÈ


§Áø §¿¡ì¸¢

ÌÆ¢Å¡¸×õ

þÕìÌõ.

0,0 ±ýÀÐ

Å¨Ç× Á¡üÚô

ÒûÇ¢.

0,0 ±ýÀÐ

Å¨Ç× Á¡üÚô

ÒûÇ¢ÂøÄ.

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Å¨ÃôÀ¼õ

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¡ÚòÐ Å¨¸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¡ÚòÐ Å¨¸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 ¦ÀÚÅ¾¡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


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


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


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)


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 ÒûÇ¢Â¢Öõ º¡ö× 𝒚 + 𝟐𝒙 ±Éì ¦¸¡ñÎ ¬¾¢ÅÆ¢Â¡¸î ¦ºøÖõ

Å¨ÇÅ¨ÃÂ¢ý ºÁýÀ¡Î 𝒚 = 𝟐(𝒆𝒙 − 𝒙 − 𝟏) ±Éì ¸¡ðÎ¸. M-2010

¾£÷×:

º¡ö× = 𝑦 + 2𝑥

𝑑𝑦

𝑑𝑥= 𝑦 + 2𝑥

𝑑𝑦

𝑑𝑥− 𝑦 = 2𝑥


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

Å¨ÇÅ¨Ã ¬¾¢ ÅÆ¢î ¦ºøÅ¾¡ø , 𝑥 = 0 ±É¢ø 𝑦 = 0

0 = −2 0 + 1 + 𝑐𝑒0

0 = −2 + 𝑐

𝑐 = 2 Å¨ÇÅ¨ÃÂ¢ý 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


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


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 «ÇÅ¢Ä¢ÕóÐ 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 þÃº¡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 ¿¢¨ÄÂ¢ø «ô¦À¡ÕÇ¢ý ±¨¼Â¢¨Éì ¸¡ñ¸.

¾£÷×:𝑡 ±Ûõ §¿Ãò¾¢ø ¦À¡ÕÇ¢ý þÕôÒ 𝐴 ±ý¸.

¸½ì¸¢ý ÀÊ,

𝑑𝐴


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 ¿¢¨ÄÂ¢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¨ÈÂ¢ø ÅðÊ¨Âì ¸½ì ¢̧Î¸¢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÷îº¢ Â¡É ÜðÎ ÅðÊ ãÄõ

¬ñ¦¼¡ýÚìÌ 𝟖% ÅðÊ ¦ÀÕÌ¸¢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


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÷îº¢ ÜðÎ ÅðÊ ¸½ì¸¢¼ôÀÎ¸¢ÈÐ. ÅðÊ

Å£¾õ ¬ñ¦¼¡ýÚìÌ 𝟒 º¾Å£¾Á¡¸ þÕôÀ¢ý «ò¦¾¡¨¸ ±ò¾¨É

¬ñÎ¸Ç¢ø ¬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


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¡¸) ¬ñÎ¸Ç¢ø ¬ÃõÀò ¦¾¡¨¸¨Âô §À¡ø þÕ Á¼í¸¡Ìõ.

24. ´Õ ¿¸Ãò¾¢ø ¯ûÇ Áì¸û ¦¾¡¨¸Â¢ý ÅÇ÷îº¢ 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


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¢. ¢̧Ã¡õ ¬¸ 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 Ì¨ÈÂ ±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


𝑑𝐴

𝑑𝑡= 𝑘𝐴


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 Ì¨ÈÂ ±ÎòÐì

¦¸¡ûÙõ ¸¡Ä «Ç× 136 ¿¡ð¸û.(§¾¡Ã¡ÂÁ¡¸)

26. ¦ÅôÀ ¿¢¨Ä 𝟏𝟓𝟎𝑪 ¯ûÇ ´Õ «¨ÈÂ¢ø ¨Åì¸ôÀðÎûÇ §¾¿£Ã¢ý ¦ÅôÀ

¿¢¨Ä 𝟏𝟎𝟎𝟎𝑪 ¬Ìõ. «Ð 𝟓 ¿¢Á¢¼í¸Ç¢ø 𝟔𝟎𝟎𝑪 ¬¸ Ì¨ÈóÐ Å¢Î¸¢ÈÐ.

§ÁÖõ 𝟓 ¿¢Á¢¼õ ¸Æ¢òÐ §¾¿£Ã¢ý ¦ÅôÀ ¿¢¨ÄÂ¢¨É ¸¡ñ¸. O-2009

¾£÷×: 𝑡 ±Ûõ §¿Ãò¾¢ø §¾¿£Ã¢ý ¦ÅôÀ ¿¢¨Ä 𝑇(𝑡) ±ý¸.

¿¢äð¼É¢ý ÌÇ¢÷îº¢ 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÷ ÀÃ¢§º¡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¡÷.

«¨ÈÂ¢ý ¦ÅôÀ ¿¢¨Ä «Ç× (¿¢¨ÄÂ¡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¢÷îº¢ 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÷îº¢Â¡¸ §¸ò§¾¼÷ ±ýÈ ¸ÕÅ¢Â¢ý ãÄõ ¸ñ¸¡½¢ì¸ôÀÎ¸¢ÈÐ.

𝒕 = 𝟎 ±ýÈ §¿Ãò¾¢ø §¿¡Â¡Ç¢ìÌ 𝟏𝟎 Á¢.¸¢Ã¡õ §Å¾¢ô ¦À¡Õû

¦¸¡Îì¸ôÀÎ¸¢ÈÐ. þÐ −𝟑𝒕𝟏

𝟐 Á¢. ¢̧Ã¡õ/ 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)ÓØ¨ÁÂ¡¸ §Å¾¢ô¦À¡Õû ¦ÅÇ¢§ÂÈ ±ÎòÐì ¦¸¡ûÙõ Ì¨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¡ÕûÓØ¨ÁÂ¡¸¦ÅÇ¢§ÂÈ¢ Å¢ð¼Ð ±Éô ¦À¡Õû.

0 = 10 − 2𝑡3

2 ⇒ 10 = 2𝑡3

2 ⇒ 5 = 𝑡3

2

𝑡3 = 52 = 25 ⇒ 𝑡 = 2.9

ÓØ¨ÁÂ¡¸ §Å¾¢ô¦À¡Õû ¦ÅÇ¢§ÂÈ ±ÎòÐì ¦¸¡ûÙõ Ì¨Èó¾Àðº ¸¡Ä

«Ç× 2 ⋅ 9 Á½¢.

29. ÑñÏÂ¢÷¸Ç¢ý ¦ÀÕì¸ò¾¢ø, À¡ìËÃ¢Â¡Å¢ý ¦ÀÕì¸Å£¾Á¡ÉÐ «¾¢ø

¸¡½ôÀÎõ À¡ìËÃ¢Â¡Å¢ý ±ñ½¢ì¨¸ìÌ Å¢¸¢¾Á¡¸ «¨ÁóÐûÇÐ.

þô¦ÀÕì ¸ò¾¡ø À¡ìËÃ¢Â¡Å¢ý ±ñ½¢ì¨¸ 𝟏 Á½¢ §¿Ãò¾¢ø

ÓõÁ¼í¸¡ ¢̧ÈÐ ±É¢ø ³óÐ Á½¢ §¿Ã ÓÊÅ¢ø À¡ìËÃ¢Â¡Å¢ý

±ñ½¢ì¨¸ ¬ÃõÀ ¿¢¨Ä¨Âì ¸¡ðÊÖõ 𝟑𝟓 Á¼í¸¡Ìõ ±Éì ¸¡ðÎ¸.

J-2006,M-2009

¾£÷×:𝑡 ±Ûõ §¿Ãò¾¢ø À¡ìËÃ¢Â¡Å¢ý ±ñ½¢ì¨¸ 𝐴 ±ý¸.

¸½ì¸¢ý ÀÊ,

𝑑𝐴


𝑑𝐴

𝑑𝑡= 𝑘𝐴


¬ÃõÀ ¿¢¨ÄÂ¢ø ¯ûÇ À¡ìËÃ¢Â¡Å¢ý ±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¡ìËÃ¢Â¡Å¢ý ±ñ½¢ì¨¸ ¬ÃõÀ ¿¢¨Ä¨Âì

¸¡ðÊÖõ 35 Á¼í¸¡Ìõ.

ÌÄí¸û

Å¨ÃÂ¨È:

𝐺 ´Õ ¦ÅüÈüÈ ¸½õ ±ý¸. ∗ ´Õ ®ÕÚôÒî ¦ºÂÄ¢, (𝐺,∗) ¬ÉÐ

ÌÄÁ¡¸Â¢Õì¸ À¢ýÅÕõ ¿¢Àó¾¨É¸û ¯ñ¨ÁÂ¡¸ §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 ¯ÚôÒ¸Ç¡Ìõ.

∴ «¨¼ôÒ Å¢¾¢ ¯ñ¨ÁÂ¡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¯ÚôÒ¸Ç¡Ìõ.

∴ «¨¼ôÒ Å¢¾¢ ¯ñ¨ÁÂ¡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

ÀÃ¢Á¡üÚ Å¢¾¢ «ð¼Å¨½Â¢Ä¢ÕóÐ ÀÃ¢Á¡üÚ Å¢¾¢Ôõ ¯ñ¨Á.

∴ (𝑮,∙𝟏𝟏) ´Õ ±À£Ä¢Âý ÌÄÁ¡Ìõ.

3) âîº¢ÂÁüÈ ¸øô¦Àñ¸Ç¢ý ¸½Á¡É 𝑪 − {𝟎} þø Å¨ÃÂÚì¸ôÀð¼

𝒇𝟏 𝒛 = 𝒛, 𝒇𝟐 𝒛 = −𝒛,𝒇𝟑 𝒛 =𝟏

𝒛, 𝒇𝟒 𝒛 = −

𝟏

𝒛, ∀𝒛 ∈ 𝑪 − {𝟎} ±ýÈ º¡÷Ò¸û

Â¡×õ «¼í¸¢Â ¸½õ {𝒇𝟏,𝒇𝟐, 𝒇𝟑, 𝒇𝟒} ¬ÉÐ º¡÷Ò¸Ç¢ý §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 ¯ÚôÒ¸Ç¡Ìõ. ∴ «¨¼ôÒ Å¢¾¢ ¯ñ¨ÁÂ¡Ìõ.

§º÷ôÒ Å¢¾¢ º¡÷Ò¸Ç¢ý §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

ÀÃ¢Á¡üÚ Å¢¾¢ «ð¼Å¨½Â¢Ä¢ÕóÐ ÀÃ¢Á¡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 ¯ÚôÒ¸Ç¡Ìõ. ∴ «¨¼ôÒ Å¢¾¢ ¯ñ¨ÁÂ¡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¨ÁÂ¡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¨ÁÂ¡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.

ÀÃ¢Á¡üÚ Å¢¾¢:

¦ÀÕì¸ø ±ô¦À¡ØÐõ ÀÃ¢Á¡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 ÁðÎ§Á ¯ûÇ¾¡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


§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 Å¨ÃÂ¨ÈôÀÊ, 𝑎 ∗ 𝑒 = 𝑎.

∗ þý Å¨ÃÂ¨ÈôÀÊ, 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 Å¨ÃÂ¨ÈôÀÊ,

𝑎 ∗ 𝑎−1 = −2.

∗ þý Å¨ÃÂ¨ÈôÀÊ, 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.


𝑎, 𝑏 ∈ ℤ ±ý¸. À¢ýÉ÷

𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 2 = 𝑏 + 𝑎 + 2 = 𝑏 ∗ 𝑎.

±É§Å ÀÃ¢Á¡üÚ â÷ò¾¢Â¡¸¢ÈÐ.

∴ (ℤ,∗) ´Õ ±À£Ä¢Âý ÌÄÁ¡Ìõ.

ℤ þø ÓÊÅüÈ ¯ÚôÒ¸û ¯ûÇ¾¡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


§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¸.


∗ þý Å¨ÃÂ¨ÈôÀÊ,

𝑎 ∗ 𝑒 =𝑎𝑒

3⇒

𝑎𝑒

3= 𝑎 ⇒ 𝑒 = 3 ∈ 𝐺


±¾¢÷Á¨È Å¢¾¢: 𝑎−1 ∈ 𝑍 ±ýÀÐ 𝑎 ∈ 𝑍 þý ±¾¢÷Á¨È ¯ÚôÒ ±ý¸.


𝑎 ∗ 𝑎−1 = 3.


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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 ¿¢ÚÅ¢É¡ø §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


§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¸.



𝑎 ∗ 𝑒 = 𝑎 + 𝑒 + 𝑎𝑒

⇒ 𝑎 + 𝑒 + 𝑎𝑒 = 𝑎

⇒ 𝑒 + 𝑎𝑒 = 𝑎 − 𝑎

⇒ 𝑒 1 + 𝑎 = 0

⇒ 𝑒 =0

1 + 𝑎= 0 ∵ 𝑎 ≠ −1


±¾¢÷Á¨È Å¢¾¢: 𝑎−1 ∈ 𝐺 ±ýÀÐ 𝑎 ∈ 𝐺 þý ±¾¢÷Á¨È ¯ÚôÒ ±ý¸.


𝑎 ∗ 𝑎−1 = 0.


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.


ÀÃ¢Á¡üÚ Å¢¾¢: 𝑎, 𝑏 ∈ 𝐺 ±ý¸. À¢ýÉ÷

𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 𝑎𝑏 = 𝑏 + 𝑎 + 𝑏𝑎 = 𝑏 ∗ 𝑎.


∴ (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 ¿¢ÚÅ¢É¡ø §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


§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¸.



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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±¾¢÷Á¨È Å¢¾¢:

𝑎−1 ∈ 𝐺 ±ýÀÐ 𝑎 ∈ 𝐺 þý ±¾¢÷Á¨È ¯ÚôÒ ±ý¸.


𝑎 ∗ 𝑎−1 = 0.


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.


ÀÃ¢Á¡üÚ Å¢¾¢: 𝑎, 𝑏 ∈ 𝐺 ±ý¸. À¢ýÉ÷

𝑎 ∗ 𝑏 = 𝑎 + 𝑏 − 𝑎𝑏 = 𝑏 + 𝑎 − 𝑏𝑎 = 𝑏 ∗ 𝑎. ±É§Å ÀÃ¢Á¡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


§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 Å¨ÃÂ¨ÈôÀÊ,

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¸.


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 } ±ýÈ¸½õ«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


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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 Å¨ÃÂ¨ÈôÀÊ,

𝐴𝐸 = 𝐴 = 𝑎 00 0

.


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¸.


𝐴𝐵 = 𝐸 = 𝑒 00 0


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 } ±ýÈ¸½õ«½¢ô¦ÀÕì¸Ä¢ý £̧úÌÄõ.

ÀÃ¢Á¡üÚ Å¢¾¢: 𝐴, 𝐵 ∈ 𝐺 ±ý¸. À¢ýÉ÷

𝐴𝐵 = 𝑎𝑏 00 0

= 𝑏𝑎 00 0

= 𝐵𝐴


∴ 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


§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 ±ýÈ¸½õÜð¼Ä¢ý £̧úÌÄõ.

ÀÃ¢Á¡üÚ Å¢¾¢: Üð¼ø, ±ô¦À¡ØÐõ ÀÃ¢Á¡üÚ Å¢¾¢ìÌðÀÎõ.


𝐺 þø ÓÊÅüÈ ¯ÚôÒ¸û ¯ûÇ¾¡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


§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 ±ýÈ¸½õ ¦ÀÕì¸Ä¢ý £̧úÌÄõ.


¦ÀÕì¸ø, ±ô¦À¡ØÐõ ÀÃ¢Á¡üÚ Å¢¾¢ìÌðÀÎõ.


(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


§º÷ôÒ Å¢¾¢:

¸Äô¦Àñ¸Ç¢ý ¦ÀÕì¸ø ±ô¦À¡ØÐõ §º÷ôÒ Å¢¾¢ìÌðÀÎõ.

ºÁÉ¢ Å¢¾¢:


±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 ±ýÈ¸½õ ¸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 Ó¨ÈÂ¢ø ¸¡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 ÀóÐ¸¨Ç ±ÎìÌõ §À¡Ð ¸¢¨¼ìÌõ º¢ÅôÒô ÀóÐ¸Ç¢ý

±ñ½¢ì¨¸ ±ý¸.

∴ 𝑋 = 0,1,2,3

𝑅 ±ýÀÐ º¢ÅôÒ ÀóÐ ±ÎìÌõ ¿¢¸úîº¢ ±É×õ, 𝑊 ±ýÀÐ ¦Åû¨Ç ÀóÐ

±ÎìÌõ ¿¢¸úîº¢ ±É×õ ¦¸¡û¸.

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(i) ¾¢ÕôÀ¢ ¨ÅìÌõ Ó¨ÈÂ¢ø

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¢ ¨Åì¸¡ Ó¨ÈÂ¢ø

´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


−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


∞

𝛽 10 𝛼

𝑑𝑡

=𝑘


−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 ÅÂÐ¨¼Â ´Õ ¯Â÷ó¾ Å¨¸ ¿¡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 ¬ñÎ¸ÙìÌ §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 ¬ñÎ¸ÙìÌ 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 ¬ñÎ¸û Å¨Ã ¯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 §ÀÕóÐ¸Ùì¸¡É

𝝀 = 𝟎 ⋅ 𝟗

∴ 𝟓 ¿¢Á¢¼ò¾¢ø ¯û§Ç ÅÕõ §ÀÕóÐ¸Ùì¸¡É ºÃ¡ºÃ¢

5𝜆 = 5 × 0 ⋅ 9 = 4 ⋅ 5

𝟓 ¿¢Á¢¼ò¾¢ø ºÃ¢Â¡¸ 𝟗 §ÀÕóÐ¸û ¯û§Ç ÅÃ ¿¢¸ú¾¸×

125𝑃 𝑋 = 9 =𝑒5𝜆 × 5𝜆 9

9! =

𝑒4⋅5 × 4 ⋅ 5

9!

9

(ii)

∴ 8 ¿¢Á¢¼ þ¨¼¦ÅÇ¢Â¢ø ¯û§Ç ÅÕõ §ÀÕóÐ¸Ùì¸¡É ºÃ¡ºÃ¢ 8𝜆 = 8 × 0 ⋅

9 = 7 ⋅ 2

8 ¿¢Á¢¼ þ¨¼¦ÅÇ¢Â¢ø 10 ìÌ Ì¨ÈÅ¡¸ §ÀÕóÐ¸û ¯û§Ç ÅÃ ¿¢¸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¦ÅÇ¢Â¢ø ¯û§Ç ÅÕõ §ÀÕóÐ¸Ùì¸¡É ºÃ¡ºÃ¢ 11𝜆 = 11 ×

0 ⋅ 9 = 9 ⋅ 9

111 ¿¢Á¢¼ þ¨¼¦ÅÇ¢Â¢ø Ì¨Èó¾Àðºõ14§ÀÕóÐ¸û¯û§Ç ÅÃ ¿¢¸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¢ÀòÐ¸û

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¢ÀòÐ¸ÙìÌ

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


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 «¨ÁÅ¾¡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¢üÚóÐ¸Ç¢ø ¦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


∞

0

𝑑𝑧

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= 𝜙 𝑧

2⋅50

0


∞

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


0⋅2667

0

𝑑𝑧

= 𝜙 𝑧

2⋅0667

0


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


0⋅4

0

𝑑𝑧

= 𝜙 𝑧

1⋅2

0


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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