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Á¡½Å÷ ¨¸§ÂÎÁ¡½Å÷ ¨¸§ÂÎÁ¡½Å÷ ¨¸§ÂÎÁ¡½Å÷ ¨¸§ÂΠި¼¸û Å¢¨¼¸û Å¢¨¼¸û Å¢¨¼¸û

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1111.... ¸½í¸Ùõ º¡÷¸½í¸Ùõ º¡÷¸½í¸Ùõ º¡÷¸½í¸Ùõ º¡÷Ò¸ÙõÒ¸ÙõÒ¸ÙõÒ¸Ùõ

((((1111)))) A = { l,m,n,o,2,3,4,7 } ÁüÚõ A = { l,m,n,o,2,3,4,7 } ÁüÚõ A = { l,m,n,o,2,3,4,7 } ÁüÚõ A = { l,m,n,o,2,3,4,7 } ÁüÚõ B = { 2,5,3,B = { 2,5,3,B = { 2,5,3,B = { 2,5,3,----2,m,n,o,p } ¬¸¢ÂÅüÈ¢üÌ ¸½í¸Ç¢ý 2,m,n,o,p } ¬¸¢ÂÅüÈ¢üÌ ¸½í¸Ç¢ý 2,m,n,o,p } ¬¸¢ÂÅüÈ¢üÌ ¸½í¸Ç¢ý 2,m,n,o,p } ¬¸¢ÂÅüÈ¢üÌ ¸½í¸Ç¢ý ¦ÅðÎ À¡¢Á¡üÚô ÀñÒ ¯¨¼ÂÐ ±ýÀ¨¾ ¦ÅðÎ À¡¢Á¡üÚô ÀñÒ ¯¨¼ÂÐ ±ýÀ¨¾ ¦ÅðÎ À¡¢Á¡üÚô ÀñÒ ¯¨¼ÂÐ ±ýÀ¨¾ ¦ÅðÎ À¡¢Á¡üÚô ÀñÒ ¯¨¼ÂÐ ±ýÀ¨¾ º¡¢À¡÷ì¸×õ.º¡¢À¡÷ì¸×õ.º¡¢À¡÷ì¸×õ.º¡¢À¡÷ì¸×õ. ¾£÷× : ¾£÷× : ¾£÷× : ¾£÷× : A = { l,m,n,o,2,3,4,7 }A = { l,m,n,o,2,3,4,7 }A = { l,m,n,o,2,3,4,7 }A = { l,m,n,o,2,3,4,7 } B = { 2,5,3,B = { 2,5,3,B = { 2,5,3,B = { 2,5,3,----2,m,n,o,p } 2,m,n,o,p } 2,m,n,o,p } 2,m,n,o,p }

(A (A (A (A ∩∩∩∩ B) = { 2,3,m,n,o } B) = { 2,3,m,n,o } B) = { 2,3,m,n,o } B) = { 2,3,m,n,o }

(B (B (B (B ∩∩∩∩ A) = { A) = { A) = { A) = { 2,3,m,n,o }2,3,m,n,o }2,3,m,n,o }2,3,m,n,o }

(A (A (A (A ∩∩∩∩ B) = (B B) = (B B) = (B B) = (B ∩∩∩∩ A)A)A)A) ±É§Å ¦ÅðÎ À¡¢Á¡üÚôÀñÒ ¯¨¼ÂÐ.±É§Å ¦ÅðÎ À¡¢Á¡üÚôÀñÒ ¯¨¼ÂÐ.±É§Å ¦ÅðÎ À¡¢Á¡üÚôÀñÒ ¯¨¼ÂÐ.±É§Å ¦ÅðÎ À¡¢Á¡üÚôÀñÒ ¯¨¼ÂÐ.

((((2222)))) A = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõA = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõA = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõA = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõ

C = { 1,2,3,4,5,6 } ±É¢ø, AC = { 1,2,3,4,5,6 } ±É¢ø, AC = { 1,2,3,4,5,6 } ±É¢ø, AC = { 1,2,3,4,5,6 } ±É¢ø, A∪∪∪∪(B(B(B(B∩∩∩∩C) ¸¡ñ¸. C) ¸¡ñ¸. C) ¸¡ñ¸. C) ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : A = { 4,6,7,8,9 }A = { 4,6,7,8,9 }A = { 4,6,7,8,9 }A = { 4,6,7,8,9 } B = { 2,4,6 }B = { 2,4,6 }B = { 2,4,6 }B = { 2,4,6 } C = { 1,2,3,4,5,6 }C = { 1,2,3,4,5,6 }C = { 1,2,3,4,5,6 }C = { 1,2,3,4,5,6 }

( B( B( B( B∩∩∩∩C ) = { 2,4,6 }C ) = { 2,4,6 }C ) = { 2,4,6 }C ) = { 2,4,6 }

AAAA∪∪∪∪( B( B( B( B∩∩∩∩C ) = { 2,4,6,7,8,9 }C ) = { 2,4,6,7,8,9 }C ) = { 2,4,6,7,8,9 }C ) = { 2,4,6,7,8,9 }

((((3333)))) A = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõA = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõA = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõA = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõ

C = { 1,2,3,4,5,6 } ±É¢ø, AC = { 1,2,3,4,5,6 } ±É¢ø, AC = { 1,2,3,4,5,6 } ±É¢ø, AC = { 1,2,3,4,5,6 } ±É¢ø, A∩∩∩∩(B(B(B(B∪∪∪∪C) ¸¡ñ¸. C) ¸¡ñ¸. C) ¸¡ñ¸. C) ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : A = { 4,6,7,8,9 }A = { 4,6,7,8,9 }A = { 4,6,7,8,9 }A = { 4,6,7,8,9 } B = { 2,4,6 }B = { 2,4,6 }B = { 2,4,6 }B = { 2,4,6 } C = { 1,2,3,4,5,6 }C = { 1,2,3,4,5,6 }C = { 1,2,3,4,5,6 }C = { 1,2,3,4,5,6 }

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

AAAA∩∩∩∩( B( B( B( B∪∪∪∪C ) = { 4,6 }C ) = { 4,6 }C ) = { 4,6 }C ) = { 4,6 } ((((4444)))) A = { 10,15,20,25,30,35,40,45,50 }, A = { 10,15,20,25,30,35,40,45,50 }, A = { 10,15,20,25,30,35,40,45,50 }, A = { 10,15,20,25,30,35,40,45,50 }, B={ 1,5,10,15,20,30 } ÁüÚõ C={ 7,8,15,20,35,45,48}B={ 1,5,10,15,20,30 } ÁüÚõ C={ 7,8,15,20,35,45,48}B={ 1,5,10,15,20,30 } ÁüÚõ C={ 7,8,15,20,35,45,48}B={ 1,5,10,15,20,30 } ÁüÚõ C={ 7,8,15,20,35,45,48}

±É¢ø, A ±É¢ø, A ±É¢ø, A ±É¢ø, A \\\\ ( B( B( B( B∩∩∩∩C ) ¸¡ñ¸.C ) ¸¡ñ¸.C ) ¸¡ñ¸.C ) ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : A = { 10,15,20,25,30,35,40,45,50 }A = { 10,15,20,25,30,35,40,45,50 }A = { 10,15,20,25,30,35,40,45,50 }A = { 10,15,20,25,30,35,40,45,50 } B = { 1,5,10,15,20,30 }B = { 1,5,10,15,20,30 }B = { 1,5,10,15,20,30 }B = { 1,5,10,15,20,30 } C = { 7,8,15,20,35,45,48 } C = { 7,8,15,20,35,45,48 } C = { 7,8,15,20,35,45,48 } C = { 7,8,15,20,35,45,48 }

( B( B( B( B∩∩∩∩C ) = { 15,20 }C ) = { 15,20 }C ) = { 15,20 }C ) = { 15,20 }

A A A A \\\\ ( B( B( B( B∩∩∩∩C ) = { 10,25,30,35,40,45,50 } C ) = { 10,25,30,35,40,45,50 } C ) = { 10,25,30,35,40,45,50 } C ) = { 10,25,30,35,40,45,50 }

((((5555)))) P = { a,b,c } , Q = { g,h,x,y } ÁüÚõ R = { P = { a,b,c } , Q = { g,h,x,y } ÁüÚõ R = { P = { a,b,c } , Q = { g,h,x,y } ÁüÚõ R = { P = { a,b,c } , Q = { g,h,x,y } ÁüÚõ R = {

±É¢ø, R ±É¢ø, R ±É¢ø, R ±É¢ø, R \\\\ ( P ( P ( P ( P ∩∩∩∩ Q ) ¸¡ñ¸.Q ) ¸¡ñ¸.Q ) ¸¡ñ¸.Q ) ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : P = { a,b,c }P = { a,b,c }P = { a,b,c }P = { a,b,c } Q = { g,h,x,y }Q = { g,h,x,y }Q = { g,h,x,y }Q = { g,h,x,y } R = { a,e,f,s }R = { a,e,f,s }R = { a,e,f,s }R = { a,e,f,s }

( P ( P ( P ( P ∩∩∩∩ Q ) = { Q ) = { Q ) = { Q ) = { ∅∅∅∅ } } } }

R R R R \\\\ ( P ( P ( P ( P ∩∩∩∩ Q ) = { a,e,f,s } Q ) = { a,e,f,s } Q ) = { a,e,f,s } Q ) = { a,e,f,s }

2

Ò¸ÙõÒ¸ÙõÒ¸ÙõÒ¸Ùõ

2,m,n,o,p } ¬¸¢ÂÅüÈ¢üÌ ¸½í¸Ç¢ý 2,m,n,o,p } ¬¸¢ÂÅüÈ¢üÌ ¸½í¸Ç¢ý 2,m,n,o,p } ¬¸¢ÂÅüÈ¢üÌ ¸½í¸Ç¢ý 2,m,n,o,p } ¬¸¢ÂÅüÈ¢üÌ ¸½í¸Ç¢ý ¦ÅðÎ À¡¢Á¡üÚô ÀñÒ ¯¨¼ÂÐ ±ýÀ¨¾ ¦ÅðÎ À¡¢Á¡üÚô ÀñÒ ¯¨¼ÂÐ ±ýÀ¨¾ ¦ÅðÎ À¡¢Á¡üÚô ÀñÒ ¯¨¼ÂÐ ±ýÀ¨¾ ¦ÅðÎ À¡¢Á¡üÚô ÀñÒ ¯¨¼ÂÐ ±ýÀ¨¾

±É§Å ¦ÅðÎ À¡¢Á¡üÚôÀñÒ ¯¨¼ÂÐ.±É§Å ¦ÅðÎ À¡¢Á¡üÚôÀñÒ ¯¨¼ÂÐ.±É§Å ¦ÅðÎ À¡¢Á¡üÚôÀñÒ ¯¨¼ÂÐ.±É§Å ¦ÅðÎ À¡¢Á¡üÚôÀñÒ ¯¨¼ÂÐ.

A = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõA = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõA = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõA = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõ

C) ¸¡ñ¸. C) ¸¡ñ¸. C) ¸¡ñ¸. C) ¸¡ñ¸.

A = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõA = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõA = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõA = { 4,6,7,8,9 }, B = { 2.4.6 } ÁüÚõ

C) ¸¡ñ¸. C) ¸¡ñ¸. C) ¸¡ñ¸. C) ¸¡ñ¸.

A = { 10,15,20,25,30,35,40,45,50 }, A = { 10,15,20,25,30,35,40,45,50 }, A = { 10,15,20,25,30,35,40,45,50 }, A = { 10,15,20,25,30,35,40,45,50 },

B={ 1,5,10,15,20,30 } ÁüÚõ C={ 7,8,15,20,35,45,48}B={ 1,5,10,15,20,30 } ÁüÚõ C={ 7,8,15,20,35,45,48}B={ 1,5,10,15,20,30 } ÁüÚõ C={ 7,8,15,20,35,45,48}B={ 1,5,10,15,20,30 } ÁüÚõ C={ 7,8,15,20,35,45,48}

A = { 10,15,20,25,30,35,40,45,50 }A = { 10,15,20,25,30,35,40,45,50 }A = { 10,15,20,25,30,35,40,45,50 }A = { 10,15,20,25,30,35,40,45,50 }

C = { 7,8,15,20,35,45,48 } C = { 7,8,15,20,35,45,48 } C = { 7,8,15,20,35,45,48 } C = { 7,8,15,20,35,45,48 }

C ) = { 10,25,30,35,40,45,50 } C ) = { 10,25,30,35,40,45,50 } C ) = { 10,25,30,35,40,45,50 } C ) = { 10,25,30,35,40,45,50 }

P = { a,b,c } , Q = { g,h,x,y } ÁüÚõ R = { P = { a,b,c } , Q = { g,h,x,y } ÁüÚõ R = { P = { a,b,c } , Q = { g,h,x,y } ÁüÚõ R = { P = { a,b,c } , Q = { g,h,x,y } ÁüÚõ R = { a,e,f,s } a,e,f,s } a,e,f,s } a,e,f,s }

((((6666)))) U = { 4,8,12,16,20,24,28 } , A = { 8,16,24 } ÁüÚõU = { 4,8,12,16,20,24,28 } , A = { 8,16,24 } ÁüÚõU = { 4,8,12,16,20,24,28 } , A = { 8,16,24 } ÁüÚõU = { 4,8,12,16,20,24,28 } , A = { 8,16,24 } ÁüÚõ

B = { 4,16,20,28 } ±É¢ø, ( AB = { 4,16,20,28 } ±É¢ø, ( AB = { 4,16,20,28 } ±É¢ø, ( AB = { 4,16,20,28 } ±É¢ø, ( A¾£÷× : ¾£÷× : ¾£÷× : ¾£÷× : U = { 4,8,12,16,20,24,28 } U = { 4,8,12,16,20,24,28 } U = { 4,8,12,16,20,24,28 } U = { 4,8,12,16,20,24,28 } A = { 8,16,24 } A = { 8,16,24 } A = { 8,16,24 } A = { 8,16,24 } B = { 4,16,20,28 } B = { 4,16,20,28 } B = { 4,16,20,28 } B = { 4,16,20,28 }

( A( A( A( A∪∪∪∪B ) = { 4,8,16,20,24,28 }B ) = { 4,8,16,20,24,28 }B ) = { 4,8,16,20,24,28 }B ) = { 4,8,16,20,24,28 }

((((7777)))) ¦ÅýÀ¼õ Ũø. (1) ( A¦ÅýÀ¼õ Ũø. (1) ( A¦ÅýÀ¼õ Ũø. (1) ( A¦ÅýÀ¼õ Ũø. (1) ( A¾£÷× :¾£÷× :¾£÷× :¾£÷× :

((((1111)))) ( A( A( A( A∪∪∪∪B ) ( AB ) ( AB ) ( AB ) ( A

((((2222)))) ( B( B( B( B∪∪∪∪C ) ( BC ) ( BC ) ( BC ) ( B

((((8888)))) n(U) = 700, n(A) = 200, n(B) = 300 ÁüÚõ n(U) = 700, n(A) = 200, n(B) = 300 ÁüÚõ n(U) = 700, n(A) = 200, n(B) = 300 ÁüÚõ n(U) = 700, n(A) = 200, n(B) = 300 ÁüÚõ

n(An(An(An(A∩∩∩∩B) = 100 ±É¢ø, n(AB) = 100 ±É¢ø, n(AB) = 100 ±É¢ø, n(AB) = 100 ±É¢ø, n(A¾£÷× :¾£÷× :¾£÷× :¾£÷× : n(U) = 700n(U) = 700n(U) = 700n(U) = 700 n(A) = 200n(A) = 200n(A) = 200n(A) = 200 n(B) = 300 n(B) = 300 n(B) = 300 n(B) = 300

n(An(An(An(A∩∩∩∩B) = 100 B) = 100 B) = 100 B) = 100

n(An(An(An(A∪∪∪∪B) = n(A) + n(B) B) = n(A) + n(B) B) = n(A) + n(B) B) = n(A) + n(B)

n(An(An(An(A∪∪∪∪B) = 200 + 300 B) = 200 + 300 B) = 200 + 300 B) = 200 + 300

n(An(An(An(A∪∪∪∪B) = 400 B) = 400 B) = 400 B) = 400

n(An(An(An(A∪∪∪∪B) B) B) B) IIII = n(U) = n(U) = n(U) = n(U) = 700 = 700 = 700 = 700 ----

n(An(An(An(A∪∪∪∪B) B) B) B) I I I I = 300 = 300 = 300 = 300

n(An(An(An(AIIII ∩∩∩∩ BBBBIIII ) = 300) = 300) = 300) = 300

((((9999)))) n(A) = 17, n(B) = 17, n(C) = 17, n(An(A) = 17, n(B) = 17, n(C) = 17, n(An(A) = 17, n(B) = 17, n(C) = 17, n(An(A) = 17, n(B) = 17, n(C) = 17, n(A

n(Bn(Bn(Bn(B∩∩∩∩C) = 6, n(AC) = 6, n(AC) = 6, n(AC) = 6, n(A∩∩∩∩C) = 5 ÁüÚõ n(AC) = 5 ÁüÚõ n(AC) = 5 ÁüÚõ n(AC) = 5 ÁüÚõ n(A

±É¢ø,±É¢ø,±É¢ø,±É¢ø, n(An(An(An(A ∪∪∪∪ BBBB ∪∪∪∪ C) ³ì ¸¡ñ¸.C) ³ì ¸¡ñ¸.C) ³ì ¸¡ñ¸.C) ³ì ¸¡ñ¸.¾£÷× :¾£÷× :¾£÷× :¾£÷× :

n(An(An(An(A∪∪∪∪BBBB∪∪∪∪C) = n(A) + n(B) + n(C) C) = n(A) + n(B) + n(C) C) = n(A) + n(B) + n(C) C) = n(A) + n(B) + n(C)

= 17 + 17 + 17 = 17 + 17 + 17 = 17 + 17 + 17 = 17 + 17 + 17

n(An(An(An(A∪∪∪∪BBBB∪∪∪∪C) = 35 C) = 35 C) = 35 C) = 35

( ( ( ( ((((AAAA∪∪∪∪B) B) B) B) IIII = { 12 }= { 12 }= { 12 }= { 12 }

U = { 4,8,12,16,20,24,28 } , A = { 8,16,24 } ÁüÚõU = { 4,8,12,16,20,24,28 } , A = { 8,16,24 } ÁüÚõU = { 4,8,12,16,20,24,28 } , A = { 8,16,24 } ÁüÚõU = { 4,8,12,16,20,24,28 } , A = { 8,16,24 } ÁüÚõ

B = { 4,16,20,28 } ±É¢ø, ( AB = { 4,16,20,28 } ±É¢ø, ( AB = { 4,16,20,28 } ±É¢ø, ( AB = { 4,16,20,28 } ±É¢ø, ( A∪∪∪∪B ) B ) B ) B ) IIII ¸¡ñ¸.¸¡ñ¸.¸¡ñ¸.¸¡ñ¸.

U = { 4,8,12,16,20,24,28 } U = { 4,8,12,16,20,24,28 } U = { 4,8,12,16,20,24,28 } U = { 4,8,12,16,20,24,28 }

B = { 4,16,20,28 } B = { 4,16,20,28 } B = { 4,16,20,28 } B = { 4,16,20,28 }

B ) = { 4,8,16,20,24,28 }B ) = { 4,8,16,20,24,28 }B ) = { 4,8,16,20,24,28 }B ) = { 4,8,16,20,24,28 }

¦ÅýÀ¼õ Ũø. (1) ( A¦ÅýÀ¼õ Ũø. (1) ( A¦ÅýÀ¼õ Ũø. (1) ( A¦ÅýÀ¼õ Ũø. (1) ( A∪∪∪∪B ) B ) B ) B ) IIII (2) ( B(2) ( B(2) ( B(2) ( B∪∪∪∪C ) C ) C ) C ) \\\\ AAAA

B ) ( AB ) ( AB ) ( AB ) ( A∪∪∪∪B ) B ) B ) B ) IIII

C ) ( BC ) ( BC ) ( BC ) ( B∪∪∪∪C ) C ) C ) C ) \\\\ AAAA

n(U) = 700, n(A) = 200, n(B) = 300 ÁüÚõ n(U) = 700, n(A) = 200, n(B) = 300 ÁüÚõ n(U) = 700, n(A) = 200, n(B) = 300 ÁüÚõ n(U) = 700, n(A) = 200, n(B) = 300 ÁüÚõ

B) = 100 ±É¢ø, n(AB) = 100 ±É¢ø, n(AB) = 100 ±É¢ø, n(AB) = 100 ±É¢ø, n(Aiiii ∩∩∩∩ BBBBiiii ) ³ì ¸¡ñ¸.) ³ì ¸¡ñ¸.) ³ì ¸¡ñ¸.) ³ì ¸¡ñ¸.

B) = 100 B) = 100 B) = 100 B) = 100

B) = n(A) + n(B) B) = n(A) + n(B) B) = n(A) + n(B) B) = n(A) + n(B) ---- n(An(An(An(A∩∩∩∩B)B)B)B)

B) = 200 + 300 B) = 200 + 300 B) = 200 + 300 B) = 200 + 300 ---- 100100100100

B) = 400 B) = 400 B) = 400 B) = 400

= n(U) = n(U) = n(U) = n(U) ---- n(An(An(An(A∪∪∪∪B)B)B)B) ---- 400 400 400 400

= 300 = 300 = 300 = 300

n(A) = 17, n(B) = 17, n(C) = 17, n(An(A) = 17, n(B) = 17, n(C) = 17, n(An(A) = 17, n(B) = 17, n(C) = 17, n(An(A) = 17, n(B) = 17, n(C) = 17, n(A∩∩∩∩B) = 7,B) = 7,B) = 7,B) = 7,

C) = 5 ÁüÚõ n(AC) = 5 ÁüÚõ n(AC) = 5 ÁüÚõ n(AC) = 5 ÁüÚõ n(A∩∩∩∩ BBBB∩∩∩∩ C) = 2 C) = 2 C) = 2 C) = 2

C) ³ì ¸¡ñ¸.C) ³ì ¸¡ñ¸.C) ³ì ¸¡ñ¸.C) ³ì ¸¡ñ¸.

C) = n(A) + n(B) + n(C) C) = n(A) + n(B) + n(C) C) = n(A) + n(B) + n(C) C) = n(A) + n(B) + n(C) ---- n(An(An(An(A∩∩∩∩B) B) B) B) ---- n(Bn(Bn(Bn(B∩∩∩∩C)C)C)C)

---- n(An(An(An(A∩∩∩∩C) + n(AC) + n(AC) + n(AC) + n(A∩∩∩∩BBBB∩∩∩∩C)C)C)C) = 17 + 17 + 17 = 17 + 17 + 17 = 17 + 17 + 17 = 17 + 17 + 17 ---- 7 7 7 7 ---- 6 6 6 6 ---- 5 + 5 + 5 + 5 + 2222

C) = 35 C) = 35 C) = 35 C) = 35

Ê Á¡÷¸ý Å¢¾¢ôÀÊÊ Á¡÷¸ý Å¢¾¢ôÀÊÊ Á¡÷¸ý Å¢¾¢ôÀÊÊ Á¡÷¸ý Å¢¾¢ôÀÊ

(A(A(A(A∪∪∪∪B) B) B) B) IIII = A= A= A= AIIII ∩∩∩∩ BBBBIIII

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((((10101010)))) f = { (12,2), (13,3), (15,3), (14,2), f = { (12,2), (13,3), (15,3), (14,2), f = { (12,2), (13,3), (15,3), (14,2), f = { (12,2), (13,3), (15,3), (14,2), (17,17) }(17,17) }(17,17) }(17,17) } ±ýÈ º¡÷À¢ø 2 ÁüÚõ 3 ¬¸¢ÂÅüÈ¢ý Óý ±ýÈ º¡÷À¢ø 2 ÁüÚõ 3 ¬¸¢ÂÅüÈ¢ý Óý ±ýÈ º¡÷À¢ø 2 ÁüÚõ 3 ¬¸¢ÂÅüÈ¢ý Óý ±ýÈ º¡÷À¢ø 2 ÁüÚõ 3 ¬¸¢ÂÅüÈ¢ý Óý ¯Õ츨Çì ¸¡ñ¸.¯Õ츨Çì ¸¡ñ¸.¯Õ츨Çì ¸¡ñ¸.¯Õ츨Çì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : 2 ý Óý ¯Õì¸û = 12 , 142 ý Óý ¯Õì¸û = 12 , 142 ý Óý ¯Õì¸û = 12 , 142 ý Óý ¯Õì¸û = 12 , 14 3 ý Óý ¯Õì¸û = 13 , 15. 3 ý Óý ¯Õì¸û = 13 , 15. 3 ý Óý ¯Õì¸û = 13 , 15. 3 ý Óý ¯Õì¸û = 13 , 15.

((((11111111)))) ¦¸¡Îì¸ôÀðÎûÇ f = { (1,3), (2,5), (4,7), ¦¸¡Îì¸ôÀðÎûÇ f = { (1,3), (2,5), (4,7), ¦¸¡Îì¸ôÀðÎûÇ f = { (1,3), (2,5), (4,7), ¦¸¡Îì¸ôÀðÎûÇ f = { (1,3), (2,5), (4,7), (5,9), (3,1) } ±Ûõ º¡÷ÒìÌ, Á¾¢ôÀ¸õ ÁüÚõ (5,9), (3,1) } ±Ûõ º¡÷ÒìÌ, Á¾¢ôÀ¸õ ÁüÚõ (5,9), (3,1) } ±Ûõ º¡÷ÒìÌ, Á¾¢ôÀ¸õ ÁüÚõ (5,9), (3,1) } ±Ûõ º¡÷ÒìÌ, Á¾¢ôÀ¸õ ÁüÚõ ţõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸.ţõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸.ţõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸.ţõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : f f f f = { (1,3), (2,5), (4,7), (5,9), (3,1) }= { (1,3), (2,5), (4,7), (5,9), (3,1) }= { (1,3), (2,5), (4,7), (5,9), (3,1) }= { (1,3), (2,5), (4,7), (5,9), (3,1) } Á¾¢ôÀ¸õ Á¾¢ôÀ¸õ Á¾¢ôÀ¸õ Á¾¢ôÀ¸õ = { 1,2,3,4,5 }= { 1,2,3,4,5 }= { 1,2,3,4,5 }= { 1,2,3,4,5 } ţõ = ţõ = ţõ = ţõ = { 1,3,5,7,9 }{ 1,3,5,7,9 }{ 1,3,5,7,9 }{ 1,3,5,7,9 }

((((12121212)))) A = { A = { A = { A = { 1,2,3,4,5 } , B = N ÁüÚõ 1,2,3,4,5 } , B = N ÁüÚõ 1,2,3,4,5 } , B = N ÁüÚõ 1,2,3,4,5 } , B = N ÁüÚõ ffff : A : A : A : A →→→→ B B B B ¬ÉÐ f¬ÉÐ f¬ÉÐ f¬ÉÐ f (x) = x (x) = x (x) = x (x) = x 2222 ±É ŨÃÂÚì¸ôÀðÎûÇÐ. f ±É ŨÃÂÚì¸ôÀðÎûÇÐ. f ±É ŨÃÂÚì¸ôÀðÎûÇÐ. f ±É ŨÃÂÚì¸ôÀðÎûÇÐ. f ----ýýýý

ţò¨¾ì ¸¡ñ¸. §ÁÖõ º¡÷À¢ý Ũ¸¨Âì ¸¡ñ¸.ţò¨¾ì ¸¡ñ¸. §ÁÖõ º¡÷À¢ý Ũ¸¨Âì ¸¡ñ¸.ţò¨¾ì ¸¡ñ¸. §ÁÖõ º¡÷À¢ý Ũ¸¨Âì ¸¡ñ¸.ţò¨¾ì ¸¡ñ¸. §ÁÖõ º¡÷À¢ý Ũ¸¨Âì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : A = { 1,2,3,4,5 }A = { 1,2,3,4,5 }A = { 1,2,3,4,5 }A = { 1,2,3,4,5 } f (x) = x f (x) = x f (x) = x f (x) = x 2222 f (1) = 1 f (1) = 1 f (1) = 1 f (1) = 1 2222 = 1= 1= 1= 1 f (2) = 2 f (2) = 2 f (2) = 2 f (2) = 2 2222 = 4= 4= 4= 4 f (3) = 3 f (3) = 3 f (3) = 3 f (3) = 3 2222 = 9 = 9 = 9 = 9

f (4) = 4 f (4) = 4 f (4) = 4 f (4) = 4 2222 = 16= 16= 16= 16 f (5) = 5 f (5) = 5 f (5) = 5 f (5) = 5 2222 = 25= 25= 25= 25

(i) (i) (i) (i) ţõ = { 1,4,9,16,25 }ţõ = { 1,4,9,16,25 }ţõ = { 1,4,9,16,25 }ţõ = { 1,4,9,16,25 } (ii)(ii)(ii)(ii) º¡÷À¢ý Ũ¸ = ´ýÚìÌ ´ýÈ¡É º¡÷Òº¡÷À¢ý Ũ¸ = ´ýÚìÌ ´ýÈ¡É º¡÷Òº¡÷À¢ý Ũ¸ = ´ýÚìÌ ´ýÈ¡É º¡÷Òº¡÷À¢ý Ũ¸ = ´ýÚìÌ ´ýÈ¡É º¡÷Ò

((((13131313)))) A = { A = { A = { A = { ----2,2,2,2,----1,1,2 } , ÁüÚõ f 1,1,2 } , ÁüÚõ f 1,1,2 } , ÁüÚõ f 1,1,2 } , ÁüÚõ f = { = { = { = { ( x, 1 ( x, 1 ( x, 1 ( x, 1 \\\\ x), xx), xx), xx), x∈∈∈∈ A } A } A } A } ±É¢ø, f ý ţò¨¾ì ¸¡ñ¸. §ÁÖõ f ±ýÀÐ ±É¢ø, f ý ţò¨¾ì ¸¡ñ¸. §ÁÖõ f ±ýÀÐ ±É¢ø, f ý ţò¨¾ì ¸¡ñ¸. §ÁÖõ f ±ýÀÐ ±É¢ø, f ý ţò¨¾ì ¸¡ñ¸. §ÁÖõ f ±ýÀÐ A ¢ĢÕóÐ A ìÌ ´Õ º¡÷À¡ÌÁ¡ ?A ¢ĢÕóÐ A ìÌ ´Õ º¡÷À¡ÌÁ¡ ?A ¢ĢÕóÐ A ìÌ ´Õ º¡÷À¡ÌÁ¡ ?A ¢ĢÕóÐ A ìÌ ´Õ º¡÷À¡ÌÁ¡ ?

¾£÷× :¾£÷× :¾£÷× :¾£÷× : A = { A = { A = { A = { ----2,2,2,2,----1,1,2 }1,1,2 }1,1,2 }1,1,2 } f = { ( x, 1 f = { ( x, 1 f = { ( x, 1 f = { ( x, 1 //// x)x)x)x) f = { ( f = { ( f = { ( f = { ( ----2, 2, 2, 2, ----1111//// 2 ), ( 2 ), ( 2 ), ( 2 ), ( ----1,1,1,1,----1 ), ( 1,1 ), ( 2, 1 /1 ), ( 1,1 ), ( 2, 1 /1 ), ( 1,1 ), ( 2, 1 /1 ), ( 1,1 ), ( 2, 1 / 2 ) }2 ) }2 ) }2 ) } ţõ = { ţõ = { ţõ = { ţõ = { ----1 /1 /1 /1 / 2,2,2,2,----1,1,1 /1,1,1 /1,1,1 /1,1,1 / 2 }2 }2 }2 } F ±ýÀÐ A F ±ýÀÐ A F ±ýÀÐ A F ±ýÀÐ A ¢ĢÕóÐ A ìÌ ´Õ º¡÷À¡Â¢Ä¢ÕóÐ A ìÌ ´Õ º¡÷À¡Â¢Ä¢ÕóÐ A ìÌ ´Õ º¡÷À¡Â¢Ä¢ÕóÐ A ìÌ ´Õ º¡÷À¡¸¡Ð. ¸¡Ð. ¸¡Ð. ¸¡Ð. ²¦ÉÉ¢ø A ¢ø ¯ûÇ ²¦ÉÉ¢ø A ¢ø ¯ûÇ ²¦ÉÉ¢ø A ¢ø ¯ûÇ ²¦ÉÉ¢ø A ¢ø ¯ûÇ ----2 ,2 ¬¸¢Â Óý 2 ,2 ¬¸¢Â Óý 2 ,2 ¬¸¢Â Óý 2 ,2 ¬¸¢Â Óý ¯Õì¸ÙìÌ ¿¢Æø ¯Õì¸û þø¨Ä.¯Õì¸ÙìÌ ¿¢Æø ¯Õì¸û þø¨Ä.¯Õì¸ÙìÌ ¿¢Æø ¯Õì¸û þø¨Ä.¯Õì¸ÙìÌ ¿¢Æø ¯Õì¸û þø¨Ä. ((((14141414)))) X = { 1,X = { 1,X = { 1,X = { 1,2,3,4 } ±ý¸. g = { (3,1), (4,2), (2,1) } 2,3,4 } ±ý¸. g = { (3,1), (4,2), (2,1) } 2,3,4 } ±ý¸. g = { (3,1), (4,2), (2,1) } 2,3,4 } ±ý¸. g = { (3,1), (4,2), (2,1) }

±ýÈ ¯È× x Ä¢ÕóÐ x ìÌ ´Õ º¡÷À¡ÌÁ¡ ? ±ýÈ ¯È× x Ä¢ÕóÐ x ìÌ ´Õ º¡÷À¡ÌÁ¡ ? ±ýÈ ¯È× x Ä¢ÕóÐ x ìÌ ´Õ º¡÷À¡ÌÁ¡ ? ±ýÈ ¯È× x Ä¢ÕóÐ x ìÌ ´Õ º¡÷À¡ÌÁ¡ ? ¯ý Å¢¨¼ìÌ ²üÈ Å¢Çì¸õ ¾Õ¸.¯ý Å¢¨¼ìÌ ²üÈ Å¢Çì¸õ ¾Õ¸.¯ý Å¢¨¼ìÌ ²üÈ Å¢Çì¸õ ¾Õ¸.¯ý Å¢¨¼ìÌ ²üÈ Å¢Çì¸õ ¾Õ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : g = { (3,1), (4,2), (2,1) } ±Éì g = { (3,1), (4,2), (2,1) } ±Éì g = { (3,1), (4,2), (2,1) } ±Éì g = { (3,1), (4,2), (2,1) } ±Éì ¦¸¡Îì¸ôÀðÎûÇÐ. 1 ±Ûõ x ¦¸¡Îì¸ôÀðÎûÇÐ. 1 ±Ûõ x ¦¸¡Îì¸ôÀðÎûÇÐ. 1 ±Ûõ x ¦¸¡Îì¸ôÀðÎûÇÐ. 1 ±Ûõ x ----ý ¯ÚôÀ¢üÌ ¿¢Æø ý ¯ÚôÀ¢üÌ ¿¢Æø ý ¯ÚôÀ¢üÌ ¿¢Æø ý ¯ÚôÀ¢üÌ ¿¢Æø ¯Õ x ¯Õ x ¯Õ x ¯Õ x ----ø þø¨Ä. «¾¡ÅÐ, g ø þø¨Ä. «¾¡ÅÐ, g ø þø¨Ä. «¾¡ÅÐ, g ø þø¨Ä. «¾¡ÅÐ, g ----ý Á¾¢ôÀ¸õ ý Á¾¢ôÀ¸õ ý Á¾¢ôÀ¸õ ý Á¾¢ôÀ¸õ

{ 2,3,4 } { 2,3,4 } { 2,3,4 } { 2,3,4 } ≠≠≠≠ x. ±É§Å g ±ýÀÐ ´Õ º¡÷ÀøÄ. x. ±É§Å g ±ýÀÐ ´Õ º¡÷ÀøÄ. x. ±É§Å g ±ýÀÐ ´Õ º¡÷ÀøÄ. x. ±É§Å g ±ýÀÐ ´Õ º¡÷ÀøÄ. ((((15151515)))) R = { (a,R = { (a,R = { (a,R = { (a,----2), (2), (2), (2), (----5,b), (8,c), (d,5,b), (8,c), (d,5,b), (8,c), (d,5,b), (8,c), (d,----1) } ±ýÀÐ 1) } ±ýÀÐ 1) } ±ýÀÐ 1) } ±ýÀÐ

ºÁÉ¢î º¡÷¨Àì ÌÈ¢ì̦ÁÉ¢ø, a,b,c ÁüÚõ d ºÁÉ¢î º¡÷¨Àì ÌÈ¢ì̦ÁÉ¢ø, a,b,c ÁüÚõ d ºÁÉ¢î º¡÷¨Àì ÌÈ¢ì̦ÁÉ¢ø, a,b,c ÁüÚõ d ºÁÉ¢î º¡÷¨Àì ÌÈ¢ì̦ÁÉ¢ø, a,b,c ÁüÚõ d ¬¸¢ÂÅüÈ¢ý ¬¸¢ÂÅüÈ¢ý ¬¸¢ÂÅüÈ¢ý ¬¸¢ÂÅüÈ¢ý Á¾¢ôÒ¸¨Çì ¸¡ñ¸.Á¾¢ôÒ¸¨Çì ¸¡ñ¸.Á¾¢ôÒ¸¨Çì ¸¡ñ¸.Á¾¢ôÒ¸¨Çì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : R ±ýÀÐ ´Õ ºÁÉ¢î º¡÷¨Àì ÌÈ¢ì̦ÁÉ¢ø, R ±ýÀÐ ´Õ ºÁÉ¢î º¡÷¨Àì ÌÈ¢ì̦ÁÉ¢ø, R ±ýÀÐ ´Õ ºÁÉ¢î º¡÷¨Àì ÌÈ¢ì̦ÁÉ¢ø, R ±ýÀÐ ´Õ ºÁÉ¢î º¡÷¨Àì ÌÈ¢ì̦ÁÉ¢ø, ´ù¦Å¡Õ ¯ÚôÀ¢ý ¿¢Æø ¯Õ «§¾ ¯ÕôÀ¡Ìõ. ´ù¦Å¡Õ ¯ÚôÀ¢ý ¿¢Æø ¯Õ «§¾ ¯ÕôÀ¡Ìõ. ´ù¦Å¡Õ ¯ÚôÀ¢ý ¿¢Æø ¯Õ «§¾ ¯ÕôÀ¡Ìõ. ´ù¦Å¡Õ ¯ÚôÀ¢ý ¿¢Æø ¯Õ «§¾ ¯ÕôÀ¡Ìõ. ±É§Å, a = ±É§Å, a = ±É§Å, a = ±É§Å, a = ----2, b = 2, b = 2, b = 2, b = ----5, c = 8, d = 5, c = 8, d = 5, c = 8, d = 5, c = 8, d = ----1 ¬Ìõ. 1 ¬Ìõ. 1 ¬Ìõ. 1 ¬Ìõ.

((((16161616)))) A = { 1,2,3,4 }, ÁüÚõ B = { A = { 1,2,3,4 }, ÁüÚõ B = { A = { 1,2,3,4 }, ÁüÚõ B = { A = { 1,2,3,4 }, ÁüÚõ B = { ----1,2,3,4,5,6,7,9,10,1,2,3,4,5,6,7,9,10,1,2,3,4,5,6,7,9,10,1,2,3,4,5,6,7,9,10, 11,12 } 11,12 } 11,12 } 11,12 } ±ý¸. R = { (1,3), (2,6), (3,10), (4,9) } ±ý¸. R = { (1,3), (2,6), (3,10), (4,9) } ±ý¸. R = { (1,3), (2,6), (3,10), (4,9) } ±ý¸. R = { (1,3), (2,6), (3,10), (4,9) } A X B ´Õ ¯È× ±É¢ø, R ³ ´Õ º¡÷Ò ±Éì A X B ´Õ ¯È× ±É¢ø, R ³ ´Õ º¡÷Ò ±Éì A X B ´Õ ¯È× ±É¢ø, R ³ ´Õ º¡÷Ò ±Éì A X B ´Õ ¯È× ±É¢ø, R ³ ´Õ º¡÷Ò ±Éì ¸¡ðθ. ÁüÚõ «¾ý ţò¨¾ì ¸¡ñ¸.¸¡ðθ. ÁüÚõ «¾ý ţò¨¾ì ¸¡ñ¸.¸¡ðθ. ÁüÚõ «¾ý ţò¨¾ì ¸¡ñ¸.¸¡ðθ. ÁüÚõ «¾ý ţò¨¾ì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : R ±ýÈ ¯ÈÅ¢ø Á¾¢ôÀ¸ò¾¢ø ¯ûÇ ´ù¦Å¡Õ R ±ýÈ ¯ÈÅ¢ø Á¾¢ôÀ¸ò¾¢ø ¯ûÇ ´ù¦Å¡Õ R ±ýÈ ¯ÈÅ¢ø Á¾¢ôÀ¸ò¾¢ø ¯ûÇ ´ù¦Å¡Õ R ±ýÈ ¯ÈÅ¢ø Á¾¢ôÀ¸ò¾¢ø ¯ûÇ ´ù¦Å¡Õ Óý ¯Õ×õ ¦Åù§ÅÚ ¿¢Æø ¯Õ츨Çì Óý ¯Õ×õ ¦Åù§ÅÚ ¿¢Æø ¯Õ츨Çì Óý ¯Õ×õ ¦Åù§ÅÚ ¿¢Æø ¯Õ츨Çì Óý ¯Õ×õ ¦Åù§ÅÚ ¿¢Æø ¯Õ츨Çì ¦¸¡ñÎûÇÉ. ¦¸¡ñÎûÇÉ. ¦¸¡ñÎûÇÉ. ¦¸¡ñÎûÇÉ. ±ý§Å R ±ýÀÐ ´ýÚìÌ ´ýÈ¡É ±ý§Å R ±ýÀÐ ´ýÚìÌ ´ýÈ¡É ±ý§Å R ±ýÀÐ ´ýÚìÌ ´ýÈ¡É ±ý§Å R ±ýÀÐ ´ýÚìÌ ´ýÈ¡É º¡÷À¡Ìõ.º¡÷À¡Ìõ.º¡÷À¡Ìõ.º¡÷À¡Ìõ. ţõ = { 3,6,9,10 }ţõ = { 3,6,9,10 }ţõ = { 3,6,9,10 }ţõ = { 3,6,9,10 }

((((17171717)))) A = { 1,4,9,16 }, Ä¢ÕóÐ B = { A = { 1,4,9,16 }, Ä¢ÕóÐ B = { A = { 1,4,9,16 }, Ä¢ÕóÐ B = { A = { 1,4,9,16 }, Ä¢ÕóÐ B = { ----1,2,1,2,1,2,1,2,----3,3,3,3,----4,5,6 } ìÌ 4,5,6 } ìÌ 4,5,6 } ìÌ 4,5,6 } ìÌ f = { (1,f = { (1,f = { (1,f = { (1,----1), (4,2), (9,1), (4,2), (9,1), (4,2), (9,1), (4,2), (9,----3), (16,3), (16,3), (16,3), (16,----4) } ±ýÈ4) } ±ýÈ4) } ±ýÈ4) } ±ýÈ ¯È× ¯È× ¯È× ¯È× º¡÷À¡ÌÁ¡? º¡÷Ò ±É¢ø, Å£º¡÷À¡ÌÁ¡? º¡÷Ò ±É¢ø, Å£º¡÷À¡ÌÁ¡? º¡÷Ò ±É¢ø, Å£º¡÷À¡ÌÁ¡? º¡÷Ò ±É¢ø, ţò¨¾ì ¸¡ñ¸.ò¨¾ì ¸¡ñ¸.ò¨¾ì ¸¡ñ¸.ò¨¾ì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : f = { (1,f = { (1,f = { (1,f = { (1,----1), (4,2), (9,1), (4,2), (9,1), (4,2), (9,1), (4,2), (9,----3), (16,3), (16,3), (16,3), (16,----4) } 4) } 4) } 4) } A BA BA BA B ´ýÚìÌ ´ýÈ¡É º¡÷Ò´ýÚìÌ ´ýÈ¡É º¡÷Ò´ýÚìÌ ´ýÈ¡É º¡÷Ò´ýÚìÌ ´ýÈ¡É º¡÷Ò Å£îº¸õ = { ţõ = { ţõ = { ţõ = { ----1,2,1,2,1,2,1,2,----3,3,3,3,----4 }4 }4 }4 }

((((18181818)))) X = { 1,2,3,4,5 }, y = { 1,3,5,7,9 } ±ý¸. x Ä¢ÕóÐ X = { 1,2,3,4,5 }, y = { 1,3,5,7,9 } ±ý¸. x Ä¢ÕóÐ X = { 1,2,3,4,5 }, y = { 1,3,5,7,9 } ±ý¸. x Ä¢ÕóÐ X = { 1,2,3,4,5 }, y = { 1,3,5,7,9 } ±ý¸. x Ä¢ÕóÐ y ì¸¡É ¯È× { (1,1), (1,3), (3,5), (3,7), (5,7) } y ì¸¡É ¯È× { (1,1), (1,3), (3,5), (3,7), (5,7) } y ì¸¡É ¯È× { (1,1), (1,3), (3,5), (3,7), (5,7) } y ì¸¡É ¯È× { (1,1), (1,3), (3,5), (3,7), (5,7) } ±É ŨÃÂÚì¸ôÀðÎûÇÐ. þÐ º¡÷À¡ÌÁ¡? ±É ŨÃÂÚì¸ôÀðÎûÇÐ. þÐ º¡÷À¡ÌÁ¡? ±É ŨÃÂÚì¸ôÀðÎûÇÐ. þÐ º¡÷À¡ÌÁ¡? ±É ŨÃÂÚì¸ôÀðÎûÇÐ. þÐ º¡÷À¡ÌÁ¡? º¡÷ÀøÄ ±É¢ø «¾ü¸¡É ¸¡Ã½õ ÜÚ¸.º¡÷ÀøÄ ±É¢ø «¾ü¸¡É ¸¡Ã½õ ÜÚ¸.º¡÷ÀøÄ ±É¢ø «¾ü¸¡É ¸¡Ã½õ ÜÚ¸.º¡÷ÀøÄ ±É¢ø «¾ü¸¡É ¸¡Ã½õ ÜÚ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : ¯È× { (1,1), (1,3), (3,5), (3,7), (5,7) } ¯È× { (1,1), (1,3), (3,5), (3,7), (5,7) } ¯È× { (1,1), (1,3), (3,5), (3,7), (5,7) } ¯È× { (1,1), (1,3), (3,5), (3,7), (5,7) } XXXX YYYY X Ä¢ÕóÐ Y ì¸¡É ¯È× ´Õ X Ä¢ÕóÐ Y ì¸¡É ¯È× ´Õ X Ä¢ÕóÐ Y ì¸¡É ¯È× ´Õ X Ä¢ÕóÐ Y ì¸¡É ¯È× ´Õ º¡÷ÀøÄ.º¡÷ÀøÄ.º¡÷ÀøÄ.º¡÷ÀøÄ. ²¦ÉÉ¢ø, Á¾¢ôÀ¸ò¾¢ø 2, ²¦ÉÉ¢ø, Á¾¢ôÀ¸ò¾¢ø 2, ²¦ÉÉ¢ø, Á¾¢ôÀ¸ò¾¢ø 2, ²¦ÉÉ¢ø, Á¾¢ôÀ¸ò¾¢ø 2, 4444 ¬¸¢Â¬¸¢Â¬¸¢Â¬¸¢Â Óý ¯Õì¸ÙìÌ ¿¢Æø ¯Õì¸û Óý ¯Õì¸ÙìÌ ¿¢Æø ¯Õì¸û Óý ¯Õì¸ÙìÌ ¿¢Æø ¯Õì¸û Óý ¯Õì¸ÙìÌ ¿¢Æø ¯Õì¸û þø¨Ä.þø¨Ä.þø¨Ä.þø¨Ä. 1 ÁüÚõ 3 ¬¸¢Â ÓýÛÕì¸ÙìÌ1 ÁüÚõ 3 ¬¸¢Â ÓýÛÕì¸ÙìÌ1 ÁüÚõ 3 ¬¸¢Â ÓýÛÕì¸ÙìÌ1 ÁüÚõ 3 ¬¸¢Â ÓýÛÕì¸ÙìÌ

þÃñÎ ¿¢Æø ¯Õì¸û ¯ûÇÉþÃñÎ ¿¢Æø ¯Õì¸û ¯ûÇÉþÃñÎ ¿¢Æø ¯Õì¸û ¯ûÇÉþÃñÎ ¿¢Æø ¯Õì¸û ¯ûÇÉ....

((((19191919)))) À¢ýÅÕõ «õÒìÌÈ¢ô À¼í¸û º¡÷¨Àì À¢ýÅÕõ «õÒìÌÈ¢ô À¼í¸û º¡÷¨Àì À¢ýÅÕõ «õÒìÌÈ¢ô À¼í¸û º¡÷¨Àì À¢ýÅÕõ «õÒìÌÈ¢ô À¼í¸û º¡÷¨Àì ÌȢ츢ýÈÉÅ¡ ±Éì ÜÚ¸. ¯ý Å¢¨¼ìÌò ÌȢ츢ýÈÉÅ¡ ±Éì ÜÚ¸. ¯ý Å¢¨¼ìÌò ÌȢ츢ýÈÉÅ¡ ±Éì ÜÚ¸. ¯ý Å¢¨¼ìÌò ÌȢ츢ýÈÉÅ¡ ±Éì ÜÚ¸. ¯ý Å¢¨¼ìÌò ¾Ìó¾ ¸¡Ã½õ ÜÚ¸.¾Ìó¾ ¸¡Ã½õ ÜÚ¸.¾Ìó¾ ¸¡Ã½õ ÜÚ¸.¾Ìó¾ ¸¡Ã½õ ÜÚ¸.

(1) P Q (2) L M(1) P Q (2) L M(1) P Q (2) L M(1) P Q (2) L M

¾£÷× :¾£÷× :¾£÷× :¾£÷× : À¼õ (1) À¼õ (1) À¼õ (1) À¼õ (1) º¡÷ÀøÄ. ²¦ÉÉ¢ø c ±ýÈ Óý ¯ÕìÌ º¡÷ÀøÄ. ²¦ÉÉ¢ø c ±ýÈ Óý ¯ÕìÌ º¡÷ÀøÄ. ²¦ÉÉ¢ø c ±ýÈ Óý ¯ÕìÌ º¡÷ÀøÄ. ²¦ÉÉ¢ø c ±ýÈ Óý ¯ÕìÌ ¿¢Æø ¯Õ þø¨Ä.¿¢Æø ¯Õ þø¨Ä.¿¢Æø ¯Õ þø¨Ä.¿¢Æø ¯Õ þø¨Ä. À¼õ (2) µ÷ º¡÷À¡Ìõ. «¨ÉòÐ Óý ¯Õì¸ÙìÌõ À¼õ (2) µ÷ º¡÷À¡Ìõ. «¨ÉòÐ Óý ¯Õì¸ÙìÌõ À¼õ (2) µ÷ º¡÷À¡Ìõ. «¨ÉòÐ Óý ¯Õì¸ÙìÌõ À¼õ (2) µ÷ º¡÷À¡Ìõ. «¨ÉòÐ Óý ¯Õì¸ÙìÌõ ¿¢Æø ¯Õì¸û ¯ûÇÉ. ´ýÚìÌ ´ýÈ¡É º¡÷Ò.¿¢Æø ¯Õì¸û ¯ûÇÉ. ´ýÚìÌ ´ýÈ¡É º¡÷Ò.¿¢Æø ¯Õì¸û ¯ûÇÉ. ´ýÚìÌ ´ýÈ¡É º¡÷Ò.¿¢Æø ¯Õì¸û ¯ûÇÉ. ´ýÚìÌ ´ýÈ¡É º¡÷Ò.

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2222.... ¦Áö¦Âñ¸Ç¢ý¦Áö¦Âñ¸Ç¢ý¦Áö¦Âñ¸Ç¢ý¦Áö¦Âñ¸Ç¢ý ¦¾¡¼÷Å¡¢¨º¸Ùõ ¦¾¡¼÷¸Ùõ¦¾¡¼÷Å¡¢¨º¸Ùõ ¦¾¡¼÷¸Ùõ¦¾¡¼÷Å¡¢¨º¸Ùõ ¦¾¡¼÷¸Ùõ¦¾¡¼÷Å¡¢¨º¸Ùõ ¦¾¡¼÷¸Ùõ ((((1111)))) 4, 9, 14, .......... ±ýÈ ÜðÎò4, 9, 14, .......... ±ýÈ ÜðÎò4, 9, 14, .......... ±ýÈ ÜðÎò4, 9, 14, .......... ±ýÈ ÜðÎò ¦¾¡¼÷¦¾¡¼÷¦¾¡¼÷¦¾¡¼÷Å¡¢¨ºÂ¢ý Å¡¢¨ºÂ¢ý Å¡¢¨ºÂ¢ý Å¡¢¨ºÂ¢ý

17 ¬ÅÐ ¯Úô¨Àì ¸¡ñ¸.17 ¬ÅÐ ¯Úô¨Àì ¸¡ñ¸.17 ¬ÅÐ ¯Úô¨Àì ¸¡ñ¸.17 ¬ÅÐ ¯Úô¨Àì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : 4, 9, 14, ..4, 9, 14, ..4, 9, 14, ..4, 9, 14, .......... ±ýÈ ÜðÎò ¦¾¡¼÷Å¡¢¨ºÂ¢ø........ ±ýÈ ÜðÎò ¦¾¡¼÷Å¡¢¨ºÂ¢ø........ ±ýÈ ÜðÎò ¦¾¡¼÷Å¡¢¨ºÂ¢ø........ ±ýÈ ÜðÎò ¦¾¡¼÷Å¡¢¨ºÂ¢ø a = 4 d = 5a = 4 d = 5a = 4 d = 5a = 4 d = 5 t t t t nnnn = a + ( n = a + ( n = a + ( n = a + ( n ---- 1 ) d1 ) d1 ) d1 ) d tttt 17171717 = 4 + ( 17 = 4 + ( 17 = 4 + ( 17 = 4 + ( 17 ---- 1 ) 51 ) 51 ) 51 ) 5 = 4 + ( 16 x 5 ) = 4 + 80= 4 + ( 16 x 5 ) = 4 + 80= 4 + ( 16 x 5 ) = 4 + 80= 4 + ( 16 x 5 ) = 4 + 80

t t t t 17171717 = 84 = 84 = 84 = 84

((((2222)))) ´Õ ÜðÎò ¦¾¡¼¡¢ø S´Õ ÜðÎò ¦¾¡¼¡¢ø S´Õ ÜðÎò ¦¾¡¼¡¢ø S´Õ ÜðÎò ¦¾¡¼¡¢ø Snnnn = 1275, Ó¾ø ¯ÚôÒ = 1275, Ó¾ø ¯ÚôÒ = 1275, Ó¾ø ¯ÚôÒ = 1275, Ó¾ø ¯ÚôÒ a = 5, ¦À¡Ð Å¢ò¾¢Â¡ºõ d = 4 ±É¢ø ¸¡ñ¸.a = 5, ¦À¡Ð Å¢ò¾¢Â¡ºõ d = 4 ±É¢ø ¸¡ñ¸.a = 5, ¦À¡Ð Å¢ò¾¢Â¡ºõ d = 4 ±É¢ø ¸¡ñ¸.a = 5, ¦À¡Ð Å¢ò¾¢Â¡ºõ d = 4 ±É¢ø ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : a = 5, d = 4 , Sa = 5, d = 4 , Sa = 5, d = 4 , Sa = 5, d = 4 , Snnnn = 1275= 1275= 1275= 1275 SSSSn n n n = n ( 2a + ( n = n ( 2a + ( n = n ( 2a + ( n = n ( 2a + ( n ---- 1 ) d ) 1 ) d ) 1 ) d ) 1 ) d ) 2222 1275 = n 1275 = n 1275 = n 1275 = n ( 2 x 5 + (n ( 2 x 5 + (n ( 2 x 5 + (n ( 2 x 5 + (n ---- 1)4 ) 1)4 ) 1)4 ) 1)4 ) 2222 1275 x 2 = n ( 10 + 4n 1275 x 2 = n ( 10 + 4n 1275 x 2 = n ( 10 + 4n 1275 x 2 = n ( 10 + 4n ---- 4 )4 )4 )4 ) 2550 = n ( 6 + 4n )2550 = n ( 6 + 4n )2550 = n ( 6 + 4n )2550 = n ( 6 + 4n ) 2550 = 6n + 4n 2550 = 6n + 4n 2550 = 6n + 4n 2550 = 6n + 4n 2222

4n 4n 4n 4n 2 2 2 2 +6n +6n +6n +6n ---- 2550 = 0 ( 2550 = 0 ( 2550 = 0 ( 2550 = 0 ( ÷÷÷÷ 2 2 2 2 )))) 2550255025502550

2n 2n 2n 2n 2222 + 3n + 3n + 3n + 3n ---- 1275 = 01275 = 01275 = 01275 = 0 50 50 50 50 51515151 ( n ( n ( n ( n ---- 25 ) ( 2n + 51 )25 ) ( 2n + 51 )25 ) ( 2n + 51 )25 ) ( 2n + 51 ) =0=0=0=0 2 22 22 22 2 n = 25 ( or ) n = n = 25 ( or ) n = n = 25 ( or ) n = n = 25 ( or ) n = ----51515151 //// 2222

n = 25n = 25n = 25n = 25

((((3333)))) 13 ¬ø ÅÌÀÎõ ®¡¢Äì¸ Á¢¨¸ ÓØ 13 ¬ø ÅÌÀÎõ ®¡¢Äì¸ Á¢¨¸ ÓØ 13 ¬ø ÅÌÀÎõ ®¡¢Äì¸ Á¢¨¸ ÓØ 13 ¬ø ÅÌÀÎõ ®¡¢Äì¸ Á¢¨¸ ÓØ

±ñ¸Ç¢ý ±ñ½¢ì¨¸¨Âì ¸¡ñ¸.±ñ¸Ç¢ý ±ñ½¢ì¨¸¨Âì ¸¡ñ¸.±ñ¸Ç¢ý ±ñ½¢ì¨¸¨Âì ¸¡ñ¸.±ñ¸Ç¢ý ±ñ½¢ì¨¸¨Âì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ®¡¢Äì¸ Á¢¨¸ ±ñ¸û : 10,11,12,13,...............,99®¡¢Äì¸ Á¢¨¸ ±ñ¸û : 10,11,12,13,...............,99®¡¢Äì¸ Á¢¨¸ ±ñ¸û : 10,11,12,13,...............,99®¡¢Äì¸ Á¢¨¸ ±ñ¸û : 10,11,12,13,...............,99 a = 13, d = 13 , L = 91a = 13, d = 13 , L = 91a = 13, d = 13 , L = 91a = 13, d = 13 , L = 91 ¯ÚôҸǢý ±ñ½¢ì¨¸ ¯ÚôҸǢý ±ñ½¢ì¨¸ ¯ÚôҸǢý ±ñ½¢ì¨¸ ¯ÚôҸǢý ±ñ½¢ì¨¸ n = L n = L n = L n = L ---- a + 1a + 1a + 1a + 1 dddd n = 91 n = 91 n = 91 n = 91 ---- 13 + 113 + 113 + 113 + 1 13131313 = 78 + 1 = 6 + 1= 78 + 1 = 6 + 1= 78 + 1 = 6 + 1= 78 + 1 = 6 + 1 13131313 ¯ÚôҸǢý ±ñ½¢ì¨¸ n = 7 ¯ÚôҸǢý ±ñ½¢ì¨¸ n = 7 ¯ÚôҸǢý ±ñ½¢ì¨¸ n = 7 ¯ÚôҸǢý ±ñ½¢ì¨¸ n = 7

((((4444)))) 1 1 1 1 3333 + 2 + 2 + 2 + 2 3333 + 3 + 3 + 3 + 3 3333 + .........+ n + .........+ n + .........+ n + .........+ n 3333 = 36100 ±É¢ø, = 36100 ±É¢ø, = 36100 ±É¢ø, = 36100 ±É¢ø,

1 + 2 + 3 + .........+ n ý Á¾¢ô¨Àì ¸¡ñ¸.1 + 2 + 3 + .........+ n ý Á¾¢ô¨Àì ¸¡ñ¸.1 + 2 + 3 + .........+ n ý Á¾¢ô¨Àì ¸¡ñ¸.1 + 2 + 3 + .........+ n ý Á¾¢ô¨Àì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : 1 1 1 1 3333 + 2 + 2 + 2 + 2 3333 + 3 + 3 + 3 + 3 3333 + .........+ n + .........+ n + .........+ n + .........+ n 3 3 3 3 = 36100 = 36100 = 36100 = 36100 n ( n + 1 ) n ( n + 1 ) n ( n + 1 ) n ( n + 1 ) 2222 = 36100= 36100= 36100= 36100 2222

n ( n + 1 ) = n ( n + 1 ) = n ( n + 1 ) = n ( n + 1 ) = √√√√ 36100361003610036100 2222

1 + 2 + 3 + .........+ n = 1901 + 2 + 3 + .........+ n = 1901 + 2 + 3 + .........+ n = 1901 + 2 + 3 + .........+ n = 190

((((5555)))) 5 + 11 + 5 + 11 + 5 + 11 + 5 + 11 + 17 + ...........+ 95 ±ýÈ ÜðÎò ¦¾¡¼¡¢ý 17 + ...........+ 95 ±ýÈ ÜðÎò ¦¾¡¼¡¢ý 17 + ...........+ 95 ±ýÈ ÜðÎò ¦¾¡¼¡¢ý 17 + ...........+ 95 ±ýÈ ÜðÎò ¦¾¡¼¡¢ý Üξø ¸¡ñ¸.Üξø ¸¡ñ¸.Üξø ¸¡ñ¸.Üξø ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : 5 + 11 + 17 + ...........+ 95 ±ýÈ ¦¾¡¼¡¢ø5 + 11 + 17 + ...........+ 95 ±ýÈ ¦¾¡¼¡¢ø5 + 11 + 17 + ...........+ 95 ±ýÈ ¦¾¡¼¡¢ø5 + 11 + 17 + ...........+ 95 ±ýÈ ¦¾¡¼¡¢ø a = 5 d = 6 L = 95a = 5 d = 6 L = 95a = 5 d = 6 L = 95a = 5 d = 6 L = 95 ¯ÚôҸǢý ±ñ½¢ì¨¸ n = L ¯ÚôҸǢý ±ñ½¢ì¨¸ n = L ¯ÚôҸǢý ±ñ½¢ì¨¸ n = L ¯ÚôҸǢý ±ñ½¢ì¨¸ n = L ---- a + 1a + 1a + 1a + 1 dddd n = 95 n = 95 n = 95 n = 95 ---- 5 + 15 + 15 + 15 + 1 6666 = 90 + 1 = 15 + 1= 90 + 1 = 15 + 1= 90 + 1 = 15 + 1= 90 + 1 = 15 + 1 6666

n = 16n = 16n = 16n = 16

¦¾¡¼¡¢ý Üξø S¦¾¡¼¡¢ý Üξø S¦¾¡¼¡¢ý Üξø S¦¾¡¼¡¢ý Üξø Snnnn = n ( L + a ) = n ( L + a ) = n ( L + a ) = n ( L + a ) 2222 = 16 ( 95 + 5 ) = 8 x 100 = 16 ( 95 + 5 ) = 8 x 100 = 16 ( 95 + 5 ) = 8 x 100 = 16 ( 95 + 5 ) = 8 x 100 2222

SSSSn n n n = 800= 800= 800= 800

((((6666)))) FFFF1111 = F= F= F= F2222 = 1 ÁüÚõ= 1 ÁüÚõ= 1 ÁüÚõ= 1 ÁüÚõ FFFFnnnn = F= F= F= Fn n n n ---- 1111 + F+ F+ F+ Fn n n n ---- 2222 ............................ n = 2,3,....... ¦¸¡Îì¸ôÀð¼ ¦¾¡¼÷Å¡¢¨ºÂ¢ý Ó¾ø 5 n = 2,3,....... ¦¸¡Îì¸ôÀð¼ ¦¾¡¼÷Å¡¢¨ºÂ¢ý Ó¾ø 5 n = 2,3,....... ¦¸¡Îì¸ôÀð¼ ¦¾¡¼÷Å¡¢¨ºÂ¢ý Ó¾ø 5 n = 2,3,....... ¦¸¡Îì¸ôÀð¼ ¦¾¡¼÷Å¡¢¨ºÂ¢ý Ó¾ø 5 ¯ÚôÒ¸¨Çì ¸¡ñ¸.¯ÚôÒ¸¨Çì ¸¡ñ¸.¯ÚôÒ¸¨Çì ¸¡ñ¸.¯ÚôÒ¸¨Çì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

FFFF1111 = F= F= F= F2222 = 1 = 1 = 1 = 1 FFFFn n n n = F= F= F= Fn n n n ---- 1111 + F+ F+ F+ Fn n n n ---- 2 2 2 2 ( n = 3 ) F( n = 3 ) F( n = 3 ) F( n = 3 ) F3333 = F= F= F= F2222 + F+ F+ F+ F1111 = 1 + 1 = 2= 1 + 1 = 2= 1 + 1 = 2= 1 + 1 = 2 ( n = 4 ) F( n = 4 ) F( n = 4 ) F( n = 4 ) F4444 = F= F= F= F3333 + F+ F+ F+ F2222 = 2 + 1 = = 2 + 1 = = 2 + 1 = = 2 + 1 = 3333 ( n = 5 ) F( n = 5 ) F( n = 5 ) F( n = 5 ) F5555 = F= F= F= F4444 + F+ F+ F+ F3333 = 3 + 2 = 5= 3 + 2 = 5= 3 + 2 = 5= 3 + 2 = 5

∴∴∴∴ ¦¾¡¼¡¢ý Ó¾ø 5 ¯ÚôÒ¸û = 1, 1, 2, 3, 5, .......... ¦¾¡¼¡¢ý Ó¾ø 5 ¯ÚôÒ¸û = 1, 1, 2, 3, 5, .......... ¦¾¡¼¡¢ý Ó¾ø 5 ¯ÚôÒ¸û = 1, 1, 2, 3, 5, .......... ¦¾¡¼¡¢ý Ó¾ø 5 ¯ÚôÒ¸û = 1, 1, 2, 3, 5, .......... ((((7777)))) ãýÚ ±ñ¸Ç¢ý Å¢¸¢¾õ 2 : 5 : 7 ±ý¸. Ӿġõ ãýÚ ±ñ¸Ç¢ý Å¢¸¢¾õ 2 : 5 : 7 ±ý¸. Ӿġõ ãýÚ ±ñ¸Ç¢ý Å¢¸¢¾õ 2 : 5 : 7 ±ý¸. Ӿġõ ãýÚ ±ñ¸Ç¢ý Å¢¸¢¾õ 2 : 5 : 7 ±ý¸. Ӿġõ

±ñ, þÃñ¼¡õ ±ñ½¢Ä¢ÕóÐ 7 ³ì ¸Æ¢òÐô ±ñ, þÃñ¼¡õ ±ñ½¢Ä¢ÕóÐ 7 ³ì ¸Æ¢òÐô ±ñ, þÃñ¼¡õ ±ñ½¢Ä¢ÕóÐ 7 ³ì ¸Æ¢òÐô ±ñ, þÃñ¼¡õ ±ñ½¢Ä¢ÕóÐ 7 ³ì ¸Æ¢òÐô ¦ÀÈôÀÎõ ±ñ ÁüÚõ ãýÈ¡õ ±ñ ¬¸¢ÂÉ ´Õ ¦ÀÈôÀÎõ ±ñ ÁüÚõ ãýÈ¡õ ±ñ ¬¸¢ÂÉ ´Õ ¦ÀÈôÀÎõ ±ñ ÁüÚõ ãýÈ¡õ ±ñ ¬¸¢ÂÉ ´Õ ¦ÀÈôÀÎõ ±ñ ÁüÚõ ãýÈ¡õ ±ñ ¬¸¢ÂÉ ´Õ ÜðÎò¦¾¡¼÷ÜðÎò¦¾¡¼÷ÜðÎò¦¾¡¼÷ÜðÎò¦¾¡¼÷Å¡¢¨º¨Â Å¡¢¨º¨Â Å¡¢¨º¨Â Å¡¢¨º¨Â ²²²²üÀüÀüÀüÀÎò¾¢É¡ø, Îò¾¢É¡ø, Îò¾¢É¡ø, Îò¾¢É¡ø, «ù¦Åñ¸¨Çì ¸¡ñ¸.«ù¦Åñ¸¨Çì ¸¡ñ¸.«ù¦Åñ¸¨Çì ¸¡ñ¸.«ù¦Åñ¸¨Çì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : ãýÚ ±ñ¸û 2x, 5x, 7x ±ý¸.ãýÚ ±ñ¸û 2x, 5x, 7x ±ý¸.ãýÚ ±ñ¸û 2x, 5x, 7x ±ý¸.ãýÚ ±ñ¸û 2x, 5x, 7x ±ý¸. Ó¾ø ±ñ = 2 xÓ¾ø ±ñ = 2 xÓ¾ø ±ñ = 2 xÓ¾ø ±ñ = 2 x þÃñ¼¡õ ±ñ = 5 x þÃñ¼¡õ ±ñ = 5 x þÃñ¼¡õ ±ñ = 5 x þÃñ¼¡õ ±ñ = 5 x ---- 7777 ãýÈ¡õ ±ñ = 7 xãýÈ¡õ ±ñ = 7 xãýÈ¡õ ±ñ = 7 xãýÈ¡õ ±ñ = 7 x ( 5 x ( 5 x ( 5 x ( 5 x ---- 7 ) 7 ) 7 ) 7 ) ---- 2 x = 7 x 2 x = 7 x 2 x = 7 x 2 x = 7 x ---- ( 5 x ( 5 x ( 5 x ( 5 x ---- 7 )7 )7 )7 ) 3 x 3 x 3 x 3 x ---- 7 = 2 x + 77 = 2 x + 77 = 2 x + 77 = 2 x + 7 3 x 3 x 3 x 3 x ---- 2 x = 2 x = 2 x = 2 x = 7 + 77 + 77 + 77 + 7

x = 14x = 14x = 14x = 14 «ó¾ ãýÚ ±ñ¸û : 2 x 14, 5 x 14, 7 x 14 «ó¾ ãýÚ ±ñ¸û : 2 x 14, 5 x 14, 7 x 14 «ó¾ ãýÚ ±ñ¸û : 2 x 14, 5 x 14, 7 x 14 «ó¾ ãýÚ ±ñ¸û : 2 x 14, 5 x 14, 7 x 14 : 28 , 70 , 98: 28 , 70 , 98: 28 , 70 , 98: 28 , 70 , 98

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3333.... þÂü¸½¢¾õþÂü¸½¢¾õþÂü¸½¢¾õþÂü¸½¢¾õ ((((1111)))) Á£.¦À¡.Á. ¸¡ñ¸ : xÁ£.¦À¡.Á. ¸¡ñ¸ : xÁ£.¦À¡.Á. ¸¡ñ¸ : xÁ£.¦À¡.Á. ¸¡ñ¸ : x2 2 2 2 y + x yy + x yy + x yy + x y2222 , x, x, x, x2222 + x y.+ x y.+ x y.+ x y. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : xxxx2 2 2 2 y + x yy + x yy + x yy + x y2222 , x, x, x, x2222 + x y+ x y+ x y+ x y x y ( x + y ) , x ( x + y )x y ( x + y ) , x ( x + y )x y ( x + y ) , x ( x + y )x y ( x + y ) , x ( x + y ) Á£.¦À¡.Á. = x y ( x + y )Á£.¦À¡.Á. = x y ( x + y )Á£.¦À¡.Á. = x y ( x + y )Á£.¦À¡.Á. = x y ( x + y ) ((((2222)))) ¾£÷ì¸ : 3 x ¾£÷ì¸ : 3 x ¾£÷ì¸ : 3 x ¾£÷ì¸ : 3 x ---- 8 = 28 = 28 = 28 = 2

xxxx ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

3 x 3 x 3 x 3 x ---- 8 8 8 8 = 2= 2= 2= 2 ----24 24 24 24 xxxx 3 x3 x3 x3 x2222 ---- 8 8 8 8 = 2 x= 2 x= 2 x= 2 x ----6 4 6 4 6 4 6 4 3 x3 x3 x3 x2222 ---- 2 x 2 x 2 x 2 x ---- 8 = 0 3 3 8 = 0 3 3 8 = 0 3 3 8 = 0 3 3

( x ( x ( x ( x ---- 2 )( 3x + 4 ) = 02 )( 3x + 4 ) = 02 )( 3x + 4 ) = 02 )( 3x + 4 ) = 0

¾£÷× : 2 , ¾£÷× : 2 , ¾£÷× : 2 , ¾£÷× : 2 , ---- 4444 3333

((((3333)))) ¾£÷ì¸ : x ¾£÷ì¸ : x ¾£÷ì¸ : x ¾£÷ì¸ : x ++++ 1 = 261 = 261 = 261 = 26

x x x x 5 5 5 5 ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

x + 1 = 26x + 1 = 26x + 1 = 26x + 1 = 26 x x x x 5 255 255 255 25 5 x5 x5 x5 x2 2 2 2 + 5 = 26 x + 5 = 26 x + 5 = 26 x + 5 = 26 x

5 x5 x5 x5 x2222 ---- 26 x + 5 = 0 26 x + 5 = 0 26 x + 5 = 0 26 x + 5 = 0 ----25 25 25 25 ----1 1 1 1 ( x ( x ( x ( x ---- 5 )( 5x 5 )( 5x 5 )( 5x 5 )( 5x ---- 1 ) = 0 5 51 ) = 0 5 51 ) = 0 5 51 ) = 0 5 5

5 5 5 5 ----25 25 25 25 3 53 53 53 5 6 6 6 6 ----30 7 630 7 630 7 630 7 6 x = y = 1x = y = 1x = y = 1x = y = 1 ----150 + 150 150 + 150 150 + 150 150 + 150 ----175 + 90 18 175 + 90 18 175 + 90 18 175 + 90 18 ---- 35353535 x = yx = yx = yx = y = 1= 1= 1= 1 0 0 0 0 ----85 85 85 85 ----17171717 x = 0 ; y = x = 0 ; y = x = 0 ; y = x = 0 ; y = ----85858585 ----17 17 17 17 ----17171717 ¾£÷× : x = 0 ¾£÷× : x = 0 ¾£÷× : x = 0 ¾£÷× : x = 0 y = 5y = 5y = 5y = 5 ((((5555)))) x = 1 x = 1 x = 1 x = 1 \\\\ 4 ÁüÚõ x = 4 ÁüÚõ x = 4 ÁüÚõ x = 4 ÁüÚõ x = ----1 ±ýÈ âÂí¸¨Çì 1 ±ýÈ âÂí¸¨Çì 1 ±ýÈ âÂí¸¨Çì 1 ±ýÈ âÂí¸¨Çì

¦¸¡ñ¼ þÕÀÊ ÀøÖÚôÒì §¸¡¨Å¨Âì ¸¡ñ¸.¦¸¡ñ¼ þÕÀÊ ÀøÖÚôÒì §¸¡¨Å¨Âì ¸¡ñ¸.¦¸¡ñ¼ þÕÀÊ ÀøÖÚôÒì §¸¡¨Å¨Âì ¸¡ñ¸.¦¸¡ñ¼ þÕÀÊ ÀøÖÚôÒì §¸¡¨Å¨Âì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : âÂí¸Ç¢ý Üξø = 1 âÂí¸Ç¢ý Üξø = 1 âÂí¸Ç¢ý Üξø = 1 âÂí¸Ç¢ý Üξø = 1 ---- 1 = 1 = 1 = 1 = ----3333 4 4 4 4 4 4 4 4 âÂí¸Ç¢ý ¦ÀÕì¸üÀÄý = âÂí¸Ç¢ý ¦ÀÕì¸üÀÄý = âÂí¸Ç¢ý ¦ÀÕì¸üÀÄý = âÂí¸Ç¢ý ¦ÀÕì¸üÀÄý = 1 x 1 x 1 x 1 x ----1 = 1 = 1 = 1 = ----1111 4 44 44 44 4 §¾¨ÅÂ¡É ºÁýÀ¡Î :§¾¨ÅÂ¡É ºÁýÀ¡Î :§¾¨ÅÂ¡É ºÁýÀ¡Î :§¾¨ÅÂ¡É ºÁýÀ¡Î : x x x x 2222 ---- ( ( ( ( âÂí¸Ç¢ý âÂí¸Ç¢ý âÂí¸Ç¢ý âÂí¸Ç¢ý ÜÜÜÜξøξøξøξø ) x + â) x + â) x + â) x + âÂí¸Ç¢ýÂí¸Ç¢ýÂí¸Ç¢ýÂí¸Ç¢ý ¦À¦À¦À¦ÀÕì¸üÀÄýÕì¸üÀÄýÕì¸üÀÄýÕì¸üÀÄý = 0= 0= 0= 0

x x x x 2222 + 3 + 3 + 3 + 3 //// 4 4 4 4 ---- 1 1 1 1 //// 4 = 04 = 04 = 04 = 0

4 x 4 x 4 x 4 x 2222 + 3 x + 3 x + 3 x + 3 x ---- 1 = 01 = 01 = 01 = 0

((((6666)))) 3 , 4 ¬¸¢ÂÅü¨È ãÄí¸Ç¡¸ì ¦¸¡ñ¼ þÕÀÊî 3 , 4 ¬¸¢ÂÅü¨È ãÄí¸Ç¡¸ì ¦¸¡ñ¼ þÕÀÊî 3 , 4 ¬¸¢ÂÅü¨È ãÄí¸Ç¡¸ì ¦¸¡ñ¼ þÕÀÊî 3 , 4 ¬¸¢ÂÅü¨È ãÄí¸Ç¡¸ì ¦¸¡ñ¼ þÕÀÊî ºÁýÀ¡ð¨¼ì ¸¡ñ¸.ºÁýÀ¡ð¨¼ì ¸¡ñ¸.ºÁýÀ¡ð¨¼ì ¸¡ñ¸.ºÁýÀ¡ð¨¼ì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : ãÄí¸Ç¢ý ÜξøãÄí¸Ç¢ý ÜξøãÄí¸Ç¢ý ÜξøãÄí¸Ç¢ý Üξø = 3 + 4 = 7= 3 + 4 = 7= 3 + 4 = 7= 3 + 4 = 7 ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = 3 x 4 = 12ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = 3 x 4 = 12ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = 3 x 4 = 12ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = 3 x 4 = 12 §¾¨ÅÂ¡É ºÁýÀ¡Î :§¾¨ÅÂ¡É ºÁýÀ¡Î :§¾¨ÅÂ¡É ºÁýÀ¡Î :§¾¨ÅÂ¡É ºÁýÀ¡Î : x x x x 2222 ---- (ãÄí¸Ç¢ý Üξø ) x + ãÄí¸Ç¢ý (ãÄí¸Ç¢ý Üξø ) x + ãÄí¸Ç¢ý (ãÄí¸Ç¢ý Üξø ) x + ãÄí¸Ç¢ý (ãÄí¸Ç¢ý Üξø ) x + ãÄí¸Ç¢ý

¦ÀÕì¸üÀÄý = 0¦ÀÕì¸üÀÄý = 0¦ÀÕì¸üÀÄý = 0¦ÀÕì¸üÀÄý = 0 x x x x 2222 ---- 7 x + 12 = 07 x + 12 = 07 x + 12 = 07 x + 12 = 0

((((7777)))) 4 + 4 + 4 + 4 + √√√√7, 4 7, 4 7, 4 7, 4 ---- √√√√7 ¬¸¢ÂÅü¨È ãÄí¸Ç¡¸ì 7 ¬¸¢ÂÅü¨È ãÄí¸Ç¡¸ì 7 ¬¸¢ÂÅü¨È ãÄí¸Ç¡¸ì 7 ¬¸¢ÂÅü¨È ãÄí¸Ç¡¸ì 2 2 2 2 2 2 2 2

¦¸¡ñ¼ þÕÀÊî ºÁýÀ¡ð¨¼ì ¸¡ñ¸.¦¸¡ñ¼ þÕÀÊî ºÁýÀ¡ð¨¼ì ¸¡ñ¸.¦¸¡ñ¼ þÕÀÊî ºÁýÀ¡ð¨¼ì ¸¡ñ¸.¦¸¡ñ¼ þÕÀÊî ºÁýÀ¡ð¨¼ì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

ãÄí¸Ç¢ý Üξø = 4+ãÄí¸Ç¢ý Üξø = 4+ãÄí¸Ç¢ý Üξø = 4+ãÄí¸Ç¢ý Üξø = 4+√√√√7 +47 +47 +47 +4----√√√√7 = 4+7 = 4+7 = 4+7 = 4+√√√√7+47+47+47+4----√√√√7777 2 22 22 22 2 2222 = 8 = 4= 8 = 4= 8 = 4= 8 = 4 2222

ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = 4+ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = 4+ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = 4+ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = 4+√√√√7x47x47x47x4----√√√√7 = 7 = 7 = 7 = (4+(4+(4+(4+√√√√7)(47)(47)(47)(4----√√√√7)7)7)7) 2 2 2 2 2 2 2 2 4444

= 4 = 4 = 4 = 4 2222 ---- ( ( ( ( √√√√7 ) 7 ) 7 ) 7 ) 2222 = 16 = 16 = 16 = 16 ---- 7 7 7 7 4 4 4 4 4 4 4 4 ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = 9ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = 9ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = 9ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = 9 4444 §¾¨ÅÂ¡É ºÁýÀ¡Î§¾¨ÅÂ¡É ºÁýÀ¡Î§¾¨ÅÂ¡É ºÁýÀ¡Î§¾¨ÅÂ¡É ºÁýÀ¡Î :::: x x x x 2222 ---- (ãÄí¸Ç¢ý Üξø ) x + ãÄí¸Ç¢ý (ãÄí¸Ç¢ý Üξø ) x + ãÄí¸Ç¢ý (ãÄí¸Ç¢ý Üξø ) x + ãÄí¸Ç¢ý (ãÄí¸Ç¢ý Üξø ) x + ãÄí¸Ç¢ý

¦ÀÕì¸üÀÄý = ¦ÀÕì¸üÀÄý = ¦ÀÕì¸üÀÄý = ¦ÀÕì¸üÀÄý = 0000 x x x x 2222 ---- 4 x + 9 / 4 = 04 x + 9 / 4 = 04 x + 9 / 4 = 04 x + 9 / 4 = 0

4444x x x x 2222 ---- 16 x + 9 = 016 x + 9 = 016 x + 9 = 016 x + 9 = 0

((((8888)))) 7 + 7 + 7 + 7 + √√√√3 , 7 3 , 7 3 , 7 3 , 7 ---- √√√√3 ¬¸¢ÂÅü¨È ãÄí¸Ç¡¸ì 3 ¬¸¢ÂÅü¨È ãÄí¸Ç¡¸ì 3 ¬¸¢ÂÅü¨È ãÄí¸Ç¡¸ì 3 ¬¸¢ÂÅü¨È ãÄí¸Ç¡¸ì ¦¸¡ñ¼ þÕÀÊî ºÁýÀ¡ð¨¼ì ¸¡ñ¸.¦¸¡ñ¼ þÕÀÊî ºÁýÀ¡ð¨¼ì ¸¡ñ¸.¦¸¡ñ¼ þÕÀÊî ºÁýÀ¡ð¨¼ì ¸¡ñ¸.¦¸¡ñ¼ þÕÀÊî ºÁýÀ¡ð¨¼ì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

ãÄí¸Ç¢ý Üξø = ãÄí¸Ç¢ý Üξø = ãÄí¸Ç¢ý Üξø = ãÄí¸Ç¢ý Üξø = 7 + 7 + 7 + 7 + √√√√3 + 7 3 + 7 3 + 7 3 + 7 ---- √√√√3 3 3 3 = 14= 14= 14= 14

ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = ( ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = ( ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = ( ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = ( 7 + 7 + 7 + 7 + √√√√3 ) x (7 3 ) x (7 3 ) x (7 3 ) x (7 ---- √√√√3 )3 )3 )3 )

= 7 = 7 = 7 = 7 2222 ---- (((( √√√√3333 ) ) ) ) 2222 = 49 = 49 = 49 = 49 ---- 3333 ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý ãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý = 46= 46= 46= 46 §¾¨ÅÂ¡É ºÁýÀ¡Î :§¾¨ÅÂ¡É ºÁýÀ¡Î :§¾¨ÅÂ¡É ºÁýÀ¡Î :§¾¨ÅÂ¡É ºÁýÀ¡Î : x x x x 2222 ---- (ãÄí¸Ç¢ý Üξø ) x + ãÄí¸Ç¢ý (ãÄí¸Ç¢ý Üξø ) x + ãÄí¸Ç¢ý (ãÄí¸Ç¢ý Üξø ) x + ãÄí¸Ç¢ý (ãÄí¸Ç¢ý Üξø ) x + ãÄí¸Ç¢ý

¦ÀÕì¸üÀÄý =¦ÀÕì¸üÀÄý =¦ÀÕì¸üÀÄý =¦ÀÕì¸üÀÄý = 0000 x x x x 2222 ---- 14 x + 4614 x + 4614 x + 4614 x + 46 = 0= 0= 0= 0

((((9999)))) «Îò¾Îò¾ þÕ ´ü¨ÈôÀ¨¼ ÓØì¸Ç¢ý «Îò¾Îò¾ þÕ ´ü¨ÈôÀ¨¼ ÓØì¸Ç¢ý «Îò¾Îò¾ þÕ ´ü¨ÈôÀ¨¼ ÓØì¸Ç¢ý «Îò¾Îò¾ þÕ ´ü¨ÈôÀ¨¼ ÓØì¸Ç¢ý Üξø 20, ±É¢ø «ó¾ ±ñ¸¨Çì ¸¡ñ¸.Üξø 20, ±É¢ø «ó¾ ±ñ¸¨Çì ¸¡ñ¸.Üξø 20, ±É¢ø «ó¾ ±ñ¸¨Çì ¸¡ñ¸.Üξø 20, ±É¢ø «ó¾ ±ñ¸¨Çì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : «Îò¾Îò¾ þÕ ´ü¨ÈôÀ¨¼ ±ñ¸û x, x+2 ±ý¸.«Îò¾Îò¾ þÕ ´ü¨ÈôÀ¨¼ ±ñ¸û x, x+2 ±ý¸.«Îò¾Îò¾ þÕ ´ü¨ÈôÀ¨¼ ±ñ¸û x, x+2 ±ý¸.«Îò¾Îò¾ þÕ ´ü¨ÈôÀ¨¼ ±ñ¸û x, x+2 ±ý¸. ±ý§Å x + x + 2 = 20±ý§Å x + x + 2 = 20±ý§Å x + x + 2 = 20±ý§Å x + x + 2 = 20 2 x = 20 2 x = 20 2 x = 20 2 x = 20 ---- 2222 x = 18 / 2x = 18 / 2x = 18 / 2x = 18 / 2 x = 9x = 9x = 9x = 9 «ó¾ þÕ ´ü¨ÈôÀ¨¼ ±ñ¸û 9 , 11.«ó¾ þÕ ´ü¨ÈôÀ¨¼ ±ñ¸û 9 , 11.«ó¾ þÕ ´ü¨ÈôÀ¨¼ ±ñ¸û 9 , 11.«ó¾ þÕ ´ü¨ÈôÀ¨¼ ±ñ¸û 9 , 11.

¾£÷× : 5 , 1¾£÷× : 5 , 1¾£÷× : 5 , 1¾£÷× : 5 , 1 5555

((((4444)))) ¾£÷ì¾£÷ì¾£÷ì¾£÷ì¸ ¸ ¸ ¸ : 3x + 5y = 25 ; 7x + 6y = 30: 3x + 5y = 25 ; 7x + 6y = 30: 3x + 5y = 25 ; 7x + 6y = 30: 3x + 5y = 25 ; 7x + 6y = 30 ¾£÷× :¾£÷× :¾£÷× :¾£÷× : 3x + 5y 3x + 5y 3x + 5y 3x + 5y ---- 25 = 0 25 = 0 25 = 0 25 = 0 7x + 6y 7x + 6y 7x + 6y 7x + 6y ---- 30 = 030 = 030 = 030 = 0 x y 1x y 1x y 1x y 1

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((((10101010)))) ÅÌòÐ ®× ÅÌòÐ ®× ÅÌòÐ ®× ÅÌòÐ ®× ,,,, Á£¾¢ ¸¡ñ¸.Á£¾¢ ¸¡ñ¸.Á£¾¢ ¸¡ñ¸.Á£¾¢ ¸¡ñ¸.

( x ( x ( x ( x 3333 + x+ x+ x+ x 2222 ---- 3 x + 5 ) 3 x + 5 ) 3 x + 5 ) 3 x + 5 ) ÷÷÷÷ ( x ( x ( x ( x ---- 1 )1 )1 )1 ) ¾£÷× :¾£÷× :¾£÷× :¾£÷× : P ( x ) = x P ( x ) = x P ( x ) = x P ( x ) = x 3333 + x+ x+ x+ x 2222 ---- 3 x + 53 x + 53 x + 53 x + 5 1 1 1 1 1 1 1 1 1 1 1 1 ----3 53 53 53 5 0 1 2 0 1 2 0 1 2 0 1 2 ----1111

1 2 1 2 1 2 1 2 ----1 41 41 41 4 ®× = x ®× = x ®× = x ®× = x 2222 + + + + 2 x 2 x 2 x 2 x ---- 1111 Á£¾¢ = 4Á£¾¢ = 4Á£¾¢ = 4Á£¾¢ = 4 ((((11111111)))) ÍÕì̸. 6 x ÍÕì̸. 6 x ÍÕì̸. 6 x ÍÕì̸. 6 x 2222 ---- 54545454

x x x x 2222 + 7 x + 12+ 7 x + 12+ 7 x + 12+ 7 x + 12 ¾£÷× :¾£÷× :¾£÷× :¾£÷× : 6 x 6 x 6 x 6 x 2222 ---- 54 = 6 ( x 54 = 6 ( x 54 = 6 ( x 54 = 6 ( x 2222 ---- 9 ) 9 ) 9 ) 9 ) = 6 ( x = 6 ( x = 6 ( x = 6 ( x 2222 ---- 3 3 3 3 2222 )))) x x x x 2222 + 7 x + 12+ 7 x + 12+ 7 x + 12+ 7 x + 12 (x + 3)(x + 4) (x + 4)(x + 3)(x + 3)(x + 4) (x + 4)(x + 3)(x + 3)(x + 4) (x + 4)(x + 3)(x + 3)(x + 4) (x + 4)(x + 3) = 6 (x + 3)( x = 6 (x + 3)( x = 6 (x + 3)( x = 6 (x + 3)( x ---- 3)3)3)3) (x + 4)(x + 3)(x + 4)(x + 3)(x + 4)(x + 3)(x + 4)(x + 3) = 6 ( x = 6 ( x = 6 ( x = 6 ( x ---- 3 )3 )3 )3 ) ( x + 4 ) ( x + 4 ) ( x + 4 ) ( x + 4 )

((((12121212)))) Å÷ì¸ãÄõ ¸¡ñ¸. (x + 1) Å÷ì¸ãÄõ ¸¡ñ¸. (x + 1) Å÷ì¸ãÄõ ¸¡ñ¸. (x + 1) Å÷ì¸ãÄõ ¸¡ñ¸. (x + 1) 6666 + 1 + 2+ 1 + 2+ 1 + 2+ 1 + 2 (x + 1) (x + 1) (x + 1) (x + 1) 6666

¾£÷× :¾£÷× :¾£÷× :¾£÷× :

(x + 1) (x + 1) (x + 1) (x + 1) 6666 + 1 + 2+ 1 + 2+ 1 + 2+ 1 + 2 ((((a+b)a+b)a+b)a+b)2222=a=a=a=a2222+b+b+b+b2222+2ab+2ab+2ab+2ab

√√√√ (x + 1) (x + 1) (x + 1) (x + 1) 6666

(x + 1) (x + 1) (x + 1) (x + 1) 3333 + 1 + 1 + 1 + 1 2222

= = = = √ √ √ √ (x + 1) (x + 1) (x + 1) (x + 1) 3333

= = = = (x + 1) (x + 1) (x + 1) (x + 1) 3333 + 1 + 1 + 1 + 1

(x + 1) (x + 1) (x + 1) (x + 1) 3333 ((((13131313)))) Å÷ì¸ãÄõ ¸¡ñ¸. x Å÷ì¸ãÄõ ¸¡ñ¸. x Å÷ì¸ãÄõ ¸¡ñ¸. x Å÷ì¸ãÄõ ¸¡ñ¸. x 6666 + 1 + 1 + 1 + 1 ---- 2222

x x x x 6666

¾£÷× :¾£÷× :¾£÷× :¾£÷× : xxxx 6666 + 1 + 1 + 1 + 1 ---- 2 2 2 2 = = = = x x x x 3333 ---- 1 1 1 1 3333

√√√√ x x x x 6666 √√√√ x x x x 3333 (a(a(a(a----b)b)b)b)2222=a=a=a=a2222+b+b+b+b2222----2ab 2ab 2ab 2ab = x = x = x = x 3333 ---- 1111 x x x x 3333 ((((14141414)))) x x x x 2222 ---- 8 x + 12 = 0 ±ýÈ ºÁýÀ¡ðÊý 8 x + 12 = 0 ±ýÈ ºÁýÀ¡ðÊý 8 x + 12 = 0 ±ýÈ ºÁýÀ¡ðÊý 8 x + 12 = 0 ±ýÈ ºÁýÀ¡ðÊý

ãÄí¸Ç¢ý ¾ý¨Á¨Â ¬Ã¡ö¸.ãÄí¸Ç¢ý ¾ý¨Á¨Â ¬Ã¡ö¸.ãÄí¸Ç¢ý ¾ý¨Á¨Â ¬Ã¡ö¸.ãÄí¸Ç¢ý ¾ý¨Á¨Â ¬Ã¡ö¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x 2222 + b x + c = 0+ b x + c = 0+ b x + c = 0+ b x + c = 0 ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = x ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = x ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = x ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = x 2222 ---- 8 x + 12 = 08 x + 12 = 08 x + 12 = 08 x + 12 = 0 a = 1 ; b = a = 1 ; b = a = 1 ; b = a = 1 ; b = ----8 ; c = 128 ; c = 128 ; c = 128 ; c = 12

ãÄí¸Ç¢ý ¾ý¨Á ãÄí¸Ç¢ý ¾ý¨Á ãÄí¸Ç¢ý ¾ý¨Á ãÄí¸Ç¢ý ¾ý¨Á ∆∆∆∆ = = = = b b b b 2222 ---- 4 a c4 a c4 a c4 a c = (= (= (= (----8) 8) 8) 8) 2222 ---- 4 x 1 x 12 4 x 1 x 12 4 x 1 x 12 4 x 1 x 12 = 64 = 64 = 64 = 64 ---- 48 = 1648 = 1648 = 1648 = 16

∆∆∆∆ >>>> 0000 ãÄí¸Ç¢ý ¾ý¨Á :ãÄí¸Ç¢ý ¾ý¨Á :ãÄí¸Ç¢ý ¾ý¨Á :ãÄí¸Ç¢ý ¾ý¨Á : ¦Áö¦Áö¦Áö¦Áö¦Âñ¸û ÁüÚõ ºÁÁ¢ø¨Ä¦Âñ¸û ÁüÚõ ºÁÁ¢ø¨Ä¦Âñ¸û ÁüÚõ ºÁÁ¢ø¨Ä¦Âñ¸û ÁüÚõ ºÁÁ¢ø¨Ä....

((((15151515)))) 9x 9x 9x 9x 2222 +12 x + 4 = 0 ±ýÈ ºÁýÀ¡ðÊý ãÄí¸Ç¢ý +12 x + 4 = 0 ±ýÈ ºÁýÀ¡ðÊý ãÄí¸Ç¢ý +12 x + 4 = 0 ±ýÈ ºÁýÀ¡ðÊý ãÄí¸Ç¢ý +12 x + 4 = 0 ±ýÈ ºÁýÀ¡ðÊý ãÄí¸Ç¢ý ¾ý¨Á¨Â ¬Ã¡ö¸.¾ý¨Á¨Â ¬Ã¡ö¸.¾ý¨Á¨Â ¬Ã¡ö¸.¾ý¨Á¨Â ¬Ã¡ö¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x 2222 + b x + c = 0+ b x + c = 0+ b x + c = 0+ b x + c = 0 ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = 9 x ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = 9 x ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = 9 x ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = 9 x 2222 + 12 x + 4 = 0+ 12 x + 4 = 0+ 12 x + 4 = 0+ 12 x + 4 = 0 a = 9 ; b = 12 ; c = 4a = 9 ; b = 12 ; c = 4a = 9 ; b = 12 ; c = 4a = 9 ; b = 12 ; c = 4

ãÄí¸Ç¢ý ¾ý¨Á ãÄí¸Ç¢ý ¾ý¨Á ãÄí¸Ç¢ý ¾ý¨Á ãÄí¸Ç¢ý ¾ý¨Á ∆∆∆∆ = = = = b b b b 2222 ---- 4 a c4 a c4 a c4 a c = (12) = (12) = (12) = (12) 2222 ---- 4 x 9 x 4 4 x 9 x 4 4 x 9 x 4 4 x 9 x 4 = 144 = 144 = 144 = 144 ---- 111144 44 44 44

∆∆∆∆ ==== 0000 ãÄí¸Ç¢ý ¾ý¨Á :ãÄí¸Ç¢ý ¾ý¨Á :ãÄí¸Ç¢ý ¾ý¨Á :ãÄí¸Ç¢ý ¾ý¨Á : ¦Áö¦Âñ¸û ÁüÚõ ºÁÁ¡É¨Å.¦Áö¦Âñ¸û ÁüÚõ ºÁÁ¡É¨Å.¦Áö¦Âñ¸û ÁüÚõ ºÁÁ¡É¨Å.¦Áö¦Âñ¸û ÁüÚõ ºÁÁ¡É¨Å.

((((16161616)))) 2x 2x 2x 2x 2222 ----10 x + k = 0 ±ýÈ ºÁýÀ¡ðÊý ãÄí¸û 10 x + k = 0 ±ýÈ ºÁýÀ¡ðÊý ãÄí¸û 10 x + k = 0 ±ýÈ ºÁýÀ¡ðÊý ãÄí¸û 10 x + k = 0 ±ýÈ ºÁýÀ¡ðÊý ãÄí¸û ¦Áö¦Âñ¸û ÁüÚõ ºÁõ ±É¢ø, k ý Á¾¢ôÒ ¸¡ñ¸.¦Áö¦Âñ¸û ÁüÚõ ºÁõ ±É¢ø, k ý Á¾¢ôÒ ¸¡ñ¸.¦Áö¦Âñ¸û ÁüÚõ ºÁõ ±É¢ø, k ý Á¾¢ôÒ ¸¡ñ¸.¦Áö¦Âñ¸û ÁüÚõ ºÁõ ±É¢ø, k ý Á¾¢ôÒ ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x 2222 + b x + c = 0+ b x + c = 0+ b x + c = 0+ b x + c = 0 ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = 2 x ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = 2 x ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = 2 x ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = 2 x 2222 ---- 10 x + k = 010 x + k = 010 x + k = 010 x + k = 0 a = 2 ; b = a = 2 ; b = a = 2 ; b = a = 2 ; b = ----10 ; c = k10 ; c = k10 ; c = k10 ; c = k

ãÄíì¸û ¦Áö¦Âñ¸û ÁüÚõ ºÁõ ±É¢ø,ãÄíì¸û ¦Áö¦Âñ¸û ÁüÚõ ºÁõ ±É¢ø,ãÄíì¸û ¦Áö¦Âñ¸û ÁüÚõ ºÁõ ±É¢ø,ãÄíì¸û ¦Áö¦Âñ¸û ÁüÚõ ºÁõ ±É¢ø, b b b b 2222 ---- 4 a c = 04 a c = 04 a c = 04 a c = 0 ((((----10) 10) 10) 10) 2222 ---- 4 x 2 x k = 04 x 2 x k = 04 x 2 x k = 04 x 2 x k = 0 100 100 100 100 ---- 8 k = 08 k = 08 k = 08 k = 0

- 8 k = 8 k = 8 k = 8 k = ---- 100100100100 K = K = K = K = ---- 100 / 100 / 100 / 100 / ----8888 K = 25 / 2K = 25 / 2K = 25 / 2K = 25 / 2

((((17171717)))) ax ax ax ax 2222 ---- 5 x + c = 0 ±ýÈ ºÁýÀ¡ðÊý 5 x + c = 0 ±ýÈ ºÁýÀ¡ðÊý 5 x + c = 0 ±ýÈ ºÁýÀ¡ðÊý 5 x + c = 0 ±ýÈ ºÁýÀ¡ðÊý

ãÄí¸Ç¢ý Üξø 10, ÁüÚõ ¦ÀÕì¸üÀÄý 10 ãÄí¸Ç¢ý Üξø 10, ÁüÚõ ¦ÀÕì¸üÀÄý 10 ãÄí¸Ç¢ý Üξø 10, ÁüÚõ ¦ÀÕì¸üÀÄý 10 ãÄí¸Ç¢ý Üξø 10, ÁüÚõ ¦ÀÕì¸üÀÄý 10 ±É¢ø, a ÁüÚõ b ¬¸¢ÂÅüÈ¢ý Á¾¢ôÒ¸¨Çì ¸¡ñ¸.±É¢ø, a ÁüÚõ b ¬¸¢ÂÅüÈ¢ý Á¾¢ôÒ¸¨Çì ¸¡ñ¸.±É¢ø, a ÁüÚõ b ¬¸¢ÂÅüÈ¢ý Á¾¢ôÒ¸¨Çì ¸¡ñ¸.±É¢ø, a ÁüÚõ b ¬¸¢ÂÅüÈ¢ý Á¾¢ôÒ¸¨Çì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x ºÁýÀ¡ðÊý ¦À¡Ð ÅÊÅõ = a x 2222 + b x + c = 0+ b x + c = 0+ b x + c = 0+ b x + c = 0 ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = a x ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = a x ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = a x ¦¸¡Îì¸ôÀð¼ ºÁýÀ¡Î = a x 2222 ---- 5 x + c = 05 x + c = 05 x + c = 05 x + c = 0 a = a ; b = a = a ; b = a = a ; b = a = a ; b = ----5 ; c = c5 ; c = c5 ; c = c5 ; c = c ãÄí¸Ç¢ý Üξø ãÄí¸Ç¢ý Üξø ãÄí¸Ç¢ý Üξø ãÄí¸Ç¢ý Üξø ---- b = 5 = 10b = 5 = 10b = 5 = 10b = 5 = 10 a a a a a a a a a = 5 / 10a = 5 / 10a = 5 / 10a = 5 / 10

a = a = a = a = ½½½½ ãÄí¸Ç¢ý ¦ÀãÄí¸Ç¢ý ¦ÀãÄí¸Ç¢ý ¦ÀãÄí¸Ç¢ý ¦ÀÕì¸üÀÄý Õì¸üÀÄý Õì¸üÀÄý Õì¸üÀÄý = c = 10= c = 10= c = 10= c = 10 aaaa

c = 10 x c = 10 x c = 10 x c = 10 x ½½½½ ¾£÷× :¾£÷× :¾£÷× :¾£÷× : c = 5c = 5c = 5c = 5

a = a = a = a = ½½½½ c = 5c = 5c = 5c = 5

((((18181818)))) ÍÕì̸. x ÍÕì̸. x ÍÕì̸. x ÍÕì̸. x 3333 ---- 27272727 x x x x 2222 ---- 9999 ¾£÷× ¾£÷× ¾£÷× ¾£÷× :::: aaaa3333----bbbb3333 = (a= (a= (a= (a----b)(ab)(ab)(ab)(a2222+ab+b+ab+b+ab+b+ab+b2222)))) aaaa2222----bbbb2222 = (a+b)(a= (a+b)(a= (a+b)(a= (a+b)(a----b)b)b)b) x x x x 3333 ---- 27 = x 27 = x 27 = x 27 = x 3333 ---- 3 3 3 3 3333 x x x x 2222 ---- 9 x 9 x 9 x 9 x 2222 ---- 3333 2222 = (x = (x = (x = (x ---- 3)(x 3)(x 3)(x 3)(x 2222 + 3x + 3 + 3x + 3 + 3x + 3 + 3x + 3 2222)))) (x + 3)(x (x + 3)(x (x + 3)(x (x + 3)(x ---- 3) 3) 3) 3)

= (x = (x = (x = (x 2222 + 3x + 9)+ 3x + 9)+ 3x + 9)+ 3x + 9) (x + 3)(x + 3)(x + 3)(x + 3)

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4444.... «½¢¸û«½¢¸û«½¢¸û«½¢¸û ((((1111)))) A = 2 3 A = 2 3 A = 2 3 A = 2 3 ---- 1 5 ±É¢ø, A ý Üð¼ø 1 5 ±É¢ø, A ý Üð¼ø 1 5 ±É¢ø, A ý Üð¼ø 1 5 ±É¢ø, A ý Üð¼ø

----9 5 7 9 5 7 9 5 7 9 5 7 ----1111 §¿÷Á¡Ú «½¢¨Âì ¸¡ñ¸.§¿÷Á¡Ú «½¢¨Âì ¸¡ñ¸.§¿÷Á¡Ú «½¢¨Âì ¸¡ñ¸.§¿÷Á¡Ú «½¢¨Âì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : A = 2 3 A = 2 3 A = 2 3 A = 2 3 ---- 1 5 1 5 1 5 1 5 ----9 5 7 9 5 7 9 5 7 9 5 7 ----1111 = 2= 2= 2= 2----1 31 31 31 3----5 = 1 5 = 1 5 = 1 5 = 1 ----2222 ----9999----7 5+1 7 5+1 7 5+1 7 5+1 ----16 6 16 6 16 6 16 6

A ý Üð¼ø §¿÷Á¡Ú «½¢ = A ý Üð¼ø §¿÷Á¡Ú «½¢ = A ý Üð¼ø §¿÷Á¡Ú «½¢ = A ý Üð¼ø §¿÷Á¡Ú «½¢ = ----1 21 21 21 2 16 16 16 16 ----6666 ((((2222)))) «½¢ô¦ÀÕ츨Çì ¸¡ñ¸. 3 «½¢ô¦ÀÕ츨Çì ¸¡ñ¸. 3 «½¢ô¦ÀÕ츨Çì ¸¡ñ¸. 3 «½¢ô¦ÀÕ츨Çì ¸¡ñ¸. 3 ----2 4 2 4 2 4 2 4 1 1 1 1 5 1 2 75 1 2 75 1 2 75 1 2 7 ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

3 3 3 3 ----2 4 1 = 12 2 4 1 = 12 2 4 1 = 12 2 4 1 = 12 ---- 4 34 34 34 3----14 14 14 14 5 1 2 7 20+2 5+75 1 2 7 20+2 5+75 1 2 7 20+2 5+75 1 2 7 20+2 5+7

= 8 = 8 = 8 = 8 ----11111111 22 1222 1222 1222 12 ((((3333)))) 2 9 2 9 2 9 2 9 ----3 4 2 ±ýÀ¾ü3 4 2 ±ýÀ¾ü3 4 2 ±ýÀ¾ü3 4 2 ±ýÀ¾üÌ «½¢Ì «½¢Ì «½¢Ì «½¢¸Ç¢ý¸Ç¢ý¸Ç¢ý¸Ç¢ý

4 4 4 4 ----1 0 1 0 1 0 1 0 ----6 7 ¦ÀÕì¸ø 6 7 ¦ÀÕì¸ø 6 7 ¦ÀÕì¸ø 6 7 ¦ÀÕì¸ø ----2 1 2 1 2 1 2 1 ŨÃÂÚì¸ôÀÎÁ¡É¡øŨÃÂÚì¸ôÀÎÁ¡É¡øŨÃÂÚì¸ôÀÎÁ¡É¡øŨÃÂÚì¸ôÀÎÁ¡É¡ø «¾¨Éì ¸¡ñ¸.«¾¨Éì ¸¡ñ¸.«¾¨Éì ¸¡ñ¸.«¾¨Éì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : «½¢¸Ç¢ý ¦ÀÕì¸ø ŨÃÂÚì¸ôÀðÎûÇÐ.«½¢¸Ç¢ý ¦ÀÕì¸ø ŨÃÂÚì¸ôÀðÎûÇÐ.«½¢¸Ç¢ý ¦ÀÕì¸ø ŨÃÂÚì¸ôÀðÎûÇÐ.«½¢¸Ç¢ý ¦ÀÕì¸ø ŨÃÂÚì¸ôÀðÎûÇÐ. AB = 2 9 AB = 2 9 AB = 2 9 AB = 2 9 ----3 4 2 3 4 2 3 4 2 3 4 2

4 4 4 4 ----1 0 1 0 1 0 1 0 ----6 7 6 7 6 7 6 7 ----2 1 2 1 2 1 2 1 = 8= 8= 8= 8----54+6 4+6354+6 4+6354+6 4+6354+6 4+63----3333 16+6+016+6+016+6+016+6+0 8888----7+07+07+07+0

AB = AB = AB = AB = ----40 6440 6440 6440 64 22 122 122 122 1 ((((4444)))) A = 8 5 2 ±É¢ø, A A = 8 5 2 ±É¢ø, A A = 8 5 2 ±É¢ø, A A = 8 5 2 ±É¢ø, A TTTT ÁüÚõ (A ÁüÚõ (A ÁüÚõ (A ÁüÚõ (A TTTT) ) ) ) TTTT 1 1 1 1 ----3 4 3 4 3 4 3 4 ¬¸¢ÂÅü¨Èì ¸¡ñ¸.¬¸¢ÂÅü¨Èì ¸¡ñ¸.¬¸¢ÂÅü¨Èì ¸¡ñ¸.¬¸¢ÂÅü¨Èì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

A = 8 5 2 A = 8 5 2 A = 8 5 2 A = 8 5 2 1 1 1 1 ----3 4 3 4 3 4 3 4

A A A A TTTT = 8 1= 8 1= 8 1= 8 1 5 5 5 5 ----3333 2 4 2 4 2 4 2 4

(A (A (A (A TTTT) ) ) ) TTTT = = = = 8 5 2 8 5 2 8 5 2 8 5 2 1 1 1 1 ----3 4 3 4 3 4 3 4 ((((5555)))) A = 1 2 3 ±É¢ø, (A A = 1 2 3 ±É¢ø, (A A = 1 2 3 ±É¢ø, (A A = 1 2 3 ±É¢ø, (A TTTT) ) ) ) TTTT = A ±ýÀ¨¾= A ±ýÀ¨¾= A ±ýÀ¨¾= A ±ýÀ¨¾

2 4 2 4 2 4 2 4 ----5 º¡¢À¡÷ì¸.5 º¡¢À¡÷ì¸.5 º¡¢À¡÷ì¸.5 º¡¢À¡÷ì¸. 3 3 3 3 ----5 65 65 65 6

¾£÷× :¾£÷× :¾£÷× :¾£÷× :

A = 1 2 3 A = 1 2 3 A = 1 2 3 A = 1 2 3 2 4 2 4 2 4 2 4 ----5 5 5 5 3 3 3 3 ----5 65 65 65 6

A A A A TTTT = 1 2 3 = 1 2 3 = 1 2 3 = 1 2 3 2 4 2 4 2 4 2 4 ----5 5 5 5 3 3 3 3 ----5 65 65 65 6 ( A ( A ( A ( A TTTT) ) ) ) TTTT = 1 2 3 = 1 2 3 = 1 2 3 = 1 2 3 2 4 2 4 2 4 2 4 ----5 5 5 5 3 3 3 3 ----5 65 65 65 6

(A (A (A (A TTTT) ) ) ) TTTT = A º¡¢À¡÷ì¸ôÀð¼Ð.= A º¡¢À¡÷ì¸ôÀð¼Ð.= A º¡¢À¡÷ì¸ôÀð¼Ð.= A º¡¢À¡÷ì¸ôÀð¼Ð. ((((6666)))) 30 ¯ÚôÒ¸û ¦¸¡ñ¼ «½¢ìÌ ±ùŨ¸ 30 ¯ÚôÒ¸û ¦¸¡ñ¼ «½¢ìÌ ±ùŨ¸ 30 ¯ÚôÒ¸û ¦¸¡ñ¼ «½¢ìÌ ±ùŨ¸ 30 ¯ÚôÒ¸û ¦¸¡ñ¼ «½¢ìÌ ±ùŨ¸

Å¡¢¨º¸û þÕì¸ þÂÖõ?Å¡¢¨º¸û þÕì¸ þÂÖõ?Å¡¢¨º¸û þÕì¸ þÂÖõ?Å¡¢¨º¸û þÕì¸ þÂÖõ? ¾£÷× :¾£÷× :¾£÷× :¾£÷× : 30 ¯ÚôÒ¸û ¦¸¡ñ¼ «½¢ì¸¡É Å¡¢¨º¸û :30 ¯ÚôÒ¸û ¦¸¡ñ¼ «½¢ì¸¡É Å¡¢¨º¸û :30 ¯ÚôÒ¸û ¦¸¡ñ¼ «½¢ì¸¡É Å¡¢¨º¸û :30 ¯ÚôÒ¸û ¦¸¡ñ¼ «½¢ì¸¡É Å¡¢¨º¸û : 1 x 30 6 x 51 x 30 6 x 51 x 30 6 x 51 x 30 6 x 5 2 x 15 10 x 32 x 15 10 x 32 x 15 10 x 32 x 15 10 x 3 3 x 10 15 x 23 x 10 15 x 23 x 10 15 x 23 x 10 15 x 2 5 x 6 30 x 15 x 6 30 x 15 x 6 30 x 15 x 6 30 x 1 ((((7777)))) A = 4 A = 4 A = 4 A = 4 ----2 ÁüÚõ B = 8 2 2 ÁüÚõ B = 8 2 2 ÁüÚõ B = 8 2 2 ÁüÚõ B = 8 2

5 5 5 5 ----9 9 9 9 ----1 1 1 1 ----3333 ±É¢ø, 6 A ±É¢ø, 6 A ±É¢ø, 6 A ±É¢ø, 6 A ---- 3 B ³ì ¸¡ñ¸.3 B ³ì ¸¡ñ¸.3 B ³ì ¸¡ñ¸.3 B ³ì ¸¡ñ¸.

¾£÷¾£÷¾£÷¾£÷× :× :× :× : 6 A 6 A 6 A 6 A ---- 3 B = 6 4 3 B = 6 4 3 B = 6 4 3 B = 6 4 ----2 2 2 2 ---- 3 8 2 3 8 2 3 8 2 3 8 2 5 5 5 5 ----9 9 9 9 ----1 1 1 1 ----3333 = 24 = 24 = 24 = 24 ----12 12 12 12 ---- 24 6 24 6 24 6 24 6 30 30 30 30 ----54 54 54 54 ----3 3 3 3 ----9999 = 24 = 24 = 24 = 24 ---- 24 24 24 24 ----12 12 12 12 ---- 6666 30 + 3 30 + 3 30 + 3 30 + 3 ----54 + 9 54 + 9 54 + 9 54 + 9 6 A 6 A 6 A 6 A ---- 3 B = 0 3 B = 0 3 B = 0 3 B = 0 ----18181818 33 33 33 33 ----45454545

((((8888)))) A = 3 2 ÁüÚõ B = 8 A = 3 2 ÁüÚõ B = 8 A = 3 2 ÁüÚõ B = 8 A = 3 2 ÁüÚõ B = 8 ----1111

5 1 4 35 1 4 35 1 4 35 1 4 3 ±É¢ø, C = 2 A + B ³ì ¸¡ñ¸.±É¢ø, C = 2 A + B ³ì ¸¡ñ¸.±É¢ø, C = 2 A + B ³ì ¸¡ñ¸.±É¢ø, C = 2 A + B ³ì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : C = 2 A + B C = 2 A + B C = 2 A + B C = 2 A + B ((((1111)))) C = 2 3 2 + 8 C = 2 3 2 + 8 C = 2 3 2 + 8 C = 2 3 2 + 8 ----1111

5 1 4 3 5 1 4 3 5 1 4 3 5 1 4 3

= 6 4 + 8 = 6 4 + 8 = 6 4 + 8 = 6 4 + 8 ----1111 10 2 4 310 2 4 310 2 4 310 2 4 3

= 6 + 8 4 = 6 + 8 4 = 6 + 8 4 = 6 + 8 4 ---- 1 1 1 1 10+ 4 10+ 4 10+ 4 10+ 4 2 + 32 + 32 + 32 + 3

C = 14 3C = 14 3C = 14 3C = 14 3 14 5 14 5 14 5 14 5 ((((9999)))) A = ( 1 2 A = ( 1 2 A = ( 1 2 A = ( 1 2 ----1 ) , B = 21 ) , B = 21 ) , B = 21 ) , B = 2 ±É¢ø, ( AB )±É¢ø, ( AB )±É¢ø, ( AB )±É¢ø, ( AB ) TTTT ¸¡ñ¸. 1¸¡ñ¸. 1¸¡ñ¸. 1¸¡ñ¸. 1 4444 ¾£÷× :¾£÷× :¾£÷× :¾£÷× : A B = ( 1 2 A B = ( 1 2 A B = ( 1 2 A B = ( 1 2 ----1 ) 21 ) 21 ) 21 ) 2 1111 4 4 4 4

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A B = A B = A B = A B = ( 2 + 2 ( 2 + 2 ( 2 + 2 ( 2 + 2 ---- 4 )4 )4 )4 ) A B = ( 0 )A B = ( 0 )A B = ( 0 )A B = ( 0 ) ( AB ) ( AB ) ( AB ) ( AB ) T T T T = ( 0 )= ( 0 )= ( 0 )= ( 0 ) ((((10101010)))) A = 8 A = 8 A = 8 A = 8 ----7 ÁüÚõ B = 9 7 ÁüÚõ B = 9 7 ÁüÚõ B = 9 7 ÁüÚõ B = 9 ----3 2 ±É¢ø 3 2 ±É¢ø 3 2 ±É¢ø 3 2 ±É¢ø

----2 4 6 2 4 6 2 4 6 2 4 6 ----1 1 1 1 ----5555 0 3 0 3 0 3 0 3 BA «½¢¨Âì ¸¡ñ¸. ( BA «½¢ þÕôÀ¢ý )BA «½¢¨Âì ¸¡ñ¸. ( BA «½¢ þÕôÀ¢ý )BA «½¢¨Âì ¸¡ñ¸. ( BA «½¢ þÕôÀ¢ý )BA «½¢¨Âì ¸¡ñ¸. ( BA «½¢ þÕôÀ¢ý )

¾£÷× :¾£÷× :¾£÷× :¾£÷× : BA = 9 BA = 9 BA = 9 BA = 9 ----3 2 8 3 2 8 3 2 8 3 2 8 ----7777 6 6 6 6 ----1 1 1 1 ----5 5 5 5 ----2 42 42 42 4 0 30 30 30 3 = 72+6+0 = 72+6+0 = 72+6+0 = 72+6+0 ----63636363----12+612+612+612+6 48+2+0 48+2+0 48+2+0 48+2+0 ----42424242----4444----15151515

BA = 78 BA = 78 BA = 78 BA = 78 ----69696969 50 50 50 50 ----61616161

((((11111111)))) a a a a I jI jI jI j = = = = 2 i 2 i 2 i 2 i ---- 3 j ±ýÈ ¯ÚôÒ¸¨Çì ¦¸¡ñÎ3 j ±ýÈ ¯ÚôÒ¸¨Çì ¦¸¡ñÎ3 j ±ýÈ ¯ÚôÒ¸¨Çì ¦¸¡ñÎ3 j ±ýÈ ¯ÚôÒ¸¨Çì ¦¸¡ñÎ Å¡¢¨º 2 x 3 ¯ûÇ «½¢ A = | a Å¡¢¨º 2 x 3 ¯ûÇ «½¢ A = | a Å¡¢¨º 2 x 3 ¯ûÇ «½¢ A = | a Å¡¢¨º 2 x 3 ¯ûÇ «½¢ A = | a ijijijij | «¨Áì¸×õ.| «¨Áì¸×õ.| «¨Áì¸×õ.| «¨Áì¸×õ.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : 2 x 3 Å¡¢¨ºÔ¨¼Â «½¢ A = 2 x 3 Å¡¢¨ºÔ¨¼Â «½¢ A = 2 x 3 Å¡¢¨ºÔ¨¼Â «½¢ A = 2 x 3 Å¡¢¨ºÔ¨¼Â «½¢ A = a a a a 1 11 11 11 1 a a a a 1 21 21 21 2 aaaa 1 31 31 31 3 a a a a 2222 1111 a a a a 2 22 22 22 2 a a a a 2 32 32 32 3 a a a a I jI jI jI j = 2 i = 2 i = 2 i = 2 i ---- 3 j3 j3 j3 j a a a a 1 1 1 1 1 1 1 1 = 2 = 2 = 2 = 2 ---- 3 = 1 a 3 = 1 a 3 = 1 a 3 = 1 a 2 1 2 1 2 1 2 1 = = = = 4 4 4 4 ---- 3 = 13 = 13 = 13 = 1 a a a a 1 21 21 21 2 = 2 = 2 = 2 = 2 ---- 6 = 4 a 6 = 4 a 6 = 4 a 6 = 4 a 2 22 22 22 2 = 4 = 4 = 4 = 4 ---- 6 = 26 = 26 = 26 = 2 a a a a 1 31 31 31 3 = 2 = 2 = 2 = 2 ---- 9 = 7 a 9 = 7 a 9 = 7 a 9 = 7 a 2 32 32 32 3 = 4 = 4 = 4 = 4 ---- 9 = 59 = 59 = 59 = 5

A = 1 4 7A = 1 4 7A = 1 4 7A = 1 4 7 1111 2 5 2 5 2 5 2 5 ((((12121212)))) a a a a I jI jI jI j = i = i = i = i ---- j ±ýÈ ¯ÚôÒ¸¨Çì ¦¸¡ñÎj ±ýÈ ¯ÚôÒ¸¨Çì ¦¸¡ñÎj ±ýÈ ¯ÚôÒ¸¨Çì ¦¸¡ñÎj ±ýÈ ¯ÚôÒ¸¨Çì ¦¸¡ñÎ

i + ji + ji + ji + j Å¡¢¨º 2 x 2 ¯ûÇ «½¢ A = | a Å¡¢¨º 2 x 2 ¯ûÇ «½¢ A = | a Å¡¢¨º 2 x 2 ¯ûÇ «½¢ A = | a Å¡¢¨º 2 x 2 ¯ûÇ «½¢ A = | a ijijijij | «¨Áì¸×õ. | «¨Áì¸×õ. | «¨Áì¸×õ. | «¨Áì¸×õ.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : 2 x 2 Å¡¢¨ºÔ¨¼Â «½¢ A = a 2 x 2 Å¡¢¨ºÔ¨¼Â «½¢ A = a 2 x 2 Å¡¢¨ºÔ¨¼Â «½¢ A = a 2 x 2 Å¡¢¨ºÔ¨¼Â «½¢ A = a 1 11 11 11 1 a a a a 1 21 21 21 2 a a a a 2 12 12 12 1 a a a a 2 22 22 22 2

a a a a I jI jI jI j = i = i = i = i ---- j j j j i + ji + ji + ji + j

aaaa 1 1 1 1 1 1 1 1 = 1 = 1 = 1 = 1 ---- 1 = 0 1 = 0 1 = 0 1 = 0 a a a a 2 12 12 12 1 = 2 = 2 = 2 = 2 ---- 1 = 11 = 11 = 11 = 1 1 + 1 2 1 + 1 2 1 + 1 2 1 + 1 2 + 1 3+ 1 3+ 1 3+ 1 3 a a a a 1 21 21 21 2 = 1 = 1 = 1 = 1 ---- 2 = 2 = 2 = 2 = ----1 a 1 a 1 a 1 a 2 22 22 22 2 = 2 = 2 = 2 = 2 ---- 2 = 02 = 02 = 02 = 0 1 + 2 3 2 + 21 + 2 3 2 + 21 + 2 3 2 + 21 + 2 3 2 + 2 A = 0 A = 0 A = 0 A = 0 1111 3333 ----1 01 01 01 0 3333

5555.... ¬Âò¦¾¡¨Ä ¬Âò¦¾¡¨Ä ¬Âò¦¾¡¨Ä ¬Âò¦¾¡¨Ä ÅÊÅ¢Âø ÅÊÅ¢Âø ÅÊÅ¢Âø ÅÊÅ¢Âø

((((1111)))) º¡ö×째¡½õ 45º¡ö×째¡½õ 45º¡ö×째¡½õ 45º¡ö×째¡½õ 45°°°° ÁüÚõ y ¦ÅðÎòÐñÎ 2 /5ÁüÚõ y ¦ÅðÎòÐñÎ 2 /5ÁüÚõ y ¦ÅðÎòÐñÎ 2 /5ÁüÚõ y ¦ÅðÎòÐñÎ 2 /5 ¬¸¢ÂÅü¨Èì ¦¸¡ñ¼ §¿÷째¡ðÊý ºÁýÀ¡ð¨¼ì ¬¸¢ÂÅü¨Èì ¦¸¡ñ¼ §¿÷째¡ðÊý ºÁýÀ¡ð¨¼ì ¬¸¢ÂÅü¨Èì ¦¸¡ñ¼ §¿÷째¡ðÊý ºÁýÀ¡ð¨¼ì ¬¸¢ÂÅü¨Èì ¦¸¡ñ¼ §¿÷째¡ðÊý ºÁýÀ¡ð¨¼ì ¸¡ñ¸.¸¡ñ¸.¸¡ñ¸.¸¡ñ¸. ¾£÷×¾£÷×¾£÷×¾£÷× ::::

º¡ö×째¡½õ tan º¡ö×째¡½õ tan º¡ö×째¡½õ tan º¡ö×째¡½õ tan θθθθ = 45= 45= 45= 45°°°°

m = tan 45m = tan 45m = tan 45m = tan 45°°°° ±É¢ø,±É¢ø,±É¢ø,±É¢ø, m = 1m = 1m = 1m = 1 y ¦ÅðÎò ÐñÎ c = 2 / 5y ¦ÅðÎò ÐñÎ c = 2 / 5y ¦ÅðÎò ÐñÎ c = 2 / 5y ¦ÅðÎò ÐñÎ c = 2 / 5 §¾¨ÅÂ¡É §¿÷째¡ðÊý ºÁýÀ¡Î§¾¨ÅÂ¡É §¿÷째¡ðÊý ºÁýÀ¡Î§¾¨ÅÂ¡É §¿÷째¡ðÊý ºÁýÀ¡Î§¾¨ÅÂ¡É §¿÷째¡ðÊý ºÁýÀ¡Î y = m x + cy = m x + cy = m x + cy = m x + c y = 1 (x) + 2 / 5y = 1 (x) + 2 / 5y = 1 (x) + 2 / 5y = 1 (x) + 2 / 5 5 y = 5 x + 25 y = 5 x + 25 y = 5 x + 25 y = 5 x + 2 5 x 5 x 5 x 5 x ---- 5 y + 2 = 05 y + 2 = 05 y + 2 = 05 y + 2 = 0

((((2222)))) ( 1,2 )( 1,2 )( 1,2 )( 1,2 ) ±ýÈ Ò±ýÈ Ò±ýÈ Ò±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøÅÐõ y = 7 ±ýÈ ûÇ¢ ÅÆ¢î ¦ºøÅÐõ y = 7 ±ýÈ ûÇ¢ ÅÆ¢î ¦ºøÅÐõ y = 7 ±ýÈ ûÇ¢ ÅÆ¢î ¦ºøÅÐõ y = 7 ±ýÈ §¿÷째¡ðÊüÌ ¦ºíÌò¾¡ÉÐÁ¡É §¿÷째¡ðÊý §¿÷째¡ðÊüÌ ¦ºíÌò¾¡ÉÐÁ¡É §¿÷째¡ðÊý §¿÷째¡ðÊüÌ ¦ºíÌò¾¡ÉÐÁ¡É §¿÷째¡ðÊý §¿÷째¡ðÊüÌ ¦ºíÌò¾¡ÉÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡ð¨¼ì ¸¡ñ¸.ºÁýÀ¡ð¨¼ì ¸¡ñ¸.ºÁýÀ¡ð¨¼ì ¸¡ñ¸.ºÁýÀ¡ð¨¼ì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : §¾¨ÅÂ¡É §¿÷째¡Î x «îÍìÌ ¦ºíÌò¾¡¸×õ §¾¨ÅÂ¡É §¿÷째¡Î x «îÍìÌ ¦ºíÌò¾¡¸×õ §¾¨ÅÂ¡É §¿÷째¡Î x «îÍìÌ ¦ºíÌò¾¡¸×õ §¾¨ÅÂ¡É §¿÷째¡Î x «îÍìÌ ¦ºíÌò¾¡¸×õ y «îÍìÌy «îÍìÌy «îÍìÌy «îÍìÌ þ¨½Â¡¸×õ þÕôÀ¾¡ø x = c ¬Ìõ.þ¨½Â¡¸×õ þÕôÀ¾¡ø x = c ¬Ìõ.þ¨½Â¡¸×õ þÕôÀ¾¡ø x = c ¬Ìõ.þ¨½Â¡¸×õ þÕôÀ¾¡ø x = c ¬Ìõ. §ÁÖõ ( 1,2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøž¡ø §ÁÖõ ( 1,2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøž¡ø §ÁÖõ ( 1,2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøž¡ø §ÁÖõ ( 1,2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøž¡ø §¾¨ÅÂ¡É ºÁýÀ¡Î§¾¨ÅÂ¡É ºÁýÀ¡Î§¾¨ÅÂ¡É ºÁýÀ¡Î§¾¨ÅÂ¡É ºÁýÀ¡Î x = 1 ( or ) x x = 1 ( or ) x x = 1 ( or ) x x = 1 ( or ) x ---- 1 = 01 = 01 = 01 = 0

((((3333)))) ( ( ( ( ----5,5,5,5,----2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøÅÐõ 2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøÅÐõ 2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøÅÐõ 2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøÅÐõ ¬Â«î͸ÙìÌ þ¨½Â¡ÉÐÁ¡É §¿÷째¡Î¸Ç¢ý¬Â«î͸ÙìÌ þ¨½Â¡ÉÐÁ¡É §¿÷째¡Î¸Ç¢ý¬Â«î͸ÙìÌ þ¨½Â¡ÉÐÁ¡É §¿÷째¡Î¸Ç¢ý¬Â«î͸ÙìÌ þ¨½Â¡ÉÐÁ¡É §¿÷째¡Î¸Ç¢ý ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.ºÁýÀ¡Î¸¨Çì ¸¡ñ¸.ºÁýÀ¡Î¸¨Çì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : x «îÍìÌ þ¨½Â¡¸×õ ( x «îÍìÌ þ¨½Â¡¸×õ ( x «îÍìÌ þ¨½Â¡¸×õ ( x «îÍìÌ þ¨½Â¡¸×õ ( ----5,5,5,5,----2 ) ±ýÈ ÒûÇ¢ 2 ) ±ýÈ ÒûÇ¢ 2 ) ±ýÈ ÒûÇ¢ 2 ) ±ýÈ ÒûÇ¢ ÅƢøÅÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡Î y = ÅƢøÅÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡Î y = ÅƢøÅÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡Î y = ÅƢøÅÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡Î y = ----2.2.2.2. y «îÍìÌ þ¨½Â¡¸×õ ( y «îÍìÌ þ¨½Â¡¸×õ ( y «îÍìÌ þ¨½Â¡¸×õ ( y «îÍìÌ þ¨½Â¡¸×õ ( ----5,5,5,5,----2 ) ±ýÈ ÒûÇ¢ 2 ) ±ýÈ ÒûÇ¢ 2 ) ±ýÈ ÒûÇ¢ 2 ) ±ýÈ ÒûÇ¢ ÅƢøÅÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡Î x = ÅƢøÅÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡Î x = ÅƢøÅÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡Î x = ÅƢøÅÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡Î x = ----5.5.5.5. ((((4444)))) ( ( ( ( ----2,3 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøÅÐõ, º¡ö× 1 /32,3 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøÅÐõ, º¡ö× 1 /32,3 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøÅÐõ, º¡ö× 1 /32,3 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøÅÐõ, º¡ö× 1 /3 ¯¨¼ÂÐÁ¡É ¯¨¼ÂÐÁ¡É ¯¨¼ÂÐÁ¡É ¯¨¼ÂÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡ð¨¼ì ¸¡ñ¸.§¿÷째¡ðÊý ºÁýÀ¡ð¨¼ì ¸¡ñ¸.§¿÷째¡ðÊý ºÁýÀ¡ð¨¼ì ¸¡ñ¸.§¿÷째¡ðÊý ºÁýÀ¡ð¨¼ì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

( x( x( x( x 1111 , y, y, y, y1111 ) ÁüÚõ º¡ö× m ¦¸¡Îì¸ôÀð¼¡ø ) ÁüÚõ º¡ö× m ¦¸¡Îì¸ôÀð¼¡ø ) ÁüÚõ º¡ö× m ¦¸¡Îì¸ôÀð¼¡ø ) ÁüÚõ º¡ö× m ¦¸¡Îì¸ôÀð¼¡ø §¿÷째¡ðÊý ºÁýÀ¡Î§¿÷째¡ðÊý ºÁýÀ¡Î§¿÷째¡ðÊý ºÁýÀ¡Î§¿÷째¡ðÊý ºÁýÀ¡Î y y y y ---- yyyy1 1 1 1 = m ( x = m ( x = m ( x = m ( x ---- xxxx1111 )))) y y y y ---- 3 = 1 ( x + 2 )3 = 1 ( x + 2 )3 = 1 ( x + 2 )3 = 1 ( x + 2 ) 3333 3 y 3 y 3 y 3 y ---- 9 = x + 29 = x + 29 = x + 29 = x + 2 x x x x ---- 3 y + 2 + 9 = 03 y + 2 + 9 = 03 y + 2 + 9 = 03 y + 2 + 9 = 0

x x x x ---- 3 y + 11 = 03 y + 11 = 03 y + 11 = 03 y + 11 = 0 ((((5555)))) x x x x ---- 2 y + 3 = 0 ±ýÈ §¿÷째¡ðÊüÌ 2 y + 3 = 0 ±ýÈ §¿÷째¡ðÊüÌ 2 y + 3 = 0 ±ýÈ §¿÷째¡ðÊüÌ 2 y + 3 = 0 ±ýÈ §¿÷째¡ðÊüÌ ¦ºíÌò¾¡¸×õ ( 1, ¦ºíÌò¾¡¸×õ ( 1, ¦ºíÌò¾¡¸×õ ( 1, ¦ºíÌò¾¡¸×õ ( 1, ----2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î 2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î 2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î 2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºøÅÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡ð¨¼ì ¸¡ñ¸.¦ºøÅÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡ð¨¼ì ¸¡ñ¸.¦ºøÅÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡ð¨¼ì ¸¡ñ¸.¦ºøÅÐÁ¡É §¿÷째¡ðÊý ºÁýÀ¡ð¨¼ì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : x x x x ---- 2 y + 3 = 0 ±ýÈ §¿÷째¡ðÊüÌ 2 y + 3 = 0 ±ýÈ §¿÷째¡ðÊüÌ 2 y + 3 = 0 ±ýÈ §¿÷째¡ðÊüÌ 2 y + 3 = 0 ±ýÈ §¿÷째¡ðÊüÌ ¦ºí̦ºí̦ºí̦ºíÌò¾¡É ò¾¡É ò¾¡É ò¾¡É §¸¡ðÊý§¸¡ðÊý§¸¡ðÊý§¸¡ðÊý ºÁýÀ¡Î 2 x + y + k = 0ºÁýÀ¡Î 2 x + y + k = 0ºÁýÀ¡Î 2 x + y + k = 0ºÁýÀ¡Î 2 x + y + k = 0 «Ð ( 1,«Ð ( 1,«Ð ( 1,«Ð ( 1,----2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºø¸¢ÈÐ. ±É¢ø,2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºø¸¢ÈÐ. ±É¢ø,2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºø¸¢ÈÐ. ±É¢ø,2 ) ±ýÈ ÒûÇ¢ ÅÆ¢î ¦ºø¸¢ÈÐ. ±É¢ø,

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2 (1) + ( 2 (1) + ( 2 (1) + ( 2 (1) + ( ----2 ) + k = 02 ) + k = 02 ) + k = 02 ) + k = 0 2 2 2 2 ---- 2 + k = 02 + k = 02 + k = 02 + k = 0 K = 0K = 0K = 0K = 0 §¾¨ÅÂ¡É §¿÷째¡ðÊý ºÁýÀ¡Î§¾¨ÅÂ¡É §¿÷째¡ðÊý ºÁýÀ¡Î§¾¨ÅÂ¡É §¿÷째¡ðÊý ºÁýÀ¡Î§¾¨ÅÂ¡É §¿÷째¡ðÊý ºÁýÀ¡Î

2 x + y = 02 x + y = 02 x + y = 02 x + y = 0 ((((6666)))) ´Õ ´Õ ´Õ ´Õ §¿÷째¡ðÊý x ÁüÚõ y ¦ÅðÎòÐñθû§¿÷째¡ðÊý x ÁüÚõ y ¦ÅðÎòÐñθû§¿÷째¡ðÊý x ÁüÚõ y ¦ÅðÎòÐñθû§¿÷째¡ðÊý x ÁüÚõ y ¦ÅðÎòÐñθû ӨȧÂӨȧÂӨȧÂӨȧ ----2 / 7 ÁüÚõ 3 / 5 ±É¢ø, «ì§¸¡ðÊý 2 / 7 ÁüÚõ 3 / 5 ±É¢ø, «ì§¸¡ðÊý 2 / 7 ÁüÚõ 3 / 5 ±É¢ø, «ì§¸¡ðÊý 2 / 7 ÁüÚõ 3 / 5 ±É¢ø, «ì§¸¡ðÊý ºÁýÀ¡ð¨¼ì ¸¡ñ¸.ºÁýÀ¡ð¨¼ì ¸¡ñ¸.ºÁýÀ¡ð¨¼ì ¸¡ñ¸.ºÁýÀ¡ð¨¼ì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : x ¦ÅðÎòÐñÎ a = x ¦ÅðÎòÐñÎ a = x ¦ÅðÎòÐñÎ a = x ¦ÅðÎòÐñÎ a = ----2 / 72 / 72 / 72 / 7 y ¦ÅðÎòÐñÎ b = 3 / 5y ¦ÅðÎòÐñÎ b = 3 / 5y ¦ÅðÎòÐñÎ b = 3 / 5y ¦ÅðÎòÐñÎ b = 3 / 5 §¾¨ÅÂ¡É §¿÷째¡ðÊý ºÁýÀ¡Î §¾¨ÅÂ¡É §¿÷째¡ðÊý ºÁýÀ¡Î §¾¨ÅÂ¡É §¿÷째¡ðÊý ºÁýÀ¡Î §¾¨ÅÂ¡É §¿÷째¡ðÊý ºÁýÀ¡Î x + y x + y x + y x + y = 1= 1= 1= 1 a ba ba ba b x + y x + y x + y x + y = 1= 1= 1= 1 ----2 / 7 3 / 52 / 7 3 / 52 / 7 3 / 52 / 7 3 / 5

7 x + 5 y 7 x + 5 y 7 x + 5 y 7 x + 5 y = 1= 1= 1= 1 ----2 32 32 32 3

----21 x + 10 y 21 x + 10 y 21 x + 10 y 21 x + 10 y = 6= 6= 6= 6

21212121 x x x x ---- 10 y 10 y 10 y 10 y + 6 = 0+ 6 = 0+ 6 = 0+ 6 = 0 ((((7777)))) ( ( ( ( ----3,5 ) ÁüÚõ ( 4,3,5 ) ÁüÚõ ( 4,3,5 ) ÁüÚõ ( 4,3,5 ) ÁüÚõ ( 4,----9 ) ¬¸¢Â ÒûÇ¢¸¨Ç 9 ) ¬¸¢Â ÒûÇ¢¸¨Ç 9 ) ¬¸¢Â ÒûÇ¢¸¨Ç 9 ) ¬¸¢Â ÒûÇ¢¸¨Ç þ¨½ìÌõ §¸¡ðÎòÐñ¨¼ 1 : 6 ±ýÈ Å¢¸¢¾ò¾¢ø þ¨½ìÌõ §¸¡ðÎòÐñ¨¼ 1 : 6 ±ýÈ Å¢¸¢¾ò¾¢ø þ¨½ìÌõ §¸¡ðÎòÐñ¨¼ 1 : 6 ±ýÈ Å¢¸¢¾ò¾¢ø þ¨½ìÌõ §¸¡ðÎòÐñ¨¼ 1 : 6 ±ýÈ Å¢¸¢¾ò¾¢ø ¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¨Âì ¸¡ñ¸.¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¨Âì ¸¡ñ¸.¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¨Âì ¸¡ñ¸.¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¨Âì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

( x( x( x( x1111 , y, y, y, y1111 ) , ( x) , ( x) , ( x) , ( x2222 , y, y, y, y2222 ) ) ) ) l : m: m: m: m ( ( ( ( ----3 , 5 ) , ( 4 , 3 , 5 ) , ( 4 , 3 , 5 ) , ( 4 , 3 , 5 ) , ( 4 , ----9 ) 1 : 69 ) 1 : 69 ) 1 : 69 ) 1 : 6 ¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢

P P P P = = = = l xxxx2222 + m x+ m x+ m x+ m x1 1 1 1 , , , , l yyyy2222 + m y+ m y+ m y+ m y1111

l + m + m + m + m l + mmmm = (1 x 4) + (6 x = (1 x 4) + (6 x = (1 x 4) + (6 x = (1 x 4) + (6 x ----3) , (1 x 3) , (1 x 3) , (1 x 3) , (1 x ----9) + (6 x 5)9) + (6 x 5)9) + (6 x 5)9) + (6 x 5) 1 + 6 1 + 61 + 6 1 + 61 + 6 1 + 61 + 6 1 + 6 = 4 = 4 = 4 = 4 ---- 18 , 18 , 18 , 18 , ----9 + 309 + 309 + 309 + 30 7 77 77 77 7 = = = = ----14 , 2114 , 2114 , 2114 , 21 7 77 77 77 7 P = ( P = ( P = ( P = ( ----2 , 3 )2 , 3 )2 , 3 )2 , 3 ) ((((8888)))) ( 3,5 ) ÁüÚõ ( 8,( 3,5 ) ÁüÚõ ( 8,( 3,5 ) ÁüÚõ ( 8,( 3,5 ) ÁüÚõ ( 8,10101010 ) ¬¸¢Â ÒûÇ¢¸¨Ç ) ¬¸¢Â ÒûÇ¢¸¨Ç ) ¬¸¢Â ÒûÇ¢¸¨Ç ) ¬¸¢Â ÒûÇ¢¸¨Ç þ¨½ìÌõ §¸¡ðÎòÐñ¨¼ þ¨½ìÌõ §¸¡ðÎòÐñ¨¼ þ¨½ìÌõ §¸¡ðÎòÐñ¨¼ þ¨½ìÌõ §¸¡ðÎòÐñ¨¼ 2222 : : : : 3333 ±ýÈ Å¢¸¢¾ò¾¢ø ±ýÈ Å¢¸¢¾ò¾¢ø ±ýÈ Å¢¸¢¾ò¾¢ø ±ýÈ Å¢¸¢¾ò¾¢ø ¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¨Âì ¸¡ñ¸.¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¨Âì ¸¡ñ¸.¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¨Âì ¸¡ñ¸.¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¨Âì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

( x( x( x( x1111 , y, y, y, y1111 ) , ( x) , ( x) , ( x) , ( x2222 , y, y, y, y2222 ) ) ) ) l : m: m: m: m ( 3 , 5 ) , ( 8 , 10 ) 2 : 3( 3 , 5 ) , ( 8 , 10 ) 2 : 3( 3 , 5 ) , ( 8 , 10 ) 2 : 3( 3 , 5 ) , ( 8 , 10 ) 2 : 3 ¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢¯ðÒÈÁ¡¸ À¢¡¢ìÌõ ÒûÇ¢

P P P P = = = = l xxxx2222 + m x+ m x+ m x+ m x1 1 1 1 , , , , l yyyy2222 + m y+ m y+ m y+ m y1111

l + m + m + m + m l + mmmm

= (2 x 8) + (3 x 3) , (2 x 10= (2 x 8) + (3 x 3) , (2 x 10= (2 x 8) + (3 x 3) , (2 x 10= (2 x 8) + (3 x 3) , (2 x 10) + (3 x 5)) + (3 x 5)) + (3 x 5)) + (3 x 5) 2 + 3 2 + 32 + 3 2 + 32 + 3 2 + 32 + 3 2 + 3 = 16 + 9 , 20 + 15= 16 + 9 , 20 + 15= 16 + 9 , 20 + 15= 16 + 9 , 20 + 15 5 55 55 55 5 = 25 , 35= 25 , 35= 25 , 35= 25 , 35 5 55 55 55 5 P = ( 5 , 7 )P = ( 5 , 7 )P = ( 5 , 7 )P = ( 5 , 7 )

((((9999)))) A (2,3), B (4,0) ÁüÚõ C (6,A (2,3), B (4,0) ÁüÚõ C (6,A (2,3), B (4,0) ÁüÚõ C (6,A (2,3), B (4,0) ÁüÚõ C (6,----3) ¬¸¢Â ÒûÇ¢¸û 3) ¬¸¢Â ÒûÇ¢¸û 3) ¬¸¢Â ÒûÇ¢¸û 3) ¬¸¢Â ÒûÇ¢¸û

´§Ã §¿÷째¡ðÊø «¨ÁóÐûÇÉ ±É ¿¢åÀ¢.´§Ã §¿÷째¡ðÊø «¨ÁóÐûÇÉ ±É ¿¢åÀ¢.´§Ã §¿÷째¡ðÊø «¨ÁóÐûÇÉ ±É ¿¢åÀ¢.´§Ã §¿÷째¡ðÊø «¨ÁóÐûÇÉ ±É ¿¢åÀ¢. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : A (2,3), B A (2,3), B A (2,3), B A (2,3), B (4,0) ÁüÚõ C (6,(4,0) ÁüÚõ C (6,(4,0) ÁüÚõ C (6,(4,0) ÁüÚõ C (6,----3) ¬¸¢Â ÒûÇ¢¸¨Çì 3) ¬¸¢Â ÒûÇ¢¸¨Çì 3) ¬¸¢Â ÒûÇ¢¸¨Çì 3) ¬¸¢Â ÒûÇ¢¸¨Çì ¦¸¡ñ¼ Ó째¡½ò¾¢ý ÀÃôÀÇצ¸¡ñ¼ Ó째¡½ò¾¢ý ÀÃôÀÇצ¸¡ñ¼ Ó째¡½ò¾¢ý ÀÃôÀÇצ¸¡ñ¼ Ó째¡½ò¾¢ý ÀÃôÀÇ× xxxx1111 xxxx2222 xxxx3333 xxxx1111

= = = = ½ ½ ½ ½ º.«.º.«.º.«.º.«. YYYY1111 yyyy2222 yyyy3333 yyyy1111

2222 4444 6666 2 2 2 2

==== ½ ½ ½ ½ º.«.º.«.º.«.º.«. 3333 0000 ----3333 3 3 3 3

==== ½ ½ ½ ½ ( ( ( ( 0 0 0 0 ---- 12 + 1812 + 1812 + 1812 + 18) ) ) ) ---- (((( 12 + 0 12 + 0 12 + 0 12 + 0 ---- 6 6 6 6 ) ) ) ) º«. º«. º«. º«.

= ½ = ½ = ½ = ½ (((( 6 6 6 6 ---- 6666 )))) º.«.º.«.º.«.º.«.

= = = = ½ ½ ½ ½ x x x x 0000 º.«.º.«.º.«.º.«. = 0= 0= 0= 0 ±É§Å ¦¸¡Îì¸ôÀð¼ ÒûÇ¢¸û ´§Ã §¿÷째¡ðÊø ±É§Å ¦¸¡Îì¸ôÀð¼ ÒûÇ¢¸û ´§Ã §¿÷째¡ðÊø ±É§Å ¦¸¡Îì¸ôÀð¼ ÒûÇ¢¸û ´§Ã §¿÷째¡ðÊø ±É§Å ¦¸¡Îì¸ôÀð¼ ÒûÇ¢¸û ´§Ã §¿÷째¡ðÊø «¨Á¸¢ýÈÉ.«¨Á¸¢ýÈÉ.«¨Á¸¢ýÈÉ.«¨Á¸¢ýÈÉ.

((((10101010)))) A (6,7), B (A (6,7), B (A (6,7), B (A (6,7), B (----4,1) ÁüÚõ C (a,4,1) ÁüÚõ C (a,4,1) ÁüÚõ C (a,4,1) ÁüÚõ C (a,----9) ¬¸¢Â 9) ¬¸¢Â 9) ¬¸¢Â 9) ¬¸¢Â

ÒûÇ¢¸¨Ç ¯îº¢¸Ç¡¸ì ¦¸¡ñ¼ ÒûÇ¢¸¨Ç ¯îº¢¸Ç¡¸ì ¦¸¡ñ¼ ÒûÇ¢¸¨Ç ¯îº¢¸Ç¡¸ì ¦¸¡ñ¼ ÒûÇ¢¸¨Ç ¯îº¢¸Ç¡¸ì ¦¸¡ñ¼ ∆∆∆∆ ABC ý ÀÃôÒ ABC ý ÀÃôÒ ABC ý ÀÃôÒ ABC ý ÀÃôÒ 68 º.«. ±É¢ø, a ý Á¾¢ô¨Àì ¸¡ñ¸.68 º.«. ±É¢ø, a ý Á¾¢ô¨Àì ¸¡ñ¸.68 º.«. ±É¢ø, a ý Á¾¢ô¨Àì ¸¡ñ¸.68 º.«. ±É¢ø, a ý Á¾¢ô¨Àì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× :

∆∆∆∆ ABC ý ÀÃôÒABC ý ÀÃôÒABC ý ÀÃôÒABC ý ÀÃôÒ = 68 º.«.= 68 º.«.= 68 º.«.= 68 º.«. xxxx1111 xxxx2222 xxxx3333 xxxx1111

½ = ½ = ½ = ½ = 68 º.«.68 º.«.68 º.«.68 º.«. YYYY1111 yyyy2222 yyyy3333 yyyy1111

6666 ----4 4 4 4 aaaa 6666

½ = ½ = ½ = ½ = 68686868 º.«.º.«.º.«.º.«. 7777 1111 ----9 79 79 79 7

( ( ( ( 6 +6 +6 +6 + 36363636 + + + + 7 a7 a7 a7 a) ) ) ) ---- ( ( ( ( ----28282828 + + + + aaaa ---- 54 54 54 54 ) ) ) ) = 68 x 2= 68 x 2= 68 x 2= 68 x 2 º«. º«. º«. º«.

(((( 42 + 7 a 42 + 7 a 42 + 7 a 42 + 7 a + 82 + 82 + 82 + 82 ---- aaaa ) ) ) ) = 136 º.«.= 136 º.«.= 136 º.«.= 136 º.«. 124 124 124 124 ---- 6 a = 1366 a = 1366 a = 1366 a = 136 ----6 a = 136 6 a = 136 6 a = 136 6 a = 136 ---- 124124124124 a = a = a = a = 12121212 ----6 6 6 6 a = a = a = a = ----2222

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((((11111111)))) (a,1),(a,1),(a,1),(a,1), (1,2), ÁüÚõ (0, b+1) ¬¸¢Â ÒûÇ¢¸û (1,2), ÁüÚõ (0, b+1) ¬¸¢Â ÒûÇ¢¸û (1,2), ÁüÚõ (0, b+1) ¬¸¢Â ÒûÇ¢¸û (1,2), ÁüÚõ (0, b+1) ¬¸¢Â ÒûÇ¢¸û ´§Ã §¿÷째¡ðÊø «¨Áó¾¡ø 1 + 1 = 1´§Ã §¿÷째¡ðÊø «¨Áó¾¡ø 1 + 1 = 1´§Ã §¿÷째¡ðÊø «¨Áó¾¡ø 1 + 1 = 1´§Ã §¿÷째¡ðÊø «¨Áó¾¡ø 1 + 1 = 1 ±É ¿¢Ú׸. ±É ¿¢Ú׸. ±É ¿¢Ú׸. ±É ¿¢Ú׸. a b a b a b a b

¾£÷× :¾£÷× :¾£÷× :¾£÷× : (a,1), (1,2), ÁüÚõ (0, b+1) ¬¸¢Â ÒûÇ¢¸û ´§Ã (a,1), (1,2), ÁüÚõ (0, b+1) ¬¸¢Â ÒûÇ¢¸û ´§Ã (a,1), (1,2), ÁüÚõ (0, b+1) ¬¸¢Â ÒûÇ¢¸û ´§Ã (a,1), (1,2), ÁüÚõ (0, b+1) ¬¸¢Â ÒûÇ¢¸û ´§Ã §¿÷째¡ðÊø «¨ÁÔõ ±É¢ø,§¿÷째¡ðÊø «¨ÁÔõ ±É¢ø,§¿÷째¡ðÊø «¨ÁÔõ ±É¢ø,§¿÷째¡ðÊø «¨ÁÔõ ±É¢ø,

∆∆∆∆ ý ÀÃôÀÇ× = 0ý ÀÃôÀÇ× = 0ý ÀÃôÀÇ× = 0ý ÀÃôÀÇ× = 0 xxxx1111 xxxx2222 xxxx3333 xxxx1111

½ ½ ½ ½ = 0= 0= 0= 0 YYYY1111 yyyy2222 yyyy3333 yyyy1111

a a a a 1111 0000 aaaa

½ = 0½ = 0½ = 0½ = 0 1111 2 2 2 2 b+1b+1b+1b+1 1111

((((2a + b + 1 + 0) 2a + b + 1 + 0) 2a + b + 1 + 0) 2a + b + 1 + 0) ---- ( 1 + 0 + ab + a( 1 + 0 + ab + a( 1 + 0 + ab + a( 1 + 0 + ab + a) = 0 ) = 0 ) = 0 ) = 0

(((( 2a + b + 1 2a + b + 1 2a + b + 1 2a + b + 1 ---- 1 1 1 1 ---- ab ab ab ab ---- a a a a )))) = 0= 0= 0= 0 a + b a + b a + b a + b ---- abababab = = = = 0000 a + b = aba + b = aba + b = aba + b = ab

( ( ( ( ÷÷÷÷ ab) a + b = abab) a + b = abab) a + b = abab) a + b = ab ab ab abab ab abab ab abab ab ab

1 + 1 = 11 + 1 = 11 + 1 = 11 + 1 = 1 a ba ba ba b

((((12121212)))) A (a,A (a,A (a,A (a,----3), B (3,a) ÁüÚõ C (3), B (3,a) ÁüÚõ C (3), B (3,a) ÁüÚõ C (3), B (3,a) ÁüÚõ C (----1,5) ¬¸¢Â ÒûÇ¢¸¨Ç 1,5) ¬¸¢Â ÒûÇ¢¸¨Ç 1,5) ¬¸¢Â ÒûÇ¢¸¨Ç 1,5) ¬¸¢Â ÒûÇ¢¸¨Ç

¯îº¢¸Ç¡¸ì ¦¸¡ñ¼ ¯îº¢¸Ç¡¸ì ¦¸¡ñ¼ ¯îº¢¸Ç¡¸ì ¦¸¡ñ¼ ¯îº¢¸Ç¡¸ì ¦¸¡ñ¼ ∆∆∆∆ ABC ý ÀÃôÒ 12 º.«. ABC ý ÀÃôÒ 12 º.«. ABC ý ÀÃôÒ 12 º.«. ABC ý ÀÃôÒ 12 º.«. ±É¢ø, a ý Á¾¢ô¨Àì ¸¡ñ¸.±É¢ø, a ý Á¾¢ô¨Àì ¸¡ñ¸.±É¢ø, a ý Á¾¢ô¨Àì ¸¡ñ¸.±É¢ø, a ý Á¾¢ô¨Àì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× :

∆∆∆∆ ABC ý ÀÃôÒ = 12 º.«.ABC ý ÀÃôÒ = 12 º.«.ABC ý ÀÃôÒ = 12 º.«.ABC ý ÀÃôÒ = 12 º.«. xxxx1111 xxxx2222 xxxx3333 xxxx1111

½ = ½ = ½ = ½ = 12 º.«.12 º.«.12 º.«.12 º.«. YYYY1111 yyyy2222 yyyy3333 yyyy1111

a 3 a 3 a 3 a 3 ----1 a 1 a 1 a 1 a

½ = ½ = ½ = ½ = 12121212 º.«.º.«.º.«.º.«. ----3 a 5 3 a 5 3 a 5 3 a 5 ----3 3 3 3

( a( a( a( a2222 + 15 + 3) + 15 + 3) + 15 + 3) + 15 + 3) ---- ( ( ( ( ----9 9 9 9 ---- a + 5aa + 5aa + 5aa + 5a ) ) ) ) = = = = 12121212 x x x x 2 º«. 2 º«. 2 º«. 2 º«.

(((( a a a a 2222 + 18 + 9 + 18 + 9 + 18 + 9 + 18 + 9 ---- 4a 4a 4a 4a ) ) ) ) = 24= 24= 24= 24 º.«.º.«.º.«.º.«. aaaa 2222 ---- 4a + 27 4a + 27 4a + 27 4a + 27 ---- 24242424 = = = = 0000 aaaa 2222 ---- 4a + 3 =4a + 3 =4a + 3 =4a + 3 = 0000 (a (a (a (a ---- 3)(a 3)(a 3)(a 3)(a ---- 1)1)1)1) = 0= 0= 0= 0

a = a = a = a = 3 , 13 , 13 , 13 , 1

((((13131313)))) ´Õ Åð¼ò¾¢ý ¨ÁÂõ (´Õ Åð¼ò¾¢ý ¨ÁÂõ (´Õ Åð¼ò¾¢ý ¨ÁÂõ (´Õ Åð¼ò¾¢ý ¨ÁÂõ (----6,4). «ùÅð¼ò¾¢ý 6,4). «ùÅð¼ò¾¢ý 6,4). «ùÅð¼ò¾¢ý 6,4). «ùÅð¼ò¾¢ý ´Õ  ¬¾¢ôÒûÇ¢ ±É¢ø, Áü¦È¡Õ ´Õ  ¬¾¢ôÒûÇ¢ ±É¢ø, Áü¦È¡Õ ´Õ  ¬¾¢ôÒûÇ¢ ±É¢ø, Áü¦È¡Õ ´Õ  ¬¾¢ôÒûÇ¢ ±É¢ø, Áü¦È¡Õ ӨɨÂì ¸¡ñ¸.ӨɨÂì ¸¡ñ¸.ӨɨÂì ¸¡ñ¸.ӨɨÂì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : Åð¼ò¾¢ý ¨ÁÂõ ±ýÀРŢð¼ò¾¢ý ¿ÎôÒûÇ¢. Åð¼ò¾¢ý ¨ÁÂõ ±ýÀРŢð¼ò¾¢ý ¿ÎôÒûÇ¢. Åð¼ò¾¢ý ¨ÁÂõ ±ýÀРŢð¼ò¾¢ý ¿ÎôÒûÇ¢. Åð¼ò¾¢ý ¨ÁÂõ ±ýÀРŢð¼ò¾¢ý ¿ÎôÒûÇ¢. ´Õ  (0,0). ¿ÎôÒûÇ¢ (´Õ  (0,0). ¿ÎôÒûÇ¢ (´Õ  (0,0). ¿ÎôÒûÇ¢ (´Õ  (0,0). ¿ÎôÒûÇ¢ (----6,4).6,4).6,4).6,4). ¿ÎôÒûÇ¢ = x¿ÎôÒûÇ¢ = x¿ÎôÒûÇ¢ = x¿ÎôÒûÇ¢ = x 1111 + x+ x+ x+ x2222 , y , y , y , y 1111 + y+ y+ y+ y 2222 2 22 22 22 2

( ( ( ( ----6 , 4 ) = 0 + x 6 , 4 ) = 0 + x 6 , 4 ) = 0 + x 6 , 4 ) = 0 + x 2222 , 0 + , 0 + , 0 + , 0 + yyyy 2222 2 22 22 22 2 X X X X 2222 = = = = ----6 ; y 6 ; y 6 ; y 6 ; y 2222 = 4= 4= 4= 4 2 22 22 22 2 X X X X 2222 = = = = ----12 ; y 12 ; y 12 ; y 12 ; y 2222 = 8= 8= 8= 8

Áü¦È¡Õ  = ( Áü¦È¡Õ  = ( Áü¦È¡Õ  = ( Áü¦È¡Õ  = ( ----12 , 8 )12 , 8 )12 , 8 )12 , 8 )

((((14141414)))) ( 7,3 ), ( 6,1 ),( 7,3 ), ( 6,1 ),( 7,3 ), ( 6,1 ),( 7,3 ), ( 6,1 ), ( 8,2 ) ÁüÚõ ( p,4 ) ±ýÀÉ ( 8,2 ) ÁüÚõ ( p,4 ) ±ýÀÉ ( 8,2 ) ÁüÚõ ( p,4 ) ±ýÀÉ ( 8,2 ) ÁüÚõ ( p,4 ) ±ýÀÉ µ÷ þ¨½¸Ãò¾¢ý Å¡¢¨ºôÀÊ «¨Áó¾ ¯îº¢¸û µ÷ þ¨½¸Ãò¾¢ý Å¡¢¨ºôÀÊ «¨Áó¾ ¯îº¢¸û µ÷ þ¨½¸Ãò¾¢ý Å¡¢¨ºôÀÊ «¨Áó¾ ¯îº¢¸û µ÷ þ¨½¸Ãò¾¢ý Å¡¢¨ºôÀÊ «¨Áó¾ ¯îº¢¸û ±É¢ø, p ý Á¾¢ô¨Àì ¸¡ñ¸.±É¢ø, p ý Á¾¢ô¨Àì ¸¡ñ¸.±É¢ø, p ý Á¾¢ô¨Àì ¸¡ñ¸.±É¢ø, p ý Á¾¢ô¨Àì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : A ( 7,3 ), B ( 6,1 ), C ( 8,2 ) ÁüÚõ D ( p,4 ) ±ýÀÉ A ( 7,3 ), B ( 6,1 ), C ( 8,2 ) ÁüÚõ D ( p,4 ) ±ýÀÉ A ( 7,3 ), B ( 6,1 ), C ( 8,2 ) ÁüÚõ D ( p,4 ) ±ýÀÉ A ( 7,3 ), B ( 6,1 ), C ( 8,2 ) ÁüÚõ D ( p,4 ) ±ýÀÉ µ÷ þ¨½¸Ãò¾¢ý Å¡¢¨ºôÀÊ «¨Áó¾ ¯îº¢¸û µ÷ þ¨½¸Ãò¾¢ý Å¡¢¨ºôÀÊ «¨Áó¾ ¯îº¢¸û µ÷ þ¨½¸Ãò¾¢ý Å¡¢¨ºôÀÊ «¨Áó¾ ¯îº¢¸û µ÷ þ¨½¸Ãò¾¢ý Å¡¢¨ºôÀÊ «¨Áó¾ ¯îº¢¸û ±É¢ø, AC, BD ¬¸¢Â ã¨ÄÅ¢ð¼í¸Ç¢ý ±É¢ø, AC, BD ¬¸¢Â ã¨ÄÅ¢ð¼í¸Ç¢ý ±É¢ø, AC, BD ¬¸¢Â ã¨ÄÅ¢ð¼í¸Ç¢ý ±É¢ø, AC, BD ¬¸¢Â ã¨ÄÅ¢ð¼í¸Ç¢ý ¿ÎôÒûÇ¢¸û ºÁõ.¿ÎôÒûÇ¢¸û ºÁõ.¿ÎôÒûÇ¢¸û ºÁõ.¿ÎôÒûÇ¢¸û ºÁõ. ¿ÎôÒûÇ¢ = x¿ÎôÒûÇ¢ = x¿ÎôÒûÇ¢ = x¿ÎôÒûÇ¢ = x 1111 + x+ x+ x+ x2222 , y , y , y , y 1111 + y+ y+ y+ y 2222 2 22 22 22 2 AC ¢ý ¿ÎôÒûÇ¢ = BD ¢ý ¿ÎôÒûÇ¢AC ¢ý ¿ÎôÒûÇ¢ = BD ¢ý ¿ÎôÒûÇ¢AC ¢ý ¿ÎôÒûÇ¢ = BD ¢ý ¿ÎôÒûÇ¢AC ¢ý ¿ÎôÒûÇ¢ = BD ¢ý ¿ÎôÒûÇ¢

7+8 , 3+2 = 6+P , 1+47+8 , 3+2 = 6+P , 1+47+8 , 3+2 = 6+P , 1+47+8 , 3+2 = 6+P , 1+4 2 2 2 22 2 2 22 2 2 22 2 2 2

15 , 5 = 6+P , 515 , 5 = 6+P , 515 , 5 = 6+P , 515 , 5 = 6+P , 5 2 2 2 22 2 2 22 2 2 22 2 2 2

6 + P6 + P6 + P6 + P ==== 15151515 2222 2222 6 + P = 15 X 26 + P = 15 X 26 + P = 15 X 26 + P = 15 X 2 2222 P = 15 P = 15 P = 15 P = 15 ---- 6666

P = 9P = 9P = 9P = 9 ((((15151515)))) 5x 5x 5x 5x ---- 2y 2y 2y 2y ---- 9 = 0 ÁüÚõ ay + 2x 9 = 0 ÁüÚõ ay + 2x 9 = 0 ÁüÚõ ay + 2x 9 = 0 ÁüÚõ ay + 2x ---- 11 = 0 ¬¸¢Â 11 = 0 ¬¸¢Â 11 = 0 ¬¸¢Â 11 = 0 ¬¸¢Â §¿÷째¡§¿÷째¡§¿÷째¡§¿÷째¡Î¸û ´ýÚ즸¡ýÚ ¦ºíÌòÐ ±É¢ø, a ý θû ´ýÚ즸¡ýÚ ¦ºíÌòÐ ±É¢ø, a ý θû ´ýÚ즸¡ýÚ ¦ºíÌòÐ ±É¢ø, a ý θû ´ýÚ즸¡ýÚ ¦ºíÌòÐ ±É¢ø, a ý Á¾¢ô¨Àì ¸¡ñ¸.Á¾¢ô¨Àì ¸¡ñ¸.Á¾¢ô¨Àì ¸¡ñ¸.Á¾¢ô¨Àì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : 5x 5x 5x 5x ---- 2y 2y 2y 2y ---- 9 = 0 ±ýÈ §¿÷째¡ðÊý º¡ö×9 = 0 ±ýÈ §¿÷째¡ðÊý º¡ö×9 = 0 ±ýÈ §¿÷째¡ðÊý º¡ö×9 = 0 ±ýÈ §¿÷째¡ðÊý º¡ö× m m m m 1111 ==== ---- x ý ¦¸Ø = x ý ¦¸Ø = x ý ¦¸Ø = x ý ¦¸Ø = ---- 5555 y ý ¦¸Ø y ý ¦¸Ø y ý ¦¸Ø y ý ¦¸Ø ---- 2222 m m m m 1111 = 5= 5= 5= 5 2222 ay + 2x ay + 2x ay + 2x ay + 2x ---- 11 = 0 11 = 0 11 = 0 11 = 0 ±ýÈ §¿÷째¡ðÊý º¡ö×±ýÈ §¿÷째¡ðÊý º¡ö×±ýÈ §¿÷째¡ðÊý º¡ö×±ýÈ §¿÷째¡ðÊý º¡ö× m m m m 2222 = = = = ---- 2222 aaaa þÕ §¿÷째¡Î¸û ´ýÚ즸¡ýÚ ¦ºíÌòÐ ±É¢øþÕ §¿÷째¡Î¸û ´ýÚ즸¡ýÚ ¦ºíÌòÐ ±É¢øþÕ §¿÷째¡Î¸û ´ýÚ즸¡ýÚ ¦ºíÌòÐ ±É¢øþÕ §¿÷째¡Î¸û ´ýÚ즸¡ýÚ ¦ºíÌòÐ ±É¢ø m m m m 1111 x m x m x m x m 2222 = = = = ----1111 5 x 5 x 5 x 5 x ----2 = 2 = 2 = 2 = ----1111 2 a 2 a 2 a 2 a

----5 = 5 = 5 = 5 = ----aaaa

a = 5 a = 5 a = 5 a = 5

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((((16161616)))) x + 2 y + x + 2 y + x + 2 y + x + 2 y + 1111 = 0 ÁüÚõ 2 x = 0 ÁüÚõ 2 x = 0 ÁüÚõ 2 x = 0 ÁüÚõ 2 x ---- y + 5 = 0y + 5 = 0y + 5 = 0y + 5 = 0 ±ýÈ ±ýÈ ±ýÈ ±ýÈ §¿÷째¡Î¸û ´ýÚ즸¡ýÚ ¦ºíÌò¾¡É¨Å ±É §¿÷째¡Î¸û ´ýÚ즸¡ýÚ ¦ºíÌò¾¡É¨Å ±É §¿÷째¡Î¸û ´ýÚ즸¡ýÚ ¦ºíÌò¾¡É¨Å ±É §¿÷째¡Î¸û ´ýÚ즸¡ýÚ ¦ºíÌò¾¡É¨Å ±É ¿¢Ú׸.¿¢Ú׸.¿¢Ú׸.¿¢Ú׸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ¦¸¡Îì¸ôÀð¼ §¿÷째¡Î¸Ç¢ý ºÁýÀ¡Î¸û¦¸¡Îì¸ôÀð¼ §¿÷째¡Î¸Ç¢ý ºÁýÀ¡Î¸û¦¸¡Îì¸ôÀð¼ §¿÷째¡Î¸Ç¢ý ºÁýÀ¡Î¸û¦¸¡Îì¸ôÀð¼ §¿÷째¡Î¸Ç¢ý ºÁýÀ¡Î¸û x x x x ++++ 2y +2y +2y +2y +1111 = 0 , = 0 , = 0 , = 0 , 2222 x x x x ---- y y y y + 5+ 5+ 5+ 5 = 0= 0= 0= 0 aaaa 1111 = = = = 1111 , b , b , b , b 1111 = 2 , a = 2 , a = 2 , a = 2 , a 2222 = = = = 2222 , b, b, b, b 2222 = = = = ----1111 a a a a 1111 aaaa 2222 + b + b + b + b 1111 b b b b 2222 = 0= 0= 0= 0 1 1 1 1 xxxx 2 + 2 x 2 + 2 x 2 + 2 x 2 + 2 x ----1 1 1 1 = 0= 0= 0= 0 2 + 2 + 2 + 2 + ----2 = 02 = 02 = 02 = 0 0 = 00 = 00 = 00 = 0 ±ý§Å, þùÅ¢Õ §¿÷째¡Î¸Ùõ ´ýÚ즸¡ýÚ ±ý§Å, þùÅ¢Õ §¿÷째¡Î¸Ùõ ´ýÚ즸¡ýÚ ±ý§Å, þùÅ¢Õ §¿÷째¡Î¸Ùõ ´ýÚ즸¡ýÚ ±ý§Å, þùÅ¢Õ §¿÷째¡Î¸Ùõ ´ýÚ즸¡ýÚ ¦ºíÌò¾¡É¨Å.¦ºíÌò¾¡É¨Å.¦ºíÌò¾¡É¨Å.¦ºíÌò¾¡É¨Å. ((((17171717)))) A (4,A (4,A (4,A (4,----6), B (3,6), B (3,6), B (3,6), B (3,----2) , C 2) , C 2) , C 2) , C (5,2) ¬¸¢ÂÅü¨È (5,2) ¬¸¢ÂÅü¨È (5,2) ¬¸¢ÂÅü¨È (5,2) ¬¸¢ÂÅü¨È ¯îº¢¸Ç¡¸ì ¦¸¡ñ¼ Ó째¡½ò¾¢ý ¿Î째¡ðÎ ¯îº¢¸Ç¡¸ì ¦¸¡ñ¼ Ó째¡½ò¾¢ý ¿Î째¡ðÎ ¯îº¢¸Ç¡¸ì ¦¸¡ñ¼ Ó째¡½ò¾¢ý ¿Î째¡ðÎ ¯îº¢¸Ç¡¸ì ¦¸¡ñ¼ Ó째¡½ò¾¢ý ¿Î째¡ðÎ ¨ÁÂõ ¸¡ñ¸.¨ÁÂõ ¸¡ñ¸.¨ÁÂõ ¸¡ñ¸.¨ÁÂõ ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : Ó째¡½ò¾¢ý ¿Î째¡ðÎ ¨ÁÂõÓ째¡½ò¾¢ý ¿Î째¡ðÎ ¨ÁÂõÓ째¡½ò¾¢ý ¿Î째¡ðÎ ¨ÁÂõÓ째¡½ò¾¢ý ¿Î째¡ðÎ ¨ÁÂõ G = x G = x G = x G = x 1111 + x + x + x + x 2222 + x+ x+ x+ x 3333 , y , y , y , y 1111 + y + y + y + y 2 2 2 2 + y + y + y + y 3333 3 33 33 33 3

= 4 + 3 + 5 , = 4 + 3 + 5 , = 4 + 3 + 5 , = 4 + 3 + 5 , ----6 6 6 6 ---- 2 + 22 + 22 + 22 + 2 3 33 33 33 3

= 12 , = 12 , = 12 , = 12 , ----6666 3 33 33 33 3 GGGG = = = = ( 4 , ( 4 , ( 4 , ( 4 , ----2 )2 )2 )2 ) ((((18181818)))) ´Õ Ó째¡½ò¾¢ý ¿Î째¡ðÎ´Õ Ó째¡½ò¾¢ý ¿Î째¡ðÎ´Õ Ó째¡½ò¾¢ý ¿Î째¡ðÎ´Õ Ó째¡½ò¾¢ý ¿Î째¡ðÎ ¨ÁÂõ (1,3) ¨ÁÂõ (1,3) ¨ÁÂõ (1,3) ¨ÁÂõ (1,3) ÁüÚõ «¾ý ²§¾Ûõ þÕ Ó¨É¸û (ÁüÚõ «¾ý ²§¾Ûõ þÕ Ó¨É¸û (ÁüÚõ «¾ý ²§¾Ûõ þÕ Ó¨É¸û (ÁüÚõ «¾ý ²§¾Ûõ þÕ Ó¨É¸û (----7,6), (8,5) 7,6), (8,5) 7,6), (8,5) 7,6), (8,5) ±É¢ø, ãýÈ¡ÅÐ ¯îº¢¨Âì ¸¡ñ¸. ±É¢ø, ãýÈ¡ÅÐ ¯îº¢¨Âì ¸¡ñ¸. ±É¢ø, ãýÈ¡ÅÐ ¯îº¢¨Âì ¸¡ñ¸. ±É¢ø, ãýÈ¡ÅÐ ¯îº¢¨Âì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : Ó째¡½ò¾¢ý ¿Î째¡ðÎ ¨ÁÂõÓ째¡½ò¾¢ý ¿Î째¡ðÎ ¨ÁÂõÓ째¡½ò¾¢ý ¿Î째¡ðÎ ¨ÁÂõÓ째¡½ò¾¢ý ¿Î째¡ðÎ ¨ÁÂõ G = x G = x G = x G = x 1111 + x + x + x + x 2222 + x+ x+ x+ x 3333 , y , y , y , y 1111 + y + y + y + y 2 2 2 2 + y + y + y + y 3333 3 33 33 33 3

( 1,3 ) = ( 1,3 ) = ( 1,3 ) = ( 1,3 ) = ----7 + 8 + x 7 + 8 + x 7 + 8 + x 7 + 8 + x 3333 , 6 + 5 + y , 6 + 5 + y , 6 + 5 + y , 6 + 5 + y 3333 3 33 33 33 3

= 1 + x= 1 + x= 1 + x= 1 + x 3333 , 11 + y , 11 + y , 11 + y , 11 + y 3333 3 33 33 33 3 1 + x 1 + x 1 + x 1 + x 3333 = 1 ; 11 + y = 1 ; 11 + y = 1 ; 11 + y = 1 ; 11 + y 3333 = 3= 3= 3= 3 3 3 3 3 3 3 3 3 x x x x 3333 = 3 = 3 = 3 = 3 ---- 1 ; y 1 ; y 1 ; y 1 ; y 3333 = 9 = 9 = 9 = 9 ---- 11111111 x x x x 3333 = 2 ; y = 2 ; y = 2 ; y = 2 ; y 3333 = = = = ----2222

ãýÈ¡ÅÐ ¯îº¢ ãýÈ¡ÅÐ ¯îº¢ ãýÈ¡ÅÐ ¯îº¢ ãýÈ¡ÅÐ ¯îº¢ = ( 2 , = ( 2 , = ( 2 , = ( 2 , ----2 )2 )2 )2 )

6666.... ÅÊÅ¢ÂøÅÊÅ¢ÂøÅÊÅ¢ÂøÅÊÅ¢Âø ((((1111)))) À¼ò¾¢ø, TP ´Õ ¦¾¡Î§¸¡Î. A, B À¼ò¾¢ø, TP ´Õ ¦¾¡Î§¸¡Î. A, B À¼ò¾¢ø, TP ´Õ ¦¾¡Î§¸¡Î. A, B À¼ò¾¢ø, TP ´Õ ¦¾¡Î§¸¡Î. A, B ±ýÀÉ Åð¼ò¾¢ý Á£ÐûÇ ÒûÇ¢¸û. ±ýÀÉ Åð¼ò¾¢ý Á£ÐûÇ ÒûÇ¢¸û. ±ýÀÉ Åð¼ò¾¢ý Á£ÐûÇ ÒûÇ¢¸û. ±ýÀÉ Åð¼ò¾¢ý Á£ÐûÇ ÒûÇ¢¸û.

BTP = 72BTP = 72BTP = 72BTP = 72°°°° ÁüÚõ ATB = 43ÁüÚõ ATB = 43ÁüÚõ ATB = 43ÁüÚõ ATB = 43°°°° ±É¢ø,±É¢ø,±É¢ø,±É¢ø, ABT ³ì ¸¡ñ¸. ABT ³ì ¸¡ñ¸. ABT ³ì ¸¡ñ¸. ABT ³ì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

BTP = BAT = 72BTP = BAT = 72BTP = BAT = 72BTP = BAT = 72°°°° ( ( ( ( Á¡üÚ Åð¼òÐñθǢø Á¡üÚ Åð¼òÐñθǢø Á¡üÚ Åð¼òÐñθǢø Á¡üÚ Åð¼òÐñθǢø «¨Áó¾ «¨Áó¾ «¨Áó¾ «¨Áó¾

∆∆∆∆ ABT ¢ĢÕóÐABT ¢ĢÕóÐABT ¢ĢÕóÐABT ¢ĢÕóÐ ,,,, §¸¡½í¸û ºÁõ§¸¡½í¸û ºÁõ§¸¡½í¸û ºÁõ§¸¡½í¸û ºÁõ) ) ) )

ATB + BAT + ABT = 180ATB + BAT + ABT = 180ATB + BAT + ABT = 180ATB + BAT + ABT = 180°°°°

43434343°°°° + 72+ 72+ 72+ 72°°°° + ABT = 180+ ABT = 180+ ABT = 180+ ABT = 180°°°°

ABT = 180ABT = 180ABT = 180ABT = 180°°°° ---- 115115115115°°°°

ABT = 65ABT = 65ABT = 65ABT = 65°°°°

((((2222)))) MP ±ýÀÐMP ±ýÀÐMP ±ýÀÐMP ±ýÀÐ ∆∆∆∆ MNO ø M ý ¦ÅÇ¢ôÒÈMNO ø M ý ¦ÅÇ¢ôÒÈMNO ø M ý ¦ÅÇ¢ôÒÈMNO ø M ý ¦ÅÇ¢ôÒÈ þÕºÁ ¦ÅðÊ. §ÁÖõ NO ý þÕºÁ ¦ÅðÊ. §ÁÖõ NO ý þÕºÁ ¦ÅðÊ. §ÁÖõ NO ý þÕºÁ ¦ÅðÊ. §ÁÖõ NO ý ¿£ðº¢Â¢¨É P ø ¿£ðº¢Â¢¨É P ø ¿£ðº¢Â¢¨É P ø ¿£ðº¢Â¢¨É P ø ºó¾¢ì¸¢ÈÐ . ºó¾¢ì¸¢ÈÐ . ºó¾¢ì¸¢ÈÐ . ºó¾¢ì¸¢ÈÐ . MN = 10 ¦º.Á£, MO = 6 ¦º.Á£., NO = 12 MN = 10 ¦º.Á£, MO = 6 ¦º.Á£., NO = 12 MN = 10 ¦º.Á£, MO = 6 ¦º.Á£., NO = 12 MN = 10 ¦º.Á£, MO = 6 ¦º.Á£., NO = 12 ¦º.Á£. ±É¢ø, ¦º.Á£. ±É¢ø, ¦º.Á£. ±É¢ø, ¦º.Á£. ±É¢ø, OP ¨Âì ¸¡ñ¸.OP ¨Âì ¸¡ñ¸.OP ¨Âì ¸¡ñ¸.OP ¨Âì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

∆∆∆∆ MNO ø MNO ø MNO ø MNO ø §¸¡½ þÕºÁ¦ÅðÊ §¾üÈôÀʧ¸¡½ þÕºÁ¦ÅðÊ §¾üÈôÀʧ¸¡½ þÕºÁ¦ÅðÊ §¾üÈôÀʧ¸¡½ þÕºÁ¦ÅðÊ §¾üÈôÀÊ NP = MNNP = MNNP = MNNP = MN OP MOOP MOOP MOOP MO

12 + OP = 1012 + OP = 1012 + OP = 1012 + OP = 10 OP 6OP 6OP 6OP 6

6 6 6 6 ( 12 + OP) = 10 X OP( 12 + OP) = 10 X OP( 12 + OP) = 10 X OP( 12 + OP) = 10 X OP 72 + 6 OP 72 + 6 OP 72 + 6 OP 72 + 6 OP ---- 10 OP = 010 OP = 010 OP = 010 OP = 0 ----4 OP4 OP4 OP4 OP = = = = ----72727272 OP = 72OP = 72OP = 72OP = 72 4444 OP = 18 ¦º.Á£.OP = 18 ¦º.Á£.OP = 18 ¦º.Á£.OP = 18 ¦º.Á£.

((((3333)))) ∆∆∆∆ ABC ø A ý ¯ðÒÈ §¸¡½ þÕºÁ ¦ÅðÊABC ø A ý ¯ðÒÈ §¸¡½ þÕºÁ ¦ÅðÊABC ø A ý ¯ðÒÈ §¸¡½ þÕºÁ ¦ÅðÊABC ø A ý ¯ðÒÈ §¸¡½ þÕºÁ ¦ÅðÊ AD ¬ÉÐ , Àì¸õ BCAD ¬ÉÐ , Àì¸õ BCAD ¬ÉÐ , Àì¸õ BCAD ¬ÉÐ , Àì¸õ BC ³ ³ ³ ³ DDDD ø ºó¾¢ì¸¢ÈÐ.ø ºó¾¢ì¸¢ÈÐ.ø ºó¾¢ì¸¢ÈÐ.ø ºó¾¢ì¸¢ÈÐ. BD = 2.5 BD = 2.5 BD = 2.5 BD = 2.5 ¦º.Á£, AB = 5 ¦º.Á£, ÁüÚõ AC = 4.2 ¦º.Á£. ±É¢ø, ¦º.Á£, AB = 5 ¦º.Á£, ÁüÚõ AC = 4.2 ¦º.Á£. ±É¢ø, ¦º.Á£, AB = 5 ¦º.Á£, ÁüÚõ AC = 4.2 ¦º.Á£. ±É¢ø, ¦º.Á£, AB = 5 ¦º.Á£, ÁüÚõ AC = 4.2 ¦º.Á£. ±É¢ø, DC ³ì ¸¡ñ¸.DC ³ì ¸¡ñ¸.DC ³ì ¸¡ñ¸.DC ³ì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× :

∆∆∆∆ ABC ø §¸¡½ þÕºÁ ¦ÅðÊ §¾üÈôÀÊ,ABC ø §¸¡½ þÕºÁ ¦ÅðÊ §¾üÈôÀÊ,ABC ø §¸¡½ þÕºÁ ¦ÅðÊ §¾üÈôÀÊ,ABC ø §¸¡½ þÕºÁ ¦ÅðÊ §¾üÈôÀÊ, BDBDBDBD = AB= AB= AB= AB DC ACDC ACDC ACDC AC

DC = BD X ACDC = BD X ACDC = BD X ACDC = BD X AC ABABABAB

DC = 2.5 X 4.2DC = 2.5 X 4.2DC = 2.5 X 4.2DC = 2.5 X 4.2 5555

DC = 2.1 ¦º.Á£. DC = 2.1 ¦º.Á£. DC = 2.1 ¦º.Á£. DC = 2.1 ¦º.Á£.

((((4444)))) ∆∆∆∆ ABC ø DE || BC ÁüÚõ AD = 2 , ABC ø DE || BC ÁüÚõ AD = 2 , ABC ø DE || BC ÁüÚõ AD = 2 , ABC ø DE || BC ÁüÚõ AD = 2 , DB 3DB 3DB 3DB 3 AE = 3.7AE = 3.7AE = 3.7AE = 3.7 ¦º.Á£,¦º.Á£,¦º.Á£,¦º.Á£, ±É¢ø, EC ³ì ¸¡ñ¸.±É¢ø, EC ³ì ¸¡ñ¸.±É¢ø, EC ³ì ¸¡ñ¸.±É¢ø, EC ³ì ¸¡ñ¸. ¾£÷× : ¾£÷× : ¾£÷× : ¾£÷× : AD = AEAD = AEAD = AEAD = AE BD CE ( BD CE ( BD CE ( BD CE ( §¾øŠ §¾üÈôÀÊ )§¾øŠ §¾üÈôÀÊ )§¾øŠ §¾üÈôÀÊ )§¾øŠ §¾üÈôÀÊ )

T P

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CE = AE X BDCE = AE X BDCE = AE X BDCE = AE X BD ADADADAD CE = 3.7 X 3CE = 3.7 X 3CE = 3.7 X 3CE = 3.7 X 3 2222

CE = 5.55 ¦º.Á£.CE = 5.55 ¦º.Á£.CE = 5.55 ¦º.Á£.CE = 5.55 ¦º.Á£. ((((5555)))) AB AB AB AB ÁüÚõ CD ±ÁüÚõ CD ±ÁüÚõ CD ±ÁüÚõ CD ±ýÈýÈýÈýÈ þÕ ¿¡ñ¸û Åð¼ò¾¢üÌ þÕ ¿¡ñ¸û Åð¼ò¾¢üÌ þÕ ¿¡ñ¸û Åð¼ò¾¢üÌ þÕ ¿¡ñ¸û Åð¼ò¾¢üÌ ¦ÅÇ¢§Â ¦ÅÇ¢§Â ¦ÅÇ¢§Â ¦ÅÇ¢§Â PPPP ±Ûõ ÒûǢ¢ø ¦ÅðÊ즸¡û¸¢ýÈÉ.±Ûõ ÒûǢ¢ø ¦ÅðÊ즸¡û¸¢ýÈÉ.±Ûõ ÒûǢ¢ø ¦ÅðÊ즸¡û¸¢ýÈÉ.±Ûõ ÒûǢ¢ø ¦ÅðÊ즸¡û¸¢ýÈÉ. AB = 4 AB = 4 AB = 4 AB = 4 ¦º.Á£, ¦º.Á£, ¦º.Á£, ¦º.Á£, BP = 5 BP = 5 BP = 5 BP = 5 ¦º.Á£, ÁüÚõ ¦º.Á£, ÁüÚõ ¦º.Á£, ÁüÚõ ¦º.Á£, ÁüÚõ PD = 3 PD = 3 PD = 3 PD = 3 ¦º.Á£. ¦º.Á£. ¦º.Á£. ¦º.Á£. ±É¢ø ±É¢ø ±É¢ø ±É¢ø CDCDCDCD ³ì ¸¡ñ¸. ³ì ¸¡ñ¸. ³ì ¸¡ñ¸. ³ì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : Åð¼ò¾¢üÌ ¦ÅÇ¢§Â þÕ ¿¡ñ¸û ¦ÅðÊì ¦¸¡ñ¼¡øÅð¼ò¾¢üÌ ¦ÅÇ¢§Â þÕ ¿¡ñ¸û ¦ÅðÊì ¦¸¡ñ¼¡øÅð¼ò¾¢üÌ ¦ÅÇ¢§Â þÕ ¿¡ñ¸û ¦ÅðÊì ¦¸¡ñ¼¡øÅð¼ò¾¢üÌ ¦ÅÇ¢§Â þÕ ¿¡ñ¸û ¦ÅðÊì ¦¸¡ñ¼¡ø PA X PB = PC X PDPA X PB = PC X PDPA X PB = PC X PDPA X PB = PC X PD 9 X 5 = (3 + x) X 39 X 5 = (3 + x) X 39 X 5 = (3 + x) X 39 X 5 = (3 + x) X 3 3 + x = 9 X 53 + x = 9 X 53 + x = 9 X 53 + x = 9 X 5 3333 x = 15 x = 15 x = 15 x = 15 ---- 3 = 123 = 123 = 123 = 12 CD = 12 ¦º.Á£. CD = 12 ¦º.Á£. CD = 12 ¦º.Á£. CD = 12 ¦º.Á£. ((((6666)))) ¦¸¡Îì¸ôÀð¼ À¼ò¾¢ø x ý Á¾¢ô¨Àì ¸¡ñ¸.¦¸¡Îì¸ôÀð¼ À¼ò¾¢ø x ý Á¾¢ô¨Àì ¸¡ñ¸.¦¸¡Îì¸ôÀð¼ À¼ò¾¢ø x ý Á¾¢ô¨Àì ¸¡ñ¸.¦¸¡Îì¸ôÀð¼ À¼ò¾¢ø x ý Á¾¢ô¨Àì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : À¼ò¾¢Ä¢ÕóÐ, À¼ò¾¢Ä¢ÕóÐ, À¼ò¾¢Ä¢ÕóÐ, À¼ò¾¢Ä¢ÕóÐ, PA = 4, PC = 8, PD = 3PA = 4, PC = 8, PD = 3PA = 4, PC = 8, PD = 3PA = 4, PC = 8, PD = 3 PB = x , ±É§Å,PB = x , ±É§Å,PB = x , ±É§Å,PB = x , ±É§Å, PA X PB = PC X PDPA X PB = PC X PDPA X PB = PC X PDPA X PB = PC X PD 4 X x = 8 X 34 X x = 8 X 34 X x = 8 X 34 X x = 8 X 3 x = 8 X 3x = 8 X 3x = 8 X 3x = 8 X 3 4444 x = 6x = 6x = 6x = 6

7777.... Ó째¡½Å¢ÂøÓ째¡½Å¢ÂøÓ째¡½Å¢ÂøÓ째¡½Å¢Âø ((((1111)))) Sin Sin Sin Sin θθθθ + cos + cos + cos + cos θθθθ = 1 ±ýÈ = 1 ±ýÈ = 1 ±ýÈ = 1 ±ýÈ Óü¦È¡Õ¨Á¨Â Óü¦È¡Õ¨Á¨Â Óü¦È¡Õ¨Á¨Â Óü¦È¡Õ¨Á¨Â

Cosec Cosec Cosec Cosec θθθθ sec sec sec sec θθθθ ¿¢Ú׸.¿¢Ú׸.¿¢Ú׸.¿¢Ú׸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

ssssin in in in θθθθ + cos + cos + cos + cos θθθθ = sin= sin= sin= sin θθθθ + cos + cos + cos + cos θθθθ

ccccosec osec osec osec θθθθ sec sec sec sec θ θ θ θ 1 1 1 1 1111

sinsinsinsin θθθθ cos cos cos cos θθθθ

= sin = sin = sin = sin 2222 θθθθ + cos + cos + cos + cos 2222 θθθθ

ssssin in in in θθθθ + cos + cos + cos + cos θθθθ = 1= 1= 1= 1

ccccosec osec osec osec θθθθ sec sec sec sec θ θ θ θ

((((2222)))) ( sin ( sin ( sin ( sin 6666 θθθθ + cos + cos + cos + cos 6666 θθθθ ) = 1 ) = 1 ) = 1 ) = 1 ---- 3 sin 3 sin 3 sin 3 sin 2222 θθθθ cos cos cos cos 2222 θθθθ ¿¢Ú׸.¿¢Ú׸.¿¢Ú׸.¿¢Ú׸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

(sin sin sin sin 6666 θθθθ + cos + cos + cos + cos 6666 θθθθ ) ) ) ) aaaa3333+b+b+b+b3333=(a+b)=(a+b)=(a+b)=(a+b)3333----3ab(a+b)3ab(a+b)3ab(a+b)3ab(a+b)

sin sin sin sin 2222 θθθθ + cos + cos + cos + cos 2222 θθθθ = 1= 1= 1= 1

= (sin = (sin = (sin = (sin 2222 θθθθ) ) ) ) 3333 + (cos + (cos + (cos + (cos 2222 θθθθ ) ) ) ) 3333

= ( sin = ( sin = ( sin = ( sin 2222 θθθθ + cos + cos + cos + cos 2222 θθθθ) ) ) ) 3333 ---- 3sin3sin3sin3sin2222 θθθθ coscoscoscos2222 θθθθ (sin(sin(sin(sin2222 θθθθ +cos+cos+cos+cos 2222 θθθθ))))

= 1 = 1 = 1 = 1 ---- 3 sin 3 sin 3 sin 3 sin 2222 θθθθ cos cos cos cos 2222 θθθθ ¿¢ÚÅôÀð¼Ð.¿¢ÚÅôÀð¼Ð.¿¢ÚÅôÀð¼Ð.¿¢ÚÅôÀð¼Ð.

((((3333)))) 1 1 1 1 ---- sin sin sin sin θθθθ = sec = sec = sec = sec θθθθ ---- tan tan tan tan θθθθ ¿¢Ú׸.¿¢Ú׸.¿¢Ú׸.¿¢Ú׸.

√ √ √ √ 1 + sin1 + sin1 + sin1 + sin θθθθ ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

1 1 1 1 ---- sin sin sin sin θθθθ = 1 = 1 = 1 = 1 ---- sinsinsinsin θθθθ X 1 X 1 X 1 X 1 ---- sinsinsinsin θθθθ

√√√√ 1 + sin1 + sin1 + sin1 + sin θθθθ √√√√ 1 + sin 1 + sin 1 + sin 1 + sin θθθθ 1 1 1 1 ---- sinsinsinsin θθθθ

= ( 1 = ( 1 = ( 1 = ( 1 ---- sinsinsinsin θθθθ ) ) ) ) 2222

√√√√ 1 1 1 1 ---- sin sin sin sin 2222 θθθθ 1 1 1 1 ---- sin sin sin sin 2222 θ = θ = θ = θ = cos cos cos cos 2222 θθθθ

= 1 = 1 = 1 = 1 ---- sin sin sin sin θθθθ = 1 = 1 = 1 = 1 ---- sinsinsinsin θθθθ

cos cos cos cos θθθθ cos cos cos cos θθθθ coscoscoscos θθθθ

1 1 1 1 ---- sin sin sin sin θθθθ = sec= sec= sec= sec θθθθ ---- tantantantan θθθθ

√√√√ 1 + sin1 + sin1 + sin1 + sin θθθθ

((((4444)))) ¿¢Ú׸. sec ¿¢Ú׸. sec ¿¢Ú׸. sec ¿¢Ú׸. sec θθθθ (1 (1 (1 (1 ---- sin sin sin sin θ θ θ θ )))) (((( sec sec sec sec θθθθ + tan+ tan+ tan+ tan θθθθ ) = 1) = 1) = 1) = 1 ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

sec sec sec sec θθθθ (1 (1 (1 (1 ---- sinsinsinsin θθθθ )(sec )(sec )(sec )(sec θθθθ + tan + tan + tan + tan θθθθ))))

= ( sec = ( sec = ( sec = ( sec θθθθ ---- secsecsecsec θθθθ sinsinsinsin θθθθ ) ( sec) ( sec) ( sec) ( sec θθθθ + tan + tan + tan + tan θθθθ) ) ) )

= ( sec = ( sec = ( sec = ( sec θθθθ ---- 1 x sin 1 x sin 1 x sin 1 x sin θθθθ)( sec)( sec)( sec)( sec θθθθ + tan + tan + tan + tan θθθθ ))))

ccccosososos θθθθ

= ( sec = ( sec = ( sec = ( sec θθθθ ---- tan tan tan tan θθθθ )( sec )( sec )( sec )( sec θθθθ + tan + tan + tan + tan θθθθ ))))

= sec = sec = sec = sec 2222 θθθθ ---- tan tan tan tan 2222 θθθθ = 1 ¿¢ÚÅôÀð¼Ð.= 1 ¿¢ÚÅôÀð¼Ð.= 1 ¿¢ÚÅôÀð¼Ð.= 1 ¿¢ÚÅôÀð¼Ð.

((((5555)))) À¢ýÅÕõ Óü¦È¡Õ¨Á¨Â ¿¢Ú׸.À¢ýÅÕõ Óü¦È¡Õ¨Á¨Â ¿¢Ú׸.À¢ýÅÕõ Óü¦È¡Õ¨Á¨Â ¿¢Ú׸.À¢ýÅÕõ Óü¦È¡Õ¨Á¨Â ¿¢Ú׸.

1 + cos 1 + cos 1 + cos 1 + cos θθθθ ---- sin sin sin sin 2222 θθθθ = cot = cot = cot = cot θθθθ

sin ( 1 + cos sin ( 1 + cos sin ( 1 + cos sin ( 1 + cos θθθθ) ) ) ) ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

1 1 1 1 + cos + cos + cos + cos θθθθ ---- sin sin sin sin 2222 θθθθ = cos = cos = cos = cos θθθθ + ( 1 + ( 1 + ( 1 + ( 1 ---- sin sin sin sin 2222 θθθθ ))))

sin sin sin sin θθθθ ( 1 + cos ( 1 + cos ( 1 + cos ( 1 + cos θθθθ) sin ) sin ) sin ) sin θθθθ ( 1 + cos ( 1 + cos ( 1 + cos ( 1 + cos θθθθ))))

= cos = cos = cos = cos θθθθ + cos + cos + cos + cos 2222 θθθθ

sin sin sin sin θθθθ ( 1 + cos ( 1 + cos ( 1 + cos ( 1 + cos θθθθ ))))

= cos = cos = cos = cos θθθθ (1 + cos (1 + cos (1 + cos (1 + cos θ θ θ θ ))))

sin sin sin sin θθθθ ( 1 + cos ( 1 + cos ( 1 + cos ( 1 + cos θθθθ))))

1 + cos 1 + cos 1 + cos 1 + cos θθθθ ---- sin sin sin sin 2222 θθθθ = cot = cot = cot = cot θθθθ

sin sin sin sin θθθθ ( 1 + cos ( 1 + cos ( 1 + cos ( 1 + cos θθθθ ))))

((((6666)))) 1 1 1 1 ---- 1 1 1 1 ---- sin sin sin sin 2222 θθθθ = 1 ±É ¿¢Ú׸.= 1 ±É ¿¢Ú׸.= 1 ±É ¿¢Ú׸.= 1 ±É ¿¢Ú׸.

sinsinsinsin 2222 θθθθ 1 1 1 1 ---- cos cos cos cos 2222 θθθθ ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

1 1 1 1 ---- 1 1 1 1 ---- sin sin sin sin 2222 θθθθ = = = = 1 1 1 1 ---- cos cos cos cos 2222 θθθθ

sinsinsinsin 2222 θθθθ 1 1 1 1 ---- cos cos cos cos 2222 θθθθ sin sin sin sin 2222 θθθθ sin sin sin sin 2222 θθθθ

= 1 = 1 = 1 = 1 ---- cos cos cos cos 2222 θθθθ

sin sin sin sin 2222 θθθθ

= sin = sin = sin = sin 2222 θθθθ 1111----cos cos cos cos 2222 θθθθ = sin = sin = sin = sin 2222 θθθθ

sin sin sin sin 2222 θθθθ 1111----sinsinsinsin2 2 2 2 θ = θ = θ = θ = cos cos cos cos 2222 θθθθ

1 1 1 1 ---- 1 1 1 1 ---- sin sin sin sin 2222 θθθθ = 1= 1= 1= 1

sinsinsinsin 2222 θθθθ 1111 ---- cos cos cos cos 2222 θθθθ

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((((7777)))) cos cos cos cos θθθθ = 1 + sin = 1 + sin = 1 + sin = 1 + sin θθθθ ±É ¿¢Ú׸.±É ¿¢Ú׸.±É ¿¢Ú׸.±É ¿¢Ú׸.

sec sec sec sec θθθθ ---- tantantantan θθθθ ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

cos cos cos cos θθθθ

sec sec sec sec θθθθ ---- tan tan tan tan θθθθ

= cos = cos = cos = cos θθθθ X X X X sec sec sec sec θθθθ + tan+ tan+ tan+ tan θθθθ

sec sec sec sec θθθθ ---- tantantantan θθθθ sec sec sec sec θθθθ + tan+ tan+ tan+ tan θθθθ

= cos = cos = cos = cos θθθθ X secX secX secX sec θθθθ + cos+ cos+ cos+ cos θθθθ X tanX tanX tanX tan θθθθ

sec sec sec sec 2222 θθθθ ---- tan tan tan tan 2222 θθθθ secsecsecsec2222 ---- tantantantan2222 =1=1=1=1

= cos = cos = cos = cos θθθθ X 1 + cos X 1 + cos X 1 + cos X 1 + cos θθθθ X sinX sinX sinX sin θθθθ

cos cos cos cos θθθθ coscoscoscos θθθθ

= 1 + sin= 1 + sin= 1 + sin= 1 + sin θθθθ ¿¢ÚÅôÀð¼Ð ¿¢ÚÅôÀð¼Ð ¿¢ÚÅôÀð¼Ð ¿¢ÚÅôÀð¼Ð

((((8888)))) 1 + sec 1 + sec 1 + sec 1 + sec θθθθ = sin = sin = sin = sin 2222 θθθθ ±É ¿¢Ú׸. ±É ¿¢Ú׸. ±É ¿¢Ú׸. ±É ¿¢Ú׸.

sec sec sec sec θθθθ 1 1 1 1 ---- coscoscoscos θθθθ ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

sin sin sin sin 2222 θθθθ = = = = sin sin sin sin 2222 θθθθ X 1 + cosX 1 + cosX 1 + cosX 1 + cos θθθθ

1 1 1 1 ---- cos cos cos cos θθθθ 1 1 1 1 ---- cos cos cos cos θθθθ 1 + cos1 + cos1 + cos1 + cos θθθθ

==== sin sin sin sin 2222 θθθθ ( 1 + cos ( 1 + cos ( 1 + cos ( 1 + cos θ θ θ θ ))))

1 1 1 1 ---- cos cos cos cos 2222 θθθθ

= sin = sin = sin = sin 2222 θθθθ ( 1 + cos ( 1 + cos ( 1 + cos ( 1 + cos θθθθ ))))

sin sin sin sin 2222 θθθθ = 1 + 1= 1 + 1= 1 + 1= 1 + 1

sec sec sec sec θθθθ

sin sin sin sin 2 2 2 2 θθθθ = 1 + sec= 1 + sec= 1 + sec= 1 + sec θθθθ

1 1 1 1 ---- cos cos cos cos θθθθ secsecsecsec θθθθ

((((9999)))) 1 1 1 1 ---- coscoscoscos θθθθ = = = = cocococosec sec sec sec θθθθ ---- cotcotcotcot θθθθ ¿¢Ú׸.¿¢Ú׸.¿¢Ú׸.¿¢Ú׸.

√ √ √ √ 1 + cos1 + cos1 + cos1 + cos θθθθ ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

1 1 1 1 ---- cos cos cos cos θθθθ = 1 = 1 = 1 = 1 ---- coscoscoscos θθθθ X 1 X 1 X 1 X 1 ---- coscoscoscos θθθθ

√√√√ 1 + cos 1 + cos 1 + cos 1 + cos θθθθ √√√√ 1 + cos1 + cos1 + cos1 + cos θθθθ 1 1 1 1 ---- coscoscoscos θθθθ

= ( 1 = ( 1 = ( 1 = ( 1 ---- coscoscoscos θθθθ ) ) ) ) 2222

√√√√ 1 1 1 1 ---- coscoscoscos 2222 θ θ θ θ 1 1 1 1 ---- coscoscoscos 2222 θ = θ = θ = θ = sinsinsinsin 2222 θθθθ

= 1 = 1 = 1 = 1 ---- coscoscoscos θθθθ = 1 = 1 = 1 = 1 ---- cocococossss θθθθ

sinsinsinsin θθθθ sinsinsinsin θθθθ sinsinsinsin θθθθ

1 1 1 1 ---- cocococos s s s θθθθ = = = = cocococosecsecsecsec θθθθ ---- cotcotcotcot θθθθ

√√√√ 1 + cos1 + cos1 + cos1 + cos θθθθ

((((10101010)))) ( sin ( sin ( sin ( sin θθθθ + cosec + cosec + cosec + cosec θθθθ ) ) ) ) 2222 + ( cos + ( cos + ( cos + ( cos θθθθ +sec +sec +sec +sec θθθθ) ) ) ) 2222

= 7 + tan = 7 + tan = 7 + tan = 7 + tan 2222 θθθθ + cot + cot + cot + cot 2222 θθθθ ±ýÈ Óü¦È¡Õ¨Á¨Â ±ýÈ Óü¦È¡Õ¨Á¨Â ±ýÈ Óü¦È¡Õ¨Á¨Â ±ýÈ Óü¦È¡Õ¨Á¨Â ¿¢Ú׸¿¢Ú׸¿¢Ú׸¿¢Ú׸

¾£÷× :¾£÷× :¾£÷× :¾£÷× :

( sin ( sin ( sin ( sin θθθθ + cosec + cosec + cosec + cosec θθθθ ) ) ) ) 2222 + ( cos + ( cos + ( cos + ( cos θθθθ +sec +sec +sec +sec θθθθ) ) ) ) 2222

= sin = sin = sin = sin 2222 θθθθ + 2 sin+ 2 sin+ 2 sin+ 2 sin θθθθ cosec cosec cosec cosec θθθθ + cosec + cosec + cosec + cosec 2222 θθθθ + cos + cos + cos + cos 2222 θθθθ

+ 2cos + 2cos + 2cos + 2cos θθθθ sec sec sec sec θθθθ + sec + sec + sec + sec 2222 θθθθ ( (a+b)( (a+b)( (a+b)( (a+b)2222=a=a=a=a2222+2ab+b+2ab+b+2ab+b+2ab+b2222 ))))

= sin = sin = sin = sin 2222 θθθθ+ cos + cos + cos + cos 2222 θθθθ + 2 sin+ 2 sin+ 2 sin+ 2 sin θθθθ X 1 + 2 cos X 1 + 2 cos X 1 + 2 cos X 1 + 2 cos θθθθ X 1 X 1 X 1 X 1

sin sin sin sin θθθθ coscoscoscos θθθθ

+ ( 1 + cot + ( 1 + cot + ( 1 + cot + ( 1 + cot 2222 θθθθ) + ( 1 + tan ) + ( 1 + tan ) + ( 1 + tan ) + ( 1 + tan 2222 θθθθ ))))

= 1 + 2 + 2 + 1 + 1 + tan = 1 + 2 + 2 + 1 + 1 + tan = 1 + 2 + 2 + 1 + 1 + tan = 1 + 2 + 2 + 1 + 1 + tan 2222 θθθθ + cot + cot + cot + cot 2222 θθθθ

= 7 + tan = 7 + tan = 7 + tan = 7 + tan 2222 θθθθ + cot + cot + cot + cot 2222 θθθθ ¿¢ÚÅôÀð¼Ð. ¿¢ÚÅôÀð¼Ð. ¿¢ÚÅôÀð¼Ð. ¿¢ÚÅôÀð¼Ð.

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100 «Ê àÃò¾¢ø 100 «Ê àÃò¾¢ø 100 «Ê àÃò¾¢ø 100 «Ê àÃò¾¢ø ¿¢ýÚ «¾ý ¯îº¢¨Â 45¿¢ýÚ «¾ý ¯îº¢¨Â 45¿¢ýÚ «¾ý ¯îº¢¨Â 45¿¢ýÚ «¾ý ¯îº¢¨Â 45°°°° ²üÈ째¡½ò¾¢ø À¡÷ò¾¡ø རý ¯ÂÃõ ±ýÉ?²üÈ째¡½ò¾¢ø À¡÷ò¾¡ø རý ¯ÂÃõ ±ýÉ?²üÈ째¡½ò¾¢ø À¡÷ò¾¡ø རý ¯ÂÃõ ±ýÉ?²üÈ째¡½ò¾¢ø À¡÷ò¾¡ø རý ¯ÂÃõ ±ýÉ? ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

¦ºí§¸¡½ ¦ºí§¸¡½ ¦ºí§¸¡½ ¦ºí§¸¡½ ∆∆∆∆ ABC ABC ABC ABC ¢ø ¢ø ¢ø ¢ø h

tan tan tan tan θ = θ = θ = θ = ±À = AE ±À = AE ±À = AE ±À = AE «À EC«À EC«À EC«À EC

tan 45tan 45tan 45tan 45° ° ° ° = h = h = h = h 100100100100 1 = h1 = h1 = h1 = h 100100100100 h = 100 «Ê h = 100 «Ê h = 100 «Ê h = 100 «Ê རý ¦Á¡ò¾ ¯ÂÃõ = 100 + 5 = 105 «Êརý ¦Á¡ò¾ ¯ÂÃõ = 100 + 5 = 105 «Êརý ¦Á¡ò¾ ¯ÂÃõ = 100 + 5 = 105 «Êརý ¦Á¡ò¾ ¯ÂÃõ = 100 + 5 = 105 «Ê

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¾¨ÃÔ¼ý 60¾¨ÃÔ¼ý 60¾¨ÃÔ¼ý 60¾¨ÃÔ¼ý 60°°°° §¸¡½ò¨¾ ²üÀÎòи¢ÈÐ. ²½¢Â¢ý §¸¡½ò¨¾ ²üÀÎòи¢ÈÐ. ²½¢Â¢ý §¸¡½ò¨¾ ²üÀÎòи¢ÈÐ. ²½¢Â¢ý §¸¡½ò¨¾ ²üÀÎòи¢ÈÐ. ²½¢Â¢ý «Ê ÍÅüȢĢÕóÐ 3.5 Á£. àÃò¾¢ø ¯ûÇÐ ±É¢ø, «Ê ÍÅüȢĢÕóÐ 3.5 Á£. àÃò¾¢ø ¯ûÇÐ ±É¢ø, «Ê ÍÅüȢĢÕóÐ 3.5 Á£. àÃò¾¢ø ¯ûÇÐ ±É¢ø, «Ê ÍÅüȢĢÕóÐ 3.5 Á£. àÃò¾¢ø ¯ûÇÐ ±É¢ø, ²½¢Â¢ý ¿£Çò¨¾ì ¸¡ñ¸.²½¢Â¢ý ¿£Çò¨¾ì ¸¡ñ¸.²½¢Â¢ý ¿£Çò¨¾ì ¸¡ñ¸.²½¢Â¢ý ¿£Çò¨¾ì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : AC = ²½¢Â¢ý ¿£ÇõAC = ²½¢Â¢ý ¿£ÇõAC = ²½¢Â¢ý ¿£ÇõAC = ²½¢Â¢ý ¿£Çõ

¦ºí§¸¡½ ¦ºí§¸¡½ ¦ºí§¸¡½ ¦ºí§¸¡½ ∆∆∆∆ ABC ¢øABC ¢øABC ¢øABC ¢ø

cos cos cos cos θθθθ = «À = BC= «À = BC= «À = BC= «À = BC ¸ AC¸ AC¸ AC¸ AC

cos 60cos 60cos 60cos 60°°°° = 3.5= 3.5= 3.5= 3.5 x x x x 1 = 3.51 = 3.51 = 3.51 = 3.5 2 x2 x2 x2 x x = 3.5 X 2 = x = 3.5 X 2 = x = 3.5 X 2 = x = 3.5 X 2 = 7777 ²½¢Â¢ý ¿£Çõ = 7 Á£. ²½¢Â¢ý ¿£Çõ = 7 Á£. ²½¢Â¢ý ¿£Çõ = 7 Á£. ²½¢Â¢ý ¿£Çõ = 7 Á£.

((((13131313)))) ¯ÂÃõ 150 ¦º.Á£. ¯ûÇ ´Õ º¢ÚÁ¢ ´Õ ¯ÂÃõ 150 ¦º.Á£. ¯ûÇ ´Õ º¢ÚÁ¢ ´Õ ¯ÂÃõ 150 ¦º.Á£. ¯ûÇ ´Õ º¢ÚÁ¢ ´Õ ¯ÂÃõ 150 ¦º.Á£. ¯ûÇ ´Õ º¢ÚÁ¢ ´Õ

Å¢ÇìÌì ¸õÀò¾¢ý Óý ¿¢ýÈÅ¡Ú 150 Å¢ÇìÌì ¸õÀò¾¢ý Óý ¿¢ýÈÅ¡Ú 150 Å¢ÇìÌì ¸õÀò¾¢ý Óý ¿¢ýÈÅ¡Ú 150 Å¢ÇìÌì ¸õÀò¾¢ý Óý ¿¢ýÈÅ¡Ú 150 √√√√3 ¦º.Á£. 3 ¦º.Á£. 3 ¦º.Á£. 3 ¦º.Á£. ¿£ÇÓûÇ ¿¢Æ¨Ä ²üÀÎòи¢È¡û ±É¢ø, Å¢ÇìÌì ¿£ÇÓûÇ ¿¢Æ¨Ä ²üÀÎòи¢È¡û ±É¢ø, Å¢ÇìÌì ¿£ÇÓûÇ ¿¢Æ¨Ä ²üÀÎòи¢È¡û ±É¢ø, Å¢ÇìÌì ¿£ÇÓûÇ ¿¢Æ¨Ä ²üÀÎòи¢È¡û ±É¢ø, Å¢ÇìÌì ¸õÀò¾¢ý ²üÈ째¡½ò¨¾ì ¸¡ñ¸.¸õÀò¾¢ý ²üÈ째¡½ò¨¾ì ¸¡ñ¸.¸õÀò¾¢ý ²üÈ째¡½ò¨¾ì ¸¡ñ¸.¸õÀò¾¢ý ²üÈ째¡½ò¨¾ì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : AB º¢ÚÁ¢Â¢ý ¯ÂÃõ = 150 ¦º.Á£.AB º¢ÚÁ¢Â¢ý ¯ÂÃõ = 150 ¦º.Á£.AB º¢ÚÁ¢Â¢ý ¯ÂÃõ = 150 ¦º.Á£.AB º¢ÚÁ¢Â¢ý ¯ÂÃõ = 150 ¦º.Á£.

¦ºí§¸¡½ ¦ºí§¸¡½ ¦ºí§¸¡½ ¦ºí§¸¡½ ∆∆∆∆ ABC ¢ø,ABC ¢ø,ABC ¢ø,ABC ¢ø,

tan tan tan tan θθθθ = ±À = AB= ±À = AB= ±À = AB= ±À = AB

C B

A

600

3.5 Á£.Á£.Á£.Á£.

B

A

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150√ 3

A

E

B D

C 450

100

100

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«À BC«À BC«À BC«À BC

tan tan tan tan θθθθ = 150 = 1= 150 = 1= 150 = 1= 150 = 1

150150150150√√√√3 3 3 3 √√√√3333

θ θ θ θ = 30= 30= 30= 30°°°°

8888.... «ÇÅ¢Âø«ÇÅ¢Âø«ÇÅ¢Âø«ÇÅ¢Âø ((((1111)))) ´Õ ¾¢ñÁ §¿÷Å𼠯ըǢý ¬Ãõ 14 ´Õ ¾¢ñÁ §¿÷Å𼠯ըǢý ¬Ãõ 14 ´Õ ¾¢ñÁ §¿÷Å𼠯ըǢý ¬Ãõ 14 ´Õ ¾¢ñÁ §¿÷Å𼠯ըǢý ¬Ãõ 14 ¦º.Á£. ¦º.Á£. ¦º.Á£. ¦º.Á£. ÁüÚõ ¯ÂÃõ 8 ¦º.Á£. ±ÉÁüÚõ ¯ÂÃõ 8 ¦º.Á£. ±ÉÁüÚõ ¯ÂÃõ 8 ¦º.Á£. ±ÉÁüÚõ ¯ÂÃõ 8 ¦º.Á£. ±É¢ø, «¾ý ŨÇÀÃôÒ ¢ø, «¾ý ŨÇÀÃôÒ ¢ø, «¾ý ŨÇÀÃôÒ ¢ø, «¾ý ŨÇÀÃôÒ ÁüÚõ ¦Á¡ò¾ôÀÁüÚõ ¦Á¡ò¾ôÀÁüÚõ ¦Á¡ò¾ôÀÁüÚõ ¦Á¡ò¾ôÀÃô¨Àì ¸¡ñ¸.Ãô¨Àì ¸¡ñ¸.Ãô¨Àì ¸¡ñ¸.Ãô¨Àì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ¾¢ñÁ ¯Õ¨Ç¢ý ¬Ãõ r ¾¢ñÁ ¯Õ¨Ç¢ý ¬Ãõ r ¾¢ñÁ ¯Õ¨Ç¢ý ¬Ãõ r ¾¢ñÁ ¯Õ¨Ç¢ý ¬Ãõ r = 14 ¦º.Á£.= 14 ¦º.Á£.= 14 ¦º.Á£.= 14 ¦º.Á£. ¯ÂÃõ h ¯ÂÃõ h ¯ÂÃõ h ¯ÂÃõ h = 8 ¦º.Á£.= 8 ¦º.Á£.= 8 ¦º.Á£.= 8 ¦º.Á£.

ŨÇÀÃôÒ Å¨ÇÀÃôÒ Å¨ÇÀÃôÒ Å¨ÇÀÃôÒ = 2 = 2 = 2 = 2 π π π π r h º.«.r h º.«.r h º.«.r h º.«. = 2= 2= 2= 2 X 22 X 14 X 8X 22 X 14 X 8X 22 X 14 X 8X 22 X 14 X 8 7777

¾¢ñÁ ¯Õ¨Ç¢ý ŨÇÀÃôÒ = 704 ¾¢ñÁ ¯Õ¨Ç¢ý ŨÇÀÃôÒ = 704 ¾¢ñÁ ¯Õ¨Ç¢ý ŨÇÀÃôÒ = 704 ¾¢ñÁ ¯Õ¨Ç¢ý ŨÇÀÃôÒ = 704 º.º.º.º.¦º.Á£ ¦º.Á£ ¦º.Á£ ¦º.Á£

¦Á¡ò¾ôÀÃôÒ = 2 ¦Á¡ò¾ôÀÃôÒ = 2 ¦Á¡ò¾ôÀÃôÒ = 2 ¦Á¡ò¾ôÀÃôÒ = 2 ππππ r ( h + r ) º.«.r ( h + r ) º.«.r ( h + r ) º.«.r ( h + r ) º.«. = 2 X 22 X 14 ( 8 + 14 )= 2 X 22 X 14 ( 8 + 14 )= 2 X 22 X 14 ( 8 + 14 )= 2 X 22 X 14 ( 8 + 14 ) 7777 ¾¢ñÁ ¯Õ¨Ç¢ý ¦Á¡ò¾ôÀÃôÒ= 1936 ¾¢ñÁ ¯Õ¨Ç¢ý ¦Á¡ò¾ôÀÃôÒ= 1936 ¾¢ñÁ ¯Õ¨Ç¢ý ¦Á¡ò¾ôÀÃôÒ= 1936 ¾¢ñÁ ¯Õ¨Ç¢ý ¦Á¡ò¾ôÀÃôÒ= 1936 º.¦º.Á£.º.¦º.Á£.º.¦º.Á£.º.¦º.Á£. ((((2222)))) ÁÃò¾¢É¡Ä¡É ´Õ ¾¢ñÁì ÜõÀ¢ý «ÊîÍüÈÇ× ÁÃò¾¢É¡Ä¡É ´Õ ¾¢ñÁì ÜõÀ¢ý «ÊîÍüÈÇ× ÁÃò¾¢É¡Ä¡É ´Õ ¾¢ñÁì ÜõÀ¢ý «ÊîÍüÈÇ× ÁÃò¾¢É¡Ä¡É ´Õ ¾¢ñÁì ÜõÀ¢ý «ÊîÍüÈÇ× 44 44 44 44 ¦º.Á£. ÁüÚõ «¾ý ¯ÂÃõ 12 ¦º.Á£. ±É¢ø, ¦º.Á£. ÁüÚõ «¾ý ¯ÂÃõ 12 ¦º.Á£. ±É¢ø, ¦º.Á£. ÁüÚõ «¾ý ¯ÂÃõ 12 ¦º.Á£. ±É¢ø, ¦º.Á£. ÁüÚõ «¾ý ¯ÂÃõ 12 ¦º.Á£. ±É¢ø, «ò¾¢ñÁì ÜõÀ¢ý ¸É «Ç¨Åì ¸¡ñ¸.«ò¾¢ñÁì ÜõÀ¢ý ¸É «Ç¨Åì ¸¡ñ¸.«ò¾¢ñÁì ÜõÀ¢ý ¸É «Ç¨Åì ¸¡ñ¸.«ò¾¢ñÁì ÜõÀ¢ý ¸É «Ç¨Åì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

ÜõÀ¢ý «ÊîÍüÈÇ× 2ÜõÀ¢ý «ÊîÍüÈÇ× 2ÜõÀ¢ý «ÊîÍüÈÇ× 2ÜõÀ¢ý «ÊîÍüÈÇ× 2ππππr = 44 ¦º.Á£.r = 44 ¦º.Á£.r = 44 ¦º.Á£.r = 44 ¦º.Á£. r = 44 X 7 = 7 ¦º.Á£.r = 44 X 7 = 7 ¦º.Á£.r = 44 X 7 = 7 ¦º.Á£.r = 44 X 7 = 7 ¦º.Á£. 2 X 22 X 22 X 22 X 22222 ¯ÂÃõ h = 12 ¦º.Á£.¯ÂÃõ h = 12 ¦º.Á£.¯ÂÃõ h = 12 ¦º.Á£.¯ÂÃõ h = 12 ¦º.Á£.

¸É «Ç× = ¸É «Ç× = ¸É «Ç× = ¸É «Ç× = 1 1 1 1 ππππ r r r r 2 2 2 2 h ¸.«.h ¸.«.h ¸.«.h ¸.«. 3333 = 1 X 22 X 7 X 7 X 12= 1 X 22 X 7 X 7 X 12= 1 X 22 X 7 X 7 X 12= 1 X 22 X 7 X 7 X 12 3 73 73 73 7 ¾¢ñÁì ÜõÀ¢ý ¸É «Ç× = 616 º.¦º.Á£. ¾¢ñÁì ÜõÀ¢ý ¸É «Ç× = 616 º.¦º.Á£. ¾¢ñÁì ÜõÀ¢ý ¸É «Ç× = 616 º.¦º.Á£. ¾¢ñÁì ÜõÀ¢ý ¸É «Ç× = 616 º.¦º.Á£. ((((3333)))) ´Õ ¾¢ñÁ §¿÷Åð¼ ÜõÀ¢ý «ÊîÍüÈÇ× 236 ¦º.Á£ ´Õ ¾¢ñÁ §¿÷Åð¼ ÜõÀ¢ý «ÊîÍüÈÇ× 236 ¦º.Á£ ´Õ ¾¢ñÁ §¿÷Åð¼ ÜõÀ¢ý «ÊîÍüÈÇ× 236 ¦º.Á£ ´Õ ¾¢ñÁ §¿÷Åð¼ ÜõÀ¢ý «ÊîÍüÈÇ× 236 ¦º.Á£ ÁüÚõ «¾ý º¡ÔÂÃõ 12 ¦º.Á£. ±É¢ø, «ìÜõÀ¢ý ÁüÚõ «¾ý º¡ÔÂÃõ 12 ¦º.Á£. ±É¢ø, «ìÜõÀ¢ý ÁüÚõ «¾ý º¡ÔÂÃõ 12 ¦º.Á£. ±É¢ø, «ìÜõÀ¢ý ÁüÚõ «¾ý º¡ÔÂÃõ 12 ¦º.Á£. ±É¢ø, «ìÜõÀ¢ý ŨÇÀÃô¨Àì ¸¡ñ¸. ŨÇÀÃô¨Àì ¸¡ñ¸. ŨÇÀÃô¨Àì ¸¡ñ¸. ŨÇÀÃô¨Àì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

ÜõÀ¢ý «ÜõÀ¢ý «ÜõÀ¢ý «ÜõÀ¢ý «ÊîÍüÈÇ× 2 ÊîÍüÈÇ× 2 ÊîÍüÈÇ× 2 ÊîÍüÈÇ× 2 ππππ r = 236 ¦º.Á£.r = 236 ¦º.Á£.r = 236 ¦º.Á£.r = 236 ¦º.Á£.

ππππ r = 118 ¦º.Á£.r = 118 ¦º.Á£.r = 118 ¦º.Á£.r = 118 ¦º.Á£.

º¡ÔÂÃõ º¡ÔÂÃõ º¡ÔÂÃõ º¡ÔÂÃõ l = 12 ¦º.Á£.12 ¦º.Á£.12 ¦º.Á£.12 ¦º.Á£.

ŨÇÀÃôÒ = ŨÇÀÃôÒ = ŨÇÀÃôÒ = ŨÇÀÃôÒ = ππππ r r r r l º.«.º.«.º.«.º.«. = 118 X 12= 118 X 12= 118 X 12= 118 X 12 ¾¢ñÁì ÜõÀ¢ý ŨÇÀÃôÒ = 1416 º.¦º.Á£. ¾¢ñÁì ÜõÀ¢ý ŨÇÀÃôÒ = 1416 º.¦º.Á£. ¾¢ñÁì ÜõÀ¢ý ŨÇÀÃôÒ = 1416 º.¦º.Á£. ¾¢ñÁì ÜõÀ¢ý ŨÇÀÃôÒ = 1416 º.¦º.Á£. ((((4444)))) 62.37 62.37 62.37 62.37 ¸.¦º.Á£. ¸É «Ç× ¦¸¡ñ¼ ´Õ ¾¢ñÁ¸.¦º.Á£. ¸É «Ç× ¦¸¡ñ¼ ´Õ ¾¢ñÁ¸.¦º.Á£. ¸É «Ç× ¦¸¡ñ¼ ´Õ ¾¢ñÁ¸.¦º.Á£. ¸É «Ç× ¦¸¡ñ¼ ´Õ ¾¢ñÁ §¿÷Å𼠯ըǢý ¯ÂÃõ 4§¿÷Å𼠯ըǢý ¯ÂÃõ 4§¿÷Å𼠯ըǢý ¯ÂÃõ 4§¿÷Å𼠯ըǢý ¯ÂÃõ 4....5 ¦º.Á£. ±É¢ø 5 ¦º.Á£. ±É¢ø 5 ¦º.Á£. ±É¢ø 5 ¦º.Á£. ±É¢ø «ù×ըǢý «ù×ըǢý «ù×ըǢý «ù×ըǢý ¬Ãò¨¾ì ¸¡ñ¸.¬Ãò¨¾ì ¸¡ñ¸.¬Ãò¨¾ì ¸¡ñ¸.¬Ãò¨¾ì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ¯Õ¨Ç¢ý ¯Â¯Õ¨Ç¢ý ¯Â¯Õ¨Ç¢ý ¯Â¯Õ¨Ç¢ý ¯ÂÃÃÃÃõ h = 4õ h = 4õ h = 4õ h = 4....5 ¦º.Á£. 5 ¦º.Á£. 5 ¦º.Á£. 5 ¦º.Á£.

¸É «Ç× ¸É «Ç× ¸É «Ç× ¸É «Ç× ππππ r r r r 2222 h = 62.37 ¸.¦º.Á£.h = 62.37 ¸.¦º.Á£.h = 62.37 ¸.¦º.Á£.h = 62.37 ¸.¦º.Á£. r r r r 2222 = 62.37= 62.37= 62.37= 62.37

ππππ h h h h = 62.37 X 7= 62.37 X 7= 62.37 X 7= 62.37 X 7 22 X 422 X 422 X 422 X 4....5555 r r r r 2222 = 4.41= 4.41= 4.41= 4.41 ¬Ãõ r = 2.1 ¦º.Á£. ¬Ãõ r = 2.1 ¦º.Á£. ¬Ãõ r = 2.1 ¦º.Á£. ¬Ãõ r = 2.1 ¦º.Á£. ((((5555)))) 8.4 ¦º.Á£. 8.4 ¦º.Á£. 8.4 ¦º.Á£. 8.4 ¦º.Á£. Å¢ð¼õ ¦¸¡ñ¼ ´Õ §¸¡ÇÅÊŠŢð¼õ ¦¸¡ñ¼ ´Õ §¸¡ÇÅÊŠŢð¼õ ¦¸¡ñ¼ ´Õ §¸¡ÇÅÊŠŢð¼õ ¦¸¡ñ¼ ´Õ §¸¡ÇÅÊÅ ¾¢ñÁ ¯§Ä¡¸ ±È¢ÌñÊý ¸É «Ç¨Åì ¸¡ñ¸.¾¢ñÁ ¯§Ä¡¸ ±È¢ÌñÊý ¸É «Ç¨Åì ¸¡ñ¸.¾¢ñÁ ¯§Ä¡¸ ±È¢ÌñÊý ¸É «Ç¨Åì ¸¡ñ¸.¾¢ñÁ ¯§Ä¡¸ ±È¢ÌñÊý ¸É «Ç¨Åì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : §¸¡Ç ÌñÊý Å¢ð¼õ d = 8.4 ¦º.Á£.§¸¡Ç ÌñÊý Å¢ð¼õ d = 8.4 ¦º.Á£.§¸¡Ç ÌñÊý Å¢ð¼õ d = 8.4 ¦º.Á£.§¸¡Ç ÌñÊý Å¢ð¼õ d = 8.4 ¦º.Á£. ¬Ãõ r = 4.2 ¦º.Á£.¬Ãõ r = 4.2 ¦º.Á£.¬Ãõ r = 4.2 ¦º.Á£.¬Ãõ r = 4.2 ¦º.Á£.

¸É «Ç× v = 4 ¸É «Ç× v = 4 ¸É «Ç× v = 4 ¸É «Ç× v = 4 ππππ r r r r 3333 º.«.º.«.º.«.º.«. 3333 = 4 X 22 X 4.2 X 4.2 X 4.2= 4 X 22 X 4.2 X 4.2 X 4.2= 4 X 22 X 4.2 X 4.2 X 4.2= 4 X 22 X 4.2 X 4.2 X 4.2 3 X 73 X 73 X 73 X 7 §¸¡Ç ÌñÊý ¸É«Ç× = 310.464§¸¡Ç ÌñÊý ¸É«Ç× = 310.464§¸¡Ç ÌñÊý ¸É«Ç× = 310.464§¸¡Ç ÌñÊý ¸É«Ç× = 310.464 ¸.¦º.Á£.¸.¦º.Á£.¸.¦º.Á£.¸.¦º.Á£. ((((6666)))) 98.56 º.¦º.Á£. ÒÈôÀÃôÒ ¦¸¡ñ¼ ´Õ 98.56 º.¦º.Á£. ÒÈôÀÃôÒ ¦¸¡ñ¼ ´Õ 98.56 º.¦º.Á£. ÒÈôÀÃôÒ ¦¸¡ñ¼ ´Õ 98.56 º.¦º.Á£. ÒÈôÀÃôÒ ¦¸¡ñ¼ ´Õ ¾¢ñÁì ¾¢ñÁì ¾¢ñÁì ¾¢ñÁì §¸¡Çò¾¢ý ¬Ãò¨¾ì ¸¡ñ¸.§¸¡Çò¾¢ý ¬Ãò¨¾ì ¸¡ñ¸.§¸¡Çò¾¢ý ¬Ãò¨¾ì ¸¡ñ¸.§¸¡Çò¾¢ý ¬Ãò¨¾ì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× :

§¸¡Çò¾¢ý ÒÈôÀÃôÒ 4 §¸¡Çò¾¢ý ÒÈôÀÃôÒ 4 §¸¡Çò¾¢ý ÒÈôÀÃôÒ 4 §¸¡Çò¾¢ý ÒÈôÀÃôÒ 4 ππππ r r r r 2222 = 98.56 º.¦º.Á£.= 98.56 º.¦º.Á£.= 98.56 º.¦º.Á£.= 98.56 º.¦º.Á£. r r r r 2222 = 98.56= 98.56= 98.56= 98.56

4 4 4 4 ππππ = 98.56 X 7= 98.56 X 7= 98.56 X 7= 98.56 X 7 4 X 224 X 224 X 224 X 22 r r r r 2222 = 7.84= 7.84= 7.84= 7.84 §¸¡Çò¾¢ý ¬Ãõ r = 2.8 ¦º.Á£.§¸¡Çò¾¢ý ¬Ãõ r = 2.8 ¦º.Á£.§¸¡Çò¾¢ý ¬Ãõ r = 2.8 ¦º.Á£.§¸¡Çò¾¢ý ¬Ãõ r = 2.8 ¦º.Á£. ((((7777)))) þÃñÎ þÃñÎ þÃñÎ þÃñÎ §¿÷Å𼠯ըǸǢý ¬Ãí¸Ç¢ý Å¢¸¢¾õ §¿÷Å𼠯ըǸǢý ¬Ãí¸Ç¢ý Å¢¸¢¾õ §¿÷Å𼠯ըǸǢý ¬Ãí¸Ç¢ý Å¢¸¢¾õ §¿÷Å𼠯ըǸǢý ¬Ãí¸Ç¢ý Å¢¸¢¾õ 3 : 2 ±ý¸. §ÁÖõ «ÅüÈ¢ý ¯ÂÃí¸Ç¢ý Å¢¸¢¾õ 5 : 3 3 : 2 ±ý¸. §ÁÖõ «ÅüÈ¢ý ¯ÂÃí¸Ç¢ý Å¢¸¢¾õ 5 : 3 3 : 2 ±ý¸. §ÁÖõ «ÅüÈ¢ý ¯ÂÃí¸Ç¢ý Å¢¸¢¾õ 5 : 3 3 : 2 ±ý¸. §ÁÖõ «ÅüÈ¢ý ¯ÂÃí¸Ç¢ý Å¢¸¢¾õ 5 : 3 ±É¢ø, «ÅüÈ¢ý ŨÇÀÃôҸǢý Å¢¸¢¾ò¨¾ì ¸¡ñ¸.±É¢ø, «ÅüÈ¢ý ŨÇÀÃôҸǢý Å¢¸¢¾ò¨¾ì ¸¡ñ¸.±É¢ø, «ÅüÈ¢ý ŨÇÀÃôҸǢý Å¢¸¢¾ò¨¾ì ¸¡ñ¸.±É¢ø, «ÅüÈ¢ý ŨÇÀÃôҸǢý Å¢¸¢¾ò¨¾ì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ¯Õ¨Ç¸Ç¢ý ¯Õ¨Ç¸Ç¢ý ¯Õ¨Ç¸Ç¢ý ¯Õ¨Ç¸Ç¢ý ¬Ãí¸Ç¢ý Å¢¸¢¾õ r ¬Ãí¸Ç¢ý Å¢¸¢¾õ r ¬Ãí¸Ç¢ý Å¢¸¢¾õ r ¬Ãí¸Ç¢ý Å¢¸¢¾õ r 1111 : r : r : r : r 2222 = 3 : 2= 3 : 2= 3 : 2= 3 : 2 ¯ÂÃí¸Ç¢ý Å¢¸¢¾õ h ¯ÂÃí¸Ç¢ý Å¢¸¢¾õ h ¯ÂÃí¸Ç¢ý Å¢¸¢¾õ h ¯ÂÃí¸Ç¢ý Å¢¸¢¾õ h 1111 : h : h : h : h 2222 = 5 : 3= 5 : 3= 5 : 3= 5 : 3

ŨÇÀÃôҸǢý Å¢¸¢¾õ 2 ŨÇÀÃôҸǢý Å¢¸¢¾õ 2 ŨÇÀÃôҸǢý Å¢¸¢¾õ 2 ŨÇÀÃôҸǢý Å¢¸¢¾õ 2 ππππ rrrr1 1 1 1 hhhh1111 : 2 : 2 : 2 : 2 π π π π rrrr2222 hhhh2222 rrrr1 1 1 1 hhhh1111 : : : : rrrr2222 hhhh2222 3 X 5 : 3 X 5 : 3 X 5 : 3 X 5 : 2 X 32 X 32 X 32 X 3 ŨÇÀÃôҸǢý Å¢¸¢¾õ = 5 : 2ŨÇÀÃôҸǢý Å¢¸¢¾õ = 5 : 2ŨÇÀÃôҸǢý Å¢¸¢¾õ = 5 : 2ŨÇÀÃôҸǢý Å¢¸¢¾õ = 5 : 2 ((((8888)))) ´Õ ¾¢ñÁ §¿÷Å𼠯ըǢý ¦Á¡ò¾ôÀÃôÒ ´Õ ¾¢ñÁ §¿÷Å𼠯ըǢý ¦Á¡ò¾ôÀÃôÒ ´Õ ¾¢ñÁ §¿÷Å𼠯ըǢý ¦Á¡ò¾ôÀÃôÒ ´Õ ¾¢ñÁ §¿÷Å𼠯ըǢý ¦Á¡ò¾ôÀÃôÒ

660 660 660 660 º.¦º.Á£. «¾ý «ÊôÀì¸ò¾¢ý Å¢ð¼õ 14 ¦º.Á£. º.¦º.Á£. «¾ý «ÊôÀì¸ò¾¢ý Å¢ð¼õ 14 ¦º.Á£. º.¦º.Á£. «¾ý «ÊôÀì¸ò¾¢ý Å¢ð¼õ 14 ¦º.Á£. º.¦º.Á£. «¾ý «ÊôÀì¸ò¾¢ý Å¢ð¼õ 14 ¦º.Á£. ±É¢ø, «ù×ըǢý ¯ÂÃò¨¾ì ¸¡ñ¸.±É¢ø, «ù×ըǢý ¯ÂÃò¨¾ì ¸¡ñ¸.±É¢ø, «ù×ըǢý ¯ÂÃò¨¾ì ¸¡ñ¸.±É¢ø, «ù×ըǢý ¯ÂÃò¨¾ì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ¯Õ¨Ç¢ý Å¢ð¼õ d = 14 , ¬Ãõ r = 7 ¦º.Á£. ¯Õ¨Ç¢ý Å¢ð¼õ d = 14 , ¬Ãõ r = 7 ¦º.Á£. ¯Õ¨Ç¢ý Å¢ð¼õ d = 14 , ¬Ãõ r = 7 ¦º.Á£. ¯Õ¨Ç¢ý Å¢ð¼õ d = 14 , ¬Ãõ r = 7 ¦º.Á£.

¦Á¡ò¾ô ÒÈôÀÃôÒ ¦Á¡ò¾ô ÒÈôÀÃôÒ ¦Á¡ò¾ô ÒÈôÀÃôÒ ¦Á¡ò¾ô ÒÈôÀÃôÒ 2222ππππr(h+r)r(h+r)r(h+r)r(h+r) = = = = 660 º.¦º.Á£.660 º.¦º.Á£.660 º.¦º.Á£.660 º.¦º.Á£. h + r = 660h + r = 660h + r = 660h + r = 660

2222ππππrrrr h + 7 = 660 X 7h + 7 = 660 X 7h + 7 = 660 X 7h + 7 = 660 X 7 2 X 22 X 72 X 22 X 72 X 22 X 72 X 22 X 7 h = 15 h = 15 h = 15 h = 15 ---- 7777 ¯ÂÃõ¯ÂÃõ¯ÂÃõ¯ÂÃõ h = 8h = 8h = 8h = 8 ¦º.Á£.¦º.Á£.¦º.Á£.¦º.Á£.

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((((9999)))) ´Õ ¾¢ñÁ §¿÷Åð¼ì ÜõÀ¢ý ¬Ãõ ÁüÚõ ¯ÂÃõ ´Õ ¾¢ñÁ §¿÷Åð¼ì ÜõÀ¢ý ¬Ãõ ÁüÚõ ¯ÂÃõ ´Õ ¾¢ñÁ §¿÷Åð¼ì ÜõÀ¢ý ¬Ãõ ÁüÚõ ¯ÂÃõ ´Õ ¾¢ñÁ §¿÷Åð¼ì ÜõÀ¢ý ¬Ãõ ÁüÚõ ¯ÂÃõ Өȧ 7 ¦º.Á£. ÁüÚõ 24 ¦º.Á£. ±É¢ø, «¾ý Өȧ 7 ¦º.Á£. ÁüÚõ 24 ¦º.Á£. ±É¢ø, «¾ý Өȧ 7 ¦º.Á£. ÁüÚõ 24 ¦º.Á£. ±É¢ø, «¾ý Өȧ 7 ¦º.Á£. ÁüÚõ 24 ¦º.Á£. ±É¢ø, «¾ý ŨÇÀÃô¨Àì ¸¡ñ¸.ŨÇÀÃô¨Àì ¸¡ñ¸.ŨÇÀÃô¨Àì ¸¡ñ¸.ŨÇÀÃô¨Àì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ÜõÀ¢ý ÜõÀ¢ý ÜõÀ¢ý ÜõÀ¢ý ¬Ãõ r = 7 ¦º.Á£.¬Ãõ r = 7 ¦º.Á£.¬Ãõ r = 7 ¦º.Á£.¬Ãõ r = 7 ¦º.Á£. ¯ÂÃõ h = 24 ¦º.Á£.¯ÂÃõ h = 24 ¦º.Á£.¯ÂÃõ h = 24 ¦º.Á£.¯ÂÃõ h = 24 ¦º.Á£.

º¡ÔÂÃõ º¡ÔÂÃõ º¡ÔÂÃõ º¡ÔÂÃõ l = = = = √√√√ h h h h 2222 + r + r + r + r 2222 = = = = √√√√ 24 24 24 24 2222 + 7 + 7 + 7 + 7 2222

= = = = √√√√ 576 + 49 = 576 + 49 = 576 + 49 = 576 + 49 = √√√√ 625625625625

l = 25 ¦º.Á£.= 25 ¦º.Á£.= 25 ¦º.Á£.= 25 ¦º.Á£.

ÜõÀ¢ý ŨÇÀÃôÒ = ÜõÀ¢ý ŨÇÀÃôÒ = ÜõÀ¢ý ŨÇÀÃôÒ = ÜõÀ¢ý ŨÇÀÃôÒ = ���� r r r r l º.«.º.«.º.«.º.«. = 22 X22 X22 X22 X 7 X 257 X 257 X 257 X 25 7777 ÜõÀ¢ý ŨÇÀÃôÒ = 550 º.¦º.Á£. ÜõÀ¢ý ŨÇÀÃôÒ = 550 º.¦º.Á£. ÜõÀ¢ý ŨÇÀÃôÒ = 550 º.¦º.Á£. ÜõÀ¢ý ŨÇÀÃôÒ = 550 º.¦º.Á£. ((((10101010)))) þÃñÎ §¸¡Çí¸Ç¢ýþÃñÎ §¸¡Çí¸Ç¢ýþÃñÎ §¸¡Çí¸Ç¢ýþÃñÎ §¸¡Çí¸Ç¢ý ¬Ãí¸Ç¢ý¬Ãí¸Ç¢ý¬Ãí¸Ç¢ý¬Ãí¸Ç¢ý Å¢¸¢¾õ Å¢¸¢¾õ Å¢¸¢¾õ Å¢¸¢¾õ 3 : 4 3 : 4 3 : 4 3 : 4 ±É¢ø, «ÅüÈ¢ý ¸É «Ç׸Ǣý Å¢¸¢¾õ ¸¡ñ¸.±É¢ø, «ÅüÈ¢ý ¸É «Ç׸Ǣý Å¢¸¢¾õ ¸¡ñ¸.±É¢ø, «ÅüÈ¢ý ¸É «Ç׸Ǣý Å¢¸¢¾õ ¸¡ñ¸.±É¢ø, «ÅüÈ¢ý ¸É «Ç׸Ǣý Å¢¸¢¾õ ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : þÃñÎ §¸¡Çí¸Ç¢ý ¬Ãí¸Ç¢ý Å¢¸¢¾õ þÃñÎ §¸¡Çí¸Ç¢ý ¬Ãí¸Ç¢ý Å¢¸¢¾õ þÃñÎ §¸¡Çí¸Ç¢ý ¬Ãí¸Ç¢ý Å¢¸¢¾õ þÃñÎ §¸¡Çí¸Ç¢ý ¬Ãí¸Ç¢ý Å¢¸¢¾õ r r r r 1111 : r : r : r : r 2222 = 3 : 4 = 3 : 4 = 3 : 4 = 3 : 4

¸É «Ç׸Ǣý Å¢¸¢¾õ = 4 ¸É «Ç׸Ǣý Å¢¸¢¾õ = 4 ¸É «Ç׸Ǣý Å¢¸¢¾õ = 4 ¸É «Ç׸Ǣý Å¢¸¢¾õ = 4 ���� rrrr1111 3333 : 4 : 4 : 4 : 4 ���� rrrr2222

3333 3 33 33 33 3 rrrr1111

3333 : r: r: r: r2222 3333 = 3 = 3 = 3 = 3 3333 : 4: 4: 4: 4 3333

¸É «Ç׸Ǣý Å¢¸¢¾õ = 27 : 64¸É «Ç׸Ǣý Å¢¸¢¾õ = 27 : 64¸É «Ç׸Ǣý Å¢¸¢¾õ = 27 : 64¸É «Ç׸Ǣý Å¢¸¢¾õ = 27 : 64

((((11111111)))) ´Õ ¾¢ñÁ ¯Õ¨Ç¢ý ¬Ãõ 14 ¦º.Á£. ¯ÂÃõ ´Õ ¾¢ñÁ ¯Õ¨Ç¢ý ¬Ãõ 14 ¦º.Á£. ¯ÂÃõ ´Õ ¾¢ñÁ ¯Õ¨Ç¢ý ¬Ãõ 14 ¦º.Á£. ¯ÂÃõ ´Õ ¾¢ñÁ ¯Õ¨Ç¢ý ¬Ãõ 14 ¦º.Á£. ¯ÂÃõ 30 ¦º.Á£. ±É¢ø, «ù×ըǢý ¸É «Ç× ¸¡ñ¸.30 ¦º.Á£. ±É¢ø, «ù×ըǢý ¸É «Ç× ¸¡ñ¸.30 ¦º.Á£. ±É¢ø, «ù×ըǢý ¸É «Ç× ¸¡ñ¸.30 ¦º.Á£. ±É¢ø, «ù×ըǢý ¸É «Ç× ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ¯Õ¨Ç¢ý ¬Ãõ r = 14 ¦º.Á£.¯Õ¨Ç¢ý ¬Ãõ r = 14 ¦º.Á£.¯Õ¨Ç¢ý ¬Ãõ r = 14 ¦º.Á£.¯Õ¨Ç¢ý ¬Ãõ r = 14 ¦º.Á£. ¯ÂÃõ h = 30 ¦º.Á£.¯ÂÃõ h = 30 ¦º.Á£.¯ÂÃõ h = 30 ¦º.Á£.¯ÂÃõ h = 30 ¦º.Á£.

¸É «Ç× = ¸É «Ç× = ¸É «Ç× = ¸É «Ç× = ���� r r r r 2222 h º.«.h º.«.h º.«.h º.«. = 22 X 14 X 14 X 30= 22 X 14 X 14 X 30= 22 X 14 X 14 X 30= 22 X 14 X 14 X 30 7 7 7 7 ¯Õ¨Ç¢ý ¸É«Ç× = 18480 ¯Õ¨Ç¢ý ¸É«Ç× = 18480 ¯Õ¨Ç¢ý ¸É«Ç× = 18480 ¯Õ¨Ç¢ý ¸É«Ç× = 18480 ¸̧̧̧.¦º.Á£. .¦º.Á£. .¦º.Á£. .¦º.Á£.

11.11.11.11. ÒûǢ¢ÂøÒûǢ¢ÂøÒûǢ¢ÂøÒûǢ¢Âø (1)(1)(1)(1) Ó¾ø 13 þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸ò¨¾ì Ó¾ø 13 þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸ò¨¾ì Ó¾ø 13 þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸ò¨¾ì Ó¾ø 13 þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸ò¨¾ì ¸½ì¸¢Î¸.¸½ì¸¢Î¸.¸½ì¸¢Î¸.¸½ì¸¢Î¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : Ó¾ø n þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸õÓ¾ø n þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸õÓ¾ø n þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸õÓ¾ø n þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸õ

���� = = = = n n n n 2222 ---- 1111

√ √ √ √ 12121212

n = 13 ±É¢ø, n = 13 ±É¢ø, n = 13 ±É¢ø, n = 13 ±É¢ø, ���� = 13 = 13 = 13 = 13 2222 ---- 1 = 169 1 = 169 1 = 169 1 = 169 ---- 1111

√√√√ 12 12 12 12 √√√√ 12121212

= 168 = = 168 = = 168 = = 168 = √√√√ 14141414

√√√√ 12121212

���� = 3.74= 3.74= 3.74= 3.74 ((((2222)))) Ó¾ø 1Ó¾ø 1Ó¾ø 1Ó¾ø 10000 þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸ò¨¾ì þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸ò¨¾ì þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸ò¨¾ì þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸ò¨¾ì

¸½ì¸¢Î¸.¸½ì¸¢Î¸.¸½ì¸¢Î¸.¸½ì¸¢Î¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : Ó¾ø n þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸õÓ¾ø n þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸õÓ¾ø n þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸õÓ¾ø n þÂø ±ñ¸Ç¢ý ¾¢ð¼Å¢Äì¸õ

���� = = = = n n n n 2222 ---- 1111

√ √ √ √ 12121212

n = 10 ±É¢ø, n = 10 ±É¢ø, n = 10 ±É¢ø, n = 10 ±É¢ø, ���� = 10 = 10 = 10 = 10 2222 ---- 1 = 1 = 1 = 1 = 100100100100 ---- 1111

√√√√ 12 12 12 12 √√√√ 12121212 = 99= 99= 99= 99

√√√√ 12121212

���� = = = = 2222....87878787 ((((3333)))) ´Õ ÒûÇ¢ Å¢ÅÃò ¦¾¡ÌÀ¢ý Á£îº¢Ú Á¾¢ôÒ 12 ´Õ ÒûÇ¢ Å¢ÅÃò ¦¾¡ÌÀ¢ý Á£îº¢Ú Á¾¢ôÒ 12 ´Õ ÒûÇ¢ Å¢ÅÃò ¦¾¡ÌÀ¢ý Á£îº¢Ú Á¾¢ôÒ 12 ´Õ ÒûÇ¢ Å¢ÅÃò ¦¾¡ÌÀ¢ý Á£îº¢Ú Á¾¢ôÒ 12 «¾ý Å£îÍ 59 ±É¢ø, «ôÒûÇ¢ Å¢ÅÃò¾¢ý «¾ý Å£îÍ 59 ±É¢ø, «ôÒûÇ¢ Å¢ÅÃò¾¢ý «¾ý Å£îÍ 59 ±É¢ø, «ôÒûÇ¢ Å¢ÅÃò¾¢ý «¾ý Å£îÍ 59 ±É¢ø, «ôÒûÇ¢ Å¢ÅÃò¾¢ý Á£ô¦ÀÕ Á¾¢ô¨Àì ¸¡ñ¸.Á£ô¦ÀÕ Á¾¢ô¨Àì ¸¡ñ¸.Á£ô¦ÀÕ Á¾¢ô¨Àì ¸¡ñ¸.Á£ô¦ÀÕ Á¾¢ô¨Àì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ÒûÇ¢ Å¢ÅÃò¾¢ý Á£îº¢Ú Á¾¢ôÒÒûÇ¢ Å¢ÅÃò¾¢ý Á£îº¢Ú Á¾¢ôÒÒûÇ¢ Å¢ÅÃò¾¢ý Á£îº¢Ú Á¾¢ôÒÒûÇ¢ Å¢ÅÃò¾¢ý Á£îº¢Ú Á¾¢ôÒ S = 12S = 12S = 12S = 12 ÅÅÅÅ£îÍ = 59£îÍ = 59£îÍ = 59£îÍ = 59 Å£îÍ = Á£ô¦ÀÕ Á¾¢ôÒ Å£îÍ = Á£ô¦ÀÕ Á¾¢ôÒ Å£îÍ = Á£ô¦ÀÕ Á¾¢ôÒ Å£îÍ = Á£ô¦ÀÕ Á¾¢ôÒ ---- Á£îº¢Ú Á¾¢ôÒ Á£îº¢Ú Á¾¢ôÒ Á£îº¢Ú Á¾¢ôÒ Á£îº¢Ú Á¾¢ôÒ 59 = L 59 = L 59 = L 59 = L ---- 12121212 59 + 12 = L59 + 12 = L59 + 12 = L59 + 12 = L L = 71 L = 71 L = 71 L = 71

((((4444)))) ´Õ ÒûÇ¢ Å¢ÅÃò ¦¾¡ÌôÀ¢ý ´Õ ÒûÇ¢ Å¢ÅÃò ¦¾¡ÌôÀ¢ý ´Õ ÒûÇ¢ Å¢ÅÃò ¦¾¡ÌôÀ¢ý ´Õ ÒûÇ¢ Å¢ÅÃò ¦¾¡ÌôÀ¢ý Á£ô¦ÀÕ Á¾¢ôÒ Á£ô¦ÀÕ Á¾¢ôÒ Á£ô¦ÀÕ Á¾¢ôÒ Á£ô¦ÀÕ Á¾¢ôÒ 7.44 ÁüÚõ «¾ý Å£îÍ 2.26 ±É¢ø, 7.44 ÁüÚõ «¾ý Å£îÍ 2.26 ±É¢ø, 7.44 ÁüÚõ «¾ý Å£îÍ 2.26 ±É¢ø, 7.44 ÁüÚõ «¾ý Å£îÍ 2.26 ±É¢ø, «ò¦¾¡ÌôÀ¢ý Á£îº¢Ú Á¾¢ô¨Àì ¸¡ñ¸.«ò¦¾¡ÌôÀ¢ý Á£îº¢Ú Á¾¢ô¨Àì ¸¡ñ¸.«ò¦¾¡ÌôÀ¢ý Á£îº¢Ú Á¾¢ô¨Àì ¸¡ñ¸.«ò¦¾¡ÌôÀ¢ý Á£îº¢Ú Á¾¢ô¨Àì ¸¡ñ¸.

¾£÷× :¾£÷× :¾£÷× :¾£÷× : ÒûÇ¢ Å¢ÅÃò¾¢ý Á£ô¦ÀÕ Á¾¢ôÒÒûÇ¢ Å¢ÅÃò¾¢ý Á£ô¦ÀÕ Á¾¢ôÒÒûÇ¢ Å¢ÅÃò¾¢ý Á£ô¦ÀÕ Á¾¢ôÒÒûÇ¢ Å¢ÅÃò¾¢ý Á£ô¦ÀÕ Á¾¢ôÒ L = 7.44L = 7.44L = 7.44L = 7.44 Å£îÍ = 2.26Å£îÍ = 2.26Å£îÍ = 2.26Å£îÍ = 2.26 Å£îÍ = Á£ô¦ÀÕ Á¾¢ôÒ Å£îÍ = Á£ô¦ÀÕ Á¾¢ôÒ Å£îÍ = Á£ô¦ÀÕ Á¾¢ôÒ Å£îÍ = Á£ô¦ÀÕ Á¾¢ôÒ ---- Á£îº¢Ú Á¾¢ôÒ Á£îº¢Ú Á¾¢ôÒ Á£îº¢Ú Á¾¢ôÒ Á£îº¢Ú Á¾¢ôÒ 2.26 = 7.44 2.26 = 7.44 2.26 = 7.44 2.26 = 7.44 ---- SSSS S = 7.44 S = 7.44 S = 7.44 S = 7.44 ---- 2.262.262.262.26 S = 5.18 S = 5.18 S = 5.18 S = 5.18

((((5555)))) 43,24,38,56,22,39,45 ¬¸¢Â ÒûÇ¢ 43,24,38,56,22,39,45 ¬¸¢Â ÒûÇ¢ 43,24,38,56,22,39,45 ¬¸¢Â ÒûÇ¢ 43,24,38,56,22,39,45 ¬¸¢Â ÒûÇ¢ Å¢ÅÃí¸Ç¢ý ţŢÅÃí¸Ç¢ý ţŢÅÃí¸Ç¢ý ţŢÅÃí¸Ç¢ý Å£îÍîÍîÍîÍ ÁüÚõ Å£îÍì ¦¸Ø ¸¡ñ¸.ÁüÚõ Å£îÍì ¦¸Ø ¸¡ñ¸.ÁüÚõ Å£îÍì ¦¸Ø ¸¡ñ¸.ÁüÚõ Å£îÍì ¦¸Ø ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : 43,24,38,56,22,39,45 ¬¸¢Â ÒûÇ¢ Å¢ÅÃí¸Ç¢ý43,24,38,56,22,39,45 ¬¸¢Â ÒûÇ¢ Å¢ÅÃí¸Ç¢ý43,24,38,56,22,39,45 ¬¸¢Â ÒûÇ¢ Å¢ÅÃí¸Ç¢ý43,24,38,56,22,39,45 ¬¸¢Â ÒûÇ¢ Å¢ÅÃí¸Ç¢ý Á£ô¦ÀÕ Á¾¢ôÒ L = 56Á£ô¦ÀÕ Á¾¢ôÒ L = 56Á£ô¦ÀÕ Á¾¢ôÒ L = 56Á£ô¦ÀÕ Á¾¢ôÒ L = 56 Á£îº¢Ú Á¾¢ôÒ S = 22Á£îº¢Ú Á¾¢ôÒ S = 22Á£îº¢Ú Á¾¢ôÒ S = 22Á£îº¢Ú Á¾¢ôÒ S = 22 Å£îÍ = L Å£îÍ = L Å£îÍ = L Å£îÍ = L ---- SSSS = 56 = 56 = 56 = 56 ---- 22222222

Å£îÍÅ£îÍÅ£îÍÅ£îÍ = 34 = 34 = 34 = 34

Å£îÍì ¦¸Ø = L Å£îÍì ¦¸Ø = L Å£îÍì ¦¸Ø = L Å£îÍì ¦¸Ø = L ---- SSSS L + S L + S L + S L + S = 34= 34= 34= 34 78787878

Å£îÍì ¦¸Ø = 0.436Å£îÍì ¦¸Ø = 0.436Å£îÍì ¦¸Ø = 0.436Å£îÍì ¦¸Ø = 0.436

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((((6666)))) 50 «Ç׸Ǣø Á¢¸ô¦À¡¢Â Á¾¢ôÒ50 «Ç׸Ǣø Á¢¸ô¦À¡¢Â Á¾¢ôÒ50 «Ç׸Ǣø Á¢¸ô¦À¡¢Â Á¾¢ôÒ50 «Ç׸Ǣø Á¢¸ô¦À¡¢Â Á¾¢ôÒ 3.84 ¸¢.¸¢. 3.84 ¸¢.¸¢. 3.84 ¸¢.¸¢. 3.84 ¸¢.¸¢. «¾ý Å£îÍ 0.46 ¸¢.¸¢. ±É¢ø, «ÅüÈ¢ý Á£îº¢Ú «¾ý Å£îÍ 0.46 ¸¢.¸¢. ±É¢ø, «ÅüÈ¢ý Á£îº¢Ú «¾ý Å£îÍ 0.46 ¸¢.¸¢. ±É¢ø, «ÅüÈ¢ý Á£îº¢Ú «¾ý Å£îÍ 0.46 ¸¢.¸¢. ±É¢ø, «ÅüÈ¢ý Á£îº¢Ú Á¾¢ô¨Àì ¸¡ñ¸.Á¾¢ô¨Àì ¸¡ñ¸.Á¾¢ô¨Àì ¸¡ñ¸.Á¾¢ô¨Àì ¸¡ñ¸. ¾£÷× : ¾£÷× : ¾£÷× : ¾£÷× : 50 «Ç׸Ǣý50 «Ç׸Ǣý50 «Ç׸Ǣý50 «Ç׸Ǣý Á£ô¦ÀÕ Á¾¢ôÒÁ£ô¦ÀÕ Á¾¢ôÒÁ£ô¦ÀÕ Á¾¢ôÒÁ£ô¦ÀÕ Á¾¢ôÒ

L = 3.84L = 3.84L = 3.84L = 3.84 Å£îÍ = 0.46Å£îÍ = 0.46Å£îÍ = 0.46Å£îÍ = 0.46 Å£îÍ = Á£ô¦ÀÕ Å£îÍ = Á£ô¦ÀÕ Å£îÍ = Á£ô¦ÀÕ Å£îÍ = Á£ô¦ÀÕ Á¾¢ôÒ Á¾¢ôÒ Á¾¢ôÒ Á¾¢ôÒ ---- Á£îº¢Ú Á¾¢ôÒ Á£îº¢Ú Á¾¢ôÒ Á£îº¢Ú Á¾¢ôÒ Á£îº¢Ú Á¾¢ôÒ 0.46 = 3.84 0.46 = 3.84 0.46 = 3.84 0.46 = 3.84 ---- SSSS S = 3.84 S = 3.84 S = 3.84 S = 3.84 ---- 0.460.460.460.46 S = 3.38 S = 3.38 S = 3.38 S = 3.38

((((7777)))) ´Õ ÒûÇ¢ Å¢ÅÃò¾¢ý Á¡ÚÀ¡ðÎì ¦¸Ø 57 ´Õ ÒûÇ¢ Å¢ÅÃò¾¢ý Á¡ÚÀ¡ðÎì ¦¸Ø 57 ´Õ ÒûÇ¢ Å¢ÅÃò¾¢ý Á¡ÚÀ¡ðÎì ¦¸Ø 57 ´Õ ÒûÇ¢ Å¢ÅÃò¾¢ý Á¡ÚÀ¡ðÎì ¦¸Ø 57 ÁüÚõ ¾¢ð¼Å¢Äì¸õ 6.84 ±É¢ø, «¾ý ÜðÎî ÁüÚõ ¾¢ð¼Å¢Äì¸õ 6.84 ±É¢ø, «¾ý ÜðÎî ÁüÚõ ¾¢ð¼Å¢Äì¸õ 6.84 ±É¢ø, «¾ý ÜðÎî ÁüÚõ ¾¢ð¼Å¢Äì¸õ 6.84 ±É¢ø, «¾ý ÜðÎî ºÃ¡º¡¢¨Âì ¸¡ñ¸.ºÃ¡º¡¢¨Âì ¸¡ñ¸.ºÃ¡º¡¢¨Âì ¸¡ñ¸.ºÃ¡º¡¢¨Âì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : Á¡Á¡Á¡Á¡ÚÀ¡ðÎì ¦¸Ø CV = 57ÚÀ¡ðÎì ¦¸Ø CV = 57ÚÀ¡ðÎì ¦¸Ø CV = 57ÚÀ¡ðÎì ¦¸Ø CV = 57

¾¢ð¼Å¢Äì¸õ ¾¢ð¼Å¢Äì¸õ ¾¢ð¼Å¢Äì¸õ ¾¢ð¼Å¢Äì¸õ ���� = 6.84= 6.84= 6.84= 6.84

Á¡ÚÀ¡ðÎì ¦¸Ø CV = Á¡ÚÀ¡ðÎì ¦¸Ø CV = Á¡ÚÀ¡ðÎì ¦¸Ø CV = Á¡ÚÀ¡ðÎì ¦¸Ø CV = ���� X 100X 100X 100X 100 x x x x

x = x = x = x = ���� X 100X 100X 100X 100 CVCVCVCV x = 6.84 X 100x = 6.84 X 100x = 6.84 X 100x = 6.84 X 100 57575757 ÜðÎî ºÃ¡º¡¢ x = 12 ÜðÎî ºÃ¡º¡¢ x = 12 ÜðÎî ºÃ¡º¡¢ x = 12 ÜðÎî ºÃ¡º¡¢ x = 12

12. ¿¢¸ú¾¸×12. ¿¢¸ú¾¸×12. ¿¢¸ú¾¸×12. ¿¢¸ú¾¸× (1) (1) (1) (1) þÕ ¿¡½Âí¸¨Ç ´§Ã ºÁÂò¾¢ø ÍñÎõ§À¡Ð þÕ ¿¡½Âí¸¨Ç ´§Ã ºÁÂò¾¢ø ÍñÎõ§À¡Ð þÕ ¿¡½Âí¸¨Ç ´§Ã ºÁÂò¾¢ø ÍñÎõ§À¡Ð þÕ ¿¡½Âí¸¨Ç ´§Ã ºÁÂò¾¢ø ÍñÎõ§À¡Ð «¾¢¸ÀðºÁ¡¸ ´Õ ¾¨Ä ¸¢¨¼ôÀ¾ü¸¡É «¾¢¸ÀðºÁ¡¸ ´Õ ¾¨Ä ¸¢¨¼ôÀ¾ü¸¡É «¾¢¸ÀðºÁ¡¸ ´Õ ¾¨Ä ¸¢¨¼ôÀ¾ü¸¡É «¾¢¸ÀðºÁ¡¸ ´Õ ¾¨Ä ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : þÕ ¿¡½Âí¸¨Ç ÍñÎõ§À¡Ð ÜÚ¦ÅÇ¢þÕ ¿¡½Âí¸¨Ç ÍñÎõ§À¡Ð ÜÚ¦ÅÇ¢þÕ ¿¡½Âí¸¨Ç ÍñÎõ§À¡Ð ÜÚ¦ÅÇ¢þÕ ¿¡½Âí¸¨Ç ÍñÎõ§À¡Ð ÜÚ¦ÅÇ¢ S = { HH, HT, TH, TT }S = { HH, HT, TH, TT }S = { HH, HT, TH, TT }S = { HH, HT, TH, TT } n (S) = 4n (S) = 4n (S) = 4n (S) = 4 A = «¾¢¸ÀðºÁ¡¸ ´Õ ¾¨Ä ¸¢¨¼ò¾øA = «¾¢¸ÀðºÁ¡¸ ´Õ ¾¨Ä ¸¢¨¼ò¾øA = «¾¢¸ÀðºÁ¡¸ ´Õ ¾¨Ä ¸¢¨¼ò¾øA = «¾¢¸ÀðºÁ¡¸ ´Õ ¾¨Ä ¸¢¨¼ò¾ø A = { HT, TH, TT }A = { HT, TH, TT }A = { HT, TH, TT }A = { HT, TH, TT } n(A) = 3n(A) = 3n(A) = 3n(A) = 3 P(A) = n(A)P(A) = n(A)P(A) = n(A)P(A) = n(A) n(S) n(S) n(S) n(S) P(A) = 3P(A) = 3P(A) = 3P(A) = 3 4444 ((((2222)))) ãýÚ ¿¡½Âí¸û ´§Ã §¿Ãò¾¢ø Íñ¼ôÀθ¢ýÈÉ. ãýÚ ¿¡½Âí¸û ´§Ã §¿Ãò¾¢ø Íñ¼ôÀθ¢ýÈÉ. ãýÚ ¿¡½Âí¸û ´§Ã §¿Ãò¾¢ø Íñ¼ôÀθ¢ýÈÉ. ãýÚ ¿¡½Âí¸û ´§Ã §¿Ãò¾¢ø Íñ¼ôÀθ¢ýÈÉ. ̨Èó¾Ð þÕ ¾¨Ä¸û ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× ̨Èó¾Ð þÕ ¾¨Ä¸û ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× ̨Èó¾Ð þÕ ¾¨Ä¸û ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× ̨Èó¾Ð þÕ ¾¨Ä¸û ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× ¡Ð?¡Ð?¡Ð?¡Ð?

¾£÷× :¾£÷× :¾£÷× :¾£÷× : ãýÚ ¿¡½Âí¸û ÍñÎõ§À¡Ð ÜÚ¦ÅÇ¢ãýÚ ¿¡½Âí¸û ÍñÎõ§À¡Ð ÜÚ¦ÅÇ¢ãýÚ ¿¡½Âí¸û ÍñÎõ§À¡Ð ÜÚ¦ÅÇ¢ãýÚ ¿¡½Âí¸û ÍñÎõ§À¡Ð ÜÚ¦ÅÇ¢ S = { HHH, HHT, HTH, HTT,S = { HHH, HHT, HTH, HTT,S = { HHH, HHT, HTH, HTT,S = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }THH, THT, TTH, TTT }THH, THT, TTH, TTT }THH, THT, TTH, TTT } n(S) = 8n(S) = 8n(S) = 8n(S) = 8 A = ̨Èó¾Ð þÕ ¾¨Ä¸û ¸¢¨¼ò¾øA = ̨Èó¾Ð þÕ ¾¨Ä¸û ¸¢¨¼ò¾øA = ̨Èó¾Ð þÕ ¾¨Ä¸û ¸¢¨¼ò¾øA = ̨Èó¾Ð þÕ ¾¨Ä¸û ¸¢¨¼ò¾ø A = { HHH, HHT, HTH, THH }A = { HHH, HHT, HTH, THH }A = { HHH, HHT, HTH, THH }A = { HHH, HHT, HTH, THH } n(A) = 4n(A) = 4n(A) = 4n(A) = 4

P(A) = n(A) P(A) = n(A) P(A) = n(A) P(A) = n(A) = 4 = 4 = 4 = 4 n(S) n(S) n(S) n(S) 8888 P(A) = 1P(A) = 1P(A) = 1P(A) = 1 2222

((((3333)))) ´́́́Õ º£Ã¡É À¸¨¼ þÃñÎ Ó¨È ¯Õð¼ôÀÎÕ º£Ã¡É À¸¨¼ þÃñÎ Ó¨È ¯Õð¼ôÀÎÕ º£Ã¡É À¸¨¼ þÃñÎ Ó¨È ¯Õð¼ôÀÎÕ º£Ã¡É À¸¨¼ þÃñÎ Ó¨È ¯Õð¼ôÀθ¢Èи¢Èи¢Èи¢ÈÐ Ó¸ ±ñ¸Ç¢ý Üξø 9 ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× Ó¸ ±ñ¸Ç¢ý Üξø 9 ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× Ó¸ ±ñ¸Ç¢ý Üξø 9 ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× Ó¸ ±ñ¸Ç¢ý Üξø 9 ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× ¸¡ñ¸.¸¡ñ¸.¸¡ñ¸.¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ´Õ À¸¨¼ þÕ Ó¨È ¯Õð¼ôÀÎõ§À¡Ð ÜÚ¦ÅÇ¢´Õ À¸¨¼ þÕ Ó¨È ¯Õð¼ôÀÎõ§À¡Ð ÜÚ¦ÅÇ¢´Õ À¸¨¼ þÕ Ó¨È ¯Õð¼ôÀÎõ§À¡Ð ÜÚ¦ÅÇ¢´Õ À¸¨¼ þÕ Ó¨È ¯Õð¼ôÀÎõ§À¡Ð ÜÚ¦ÅÇ¢ S = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6)S = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6)S = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6)S = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6) (2,1), (2,2), (2,3), (2,4), (2,5), (2,6)(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)(2,1), (2,2), (2,3), (2,4), (2,5), (2,6) (3,1), (3,2), (3,3), (3,4), (3,5), (3,6)(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)(3,1), (3,2), (3,3), (3,4), (3,5), (3,6) (4,1), (4,2), (4,3), (4,4), (4,5), (4,6)(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)(4,1), (4,2), (4,3), (4,4), (4,5), (4,6) (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)(5,1), (5,2), (5,3), (5,4), (5,5), (5,6) (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)(6,1), (6,2), (6,3), (6,4), (6,5), (6,6) }}}} n(S) = 36n(S) = 36n(S) = 36n(S) = 36 A = Ó¸ ±ñ¸Ç¢ý Üξø 9 ¸¢A = Ó¸ ±ñ¸Ç¢ý Üξø 9 ¸¢A = Ó¸ ±ñ¸Ç¢ý Üξø 9 ¸¢A = Ó¸ ±ñ¸Ç¢ý Üξø 9 ¸¢¨¼¨¼¨¼¨¼ò¾øò¾øò¾øò¾ø A = { (3,6), A = { (3,6), A = { (3,6), A = { (3,6), (4,5), (5,4), (6,3) }(4,5), (5,4), (6,3) }(4,5), (5,4), (6,3) }(4,5), (5,4), (6,3) } n(A) = 4n(A) = 4n(A) = 4n(A) = 4

P(A) = n(A) = 4 P(A) = n(A) = 4 P(A) = n(A) = 4 P(A) = n(A) = 4 n(S) n(S) n(S) n(S) 36363636 P(A) = 1P(A) = 1P(A) = 1P(A) = 1 9999

((((4444)))) ãýÚ À¸¨¼¸û ´§Ã §¿Ãò¾¢ø ãýÚ À¸¨¼¸û ´§Ã §¿Ãò¾¢ø ãýÚ À¸¨¼¸û ´§Ã §¿Ãò¾¢ø ãýÚ À¸¨¼¸û ´§Ã §¿Ãò¾¢ø ¯Õð¼ôÀθ¢ýÈÉ. ãýÚ À¸¨¼¸Ç¢Öõ ´§Ã ±ñ ¯Õð¼ôÀθ¢ýÈÉ. ãýÚ À¸¨¼¸Ç¢Öõ ´§Ã ±ñ ¯Õð¼ôÀθ¢ýÈÉ. ãýÚ À¸¨¼¸Ç¢Öõ ´§Ã ±ñ ¯Õð¼ôÀθ¢ýÈÉ. ãýÚ À¸¨¼¸Ç¢Öõ ´§Ã ±ñ ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× ¡Р?¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× ¡Р?¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× ¡Р?¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× ¡Р? ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ãýÚ À¸¨¼¸û ¯ÕðÎõ§À¡Ð ÜÚ¦ÅÇ¢ ãýÚ À¸¨¼¸û ¯ÕðÎõ§À¡Ð ÜÚ¦ÅÇ¢ ãýÚ À¸¨¼¸û ¯ÕðÎõ§À¡Ð ÜÚ¦ÅÇ¢ ãýÚ À¸¨¼¸û ¯ÕðÎõ§À¡Ð ÜÚ¦ÅÇ¢ S = { (1,1,1), (1,1,2),. . . . . .. S = { (1,1,1), (1,1,2),. . . . . .. S = { (1,1,1), (1,1,2),. . . . . .. S = { (1,1,1), (1,1,2),. . . . . .. . . . . . . .(2,2,2), . . . . . . . . . . . . .(2,2,2), . . . . . . . . . . . . .(2,2,2), . . . . . . . . . . . . .(2,2,2), . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(6,6,6) }. . . . . . . . . . . . . .(6,6,6) }. . . . . . . . . . . . . .(6,6,6) }. . . . . . . . . . . . . .(6,6,6) } n(S) = 216n(S) = 216n(S) = 216n(S) = 216 A = ãýÚ À¸¨¼A = ãýÚ À¸¨¼A = ãýÚ À¸¨¼A = ãýÚ À¸¨¼¸Ç¢Öõ ´§Ã ±ñ ¸¢¨¼ò¾ø¸Ç¢Öõ ´§Ã ±ñ ¸¢¨¼ò¾ø¸Ç¢Öõ ´§Ã ±ñ ¸¢¨¼ò¾ø¸Ç¢Öõ ´§Ã ±ñ ¸¢¨¼ò¾ø A = { (1,1,1), (2,2,2), (3,3,3), (4,4,4), A = { (1,1,1), (2,2,2), (3,3,3), (4,4,4), A = { (1,1,1), (2,2,2), (3,3,3), (4,4,4), A = { (1,1,1), (2,2,2), (3,3,3), (4,4,4), (5,5,5),(6,6,6) }(5,5,5),(6,6,6) }(5,5,5),(6,6,6) }(5,5,5),(6,6,6) } n(A) = 6n(A) = 6n(A) = 6n(A) = 6 P(A) = n(A) = 6P(A) = n(A) = 6P(A) = n(A) = 6P(A) = n(A) = 6 n(S) 216n(S) 216n(S) 216n(S) 216 P(A) P(A) P(A) P(A) = 1= 1= 1= 1 36363636 ((((5555)))) Ó¾ø þÕÀÐ þÂø ±ñ¸Ç¢Ä¢ÕóÐ ´Õ ÓØ Ó¾ø þÕÀÐ þÂø ±ñ¸Ç¢Ä¢ÕóÐ ´Õ ÓØ Ó¾ø þÕÀÐ þÂø ±ñ¸Ç¢Ä¢ÕóÐ ´Õ ÓØ Ó¾ø þÕÀÐ þÂø ±ñ¸Ç¢Ä¢ÕóÐ ´Õ ÓØ ±ñ ºÁÅ¡öôÒ Ó¨È¢ø §¾÷ó¦¾Îì¸ôÀθ¢ÈÐ. «ó¾ ±ñ ºÁÅ¡öôÒ Ó¨È¢ø §¾÷ó¦¾Îì¸ôÀθ¢ÈÐ. «ó¾ ±ñ ºÁÅ¡öôÒ Ó¨È¢ø §¾÷ó¦¾Îì¸ôÀθ¢ÈÐ. «ó¾ ±ñ ºÁÅ¡öôÒ Ó¨È¢ø §¾÷ó¦¾Îì¸ôÀθ¢ÈÐ. «ó¾ ±ñ ´Õ À¸¡ ±ñ½¡¸ þÕôÀ¾ü¸¡É ¿¢¸ú¾¸Å¢¨Éì ±ñ ´Õ À¸¡ ±ñ½¡¸ þÕôÀ¾ü¸¡É ¿¢¸ú¾¸Å¢¨Éì ±ñ ´Õ À¸¡ ±ñ½¡¸ þÕôÀ¾ü¸¡É ¿¢¸ú¾¸Å¢¨Éì ±ñ ´Õ À¸¡ ±ñ½¡¸ þÕôÀ¾ü¸¡É ¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.¸¡ñ¸.¸¡ñ¸.¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ÜÚ¦ÅÇ¢ ÜÚ¦ÅÇ¢ ÜÚ¦ÅÇ¢ ÜÚ¦ÅÇ¢ n(S) = 20n(S) = 20n(S) = 20n(S) = 20 A = À¸¡ ±ñ ¸¢¨¼ò¾øA = À¸¡ ±ñ ¸¢¨¼ò¾øA = À¸¡ ±ñ ¸¢¨¼ò¾øA = À¸¡ ±ñ ¸¢¨¼ò¾ø A = { 2,3,5,7,11,13,17,19 }A = { 2,3,5,7,11,13,17,19 }A = { 2,3,5,7,11,13,17,19 }A = { 2,3,5,7,11,13,17,19 } n(A) = 8n(A) = 8n(A) = 8n(A) = 8

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P(A) = n(A) = 8 P(A) = n(A) = 8 P(A) = n(A) = 8 P(A) = n(A) = 8 n(S) 20n(S) 20n(S) 20n(S) 20 P(A) = 2P(A) = 2P(A) = 2P(A) = 2 5555 ((((6666)))) ´Õ ¨À¢ø ¯ûÇ 1 Ó¾ø 100 Ũà ´Õ ¨À¢ø ¯ûÇ 1 Ó¾ø 100 Ũà ´Õ ¨À¢ø ¯ûÇ 1 Ó¾ø 100 Ũà ´Õ ¨À¢ø ¯ûÇ 1 Ó¾ø 100 Ũà ±ñ¸Ç¡ø ÌÈ¢ì¸ôÀð¼ 100 º£ðθǢĢÕóÐ ´Õ ±ñ¸Ç¡ø ÌÈ¢ì¸ôÀð¼ 100 º£ðθǢĢÕóÐ ´Õ ±ñ¸Ç¡ø ÌÈ¢ì¸ôÀð¼ 100 º£ðθǢĢÕóÐ ´Õ ±ñ¸Ç¡ø ÌÈ¢ì¸ôÀð¼ 100 º£ðθǢĢÕóÐ ´Õ º£ðÎ ±Îì¸ôÀθ¢ÈÐ. «ùÅ¡Ú ±Îì¸ôÀÎõ º£ðÊý º£ðÎ ±Îì¸ôÀθ¢ÈÐ. «ùÅ¡Ú ±Îì¸ôÀÎõ º£ðÊý º£ðÎ ±Îì¸ôÀθ¢ÈÐ. «ùÅ¡Ú ±Îì¸ôÀÎõ º£ðÊý º£ðÎ ±Îì¸ôÀθ¢ÈÐ. «ùÅ¡Ú ±Îì¸ôÀÎõ º£ðÊý ±ñ 10 ¬ø ÅÌÀÎõ ±ñ½¡¸ þÕôÀ¾ü¸¡É ±ñ 10 ¬ø ÅÌÀÎõ ±ñ½¡¸ þÕôÀ¾ü¸¡É ±ñ 10 ¬ø ÅÌÀÎõ ±ñ½¡¸ þÕôÀ¾ü¸¡É ±ñ 10 ¬ø ÅÌÀÎõ ±ñ½¡¸ þÕôÀ¾ü¸¡É ¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ÜÚ¦ÅÇ¢ n(S) = 100ÜÚ¦ÅÇ¢ n(S) = 100ÜÚ¦ÅÇ¢ n(S) = 100ÜÚ¦ÅÇ¢ n(S) = 100 A = 10 ¬ø ÅÌÀÎõ ±ñ½¡¸ þÕò¾øA = 10 ¬ø ÅÌÀÎõ ±ñ½¡¸ þÕò¾øA = 10 ¬ø ÅÌÀÎõ ±ñ½¡¸ þÕò¾øA = 10 ¬ø ÅÌÀÎõ ±ñ½¡¸ þÕò¾ø A = { 10,20,30,40,50,60,70,80,90,100 }A = { 10,20,30,40,50,60,70,80,90,100 }A = { 10,20,30,40,50,60,70,80,90,100 }A = { 10,20,30,40,50,60,70,80,90,100 } n(A) = 10n(A) = 10n(A) = 10n(A) = 10

P(A) = n(A) = 10 P(A) = n(A) = 10 P(A) = n(A) = 10 P(A) = n(A) = 10 n(S) 100n(S) 100n(S) 100n(S) 100 P(A) = 1P(A) = 1P(A) = 1P(A) = 1 10101010 ((((7777)))) 35 ¦À¡Õð¸û «¼í¸¢Â ¦¾¡ÌôÒ ´ýÈ¢ø35 ¦À¡Õð¸û «¼í¸¢Â ¦¾¡ÌôÒ ´ýÈ¢ø35 ¦À¡Õð¸û «¼í¸¢Â ¦¾¡ÌôÒ ´ýÈ¢ø35 ¦À¡Õð¸û «¼í¸¢Â ¦¾¡ÌôÒ ´ýÈ¢ø 7 7 7 7 ¦À¡Õð¸û ̨ÈÀ¡Î¨¼ÂÉ. «ò¦¾¡ÌôÀ¢Ä¢ÕóÐ ´Õ ¦À¡Õð¸û ̨ÈÀ¡Î¨¼ÂÉ. «ò¦¾¡ÌôÀ¢Ä¢ÕóÐ ´Õ ¦À¡Õð¸û ̨ÈÀ¡Î¨¼ÂÉ. «ò¦¾¡ÌôÀ¢Ä¢ÕóÐ ´Õ ¦À¡Õð¸û ̨ÈÀ¡Î¨¼ÂÉ. «ò¦¾¡ÌôÀ¢Ä¢ÕóÐ ´Õ ¦À¡Õû ºÁÅ¡öôÒ Ó¨È¢ø §¾÷ó¦¾ÎìÌõ§À¡Ð «Ð ¦À¡Õû ºÁÅ¡öôÒ Ó¨È¢ø §¾÷ó¦¾ÎìÌõ§À¡Ð «Ð ¦À¡Õû ºÁÅ¡öôÒ Ó¨È¢ø §¾÷ó¦¾ÎìÌõ§À¡Ð «Ð ¦À¡Õû ºÁÅ¡öôÒ Ó¨È¢ø §¾÷ó¦¾ÎìÌõ§À¡Ð «Ð ̨ÈÀ¡¼üÈ ¦À¡ÕÇ¡¸ þÕôÀ¾ü¸¡É ¿¢¸ú¾¸× ¡Р?̨ÈÀ¡¼üÈ ¦À¡ÕÇ¡¸ þÕôÀ¾ü¸¡É ¿¢¸ú¾¸× ¡Р?̨ÈÀ¡¼üÈ ¦À¡ÕÇ¡¸ þÕôÀ¾ü¸¡É ¿¢¸ú¾¸× ¡Р?̨ÈÀ¡¼üÈ ¦À¡ÕÇ¡¸ þÕôÀ¾ü¸¡É ¿¢¸ú¾¸× ¡Р? ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ¦¾¡ÌôÀ¢ÖûÇ ¦À¡Õð¸û n(S) = 35¦¾¡ÌôÀ¢ÖûÇ ¦À¡Õð¸û n(S) = 35¦¾¡ÌôÀ¢ÖûÇ ¦À¡Õð¸û n(S) = 35¦¾¡ÌôÀ¢ÖûÇ ¦À¡Õð¸û n(S) = 35 ̨ÈÀ¡Î¨¼Â ¦À¡Õð¸û = 7̨ÈÀ¡Î¨¼Â ¦À¡Õð¸û = 7̨ÈÀ¡Î¨¼Â ¦À¡Õð¸û = 7̨ÈÀ¡Î¨¼Â ¦À¡Õð¸û = 7 ̨ÈÀ¡¼üĘ̀ÈÀ¡¼üĘ̀ÈÀ¡¼üĘ̀ÈÀ¡¼üȨŠn(A) = 28¨Å n(A) = 28¨Å n(A) = 28¨Å n(A) = 28 P(A) = n(A) = 28 P(A) = n(A) = 28 P(A) = n(A) = 28 P(A) = n(A) = 28 n(S)n(S)n(S)n(S) 35353535 P(A) = 4P(A) = 4P(A) = 4P(A) = 4 5555

((((8888)))) ¿¿¿¿ýÌ ¸¨ÄòÐ «Î츢 ¨Åì¸ôÀð¼ 52 ýÌ ¸¨ÄòÐ «Î츢 ¨Åì¸ôÀð¼ 52 ýÌ ¸¨ÄòÐ «Î츢 ¨Åì¸ôÀð¼ 52 ýÌ ¸¨ÄòÐ «Î츢 ¨Åì¸ôÀð¼ 52 º£ðθ¨Çì ¦¸¡ñ¼ º£ðÎì¸ðÊÄ¢ÕóÐ ºÁÅ¡öôÒ º£ðθ¨Çì ¦¸¡ñ¼ º£ðÎì¸ðÊÄ¢ÕóÐ ºÁÅ¡öôÒ º£ðθ¨Çì ¦¸¡ñ¼ º£ðÎì¸ðÊÄ¢ÕóÐ ºÁÅ¡öôÒ º£ðθ¨Çì ¦¸¡ñ¼ º£ðÎì¸ðÊÄ¢ÕóÐ ºÁÅ¡öôÒ Ó¨È¢ø ´Õ º£ðΠӨȢø ´Õ º£ðΠӨȢø ´Õ º£ðΠӨȢø ´Õ º£ðÎ ±Îì¸ôÀθ¢ÈÐ. «ó¾ º£ðÎ ±Îì¸ôÀθ¢ÈÐ. «ó¾ º£ðÎ ±Îì¸ôÀθ¢ÈÐ. «ó¾ º£ðÎ ±Îì¸ôÀθ¢ÈÐ. «ó¾ º£ðÎ À¢À¢À¢À¢ýýýýÅÕÅÉÅ¡¸ þÕì¸ ¿¢¸ú¾¸×¸¨Çì ¸¡ñ¸.ÅÕÅÉÅ¡¸ þÕì¸ ¿¢¸ú¾¸×¸¨Çì ¸¡ñ¸.ÅÕÅÉÅ¡¸ þÕì¸ ¿¢¸ú¾¸×¸¨Çì ¸¡ñ¸.ÅÕÅÉÅ¡¸ þÕì¸ ¿¢¸ú¾¸×¸¨Çì ¸¡ñ¸.

((((1111)))) Š§À¼¡¸ (2) þạš¸Š§À¼¡¸ (2) þạš¸Š§À¼¡¸ (2) þạš¸Š§À¼¡¸ (2) þạš¸ ¾£÷× :¾£÷× :¾£÷× :¾£÷× : ÜÚ¦ÅÇ¢ n(S) = 52ÜÚ¦ÅÇ¢ n(S) = 52ÜÚ¦ÅÇ¢ n(S) = 52ÜÚ¦ÅÇ¢ n(S) = 52 A = Š§ÀÎ º£ð¼¡¸ þÕò¾øA = Š§ÀÎ º£ð¼¡¸ þÕò¾øA = Š§ÀÎ º£ð¼¡¸ þÕò¾øA = Š§ÀÎ º£ð¼¡¸ þÕò¾ø n(A) = n(A) = n(A) = n(A) = 13131313 P(A) = n(A) = 13 P(A) = n(A) = 13 P(A) = n(A) = 13 P(A) = n(A) = 13 n(S) 52n(S) 52n(S) 52n(S) 52 P(A) = 1P(A) = 1P(A) = 1P(A) = 1 4444 B = þạš¸ þÕò¾øB = þạš¸ þÕò¾øB = þạš¸ þÕò¾øB = þạš¸ þÕò¾ø n(B) = 4 n(B) = 4 n(B) = 4 n(B) = 4 P(B) = n(B) = 4 P(B) = n(B) = 4 P(B) = n(B) = 4 P(B) = n(B) = 4 n(S) 52n(S) 52n(S) 52n(S) 52 P(A) = 1P(A) = 1P(A) = 1P(A) = 1 13131313

((((9999)))) ¿ýÌ ¸¨ÄòÐ «Î츢 ¨Åì¸ôÀð¼ 52 ¿ýÌ ¸¨ÄòÐ «Î츢 ¨Åì¸ôÀð¼ 52 ¿ýÌ ¸¨ÄòÐ «Î츢 ¨Åì¸ôÀð¼ 52 ¿ýÌ ¸¨ÄòÐ «Î츢 ¨Åì¸ôÀð¼ 52 º£ðθ¨Çì ¦¸¡ñ¼ º£ðÎì¸ðÊÄ¢ÕóÐ ºÁÅ¡öôÒ º£ðθ¨Çì ¦¸¡ñ¼ º£ðÎì¸ðÊÄ¢ÕóÐ ºÁÅ¡öôÒ º£ðθ¨Çì ¦¸¡ñ¼ º£ðÎì¸ðÊÄ¢ÕóÐ ºÁÅ¡öôÒ º£ðθ¨Çì ¦¸¡ñ¼ º£ðÎì¸ðÊÄ¢ÕóÐ ºÁÅ¡öôÒ Ó¨È¢ø ӨȢø ӨȢø ӨȢø ´Õ º£ðÎ ±Îì¸ôÀθ¢ÈÐ. «ó¾ º£ðÎ´Õ º£ðÎ ±Îì¸ôÀθ¢ÈÐ. «ó¾ º£ðÎ´Õ º£ðÎ ±Îì¸ôÀθ¢ÈÐ. «ó¾ º£ðÎ´Õ º£ðÎ ±Îì¸ôÀθ¢ÈÐ. «ó¾ º£ðÎ À¢À¢À¢À¢ýýýýÅÕÅÉÅ¡¸ þÕì¸ ¿¢¸ú¾¸×¸¨Çì ÅÕÅÉÅ¡¸ þÕì¸ ¿¢¸ú¾¸×¸¨Çì ÅÕÅÉÅ¡¸ þÕì¸ ¿¢¸ú¾¸×¸¨Çì ÅÕÅÉÅ¡¸ þÕì¸ ¿¢¸ú¾¸×¸¨Çì ¸¡ñ¸¸¡ñ¸¸¡ñ¸¸¡ñ¸.... ((((1111)))) ¸ÕôÒ þạ (2) ¼ÂÁñð 10¸ÕôÒ þạ (2) ¼ÂÁñð 10¸ÕôÒ þạ (2) ¼ÂÁñð 10¸ÕôÒ þạ (2) ¼ÂÁñð 10

¾£÷× :¾£÷× :¾£÷× :¾£÷× : ÜÚ¦ÅÇ¢ n(S) = 52ÜÚ¦ÅÇ¢ n(S) = 52ÜÚ¦ÅÇ¢ n(S) = 52ÜÚ¦ÅÇ¢ n(S) = 52 A = ¸ÕôÒ þạš¸ þÕò¾øA = ¸ÕôÒ þạš¸ þÕò¾øA = ¸ÕôÒ þạš¸ þÕò¾øA = ¸ÕôÒ þạš¸ þÕò¾ø n(A) = 2n(A) = 2n(A) = 2n(A) = 2 P(A) = n(A) = 2 P(A) = n(A) = 2 P(A) = n(A) = 2 P(A) = n(A) = 2 n(S) 52n(S) 52n(S) 52n(S) 52 P(A) = 1P(A) = 1P(A) = 1P(A) = 1 26262626 B = ¼ÂÁñð 10 º£ð¼¡¸ þÕò¾øB = ¼ÂÁñð 10 º£ð¼¡¸ þÕò¾øB = ¼ÂÁñð 10 º£ð¼¡¸ þÕò¾øB = ¼ÂÁñð 10 º£ð¼¡¸ þÕò¾ø n(B) = 1 n(B) = 1 n(B) = 1 n(B) = 1 P(B) = n(B) P(B) = n(B) P(B) = n(B) P(B) = n(B) n(S) n(S) n(S) n(S) P(A) = 1P(A) = 1P(A) = 1P(A) = 1 52525252 ((((10101010)))) ºÁÅ¡öôÒ Ó¨È¢ø §¾÷ó§¾Îì¸ôÀÎõ ºÁÅ¡öôÒ Ó¨È¢ø §¾÷ó§¾Îì¸ôÀÎõ ºÁÅ¡öôÒ Ó¨È¢ø §¾÷ó§¾Îì¸ôÀÎõ ºÁÅ¡öôÒ Ó¨È¢ø §¾÷ó§¾Îì¸ôÀÎõ ¦¿ð¼¡ñÊø¦¿ð¼¡ñÊø¦¿ð¼¡ñÊø¦¿ð¼¡ñÊø 53 ¦ÅûǢ츢ƨÁ¸û ¸¢¨¼ôÀ¾ü¸¡É 53 ¦ÅûǢ츢ƨÁ¸û ¸¢¨¼ôÀ¾ü¸¡É 53 ¦ÅûǢ츢ƨÁ¸û ¸¢¨¼ôÀ¾ü¸¡É 53 ¦ÅûǢ츢ƨÁ¸û ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸. ¾£÷× :¾£÷× :¾£÷× :¾£÷× : 366 ¿¡ð¸û ¯ûÇ ¦¿ð¼¡ñÊø 52 Å¡Ãí¸û 366 ¿¡ð¸û ¯ûÇ ¦¿ð¼¡ñÊø 52 Å¡Ãí¸û 366 ¿¡ð¸û ¯ûÇ ¦¿ð¼¡ñÊø 52 Å¡Ãí¸û 366 ¿¡ð¸û ¯ûÇ ¦¿ð¼¡ñÊø 52 Å¡Ãí¸û ÁüÚõ 2 ¿¡ð¸û ¯ûÇÉ.ÁüÚõ 2 ¿¡ð¸û ¯ûÇÉ.ÁüÚõ 2 ¿¡ð¸û ¯ûÇÉ.ÁüÚõ 2 ¿¡ð¸û ¯ûÇÉ. ±É§Å ÜÚ¦ÅÇ¢±É§Å ÜÚ¦ÅÇ¢±É§Å ÜÚ¦ÅÇ¢±É§Å ÜÚ¦ÅÇ¢ S = { ( »¡Â¢Ú,¾¢í¸û ), ( ¾¢,¦º), (¦º,Ò ), ( Ò,Å¢ ),S = { ( »¡Â¢Ú,¾¢í¸û ), ( ¾¢,¦º), (¦º,Ò ), ( Ò,Å¢ ),S = { ( »¡Â¢Ú,¾¢í¸û ), ( ¾¢,¦º), (¦º,Ò ), ( Ò,Å¢ ),S = { ( »¡Â¢Ú,¾¢í¸û ), ( ¾¢,¦º), (¦º,Ò ), ( Ò,Å¢ ), ( Å¢,¦Å), ( ¦Å,º ), ( º,»¡ ) }( Å¢,¦Å), ( ¦Å,º ), ( º,»¡ ) }( Å¢,¦Å), ( ¦Å,º ), ( º,»¡ ) }( Å¢,¦Å), ( ¦Å,º ), ( º,»¡ ) } n(S) = 7 n(S) = 7 n(S) = 7 n(S) = 7 A = 53 ¦ÅûǢ츢ƨÁ ¸¢¨¼ò¾øA = 53 ¦ÅûǢ츢ƨÁ ¸¢¨¼ò¾øA = 53 ¦ÅûǢ츢ƨÁ ¸¢¨¼ò¾øA = 53 ¦ÅûǢ츢ƨÁ ¸¢¨¼ò¾ø A = { ( Å¢, ¦Å ), ( ¦Å, º ) } A = { ( Å¢, ¦Å ), ( ¦Å, º ) } A = { ( Å¢, ¦Å ), ( ¦Å, º ) } A = { ( Å¢, ¦Å ), ( ¦Å, º ) } n(A) = 2n(A) = 2n(A) = 2n(A) = 2 P(A) = n(A) P(A) = n(A) P(A) = n(A) P(A) = n(A) n(S) n(S) n(S) n(S) P(A) = 2P(A) = 2P(A) = 2P(A) = 2 7777

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