高校ゼミ 数学A
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1 U n(u)=0n()=45n()=85n()=4 n() n()=n(u)-n() =0-45=75 75 n() n()=n()+n()-n() = =87 87 n() = n()=n()=0-4= U4 6 ={ } n()=50-7+=4 4 4 n(u)=00-00+=0 ={ } n()=75-5+=5 n()=0-5= n() ={905} n()=5-9+=7 n()=n()+n()-n()=5+4-7= n() n()=n(u)-n()=0-68= U n()=n(u)-n()=n(u)-{n()+n()-n()} =50-( )=5 5 n()=n()-n()=00-5=75 75
2 =40() (a+b+c)(m+n)(+ ) 40 =() 6 5 6= 5=5 (+)(+)=9() +=4() 9 4
3 FG 7 4 P 7 P 4 =7654= P P 6 =6!=654= P 0 P =8() =8() ! 4 P! 4 4! 5!!=54=40() 4 P 4!=44=88() (4-)!=!=6() =6() 6
4 = 9 = = = 7-7= = 5 FRVR 7 R
5 5 6 5 F 6 F FF
6 P()=P()+P()-P() P()=P()+P()-P()= 6
7 []
8 8 0 0 P () { } 0 {68} P ()= 0 P () {6958} 6 P ()= 40 0 P()= P () P ()= 4
9 9 F F == = 4 F 8 7 =80=45 =49 0 7=49= 8 9 F FF==4 ==87 F =FF 6== =85 0=85 = 6 6 =6=5=4 F 6 F 4 5 ==64= =(-)= ==5=0 0 FF FF= == = F F=(+)= F= 5 P P<P P P= P P<P < = P < P P< P<P P P
10 0 I = = =80 0 =40 94 I =80 =8 0 I 8 40 =80 -(5 +8 )=7 7 G G G F G G F G GG= G G=G= GGF G H GGH= GF GFH=GH= GF= G =G= G= = H 8 I III I I=80 -(I+ ) II I I+ = (+) +=80 - (+)=90 - I=90 + >90 I I II III
11 PP QQ R 4 Q 7 R Q P 5 6 P 45 QQ R Q 4 P RR R Q 5 7 P 5 7 P Q P Q R RR R Q P P 4 R 5 7
12 x =80-70 =0 =80 -(0 +8 )= =80-0 =70 =80 -(85 + )= =80 =80 -=80-6 =0 54 =7 7 0 F F F = F = F F= F F F F F = =F =F F F
13 T 70 y 56 T 65 T T =T=56 =80 -( )=54 =T=0 =(80-0 )= PT T 4 8 P 5 P T 6 4 P ==90 =8 =0 S R P=SP=QQ=RR=S P (P+P)+(Q+Q)+(R+R)+(S+S) =(P+P)+(R+R)=(+) Q H =8=6 6 S S= H H H = -H H =0-8 =6H=6 PSPQ S=Q=4 +=(S+)+(Q+Q)=(4+)+(4++6)=8 =
14 4 rr ' d ( ) r=8r '=5d= r=4r '=d=8 r-r '=8-5= d=r-r ' r+r '=4+=6 d>r+r ' r=8r '=5d= r+r '=8+5= d=r+r ' r=6r '=d=5 r-r '=6-=r+r '=6+=9 r-r '<d<r+r ' 4 4 '=6 ' ' ' r r r=6-4= ' r r=6+4=0 r0 r ' '=0 ' H ' 55 m ' ' 5 7 M m m ' 5(5-7 ) m
15 5 m m m = P Q = P abc bc a a a =c X =b X bc a a a P b = Q c Q a = a= =a ==b a G F c = FG H b F =F F =b=b F=a G F FGH
16 6 FGH 0 i 90 FH H FG FFG H 90 F G G F FFG FG G == (64+86)=4 (64+86)=6 v-e+f= 4-e+(6+8)=e=6 4 6
17 =95 = ab 5 9 ab (ab) 5 b= b=0 +6+a+0=7+a=9 a= b=5 +6+a+5=+a=8 a=6 (ab)=(0)(65) n 4n 78n 4= 4 78= =( ) 4 7 =( 7) n==6 n=7=4 n=4 468 a a a= = 5 (+)(+)(+)=4() = 57 (+)(+)(+)(+)=() =70 ( )< 70 70= 5 70 (+)(+)(+)=6() < 6=8() 8
18 = 90= 5 = =55= 57 5= = = = 7= 0= 5 75=5 05=5775=5 = 5 5 5= = ab l a a'b b' []a'b' []l a'b'[]ab l n n n 4= 504= 7 n= a 7(a=0) n= n= n= ab a<b a=6a'b=6b'(a'b' a'<b')l= a'b' 7=6a'b'a'b'= (a'b')=()(4) (ab)=(67)(84) (ab)=(67)(84) ab a<b ab= l 40=60 =4 l= a'b' 60=4a'b'a'b'=5(a'b')=(5)(5) (ab)=(460)(0) (ab)=(460)(0) a a+ 7 a+7 9 a+6 6 a+a+7 mn a+=7ma+7=9n a+6=(a+)+4=7m+4=7(m+) a+6=(a+7)+9=9n+9=9(n+) a+6 7 a a
19 9 ab a 9 b a=9k+b=9l+4(kl ) a+b ab a+b=(9k+)+(9l+4)=9(k+l)+7 ab=(9k+)(9l+4)=9 kl+9k4+9l+4 7 =9(9kl+4k+l+) k k+k+ (k+) +(k+) -=(4k +4k+)+(4k +k+9)- =8k +6k+8=8(k +k+) 8 n n -n n -n=n(n -)=n(n-)(n+) n-nn+ n -n 5 5 5= 55 5= = = (45+7-9)=7 N=50 N > N =7 7 7
20 = =66+4 6= = = = = = = = = = = 7=6+55=7-6 6=5+=6-5 =6-(7-6) =7(-)+6 7(-)+6= =- = 9- = 9=+66=9- =6+=-6 =-(9-) =9(-)-(-) 5+4 = 5=4+=5-4 4=+=4- =5+=-5 =-(4-)5=+4(-5) =(5-4)+4(-5) =5+4(-6) 5+4(-6)= = =-6 = =-6-7 = =7+55=-7 7=55+=7-55 5=+=5- =5-(7-55)=5-7 9(-)-(-)= =(-7)-7=-7 =- =- -7= = = = = a b a+7b a+b a+7b=(a+b)+ a+7b a+b a+b a+b=b+ a+b b a b a+7b a+b a+7b a+b b a
21 +9 =0 4-7 = 5- = 7+ =- =7k+ =4k+(k ) =-k- =7k+6(k ) +7 = 7 =-7 =5 (-7)+75= - (+7)+7( -5)=0 7 +7=-7k -5=k(k ) =-7k-7 =k+5(k ) 6-9 =6 9 =- = = 6(-)-9(-0)= 6(-6)-9(-0)= - 6(+6)-9( +0)= =9k +0=6k(k ) =9k-6 =6k-0(k ) 5 7 n n=5+n=7 +( )5+= = = = 5-7= - 5(-)-7( -)=05 7 -=7k(k ) =7k+n=5(7k+)+=5k+75k+7 k= n=5+7=
22 [] 9 [ 0 ] 7 0 [ 60 ] 5 [ 00 ] =6= 4 8= 740= 55=5 75=5 5 n 0 n n= a 5 b n0 n =8() 8
23 n 0 0 () 45 (6) = = =8++= 76 0 [] 47[ ] 506[ 7 ] =0 () =5 (7) ) 0 5 (7) () = = (5) =4 5 = [] 0.58[ 5 ] [ 4 ] =0. (5) =0. (4) 0. (4) () 0 () +0 () 00 () - () =00 () =000 () 00 () 0 () 00 () () =000 () = () ) () () X
24 U n(u)=0n()=45n()=85n()=4 n() n()=n(u)-n() =0-45=75 75 n() n()=n()+n()-n() = =87 87 n() = n()=n()=0-4= U4 6 ={ } n()=50-7+=4 4 4 n(u)=00-00+=0 ={ } n()=75-5+=5 n()=0-5= n() ={905} n()=5-9+=7 n()=n()+n()-n()=5+4-7= n() n()=n(u)-n()=0-68= U n()=n(u)-n()=n(u)-{n()+n()-n()} =50-( )=5 5 U n()=n()-n()=00-5=75 75
25 (66) =9() () =0() =40() (a+b+c)(m+n)(+ ) 40 =() 6 5 6= 5=5 (+)(+)=9() +=4() 9 4
26 FG 7 4 P 7 P 4 =7654= P P 6 =6!=654= P 0 P =8() =8() ! 4 P! 4 4! 5!!=54=40() 4 P 4!=44=88() (4-)!=!=6() =6() 6
27 = 9 = 98 =6() = =6() = 54 4 =60() = = 5 =5() =() 6-5=() 7-7= 76-7=4() = () =60() 6 4 = 65 4! =5() 60 5 FRVR 7 R 7!!!!!! =60() 7 5!=06=60() ! 4!! =5() 5
28 = = F 6 F FF 4! 4! = 6! 5 5 4! 4!! 4! 6! = = 6 = 7 4 = 4 7 = = = 6 0 = 0 0 = 6 6
29 = = = = ()()()(4)()()()(4) = = = = = P()=P()+P()-P() P()=P()+P()-P()= =
30 = = [] [] 9 6 [][] = =5 6 8 = = = = 64 64
31 8 0 0 P () { } 0 {68} P ()= n() = n() 0 P () {6958} 6 P ()= n() = n() 6 = 40 0 P()= 40 0 P()= P ()= P() = P() [][] 4 9 [] = = [] = = P()=P()+P()= = + = 4 = P () P ()= P() = P() 4 = 4 4
32 9 F F == = 4 F 8 7 =80=45 =49 0 7=49= 8 9 F FF==4 ==87 F =FF 6== =85 0=85 = =6=5=4 F 6 F 4 5 ==64= =(-)= ==5=0 0 FF FF= == = + = 5 5= FF==6= F F=(+)= F= = 5 = 5 F=5 5 P P<P P P= P P<P < = P < P P< P<P P P
33 0 I = = =80 0 =40 94 I =80 =8 0 I 8 40 =80 -(5 +8 )=7 7 G G G F G G F G GG= G G=G= H GGF G H GGH= GF GFH=GH= GF= H= GGF= = G =G= G= G= = 9 = 8 8 I III I I=80 -(I+ ) II I I+ = (+) +=80 - (+)=90 - I=90 + >90 I I II III
34 PP QQ R 4 Q P P Q Q R R = 7 R Q P P 4 P = P = 9 8 P PP=98 5 P Q Q Q = 7 Q = QQ=45 45 QQ RR PR R R Q Q P 5 4 P P Q Q R R = 7 P 5 7 Q Q Q = Q = 6 7 QQ= R R = R R = 7 RR= P Q P Q R RR P P Q Q R R = R R = R R = RR= R P Q P P R P P R R = P P = = 4 P=4 4 R R Q R R Q Q = 5 R R = = 5 R=5 5 P=4 = 4 7 P= 4 7 = 7 =7 7
35 x = = (60-50 ) =0-60 =50 ==4 ==50 =00 =- =55 =90-4 = = = ( ) =54 =80-70 =0 =80-0 =70 =80 -(85 + )= =80 =80 -(0 +8 )= =80 -=80-6 =0 54 =7 7 0 F F F = F = F F= F F F F F = =F =F F F
36 T 70 y 56 T 65 T T =T=56 =T=0 =80 -=80-00 =80 =80 -( )=54 =(80-0 )=5 T T+T=80 T=80-50 =0 56 =T= PT T 4 8 P 5 P T 6 4 P PP=PP 85=4=0 PP=PP 7(7+)=6(6+8)=5 PP=PT 4(4+6)= =40>0 = ==90 =8 =0 S R P=SP=QQ=RR=S P (P+P)+(Q+Q)+(R+R)+(S+S) =(P+P)+(R+R)=(+) Q H =8=6 6 S S= H H H = -H H =0-8 =6H=6 PSPQ S=Q=4 +=(S+)+(Q+Q)=(4+)+(4++6)=8 =
37 4 rr ' d ( ) r=8r '=5d= r=4r '=d=8 r-r '=8-5= d=r-r ' r+r '=4+=6 d>r+r ' 4 r=8r '=5d= r=6r '=d=5 r+r '=8+5= d=r+r ' r-r '=6-=r+r'=6+=9 r-r '<d<r+r' 4 '=6 ' ' ' r r r=6-4= ' r r=6+4=0 r0 r ' '=0 ' ' H =H 'H H=' -'H =0 -(0-5) =75 =55 =55 55 m ' H ' ' ' M =M 'M M=' -'M =0 -(0+5) =75 =5 7 = M m m = === =+=+=+(+)=+ 55 =+5 7 ' 5(5-7 ) = 5(5-7 ) m
38 5 m m m = P Q = P abc bc a a a =c X =b X ac=b = bc a a a ==a a F = a= =a ==b G a c = FG F =F F =b=b F=a F FGH P b = Q c Q b a H c F G X F b a
39 6 FGH 0 i 90 FH H FG FFG H 90 F G G F FFG 45 FG G 60 G FH GHG H FGG H H=60 H= H= =6=H= == (64+86)=4 (64+86)=6 v-e+f= 4-e+(6+8)=e=6 4 6
40 =95 = ab 5 9 ab (ab) 5 b= b=0 +6+a+0=7+a=9 a= b=5 +6+a+5=+a=8 a=6 (ab)=(0)(65) n 4n 78n 4= 4 78= =( ) 4 7 =( 7) n==6 n=7=4 n=6 n=4 468 a a 468 a 468= a a=a= =7 a= = 5 890= 57 (+)(+)(+)=4() (+)(+)(+)(+)=() 4 =70 ( )< 70 70= 5 70 (+)(+)(+)=6() < 6=8() 8
41 = 90= 5 = =55= 57 5= = = = 7= 0= 5 75=5 05=5775=5 = 5 5 5= = ab l a a'b b' []a'b' []l a'b'[]ab l n n n 4= 504= 7 n= a 7(a=0) n= n= n= ab a<b a=6a'b=6b'(a'b' a'<b')l= a'b' 7=6a'b'a'b'= (a'b')=()(4) (ab)=(67)(84) (ab)=(67)(84) ab a<b ab= l 40=60 =4 l= a'b' 60=4a'b'a'b'=5(a'b')=(5)(5) (ab)=(460)(0) (ab)=(460)(0) a a+ 7 a+7 9 a+6 6 a+a+7 mn a+=7ma+7=9n a+6=(a+)+4=7m+4=7(m+) a+6=(a+7)+9=9n+9=9(n+) a+6 7 a a
42 9 ab a 9 b a=9k+b=9l+4(kl ) a+b ab a+b=(9k+)+(9l+4)=9(k+l)+7 ab=(9k+)(9l+4)=9 kl+9k4+9l+4 7 =9(9kl+4k+l+) =9 8 =( ) 0 = = = 8 k k+k+ (k+) +(k+) -=(4k +4k+)+(4k +k+9)- =8k +6k+8=8(k +k+) 8 n n -n n -n=n(n -)=n(n-)(n+) n-nn+ n -n 5 5 5= 55 5= = = (45+7-9)=7 N=50 N > N =7 7 7
43 = =66+4 6= = = = = = = = = = = = 7=6+55=7-6 6=5+=6-5 =6-(7-6) =7(-)+6 7(-)+6= =- = 5+4 = 5=4+=5-4 4=+=4- =5+=-5 =-(4-)5=+4(-5) =(5-4)+4(-5) =5+4(-6) 5+4(-6)= = =-6 =- = = =-6 9- = -7 = 9=+66=9- =7+55=-7 =6+=-6 7=55+=7-55 =-(9-) 5=+=5- =9(-)-(-) =5-(7-55)=5-7 9(-)-(-)= =(-7)-7=-7 =- =- -7= = = =- =- = = a b a+7b a+b a+7b=(a+b)+ a+7b a+b a+b a+b=b+ a+b b a b a+7b a+b a+7b a+b b a
44 +9 =0 =-9 9 =-9k =k(k ) 4-7 = = = 4-7= -4(-)-7( -)= =7k -=4k(k ) =-9k =k(k ) =7k+ =4k+(k ) 5- = 7+ =- = = 5 - = = =- 7 + = 5-= 7+(-)= 54-6= - 7(-)+6=- - 5(-4)-( -6)=0-7(+)+( -6)= =k -6=5k(k ) +=-k -6=7k(k ) =k+4 =5k+6(k ) =-k- =7k+6(k ) +7 = 7 =-7 =5 (-7)+75= - (+7)+7( -5)=0 7 +7=-7k -5=k(k ) 6-9 =6 9 =- = = 6(-)-9(-0)= 6(-6)-9(-0)= - 6(+6)-9( +0)= =9k +0=6k(k ) 7=+5 5=7- =5+ =-5 5=+ =5- =5-(-5) =55+(-) =(7-)5+(-) =(-7)+75 =-7k-7 =k+5(k ) 6=9+6 6=6-9 9=6+ =9-6 =9-(6-9) =6(-)-9(-0) =9k-6 =6k-0(k ) 5 7 n n=5+n=7 +( )5+= = = = 5-7= - 5(-)-7( -)=05 7 -=7k(k ) =7k+n=5(7k+)+=5k+75k+7 k= n=5+7=
45 = =.8585= =0.4666= = [] 9 [ 0 ] 7 0 [ 60 ] 5 [ 00 ] 9 7 = = = =6+ 60= = =6= 4 8= 740= 55=5 75= n 0 n n= a 5 b n0 n =8() 8
46 n 0 0 () 45 (6) = = =8++= =44+0+= [] 47[ ] ) [ 7 ] 7 ) ) 7 ) 745 ) ) 06 =0 () ) 5 =5 (7) 7 ) 5 ) 7 ) ) () 5 (7) () = + = (5) = = [] 0.58[ 5 ] [ 4 ] * 5 * =0. (5) * 5 =0. (4) * * (5) 0. (4) 9 9 = 9 =0.0 () 0.0 () 0 () +0 () 0 00 () - () =00 () =000 () () 000 () 00 () 0 () () () =000 () * 0 ) = () () () X
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