Ñóïðóí Â.Ï. - Ìàòåìàòèêà äëÿ ñòàðøåêëàññíèêîâ Çàäà÷è...
99
-
Upload
emanuelsaint -
Category
Documents
-
view
242 -
download
5
description
algebra
Transcript of Ñóïðóí Â.Ï. - Ìàòåìàòèêà äëÿ ñòàðøåêëàññíèêîâ Çàäà÷è...
-
U
R S
S
U R
S S
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1. . . . . . . . . . . . . . . 7
1.1. . . . . . . . . . . . . . . . . . . . . . . 7 1.2. . . . . . . . . . . . . . . . . . . . . 8 1.3. . . . . . . . . . . . 9 1.4. . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6. . . . . . . . . . . 11 1.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2. , . . . . . . . . . . . . . . . . . . . 13
2.1. . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2. . . . . . . . . . . . . . . . . . . . . . . 15 2.3. . . . . . . . . . . . . . . . 18 2.4.
. . . . . . . . . . . . . . . . . . . . . . . . 22 2.5. . . . . . . . . . . . . . . . . . 25 2.6. . . . . . . . . . . . . . . . . . . 41 2.7. . . . . . . . . . . . . . . . . 56 2.8. . . . . . . . . . . . . . . . . . . . . 84 2.9. . . . . . . . . . . . . . . . . . . . . . 88 2.10. . . . . . . . . . . . . . . . . . . . . . 113 2.11. . . . 118 2.12. . . 128 2.13. . . . . . 133 2.14. . . . . . . 136 2.15. . . . . . . . . . . . . 154 2.16. . . . . . . . . . 156 2.17. . . . . . . . . . . . . . . . . . . 160
style/locdef.clo
-
U
R S
S
U R
S S
4
2.18. . . . . . . . . . . . . . . . . . . . . 170 2.19. . . . . . . . . . . . 173
3. . . . . . . . . . . . . . 179
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
style/locdef.clo
, - , - , - . , ( - ). , - - .
, , - , - , , - . , , - , , - , .
300 , - -. - , -, .
- - : (., , 2002). - , - - - (.) .
gl01.tex
-
U
R S
S
U R
S S
6
- , - . (, , ). - .
1
- , () -. , , -, , , -. ( ) -, , , - . , - , , . , () - .
-, - , , - , . , - , , . - ( ) 300 .
1.1.
1 0, 2 0, , 0, 1 + 2 + +
12 , (1)
gl01.tex
-
U
R S
S
U R
S S
8 1.
2. (1) , 1 = 2 = = ( , . 299).
, = 2, (1)
1 + 22
12. (2)
1 = 2 =1
, (2)
+1
2, (3)
0. (3) = 1. , 0,
+1
2, (4)
, =1. 1 0, 2 0, , 0,
1
1+
1
2+ +
1
2
1 + 2 + + . (5)
(5) - (1) .
1
1+
1
2+ +
1
12
1 + 2 + +
=
2
1 + 2 + +
.
1.2.
- : 1
(1+ ) 1+ . (6)
(6) = 0 = 1. (6) (. 293).
gl01.tex
1.4. 9
- , - :
0 1,
(1+ ) 1+ , (7) 0 1,
(1+ ) 1+ , (8) 1.
, (7) (8) , = 0.
1.3.
1, 2, ,
1, 2, ,
(11 + 22 + + )2
21 +
22 + +
2
21 +
22 + +
2
, (9)
2. (9) ,
, . . - ( = 0) , = 1, 2, ,
=
. (9)
22. -
(+ ) 21( + ), (10) 0, 0 1. (10) , = 1 = . (10) (. 295).
1.4.
-
(+ ) = 0
0 + 1
1
1 + 2
2
2 + +
0
, (11)
gl01.tex
-
U
R S
S
U R
S S
10 1.
=
!
!( )!.
! = 1 2 , 0! = 1! = 1 0 . , 0
=
= 1 1
= 1
= . (11) = = 1,
(+ 1) = 0
+ 1
1 + + 2
2 + 1
+
.
1
= 1
= ,
(+ 1) = + 1 = 2 + + 1, (12)
= 0
1 + 1
2 + + 1
= 0
2 + 1
3 + + 2
.
(12) , ,
.
1.5.
, .
. .
=
, 0;
, 0.
, = , =.
- . . , .
- :
gl01.tex
1.6. 11
1) () () ,
() ();
() ().2) () () ,
() () () ().3) () () ,
() ();
() (),
() ();
() ().
, () () -
()+ () () (). :
()+ () = () + (), () () 0; ()+ () = () (), () () 0; ()+ () = ()+ (), () 0 () 0; ()+ () = () (), () 0 () 0.
1.6.
sin cos
sin cos =
2 + 2 sin ( ), (13)
sin =
2 + 2, cos =
2 + 2
2, . ,
sin cos =
2 sin
4
.
gl01.tex
-
U
R S
S
U R
S S
12 1.
,
2 sin cos
2. -
:
2 arcsin
2, 0 arccos (1 1),
2
arctg
2, 0 arcctg ( ),
arcsin () = arcsin , arccos () = arccos ,arctg () = arctg , arcctg () = arcctg ,
arcsin + arccos =
2, arctg + arcctg =
2.
1.7.
- ,
log
(() ()) = log
()+ log
(), (14)
log
()
()= log
() log
(), (15)
log
() = log
() ( ). (16)
(14), (15) , . (14), (15) - .
, (16) .
()log () = ()log (), (17)
, () 0, () 0, 0 = 1.
2
,
2.1.
1. , 55+1+45+2+35
11.
. - :
55+1 + 45+2 + 35 = 5 (3125) + 16 (1024) + (243) =
= 5 (11 284+ 1) + 16 (11 93+ 1) + (11 22+ 1).
, (12),
55+1 + 45+2 + 35 == 5 (11+ 1)+ 16 (11 + 1)+ (11 + 1) == 11 (5+ 16 + )+ 5+ 16+ 1 = 11 + 22 = 11.
, , , , , .
2. , 4 + 15 1 9.
. (12),
4 + 15 1 = (3+ 1) + 15 1 == (9+ 3+ 1)+ 15 1 = 9+ 18,
. , 9.
3. , 25+3 + 5 3+2 17.
gl02.tex
-
U
R S
S
U R
S S
14 2.
. :
25+3 + 5 3+2 = 8 32 + 9 15 =
= 8 (17+ 15) + 9 15 =
= 8 (17+ 15)+ 9 15 =
= 8 17+ 17 15 = 17,
, . , 25+3+5 3+2
17 . - (11).
4. , 32+3+4027 64.
.
32+3 + 40 27 = 27 9 + 40 27 =
= 27 (8+ 1) + 40 27 == 27 (64+ 8+ 1)+ 40 27 == 27 64+ 27 8+ 27+ 40 27 == 27 64+ 256,
. , 64.
5. , 5(5 + 1) 6(3 + 2) 91.
. , - 7 13. 7 13, , - 91 .
5(5 + 1) 6(3 + 2) = 25 + 5 18 12.
:
1) 25 + 5 18 12 = (7+ 18) + 5 18 (7+ 5) =
= (7+ 18)+ 5 18 (7 + 5) == 7 7, , .
gl02.tex
2.2. 15
2) 25 + 5 18 12 = (13+ 12) + 5 (13+ 5) 12 =
= (13 + 12)+ 5 (13+ 5) 12 == 13 13, , .
5(5+1)6(3+2) 7 13, 7 13 = 91.
. 15 (., , 290).
2.2.
6.
1
1 5+
1
5 9+
1
9 13+ +
1
(4 3) (4+ 1).
.
, . .
1
1 5=
1
4
1
1
1
5
,1
5 9=
1
4
1
5
1
9
,
1
9 13=
1
4
1
9
1
13
, ,
1
(4 3) (4+ 1)=
1
4
1
4 3
1
4+ 1
.
=1
4
1
1
5+
1
5
1
9+
1
9
1
13+ +
1
4 3
1
4+ 1
=
=1
4
1
1
4+ 1
=
4+ 1.
7.
1 2+ 2 3+ + ( 1) , 2.
.
,
- 1+ 2+ + .
= 1 2+ 2 3+ + ( 1) =
gl02.tex
-
U
R S
S
U R
S S
16 2.
= (2+ 3+ 4+ + )+ (3+ 4+ + ) ++ (4+ + ) + + ( 1+ )+ =
= (
1) + ( 2) ++ (
3)+ + ( 2) + ( 1) == ( 1)
(1 + 2 + 3 + + 2 + 1).
,
=1
2(+ 1) =
1
2(2 + ).
=1
2( 1)(2 + )
1
2
12 + 22 + 32 + + ( 1)2 + 1+ 2+ 3+ + 1
.
(. 281),
12 + 22 + + 2 =1
6
(+ 1)(2+ 1).
=1
2( 1)(2 + )
1
2
1
6( 1)(2 1) +
1
2( 1)
=
=1
2( 1)
2 +
22
6
2
=
=( 1)(62 + 6 22 + 3)
12=
(2 1)
3.
. . -
( 1) =1
3
( 1) (+ 1) ( 2) ( 1)
,
.
=
=2
( 1)
=1
3
=2
( 1) (+ 1) ( 2) ( 1)
=
=1
3
=2
( 1) (+ 1)
1
=1
( 1) (+ 1)
=
=1
3
( 1) (+ 1) =
2
1
3.
gl02.tex
2.2. 17
8. =1
5+
2
52+
3
53+ +
5
+ .
. 5 ,
5 = 1+2
5+
3
52+ +
51+ .
,
4=1+
2
5
1
5
+
3
52
2
52
+ +
51
1
51
+ =
=1+1
5+
1
52+ +
1
51+ =
1
1
1
5
=5
4.
= 516.
. 8 -
1, 2, , , , 1 = 1 = 15, . .
1 + 2 + + + =
1
1
=1
1
1
5
=5
4.
9.
=1
cos cos 2
+1
cos 2 cos 3
+ +1
cos 9 cos 10
.
. :
=1
cos cos 2
+1
cos 2 cos 3
+ +1
cos 9 cos 10
=
=1
sin
sin
cos cos 2
+sin
cos 2 cos 3
+ +sin
cos 9 cos 10
=
=1
sin
sin (2)
cos cos 2
+sin (3 2)
cos 2 cos 3
+ +sin (10 9)
cos 9 cos 10
=
=1
sin
tg 2 tg+ tg 3 tg 2+ + tg 10 tg 9
=
=1
sin (tg 10 tg)=
sin 9
sin cos cos 10
=2 sin 9
sin 2 cos 10
.
gl02.tex
-
U
R S
S
U R
S S
18 2.
. -
sin ( ) = sin cos cos sin tg tg =sin ( )
cos cos
.
2.3.
10.3
50+ 19
7.
. 3
50+ 19
7 .
1. 50+ 19
7 - , . . 50 + 19
7 = (+
7 )3 .
50+ 19
7 = 3 + 3
7 2+ 212 + 7
7 3,
50+ 19
7 = (3 + 212) +
7 (32+ 73).
-
3 + 212 = 50,
32+ 73 = 19
3 + 212
32+ 73=
50
19.
3
=
,
3 + 21
32 + 7=
50
19 193 1502 + 399 350 = 0.
= 2. =
= 2, = 2.
83 + 423 = 50. -, = 1. = 2, = 2 50+ 19
7 = (2+
7)3 .
3
50+ 19
7 = 2+
7.
gl02.tex
2.3. 19
2. 3
50+ 19
7 +3
50 19
7.
3
50+ 19
7 +3
50 19
7 =
. (+)3 = 3+3+3(+).
50+ 19
7+ 50 19
7+ 3 3
27 = 3
3 + 9 100 = 0. 1 = 4, .
,
3
50+ 19
7 +3
50 19
7 = 4.
3
50+ 19
7 = .
3
50+ 19
7
3
50 19
7
= 3,
3
50 19
7 =
3
3
50+ 19
7
3
= 4 2 4 3 = 0.
1,2 = 2
7. 0, = 2+
7 3
50+ 19
7 = 2+
7.
, 3
50 19
7 = 2
7.
11.
4444488889.
. :
4444488889=
4444444444+44444+1=
=
4 1111111111+4 11111+1=1
3
4 9999999999+4 99999+9=
=1
3
4 (10101)+4 (1051)+9=1
3
4 1010+4 105+1=
=1
3(2 105+1)=
200001
3=66667.
gl02.tex
-
U
R S
S
U R
S S
20 2.
12.3
999700029999.
. :
3
999700029999 =3
999 109 + 7 108 + 2 104 + 9999 =
= 3
103 1
109 + 7 108 + 2 104 + 104 1 =
=3
1012 109 + 7 108 + 3 104 1 =
=3
1012 3 108 + 3 104 1 =
= 104 1 = 9999.
13. 3
370370 037
89
11 1
30
00 0
30
.
.
3
370370 037
89
11 1
30
00 0
30
=
= 3
37 (1087 + 1084 + + 100) 11 1
30
1030 =
=3
37
1090 1
103 1
1
9
(1030 1) 1030 =
=3
1090 1
27
1060 1030
9=
=3
1090 3 1060 + 3 1030 127
=
=1030 1
3= 33 3
30
.
14.
44 4
2
+ 11 1
+1
66 6
, 3.
gl02.tex
2.3. 21
. :
44 4
2
+ 11 1
+1
66 6
=
=
4
9
99 9
2
+1
9
99 9
+1
6
9
99 9
=
=1
3
4 (102 1) + (10+1 1) 6 (10 1) =
=1
3
4 102 + 4 10 + 1 =
=1
3
(2 10 + 1) =
=1
3
2 00 0
1
1 = 66 6
1
7.
15.
2+
2+
2+ .
. =
2+
2+
2+ . , 0. ,
2 = 2+
2+
2+
2+ .
- 2 = 2 + 2 2 = 0. - 1 = 2 2 = 1. 0, 1 = 2
2+
2+
2+ = 2.
16.
+
, 2 + 2 = 3 0 .
. 0 , = , 1. , 2 + 2 = 3 2 ,
2
3+ 1 = 0.
gl02.tex
-
U
R S
S
U R
S S
22 2.
1, 2 3 + 1 = 0
=
3+
5
2.
+
,
+
=
+ 1
1=
3+
5
2+ 1
3+
5
2
1
=5+
5
1+
5=
5 (
5+ 1)1+
5=
5.
2.4.
17.
(+ + )3 (+ )3 ( + )3 (+ + )3.
. + = = . , - () ,
(+ + )3 (+ )3 ( + )3 (+ + )3 =
=
(+ )3 ( )3
(+ )3 ( )3
=
= 2 (32 2) 2 (32 2)= 6 (2 2)= 24.
2 2 = 4 .
18. , 3
+ 3
+ 3
= 0, (+ + )3 = 27 .
. 3
+ 3
= 3
, (+ )3 = 3 + 3 + 3(+ ),
+ + 3 3
( 3
+ 3
) = ,
+ + 3 3
( 3
) = ,
+ + = 3 3
(+ + )3 = 27.
19. , 1
1+ +
+1
1+ +
+1
1+ +
= 1, , , = 1.
gl02.tex
2.4. 23
. ( - , = 1), :
1
1+ +
+
(1+ + )+
(1+ + )=
=1
1+ +
+
+ +
+
+ + 2
=
=1
1+ +
+
+ + 1+
+ 1+
=1+ +
1+ +
= 1.
20. tg , 2 tg sin + 5 cos = 10.
. :
2 tg sin + 5 cos = 10, 5 cos sin 10+ 2 tg = 0,
5 cos sin 2
5
sin
cos
= 0,
cos (5 cos sin ) 2 (5 cos sin ) = 0,(5 cos sin )(cos 2) = 0. ()
cos 2 = 0, ()
5 cos sin = 0.
2 tg sin + 5 cos = 10
, 2 tg = 10 tg = 5.
21. , ctg 70 + 4 cos 70 =
3.
. , - ctg 70 + 4 cos 70 =
3 tg 20 + 4 sin 20 =
3
sin 20 =
3 tg 20
4. ()
().
tg tg =sin ( )
cos cos
,
gl02.tex
-
U
R S
S
U R
S S
24 2.
:
3 tg 20
4=
tg 60 tg 20
4=
=sin 40
4 cos 60 cos 20
=2 sin 20 cos 20
4
1
2
cos 20
= sin 20.
, (), , .
22. , cos = tg , cos = tg , cos = tg
,, (0, 2). ,
sin = sin = sin =
5 1
2.
. ,
cos2 =sin2
cos2
=1 cos2
cos2
=1
cos2 1 =
=1
1
cos2 1
1 =1
1
1
cos2 1
1
1.
cos2 = , -
=1
2 1
1 2+1 = 0.
, 0 1, -
=
5 1
2, . . cos2 =
5 1
2.
sin2 = 1 cos2 = 1
5 1
2=
3
5
2
sin =
3
5
2=
6 2
5
4=
=
(
5 1)2
4=
5 1
2=
5 1
2.
gl02.tex
2.5. 25
cos , cos ,
sin = sin = sin =
5 1
2.
. , ,,
(0, 2). cos = tg ,cos = tg , cos = tg - sin, sin
sin .
2.5.
23. , -
1+1
2+ +
1
+ 1 1
.
. 1 +1
2+ +
1
=
, -
+
+ 1. +
(1) :
+
= +
1+1
2+ +
1
=
= (1+ 1)+
1+1
2
+ +
1+1
=
=2
1+
3
2+ +
+ 1
2
1
3
2
+ 1
=
=
+ 1.
.
24. , 1
2
3
4
5
6
99
100
1
10.
. 1
2
3
4
5
6
99
100= . ,
:1
2
2
3,3
4
4
5,5
6
6
7, ,
99
100
100
101.
gl02.tex
-
U
R S
S
U R
S S
26 2.
1
2
3
4
5
6
99
100
2
3
4
5
6
7
100
101
2
3
4
5
6
7
100
101. ()
() , .
2
1
2
3
4
5
6
99
100
2
3
4
5
6
7
100
101=
1
101
1
100,
. . 2
1
100
1
10.
25. , 3 + 3 = 5 + 5 . -, 2 + 2 1+ .
. -. , , 3 + 3 = 5 + 5
2 + 2 1+ . 3 3 .
3(2 + 2) 3(1+ ) 3(2 + 2) 3(1+ ).
5 + 32 + 23 + 5 3 + 4+ 3 + 4. ()
3 + 3 = 5 + 5 , ()
3
2 + 23 4+ 4, 2+ 2 3 + 3,
(+ ) (+ )(2 + 2),
2
+ 2, ( )2 0.
. , 3 + 3 = 5 + 5 , .
gl02.tex
2.5. 27
26. + + = 1. , 2 + 2 + 2 13.
. . 1. (9)
1 = (+ + )2 = (1 + 1 + 1 )2 (12 + 12 + 12)(2 + 2 + 2) = 3(2 + 2 + 2).
2 + 2 + 2 13.
2.
1 = (+ + )2 = 2 + 2 + 2 + 2+ 2+ 2. ()
2 2+2 , 2 2+2 2 2+2 (- , , (2),
() 1 32+32+32 2+2+2 13.
. (9), .
1 + 2 + + = 1,
21 +
22 + +
2
1
.
27. , ,
2 + 2 + 2 = 1. ,
3 + +
3.
. (9),
(+ + )2 = (1 + 1 + 1 )2 (12 + 12 + 12)(2 + 2 + 2) = 3(2 + 2 + 2).
2 + 2 + 2 = 1,
(+ + )2 3
3 + +
3.
. (9), .
21 + 22 + +
2
= 1,
1 + 2 + +
.
28. ,
4+ 1+
4+ 1+
4+ 1 5 , + + = 1 4+ 1 0, 4+ 1 0, 4+ 1 0.
gl02.tex
-
U
R S
S
U R
S S
28 2.
. (2),
4+ 1 =
1 (4+ 1) 1+ 4+ 12
=4+ 2
2= 2+ 1,
4+ 1 2+ 1
4+ 1 2+ 1. + + = 1,
4+ 1+
4+ 1+
4+ 1 2(+ + ) + 3 = 5. . , -
(2) , 1 = 2 . , , 4+1 = 1, 4+1 = 1 4+1 = 1, . . = = = 0. ,
+ + = 1.
. - (9).
4+ 1+
4+ 1+
4+ 1
2
(12+12+12) (4+1+4+1+4+1).
+ + = 1,
4+ 1+
4+ 1+
4+ 1
2
3 7 = 21 25, . .
4+ 1+
4+ 1+
4+ 1 5.
29. , , , , - + + = 5 + + = 8,
1 73, 1 7
3 1 7
3.
. + = 5 . (+ )+ = 8, = 8(+ ),
= 8 (5 )
= 2 5+ 8. ()
2
5+ 8 0
( 2 5 + 8 = 0 ), () , 0. -, , 0 0. ,
gl02.tex
2.5. 29
, - .
+ + = 5. 0, 0 0. , + + = 5 0, 0, 0,
5, 5 5., (2), ,
+ 2 . 5 2 4 2 10+ 25. ()
4(2 5+ 8) 2 10+ 25 32 10+ 7 0,
1 73.
, , .
30.
+ 2
+ 3 3
6, 0,
0 0.
. (1) = 6,
+ 2
+ 3 3
=
+
+
+ 3
+ 3
+ 3
6 6
3
3
3
= 6 6
= 6.
, .
. , , . .
1
2+ 2
2
3+ +
1 (+ 1)
2,
1. , 1 = 2 = = = 1.
31. , - 2
+ 3 3
5 5
.
gl02.tex
-
U
R S
S
U R
S S
30 2.
. (1) = 5,
2
+ 3 3
=
+
+ 3
+ 3
+ 3
5
5
3
3
3 = 5 5
,
. . .
32. 3+3+3+3 +++, 0, 0, 0, 0.
.
3 + 3 + 3 + 3 =1
3(3 + 3 + 3)+
1
3(3 + 3 + 3) +
+1
3(3 + 3 + 3)+
1
3(3 + 3 + 3).
( ) - (1) = 3 3+3+3 3,
3+3+3 3, 3+3+3 3, 3+3+3 3. .
33. , 1
2+
1
1+
1
+ 1+
1
+ 2
4
, 2.
. 2, 1
2
0,1
1
0,
1
+ 1
0,1
+ 2
0
(5).
1
2+
1
1+
1
+ 1+
1
+ 2 16
2+ 1+ + 1+ + 2=
4
.
, 1
2+
1
1+
1
+ 1+
1
+ 2
=4
.
, 1
2
=1
1
=1
+ 1
=1
+ 2.
34. 1+ 24 2 + 23 .
gl02.tex
2.5. 31
. () (1) = 2 = 4, 1 + 4 22 1 + 4 + 4 + 4 43 . , 2+ 44 22 + 43. .
. -, , , 1+ 22 + 6 3+ 3 , , .
() (1) :
2 + 6 2
2
6 = 23,
1+ 1+ 2 + 2 + 2 + 6 6 6
1 1 2 2 2 6 = 6,
2 + 6 23 2+ 32 + 6 6. , 2+ 42 + 26 6+ 23 .
, .
35. ,
1+1
1+1
1+1
1,
0, 0, 0 = 1.
. = 1, , , - =
, =
=
.
( + )( + )( + ) , () 0, 0, 0.
= + , = +
= + , =
+
2, =
+
2, =
+
2 ()
(+ )( + )(+ ) 8, () 0, 0, 0.
() - (2),
+ 2
, + 2
+ 2
.
, .
gl02.tex
-
U
R S
S
U R
S S
32 2.
. () () - .
36. ,
(11+22+ +)2
21+
22+ +
2
21+
22+ +
2
,
2.
. - (1+1)
2+ (2+2)2+ + (
+
)20. -
2
21 +
22 + +
2
+ 2
11 + 22 + +
+
+
21 +
22 + +
2
0. -
, 0, . .
11+22+ +2
21+
22+ +
2
21+
22+ +
2
0. -
(9).
37.
+
+ +
1+
4+ 12
,
0.
. ,
+
+ +
+
+
+ ,
0. =
+
+
+
, 2 = 0. 0, -
1 =1+
1+ 4
2.
.
gl02.tex
2.5. 33
38. ,
1
2
(+ )(1 )(1+ 2)(1+ 2)
1
2, , -
.
. = tg = tg ,
2
2
2
2.
(+ )(1 )(1+ 2)(1+ 2)
=(tg + tg )(1 tg tg )
(1+ tg2 )(1+ tg2 )=
=
sin (+ )cos cos
cos (+ )cos cos
1
cos2
1
cos2
=
= sin (+ ) cos (+ ) =1
2sin 2+ .
1
2 1
2sin 2(+ ) 1
2,
.
39. , , 2 + 2 + 32 + 2+ 6 + 3 0.
. .
2 + 2 + 32 + 2+ 6 + 3 =
= 2 + 2( + 1) + ( + 1)2 ( + 1)2 + 32 + 6 + 3 =
= (+ + 1)2 2 2 1+ 32 + 6 + 3 =
= (+ + 1)2 + 2( + 1)2 0.
40. 2+ 2 cos sin2 + cos2( 1) 0.
.
2+ 2 cos sin2 + cos2( 1) =
= 2+ 2 cos 1+ cos2 + cos2( 1) =
= (cos + 1)2 + cos2( 1) 0. .
gl02.tex
-
U
R S
S
U R
S S
34 2.
,
cos = 1,cos ( 1) = 0.
= + 2,
1 =
2+ ,
, .,
+ 2 =
2+ + 1.
=2
4 2+ 1, . . , -
. , ,
2+ 2 cos sin2 + cos2( 1) 0.
41. , 0 ,
+
+
+
+
.
. 0 ,
1,
1,
1.
1 0,
1 0,
1 0
1
1
1
0.
-.
42. 0 . ,
2
+ 2+ 2 2+ 2+ 2.
. 0 , 0, 0 0. ( ) ( ) ( ) 0. ,
2
+ 2+ 2 2+ 2+ 2.
gl02.tex
2.5. 35
43. , 0, 0, 0, 0,
(+ )(+ )
+
.
. (9),
(
+
)2
(
)2 + (
)2
(
)2 + (
)2
= (+ )(+ ).
.
44. , 0, 0, 0, + 0, + 0,
+ 0, 2
+
+2
+
+2
+
3.
. = + , = + ,
= + .
2+ 2+ 2 = + + , 2 = + + 2,(+ ) = + + , 2 = + , 2 = +
+ +
+
+
+
+
3. ()
()
+
+
+
+
+
6. ()
+ 0, + 0, + 0, 0, 0,
0. (3), () :
+
+
+
+
+
2+ 2+ 2 = 6.
45. , + + = 1 0, 0, 0,
+ +
+
+
+
32.
gl02.tex
-
U
R S
S
U R
S S
36 2.
. (2) , + + = 1,
+
=
+ (1 )=
(1 )(1 )=
=
(+ )(+ )=
+
+
12
+
+
+
.
,
+
12
+
+
+
+
12
+
+
+
.
,
+
+
+
+
+
12
+
+
+
+
+
+
+
+
+ +
+
=3
2.
46. , + + 3 0, 0, 0,
+ 1+
+ 1+
+ 1 3
2.
. , -
1
+ 1+
1
+ 1+
1
+ 1 3
2. ()
:
+ 1 1
+ 1+
+ 1 1
+ 1+
+ 1 1
+ 1 3
2,
1
1
+ 1+ 1
1
+ 1+ 1
1
+ 1 3
2.
(). ()
(5), , + + 3,
gl02.tex
2.5. 37
1
+ 1+
1
+ 1+
1
+ 1 9
+ + + 3 9
6=
3
2.
, () - (1) = 3.
47. ,
0, 0, 0, + +
3
2 + 2 + 2
3.
. . , - , ,
+ +
3
2 + 2 + 2
3. ()
(),
(+ + )2
3
2 + 2 + 2,
2 + 2 + 2 + 2+ 2+ 2 32 + 32 + 32,
22 + 22 + 22 2 2 2 0,
( )2 + ( )2 + ( )2 0. -
, , .
48. , 0, 0, 0,
+ 2
2
+
+ 2
2
+
+ 2
2
27.
. -, (1) = 3.
+ 2
2
=
+ +
2
3
3
2
2
=9
3
2
4
2.
gl02.tex
-
U
R S
S
U R
S S
38 2.
+ 2
2
9
3
2
4
2
+ 2
2
9
3
2
4
2.
+ 2
2
+
+ 2
2
+
+ 2
2
9
3
2
4
2 +9
3
2
4
2 +9
3
2
4
2
3
3
9 9 9 3
2
4
3
2
4
3
2
4
2
2
2= 27.
, .
49.
42 + 1+
42 + 1
2
2,
0 12 0 1
2.
. = 2 = 2, -
2 + 1+
2 + 1
2, ()
0 1 0 1. 0 1, 0 1,
2 + 1+
2 + 1
2 + 2+
2 + 2=
+
2 + 2. ()
(10) = 2
(+ )2 2(2 + 2),
+
2 + 2
2.
() .
gl02.tex
2.5. 39
50.
22 + 3+
22 + 3
2
5
0 1 0 1.. 0 1 0 1,
22 + 3+
22 + 3
22 + 22 + 1+
22 + 22 + 1=
+
22 + 22 + 1. ()
( + )2 2(2 + 2) = 22 + 22 , (10) = 2, , 1 1, ()
22 + 3+
22 + 3
2
(+ )2
22 + 22 + 1 2
2 + 22
22 + 22 + 1=
= 1
1
22 + 22 + 1 1 1
5=
4
5.
.
51.
1+
1+
1 6,
1, 1 1.
. 1, 1 0. , - (2), :
1=
1
1+
1
1
2
1
1
1
1= 2
1
1.
,
1 2
1
1
1 2
1
1.
gl02.tex
-
U
R S
S
U R
S S
40 2.
, (1) = 3,
1+
1+
1
2
1
1+ 2
1
1+ 2
1
1
3 3
2
1 1 2
1
1
2
1
1= 6.
52.
+ 1 1+
2
2
8, 0.
. () =
+ 1 1
2+
2
8.
, (0) = 0. - , 0
= () , . .
0 () (0) () 0. = ()
, . .
=1
2
+ 1
1
2+
4. -
(2),
+ 1 =
1 (+ 1) 1+ + 12
=
+ 22
1
+ 2
1
2+
4=
1
2
2
+ 2+
+ 22
2
.
0, 0
2
+ 2
1
(3) 2
+ 2+
+ 22
2. ,
0.
, 0, () 0, . . .
gl02.tex
2.6. 41
2.6.
53. (3+ 5)2 + (+ 6)3 = 42 + 1.
.
(3+ 5)2 (2)2
+
(+ 6)3 13
= 0,
(+ 5)(5+ 5)+ (+ 5)(2 + 13+ 43) = 0,
(+ 5)(2 + 18+ 48) = 0.
+5=0 2 + 18+ 48 = 0, - 1 =5, 2 = 9
33 3 =9+
33.
54. (2 1)2 3 = 5.
. -:
(2 1)2 22
(3 + 13) = 0,
(2 + 1)(2 3) (+ 1)(2 + 1) = 0,
(2 + 1)(2 2 4) = 0.
2 + 1 0, 2 2 4 = 0, 1 = 1 +
5 2 = 1
5.
55. 2( 1)2 + ( 2)3 = 76.
. :
2( 1)2 72 + ( 2)3 33 = 0
,
(1)7
(1)+7
+ (5)
(2)2+3(2)+9
=0,
(27) (2+7)+ (5) (2+7)=0,
(2+7) (212)=0.
2 + 7 0,
2
12 = 0, 1 = 2
3 2 = 2
3.
gl02.tex
-
U
R S
S
U R
S S
42 2.
56. ( 5)2 + ( 4)3 + ( 3)4 = 2.
. :
( 5)2 12
+ ( 4)3 +
( 3)4 14
= 0,
( 4)( 6)+ ( 4)(2 8+ 16)+
+ ( 4)( 2)(2 6+ 10) = 0,
( 4)(2 7+ 10)+ ( 4)( 2)(2 6+ 10) = 0,
( 4)( 2)(2 5+ 5
= 0.
1 = 2, 2 = 4, 3,4 =5
5
2.
57. (2 + + 4)2 + 8(2 + + 4) + 152 = 0.
. = 0 ( - = 0), 2
2 + 8 + 15 = 0, ()
=
2 + + 4
.
() 1 = 5 2 = 3.
2 + + 4
= 5
2 + + 4
= 3.
1 = 3+
5 2 = 3
5,
3 = 2.
58. 2(4 + 36) 6
3(4 + 4) = 0.
. -
6
6
34 + 362 24
3 = 0
2
2
3
3= 0.
2 = 2
3 1 =4
12, 2 =
4
12.
gl02.tex
2.6. 43
59. (2 + 2 1)2 + 22 + 3 = 3.
.
(2 + 2 1)2 + 2(2 + 2 1)+ 1 = + 2. ()
() , -
(2 + 2)2 = + 2,
2
(+ 2)2 = + 2,
(+ 2)(3 + 22 1) = 0
(+ 1)(+ 2)(2 + 1) = 0.
1 = 1, 2 = 2, 3 =
1+
5
2 4 =
1
5
2.
60. 2 + 2+ 3 = (2 + + 1)(4 + 2 + 4).
.
2 + + 1 0 4 + 2 + 4 4,
2 + 2+ 3 4(2 + + 1). 32 + 2+ 1 0.
,
32 + 2+ 1 = 3
+1
3
2
+2
3 2
3.
, .
61. (6+ 7)2(3+ 4)(+ 1) = 1.
. 12, 6 + 7 = . - : (6 + 7)2(6 + 8)(6 + 6) = 12,
2( + 1)( 1) = 12 4 2 12 = 0.
gl02.tex
-
U
R S
S
U R
S S
44 2.
4 2 12 = 0,
2 = 4 (, 2 0) 1 = 2, 2 = 2. 6+ 7 = , =
7
6.
1 =
1 7
6=
2 7
6=
5
6
2 =2 7
6=
2 7
6=
3
2.
62. ( 1)( 2)( 4)( 8) = 102 .
. :
( 1)( 2)( 4)( 8) =
( 1)( 8)
( 2)( 4)
=
= (2 9+ 8)(2 6+ 8).
,
(2 9+ 8)(2 6+ 8) = 102. ()
= 0 (),
2 ,
9+8
6+8
=10.
9+8
= , (+3) = 10 2+310 = 0. 1 = 2 2 = 5.
9+8
=, -
9+8
= 2 9+8
= 5.
2 11 + 8 = 0 2 4 + 8 = 0. 1 =11+
89
2 2 =
11
89
2, .
63.
+ 1
2 + 2
+
+ 6
2 + 12+ 35=
+ 2
2 + 4+ 3+
+ 5
2 + 10+ 24.
gl02.tex
2.6. 45
. :
+ 1
(+ 2)+
+ 6(+ 5)(+ 7)
=
+ 2(+ 1)(+ 3)
+
+ 5(+ 4)(+ 6)
,
+ (+ 2)
(+ 2)+
(+ 5)+ (+ 7)(+ 5)(+ 7)
=
=(+ 1)+ (+ 3)(+ 1)(+ 3)
+(+ 4)+ (+ 6)(+ 4)(+ 6)
,
1
+1
+ 2+
1
+ 5+
1
+ 7=
1
+ 1+
1
+ 3+
1
+ 4+
1
+ 6,
1
+1
+ 7
+
1
+ 2+
1
+ 5
=
=
1
+ 1+
1
+ 6
+
1
+ 3+
1
+ 4
,
2+ 7
2 + 7
+2+ 7
2 + 7+ 10=
2+ 7
2 + 7+ 6+
2+ 7
2 + 7+ 12. ()
() , 2 + 7 = 0, . . 1 = 31
2 ().
2+ 7 = 0. () 2+7, 2+7 = .
1
+1
+ 10=
1
+ 6+
1
+ 12. ()
() - , . . 2 + 18 + 90 = 0. .
,
1 = 31
2.
64.1
+1
+ 1+
1+ 2
+1
+ 3+
1
+ 4= 0.
gl02.tex
-
U
R S
S
U R
S S
46 2.
.
1
+1
+ 4
+
1
+ 1+
1
+ 3
+1
+ 2= 0,
2(+ 2)
2 + 4
+2(+ 2)
2 + 4+ 3+
1
+ 2= 0. ()
= 2, () +2.
2
2 + 4
+2
2 + 4+ 3+
1
2 + 4+ 4= 0. ()
2+4 = , ()
2
+2
+ 3+
1
+ 4= 0 52 + 25 + 24 = 0.
1,2 =
25
145
10.
= 2 + 4, -
2 + 4 =
25
145
10 2 + 4 =
25+
145
10.
1 = 2+
15
145
10, 2= 2
15
145
10,
3 = 2+
15+
145
10, 4= 2
15 +
145
10.
65. 4 =11 6
6 11.
. -
611
11
6.
gl02.tex
2.6. 47
:
65 114 11+ 6 = 0,
6(5 + 1) 11(3 + 1) = 0,
6(+ 1)(4 3 + 2 + 1) 11(+ 1)(2 + 1) = 0,
(+ 1)(64 173 + 172 17+ 6) = 0.
1 = 1.,
64 173 + 172 17+ 6 = 0. ()
, = 0 (). ,
2 +1
= .
2 +1
2= 2 2
() -
62 17 + 5 = 0,
1 =5
2 2 =
1
3.
= +1
, 2 = 52, . . +
1
=5
2
22 5+ 2 = 0, 2 = 2 3 =1
2.
66.
4 + 1
(2 + 1)=
41
15
. = 0, 2 .
2 +1
2
+1
=41
15. ()
gl02.tex
-
U
R S
S
U R
S S
48 2.
= +1
, 2 +1
2= 2 2 -
()
2
2
=41
15 152 41 30 = 0,
1 =10
3 2 =
3
5.
= +1
, 2.
+1
=10
3, . . 32 10+ 3 = 0,
1 = 3 2 =1
3.
67. (2 + + 1)2 = 2(32 + + 1).
. , = 0. , 4
1+1
+1
2
2
= 3+1
+1
2. ()
1 +1
+1
2= , ()
2 = + 2 2 2 = 0. 1 = 2 2 = 1.
1+1
+1
2 = 2 1+1
+1
2 = 1,
2
1 = 0 22 + + 1 = 0.
1 =1+
5
2 2 =
1
5
2,
.
gl02.tex
2.6. 49
68. = 1 5(1 52)2 .
. = 152 .
= 1 52,
= 1 52,()
1 1. () -
, = 5(2 2) ( )(5+ 5 1) = 0. = 0, = ()
52 + 1 = 0.
1 =
1+
21
10 2 =
1
21
10.
5+51 = 0. =1 5
5 -
() 25254 = 0,
3 =1+
17
10 3,4 =
1
17
10.
, 1.
69.
2
1
22 + 2=
32 3.
. - , = 0.
,
1
1
2+ 1
2
=1
3 1
3
. ()
1
= , ()
1
2 + 1=
1
3 1, . .
32 4 + 1 = 2 + 1, ( 2) = 0 1 = 0, 2 = 2.
gl02.tex
-
U
R S
S
U R
S S
50 2.
1 = 0,
1
= 0 1 = 1, 2 = 1. 2 = 2,
1
= 2 3 = 1+
2, 4 = 1
2.
70.24
2
2=
12
2
+ 2 .
. - = 0, = 1 = 2. , :
24
2
2
12
2
= 2 ,12
( 1)( 2)= 2 ,
( 1)2( 2) = 12, (2 2)(2 2+ 1) = 12.
2 2 = , ( + 1) = 12 2 + 12 = 0. 1 = 4 2 = 3.
2 2 = ,
2
2+ 4 = 0 2 2 3 = 0.
,
1 = 1 2 = 3.
71. 2 +812
(+ 9)2= 40.
. , - 2 + 2 = ( )2 + 2.
2 +812
(+ 9)2=
9+ 9
2
+182
+ 9=
2
+ 9
2
+182
+ 9.
=
2
+ 9. -
2+1840 = 0.
1 = 2 2 = 20. - .
1)
2
+ 9= 2, 2 2 18 = 0
1 = 1+
19, 2 = 1
19.
gl02.tex
2.6. 51
2)
2
+ 9= 20, 2 + 20+ 180 = 0. -
.
72. 102( 2)2 = 9(2 + ( 2)2).
. 71, - 2 + 2 = ( )2 + 2. 102( 2)2 = 9
( 2)
2+ 2( 2)
52( 2)2 = 9
2+ ( 2)
. ( 2) = , -
52 9 18 = 0, 1 = 3
2 =
6
5.
( 2) = , 2 2
3 = 0 52 10+ 6 = 0. 1 = 3 2 = 1, .
73. 24 + 24 = 4 1.
. :
24 + 24 422 = 422 + 4 1,
2(2 2)2 + (2 12) = 0.
2 = 2,2 = 1,
1 =1=
22 2 =2=
22.
74.3
2 + 3+
4
2 + 4=
(+ 1)4 + 1(+ 1)2
.
. 2 0, - 2. (3), . .
(+ 1)4 + 1(+ 1)2
= (+ 1)2 +1
(+ 1)2 2.
gl02.tex
-
U
R S
S
U R
S S
52 2.
, , 2. 1 = 0.
75. (2+ 3 6)4 + 24 + 34 = 6.
. = 623, , - , , , :
2+ 3 + = 6,
24 + 34 + 4 = 6.()
() () - (9),
36 = (2+ 3 + )2 = (
2
2+
3
3 + 1 )2 (2+ 3+ 1)(22 + 32 + 2) = 6(22 + 32 + 2),
36 (22 + 32 + 2)2 = (
2
22 +
3
32 + 1 2)2 (2+ 3+ 1)(24 + 34 + 4) = (6+ 34 + 4).
, 24 + 34 + 4 6. () , -
(9) -
.
2
2=
3
3=
1, . . = = .
() 6 = 6 1 = 1. , 1 = 1, 1 = 1 1 = 1.
76. 42 + 12+12
+4
2= 47.
. = +1
, 2+1
2= 22
4(2 2)+ 12 = 47.
42 +1255 = 0, 1 =5
2
2 =
11
2. ,
.
gl02.tex
2.6. 53
1) +1
=5
2, 22 5+ 2 = 0 1 = 2, 2 =
1
2.
2) +1
=
11
2. 22 + 11+ 2 = 0,
3 =
11
105
4 4 =
11+
105
4.
77. 3 + 2 + =
1
3.
.
33 + 32 + 3+ 1 = 0,
(+ 1)3 = 23, + 1 = 3
2.
1 =
1
1+ 3
2.
78. 4 23 + 22 + 4 8 = 0.
.
4
23 + 22 + 4 8 = 4 22 (23 4)+ 42 8 =
= 2(2 2) 2(2 2) + 4(2 2) =
= (2 2)(2 2+ 4).
2 2+ 4 0, 2 2 = 0 1 =
2, 2 =
2.
79. 8(4 + 4) 4(2 + 2)+ 1 = 0.
. 2,
164 +1648282 +2 = 0, (42 1)2 + (42 1)2 = 0.
(42 1)2 0 (42 1)2 0, 42 = 1 42 = 1. ,
1 =1
2, 1 =
1
2; 2 =
1
2, 2 =
1
2; 3 =
1
2, 3 =
1
2 4 =
1
2,
4 =
1
2.
gl02.tex
-
U
R S
S
U R
S S
54 2.
80. 3 + 3+ 5
2 = 0.
. , .
3 + 3+ 5
2 , . .
(+ )(1
2 + 2+ 3) =
3 + 3+ 5
2. ()
() - - , , 1 , 2 , 3 , . .
1 = 1,
2 + 1 = 0,
2 + 3 = 3,
3 = 5
2.
=
2, 1 = 1,
2 =
2 3 = 5.
(+
2)(2
2+ 5) = 0.
2
2 + 5 0, 1 =
2.
81. ( 2)6 + ( 4)6 = 64.
. = 3, (+ 1)6 + ( 1)6 = 64. (11) = 6,
(+1)6+ (1)6=6+65+154+203+152+6+1+6
65+154203+1526+1=26+304+302+2.
6 + 154 + 152 31 = 0. = 2 ,
3 + 152 + 15 31 = 0 ( 1)(2 + 16 + 31) = 0, 0.
gl02.tex
2.6. 55
2 + 16 + 31 = 0 , ( 1)(2 + 16 + 31) = 0 1 = 1.
2 = , 1 = 1 2 = 1. = + 3, 1 = 4 2 = 2.
82.
2 1
3
=
1
2.
.
1
2= . = 2 + 1
4 + 13
= , ()
.
= []+ 0 1, 0 [] 1.
() , 0 4 + 13
1
1 2. 1 2, 1 = 1, 2 = 0
3 = 1. = 2 + 1, 1 = 1, 2 = 1 3 = 3.
83. 3+ 4[] = 5+ 6.
. = [] + , 3
[]+
+ 4[] = 5+ 6
=7[] 6
2. ()
0 1, 0 7[] 62
1. ,
0 7[] 6 2 67 [] 8
7. []
, 6
7 [] 8
7 [] = 1.
gl02.tex
-
U
R S
S
U R
S S
56 2.
[] = 1 (), =1
2. ,
= []+ . 1 = 1+1
2=
3
2, . .
1 =3
2.
84. 22
= .
. , 0 1. - .
1) 0 14. 0 2 1
2, 2 = 2, 22 = 4 = 4
4 = . 1 = 0.
2) 1
4 1
2.
1
2 2 1, 2 = 2
22 = 4. 1 4 2, 4 = 4 1 4 1 = , . . 2 =
1
3.
, 2 .
3) 1
2 3
4. 1 2 3
2, 2 = 2 1 22 =
4 2. , 2 4 3, - 0 42 1, 4 2 = 42 4 2 = . 3 =
2
3.
1
2 3
3
4,
3 .
4) 3
4 1. 3
2 2 2, 2 = 2 1
22 = 4 2. 3 4 4 1 4 2 2, 4 2 = 4 3 4 3 = . = 1. , 0 1. = 1 .,
1 = 0, 2 = 13 3 = 23.
2.7.
85.
1+
2 =3
1.
gl02.tex
2.7. 57
. - 1. () =
1 +
2 () =3
1. = ()
, = () - , . , 1 = 2.
86.
(+ 4)(2+ 3)3
+ 8 = 4
(+ 8)(2+ 3)+3
+ 4.
. -
3
2.
+ 4 = ,
2+ 3 =
+ 8 = . 3 = 4 + 3, 0, 0 0. :
3+ 3 = 4,
( 3)+ ( 3) = 4 ( 3) (+ ) = 4. ()
+ 0, () , 3 0,. .
2+ 3 3 3. ()
2+ 3 3
+ 4+
+ 8
= 4. ()
() =
2+ 3 3
+ 4+
+ 8
. , = ()
3. () . 1 = 5.
1 = 5 , - () . .
4
+ 4+
+ 8=
2+ 3 3,
4
+ 8
+ 4
+ 8 4=
2+ 3 3,
+ 8
+ 4 =
2+ 3 3.
gl02.tex
-
U
R S
S
U R
S S
58 2.
, ,
(+ 8)(+ 4) = 3
2+ 3. -
, 2 6+ 5 = 0. 3, 1 = 5.
87. 2 4 2 = 2
3 + 1.
. -
3 + 1 0,
2 4 2 0,
, 1 2
6 2+
6.
3 + 1 = (+ 1)(2 + 1)
2
4 2 = (2 + 1
3(+ 1),
3(+ 1) + 2
+ 1
2
+ 1
2
+ 1
= 0. ()
, , 1 + 1 0, - 2 + 1 0 ! .
, ()
2
+ 1
+ 1
2
+ 1= .
32 + 2 1 = 0
1 =1
3.
+ 1
2
+ 1=
1
3.
2 10 8 = 0,
1 = 5 +
33 2 = 5
33. 1 2 , .
gl02.tex
2.7. 59
88.4
2 1 =
2
4+
3
4.
. (1) = 4,
4
2 1 = 4
1 1 1 (2 1) 1+ 1+ 1+ 2 1
4=
+ 12
.
4
2 1 =
2
4+3
4,
2
4+
3
4 + 1
2, 2 + 3 2+ 2 ( 1)2 0.
( 1)2 0, ( 1)2 = 0 1 = 1. - , 1 = 1 .
89.35+ 7 3
5 12 = 1.
. . 1. ,
( )3 = 3 3 3( ),
5+ 7 5+ 12 3 3
(5+ 7)(5 12) = 1.
3
(5+ 7)(5 12) = 6,
52 25 300 = 0
2
12 = 0.
1 = 3 2 = 4. ,
. 2. = 3
5+ 7 = 3
5 12.
= 1,
3
3 = 19
= 1,
2 + + 2 = 19.
= 1, 2 +(1)+ (1)2 = 19 26 = 0. 2 6 = 0 1 = 2 2 = 3.
gl02.tex
-
U
R S
S
U R
S S
60 2.
3
5+ 7 = , 3
5+ 7 =2 3
5+ 7 = 3. - , 1 = 3 2 = 4.
90.30
3
35 3= +
3
35 3 .
. 3
35 3 = . -
3 + 3 = 35,30
= + .()
+ = = , ()
3
3 = 35,
= 30.
1 = 5 1 = 6., ,
+ = 5,
= 6.
1 = 2 2 = 3.
91. 2 + + 12
+ 1 = 36.
. =
+ 1, 0 = 2 1. :
(+ 1)+ 12
+ 1 = 36,
(2 1)2 + 12 = 36,
4
2 + 12 36 = 0,
4
( 6)2 = 0,
(2 + 6)(2 + 6) = 0.
2 + 6 = 0 (), 2 + 6 = 0
gl02.tex
2.7. 61
1 = 2. ,
+ 1 = 2 1 = 3.
92. 2 5 4
+ 13 = 0.
. - (2 6+ 9) + ( 4
+ 4) = 0 ( 3)2 + (
2)2 = 0., -
, - , . , .
93.
22 4+ 3+
32 6+ 7 = 2+ 2 2 .
. .
22 4+ 3 = 2( 1)2 + 1 1
32 6+ 7 = 3( 1)2 + 4 4,
22 4+ 3 1
32 6+ 7 2.
22 4+ 3+
32 6+ 7 3. .
, 2+ 2 2 = 3 ( 1)2 3.,
, 3, , 1 = 1.
94.
+
1 = 1.
. -
0,
1 0,1 0,
gl02.tex
-
U
R S
S
U R
S S
62 2.
5 1
2 1.
, , :
= ,
1 = ,
1 = .
()
() - , ,
+ = 1,
2 + 2 = 1,
2
= 2, 0, 0 0. 1 =
4
5, 1 =
3
5 1 =
1
5.
=
,
=4
5 1 =
16
25.
95.
2 + 1+
2 + 1 = 2 + 2.
. - (2),
2 + 1 =
1 (2 + 1) 1+
2 + 12
=
2 +
2,
2 + 1 =
1 ( 2 + 1) 1+
2 + 12
=
2 + 22
.
2 + 1+
2 + 1
2 +
2+
2 + 22
= + 1
2+2 +1. ( 1)2 0 = 1.
, 1 = 1. 1 = 1 , .
gl02.tex
2.7. 63
. -. (9).
(
2 + 1+
2 + 1)2 (12 + 12)(2 + 1+ 2 + 1) = 4
2 + 1+
2 + 1 2
.
,
2
+ 2 2
. 2 2+ 1+ 2
+ 1 0 ( 1)2 + (
1)2 0. ( 1)2 + (
1)2 0, ( 1)2 + (
1)2 = 0 1 = 1.
96. 3
25 (22 + 9) = 4+3
.
. , = 0.
3
254(22 + 9) = 42 + 3. ()
(1) = 3, () :
3
254(22 + 9) = 3
52 52 (22 + 9)
5
2 + 52 + 22 + 93
= 42 + 3.
, - . , 52 = 22 + 9 2 = 3. 1 =
3 2 =
3.
97. 2
1+
1+ (+ 1)
1+ (+ 2)
1+ (+ 3)(+ 5) = .
. , 0. :
2
1+
1+ (+ 1)
1+ (+ 2)
1+ (+ 3)(+ 5) =
gl02.tex
-
U
R S
S
U R
S S
64 2.
= 2
1+
1+ (+ 1)
1+ (+ 2)
(+ 4)2 =
= 2
1+
1+ (+ 1)
1+ (+ 2)(+ 4) =
= 2
1+
1+ (+ 1)
(+ 3)2 =
= 2
1+
1+ (+ 1)(+ 3) =
= 2
1+
(+ 2)2 =
= 2
1+ (+ 2) = 2
(+ 1)2 = 2(+ 1) = 2+ 2.
2 + 2 = = 2. 0, .
98. 2
+ 15
= 3
+ 3
1
.
. - 1.
2
+ 15+ 3
1 = 3
+ 3+ 2
,
.
2+
2 + 14 15 =
2 + 3. ()
(),
4
2 + 14 15 = 11 11. ()
() - , 11 11 0 1. , 1, = 1. - = 1 , 1 = 1 .
99.
+ 1+
3 = 2
2 + 1.
gl02.tex
2.7. 65
. - 1 3.
(9),
(
+ 1+
3 )2 (2 + 1)(+ 1+ 3 ) = 4(2 + 1).
+ 1 +
3 2
2 + 1. , - , .
+ 1=
1
3
.
-, 0. 3 32 + + 1 = 0, (1)(221) = 0. 1 = 1 2 = 1+
2.
100.4
1 + 4
1+ = 4.
. - . , 1 1., , . .
4
1 4
2 4
1+ 4
2. ,
24
2, , , 4. , .
. - (8) (10) = 4. :
4
1 + 4
1+ = (1 )14 + (1+ )
14 1
4+ 1+
4= 2,
4
1 + 4
1+ 4
8(1 + 1+ ) = 2.
101.413+ 1 + 4
4 1 = 3 4
.
. -
14.
gl02.tex
-
U
R S
S
U R
S S
66 2.
4
, -
4
13+1
+4
4
1
= 3.
=4
13+1
=4
4
1
,
+ = 3,
4 + 4 = 17,()
0 0. 4 + 4 =
( + )2 22
222 ,
() (9 2)2 222 = 17, 22 18 + 32 = 0. - = 2 = 16.
+ = 3,
= 2
+ = 3, = 16.
1 = 1,
1 = 2 2 = 2, 2 = 1, .
4
13+1
= 1 4
13+1
= 2.
1 =
1
12, 2 =
1
3. 1
4,
2 =1
3.
102.4
2 + 4
4 = 2.
. - 2 4.
, (10) = 4, . .
+ 4
8(4 + 4). ()
()
4 2 + 4
4 4
8( 2+ 4 ) = 2.
gl02.tex
2.7. 67
, () , , = . ,
4 2 = 4
4 1 = 3.
, 1 = 3 .
. - (9).
4 2+
4
4 4
(12 + 12) (
2+
4 )
2=
= 4
2+
4 2 4 (12 + 12) ( 2+ 4 ) = 16.
4
2 +4
4 2. - , , - . ,
4 2 =
4
4 , . . 1 = 3.
103.
32 5+ 7+
32 7+ 2 = 3.
.
32 5+ 7
32 7+ 2,
,
32 5+ 7 =
32 7+ 2, . . =
5
2.
32 5+ 7 32 + 7 2
32 5+ 7
32 7+ 2= 3,
32 5+ 7
32 7+ 2 =2+ 5
3. ()
() ,
32 5+ 7 =
+ 73
. , + 7 0
7. -
262 59 + 14 = 0, -
1 = 2 2 =7
26. -
gl02.tex
-
U
R S
S
U R
S S
68 2.
, .
, , =
5
2 -
.
104.
17 2 =
3
2.
. =
= 3
,
+ = 3
17 4 = 2 , . .
4 + 4 = 17. ,
+ = 3,
4 + 4 = 17,()
0 3. ()
4 + 4 =
(+ )2 22
222.
, (), - ()2 18 + 32 = 0, = 2 = 16.
,
+ = 3,
= 2
+ = 3,
= 16.
1 = 1,
1 = 2 2 = 2, 2 = 1, . = 2 , 1 = 1 2 = 4. ,
.
105. +
3+
= 3.
. , 0 3. 3 +
= , 3
+
= 3,
= 3+
.()
gl02.tex
2.7. 69
() ,
+
=
(
+
)(
+ 1) = 0.
0 0,
+
0
+1 = 0,. .
=
+ 1. (), +
2 = 0,
= 1 1 = 1.
. ,
() = +
3+
.
= () 0,
() = 3 .
1 = 1 .
106.
17 2
45 2 = 14.
. -
135
8 22
1
2, -
14 0 14 2212.
=
45 2, =45 2
2
2
17 2 = 17 2, ()
0
17. =
17 2, () -
2 = 17 2,
2 = 17 2.()
() -, 2 2 = 2 + 2 , 2 2 2( ) = 0 ( )( + 2) = 0.
, = = 2 . () 2 + 2 17 = 0
gl02.tex
-
U
R S
S
U R
S S
70 2.
2 2 13 = 0. 1 = 1+ 3
2, 2 = 1 3
2, 3 = 1+
14 4 = 1
14. , 0
17. ()
1 = 3
2 1. =45 2
2,
1 =45 (3
2 1)2
2=
45 18 + 6
2 1
2= 13 + 3
2.
, 1 =13 + 3
2
107.
1 +
1+ +4
1 2 +4
1+ 2 = 4.
.
, 1 1,
= 1 = 1 , , 1 1. (8), . .
1 +
1+ +4
1 2 +4
1+ 2 =
= (1 )12 + (1+ )12 + (1 2)14 + (1+ 2)
14
1
2+ 1+
2+ 1
2
4+ 1+
2
4= 4.
,
1 +
1+ +4
1 2 +4
1+ 2 4.
-, , -, 1 = 0. , 1 = 0 .
108.
1
3+ 6
1+ =
1
24
4
+
1+
36
6
.
. - 1 3. = 1 = 3 - , , 1 3.
gl02.tex
2.7. 71
(8), (7),
1
3+ 6
1+=
1
3
12
+ (1+)16 1
6+ 1+
6= 2
1
24
4
+
1+
36
6
1
6+ 1+
6= 2.
, , 2. , , , . ,
1 = 0.
109.5
1+
1 2 +5
1
1 2 = 2.
. - 1 1. -, = 0 , ,
1 2 1. (8),
5
1+
1 2 +5
1
1 2 =
=
1+
1 2
15+
1
1 2
15
1+
1 2
5+ 1
1 2
5= 2.
5
1+
1 2 +5
1
1 2 2 , , (8) , ,
1 2 = 0 1 = 1, 2 = 1.
110. 4
2+
1+36
2+
4
1= 28.
gl02.tex
-
U
R S
S
U R
S S
72 2.
.
2 = 2
1 = 2 , -
42 + 2 +36
2+
4
2= 28. ,
2
6
2
+
2
2
, 2
6
= 0,
2
= 0 2 = 3, 2 = 2.
,
2 = 3
1 = 2,
1 = 11 1 = 5.
111.
1+
1 = .
. , 1 1. - (2),
1 =
( 1) 1 1+ 12
=
2.
,
1
2. -
,
1 =
( 1) 1 2. -
,
1+
1 . ,
() (2) . , 1 = 1 1 = 1, . . 1 = 1 = 2.
112.
22 + 22 = + 1.
. (10) = 2,
+
22 + 22.
1, + 1
22 + 22 , , .
113.
3 + 2 + 4+ 4 =
+ 2.
. , 0. - (9) ,
(
+ 2)2 = (
+ 2 1)2 (2 + 4)(+ 1) = 3 + 2 + 4+ 4.
gl02.tex
2.7. 73
,
+ 2
3 + 2 + 4+ 4.
-, , (9) , , ( = 0) , =
2 = 1. = 2
1 = 0, 2 = 4.
114.
2
3+
9 2+
1
4
3
9 2
= 1.
. - 3 0 0 3. - =
9 2 . 2 =92
9 2
3+
+1
4(3 )= 1. ()
(2), ().
9 2
3+
+1
4(3 ) 2
9 2
3+
1
4(3 )= 1.
, (2) - , .
9 2
3+
=1
4(3 ). ()
0, 3 + 0 () - 3 =
1
4(3 ) 4(3 )2 = 1.
1 =5
2 2 =
7
2.
gl02.tex
-
U
R S
S
U R
S S
74 2.
2 = 9 2 , 2 =11
4 2 =
13
4.
2 0, 1 =
11
2 2 =
11
2.
115.4
+
2 +
1
2
=3
.
. -
2 + 0,
2
0, = 0,
, 1 1.
, . .
4(
2 + )
2
(2 + )
+
2
2
(2 )=
3
.
4
2 +
2
= 5+ 3. , - ,
162 + 16 8
4
2 + 2 = 252 + 30+ 9,
8
4
2 + 82 + 15+ 9 = 0. ()
8
4
2 0 82 + 15 + 9 0 ( - , 82+15+9 = 0), () .
116.
13 5+ 2 =
5 3.
. -
13 5 0,5 3 0,
53.
gl02.tex
2.7. 75
13 5+
5 3 = 2. ()
() , 2 0 2. , 5
3.
() .
117.
10+ 3
2
1+ 4
5 = 3.
.
2
1 0,
10+ 3
2
1
10 3. , 4
5 0, 3, , .
118.
2 + 3 =
1.
. , 1 2. - .
2 1 3 1,
2 + 3 0,. . 0. ,
1 0. - , 0.
1 = 0 1 = 1, - , 1 . , - .
119. 2 + 2+ 3 =
4 2 .
. 2 + 2+ 3 = (+ 1)2 + 2 2
4 2 2,
2 + 2+ 3 = 2,
4 2 = 2.
gl02.tex
-
U
R S
S
U R
S S
76 2.
= 1, = 0.
, -, .
120. 2 + 2 = 2 4
2 1.
. -
12.
(1) = 4,
24
2 1 = 2 4
1 1 1 (2 1) 2 1+ 1+ 1+ 2 14
= + 1.
2 + 2 + 1 ( 1)2 0.
(1)2 0, (1)2 = 0 1 = 1. 1 = 1 ,
1 .
121.4
82 2 + 2
2
+ 1 =
3.
. 4
82 2 0
2
2
+ 1 =
42 4+ 4 =
(2 1)2 + 3
3,
3.
3,
4
82 2 = 0,
(2 1)2 + 3 =
3.
-
1 =1
2.
122.
2+
4 = 2 6+ 11.
.
(10) = 2, . . +
2(2 + 2).
gl02.tex
2.7. 77
2+
4
2
2+ 4 ) = 2.
2
6+ 11 = ( 3)2 + 2 2.,
, 2. 1 = 3.
1 , 1 = 3 .
. , (9).
(
2+
4 )2 (12 + 12)( 2+ 4 ) = 4,
2+
4 2.
123.
1+
5 = 2 6+ 7.
. - 1 5.
+
2 + 2 , - . ,
1+
5 2. () 2,
1+
5 2 = 2 6+ 5
1+
5 2 = ( 1)( 5). ()
(), , () . 1 5 . , () , 0. ( 1)( 5) = 0 1 = 1 2 = 5. , 1 = 1 2 = 5 .
gl02.tex
-
U
R S
S
U R
S S
78 2.
124.3
3 + 62 6 1 =
2 + 4+ 1.
. -
3
(+ 2)3 9(2+ 1) =
(+ 2)2 3. ()
() 6- ,
((+ 2)3 9(2+ 1))2 = ((+ 2)2 3)3,
(+ 2)6 18(+ 2)3(2+ 1)+ 81(2+ 1)2 =
= (+ 2)6 9(+ 2)4 + 27(+ 2)2 27,
34 + 183 + 32 = 0, 2(2 + 6+ 1) = 0.
2(2 + 6+ 1) = 0 1 = 0,
2 = 3 + 2
2 3 = 3 2
2. -, 1 = 0 .
125.
1 2 = 43 3.
. - 1 1. , . . = cos , 0 . -
1 cos2 = 4 cos3 3 cos
sin = 4 cos3 3 cos . () 0 , sin 0, sin = sin ()
:
sin = 4 cos3 3 cos , sin = cos 3,
cos 3 cos
2
= 0, sin
+
4
sin
2
4
= 0.
=3
4
+ =
8+
2, , .
0 , - , , 1 =
8, 2 =
5
8 3 =
3
4.
gl02.tex
2.7. 79
= cos , ,
1 = cos 1 = cos
8=
=
1
2
1+ cos
4
=
1
2
1+
2
2
=1
2
2+
2,
2 = cos 2 = cos5
8=
1
2
1+ cos5
4
=
=
1
2
1 cos
4
=
1
2
1
2
2
=
1
2
2
2,
3 = cos 3 = cos3
4= cos
4=
2
2.
,
1 =1
2
2+
2, 2 =
1
2
2
2, 3 =
2
2.
126.
1 = 22 1+ 2
1 2 .
. , 1 1.
= cos , 0 .
1 cos = 2 cos2 1+ 2 cos
1 cos2 ,
2
sin
2
= 2 cos2 1+ 2 cos sin . ()
0 ,
sin
2
= sin
2, sin = sin
()
2 sin
2= cos 2 + sin 2,
:
2 sin
2=
2 sin
2 +
4
,
gl02.tex
-
U
R S
S
U R
S S
80 2.
sin
2 +
4
sin
2= 0,
cos
5
4+
8
sin
3
4+
8
= 0. ()
() =
310
+4
5 =
6+
4
3,
, .
() 0 , . . 1 =3
10.
, -
1 = cos3
10= cos 54 .
. , . ,
sin 18 =
6 2
5
4, cos 18 =
10+ 2
5
4 sin 36 =
10 2
5
4.
1 = cos 54
= sin 36 =
10 2
5
4.
127.
2+
2
2+ = .
. -
2,
2
2+ 0,
,
2 2.
2 2,
= 2 cos , 0
4.
gl02.tex
2.7. 81
2+
2
2+=
2+
2
2+ 2 cos =
=
2+
2
4 cos2
2=
2+
2 2
cos
2
=
=
2+
2 2 cos
2=
2+
4 sin2
4=
2+ 2
sin
4
=
=
2+ 2 sin
4=
2
sin
8+ cos
8
=
=
2
sin
8+ cos
8
= 2 cos
8
4
.
, 0
4. ,
cos
2 0, sin
4 0 sin
8+ cos
8 0.
- ,
cos
8
4
= cos .
, =
8
4
+2, .
0
4, =
8
4
+ 2
1 =2
9 1 = 2 cos 1 = 2 cos
2
9.
128.4
1 + 2 3
3+ 2 = 4+
3 .
. , - 1 3. = 4
1+2 3
3+ 2 -, = 4+
3 .
gl02.tex
-
U
R S
S
U R
S S
82 2.
, . 1 = 2 .
129.
1+
+ 3+ 2
( 1)(+ 3) = 4 2.
. - 1. -, 4 2 0 2. 1 2.
, =
1+
+ 3.
2 = 1+2
( 1)(+ 3)++3 = 2+2+2
( 1)(+ 3)
2 + 6 = 0, 1 = 2 2 = 3. 0, = 2.
=
1 +
+ 3 = 2,
1 +
+ 3 = 2
1 = 2
+ 3. ,
2
+ 3 0 1. , 1 2, = 1.
= 1 , 1 = 1 .
130.
22 + 3+ 22 3 = .
. =
22 + 3.
22 + 3 = ,
+ 22 3 = .()
() , 0 0. (), , = 2 2 ( )(+ + 1) = 0.
0 0, + + 1 0. - ( )(+ + 1) = 0 = .
gl02.tex
2.7. 83
()
22 + 3 =
2 + 3 = 0. ,
, 1 =
13 1
2.
131.
+ 1
+4( 1) 3
1 + 43
( 1)2= 10.
. , - , . .
+1
+ 4 3
( 1)2 +4
3
( 1)2= 10,
4
14
2
+ 4
3
1
13
1
2
= 0.
, - , , . . 4
= 1 3
1 = 1. - 1 = 1, 1 = 2 2 = 1, 2 = 0.
132.
+ = .
.
+ = .
+ = ,
= ,()
, 0 0., () ,
.
+ = 2 2 (+ )( + 1) = 0.
.
1) + = 0, . . = . 0 0, 1 = 0,
1 = 0 (). ,
1 = 0 = 0.
gl02.tex
-
U
R S
S
U R
S S
84 2.
2) + 1 = . ()
+1 =
+ , 2+2+1 = + 2++1 = 0.
2,3 =
1
43
2.
,
34.
0, 3 =
1
4 3
2 -
.
2 0, . .
1+
4 3
2 0,
4 3 1, 4 4 1.
, = 0, 1 = 0; 1, 2 =
1+
43
2.
2.8.
133. 1+ 2 6 = 5.
. . .
1) 1, 1 0, 2 6 0 1 = + 1,
2 6 = 2+ 6. 3 = 2, . . = 23. 1 - , 1 =23 .
2) 1 3, ( 1) (2 6) = 5 = 0.
-, , 1 3.
3) 3. ( 1) + (2 6) = 5 = 4. 3, 2 = 4 .
,
1 =2
3 2 = 4.
gl02.tex
2.8. 85
134. 2 4 2 9 = 5.
.
2
9+ 5 = 2 4. ()
, ()+ () = ()+ () , () () 0.
(2 9) + 5 = 2 4, () - 5(2 9) 0 2 9. , () , - 3 3.
135. 2 3 10+ 2 7+ 10 = 20 4.
. () = 2 3 10 () = 2 7+ 10, () + () = () + (). , () = () () = (), . .
() 0 () 0. ,
2
3 10 0,
2
7+ 10 0,
( 5)(+ 2) 0,( 5)( 2) 0.
1 = 5 2 2.
136. + 4 2 = 3 1.
. ,
3 1 0 13. , + 4 0. + 4 = + 4
+ 4 2 = 3 1, 4 = 3 1.
, 4 = (31). 1 = 54 2 = 32. 13, - 1 = 54.
gl02.tex
-
U
R S
S
U R
S S
86 2.
137.
2+ + 3 5
6+ + 1 7= 0.
. , :
2+ + 3 5 = 0,
6+ + 1 7 = 0.()
, () + () = () () - () () 0. 2+ + 3 = 5 6+ + 1 = 7 , -, 3 2 1 6.
() , 3 2 , 1 6. , 3 1.
138. 2+ 1+ + + 1+ + 2 = 2 4.
. ,
2
4 0, . . 2 2. .
1) 2, . , ,
( 2) ( 1) (+ 1) (+ 2) = 2 4
2 + 5 4 = 0.
2, -
1 =
5
41
2.
2) 2, ( 2)+ ( 1)+ + (+ 1)+ (+ 2) = 2 4
2
5 4 = 0.
2, 2 =5+
41
2.
gl02.tex
2.8. 87
139. 4 72 + 2+ 2 = 4 1 22 3.
. 4 1 = 22 3 = , -
2
2 = (22 3)2 (4 1)2,
2
2 = 44 122 + 9 162 + 8 1,
2
2 = 44 282 + 8+ 8,
2
2 = 4(4 72 + 2+ 2).
2 2 = 4( ) ( )( + + 4) = 0. , = 0
+ = 4. .
1) = . 41= 223 41=(223). 2 2 1 = 0 2 + 2 2 = 0, 1 = 1+
2,
2 = 1
2, 3 = 1+
3 4 = 1
3.
2) + = 4, 4 1 + 22 3 = 4. , .
140. 3 2 2 = 3
3+ 18 2
3+ 18 2
.
. - 6.
() = 3 2 2.
, - () = (
3+ 18). , = () -
,
() =
5 4, 2;
+ 4, 2. , = () ! .
, - () = (
3+ 18) =
3+ 18.
gl02.tex
-
U
R S
S
U R
S S
88 2.
0. ,
2
3 18 = 0,
1 = 6.
141.
2
9+ 24
62 59+ 149 = 5 .
. ,
2
9+ 24
62 59+ 149 0.
2
9+ 24 62 59+ 149,52 50+ 125 0,
( 5)2 0 = 5., -
= 5. , , , - . = 5 . 0 1 = 5 .
2.9.
142.
7+ +
2+ = 5,
2+ + = 1.
.
7+ =
2+ = , + = 5.
, ,
7+ = 2,
2+ = 2,
, =
2
2
5 =
22 72
5.
gl02.tex
2.9. 89
-
+
2
2
5+
22 72
5= 1, 32 82 + 5 = 5.
, -
+ = 5,
32 82 + 5 = 5,
0 0. + = 5, = 5 -
2 + 5 14 = 0, - 1 = 2. = 5 , 1 = 3. ,
7+ =
2+ = ,
7+ = 9,2+ = 4,
1 = 1 1 = 2.
143.
1+ =18
+
,
1+ 22 =20822
2 + 2.
.
(1+ )(+ )
= 18,
(1+ 22)(2 + 2)
2
2= 208.
,
1
+ 1
(+ ) = 18,
1
2
2+ 1
(2 + 2) = 208,
gl02.tex
-
U
R S
S
U R
S S
90 2.
+1
+ +1
= 18,
2 +1
2+ 2 +
1
2= 208.
()
= +1
= +1
.
2 +1
2= 22, 2 +
1
2= 22 ()
,
+ = 18,
2 + 2 = 212.
1 = 4, 1 = 14 2 = 14, 2 = 4. -
- :
1)
+1
= 4,
+1
= 14,. .
2
4+ 1 = 0,
2
14 + 1 = 0;
2)
+1
= 14,
+1
= 4,. .
2
14+ 1 = 0,
2
4 + 1 = 0.
- , . .
1=2+
3,
1=7+4
3,
2 =2
3,
2 =7+4
3,
3 =2+
3,
3 =74
3,
4 =2
3,
4 =74
3,
5 =7+4
3,
5 =2+
3,
6 =74
3,
6 =2+
3,
7 =7+4
3,
7 =2
3,
8 =74
3,
8 =2
3.
144.
+ + = 12,
= 2+ + + , 0, 0, 0.
gl02.tex
2.9. 91
. (1) = 3 , 0, 0, 0, .
12 = + + 3 3
()2 .
, 8. (1) = 4 -
, .
= 2+ + + 4 4
2 8. 8 8, = 8. ,
-, , 1 = 1 = 1 = 2. , - 1, 1, 1 .
145.
+ = 6,
2 + 2 + 2 = 14,
3
3 + 3 = 36.
. = = .
+ = 6,
2 + 2 + 2 = 14,
(2 + 3)+ 3 = 36.
()
() = 6 , , (6 )2 + 2 + 2 = 14 = 6 2 11.
().
(6 )((6 )2 + 3(6 2 11)) + 3 = 36,
(6 )(3+ 6 22) + 3 = 36,
3
62 + 11 6 = 0.
1 = 1, 2 = 2 3 = 3.
gl02.tex
-
U
R S
S
U R
S S
92 2.
= 6 = 6 2 11, 1 = 5, 1 = 6,2 = 4, 2 = 3 3 = 3, 3 = 2.
.
1)
= 5, = 6,
2
5+ 6 = 0,
1 = 2 2 = 3. 1 = 3 2 = 2.
2)
= 4,
= 3,, 2 4 + 3 = 0,
3 = 1 4 = 3. 3 = 3 4 = 1.
3)
= 3,
= 2, , 2 3 + 2 = 0,
5 = 1, 6 = 2. 5 = 2 6 = 1.,
:
1 = 2, 1 = 3, 1 = 1; 4 = 3, 4 = 1, 4 = 2;
2 = 3, 2 = 2, 2 = 1; 5 = 1, 5 = 2, 5 = 3;
3 = 1, 3 = 3, 3 = 2; 6 = 2, 6 = 1, 6 = 3.
146.
+ = 8,
+ = 9,
+ = 5.
. + + = 11. ,
= 2, = 3 = 6. , 222 = 36 = 6.
= 6, 1 =6
= 1, 1 =6
= 2 1 =6
= 3.
=6, 2 =
6
=1, 2 =
6
=2 2 =
6
=3.
gl02.tex
2.9. 93
147.
+ + = 1,
+ + = 2,
+ + = 5.
. 1,
+ + + 1 = 2,
+ + + 1 = 3,
+ + + 1 = 6
(+ 1)( + 1) = 2,(+ 1)( + 1) = 3,( + 1)( + 1) = 6.
(+ 1)2( + 1)2( + 1)2 = 36 (+ 1)( + 1)( + 1) = 6.
(+ 1)( + 1)( + 1) = 6,
+ 1 =6
( + 1)( + 1)=
6
6= 1,
+ 1 =6
(+ 1)( + 1)=
6
3= 2,
+ 1 =6
(+ 1)( + 1)=
6
2= 3
1 = 0, 1 = 1, 1 = 2. (+1)(+1)(+1) = 6,
2 = 2, 2 = 3, 2 = 4.
148.
+ []+ = 3,9,
+ []+ = 3,5,
+ []+ = 2.
. ,
= []+ , = []+ = []+ .
[]+ + []+ = 3,9,[]+ + []+ = 3,5,[]+ + []+ = 2.
()
gl02.tex
-
U
R S
S
U R
S S
94 2.
(),
[]+ + []+ + []+ = 4,7. -
, (),
[]+ = 0,8,[]+ = 1,2,[]+ = 2,7.
()
[], [], [] 0 1, 0 1, 0 1, () , [] = 1, = 0,7;[] = 2, = 0,8 [] = 0, = 0,2.
= [] + , = [] + = [] + ,
1 = 1,7; 1 = 2,8 1 = 0,2.
149.
2 + 2 + 2 = 14,
+ + = 11,
= 6,
0, 0, 0.
. 2 , (+ + )2 = 36 + + = 6. .1) ++ = 6. + = 6,
=6
. ,
( + ) + = 11.
, . . (6 )+6
= 11,
362+116 = 0 ( 1)(2 5+ 6) = 0.
1 = 1, 2 = 2 3 = 3.
+ = 6 =6
,
, -
+ = 5,
= 6,
+ = 4,
= 3,
+ = 3,
= 2.
gl02.tex
2.9. 95
1 = 2, 1 = 3, 2 = 3, 2 = 2,
3 = 1, 3 = 3, 4 = 3, 4 = 1,
5 = 1, 5 = 2, 6 = 2, 6 = 1.
2) ++ = 6. , , .,
1 = 1, 1 = 2, 1 = 3; 2 = 1, 2 = 3, 2 = 2;
3 = 2, 3 = 1, 3 = 3; 4 = 2, 4 = 3, 4 = 1;
5 = 3, 5 = 1, 5 = 2; 6 = 3, 6 = 2, 6 = 1.
150.
+ 3+
2 + 7+ 11 2,
2 + 8+ 12 0.
.
+ 3 =
2 + 7+ 11 = .
+ 2,
2 + 2 2,()
0 0. () 2, -
,
2
2+ 2 2 2 ( 1)2 + ( 1)2 0.
, = = 1.
+ 3 = ,
+ 3 = 1 1 = 2.
, 1 = 2 , .
151.
+ = 6,
+ = 6,
+ = 6.
gl02.tex
-
U
R S
S
U R
S S
96 2.
.
,
+ = 6;( )( 1) = 0;( )( 1) = 0.
,
+ = 6;
= 0;
= 0,
+ = 6; = 0;
1 = 0,
+ = 6;
1 = 0;
= 0,
+ = 6;
1 = 0;
1 = 0.
:
1 = 3, 1 = 3, 1 = 3;
2 = 2, 2 = 2, 2 = 2;
3 = 1, 3 = 1, 3 = 5;
4 = 1, 4 = 5, 4 = 1
5 = 5, 5 = 1, 5 = 1.
152.
2 + 42 + 5 = 4,
= .
. = ,
2 + 42 + 5 = 4( ),
2 + 42 + 5 4+ 4 = 0,
( 2)2 + (2 + 1)2 = 0. ()
( 2)2 0 (2 + 1)2 0, (), 2 = 0 2 1 = 0. , 1 = 2
1 =
1
2.
1 1 -
, 1 = 1 1 = 2
1
2
=5
2.
, -
1 = 2, 1 =
1
2 1 =
5
2.
gl02.tex
2.9. 97
153.
+
+ 2 =
2,
4 + 19 = 20(+ ).
.
0
+ 2 0, ,
2 0 2. ,
+ 2
(+ 2)+ + 2 = 2,
+ +
2 + 2 = 1,
2 + 2 = 1 (+ ). ()
1 ( + ) 0 + 1. () :
(+ )2 2 = 1 (+ ),
(+ )2 2 = 1 2(+ ) + (+ )2,
2(+ ) 2 = 1,
+ =
2 + 12
.
+ 1 + =
2 + 12
,
2 + 12
1
1 1.
+ =
2 + 12
-
,
4 + 19 = 10(2 + 1), 4 102 + 9 = 0.
1 = 1, 2 = 1,3 = 3 4 =3. , 1 1,
1 =1 2 = 1. -, 1 = 0 2 = 2.
154.
3
+ 32 + 222 + 23 + 3 = 30,
2
+ + + + 2 = 11.
gl02.tex
-
U
R S
S
U R
S S
98 2.
. :
(+ )2 + (+ )
= 30,
(+ ) + + + = 11.
= + = , -
(+ ) = 30,
+ + = 11.()
() = +, =
= 30,
+ = 11,
1 = 5, 1 = 6 2 = 6, 2 = 5.
+ = 5,
= 6
+ = 6,
= 5.
1 = 2, 1 = 3; 2 = 3, 2 = 2, 3 = 1, 3 = 5 4 = 5, 4 = 1.
, - , . .
+ = 2,
= 3,
+ = 3,
= 2,
+ = 1,
= 5,
+ = 5,
= 1.
- . , .
-
: 1 = 1, 1 = 2, 2 = 2, 2 = 1, 3 =5+
21
2,
3 =5
21
2 4 =
5
21
2, 4 =
5+
21
2.
gl02.tex
2.9. 99
155.
102 + 52 2 38 6 + 41 = 0,
32 22 + 5 17 6 + 20 = 0.
. 5, 2.
562 + 212 224 42 + 245 = 0,
82 + 32 32 6 + 35 = 0,
8( 2)2 + 3( 1)2 = 0.
, 1 = 2, 1 = 1. , 1=2, 1=1 .
156.
22 3 + 2 = 3,
2 + 2 22 = 6.
. , = 0 = 0.
2 . - 32 8 + 42 = 0. = 0, 2
= .
32 8 + 4 = 0,
1 = 2 2 =2
3.
.
1) = 2, . . = 2 . 2(2)2 3(2) + 2 = 3 2 = 1. 1 = 2, 1 = 1 2 = 2, 2 = 1.
2) =2
3, . . =
2
3.
2
2
3
2
3
2
3
+2 = 3 2 = 27.
, .
gl02.tex
-
U
R S
S
U R
S S
100 2.
, 1 = 2, 1 = 1 2 = 2, 2 = 1.
157.
3
32 + 2 = 2,
3 32 + 2 = 2.
. -
,
3 32 + 2 0;
3
32 + 2 0
( 1)( 2) 0;
( 1)( 2) 0., ,
0 1 2, 0 1 2.
,
3
3
3(2 2)+ 2( ) = 2 2,
( )(2 + + 2) ( )(3+ 3)++ 2( ) + ( )(+ ) = 0,
( )(2 + + 2 2 2 + 2) = 0
( )
( 1)2 + ( 1)2 +
= 0. ()
, 0 0,
( 1)2 + ( 1)2 + 0.
() , =0 = . = .
3 42 + 2 = 0 (2 4+ 2) = 0.
1 = 0, 2 = 2
2 3 = 2+
2., 1, 2, 3
. = , 1 = 0, 2 = 2
2 3 = 2+
2.
158.
+ + = 0,1
+1
+1
= 0.
gl02.tex
2.9. 101
. , = 0. ,
+ + = 0. ()
+ = . + (), + ( + ) = 0,
2 = 0 = 2 . = 2 = 2 ., ()
2 + 2 + 2 = 0,
, = = = 0. = 0, .
159.
+ + 2 =3
2,
4
5
2 + 2 =5
4.
.
+ =3
2
2, (+ )2 =
3
2
22
2 + 2 + 2 = 42 6 +9
4. ()
4 = 52 10 +25
4. ()
() (),
2
2 + 2 = 2 + 4 4, ( )2 = ( 2)2.
( )2 0 ( 2)2 0, ()2 = ( 2)2 , = 0 2 = 0 1 = 1 1 = 2.
1 + 1 + 4 =3
2 1 =
5
4.
1 =
5
4, 1=
5
4,
1 = 2 ,
, , .
gl02.tex
-
U
R S
S
U R
S S
102 2.
160.
+1
+ 2 = 3,
+1
= 2, 0, 0 0.
. , ,
+1
+ 2 + +1
= 3+ 2,
+1
+ +1
+ 2 2 + 1 = 4
+1
+
+1
+ ( 1)2 = 4. ()
0, 0 0,
(3), +1
2 + 1
2.
( 1)2 0,
+1
+
+1
+ ( 1)2 4.
, ()
, +1
= 2, +1
= 2 (1)2 = 0.
1 = 1, 1 = 1 1 = 1.
161.
2 + 1 = 2
,
2
1 = 2
1 4.
. 2
= 2 + 1 1 4 1. , 14 0, . . 4 1. , , 4 = 1.
=1
4 ,
= 0 = 1.
gl02.tex
2.9. 103
=1
4
,
1 = 0, 1 = 1, 1 =1
4 2 = 0, 2 = 1, 2 =
1
4,
. , , -
, .
162.
3 ( + 1)2 =
,
+ 8 =
9.
. ,
= 3 ( + 1)2 3
9.
9 0, . . 9. ,
= 9.
3 ( + 1)2 = 3,
+ 8 = 0,
= 9,
1 = 8 1 = 1.
163.
2
= 6 2,
32 + 2 = 2.
. , 6 2 0 2 6. , .
, 322+2 = 0 - , -, . . = 26 0 2 6. , 2 6. 2 = 6 = 2 ( ), 32 = 2 ( ).
gl02.tex
-
U
R S
S
U R
S S
104 2.
2 = 6,
= 2,
32 = 2
1 =
6, 1 =
2
3 2 =
6, 2 =
2
3.
164.
2 + 2 +
2 = 8
2,
+
= 4.
.
0 0.
2, - ,
22 + 22 + 2
= 16,
+ + 2
= 16.
,
22 + 22 = + . , , 22+22 = 2+2+2 ()2 = 0.
, = . - 2
= 4 1 = 4. = ,
1 = 4 1 = 4.
165.
= + 2,
+ = 2
+
.
. , - , 0, 0, 0.
:
+ = 2
2 + 2
( 2
+ )+ ( 2
+ ) = 0,
(
)2 + (
)2 = 0. , = = . -
22 = 0,
gl02.tex
2.9. 105
1 = 2 2 = 1. 2 = 1 . = = , 1 = 2, 1 = 2 1 = 2.
166.
(+ 3)3 = 3 2,
42 + 2 = 8,(+ 3)(2 ) = 5+ 16,
0.
. 2+4(1)2 = 4, 2 4, . . 2 2. , - , . .
2
2( 4)+ 16 6 = 0,
, , . . = ( 4)2 + 6 16 0 ( 2) 0.
,
0,
2 2,
( 2) 0, 1 = 0 2 = 2.
1 = 0,
(+ 3)3 = 3 2,
42 = 8,
(+ 3) = 5+ 16,
1 = 4 1 = 2. 2 = 2,
(+ 3)3 = 3 2,
42 + 4 = 8,(+ 3)(4 ) = 5+ 16,
2 = 2 2 = 1. , -
: 1 = 4, 1 = 2, 1 = 0 2 = 2, 2 = 1, 2 = 2.
gl02.tex
-
U
R S
S
U R
S S
106 2.
167.
+
+
= 3,
+
+
= 3,+ + = 3,
0, 0, 0.
. , -
+
+
+
+
+
= 6. ()
0, 0, 0, (3)
,
+
2,
+
2
+
2.
,
+
+
+
+
+
6. (), , (3) , , = = .
3 = 3 1 = 1. , 1 = 1 1 = 1. , , , 1 = 1, 1 = 1 1 = 1 .
168.
+
+ 1 = 1,
+ 1+
= 1.
. , - 0 0.
0, + 1 1
+ 1 1. ,
0,
+
+ 1 1. - ,
= 0
+ 1 = 1. , 1 = 0 1 = 0. , 1 = 0, 1 = 0 - .
169.
2 + 22 + 32 = 4,
4 + 4 + 4 = 1.
gl02.tex
2.9. 107
. (9),
(2 + 22 + 32)2 (12 + 22 + 32)(4 + 4 + 4)= 14(4 + 4 + 4).
4 + 4 + 4 = 1, 2 + 22 + 32
14.
2 + 22 + 32 = 4, . . 2 + 22 + 32
14. .
170.
2 + 2 + 2 = 1,
+ 2 + 2 = 3.
. (9) - ,
(+ 2 + 2)2 (12 + 22 + 22)(2 + 2 + 2) = 9.
, + 2 + 2 3.
+ 2 + 2 = 3, . , , = , = 2 = 2. + 2 + 2 = 3
+2(2)+2(2) = 3, , =1
3. 1 =
1
3
1 = 1 =2
3.
, , , - .
171.
1 + 2 + 3 + 4 = 1,
51 +
52 +
53 +
54 = 1234,
10, 20, 30, 40.
. (1) = 5 , 1 + 2 + 3 + 4 = 1,
51 +
52 +
53 +
54 =
gl02.tex
-
U
R S
S
U R
S S
108 2.
=1234 =1234
1+2+3+4
==11234+12234+12334+12344
15
51+
51+
52+
53+
54
+1
5
51+
52+
52+
53+
54
+
+1
5
51+
52+
53+
53+
54
+1
5
51+
52+
53+
54+
54
=
=51+
52+
53+
54.
, , , 1 = 2 = 3 = 4 .
1 = 2 = 3 = 4 =1
4.
172.
2 = 1,
2 = 1,
2 = 1.
.
= 2 + 1,
= 2 + 1,
= 2 + 1.
()
, 1. () = 2 + 1, - () = () = (()) = ((())). () = 2+1 1, ((())) = () = , . . 2+1 = . - , .
. 172174 - : = () , (( (()) ))
=
() = .
gl02.tex
2.9. 109
173.
= 1,
= 1,
= 1.
. :
=
+ 1,
=
+ 1,
=
+ 1.
()
, 1.
() =
+ 1, 0., = () = (()) = ((())), . . = ((())).
() =
+ 1 , =
()
- = () =
+ 1. 1 =
2 3 + 1 = 0,
1 =3+
5
2 2 =
3
5
2. 1, -
.
, 1 = 1 =3+
5
2.
174.
2 + 3 = 5,
2 + 3 = 5,
2 + 3 = 5.
.
= log5(2
+ 3),
= log5(2
+ 3),
= log5(2
+ 3).()
()= log5(2
+ 3). = (), = (), = () () -
()
= . = ()
! ,
()
=
gl02.tex
-
U
R S
S
U R
S S
110 2.
() = , . . log5(2
+ 3) = 2 + 3 = 5 .
2
5
+
3
5
= 1.
,
2
5
+
3
5
= 1 -
1 = 1. , - , .
, , 1 = 1 = 1 = 1.
175.
= log3
,
2+2 + 8 = 5 4.
. 0 0.,
+ log3 =
+ log3 . ()
() =
+ log3 , () () = ().
= () 0, - () = () = . 2+2 + 8 = 5 4 . = 2 , 352+4 = 0, 1 = 0, 2 = 1 3 = 4.
= 2 0, 1. 2 = 4 1 = 2. = , 1 = 2.
176.
+3
2 + 2= 3,
+ 3
2 + 2= 0.
. , - , = 0 = 0.
gl02.tex
2.9. 111
, ,.
2 +(3 ) (+ 3)
2 + 2= 3 2 1 = 3 .
= 0, =3
2+
1
2
.
(2 + 2) 3 = 0,
3
2+
1
22
+ 2
3
2+
1
2
3 = 0
44 32 1 = 0.
2 = 1 1 = 1, 2 = 1.
=3
2+
1
2
, 1 = 2 2 = 1.
177.
= (1+ ),
+ + + 2 = 0 ?
. =
2
+ 1.
= 1 ( = 1 ), , = 1.
=
2
+ 1 -
- , . .
2(+ 1) + (+ 2)+ 2 = 0. ()
() ( 1), - . , () 0. = (+ 2)2 4(+ 1)(2 ) = 52 4, -
52 4 = 0, =
2
5.
gl02.tex
-
U
R S
S
U R
S S
112 2.
() ( ) , () , = 1.
, , , () , -, . . = 1.
, () , 1.
= 1 () . ,
1 (+ 1) 1 (+ 2)+ 2 = 0,
+ 1 2+ 2 = 0, = 1.
,
, =
2
5 = 1.
178. ,
+ = ,
2 + = ,
2 + 2 + 2 = 4
?
. , - , , , , , , - (0, 0, ).
= = 0 = , = = = 2. , , - () , = = 2 = = 2.
= = 2,
+ = 2,
2 + = 2,
2 + 2 + 2 = 4.
()
gl02.tex
2.10. 113
() :
(0; 0; 2),
5+ 12
;
5 1
2; 1
,
5 1
2;
5+ 12
; 1
,
5+ 12
;
5 1
2; 1
5 1
2;
5+ 12
; 1
.
= = 2,
+ = 2,
2 + = 2,
2 + 2 + 2 = 4.
()
()
1 = 1 = 0 1 = 2, , = = 2.
2.10.
179.
1 2+ 8.
. . 1. -
1. -, ( ), . - 2+8 0, ,
4 . 4 1.
,
1 (2+ 8)2,
42 + 33+ 63 0
51
4
3.
4 1,
4 3.
gl02.tex
-
U
R S
S
U R
S S
114 2.
,
1 2 + 8 3.
2.
1 2(1 )+ 10. ()
1 = , () 22 + 10 22 + 10 0, 0. , 0, 2.
1 = ,
1 2 3. 3.
1 = 2+ 8 ()
, 1 = 3 . () =
1 () = 2+ 8. ( , 1), - ! . 1 = 3 ().
, 3, () (3) () (3). (3) = (3), () () 3.
. .
180.
+ 3+ 4
9
3.
. - 3 9.
+ 3 0 4
9 0, ,
4
9
3
+ 3
3. 0, , 0. , -, , .
181.1
1 42
3.
. 1 42 0 = 0, - 0 1
2.
gl02.tex
2.10. 115
1 +
1 42 , ,
4
1+
1 42
3. ()
12, ()
2, 1. () - .
, - , 0
12, . .
1
2 0 0 1
2.
182.3
2
4 3.
. - , 4. -
3
2
4 + 3, .
2 (4 )
4 + 9(4 )+ 27
4 + 27.
4
8 61
31
. ()
4, () . ,
4 0, () .
, 4.
183.
1
1+
+ 2 4.
. - 2 1.
1
1+
+ 2 + 4.
gl02.tex
-
U
R S
S
U R
S S
116 2.
,
1 1+
+ 2 + 8
1+
+ 2 + 16,
+ 2+ 8
1+
+ 2 + 16 0.
, 2 1.
184.
2+
4 2
2+
2
4 2
2 .
. - 2 2. - = 0 ( , = 0 ), , 0 2.
1+
4 2
2
12
+
1
4 2
212
. ()
0 2,
4 2
2
1. -
() (8).
1+
42
2
12
+
1
42
2
12
1+
42
4
