Matematika Za 1 Razred Gimnazije Banja Luka
April 7, 2017 | Author: VinkaGajic | Category: N/A
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Садржај I
.............................................................................................................................. 5 .......................................................................................................... 5 1.1 .............................................................................................................. 6 1.2 ............................................................................................................. 6 1.3 .............................................................................................................. 8 1.4 ...................................................................................... 10 1.5 .......................................................................................................... 11 2. .......................................................................................................................... 13 2.1 .......................................................................................... 13 2.2 ............................................................................................................ 16 2.3 ................................................................................................................. 17 3. .................................................................................................................... 18 3.1 ............................................................................................................ 18 4. .................................................................................................................. 21 4.1 ............................................................................................................ 21 4.2 ........................................................................................... 24 4.3 ..................................................................................... 25 4.4 ................................................................................................................. 26 II .......................................................................................................................... 28 5. ......................................................................................................... 28 5.1 ................................................................................... 28 5.2 .......................................................................................................... 30 5.3 ..................................................................................................... 34 6. .................................................................................................................. 36 6.1. ....................................................................................................... 36 6.2. ....................................................................................................... 38 6.3. ....................................................................................... 40 6.4. ................................................................................................. 44 7. ....................................................................................................................... 46 7.1. ...................................................................................... 46 7.2. ................................................................................... 48 7.3. ......................................................................................... 49 8. ....................................................................................................................... 51 III .................................................................................................................. 55 λ. ................................................................................................................... 56 λ.1. , ............................................................................. 57 λ.2. .......................................................................................... 59 λ.3. .................................................................................... 60 λ.4. ........................................................................................ 61 λ.5. .................................................................................. 62 10. .................................................................................................................... 62 10.1. .................................................................... 63 10.2. ....................................................................... 64 1.
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10.3. ...................................................................................... 66 10.4. .............................................................................................................. 67 11. .......................................................................................................... 67 11.1. ................................................................................. 68 11.2. .......................................................................... 70 11.3. ....................................................................................... 73 11.4. .................................................................................. 75 11.5. ...................................................................................... 77 IV ................................................................................................................... 80 12. ......................................................................................................... 80 12.1. .................................................................................... 81 12.2. ........................................................................ 83 13. ............................................................................................................. 86 13.1. ............................................................................................... 87 13.2. ..................................................................................... 89 13.3. ........................................................................................................... 92 13.4. .................................................................................................... 93 14. ................................................................................................................ 93 14.1. ........................................................................................... 94 14.2. ...................................................................................................... 97 14.3. ................................................................................................ 98 15. ............................................................................................ 102 15.1. ....................................................................................................... 102 15.2. ............................................................................................................ 105 V ....................................................................................................................... 107 16. Ј ............................................................................................................ 108 16.1. ..................................................................................... 109 16.2. ............................................................................................... 113 16.3. ................................................................................. 118 17. ....................................................................................................... 121 17.1 ................................................................................................... 121 17.2. .................................................................................. 123 17.3. ............................................................................. 126 VI ..................................................................................................................... 130 18. ................................................................................................. 130 18.1. ........................................................................................ 130 18.2. ................................................................................. 133 18.3. Ј ............................................................................................ 137 1λ. ............................................................................................................... 139 1λ.1. ....................................................................................... 139 1λ.2. ........................................................................................ 145 1λ.3. Ј .......................................................................... 148 VII ................................................................................................................ 153 20. .......................................................................... 154 20.1. Ј ..................................................................................... 154 20.2. ..................................................................... 156
3
: 20.3. 21. 21.1. 21.2. 21.3. VIII 22. 22.1. 22.2. 22.3. 23. 23.1. 23.2. 24. 24.1. 24.2.
I
.................................................................................................... 159 .................................................................................... 162 ................................................................................. 162 ....................................................................................... 163 ........................................................................................ 164 ..................................................................................................... 170 .............................................................................................. 170 ................................................ 170 ................................................................... 173 ...................................................................................................... 175 .......................................................................................... 178 .......................................................................................... 179 ....................................................................................... 181 ........................................................................................... 184 ............................................................................... 184 ................................................................................................. 188
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2.3.2 M, N, P μ M N = {1, 2, 3, 4, 5, 6}, N P = {2, 4, 5, 6, 7, 8}, P M = {1, 3, 5, 6, 7, 8}, M N = {5, 6}, N \ P = {7, 8}.
2.3.3 B μ (a) A B A B = A; (b) A B A B = B; (c) A (A B) = A; (d) A (A B) = A. 2.3.4 A B ( ) AB= |A B| = |A| + |B|; (b) |A B| = |A| + |B| - |A B|; (c) |A \ B| = |A| - |A B|. 2.3.5
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B
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: A B = {(a,1), (a,2), (b,1), (b,2), (c,1), (c,2)}. |A B| = |A||B| = 32 = 6 , ђ . 26 = 64 , , 63 ( ). BA 63 . , 126 .
18
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(xA)(!yB) y = f(x), .
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f D(f) = X,
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4.1.2
ZZ f(n) = 2n - 3, 2n – 3
4.1.3 , ,
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n = (m – 3)/2,
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f = x2 ,
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f(x+1) = x - 2 !
ђ
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ο1 ο2 ο3 ο4 ο5 ... . x+1 = t,
f(2) = -1, f(3) = 0, f(4) = 1, f(5) = 2, f(6) = 3, x = t – 1,
f(t) = t – 3, 23
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I
f(x) = x – 3. , ,
t.
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(4.1.7)
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B A,
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X Y {f(x) | x S} A = {a, b, c}
4.2.2
S X,
f(A) = {1, 2}. f(A B) = f(A) f(B),
f A
: y f(AB).
), (b, c ). .
B f y = f(x)
D. .
x AB.
B
y f(A)f(B).
D.
24
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I
f(AB) f(A) f(B). , y f(A) f(B). y = f(a) a A, y = f(b) b B. f(b), a = b. , y f(AB). f(AB) = f(A) f(B). , f(x) = f(y) x y. f(B) = {f(x)} .
.
y f(A)
y f(B). , f(a) = f(A) f(B) f(AB).
f(AB) = f(A) f(B) D. . , x, y D ο {x} B = {y}. AB = , f(AB) = . ђ , f(A) f(AB) f(A) f(B), . ,
4.3
f:AB
g : B C,
.
h : A C, h(x) = (g o f)(x) = g(f(x)). ,
g.
μ h(a) = (g o f)(a) = g(f(a)) = g(1) = , h(b) = (g o f)(b) = g(f(b)) = g(2) = , h(c) = (g o f)(c) = g(f(c)) = g(2) = .
4.3.1
1 2 3 4 1 2 3 4 , g . f 3 1 4 2 2 3 4 1 ) f o g ν ) g o f ν ) f2 = f o f ν ) g2.
)
) μ fog gof (f o g)(1) = f(g(1)) = f(3) = 4 (g o f)(1) = g(f(1)) = g(2) = 1 (f o g)(2) = f(g(2)) = f(1) = 2 (g o f)(2) = g(f(2)) = g(3) = 4 (f o g)(3) = f(g(3)) = f(4) = 1 (g o f)(3) = g(f(3)) = g(4) = 2 (f o g)(4) = f(g(4)) = f(2) = 3 (g o f)(4) = g(f(4)) = g(1) = 3 1 2 3 4 1 2 3 4 1 2 3 4 ν ) g f ; ) f 2 . ) f g 4 2 1 3 1 4 2 3 3 4 1 2 . , . 4.3.2
(x R). ) f o (g o h) = (f o g) o h. )
f(x) = 2x + 1, g(x) = 2x – 3, h(x) = 3x – 1,
,
f o g o h. 25
:
I
: ) [f o (g o h)](x) = f((g o h)(x)) = f(g(h(x))) [(f o g) o h](x) = (f o g)(h(x)) = f(g(h(x))). ) f(g(h(x))) = f(g(3x – 1)) = f(2(3x – 1) – 3) = 2(2(3x – 1) – 3) + 1 = 12x – 9.
4.4
f:RR
,
x2 ) f(x) = 2x2 – x + 1 ; b) f(x) = x3 + 2x ; c) f(x) = |x| – 3x + 2 ; d) f ( x) . x3
4.4.1
4.4.2 a) b)
A = {1, 2, 3}. .
:
.
,f:RR: x 2 x 1 x 1 3x 2 2x 3 2x 1 ; b) f ( ; c) f ( ; a) f ( ) ) ) x 2 x 2 x 1 3x 2 2x 3 3x 2 1 1 1 1 d) 2 f ( x) f ( ) 3x ; e) 3 f ( x) 2 f ( ) ; f) f ( x) 3 f ( ) x2 . x x x x 1 2t 2 1 1 2t 1 5t x 1 1 t f (t ) , , : ) x t 1 2t 1 t 5 4t x 2 3 2 1 t 1 1 5x . d) x = t, 2 f (t ) f ( ) 3t ; f ( x) t 5 4x 1 3 x = 1/t, 2 f ( ) f (t ) . t t 3 1 t 2u v 3t 2v u , u f (t ) , v f ( ) t t 1 1 . u 2t , f ( x) 2 x . t x 4.4.3
4.4.4 a) b) c)
1 2 3 4 1 2 3 4 1 2 3 4 , g h . f 3 3 4 1 2 4 3 1 4 3 2 1 f2,g3 h3; ς e je u v, fo u =h vo g =h . (f o g)-1 = g -1 o f -1.
4.4.5 f : X Y A, B X. a) f(AB) = f(A) f(B); b) f(AB) f(A) f(B); c) f(A\B) f(A) \ f(B),
: .
26
: f : X Y,
4.4.6
SY
μ a) f (AB) = f -1(A) f -1(B); b) f -1(AB) = f -1(A) f -1(B); c) f -1(A\B) = f -1(A) \ f -1(B).
I
f -1(S) = {x X | f(x) S}.
A, B Y.
-1
A B,
4.4.7
A, B |A| = |B|.
a) b) c) d) : d)
,
f : A B.
ђ :
ν , . Z N; , . Q N; -1 1, . R (-1, 1).
f ( x)
x 1 x
R
(-1, 1).
27
:
I
II .
,
(R), , .
μ (I),
(Q).
(Z), (N). .
5. Ц ј
ј
N ο д1, 2, 3, 4, … ж Z = {0, 1, -1, 2, -2, 3, -3, … ж, f : N Z, (
)
0 –
N Z,
(Z ) #
ђ
,
. |N| = |Z| = 0.
kard N = kard Z,
,
5.1 x y
.
„ „
“
5.1.1: y S.
, 5.1.2:
, .
“
.
,
S Z#, S Z#,
S. S 1. n no , n Z#, nS n + 1 S; 2. no m < n , n Z#, mS n S. # Tada {n Z | n no } S.
xS
μ
.
no
28
: 5.1.3
μ 1 2 3 ... n
n
1,
1 (1 1) , . 2
ђ 1.
1 2 3 4
10, !
ђ
4 (4 1) , 2
!
I
n(n 1) . 2 .
,
n = 1.
1, ,
,
.
,
n = 4.
,
(n no) S(n) S(n+1),
. .
μ
n(n 1) 2 (n 1)(n 2) . S(n+1) : 1 2 3 ... n (n 1) 2 S(n+1) S(n) n(n 1) (n+1). ђ (n 1) , 2
S(n) : 1 2 3 ... n
(n 1)(n 2) , 2
. ,
,
.
.
(n 3) 2n + 1 2n .
5.1.4
n = 3, n = 4, n = 5,
,
23 + 1 2 , 24 + 1 24 , 25 + 1 25 , 3
, , ,
7 8. λ 16. 11 32.
μ
(n no = 3) S(n) S(n+1), μ n+1 S(n) : 2n + 1 2 , S(n + 1) : 2(n+1)+ 1 2 . S(n+1) S(n). 2(n+1)+ 1 = 2n + 2 + 1 = (2n + 1) +2 2n + 2 , S(n) 2n +2n , n 3 2 2n = 2n+1 , S(n+1) . , 2n + 1 2n n 3. n
29
:
) ) ) )
I
5.1.5 μ n {1, 2, 3, ...} 2 n 1 n 3 (1) 2 0 (mod 5) , . n(n 1)(2n 1) ; 12 2 2 32 ... n 2 6 6ν 7n 1 n n2 .
5.2
(
)
2
4
5.2.1. 8.
2n. 3 6+λ+λ+λ+3 3n.
,
λ.
75
,
λ.
5n . 7
14
25.
7.
,
7.
,
2,
4.
5 25,
5ν
2
6λλλ3
,
3.
5n
203
λ·1 ο 26, 26 . 57575736
13. ,
, 57575746 5.2.2.
13.
.
4
,
nN λ
λ
3n
11 11. , 1 358 024 67λ (3+8+2+6+λ) – (1+5+0+4+7) = 28 – 17 ο 11 13
8
n
5.
25.
.
25
,
123...λ75 n
7
20 – 2·3 ο 14,
11 11. 351 .
13 ђ
,
4
6 2 4 14 4 6 2 3 12 .
4,
n
35 –
4.
.
30
: ) 23
ν ) 47 10, 11, 12, ... : )
.
23
,
23 : 2 = 12 12 : 2 = 6 6:2=3 3:2=1 1:2=0
3ν ) 56, n = 7; ) 98, n ο 12. A, B, C, … . 2
q
,
x – qy
(5.1.1) q Z.
r
,
S. r py + r , ђ .
7 ο 23 + 1, .
x = qy + r . ,
0r, r |y| = y .
, r – y S, r |y| = -y , . x = p|y| + r, 0 r < |y| , q=-p .
0.
y 0. 0 r < |y| . 7
x = qy + r,
, .
y > 0. je y 1. S = {x – uy | u Z, x – uy 0}.
S
r S .
r=
r < |y|. x – (q+1)y = (x – qy) – y = r – y r , p . x= 0 r < |y| ,
q r. x = qy + r = q’y + r’, 0 r , r’ ξ еy|. r – r’ ο (q – |r – r’е ο еq – q’ееy| . -|y| < r – r’ ξ еy|, |r’ – r| < |y|. 0 |q – q’е ξ 1. q – q’ , , |r’ – r| ο 0. , q – q’ ο 0 r’ – r ο 0, . q = q’ r’ = r . q r .
q’ q’)y .
r’
5.2.4 q , qy + r,
x, y Z x = qy + r,
7, 3 Z
, 1
. ђ
2
).
, ,3
,
10 . 23 110012 . 110012 1 2 4 1 23 0 2 2 0 21 1 2 0 16 8 0 0 1 23
. 1 0 0 1 1
5.2.3. (
2
I
0 r < |y|.
r
|y| = y ,
x , x = qy + r,
0 r ,
ABD,
BC > AC, . a > b.
ABC,
D, ADC
A D BAC =
= ABD, . > > .
10.1.7.
.
> ,
,
a > b. b > a. ( .
.
10.2. 1. 2. 3.
10.2.1. AB CD 2 AB CD 2 AB CD 2
: 1. AD > CD DB > CD AD CD ( ACD = (
.
) 2:1
CD
ABC:
C > 90. C = 90. C < 90.
ABC ACD > , DCB > .
> + , 180, DB CD) DCB > ). = 90.
.
> 180 - ,
> 90. 2.
= + ,
, .
64
:
λ0 (
,
.
I
)
.
λ0,
λ0. .
.
,
10.2.2. . : B
C
D.
1 + 2 = 180. 10.2.3.
) )
, + 2 = 180.
ABCD 360,
,
ABC C
.
)
ђ ,
ς
3645'30''
ђ
6225'30'' .
. B.
: )
,
,
ђ
60. , )
,
. . . 10.2.4. : A B + = 90. ABS,
S 135,
S .
135.
45. 135.
.
ABC , . , ,
/2 /2, , λ0.
B(
)
45, C
65
: 10.2.5. Ј
5
I 12
: ο 13
.
-
.
,Ј ABC. ABC = = 90 x2 5 2 12 2 , .
.
Ј 45 + = 112,38.
10.3. ,
.Ј
.
.
)
,
.
.
10.3.1. . 2.
1.
1.
=1+2 AD
(
,
. ,
OBC = , AD = 2. 3.
3.
AOB = 2 DB,
BOC, . 2. 1 = 2 1 2= 2 2 , = 1 – 2,
BD. 10.3.2. (
)
.
66
:
(
) )
I
. ), ђ
)
(
)
.
.
: )
2. ACD )
180.
.
BCD. tAB ACB
,
.
,
CAD
10.4. 1.
.
.
2.
.
3. λ. 4.
.
ς
5. 6.
ABC
ђ
CD
. 105.
AB CD = BC.
C.
7.
AA’
8. H AHB + C = 180.
11.
,
BB’
ABC
(
ha
(
,
,
hb)
ABC,
)
, ,
ђ
AB
,
.
. ,
,
,
67
: , ) ) ) )
.
ђ
,
I
,
(
)μ ν
ν ν ,
. ( AB1C
1
11.0.1. М Op OM
)
.
AB2C
(
),
,
b.
2
1
Opq Q
M
2.
M. Oq
P MP = MQ,
Opq.
:
, OMP OMQ (P = Q = 90), , OM. MP = MQ, OMP OMQ, POM = QOM.
11.1. 11.1.1. :
Q
ђ
.
,
ADC A1D1C1 a = a 1 h = h1,
ABC
,
.
A1B1C1 μ BC B1C1 , AC A1C1 CD C1 D1 , . a a 1, b b1 h h1. , . D = 90. a b. ,
D = D1,
68
: ADC A1D1C1. , ABC A1B1C1.
(
10.1.7.), . A = A1.
A = A1,
,
(
.
.
AE
= .
ABC
ab
.
(
) ,
. ,
ABC, ==
CD. AD = DB.
E ADE (
h,
)
: C/2,
b
a = a 1 , b = b1
.
2:1 11.1.2.
I
, C-D-E CD = DE. ), BDC ADE. AEC , . AC = AE. AC = BC, . .
CD BDC = BC = ,
11.1.3. . :
AC = BC , . AEC BDC, AE = BD, . , = BT ο 2/3
AD = BE,
,
AC = BC, .
EC = CD = C .
, AEC BDC,
, . AE = BD, DT = ET ο 1/3 ATD = BTE. ATD BTE. ABC
,
AT
69
: 11.1.4. ( ABC A1B1C1 AB A1B1. BCD = B1C1D1, :
(
),
.
I , 17. D D1 AD = A1D1.
1λ83.
)
BCD B1C1D1 AB – DB = A1B1 – D1B1,
DB = D1B1
AD =
A1D1.
11.1.5. 1. ABC A’B’C’ ) b = b’, ha = ha ’, = ’; ) hc = hc’, = ’, = ’; ) a = a’, c = c’, tc = tc’; ) hc = hc’, tc = tc’, c = c’. 2. )
ν )
.
3.
4.
μ
μ
. M
AB
5. ) )
. ν
.
6. (
(
)
.
11.2. ,
,
.
.
AM = BM.
)
μ
,
,
.
11.2.1.
,
. ,
70
:
I
.
,
ABCD
AC. ACB = CAD,
CAB = ACD, ABC CDA.
.
.
11.2.2. , DA . E CDE (
. AB CD ABCD,
BC
ABE
)
.
,
(
, .
).
11.2.3.
.
11.2.4.
.
:
ABC
M N
AC
.
MN . P-M-N PM = MN. ANCP (2) CN PA . ,
11.2.5. ABCD. :
M N DM BN ,
AME = FC.
CNF
EG || MB EGF AME ( ђ AE = EF. AC .
11.2.6.
.
DM BN ( ) MBGE ), DM BN .
P (3)
(2)
, MN AB AB = 2MN.
.
ABNP
MN
BC,
AB CD
.
AC E
F.
AE
2μ1
71
:
:
ABC
,
(
.
ABDC. CM BN (N T V S
,
TV A.
) AS CM
AS
D. CD)
I
AD AT = TV = VD. AT = 2TS,
T
2μ1
AS
11.2.7.
.
.
. ,
. .
11.2.8. ( ABCD AC, E .
, 15. AB > BC. AB1 CD. : DCA AB1
,
AEC ђ
11.2.9. BAM
DC BC
:
B1
,
.
CB1E CAB
CAB CAB1 AB
AC. DCA CAB1, (AE = EC). , CB1 = CB = DA DEA CEB1, ADE CB1E .
ABCD N.
DC
1λ84.) B ADE
M. AM = DM + BN.
P
P-D-M PD = BN, . ADP ABN, APD PAM, APD = ANB = 90 - , PAM = PAD + DAM = + (90 , AMP 2) = 90 - . . AM = MP = DM + DP = MD + BN.
72
:
.
,
.
,
I
.Ј
,
–
, . .
. 11.2.10.1.
) )
ν
2.
ђ
3.
ν
ν
.
.
. AC ABC ACD.
4. C. 5.
ABCD
A ,
.
6. M N + CD = 2MN.
AD
7.
8. CD BD
μ
) )
BC
ABCD,
AB ,
. (
ABCD BC AD,
) M N AC
11.3.
10.3.
(
)
(
,
. ( (
AB AC
.
) )
BD.
( 10.3.2.)
10.3.1.). ђ
.
.
73
: 11.3.1. CD = BC + AD. еAB – CD| = |AD – BC|.
I
ABCD
AB +
. ABCD M, N, P
CD
DA.
DQ.
AB, BC, A, B, C D , . AM = AQ, BN = BM, CN = CP DP = AB + CD = BC + AD. 11.3.2.
Q,
(180).
: ABCD. BD, . + = 180.
(
( A C)
)
11.3.3.
A. B C.
(λ0).
BAC :
, AS SA SB
SA SB SA = SB = SC. S
,
.
A ,
A, B C, A
BC 11.3.4. 1.
65, 105
ς
2.
115. 1μ2μ3.
A, B C ABC.
3. s – c,
s
.
a bc 2
s – a, s – b .
74
: 4.
.
5.
I
.
.
. А
6. D.
ABC
BC
ς
ABD
[30] 7.
O
AB. C D.
A B COD .
11.4. 180 ( 360 (
10.1.2.), ( 10.1.4.). ђ
,
ђ ,
10.1.3.), .
,
. 11.4.1.
(&10.).
,
.
:
ABC
AO sA sB.
ON OP CA AB.
APO /2. /2,
BPO OP = OM.
OP = ON.
OCM,
.
ђ ,
OP = ON = OM. CNO, OM = ON.
OCN
11.4.2.
ANO
BMO
CMO
,
BO OM, BC,
CO
,
OP = ON = OM .
.
75
: :
ABC
S BC
I sa CA.
M N
. ,
MBS MCS . NCS NAS. ,
sb
BS = CS.
CS = AS. AS = BS = CS. ABS sc
S
ђ ,
AB. S
ABC. . 11.4.3. :
, ,
A, B C
. ABC NP, PM MN BC, CA AB. BMC, CNA
APB
, ,
. A, B C NP, PM MN. MNP
(11.1.2.), H,
ABC. ,
.
(&11.1.). ђ ,
,
11.4.4. 1. 2.
2:1 (
11.2.4.)
2μ1
,
. (
11.2.6) , .
.
. . .
.
76
:
. 1. T. ,
I
,
AN BM
ABC
MN .
AB ABT
ђ
.
PQ, AB
MN || PQ MN = PQ. MNT PQT MNT PQT, AT Q BT, A B. 2. AN BM ABC
PT = TN QT = TM.
T. CP
BM . , BS = 2SM,
,
BS
2 BM . 3 2 BT BM . 3
.
,
S
, BT = 2TM BS = BT, S T .
,
CP
P 2μ1
11.4.5. 1.
ђ
. .
2. 3.
.
4. BC
D
[
DE BC.
AB F
CD BED GF || BD, DE GF.
ABC. DE. . G GF CD. , CF
BE F ABE
11.5. 10.1.6.
7.).
E AE CF. GE
CF DG. AE || DG
CDG, ђ , DG AE CF.] (
77
:
I ,
. 11.5.1. (
)
,
:
BC . ACD BAC = DAC = ADC = . < BAD, BDA < BAD. 10.1.7.) a ’
a = a’, b = b’
c > c’. A= CB’ AB
DB = DB’. AD + DB’ ρ AB’,
c > c’
μ = ’, < ’,
> ’.
> ’. 1)
.
11.5.4.
ђ
1.
. 2)
. .
2.
.
3. [
(
,
: C = C’, . s BCB’ D. DCB DCB’ ( ), , AB > AB’.
11.5.3. :
“
O
ABC BO AO, BOC: /2 < /2
. CO < BO.
, ABO: /2 /2 , CO.]
78
:
I .
4. 5.
ђ
. .
6. [ ,
7.
μ
.
10. ABC ) ACB < AMB; ) AM + MB < AC + CB.
M.
:
11. ABC s < AM + BM + CM < a + b + c,
M.
:
c
. .
ν
a, b
tc < a + c/2] .
¾
8. 9. ) )
tc < b + c/2
, s
.
79
:
I
IV . , ,
,
(
(&13), ђ ,
(
,
ῖ "–
μ" υ .
ђ
) ,
,
12. К
,
ђ
.
). (&14.).
,
ј 2
,
.
.
. : 1μ 2: 3μ 4μ
ν ν
5μ 6μ
ν ν
ν
. :
( - - ) ( - - )
ν
ν
ν
( - - ) ( - - )
ђ ν . ,
ђ , . 2
,
ђ ,
(42λ-384. . . . .)μ „
,
.
,
,
,
,
μ
.“
80
:
,
ђ
.
. .
,
.
.
.
,
a, b (
.
)
ha ,
ABC.
A
.
.
,
12.1.1. .
B C BC
ν
.
μ
12.1.
ν
ν
,
.
μ
,
,
I
a, b,
C
ha . . B
,
k(B,a)
a
Bx
BC.
.
BC : Bx; C = Bx k(B,a). k1(C,b). n BC C; P, Q n k2(C, ha ). p q P Q BC. А: p k1(C,b), q k1(C,b), AiBC (i = 1, 2, 3, 4 ),
.
.
.
81
: . < ha , b = ha ,
,
I
2,
4
,
b .
b > ha .
12.1.2.
.
. ABC,
a + c.
. AB D.
B-D, AD = c + a.
B A-
BDC s
DC
B. . ADC,
, .
. B = AD s. .
s(D,C) ABC.
DC ADC, BDC D = /2. , AD = AB + BD = c + a, AB + BC = a + c.
s
, BD = BC = a, ADC ( B,
.
ADC, DC.
, a + c, /2 ).
. < 90
s
90
AD .
. 12.1.3. 1. ha , hb
.
2. hb, tb 3.
4.
.
ђ
.
,
.
. ђ
a, b
tc. .
ta , tb , tc.
82
: 5.
,
.
6.
.
AB = c, hc
AC = b, BC = c
7. )
I
tc
ABC.
ha
ABC. μ
ν )
μ ) b – c, , hb (b > c); ) a + c, b, .
8. ) c – a, b, (c > a); ) b + c, hb, ; 9.
,
– .
a, b – c
. 10.
.
.
12.2. ,
12.2.1. ,
3
.А
:
k
. M, N P.
CE = h N,
=s
.
P, CN AN = NB. AB К h AB ј μ
ON. G
M P .
G O, G
C = h k. ON. :
NO,
3
ј :
, M, CG = t
ABC CF
ACB , ђ
ANB.
N M
.
G = CP ON . ABC. NO ,
k
M
, 1λ60.
83
: 12.2.2.
(
,
).
.
.А
,
: AB, BC, CD DA A. = 180 – . PQ
k
I
P, Q, R S
, B
,
.
p
BD PQ
k K. К
ј μ PQ
l
SR PQ.
.
. A C
RS DR, DS, BP
μ
BQ.
ABC.
BK
K
RS L
k
.
, DL
Q
S R
. S P, .
.
ν
12.2.3. .А
PBK = CBK, . ADC. , BD ABCD
)
ј μ
( :
k1
,
ђ
.
= K L )
k2
.
K L. D = k p, B = l p. DS BP DR BQ.
p
PK = QK (
.
PQ
S ,
. 4
O1
O2,
. d.
4
1λ5λ.
84
: ,
t1 , AB = d
t2
CD =
d. O2P k3
I
.
O2Q O2 O2P = O2Q.
К
ј :
O2.
d
.
,
k2. k3
k1
ј :
.
.
(d)
O2
k3.
,
,
(O2)
,
.
,
k1
k3.
,
.
ς
. 12.2.4. .
. AB = a.
ABC B
BC = a/2, .
C
a/2, AC
D
E. ,
DE = a. AD . AD = x, M AB M x : (a – x). .
a a a x , 2 2 2
2
2
AB + BC = AC
2
a : x x : (a x) .
2
2
AM =
x2 a (a x) ,
12.2.5. 1.
.
85
:
I .
2.
.
3. 4. )
μ
ν )
5. )
ν )
6. )
ν )
7. )
. μ
.
μ
. μ
ν ) .
8.
9.
k
. .
d ( d < k).
k
d. ,
10.
13.
.
.
ј m
(A,B )(! m R m 0) d(A,B) = m.
: ,
A B
ђ
m m.
m
.
A, B
d(A,B) A B.
μ 1. d(A,A) = 0; 2. d(A,B) = d(B,A); 3. d(A,B) d(A,C) + d(C,B). .
A, B C .
(3) .
μ
(
)
,
.
86
:
I
(f : ) : d(A,B) = d(f(A),f(B)), . d(f(A),f(B))
A B d(A,B)
B,
f(A) ,
,
f(B)
f
f(A) = A. . 1)
ν
2)
A .
A
μ
, ,
ν
ν
3) 4) 5) 6) 7)
ν .
ν
.
13.1. A A’
p
p
А
AA’
Ss : A’
, .
s
13.1.1. :
. AB,
.
AB
A’B’
s, Ss : AB A’B’. , .
A’B’ AB’, AA’
BSS2 B’SS2 AB = A’B’.
S1 SA = SA’
13.1.2.
5
.
S2. SB = SB’.
,
5
d(A,B) = d(A’,B’).
BB’ s ASS1 A’SS1
S, SAB SA’B’
.
, 1λ68.
87
: .А
μ
,
M, N P O
k ABC. BN CP –
I
AM, S,
.
CAM = BAM CM = BM OM BC. CBN = ABN CN = AN ON AC. ACP = BCP AP = BP OP AB.
BAC = 180 – NOP, . , BOC = 2BAC.
BAC К
ј μ OM, ON OP. MOC = . B ON. μ
NOP. OM. A
C
AM, BN CP C AM
. , B CM = BM, . CAM = BAM .
BC. CAB. ј μ Qq
OC C OM =
OM
M, N P .
,
13.1.3.
Opq
.
PQ :
1
Pp
.
2
p
.
q, p
1 2
P
Q.
q
PQ .
TP + PQ + QT = T1P + PQ + QT2 = T1T2. , . TP’ + P’Q’ + Q’T = T1P’ + P’Q’ + Q’T2 T1PQT2 T1P’Q’T2. , .
. , .
.
ђ
,
.
μ
88
:
I
13.1.4. 1. )
ν )
μ
ν )
2.
.
AB :
)
s; )
A B
s.
A B
s.
: a) a, hb, b + c; ) c, ha , a + c.
3. 4.
μ
a,
) ) )
d
b p
5.
c; k; k1
k2.
A B APB
6.
ABC D
P p
p. .
BC
AC = BC E.
AB = c. ABE
AC
3c.
ABC.
13.2.
13.2.1. S .
,A B
SS : A B
μ
AB, A
B 13.2.2.
S. F1
F2
S, 1 2
S,
.
.
)
13.2.3.
Sp
p
p'
p;
) m||p (
M ђ m||p').
QQ'
Q
Q'
p
p'
89
: : ) B'
А'
I Аp
Bp
S,
S, p' ђ А' B', . AS = SA', BS = SB' ASB = A'SB' SAB SA'B' , = '. ђ , , p||p'. c S p p' A'SC'
,
C C', CS = SC',
C p' n p'
C'
) np N, SAN SA'N', NS = SN', . Q Q' q QQ' S, M m. Q rp P M PM = MP'. NN'PP' NP SM || NP, . m || p. c,
p
S. N'. S
p p
Q' ,
13.2.4. )
)
ASC S.
.
. M
p'
p',
ђ r p' ђ NSMP
.
M. P'. SM = .
.
)
. 13.2.5. ( :
.
).
AB A'B' S . , AS = SA’ SB = SB’ , ASB = A’SB’ SAB SA’B’ AB = A’B’ = ’ AB || A’B’, . .
SAB SA’B’ S.
S.
,
AB A’B’
S. , AB A’B’ AA’ BB’ , = ’, = ’,
90
:
(
)
,
.
13.2.6.
.
13.2.7. : a’, b’ ’ a’b’ ο T’ ’
I
.
,
,
a, b
S. ђ a || a’, b || b’. М M-S-M’ M’ S. STM ST’M’ MS = SM’, . ' S. 13.2.8. .
.
:
A, B
A’, B’ ASB A’SB’ .
AB A’B’.
S
’.
,
13.2.9. 1. ) ) ) 2. )
(1.
(1. (1.
ν 2.
ν 3.
, .
);
). μ
ν )
,
5)
ν 3.
ν )
3)
4)
ν 2. ν 2.
μ
)ν
. ђ
.
,
. .
91
: 6)
I
ABCD. A1, B1, C1 DD1.
CC1
13.3. MOM’ ο
D1
B, C, D A1B1C1D1
M OM = OM’.
( = aOb)
.
A ђ
S , .
(a, b).
13.3.1.
, : M M’.
(Oa
AOA’ = BOB’ =
M'
ђ
.
,
13.3.2.
OA = OA’, OB = OB’. AB = A’B’. ) A B . ,
.
= 120.
2. 3.
ђ
30.
5.
.
: ) = 30; ) = 45; ) = 60.
1.
4.
O μ
AOB A’OB’ (
,
AA1, BB1,
S :
Ob)
AB ,
:
.
.
. .
92
: 6.
I
ђ
,
.
13.4.
Sv :
,
M’
v
M
MM ' Sv M M ' .
.
13.4.1.
. v
:
Sv : AB A' B' ,
.
A B
ABB’A’, , .
.
13.4.2.
: )
1. 2.
aOb,
ђ
p
ν )
.
d. d.
q || p
k1(C1, r 1) k2(C2, r 2), r 1 r 2 A k1, B k2 AB || C1C2.
3.
4.
.
5.
AB, ABCD
p
d.
ђ
AB = d
. Ck Dp
k. .
6.
. [
. . .]
14. .
(
)
А', B'
' ',
А, B АB : А'B' .
93
: '
.
I
Ф ~ Ф'.
' ,
,
.
.
14.1. (
ђ
(
.
, . . Intercept theorem),
624. – 546. . . . .)
, „
„ , 14.1.1.
“.
,
.
,
.
. “.
ABC
DE (D AB, E AC).
.
μ
DE || BC
94
:
I
AD / BD = k, AE / CE = k. : DE , DG BE
CD.
CD.
1 AD EF 2 1 P(ADE) = AE DG 2
P(ADE) =
.
,
k ,
ABC
BC. AD / BD = AD / CE. E D EF AB AC . , μ EF AB DG AC, BE,
1 BD EF , 2 1 P(CED) = CE DG . 2
P(BDE) =
P(ADE) / P(BDE) = AD / BD P(ADE) / P(CED) = AE / CE.
ђ
P(BDE) = P(CED) BC, . AD / BD = AE / CE
DE B C. .
ADE = ADC AED = ACB. , AD / BD = AE / CE BD / AD = CE / AE, BD / AD + 1 = CE / AE + 1, (BD + AD)/AD = (CE + AE)/AE, . AB / AD = AC / AE, AD / AB = AE / AC. 14.1.2. :
,
. ABC
l
AB AC AD / DB = AL / LC, DL || BC.
D
l
L
. BC
p, DP. (14.1.1.) AD / DB = AP / PC. AP / PC = AL / LC.
AD / DB = AL / LC
. 14.1.3.
.
,
95
:
: AE BE , EC ED ABCD
I
ABCD . .
AB. CE CF , EA FB DE CF CE DE , . EF EB FB EA EB ABCD .
E
EF (F BC) (14.1.1.)
CD.
AB || DC
14.1.4.
.
:
p ,
c
q
a, b A, B C,
. p || q a, b c
(p) AGED ђ , GHFE
ACH je BG || CH, AB DE . BC EF
1.
14.1.5. M, N P
AB, BC :
) P(AMP) = P(BNM); ) P(MNP) =
CA
4. ) ab; )
F,
A, G H. , AG = DE. , GH = EF. AB AG , BC GH
ABC,
1 P (ABC ) . 4
2.
3. ) 5, ) 7
D, E AB DE . BC EF А.
. ; )
a : b.
a b a a b a b ; ) ; ) ; ) ; ђ) . ab b a b a b a b a
b
2
μ
2
96
: 5.
I μ
DC || AB
6.
ABC
14.2.
h: А OA' k OA, k R\{0}.
A’ = h(A). k
.
.
, k0
. h
μ 1. AB
AB’ (
2. 14.2.1. : OC),
C’A’.
k = 2. k(C,r). .
Ф’(C’,2r),
,
)
, AB = kAB’; .
.
k’ k’
k OC : OC’ = CA : C’T’,
OC’ = 2OC (C’
OC' 2 OC
CA ||
k.
97
: (k > 0) (k < 0)
h
,
.
I
14.2.2. ABC
A’B’C’.
:
1 , k , ABC. 2
k = – 0,5
A’, B’
C’
A’–O–A OA = 2OA’, B’–O–B OB = 2OB’ C’–O–C OC = 2OC’. A’B’C’ ABC 1 1 OA' OA, OB' OB 2 2
k = – 0,5 14.2.3. 1.
(
3 2 μ ) k ; ) k . 2 3
2.
7 5 : ) k ; ) k . 3 3
30
,
1 OC' OC . 2
)
ABC
15, 9
3. 45. 4.
.
5.
.
6.
14.3.
, .
ς 14.3.1.
(A, B ) A’B’ = kAB,
f: A’ = f(A), B’ = f(B), k > 0.
98
: (
μ
~
')
I A, B
' f ( A) f ( B) k AB .
k (
)
. ,
.
14.3.2.
.
, 1. (
. :
)
.
2. ( ) 3. (
)
4. (
)
,
. . . .
3.
,
(
)
, BC : B’C’ = AC : A’C’ A = A’ ADC A’B’C’ , ABC ~ A’B’C’
BC < AC, ,
.
. ђ B = B’.
14.3.3. :
ABC
tc’ = CD’
99
: A’B’C
tc’,
I
. ,
.
,
. AB || A’B’. К ј μ ABC AD ( C D ( ) tc’ = CD’. AB A’ AC B’ BC. 2. ( ) ABC ~ A’B’C. ,
14.3.4.
A’B’C. .
,
.
:
AB), D’
,
ABC
, ђ
p
AA’ CC’ AD CE. .
,
AD H. 2.( ) A’OT ~ AHT OT : TH = A’T : TA = 1 : 2. , OT : TH = 1 : 2. p
p H .
H’ ђ
p
OT : TH’ = 1 : 2,
p
ђ
14.3.5. 1. 2. (
. .
H
.
,
2. ( ).
,
AD CE H’ ђ TH = TH’, . H ђ
CD
. ABC ABC. μ
)
100
: 3.
I
. :
ABC a
b.
C h CD AD = p DB = q AB = c.
1) ACD CBD ( 2. ) p:h=h:q . 2) CBD ~ ABC a : q = c : a, ACD ~ ABC b : p = c : b, . 3) , a2 = cq b2 = cp a 2 + b2 = cq + cp = c(q + p) = c2. , a 2 + b2 = c2, ABC C. B’ CB’ = CB = a ACB’ = 90. a 2 + b2 = d2, d = AB’. ђ , 2 2 2 a +b =c d = c. ( ) ABC AB’C, ACB = ACB’ = 90. 14.3.6.
a
:
( AD = a, DB = b
AB.
14.3.7. ђ : CDP P. A C ,
k ,
P
b.
10.3.2.). A–D–B (a + b = AB) .
ђ CD
.
OA = OB. C ABC a
CD b.
.
ABP
k
BD ADP ~ CBP, AP : DP = CP : BP. APBP = CPDP .
14.3.8. p2 = PAPB = PCPD.
P
k
101
: 14.3.9.
6, 7
1. ) ) 2.
I
λ.
,
ς
.
3. )
ABC C; )
4.
A B ) C.
(
ABC
.
AB, 5.
.
6. ) 15 ; ) 10 ; ) 7. )
13?
μ
7; )
a, b, c
d.
a b ; ) a bc ; ) 2
2
5.
a b ; ) ab cd . 2
13, 14
8. 9.
k
10.
15.
ј
15.
. .
P
P
μ
2
. ,
k,
ђ
.
ј
15.1. AB, .
1. AB:AF = AF:FB. :
,
F
AB = a,
102
:
I
k1(O, a/2) BC = a, BCAB, A-D-O-E, D, E k1, k2(A, AD), AD = x = AF, F AB. F AB . . : ABO, ABO = 90 AB BAO = 30. k1(O, ). 2 A-D-O-E, D, E k1. k2(A, AD). k2 AB = {F}. F AB . : A k1 ADAE = AB2. x2 a 2 ax , . x2 a (a x) , x2 ax a 2 , x (x a) a 2 , ( ђ , 12.2.4.). a : x x : (a x) , 6
2.
O
ABC.
1,
. :
D, E
2,
3
BC, CA, AB 1,
F
2,
3.
BC, AC AB, . O2AC = 3, CAO = 2, OAB = 1, OBC = 1. 1,
2
3
ADC ~ BEC,
ACB. 2 1, O1C O2C, 3 OO2 AC 2 AC.
2. ,
1
BC . 1
3,
1
a 7
:
7
1,
3,
2,
A, B, C
2
3,
2.
3.
6
K
3
AB.
b
C
.
,
. a'
a
C
λ0.
, 1λ72. , 1λ71.
103
: b, B = b a' A a.
a'
I
a . C
(
ђ b 4. (
ABC
b a' , C 90). C b a'. ba
a ba b a, b || a' ,
ђ
A B, λ0 . ba
C
a. (
) :
2μ1
) 2μ1 .
AA1
,
.. ,
ђ
.
BB1
А А1. A1B1 . T BB1 BT : B1T = 2 : 1, CT : C1T = 2 : 1, –1/2
AB CC1
CC1
, T .
5. . :
, AE BE CE
DB = q,
p
q pq . 2 CD pq
pq . 2
pq ,
,
, ABC CD
AB
AE CDE
CD < CE .
pq pq , 2
AD = p
.
p = q.
104
: 6.
I .
A, B C C C.
AB
8
A B
ACt = ’, BCt’ ο ’, CBA = , AC = x, BC = y,
:
P ђ
Q. d
t
.
D AB
A B C. ,
= ’
= ’. ACD ~ CQB d : q = x : y. , BDC ~ CPA p : d = x : y. d : q = p : d, . d2 = pq,
.
15.2. 1. .) [
.
ς(
μ
c, tc
tb tc,
AC c 5 t c c .] 4 4
tc. 2.
C
k
A B . [
8 9
μ ђ A B
D
, 1λ70. , 1λ71.
AD : BD = m : n. , m=n
.
AC
BC
m . n 9 ?
D .]
.
105
: 10
3.
I
M, N P , .
ABC
M, N P. 4.
ABC ABC 11
ABC. 5.
. a 2 + b2 = c2, a b c 2 .
a, b, c .
c
ђ
,
a, b .
6. .
12
( .
.]
[
10 11 12
Ј
)
, 1λ6λ. , 1λ71. , 1λ73.
.
.
106
:
I
V .
, ,
,
. .
„
“
,
.
,
,
.
, (ax = b), bx + c ο 0) ђ (x2 + y2 = z2).
(ax2 +
(Ἥ ω ὁ Ἀλ α ύ , 10 – 70. .) (Δι φα Ἀ α , 210 – 2λ0. .)
ὁ
. „ -џ
.
)“
(
μ
(al-Khwarizmi, 780 – 840. .) (Omar Khayyam, 1048 – 1131.) ( ) (Abū KāmТХ, 850 – λ30.). x + y + z = 10, x2 + y2 = z2
xz = y2.
(Leonardo Fibonacci, 1170 – 1250.) x3 + 2x2 + cx = d, (Scipione del Ferro, 1465 – 1526.), (Niccolò Fontana Tartaglia, 1500 – 1557.) (Gerolamo Cardano, 1501 – 1576.). (Lodovico Ferrari, 1522 – 1565.) , 1λ. (Niels Henrik Abel, 1802 – 1829.) (Evariste Galois, 1811 – 1832.) .
107
: ,
16.
(René Descartes, 1596 – 1650.)
,
Њ
I .
(Carl Friedrich Gauss, 1777 – 1855.),
. Cauchy, 1789 – 1857.), (Sophus Lie, 1842 – 1899.).
,
(Arthur Cayley, 1821 – 1895.)
.
(Augustin
(William Rowan Hamilton, 1805 – , . (Hermann Grassmann, 1809 – 1877.) (Josiah Willard Gibbs, 1839 – 1903.) . ,
1865.) .
(George Boole, 1815 – 1864.) .
16. Ј Ј
. ,
, 5x + 7 = 22, x
. (
3. .
= 3) . ђ
(
ђ
. , |x + 2| + |x – 3| = 5 (
,
) x [-2, 3], ,
,
.
) . x + 1 = x + 2. ,
x
2x 6 2, x3
„ο“ .
3.
„“
,
.
.
,
108
:
x– 1 =0
.
x2 – 1 = 0 {1}, .
,
{1, -1}.
(
(
)
.
I
) μ 5x + 7 = 22, / -7 5x + 7 – 7 = 22 – 7, 5x = 15, /:5 5 x 15 , 5 5 x = 3.
,
1510. – 1558.).
14x + 15 = 71,
16.1.1. :
. 8x – 5 = 2x + 7
.
, je 82 – 5 = 22 + 7 ? 16 – 5 = 4 + 7 11 = 11 T
8x – 5 = 2x + 7 , /+5 8x = 2x + 12 , /-2x 6x = 12 , /:6 x = 2.
16.1.2. Ј (F)
(C) 212F.
,
1557.
(Robert Recorde,
16.1.
.
.
F = 1,8C + 32.
: 212 = 1,8C + 32 212 – 32 = 18C 18C = 180 C = 100. 30%
16.1.3. :
21). λ
.
,
. x
,
21
ς + 21.
30 ( x 21) , . 10 x 3x 63 , 100
. x = 30%(x + ο λ.
109
: 16.1.4. ).
I
&6.2. (
)
&6.3. (
16.1.5.
: 2x + 5y = 11, y = 3x + 2.
. 1 . x 17
:
( ) 2x + 5(3x + 2) = 11, 37 1 , y 3 2 , . y . 17 17
.
16.1.6.
( 2
) 2, 10x + 6y = 8, 21x – 6y = 9. 17 . 31x = 17, x 31
.
5
13 . 31
16.1.7.
(
)
7
17 13 2 3, 31 31
μ
a(x + y) + b(x – y) = a, a(x – y) – b(x + y) = b.
μ (a + b)x + (a – b)y = a
– (a – b)y = b. ( a b) y
3.
17 13 3 4 31 31
:Ј a b . 2
( a = -b 0, b)
)
(
y
1 37 5 11 , 17 17
(
μ 5x + 3y = 4, 7x – 2y = 3.
:
)
(a + b)x a b ( a b) x 2
,
0x = 0, 0y = 0, .
a= b=0 a = b 0,
). xR x
y
1 . 2
x
1 2
y R. (a
1 1 , y . 2 2
110
:
I
,
k, n
,
.
y = kx + n,
,
y = 2x – 3 x -2, -1, 0, 1, 2, ... y = -7, -5, -3, -1, +1, ... . ђ ( , ) (-2, -7), (-1, -5), (0, -3), (1, -1), (2, 1), ..., .
,
. ,
k ρ 0, X . Y .
kξ0
. k
,
.
„ .
16.1.8.
,
“ kο 0
.
n μ
5x + 3y = 4, 7x – 2y = 3. : 7 5 4 3 y x , y x . 2 3 3 2
μ
А(17/31, 13/31) .
(1)
, . (2)
њ μ ? 16.1.9. 1. ) 3 – 2(4 – 3(5 – 4(6 – x))) = 20x + 127; 5 x 2 x 2 x 3x ) 3 5 6 x 43 ; 5 3 4 2 ) 1 2 3 4 7 1 x x x x x ; 2 3 4 5 12 5 2. x, y ) a (ax 1) 2(2 x 1) ;
: ) x – 5(x – 4(x – 3(2x – 1))) = 4x – 140; 4 x 3x 5 x 2 x ) 5 3 4 x 37 ; 4 2 5 3 ђ) 1 12 34 5 3 5 1 x x . 3 23 45 6 7 16 a, b, p, ) b y 4 b( y 4) ;
q
μ
2
111
: )
ay ay 2ay ; 2 b a a b a b2
)
3. ) |2x – 1| + |x + 2| = -3x – 1; ) |x + 1| + |x| + |x – 1| = x + 1; 4.
145
.
1λλ
ς 5
. 7
6. 35 3 7. )
)
)
3
. 30
3
џ
ς
5
џ
,
)
1 2 y 3x 2 y 2 x 3 y , 2 3 9 x 23 y 621 ;
)
5x 3 y 3 4 x 7 y 1 4, 6 9 7 x 3 y 5 5x 2 y 1 3; 12 14
ђ)
ς
μ
x
3 5x 3 y 1 3x 5 y , 3 5 15x 11y 195 ;
3x 2 y 2 5 x 6 y 2 2; 10 12 6x 2 y 2 4x 3 y 2 3; 15 21
7x + 4y = 27, 6y + 7z = 9, 4z + 6x = 42. μ
5x 3 y 1 2x 7 y 3 2 1 , 3 x + 5y + 1 = -6y;
9. ) y = 2x – 3 y = -x + 3; ) 3x – 5y = -1 2x + 3y = 12; 10.
џ
.
5x + 3y = 1, 2y + 5z = 14, 3z + 2x = 16; 8. )
2 x p x q 3 px ( p q) 2 . q p pq
: ) |3x + 2| – |2x – 1| = 5x + 1; ) |x – 2| – |x| + |x + 2| = x + 4.
.
5.
I
)
4x 2 y 1 5x 3 y 1 3 1, 7 10x – 7y + 2 = 21 – 9x;
: ) y = 0,5x – 1 y = -1,5x + 3; ) 5x + 2y = 3 -7x + 3y = 19. m 112
: )
I
) -3x + (m+ 1)y = 3, (m+ 1)x – 3y = 3; ς ς
) ) 11. )
-2x + (m -1)y = 5, (m + 1)x – 4y = 5;
μ
)
|x + 1| + |y – 2| = 3, 2|x + 1| = 3y – 1;
12. ) y = |2x – 1| + 3; ) y = |3x – 1| + 2x – 5;
:
|x + y| + |x – y| = 5, 3|x – y| = 2x + 1.
) y = |2x + 1| – 3; ) y = -0,5|x + 2| + 1,5(x – 1).
13. )
)
μ
16.2. ,
(
.
μ a11x1 a12 x2 b1 a 21x1 a 22 x2 b2 , )
x1 , 1
x2 2
.
,
. x1 1 , x2 2 ,
a11 a12 a11a 22 a12 a 21 , a 21 a 22
1
2
b1 b2 a11
a12 b1a 22 a12b2 , a 22 b1
a 21 b2
a11b2 b1 a 21 .
113
:
I 1
. 2
x1.
.
x2. ( (
) ).
16.2.1.
a11x1 a12 x2 b1 a 21x1 a 22 x2 b2 , .
:
a 22 a 12 . x2 (a11a 22 a 21a12 ) x1 b1a 22 b2 a12 , . . , x1 1 a 21 a 11 . x1 , (a11a 22 a 21a12 ) x2 b1a 21 b2 a11 , . x2 2 . . ,
.
16.2.2.
m m
μ
x – my = m, x + ny = n.
:
x
,
mn (mn) 2mn
n n x x, y y,
1 m 1
y
n
n m,
1 m 1 n
n m.
,
μ
(n m) x 2mn , (n m) y n m . (n + m ο 0), , . n = -m 0,
, . n = m = 0, x = 0, y – .
. ,
,
114
:
I
(n + m 0)
x
nm 2mn , y nm nm
m n. .
,
a11x1 a12 x2 a13 x3 b1 , a 21x1 a 22 x2 a 23 x3 b2 , a 31x1 a 32 x2 a 33 x3 b3 , x1 , x2 x3 , ( x1 1 , x2 2 , x3 3 , . , μ a 11 a 12 a 13
.
,
,
, .
b1 1 b2
a 12 a 22
b3
a 32
a 21 a 22 a 31 a 32 ,
a 11 b1 a 13 a 23 , 2 a 21 b2 a 31 b3 a 33
μ
)
, 1 , 2
3
a 23 , a 33 ,
a 11 a 12 a 13 a 23 , 3 a 21 a 22 a 31 a 32 a 33
,
b1 b2 . b3 .
.
,
a11
a 12
a 21 a 22 a 31 a 32
a 13 a11
a12
a 23 a 21 a 22 = a 33 a 31 a 32
= a11a 22a 33 a12a 23a 31 a13a 21a 32 – ( a 31a 22a13 a 32a 23a11 a 33a 21a12 ). 16.2.3.
:
2 x 3 y z 9 5 x y 2 z 12 x 2 y 3z 1. ,
:
115
:
2 3
I
1 2 3
5 1 2 5 1 = -6 + 6 – 10 – (1 + 8 + 45) = -64. 1 2 3 1 2 μ 9 3 1 9 3
x 12 1
1 2 12 2 3 1
2 9
y 5 12 1 1
1 = 27 + 6 – 24 – (1 – 36 + 108) = -64, 2
1 2 9
2 5 12 = -72 + 18 + 5 – (12 – 4 + 135) = -192, 3 1 1
2 3 9 2 3
z 5 1 12 5 1 = 2 – 36 + 90 – (-9 – 48 – 15) = 128. 1 2 1 1 2 x x , y y, z z , -64x = -64, -64y = -192, -64z = 128, . x = 1, y = 3, z = -2.
, : 21 – 33 + 1(-2) = -λ ( !).
ο1(
), 51 + 13 – 2(-2) ο 12 (
(
),
.
ђ
a 11
a 12
a 21 a 22 a 31 a 32
a a 23 a11 22 a 32 a 33
(
)
. -
μ
a 13
.
,
)
.
), 1 – 6 + 6
.
,
, (
a 23 a a 23 a a 22 a12 21 a13 21 a 33 a 31 a 33 a 31 a 32
a11 (a 22a 33 a 23a 32 ) a12 (a 21a 33 a 23a 31 ) a13 (a 21a 32 a 22a 31 ) , ђ ,
16.2.4.
μ
116
:
I
1 2 1 x y z 8 4 3 5 13 x y z 7 4 5 21. y z x 1 1 1 u , v , w, x z y u + 2v + w = 8 4u – 3v + 5w = 13 -5u + 7v + 4w = 21. ( ) μ 1 2 1 4 5 3 5 4 3 4 3 5 = 1 2 1 = -116. 7 4 5 4 5 7 5 7 4 ( ) u, v w μ 8 2 1 13 3 8 2 8 2 u 13 3 5 = = 1 = -116, 5 4 21 7 21 7 13 3 21 7 4
:
1
8
1
v 4 13 5 ο 5 21 4
ο 1
1 w 4
2 8 3 13 ο 5 7 21
u = 1, v = 2, w = 3. 1 2 1 8 ( 1 1/ 2 1/ 3 !).
ο 4
13 5 21 4 2
8
7 21
4 3
8
21 4 1
2. )
5
8
5 21
8
1
13 5
13
1
= -232,
2
5 7
= -348.
-116u = -116, -116v = -232, -116w = -348, 1 1 x = 1, y , z . μ 2 3 7 4 5 4 3 5 ), ), 21 ( 13 ( 1/ 2 1/ 3 1 1 1/ 2 1/ 3
16.2.5. 1. )
1
. :
2 x 3 y 12 3x 5 y 1;
)
)
2 x 3 y 17 5 x y 10.
,
ђ
μ
117
: 3x 7 y 11 9 x 21y 22;
3. )
4x 6 y 8 6 x 9 y 12.
: )
x ay 5 3x ay 6;
4. )
:
)
2 x 3 y 4 z 3 x y z 4 3x 2 y z 6;
5.
I
x y 1 2 x a y a.
x y 2z 1 x 2y z 9 2 x y z 6. ,
μ
4 x 8 y 4 z 2 3x 6 y 3z 1 x 2 y z 1.
16.3. ,
,
.
16.3.1.
7.2. (
16.3.2.
7.3. (
16.3.3.
). ).
1 1 a, x y x y 1 1 b. x y x y
1 1 , v x y x y μ u + v = a u – v = b. , a b a b u v , 2 2 :
,
μ
u
ђ
x y
2 2 x y . a b a b 2a x 2 a b2
118
: y
2b . a b2
,
a = b.
2
, .
16.3.4.
μ
1.
9 1 ) a ; 4 2 2 ) x + 4x – 60; ) 2u2 – 31u + 42;
4 1 ) b ; 9 3 2 ) y – 28y + 180; ђ) 3v2 + 43v – 30.
2
2. )
)
)
ђ)
I
2
y y 1 x4 2 x2 9 ; ) 4 ; x 4 x3 10 x2 12 x 9 y 4 3 y 2 1 2 y3 2 y 9 4 1 2 1 2 a 3 x2 a x : : ; ) ; 8 15 a 5 1 6 x3 1 2 1 2 a a x x 2 z4 z2 2z 1 z2 1 z2 z2 2z ; 2 : 2 z 4 2 z 3 z 2 1 z 3 1 z 1 z 2 a 1 b 1 c 1 3abc a b c . 1 1 1 bc ac ab a b c
:
4
2
μ
3.
1 1 1 a 5 5 1; a a x y z ) a b c 1 a 2 b 2 c 2 1 xy + xz + yz = 0; a b c
) a
[ . 2,
4.
5.
1/a = b, a 2 + b2 = (a + b)2 – 2ab = -1 a 3 + b3 = (a + b)(a 2 – ab + b2) = 5 2 2 a + b = (a + b )(a 3 + b3) – a 2b2(a + b) = 1. . (a + b + c)2 = a 2 + b2 + c2 + 2ab x y z 1 + 2ac + 2bc, a kx , b ky , c kz .] a b c k 5
.
35
.
2 1,89
10
6, 10,
.
5
15 6 500
,
5
2
,
10
,
? .
15
.
119
: 84450
ς 6.
3 21 4 ) ; 2 14 2 x 49 x 14 2 x x x2 ; ) 2 x 1 1 x
) ђ)
10.
x 0
x m,
11.
7 7 2 ; 2 3x 9 3x 9 x 6 x 9 3 5 5 ) x3 x3 ; 5 3 14 x3 x3
)
2x b x 2 x a ; 2 2 2 x 3x x 3x x 9 x a xb xc 1 1 1 ) 2 . bc ac ab a b c
)
mx n mx n 2mx 4 2 2 2 x mx x mx x m2 m n .
m2 x 2 x 2 1 x 1 1 mx x 2 m m Z , m 0.
5 4 2, x 2 y 2x y 4 5 41 ; x 2 y 2 x y 20 3 2 11 2 , ) 2 2 2 65 x y x y 2 3 49 2 ; 2 2 2 65 x y x y )
μ
x2 4 x 4 x2 1 2 x 4 4 x x2 .
μ ax bx ) a b; a x xb x ab x ac x bc ) a bc; a b a c bc
9.
,
x2 4 x 4 x2 6 x 9 x2 2 x 1 ;
7.
8.
I
μ
4 1 1, x y 1 x y 1 18 2,5 1; x y 1 x y 1 ) x2 15 4 y 2 5 3. y 2 4 3x2 20 )
μ
120
: )
I
x y a , x y b c xc a b ; yb a c
a b 1, x y x y
)
a2 b2 a b ; 2 x y x y mZ
12.
2m 1 1, x my x my 10m 3 1, x my x my .
ј
17. 17.1
a 2x + b2,
.
, x a 1x + b1 > a 2x + b2, a 1x + b1 < a 2x + b2, a 1x + b1 a 2x + b2, a 1, b1, a 2 b2 ,
, a 1x + b1 ,
. ,
. 2 x 1 3x 1 x 1 . 1 3 2 6
17.1.1. :
(
,
5+x
a
) 2(2x – 1) – 3(3x + 1) 6 – (x – 1), -5x – 5 7 – x. , -4x 12, -4 ( , x -3. x [-3, + ). -x < 3 {, , }.
.
6
)μ
x > -3,
121
:
I „
, 17.1.2.
.
mx > 3m.
: (i) (ii) (iii)
“.
μ m>0 m=0 m 30
ν x < 3.
,
.
,
.
,
x2 + x – 2 4.
17.1.3. :
x > 3;
μ
x x6 0, (x + 3)(x – 2) 0. μ , x + 3 0 x – 2 0, x -3 x 2, . x x + 3 0 x – 2 0, x -3 x 2, . x (-, -3. x (-, -3 [2, +. 2
, [2, +; , 17.1.4. :
μ
3 4x x. x2
( μ 3 4x x 0, x2 x2 2 x 3 0, x2 x2 2 x 3 0, . x2 ( x 1)( x 3) 0, x2 ( x 3)( x 1)( x 2) 0 . ђ ! f(x) = (x + 3)(x – 1)(x – 2) . , x {-3, 1, 2ж , x = -10 f(-10) = (-7)(-11)(-12) < 0. f(x) < 0 x < -3, f(x) > 0 -3 < x ξ 1, f(x) < 0 1 ξ x ξ 2, f(x) > 0 2 ξ x < +. , (-, -3) (1, 2). . )
ђ
122
: 17.1.5.
μ
1.
i. 2x + 5 -3; 2. i. 3x – m -mx + 3;
μ
3. i. |2x – 5| < 7;
ii. 3
I
1 x 5; 2
iii.
2x 3 3x 1 . 4
μ ii. 5x – a > -ax + 5;
iii. kx + 1 < x + k2.
ii. |3x – 1| > 2;
iii. |x – 1| + |x + 1| 2. μ
4. i.
3x – 5 0 2x – 3 0;
5. i.
ii.
2x – 5 0 3x – 13 < 0;
iii. -x + 7 < 0 5x + 15 0.
:
iv.
2x 3 1; x5
ii.
3 2x 1; 3x 2
iii.
x + x -2x 0;
v.
x + 2x – x 2;
vi.
mx y 3 x my 5; (x, y)
ii.
3
6. i.
2
17.2.
a a a a a
, b + c;
,
.
ђ .
|a + b| |a| + |b|.
123
: ,
a
.
I
b a
b
.
. 17.2.2.
a
a
b
|a – b| ||a| – |b||.
b
|a| = |a – b + b| |a – b| + |b| |a – b| |a| - |b|, |b| = |b – a + a| |b – a| + |a| |b – a| |b| – |a|. |a – b| = |b – a|, |a| - |b| |b| – |a|.
: |a – b|
.
,
17.2.3. (xR) x2 0. ,
. 17.2.4. i. x x 1 0 ; ii. x2 – xy + y2 0; iii. x2 y2 z 2 xy yz zx 0 .
μ
, y, z
2
1 3 x x 1 x , 2 4 2
: (i)
2
1 3 x2 xy y 2 x y y 2 ; (iii) 2 4 2
(ii)
1 x y2 y z2 z x2 0 . 2
17.2.5.
;
x2 y2 z 2 xy yz zx =
a, b 0
, .
ab
ђ :
( a b)2 0
2 ab a b ,
17.2.6. i.
a 2 ab b 0 , 2
ab
a 2 b2 , 2
ђ
a b . 2 .
a b 2 ab , .
a
b.
124
: a
ii.
I
1 2, a
a.
: (i)
b2,
a2 a
ν (ii)
1/a.
17.2.7.
a, b, c 0
, .
3
abc
ђ :
abc
b3
.
a bc . 3
17.2.4.iii. a 2 b 2 c 2 ab bc ca 0 . a+ b+ c a 3 + b3 + c3 – 3abc 0, .
ђ
a 3 b3 c3 , 3 y, c 3 z
a 3 x,
x = a 3, y = b3, z = c3. ђ
.
17.2.8.
μ
x, y z x y z2 x y z. 2
1.
2
2
(w – 1)2.]
[ 2.
μ
x + y > 0,
x y3 x y . 2 2 [ 3
3.
a, b
c [
4.
[ 5.
3
, ab + bc + ca 3abc. ab abc
0 ( x y)( x y) 2 .] μ ,
a, b c μ ab bc ca a bc. c a b ab bc 17.2.5. , c a
a, b, c 0 μ (a b c)( a b c ) 9abc . [ ђ ,
.]
,
17.2.7.
a, b, c,
.]
.]
125
:
6.
a, b
7.
2
1 1 a b
I
. .
a 2 b2 . 2
a, b
.
23, 40
8. < 40?
31. [30 < x < 40]
9. Ј km/h.
Ј
20
30
,
,Ј
ђ
.Ј ς
,
15
.
[25 km/h]
17.3.
(
,
, 1957.) К
, ,
(1λ36.) , 1912 – 1986.) (George Bernard Dantzig, 1914 – 2005.)
. Koopmans, 1910 – 1985.)
17.3.1. 1 2. 2 . :
М1
1 2
1λ55. .
.
1λ47.
(John von Neumann, 1903 – ( (1λ3λ.). (Tjalling Charles
1λ75. .
,
4
), 1λ35.
М1 М2 2 4 М1, 500
300 ђ ђ
М1 М2
М2
.
М2,
.
. .
F(x1,x2) = 300x1 + 500x2.
126
:
2 x1 4 x2 24 4 x1 2 x2 24,
(1)
(2)
.
(0,6)
(12,0)
.
x1 0 x2 0. ,
,
.
I
, (0,0)
,
,
(0,12) (6,0), ђ (0,0). (4,4).
, (0,0), (6,0), (4,4), (0,6), μ 0, 1800, 3200, 3000. ,
3200
М1,
.
2
F(x1,x2)
4
1
М2.
ђ 4
,
(4,4).
(
)μ
F = ax + by + cz + ... , (
)
μ
Ax + By + Cz + ... N, ,
Ax + By + Cz + ... N. ,
F x, y, z, ...
17.3.2. :
.
.
2x – 3y 6. ,
(x, y)
127
:
.
.
,
, ,
I
(
2x – 3y = 6,
(0, 0) 20 – 30 6),
. 17.3.3.
2 x 3 y 6, x 2 y 3, x 1.
:
2x – 3y = 6, x + 2y = 3 . , .
,
x=1
,
.
(
μ
)
(1, -4/3), (3, 0), (1, 1).
17.3.4.
2 x 3 y 6, x 2 y 3, x 1.
:
,
3x + 4y = 6 (0ν 1,5)
(2, 0),
,
. 6 .
, ,
μ
F = 3x + 4y,
-7/3
3x + 4y = -7/3
F F
.
F F (1, -4/3).
, ,
, F = -7/3.
128
: 17.3.5. ђ
1. 1λ20
10
12 .
780 $1,00
I
.
5 ђ
3
, ђ
$1,20
. ,
. / / (
/
10 5 1,20
. )
(
12 3 1,00
F = 1,2x + 1,0y 5x + 3y 780 ( ), ) (
2. ) x < 3;
μ ) 3x – y 2;
3. )
)
3x y 2 2 x y 5;
4. )
5. )
. )
μ
[
μ
1920 789 μ 10x + 12y 1920 )
.] ) 2x + y 5.
μ 2 x 3 y 5 3x 2 y 1;
μ x 2 y 5 x y 1 x y 4.
. )
)
x 2y 3 4 x 3 y 2.
F = 2x + y
F = 100 – 2x – y
0 x 4, 0 y 4 x y 6 x 4 y 8.
129
:
I
VI , ,
).
,
. ,
(384 – 322. . . . .) (1. ) Њ (Isaac Newton, 1643 – 1727.)
ђ
.
.
,
1818), (Jean Robert Argand, 1768 – 1822), 1855) , 1837. 1843. Hamilton, 1805 – 1865). 1850(Peter Guthrie Tait, 1831 – 1901) . , 1809 – 1877) 1832. „ (Ausdehnungslehre), .
,
(1687.
1λ. . (Caspar Wessel, 1745 – (Carl Friedrich Gauss, 1777 – (William Rowan (Hermann Grassmann, “ 1844.
(James Clerk Maxwell, 1831 – 1879)
. (William Kingdon Clifford, 1845 – 1879) (1878.), (J. Willard Gibbs, 1839 (Oliver Heaviside, 1850 – 1925).
– 1903)
ј
18. 18.1. ђ
.
16.1.
, . 18.1.2. .
, .
, ђ
. AB,
AB
AB CD AB = CD. ,
|AB|
,
, .
EF
GH EF = -GH.
130
:
I
AB ,
AB a,
a.
18.1.3. Ј
V.
: . (a V) a = a; . (a, b V) a = b b = a; . (a, b, c V) a = b b = c a = c,
μ
.
18.1.4.
AB
AC, . AB +
BC
BC = AC. , .
b b:
a b, .
a
a b a (b) . 18.1.5.
ABCD AB, BC, CD
MN
:
18.1.6.
1. 2. 3. 4.
M, N, P Q MNPQ
.
MN PQ
1 1 1 1 AB BC , QP CD DA. 2 2 2 2
1 AB BC CD DA 0 , . MN QP , 2 . ,
.
DA.
MN QP .
MNPQ
(a, b, c V): : a + b = b + a; : (a + b) + c = a + (a + b) = a + b + c. (k, m R):
k(ma) = (km)a; (k + m)a = k a + m a; k(a + b) = k a + k b; 1a = a.
18.1.7. ( A A1,
b
) B B1 C μ AB CA CB . A1 B1 CA1 CB1
p
q a
a b,
131
: :
I
,
AB
(
A1 B1
AB k A1 B1 ,
),
AB | k | . A1 B1
ABC
AB CB CA
A1B1C
A1 B1 CB1 CA1 .
CB CA k CB1 CA1 ,
CB k CB1 CA k CA1 . , , AB CA CB μ , | k | A1 B1 CA1 CB1
CA k CA1 , 18.1.8. Ј
1.
,
18.1.9. AB CD
2. )
AC
AB
.
a . |a|
BD AD
ABCD AB. AD, BC AB, BC CD. Ј
AD . BD .
AB AC AD
60 M, N P
BC CD
3. AB AD a μ AB , BC , AD , AM , AN ,
NM .
CA u
4. D,
CB 2v . DE E. μ ) AB ; ) AE ; ) CE . [ 2v u ;
5.
CB k CB1
ABCD.
b.
AP , NP , MP
b
AC BD .
) 3.
a0
a
a
AB
u
v
1 1 3 1 v u ; u v] 2 4 4 2
.
132
: 6.
I
.
7.
AB CD AD CB .
A, B, C, D
8.
T
TA TB TC 0 .
ABC,
18.2. ,
ђ
.
(3,2), (-3,1), (1,5ν-4) ( – ) , ( ),
( – .
3,
.
3 , 2
3 , 1 , .
1,5 4
(z – ,
).
)
ђ μ
ђ
,
(3,2), (-3,1), (1,5;-4),
.
, A = (3,2), B = (-3,1), C = (1,5;-4), A(3,2), B(-3,1), C(1,5;-4). (
.
) μ OA 3i 2 j ,
OB 3i j , OC 1,5i 4 j , i, j ( k ) ( .
,
,
). –
–
ђ
,
133
: ,
I
.
.
( . René Descartes, . Renatus Cartesius, 1596– 1650.) (La Géométrie,
1637.) .
ђ
, . , .
.
18.2.1.
.
:
A(3,2), B(-3,1) 18,25 , .
13 , 10 , P1 ( x1 , y1 )
P2 ( x2 , y2 ) ,
,
x x2 x1 , y y2 y1 .
P1 P2T , P1 P2 ( x2 x1 ) 2 ( y2 y1 ) 2 . (0,0),
ђ
C(1,5;-4)
P1 P2 x2 y 2 , .
,
OA x y . 2
2
OA 32 2 2 , . OA 13 .
А( , ) ,
A(3,2)
134
:
I
, (
,
). ђ ,
.
(16.1.6.)
18.2.2.
, .
P2 ( x2 , y2 ) ,
P1 ( x1 , y1 ) .
.
OP2
OP1
:
.
OP1
OP2 T ( x1 x2 , y1 y2 ) . OT .
, . ( x1 , y1 ) ( x2 , y2 ) ( x1 x2 , y1 y2 ) . 18.2.3. P0 :
P 1P 2
P1 ( x1 , y1 ) a:b.
P2 ( x2 , y2 ) .
P 1P 0 : P 0P 2 = a : b. ђ ( x0 x1 ) : ( x2 x0 ) a : b ( y0 y1 ) : ( y2 y0 ) a : b . , by ay2 bx1 ax2 , y0 1 x0 a b a b OP0 . , a : b = 1 : 1, P0 (
x1 x2 y1 y2 , ) 2 2
P 1P 2.
k1, k2, ..., kn ,
18.2.4.
x1, x2, ..., xn k1x1 + k2x2 + ... + knxn, , n = 1, 2, 3, ... . k1, k2, ..., kn k1x1 + k2x2 + ... + knxn = 0, x1, x2, ..., xn k1 = k2 = ... = kn = 0 , .
. x1, x2, ..., xn
135
: 18.2.5. = 3i – 2j, y = j – 3k, z = 3i – 6k
i, j, k
I ,
.
: , 3i – 6k = a(3i – 2j) + b(j – 3k), + (-6 + 3b)k = 0, a = 1, b = 2.
a
x
b
z = ax + by. (3 – 3a)i + (2a – b)j
. х у ax + by ,
V2
2b
OX OY,
3-
x, y z ax + by + cz ,
a
a, b
i
. j. V3
.
c OX, OY OZ
i,
j k. 18.2.6.
v V3
V3,
x, y, z a, b, c
v = ax
+ by + cz. : a 2, b2, c2 ђ
,
0, c2 – c1 = 0, .
v v = a 1x + b1y + c1z = a 2x + b2y + c2z, (a 2 – a 1)x + (b2 – b1)y + (c2 – c1)z = 0.
a 1, b1, c1 x, y, z a 2 – a 1 = 0, b2 – b1 =
a 2 = a 1, b2 = b1, c2 = c1.
18.2.7. 1. A(3,1) B(-1,3). ) 5 OA 3 OB ; ) i ; ) j ; ) OC , 2. A(-2,3) B(1,2). ) 2:3; ) 3:4; ) 5:1; ) 4:5. 3. ) (7, -5, 1), (4, 2, 3) , 4. ) ) )
OA C(5,4).
AB
OC
μ (-13, -11, -2); ) (-1, 5, 2), 2, 1, -3) .
A(-1, -3), B(5, 1), C(-2, 2). ABC. ABC. OA,
μ
OB
AB,
μ
(3, -2, -4);
t A.
136
:
I
А, B, C, D
5. ) )
AC BD AC BD
.
P n(xn, yn), n = 1, 2, 3, 4. n, n.
6. u(2,-1) v(1,3). ) (4,5); ) (-3,2); ) (7,-1); ) (-2,-5).
18.3. Ј p , P , x x2 y1 y2 , P 1 . 2 2 1 OP OA1 A1 A2 . 2 P А1А2, OP OA1 A1 A2 , R. . 18.3.1.
r (
,
, A1 A2 ,
μ 18.3.2. А(a 1, a 2) B(b1, b2) :
А1(x1, y1) А2(x2, y2) , . .
А1А2,
,
r a b , R,
a OA .
).
:
.
b
),
(18.3.1.)
b x a1 1 y a 2 b2 ( R)
(
, ,
x a1 b1 . y a 2 b2
x a1 b1 , y a 2 b2 .
137
:
x y
18.3.3. ( А(a 1, a 2) B(b1, b2) :
ђ
x r . y y a2 , . b2
x a1 b1
x a1 y a 2 . b1 b2
18.3.4.
2 3 r t , 1 4
t=0
.
1 0
,
B ο (5, 5), .
3 AB . 4
.
b1b2 ax + by = c.
0 1 t
1 .
A = (2, 1),
) x – y = 3;
5
A(2, 1), .
/
A ) A(-1, 2), v = 3i + j;
3. ) ) )
32 4 2 = 5
μ ) 2x + 3y = 5.
2. v: ) A(2, 3), v = 5i – 4j;
.Ј
3 AB = 4
B(5, 5) 18.3.5. 1. ) x + y = 5;
,
)
b2x – b1y = a 1b2 – a 2b1,
:
I
.
ν
) A(2, -3), v = -2i -3j.
2 3 , R. r 1 4 r a b ;
138
: 4. )
)
v1 = 2i + 3j + (i + 2j), v2 = 5i – 2j + (2i + j);
I
ђ
v1
v1 = (3, 5) + (-2, 1), v2 = (4, -3) + (-2, -5);
5 3 r 4 2
5. 6. a) ) ) )
(-35, 23)
7. (
A B
) )
x = 4 – 3t, y = -4 + 2t. ν
.
/
v
μ 10μ00 10μ30
5 3 , v1 4 3 2 . v2 4 3
x4 y 2 . 3 5
ν (-47, 30);
t
r
)
v2: )
A B
, rA = 7i + 2j, vA = 3i + 2j, , rB = 8i + 5j, vB = 4i – j. , ς .
, B
19.
18.
1λ.
.
.
ν
. ν
ν ν (
(
,
: ν
ν
.
ν
)ν ν
. Leontief model);
19.1. a j1
, , a j2
1
2
. a j3
j (j = 1, 2) : 139
:
Ф1 Ф2
I
a 11 a 21
Ј a 13 a 23.
a 12 a 22
.
a13 . a 23
a a ο 11 12 a 21 a 22
(
М23, . 3)
М
(
2 М2,3.
.
3
.
3).
,
,
,
19.1.1.
ђ
A = ||a jk|| mn
2 3.
, (
.
,
)
(
.
)
B = ||bjk|| mn, A= B a jk = bjk (j = 1, 2, ..., m; k = 1, 2, ..., n). A B,
40 35 280 A 50 42 320
38 32 175 , B 46 38 316
μ 40 38 35 32 280 175 78 67 455 = . A B 50 46 42 38 320 316 96 80 636
(
19.1.2. , A = ||a jk|| C = A + B = ||cjk||
B = ||bjk||
)
.
mn, cjk = a jk + bjk (j = 1, 2, ..., m; k = 1, 2, ...,
n). ђ ,
.
,
.
140
:
I ,
, R
.
μ
a11 a12 a 21 a 22
.
,
a13 a11 a12 a13 . = a 23 a 21 a 22 a 23
19.1.3. . ,
A
R,
A = ||a jk||
.
||a jk||. ,
,
19.1.4.
.
2 7 2 9 7 3 1 3 2 1 1 9 = 9 8 . 9 = 5 4 . 5 1 4 0 13 12 6 0 (12) 13 (6) 12 19 А
,
,
. 19.1.5. 1. 2. 3. 4.
μ (A,B,C) A + (B + C) = (A + B) + C ( (A,B) A + B = B + A ( )ν (1О)(А) А + О = О + А = А ( (A)(A’) A + A’ = A’ + A = O (
,
,
A’
)ν )ν ).
М
. , P
,
. p1, p2
p3
,
:
141
:
I
p1 P = p2 . p 3 μ
a a M P = 11 12 a 21 a 22 M2,3 P3,1 = T2,1 .
p1 a 13 a11 p1 a12 p 2 a13 p3 p2 = = T. a 23 a 21 p1 a 22 p 2 a 23 p3 p3 ,
.
T2,1 ,
μ
,
q1 Q = q2 , q 3 M Q.
p1 S = || P Q || = p 2 p 3
a1 b1 a 2 b2
.
n
p1 c1 p2 c 2 p3
.
.
,
μ
q1 q2 , q3
,MS=
q1 a p b p c p q 2 = 1 1 1 2 1 3 a p b2 p 2 c2 p3 q3 2 1
a1q1 b1 q 2 c1q3 . a 2 q1 b2 q 2 c2 q3
,
Am,n Bn,p = Cm,p.
19.1.6. m n n p (m, n, p N) j(j = 1, 2, ..., m) k- (k = 1, 2, ..., p) , n j, k-
n .
142
:
,
A = ||a jk||
I
mn
B = ||bjk||
c jk a ji bik .
np,
n
C = AB = ||cjk|| ,
Am,n cjk = a jibik (
ђ
mp,
/
)
i 1
Bn,p
(
Cm,p i = 1, 2, ..., n.
)
.
(
,
).
, .
.
, 19.1.7. A(BC) = (AB)C – A(B + C) = AB + AC – (A + B)C = AC + BC –
,
A, B C
μ
; ; . .
19.1.8. 20%
P1 P2 30%
.
x y
.
.
.
:
, x1 = 0,8x + 0,3y, y1 = 0,2x + 0,7y. 0,8 0,3 x x1 , μ 0,2 0,7 y y1
P1
P2 μ
MX = X1.
,
μ x2 = 0,8x1 + 0,3y1, y2 = 0,2x1 + 0,7y1. 0,8 0,3 x1 x2 , μ MX1 = X2, M2X = X2, y y 0 , 2 0 , 7 1 2 0,8 0,3 0,8 0,3 . , M2 = MM = 0,2 0,7 0,2 0,7 0,8 0,8 0,3 0,2 0,8 0,3 0,3 0.7 0,70 0,45 = . M2 = 0,2 0,8 0,7 0,2 0,2 0,3 0,7 0,7 0,30 0,55 M3 = M2M, M4 = M3M, ..., Mn = Mn-1M, n = 1, 2, 3, ... . ,
n-
(n = 1, 2, 3, ...), xn = 0,8xn-1 + 0,3yn-1,
:
143
:
I
yn = 0,2xn-1 + 0,7yn-1. 0,8 0,3 xn xn 1 , MXn-1 = Xn, μ 0,2 0,7 yn yn 1 19.1.9. 1.
MnX = Xn.
11 3 5 3 2 2 5 12 , B 8 17 , C . A 13 1 4 4 7 6 9 1 2B, A + C, 3A – 2C, A B, B A.
2.
.
μ
a 21 b 4b 5c . 5 1 a b c
3 7a 2b 4b 5c ; 5 9a 3a 0 8 1 2 4 3 3 6 18 . x 1 2 5 5 6 7 30
2 ) x 5 1 ) x2 2
3.
4.
1 2 A 4 1 32n 0 A2 n 2n 0 3
A A = A2, A2 A = A3 .
n
1 0 A 1 1
5.
A3 A = A4.
1 0 , An n 1
.
n 1λ.1.7.
6. 3 7. 3.
P1
a
b
,
,
a
b
2 ?
, 10 4, 5
P2
6
1, 7 2 4 1, 7 .
3
2
3. 8 ,
3. [
4
ђ
1, 2 3. 1, 8 2 10 1, 2 3 ο λ3, 116, λλ]
144
: 8.
A, B C
A B C )
μ
P 2 4 2
,
μ
5, 10
5
Q 3 2 4
P, Q
R.
R 1 5 2 100
ν
) )
I
,
ν
P, Q
R
200
.
[(800, 900, 800); (45, 65, 60); 34000]
19.2. ,
. m=n
A = ||a jk|| ,
An.
a 12 a , (a 11), 11 a 21 a 22
a 12 a 12 a 21 a 22 a 31 a 32
a 13 a 23 , ..., a 33
a 11 a 12 a 21 a 22 ... ... a n1 a n 2
. (
) mn
... a 1n ... a 2 n , ... ... ... a nn
1, 2, 3, ... , n. a ii (i = 1, 2, ..., n). AB BA
A B .
,
. Ј
I
,
.
I = ||jk||,
Ј
jk
1, 0,
jk =
j k. 1 0 . μ I 2 0 1
, A
,
I
145
:
I
AI = IA = A. А
n
X
AX = XA = In A-1.
a b A c d
19.2.1.
А
X A1
1 d b . ad bc c a
0 1 ad bc 1 a b d b = = I. ad bc ad bc c d c a ad bc 0 0 1 ad bc 1 d b a b = = I. A1 A ad bc ad bc c a c d ad bc 0
: AA1
A (1λ.2.1.) det A = .
(
) det A 0 (1λ.2.1.).
ν
19.2.2.
,
det A.
a
ad bc ,
b
c d
.
,
А
A-1
det A = 0
μ
2x y 7 5 x 3 y 6.
: 2 1 x 7 . 5 3 y 6 1
B,
,
AX = B, . 2 1 3 1 . det A = = 1, A-1 = 5 3 5 2 A-1 A-1(AX) = A-1B, (A-1A)X = A-1 -1 IX = A B, . X = A B. , x 3 1 7 21 6 27 = . = X = = y 5 2 6 35 12 47 x = 27, y = 47. μ 227 – 47 = 7 ( ), -527 + 347 ο 6 ( !). 2.
3.
А
3.
2.
.
А.
146
:
I
1 4 7 1 2 3 4 5 6 2 5 8 . 3 6 9 7 8 9 T
, A = ||a jk|| bjk = a kj (j = 1, 2, ..., n; k = 1, 2, ..., m) АT, A’.
a jk j-
19.2.3.
mn,
B = ||bjk||
( 16.2.4.). A = ||a jk|| Ajk = (-1)j+ kDjk, Djk k. ||Ajk||T n А adj A. det A 0,
A1
nm,
А
n det A
adj A . det A
19.2.4. 1. )
2. )
3. )
3 1 ; 5 2
)
3 k ; 2 k 1
)
i. A(a)A(b); 5.
m 5 2 ; m 3
)
)
4x – 5y =2 3x + y = 6;
4.
4 2 ; 5 3
)
:
ς 1 n . 1 n 1
:
) 7x + 3y = 10 6x – 5y = 4;
x 1 2 x , x R. A( x) 2(1 x) 2 x 1 ii. A2(a);
X
2 6 . 1 3
-9x + 6y = 15 6x – 4y = -10; :
iii. An(a), n N.
μ 3 4 1 2 2 1 X . 5 1 1 3 3 4
147
:
6.
I
1 2 1 5 X . μ 1 2 1 5 2a 1 2b 5 , a [ b a
X
.
7.
8. )
a b 2 , a = -bc] [ c a
x y z 1 x y z 1 x y z 1. [(1, -2, -1);
9. 1. AX = B, 2. XA = B, 1 A 1 2 10.
1, 2
,
( 1)(1,1,1) ] ( 1)( 2)
μ 3. AX = BA-1B, 4. XA = BA-1B, 2 1 1 1 1 1 0 , B 0 0 0 . 0 0 0 0 1 1,
3.
.
1 2 1 3
1 2 3
]
:
)
10 x 9 z 19 8 x y 10 y 12 z 10;
b
2
2 3 1 2
3 3 4 2 1 2λ, 13
.
μ 2, 3
[
19.3. Ј
( ,
,
. 19.3.1.
A B
mn
16 4]
R), , R.
, R, 148
:
I
( + )A = A + A, (A + B) = A + B, ()A = (A), 1A = A.
A1, A2, ..., Ak
(mn)
(18.2.4.)
1 A1 2 A2 ... k Ak r Ar , k
r 1
1, 2, ..., k R.
mn.
.
,
ђ
1 2 3 1 7 1 , 1 1 2 1 1 2
,
ν
. A, B, C
In = (jk)n , , (AB)C
3 1 1 2 4 5 . 2 1 1 1 1 5
, .
,
A(BC)
19.3.2. A = (a jk)m,n , B = (bjk)n,p , C = (cjk)p,q (AB)C = A(BC). : (AB)C = (djk)m,pC =
n a ji bik c kr = k 1 i 1 p
p a ji b jk c kr = A(ejr )n,q = A(BC). i 1 k 1
R.
a ji b jk ckr = p
.
n
k 1 i 1
a n
p
i 1 k 1
ji
b jk c kr =
n
1λ.3.1. 19.3.3.
,
ђ
A, B, C, D,
R
mn
R,
.
C(A + B) = CA + CB, (A + B)C = AC + BC, (AB) = (A)B = A(B).
А AA, A = AA2, A4 = AA3, 3
19.3.4.
n. . (5.1.2.)
s+1
A
s
= AA ,
s
A0 = In, A1 = A, A2 = . . r
s
μ
149
:
I
1. Ar As = Ar+ s, 2. (Ar )s = Ars. 19.3.5.
P(A) = А – I
Q(A) = A2
+ A + I. : P(A)Q(A) = (A – I)(A2 + A + I) = A3 + A2 + A – A2 – A – I = A3 – I. , ђ ,
,
(
.
)μ
ђ
. ,
1. ab = ba; 2. ab = 0 a = 0 b = 0; 3. (ab = ac a 0) b = c. 19.3.5.
.
1 1 1 1 0 0 . : 1 1 1 1 0 0
3.
,
.
1 2 3 1 2 3 1 2 0 A 1 1 0 , B 1 1 1 , C 1 1 1 , 2 2 2 1 4 0 1 1 1 3 4 1 AB AC 2 3 2 . 3 2 7
, AB = AC
A 0
B = C.
, 19.3.6.
AB T T
T
BA :
. T T BA
. T T
,
(AB) = B A .
(AB)m,p (AB)T
pm, BT je pm.
,
A = (a jk) B = (bjk) pn, AT nm, T (AB) BTAT
mn
np .
150
:
(AB)j,k , . (k, j)
1.
19.3.7. a b A c d
2.
f 1 1 n 1 0 f n1
bik a ji , . n
i 1
a n
i 1
i. AX = XA.
A = 0. f n 1 , fn f n 2 f-1 = 0, f0 = 1, fn = fn-1 + fn-2.
AC = CB. Bn = C-1AnC]
x y a b , . X A u v c d AX = XA = 0, (A + X)2 = A2 + X2?
ii. AX = XAT. [(ii)
y x c d a X y x b b
b .
ji ik
2
4 7 2 1 1 1 , C , , B A 9 5 3 2 0 1 Bn, n . [ B = C-1AC B2 = C-1A2C,
4.
(AB)T.
(k, j)
A2 = 0.
k
n
6. Ak.
i 1
b
ji ik
A2 – (a + d)A + (ad – bc)I = 0.
A = 0 (k N)
5.
n
BTAT
i. ii.
3.
a
I
; y
1 0 0 A 1 0 1 0 1 0
:
AkX = X(AT)k? b = c = 0, a = d
X ν b0 b a d y v y c0 ν ad X c c y v b c v x x a d . , a d X c x b v v a d a d
.]
An = An-2 + A2 + I, n N.
I + A + A2 + ... + Ak = 0,
A [
А
Ak+1 = I,
A-1 = .]
151
: 7.
А
I
k , Ak = 0 -1 2 k-1 (I – A) = I + A + A + ... + A . [ (Ak – Ik) = (A – I)(Ak-1 + Ak-2 + ... + A + I).]
152
:
I
VII , x2 3xy
. 2 x 3xy y 3 5
2 2 y 5
2
,
.
2
1, 17x3y, 25x – 3y, .
,
,
,
,
(4
.
– 4xy + 3), 3xy-2, ,
)
(
(x + y ), 2
1 , x 2
15.
,
7 xy ay 2 bxyz , x4 – 2x3y + 3x2y2 – 4xy3, ... 13 , , , .
3
.
(5
),
), ,
(x – 2y + 3),
(5x
x + y, ..., ,
,
.
.
2. . . . (九章算術, . JТǔzСāng SuрnsСù) , 5x + 4y + z = 37.
(Scipione del Ferro, 1465-1526.)
„
“
„ 37
(Nicolas Chuquet, 1445 – 1500?) . 1515. ,
“. 1484.
153
: .
I
(Niccolo Fontana Tartaglia, (Girolamo Cardan, 1501-1576.) .
1500?-1557.)
16. (The Whetstone of Witte, 1557.) (Robert Recorde, 1510 – 1558.) „+“ „ –„ integra, 1544.) (La geometric, 1637) , (x, y, z, ...) ,
„ο“
(Arithmetica (Michael Stifel, 1487 – 1567.). , (a, b, c, ...) , ( . 2).
.
,
.
20. А 20.1. Ј
20.1.1.
, fa (x) = a 0 + a 1x + a 2x + ... + a mxm, a = (a 0, a 1, a 2, ..., a m) m ,m ,a m+1- ,
.
a k (k = 0, 1, 2, ..., m) (
,
(
. x
,
)
.
fa (x) = 0
,
ak ο 0
k = 0, 1, 2, ..., m.
a k 0. a n 0. fa (x) = a 0 + a 1x + a 2x + ... + a nxn. ak ak , k = 0, 1, 2, ..., n-1. nk x x
)
,
2
|xn-k| |x|, 17.2.1.)
an
.
.
a kxk
ђ
(
.
.
20.1.2. :
, a 1 , a 2, ..., a m ,
, fa
n
|x| 1,
a0 a a 1 n11 n22 ... a n a 0 a1 a 2 ... a n . n x x x x 1 a 0 a1 a 2 ... a n1 , ђ x an
a0 a a n11 ... n 1 . n x x x
a a a f a ( x) xn 0n n11 n22 ... a n , x x x
154
:
I
fa (x) 0, .
.
fa ( x) fb ( x) , . a = b.
20.1.3. :
, b, . a k = bk
k = 0, 1, 2, ..., m. (20.1.2.) c
fc(x) = 0 ,
a=
k. ,
. x – 1.
c = a – b, . ck = a k – bk
,
20.1.4. :
p(x) = x3 – 2x2 + 3x – 4
f(x) = b0 + b1(x – 1) + b2(x – 1)2 + b3(x – 1)3 = b0 + b1(x – 1) + b2(x2 – 2x + 1) + b3(x3 – 3x2 + 3x – 1) = b3x3 + (b2 – 3b3)x2 + (b1 – 2b2 + 3b3)x + (b0 – b1 + b2 – b3). b3 = 1, b2 – 3b3 = -2, b1 – 2b2 + 3b3 = 3, b0 – b1 + b2 – b3 = -4, b3 = 1, b2 = 1, b1 = 2, b0 = -2. , 2 p(x) = -2 + 2(x – 1) + (x – 1) + (x – 1)3.
20.1.5. 1. ) (x – 1)3 + (x – 2)(x2 + 3x + 4);
μ ) (x + 1)3 – (x + 2)(x2 – 2x – 3).
2. ) p(x) = 3x3 – 4x2 – 5x + 2 q(x) = (x – 2)(ax2 + bx + c);
p(x) q(x) μ ) p(x) = -2x3 + 3x2 + 4x + 15 q(x) = (x – 3)(ax2 + bx + c).
3. ) x – 1; 4. r(x)/q(x): ) (6x2 + 13x – 5)/(3x – 1);
p(x) = x3 + 2x2 + 3x + 4 ) x + 1.
μ
r(x) = p(x)q(x), ) (10x2 + 3x – 12)/(2x – 3).
p(x) = q(x)(x – c) + r, q(x) r , μ ) (2x2 – 3x + 1)/(x – 2); ) (3x2 +2x – 1)/(x + 2). 2 [a 2x – 3x + 1 – r = (ax + b)(x – 2) ..., a = 2, b = 1, r = 3; 3, -4, 7]
5.
155
:
I
20.2. (
.
p(x)
r 1(x) ο 0 q(x) , p(x) = x2 – 4, q(x) = x – 2
r 1(x). q(x).
5.2.3.)
q(x)
q1(x) p(x) q1(x) = x + 2
r 1(x) = 0:
(x2 – 4):(x – 2) = x + 2 . x2 – 2x /2x – 4 2x – 4 /0 ђ
,
1
r 1(x) 0 q2(x) 2 q2(x) = x – 4x + 1
. r 2(x). r 2(x) = 2:
q(x) r 1(x) x3 – 3x2 + 5x – 3
,
x–
(x3 – 5x2 + 5x – 3):(x – 1) = x2 – 4x + 1. x3 – x2 /-4x2 + 5x – 3 -4x2 + 4x /x–3 x – 1 /2 ,
k = 1, 2, 3, ... r k-1(x). , p(x) = q(x)q1(x) + r 1(x), q(x) = r 1(x)q2(x) + r 2(x), r 1(x) = r 2(x)q3(x) + r 3(x), ... r k-2(x) = r k-1(x)qk(x) + r k(x), r k-1(x) = r k(x)qk+1(x).
r k(x)
20.2.1.
p(x) r(x)
20.2.1.
1 .
r(x) | q(x),
20.2.3. p(x)
NZD, r(x)
q(x) p(x)
μ
r(x) | p(x) ( . r p(x) q(x).
q(x)
p(x)
D(x) D(x) | p(x) q(x) ђ
p)
q(x), ,
D(x) | q(x) D(x). .
156
: 20.2.4.
(
)
.
I p(x)
q(x)
,
.
NZD.
.
(nzd)
. 299 = 1323 360 = 1320, 23 20 nzd(260, 39) = 13, 299 – 260 = 39. ( m = 7 n = -8)
, nzd(299, 260) = 13 . ђ ( 5.2.5.), m299 + n360. 20.2.5.
:
D(x) p(x)
(NZD) q(x) D(x) = p(x)f(x) + q(x)g(x).
f(x)
13 =
g(x)
r 1(x) = f(x) – q1(x)g(x), r 2(x) = -q2(x)f(x) + (1 + q1(x)q2(x))g(x), r 3(x) = (1 + q2(x)q3(x))f(x) – (q1(x) + q3(x) + q1(x)q2(x)q3(x))g(x), ... D(x) = r k(x) = p(x)f(x) + q(x)g(x). )
20.2.6.
:
2322
2322 = 6543 + 360 654 = 3601 + 294 360 = 2941 + 66 294 = 664 + 30 66 = 302 + 6 30 = 65 , nzd(2322, 654) = 6.
nzd(2322, 654) = nzd(654, 360) nzd(654, 360) = nzd(360, 294) nzd(360, 294) = nzd(294, 66) nzd(294, 66) = nzd(66, 30) nzd(66, 30) = nzd(30, 6) nzd(30, 6) = 6
20.2.7. f(x) = x4 + 3x3 – x2 – 4x – 3 :
3f(x)
nzd (
654.
NZD g(x) = 3x + 10x2 + 2x – 3. 3
g(x):
1 (3x4 + 9x3 – 3x2 – 12x – 9):(3x3 + 10x2 + 2x – 3) = x , 3 4 3 2 3x + 10x + 2x – 3x /-x3 – 5x2 – 9x – 9 10 2 x3 x2 x 1 /3 3
157
:
I
5 25 x2 x 10 3 3 5 25 5 1 q1(x) = x r 1(x) = x2 x 10 = x2 5 x 6 . 3 3 3 3 ,
.
NZD
g(x) q1(x), . (3x3 + 10x2 + 2x – 3):(x2 + 5x + 6) = 3x – 5. 3x3 + 15x2 + 18x /-5x – 16x – 3 -5x – 25x – 30 /9x + 27 x2 + 5x + 6 9x + 27, x + 3. x+2 , x+3 f(x) g(x).
. – 1)(x + 3), ,
ђ
f(x) g(x) , NZD f(x) g(x) f(x) = (x3 – x – 1)(x + 3) x3 – x – 1 3x2 + x – 1 , x+3
, .
,
.
g(x) = (3x2 + x
(NZD). 20.2.8. 1. μ ) 5x2 + 5y2 + 8xy + 2y – 2x + 2; ) 4x2 + 4y2 – 6xy – 4x + 4y + 4; 2 2 ) 6x + 6y + 8xy – 10x – 10y + 1; ) 6x2 + 6y2 – 8xy + 10x – 10y + 1. [4(x + y)2 + (x – 1)2 + (y + 1)2; 3(x – y)2 + (x – 2)2 + (y + 2)2; 5(x + y – 1)2 + (x – y)2 + 1; 5(x – y + 1)2 + (x + y)2 + 1.] 2. ) x4 + 3x2 + 4; ) x4 +x2y2 + y4;
: ) x4 + 4y4; ) x8 + x4 +1. [(x2 – x + 2)(x2 + x +2); (x2 + 2y2 – 2xy)(x2 + 2y2 + 2xy); 2 2 (x + y – xy)(x2 + y2 + xy); (x2 + 1 – x)(x2 + 1 + x)(x4 + 1 – x2).]
3. ) (x + y + z)3 – x3 – y3 – z3; ) (x – y)3 + (y – z)3 + (z – x)3;
μ
) x2(y – z) + y2(z – x) + z2(x – y); ) (x + y + z)(xy + yz + zx) – xyz.13 [3(x + y)(y + z)(z + x); (y – z)(x – y)(x – z)] μ ) x3 – 6x2 – x + 30 x3 – 19x – 30; ) x3 + 13x2 + 47x + 35
4. NZD ) x3 – 7x + 6 x3 + 2x2 – 5x – 6; ) x3 + 10x2 + 31x + 30 13
ђ
,
1.
, 15.
1980.
158
: 2x3 + 9x2 – 7x – 6; 5. ) 2322
3x3 + 23x2 + 19x – 7. μ
nzd 763ν
) 3λλ7
654ν
I
) 1735
635.
6. ) (3x3 – 4x +5):(x + 2); ) (2x4 + 3x3 – 4x + 5):(x2 + x + 1);
μ ) (4x + x – 1):(x – 2); ) (3x4 – 2x2 + 4x – 5):(x2 – x + 1).
7. NZD ) 2x3 – x2 – 2x + 1 2x2 + 3x + 1; ) x4 – x3 – 2x2 + x – 2 x3 – x2 – x – 2;
μ ) 2x3 – x2 – 2x + 1 2x2 – x – 1; ) x4 + x3 – 2x2 + x + 2 x3 + x2 – x + 2.
8. )
3
NZD )
3x3 10 x2 2 x 3 ; x 4 3 x3 x 2 4 x 3
)
:
3x3 10 x2 4 x 3 ; x 4 3 x3 x 2 2 x 3
)
x 2x 2x 1 ; 4 x 2 x3 3 x 2 2 x 1 3
2
x3 2 x 2 2 x 1 . x 4 2 x3 3 x 2 2 x 1
2
20.3. ( 20.3.1. q(x) r(x) g(x).
ђ 5.2.3.).
,
f(x) g(x) f(x) = g(x)q(x) + r(x),
r(x)
f(x) = a 0 + a 1x + a 2x + ... + a mxm, g(x) = b0 + b1x + b2x + ... + bnxn mn 0. n (&5.1.) m. m < n ђ (q 0, r f). , ђ f(x) m0 n. m = m0, a f1 ( x) f ( x) m xmn g ( x) m, xm . bn q1(x) r(x) f1(x) = g(x)q1(x) + a q( x) m xmn q1 ( x) . r(x). ђ , f(x) = g(x)q(x) + r(x), bn :
.
, f(x) = g(x)q(x) + r(x), f(x)
q(x)
r(x) g(x)
159
:
I
, f(x) = x3 + 1 q(x) = x + 1 , . r(x) ο 0,
g(x) = x2 – x + 1 (f)
(g). 20.3.2. (
)
:
f(x) (20.3.1.)
R
x=
.
20.3.3. x3 + kx2 – 7x + 6
x-
f(x) = (x - )q(x) + c, f() = c.
k g(x) = x – 2
: f(2). Ј
,
f(2) ο 3 f(x)
a
f(x) = x– 2 k = ¾.
f(a) = 0.
a
.
20.3.4.
) ђ
(
,
1, 2
(j = 0, 1, 2, ..., m)
,
, 3.
f(x)
. f(1) =
.
f a 0 , (20.3.4.)
20.1.1.
ђ
(20.3.2.)
.
ђ
ђ
.
aj
20.3.5. g(x)
f(x)
x {1, 2, -3}, f(x) ,
,
|a 0 |a m.
,
x3 – 7x + 6.
: f(x) = x3 – 7x + 6 0, f(2) = 0 f(-3) = 0. , x – 1, x – 2 x + 3. f(x) , . f(x) = x3 – 7x + 6 = (x – 1)(x – 2)(x + 3).
ђ
c
3.
f(x) 8 + 4k – 14 + 6 = 3,
a (x – a). f(a) ο 0
f().
, Z
.
g(x), f(x). (20.3.1.)
.
160
:
I
20.3.6. g(x) = x – 1.
f(x) = 1 + x3 + x10 + x101
2
:
(20.3.1.)
= ax + b = ((+1)2 – 1)0 + a(+1) + b + b, . a = b = 2.
f(x) = (x2 – 1)q(x) + r(x), ξ 2. , x = 1 2 f(-1) = ((-1) – 1)0 + a(-1) + b, , r(x) = 2x + 2. ,
,
ђ
.
.
20.3.7.
r(x) f(+1) 4 ο a + b, 0 = -a
mN
f(x)
f(x) = a(x – x1)(x – x2)...(x – xm), a 0,
, ,
m
μ
1. ) x3 – 2x2 + 3x + 4 x – 2; ) x2 – 3x + 4 2x + 3;
4.
3
a –b ?
x8 + 3x3 + ax + b
x2 – 1
x3 + ax2 + x + b
x2 + x – 2
2
5. + 2,
a
x3 + 2x2 + ax + b 7x + 7,
3.
x,
3x
b.
6.
14
2
f(x) = x4 + ax3 + 3x2 – 3x + 2
g(x) = x – 1. 2
) x3 + 2x2 – 3x – 4 x + 2; ) x2 + 2x – 3 3x – 2. μ ) x + x – 17x + 15; ) x4 + 2x3 – 3x2 – 4x + 4; ђ) 2x4 + 3x3 – 10x2 – 5x – 6.
2. ) x3 + 9x2 + 23x + 15; ) x4 – 2x3 – 3x2 + 4x + 4; ) 2x4 – 3x3 – 10x2 + 5x – 6; 3.14
x1, x2 , ..., xm
x1, x2 , ..., xm, .
m
20.3.8.
.
x2 – x – 2
3a + 3b ?
10.
2001–0λ.
.
161
:
I
7.
x2008 + x1007 + 1
x2 + 1.
8.
x2004 – x2000 + x
x2 – 1.
9. x+1
2
f(x) -1.
x–1
x5 – 3x4 + ax3 + x2 + b
10.
21.
1, f(x)
x2 – 1.
(x – 2)2,
a 2 – b2.
ј
21.1. 21.1.1.
210. ђ
5, 6 7. ..., -3, -2, -1, ..., 4, 5, 6, 7, 8, ...
ς .
210 ο 2357 = 567. .
210, . x3 – x = 210. , 216) – (x – 6) = 0, (x – 6)(x2 + 6x + 36) – (x – 6) = 0, 0. μ x – 6 ο 0, x2 + 6x + 35 ο 0. μ 5, 6 7. x2 + 6x + 35 = 0 26 = 0 . 21.1.2.
x = 6,
, (x + 3)2 + 729 m3.
x + 2. :
,
a = x + 2. x + 2 = 9, x = 7 m.
3
(x + 2) , 21.1.3.
x
.
2
66 m . : ο , x2 + 5x – 66 = 0. x(x + 11) – 6(x + 11) = 0, : μ
ο + 5,
,
ο 66.
(x + 11)(x – 6) = 0. x = -11, ο -6, . x = 6 m, ο 11 m.
21.1.4. 1. ) 156;
,
(x – 1)x(x + 1) = (x3 – 2 (x – 6)(x + 6x + 35) =
) 210.
V = a 3,
x+5
. 729 =
.
x(x + 5) = 66, x2 + 11x – 6x – 66 = 0, ,
μ
162
:
I .
2.
3.
.
7.
3.
.
30.
ς
. 3. ) S = t2 – 15t + 5036; 4. μ ) m = 4, p = 30;
t ) S = t2 – 16t + 5060. 5000?
ς m
.
.
,
p
,
) m = 3, p = 14. [3, 10; 2, 7 m]
5.
.
6.
24
24 m
.
. ђ
ђ
2 cm
. .
70 m2.
107 m2]
[145,
21.2.
. 3 , 5
21.2.1. 7 . 9
ς :
8
3 x 7 . 5 x 9
.
21.2.2.
.
:
Ј ,
/3 + /8 ο 1,
21.2.3.
3 1/3,
24/11
6 ς
4
.
= 4. , ς
1/8.
. ,
2 Ј
.
.
163
: :
6 + x, Ј
4 4 1, 6 x x
I 4
x,
μ
12
,Ј
.Ј 6
.
21.2.4. 2 . 3
4 , 9
1. ς
[6] 25 . 6
2. ς
[ 3. 12
10
.
ς
1 4
6]
, [5,5 min]
4.
,
,
2
3
.
.
ς [3
5.
,
3
ς
2
.
6
]
[6
]
.
21.3. f(x) = a 0 + a 1x + a 2x + ... + a mxm m = 0, 1, 2, 3, 4, ... . , f3 ( x)
12.3.1. 1 3 x 2
f 4 ( x)
,
,
m, ,
,
.
m .
f0(x) = 2, f1(x) = 2x, f2(x) = x2, 1 4 x . 2
: x f0 f1
. -2 2 -4
-1 2 -2
0 2 0
1 2 2
2 2 4
164
: 4 -4 8
f2 f3 f4
1 -0,5 0,5
I 0 0 0
1 0,5 0,5
4 4 8 μ
ђ
f0
(
(
) ) .
12.3.2. x. i.
ς
( ab ,
(0, 0) (1, 2). (0, 0), ) 4. - .
2. f2 (
(
f1 ) -
) ,
a = 50 cm, b = 40 cm.
165
: ii. Ј iii.
I ?
ς :
. (i.) V = (a – 2x)(b – 2x)x = 4x(25 – x)(20 – x) cm2. (ii.)
, 0 ξ < 20 cm.
a, b. (iii.) , 0, b/2
a/2
V
.
.
, , , a < b. , 20
-
25 ( ђ
0,
). , (
7,4 cm)
. ( 12.3.3.
m-
)
(m = 0, 1, 2, ...) ђ m+1-
.
.
166
:
I
3
:
(-1, 0), (0, -3), (3, 0). 2.
y = ax2 + bx + c,
.
μ a(-1)2 + b(-1) + c = 0, a02 + b0 + c = -3, a32 + b3 + c = 0. a = 1, b = -2, c = -3. y = x2 – 2x – 3. 12.3.4. 1.
2.
3.
4.
y = -1, y = 3
. . .
1.
(
2.
(
ђ
y = x + 1. )
(-1, -3) )
(2, 3).
(-1, 3), (1, 1), (3, 7)
μ
)
)
)
)
167
:
)
I
ђ)
5. t
1.
,
250
. (
,
100 ML
) .
[
V(t) = -t3 + 30t2 – 131t + .
ς Ж
168
:
6.
64
40
I
.
,
cm cm .
.
. [Vmax(8) = 4608 cm3]
169
:
I
VIII . trТgōnon – ђ
(
,
(
, metron -
(
(
(
(
ђ
1λ00-1600. . . .) . ђ 430. . . .). , 140. . . .) , 100. . ).
1650. . . .),
,
) .
,
,
(
, IV
V
).
БII
, ,
.
(De triangulis omni modis, 1464.) . (De revolutionibus orbium coelestium, (Opus palatinum de trianulis, 1596.) ,
.
1543.)
.
ђ
-
.
22.
ђ
.
,
.
22.1. (C1 = C = 90), (1 = ).
( B
)
A1B1C1
ABC
А
180,
(1 = ).
,
λ0.
,
,
170
: , .
,
.
,
,
I
,
. 22.1.1 sin
.
ђ
a b a b , cos , tg = , ctg = , c c b a 1 1 , csc . sec cos sin
sin
ђ ,
.
.
b b a a , cos , tg = , ctg = . c c a b
,
.
.
,
22.1.2. i. sin 2 cos 2 1 ; ii. 1 + tan2 = sec2 ; iii. 1 + ctg2 = csc2.
a 2 b2 a b : ) sin cos c2 c c 2
2
c2 1 . c2
22.1.3. i. tg = : tg =
2
ο
2
cos sin ; ii. ctg = . cos sin
a a / c sin . b b / c cos
22.1.4. tg ctg .= 1. : tg ctg .= .
a b 1 . b a ,
.
,
.
171
:
I
22.1.5. + = 90 sin cos tg = ctg . .
t
22.1.6. sin
cos
1 t2
sin cos 2 2 2 t (1 sin ) sin . Ј (1 t 2 ) sin 2 t 2 , t cos sin 2
2
. 22.1.2.)
(
2
μ ) 7, 24, 25; ђ) 2,8; 4,5; 5,3.
) 5, 12, 13; ) 1,2; 3,5; 3,7; ) cos = 0,96; 1 ) cos ; 1 a 2
1 (sin x cos x) ; cos 2 x 2
(1 t 2 ) cos 2 1
.
2. ) sin = 0,6; 2a ) sin 2 ; a 1
4.
t = tg .
,
t 2 cos 2 sin 2 ,
22.1.7. 1. ) 3, 4, 5; ) 2,0; 2,1; 29;
)
1 t 2
t = tg =
:
3.
1
: )
1 tg 2 ; 1 ctg 2
μ
ђ) ctg
1 a 2 . a
csc 2 sin 2 ; tg 2 [2tgx; tg2; ctg4]
) ctg2 = cos2 + (ctg cos )2; 1 ) sec 2 csc 2 . 2 sin cos 2
) ctg sec = csc ;
6. ) tg2 sin 2 tg2 sin 2 ;
) tg = 2,4;
)
μ ) sin215 + cos225 + cos265 + sin275 = 2; ) tg 1 tg 2 tg 3 ... tg 89 = ctg 45.
5. ) tg + ctg = sec csc ;
μ
μ )
sin 1 cos ; 1 cos sin
172
)
sin 3 x cos 3 x sin x cos x ; 1 sin x cos x
tg x μ sin x 2 cos x ) 5; 3 sin x 4 cos x
:
I
)
1 1 ctg 2 x . sin 2 x cos 2 x 1 ctg 2 x
)
3 sin x 2 cos x 4. 5 sin x cos x
7.
μ
8. sin x, cos x, tg x ctg x ) 3 sin x + 4 cos x = 5;
) 5 sin x + 12 cos x = 13.
22.2. 30, 45
.
60, 22.2.1.
30
sin 1 2
cos 3 2 2 2 1 2
2 2 3 2
45 60
tg 3 3
:
1
3
:
ABC CD
AC = = h.
ACD,
3a 2 a , h a = 4 2 2
2
2
h
a 3 . 2 1 a /2 h = , cos 30 sin 30 a 2 a
=
3 2
tg 30
1 a/2 = = h 3
3.
ACD
173
:
I
3 a/2 1 h = , cos 60 = a a 2 2 h tg 60 = , 3. a /2 MNPQ s d, . MNP sin 60
d MP =
sin 45
„
.
0
s2 s2 = s 2 .
2 2 s s = , cos 45 = d d 2 2
( ,
ђ
1, )
5
“
tg 45
s = 1. s
0,001 λ0.
.
22.2.2.
sin 0 1
0 90
cos 1 0
tg 0
ctg 0
22.2.3.
1. μ 2 2 2 ) cos 30 + sin 45 + ctg 60; sin 60 tg 30 ) ; ctg 45 cos 60
) sin2 60 – cos2 45 + tg2 30; cos 30 ctg 60 ) . tg 45 sin 30 10 cm.
2. 3.
30
. 5
4. 45 5.
6. 7.
60. AC
2 cm.
+ = 90 sin + cos = 1, < 45,
10 cm.
3, BAC
ABCDE
.
174
:
I
a) 2sin(45 – ) – 5cos(45 + ) + 1 = 0; ) tg( + 30) = 4 – 3ctg(60 - ).
8.
,
,
) sin cos ; 2 2 2
μ
) tg ctg . 2 2 2
22.3. ABCD
a b AD,
P 1 = ab.
1 a P2 ab . 2 CDA a = AD.
, ABC
CDA, a CD = b
ABCD
h
,
P 3 = ah.
DD1C A1BCD1
,
AA1B,
. ,
AC
ABC CDA, , ABC ABCD. 1 h P3 ah . 2
, CDA a b
P4
22.3.1.
a b h. 2
P
ab sin . 2
,
h P 4 = mh, a b m 2 . ,
ABCD ABC
.
AC 1 1 ah bh 2 2
ACD
. a
b
C,
175
:
: h = BD CDB.
ђ
= ACB,
ABC,
22.3.2. (
.
a b c 2r , sin sin sin a, b c :
ABD
, r
P
: sin
c 2r
a bc 2
, r
ABC . AD ABD = 90. , ADB = ACB = , c AB , . sin 2r . AD sin A B P
22.3.3.
s
b = AC,
)
,
,
a = BC
, h = asin . ah P 2 ab P sin . 2
,
22.3.4.
I
abc , 4r
a, b
,
k(O, r),
.
c .
ab sin 2
(22.3.1.)
(22.3.2.).
P s ,
.
.
176
: :
k(O, ),
.
P
I
ABC
ABO, BCO CAO c a b , 2 2 2
a bc . 2
22.3.5. ( s
:
a bc 2
)
P s(s a )(s b)(s c) ,
a, b, c
. ABC
CD p + q = c. h2 = a 2 – q2 = b2 – p2, a 2 – b2 = q2 – p2 = (q – p)(q + p) = (q – p)c. q+ p 2 2 = c, q – p = (a – b )/c, c2 a 2 b2 c2 a 2 b2 , p . q 2c 2c c 2 a 2 b 2 c2 a 2 b2 = a h2 = a 2 – q2 = (a – q)(a + q) = a 2c 2c
2ac c 2 a 2 b 2 2ac c 2 a 2 b 2 b 2 (a c) 2 (a c) 2 b 2 = = 2c 2c 4c 2 [b (a c)][b (a c)][(a c) b][(a c) b] = 4c 2 (b a c)(b a c)(a c b)(a c b) 2( s a ) 2( s c) 2( s b) 2s = . 2 4c 4c 2 hc 4s( s a )(s b)(s c) hc 2 s(s a )(s b)(s c) . , P = h2 2 2 c s(s a )(s b)(s c) . 22.3.6.
12
1. 30. 2.
42,
ς
,
ђ 2μ3,
ђ
60.
177
: 3.
12,
ς
4.
.
5. 6.
10
12,
24
10,
7, 24
.
7. ) = 45, = 60, a = 4;
8.
I 5
3.
45. 13
15.
.
25. μ ) = 30, = 45, c = 6.
) = 60, a = 3, b = 2. 3 3 2 [ 2(3 3 ) ; 9( 3 1) ; ] 2
a, b
c
,
.
ς
[2(a 2 + ab + b2)]
a 2 b2 , 4
9. .
[ P
10.
a
b
a 2 b2 a 2 b2 ab sin = sin = 1, 2ab 4 2 sin = 1. = 90 (a – b)2 = 0] ta , tb tc ? 4 [ t (t t a )(t t b )(t t c ) , 2t t a t b t c ] 3
23. ј
њ
,
, .
( i.
,
.
) .
μ
(
). .Ј
. 178
: ii. Ј
(
I .
λ0).
.
. μ i.
,
ђ
ii. iii.
.
.
180,
.
(
).
.
iv.
.
.
. ,
.
,
.
23.1. 1.
ABC.
CD
ABC h ADC : h = bsin ( sin b h h = asin . CDB : sin a a b . bsin = asin , . sin sin
ADC
sin(180-) = sin )
ABC
b c . sin sin
А (22.3.2.)
CDB.
:
a b c . sin sin sin
179
: 2.
,
.
, ABC, BC = 6 cm, A = 44
3.
AC
C = 45. AB 19 , sin 45 sin 25 31,78999 cm, AB 31,8 cm ( ).
sin 59 BC 7,40364 sin 44 .
.
ђ
( ),
(
AC BC , sin B sin A
,
AC 7,40 cm (
.
,
sin B
12 sin 22 9
23.1.7. ABC, BC = 8 cm, A = 46
,
.
B. AB1C
B2, 12 12 9 . sin B1 sin B2 sin 22
ς
ђ
(&11.).
, ABC, A = 22, BC = 9 cm, AC = 12 cm.
B1
).
AB.
)
,
1.
AC.
B = 25.
A + B + C = 180, , sin 45 AB 19 sin 25 5 , 3
, 6.
B = 59.
, ABC, AC = 19 cm, A = 110
4.
5.
I
,
: B1 30 B = 58.
AB2C, AB1 AB2. ,
B2 150.
AC.
180
:
2.
ABC, AC = 15 cm, A = 105 ABC,
3. 4.
I
B = 31.
AB.
B = 55, BC = 10 cm, AC = 13 cm.
ABC, A = 25, BC = 8 cm, AC = 11 cm.
5. 4742’.
. A =
AC = 26,6 cm .
6.
15,5 cm,
.
ђ
5334’.
23.2. , 1. (
,
.
) a, b, c 2 a = b + c – 2bccos A, b2 = c2 + a 2 – 2cacos B, c2 = a 2 + b2 – 2accos C.
2
. ABC,
:
2
. ,
).
(
:
,
ABC,
CD = h – x, x – c.
(
)
AB.
ADC h2 = b2 – x2,
x = AD, , .
DB
c
h2 = a 2 – (x – c)2.
, b2 – x2 = a 2 – (x – c)2 b2 – x2 = a 2 – (x2 – 2cx + c2) a 2 = b2 + c2 – 2cx.
181
: x = bcos A, . C ο 0, 2.
I
a 2 = b2 + c2 – 2cbcos A. A
B
. .
cos C ο 0,
ABC, BC = 9 cm, AC = 10 cm
.
.
C = 56.
,
,
AB A.
c2 = a 2 + b2 – 2abcos C c2 = 92 + 102 - 2910cos 56 c = 8,96355 ... . , AB = 8,96 cm 3
c2,
.
b2 c2 a 2 = 0,55416... 2bc A = 56,3, . cos A
3.
XYZ, Y = 124, XY = 12 cm
YZ = 21,6 cm.
XZ X.
y2 = z2 + x 2 – 2zx cos Y y2 = (12)2 + (21,6)2 -2(12)(21,6)cos 124, y = 30,00742 ... , . XZ = 30,0 cm, 3 .
y 2 z 2 x2 = 0,80242 cos X 2 yz X = 36,6, 0,6 60 X = 3636’.
4.
.
ABC, a = 8, b = 15
.
c = 12, (
10.1.7.).
182
:
I
, cos
b = 15.
a 2 c2 b2 = -0,08854... . 2ac .
= 95,
.
5.
4μ5μ6. .
, a, b .
,
c
cos
, k a = 4k.
4k, 5k 6k
b2 c2 a 2 (5k) 2 (6k) 2 (4k) 2 25 36 16 = = = 0,75. 2bc 60 2(5k)(6k) . ο 41,40λ62211... , 8 .
Ј 60 , 0,40962211... 60 ο 24,5773266...
60,
. ђ ,1 60 60, . 0,5773266... 60 = 34,639596... = 41 24’ 35” .
= 41 24’ 35”
41
1. 2.
24 35 = 41,410, 60 60 2
XYZ, Y = 112, XY = 15,4 cm
4.
5. 135.
ABC, a = 9, b = 17
. ,
,
μ
.
23.2.6. ABC, BC = 21 cm, AC = 18 cm
3.
.
C = 62.
AB B.
XZ Z.
YZ = 17 cm.
c = 8,
7μ5μ6. . 32 .
,
ђ
183
:
I [ 16 2 2 , 16 2 2 cm]
ABCD, AB = 6 cm, BC = 7 cm, AD = 12 cm, ABC = 120, ACD = ADC, 3 .
6. 70.
24. Г
ј
24.1. ,
ABC D AB,
c .
ADC BDC. 1 = ACD c = AB c1 = AD
Њ
= ACB 2 = DCB, c2 = DB. ђ ,
1 = CDA 2 = BDC. ,
c1 b sin 1 . c2 a sin 2
24.1.1.
a sin 2 . c 2 sin 2
c1 sin 1 b sin 1
: ђ
ђ
sin 1 sin 2 ,
. 24.1.2. T B C.
: A B
C
CD
,
24.1.4.
ACP
C
1 = 2 QBC
sin 1 = sin 2, P
je
c1 = c2 bsin 1 = asin 2, ADC BDC .
, 24.1.3.
:
ABC
Q
. c1 : c2 = b : a. AB
AP AQ AC . BP BQ BC
ABC
2
,
184
:
I
: = , PCQ = .
AP AC sin BP BC sin( )
ACP = QBC (24.1.1.)
AQ AC sin( ) , BQ BC sin
. 24.1.5.
P 1, P 2, P 3 :
A1A2A3,
A2A3, A3A1, A1A2
P1 A2 P2 A3 P3 A1 sin P1 A1 A2 sin P2 A2 A3 sin P3 A3 A1 . P1 A3 P2 A1 P3 A2 sin P1 A1 A3 sin P2 A2 A1 sin P3 A3 A2 ,
.
:
(24.1.1.) : P1 A2 A1 A2 sin P1 A1 A2 , P1 A3 A1 A3 sin P1 A1 A3 P2 A3 A2 A3 sin P2 A2 A3 P2 A1 A1 A2 sin P2 A2 A1 P3 A1 A A sin P3 A3 A1 1 3 . P3 A2 A2 A3 sin P3 A3 A2 , . (A1A2A3)
A1)
24.1.6.
, , P1
Q1
A1A2A3.
μ
.
(A2A3) (
A2A3
P1 A2 Q1 A2 sin P1 A1 A2 sin Q1 A1 A2 : : . P1 A3 Q1 A3 sin P1 A1 A3 sin Q1 A1 A3 :
(24.1.1.)
P1 A2 A1 A2 sin P1 A3 A1 A3 sin( )
185
:
I
Q1 A2 A1 A2 sin( ) . Q1 A3 A1 A3 sin , P1 A2 Q1 A2 sin( ) sin . : : P1 A3 Q1 A3 sin( ) sin . 24.1.7. A, B, C
A’, B’, C’,
AB A' B' OB OB' . : : AC A' C ' OC OC' : OA'C ' ,
,
(24.1.1.) AB OB sin AC OC sin( ) A' B' OB' sin . A' C ' OC' sin( ) ђ
OAC
.
24.1.8. A, B, C, D
A’, B’, C’,
D’, A' C ' B' C ' AC BC = . : : A' D' B' D' AD BD
(24.1.6.),
, .
186
: : OB, OA.
BC : BD BC : BD
I
, OAD OBD
(24.1.7.) AC A' C ' OC OC' = = : : AD A' D' OD OD' AC A' C ' B' C ' , = : AD A' D' B' D' A' C ' B' C ' B' C ' AC BC , = . : : A' D' B' D' B' D' AD BD 24.1.9. ,
(
(A, B; C, D) =
. cross-ratio) ( . double ratio), A-B-C-D,
AC BC AC BD = . : AD BD BC AD
(24.1.8.)
, 24.1.10.
.
(24.1.8.) .
,
ђ
, .
,
.
A, B, C, D
3, 1 2 . (A, B; C, D) = AC AD 4 6 = = : = 2. ђ , : BC BD 1 3 2 1 CD CB = : = -1. (C, A; D, B) = : 6 3 AD AB (24.1.8.)
,
ς
. 24.1.11.
.
,
ђ
ђ ,
.
, . ,
. ,
.
ђ
187
:
I
! 24.1.12. ђ . 2 3 AC AD 4 = : = . (A, B; C, D) = : 1 2 BC BD 3 , =
ς A, B, C, D,
(A, B; C, D) =
AC AD : BC BD
20 12 5 4 12 8 12 8 4 = = . , : 8 24 84 4 3 8 12, 8 4 ! , AC AD 12 4 12 4 2 16 6 4 = = = . (A, B; C, D) = : : 4 BC BD 4 18 3 42 , 12, 4 2 !
24.2. .
,
,
.
,
24.2.1. OAOA’ = r 2. А k(O, r).
А, A’, O .
k(O, r)
A’. k. OA
А.
A’.
, k(O, r) P
Q.
.
.
А PQ
k OA
A’.
OAOA’ = r , 2
(
) ,
ђ А A’ А
OA’P, PA’A OPA . OP:OA = OA’:OP . ђ
.
24.2.2.
188
: i.
I ,
.
ii.
.
.
,
iii. 24.2.3.
. :
, .
,
B
a B’
O. OA
.Њ
A,
A’; a A’, OA’B
.
A
OA’B OB’A , r = OA’OA = OB’OB OA’:OB = OB’:OA A’OB’, A OB’ B’, B. 2
(
,
) .
24.2.4.
ђ ,
,
.
. (A)
(s)
(k). (P Q) (p q). (a)
. , Q, k, a P, Q
B А. ,
24.2.5.
.
P А
s A.
B
!
.
189
: 24.2.5. , .
,
,
15
: XOY R R
, (24.1.9.).
,
.
,P X Y OP R, OXP ORX P, . XYR RP,
I
Q
.
PQ
ARS.
PS
.
, RQ, PRQ
RQ
.
(24.1.3.),
P
,
Q
XY
. 24.2.6.
,
. , QR
Y,
А.
А RP SQ
APQ Z. 24.1.1.
YZ
Y Z
APQ M, ARS N.
MP AP MQ AQ
NR AR . NS AS (24.1.1.) MP ZP sin MZP MQ ZQ sin MZQ YQ ZQ sin YZQ , YR ZR sin YZR ZP SP sin QSP ZR SR sin QSR YR SR sin PSR . YQ SQ sin PSQ
15
ђ
, . (R. Lachlan: An Elementary Treatise on Modern Pure Geometry,1893.).
190
:
MP SP sin PSR . MQ SQ sin QSR M. А.
,
24.1.1.
I
AP SP sin PSR , AQ SQ sin QSR N.
, MN
24.2.7. . 24.2.8. .
191
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