3+2+5=0
:
ϳ :
:
, 25x-5y-13z-67=0 .
³: 25x-5y-13z-67=0.
7.
, .
1) (x0,y0) , .
2)
3) d1 :
4) d2
5) , d1=d2
6) y0; , ;
:
7)
)
)
³:
8.
, 16;
.
г -
b.
- .
:
, :
³:
9.
:
2x2+3y2+5z2-4z-6y+10z-1=0
. :
³: .
10.
ǒ : x2 = 2pz
, . , .
| |||||
2
I.
1. ; 3. ; 5.
2. 4. ; 6. ; 10.
7. ; 8. ; 9.
II. , :
1. sin3x; 2. sin4x; 3. cos5x; 4.cos6x; 5. sin3xcos5x;
6. 3sin4xcos3x; 7. 5cos3xsin2x; 8. sin7x; 9. cos4xsin3x; 10.sin5x.
III. :
1. cos x + cos 2x+ + cos nx; 5. cos +cos +cos ++cos
2. sin x + sin 2x + + sin nx; 6. sin +sin +sin ++sin
3. sin x + 2sin 2x + + nsin nx; 7. cos2 x + cos2 2x+ + cos2 nx;
4. cos x +2cos 2x+ + ncos nx; 8. sin2 x + sin2 2x + + sin2 nx;
9. + cos x + cos 2x+ + cos nx; 10. 1+acosφ+a2cos2φ++akcoskφ.
IV. 1. 6- .
2. , n- n.
3. 15- .
4. 10 .
5. 15- .
6. , nm- n- m- .
7. , , .
8. , 2+ , .
|
|
9. n- .
10. 24- .
V. :
(*)
1) (*) = .
2) .
3) :
) ;
) .
4) ' , ;
5) , :
) *- :
) : (/)~(/A-1).
6) (*) .
(*)
a | ||||||||||
b | ||||||||||
c | -4 | -3 | -5 | -2 | ||||||
e | -2 | -4 | -2 | -2 | -4 | |||||
f | -5 | |||||||||
d | -1 | -2 | -2 | -3 | -8 | -1 |
VI. (*).
(*)
(*)
m | ||||||||||
n | -1 | -3 | -2 | -5 | -2 |
2
1. : (1- )30.
: z=1-i n- . : z=r(cos φ+isin φ), ;
=1; b=- ; i= =2; .
; z=2(cos +isin ).
z30= .
³: 230.
2. , cos3.
': z = cos x + isin x.
cos x =; sin x =;
cos kx =; sin kx =;
cos3x = = (z3 + 3z2 * z-1 + 3z-2 * z +z-3) = ((z3 + z-3) + 3(z + z-1)) =
= +3 = (cos 3x + 3cos x);
³: (cos 3x + 3cos x).
3. : cos x + cos 2x + + cos nx.
': S = cos x + cos 2x + + cos nx,
T = sin x + sin 2x + + sin nx.
S + Ti = (cos x + isin x) + (cos 2x + isin 2x) + + (cos nx + isin nx);
cos x + isin x = a.
S+Ti = a + a2 ++ an = = = = =
= = A.
a + a-1 2=cos x +isin x +cos x isin x 2 =2cos x 2 = 2(1 cos x) = 4 sin2;
|
|
A = =
= + i;
,
S = = =
= = =
= =;
³:.
4. 15- .
': ξ1 15- :
ξ1 = cos + isin;
ϳ ξ1 0 14 15- . 15- .
15- . S.
S = Σ ξi (1 + (ξ5 + ξ10) + (ξ3 + ξ6 + ξ9 + ξ12)) = 0 (1 1 1) = 1.
³: S = 1.
5. :
x1 2x2 + 3x3 = 2
2x1 x2 3x3 = 2
x1 + 2x2 4x3 = 1
1 -2 3
1. = 2 -1 -3;
1 2 -4
2. .
3.
)
)
4. ' , .
5. , .
6. ' .
':
1) :
= ~ ~ ~ ~; r(A) = 3.
2) -. .
=;.
~ ~ ~ ~; r(A) = 3;
r(A) = 3.
, , .
3) :
)
| A | = = = = = 3 = 3 = 15
)
| A | = = 1 + 2 + 3 = 10 + 2 * (-5) + 3 * 5 = 15.
4) ' . , , :
~.
x1 2x2 + 3x3 = 2 x1 = 1;
x2 + 2x3 = 3 ~ x2 = 1;
5x3 = 5 x3 = 1;
³: (1, 1, 1).
' , :
x1 =; x2 =; x3 =.
, '.
∆x1 = = 8 12 6 3 + 12 + 16 = 15;
∆ = 15; ∆x2 = = 8 6 6 + 6 3 + 16 = 15;
∆x3 = = 1 + 8 + 4 + 2 + 4 4 = 15;
x1 =; x1 = 1; x2=; x2 = 1; x3=; x3 = 1.
³: (1, 1, 1).
5) , . (|| ¹ 0), .
) *, .
A11 = 10; A21 = 2; A31 = 9;
=; A12 = 5; A22 = 7; A32 = 9;
A13 = 5; A23 = 4; A33 = 3.
|
|
A* =. A-1 =; ∆ = 15.
A-1 =.
) . :
( / ) =
( / -1).
( / ) ~ ~ ~ ~
~ ~ ~
~
-1 =.
6) :
x1 2x2 + 3x3 = 2
2x1 x2 3x3 = 2 =; =; =.
x1 + 2x2 4x3 = 1
: * =
* =, = -1 * ' .
= * =
=(+ ; + ; + ) = (1, 1, 1).
³: = (1, 1, 1).
6. ' ' :
x1 + 2x2 + x3 + 3x4 = 9
2x1 x2 3x3 + x4 = 2
|
|
=.
:
~; r(A) = 2; r(A) = 2.
, ' (r < n), n .
:
|
5x2 5x3 5x4 = -20 x2 + x3 + x4 = 4
3 4 . : 3 = 4 = 0,
|
x2 = 4 2 = 4;
, 0 = (1, 4, 0, 0) ' .
, :
x1 + 2x2 +x3 + 3x4 = 0;
x2 + x3 + x4 = 0.
' ;
4 2 = 2 '. R2 : 1 = (1, 0); 2 = (0, 1) 3 4 <1, 2>.
|
4 = 0; 2 = 1; 2 = 1,
|
' = (-1, -1, 1, 0) ' .
3 = 0, 4 = 1, :
|
4 = 1; 2 = 1; 2 = 1
|
'' = (-1, -1, 0, 1) '.
' ':
|
|
| |||||||
| |||||||||
= α' + β''; = (α, -α, α, 0) + (-β, -β, 0, β) = (α β, -α β, α, β); α, β R.
' :
|
= 0 + = (1, 4, 0, 0) + (α β, -α β, α, β) = (1 α β; 4 α β; α, β); α, β R.
|
|
³: (1 α β; 4 α β; α, β); α, β R.
3
1. R
1. , , ;
2. ;
3. ,
4. ;
5. n .
6. f (), f(0)= 1 ;
7. (, );
8. , ;
9. P.
10. P : - - +=, K0- - .
. , .
:
a | -1 | -3 | -2 | -2 | ||||||
b | -2 | -3 | -5 | |||||||
c | -1 | -3 | -3 | -1 |
. , .
1.
2.
3.
4.
5.
6.
7
8.
9.
10.
V. V U, a 1, a 2,... ak b1, b2,.,b .
e1 | -2 | -4 | -5 | -2 | ||||||
e2 | -2 | -2 | -2 | -4 | -7 | |||||
e3 | -1 | -5 | -8 | -2 | ||||||
t1 | -2 | -2 | -2 | -4 | -7 | |||||
t2 | -3 | -2 | -5 | -9 | -2 | |||||
t3 | -4 | -2 | -8 | -14 | -6 |
V. , , L
) L;
) ;
)
) ;
) ;
1 | ||||||||||
2 | -1 | |||||||||
3 | -2 | -1 | ||||||||
4 | -1 | -2 | -1 | -2 | -1 | -1 | ||||
b1 | ||||||||||
b2 | -5 | |||||||||
b3 | -1 | -5 | -2 | |||||||
b4 | -1 | -1 | -3 | -3 | ||||||
c1 | ||||||||||
c2 | -1 | |||||||||
c3 | -1 | -1 | -6 | |||||||
c4 | -1 | -7 | -4 | |||||||
x1 | -1 | |||||||||
x2 | -2 | -1 | -2 | -1 | ||||||
x3 | -1 | -2 | -3 | -3 | -3 | -1 | -3 | |||
x4 |
|
|
V. ,
R. , :
- E1 = ; = ; = ;
V. A = < 1, 2 > , B b=<b1,b2,> . AB , .
1 | (-3,-1) | (1,3) | (1,-1) | (-1,2) | (2,3) | (1,7
: 2017-01-28; !; : 716 | : : , . |
: 0.279 .