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)
104
127
400
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Copyright 2015
[email protected]
3 , , 2015
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1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 7
1.1 71.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 11
1.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 16
1.2 211.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 23
2.1 -- 232.1.1 . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.1.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 26
2.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 28
2.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 31
2.2 - 332.2.1 . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 35
3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 37
3.1 373.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 42
3.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 46
3.2 1 483.2.1 . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 49
-
3.3 513.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 53
3.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 56
3.4 583.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 62
3.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 65
3.5 713.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 72
3.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 76
3.6 2 783.6.1 . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
3.6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 80
3.6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 86
3.7 2 883.7.1 . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 89
3.7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 93
4 - . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
4.1 974.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 100
4.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 103
4.2 1084.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 110
4.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 112
4.3 1174.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.3.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 118
4.3.3 - . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 132
4.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 136
4.4 1404.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 140
4.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 145
5 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
147
5.1 1475.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 150
5.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 160
-
5.2 1625.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 164
5.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 170
5.3 1725.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 178
5.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 187
6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 191
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 191
6.1 191
191
6.2 191
191
-
-
1.
1.1
1.1.1 1.1 ;
x+ y = . (x, y) .
1.2 y = ;
y = xx A(0, )
1.1: y =
1.3 x = ;
x = yy A(, 0)
: x = .
-
1.2: x =
1.4 x + y = 6= 0 6= 0 .
:
i. 6= 0 = 0. :x+ y = 0x+ y = y = y =
xx A(0,
).
ii. 6= 0 6= 0. : x+y = y = x+ y = x+
.
iii. = 0 6= 0. :x+ y = x = x =
xx A(
, 0).
1.5 ;
-
:{1x+ 1y = 12x+ 2y = 2
1.6 ;
(x, y) - . .
1.7 ;
.
1.8 ;
- 8
-
:
i. .ii. .iii. .iv.
1.9 ;
x, y . , .
1.3:
1.10 ( ) ;
( ), (x, y) . , .
1.4:
- 9
-
1.11 ;
.
1.12 {1x+ 1y = 12x+ 2y = 2
. -
D,Dx, Dy;
{1x+ 1y = 12x+ 2y = 2
. :
i. D =1 12 2
= 1 2 2 1
ii. Dx =1 12 2
=1 2 2 1
iii. Dy =1 12 2
= 1 2 2 1
1.13 ;
{1x+ 1y = 12x+ 2y = 2
:
i. D 6= 0, , (x, y) x = DxD
y =DyD.
ii. D = 0, .
1.14 ;
x + y + z = 0, , , , . .
1.15 ;
:
1x+ 1y + 1z = 12x+ 2y + 2z = 23x+ 3y + 4z = 3
1.16 ;
3x3, , 2x2, 3x3 .
- 10
-
1.1.2 -
1.1 . :
) ( - ) .
) - .
1.1 :{x 2y = 63x+ 4y = 8
1.1 :{x 2y = 63x+ 4y = 8
{x = 2y + 63x+ 4y = 8
{x = 2y + 63(2y + 6) + 4y = 8
{x = 2y + 66y + 18 + 4y = 8
{x = 2y + 66y + 4y = 8 18
{x = 2y + 610y = 10
{x = 2y + 6
y =1010
{x = 2y + 6y = 1
{x = 2(1) + 6y = 1
{x = 2 + 6y = 1
{x = 4y = 1
(x, y) = (4,1).
1.2 . :
i. ... . ...
ii. - ... .
iii. .iv. -
.
1.2 :{x 2y = 63x+ 4y = 8
1.2 :{x 2y = 6 23x+ 4y = 8
{
2x 4y = 123x+ 4y = 8
{
2x 4y + 3x+ 4y = 12 + 8x 2y = 6
{5x = 20x 2y = 6
{x =
20
5x 2y = 6
{x = 4x 2y = 6
{x = 44 2y = 6
{x = 42y = 6 4
{x = 4y = 1
(x, y) = (4,1).
1.3 .
- 11
-
:
i. {1x+ 1y = 12x+ 2y = 2
ii. Diii. D 6= 0 Dx, Dy.
, (x, y) x =DxD
y =DyD.
iv. D = 0 ( ) . , .
1.3 :{x 2y = 63x+ 4y = 8
1.3 : D =1 23 4
= 1 4 3 (2) = 4 + 6 = 10 6= 0,
. :
Dx =6 28 4
= 6 4 8 (2) = 24 + 16 = 40
Dy =1 63 8
= 1 8 3 6 = 8 18 = 10.
, x =DxD
=40
10= 4 y =
DyD
=1010
= 1. (x, y) = (4,1).
1.4 :{
2x 3y = 404x+ 6y = 80
1.4 : D =2 34 6 = 2 6 (4) (3) = 1212 = 0,
.
2 :{
2x 3y = 402x 3y = 40
, 2x 3y = 40. 2x3y = 40 2x = 40+3y x =
40 + 3y
2.
(x, y) = (40 + 3
2, ), R.
1.5 :{
2x 3y = 404x+ 6y = 100
1.5 : D =2 34 6 = 2 6 (4) (3) = 1212 = 0,
- 12
-
. 2 :{
2x 3y = 402x 3y = 50 .
1.4 22. D,Dx, Dy . :
i. D 6= 0 ( D 6= 0) :
x =DxD
y =DyD.
ii. D = 0, .
1.6 {x+ y = x+ y = + 2
1.6 :
D = 11
= 2 1 = ( 1)(+ 1).
Dx = 1
+ 2 = 2 2 = (+ 1)( 2).
Dy = 1 + 2
= (+ 2) = 2 + 2 = 2 + = (+ 1).
:1. D 6= 0 ( 1)(+ 1) 1 6= 0 + 1 6= 0 6= 1 6= 1,
:
x =DxD
=(+ 1)( 2)( 1)(+ 1)
= 2 1
y =DyD
=(+ 1)
( 1)(+ 1)=
1.
2. D 6= 0 = 1 = 1, :i. = 1 :{
1x+ y = 1x+ 1y = 1 + 2
{x+ y = 1x+ y = 3
.
ii. = 1 :{1x+ y = 1x 1y = 1 2
{x y = 1x y = 1 x y = 1 x = y + 1
(x, y) = (+ 1, ) R.
1.5 - . -
- 13
-
2x2, .
1.7
2x y + 3z = 9 (1)x+ 3y z = 10 (2)3x+ y z = 8 (3)
1.7 i. (2) x (1) (3).
2x y + 3z = 9x = 3y + z + 103x+ y z = 8
2(3y + z + 10) y + 3z = 9x = 3y + z + 103(3y + z + 10) + y z =
8
6y + 2z + 20 y + 3z = 9x = 3y + z + 109y + 3z + 30 + y z = 8
7y + 5z + 20 = 29x = 3y + z + 108y + 2z = 22
7y + 5z = 11 2x = 3y + z + 108y + 2z = 22 (5)
14y + 10z = 58x = 3y + z + 1040y 10z = 110
14y + 10z + 40y 10z = 58 + 110x = 3y + z + 1040y 10z = 110
26y = 52x = 3y + z + 1040y 10z = 110
y = 2x = 3 2 + z + 1040 2 10z = 110
y = 2x = 6 + z + 1080 10z = 110
y = 2x = 6 + (3) + 10z = 3
y = 2x = 1z = 3
(x, y, z) = (1, 2,3).
ii. z (2), (3). :
2x y + 3z = 9x+ 3y z = 10 33x+ y z = 8 3
2x y + 3z = 93x+ 9y 3z = 309x+ 3y 3z = 24
(1) :
- 14
-
2x y + 3z = 95x+ 8y = 2111x+ 2y = 15
2x y + 3z = 95x+ 8y = 2111x+ 2y = 15 (4)
2x y + 3z = 95x+ 8y = 2144x 8y = 60
2x y + 3z = 95x+ 8y = 2139x = 39
2 1 y + 3z = 95 1 + 8y = 21x = 1
2 y + 3z = 95 + 8y = 21x = 1
2 2 + 3z = 9y = 2x = 1
z = 1y = 2x = 1
(x, y, z) = (1, 2,3).
- 15
-
1.1.3
1. : 2x+ 3y = 7i. (1, 2), (3,1), (7,6) .ii. (, ) R (, 2), (1,
).iii. .
2. ;i. x 3y = 1 (4, 1), (0, 1).ii. x 3 +y = 3 (1, 1), (0,
4).
3. :
i.{
2x y = 6x+ y = 3 ii.
{2x 3y = 15x+ 4y = 37
iii.{
3x+ 4y = 72x 3y = 1
4. :
i.{
2x y = 13x+ y = 7
ii.{
2x y = 14x+ 2y = 3 iii.
{3x 4y = 79x+ 6y = 6
5. :
i.
{ x 2y3 2x+ y
4= 2
4(x+ y) 3(x y) = 2(3 x+ y) 3x
ii.
{1 =
5x+ 3y
2 4(x 1) y
37(x 2) 5(y 4) = 0
6. , R x+y = 5 A(1, 2) B(1, 3)
7. A(3, 2) B(4,2)
8. .(1) : 2x+ 3y = 7, (2) : x+ y = 4, (3) : 7x 11y = 40
9. 3. , 9. .
10. A(3, 2) B(4, 5) . M(3, |+ 2|) , R.
11. :
5
11, 4
2
5.
12. 24 -
2 3.13. : -
. 48 , 130 .
14. |3x+ 2y 12 = 0|+ |4x 3y + 1| = 015. (2x+ 3y 2)2 + (x+ 2y +
3)2 = 016. :
i.x+ 3 x 1x 2 1 x = 0 ii.
|x| 11 |x| = 0
iii.x+ 1 x2x 1
= 0 iv.x 11 x
+ 21 xx 1
=x 31 x
17. :
- 16
-
i. D1 =x+ 3 x 1x 2 1 x ii. D2 =
2
iii. D3 =x+ y x yx2 y2 x2 + y2
18. (1) :1 x 2 y3 x 1 y
(2) :2 41 3
x 5 = y.
19. , :
i.{x 3y = 12x+ y = 4
ii.{x+ 2y = + 13x y = iii.
{5x 2y = 110x+ 4y = 10
20. :
i.{x y = 1x+ 3y = 3+ 3
ii.{
3x+ 2y = 5x+ y = 1
iii.{
( 1)x+ 4y = ( 1)x+ 2y = 2
21. :
i.{
( 1)x y = 4x 2y = 4 ii.
{4x+ y = 92x+ 18y = 27 iii.
{x+ y = 5x+ y = 5
iv.{x y = 2x y = 4 v.
{(+ 1)x 2y = x y = 2 3
222. R : ) , )
;
i.{x 4y = + 13x+ 12y = 5 ii.
{4x+ y = 92x+ 18y = 27
23. :
i.{x y = x+ y = 1
ii.{x+ y = 2+ 12x+ y = 2
iii.{x+ y = + 1x+ y = 2+ 3
iv.{x+ y = 1x+ y = +
24. , R, :
i.{x+ y = 1x 2y = 2 ii.
{x+ x = x+ yx+ 3y = 1
iii.{
(+ 1)x y = 4( 1)x+ (+ 2)y = 3
25. R :{(+ 3)x+ ( 1)y = 2+ 1( 2)x ( 1)y = 3+ 7
26. R :
i.{x y = x y = 2 ii.
{( 2)x y = 24x+ (+ 3)y = 3
27. R :
i.{
(2 1)x+ 10y = 32x+ 4y = 5
ii.{
( 2)x (+ 1)y = 73x 6y = 5
28. R :
i.{
( 1)x y = 2x+ y = 0
ii.{x+ 3y = 1x+ y = 2
29. :
i.{
(+ 1)x y = 1x+ y = 1 ii.
{x+ ( + 2)y = 2 + 1x ( 1)y = 2
, .
- 17
-
30. : f(x) =x+ 2
+ x. , R, :
f(0) = 1, f(4) = 3.
31.
2x+ y = + 2
3x 2y + = 5i. Rii. x, yiii. (x, y)
x+ y = 5;32. (1) : x 2y = 2, (2) : x 2y = 2
i. (1), (2) Rii. iii.
xx
33. {
(+ 2)x 3y = 2+ 3x+ y = 3
(x0, y0), R, : x0 + 3y0 = 7.
34. {2x+ y = 1x+ y =
i. R, (x0, y0);ii. R , : 2x0 + 3y0 = 3.
35.
x+ y = 1
x+ (1 )y = 0 R.
xo, yo xo < yo + 1
36. : (429)x = 23 , {x+ y = 13x+ 2y = 2
.37. : (x 1) = x + 1, ,
{x+ 3y = yx+ 2y = 4 .
38. 2x2 Dx +Dy = 9D
Dx Dy = 5D
.39. 2x2
Dx +Dy = 7D
Dx Dy = 5D
.40. 2x2
2Dx + 3Dy = D
4D + 7D = 11D
- 18
-
.41. 2x2
D2 +D2x +D2y = 2D 6Dx + 4Dy 14
.42. 2x2 xo, yo
D = Dx + 3Dy
:i. xo + 3yo = 1
ii. x2o 9y2o =Dx 3Dy
D43. 2x2 : D2x +D2y +D 2(Dx + 2D)
i. ii.
44. :
x+ y = 1
x 2y = i. ii. (xo, yo)iii. : (xo + 2)2 + (yo 1)2 < 8
45.
( 1)x+ 2y = 2
2x+ ( 1)y = 1 x2 + 5(D 1)x 6(D 1)2 = 0
i. ii.
46. :
2x+ 3y = 2x+ 5y + 2 = 33x 4y + 5 = 25
47. :
x 2y + 3 = 02x 3y = 03x 5y + 2 = 0
48. :
x+ y = 1y + = 2 + x = 7
49. :
x+ y = 512x 2y + 9 = 604x+ 3y = 18
50. :
2x+ 6y + = 18x+ 2y + 3 = 143x+ 8y + 4 = 20
51. :
x y + = 52x y = 11x 3y + 7 = 3
- 19
-
52. :
x 2y + 3 = 52x+ 4y 4 = 103x 6y + 9 = 15
53. :
2x 3y + = 1x y + 3 = 83x+ 2y + 5 = 16
- 20
-
1.2
1.2.1 1.17 ;
.
1.6 , , -
- 21
-
1.2.2
1. :
i.{x+ 2y = 3x(y 1) = 0 ii.
{x 2y = 5(x 3)(x 5) = 0
iii.{
2x+ y = 5x2 y2 = 0 iv.
{xy + x+ y + 1 = 0x+ 2y = 2
2. :
i.
2
x+
3
y= 9
1
x 2y
= 1ii.
3
x+
7
y= 4
1
x+
2
y= 1
iii.{
3(x+ y) 2(x y) = 132(x+ y) + 3(x y) = 13 iv.
4
x+ y 5x y
= 3
5
x+ y 4x y
= 6
3.
i.{
8x2 y2 = 16y = 2x
ii.{x2 2xy + y2 = 1x 2y = 2
- 22
-
-- -
-
2.
2.1 --
2.1.1
2.1 ;
f , x1, x2 x1 < x2 :
f(x1) < f(x2)
, [4,16] - .
2.1:
2.2 ;
f , x1, x2 x1 < x2 :
f(x1) > f(x2)
-
, [16,24] .
2.2:
2.3 ;
.
2.4 x0 ;
f, A, x0 A() :f(x0) f(x) x A. x0 A , f(x0) minf(x).
2.3:
2.5 x0 ;
f, A, x0 A() :f(x0) f(x) x A. x0 A , f(x0) f maxf(x).
- 24
-
2.6 ;
.
2.7 ; ;
f , , x A : x A f(x) = f(x) . yy.
2.4:
2.8 A ; ;
f, A, , x A : x A f(x) = f(x). .
2.5:
- 25
-
2.1.2 -
:
1. - , x1, x2 , :x1 < x2 f(x1) < f(x2)
2. f(x) M x A, f M.
3. f - , , x1, x2 , - : f(x1) < f(x2) x1 > x2.
4. f(x) f(x0) x A, f x0, f(x0).
5. - yy, .
6. f f(x) = f(x) x A, , f(x) = f(x) x A, .
7. f : A R , .
8. f : A R, 0 R , - f 0(0, 0).
9. f : A R , R < f() < f().
10. f : A R , R f() f().
11. f : A R , R f() < f() < .
12. f : A R , R f() = f() = .
13. f R f(1) >f(2).
14. f R f(
2) >f(2).
15. f R f(10) >f( 220).
16. f R f(1) >f(
3).
17. f , x - y.
18. f , x - y.
19. f(0) = 0.
- 26
-
20. , .
21. f , f(2006) > f(2007).
22. x1 < x2 f(x1) f(x2) < 0, .
23. x1 < x2 f(x) :
f(x) > 0 f(x1)
f(x2)< 1, f
.
24. f : R R f(0) = 5 f(5) = 0 f .
25. f : R R f(0) = 5 f(5) = 0 f .
26. f : R R f(0) = 5 f(5) = 0 f .
27. f : R R f(0) = 5 f(5) = 0 f f(10) < 0.
28. f : R R f(0) = 5 f(5) = 0 f f(2) > 6.
29. f : R R f(1) 0 f(x1) > f(x2) f .
2.1 f(x) = x25x = (, 0).
2.1 x1, x2 R x1 < x2. :x1 < x2 x21 > x22 ( 1)x1 < x2
5x1 > 5x2 ( 2) ( 1 2) :x21 5x1 > x22 5x2 f(x1) > f(x2), f
= (, 0).
2.2 f(x) =1
x.
2.2 x1 < x2 f(x1) f(x2) =1
x1 1x2
=x2 x1x1x2
x2 x1 > 0 : x1, x2 < 0 x1x2 > 0 f(x1) f(x2) > 0.
f
(, 0). x1, x2 > 0 x1x2 > 0 f(x1) f(x2) > 0. f
(0,+). f (, 0) (0,+).
- 28
-
2.2 f A, f :
f(x) f(x0) f(x) f(x0).
x0 f.
2.3 f(x) = 2|x 1|+ 5.
2.3 f A = R. x R :|x 1| 0 2|x 1| 0 2|x 1|+ 5 5 f(x) 5 = f(1) f
1, f(1) = 5.
2.4 f(x) = x2 + 1.
2.4 f A = R. x R :x2 0 x2 + 1 1 f(x) 1 = f(0). f 0, f(0) =
1.
2.5 f(x) = (x+ 2)2 1.
2.5 f A = R. x R :(x+ 2)2 0 (x+ 2)2 1 1 f(x) 1 = f(2). f 2, f(2)
= 1.
- 29
-
2.3 - f A, :
i. x A x A. ( A ).
ii. f(x) : f(x) = f(x) ( f ) f(x) = f(x) ( f )
2.6 f(x) = 2x6 + |x| 2 .
2.6 f A = R. x A : x A f(x) = 2(x)6 + | x| 2 = 2x6 + |x| 2 =
f(x)
f .
f(x) = 2x5 x3 + x .: f A = R. x A : x A f(x) = 2(x)5 (x)3 + (x)
= 2x5 + x3 x = (2x5 x3 + x) = f(x)
f .
- 30
-
2.1.4
1. :iii. f(x) = 3x 4 iv. g(x) = 2x+ 5v. h(x) = 5 + 4x vi. (x) =
1 3x
2. R f R :
i. f(x) = ( 1)x+ 4 ii. f(x) = (2 9)x 5
iii. f(x) = (|| 3)x 1 iv. f(x) = (4 ||)x 10
v. f(x) = (2 5+ 6)x+ 2 vi. f(x) = (2 + 3 2)x 73. f(x) = (1 )x+
3x 1 g(x) = (4 2)x+ 2 3
, i. f ,ii. g .
4. :i. f(x) = x3 1 ii. g(x) = x3 + 5iii. h(x) =
x 2 + 1 iv. (x) = (x 1)2 + 1, x 1
5. f, g : R R . :i. f(x2) > f(x+ 6), f ,ii. g(x2 + x) g(2x+
12), g .
6. f(x) =
5
4 x.i. Afii. f Afiii. Cf xx yy.iv. f(x) =
2.
7. f : R R , f(2014) < f(2004).
i. fii. f(3x 2) > f(8 2x).
8. f : R R , A(2, 1) B(3, 5) Cf
i. fii. f(|x| 3) 1.
9. f(x) = x +4
x, A = (0,+),
x0 = 2.10. :
i. f(x) = 3 |x 2| x0 = 2 ,ii. f(x) =
2x
x2 + 1 x1 = 1 x2 = 1
,iii. f(x) = x6 2x3 + 3 x0 = 1 .
11. :i. f(x) = 2x4 1ii. f(x) = 2(x 1)2 + 1
12. :i. f(x) = 2 + x2 ii. g(x) = 5 x4iii. h(x) = 1 (x 3)2 iv.
(x) = |x 2|+ 4
13. f : A R, x0 6= 0
- 31
-
. f x0 .
- 32
-
2.2 -
2.2.1
2.9 (x). - f(x) = (x) + c, c > 0;
f, : f(x) = (x) + c, c > 0, c .( )
2.6:
2.10 (x). f(x) = (x) c, c > 0;
f, : f(x) = (x) c, c > 0, c ( )
2.7:
2.11 (x). f(x) = (x c), c > 0;
f : c > 0, c ( ).
- 33
-
2.8:
2.12 (x). f(x) = (x+ c), c > 0;
, : f(x) = (x + c), c > 0, c ( ).
2.9:
2.13 (x). f(x) = (x c1) c2, c1, c2 > 0;
, - :f(x) = (x c1) c2, c1, c2 > 0, .
- 34
-
2.2.2
1. :
f(x) = |x|, g(x) = |x|+ 3, h(x) = |x| 3
2. :
f(x) = |x|, g(x) = |x|, h(x) = |x|+ 2, (x) = |x| 2
3. :
f(x) = |x|, g(x) = |x 1|+ 2, h(x) = |x+ 3| 4, (x) = |x 1|+ 3
4. f, - g, :
i. g(x) = f(x 1) + 2 ii. g(x) = f(x+ 2) + 3iii. g(x) = f(x 2) 4
iv. g(x) = f(x+ 3) 5
5. g, f 2 , :
i. f(x) = 3x 4 ii. f(x) = 2x+ 6iii. f(x) = x2 1 iv. f(x) = x+
2
x 1v. f(x) =
x 1 vi. f(x) =
2 x+ 1
6. g, f 3 , :
i. f(x) = 2x+ 3 ii. f(x) = 2x2 9iii. f(x) =
2x+ 3
3x 2iv. f(x) =
x 2 + 1
v. f(x) = |x 2|+ 3 vi. f(x) = x3 9x2 + 27x 27.7. g, f
5 , , :i. f(x) = x2 + 4x ii. f(x) =
x2 + 1 4
iii. f(x) = x3 + x2 + 1 iv. f(x) =x+ 3
x 1v. f(x) = |x| 5 vi. f(x) = 6 5x
2
x2 18. g, f
5 , , :i. f(x) = x2 + 6 ii. f(x) =
x2 + 2 + 2
iii. f(x) =2x 11 x
iv. f(x) =x2 + 4x+ 6
x+ 1
v. f(x) =5|x|+ 4|x|+ 1
vi. f(x) = x2 6x+ 13.
- 35
-
1
-
- 2
2
3.
3.1
3.1.1
3.1 . , , ;
. :
3.1:
=
=
(MM1)
(OM)
=
=
(0M1)
(OM)
=
=
(MM1)
(OM1)
=
=
(OM1)
(MM1)
3.2 , 0 360;
Oxy , Ot Ox
-
O Ot (. ,). Ox , Ot w. :
3.2: , 0o 360o
=(MM1)
(OM)=y
=
(0M1)
(OM)=x
=(MM1)
(OM1)=y
x( x 6= 0)
=(OM1)
(MM1)=x
y( y 6= 0)
=x2 + y2 > 0
3.3 360 ;
:
=y
=
x
=y
x( x 6= 0)
=x
y( y 6= 0)
=x2 + y2 > 0
. 360 :
k 360 + , k Z
k Z:(k 360 + ) = (k 360 + ) = ,(k 360 + ) = (k 360 + ) =
3.4 ;
O(0, 0) = 1 . .
- 38
-
3.5 ;
, M(x, y), :
3.3:
= x = = y =
xx , yy .
3.6 ;
, 1. :
1 1 1 1
3.7 ;
(, ). .
:
= yE =
, x = 1, .
- 39
-
3.4:
3.8 ;
, , .
1 2 3 4
+ + - - + - - + + - + - + - + -
3.9 ;
( 1 rad ) , , ( 1 rad). .
3.10 ;
rad. :
=
180
- 40
-
3.5:
3.11 ;
0 30 45 60 90
rad 0
6
4
3
2
01
2
2
2
3
2+1
+1
3
2
2
2
1
20
0
3
31
3 -
-3 1
3
30
: , x rad x, (x rad), (x rad), (x rad) (x rad), x, x, x, x.
- 41
-
3.1.2
3.1 360 (2)
1. : = 360 + , Z
. , 360 -
.2. (rad):
= 2 + , Z .
2 ( )
2 .
3.1 1125
3.1 360 360 + , R. 1125 360 : 1125 =3 360 + 45. :
1125 = (3 360 + 45) = 45 =
2
2.
1125 = (3 360 + 45) = 45 =
2
2.
1125 = 45 = 1.
1125 = 45 = 1.
3.2 25
3rad.
3.2 25
3 ( 2 + ) .
25
3=
25
6 2 = (4 6 + 1)
6 2 =(4 6
6+
1
6
)2 = 4 2 + 2
6= 4 2 +
3.
- 42
-
25
3
25
3= (4 2 +
3) =
3=
3
2.
25
3=
3=
1
2.
25
3=
3=
3.
25
3=
3=
3
3.
3.2 - 360 (2)
, 90 (
2),
180 (), 270o3
2 360o 2 = 90 + ( =
2+ )
= 180 ( = ) = 270 ( = 3grp
2 )
= 3600 ( = 2 ) 1 . 2
2
3
2.
( ) ( ). . 1 , 2 , 3 4 .
3.3 225
3.3
225 = (180 + 45) = 45 =
2
2
225 = (180 + 45) = 45 =
2
2.
225 = (180 + 45) = 45 = 1.
225 = (180 + 45) = 45 = 1.
- 43
-
3.4 :
( + x) (2 x) (132
+ x)
( x) (x) (172
+ x)= 1.
3.4 .
( + x) = x
(2 x) = [2 + (x)] = (x) = x.
(13
2+ x) = (6 +
2+ x) = (
2+ x) = x.
( x) = x.(x) = x.
(17
2+ x) = (8 +
2+ x) = (
2+ x) = x.
:
( + x) (2 x) (132
+ x)
( x) (x) (172
+ x)=
x (x) x(x) (x) (x)
= 1
3.3
, A+ B + =180, A = 180 (B + ) B = 180 (A+ ) = 180 (A+ B)
B +
2= 90 A
2A+
2= 90 B
2
A+ B
2= 90
2 - 90(
2) 180().
3.5 AB :i. A+ (B + ) = 0.
ii. 2A
2+ 2
B +
2= 1
3.5 i. AB A+ B + = 180, A = 180 (B + ) A = [180 (B + )] A = (B +
) A+ (B + ) = 0.
ii. A+ B + = 180,
- 44
-
A
2+B
2+
2= 90 B +
2= 90 A
2
B +
2= (90 A
2) =
A
2.
:
2A
2+ 2
B +
2= 2
A
2+ 2
A
2= 1.
- 45
-
3.1.3
1. :i.
6,
4,
3,
2
ii.2
3,
5
6,
3
4, 2
iii.
12,
18,
5
12,
5
182. :
i. 120, 150, 135, 270
ii. 10, 15, 12
3. :
i. A =230 + 230 60 + 45
260 230 30 + 45
ii. B =26 +
2 4 + 2
3
4 +
3 3 + 4
4. :
i. A =9
4260+
2
245+
3
260
ii. B = 3230 +8
3230 +
1
2245 4
3260
iii. = 3245 260 12260 +
1
8(245)1
iv. =230 +
2 45 + 260
45 +
3 30 + 15. :
i. A =26 +
2 4 + 2
3
4 +
3 3 + 4
ii. B =90 0 + (+ )45 2+ 1
245 2180 + 290 (+ )245 + 1
iii. =360 + 260 330 + 6
3 60
230 + 260 2
3 30 + 645 456.
i) =73
6ii) =
93
4iii) t = 65
3iv) s = 185
67. :
i) 20 ii) 50 iii) 71
iv) 60 +
2v) 30 +
4vi) 10
6
vii) 2 +
3, Z viii) (2+ 1)
2, Z ix) +
2, Z
8. :
i) =41
4ii) =
601
6iii) =
901
3
iv) =61
6v) =
49
4vi) =
301
39. :
- 46
-
i. A =6(2 +
6) + 2(4 +
3)
3 (6 +
3) + (8 +
4)
ii. B =2
13
6+
2 174
+ 61
3
17
4+
3 253
+ 33
4
iii. =2
13
6+
2 174
+ 261
3
17
4+
3 253
+ 33
410. , [0,
2] :
i. + 3 = 4ii. 2+ 3 = 5
11. + 2 = 3.
- 47
-
3.2 1
3.2.1 3.12 ;
. :
() = () = () = () =
3.13 180;
180 - .
(180 ) = (180 ) = (180 ) = (180 ) =
3.14 180;
180 , .
(180 + ) = (180 + ) = (180 + ) = (180 + ) =
3.15 90;
90 , .
(90 ) = (90 ) = (90 ) = (90 ) =
- 48
-
3.2.2
1. , :
i.) = 330
ii.) = 660
iii.) = 405
iv.) = 30
2. :
i) 80 ii) 101 iii) 2011
iv)121
2v)
43
2vi)
181
6
3. :
i)20 50
40 70= 1 ii)
100 40
800 140= 1
iii)3 85
5 87= 1 iv)
110 145
70 35= 1
4. :() A = 50 50 130 130() B = 220 220 + 2160() = 10 80 + 10
80() = 25 155 25 155
5. :() 120 150 135 + 120 150 135 = 0() 120 150 135 210 225 240 =
2
6. :
(90 ) 132 312 90
(180 + ) 222 48 180= 1.
7. :
40 70
20 50+10 55
35 80+15 65
2575= 3
8. :i. (90 ) = 1ii. 1 2 3 89 = 1.
9. :i. 1 + 2 + 3 + + 89 = 1 + 2 + 3 + + 89ii. 1 2 3 89 = 1.
10. :i. (72 + ) (18 + ) = 0ii. (110 + ) ( 20) = 0.iii. (171 + )
(9 + ) = 0iv. (171 + ) + (9 + ) = 0v. (192 + ) + ( 12) = 0
- 49
-
vi. (205 + ) + (25 + ) = 0vii. (190 + ) (10 + ) = 0
11. :
(270 ) (180 )(270 ) (540 + )
+(450 ) (90 )(180 + ) (540 + )
= 2
12. :
73 3112 (
154 )
2310
( 12) 173
265 (
196 )
= 1
- 50
-
3.3
3.3.1
3.16 2 + 2 = 1.
M(x, y) - , : , (OM) = 1 (OM)2 = |x|2 + |y|2 = x2 + y2
3.6: x = y =
:x2 + y2 = 1
: 2 + 2 = 1.
3.17 =
=
: =
y
x=
( x = 6= 0)
=x
y=
( y = 6= 0)
3.18 = 1.
: =
=
( 6= 0 6= 0)
: =
= 1.
3.19 2 =1
1 + 2
2 + 2 = 1 2 6= 0 :2
2+2
2 1
2= 2 + 1 =
1
2 2 = 1
2 + 1.
- 51
-
3.20 2 =2
1 + 2
2 + 2 = 1 2 =1
1 + 2:
2 + 2 = 1 2 + 11 + 2
= 1 2 = 1 11 + 2
2 =
1 + 2 11 + 2
2 = 2
1 + 2
- 52
-
3.3.2
3.4 , : 2 + 2 = 1, = 1
2 =1
1 + 2 2 =
2
1 + 2
.
3.6 x = 45
< x 0 , x A :
i. x+ T A, x T Aii. f(x+ T ) = f(x T ) = f(x)
.
3.22
:
i. x (xrad) .
x = (x rad)
ii. x (xrad) .
x = (x rad)
iii. , : x =x
x.
: R1 = {x|x 6= 0}iv. , : x =
x
x.
: R2 = {x|x 6= 0}
3.23 f(x) = x;
f(x) = x :
i. A = R.ii. T = 2
: x+ T, x T A = R x R f(x+ T ) = f(x+ 2) = (x+ 2) = x = f(x), x
Af(x T ) = f(x 2) = (x 2) = x = f(x), x A
iii. [0,
2] [
3
2, 2].
[
2,3
2].
iv. : x =
2
2= 1
x =3
2
3
2= 1
:v. f(x) = x
:
- 58
-
3.7: f(x) = x
3.8: f(x) = x
vi. , f(x) = x ( (x) = x x R) O(0, 0) .
3.24 f(x) = x;
f(x) = x :
i. A = R.ii. T = 2
: x+ T, x T A = R x R f(x+ T ) = f(x+ 2) = (x+ 2) = x = f(x), x
Af(x T ) = f(x 2) = (x 2) = x = f(x), x A
iii. [0, ]. [, 2].
iv. : x = 0 0 = 1 x = 2 2 = 1 x = = 1
:v. f(x) = x :vi. , f(x) = x ( (x) = x
x R) yy.
- 59
-
3.9: f(x) = x
3.10: f(x) = x
3.25 f(x) =x f(x) = x;
f(x) = x, , > 0:
i. , .
ii. 2
f(x) = x, , > 0.
3.26 f(x) = x;
f(x) = x :
i. A = R x : x = + 2, k Z
ii. =
iii. (2,
2).
iv. ( ). - f(x) = x :
- 60
-
3.11: f(x) = x
v. , ( f(x) = (x) = x = f(x), x A O(0, 0) .
vi. x 2
x .
x = 2
f.
x
2 x +.
x =
2 f.
- 61
-
3.4.2 -
3.6 (x) (x) > 0,
||, || T = 2
3.12 f(x) = 2(3x
2), g(x) = 3(
2x)
h(x) =
3x
2.
.
3.12 :
f(x) = 2(3
2x) 2 2 T = 23
2
=4
3
g(x) = 3(
2x) 3 3 T = 22
=
2
h(x) =
3x
2
3
3 T =
21
2
= 42
3.7 , - , .
1. (x). - f(x) = (x) + c, c > 0; f, : f(x) = (x) + c, c >
0, - c .( )
2. (x). - f(x) = (x) c, c > 0; f, : f(x) = (x) c, c > 0, -
c ( )
- 62
-
3. (x). f(x) = (x c), c > 0; f : c > 0, c ( ).
4. (x). - f(x) = (x+ c), c > 0; , : f(x) = (x + c), c > 0,
c ( ).
5. (x). - f(x) = (x c1) c2, c1, c2 > 0; , :f(x) = (x c1) c2,
c1, c2 > 0
- 63
-
, .
- 64
-
3.4.3
1. :i. f(x) = 2x ii. f(x) = (x+
3)
iii. f(x) = 3x 54x+ 2 iv. f(x) = 2x 1v. f(x) = (x
3)
2. :
i. f(x) = 3x
4ii. g(x) = (3x+
4)
iii. h(x) = x
2iv. f(x) = (4x)
3. :
i. f(x) =1
32x ii. f(x) = 5
x
2iii. f(x) = 3
5
2x
7iv. f(x) =
x
2v. f(x) = 2x vi. f(x) = 2x
4. :i. f(x) = 2(x
3) ii. iii. g(x) = 4x 3
5. , , :
i. f(x) = 83x ii. f(x) = 32x iii. 12x
6. :i. f(x) = 3x 2 ii. f(x) = 5 x iii. f(x) = 22x+ 1
7. f(x) = 3x
3 3. :
i. ,ii.
8. :i. f(x) = x 3x ii. f(x) = 5 x
2iii. f(x) = |x| x iv. f(x) = x2 + xxv. f(x) = (x2 + 1)x xx vi.
f(x) = (x+
6) + (x
6)
9. :i. f(x) = 23x ii. f(x) = 3x iii. f(x) = 2x 5.
10.
f(x) = 3(x+ 2
)
11. f(x) = x
2, , > 0 4 2,
, .
12. g(x) = (3
4x), R, , .
13. :
f(x) = 2(3x 4
) 3(34 3x).
14. :
f(x) = 2x+ 1.
15. :
f(x) = 2x.
- 65
-
16. :
f(x) = 2x
2.
17. f(x) = (3 2)2x, R A(
12, 5). :
i. ,ii. f .
18. f(x) = +2x
A(
2, 0) B(
6,3
2), (0, 2).
19. :i. 128 147
ii. 200 220
iii. 300 340
20. f(x) = 2x, > 0 2.i. ii. f(x) = 4 .
21.
f(x) = (3 11)( 2)x, > 113, > 2,
T =
3 4. , .
22. :
- 66
-
- 67
-
23. :
- 68
-
- 69
-
- 70
-
3.5
3.5.1 3.27 x = ;
:
x = 2 + , Z
x = 2 + ( )
3.28 x = ;
:
x = 2 + , Z
x = 2
3.29 x = ;
:
x = + , Z
3.30 x = ;
:
x = + , Z
- 71
-
3.5.2
3.8 , , .
:
x = x =
{2 +
2 + , Z
x = x = 2 , Z
x = x = + , Z
x = x = + , Z
, , . , .
rad 0 0 0 1 0
30 612
3
2
3
3
3
45 4
2
2
2
2 1 1
60 3
3
212
3
3
3
90 2 1 0 0
2 , 1 .:
(x) = (x)
(x) = ( x)
(x) = (x)
(x) = (x)
, , - , -
3.13 : (x) = 12
- 72
-
3.13 (x) = 12 (x) = (6 ) x =
{2 + 62 + 6 , Z
x =
{2 + 62 + 56 , Z
3.14 : (x) = 12
3.14 (x) = 12 (x) = (3 ) x =
{2 + 32 3 , Z
3.15 : (x) = 1
3.15 (x) = 1 (x) = (4 ) x = +4 , Z
3.16 : (x) =
3
3.16 (x) =
3 (x) = (6 ) x = +6 , Z
3.17 : (x) = 12
3.17 (x) = 12 (x) = (6 ) (x) = (
6 )
x =
{2 + (6 )2 + (6 ), Z
x ={
2 6 2 +76 , Z
3.18 : (x) = 12
3.18 (x) = 12 (x) = ( 3 ) (x) = (
23 )
x =
{2 + 232 23 , Z
3.19 : (x) = 1
3.19 (x) = 1 (x) = (4 ) (x) = (4 ) x =
4 , Z
3.20 : (x) =
3
3.20 (x) =
3 (x) = (6 ) (x) = (6 ) x =
6 , Z
3.21 :
- 73
-
i. x =
3
2.
ii.
3 + x = 0.
3.21 i.
x =
3
2 x =
6
x = 6 x = (
6)
x = 2 +5
6
2 56
Z
ii.
3 + x = 0 x =
3
x = 3 x = (
3)
x = 3, Z.
3.22 :i. (x
3) + x = 0.
ii. 2(2x 5
) = 1. 0 x <
3.22 i.
(x 3
) + x = 0 (x 3
) = x
(x 3
) = (x) x 3
= x
2x = +
3 x =
2+
6 Z.
- 74
-
ii.2(2x
5) = 1 (2x
5) =
1
2
(2x 5
) =
3
2x 5
= 2 +
3
2x 5
= 2 3
x = +4
15
x = 15, Z
, 0 x < :0 x < 0 + 4
15<
0 (+ 415
) <
0 + 415
< 1
415 1 4
15
415 11
15 = 0 ( Z)
x =4
15
3.23 : 22(x) 5(x) + 2 = 0
3.23 22(x) 5(x) + 2 = 0
(x) = w :2w2 5w + 2 = 0 : 2 12 : (x) = 2 (x) = 12
(x) = 12 (x) = (6 ) x =
{2 + 62 + 6 , Z
x =
{2 + 62 + 56 , Z
- 75
-
3.5.3
1. :i. x =
7ii. x =
10
iii. x =
9iv. x =
3
82. :
i. (2x 3
) = (x+
4) ii. (x+
3) = (3x
6)
iii. (5x+
4) (2x
12) = 0 iv. 2x = (x
4)
3. :
i. x =1
2ii. x =
2
2
iii. x =
3 iv. x =
3
34. :
i. 2x
3 = 0 ii.
2x = 1
iii.
2x =
6 iv.
27x 3 = 05. :
i. x =
2
2ii. x = 1
iii. x =
3 iv. x = 12
6. :i. x x = 0 ii. (x
3)(1 x) = 0
iii. (x 1)(2x 1) = 0 iv. x (1 x) 0v. (x
3) (3x
3)x = 0
7. :i. (x+ 1)(2x+
3) = 0 ii. x 2x = x
iii. 2x = x iv. (2x 1)2 + 2(2x 1) = 08. :
i. 2x =1
4ii. 22x 1 = 0 iii. 32x = 1.
9. :i. 22x 5x+ 2 = 0 ii. 42x 2(
2 +
3)x+
6 = 0
iii. 32x 4
3x+ 3 = 0 iv. 2x 2
3x+ 3 = 010. :
i. 32x = 2x ii. 3x 2x+ 2x = 1iii. x
3x = 1
3 iv. 4x 1 = 0
v. 3x = x vi. 24x 72x+ 3 = 0vii. 3x 2x x+ 1 = 0 viii. 3x+ 3x =
1
11. :i. x = 1 [, 3]ii. 22x
2 = 0 [0, 2]
iii.x 1
2 x+ 1
5= 11
20 [0, ]
12. :
i.
3(4x 3
) = 3 [2,3
2]
ii. 2(x+
3) + 2(2x
3) = 0 [0, 2]
iii. 2x+ 22x = 1 [, 3]13. :
- 76
-
i. 2x+ x = 0 ii. 2x+ x = 0iii. x+ 2x = 0 iv. 2x = (
6 x)
v. 2x+ 3x = 0 vi. (
3 3x) + (
6 x) = 0.
14. :
i. x = x ii. x =2
3x
iii. 2x = x iv. x (1 + 2x) = x+ x
v.x
1 x=
x
x 1vi. x+ x =
1
x
vii. 4x = x viii. 2x x x 2x+ 1 = 015. :
ix. 2x
2+ 2
x
3= 1 x. ( x) = 1
xi. 3x+ 3x = x xii.22x
x 4(1 x) = 1
- 77
-
3.6 2
3.6.1
3.31 ;
, .
( ) = +
, .
3.32 (+ ) = .
: ( ) = + , :( ()) = () + () :
(+ ) =
3.33 (+ ) = + .
: ( ) = + ( 1) (
2 x) = x (
2 x) = x, 1
:(+ ) = (
2 (+ )) = (
2 ) + ))
= (
2 ) + (
2 ) = + .
3.34 ( ) = .
(+ ) = + . :(+ ()) = () + () :
( ) = .
3.35 (+ ) =+
1 .
(+ ), , :
(+ ) 6= 0, 6= 0 6= 0.
:
(+ ) =(+ )
(+ )= +
- 78
-
6= 0 :
=
+
(+ ) = + 1
3.36 ( ) = 1 +
.
(+ ) =+
1 .
:
(+ ()) = + ()1 ()
.
:
( ) = 1 +
.
3.37 ;
(+ ) = 1 +
( ) = + 1
- 79
-
3.6.2
3.9 1
3.24 75o
3.24 :
75o = (30o + 45o)
= 30o45o + 30o45o
=1
2
2
2+
3
2
2
2
=
2 +
6
4
75o = (30o + 45o)
= 30o45o 30o45o
=
3
2
2
2+
1
2
2
2
=
6
2
4
75o =75o
75o
2 +
6
46
2
4
=2 +
66
2
= 2 +
3
75o =1
75o
=1
2 +
3= 2
3
3.10 ( ) = + (+ ) = (+ ) = + ( ) =
- 80
-
(+ ) =+
1 ( ) =
1 + (+ ) =
1 +
( ) = + 1
3.25 : (+ )( ) = 2 2
3.25
(+ )( ) = ( + )( )= ()2 ()2
= 22 22= 2(1 2) (1 2)2= 2 22 2 + 22= 2 2
3.11 , , , , , ,1
2 = 1 2 2 = 1 2 =
=1
1 + 2 =1
2 , - 3 ,
3.26 = 35,
2< < =
15
17 0 < 0 , x =
iii. < 0 , x = ||
iv. < 0 ,
4.15 i. x4 + 8x = 0ii. x5 = 16xiii. x6 = 32xiv. (2x+ 6)3 = 8
4.15 i.x4 + 8x = 0 x(x3 + 8 = 0)
x = 0
x3 = 8
x = 0
x = 3| 8|
x = 0
x = 2
- 121
-
ii.x5 = 16x x5 16x = 0
x(x4 16) = 0
x = 0
x4 = 16
x = 0
x = 4
16
x = 0
x = 2iii.
x6 = 32x x6 + 32x = 0 x(x5 + 32) = 0
x = 0
x5 = 32
x = 0
x = 5| 32|
x = 0
x = 2iv.
(2x+ 6)3 = 8 2x+ 6 = 3
8
2x+ 6 = 2 2x = 4 x = 2
4.9 > 0 :
i. |x| = x = x = ii. |x| = || x = x =
4.16 :i. 2|x+ 1| 6 = 0ii. |2x+ 7|+ 9 = 0iii. |x 1| 3|x+ 5| =
0iv. |x+ 1| = x+ 1v. |2x 4| = 2x+ 4vi. |3x 6| = 2x 2vii. |x2 4|+
|3x 6| = 0
- 122
-
4.16 i. |f(x)| = > 0 x = , :
2|x+ 1| 6 = 0 2|x+ 1| = 6 |x+ 1| = 3
x+ 1 = 3
x+ 1 = 3
x = 2
x = 4ii. |f(x)| = < 0
: |2x+ 7|+ 9 = 0 |2x+ 7| = 9 .
iii. |f(x)| = |g(x)|
f(x) = g(x)
f(x) = g(x)|x 1| 3|x+ 5| = 0 |x 1| = 3|x+ 5|
x 1 = 3(x+ 5)
x 1 = 3(x+ 5)
x 1 = 3x+ 15
x 1 = 3x 15
2x = 16
4x = 14
x = 8
x = 72
iv. : |f(x)| = f(x) f(x) > 0 |x+ 1| = x+ 1 x+ 1 > 0 x >
1
v. : |f(x)| = f(x) f(x) < 0 |2x 4| = 2x+ 4 2x 4 < 0 2x
< 4 x < 2
vi. |f(x)| = g(x), |f(x)| 0 g(x) 0
|f(x)| = g(x)
g(x) 0
f(x) = g(x)
f(x) = g(x)
- 123
-
|3x 6| = 2x 2
2x 2 0
3x 6 = 2x 2
3x 6 = 2x+ 2
x 1
x = 4
x =8
5
x = 4
x =8
5vii. |f(x)| + |g(x)| = 0 |f(x)| 0 |g(x)| 0 : f(x) = g(x) =
0
:
|x2 4|+ |3x 6| = 0
x2 4 = 0
3x 6 = 0
x = 2
x = 2
x = 2
4.10 1 x + > 0 , R, x+ > 0 x >
i. > 0x > x >
ii. < 0
x > x <
iii. = 0 0x > x R , > 0 , 0
4.17 :
i.5x
3 1
9> (x 3)5
9ii. (x 1)2 + 3 < x2 3(x+ 1)
- 124
-
4.17 i.5x
3 1
9> (x 3)5
9 15x 1 > 5(x 3) 10x > 14
x > 1410
x > 75ii.
(x 1)2 + 3 < x2 3(x+ 1) x2 2x+ 1 + 3 < x2 3x 3 x <
7
4.11 x+ , R x+ = 0 x =
x
+x+ 0
4.12 , -
i. |x| < < x < ii. |x| > x < x >
4.18 :i. |x 3| < 2ii. |2x 3| > 5iii. |x 7| < 4iv. |2x
6| 6 0v. |x+ 7| 0
4.18 i.|x 3| < 2 2 < x 3 < 2
2 + 3 < x < 3 + 2 1 < x < 5 x (1, 5)
ii.|2x 3| > 5 2x 3 < 5 2x 3 > 5
2x < 5 + 3 2x > 5 + 3 2x < 2 2x > 8 x < 1 x >
4 x (,1) (4,+)
iii. |x 7| < 4
- 125
-
iv.|2x 6| 6 0 2x 6 = 0
2x = 6 x = 3
v. |x+ 7| 0 x R
4.19 : |x 1| < |x 2|
: |A(x)| < |B(x)| .
4.19
|x 1| < |x 2| |x 1|2 < |x 2|2
(x 1)2 < (x 2)2
x2 2x+ 1 < x2 4x+ 4 4x 2x < 4 1
x < 32
4.20 : 2|x 1| < |x 2|+ 3
: |A(x)|+ |B(x)|+ |(x)|+ < 0 .
4.20 .
x 1 2 +x 1 0 +
+x 2
0 + :
i. x (, 1),
2(x+ 1) < (x+ 2) + 3 2x+ 2 < x+ 2 + 3 x < 3 x >
3
x (, 1) x (, 1)
ii. x (1, 2),
2(x 1) < (x+ 2) + 3 2x 2 < x+ 2 + 3 3x < 7
x < 73
x (1, 2) x (1, 2)
- 126
-
iii. x (2,+) ,2(x 1) < (x 2) + 3 2x 2 < x 2 + 3
x < 3 x (2,+) x (2, 3)
4.13 x2 + x+ , 6= 0 .
i.x 1 2 +
x2 + x+ 0 0 > 0
1, 2 x2 + x+ = 0
ii.x +
x2 + x+ 0 = 0
x2 + x+ = 0
iii.x +
x2 + x+ < 0
: 2 , !!!
4.21 :i. x2 5x+ 6ii. x2 + 2x+ 1iii. 3x2 + 2x+ 4
4.21 i. x25x+6 = 0 2 3, :
x 2 3 +x2 5x+ 6 + 0 0 +
:x2 5x+ 6 > 0 x (, 2) (3.+)x2 5x+ 6 < 0 x (2, 3)x2 5x+ 6 =
0 x = 2, x = 3
ii. x2 2x+ 1 = 0 1, :
x 1 +x2 + 2x+ 1 + 0 +
- 127
-
:x2 + 2x+ 1 > 0 x R {1}x2 + 2x+ 1 = 0 x = 1
iii. 3x2 +2x+4 = 0 , :
x +3x2 + 2x+ 4 +
:3x2 + 2x+ 4 > 0 x R
4.14 , , .
: , , .
4.22 :
(x2 5x+ 6)(x2 + 2x+ 1)
4.22 : x2 5x+ 6 = 0 2 3, :
x 2 3 +x2 5x+ 6 + 0 0 +
x2 2x+ 1 = 0 1, :
x 1 +x2 + 2x+ 1 + 0 +
:x 1 2 3 +
x2 5x+ 6 + | + 0 0 +x2 + 2x+ 1 + 0 + | + | +
(x2 5x+ 6)(x2 + 2x+ 1) + 0 + 0 0 + :
(x2 5x+ 6)(x2 + 2x+ 1) > 0 x (,1) (1, 2) (3,+)(x2 5x+ 6)(x2 +
2x+ 1) < 0 x (2, 3)(x2 5x+ 6)(x2 + 2x+ 1) = 0 x = 1, 2, 3
- 128
-
4.15 , ,
4.16 i. :
A(x)
B(x)> 0 A(x)B(x) > 0
.ii. :
A(x)
B(x)> (x) A(x)
B(x) (x) > 0
A(x) B(x)(x)B(x)
> 0
[A(x) B(x)(x)]B(x) > 0 , , B(x) 6= 0.
- 129
-
, . :
1. , . : + = ( + ) : ...
. .: :
2x5 + 4x4y + 6x2
... 2, 4, 6 2. x 2. :
2x5 + 4x4y + 6x2 = 2x2(x3 + 2x2y + 3)
2. () , , :
: :
3x+ x+ 3y + y
1 : 3x+ x+ 3y + y = 3(x+ y) + (x+ y) = (x+ y)(3 + )2 : 3x+ x+ 3y
+ y = x(3 + ) + y(3 + ) = (3 + )(x+ y)
3. : : x2 6x+ 9 = x2 2 3 x+ 32 = (x 3)2 x2 4 = x2 22 = (x 2)(x+
2) x3 8 = x3 23 = (x 2)(x2 + 2x+ 22) = (x 2)(x2 + 2x+ 4)
4. 2 1 :(+ )2 = 2 + 2 + 2,( )2 = 2 2 + 2(+ )3 = 3 + 32 + 32 +
3,( )3 = 3 32 + 32 3
5. 2 2 = ( )(+ ) :(+ )2 = 2 + 2 + 2,( )2 = 2 2 + 2 .
- 130
-
6. 3 + 3 = (+ )(2 + 2),3 3 = ( )(+ + 2) :(+ )3 = 3 + 32 + 32 +
3,( )3 = 3 32 + 32 3 .
7. x2 x+ = 24. :
> 0, x2 x+ = (x x1)(x x2) x1,2 = +
2
= 0, x2 x+ = (x x1)2 x1 =2
< 0, . :
x2 (+ )x+ = (x )(x )
x2 + (+ )x+ = (x+ )(x+ )
1: x2 5x+ 6. = 2 4 = (5)2 4 1 6 = 25 24 = 1 > 0. :
x1 = +
2=(5) +
1
2 1=
5 + 1
2=
6
2= 3
x1 =
2=(5)
1
2 1=
5 12
=4
2= 2
x2 5x+ 6 = (x 3)(x 2).
2: x2 6x+ 9. = 2 4 = (6)2 4 1 9 = 36 36 = 0,
x1 =2
=(6)2 1
=6
2= 3.
x2 6x+ 9 = (x 3)2.
8. Horner
- 131
-
4.3.3 -
4.17 3 , , , 1 2.
4.23 , 3x3 + 8x2 15x+ 4 = 0
: ( ). 0, 1 ( Horner.
4.23 0, 1. 3 , .3 8 15 4 1
3 11 43 11 4 0
Horner , 3x3 + 8x215x+ 4 x 1 3x2 + 11x 4 0. : 3x3 + 8x2 15x+ 4 =
(x 1)(3x2 +11x 4)
x3 + 8x2 15x+ 4 = 0 (x 1)(3x2 + 11x 4) = 0
x 1 = 0
3x2 + 11x 4 = 0
x = 1x = 4
x = 13
4.18 1 3 1, 2 3 x3 + x2 + x+ = 01 = 1 + 2 + 3 =
2 = 12 + 13 + 23 =
3 = 123 =
3 1, 2 3 x3 1x2 + 2x 3 = 0
4.19 2 , ,
- 132
-
1
. (
.) : , , .
4.20 1
x3 + x2 + x+ = 0 (x3 + 1) + x(x+ 1) = 0 (x+ 1)(x2 + x+ 1) + x(x+
1) = 0 (x+ 1)[(x2 + x+ 1) + x] = 0
, 1 2.
2
x4 + x3 + x2 + x+ = 0 x2(x2 + x+ + 1x
+ 1
x2) = 0, x 6= 0
( )
(x2 + 1x2
) + (x+1
x) + = 0
, x+1
x= y, x2 +
1
x2= (x+
1
x)2 2x1
x= y2 2
:(y2 2) + y + = 0 2
4.21 3( ). , :
x + 1x
1 + + 1x+ 0 = 0,
, 1, . . . , 1, 0 Z p =
6= 0
0. .
4.24 2x3 + x2 + x 1 = 0
4.242 1 15 1 1
21 1 1
2 2 2 0
Horner , 2x2 +2x+2 0.
- 133
-
:
2x3 + x2 + x 1 = (x 12
)(2q2 + 2q + 2)
2x3 + x2 + x 1 = 0 (x 12
)(2x2 + 2x+ 2) = 0
x 1
2= 0
2x2 + 2x+ 2 = 0
x =
1
2
4.22 3 , , , 1 2. , , .
4.25 :
x4 3x3 3x2 + 7x+ 6 > 0
4.25 0, 1, , Horner , 6. 1,2,3,6 Horner -11 3 3 7 6 11 41 1
6
1 4 1 6 0 Horner , x4 3x3 3x2 + 7x+ 6 x+ 1 x3 4x2 + x+ 6 0. :x4
3x3 3x2 + 7x+ 6 = (x+ 1)(x3 4x2 + x+ 6) x3 4x2 + x+ 61 4 1 6 11 51
6
1 5 6 0 Horner , x34x2+x+6 x+ 1 x2 5x+ 6 0. :x3 4x2 + x+ 6 = (x+
1)(x2 5x+ 6) x4 3x3 3x2 + 7x+ 6 = (x+ 1)(x+ 1)(x2 5x+ 6) = (x+
1)2(x2 5x+ 6)
- 134
-
:
(x2 5x+ 6)(x2 + 2x+ 1) > 0
: x2 5x+ 6 = 0 2 3, :
x 2 3 +x2 5x+ 6 + 0 0 +
x2 2x+ 1 = 0 1, :
x 1 +x2 + 2x+ 1 + 0 +
:x 1 2 3 +
x2 5x+ 6 + | + 0 0 +x2 + 2x+ 1 + 0 + | + | +
(x2 5x+ 6)(x2 + 2x+ 1) + 0 + 0 0 + :
(x2 5x+ 6)(x2 + 2x+ 1) > 0 x (,1) (1, 2) (3,+)(x2 5x+ 6)(x2 +
2x+ 1) < 0 x (2, 3)(x2 5x+ 6)(x2 + 2x+ 1) = 0 x = 1, 2, 3 x (,1)
(1, 2) (3,+)
- 135
-
4.3.4
1. :i. x3 + 3x2 10x = 0ii. x5 = 9x3
iii. x4 + 8x = 0iv. x3 + 2x2 x 2 = 0v. (x 1)3 3(x2 1) + 2(x 1) =
0.
2. :i. x6 + 7x3 8 = 0ii. (2x 5)4 10(2x 5)2 + 9 = 0.iii. (x+ 4)6
+ 16(x+ 4)3 + 64 = 0iv. (x2 5)2 3(x2 5) = 4
3. :i. 2x3 x2 7x+ 6 = 0ii. 3x3 5x2 11x 3 = 0.iii. x3 3x2 10x+ 24
= 0iv. 2x3 + 9x2 + 7x 6 = 0
4. :i. x4 + 2x3 7x2 8x+ 12 = 0.ii. x4 3x3 6x2 + 6x+ 8 = 0.iii.
x4 8x2 4x+ 3 = 0.iv. x4 + 4x3 23x2 + 18 = 0.
5. .i. x3 5x2 + x 2 = 0.ii. x4 3x2 + 7x+ 1 = 0.
6. :
i.1
2x3 +
17
6x2 + 3x 4
3= 0.
ii.1
12x4 +
1
4x3 1
12x2 3
4x 1
2= 0.
7. P (x) = x3 + x2 + (3 1)x 8 x 2i. R.ii. P (x) = 0.
8. P (x) = x4 x3 x2 +x+ 1. P (x) x+ 1 12() R.() P (x) = 0.
9. P (x) = x4 + x2 + x+ x+ 2 P (x) x 1 18
i. , Rii. P (x) = 0.
10. :i. 5x4 = 10x2
ii. x3 + 2x2 9x 18 = 0iii. 3x5 + 5x4 = 3x3 + 5x2
iv. x6 64 = 0v. x3 + x2 2 = 0vi. x3 7x+ 6 = 0vii. (x+ 1)3 + 1 =
0viii. 7(3x+ 2)2(1 x)2 (3x+ 2)(1 x)2 = 0
- 136
-
ix. x3 + 8 = 7(x2 + 5x+ 6) + 9x2 36x. x4 3x3 + 6x 4 = 0
11. i. x4 = 3x2
ii. x3 3x2 4x+ 12 = 0iii. 2x5 + 9x2 = x4 + 18x3
iv. x4 5x3 + 5x 1 = 0v. x3 4x2 + 4x = 3(x2 4) (2 x)(x 5)vi. x4 +
x3 31x2 25x+ 150vii. 24x4 + 4x3 66x2 x+ 15 = 0viii. x(x 1)(x 2)(x
3) = 0ix. (x+ 6)(x+ 4)(x 3)(x 5) = 280x. 3x3 13x2 + 13x 3 = 0xi.
6x4 25x3 + 38x2 25x+ 6 = 0xii. 6x5 19x4 + 13x3 + 13x2 19x+ 6 =
0xiii. x2(x 1)2 + (x+ 1)(x+ 4) = 6x2
xiv. (x
x+ 1)6 7( x
x+ 1)3 8 = 0
xv. 6x3 7x2 7x+ 6 = 0xvi. 6x4 35x3 + 62x2 35x+ 6 = 0xvii. 6x5
41x4 + 97x3 97x2 + 41x 6 = 0xviii. x4 12x2 + 27 = 0xix. x6 13x4 48
= 0xx. (x2 + x)6 7(x2 + x)3 8 = 0xxi. (x2 1)3 2(x2 1)2 5(x2 1) + 6
= 0
12. i. x3 3x2 + x+ 2 = 0ii. 3x3 + 8x2 15x+ 4 = 0iii. x3 10x 12 =
0iv. x3 + 2x2 + 7x+ 6 = 0
13. () x4 + 3x 2 = 0() 2x4 3x3 + 6x2 24x+ 5 = 0() 4x4 15x2 + 13x
3 = 0() x4 3x3 + x2 6x+ 1 = 0
+ = 5, , Z14. , f xx
() f(x) = 3x3 3x2 5x 2() f(x) = 4x3 3x 1
15. f(x) = x3 2x2 5x+ 6, xx
16. ,
f(x) = x3 + 4x2 + 4x g(x) =9
x.
17. () x8 15x4 16 = 0() (x 1)6 9(x 1)3 + 8 = 0() 6(x+
1
x)2 + 5(x+
1
x) 6 = 0
18.
- 137
-
()1
10x3 +
1
10x2 +
1
5x 4
5= 0
() x3 56x2 22
3x+
5
2= 0
19. , R P (x) = x4 + x3 + x2 16x 12 x+ 1 x 2. P (x) = 0
20. P (x) = x6 5x4 10x2 + R x 1 P (x). , P (x) = 0
21.
A(1, 2) B(1
2,1
2).
y =x3+x2 , x3+x25x+3 = 0 .
22. A(1,2) B(2, 2). y = x32x2 , x33x26x+8 = 0 .
23. x3 + x2 + 23x 15 = 0, R 1, .
24. x5+x4+x+ = 0 , R -1, , R .
25. :i. (x+ 3)(2x+ 10)(x 2) > 0ii. (x2 + 3x 10)(x2 + 2x 1)(x2
x+ 1) 0iii. x3(x+ 2)4(x 3)6(x 4)7 > 0
26. :i. 2x3 x2 7x+ 6 > 0.ii. x3 + 2x2 11x 12 < 0.iii. 3x3
+ 5x2 26x+ 8 0iv. x3 + 3x+ 4 0.
27. :i. x3 4x2 + 5x 2 0.ii. x3 + 3x2 4 > 0.iii. 4x3 4x2 + 7x
2 0iv. 2x3 5X2 + 4x 1 < 0.
28. :i. x3 + 2x2 + 3x+ 6 > 0ii. x4 6x3 + 22x2 30x+ 13 0iii.
x3 3x+ 2 < 0iv. x4 x3 + x2 3x 6 0
29. Cf xx f(x) = x4 7x3 + 17x2 17x+ 6
30. Cf xx f(x) = x4 + x3 x2 + 3x+ 6
31. , f(x) = x4 5x3 + 3x2 + x xx
32. g(x) = x6 +2x5 +2x4 +2x3 +x2 +(x+1)2
- 138
-
, xx
33. , f(x) = 2x3 + 5x2 4x 3 xx
34. P (x) = 3x3 5x2 11x+ 3x+ 1.i. R.ii. P (x) 3x+ 1.
35. P (x) = x3 + x2 13x 5 x 1i. R.ii. P (x) = 0iii. :
A = P (99) P (
3) P (
2) P (
7) P (2014)
- 139
-
4.4
4.4.1
4.23 :
i. ii. iii. ... iv. v. .
4.1:
4.26 :2
x+
2x 3x 2
+2 x2
x2 2x= 0
4.26 :2
x+
2x 3x 2
+2 x2
x2 2x= 0 x 6= 0 x 6= 2
2
x+
2x 3x 2
+2 x2
x2 2x= 0 2
x+
2x 3x 2
+2 x2
x(x 2)= 0
x(x 2) 2x
+ x(x 2)2x 3x 2
+ x(x 2) 2 x2
x(x 2)= 0
2(x 2) + x(2x 3) + 2 x2 = 0 2x 4 + 2x2 3x+ 2 x2 = 0 x2 x 2 = 0,
= 9
x1 = 2
x2 = 1
4.24
i. :
- 140
-
A(x)
B(x)> 0 A(x)B(x) > 0
.ii. :
A(x)
B(x)> (x) A(x)
B(x) (x) > 0
A(x) B(x)(x)B(x)
> 0
[A(x) B(x)(x)]B(x) > 0 , , B(x) 6= 0.
4.27 x
x 1 2x+ 1
2 x (,4) (0,+)
- 144
-
4.4.2
1. :
i. x2 2xx 1
= x.
ii. 2x2 14(x 1)x
=2
x 3(x 1)
iii. 3x+ 2 x(3x2 8)
x 2= 2x 8
x 2iv. x2 + 4 7x+ 10
x+ 2= 2x+
4
x+ 2.
2. :
i. x 3(x+ 2)x
=4
x2 2x 2x 2
ii.14
x2 1 7x 1
= 4 x2 + x+ 1
x+ 1
iii.3x
x 3 2x
2
x+ 1 10x+ 6x2 2x 3
= 0
iv.3x2
x 2 9xx2 4
10x2 + 20
x3 4x= 4 +
10
x2 + 2x.
3. :i. x
5 + 4 = 0.
ii. x 2x 3 = 0.
iii. 3x
2 3 3x 4 = 0.
4. :
i.( x2x 2
)2 7 x2x 2
8 = 0.
ii.( xx+ 3
)3 7( xx+ 3
)2+
14x
x+ 3 8 = 0.
5. :
i.x 3 = 2. ii.
x2 + 4x 5 = 4.
iii.
3x+ 18 = x. iv.
20 x2 3 = 0.6. :
i. x
14 2x = 3. ii. 2 +
12 2x = x.
iii. x
2x+ 5 = 5. iv. x 1x+ 11 = 0.
7. :
i.x+ 8
x 4 = 2 ii.
x 5 3 =
x 8
iii.
2x+ 12
3 x = 3 iv.
2x 3x 2 = 1.
8. :
i.
4 x2 +
3x 6 = 8 4x ii.
1 x2 +x 1 = x
9. :
i.
1 + 4x = 1 +
x ii.
12
x = 6
x
10. :
- 145
-
i.x2 + 1 0 ii.
x 5 0
iii.
3x 5 < 0 iv.x2 + 3x 4 0
v.x2 + 4 < 3 vi.
x2 9 2
11. :
i.
2x 1 x 1 ii.x 4 > 2 x
iii.x 6 3 x iv. 2
x 2 < x 1
13. f(x) = x+ x 43, , R
i. , -
xx x1 =
6 x2 =
5
3ii. M(x, + ) , -
, , , x.
14. i. (2x 1)4 + 6(2x 1)2 7 = 0ii. 23x+ 52x+ 5x+ 2 = 0iii. 24x
53x+ 5x 2 = 0
15.
i.x3 + 2x 4
x 2< 1
ii.x2
x+ 1 4x 1
2x2 1
- 146
-
5.
5.1
5.1.1
5.1 , ;
, 2,
= ,
:i. 1 = ii. 0 = 1, 6= 0iii. =
1
, 6= 0
..23 = 2 2 2 = 8
5.2 ;
= , > 0, Z, Z+
5.3 ;
x = lim
, lim
= x
-
5.4 ;
1. = + 4. = ( )
2.
= 5.
=(
)3. =
() 6. (
)=
(
)
5.5 ;
f : R R f(x) = x, 6= 1
. = 1 f(x) = 1
5.6 f(x) = x > 1;
5.1: f(x) = x > 1
R. . , x1, x2
x1 < x2 x1 < x2 f yy A(0, 1).
- 148
-
, x , Ox, . f(x) = x Ox.
5.7 f(x) = x
0 < < 1;
5.2: f(x) = x 0 < < 1
R. . , x1, x2
x1 < x2 x1 > x2 f yy A(0, 1). , x ,
Ox, . f(x) = x Ox.
5.8 , ;
> 1 0 < < 1 :
x1 6= x2 x1 > x2
:
x1 = x2 x1 = x2
- 149
-
5.1.2
5.1 f(x) = x + c y = x |c| - : c > 0. c < 0.
5.1 - f(x) = 4x + 2 g(x) = 4x 1
5.1 Ch h(x) = 4x. f(x) = 4x + 2 f(x) = h(x) + 2 Cf Ch 2 . g(x) =
4x1 g(x) = h(x)1, Cg Ch 1 .
5.3:
- 150
-
5.2 f(x) = x+c - y = x |c| : c > 0. c < 0.
5.2 - f(x) = 4x+2 g(x) = 4x1
5.2 Ch h(x) = 4x. f(x) = 4x+2 f(x) = h(x+ 2) Cf Ch 2 . g(x) =
4x1 g(x) = h(x 1), Cg Ch 1 .
5.4:
- 151
-
5.3 f(x) = x : Af = R x R > 0. > 0 6= 1. > 1. () 0 <
< 1.
5.3 f(x) =(3
1 +
)x. 6= 1
f(x) :i. Rii. iii. iv.
5.3 i. f Af = R
3 1 +
> 0.
1 3 +3 + | + 0 1 + 0 + | +
31+ || + 0
3 1 +
> 0 (1, 3)
ii. f(x) =(3
1 +
)x 3 1 +
> 0
3 1 +
6= 1 3 6= 1 + 2 6= 2 6= 1. i. (1, 3) (1, 1) (1, 3)
iii. f(x) =(3
1 +
)x () 3 1 +
> 1 3 1 +
1 > 0
3 1 +
1 + 1 +
> 0
3 1 1 +
> 0
2 21 +
> 0
2(1 )1 +
> 0
2 1 1 +
> 0
1 1 +
> 0
- 152
-
1 1 +1 + | + 0 1 + 0 + | +
11+ 0 + ||
1 1 +
> 0 1 < < 1. f (1, 1).
iv. f(x) =(3
1 +
)x Af = R3 1 +
> 0 3 1 +
< 1
i. 3 1 +
> 0 (1, 3)
iii. 3 1 +
< 1 (,1) (1,+)
0 0, 6= 1
: 0, . > 0,
.. = 1 1 :
x = x = x =
.
5.4 3x = 9
5.4 3x = 9 3x = 32 x = 2
5.5 2
f(x) = g(x) f(x) = > 0, 6= 1 > 0, -f(x) = g(x) f(x) =
g(x), > 0, 6= 1 .
5.5 52x4 = 1
- 153
-
5.5 52x4 = 1 52x4 = 50 2x 4 = 0 2x = 4 x = 2
2x23x =
1
4 2x23x = 1
22
2x23x =(1
2
)2 2x23x = 22
x2 3x = 2 x2 3x+ 2 = 0 x = 1 x = 2
5.6 3
f(x) = 0 > 0, 6= 1
x = .
5.6 : 4x 2x 12 = 0
5.6
4x 2x 12 = 0 (2x)2 2x 12 = 02x=>0==== 2 2 12 = 0
1 = 4
2 = 3, 2x=>0==== 2x = 4 2x = 22
x = 2
5.7 : 2x+1 + 2x+2 + 2x1 + 2x2 = 54
5.7
2x+1 + 2x+2 + 2x1 + 2x2 = 54 2x 2 + 2x 22 + 2x 21 + 2x 22 =
54
2 2x + 4 2x + 12 2x + 1
4 2x = 54
274 2x = 54
27 2x = 216 2x = 8 2x = 23
x = 3.
- 154
-
5.7 4
f(x) = f(x) , > 0, , 6= 1
,(
)x=
(
) x =
5.8 : 8x + 18x 2 27x = 0.
5.8
8x + 18x 2 27x = 0 (23)x + (2 9)x 2 (33)x
23x + 2x (32)x 2 33x = 0 23x + 2x 32x 2 33x = 0
23x
33x+
2x 32x
33x 2 3
3x
33x= 0
23x
33x+
2x 32x
3x 32x 2 3
3x
33x= 0
(2
3
)3x+(2
3
)x 2 = 0
((2
3
)x)3+(2
3
)x 2 = 0.
(2
3
)x= 3 + 2 = 0
Horner1 0 1 2 1 1 1 21 1 2 9
3 + 2 = 0 ( 1) (2 + + 2) = 0
= 1 2 + + 2 = 0 < 0.
(2
3)x =
(23
)x = 1
(23
)x = (2
3)0
x = 0
5.8 5
[f(x)]g(x) = 1, f(x) > 0
,
g(x) = 0, f(x) > 0
f(x) = 1
5.9 : (x2 5x+ 7)x23x+2 = 1
- 155
-
5.9 :
x2 5x+ 7 = 1
x2 3x+ 2 = 0, x2 5x+ 7 > 0
x2 5x+ 7 > 0 < 0, x R x2 5x+ 7 = 1 x1 = 2 x2 = 3 x2 3x+ 2
= 0 x1 = 1 x2 = 2 x1 = 1, x2 = 2 x3 = 3
5.9 6
[f(x)]f(x) = , f(x) > 0, > 0
, f(x) =
5.10 : (x2 7x+ 15)x27x+15 = 27
5.10 (x2 7x+ 15)x27x+15 = 27 (x2 7x+ 15)x27x+15 = 33 x2 7x+ 15 =
3 : x2 7x+ 15 > 0 , < 0 x2 7x+ 15 = 3 x1 = 3 x2 = 4
5.11 :
i. 2x23x =
1
4ii. 4x2 = 0 iii. 82x3 8x2 = 0
5.11 i. 2x23x =
1
4 2x23x = 1
22 2x23x =
(12
)2 2x23x = 22 x2 3x = 2 x2 3x+ 2 = 0 x = 1 x = 2
ii. 4x2 = 0 4x2 > 0 x R.iii. 82x3 8x2 = 0 82x3 = 8x2 = 0 2x 3
= x 2 x = 1.
5.12 :
i. 92x+1 = 27x+1 ii. 4x1 =
8
5.12 92x+1 = 27x+1 (32)2x+1
=(33)x+1 32(2x+1) = 33(x+1)
34x+2 = 33x+3 4x+ 2 = 3x+ 3 x = 1.
5.13 : 9x + 3x+1 4 = 0
5.13 9x + 3x+1 4 = 0 (32)x + 3x 3 4 = 0 (3x)2 + 3 3x 4 = 0. 3x =
, > 0, 3x > 0.
2 + 3 4 = 0
- 156
-
1 = 1 2 = 4 . = 1 3x = 1 3x = 30 x = 0.
5.14 : 2 5x2 + 2x 12 5x3 3 2x3 = 0
5.14
2 5x2 + 2x 12 5x3 3 2x3 = 0 2 5x 52 + 2x 12 5x 53 3 2x 23 =
0
2 5x 152
+ 2x 12 5x 153 3 2x 1
23= 0
225 5x + 2x 12
125 5x 3
8 2x = 0
225 5x 12
125 5x = 3
8 2x 2x
5x ( 2
25 12
125
)= 2x
(38 1)
5x ( 2
125
)= 2x
( 5
8
) 5
x
2x=58 2125
(5
2
)x=
625
16
(5
2
)x=
54
24
(5
2
)x=(5
2
)4 x = 4.
5.15 : 8x + 18x 2 27x = 0.
5.15
8x + 18x 2 27x = 0 (23)x + (2 9)x 2 (33)x
23x + 2x (32)x 2 33x = 0 23x + 2x 32x 2 33x = 0
23x
33x+
2x 32x
33x 2 3
3x
33x= 0
23x
33x+
2x 32x
3x 32x 2 3
3x
33x= 0
(2
3
)3x+(2
3
)x 2 = 0
((2
3
)x)3+(2
3
)x 2 = 0.
(2
3
)x= 3 + 2 = 0
Horner1 0 1 2 1 1 1 21 1 2 9
3 + 2 = 0 ( 1) (2 + + 2) = 0
- 157
-
= 1 2 + + 2 = 0 < 0.
(2
3)x = (2
3)x = 1 (2
3)x = (
2
3)0 x = 0
5.10 :
f(x)<
g(x)
> 1, f(x) < g(x) 0 < < 1, f(x) > g(x)
5.16 :
i. 3x < 9 ii. 52x4 > 1 iii. ex 1e< 0
iv.(
3
)1x 0 vi.
(23
)x+ 1 0
5.16 i. 3x < 9 3x < 32 x < 2. 3 > 1 .
ii. 52x4 > 1 52x4 > 50 2x 4 > 0 2x > 4 x > 2 5
> 1 .
iii. ex 1e< 0 ex < 1
e(1e
)x 1 x < 1. 1
e< 1
.
iv.(
3
)1x 0 8x > 3, 8x > 0 x R.vi.(2
3
)x+ 1 0
(23
)x 1
(23
)x> 0 x R.
5.17 : 5x + 51x < 6
5.17 5x + 51x < 6 5x + 55x
< 6 52x + 5 < 5 5x 52x 6 5x + 5 < 0. 5x = , 5x > 0
> 0, 2 6 + 5 < 0
1 5 +
2 6 + 5 + 0 - 0 +
2 6 + 5 < 0 1 < < 5 1 < 5x < 5 50 < 5x < 51
0 < x < 1.
5.18 :3x 95x 1
< 0
5.18 3x 9 > 0 3x > 9 3x > 32 x > 2 5x 1 < 0 5x
< 1 5x < 50 x < 0. :
- 158
-
x 0 2 +
3x 9 | | +5x 1 | + | +
3x95x1 + | | +
3x 95x 1
< 0 0 < x < 2.
- 159
-
5.1.3
1. :
i. f(x) = 5x ii. f(x) =(3
7
)xiii. f(x) = 0, 25x iv. f(x) = 3x.
2. , , - :
i. f(x) = 2x g(x) = 2x + 1, h(x) = 2x 3
ii. f(x) =(3
5
)x, g(x) =
(35
)x+ 2, h(x) =
(35
)x 1iii. f(x) = 3x, g(x) = 3x+2, h(x) = 3x1
iv. f(x) =(3
5
)x, g(x) =
(35
)x+1, h(x) =
(35
)x33. , , -
:
i. f(x) = 4x, g(x) = 4x
ii. f(x) = 2x, g(x) = (12)x
4. R,
f(x) = (2 5)x
i. R,ii. ,iii. R,iv. R.
5. R,
f(x) = (|| 3)x
i. R,ii. ,iii. R,iv. R.
6.
i. 5x+4 = 25 ii. 32x5 = 27 iii. 2103x = 16
iv. 7x225 = 1. v. 33x+5 =
1
81vi. 3x =
1
9
vii.(1
2
)x= 8 viii. 2x+5 = 4 ix. 35x = 0.
7.
- 160
-
i. 3x2 27 = 0 ii. 2 5x+3 50 = 0 iii. 13x23x4 1 = 0
iv. 2|x|6 14
= 0 v. 72x5 =1
7vi. 3x+2 = 0.
8.
i. 5 2x = 2x+3 3
2 ii. 3x+2 = 24 3x1 + 19
iii. 2x+2 3x1 = 48. iv. 4x+2 52x1 = 3.2009.
i. 22x 10 2x + 16 = 0 ii. 9x 2 3x 3 = 0 iii. 4x+1 + 7 2x 2 =
0
iv. 9x+1 36 3x1 + 3 = 0 v. 32x2 = 4 3x vi. 52x+1 + 5 = 130
5x110.
i. 2x + 2x1 = 24x 22x ii. 25 22x4 3x = 9 5x
iii. 2x + 2 3x+1 = 3x+3 5 2x+211. :
i. 53x7 < 5x1 ii. 72x+1 713x iii.(
1
3
)4x3(
1
3
)x+12
iv.(
5
8
)32x>
(5
8
)3x7v.(
7
6
)x+5(
7
6
)53xvi.(
1
4
)x3 0. ii. 32x1 0 iii. 53x1 5 iv. 75x+2 < 7.13.
i. 22x 5 2x + 4 > 0. ii. 9x 2 3x 3 < 0.
iii. 4x 5 2x+1 + 16 0 iv. 9x+12 2 3x+2 + 27 0
v. 4x+1 8 2x1 + 1 0. vi.(1
9
)x1 30 (13
)x+1+ 1 0.
14. :
i.{
2x + 3y = 13.3 2x 2 3y = 6 ii.
{2x 5y = 3.2x+2 5y+1 = 7 iii.
{2x 3y = 14x 9y = 7
15. :
i.{
53x+1 5yx = 2572x13 7xy = 7 ii.
{25x+1 4y2x = 852x11 52y = 1 iii.
{4x2y 8y1 = 3227yx 92x+1 = 1
3
- 161
-
5.2
5.2.1
5.9 > 0, 6= 1 ;
x = x = log, > 0
5.10 ;
10x = x = log10 = log, > 0
5.11 ;
ex = x = loge = ln, > 0
5.12 ;
1. log1 = 0, log1 = 0, ln1 = 02. log = 1, log10 = 1, lne = 13.
log = , log10 = , lne = 4. log = , 10log = , eln =
5.13
1. log(x1x2) = logx1 + logx2 x1, x2 > 0, > 0, 6= 1 logx1 =
1 logx2 = 2 : x1 = 1 x2 = 2 :
x1x2 = 12 x1x2 = 1+2
1 + 2 = log(x1x2) logx1 + logx2 = log(x1x2)
2. log(x1x2
) = logx1 logx2 x1, x2 > 0, > 0, 6= 1 logx1 = 1 logx2 = 2
: x1 = 1 x2 = 2 :
x1x2
=1
2 x1
x2= 12
1 2 = log(x1x2
)
logx1 logx2 = log(x1x2
)
3. logx = logx x > 0, > 0, 6= 1
- 162
-
logx = logx = = x
() = x
= x
= logx
logx = logx
4. : log =log
log , , > 0, 6= 1, > 0
.
- 163
-
5.2.2
5.11 1. log
> 0, 6= 1 > 02. log(x1x2x3...x) = logx1 + logx2 + logx3 +
...+ logx3. .
2 1 .i. logx+ logy = log(xy)ii. logx logy = log(
x
y)
iii. log = logx
iv.1
logx = log
x
4. log(x1x2) x1x2 > 0, > 0, 6= 1 log(x1x2) = log|x1|+
log|x2|
5. log(x1x2
) x1x2 > 0, > 0, 6= 1
log(x1x2
) = log|x1| log|x2|
6. logx2 x 6= 0, > 0, 6= 1 logx2 = 2log|x|
7. , x, - ,
x = logx
8.
i. log log = 1 log =1
log
ii. log1
= log
5.19 x
logx(x2 5x+ 6)
5.19 x > 0, x 6= 1 x2 5x+ 6 > 0. x2 5x+ 6 > 0 x2 5x+ 6
= 0 2 3, :
x 2 3 +x2 5x+ 6 + 0 0 + x (, 2) (3,+)
x (3,+)
5.20 :
i. log10 0, 001 ii. log 110
10 iii. log 1
232
iv. log9
27
3v. log2 16 vi. log 32
8
27
- 164
-
5.12 log = x x = .
5.20 i.log10 0, 001 = x log10 103 = x
10x = 103
x = 3ii.
log 110
10 = x log 1
1010
12 = x
( 110
)x = 1012
10x = 1012
x = 12iii.
log 12
32 = x (12
)x = 25
2x = 25
x = 5iv.
log9
27
3 log9
3
3
3= x
log9
3 = x
log9 312 = x
(312 )x = 9
312x = 32
12x = 2
x = 4v.
log2 16 = x (
2x
= 16
(212 )x = 24
2x2 = 24
x2
= 4
x = 8vi.
log 32
8
27= x (3
2)x =
8
27
(32
)x = (23
33)12
(32
)x = (2
3)32
(32
)x = (3
2)
32
x = 32
- 165
-
5.21 x:
i. log10 x = 3 ii. log4 x = 1
2iii. log2 x =
2
3
iv. logx 16 = 4 v. logx 8 =3
2vi. logx 0, 1 = 3
5.21 i.log10 x = 3 x = 103
x = 1000ii.
log4 x = 1
2 x = 4
1
2
x =
1
4
x = 12iii.
log2 x =2
3 x =
2
23
x = (212 )
23
x = 213
iv.logx 16 = 4 x4 = 16
x = 4
16
x = 2v.
logx 8 =3
2 x
32 = 8
x = 823
x = (23)23
x = 2323
x = 22
x = 4vi.
logx 0, 1 = 3 x3 = 0, 1 x3 = 101
x = (101)13
x = 1013
x = 3
1
10
5.22
A =1
2log16 +
1
3log8 +
1
4log81
- 166
-
5.22
A =1
2log16 +
1
3log8 +
1
4log81
= log
16 + log3
8 + log4
81
= log4 + log2 + log3
= log(4 2 3)= log24
5.23 log6(12 + 6
3) + 2log6(3
3) = 2
5.23
log6(12 + 6
3) + 2log6(3
3) = log6(12 + 6
3) + log6(3
3)2
= log6(12 + 6
3) + log6(12 6
3)
= log6(12 + 6
3)(12 6
3)
= log6(122 (6
3)2)
= log6(144 108)= log636
= 2
5.24 3log38log32 = 4
5.24
3log38log32 = 3log382
= 3log34
= 4
5.25 log2 = log3 = 0 < 6= 1, :
log(9
25) = 2 + 5
5.25
log(9
25) = log(
9
2) + log
5
= log9 log2 + 5= log3
2 log2 + 5= 2log3 log2 + 5= 2 + 5
- 167
-
5.26 0 < , 6= 1, 6= 1 :
1
log()+
1
log()= 1
5.26
log =log
log , , > 0, 6= 1, > 0
1
log()+
1
log()=
1
log()
log
+1
log()
log
=log
log()+
log
log()
=log+ log
log()
=log+ log
log+ log= 1
5.27 x > 0 0 < 6= 1 :
log 1
(1
x) = logx
5.27
log 1
(1
x) =
log(1
x)
log(1
)
=log1 logxlog1 log
=logxlog
=logx
log
= logx
5.28 x > 0 0 < 6= 1 :
logx =1
logx
- 168
-
5.28
logx =logx
log
=logx
=1
logx
5.29 log5+ log5(log5) = 1 = 1
5.29
log5+ log5(log5) = 1 log5(log5) = 1 51 = log5 5 = log5
55 =
= 5
- 169
-
5.2.3
1. :
i. log2 16 ii. log5 25 iii. log3 27 iv. log6 6 v. log8 1 vi.
log71
72. :
i. log4 32 ii. log9 27 iii. log43
4
2iv. log 1
327
3 v. log0.1
10
3. :
i. log 10 ii. log 100 iii. log1
10.000iv. log
1.000 v. log 0, 001
4. :
i. ln e ii. ln 1 iii. ln e3 iv. lne v. ln
3e2
e5. :
i. log 36 = 2 ii. log1
8= 3 iii. log 8 =
3
2iv. log 4 =
2
36. x :
i. log4 x = 3 ii. log3 x = 2 iii. log9 x =3
2iv. log3(|x|+ 1) = 2
- 170
-
1. :
i. log6 2 + log6 18 ii. log 5 + log 2 iii. log 4 + log 25iv. log
50 + log 2.000 v. log2 24 log2 3 vi. log 40 log 4vii. log 2 log 200
viii. log
9.000 log 3
2. :
i. 2 log3 6 + log3 24 +1
2log3 4 ii.
log2 15 + log216 log2 10
2 log2 6 log2 12
iii.2 log3 2 + log3 18 2
log3 12 1iv.
1
2log4 25 + log4 40
3
2
log4 101
23. :
i. 102 log 6log 12 ii. 103log 5 iii. 10212
log 16 iv.102
12
log 5
5log
1.000
- 171
-
5.3
5.3.1 5.14 > 0, 6= 1;
:
f : (0,+) R f(x) = logx
.
5.15 f(x) = logg(x)f(x);
Df = {x R : g(x) > 0, g(x) 6= 1, f(x) > 0}
- 172
-
5.16 f(x) = logx > 0, 6= 1, x > 0 g(x) = x;
f(x) = logx y = x g(x) = x
5.5:
- 173
-
5.6:
- 174
-
5.17 f(x) = logx > 0, 6= 1, x > 0 ;
> 1 :
5.7: f(x) = logx > 0, 6= 1
, .
x1 < x2 logx1 < logx2
0 < < 1 :
5.8: f(x) = logx 0 < < 1
, .
x1 < x2 logx1 > logx2
- 175
-
5.18 f(x) = logx > 0, 6= 1, x > 0
> 1 :
5.9: f(x) = logx > 0, 6= 1
1. (0,+)2. R3. xx 1, f(1) = 04. 0 < x < 1 logx < 05. x
> 1 logx > 06. .
x1 < x2 logx1 < logx2
- 176
-
0 < < 1 :
5.10: f(x) = logx 0 < < 1
1. (0,+)2. R3. xx 1, f(1) = 04. 0 < x < 1 logx > 05. x
> 1 logx < 06. .
x1 < x2 logx1 > logx2
5.19 f(x) = logx, 1 1 ;
logx1 = logx2 x1 = x2 > 0, 6= 1, x1, x2 > 0
:
5.20 f(x) = logx g(x) = lnx;
10 e , f(x) = logx > 1, x > 0
- 177
-
5.3.2
5.13 , - , -.
5.30 :i. f(x) = 3log(2x 4) 2ii. g(x) = ln(3x 81)iii. h(x) =
logx21(3x+ 9)
5.30 i. 2x 4 > 0 = x > 2 = x (2,+)ii. 3x 81 > 0 > 0
= 3x > 81 = 3x > 34 = x > 4 = x (4,+)iii.
x2 1 > 0
x2 1 6= 1
3x+ 9 > 0
=
x (,1) (1,+)
x 6=
2
x (3,+)
= x (1,+)
2
5.31 f(x) = 3ln(4x 8) + 9. :i. ii. xx yy
iii. K(
4, 9)
fiv. f xx
5.31 i. 4x 8 > 0 = x > 2 = x (2,+)ii. xx f(x) = 0
3ln(4x 8) + 9 = 0 = ln(4x 8) = 3 = 4x 8 = e3 = x = 8 + e3
4> 2
.
xx A(8 + e3
4, 0)
yy f(0) , 0 6 (2,+)iii. K(
4, 9) ,
f(
4) = 9 3ln(4
4 8) + 9 = 9
ln( 8) = 0 ln( 8) = ln1 8 = 1 = 9
iv. f(x) xx
- 178
-
f(x) > 0f(x) > 0 3ln(4x 8) + 9 > 0
x > 8 + e3
4
f(x) xx x (8 + e3
4,+)
5.32 f(x) = log211x, x > 0 .
5.14 f(x) = logx > 1 0 < < 1
5.32 f(x) = log211x, x > 0 2 11 > 1 = > 6 0 < 2 11
< 1 = 11
2< < 6
5.33 i. ln2, ln5ii. log 1
35, log 1
310
5.15 f(x) = logx > 1 0 < < 1
5.33 i. ln2, ln5 , 2 < 5 ln2 < ln5
ii. log 135, log 1
310 , ,
5 < 10 log 135 > log 1
310
5.34 xln = lnx
5.34
xln = lnx ln(xln) = ln(lnx) lnlnx = lnxln
log(x+ 1) = 2 2log(
5x)
- 179
-
, x+ 1 > 0, 5x > 0 x > 0
log(x+ 1) = 2 log
5x log(x+ 1) + 2log(
5x) = 2
log(x+ 1) + log(
5x)2 = 2
log((x+ 1)5x) = 2 (x+ 1)5x = 102
5x2 + 5x 100 = 0 x2 + x 20 = 0
x = 4
x = 5
- 180
-
5.35 log(x2 x) = log(x+ 15)
5.16 log(f(x)) = log(g(x))
f(x) = g(x)
f(x) > 0
g(x) > 0
5.35 , .
lnx = lnx
ln 1 = 0, ln e = 1 log 1 = 0, log 10 = 1 lnx1 + lnx2 = ln(x1 x2)
lnx1 lnx2 = ln
x1x2
x = elnx
= ex ln
ln(f(x)) = ln(g(x))
f(x) = g(x)
f(x) > 0
g(x) > 0
log(f(x)) = log(g(x))
f(x) = g(x)
f(x) > 0
g(x) > 0 ln(f(x)) = f(x) = e log(f(x)) = f(x) = 10
, x2 x > 0, x+ 15 > 0 x < 0 x > 1
x > 15= x (15, 0) (1,+)
log(x2 x) = log(x+ 15) x2 x = x+ 15 x2 2x 15 = 0
x = 3
x = 5
- 181
-
5.36 logx(x2 + 3x+ 2) = logx(8x 2)
5.36 , x2 + 3x+ 2 > 0, 8x 2 > 0, x > 0, x 6= 14, x 6=
1
x (1, 14
) (14,+)
logx(x2 + 3x+ 2) = logx(8x 2) x2 + 3x+ 2 = 8x+2
x2 5x+ 4 = 0
x = 1
x = 4
5.37 ln(x 1) + ln(x 2) = lnx
5.37 , x 1 > x 2 > 0, x > 0 x > 2
ln(x 1) + ln(x 2) = lnx ln(x 1(x 2) = lnx (x 1(x 2) = x x2 3x+ 2
= x x2 4x+ 2 = 0
x = 2
2
x = 2 +
2
5.38 lnx2 = 2
5.38 x2 > 0 = x 6= 01
lnx2 = 2 x2 = e2
|x| = e x = e
2
lnx2 = 2 2ln|x| = 2 ln|x| = 1 |x| = e x = e
:
lnx2 = 2lnx x > 0 .
- 182
-
5.39 6(logx+ 1)2 + 6log2x = 13logx(logx+ 1)
5.39 , x 1 > 0 x > 0, x + 1 > 0 x > 1 , w = logx
:
6(logx+ 1)2 + 6log2x = 13logx(logx+ 1) 6(w + 1)2 + 6w2 = 13w(w +
1) w2 + w 6 = 0
w = 2
w = 3
w = 2 = logx = 2 = x = 100 w = 3 = logx = 3 = x = 103
5.40 logx+1(2x2 + 8x+ 6) = 2
5.40 , x+ 1 > 0 x+ 1 6= 1, 2x2 + 8x+ 6 > 0 x > 1 x 6=
1
logx+1(2x2 + 8x+ 6) = 2 (x+ 1)2 = 2x2 + 8x+ 6
x2 + 2x+ 1 = 2x2 + 8x+ 6 x2 + 6x+ 5 = 0
x = 1
x = 5
. .
5.41 2x1 = 3
5.41
2x1 = 3 x 1 = log23 x = 1 + log23 x = log22 + log23 x = log2(2
3) x = log26
5.42 log2x log4x = 3
5.17 , - , .
- 183
-
5.42 x > 0
log2x log4x = 3 logx
log2 logxlog4
= 3
logxlog2
logxlog22
= 3
logxlog2
logx2log2
= 3
2logx logx = 3 2log2 logx = log26
x = 26 = 64
5.43 10xlogx = x2x
5.43 x > 0
10xlogx = x2x log(10xlogx) = log(x2
x)
log10 + logxlogx = logx2 + logx12
1 + log2x = 2logx+ 12logx logx = w
1 + w2 = 2w + 12w
2w2 5w + 2 = 0
w = 2
w =1
2
lox = 2
logx =1
2
x = 100
w =
10
- 184
-
5.44 52x+4 > 2
5.44
52x+4 > 2 log(52x+4) > log2 (2x+ 4)log5 > log2 2xlog5
> log2 4log5
x = log2 log52log5
5.18 logf(x) = logg(x) > 1,
f(x) > 0
g(x) > 0
f(x) > g(x)
5.45 ln(x+ 1) + ln(x 2) < ln18
5.45 , x+ 1 > 0 x 2 > 0 ln(x+ 1) + ln(x 2) < ln18 ln(x+
1)(x 2) < ln18.
x+ 1 > 0
x 2 > 0
(x+ 1)(x 2) < 18
x > 1
x > 2
4 < x < 5
2 < x < 5
5.19 logf(x) = logg(x) 0 < < 1,
f(x) > 0
g(x) > 0
f(x) < g(x)
5.46 log|x|43 < 0
5.46 log|x|43 < 0 log|x|43 < log1 3 > 1 log|x|43 <
log1
0 < x| 4 < 1
- 185
-
|x| 4 > 0
|x| 4 < 1
|x| > 4
|x| < 5
x < 4 x > 4
5 < x < 5 x (5, 4) (4, 5)
5.47 xlogx+2 < 1015, x > 0
5.47
xlogx+2 < 1015 log(xlogx+2) < log(1015) (logx+ 2)logx <
15log10 log2x+ 2logx < 15, logx = w w2 + 2w 15 < 0 5 < w
< 3 5 < logx < 3 105 < x < 103
- 186
-
5.3.3
1.
i. f(x) = log(2x 6) ii. f(x) = log(15 3x)iii. f(x) = log(x2 + 2x
8) iv. f(x) = log(|2x 5| 3)
- 187
-
1. :
i. log(x 6) = 2 ii. log3(4x 7) = 2iii. log2(x2 9) = 4 iv. ln(x
1) = 3
2. :
i. log(x+ 2) = log(4 x) ii. ln(x+ 3) = ln(2x+ 8)iii. log2(x2 6x)
= log2(18 3x) iv. ln(x 1) = ln(3x+ 5)
3. :i. log(1 + x) = 1 + log(1 x)
ii. lnx
2=
lnx
2iii. 2 log(2x 1) log(3x 2x2) = log(4x 3) log x
4. :i. log2 x log x 2 = 0.ii. ln2 x lnx2 3 = 0iii. log2 x+ 3
log
x 1 = 0.
5. :
i. log2 x2 10 log x+ 4 = 0. ii. ln2 x3 9 lnx = 546. :
log3 x2 2 log2 x3 6 log x+ 20 = 0.
7. :
i.log x+ 1
log x 2
log x+ 2=
4
log2 x+ 2 log x
ii. 1 log x2 10
log x3=
20 log xlog x2
- 188
-
1. :
i.{
log(6x 4y) = 1ln(x y) = 0 ii.
{log(x 2y) = 3 log 2.log x+ log y = log 24
iii.{
log y log x = 2 log 50log(y + 1) + log x = 1.
iv.{
log(5x+ 4y) log 2 = log 71 + log(x+ y) = log 30
2. :
i.{
log(y x) = 1 log 5log x+ log 6 = log y + log2
ii.{
log x log y = 3log x5 + log y4 = 3
iii.{
log2 x+ log2 y = 13log x+ log y = 1.
iv.{
log x+ log y = 5log2 x log2 y = 5
- 189
-
1. :
i. log3(x 12) < 2 ii. log2(2x 6) 4 iii. log 13(x 2) 1
iv. log 12(2 7x) 2 v. log5(3 x) 0 vi. log 2
5(2x 5) > 0
2. :
i. log(2x+ 6) log(18 x) ii. ln(x+ 5) ln(9 x)iii. ln(|x|+ 2) ln(6
|x|) > 0 iv. log(x2 + 2) log(3x+ 12) 0
3. :
i. log 3 + log(x+ 2) 2 log 6 ii. 1 + log(x+ 1) log(32 x)iii. lnx
ln(x+ 4) > ln 3 + 2 ln 2 iv. 2 lnx ln(x 2) + ln(x+ 6)
4. :
i. log2 x 2 log x 3 > 0 ii. 2 ln2 x 5 lnx+ 2 < 0.iii. log2
x log x3 + 2 0 iv. log2 x2 12 log x+ 8 0
- 190
-
6.
6.1
1. 2. 3. 4. 5. 6. 7. 8.
6.2
1. www.mathematica.gr2. www.mathsteki.gr3. www.study4maths.com4.
www.study4exams.gr
-
-
