Mathimatika g Gymnasiou Shmeiwseis Askiseis

5/24/2018 Mathimatika g Gymnasiou Shmeiwseis Askiseis

1/76

(+ )2=

2+ 2+

2

x2+ x+ = 0 , 0

x=

24

2


2/76


3/76

1

.

-

-

.. ...


4/76


5/76

1: 3

1.

1.1.;:

:

+ = + .= .

(+ ) + = + (+ ) .( .) = ( .).

+ 0 = .1 =

+ () = 0 1

= 1 , 0

.( +) = .+.

.0 = 0

1.2..:: :

2:

= .. .

= 1 1= .

0, 0= 1

= 1

.

( )

=

( )

, , 0

-

:

1. .

=

+

2.

=

3. .

= (.)

4.

= ( )

5. ()

=

1.3. ;.:

, , ,

.

=x , x2= ( )

2 = .

0 = 0.

1.4. = , , 0.:

:

1: ( )

2 = ( ) ( )

2 2 = .

2: ( )

2 = .


6/76

4 1:

( )2

= ( )2

= .

1.5.

=

0 >0.

:

:

1:

2

=( )

( )

2

2

=

2:2

=

2

=2

=

.

1.5.1. > 0 > 0 + + + = + 0.

___________________________________________________________________________________________

1.6.= 2003

6 10 2(4 3)

3( ) 3( )

x x y

x z y z

+ + +

2(x+ 13

) 2y,

x+ y= 2003.[.=-2003]

1.7.= ( )

115 17

24

12 18 3

236

[. =1]

1.8.:(5

5+ 5

5+ 5

5+ 5

5+ 5

5)

2.(22.2

2.2

2)

2

10.[. 10

12]

1.9.:

(33)

3: 3

3+ 3

3.3

3+

3333

3.[. 3

7]

1.10. 3 42 2

x y

x y

+

= 5 -

=2 22x y

xy

+ .

[. =3]

1.11. x= 0,03 y= 310

, -

=3

73 5 2

11( ) :

xx y

y

.

[. =1000]

1.12. x= 2 y= 101,

= 51.(x

1 y

1 2x

1y)

[. -48

25]

1.13. 31 21 13 7 4+ + + + = 6.1.14. 24 6 9 16 = 12.1.15.:) 2 3 24

) 1 12 453 5 4

[. )12 )3]

1.16.:) 20 6

15 2

)

[. )2 )]

1.17.:16 + 2 81 8

2+ 2 2 8

7.[. 14=.7]

1.18. , , -:2 249 2 4 + + .

[. 7]


7/76

1: 5

1.19.:= 4 12

2+ = 8 20

2

[. =2+ 3 , =4- 5 ]

1.20.x,y , -:

5x 3xy 2y 3x y 3 3 3x y

.

1.21.- 12 cm 3 cm.

[. =3cm2]

1.22.:i) 6 11 6 11 +

ii) 19 3 19 3+ [. i)5 ii)4]

1.23. = 3 + 2 = 12 8 , -: + , , ..

[. +=3 3 - 2 , -=3 2 - 3 , =2]

1.24.i) = 3 3 27

3

+ ii) = 8 2 5 2

8 8

+ +

[. i)=6 ii)=1

2]

1.25.:3 5

3 5

+

+ 3 5

3 5

+

[. 7]

1.26.:x 3 3 = 5 (1x)

[. x=1]

1.27.:1

5 1

x

= 1

5 1

x +

+

[. x= 5 ]

1.28.5 7 3

13 : 31 3

+ +

6.

1.29.:5 3 32 15

60 :3 5 15

+

= 1.

1.30. 27 cm 12 cm 2 3 cm.

5 cm;[. 3 ]

3.1.( )

2

1

2+

( )2

1

2 2+= 1.

3.2., ,

.

32

9 2 50

8

3.3. 10 104 11

8 4

8 4

++

[. 16]


8/76

6 1:

2.

-

2.1.;:

, .

2.1.1., : 5x , 4 1 , 32+ 9 , 2,64+ 32.2.;

:

, .

2.2.1.: = 2 32 5 8 3(2)2 5(2) 8 = 3.4 + 10 8 = 12 + 10 8 = 14.

2.3.;:

.

2.3.1.: 3 , , 35x

2 , 2

3

4 , 4,35

6

2

9.

2.4.;:

.

2.4.1.: 32

2

5 , 9

2

5 , 2,7

2

5 .

2.5.;:-

.

2.5.1.: 52 , 2 , 22 :5

2+

2+ (2

2) = (5 + 1 2)

2= 4

2.

2.6.;:

.

2.6.1.: 3x2

1 , 53

2

+ 3,22

2

, 62

53

+43

2

5

.


9/76

4: 7

2.7.;:

.

2.7.1.: 22 , 34

5

4 ,

5x :

(22).(34

54).(x) = 2(34

)(1)2.5.43.x) = 32

479x.

2.8.;:

, ,

.

2.8.1.: (1) 1252: (33) = 1252.3

1

3 =

5 2

3

12

3

=

25

3

123

= 42.

(2) (10x

2

y

2

) : (5xy

4

) = (10x

2

y

2

). 4

1

5xy =

2 2

4

10

5

x y

xy

=

22

4

10

5

yx

x y

=

2

2

2x

y .

2.8.2.: .

___________________________________________________________________________________________

2.9.

i) 33

2

ii)1,4x4

y5

iii)

2

2xy

iv) 4x6

-3 v)x 1

x vi) (1+ 5 )

2.10.x+y= 2 -:

x(x3) +y(y3) + 2xy[. 10]

2.11.

) 3x2y 7x

2y 5x

3y

5+ 7x

2y+ + = 5x

2y

) 0,632+ 32 35

32+ + 423=

23

2

2.12. (5+4)

4

3

;

2.13.2x

2 , 1

2xy

2 , x

2y , yx

2 , 2x

2y

2 ,

2

3x , 1

2x

2

.

2.14.:i) 2x

2y 3x

2yx

2y+

21 x

2y

ii) 13xy

3xy

3 2

3y

3x+

35

6

xy

iii) 43

3

2

4

2

3

2

4

1,25

3

4

2

+ 14

5

3

2

4

2.15. 3x23y+2, 2x3y4. , -

.[. =3, =1]

2.16. , :

2+1

x2+ 5

3x

+2

2.17. , :

32x

2y

+7 + 2

2x

+1y

12

2.18.:i) 2xy

2.(3x

2y

4) ii) ( )323 . ( )

234

y

iii) 2 325

x y . ( )yx 358

15

iv) 2xy2.(4x

3y) .(x

4y

6)

v)xy.(2

2) .4

3

5.(

21

3

2)

2.19.:i) ( )

321

2 .(235) .(34)2

ii) (2x3y

2)

2. ( )

32 41

2x y


10/76

8 1:

iii) (32

3)

3.(

4

2)

2. ( )

25 61

3

v) ( ) ( ) ( )3 2 44 2 5 4 3 6 23 2 x y xy ax y

2.20.:i) 12x8: (6x5) ii) (2497) : (653)iii)(18

2

3) : (9

3) iv) 10xy

4z

3: (15x

3y

4z

5)

2.21.:i) [2x

3y

4.(6x

2y)] : (4x

5y

7)

ii) [(42

3x) .

5x

3) .(

21

6x)] : (14

11

5x

4)

iii) (32

3x

5)

3: (

94

4x

4)

2

iv) ( ) ( ) ( )2 3 2

2 3 4 21 2 3: :3 3 4


11/76

3: - & 9

3.

-

3.1.;:

,

.

3.1.1.: 52 3+6+ 2 + 9 2+ 3 2 + = 42 2+ 14+ 3.

___________________________________________________________________________________________

3.2.

i) 4x2 3x

3+ 5x 2x

2+ 7 x 2 + 3x

2+ 4x

3

ii) 5x4 3x

2+ 2x 7x

4 3x

3 1 2x+ 4 5x

2

iii)x2+ 2x+ 3x

2x+ 5 4x

iv) 23x 6

2x

2+ 5

3x+

3 2

4+x

4

3.3. =x23x+1 , = x

2+2x+5 = 2x

2x3

i)+ ii)+ + iii)

3.4.:

i)3x2 [(5x

3x) + 4x

2 (2x

2+6)] + (2x

25x)

ii)53 (2

2+3) + [ (3

224) + (2+6)]

3.5.:

i)(2x3) [2x (x22)] {x2[3x+4(x21)]}

ii)3x

2

{x

2

[x(1x

3

)]+2x}{2x

3

+ [x+(x

2

3)3x

2

]1}4


12/76

10 1:

4.

4.1.;:

, -

.

4.1.1.: (22).(3 12

32+ 4) = 632 232+ 82.

4.2.;:

,

.

4.2.1.: ( 2).(2 3 4) = 3 32 4 22+ 6+ 8=

3 5

2+ 2+ 8

4.2.2.: , .

___________________________________________________________________________________________

4.3.:(x

22y)3y+ (xy+y

2)(x) + (x+y)(2xy) (x+3)2y

2.

x= 2 y= 1.

4.4.2x(

2+1) (x+1)(2) x(

2+2) +

,x.

4.5.

.

) (2x+3)(x2+x1) (x

21)(x+2) 2x

3 , x= 2.

) (x2y2xy

2)(2xy) 2x

3(x+y) (xy)(2y

3), x= 1,

y= 2.

4.6.:) 3x(x

21) 4x

2(x+2) 3x+ 4(x

21)

) 5x2(x

32x

2+4) +(12x)(4x

3) x(x1) 2x

) 2(2+2) 3()(3) 42

) 3[x2 (x+4) 3] 2x2[x+ (x2)] 5

) 2{ [2(+4)] + 2} 3(

22)

4.7.,:

) * .(427+8) = 28

3 49

2+ 56

) * .(3x2+8x7) = 36x5+ * *

) 523( * 92+ * ) = 2057 * + 49

4.8.:) (3x+1)(2x3)

) (3x22x+4)(2x5)

) (x+2)(x2)(x+5)

) (x+1)(x+2)(x+3)

4.9.:) (2)(1)(23)

) (3)(4+1)(2)) (x1)(2x5)(x

2+5x1)

) (+3)(2)(221)(4

2)

4.10.:


13/76

4: 11

i) (x3)(1x) + (x2x+2)(x2+x

2)

ii)(+1)(2) (22+3)(3+2

2)

4.11. = 2+1 , = +2 = 1 -:

. ,

. ,

.

.+

.

..

4.12. = 4x1 , =x2 =x2+x1 -:

K. , . , . K. .(. .).

4.13.:(x2y)(4x3y) ( ) ( ) ( )211 372 5 2 23 3x y x y y xy

4.14.x, 2x 2x .

x. .

[.3

4]


14/76

12 1:

5.

5.1.;:

.

5.1.1.:(i) ( 2) =

2 2

(ii) (x+ 2)(x+ 5) = x2+ 7x+10

(iii) 2+

2= (+ )

2 2

5.2. (+ )2 = 2+ 2+ 2.

:

: (+ )2 = (+ )(+ )

= 2+ + +

2

= 2+ 2+ 2.

5.2.1.: (i) (+ 3)2 = 2+ 2.3 + 32 = 2+ 6+ 9.(ii) (5x+ 4y)

2 = (5x)

2+ 2.5x.4y+ (4y)

2 = 25x

2+ 40xy+ 16y

2.

(iii) (23+ 1

4

2)

2 = (2

3)

2+ 2.2

3. 14

2+ ( 1

4

2)

2 = 4

6+

3

2+ 1

16

4.

5.3. ( )2 = 2 2+ 2.

:

: ( )2 = ( )( )

= 2 + 2

= 2 2+

2.

5.3.1.: (i) ( 11)2 = 2 2.11 + 112 = 2 22+ 121.(ii) (6x

2+ 7)

2 = (7 6x

2)

2 = (7)

2 2.7.6x

2+ (6x

2)

2 = 49

2 84x

2+ 36x

4.

(iii)24

342

=24

342

+ =

2434

2

+

=24

2

+ 2.4

2 .4

3+ (4

3)

2

=8

4 + 4

4

3+ 16

2

6.


15/76

5: 13

5.4. (+)3 = 3+ 32+ 32+3.:

: (+ )3 = (+ )(+ )

2

= (+ )(2+ 2+

2)

= 3+ 2

2+

2+

2+ 2

2+

3

= 3+ 32+ 32+ 3.

5.5. ()3 = 3 32+ 323.:

: ( )3 = ( )( )

2

= ( )(2 2+

2)

= 3 2

2+

2

2+ 2

2

3

= 3 3

2+ 3

2 3.

5.6. (+)() = 22.:

: (+ )( ) = 2 +

2

= 2

2.

5.6.1.: (i) (9x+ 2y)(9x 2y) = (9x)2 (2y)2 = 812x2 42y2.(ii) (3

5+ 32)(35+ 32) = (32 35)(32+ 35) = (32)2 (35)2 = 64 910.

5.7. (i) 3

3

= ()(2

+ +2

) .(ii) 3+3 = (+)(2 +2) .

:

(i) 2 :

( )(2+ +

2) =

3+

2+

2

2

2

3

= 3

3.

(ii) 2 :

(+ )(2 +

2) =

3

2+

2+

2

2+

3

= 3+

3.

___________________________________________________________________________________________

5.8.:

) (x+ 5)2 ) (2x 3)

2 ) (3 7)

2

) (3xy+ 2)2 ) (x

2 4y)

2 ) (2

3

2)

2

5.9.:

) (23 x

2+ 4xy)

2 ) (7

23

2)

2

) (3 1

2

)2 ) (

2

3

4a

+ 23)

2

) ( 56x

2y

4 2

xy)

2 )( 1

2

2

3

2

6

)

2

5.10.:

)(2+ 2

4)

2 )(2x

3+

2

4

y)

2

)(23

2x

3 3

4

5)

2 )(1

4x

2y

3

2

11

6x y)

2

5.11.

(+ )2+ 2(+ ) .

5.12.--

:

)x2+ 2x+ ) 4x2+ 4xy+

) 92 6+ )x2+ 4y2+


16/76

14 1:

) 252+ 4

2+ ) 81

2+ 49

2

) 40xy+ 16x2+ ) 60+ 9

2+

) 123

2+ 4

4+

5.13.-

-

:

) 2 + )

2+ 1

4+

)x2+ 6

5x+ ) 49

6+

8+

) 94x

2 6xy+ ) 25

4 a

2 5+

5.14. (+)2=

2+

2+ 2, -

, .

5.15.-

= (2+3)2 (23)2

24, .

5.16.i) 2+

2 2.

ii)

:

2+

2+

2 + + .

5.17.:

)(4x+3)(4x3) )(+2)(2)

)(523)(5+2

3) )(2x

2y

43a

3x)(2x

2y

4+3a

3x)

5.18.:

) (7a+4)(47)

) ( 25x

2y

3

2)(

2+

52 x

2y

3)

) (4x3 9

2y)(4x

3 9

2y)

) ( 2x3 a

4)(a

4 2x

3)

5.19.:

) (x+y1)(x+y+1)

) (2x+y3)(2x+y+3)

) (2+3)(2+3)

) (x2+x 2 +1)(x2x 2 +1)) (x+y)(+x++y)

5.20. = 2 1 , = 2 + 1 , :

i) . ii) + iii)2 2( )

++

[. i)1 ii)2 2 iii)6]

5.21.:

)(x+ 4)3 )(2x 3)

3 )(2x 3y)

3

)(3xy+ 2)3 )(x

2+ 2y)

3 )(2a

3 x

2)

3

5.22.:

) (+3)2 (2)2 52

) (x)2 ()(+3x) (x)

2

) (x3+2y

2)

2 (y

2+2x

3)

2+ (x

32y

2)(x

3+2y

2)

) (2x+3)2+ (2x3)

2+ (2x+3)(2x3) 3(x5)

2

5.23.:

)(x+3)2+(x3)

2+(x2)

2 (x+2)

2+ (x+3)(x3)

(x+2)(x2)

)(2x+5)2 (x5)

2+ (3x1)

2 (2x+1)

2 (2x3)(2x+3)

)(2x+1)2+ (2x+1)(2x1) (x+3)(x3)

(x3)(x3)

)(x2+1)2+ (2x23)2 (3x2+4)2+ (x22)2+

(x23)(x

2+3)

5.24.:

(3x+2)(3x2)(x1) (2x1)3x(x2)(x5) + 5

= 10(x1)2.

5.25. = 2 4 15 = 6 10 :

i) 2

2.

ii) -.

5.26.

= 7 2 6 7 2 6 + = 34 14

+ 34 14+

.

-

.[. =30, =30]

5.27.-

, , 1.

.

5.28. = 10 + 5 2

: x2 20x+ 50 = 0.

5.29.x2 6xy+ 11y

2 8y+ 8 = 0

): (x 3y)2+ 2(y 2)

2= 0.

) (x,y)

.[. )(6,2)]

5.30.:

) 3

3= ( )

3+ 3( ).

) 3+

3= (+ )

3 3(+ ).

5.31.:

) (x1)3 (3x+2)

3x(x2)(x+2)

) (x+y)3y(xy)(x+y) x(xy)

) (x+2)3 3x(x1)

2+ (x1)(x+1)(x2)

5.32.:

i) (x+1)3+ 3(x+1)

2(x1) + 3(x+1)(x1)

2+ (x1)

3

ii) (xy)3 3(xy)2(x+y) + 3(xy)(x+y)2 (x+y)3

iii) (+)3 ()

3 3(+)

2() + 3(+)()

2


17/76

5: 15

iv)3

52 5

yx

+

352 5

yx +

+ 3

25 52 5 2 5

y yx x +

+

32

5 52 5 2 5

y yx x +

5.33.:) (+)

2 ()

2= 4

) (2+

2)

2 (2)

2= (

2

2)

2

) ( )2

2

+ ( )

2

2

=

) (x+3y)2 (x2y)

2 5y2= 10xy

) (2+4)(x2+1) (x+2)2= (2x)2

) (2+2)(x2+y2) = (x+y)2+ (yx)2

5.34.:

)2

( 1)

2

+

2

( 1)

2

= 3

5.35. > , 2+

2,

2

2 2 .

5.36. > 1 , ,2 1

2

2 1

2

+ .

5.37.:

) (+)2+ (+)

2+ (+)2

2 2

2= (++)2

) (x2+y2+xy)2=x2y2+ (x+y)2(x2+y2)) (++)2+ (+)2+ (+)2+ ()2=

4(2+2+2)

5.38.:

(3x2+2)

2 (2x

2+3)

2 2(2x

21)

3+ 27x

2=

5(x41) (x

22)(4x2+1)2

5.39.:

) 9992 + 1998 + 1

) 10012 + 1 2002

) 1001.999 = (1000+1)( )

) 993 + 3.99

2 + 3.99 + 1

) 1013 3.101

2 + 303 1

5.40.:

i) (2+ )2= + + 9

2

ii) (3+ )2= + + 4x

4

iii) ( 3y3)

2= 25x

4 +

iv) ( 3)2= 24

2+

v) ( + )2= 42+ 94+

vi) ( )2= 9x

2 9x+

5.41.:

i) ( 53+ )( 8

2) =

ii) ( )( + ) = 1218x

2

49

2y

6

iii) ( )(21 x

3+ ) = 3

2

4

5.42.:

i) ( + )3= 643+ + + 1253ii) ( + )

3= 8

3+ 12

2+ +

iii) ( 3)3= 1000x

3 +

5.43.:

i) (+1)3= (3)

2+ (31)

2

ii) (+)3= (3)

2+ (3)

2

iii) (x3+y

3)

2 (x

2+y

2)

3+3x

2y

2(x+y)

2= (2xy)

3

5.44.x= 22, y= 2 = 2+2, -

x2+y

2=

2.

5.45. = (x+1)2 , = (x1)2 = +8 -

:

+ = (x+3)2

5.46.x+y= 3 xy= 1 , -

:

i)x2+y

2 ii)x

3+y

3[.i)11 ii)36]

5.47.x+y= xy= , :

i)x2+y

2=

2 2 ii)x

3+y

3=

3 3

5.48. = 5 + 2 = 5 2 ,

:

i) = 2 3+

2

ii) =2 2

+

5.49.

99.[.49 , 50]

5.50. 1

= 7 , -

:

i) 2+

2

1

ii)

3

3

1

.

[.i)51 ii)364]

5.51.x> 0 (x+ 1x

)2= 9 ,

x3+

3

1

x.

[.18]

5.52.:

i)

2

+

2

2ii) (+ )2 4

iii)2 12

x + x


18/76

16 1:

iv) + 1

2 , > 0

;

5.53.

.A

x

y

z

-

, : x2+y

2=z

2.

5.54.:

A

B

x

x

y

y

, : x2+y

2=

2.

5.55.-

: (+ 2)2=

2+ 4+4.

5.56.x,y,z

x,

= 3x+ 2y+ 2z , = 2x+y+ 2z ,

= 2x+ 2y+z .

5.57.

. ,

(+1)(+2)(+3) + 1 = (2+3+1)

2.

5.58.x.y= 6 x

2

y+xy

2

+x+y= 35 x2+y

2.

[. 13]

5.59.:

= 2 10+ 272 8+ 8.

) -

.

) , -

.[. )=10, =2]

5.60.x,y,z, w

x2+ 10y2+ 10z2+ 9w2= 6(xy+yz+zx).

)

-

0.

) x w.

[. x=27w]

5.61. , -

:

2+

2+

2 2 4 6+ 14 = 0.

[. =1, =2, =3]

5.62.:

x

y .

=x, =y, = 2x+y.

() -

25

()

. :

i) xy

ii) ()

()

[. i)2 ii)2

25]


19/76

6: 17

6.

6.1.;:

, .

____________________________________________________________________________________________

6.2.:) 2 4

) 6x3+ 3x2

) 12x2y+ 6xy2 3xy

) 152 25+ 302

6.3.:) (x1) x+1) (2) + 6 3

) (2x+3y) 7(2x+3y) (3y+2x)

6.4.:) ()

2+ (+)

2() 2(

2+

2)

) (52)(4x3y) + (3y4x)

6.5.:3

+3 3

+2+ 3

19.

6.6.-

) x+ 2ay+ 3x+ 6y

) 5y 5 + y

) 4y 2y+ 2

) 15x+ 6x 20y 8y

)xy 2x+ 3y2 6y

)x3+x

2+x+ 1

) 5+

4+

3+

2+ + 1

6.7.:) + 1

)x2+xyxy

) 33 62+ 5 10

) 3

2

2+

3

) 8xy3 24y

2 7xy+ 21

) 2x2+x 2xyy

) x y x+ y

)x3 5x

2+ 3x 15

6.8.:)x2+ (5y)x 5y

) () ()

)x(x3) +y(x) (x3)

)xy(2+

2) + (x

2+y

2)

) (22) + (22)

6.9.:) (+)(+)

) x 2y x+ 2y+ x 2y

) x x+ y+ y x y

) 2

2+ + 1

)x2y+xy+xy+y+x

2

2

6.10.x,y .

2

2

1

1

x

y

+

+

= x

y

x,y .

6.11.:) 812 492

) 3xy3 27x3y

) 45x2y

4 80

2

) 202 15

)2 2

9

x y 1

4

) 1 4x2

) 1 x8

) 166 498


20/76

18 1:

6.12.:) 1001

2 1 )

2 21540 4631077

)2 2

62,31 37,69

24,62

)2 2

225,12 72,12

297,24

)

2 2

2 2

2004 1834

1924 1914

6.13.:)

2

2+ + 1

)81x4 16y4

)x8 1

) 27x 3x3

) 20x 5x3

6.14.:)

3 52 4+ 20

) 2x2y 18y+ 3x2 27

) 2x2 42x2+ 4

) 5 1 +

4

) x2+ x

2 y

2 y

2

) x3

3x(y+z)

2

) 92x

2 4

2 9

2x

2+ 4

2

6.15.-

i) (5x6)2 9

ii) (3)2 16

2

iii) (32)2 (32)

2

iv) (x5y)2

9(x+2y)2

v) 9(2)

2 16(+2)

2

vi)x3x(yz)

2

6.16.x , (x2+3x+1)2 1 -

.

6.17.:) (4x

23x18)

2 (4x

2+3x)

2

) 5(4x2) (x2)

2

) (53x)(x+4) + (3x5)(2x3) +9x2 25

6.18.

3 -

.

6.19. = 3 2+ 1.i) .

ii) ,

.[. ii)=2]

& 6.20.

i) 3 125 ii) 8

3+ 27 iii) 27

3+ 64

3

iv) 2163 1 v)x

4 64x

6.21.i) 64x

3y

6+ 27 ii)

6

3

18

iii)6+ 64

3

6.22.:) 1 2x+x

2

) 2 4y+ 4y

2

) 9x2+ 48xy+ 64y

2

) 916

x2 3

5xy+ 4

25y

2

)x2x+ 1

4

)3

xy+

2

9

y+

2

4x

6.23.-

) (+2) + 1

) 9(x+y)2 6y(x+y) +y

2

) x+ 1 + 2 x , x> 0

) y2 2 3xy+ 3x

2

6.24.-

) 2x2 12x+ 18

) x2 4x3+ 4x4

) x3+ 2x2+x+xy+y

6.25.)x

2 10x+ 24

) 2+ 12+ 27

)y2+ 3y 10

) 2+ 2 15

) 3x2 6x 24

) 2x2 2x 40

) 14 5xx2

) x2+ 2x+ 15

)x2 17x 84

) x2+ 6x+ 5

6.26. =x2 5x+ 6, x> 3.

6.27.)x2+ 9x+ 142

) 2 4+ 32

) 2+ 4 5

2

6.28.:)x

2

2+ 6 9


21/76

6: 19

)x2 4y

2+ 4y 1

) 9 92

2+ 6

) 92 12+ 4 4

2

) 1 + 2+2 2

2

2

) 9y2 4x

2+ 4x 1

) 4x2 36x

2+ 81 y

2

) 16y2 9x2+ 12x 4

6.29.:i)

2+

2+ 2( 8)

ii) 2

2x

2y

2+

2

2 2 2xy

iii)y2 4y+ 3

2 2xx

2

6.30.:i) 4+ 2+ 1

ii) 4+ 22+ 4

iii) 4 11

2

2+

4

iv)x

4

+ 3x

2

+ 4v)y

4 2y

2+ 1

vi) 254+ 31

2

2+ 16

4

6.31.:i) 4+ 44+ 322

ii)x4+ 9y

2+ 5x

2y

2

iii) 4 13

2+ 36

6.32.:)x3+ 12x2+ 48x+ 64) 8

3 12

2+ 6 1

) 3x+ 6

2x+ 12x+ 8x

6.33.) (x+y)

3 3(x+y)

2.y+ 3(x+y).y2y

3

) (+)3+ ()3+ 3(+)2.() + 3(+).()2

6.34.:) (2x1)(x+3) + 1 2x) (+)(2) + (2)

2 (

24

2)

) (x29)

2 (x+3)

2

) (2x25x+2)

2 4x

2+ 4x 1

) 53 9

5

)y2+ 4x 16x

2y

6.35.:i)

2+

2 + 2

ii) x2+ (2+1)x+ 2+

iii) 2+ 9(

2

2) + 6

iv) 2+ 2 2 2 2+ 2v)

2+ (+1) +(+1) +

2

6.36.i) 2+ 2+ 2x2+ 4x 4

ii) 9x2 4y

2 4y 1

iii) x2y

2

2+2x 2y+ 1

iv) x4+ 5x

2y

2+ 9y

4

v) 9(x+y)2 6y(x+y) +y

2

vi) (3 1) 2(

2 1) ( 1)

2

vii) ( 1)3(

2 4) + (4

2)

6.37.:i) (x25x+6)(x1) (x3)(x21)

ii) 4(+ )2 (

2

2

2+

2)

2

iii) (52+2 3)

2 (

2 2 3)

2

iv) (3x 6)(x21) (5x 10)(x 1)

2

v) (2 9)2 (+ 3)

2

6.38.:(2+ 3+ 1)2 1.

[.(+1)(+2)(+3)]

6.39.) 4 4 (+)()3

) (2+ 1)

2 4

2

) 6 1

) (2+

2

2)

2 4

2

2

6.40.) 42 4+1 2(21) + 183 92

) 5+ 24+ 3 221

6.41.:i) 1 x

3

ii) (2+x+x

2)2 1

iii) (2+x+x2)2x2

iv) x3+ 2x2 3

6.42.i)

3+

3

2

2

ii) 2

2+

2

2+

2

2

iii) 2+

2+

2+

2+

2+

2+ 2

iv) 3

2 2

3+ 3

2 2

3+ 3

2 2

3

6.43.i)- , >

=2 2(2 1) (2 1)

4

+ +

.

ii) ,

, > 1 -

1

.[. ii)=1, =0]

6.44. (105+ 25)2 (105 25)2= 10, , .

[. =7]


22/76

20 1:

6.45.:= 2 4 6 8 ... 2002

3 6 9 12 ... 3003+ + + + +

+ + + + +.

[. =2

3]

6.46.= 3+ 3 2 2.6.47.:

= (2 +x+x2) x

3

(: 1.)[. =(1+x+x

2)(4+x

2)]

6.48.):x

4+ 4y

4.

)x,y

y 2, x4+ 4y4

.

6.49.S = 1002 992+ 982 972+ + 22 12.

) 1002 992= 100 + 99 -

S = 100 + 99 + 98 + 97 + + 2 + 1.

) S

Gauss .[. )S=5050]


23/76

7: 21

7.

____________________________________________________________________________________________

7.1.:i) (x

3

x

2

21x+ 45) : (x+ 5)ii) (18x3+ 9x

2 50x 25) : (3x 5)

iii) (2x3 3x

2 17x 12) : (2x+ 3)

iv) (153 14

2+ 7+ 6) : (5+ 2)

7.2.:i) (x

3+ 4x

2 18x+ 2) : (x

2+ 1)

ii) (3+ 4

2 11 30) : (

2 6)

iii) (2y4+ 8y

3 16y+ 8) : (2y

2 4)

iv) (2x5+ 4x

6+x

2 6 + 5x 3x

3) : (x 3 +x

2+ 2x

3)

7.3.:i) [(3x+5)

2

+ (2x+3)2

3x(2x+4) (x+1)2

] : (3x 2)ii) [(x

29)

2 (x+5)(x3)

2] : (x

2+x 12)

iii)

7.4., -x

2x+ 1

x4x

2+ 2x 1.

7.5., -x+ 3 x

3 5x

2+ 7x+ 95.

7.6. (x) = 2x2 5x+ 3, -:

[(x) + (x2) (x1)] : (x 3)

7.7. (x) =x3 2x2+ 3x 1, -:

[(x+1) + (x1) (x)] : (x 2)


24/76

22 1:

8.

... & ...

____________________________________________________________________________________________

8.1. ... ... -:) 12

3

2 , 15

2

3 , 6

4

3

) 82x

3 , 4

3x

5 , 12

3x

3

) 32( )

2 , 6

2( )

2(+ )

8.2. ... ... -:

) 4(x2y

2) , 6(x+y)

2 , 3(xy)

2

) 2

2 , ( )

2 ,

3

3

) 3 6

2+ 12 8 ,

2 4 ,

2 2

) 2 3+ 2 ,

2+ 3 4 ,

3 ,

2 2+ 1


25/76

9: 23

9.

9.1.;:

, -

.

: 1x

, 23

+

,2

4

5

3

x y

xy y

.

____________________________________________________________________________________________

9.2.oi)

233

22

ii)

2

3 2

3

6 18

iii)2

2

4

3 6

a

iv)2 2

2

4

2

a

9.3.oi)

3 3

4 4

ii)

2

2

4

12 3

x xy

xy y

iii)3

2

x x

x x

+

iv)2

2

9

4 3

x

x x

+

9.4.oi)

2

2

1

(1 )

x

x

ii)2 2

2 2

4 4

4

+

iii)

a x x

x x

+

+ iv)

3 3

2 2

+

+

9.5.oi)

2 2

2 2

9 4

9 12 4

x y

x xy y

+ ii)

2 2

2

7 7

14 49

xy y x

x x

+

+

iii)2 2

2 2

49 16

49 56 16

x y

x xy y

+ iv)

3 3 2 2

3 3

3 3 6

6 6

+

9.6.i)

4

2

81

9

ii)2

2

6 9

4 3

x x

x x

+ +

iii)2 2( 1) ( 1)

1

++ + +

iv)2 2 2

2

( 4) ( 2)

4 3

x x

x x

+

+

9.7. = 22

4 4 48

2 8 6

x x

x x

+ +

.

i) x ;

ii) ;

iii) x ;

[. i)x1 3 iii)x=-4]

9.8. -;

2

2

14 49

49

+

[. ]

9.9. = 2 22 2

(3 2 ) (2 3 )

(5 4 ) (4 5 )

, .

> 1

2

.

9.10.oi)

3 2

2

2 2

3 2

x x x

x x

+ + +

ii)2

3 4

( )

+

iii)4 4

3 2 2 32 2 2 2

x x

a a

+ + + iv)

2 2

2 2

a

9.11.:5 4 4 5

4 3 2 2 3

x x y xy y

x yx y x y x

+

+

9.12. = 2 22

( 2) 2 4

9 4

x x x

x

+

i) x;

ii) .


26/76

24 1:

iii) = 1.[. iii)x=-2]

9.13. = 22

7 6

36

x x

x

+

,

=

2

2

7 6

1

x x

x

+ +

.i) x -

;

ii) .

iii) : (+)2 ()2= 4.

9.14.:7 4 3 6

5 2 3 4 4

2x x y xy

x x y x y xy

+

+=x

2+xy+y

2


27/76

10: 25

10.

____________________________________________________________________________________________

10.1.:i)

2 2

3 2 2 3 245 9 4

x x x

x x

+

ii)2

2 2

1 1:

x x

++

10.2.:i)

2

4 2

1236 5 15

ii)2

2

16 3 12:

3 98 16

+ +

iii)2 2

2 2

3 6 12 3:

1 2 2 2 4

+ +

[. i)3

2

5ii)

-3

-4iii)

2

1-]

10.3.:A =

2 2 4 2 3

3 3 2 2:

x y y xy y

x y x xy y

+

+ + .

[. A=1]

10.4.:23

3 3 2 2 2

48 2: :

x yx x

x y x xy y xy y + + .

[. 1]

10.5. = 3 22

4

5 6

x x

x x

+

i) x ;

ii) .

10.6.:2 2

2

2 2 2 2:

5 4 16 25 40

+ + + + +

=(4 5)( 2 )

2

+ +

.

10.7.:i) 3

5

1595+ ii)

44+

1

[. i)-3

1ii)

4

1]

10.8.:i) 1 1x + + 21x 24 1x

ii) 2x y

+2 2

4y

y x

iii) 12

+ 12 +

2

4

4

iv)3

1+

+3

1

2

6

9

v)xx

+

x

x

2

( )

( )( )

x x

[. i)1x

3

+ii)

yx

2

+iii)

2

2

+iv)

3

2

+v)2]

10.9.x,y,z ,:

1( )( )x y x z

+ 1( )( )y z y x

+ 1( )( )z x z y

= 0.

10.10.:i)

x y

x y

+

2 2

4xy

x y

ii)2 2

2

+

+

2 2

2 2

2

+ +

iii)2

2

+

2 2

+ +

2

2

a +

iv)2

xy+

2

2

( ) x

x xy

+

2

2

( ) y

xy y

+

v) 12

32

+

2

4

4 a

[. i)yx

yx

+

ii)-

iii)-

iv)1 v)-

2

2

+]

10.11.:i)

223+

33

2a

+2

5 3

6 6

a

+

ii)2 2

1

+

2

1

+

2

1

2 2

iii)

2

2

+

+2

2 2

4

4

a


28/76

26 1:

iv)23

12 + xx

+2

12 +xx

+4

12 x

v)2

122

xx

x 6

122 +

+xx

x +44

422 +

+xx

x

10.12.:i)

229

6

+

+3

3 +

3

ii)2x

x +33 +x

x 22

2

xx

x

iii)

3

+1

+32

42

[. i) +

3

3 ii)

x

3(x 2)iii)

3

2)2(

]

10.13. = 1 2

6

9

x

x + 3

3x .

i)x ;ii)

.

[. i)x -3 3 ii)=x

x+3]

10.14.=

2

2

6

9

x x

x

+ 2 1xx

+ 2

2

3

3

x x

x x

+ +

)x ;

),

.

[. )x 0,-3 3 )=2(x+1)

x]

10.15.:i)

2 2

2 2

1 1 x y

x y x y

ii)2 2

1 1:

( ) ( )

+ + + + +

iii)2 2

2 21

2

+ + + + + +

iv)

2 2

4 2 44 2: 1y y xyx x

+ + +

10.16.:2

2 2

4

+

+

.

10.17.:( )2 2 2 31 1 2 4:6 4 2 4

x

x x x x x x x

+ + +

[.3

x+3]

10.18.:i) 1

11

x

x

+ ii)

2

11

11

x

x

10.19.:i)

2 2

2

1 1

+

ii)

2

1

11

+

+

+

+

[. i)-ii)-]

10.20.:i)

1

1

+

+

+

ii)2

+

+

++

[. i)

ii)

1

]

10.21.:i)

aa

a

a

+

++

11

1

1

ii) 3

1

++

+

++1

3 +

a

++1

3

[. i)

12

1

2

ii)6]

10.22.:2 2 2 2

3 3

1 1:

+ +

[. ]

10.23. =x+1

1 , =x1

1 ,

=x

x

+

10.24.: = 2 +

, x= +

,

y=

+ z=

+, :

y= 2xzx z+

.

10.25.):2

( 1)( 2) + + = ( 1

1

1 +) ( 1

1 + 1

2 +)

):

11 2 3

+ 12 3 4

+ 13 4 5

+ + 11999 2000 2001

= 14

14000

+ 14002


29/76

10: 27

10.26.:2 2

44 42 2

:4 4

2 2

+ + + + +

+ + +

.[. 5]

10.27.x,y, , :

2

( )( )x

x y x +

2

( )( )

y

y y x +

2

( )( )

x y = 1.


30/76


31/76

29

x +=0

-


32/76


33/76

1: x += 0 31

1.

x += 0

1.1.;;:

x+ y= .

.

___________________________________________________________________________________________

1.1.:)(3 x).2

1=x

)101.(x 3) = 5.10

2 .

[. )1 )3,5]

1.2., --

, ;

6 +

30 =

.

[. 5]

1.3. 15.

[. 7,8]

1.4.:3310

+ 2 13

x + 2( 3)

5

x += 3 2

6x + 2 1

10x

[. x=60]

1.5.:)x 3 3 = 3 (1x) ) 2 1

5 1

x

= 2 1

5 1

x +

+

[. )1 )5

2]

1.6. 21

xx

+ 1

1x

x

+

=

2

2

1

x .

)-

;

)

x= 3.[. )-1,1 )=5]

1.7.:

2x

1

212

x +

+

13

313

x +

+ 152

= 0 .

[. x=-1]


34/76

32 2: -

2.

2

2.1. 2.:

x2+ x+ = 0 0.

x2+ x+ = 0

4.x2+ 4.x+ 4.= 0 [ 4]

42x

2+ 4x= 4 [ 2]

(2x)2+ 2.2x.= 4

(2x)2+ 2.2x.+

2=

2 4 [ 1 2 ]

(2x+ )2=

2 4

2 4 ():

(2x+ )2= :

> 0 :2x+ =

2x=

x =

2

= 0 :

2x+ = 0

2x=

x = 2

< 0 ().

2.1.1. : x

2+ x+ = 0 0

> 0 x =

2

= 0 x = 2

< 0 ().


35/76

2: 2 33

___________________________________________________________________________________________

2.2.:i)x

2x= 0 ii)2x

2 7x= 0

iii)4x2+ 36x= 0 iv)3x

2= 5x

[. i)0,1 ii)0,7

2

iii)0,-9 iv)0,5

3

]

2.3.:i)x

2 1 = 0 ii)3x

2 27 = 0

iii)6x2+ 12 = 0 iv)x

2 12 = 0

v)5x2 10 = 0 vi)8x

2

225 = 0

[. i)-1,1 ii)-3, 3 iii)iv)-2 3 ,2 3

v)- 2 , 2 vi)-5

4,5

4]

2.4.:i)300x

2 600.2

5= 0 ii)200x

2= 2000x

[. i)-8,8 ii)0,10]

2.5.:i)x(x

2 4) = 0 ii)4x(x

2+ 6) = 0

iii)(x 5)(3x2 147) = 0 iv)(2x+ 1)(8 x

2) = 0

[. i)-2,0,2 ii)0 iii)-7,5,7 iv)-2 2 ,2 2 ,-1

2]

2.6.:i) x

2 9x+14 = 0 ii)x

2 12x+20 = 0

iii) 12x2x 6 = 0 iv) 3x

2 5x 2 = 0

[. i)2,7 ii)2, 10 iii)3

4,-

2

3iv)2,-

1

3]

2.7.:i) 1000x

2+ 4000 = 4000x

ii) 400x2+ 300x= 1000

iii) 3.210.x

2 5.2

10.x+ 212

= 0

[. i)2 ii)-2,5

4iii)]

2.8.:i) (x+ 2)

2=x+ 2

ii) (x+ 2)2+x

2 4 = 0

iii) (x 1)2+ (x+ 2)

2= 29

iv)x2+ (x+1)2= (x+ 2)2v) (2x+ 1)

2+ (3x+ 1)

2= (5x+ 2)

2

[. i)-1,-2 ii)0,-2 iii)3,-4 iv)3,-1 v)-1

3,-

1

2]

2.9.:25(x+ 2)

2 9(x 2)

2= 0.

[. -8,-1

2]

2.10.) := (2x 3)

2 8x2 7(1 xx2)

) = 0.

[. )A=3x2-5x+2 )1, 2

3]

2.11.:

i) (x+ 1)(x2+ 2x 8) = 0

ii) (2x 1)(x2+ 3)(x

2+x 2) = 0

iii)x3 2x

2 9x+ 18 = 0

iv) 2x3+x2 3x= 0

[. i)-4, -1, 2, -2 ii)-2, 1,1

2iii)-

3

2, 0, 1]

2.12.:(x

25x+4)

2+ (2x

26x+4)

2= 0.

[. 1]

2.13.:) x

2+ ( 21)x 2 = 0

)x2+ 3 3x 12 = 0

) 4x3 4 3x

2x+ 3 = 0

[

.)1, 2

) 3 ,-4 3

) 3 ,

1

2,-

1

2]

2.14.): x(x 5 ) = 5.)x1,x2 -

, : x1.x2.[. )-5]

2.15.= (x

29)

2 (x+5)(x3)

2

= (x1)2 4

i) , .

ii) x

;

iii)

.

iv) x 6

=x 3.

v) x 2

9x = 3B.

[. i)A=(x-3)2(x+1)(x+4) B=(x+1)(x-3) ii)x -1,x 3

iii)(x-3)(x+4) iv)x=2 v)x=-5/2]

2.16. = 2 4x xx

,

=2

( 3)( 9)3

x x

x

+ =

3 2

4 4x x xx

+

i) x

;

ii) , .

iii) 2 = .

[. i)x 0, x -3

ii)A=x-4 , B=(x-3)2, =(x-2)

2iii)x=-1, x=3]

2.17. = 2x+ 4x2,= 1 4x

2 K = A

B, =

2

3 2

1 2

4 8 4

x x

x x x

+ +

+ +

i) x ;

ii) .

iii) = .


36/76

34 2: -

[. i)x 1

2, x 0 -1

iii)x=1

4]

2.18.:=x

3x

2 4x+ 4 = (x3)

2 4.

i)

x -

.

ii)

= 12.

[. i)x 1 5,

=(x-2)(x+2)

x-5ii)x=-16, x=4]

2.19.:(x+ 1)

3 3(x 1)

2 2(x 1)(x+ 1) = (x 1)

3+ 8

[. -7 , 1]

2.20.x.(x )

2+ (x+ )

2= (x )(x+ ) + 4x.

[. ,3]

2.21.:(x2 4)(x 1) = (x2 1)(x 2)

[. 1, 2]

2.22. :2

2

+ 3

= 2.

i) .

ii) ,

:2

x x

= 14

.

[. i)=2 ii)-1,2]

2.23.: 2 2( 3 1)5 x x+ + = 5.[. -3,-2,-1,0]

2.24.i):=x

3 3x+ 2 =x5x3x+ 1.

ii) : 1x +

= .

[. i)=(x-1)2(x+2), B=(x-1)2(x+1)(x2+x+1) ii)x=1]


37/76

3: 2 35

3.

2

___________________________________________________________________________________________

3.1. x+ 3x 3. 40 cm

2.

i) -

.

ii) x.

iii) .[. i)4x ii)x=7 iii)28cm]

3.2.- 2xm , 3 (x+1) m

4 m. .[. 6+2 3 m]

3.3.- 5cm

12cm. -

.[. 3cm, 4cm]

3.4., 2cm -

10cm. [. 6cm , 8cm]

3.5.- 1cm , -

2cm -

.

,

.[. 3cm 4cm 5cm , 12cm , 6cm2]

3.6.- 10m ,

-

.[. 6m, 8m]

3.7. = 900 :=

2 1

+ =2 1

+ > 1.

.[. 2+2]

3.8.2cm .

60cm2,

.[. 34cm , 13cm]

3.9.-.

.[. 3, 4, 5]

3.10.

10 . ;[. 15]

3.11.- 100.

.[. 6,8]

3.12. = 2x7cm , =x5 cm

27 cm2. .

[. 9cm, 3cm]

3.13. // = =900. = =x2 cm , =x+10 cm -

112 cm2,

.

[. ==8cm, B=20cm]


38/76

36 2: -

4.

4.1.;.:

, .

___________________________________________________________________________________________

4.2.:) 4

2x + + 1

2x =

24

x

x

)2

2

2x x+ =

2

1

5 6x x

+ +

) 131x +

11 x

=2

5 3

1

x

x

[. )3/2 ))]

4.3.-: 8

10x =

10x

x 10

8x = 10 x

x

[. ]

4.4.:)

2

2

4x

2

1

2x x =

2

4

2

x

x x

+

) 53x +

12x +

=2

2 1

5 6

x

x x

+

+ +

) 2xx

=2

2

4

2

x

x x

+

) 23

x

x

+

+ 2

3x

x

+ =

2

2

2( 6)

9

x

x

+

[. )3 )))-3,3]

4.5.=

2

2

2

x

x x+, = 3

2 4x =

2

14

2( 4)

x

x

.

) x

.

) = .[. )x 0 -2. x 2, -

x 2 )0,-2,2]

4.6.:

) 22

x

x + + 2

2x

x

+ = 2

) 1xx

+2

x

x

2

1

2x x = 0

) 2 3xx

2

5 3x

x

=2

3

2 8x x

x

+ + 2

) 12x

21x +

=2

3

2x x

[. )2 )1, - 1

2

)1, -0,8 )]

4.7.:) 4

1x +

2

4 3

1

x

x

= 2

) 1 2

1+x

12 x

=2

2

4

x

x

) 123 2x

83 2x +

=2

2 33

4 9

x

x

[. )- 22, 2

2)0 )2]

4.8. 2

2

2x x+

3

2 4x + ,

x= 1;

;

[.7

6, -

12

7]

4.9.=

2

22

41

4

xx

x

x

+

+

, = 11

x

x

+

3

3

x

x x.

)

x .

) 2x

(x+1).= 0.


39/76

4: 37

[. )x 0, 2 -2. x 0, 1

-1 )4]

4.10. :x

22 1

1

+

x+ 2+ 2 = 0

1.[. -1, 2

3]

4.11.:) 1

2x

x

2

1x

x

=

2

1

3 2x x +

) 13

x

x

+ 1 =

2

2

3 4

5 6

x x

x x

+

+

) 13

x

x

+ 2

3x

x

=

2

2

3

9

x

x

+

[. ))1,4 )1,-6]

4.12.:)

2

5 4

5 6

x

x x

+ +

2

10 1

3 27

x

x

+

= 5

3 9x

)2

4

2x x + 5

2 2x

x

+

+ = 2

3 6x

x

) 23

x

x

+

+ 1

1x

x

+

=

2

4

2 3x x+

)2

2

3

4

x

x

2

1

2

x

x x

+

= 22

x

x

+

+ 2

[. )103

4)3 ))1,

5

2]

4.13.:) 2

1x

x + +

1x

x + 1 =

2

2

3 5

1

x

x

)2

2 1

3

x

x x

+

+ +

2

4

9

x

x = 4

3 x+ 2

3 x

)2

2

7

1

x

x

+ 3

1x + = 2

1x

) 2xx

+ 42x

=2

8

2x x

[. )-4 ))2 )-1]

4.14.:i)

23 11

x

x

2

2

x x =

2 3 2x xx

+

ii)[. i)-3 ii)]


40/76

38 2: -

5.

-

5.1. ;.:

, :

> 0 >

< 0 <

= 0 =

5.2., . > + > + .:

: (+ ) (+ ) = + = > 0 , > .

+ > + .

5.3., . > >0 .>..:

: . .= .( ) > 0 , > 0 > 0.

. > ..

5.4., -. > ..:

: . .= .( ) < 0 , < 0 > 0.

. < ..

5.5. , . > > + >+ .:

: (+ ) (+ ) = + = ( ) + ( ) > 0 , > 0 > 0. + > + .

5.6. > > > . ( ):

> > 0 (1). , > > 0 (2).

(1) (2), :

+ > 0

> 0

> .

5.7. >


41/76

5: - 39

: >

(1).< (1).

< .

5.8. > , , 1

0 1

> 0.

: >

1

> 1

1

> 1

.

___________________________________________________________________________________________

5.9.x>y, 34x 5z 3

4y 5z.

5.10. > , 8,3 6 8,3 6.

5.11. > 0 : + > .5.12.x


42/76


43/76

41


44/76


45/76

1: 43

1.

1.1. y = x+ ;:

y= x+ y= x.

1.2. x+ =;:

x+ = .

___________________________________________________________________________________________

1.1. ,

(52,2+1)

y= x+6.[. =1]

1.2. 1, 2

y= (2+4)x+ 5 y= 4x 3 .

,

.[. =2]

1.3. y= 32x+ 1

y= (+4)x 52 ;

[. =-1 =4/3]

1.4. 1: y= (34)x+ 2 6.

1:

) (2,2)

) x x

)

) 2: y= 8x+ 7

) x x 3

) y y 4

)

3: y=(3 1) 2 1

2

x +

[. )=2 )= 43)=3 )=4 )= 18

11)=5 )= 7

3]

1.5. 4x+ 3y= 12.i) .

ii)

.

iii)

.[.ii)A(3,0) B(0,4) iii)=12,E=6]

1.6. 2 2 x+ (1)y= 12

(3,2). :

i) .

ii)

.

iii)o .[.i)=-2 ii)A(6,0) B(0,-4) iii)E=12]

1.7. 1, 2

y= 22x+ 3+ 1 y= (53)x+ 9 ,

, . 1

(1,6).i) , ..

ii)-

.[.i)==1]


46/76

44 3:

2.

___________________________________________________________________________________________

2.1.:)

4

1

x y

y

+ =

= )

2

5

x

x y

=

+ = )

2 6

y x

x y

=

+ =

[. )(x,y)=(3,1) )(x,y)=(2,3) )(x,y)=(2,2)]

2.2.:

)2 3

y x

y x

=

= )

2

2 5

y x

x y

= +

+ =

[. )(x,y)=(3,3) )(x,y)=(2,4)]

2.3.

, .

i) 2 43

x yy x

+ == +

ii) 3 22 0

x yx y

=+ =

iii)3 2 5

4 6 7

x y

y x

=

= + iv)

12 3

2

2 8 12 0

x y

x y

= =

2.4.2( ) 4 0

5( ) 10 0

x y

x y

+ =

+ = -

; .


47/76

3: 45

3.

___________________________________________________________________________________________

2.1..

i)3 1

3 2 11

x y

x y

= +

= ii)

2 4

31

x y

x y

=

+ =

[. i)(x,y)=(-5,-2) ii) (x,y)=(22,9)]

2.2.:i)

2 2

7 13 0

x y

x y

=

+ = ii)

5 45

2 3 25

x y

y x

+ =

=

[. i)(x,y)=(8,3) ii) (x,y)=(5,20)]

2.3.: 2 13 5( 1)

x y

x y

=

= +.

[. (x,y)=(5,2)]

2.4.:i)

2[ (4 )] 20 0

4 0

x y

x y

+ =

+ = ii)

2(2 3 ) 3(2 3 ) 10

4 3 4(6 2 ) 3

x y x y

x y y x

+ = +

= +

[. i)x=-8, y=2 ii)x= 52,y=1]

2.5.:i)

1 12 2

3 2 192 3 2

x y

x y

+ = =

ii)0,3 1,5 0,3 0

0,3 0,2

x y

y x

+ =

=

[. i)(x,y)=(5,-3) ii)(x,y)=(4,1)]

2.6.:i)

41

6 4

7

x y x

x y

= + =

ii)1

4 12

2 3 0

x y

x y

+ = + =

[. i)x=2, y=5 ii)x=-2,y= 12]

2.7.:i)

2 3 6

3 4

x y

yx

=

=

ii)

3 2 1 3 2

2 3

2 2 15 3

x y x y

x y x y

+ =

+ + + =

[. i)(-3,-4) ii)(-1,-2)]

2.8.:

i)

2 23

3

221

2 3

xy

yx

+ + = ++ =

ii)

2 23 ( 1)

3( 2 ) 2

x y y

x y

+ + + =

+ =

[. i)(2,1) ii)]

2.9.:i)

2 13 6 2

72

x y x y

xy

+ = + =

ii)2 4 0

3 2 23 5 5

x y

x y y xx

+ + = +

+ = +

[. i)(x,y)=(4,5) ii)(x,y)=(-2,0)]

2.10.

(2 1) (4 1) 3

( 1) ( 2) 2

x y

x y

+ + =

+ + =

x, y. , , -

(1,1).[. =-1/2, =9/2)]

2.11.:i)

2( 2 ) 3( 3 ) 83

2( 2 ) ( 2 ) 49

x y x y

y x x y

+ =

=

ii)3 2 7

2(3 2 ) 4(2 3 ) 42

x y

x y y x

+ =

+ + + =

[. i)x=5, y=-6 ii)]

2.12.:i)

321

4 8 2

5 94 11

4 6 2

yx x

yx x

+ =

+ = +

ii)

[. i)x=2,y=3 ii)]

2.13.:i)

3 21

2 5 3

61

2 3

x y x y yx

x y xx y

+ + + = + +

+ + =

ii)

3 2 2 42 3 3

2 3 45

x y x

x yx y

+ =

+ + =

[. i)(6,4) ii)(2,3)]

2.14. 22 5 32 5 1

x y

x y

=

=.

.


48/76

46 3:

[. =2 =-2]

2.15.2

3 3 3x y

x y

=

=.

.

-

.[. =1]

2.16.: 2 3 52

2 3

x y

yx

+ =

+ =

[. x= 2 ,y= 3 ]

2.17. 5x 4y= 7. -

) , .) , .

) , .

2.18.:i)

2 29 5

3 2

x y

x y

=

+ = ii)

1 1 712

7

x y

x y

+ = + =

[. i)x=3,y=- 13ii)(3,4), (4,3)]

2.19.:i) 2 2

1

1

y x

x xy y

= + =

ii) 2 22 5

7

x y

x y xy

+ = + + =

[. i)(-1,0), (2,3) ii)(2,1), (3,-1)]

2.20.:i)

2

3 2 7

2 3

x y

x xy y

+ =

+ = ii)

2

3 5

5 9

x y

x y

+ =

=

[. i)(1,2), ( 85, 1110

) ii)(2,1), (53,-16)]

2.21.i) , y= 2x+

3

(0,1).

ii)

:2

23

12

y x

y xy

= + + =

[. i)=3 ii)(x,y)=(1,3), (- 116,- 8

3)]

2.22.:i)

1 13

2 1 2

3 24

2 1 2

x y

x y

=+

+ =

+

ii)

4 121

1 10

2 2

x y x y

x y x y

+ = +

+ =

+

[. i)(- 14 ,1) ii)(5,-15)]

2.23. := x+ 4 , = 4x y = y+ 2.

.

[. 7]

2.24.-, 10 cm

1 cm

.

[. 3cm 4cm]

2.25., 5 3 2 -

8. ;[. 17]

2.26. y= x+ (2,5) (4,1).

[. y=-3x+11]

2.27. , , x

2

+ x+ = 0 x= 3 x= 7.[. =4, =-21]

2.28. f(x) = x2+ x+ . f(1) = 2 2 f(x) = 0

,

.[. =-1, =2, ]

2.29. y= x+ (,1) (1,3).

[. y=-x-2]

2.30. 9 -, . .

9 , -

.

;[. 33 12 ]

2.31. = 3x+6 ,= x

2, = 5y+4 = y x

. .[. 20]

2.32.

= x2, = 5 y2 ,

= x+ y = 1 . -

.[. ==1 ==4]

2.33. x2+ x+ = 0 16

3 +1

xx

= 2 11

x

+

, , .

[. =1,=- 154=-3,=- 7

4]

2.34. 9. 27

-

. ;[. 36]


49/76

47

y= x2

y= x2+x+ , 0


50/76


51/76

1: y = x2 49

1.

y = x2

___________________________________________________________________________________________

1.1. f(x) = 20x2.:)

2 2

( ) ( )f f

+

+= 20.

)( ) ( )

( ) ( )

f f

f f

+ +

+= 2.

1.2. y= x2 y= x+ 6.

[. (-2,4) , (3,9)]

1.3. y= (26)x2;

[. >3]

1.4. y= 3 57

x

2

x x;

[. >3

5]


52/76

50 4:

2.

y = x2+ x +

____________________________________________________________________________________________

2.1. f(x) = x2 3x+ 1. ,, x 11.

[. x=-2, x=5]

2.2. f(x) = 3x2 (+1)x 2.) ,

(1,5);

)

1 x 4.[. )=5]

2.3. f(x) = x2 4x+ 5.i) , - (1,8).

ii)-

.

iii)-

.[. i)=-1]

2.4. y= x2+ 2x+ .i) ,

(1,5).

ii).[. i)=2, =0]

2.5. y= x+ 3 y= x

2 2x+ 3.

i)-

.

ii)

.[. (0,3) (3,6)]

2.6. y= 2x2+ x+ y= x . (1,2) :

i) , .

ii)

.

[. i)=1, =-1 ii)B(-1,0)]

2.7., -.

10cm.

E

B

x

x

.[. 5cm]

2.8.- 12 cm.

i) 2xcm,

x.

ii)-

x.

iii)-

0

x

6.iv)

.[. i)=12-2x ii)E(x)=-2x

2+12x iv)x=3, ==6cm]

2.9. f(x) = x2 9x+ 12. x :

i) f(x) = 6 ii) f(x) = x

iii)f(x) = f(x) iv)f(x) = f(x+1)[. i)x=3, x=6 ii)x=2, x=6 iii)x=0 iv)x=4]

2.10.- y= x2+ x+ ,

A(1,3).


53/76

51

-


54/76


55/76

1: 53

1.

1.1.;.:

.

.

, , , -

. :

< +

< +

< + .

1.2...:

:1) -

, .

=

=

=

.

2)

, .

=

=

=

.

3) , -

.

=

= =

.


56/76

54 1:

1.3..:

:

1)

, .

2) ,

.

___________________________________________________________________________________________

1.4. , . -

.

1.5. , , , . -

-

.

1.6. . :

+ + >2

+ +.

1.7.

.

1.8. :) 2

+ +.

1.10.-.

1.11. = . -

=31 =

31 .

, - .

1.12. = .

,

= .

, -

.

1.13. = .

,

= . = .

1.14. = . -

.

.

1.15. , -

, , = = .

) .

) .

) -

.

1.16. .

,

= = .

= .

1.17. . ,

. -

:

) = .

) .

1.18. y . x y , , -

, = , = .

:) =

) =

) x y .


57/76

1: 55

1.19. . . -

, . -

= .

1.20. .

. ,

.

1.21. . -

.

1.22. Ox y . -

x y .

:

) .

) AMB .

) .

1.23. .

1.24. . = .


58/76

56 1:

2.

2.1., .:

//. // ,

,

. .

___________________________________________________________________________________________

2.2. , , .

.

2.3. . ,

.

.

2.4. , .

, -

= 12

= 12

.

= 14

.

2.5. ,

. . -

.

2.6. , , ,

, , .

.

.

2.7. , (< )

. ,

. )

.

) = =2

.

) = =2

.

) =2

+ .

) =2

.

2.8. . , .

-

.


59/76

2: 57

2.9. - ,

, , -

.

.

2.10. = 900, = 12 cm = 2x+2 cm. -

2x1 cm, -

.


60/76

58 1:

3.

3.1. ;.:

,

.

A

B

1, 2, 3

4.

:

=

=

.

___________________________________________________________________________________________

3.2. - //.

= 2x+1,

= x, = x2+2

= 4x2

+x. x.

[. x=1

7]

3.3. = 90 = 8cm ,

= 6cm. //

= 4cm.

.

[. =3,2cm]

3.4. = .

A

B

E

:i) //

ii)

=

3.5. 1 2. -

= 2cm, = xcm , = x+1 cm

AE = 3cm, .[. 2cm]

3.6. -

= = .

B

2 +1x

x

x2

+2

4 +x x2


61/76

3: 59

E

. = 2,5 m = 8 m .

[. =10m]


62/76

60 1:

5.

5.1.;:

, .

5.2..:

1) , ,

.

2) , , -

.

___________________________________________________________________________________________

5.3. 5cm , 3cm 15cm , 9cm.

.

5.4. - = 7,2dm , = 40cm , = = 65

0 ,

= 72cm , = 4dm.

.

5.5. - = 8cm , = 2cm , = = 70

0 ,

= 10cm , = 4cm. .

5.6. 1

4. = 4cm , = 5cm

.

5.7. = 12 cm = 18 cm. -

= 8 cm

. = 10 cm,

, .[. =12cm, =6cm, =15cm]

5.8. = ,1 = 2 , = 4 cm = 9 cm.

1

2

4 cm

9 cm .

[. 6cm]

5.9. 6 cm - = 2 cm.

, .

[. =18cm]

5.10. -:

= , = , = 4cm , = x+6 cm ,

= x+2 cm , = 4x+8 cm.

i) .

ii) x.

iii)

=

1

3 .

5.11., .


63/76

5: 61

i) , , .

ii) -

.

iii) = .

5.12. = 900, - = 8 cm = 10 cm.

.

i)

.ii) ,

-

.[. i)=2,4cm , =3,2cm]

5.13. = 6 x, = x(x+3) = x

2+ 5x.

) x 2;

) x -

15 cm.

) x

, = 2.

[. ))x=1 )10cm, 8cm, 12cm]


64/76

62 1:

6.

6.1.;:

.

___________________________________________________________________________________________

6.2. 490cm2.

= 25

. -

, -

.

.

i) ,

.

ii)

.

iii)-

.[. i)40cm

2ii)250cm

2iii)200cm

2]


65/76

63


66/76


67/76

1: 0 00 180 65

1.

1.1., ;:

,

.

, =

=

.

,

.

, =

=

.

,

.

, = =

.

1.1.1.: , .

1.2.:(900 ) = (900 ) = .

:

:

=

=

, = .

, -

,

+ = 900 = 90

0 .

(900 ) = . (90

0 ) = .

, 900 :

(900 ) = . (90

0 ) = .

1.2.1.: 540= (900 360) = 360.73

0= (90

0 17

0) = 17

0.

B

B


68/76

66 7:

1.3. .:

O xx

y

M(x,y)

x

y

(x, y) , OMx = .

= , :

=y

, = x

, =

y

x (x 0).

1.4. , 1 1.: x y

> > y . 1 , 1 0.

, : 1 1 1 1.

1.5..:

. .1 2

3

4

x + +

x + +

x + +

____________________________________________________________________________________________

1.6.:= 25

0 35

0+ 55

0 65

0

= 2x+ 3(900x) 5x

=2 0 2 0

100 0 0

45 4 30 2007

2006 45 260

+

[. =0, =0, =-1]

1.7. (3,0) , (2,2) , (6,8) , (0,5).

x , , x .

1.8.:

i)= 3x 5 ii)= 4 2x

iii)= 7x+ 6x iv)= 2x 3x

1.9.- y= 2

2x+ 1.

[.1, 3]

1.10.- y= 4

2x 1.

[.-1, 3]


69/76

1: 0 00 180 67

2.

1800 :

(1800) =

(1800) =

(1800) =

: (i) 1300= (180

050

0) = 50

0 0,766

(ii) 1150= (180

065

0) = 65

0 0,423

(iii) 980= (180

082

0) = 82

0 7,115.

____________________________________________________________________________________________

2.1. 350= 0,819 :) 550

) 1450

[. )0,819 )-0,819]

2.2.:=

0 0 0 0

0 0 0

(90 ) (90 ) (180 ) 45

(180 ) (180 ) 150

[. 2]

2.3. , , , :

) (+) = ) (+) =

) (+) = B

2.4.x, :) 2x 3 = 0 , 0

0


70/76

68 7:

3.

3.1.: =

.

:

O xx

y

M(x,y)

x

y

OMx = ,

=y

= x

=

y

x

=y

x

=y

x= .

, =

.

, 0, 900 2700.

3.2.: 2+ 2= 1.:

O xx

y

M(x,y)

A

:

x2 + y

2 =

2

2

2

x

+

2

2

y

=

2

2

2x

+2

y

= 1

()2+ ()

2 = 1.

, 2+ 2= 1.

____________________________________________________________________________________________

3.3.i):= 13 13 7 4 + + = 41 29 19 9

ii) 2

=

,

: =2

2

2

.

[. i)=3,=6 ii)=3

4]

3.4. x= 35

00< x< 90

0, :

i) T x x.

ii) T:

A =2

2

3 5

x x x

x

+ .


71/76

3: 69

3.5. x= 513

900 x 1800, -

:

i) T x x.

ii) T:

A =25 3

2 3

x x x

x

+ ++

.

3.6. x 00< x< 900, -:

=2

2 2

2

4 3

x x x

x x

.

[. =1

11]

3.7. = 3 = 4 0 < < 14

,

:

= 252+ (5 1)

3.

[. =1]

3.8. x2+ .x+ 2= 0 . -

, 4-

.

[. =-2 5

5, =

5

5, =-2]

3.9. =

2 2

2 2

+

, > > 0 90

0

< x< 180

0

,:

) )

[. )=-+2 2

2

)=

2 2 -

2]

3.10.: 2(1 + 2) = 1.3.11.: ( 3) = 3.3.12.i) :

2 1 4 3

23 22 3( 1)

x y x y

x

y x

+ + = + =

ii) (x,y)

:

4(x.+2006y.)2+ (2006y.+2x.)

2= 4.

[. i)(x,y)=(1,0)]

3.13.: 1 + 2=2

1

.

3.14.:(+ )

2

( )

2

= 4..

3.15.:(3+ 4)

2+ (4 3)2= 25.

3.16.:

+= 1 + .

3.17.: x.x.x+2

1

1+ x= 1.

3.18.: (1 2

1

)(1

2

1

) = 1.

3.19.:

+

= 1

1

+

.

3.20.: 22

1

1

+=

2

2.

3.21. = 3., = 3., = 3:

2

+

2

+

2

= 9.

3.22.: 11

+ 11 +

=2

2

.

3.23.: 1+

+ 1

= 1

.

3.24.: 1

1

= 1

.

3.25.:

1

+1

=2

.

3.26.: 2 2 = 2.2.3.27.: 4 2 = 4 2.3.28.: 4x 4x= 22x 1.3.29.:

i)4 4

2

1

ii)

2

4 4

2 1

[. i)2

ii)1]

3.30.: 1 2 1+

= .

3.31.:(+

)( 1

)( 1

) = 1.

3.32.:

1 x

x+

2

x

x x= x+ x.

3.33. 00< x< 900 x= 2x 1 x.

[. x=4

5, x=

3

5, x=

4

3]


72/76

70 7:

3.34.i)= 1213

900 1800 -

.

ii) :( ) ( ) 1

( ) 1

x y

x y

=

=.

[. i)=5

13, =-

5

12ii)(

5

2,

1

24)]

3.35. x, 3x+ 4x= 5.

[. =3

5, =

4

5, =

3

4]


73/76

6: & 71

4.

&

4.1. .:

, , , :

=

=

.

.

=

= . (1)

=

= . (2)

(1) (2)

.= . .= .

=

.

=

.

,

=

=

.

4.2. .:

, , , :

2=

2+

2 2.

2=

2+

2 2.

2=

2+

2 2..

.

2=

2+ 2 (1)

=

= . (2)

=

= ..

= = . (3)

(1), (2) (3)

2= (.)2+ ( .)

2

= 2.

2+

2 2.+

2.2

= 2.(

2+

2) +

2 2.

= 2+ 2 2..

2=

2+

2 2.

2=

2+

2 2..


74/76

72 7:

4.2.1. : :

2=

2+

2 2.

2.= 2+

2

2

=2 2 2

2

+ .

=2 2 2

2

+ =

2 2 2

2

+ .

___________________________________________________________________________________________

4.3. . = x = y :

x

y =

=

.

4.4. > 900 = 450,= 10 cm = 5 2 cm.

.

[. =300, =105

0]

4.5. = 2, =3

= 600.

i) , -

.

ii) ;iii) ,

= 10cm.

iv) .

[. i) =900, =30

0iii)=5cm, =5 3 cm

iv)=25 3

2cm

2]

4.6. :

2=

2+

2.

= 900.

4.7. = 3 cm , = 1 cm = 30

0, :

i) .

ii) .[. i)=1 ii)120

0]

4.8. = 2cm , = 2 cm = 45

0. -

.

4.9. = 31 ,

= x+ 1 , = x+ 2 = 600.

.[. =5, =6]

4.10. = 2, -

.

4.11. :+ = .

4.12. : =

2

2.

4.13. : = 3, = 5 = 7. = 120

0.

[. 1200]

4.14. , , : = 2 2 + + ,

.[. 120

0]

4.15. = 4 , = 4 = 120

0.

i) .

ii) B.

iii) .

[. i)=7 ii)=3 3

14iii)=

15 3

4]


75/76


76/76

, 1 ,

, Pascal (1623-1662),

1653.

.

1 = 20

1+1 = 21

1+2+1 = 22

1+3+3+1 = 23

1+4+6+4+1 = 24

Fibonacci.

, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,

1= 1 , 2= 1 n+2= n+1+ n.

,

,

.

1+9 = 10

1+5+15 = 21

1+6+21+56 = 84

Mathimatika g Gymnasiou Shmeiwseis Askiseis

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