algebra a lukeiou askiseis
DESCRIPTION
Ασκήσεις στην ύλη της Άλγεβρας Α λυκείου. Ένα πολύ καλό βοήθημα, με ενδιαφέροντα θέματα και διαβάθμιση στη δυσκολία.TRANSCRIPT
-
1
1.1 -
1.1.1 -
1. ;
2. ;
3. ; : , , , ;
4. A B ; ;
1
-
2 1.
1.1.2
Venn
A0 A A0 A
A [B A B A [ B A;B
A \B A B A \ B - A;B
AB = A \B0 B A
A B A B
B A = B \A0 A B
B A B A
(AB) [ (B A) B A - A B
(AB) [ (B A) - - A B
(A [B)0 = A0 \B0 A B (A [ B)0 - - A ,B
(A \B)0 = A0 [B0 A B (A \ B)0 - - A B
-
1.2. 3
1.2
-
1. .
2. .
3. . :
() P () = 1
() P (;) = 0() A 0 P (A) 1
4. A B :
P (A [B) = P (A) + P (B)
5. A A0 :
P (A0) = 1 P (A)
6. A B :
P (A [B) = P (A) + P (B) P (A \B)
7. A , B A B, P (A) P (B)8. A B
P (AB) = P (A) P (A \B)
- -
9. . .
) .
) ;
. )1/3
10. . . 1; 2; 5 . 2. , - . 150 ,
) 5.
) .
. )25 ) 100
11. A;B P (A) =8
15, P (B) =
1
3 P (A\B) = 1
5,
:A [B;A0; B0; A0 \B0; A0 [B0; A \B0; A0 \B; (AB) [ (B A); A [B0; A0 \B
. 23 ;715 ;
23 ;
13 ;
45
12. A :
[P (A)]2 + [P 0(A)]2 = 1
A .
-
4 1.
13. 24 14 . 4, 3 . , .
. 1124
14. , . 25% - , 85% , 38% . :
) .
) .
. )2% )62%
15. 68% , 16% , 26% . .
. 10%
16. 65% , 20% 5% . . ;
. 75%
17. : 4%. 6%. 2%. . :
i) .
ii) .
. i)8% ii)6%
18. 50% , 35% 77% . :
i) .
ii) .
iii) .
. i)8% ii)23% iii)42%
19. , 14 , 13
112 .
:
) .
) .
) .
) .
. ) 12) 1
6) 1
2) 5
6
20. A;B P (A) =3
4; P (B) =
3
8, :
) P (A [B) 0; 75)
1
8 P (A \B) 3
8.
-
1.2. 5
21. A;B P (A) =2
3; P (B) =
1
2.
) A;B .
) : P (A0 \B) 13.
22. A;B P (A) =1
2; P (A [ B) = 3
4, :
1
4 P (B) 3
4.
-
6 1.
-
2
2.1
-
1. .
() ; x; y : x = y x = y.
() , 1.
() : (+ 1)0 = 1.
() 2 + x = 5 .
() 3 = .2. :
=
3 2 = 3 23 3 = 2 2
3( ) = 2( )3 = 2
3. , .
-
-
-
-
-
- -
1
4. x+ y = 2 : A = 2( x) 3[5(y)] + [(x+ 6y)]. 21
5. x+ y = 1 : = x(x 3) + y(y 3) + 2xy. 4
6. x; y; ! x
4=y
5=!
3 x+ y + ! = 1800.
. x = 600; y = 750; ! = 450
7. 2; 3 7. . ;
7
-
8 2.
8.
=
, :
i)
=
5 95 9 (5 9) 6= 0
ii)
=+
+ ( + ) 6= 0
9. 3.
10. .
11.
i)
ii) 4, 1.
12. : (x+ y + z)2 + (x+ y z)2 + (x y + z)2 + (x y z)2 = 4(x2 + y2 + z2).13. : ( )3 3 + (+ )3 + 3( )(+ ) = (42 + 32).14. x; y : (x+3y)2 = x2+9y2
0.
15. + + = 0, : ( + + )2 = 22 + 22 + 22.
16. : (2 + 2)2 + 4(2 2) = (2 2 + 2)2.17. 34cm 60cm2.
.: x2 + y2 = (x+ y)2 2xy
18. (2x )2 = ( )2, x = x = .19. x+ y = 2 xy = 1 :
A =1
x+
1
y; B = x2 + y2 ; =
x
y+y
x; = x3 + y3 ; E = x4 + y4
20. =5x 6
(3x 5)2 (2x 1)2 .
) x .
) .
21. ,
) A =9(2x+ 1)2 (4x 1)2
4(x2 + 4x+ 4))B =
(x+ 2)(2x+ 1)2 16(x+ 2)(2x+ 5)(7 x) + 4x2 25
22. . , .
23. . ; , 6= .
24. . 2 + 41, , .
2
25. ) Euler. ; ; , :
3 + 3 + 3 3 = (+ + )(2 + 2 + 2 )= (+ + )
( )2 + ( )2 + ( )2
) :
3 + 3 + 3 = 3 () = = + + = 0
-
2.1. 9
26. De Moivre. :
4 + 4 + 4 222 222 222 = (+ + )(+ )( + )( )
27. Newton. ; ; , :
(x )(x )(x ) = x3 (+ + )x2 + ( + + )x
, x. .
28. i) : (+ + )3 = 3 + 3 + 3 + 3(+ )( + )( + )
ii) : (x 1)3 + (x 3)3 + (x 7)3 + (11 3x)3 = 0[. 2 , 4 , 5]
29. (+ + )2 = + + , : = = = 0.
30. ; . 2 + 1
2 + 1=
; .
31. ) (+ )(+ 2)(+ 3) + 4 = (2 + 3 + 2)2.
) x(x+ 1)(x+ 2)(x+ 3) + 1 = (x2 + 3x+ 1)2. .
) 100 101 102 103 .32. 314 + 313 12, .
. 26 3 5 7 13 73
33. ; ; 2 R : x=
y
=!
x6 + y6 + !6
6 + 6 + 6=
x2 + y2 + !2
2 + 2 + 2
334. + = 1 : A = 2(3 + 3) 3(2 + 2).35. x y = 1 :B = 2(3 3) 3(2 + 2).36. + + = 1 , + + = 1 = 1 , :
A = 2 + 2 + 2 ; B = 22 + 22 + 22 ; = 3 + 3 + 3 ; = 4 + 4 + 4
37. + + = 0 :
(+ )(+ ) = ( + )( + ) = ( + )( + )
38. : (2x y !)3 + (2y ! x)3 + (2! x y)3 = 3(2x y !)(2y ! x)(2! x y).39. 3x = + + : (x )3 + (x )3 + (x )3 = 3(x )(x )(x ).40. + + = 0 :
) (+ )3 + ( + )3 + ( + )3 = 3(+ )( + )( + ).
) (+ )3 + ( + )3 + ( + )3 = 3(+ )( + )( + ).
41.
+
1
2= 3 , 3 +
1
3= 0.
42. +1
= 2 :
2 +1
2= 3 +
1
3= 4 +
1
4
-
10 2.
43. ; :
+ + 2
4=
1
+ 1+
1
+ 1
; .
44. :
) 15 244 1.) 8 32 1 ; 1.) (a 1)2 a+1 a a+ 1 ; a; 2 N.
3
45. :3 + 3
1 +
+
+
1
3 3
+
1 +
= (+ )2.
46. ; ; AB ,
+
+
= 0
.
47. ; ; : + + = 2 + 2 + 2 = 3. : = = = 1.
48. + + = 0 6= 0, :4
3 + 3 3 +4
3 + 3 3 +
4
3 + 3 3 = 0
49. + + = 0 6= 0, :2 + 2 + 2
3 + 3 + 3+
2
31
+
1
+
1
= 0
50. + + = 0 6= 0, :3 + 3 + 3 3( + 2)
2 + 2 2 = 3
-
2.2. 11
2.2
-
1. .
() x; y . x > y x > y.
() x > y 2x > 2y.
- -
1
2. x > 1 :i) x2 > x ii) x3 + x > x2 + 1
3. x > 2 : x3 > 2x2 x+ 24. > > 0 : 2 + 2 > 2 + .
5. x > y > 0 : = x3 y3 = (x y)3.6. ; , :
x = 3 + 3 y = 2 + 2
7. x y > 0 : x yx+ y
x2 y2x2 + y2
.
8. : (+ )2 4. ;9. : (+ + )2 ( + ) + ( + ) + (+ ).
;
10. :2
1 + 2 1. ;
11. , 2 + 2 = 4 4.12. x; y; ; : (x+)2+(y+)2 = 4(x+y).
x = x = .
13. 2( 2) = (+ )(+ 2) = = .
14. 43 x 1
22 < y < 5
4. A = 6x 8y B = x
2+
2y
3, :
i) 2 < A < 19 ii)1912
< B < 16
15. ; 1 2 3 < < 4. :i)10 < 2 3 < 5 ii) 3 < < 8 iii) 14 < < 23 iv) 10 < 2 + 2 < 20
16. ; : 8; 3 8; 5 5; 7 5; 8. :i) 28 28; 6.
ii) 47; 31 49; 3.
17. (O;R) (O; ) : 3; 5 < R < 4 2 < < 2; 2. 7; 41 12.
18. :
1 x 5 x 38 32 < x < 4; 8 [3; 2) (3;1)
2
-
12 2.
19. 0 < x < 1 0 < y < 1, : 0 0.22. : 2 + 2 + 2 + 3 2(+ ). ;23. Schwarz. :
(2 + 2 + 2)(x2 + y2 + z2) (x+ y + z)2
; ; ; x; y; z ,
x=
y=
z.
24. AB 20m. AB = = xm E(x) , :i) A = B = 10 xii) 0 < x < 10iii) E(x) = x2 + 10xiv) E(x) 25v) 20m .
25. x; y :x2 + y2 + 6y 2(5x 17).
26. a; b; c :a2 + 10 = 8b ; b2 + 15 = 10c ; c2 + 25 = 6a
. a = 3; b = 4; c = 5
27. ; , :x = 3 + 23 y = 32
28. ) :
2
2 + 1
2 2 A = !3 B = !2 + ! + 2.
31. : 3(4 + 2 + 1) (2 + + 1)2. ;
32.
=
=
6= 0, : 2 + 2 2 + 2.
;
33. ; > 0 + = 1, :
i) 14
ii)
1 +
1
1 +
1
9 iii) 2 + 2 1
2iv) 4 + 4 1
8 ;
-
2.3. 13
2.3
2.3.1 a jaj :
jaj =
a a 0a a < 0
2.3.1 , :
jj = j j 0
jj jj
jj2 = 2
, # > 0 , :
jxj = #() x = # x = #
jxj = jj () x = x =
:
1. ( jj jj)( jj+ jj) = (jj)2 (jj)2 = 2jj2 2jj2 = 22 22 = 0.
2.
j3x 5j = 7()3x 5 = 7 3x 5 = 7()3x = 7 + 5 3x = 7 + 5()
3x = 12 3x = 2()x =
12
3 x =
23()
x = 4 x = 23
3.
j2x 5j = j3 2xj ,2x 5 = 3 2x 2x 5 = 3 + 2x,2x+ 2x = 3 + 5 2x 2x = 3 + 5,
4x = 8 0x = 2 ,x =
8
4,
x = 2
2.3.2 ; , :
1. jj = jj jj, , .
2.
= jjjj ; 6= 0
, , .
3. j+ j jj+ jj, , - .
:
-
14 2.
1. jj jj jj , - :
jj = jj jj ,jj2 = (jj jj)2 ,()2 = jj2 jj2 ,2 2 = 2 2 ; :
2.
jjjj ,
: = jjjj ,2 = jjjj
2,
2=
jj2jj2 ,
2
2=
2
2; :
:
1. 6= : jx2 x2jj j =
jx2( )jj ( )j =
jx2jj jj j =
x2j jj j = x
2.
2.3.3 > 0, :
1. jxj < () < x < 2. jxj > () x < x >
2.3.2 A B . AB , d(; ) j j. : d(; ) = j j.
-
1. .
() x :x2 + 3 = x2 + 3.
() y : jy 4j = y 4.() :
2 2 1 = 2 + 1 2.() jjxj+ xj = jxj+ x x 2 R.() jy jyjj = y jyj y 2 R.() x; y 2 R j2x y 3j = jy 2x+ 3j.() d(;) = 2.() ; 2 R : d(; ) = d(2; 2), = .
2. maxf; g ; . maxf; g =
( <
.
2 R, max. : jaj = j aj 0 , jaj a jaj a.
3. jj2 = 2, .4. jj = jj jj, ; 2 R,
; .
- -
1
5. : A =x2 1 x2 2x+ 1
6. :2 + 6+ 9 2 6+ 9 = 12.
7. : A =2 + 2+ 1+ 2 2+ 1 2 2 + 3, .
-
2.3. 15
8. , : j j+ j j j j = 0.9. 2 < x < 1 2 < y < 1 :
A = jx+ 3y + 8j+ j2x+ y 3j+ x 2y .10.
) A = jx 1j+ x 5 )B = 2 j2 xj 3x+ 111.
i) A = 5 jx 1j+ j2 + xj x 4 ii)B = j2x 1j jx+ 3j+ x12. x, :
j2x 5j = 5 2x jx 1j = x 1.
13. ) x 6= 0 : xjxj 1.) x; y; z 6= 0 : xjxj +
y
jyj +z
jzj 3.
) ; 6= 0 : 2jj +3
jj 5.
14. x; y; !, :j2x+ 6j+ j3x 2y 1j+ jx+ 4y 3! 7j = 0.
15. ; , :j2 3 + 13j+ j5 + 4 7j = 0.. = 2; = 3
16. :) j2x 1j = jx+ 1j ) j3 xj = j7x 9j ) j4x 5j j1 2xj = 0
17. :) jx 2j = 3 ) j6 5xj = 14 ) 1 + 23x = 56
18. ; 2 R x = jj+ jj , y =
jj+ jj : jxj+ jyj = 1
19. ; 6= 0 jj = jj ; .20. ; jj + jj = 0,
.
21.
+ 25+ 1 = 5 jj = 5.
22. ; 2 R + 6= 0, +
+ + 1
23. x; y 2 R d(2x; 3y) = 3y 2x , y 2x3.
24. :) jx 1j < 2 ) jx+ 3j 7 ) j3 2xj 15) jx+ 5j > 3 ) j4x 5j 3
25. , :) j 5j < j+ 2j ) d(;4) > d(5; )
26. x 2 R, :)jx 6jj4 xj < 1 )
d(2x;3)d(1; 2x)
1
27. x; y 2 R jxj 2 jyj 3. :i) j5x 2yj 16 ii) j3x+ 7y + 1j 28
2
28. :2 + 4+ 5 3 = 2 2 2+ 6.
-
16 2.
29. 1 < < 2 : = jj+ 1j 3j j4 + j 2jj .
30. ; 2 R , : 2 jj 2 jj = (jj jj)(2 + jj+ 2).31. ; :
i) + jj jj+ jj ii) jj jj jj32. x; y 2 R : j5x+ 3y 6j = 5 jxj+ 3 jy 2j.
: x(y 2) 0.
33. ; 6= 0, +
2. ;34. x 2 R,
x+ 1x 2. ;
35. x 2 R, :)jx 6jj4 xj < 1 )
d(2x;3)d(1; 2x)
1
36. ; ; x 2 R :i) j+ 4j+ j3 j 7 ii) j+ 5j+ j 9j 14iii) jx+ 2j+ j7 4xj+ j3x 5j 4
37. :
)(x+ 2)2
jx+ 2j +(1 x)2j1 xj 3 x 6= 1 x 6= 2
)42 4+ 4j2 1j +
42 + 4+ 4
j2+ 1j 2 6=12 6= 12
38. f(x) = x2+px+q. jf(0)j ; jf(1)j ; jf(1)j,
1
2.
39. ; 2 R :i) jjj jjj j+ j ii) jjj jjj j j
40. ; ;i) j j = j+ j ii) j j = jj+ jj
41. x; y 2 R x jyj y jxjxy
= 2, x; y . 3
42. :i) jjx+ 1j+ x+ 1j+ jjx+ 1j x 1j 2 jx+ 1j = 0ii) jjx 2j 2 + xj+ jj2 xj x+ 2j j4 2xj = 0
43. ; , :j+ 3j 2 1 2
44. x, :x2 2x 4x x2 445. 2 < x < 1 : x2 3x 10 < 20.46. jxj < 1 1 < y < 3 : x2 3xy y + 1 < 14.47. jxj 6= jyj jxjjx+ yj +
jyjjx yj 1.
-
2.4. 17
2.4
-
1. .
() ; > 0 p+ =
p+
p.
() x < 1 px2 2x+ 1 = x 1.
() x :px2 1 =p(x+ 1)(x 1) = px+ 1 px 1.
- -
1
2. :
r21 +
q13 +
p7 +
p4 = 5
3. x =p7 +
p5 y =
p7p5, : x2 xy + y2.
4. :i) (3
p3 2)(3p3 + 2) (p3 2)2 2p3(1 +p3) = 10 + 2p3.
ii)q(p3 2)2 +
q(1p3)2 +
q(1p2)2 = p2
5. = 2 +p2 ; = 2 +
p2 +
p2 ; = 2
p2 +
p2, = 2.
6. =q2 +
p2 +
p3 ; =
q2
p2 +
p3 ; =
p2 +
p3, : = 1.
7. .)p8 +
p32p18
) 3p20 + 5
p80 4p45
) 5p12 2p3 + 6p27
)p32p72 +p2 +p50p8
)p6 p24 p54
8. .) 5
p8p27 +p50p300 2p2
) 3p48p18p12 +p98 +p3
9. .
)2p2
)2p6
)6p12
)1p
3p2 )2
3 +p7
10. .
)3
4p7 )4
2p3p5 )
4
2p5 3p2 )
p2
2p3 +
p5
11. :
)
p3p
3 +p2+
p2p
3p2 = 5 )p7p
7 +p5+
p5p
7p5 = 6
12. :
p3 +
p2p
3p2 +p3p2p3 +
p2= 10
13. :
)1
(3p5)2 1
(3 +p5)2
=3p5
4)
1
(2p5)3 +1
(2 +p5)3
= 76
14. ) :p
2 + 13
p
2 13.) : 3
p7 + 5
p2 3
p5p2 7 = 2
15. :2p20 + 3
p8 2p75
2p45 + 3
p18 5p27 =
2
3
-
18 2.
16. :3p12 +
p20 2p8 + 8
3p27 +
p45 2p18 + 12
.
17. :i)p3 +
p3 < 3 ii)
p6 +
p7 < 3 iii)
p13 +
p5 2 : x + y = z .
, n, 3p3n2 + 3n+ 1
.
28. : x2 1x+ y2 +
1
y x =
p3 + 1 y =
p3 1.
29. :(p80
p200
p180 +
p288
p8)(p20
p45) = 10
30. : 1 +1
1p2 1p
2p3 +1p
3p4 = 2
31. .
)1
1 +p2p3 )
1p2 +
p3 +
p6
)1p
1 +p2
p1p2
32. .
Fermat 1993 Andrew Wiles. Fermat , .
-
2.4. 19
)x+
p4 x2
xp4 x2 , 2 x 2 x 6=p2
)
p+ +
p p
+ p , > > 0
33. x > 1, :x+
px2 1
xpx2 1 xpx2 1x+
px2 1 = 4x
px2 1
34. :
)p9 4p5 )
p18 8p2 )
p8 + 2
p12
35. :3p
4 2p3 3p
4 + 2p3= 3
36. = 3p375 3p16 + 3p2 3p192 = 3p72 3p9 + 3p384 + 3p4 3p162
: = 1.
37. K =1
3p25 + 3
p30 + 3
p36
. Euler, -
, .. 3
p6 3p5
3
38. :
1 +1
1 +p2+
1p2 +
p3+ 1p
98 +p99
+1p
99 +p100
39. x2 + y2 = 4, :px4 10y2 + 65 +
py4 4x2 + 20 = 11
40. .
)1p
2 + 4p3
)x y
3px2 + 3
pxy + 3
py, x; y > 0
-
20 2.
-
3
3.1 1
-
1. .
() 0 .
() 6= ( )x = , x = 1.() 2x = 2 x .() (x+ 1)2 = x2 + 4 2 .
() x = , x = .
- -
1
2. 2(x 3) = 2(3 x) , .3. 2(x 1) = (x+ 2), .
i) x = .ii) .iii) x = 3;
4. ) : (x 1) = x 1 , .) , .
411
31
5. 5( x) 6 = ( x) 3x , 2 R6. ; (3x 2) + 13x = (5x+ 3) 4 .7. ; (x 1) = x 2+ 3 .8. :
)jxj 2
2 jxj
4= 1
) jx 3j 2 jx 3j+ 13
=3(jx 3j 1)
4
)2 j4 5xj 1
3 3 j4 5xj 2
4=
5 j4 5xj+ 46
7 j4 5xj 66
21
-
22 3.
9. :
)5 jx 2j
4+
2
3(j2 xj 2) = 3 jx 2j
6
)jx 3j 1
3+
2 j2x 6j15
= j3 xj 1
)j2x 1j 1
4 1
3
3 j1 2xj
2 1
4
=j6x 3j 6
12
2
10. A B 105 . A B, 24 . B A, 30 . ; ;
11. .i) 3x 2 4 = 4(x 1) ii) 3(x 1) = 4x+ 8
12. :) (x )2 + (x 1)2 = (x )2 + (x 1)2) (x+ )( ) + (x+ )(+ ) = (+ )2
13. x+
x +x+
x = 2, .
14. :) jjxj+ 3j = 4 ) jjx+ 1j 1j = 2 ) j2 jx 1j+ 1j = 3
15. :) jx 2j+ x = 8 ) j3x 1j = x+ 1 ) jx+ 1j+ jx 1j = 2
16. :
i)x2 + 2x+ 1 x2 + 3 5x = 3
ii)4x2 + 4x+ 1 2 x2 + 1 = 2 + 2x2
3
17. ; :+ 1
x 1 +1
x2 1 = 1x+ 1
.
-
3.2. X = 23
3.2 x =
x = :
x =
= 0 0 > 0
p p
x = p
< 0 pjj
- -
1
1. :) x3 = 1000 ) x4 = 625 ) x5 + 32 = 0
2. 64cm3. .
3. . 2058 cm3, .
4. :) 3x3 12x = 0 ) 2x5 + 16x2 = 0
5. :) x5 81x = 0 ) 64x5 x3 = 0 ) 8x4 + x = 0) 6x5 + 3x = 0 ) 135x8 + 40x5 = 0 ) x105 x5 = 0
6. :) (x+ 1)5 = 32 ) (x 2)4 216(x 2) = 0) (2x 1)4 = 2x 1
2
7. :) x6 + x4 x2 1 = 0 = 0 ) x11 32x6 + x5 32 = 0
8. :) x1962 + 128x1969 = 0) x2050 + x50 = 2x1050
) 8888x2001 + 1111x2004 = 0
-
24 3.
3.3 2
-
1. .
() , .
() 6= 0. .() x2 + x+ = 0; 6= 0, . > 0.() x2 6x+ 2 1 = 0 3. = 2.() x2+x+ = 0; 6= 0, x1; x2. x21+x22 = S22P , S
P .
() x2 + x+ = 0; 6= 0, x1; x2. x31 + x32 = S3 3P , S P .
() 52 34 8x2 + 14x 15 = 0.
- -
1
2. :) x2 + 2
p4x 1 = 0 )p2x2 (p2 +p3)x+p3 = 0
3. x, x+ 1 , 3 x .4. : (x2 2x 3)2 + (3x2 6x 9)2 = 05. :
) jx 3j+ x2 9 = 0 ) x2 + x 2+ 1 x2 = 0) jxj+ x2 x = x+ x3
6. ; ; , :x2 2x+ 2 2 + 2 2 = 0, .
7. ; ; 2 Q, : 3x2 (2+ 3)x+ 2 = 0 ; 6= 0, , .
8. :x2x+ = 0 , : (24)x2x5 = 0i) ii)
9. : x2 ( 3)x+ 2 5 2 = 0 ; 2 R.) ;) ), .
10. : ( 3)x2 2(3 )x+ 5 3 = 0 ; 2 R.i) ;ii) ;iii) , ;
11. : ( 1)x2 2( 1)x+ 4 5 = 0 ; 2 R.i) , .ii) .
12. : (2+ 1)x2 + 3( 1)x + 1 = 0 .
13. , :(1 )x2 4x 3 = 0, .
14. x2 3x+ 1 = 0, x1; x2. :) x21 + x
22 )
x1x2
+x2x1
) x31 + x32 ) jx1 x2j
-
3.3. 2 25
15. :
x2 + x+ 2
3 = 0 ; 6= 0, .
16. ; (x 2)2 (4x 3) = 0, 34 34 .. 25
16
17. 4x2 + (3 +p3)x 2(5p3) = 0.
) 2.) .
18. 2x2 ( 2)x+ 7 32 42 = 0 , ; 2 R) .) .
19. , 5x2 (2 2)x+3 2 = 0 , .
20. :) 6x2 jxj 2 = 0 ) 2(x 4)2 jx 4j 15 = 0) 6(x 1)2 5 j1 xj 6 = 0
21. :) (x2 + 2x)2 (x2 + 2x) 2 = 0 ) (x2 1)2 4(1 x2) 21 = 0
22. :) x4 3x2 4 = 0 ) 3x4 + 16x2 + 5 = 0 ) 2x4 3x2 + 1 = 0
23. :
)2x+ 1
x 1 3
x2 x =x 2x
)3x 1x+ 2
182 x =
7x2 28x2 4 +
7
2 + x
2
24. 3x2 + 5 jxj+ x = 9.25. x2 2( )x+ 2 2 1 = 0 , ; 2 R
i) = 4( 1)2ii) ; , .
26. : 3x2 2(+ + )x+ + + = 0, ; ; 2 R.i) .ii) ; ;
27. ; ; AB , .i) x2 (2 + 2 2)x+ = 0ii)
2
x+ 1
2
x= 2
28. : ( 1)x2 2( 1)x 2 2+ 3 = 0 .
29. x2 3 j+ 1jx + 2 = 2 + 2, ; 2 R. , ; .
30. ; ; 2 Q, :
( )x2 + ( )x+ ( ) = 0 ; 6= 0 6=
, .
31. x2 + x + = 0; 6= 0. ; :j j = jj+ jj, .
32. x2 + 2x+ = 0; 6= 0 , 2x2 (x 1)2 + = 1.
-
26 3.
33. a b x2 + 3x + 1 = 0.
a
b+ 1
2+
b
a+ 1
2.
. 18.
34. , :
( 1)x2 (2+ 1)x+ 2 = 0
(+ 1)x2 (4 1)x 2 = 0
. .
3
35. 2x22()x+2+22+222 = 0, ; ; 2 R, . :
) =+
2) .
36. (2+2+ 2)x2+2(++ )x+3 = 0, ; ; 2 R, .) ; ; ) .
37. x2 + ax + bc = 0 ; x2 + bx + ca = 0 , , , x2 + cx+ ab = 0.
38. A. x2 + x+ = 0; 6= 0 1; 2. S = 1 +
2 S =
1
1+
1
2.
:
i) S + S1 + S2 , = 3; 4; 5; ii) S =
S , 6= 0.
iii) 1; 2 x2 2x 1 = 0, S5 =51 +
52.
B. ( ) :
i) A = (1 +p3)6 + (1p3)6
ii) B =1
(1 +p2)7
+1
(1p2)7 .
-
4
4.1 1
-
1. .
() 0x < 5 .
() 2x > 9 .
() 8x < 7 .
- -
1
2. x :5x 2
3 4 7x
6>
1 3x12
4 x2
6x 14
1 x8
3. 12 17. .
4. 19 26.
5. :) jx+ 1j < 5 ) j2x 7j < 9 ) j6 3xj 15) j3x 2j > 7 ) j12 9xj 6
2
6. : A =x2 9jxj 3 .
) x ;
) .
) A 6.
7. : A =x2 2 jxj+ 1
jxj 1 .
) x ;
) .
) A < 3.
8. :) 2 < jx 5j < 5 ) 3 < j3 xj < 7) 1 < j2x 5j < 6 ) 5 < j4 6xj < 9
9. x :
jx 2j 14
0 - -
1
2. :) f(x) = 3x2 + 15x+ 42 ) g(x) = 6x2 7x 5) h(x) = 9x2 16x+ 8 ) '(x) = 20x2 60x+ 45
3. x, . .
) A =x2 x 62x2 + 5x+ 2
)B =9x2 + 6x 815x2 + 2x 24
4. f(x) = 6x2 5x 6.) , x.
) < , > , = .
) f(p2) 0 ) f(0; 667) 0 ) f(1; 534) 0
) f(1; 5) 0 ) f( 117 ) 0 ) f(0; 6) 05. :
) 3x x2 0 ) 5 x2 < 0 ) 3x2 + 2x 8 < 06.
K =(3x2 + 7x 5)(4x2 44x+ 121)
x2 + 3x 3 , x.
7. :
)
(2x 1 > 0x2 x 2 < 0 )
8>:4 x > 02x2 x+ 6 > 03x2 + 4x 2 < 0
)
8>:2 x2 < 02x2 5x+ 12 > 0x2 + 3x 4 < 0
8. x ;) A =
p4 + 3x x2 )B = p1 x2 +p2x2 9x+ 4
2
9. '(x) = (2 + 1)x2 3x+ 3 , 2 R x.10. g(x) = x2 x+ 1 , 2 R .11. h(x) = x2+2x (+)2 , x 2 R, ; ;
.
12. ) : f(x) = x2 3x+ 4 g(x) = 2x2 5x+ 4.) x y, :
(x2 6x+ 9)(y2 3y + 4) + (4y + 5)2(2x2 5x+ 4) = 0
13. ( 2)x2 2x+2 3 = 0. , .
14. ,
(+ 2)x2 2( 2)x+ 5( 2) < 0 x 2 R.
3
-
4.2. 2 29
15. x2 + ( 2)x + 1 = 0.) 2 R, x1; x2.) :
1 < x21 + x1 x2 + x22 < 7
-
30 4.
-
5
5.1
-
1. .
() (a) a .
() a = j3 1j j3 + 1j a = 2.
- -
1
2. :
i) a =(1)2 + 1
ii) a =
1 +
1
3. a1 = 1; a2 = 3 a2+1 = a a+28 2 N. 5 .
. 577
4. (a) a1 = 3 a+1 =5a 41 + a
.
5. Fibonacci, a1 = 1; a2 = 1 a+2 = a+1 + a . 17 .
6. (a) a1 = 1 a+1 =p1 + a2 .
i) 6 .
ii) ;
7. (a) a1 = 2 a+1 =3 + a1 a , 2 N
.
i) .
ii) ;
31
-
32 5.
5.2
-
1. .
() 10; 7; 4; 20.
() (a) a2; a4; a6 2a4 = a2 + a6.
- -
1
2. 2 , 4 6 0, 3 , 5 7 6. 30 .. 690
3. 12 . 10 3 - .
) ;
) ;
) 7 30 . ;
4. 324 : , , ...
) 12 ;
) ;
5. A. 20 , - . 1 16 7 28 .
) 10 ;
) 5 15 ;
B. 1 6 , 2 9 , 3 12 ...
) ;
) ;
. .)34 )350 .)11 )55
6. , 70 250 . . 140 .
i) 20 .
ii) .
iii) 100 , . ;
. )1600 )5
7. ; ; 2 R (+)2; 2+2 ()2 .
8. ) x x4; x + 4 3x4 .
) x4 (), .
-
5.2. 33
) 10 .
. ) x = 8 ) 1 = 36 ) S10 = 0
9. :
) (x+ 2) + (x+ 5) + (x+ 8) + + (x+ 53) = 459) 1 + 7 + 13 + + x = 280 , x > 0. )2 ) 55
10. 1 ;1 ;
1
,
:
i) ; ;
ii) ( + ) ; ( + ) ; (+ )
iii) ( + )2 ; ( + )2 ; (+ )2
11. ; ; + ; + ; + .
12. , , 3 8..2; 1; 4
13. , , 3; 4 5.
2
14. a =32 3 + 1
5.
) .
) A = a15 + a16 ++ a30.
) B = a1 + a3 + a5 ++ a53.
15. AB , > > , ; ; ; . . ( ).
16. 620 640 . :
120 .
10 .
! , ! .
48 .
) !.
) !.
) !.
) .
) .
. ) 48 3! ) 600 + 15! )2 )60 )630
17. 2290 33 . . - , 2 , , 3 .
) ;
) i) ;
ii) ;
. ) 90 )i) 32 ii)48
-
34 5.
5.3
-
1. .
() .
() 1, a1.
- -
1
2. 4 108 8 8748. 4; 12; 36; 108; ::: , 4; 12;36; 108; :::
3. 3 24 8 768.. a1 = 6; = 2
4. :
A. 10 , 6 .
B. , - 33 10 .B1. 20 .
B2. ;
. A. 7290 B1. 1890 B2. 27
5. x x+ 4; 3x; 47x ..2; 1
2
6. x; 10; y , x; 6; y - , x; y.. 2; 18
7. ; ; , , - = = .
8. ; ; :
22 4
+22 4
+22 4
= 0
9. x; y; !; z :
i) (x+ z)(y + !) (x+ !)(y + z) = (y !)2 ii) y + zx
=!2z + z3
!3
10. , 216 20..2; 6;18
11. 168, 512.. 2; 8; 32
-
6
6.1
-
1. .
() f f12
= 2 f(0; 5) = 3.
() f(x) =x3 8x2 4 =
x3 23x2 22 =
(x 2)(x2 2x+ 22)(x 2)(x+ 2) =
x2 2x+ 4x+ 2
A = R f2g. - -
1
2. :
) f(x) =x+ 2
x2 5x ) g(x) =jxj
2x2 5x 3 ) h(x) =x3 + 2
jx 2j 33. :
) f(x) =
p4 x2x
) g(x) =
px+ 1
x+
x 5p2x 1 x
2 + 3x 5
) h(x) =
p3 j2 xj
x2 5x+ 64.
) f(x) =(2x+ 3)(x 2) + 4 x2
x2 2x ) g(x) =x(x+ 1) + x3 + 1
3x2 + 2x 1 2
5. f(x) =x2 3jxjx2 9 .
) f .
) .
) jf(x)j = 25 .) jf(x)j 12 .
6. f(x) =
(x
x2+1 3 x < 05x2 2x+ x 0
i) ; :f(1) = 3f(0) 1 5f(2) + 10f(1) = 1.
ii) =1
f(0) f( 12 )+ [4 + f(3)]2 + 4; 84.
. i) = 2; = 1 ii) = 1
35
-
36 6.
6.2
-
1. .
() A(4; 1) B(4; 1) xx.
() A(2; 3) B(2; 3) .() A(1;2) B(1; 2) yy.
- -
1
2. ) f(x) = j2x 3j 1 ) g(x) = x3 x
3. f(x) =x 1x2 + 1
, 2 R. M(1; 12 ) f .
) .
) f .
4. g(x) = x2 + x 2 , ; 2 R. A(1;4) B(4; 6) g.
) ; .
) f .
. ) = 1; = 4 ) (0;6); (2; 0); (3; 0)
-
6.3. F (X) = X + 37
6.3 f(x) = x+
-
1. .
() y = 3x+ 5 x0x.
() x+ y = 0 x bOy.() y =
p22 x+ 3 y =
1p2x 4 .
() 2x+ y = 5 y = 2x+ 9 .
- -
1
2. :
i) f(x) =
8>:2x+ 3 x 15 1 < x 2x 4 x > 2
ii) g(x) =
8>:3x 1 x < 23 x = 22x+ 1 x > 2
3. :i) f(x) = jx 2j+ 2x 1 ii) g(x) = j2x+ 1j jx 1j
4. :i) f(x) =
px2 + 4x+ 4 x ii) g(x) = px2 + 2x+ 1px2 2x+ 1
5. 2 R :) 1 : y = 2x 1 2 : y = (2 )x+ 3) 1 : y = j 2jx+ 2 : y = 2jjx+ 6 1
6. ) A(5; 3) 2x+ y = 3.
) , M(1; 24) - .
. )y = 2x 7 ) = 1 2
7. :
i) f(x) =x2 1jx+ 1j + 2x 1 ii) g(x) = x+
x
jxj +x 1jx 1j
-
38 6.
-
7
7.1 f(x) = x2
-
1. .
() f(x) = 3x2 .
() f(x) = 2x3 .
- -
1
2. , , A3;32
.
3. ) , f(x) = x2 g(x) = x. x2 < x.
) .
4. f(x) = x2jxj.
5. f(x) =
(1 x x 0x2 + 1 x > 0
39
-
40 7.
7.2 f(x) = x2 + x+
-
1. .
() f(x) = x2 + x+ ; 6= 0, f 2
=
4.
- -
1
2. :) f(x) = 2x2 x 6 ) f(x) = 4x2 + 4x+ 1 ) f(x) = 3x2 + 2x+ 1) f(x) = x2 + x+ 2 ) f(x) = 9x2 + 12x 4 ) f(x) = x2 + x 1
3. i) f(x) = x2 + x+ . ; ; , yy 2 xx 2 1.
ii) f .
. i) = 1; = 3; = 2
4. i) f(x) = x2 + x+ . ; ; , - A(1;4); B(2;3) (2; 5).
ii) f .
. i) = 1; = 2; = 3
5. i) f(x) = x2 + x + . ; ; , yy 5 xx 52 x = 34 .
ii) f .
. i) = 2; = 3; = 5
6. ) 2 R, f(x) = x2 6x+ xx.
) f .
. ) = 0; = 19
7. f(x) = jx2 6x+ 5j.) f .
) , y = f .
. ) 0 < < 4
8. f(x) = x26x+ 5 g(x) = 1x2.
i) , .
ii) .
iii) , y = .
. ii)A(1; 0); B(2;3) iii)4 1
9. f(x) = (x 1)2 + (2x 1)2.. x = 3
5; fmin =
15