sto-elementarnih-zadatci

1

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

**** MLADEN SRAGA ****

POTPUNO RIJEŠENI ZADACI PRIRUČNIK ZA SAMOSTALNO UČENJE

100 ELEMENTARNIH ZADATAKA koje bi svatko trebao znati riješiti

I

POTENCIJE

a a aa a a

aa

a a a

a bc a b c

a a

ab

ab

ab

ba

ba

aa a

aa

n m n m

n m n m

n

mn m n m

n n n n

n m n m

n n

n

n n n

n

⋅ =

=

= =

=

=

FHGIKJ =

FHGIKJ = FHG

IKJ =

=

=

=

+

−

−

⋅

−

−

−

:

:

b gd i

0

1

1

1

1

1aa

nn=

ALGEBARSKI IZRAZI

a b a b a b a ab b

a b b a

a b a b a b a ab b

a b b a

a b a b

a b a b a b

a b a a b ab b

a b a a b ab b

a b a b a ab b

a b

+ = + ⋅ + = + +

+ = +

− = − ⋅ − = − +

− = −

− − = +

− ⋅ + = −

+ = + + +

− = − + −

− = − ⋅ + +

+

b g b g b gb g b g

b g b g b gb g b gb g b gb g b gb gb g

b g d i

2 2 2

2 2

2 2 2

2 2

2 2

2 2

3 3 2 2 3

3 3 2 2 3

3 3 2 2

3

2

2

3 3

3 3

3 2 2

2 2 2 2 2 2 2

2

2 2

= + ⋅ − +

+ + = + + + + +

RSTUVW

RS UV

+ + =+ =

⋅ == + ⋅ +

+ =

a b a ab b

a b c a b c ab ac bc

x px qm n p

m n qx m x n

m n b

b g d ib g

a f a f

T W⋅ = ⋅+ + = = + + + =

m n a cax bx c ax mx nx c . . .

α

TEL. 01-4578-431

© M.I.M.-SRAGA 1999./2010.

2


M.I.M.-SRAGA 1999./ 2010.©

Autor: MLADEN SRAGA Grafička obrada MLADEN SRAGA Izdavač:

M.I.M.-Sraga d.o.o. Centar za poduku www.mim-sraga.com

Procijena i ocijena znanjaako se spremate :

Za polaganjeZa polaganje OSNOVNE razine VIŠE razine

0 50 0 0 50 0

51 75 1 51 81 176 85 2 82 86 286 91 3 87 90 392 95 4 91 94 4

96 100 5 95 100 5

Moj kome

− = − =

− = − =− = − =− = − =− = − =− = − =

hitno tražite instruktora

trebali bi ozbiljno poćeti vježbati

Vi znate nešto ali ima tu puno rupa

ntar:ako ste dobili ocjenu 0 -ako ste dobili ocjenu 1 - ako ste dobili ocjenu 2 - ako ste dobili ocjenu dobro možete krenut na pripreme

vi ste spremni za OZBILJNIJE zadatke

odlično- vama osnovni zadaci na mature

3 - ako ste dobili ocjenu 4 - ako ste dobili ocjenu 5 - nebi trebali predstavljat problem

Prvo riješite zadatke pa tek onda gledajte u rješenja jer su tamo kompletno riješeni....

Potpunu garanciju na zadatke i rješenja daje: centar za dopisnu poduku M.I.M.-SRAGA -dakle sve što vam se čini nejasno krivo ili sumnjivo - zovite 01-4578-431 ili 01-4579-130 i tražite dodatne upute i objašnjenja ...

Zadatci su u potpunosti naš autorski rad , ako se neki zadatak već pojavio u nekoj drugoj zbirci ...nije namjerno ponovljen s naše strane.

Centar za poduku M.I.M.-Sraga zadržava sva prava na komercijalno iskorištavanje ovih zadataka. Svako kopiranje umnažanje za vlastite potrebe i potrebe poduke i nastave je dozvoljeno i dobrodošlo. Cijena ovih zadataka je 0 kn tj. besplatni su ...i tako treba i ostati... Sve kritike i dobre i loše rado ćemo saslušati ... javite nam se na tel. 01-4578-431 ili e-mailom : [email protected]

http://www.mim-sraga.com/

mailto:[email protected]

3


M.I.M.-SRAGA 1999./ 2010.©

STO ELEMENTARNIH ZADATAKA KOJE BI SVI TREBALI ZNATI RIJEŠITI

ako imaju namjeru položiti maturu ! Koristeći ove formule trebali bi riješiti zadane zadatke:

( )

( )( )( )

0

1

1

1

1

1

POTENCIJE

:

:

1

1

1

1 1

1 1

n m n m

n m n m

nn m n m

m

n n n n

mn n m

zmn n m z

n n

n

n n n

n

nn

nn

nn

n

mn mn

mn

m n mn

a a aa a a

a a a aa

abc a b c

a a

a a

a ab b

a b bb a a

aa a

aa

aa

a a

aaa

a a

aaa

+

−

−

⋅

⋅ ⋅

−

−

−

−

−

⋅ =

=

= =

=

=

=

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

=

=

=

=

=

= =

=

= =

( )

( )( )( )( ) ( )

( )( ) ( )

( )

( )

2 3

2 4 6

2 523 2

4 2

7 2 3

7 8 9

8

3

32

543

26 52 2

35 52 4

32

42 3

322 3 2

1

2 3

1)2)

3)4) :5) : :6) :

7 )

8)

9)

10)

11) :

12) 2

213)3

214) : 23

15)

16)

x xx x x

x x xx xx x xx x xxx

x

x

x x

x x

ab

x y

a b ab

xy

x yy x

−

−

⋅ =

⋅ ⋅ =

⋅ ⋅ =

=

=

⋅ =

=

=

=

⋅ =

=

=

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞=⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞⋅ =⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

3 222 2

3 3 4

1 2 3

10 0 0 1

1 10 1 3

17 ) :

2 1 118)3 3 2

119) 27 23

1 120) 2 22 5

− − −

−

−− −

⎛ ⎞ ⎛ ⎞⎛ ⎞⋅ =⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞⋅ ⋅ ⋅ ⋅ =⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⋅ ⋅ ⋅ ⋅ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

a y yy a a

x y

y

4


M.I.M.-SRAGA 1999./ 2010.©

12

12

23

34

12

14

12

14

12

32

3 3

4 4

5 2 5 3

6 5 6 2

21)

22)

23)

24)

25) 36

26) 16

2527 )49

1628)81

6429)49

930)16

31) 10 10

32) 3 9

33) 2 8

34)

35) :

x

x

x

x

x x

x x

−

−

−

−

=

=

=

=

=

=

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞ =⎜ ⎟⎝ ⎠

⋅ =

⋅ =

⋅ =

⋅ =

=

( )

( )

( )( )

0

1

1

1

1

1

POTENCIJE

KORIJENI

:

:

1

1

1

1 1

1 1

n m n m

n m n m

nn m n m

m

n n n n

mn n m

zmn n m z

n n

n

n n n

n

nn

nn

nn

n

mn mn

mn

m n mn

n n n

n n

a a aa a a

a a a aa

abc a b c

a a

a a

a ab b

a b bb a a

aa a

aa

aa

a a

aaa

a a

aaa

a b a b

aa b

+

−

−

⋅

⋅ ⋅

−

−

−

−

−

⋅ =

=

= =

=

=

=

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

=

=

=

=

=

= =

=

= =

⋅ = ⋅

⋅ = n

b

5


M.I.M.-SRAGA 1999./ 2010.©

Koristeći formule koje smo upravo obradili i ove nove koje vam dajem ovdje riješite sljedeće zadatke: ( )

( )

( )

( )

( )

( ) ( )

( ) (

( )

( ) ( )

( )

2

2

2

23

23 4

2

2

2 3 2 3

2 3 4 5 6 2 3 4 5 6

3

2

3

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )( ) ( )

( ) ( )( )( )

( ) ( )( ) ( )

( )

2 2 2

2 2

2 2 2

2 2

2 2

2 2

3 3 2 2 3

3 3 2 2 3

3 3 2 2

3 3 2 2

2 2 2 2

ALGEBARSKI IZRAZI

2

2

3 3

3 3

2 2

a b a b a b a ab b

a b b a

a b a b a b a ab b

a b b a

a b a b

a b a b a b

a b a a b ab b

a b a a b ab b

a b a b a ab b

a b a b a ab b

a b c a b c ab a

+ = + ⋅ + = + +

+ = +

− = − ⋅ − = − +

− = −

− − = +

− ⋅ + = −

+ = + + +

− = − + −

− = − ⋅ + +

+ = + ⋅ − +

+ + = + + + + 2c bc+

)

) 3

) 2 3

) 3 4

1 2)2 3

2 340)3 4

41) 5 6

42) 2 3

43) 2 5 2 5

2 4 2 444)3 5 3 5

45) 3 4 3 4

46) 3

47 ) 2 2

48) 2 3

49

−

+ =

+ =

− =

⎛ ⎞+ =⎜ ⎟⎝ ⎠

⎛ ⎞− =⎜ ⎟⎝ ⎠

− =

− − =

− ⋅ + =

⎛ ⎞ ⎛ ⎞− ⋅ + =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

− ⋅ − =

+ =

+ ⋅ + =

− =

x

x y

x y

x y

x y

x y

a y

x x

x y x y

x y z a b x y z a b

x

x x

y z

36 37

38

39

( )

( ) ( )

3

32 3

32 3 4

3 3

1 1)2 3

2 3)3 2

) 3 2

)

⎛ ⎞+ =⎜ ⎟⎝ ⎠

⎛ ⎞− =⎜ ⎟⎝ ⎠

− =

+ − − =

x

x y

x y z

x y x y

50 51 52

6


M.I.M.-SRAGA 1999./ 2010.©

P

( ) ( )( ) ( )

3 3 2 2

3 3 2 2

ALGEBARSKI IZRAZI

a b a b a ab b

a b a b a ab b

− = − ⋅ + +

+ = + ⋅ − +

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( )

( ) ( )

( ) ( )

3 6

3 3 23 23 6 3 3 3 2 2 2 2 2

333 3 2

2

2 2 2 4

3 3 2 2Prepoznaj:

rimjer:27 8 ?

ešenje:

27 8 3 2 3 2 3 2

3 2

3 2

3 2 9 6 4

x y

x y x y x y y x x y y

a x b y

a x b y

x y x xy y

a b a b a a b b

− =

− = − = − = − ⋅ + ⋅ + =

↓ ↓

↓ ↓

= =

= =

= − ⋅ + +

− = − ⋅ + ⋅ +

rj

x 3 3 2 2

Koristi gornji primjer i ove formule riješi sljedeće zadatke: R

3

3

6

6 6

3

9 6 3

astavi na faktore:

) 8

) 8 27

) 125

)

1) 127

) 64 125

x

y

x

x y

y

x y z

− =

− =

+ =

− =

+ =

+ =

53

54

55

56

57

58

7


M.I.M.-SRAGA 1999./ 2010.©

( )2 2 2 2 2 2 2a b c a b c ab ac bc+ + = + + + + +Koristeći se formulom riješi sljedeće zadatke: ( )

( )

( )

2

2

2

) 2 3

) 2 3 4

) 3 3

x y

x y z

x y

+ + =

+ + =

+ − =

59 60

61

( ) ( )

( ) ( )

( ) ( )

2

2 2

4 6

4 4

2 4 8

2 2

22

2 2 Koristeći se formulom:

rastavi na faktore:

) x 16

) 4 36

1 25)4 49

)

) 4

) 2

) 36 16

y x

x y

x y

x y z

x y x y

x y x y

a b a b a b

− =

− =

− =

− =

− =

+ − − =

+ − − =

− = − ⋅ +

62 63 64

65 66

67

68

8


M.I.M.-SRAGA 1999./ 2010.©

( ) ( )

2 2

2 2

6 3 4 8

4 6 2 3

2 4 8

2 2 4

2 22 2 2 2 Koristeći se formulama:

slijedeće zadatke zapiši u obliku kvadrata binoma:

) 9 6

4 20 25

) 9 12 4

) 2

) 2

1) 3 94

2 2

x y y

x xy y

x x y y

x y x y

x xy y

a ab b

a ab b a b a ab b a b

+ + =

− + =

− + =

+ + =

− + − =

− + =

+ + = + − + = − 69

70)

71

72 73 74

( ) ( )2

2 2

2

2

2

...

Kvadratni trinom

Koristeći se gornjim formulama rastavi na faktore slijedeće kvadratne trinome:

75) x 2 15

76) 3 2

77 )

ax bx c ax mx nx c

m n px px q x m x n

m n q

m n b

m n a c

x

x x

x x

+ + = = + + + =

+ =+ + = = + ⋅ +

⋅ =

+ =

⋅ = ⋅

⎧ ⎫⎨ ⎬⎩ ⎭

⎧ ⎫⎨ ⎬⎩ ⎭

+ − =

+ − =

− 12− =

9


M.I.M.-SRAGA 1999./ 2010.©

Skra

( )

( )

2 2

2 2

4 4

2 2

2

2 2

2

4 4

3 3

2 2

3

Pogledaj u formule za ove brojnike u 81) i 82)

ti razlomke koristeći sve što smo izvježbali do sada:

4 6)6 9

)2

)2

)

)

)3 3

)

x yx y

x yx xy y

a ba ab b

x yy x

y xx y

x yx y

x

−=

−

−=

+ +

−=

− +

−=

−

+=

−

−=

−

( )

78

79

80

81

82

83

3

3

2

2

2 2

2 2

3 18 27)2 18

2 3)3 6 3

yx y

x xx

x xy yx xy y

+=

+

+ +=

−

− +=

+ +

84

85

86

10


M.I.M.-SRAGA 1999./ 2010.©

( )

( )

2

2

2

2 2 4 4

2 2 2

3 2)2 3 4

1 2)2 2

2 3 3 2)1 1 1

1 1)3 3

1)

)

x x x

x x

xx x x

y y

yx y x y

y yx y y x

+ −− + =

− =+ −

−+ − =

− + −

− =− −

− =+ −

− =− −

87 87 88 89 90

91

Iz

( )

2 2 2 2

3 3 2

4 4 6 3 3 6

3 3 4 2 2

4

2

računaj:

4 4 4)

2)

1 2 1)1 1 9 1

x y x xy yx y y x

x y x x y yx y x x y

xx x x

− + +⋅ =

− +

− + +⋅ =

+ +

−⎛ ⎞+ ⋅ =⎜ ⎟+ − −⎝ ⎠

92

93 94

11


M.I.M.-SRAGA 1999./ 2010.©

( )( )

2 2

2 2

2 2 2

2 22

3 2

2 2

2 3

2

3 18 27 995) :6 24 4 4

2) :

1 1 3 2 2)1 1 1 6 9

111)

111

111 2)11

1 2

00)4 4

x y xy y xx y y x x

x y y yx yxz yz zx y z

x x x xx x x x x

y

y

xxxx

x y x yx y x y

x y xyxy x y y

+ + −=

− + +

+ − +=

− −− −

− + +⎛ ⎞− + ⋅ =⎜ ⎟− + − −⎝ ⎠

+−

=−

+

+−

− =+

+−

− +−

+ −=

−− + −

96

97

98

99

1

sto-elementarnih-zadatci

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