Algebra (Horizon 2016)

7/26/2019 Algebra (Horizon 2016)

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1

01

: T P R C A E P.

M: 1. T A E . 2. T . 3. S , .

ALGEBRA

R: E. R R. R

ARITHMETIC PROGRESSIONT:

1 2 3 4 5 = + + 2 + 3 + 4

daan 4+=dnaan )1( +=

dnaan )1( +=

( )[ ]dnan

S 12

2

+=

( )naan

S +=2

1. H 3, 5, 7, 9, . . . 3,192?G:

= 3 = 2

S = 3192R:

S:

( )[ ]dnan

S 122

+=

( )[ ]dnanS 122 +=


2/22

2

[ ])2)(1()3)(2()192,3)(2( += nn

nnan 222384,6 2 +=

nnn 22)3)(2(384,6 2 +=

nnn 226384,6 2 +=

224384,6 nn +=

0192,322

=+ nn

0)57)(56( =+ nn

56=n

2. 4, 9, 14, 19, . . . 15 ?

G:

= 4

= 5

= 15

R:

S

S:( )[ ]dna

nS 12

2+=

( )[ ])5(115)4)(2(2

15+

=S

585=S

3. F 20 A 5, 8, 11, 14, . . .

G5

1=a

= 3

= 20

S:dnaa

n )1(1 +=)3)(120(5

20 +=a

62

20 =a


3/22

3

4. T A P 250 10. I 40,

.

G:

1= 10

= 40

S = 250

R:

S:

( )naan

S += 12

104010

)250)(2(2

1

=+

=+

=

naa

Sn

5. A 1 9 . T 21. F 9

M 5..

G:

1= 15= 21

= 9

R:

S

S:daa 4

15 +=

5

4

121

4

15=

=

=

aad

( )[ ]dnanS 122 1 +=

[ ] 189)5)(19()1)(2(2

9=+

=S

6. T 5 A P 17 3 10. F 8 .

G:

5= 17

3= 10

R:

8S:daa 4

15 +=

da 4171+= E.

daa 213 +=

da 210 1 += E.


4/22

4

C E :da 417 1 +=

da 210

1+=

d27=

2

7=d

F E. :da 417

1+=

da 4171 =

3

2

7)4(17

1 =

=a

2

55

2

496

2

7)7(3

8 =

+=

+=a

7. T A 2 6 12. F 12 .

G:

1= 26= 12

= 12

R:

S

S:daa 5

16 +=

2

5

212

5

16=

=

=

aad

[ ] 156)2)(112()2)(2(212

=+

=S

8. T 5 A P 15 1 3. 12 ?

G:

5= 15

1= 3

= 12

R:daa 415 +=

34

315

415 =

=

=

aad

[ ] 234)3)(112()3)(2(2

12=+

=S

9. D 6 A 8 16 4 8.

G:

8 = 16


5/22

5

4= 8

R:

6S:

daa 718 +=da 716 1 += E.

daa 314

+= E.

da 38 1 += C E. :

da 716 1 +=

da 381+=

d48 =2=d

F E. :da 716 1 +=

da 7161 =

2)2)(7(161 ==adaa 516 +=

12)2)(5()2(6 =+=a

10. 5 8 A 1 1

4 5.

G:

1= 1

4= 5

R:58

aa

daa 314 +=

2

3

)1(5

3

14=

=

=

aad

13)2)(7(17

18 =+=+= daa

7)2)(4(14

15 =+=+= daa

T:671358 == aa

11. T 7 7 39. F .

G:

1 = 7

7 39

= 9


6/22

6

R:

S

S:

( ) 207)397(2

9

2 1

=+

=+= naa

nS

12. A PP40,000 J, C D,

A . C 3 J. H ?

S:

A B C A P

L:

A = J

B = C

C = D

B A = C B

2B = A + C

A 2B + C = 0 E.

B,

A + B + C = 40,000 E.

C E. E. .

A 2B + C = 0

A + B + C = 40,000

3B = 40,000

B = PP13,333.33

S,

B = 3A

44.444,43

33.333,13

3PhP

BA ===

F E. :

C = 40,000 (A + B)

= 40,000 (4,444.44 + 13,333.33)

C = PP22,222.23

13. F 100 1,000 9.

G:

1 = 101 = 999

= 2

R:

9

S


7/22

7

S:dnaa

n )1(1 +=

4501

2

10199911 =+

=+

=

d

aan n

( ) 500,247)999101(2

450

2 1 =+

=+= naa

nS

500,279

500,247

9==

S

14. F 30 .

G:

1= 2

= 30

= 2

R:

S

S:

[ ] [ ] 930)2)(130()2)2(230)1(2

2 1

=+

=+= dnanS

1. 50 3, 5, 7, 9, . . .?

A. 50= 101

2. 50 5, 8, 11, 14, . . . ?

A. S = 2,600

3. 30 15, 13, 11, 9, . . .?

A. 30 = 43

4. A 220 10. I 30, .

A. = 11

5. T 10 A P 28 1. D 5

7.

A. P = 247

BINOMIAL THEOREM

B E F

nnnn

nnb

bannnbannbnaaba ++

+

++=+

...!3

)2)(1(

!2

)1(

!1)(

33221

T B E

))!1(

)2)...(2)(1( 11

+ +

r

barnnnn rrn


8/22

8

))!1()!1(

! 11

+

+

rrn

ban rrn

: = =

M B E

12+=nr

S E

nnS )1( += ( )

E:224

2 yyxx ++

S = 4 + 2 + 1 + 2 = 9

C:

nnS )1( += ( )

92

12)2)(12( =

++=S

S C) S =1

16)11()( 44

=+=+yx

) 65)2()21()2( 444

=+=+x

C

mmn

yx

!)!(

!

mmn

nk

=

E:12)( yx+

75yx

!7!5

!12

)!7()!712(

!12

!)!(

!=

=

=

mmn

nk

1. F 5

6

)2( x

.G: = 6 = 5 = 2 =

R:5


9/22

9

S:

4421515611

60)!4)(!2(

)()2)(!6(

)!15()!156(

)()2)(!6(

)!1()!1(

!x

xx

rrn

ban rrn

=

=+

=

+

++

2. 3 5

)26( y+ ?

G:

= 5 = 3 = 6 = 25

R:N 3

S:

8640)!2)(!3(

)2()6)(!5(

)!1()!1(

! 2311

==+

+y

rrn

ban rrn

3. 6

)( ba+ ?

S:

64)11()( 66

=+=+ba

4. F 9

)34( x , ?

G: = 9

R:S

S:nnS )1( += ( )

452

01)9)(19()1( =

++=+= nnS

5. F 222)( yx + , ?

G: = 2

R:S

S:4224222

2)( yyxxyx ++=+ ( )

S = 4 + 2 + 2 + 4 = 12C:

S nn )1( += ( )


10/22

10

122

22)2)(12( =

++=S

6. F 6

16

32

1

a .

G: = 1/6

= 6 = 1/2 = 3

R:

6

S:

11

5111616161181.192,62

)!5)(!11(

)3()2/1()!16(

)!16)()!1616(

)3()2/1)(!16(

)!1()!1(

!

a

aa

rrn

ban rrn=

=

+

=

+

++

P:

1. F 6 6

)4( x .

A. 524x

2. 4 7

)3( y+ .

A. 2,835

3. 7 7

)21( y+ .

A. 448

4. F 5

)( ba+ .

A. 32

1. T 5 1, 2, 6, 15, . . . S:

1 2 6 15 (1)2 (2)2 (3)2 (4)2

= 15 + (4)2= 31

2. D 5 13, 17, 26, 42, . . .

S:

13 17 26 42

(2)2 (3)2 (4)2 (5)2

= 42 + (5)2= 67


11/22

11

1. F 24 =x

S:

24 =x 24 =x 224 ==x 624 =+=x

2. F

1135 = x

S:

1135 = x 1135 = x

1153 =x 5113 +=x

2=x

3

16=x

1. F 52 =x

A. = 7 , = 3

2. F 1042 =x

A. = 7 , = 3

3. F 84 =+x

A. = 4 , = 12

02 =++ CBxAx

A

ACBBx

2

42

=

P R

) S R

Bxx =+ 21


12/22

12

) P R

A

Cxx =21

D:

ACB 42

) ACB 42=

,

) ACB 42> ,

) ACB 42 < ,

1. 0643 2 =+ xx

G:

A = 3B = 4

C = 6

R:

S

S:

3

4

3

421

=

==+A

Bxx

2. F 0532 =+ xx

G:A = 1B = 3C = 5

RP

S:

51

521 =

==

Cxx

3. D 0823 2

=++ kxx 4.G:

A = 3B = 2C = 8

421 ==+

A

Bxx


13/22

13

R:

K

S:

3

24

k

A

B==

6=k

3. 022 =+ kxx 8.G:

A = 1B = 2C =

A

Cxx =

21

1

821

k

A

Cxx ===

8=k

1. 0437 2 =+ xx A. 3/7

2. F 0475 2 = xx A. 4/5

I :

b

x

x

a=

bxxa :: =

=

abx =2

abx = (

Idc

ba =

d

b

c

a=

Id

c

b

a=


14/22

14

c

d

a

b=

Id

c

b

a=

d

dc

b

ba +=

+

Id

c

b

a=

dc

dc

ba

ba

+=

+

I :kxy =

=

I :

x

ky =

1. . =4 = 6,3

1=y . F = 6 = 2.

S:

2x

kty =

34

)3/1()6( 22

===

t

yx

k

5.4)2(

)6)(3(22

===x

kty

2. . = 9 = 3, = 5. F = 4 = 12.

S:

y

xkw =

59

)3)(5(===

x

wyk

33.44

125==w

D = R TVtS=


15/22

15

:S = = =

1. A B . A 8:30 .. 2.7 , B, 5 . 8:00 .. 1.8 , B, 5 . H B A?

B?

S:L: = A B

= B

60

5

7.25.8 +=+ t

x E.

60

5

8.18 =+ t

x E.

S E. E. :

6

1

8.17.25.0 =+ xx

6

15.0

7.28.1=

xx

6

15.0

)7.2)(8.1(

8.17.2=

xx

)7.2)(8.1)(6

15.0(9.0 =x

= 1.8

S E. :

60

5

7.25.8 +=+ t

x

60

5

7.2

8.15.8 +=+ t

083.960

5

7.2

8.15.8 =+=t

..05:9 mat=

2. A , 3 1,500 3.75 . () () ?

S:L: =

=

t

dv =

3

500,1=+ yx

500=+ yx E.


16/22

16

75.3

500,1= yx

400= yx E.S E. E. :

1002 =y

kphy 50= S E. :

500=+

yx 50050 =+x kphx 450=

3. A 3:00 .., 400 . T 50 . I 100 , ?

S:10050400 += tt

100350 =t 350/100=t .min14.17.2857.0 == hrst

T 3:17.14 ..

4. T A B 10,000 . I 40 A B 50 . H B A A .

S:vtS=

F A:)40(000,10 AV=

.min/250mVA =

F B:

)50(000,10 BV=

.min/200mVB =

000,10)40)(200( =+x

000,2)40)(200(000,10 ==x A


17/22

17

O 5 .

I , 12

x

O 300 .

1. T 100 . /?

G: = 100 R:

S:

rS =

32.628180

)360)(100(0

0==

xcmS


18/22

18

36.5212

32.628===

hr

cm

t

SV /

2. A 12:00 1200 ?

S:

1200

= 2020

12+=

xx

2012

= x

x

2012

12=

xx

)12)(20(11 =x 24011 =x 8.21=x .

3. A 12:00 ?

S:

3012

+= x

x


19/22

19

3012

= x

x

)12)(30(11 =x 7.32=x .

T 12:32.7 ..

P P F 5 + 4

= 5 = 5 + 4 = 4

N:I , .

1. J P . I , J . H J ?

S:

P P FJ 4 J J + 2P 4 P P + 2

4=PJ 4+=JP E.

2

2

112

++=+

P

J

2

29

++= P

J

2

218 ++=

PJ

202 +=PJ E. S E. E. :

2042 ++=JJ 24=J

2. F , K G 45. S , G 28 K . H K ?

S:

P P FK 4 K K + 6G 4 G G + 6


20/22

20

4544 =+ GK 53=+GK KG = 53 E.

628)6(2 ++=+ KG

34122 +=+ KG 222 +=KG E.

S E. E. :

22)53(2 += KK

222106 += KK 843 =K 28=K

3. A 8% B, B 9% C. B A C?

S:BA 08.1= E. CB 09.1= E.

S E. E. :

CCA 1772.1)09.1)(08.1( == A 17.72% C

I , x

1

I , .

tyx

111=+

1. J 2 . P 3 . S 6 . H ?

S:

x

1

6

1

3

1

2

1=++

x

1

6

123=

++

1=x

2. R . T 10 . H ?


21/22

21

S:L: = R

2 =

10

1

2

11=+

xx

10

1

2

12=

+

x

)10)(12(2 +=x 15=x R

2 = 30

3. A 40 120 . I 20 , 20 21 20 41, ?

S:(40)(120) = (20)(20) + (40)(20) +60 = 60 T . = 20 + 20 + 60 = 100

N. = 120 100 = 20

4. A 20 30 . T , 25 . H, 20 , 10 . F ?

S:(20)(30) = (25)(20) + 15 = 6.67 T . = 20 + 6.67 = 26.67

5. T 120 . T 10

30 46

. H ?

S:(30)(120) = (30)(30) +(20)(15) +30 = 80 T . = 30 + 15 + 80 = 125 T 125 120 = 5

6. A 25 90 . T 25 30 . T 60 . B ?

S:(25)(90) = (25)(30) + (20)(30) + 17 = 52.94 53 N. = 30 + 30 + 53 = 113 N. = 113 90 = 23


22/22

22

A% B% C% + = +

1. O 25% 3 40% . ?

S:25% 40% %

1 + 3 = 4

(25)(1) + (40)(3) = 4 = 36.25%

2. T 20% 50% . I 38% , ?

S:20% 50% 38%

2 + = 2 +

(20)(2) + 50 = (38)(2 + )40 + 50 = 76 + 3850 38 = 76 4012 = 36 = 3

3. T 20% % . I 22.5% , ?

S:20% % 22.5%3 + 1 = 4

(20)(1) + (1) = (22.5)(4) = 30%

4. A 3 25% . O . ?

S:25% 100% %2.5 + 0.5 = 3

(25)(2.5) + (100)(0.5) = (3) = 37.5%

Algebra (Horizon 2016)

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