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TRNGIHCVINHTRNGTHPTCHUYN

PNTHITHTHPTQUCGIANM2015 LN2Mn:TON Thigianlmbi180pht

Cu pn imCu1:

(2,0im)a)(1,0im)10. Tpxcnh: { }\ 1D R = .20.Sbin thin:* Giihn,timcn:Tac

1limx

y -

= + v1

limx

y +

= - .Dongthng 1x = l

timcnngcath(H).V lim lim 1

x xy y

- + = = nnngthng 1y = ltimcnngangcath(H).

* Chiu binthin:Tac 21

' 0( 1)

yx

= > -

,vimi 1x .

Suyrahmsngbintrnmikhong ( 1) - , (1 ) + .* Bngbinthin:

x - 1+y + +

y+ 1

1 -

30th:

0,5

0,5

b) (1,0im)

Tac: ( )2

1'

1y

x =

-,vimi 1x .

Vtiptuynchsgc 1k = nnhonhtipimlnghimcaphngtrnh

( )21

11x

= -

hay ( )21 1x - = 02

x

x =

= *)Vi 0x = tacphngtrnhtiptuyn 2y x = + .*)Vi 2x = tacphngtrnhtiptuyn 2y x = - .Vychaitiptuynl: 2y x = + v 2y x = - .

0,5

0,5

Cu2:(1,0im)

a)(0,5im)

Rrng cos 0 a ,chiactsvmusca Acho 3cos a tac ( )2

2 3

tan 1 tan 2 2.5 2 4

1 tan 2 tan 5 16 7A

a a a a + + +

= = = + + +

0,5

b) (0,5im)

Gis z a bi = + ( , )a b .Suyra ( ) ( )2 12 1 11 2

iz a bi a b i

i -

+ = + + = + + - +

.

Tgithit2

1z

i +

+lsthcnntac 1b = .

Khi 22 2 1 2 3z a i a a = + = + = = .0,5

th(H)cttrcOx ti(2 0),ctOy ti(0 2),nhngiaoimI(1 1)cahaingtimcnlmtmixng

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Vysphccntml 3z i = + v 3z i = - +

Cu3:(0,5 im)

Btphngtrnhchotngngvi23 12 .2 2x x x - >

23 1 22 2 3 1x x x x x x + - > + - >2 2 1 0 1 2 1 2x x x - - < - < < +

0,5

Cu 4:(1,0im)

*)iukin 24 0 2 2.x x - - Phngtrnh chotngngvi

( )22 2 234 2 2 2 2x x x x x x + - = - - - + (1)Tac ( )22 24 4 2 4 4x x x x + - = + - ,vimi [ ]2 2x - .Suyra 24 2x x + - ,vimi [ ]2 2x - . (2)Dungthc(2)xyrakhivchkhi 0x = , 2x = .

t ( )223 2x x t - = .Ddngcc [ ]12t - ,vimi [ ]2 2x - .Khivphica(1)chnhl 3 2( ) 2 2f t t t = - + , [ ]12t -

Tac 20

'( ) 3 4 0 43

tf t t t

t

= = - = =

Hnna,talic ( 1) 1f - = - , (0) 2f = ,4 223 27

f =

, ( )2 2f = .

Suyra ( ) 2f t vimi [ ]12t - .

Do ( )22 232 2 2 2 2x x x x - - - + vimi [ ]2 2x - . (3)Dungthc(3)xyrakhivchkhi 0x = , 2x = .T(2)v(3)tacnghimcaphngtrnh(1)l 0x = , 2x = .Vyphngtrnhchocnghim 0x = , 2x = .

0,5

0,5

Cu5:(1,0im)

Chrng ( )ln 3 1 0x x + ,vimi0 1x .Khidintchhnhphngcn

tnhl ( )1

0

ln 3 1S x x dx = + .

t ( )u ln 3 1x = + ,dv xdx = .Suyra 3du3 1

dxx

= +

, 21

2v x = .

Theocngthctchphntngphntac

( )1 1 2 1

2

0 00

1 3 1 1ln 3 1 ln 2 3 1

2 2 3 1 6 3 1x

S x x dx x dxx x

= + - = - - + + + 1

2

0

1 3 1 8 1ln 2 ln 3 1 ln 2 .

6 2 3 9 12x x x = - - + + = -

0,5

0,5

Cu 6:(1,0 im)

Gi HltrungimBC.Tgithitsuyra' ( )C H ABC ^ .Trong DABC tac

201 3. .sin120

2 2ABCa

S AB AC = = .

2 2 2 0 22 . .cos120 7BC AC AB AC AB a = + - =

7BC a = 7

2a

CH =0,5

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2 23

' 'C2

aC H C CH = - =

Thtchkhilngtr33

' .4ABCa

V C H S = = .

H HK AC ^ ,V ( )'C H ABC ^ ngxin'C K AC ^ ( ) ( ) ( ) , ' ' 'ABC ACC A C KH = (1)

( 'C HK D vungtiHnn 0' 90C HK

Cu 101,0im) Gis { }min , ,z x y z = .t 02

zx u + = , 0

2z

y v + = .Khitac

22 2 2

2z

x z x u + + =

,2

2 2 2

2z

y z y v + + =

(1)2 2

2 2 2 2

2 2z z

x y x y u v + + + + = +

Chrngvihaisthcdng ,u v talunc1 1 4u v u v

+ +

v ( )22 2

1 1 8u v u v

+ +

(2)

T(1)vpdng(2)tac

2 2 2 2 2 2 2 2 2 2

1 1 1 1 1 1x y y z z x u v u v

+ + + + + + + +

2 2 2 2 2 2

1 1 1 1 3 1 1

4 4u v u v u v = + + + + +

( )22 21 1 6

2u v uv u v = + +

+ +

( ) ( ) ( ) ( )2 2 2 24 6 10 10

u v u v u v x y z + = =

+ + + + +(3)

Mtkhctac ( )( )( ) ( ) ( )1 1 1 1x y z xyz xy yz zx x y z + + + = + + + + + + +

2xyz x y z = + + + + 2x y z + + + (4)

T(3)v(4)suyra ( )

( )210 5

52

P x y zx y z

+ + + + + +

.(5)

t 0x y z t + + = > .Xthms2

10 5( ) , 0

2f t t t

t = + > .

Tac3

20 5'( ) , 0

2f t t

t = - + >

Suyra '( ) 0 2f t t = = , '( ) 0 2f t t > > , '( ) 0 0 2f t t < < . (6)

T(5)v(6)tac25

2P .Dungthcxyrakhi 1, 0x y z = = = hoccc

honv.VygitrnhnhtcaP l25

2.

0,5

0,5

Ch: pnnykhngphilfilegccatrngTHPTchuynHVinhmlfilenhlithnhchppn.Cththiusttrongqutrnh nhmyli,rtmongccthycthngcm.

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