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- Ifng dung ciia tich phan: Tinh dien tich hinh phing, the tich khoi Iron xoay.. Tinh dien tich xung quanh cua hinh non tron xoay, hinh try tron xoay; tinh the tich khoi lang tru, kho

Danh cho thi sinh Icfp 12 on tap va thi Dai hoc, Cao dang

Bien soan theo ngi dung va cau true de thi cua Bg Glao due - Dao tao

0

DVL.009154

NHA XUAT BAN DAI HQC QUOC GIA HA NOI

N G U Y E N V A N N H O - LE B A Y - N G U Y I N V A N THO

T U LUA

TO^n HOC

^ Danh cho thi sinh I6p 12 on tgp va thi Dqi hoc - C a o d a n g

^ Bien soqn theo noi dung va c d u true d4 thi c u a Bp GD&DT

NHA XUAT BAN DAI HOC QUOC GIA HA NOI

Jltue lue

Lifinoiddii ^

PH AN 1:

TH

I TU YE

N SIN

H D

^I H QC

- CA

n sin

h Da

i hoc , Ca

o ding , kho

i A

6

Dap an

- than

g die

m 8

De thi tuye

n sin

h Da

i hoc , Ca

o d^ng , kho

i B , D

14

r Da

p a

n thang diem

273

PHAN 2:

TH

I T OT N GH IE

P T RU NG H QC PHO T HO NG

n 1

316

De thi to

n 2

318

Pfi^nl: THI TUY^py SINH a/^l HQQ, SAG BANG

K CMi TRUC DE THI TUYEN SINH DAI HOC, CAO DANG NAM 2009

I PHAN CHUNG CHO TAT CA THf SINH (7,0 di^m)

I

ciia ham so: Chieu bie'n thien cua ham so Cifc tri Gia tri

Idn nha't va nho nhat cua ham so' Tie'p tuyen, tiem can

(d\Jng va ngang) cua do thi ham so' Tim tren do thi nhi^ng

diem CO tinh chat cho trtfdc; ti/dng giao giCfa hai do thi (mot

trong hai do thi la du"5ng thing);

2,0

III

- Tim gidi han

- Tim nguyen ham, tinh tich phan

- Ifng dung ciia tich phan: Tinh dien tich hinh phing, the tich

khoi Iron xoay

1,0

IV

Hinh hoc khong gian (tong hcrp):Quan he song song, quan he

vuong goc cua di/dng thing, mSt phlng Tinh dien tich xung

quanh cua hinh non tron xoay, hinh try tron xoay; tinh the tich

khoi lang tru, khoi chop, khoi n6n tron xoay, khoi tru tron

xoay; tinh dien tich mSt cau va the tich khoi cau

1.0

n.PHANRlfiNG (3,0diim)

Thi sinh chi difdc l^m mot trong hai phan (phan 1 hoac phan 2)

I Theo chUtfng trinh Chudn:

VI.a - PhUcmg phdg too do trong mat phdng vd trong khong gian: 2,0

3

Cdu Ngi dung

kien thiic ^ Diem

- Du'fJn

g tron, elip, ma

t cau

g, di/ftn

g thin

- dUcJn

g thang, ma

t phan

g v

a ma

t cau

VII.a

- So pMc

- To hap, xdc

suci't, thonf>

ke

- Bat ddn}> thvtc.

Cuc tri ciia bleu

thi'fc dgi

so

1,0

2 Theo chUOitg trinh

Nang cao:

Cdu NQi dung

kien thi'tc ,

Diem

Vl b

* PhUtfng phdp toa

do trong mat phdni*

vd trong khong gian:

- Xac

djnh to

a

dp cu

a diem , vecttf

- Dxiiing

trdn, h

a diTdn

g conic , ma

t cau

- Viet

phiTPng trin

h ma

t p hin

g, dtfcJn

g thin

g

- Tinh

m de

n difdn

g thang , ma

t

ph in

g ; khoang eac

h giffa ha

t phan

g v

a

mat cau

2,0

Vll b

- Sophi'/c

, • , ^ , , , , / , +

bx +

v ~

va mot ii'x + b'

so yen

to lien quan

- Su tiep xi'tc

ciia hai dudng cong

- He phU(fng trinh

mii vd Idgarit

- To h(/p, xdc

sud't, thong

ke

- Bat dang thUc.

Cuc tri ciia hieu

U CA

N LU

U Y:

npi dun

g giijr

a hai

a can

lifii y

mot s

o va

n de

nhU"sa

u :

I PHAN CHUNG CHO TAT CA T

ve d

o th

j , cung nhi

4 ,

• Hc^m bac 3 : y-cuc'+ hx^ + cjc + J , (« ^ O)

• Ham bac 4 (dang trung phiTcfng): y - ax^ + hx^ + c , (a O)

• Ham phan thuTc dang : y = ifilA ^ ^ o, at/ - be ^ O)

cx^-d

* Khi khao sat tinh chat cua ham so , tinh loi, 16m va viec tim diem uo'n ciaa

do thi CO the bo qua khong can xet (neu can thi chi can tim diem uon cua ham bac 3 de suy ra tarn doi xiJng cua do thi , con ham bac 4 thi nen bo qua hoan toan phan nay)

* Cac bai loan ve sir tiep xuc cua hai diTfJng cong cung se khong diTdc de cap tdi trong phan chung nay

* Cac bai toan ve tiem can cung chi de cap den liem can dtfng va tiem can ngang

Cdu III:

* Viec uTng dung tich phan de tinh the tich cua khoi tron xoay chi c6 cac khoi

khi cho hinh phing quay quanh true Ox

II PHAN RIENG (3,0diem) u

1 Theo chittfitg trhih Chiidn:

Cdu Vila:

* Phan so phffc ehi c6 ctic bai toan c6 lien quan den cac phep toan ve so phffc va viec giai cac phUdng trinh bac hai co he so Ihffc , khong de cap den can bac hai cua so phuTc , cung nhu" viec giai phffdng trinh eo he so phffc va cac bai toan CO lien quan den dang lifdng giae eiia so phffc

2 Theo chUcfng trinh Ndng cao:

Cdu VIb:

Cac bai toan c6 lien quan den tie'p tuyen cua cac du'dng conic cung khong dc cap tdi trong cau true de thi mdi nay Nhff vay doi vdi cac dffdng conic chi can

on lai cac dang toan ve viet phifdng trinh ehinh tac , tim cac diem nlim trcn

conic thda tinh cha't nao do va cac bai toan ve mot so tinh chat dac trffng ci.i tifng du'dng conic ^

5

cua Bg

THI TU V^

N SIN

H 96

1 HQC , Cfl O

OANG -

K H6|

fl

(Thcfi gian Idm bdi:

trong d

o m

la^tham s

o thiTc

j {x

+ 2)-3 + log^{x

-5)^ + log, 8

m so

y = yje^

+1 , tru

yz zx

xy

II PHAN RIENG (3,0 diem)

Thi sink chi diMc lam mot trong hai phdn (phdn 1 hoqc phdn 2)

1 Theo chUcfng trinh Chudn: - ^

au VLa (2,0 diem) > i

1 Trong mat phang toa do Oxy, cho di/dng tron (C): + - bx + 5 = 0

Tim diem M thuoc true tung sao cho qua M ke difdc hai tiep tuyen cua (C)

ma goc giSa hai tiep tuyen do bang 60"

Viet phufcJng trinh tham so' cua dtfdng thang di qua diem M , cat va vuong goc

vdi dufdng thang fc?)

Cau VILa (1,0 diem) j

Tim he so cua trong khai trien thanh da thtfc ciia bieu thuTc:

P = lx^ +x-\f

2 Theo chitctng trinh ndngcao: , ' •

CSu VI.b (2,0 diem)

1 Trong mat phang toa do Oxy, cho di^dng tron [C): x^ + y^ - 6x + 5 = 0

Tim diem M thuoc true tung sao cho qua M ke di/dc hai tiep tuyen cua (C)

ma goc gii?a hai tiep tdyen do bang 60"

2 Trong khong gian vdi he toa dp Oxyz, cho dilm M(2 ; 1; O) va di/dng thang

Viet phufdng trinh chinh tic cua dtfcfng thang di qua diem M , cat va vuong

Cfiu VII.b (1,0 A\im)

Tim he so cua trong khai trien thanh da thuTc ciia bieu thtfc:

\

P = X^ +

A-7

OAP A N- TH RN

G DI^

M

Cdu Dap an

a c

6 ha

m so' y = -.v

: D

= x

Sif bie

n thien:

• Chie

u bie

n thien: y'=

'x = -2

r v

x =

0 •

[.r

>0

< =>

y'>

Do do:

+ Ham

a (()

; +cc

)

+ Ham

y' = 0

«

0,50

CiTc tri: Ha

y = -oo

0,25

Bang bien thien:

0

y

\

Xo

2 ; 0)

0,25

Phi/rtng trinh da cho tifcfng difdng vdi phi/dng trinh:

(2 sin jr - \/3) (V3 sin A- + cos A- j = 0

V(3i dieu kien do, ta c6:

Phifcfng trinh da cho tUdng dUtrng vcti phu'cJng trinh:

log2[(A- + 2)|A-5|]-log2 8

0,50

-jt = 6

;j : = —

2

Do'i chie'

u v

di dieu kie

n (*), t

a dufd

c ta

t c

a cac

ghie

m

cu

a

phtfctng trin

h d

a cho la: x = 6 v

5 la die

n tic

h can tinh

Vi Ve'+

;ln8] ne

/'-I

Kh

i x = ln3 t

hi / = 2 , k

hi

v = ln8 t

hi t

= 3 0,25

i t -

i U i

t

-i

j

2r -l

2 '

+ l '

+ l ^ =

2 + ln-

en l

a tar

n gia

c de

c de

h vuon

n cu

a mat

au ngo

ai tie

p hin

h chop

1

tron

g do

H l

a trung die

m cu

a AB.

M

=p.V -\t°—

+ Tarn gia

c OCA

vuong t

ai G / „

K i hieu R la ban kinh cua mat cau ngoai tiep hinh chop

Nhan tha'y: x^ -xy>xy , Vx,y&R

Do do: J:'^ + j " ^ > jry(jc + y ) , Vjc,y > 0

Suy ra triic tung khong c6 diem chung v d i difdng tron (C)

Vi vay qua mot diem bat k i tren true tung luon ke dtfdc hai

tiep tuye'n den (C)

0,25

X e t diem M[0;m) tuy y thuoc true tung

Qua M ke cac tiep tuyen MA va MB cua (C) (A,B la cac tiep diem) Ta c6: Goc giffa hai du'dng thang MA va MB bang

/\A/B = 6 0 " (1)

6 0 " o ^

[AMB = 120" (2)

0,25

L

Vi yV

//la

phan giac cua

o A/

/ /?

-^ -^ sin

60" 3

Dc thay, khon

g c

6 tho

a (*)

r-l;

-2 + f

2r

~l ) + l (

-2+

/) + (-!).(-/) =

-Vi the

, MH =1 - :-~

3J

0,50

Suy ra

MH la

:

A- =

2 + /

V =

1 4/

-:.^-2t 0,25

VI

La

(1,0 diem)

Theo cong

)%

ct

r(

.v -l )%

+

c tr

'(

.v -i r.

.C tv '"

va

Q '.

(A

v-^l )^

V i vay, he so cua trong khai tricn P thanh da thi?c la:

Goi H la hinh vuong goc ciia M ircn J, la c6 A/// la difrJng

He so' ciia v' trong khai iricn ciia C^,' (A - l / ' la: ~ C " Q ^

He so'ciia A' trong khai tricn ciia Q'.A" (A - 1)"^ la: + Q ' c ]

V i vay, he so ciia A ' trong khai tricn P lhanh da thuTc la:

-C^QSQ^.C^^ + IO

0,25

0,25

KH 6| B.

D

(Th&i gian lam bai:

H (7 di^m)

Cau I ( 2, 0

di^m

) ' y = m so Cho ha ^"^^^

x-2

1 Kha

o sa

t sir bien thie

C + 5

| + log , 8 = 0 '

thing

x = yle - \

Tim ta'

t ca cac gi

p hi Td ng

trinh

j

;

;

x^ + 'ix^ -\<a[4x-^x-\^

d p

Oxyz , ch

o

d iT dn g

thing d

c6

p hiT dn

g

j c - 1 y-1 z - 3

va mat phang ( P ) : 3A: - 2y - z + 5 = 0

2 1 4

1 Tinh khoang each givi& duTdng t h i n g J va mat p h i n g (P)

2 K i hieu I la hlnh chieu vuong goc cua d tren (P) V i e t phifdng trinh tham so'

cua dtfdng thang / j i ,

CSu V I L a (1, 0 Aiim )

r i m cac so thi/c x,y thoa man dang thiJc : x{3 + 5i) + y{l-2if =9 + l4i

2 Theo chumg trinh ndng cao:

Cfiu V L b ( 2,0 d i e m ) ^ - ' 1

Trong khong gian vdi he toa do Oxyz, cho difdng thang d c6 phtfcfng trinh

2 K i hieu / la hinh chieu vuong goc cua d tren (P) V i e t phi/dng trinh chinh t^c

m ca

n dtfn

g

la dirdng thing ;c = 2 , mp

+ D

o th

i ci

2;

2)

(Ik

giao Ciia hai difcJn

g ti^

m can

) Ikm

tarn do'i xtfn

y -2x + m

c

it

C C) ta

tai 66

song song vdi nha

u

2x +

3

<=> — -2x +

a ma

n

dieukien

v '(

;C

|) = 3 ''(jC2)

<::>

2J

:'+(

6)j:

m 2m-

0,50

A =

6) +8(2m

+ 3)>

0,25

II

(2,0 diem)

1 (1,0 diem)

PhiTdng trinh da cho ti/dng difdng vdi phiTdng trinh :

<=> sin jc + sin 4.x: = 1 + sin 4 J: o sin x 1

IV

(1,0 diem)

Ki hie

u A

va V tiTcfng lirn

g la chieu ca

Suyra: h = A'A.sin60"=aj3

Do do,

A

V = h^^ifQ =

f{x) = {^x^ +

3x^

-I)(

[ l;

+o o)

Suy r

a : /(

A:)>

/(1) =

3 VA:>1 V

; 7 ; 3)eva M = (

2 ; 1

; 4) l

+ n = (

3 ;

2 ; l) la mot v6

G o i d' la dtftJng th^ng di qua A va vuong goc v d i (P)

D o /i" = (3 ; - 2 ; 1) la mot vee td phap tuyen cua (P) nen n"

la m o t vee td chi phi/dng cua d' Suy ra , phi/cJng trinh cua d'

u Vl.a

(1,0 diem)

; 4 )

Gpi d'

la dU ' cJ

2 ; l ) l

Irinh cua

d'

la- -

l_

7_

i

1

Goi A'

la giao diem ciia

J ' v('J

i (Pj , t

a c

c V A ' e / To

nghiem cua he:

-^

3 -

2 -

1

[3A:

-2v f

5 =

0

, 41

40 33

14 •

7 1

4

Hcfn nSa , v

0,50

Vl.b

(1,0 diem)

i

1 Suy

41 40

Dang lifPng giac cua

r la : ;

z^2

CO S—

=32 cos — f/.sm

3 JJ

0,50

20

D HAI MUOI Dfe THI C6 idl C I A I

1

I, PHAN C H U N G C H O T A T C A T H I SINH (7 di^'m )

Cfiu I ( 2,0 di^m )

Cho ham so >• = - ( w + m x ^ + ( 3 W - 2 ) J : , trong do m la tham so thiTc

1 Khao sat sir bie'n thien va ve do thi ciia ham so' da cho, vdi m = 2

2 Tim tat ca cac gia tri cua tham so'm de ham so' da cho dong bie'n tren tap xac

dinh cua no

CSu II (2,0di^m)

1 Giai phiTcfng trinh: (2cos J: - l)(sinx + cos;c) = 1 (1)

2 Giai phi/cfng Irmh: i f t 3

Cho lang tru di^ng ABC.A'B'C c6 day la tarn giac deu Mat phang y4'BC tao

vdi day mot goc 3 0 ° v a t k m giac A'BC c6 dien tich bang 8 Tinh the tich

khoi lang tru

II PHAN RIENG (3,0 d i e m )

Thi sink chi dMc lam niQt trong haiphdn ( phdn 1 hoac phdn 2)

1 Theo chUcfitg trinh Chudn :

Cau VLa ( 2,0 diem )

1 Viet phiTdng trinh diTdng thang (A) di qua diem M{3; l ) v a cat true Ox, Oy

Ian lifdt tai B va C sao cho tam giac ABC can tai A vdi A (2 ; - 2 )

0) ,{x() > 0

Xa

c din

h

tpa dp diem

C tre

n tru

e Oz

de the tich tur die

n OABC

bang

8

CSu V IL

a (1 , 0 diem )

o 3

?

2 Theo

chtMng trinh ndng cao

:

CSu Vl.b ( 2,

a ton

g OA + OB

nho nhat

3 ), co

n din

h D nlm tre

'

CSu

V II

b

(1, 0 di§m )

o 3 ma cac chff s

MM aiai

Cfiu

I

\ Khi

m =

2 iKi

y = Y^+2x^+Ax

-c»

; h

m y = +

00

jr ->

^+

4x + 4=:(

x + 2)^

>

0, Vx

Ham so dong b i e n tren khoang (-00 ;+oo), ham so khong c6 ciTc t r i

Do thi :

-o D i e m u-on: r V

y" = 2\ 4 '

-Ta thay >^"ddi da'u tH a m sang diTcfng

k h i X d i qua d i e m x = - 2 , nen do thi cua

1 Giai phi/dng trinh

( 1 ) <=> 2 sin cos jc + 2 cos^ J: - (sin X + cos A:) = 1

<=> sin 2 + 1 + cos2X - (sin A: + cos J:) = I

< =>

72 sin

4

= x + + A:2;

u kien:

x + 2ji0

4-X

>0<:>

x + 6>0

-6<x<4 x^-2

(2)o31og

j^|;

c +

2|-3 = 31og

^ (4-A:

) +

31og, {x

+ 6)

4 4 ' 4

c:>logj^|j: + 2

lo

|g, =lo

-g, (4-x) + lo

"*

^ 4 « 4

lo

g, (4|

x + 2|) = log I [(4

t + 6)

4(

x + 2) = (4-jr)(

x + 6)

4{x + 2) = -{4-x){x +

6)

x^

+6JC-I6 = 0

2

<=>

X =

2 hay

x^-i

x = l

±V

So vdi

dieu kie

Da

t r = sinx

<5

f f

= cosx.t/

x ; D

ln

|-|t-2

|)

= l

n t-

C f i u I V

Goi H la trung d i e m cua flC => /\// 1 flC => A'// 1 B C

Cdchkhdc :

Xet ca

c vect

d a = (\

/ x ; ; b

^

'2

I '

Ta biet : a b <

o(

x + y)

4 I

= -(

II P HA

N RifiNG

f = 2y o

'-4

7 TTie o

chMiig trinh Chudn :

^

C§u VI.a ~ «

:

I

1 Da

t B(

b;

0) =

A) nO x, C(

0;

c) =

A) nO

a :

- + ^ =

1 , (be ^

A/

6 (A):=

> + (1

Tam giac

ABC ca

n ta i

A <=>AB = AC

<=>AEr = AC^

-

«(

2) '+

^-4 =

4 + (c + 2) '«

• Vd

i b = c + 4 :

(l )o c^

= 4 c>

2=

b-c+

2

b-2 = - c-

c = -2 => b = 2

• Vd i

b = -c:

(1) <»Z

? = 2

=>

c = -

Vay C O hai di/dng thang can tim: •

Dat nhom hai chff so 2, 3 1^ «

+ TriTdc tien ta tim cac so' c6 6 chff so' khac nhau thoa yeu cau bai loan, trong do chff so a, c6 the b^ng 0 hoSc khac 0

2 Theo chUcfng trinh

n&ng cao:

Cflu Vl

b

1 Go

i (A) m dirdng

thing can tim

Gia suT (A) ci

t ti

a Ox tai A(a ; 0) \h

6 dan

g + ^ = 1 ''

-a b

A/

e (A)=

+ - = 1 =>

b =-

^ (doa

>0,b

a-4

Theo ba't ding th

u Tc Cauchy, t

a c6

: OA + OB>

2 J(

a 4)

4

min

(OA + OB) =

9, da

t diTd

c khi: a

=>

b =

3

Vay(A): ^ + ^ = 1

o x + 2y-

6 = 0

2 Da

t D(

0;

>'

;0)€

0>

;

Ta c6:

AB = (l;

1;

2) A

C = (O ; -

2 ; 4) , A

D = (-2

4;

0 -2

V =

[AB, AC]

.AD| = ^

|0.(

-2) +

-4)(

y 1) + (-2) 1 =

4

- 2

y

Theo de bai t

a c6:

V =5oi

|l-2y

| = 5o!

2yl=

l-15

2y = 1

l-5

y = -

7

y =

l-Vay c6 hai die

m D can flm 1

^ :

D, (

O ;-7

; O) , D

cau VIỊ b , '

Neu goi M la so cac só tif nhien c6 ba chỉ so khac nhau va N la só cac só tif

nhien c6 ba chil so khac nhau chia het cho 3 diTcJc tao tif v4 = { 0 ; 1 ; 2 ; 3 ; 4 ; 5 } ,

thi so cac so can tim se la: M - Ậ

-So can t i m c6 dang âa2â; , a, + a2 + :3

Xet cac tap con gom 3 phan tijf cua tap A - {0;1;2;3;4;5} , ta thay chi c6

cac tap sau thoa dieu kien c6 tdng cac so chia het cho 3 la :

fi, = {0 ; 1 ; 2} , ^2 = {0 ; 1; 5} , ^ 3 - (0 ; 2 ; 4} , - { l ; 2 ; 3 }

B5={1;3;5} , = {2 ; 3 ; 4} , fl, - { 3 ; 4 ; 5}

+ Khi (3, ',0^ G B, de lap difcfc so thoa yeu cau ta thufc hien theo 2 bÚdc

- Chpn a , e B, \} c6 2 cdch

• Chpn hai so con l a i xep vao 2 v i t r i âiâ c6 2 ! cich

Vay trong triTdng hdp nay c6 2.2! = 4 so thoa yeu cau

+ Ttfdng l\i khi â e B2 hoSc e B3 m o i triTcJng hdp cung c6 4

s6'thoa yeu cau

+ K h i â ' " 3 ^ ; ; ; B-, m o i so l i mot giao hodn cua 3 phan tuT

nen mSi tri/dng hdp c6 3! so thoa yeu cau

Vay : /V = 3.4 + 4.3! = 12 + 2 4 - 3 6 so thoa yeu cau bai toan

Do do so cac so thoa yeu cau bai todn la: M - /V = 100 - 36 = 64 (so)

1 Kha

o sa

t sif bien thie

0 (1 )

2 Gia

i phiTdn

g trin

h : log , (x

-1)

^ + log

^ (2

x -1 ) = 2 (2) +

Cfiu II

I (1,

0 die

m ) '

haiphdn (phdn

1 hoacphdn

2y +

7 = 0 c AfiC n gia Vie'tphirdnglrinhcaccanhciia tar ^

m

2 Trong khong gian Oxyz cho tam giac ABC v d i A ( l ; 2 ; - l ) , B(2; - 1 ; 3),

C(-4; 7; 5) T i n h do dai duTcJng phan giac trong ke tit dinh B

CSu VII.a (1,0 6iim)

Co bao nhieu so tiT nhien c6 4 chiJ so chia het cho 4 tao b d i cac chOr so 1, 2,

3,4 trong hai trU'dng hcJp sau:

a) Cac chu" so co the triing nhau; b) Cac chu" so khac nhau

2 Theo chUffng trinh ndng cao:

Cau Vl.b (2,0 d i e m )

1 Viet phiTdng trinh diTdng lhang di qua diem A{27 : / j va ca't cac tia Ox, Oy

Ian lu'dt tai M va N sao cho do dai doan MN nho nhat

2 Cho cac vectcf a - (3 ; - I ;2) , Z)" = ( l ;] ; - 2 ) T\m vectd ddn vi dong phang

\6\ , b va tao vdi a goc 60"

Cau V n b (1,0 d i e m ) ^*

Cho cac chu* so 1, 2, 3, 4, 5 Tijf cac chfl" so da cho c6 bao nhieu each lap ra

mot so' gom 3 chiJ so' khac nhau sao cho so' tao thanh la mot so' chan be hdn

• Chieu bie'n thien:

' o G i d i han v6 ciTc, gidi han tai vo cifc va cac difdng l i e m can :

+ Ta c6: l i m y - -oo va l i m y = + Q O , do do di/dng thang x = - 1

-v->(-ir v ^ ( - i r

la tiem can diJng cua do thi ham so' da cho (khi v ("0 va khi

+ Ta c6: l i m y= Ifm v = l , nen dudng thang y = l la tiem can

ngang cua do thi ham so da cho (khi ) • + « va khi v - > - o o )

o Bang bie'n lW?n:

+ Tac

o : 3' ' = -3

X

— CO

_

1 + 0

o Ch

= ~i;x = 0=>y

bai

toan

<:=>y'<0 ,VA-e(-oo;l)

<=

>i

-2<m <2

-m>l -2<m<2 *

CS uI

a c

6 :

BD^ = DC'^ +

BC'^ IDC-.BC cosa

-oBD^ =2BC'^-2BC'^

cosa

O2A:

^

=2(x

'+

/z 2)

(l-cosor)

OOSOf COS

a

2/

r\

si n2

^

Dodo

V = x

^/

j = 2

cos«

CS

u

V

Trong mat phSn

g Oxy, t

a xe

t ca

c vcctd:

a =

1 — X ;

, b = y

;

y)

1 z; -

Ta biet

^ + '\

r '

= J81(x + y+ z)

^ + ril

>, 2.9(

x +

y + z)

ril

>

18.3

^x.y.z.-7=

-80

1

=Vl62-80 = V8

2 ^ ,

=l

3 •

' • 'H \ttL\

Cdch 2:

Ap dung bat dan

Viet phU(m^

trinh canh

AB

Taco: B ^ BH => B(t;-it-ll)

, _ X A + X

B

_t

+ 2

^ _ y A + y B

_ -3 t- i8

Ms CM

t +

2 ^

^r -ai-is'i

+ 7 = 0 ot = - 4=

3y l

3 = 0

f

Viet phU(fnfi trinh canh

BC

Toa do diem

C = ACr\CM tho

a hp :

rx -3 y- 23 = 0

[x + 2

y +

7 =

0

PhiTdng trinhcanh

C:>7A: + 9>

' + 19 = 0

2.

Ta c6:

AB

= (

l ;

-3 ; 4 ) Af

i =

V26

5C = (-

la chan du'dng phan giac Irong cua goc

B

Theo tinh cha't cu

a du

'dng phan giac, t

a c

6 :

MA BA ^

^

= -

^ M

A = — MC

2 '

Do do :

36

Vay trirdng hdp nay c6 4.4 = 16 so

t ruT

c fng hdp cung c6

2+

2 =

6 so ,

2

Theo chiMng trinh ndng

cao :

Cfiu VLb I

1 Gi

a s\jf(A ) c^

t ti

a O

x ta i

M{m ; 0)v

a c^ttiaO

y ta

i M(

0; n ) , (m

CO dan

g : — + — =

-n =

m m-

27 (do

m >

0, n > 0 =>

m >

27)

Ta c6 : :<i'J

el

m

l, m- 27

;

Dat t- ni -2

7, t>

0.

Ta c6 :

-1^+541 +

729 + 1 + — + ^

t

(2

27 27^ — + t^ +

— t t

729 + 27

t + 27t + 73

0

Theo ba

f t d^n

g thiJ

c Cauchy , t

a c6 :

27 27

^ + 27t + 27t>33

729 -^.27t.27t=:243

Dod6: MN^>2

7 + 243 + 73

0 = 1000 =:>MN>10

=>minMN-10,datdufdckhi t = 3

m =

30, n = 10

Vay phiTdng tnn

h cu

a (A )

la :

^ + ^ =

1 ha y

i

e -(^x;y;z) ,

theo gia thie

t su

y r

a