cấu trúc Đề thi Đại mọc môn toán

Li ni u

Lm sao mnh cm thy t tin, vng vng khi bc vo cc k thi? chc cc bn hc sinh rt bn khon v trn tr vi cu hi ny khi k thi tuyn sinh i hc, cao ng ang ti gn. Cc bn hc sinh rt cn mt ti liu tin cy, phong ph n luyn v kim tra kin thc ca mnh tham gia cc k thi mt cch tt nht. Nhm p ng nhu cu , cun sch Cu trc thi i Hc & B tuyn sinh xin trn trng gii thiu ti bn c, nhm gp mt phn nh s chun b kin thc cc bn c t tin khi bc vo k thi. Vi cu trc ca cun sch nh sau: Phn I: L thuyt n tp nhanh, c tc gi bin son theo cu trc thi ca B Gio Dc & o to. Nhm gip cc bn c h thng li cc kin thc v k nng gii ton mt cch n gin v hiu qu nht. Phn II: Gii thiu 15 thi i hc, cao ng mn ton khi A, B, D t nm 2008 ti 2011 ca b gio dc v o to. Phn III: tng thm s a dng v phong ph ca thi theo phng php ra mi ca B Gio Dc & o to, tc gi gii thiu ti bn c 15 thi m do tc gi bin son v cht lc rt k cng nhiu dng ton c gii thiu ti bn c. Phn IV: p n v thang im chi tit. Trong phn gii tc gi chn ra nhiu cch gii khc nhau vi mong mun c s phong ph v a dng v cch gii cho bn, cc bn c thm tham kho thm v rt ra kinh nghim cho mnh. s dng cun sch c tt v hiu qu nht, ngh cc bn c hy t mnh lm ht kh nng sau mi tham kho cch gii v t chm im cho mnh bng cch tham kho thang im m tc gi a ra. Nu mnh c s chun b tt v kin thc th nhn li thi i hc cc nm s khng c g l qu kh khn. Mc d dnh nhiu thi gian v tm huyt cho cun sch, xong khng trnh khi nhng thiu st. Tc gi rt mong nhn c s gp ca bn c ln ti bn sau c hon thin v y hn. Trn trng ! Tc Gi.

f(x)=x^ 3+3x^ 2-4

-3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

x

y

O

f(x)=-x^ 3+3x^ 2-4

-3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

x

y

O

f(x)=-x^ 3+3x^ 2-4x+2

-3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

x

y

O

f(x)=x^ 3+3x^ 2+4x+2

-3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

x

y

O

f(x)=x^ 3-3x^ 2+3x+1

-3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

x

y

O

f(x)=-x^ 3-3x^ 2-3x+1

-3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

x

y

O

PHN I. CU TRC THI TUYN SINH I HC, CAO NG. I. PHN CHUNG CHO TT C TH SINH ( 7,0 im ). Cu I ( 2,0 im ).

1. Kho st s bin thin v v th hm s.

Dng th ca hm bc ba: 3 2y ax bx cx d a 0

Tnh cht. a 0 a 0

Phng trnh y' 0 c hai

nghim phn bit.

Phng trnh y' 0 v

nghim.

Phng trnh y' 0 c

nghim kp.

f(x)=x^ 4-2x^ 2+2

-3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

x

y

O

f(x)=-x^ 4+2x^ 2+2

-3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

x

y

O

f(x)=(x+1)/(2x-1)

f(x)=1/2

x=0.5

-3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

x

y

O

f(x)=(x-1)/(2x-1)

f(x)=1/2

x=0.5

-3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

x

y

O

Dng th hm trng phng: 4 2y ax bx c a 0

Tnh cht. a 0 a 0

Phng trnh y' 0 c ba

nghim phn bit.

Phng trnh y' 0 c mt

nghim.

f(x)=x^ 4+2x^ 2+2

-3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

x

y

O

f(x)=-x^ 4-2x^ 2+2

-3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

x

y

O

Dng th hm nht bin: ax b d

y TXD : D R \cx d c

Tnh cht. ad bc 0 ad bc 0

Phng trnh o hm

ad bcy '

cx d

2

2. Nhng bi ton lin quan.

Dng 1. S tng giao ca hai th.

Cho hm s : 1 2y f x C v y = g x C a. Phng trnh honh giao im ca 1C v 2C l: f x g x * - * c 1 nghim 0x 1C v 2C ct nhau ti im 0 0M x ;f x ( tip xc nhau ti im 0 0M x ;f x ) - * v nghim 1C v 2C khng c im chung. - * c k nghim 0x 1C v 2C ct nhau ti k im. b.S tip xc ca 1C v 2C .

1C v 2C tip xc vi nhau ' '

f x g x

f x g x

c nghim l 0

x . (0

x l honh tip xc ).

Dng 2. Phng trnh tip tuyn.

Cho hm s : y f x C . a. Phng trnh tip tuyn ti.

Phng trnh tip tuyn ca th hm s C ti 0 0M x ;y c dng : '

0 0 0y f x x x y .

' 0f x l h s gc ca tip tuyn. b. Phng trnh tip tuyn i qua.

Phng trnh tip tuyn ca th hm s C i qua 1 1N x ;y c dng : 1 1y k x x y

. k l h s gc ca tip tuyn. l tip tuyn ca C

1 1'f x k x x y

1f x k

c nghim.

Gii h 1 tm k ri thay k vo l tip tuyn cn tm. c. Phng trnh tip tuyn song song.

Tip tuyn ca hm s C song song vi ng thng y k x b nn c '

0f x k

. Gii tm

0x ri thay vo hm s C tm 0y phng trnh tip tuyn cn tm. d. Phng trnh tip tuyn vung gc.

Tip tuyn cu hm s C vung gc vi ng thng dd y k x b nn c '

0 df x .k 1 . Gii

tm 0

x ri thay vo hm s C tm 0y phng trnh tip tuyn cn tm. Dng 3. Tm m hm ng bin, nghch bin.

Hm bc ba: 3 2y ax bx cx d TX:D R ' 2y Ax Bx C

- Hm s ng bin trn D ( hm tng trn tp )

'

'

A 0y 0 x D .

0 0

'y 0 ti mt s hu hn ix

- Hm s nghch bin trn D ( hm nghch trn tp )

'

'

A 0y 0 x D .

0 0

'y 0 ti mt s hu hn ix

Hm nht bin:

'

2

ax b d ad bcy TXD : D R \ , y

cx d c cx d

- Hm s ng bin trn D ( hm tng trn tp ) 'y 0 x D ad bc 0

- Hm s nghch bin trn D ( hm nghch trn tp ) 'y 0 x D ad bc 0

Hm hu t: 2ax bx c e

y TXD : D R \dx e d

2'

2

Ax Bx Cy

dx e

.

- Hm s ng bin trn D ( hm tng trn tp )

'

'

A 0y 0 x D .

0 0

- Hm s nghch bin trn D ( hm nghch trn tp )

'

'

A 0y 0 x D .

0 0

Dng 4. Cc tr ti 1 im.

Cho hm s y f x .

Du hiu 1. hm c cc tr ti '0 0x f x 0 c nghim i du qua

'f x .

Du hiu 2.

hm c cc i ti

'

0

0 ''

0

f x 0x

f x 0

hm c cc tiu ti

'

0

0 ''

0

f x 0x

f x 0

Dng 5. Tm m hm s c im un.

Hm bc ba: 3 2y ax bx cx d TX:D R

Hm s c im un nu phng trnh ''y 0 c 1 nghim.

im 0 0U x ;y l im un ca hm s

''

0

0 0

f x 0.

y f x

Hm trng phng: 4 2y ax bx c TX:D R

Hm s c im un nu phng trnh ''y 0 c 2 nghim phn bit.

Hm s khng c im un nu phng trnh ''y 0 v nghim hay c 1 nghim kp x 0 .

Dng 6. Ta im nguyn.

Cho hm s :

ax by C .

cx d

Bc 1: Thc hin php chia a thc ca C ta c B

y A cx d

Bc 2: C c ta im nguyn th B

cx d

phi nguyn B chia ht cho

cx d ( cx d l c ca B ) t tm c 1 2

x , x ... thay vo C tm c 1 2y , y ... Bc 3: Kt lun cc ta im nguyn 1 1 1 2 2 2M x ;y ,M x ;y ... Dng 7. Bin lun s nghim ca phng trnh.

Cho hm s : y f x C . Da vo C bin lun s nghim ca phng trnh F x;m 0 * . Bc 1: Bin i * sao cho v tri ging nh th C ,v phi t l ng thng d : y g x;m . Bc 2: S nghim ca * chnh l s nghim ca honh giao im ca d v C . Bc 3: Lp bng gi tr da vo th C kt lun. (c th khng cn k bng).

Dng 8. Tm im c nh ca hm s my f x C

D vo phng trnh dng: mmA B ; C qua im c nh x;y mA B tha mn

A 0m

B 0

. Gii h phng trnh trn ta tm c cc im c nh.

Dng 9. Bi ton v khong cch.

Cho 2 im A AA x ;y ) v B BB x ;y Khong cch gia AB l : 2 2

B A B AAB x x y y

Khong cch t mi im 0 0M x ;y n ng thng : Ax By C 0 c tnh theo

cng thc : 0 02 2

Ax Bx cd M,

A B

Trng hp c bit:

0: x a d M, x a

0: y b d M, y b

Tng khong cch 1 2d M, d M, ,tch khong cch 1 2d M, .d M, .Bi ton tng khong cch v tch khong cch thng c p dng cho khong ch ti cc tim cn,chng minh hng s,ngn nht,

Dng 10. Bi ton v im thuc th hm s C cch u hai trc ta .

im M C cch u hai trc ta khi M M M My x y x ta ln lt gii cc phng trnh :

f x x v f x x tm c Mx ri thay vo tm c My .

Dng 11. Tm tp hp im M.

Xc nh ta

x k mM

y h m

kh tham s m gia x v y ta c phng trnh y g x C

Tm gii hn qu tch im ( nu c).Ri kt lun qu tch im M l 1 hm s y g x C .

Dng 12. th hm s cha tr tuyt i.

th hm y f x

Ta v th y f x C .

Gi th:

- Pha trn Ox l: 1C .

- Pha di Ox l: 2C .

V 'y f x C nh sau: - Gi nguyn 1C b phn 2C .

V i xng ca 2C qua trc ox.

th hm y f x

Ta v th y f x C .

Gi th:

- Pha phi Oy l: 1C .

- Pha tri Oy l: 2C .

V 'y f x C nh sau: - Gi nguyn 1C b phn 2C .

- V i xng ca 1C qua trc oy.

th hm

0

g xy

x x

Ta v th

0

g xy f x = C

x x

.

Gi th:

- Pha phi TC l: 1C .

- Pha tri TC l: 2C .

V

'0

g xy C

x x

nh sau:

- Gi nguyn 1C b phn 2C .

- V i xng ca 2C qua trc Ox.

Dng 13. im i xng.

im 0 0M x ;y l tm i xng ca th C : y f x Tn ti hai im 1 1 1 2 2 2M x ;y ,M x ;y

thuc C tha mn 1 2 0 2 0 1

1 2 0 1 0 1 0

x x 2x x 2x x

f x f x 2y f x f 2x x 2y

( cng thc ny gi l cng

thc i trc bng php tnh tin vct ).

Vy im 0 0M x ;y l tm i xng ca th 0 0C : y f x 2y f 2x x .

Dng 14. Tm m 2

m

ax bx cC : y

dx e

tha iu kin:

Hm s mC c cc i, cc tiu nm 2 pha ca trc ox.

Bc 1: Tm m hm c cc i cc tiu 1 .

Bc 2: mC khng ct Ox y 0 v nghim2ax bx c 0 v nghim 0 2

Bc 3: Giao 1 v 2 ta tm c m.

Hm s mC c cc i,cc tiu nm cng pha ca trc Ox.

Bc 1:Tm m hm c cc i cc tiu 1 .

Bc 2: mC ct Ox ti hai im phn bit y 0 c 2 nghim phn bit 2ax bx c 0 c 2 nghim phn bit 0 2

Bc 3: Giao 1 v 2 ta tm c m.

Cu II ( 2,0 im ).

1. Phng trnh lng gic.

H thc c bn.

2 2sin x cos x 1

sin xtan x x k

cos x 2cosx

cotx x ksin x

22

2

2

tanx.cotx 1

11 tan x

cos x1

1 cot xsin x

Cung lin kt.

a. Hai cung i nhau:

cos x cos x

sin x sin x

tan x tan x

cot x cotx

b. Hai cung b nhau:

cos x cos x

sin x sin x

tan x tan x

cot x cotx

c. Hai cung ph nhau:

cos x sin x2

sin x cosx2

tan x cotx2

cot x tan x2

d. Hai cung hn km nhau :

cos x cos x

sin x sin x

tan x tan x

cot x cotx

e. Hai cung hn km nhau2

:

cos x sin x2

sin x cosx2

tan x cotx2

cot x tan x2

H qu:

k

k

cos k x 1 .cos x

sin k x 1 .sin x

tan k x tan x

cos k2 x cos x

sin k2 x sin x

cot k x cotx

Cng thc bin i:

a. Cng thc cng:

sin x y s inx.cos y sin y.cos x

sin x y s inx.cos y sin y.cos x

cos x y cos x.cos y sin x.sin y

cos x y cos x.cos y sin x.sin y

tanx tan ytan x y1 tan x.tan y

cotx.cot y 1cot x y

cotx cot y

cotx.cot y 1cot x y

cotx cot y

b. Cng thc nhn i:

2 2

2 2

sin 2x 2 sin x.cos x

cos2x cos x sin x

2cos x 1 1 2 sin x.

2

2

2 tan xtan 2x

1 tan xcot x 1

cot2x2 cotx

c. Cng thc nhn 3:

3

3

sin 3x 3 sin x 4 sin x

cos3x 4 cos x 3cos x

3

2

3

2

3 tan x tan xtan 3x

1 3 tan xcot x 3 cotx

cot3x3 cot x 1

d. Cng thc h bc:

2

2

2

1 cos2xsin x

21 cos2x

cos x2

x 1 cosxsin

2 2

2

2

2

x 1 cosxcos

2 21 cos2x

tan x1 cos2x

1 cos2xcot x

1 cos2x

e. Cng thc bin i tng thnh tch:

x y x ycos x cos y 2 cos cos

2 2x y x y

cos x cos y 2 sin sin2 2

x y x ysin x sin y 2 sin cos

2 2

x y x ysin x sin y 2 cos sin

2 2sin x y

tan x tan ycos x.cos ysin x y

cotx cot ys inx.sin y

H qu:

s inx cos x 2 sin x4

s inx cos x 2 sin x4

cosx+sin x 2 cos x4

cosx sin x 2 cos x4

f. Cng thc bin i tch thnh tng:

1cos x.cos y cos x y cos x y

21

sin x.sin y cos x y cos x y2

1sin x.cos y sin x y sin x y

2

1cos x.sin y sin x y sin x y

2tanx tan y

tan x.tan ycotx cot ycotx cot y

co tx.cot ytanx tan y

g. Cng thc chia i: tx

t tan2

2

2

2

2tsin x

1 t1 t

cosx1 t

2

2

2

2ttan x

1 t1 t

cotx1 t

H qu: Nu ta t t tan x

2

2

2

2tsin 2x

1 t1 t

cos2x1 t

2

2

2

2ttan 2x

1 t1 t

cot2x1 t

Phng trnh c bn.

a. Phng trnh sin: x k2

s inx sin k z .x k2

c bit:

s inx 1 x k22

s inx 1 x k22

s inx 0 x k .

b. Phng trnh cos: x k2

cosx cos k z .x k2

c bit:

cos x 1 x k2

cosx 1 x k2

cos x 0 x k .2

c. Phng trnh tan:

tanx tan . x k x k k z .2

c bit :

tan x 1 x k4

tanx 1 x k4

tanx 0 x k .

d. Phng trnh cotan: cotx cot . x k x k k z . c bit :

co tx 1 x k4

cotx 1 x k4

cotx 0 x k .2

Phng trnh bc n theo mt hm s lng gic.

Cch gii: t t s inx (hoc cosx, tanx,cotx ) ta c phng trnh:

n n 1 0n n 1 0

a t a t ... a t 0

(nu t s inx hoc t cosx th iu kin ca t : 1 t 1 )

Phng trnh bc nht theo sinx v cosx.

a sinx bcosx c. a.b 0 iu kin c nghim : 2 2 2a b c

Cch gii: Chia 2 v phng trnh cho 2 2a b v sau a v phng trnh lng gic c bn.

Phng trnh ng cp bc hai i vi sinx v cosx.

2 2a sin x bsin x.cox c cos x d. Cch gii:

Xt cosx 0 x k2

c phi l nghim khng ?

Xt cosx 0 Chia 2 v cho 2cos x v t t tanx .

Phng trnh dng.

a. s inx cosx b.s inx.cosx c.

Cch gii : t t s inx cos x 2 sin x ; DK : 2 t 24

2t 1

s inx.cos x2

hoc

21 ts inx.cos x

2

v gii phng trnh bc 2 theo t.

2. Phng trnh, h phng trnh v bt phng trnh.

Phng trnh Bt phng trnh cha tr tuyt i.

*a khi a 0

aa khi a 0

* a a a R.

*a b

a ba b

* a b

a b b 0a b

*a a

a R.a a

*b 0

a b b a b

*a b

a b a b

* 2

2a a a R.

* a b a b .ng thc c a.b 0. * a b a b .ng thc c a.b 0.

Phng trnh Bt phng trnh v t.

* Phng trnh: 2

g x 0f x g x

f x g x

* Bt phng trnh dng: 2

g x 0

f x g x f x 0

f x g x

* Bt phng trnh dng: f x g x TH 1 :

f x 0

g x 0

TH 2 : 2

g x 0

f x g x

H phng trnh. a. H phng trnh bc nht hai n.

' ' '

ax by c

a x by c

Trong ' ' 'a,b, c, a , b , c l cc s thc khng ng thi bng khng.

Theo nh thc Crame : ' ' ' ' ' 'x ya b c b a c

D ; D = , Da b c b a c

.

* Nu D 0 th h c nghim duy nht : yxDD

x ;yD D

* Nu x y

D D D 0 th h v s nghim : x R

c axy

b

* Nu x

y

D 0

D 0

D 0

th h cho v nghim.

b. H phng trnh i xng loi I.

Cho h phng trnh

f x;y a

Ig x;y b

Cch Gii: t 2S x y , P xy , DK: S 4P 0

F S;P 0I

G S;P 0

gii h tm c S,P . Khi x,y l nghim ca phng trnh: 2X SX P 0.

tm c nghim x,y xem xt iu kin v kt lun nghim. c. H phng trnh i xng loi II.

Cho h phng trnh:

f x;y a

IIf y;x b

Cch Gii: Tr hai phng trnh ca h cho nhau ta c :

x y

f x;y f y;x 0 x y g x;y 0g x;y 0

xt xem phng trnh c nghim

khng ri thay vo 1 trong 2 phng trnh ca II, kt lun nghim nu c. d. H phng trnh ng cp.

Cho h

f x;y a *

f y;x bTrong f x,y v g x,y ng cp bc k gi l h ng cp.

Lu : H * gi l ng cp bc k nu cc phng trnh f x,y v g x,y phi l ng cp bc

k. f x,y v g x,y ng cp bc k khi: k kf x,y m f mx,my v g x,y m g mx,my .

Cch gii: * Xt x 0 thay vo h c phi l nghim hay khng.

* Vi x 0 t y tx thay vo h ta c

k

k

f x; tx a x f 1; t a 1*

g x; tx b x g 1; t b 2

Ta th hin

1

2 th c

f 1; t a

g 1; t b v gii phng trnh ny ta c nghim t ri thay vo tm

c nghim x, y .

Cu III ( 1,0 im ). Nguyn hm tch phn.

Cng thc nguyn hm cn nh :

1xx dx C

1

1ax b

ax b dx Ca 1

Cc phng php tnh tch phn. a. Phng php tch phn tng phn.

b

a

I f x .g x dx. t

'du f x dxu f x

dv g x dx v g x dx G x

b b

bb '

a aa a

I u.v vdu f x .G x G x .f x dx

Dng 1: b

a

I f x .ln g x dx t

u ln g x

dv f x

Dng 2: b

a

I f x sin g x dx t

u f x

dv sin g x dx

b

a

I f x cos g x dx t

u f x

dv cos g x dx

1dx ln x C

x

1 1dx ln ax b C

ax b a

xx aa dx C

ln a

kx bkx b aa dx C

k.ln a

x xe dx e C

ax b ax b1e dx e C

a

sinxdx cosx C

1sin ax b dx cos ax b Ca

cosxdx sinx C

1cos ax b dx sin ax b Ca

2

1dx tanx C

cos x

2

1 1dx tan ax b C

acos ax b

2

1dx co tx C

sin x

2

1 1dx co t ax b C

asin ax b

tan xdx ln cosx C

1tan ax b dx ln cos ax b Ca

cotxdx ln sin x C

1cot ax b dx ln sin ax b Ca

adx ax C

'f xdx ln f x C

f x

1dx 2 x C

x

2 2

1 1 x adx ln C

2a x ax a

Dng 3: b

g x

a

I f x .e dx t

g x

u f x

dv e dx

Dng 4: b

g x

a

I sin f x .e dx t

g x

u sin f x

dv e dx

b

g x

a

I cos f x .e dx t

g x

u cos f x

dv e dx

Ring dng ny ta nn tnh tch phn 2 ln nh vy c tr li nh ri I . b. Phng php i bin s.

Cc dng Cch t 2

1

b

2 2

b

I a x dx hoc 2

1

b

2 2b

dxI

a x

t x a sin t hoc x acos t

2

1

b

2 2

b

I x a dx hoc 2

1

b

2 2b

dxI

x a

t

ax

sint hoc

ax

cost ;

2

1

b

2 2

b

I a x dx t x a tan t hoc x acott

2

1

b

b

a xI dx

a x

hoc

2

1

b

b

a xI dx

a x

t x acos2t

2

1

b

b

I x a b x dx t 2x a b a sin t

2

1

b

2 2b

1I dx

a x

t x a tan t

ng dng tch phn.

a. Din tch gii hn hnh phng.

Dng 1. Hnh phng gii hn bi : Hm s y f x C ,trc honh y 0 v hai ng thng

x a,x b .

b

a

S f x dx c th b du tr tuyt i da vo th.

Dng 2. Hnh phng gii hn bi : Hm s 1 2y f x C ;y g x C v hai ng thng x a,x b .

b

a

S f x g x dx c th b du tr tuyt i bng cch da vo th.

Dng 3. Hnh phng gii hn bi : Hm s 1 2y f x C ;y g x C

Gii phng trnh honh giao im ca 1C v 2 1 2 3C f x g x x ,x ,x ...

cb

a

A

B CH M

3

1

x

x

S f x g x dx c th b du tr tuyt i bng cch :

32

1 2

xx

x x

S f x g x dx f x g x dx... hoc da vo th.

b. Th tch vt trn xoay.

Vt th trn xoay gii hn bi y f x C ,y 0 ; x a,x b xoay quanh b

2

a

Ox V f x dx.

Vt th trn xoay gii hn bi x f y C ,x 0 ; y a,y b xoay quanh b

2

a

Oy V f y dy.

Cu IV ( 1,0 im ). Hnh hc khng gian.

Kin Thc C Bn V H Thc Lng.

a. H thc lng trong tam gic vung : cho ABC vung A ta c : nh l Pitago : 2 2 2BC AB AC

2 2BA BH.BC ; CA CH.CB

AB. AC BC. AH .Vi AH l ng cao.

2 2 2

1 1 1

AH AB AC

BC 2AM .Vi AM l ng trung tuyn ca cnh BC

b c b csin B , cosB , tan B , cot B

a a c b

b bb a.sinB a.cosC, c a.sinC a.cosB, a , b c.tanB c.cotC

sin B cosC

b. H thc lng trong tam gic thng: * nh l hm s Csin: 2 2 2a b c 2bc.cosA

* nh l hm s Sin: a b c

2Rsin A sin B sinC

c. Cc cng thc tnh din tch. * Cng thc tnh din tch tam gic:

a1 1 a.b.c

S a.h a.bsinC p.r p.(p a)(p b)(p c)2 2 4R

a b c

p2

l na chu vi tam gic l

c bit:

* ABC vung A : 1

S AB.AC2

* ABC u cnh a: 2a 3

S4

* Din tch hnh vung: S = cnh x cnh * Din tch hnh ch nht: S = di x rng

* Din tch hnh thoi: 1

S2

(cho di x cho ngn)

* Din tch hnh thang: 1

S2

(y ln + y nh) x chiu cao

* Din tch hnh bnh hnh: S = y x chiu cao

* Din tch hnh trn: 2S .R

Kin Thc C Bn V Hnh Hc Khng Gian. A. QUAN H SONG SONG

1. NG THNG V MT PHNG SONG SONG I.nh ngha:

ng thng v mt phng gi l song song vi nhau nu chng khng c im no chung.

a / / P a P

a

(P)

II.Cc nh l: L1: Nu ng thng d

khng nm trn mp P v

song song vi ng thng a nm trn

mp P th ng thng d

song song vi mp P

d P

d / /a d / / P

a P

d

a

(P)

L2: Nu ng thng a

song song vi mp P th

mi mp Q cha a m ct

mp P th ct theo giao

tuyn song song vi a.

a / / P

a Q d / /a

P Q d

d

a(Q)

(P)

L3: Nu hai mt phng ct nhau cng song song vi mt ng thng th giao tuyn ca chng song song vi ng thng .

P Q d

P / /a d / /a

Q / /a

a

d

QP

2.HAI MT PHNG SONG SONG I.nh ngha: Hai mt phng c gi l song song vi nhau nu chng khng c im no chung.

P / / Q P Q Q

P

II.Cc nh l:

L1: Nu mp P cha hai ng thng a, b ct nhau v cng song song

vi mp Q th

P v Q song song

vi nhau.

a,b P

a b I P / / Q

a / / Q ,b / / Q

Ib

a

Q

P

L2: Nu mt ng thng nm mt trong hai mt phng song song th song song vi mt phng kia.

P / / Q

a / / Q

a P

a

Q

P

L3: Nu hai mp P v

mp Q song song th

mi mt phng mp R

ct mp P th phi ct

mp Q v cc giao tuyn ca chng song song.

P / / Q

R P a a / /b

R Q b

b

a

R

Q

P

B.QUAN H VUNG GC

1.NG THNG VUNG GC VI MT PHNG I.nh ngha: Mt ng thng c gi l vung gc vi mt mt phng nu n vung gc vi mi ng thng nm trn mt phng .

a P a c, c P P

c

a

II. Cc nh l: L1: Nu ng thng d vung gc vi hai ng thng ct nhau a v b cng

nm trong mp P th ng thng d vung gc

vi mp P .

d a,d b

a,b P d P

a,b cat nhau

d

ab

P

L2: (Ba ng vung gc) Cho ng thng a khng vung gc vi

mp P v ng thng b

nm trong mp P . Khi , iu kin cn v b vung gc vi a l b vung gc vi hnh chiu a ca a

a P ,b P

b a b a'

a'

a

bP

trn mp P .

2.HAI MT PHNG VUNG GC I.nh ngha: Hai mt phng c gi l vung gc vi nhau nu gc gia chng bng 900.

II. Cc nh l: L1: Nu mt mt phng cha mt ng thng vung gc vi mt mt phng khc th hai mt phng vung gc vi nhau.

a P

Q P

a Q

Q

P

a

L2: Nu hai mp P v

mp Q vung gc vi nhau th bt c ng thng a no nm trong (P), vung gc vi giao tuyn ca (P) v (Q) u vung gc vi mt phng (Q).

P Q

P Q d a Q

a P ,a d

d Q

P

a

L3: Nu hai mf P

v mf Q vung gc vi

nhau v A l mt im trong (P) th ng thng a i qua im A v

vung gc vi mf Q s

nm trong mf P .

P Q

A P

a P

A a

a Q

A

Q

P

a

L4: Nu hai mt phng ct nhau v cng vung gc vi mt phng th ba th giao tuyn ca chng vung gc vi mt phng th ba.

P Q a

P R a R

Q R

a

R

QP

3.KHONG CCH

1. Khong cch t 1 im ti 1 ng thng, n 1 mt phng: Khong cch t im M n ng thng

a (hoc n mp P ) l khong cch gia hai im M v H, trong H l hnh chiu ca im M trn ng thng a (hoc

trn mp P )

d O; a OH; d O; P OH

aH

O

H

O

P

2. Khong cch gia ng thng v mt phng song song: Khong cch gia ng thng a v

mp P song song vi a l khong cch t

mt im no ca a n mp P .

d a; P OH

a

H

O

P

3. Khong cch gia hai mt phng song song: L khong cch t mt im bt k trn mt phng ny n mt phng kia.

d P ; Q OH H

O

Q

P

4. Khong cch gia hai ng thng cho nhau: L di on vung gc chung ca hai ng thng .

d a;b AB B

A

b

a

4.GC

1. Gc gia hai ng thng a v b L gc gia hai ng thng ' 'a v b cng

i qua mt im v ln lt cng phng vi a v b.

b'b

a'a

2. Gc gia ng thng a khng vung gc vi mt phng (P) L gc gia a v hnh chiu 'a ca n trn

mp P .

c bit: Nu a vung gc vi mp P th

ta ni rng gc gia ng thng a v

mp P l 900.

Pa'

a

3. Gc gia hai mt phng L gc gia hai ng thng ln lt vung gc vi hai mt phng . Hoc l gc gia 2 ng thng nm trong 2 mt phng cng vung gc vi giao tuyn ti 1 im

ba

QP

P Q

ab

4. Din tch hnh chiu: Gi S l din tch ca a gic H trong mp P v 'S l

din tch hnh chiu 'H ca (H) trn

'mp P th 'S Scos ( trong l gc gia hai mp P

'v mp P ).

C

B

A

S

B

h

a

b

c

a

a

a

B

h

C'

B'

A'

C

B

A

S

Kin thc c bn v hnh th tch.

1. TH TCH KHI LNG TR: V Bh

vi

B: Dien tch ay

h : Chieu cao

a.Th tch khi hp ch nht: V a.b.c vi a,b,c l ba kch thc

b.Th tch khi lp phng:

3V a vi a l di cnh

2. TH TCH KHI CHP:

1

V Bh3

vi

B: Dien tch ay

h : Chieu cao

3.T S TH TCH T DIN:

Cho khi t din ' ' 'SABC v A ,B ,C l cc im ty ln lt thuc SA,SB,SC ta c:

SABC

SA'B'C'

V SA SB SC

V SA' SB' SC'

* M SC , ta c:

S.ABM

S.ABC

V SA.SB.SM SM

V SA.SB.SC SC

4. TH TCH KHI CHP CT:

hV B B' BB'3

vi

B, B' : Dien tch hai ay

h : Chieu cao

BA

C

A'B'

C'

A

C

B

S

M

5.TH TCH-DIN TCH HNH TR:

xq2

S 2 Rh

V R h

R : Ban knh ay

h : Chieu cao

I o h J R O

6.TH TCH-DIN TCH HNH NN

xqS Rl .

21V R h.3

R : Ban knh ay

h : Chieu cao

l: ng sinh

l h

7.TH TCH-DIN TCH HNH NN CT:

2 2xqS R r l ,V h R r Rr

R,r : Ban knh 2 ay

h : Chieu cao

l: ng sinh

8.TH TCH-DIN TCH HNH CU:

= 2S 4 R 34

V R3

R: bn knh mt cu

Ch :

1. ng cho ca hnh vung cnh a l d a 2 ,

ng cho ca hnh lp phng cnh a l d a 3 ,

ng cho ca hnh hp ch nht c 3 kch thc a, b, c l 2 2 2d a b c .

2. ng cao ca tam gic u cnh a l a 3

h2

3. Hnh chp u l hnh chp c y l a gic u v cc cnh bn u bng nhau ( hoc c y l a gic u, hnh chiu ca nh trng vi tm ca y).

4. Lng tr u l lng tr ng c y l a gic u.

Cu V ( 1,0 im ). Bt ng thc.

Bt ng thc C-si:

a,b 0 ta c a b

ab2

du " " xy ra khi a b .

a,b R ta c 2

a bab

2


a,b,c 0 ta c 3

3a b c a b cabc abc3 3


R

R

r

h

R

l

O

O

O . R

n s

a,b,c ... 0 ta c n s nn s

a b c ...

abc ...n


Bt ng thc Bunnhiacpski :

* Vi a,b,c,x,y,z l nhng s bt k th ta lun c:

2 2 2 2 2ax by a b x y du " " xy ra khi a b

x y .

2 2 2 2 2 2 2ax by cz a b c x y z du " " xy ra khi a b c

x y z .

* Vi a,b,c R v x,y,z 0 ta lun c:

22 2 2 a b ca b c

x y z x y z

II.PHN RING ( 3,0 im ). ( phn ny v cu trc thi ca c bn v nng cao khng my g khc nhau, y tc gi s lt chung ca 2 phn vo 1) Cu VI.a(b) ( 2,0 im ).

1. Hnh ta phng.

1. TTAA IIMM VV VVEECCTT

1. Ta im: Trong khng gian vi h ta Oxy

Cho 2 im A v B : 2 im A AA x ;y ) v B BB x ;y

Vct : B A B AAB x x ;y y .

Khong cch gia AB l : 2 2

B A B AAB x x y y

Gi I l trung im ca AB : A B A Bx x y y

I ;2 2

2. Ta vc t: Trong mp ta Oxy cho : 1 2 1 2a a ;a ;b b ;b

Nu 1 2

1 2

a aa b

b b

v 1 2 1 2a b a a ;b b . 1 2 1 2ka k a ;a ka ;ka .

Tch v hng ca hai vct: 1 1 2 2a.b (a b a b )

Nu a vung gc vi b 1 1 2 2

a.b 0 a b a b 0.

di ca vect: 2 21 2

a a a , 2 21 2

b b b

Gc gia 2 vect : 1 1 2 22 2 2 2

1 2 1 2

a b a ba.bcos a.b .

a . b a a . b b

2 PHNG TRNH NG THNG:

1.Phng trnh tham s ca ng thng 00

x x at: t R

y y bt

vi 0 0M x ;y v u (a;b) l vect ch phng (VTCP)

2.Phng trnh chnh tc ca ng thng 0 0x x y y

:a b

(K: a;b 0 )

vi 0 0M (x ;y ) v u (a;b) l vect ch phng (VTCP)

3. Phng trnh tng qut ca ng thng 0 0: A x x B y y 0.

Hay Ax By C 0 (vi 0 0

C Ax By v 2 2A B 0 )trong 0 0M (x ;y ) v n A;B l vect php tuyn (VTPT) ** Ch :

* T VTCP : u a;b c th chuyn v VTPT : u a;b n b; a b;a . Hoc ngc li T

VTPT : n A;B c th chuyn v VTCP : n A;B u B; A B;A . * Mun vit c phng trnh tng qut ca ng thng cn bit c vct php tuyn v im i qua. * Mun vit c phng trnh chnh tc hay tham s ca ng thng cn bit c vct ch phng v im i qua.

* 1 2

1 2

1 2

n n song song

u u

* 1 2

21

1 2

n u vung gc

u n

4.Cc trng hp c bit:

* Phng trnh ng thng ct hai trc ta ti hai im A a;0 v B 0;b l:

x y1

a b ( phng trnh on chn ).

* Phng trnh ng thng i qua im 0 0M x ;y ) c h s gc k c dng : 0 0y y k x x

vi h s gc ca hai im B AABB A

y yAB: k

x x

.

5. Khong cch t mi im 0 0M x ;y n ng thng : Ax By C 0 c tnh theo cng

thc : 0 02 2

Ax Bx cd M,

A B

Ch : Cho im 1 1 2 2M x ;y , N x ;y

* M,N nm cng pha vi ng thng 1 1 2 2Ax By C Ax By C 0

* M,N nm khc pha vi ng thng 1 1 2 2Ax By C Ax By C 0

6. Gc gia hai ng thng 1 v 2 c vect php tuyn l 1 1 1n (a ;b ) , 2 2 2n (a ;b ) l

( 1 2n ,n ) ta c :

1 2 1 2 1 2

2 2 2 21 2 1 1 2 2

n .n a a b bcos .

n . n a b . a b

7. V tr tng i ca hai ng thng : 1 1 1 1: a x b y c 0 v 2 2 2 2: a x b y c 0.

1 ct 21 1

2 2

a b

a b

1 1 1

1 2

2 2 2

a b c/ /

a b c

1 1 11 22 2 2

a b c

a b c

8. Phng trnh ng phn gic ca hai ng thng:

1 1 1 1: a x b y c 0 v 2 2 2 2: a x b y c 0

1 1 1 2 2 2

2 2 2 21 1 2 2

a x b y c a x b y c

a b a b (tm c

2 ng phn gic)

3.PHNG TRNH NG TR N.

Phng trnh ng trn tm I a;b bn knh R c dng : = 12 2 2x a y b R

hay 2 2 2x y 2ax 2by c 0 vi 2 2 2R a b c Vi iu kin 2 2a b c 0 th phng trnh: 2 2x y 2ax 2by c 0 l

phng trnh ng trn tm I a;b bn knh R. ng trn C tm I I a;b bn knh R tip xc vi ng thng

: Ax By C 0

khi v ch khi : 2 2

A.a B.b Cd(I ; ) R.

A B

iu kin 2 ng trn 1 2C , C c tm v bn knh ln lt l 1 2 1 2I , I ,R ,R .

1 2 1 2 1 2 1 2R R I I R R C C .

1 2 1 2 1 2R R I I C , C lng nhau.

1 2 1 2 1 2R R I I C , C khng ct.

1 2 1 2 1 2R R I I C , C tip xc ngoi.

1 2 1 2 1 2R R I I C , C tip xc trong.

4.CC NG CONIC.

1. Elipse (E): 2 2

2 2 2

1 22 2

x y1 a b 0 E M / MF MF 2a , c a b .

a b

Trc ln 1 2A A 2a .nh 1 2A a;0 , A a;0 .Trc nh 1 2B B 2b . nh 1 2B 0; b ,B 0;b .

Tiu c 1 2FF 2c. Tiu im 1 2F c;0 ,F c;0 . Tm sai:c

e 1.a

Bn knh qua tiu: 1 1 2 2r MF a ex ; r MF a ex. ng chun: : a ex 0.

Phng trnh cnh hnh ch nht c s: x a ; y b .iu kin tip xc: 2 2 2 2 2a A b B C .

2. Hyperbola (H): 2 2

2 2 2

1 22 2

x y1 a b 0 E M / MF MF 2a , c a b .

a b

Trc thc 1 2A A 2a .nh 1 2A a;0 ,A a;0 .Trc o 1 2B B 2b .

Tiu c 1 2FF 2c. Tiu im 1 2F c;0 ,F c;0 .Tm sai:c

e 1.a

Nhnh phi: 1 1

2 2

F M r a ex

F M r a ex

. Nhnh tri:

1 1

2 2

F M r a ex

F M r a ex

ng tim cn bx ay 0 ng chun: : a ex 0.

Phng trnh cnh hnh ch nht c s: x a ; y b . iu kin tip xc: 2 2 2 2 2a A b B C .

Tip tuyn ti 0 00 0 0 2 2x x y x

M x ,y H : 1a b

.

3. Parabola (P): 2 P P Py 2Px , P = M / MF d F, F ;0 ;FM x ; : x 02 2 2

Tip tuyn ti 0 0 0 0 0M x ;y : y y P x x .iu kin tip xc:2PB 2AC .

2. Hnh hc ta trong khng gian.

1. TTAA IIMM VV VVEECCTT

I. Ta im: Trong khng gian vi h ta M M M M M MOxyz : M x ;y ;z OM x i y j z k

1.Cho A A AA x ;y ;z v B B BB x ;y ;z ta c:

Vct B A B A B AAB x x ;y y ;z z

di 2 2 2

B A B A B AAB x x y y z z

2. Nu M chia on AB theo t s k MA kMB th ta c :

A B A B A BM M Mx kx y ky z kz

x ; y ; z k 11 k 1 k 1 k

c bit khi M l trung im ca AB k 1 th ta c:

A BM

A BM

A BM

x xx

2

y yy

2

z zz

2

II. Ta ca vct: Trong khng gian vi h ta Oxyz .

1. 1 2 3 1 2 3a a ;a ;a a a i a j a k

2. Cho 1 2 3a a ;a ;a v 1 2 3b b ;b ;b ta c :

*

1 1

2 2

3 3

a b

a b a b

a b

v 1 1 2 2 3 3a b a b ;a b ;a b

* 1 2 3k.a ka ;ka ;ka v 1 1 2 2 3 3a.b a . b cos a;b a b a b a b .

* di 2 2 21 2 3a a a a

III. Tch c hng ca hai vect v ng dng:

1.Nu 1 2 3a a ;a ;a v 1 2 3b b ;b ;b th 2 3 3 1 1 22 3 3 1 1 2

a a a a a aa,b ; ;

b b b b b b

2.Vect tch c hng c a,b vung gc vi hai vect a v b .

3. a,b a b sin a,b .

4.Din tch tam gic ABC1

S [AB,AC]2

.

5.Th tch hnh hp ' ' ' 'ABCD.A BC DV [AB,AC].AA' .

6.Th tch t din A.BCD1

V [AB,AC].AD6

.

IV. iu kin khc:

1. a v b cng phng

1 1

2 2

3 3

a kb

a,b 0 k R : a kb a kb

a kb

2. a v b vung gc 1 1 2 2 3 3a.b 0 a .b a .b a .b 0 (tch v hng)

3.Ba vect a, b, c ng phng a,b .c 0 ( tch hn tp ca chng bng 0).

4. A,B,C,D l bn nh ca t din AB, AC, AD khng ng phng.

5.Cho hai vect khng cng phng a v b vect c ng phng vi a v b k,l R sao

cho c ka lb

6.G l trng tm ca tam gic

A B CG

A B CG

A B CG

x x xx

3

y y yABC y

3

z z zz

3

7.G l trng tm ca t din ABCD GA GB GC GD 0 .

2. MT PHNG

I. Phng trnh mt phng.

1.Trong khng gian 0xyz phng trnh dng : Ax By Cz D 0 (vi 2 2 2A B C 0 ) l

phng trnh tng qut ca mt phng, trong n A;B;C l mt vect php tuyn ca n.

2.Mt phng P i qua im 0 0 0 0M x ;y ;z v nhn vect n A;B;C lm vect php tuyn c dng : .0 0 0A x x B y y C z z 0

3.Mt phng P i qua 0 0 0 0M x ;y ;z v nhn 1 1 1a (a ;b ;c ) v 2 2 2b (a ;b ;c ) lm cp vect

ch phng th mt phng P c vect php tuyn: 1 1 1 1 1 1

2 2 1 2 2 2

b c c a a bn a,b ; ;

b c c a a b

.

4.Mt phng P ct trc Ox ti A a;0;0 , Oy ti B 0;b;0 , Oz ti C 0;0;c c dng:

x y z

1 , a,b,c 0 .a b c Gi l phng trnh mt chn cc trc ta .

II. V tr tng i ca hai mt phng.

1.Cho hai mt phng P : Ax By Cz D 0 v ' ' ' 'Q : Ax By Cz D 0

' ' '

A B CP Q

A B C .

' ' ' 'A B C D

P / / QA B C D

.

' ' ' 'A B C D

P QA B C D

.

2.Cho hai mt phng ct nhau P : Ax By Cz D 0 v ' ' ' 'Q : Ax By Cz D 0 . Phng trnh chm mt phng xc nh bi P v Q l :

' ' ' 'm Ax By Cz D n Ax By Cz D 0. ( Trong 2 2m n 0 ) III. Khong cch t mt im n mt phng:

Khong cch t 0 0 0 0M x ;y ;z n mt phng :Ax By Cz D 0 cho bi cng thc :

0 0 002 2 2

Ax By Cz Dd M ,

A B C

IV. Gc ga hai mt phng.

Gi l gc gia hai mt phng P : Ax By Cz D 0 v ' ' ' 'Q : Ax By Cz D 0 .

Ta c: P Q 0P Q 2 2 2 2 2 2P Q

n .n A.A' B.B' C.C'cos cos n ,n 0 90

n . n A B C . A' B' C'

0

P Q90 n n hai mt phng vung gc nhau.

* Trong phng trnh mt phng khng c bin x th mt phng song songOx , khng c bin y th song song Oy, khng c bin z th song song Oz.

3. NG THNG

I. Phng trnh ng thng:

1.Phng trnh tng qut ca ng thng:

' 'Ax By Cz D 0

: A : B: C A : B : CA'x B'y C'z D' 0

l giao tuyn ca hai mt phng. Ta c th

chuyn v phng trnh tham s nh sau: 1 2u n ,n a;b;c v qua im 0 0 0M x ;y ;z nn

c dng sau: 0

0

0

x x at

: y y bt t R .

z z ct

2.Phng trnh tham s ca ng thng: 0

0

0

x x at

y y bt t R

z z ct

Trong 0 0 0 0M x ;y ;z l im thuc ng thng v u a;b;c l vect ch phng ca ng thng.

3. Phng trnh chnh tc ca ung thng: 0 0 0x x y y z z

a,b,c 0 .a b c

Trong 0 0 0 0M x ;y ;z im thuc ng thng v u a;b;c l vect ch phng ca ng thng.

II. V Tr tng i ca cc ng thng v cc mt phng:

1.V tr tng i ca hai ng thng:

Cho hai ng thng i qua M c VTCP u v ' i qua 'M c VTCP u ' .

cho ' u,u ' .MM' 0

ct ' u,u ' .MM' 0 vi u,u ' 0

''

[u,u ']=0/ /

u,MM 0

''

[u,u ']=0

u,MM 0

2.V tr tng i ca ng thng v mt phng:

Cho ng thng i qua 0 0 0 0M x ;y ;z c VTCP u a;b;c v mt phng

: Ax By Cz D 0 c VTPT n (A;B;C) .

u.n 0

u.n 0/ /mp

M

nm trn mp

u.n 0mp

M

III. Khong cch:

1.Khong cch t M n ung thng i qua M0 c VTCP

0

M M,u

u a;b;c d M, .

u

2.Khong cch gia hai ng cho nhau: 1 i qua 1 1 1 1M x ;y ;z c VTCP 1 1 1 1u a ;b ;c

2 i qua 2 2 2 2M x ;y ;z c VTCP 2 2 2 2u a ;b ;c 1 2 1 2

1 2

1 2

[u ,u ].M M

d , .

[u ,u ]

IV. Gc:

1.Gc gia hai ng thng :

1 i qua 1 1 1 1M x ;y ;z c VTCP 1 1 1 1u a ;b ;c

2 i qua 2 2 2 2M x ;y ;z c VTCP 2 2 2 2u a ;b ;c

1 2 1 2 1 2 1 21 22 2 2 2 2 2

1 2 1 1 1 2 2 2

u .u a .a b .b c .ccos cos u ,u

u . u a b c . a b c

2. Gc gia ng thng v mt phng :

i qua 0M c VTCP u a;b;c , mp c VTPT n A;B;C .

Gi l gc hp bi v mp 2 2 2 2 2 2

Aa Bb Ccsin cos u,n

A B C . a b c

4. MT CU

I. Phng trnh mt cu:

Phng trnh mt cu tm I a;b;c bn knh R l 2 22 2S :(x a) y b z c =R

Phng trnh 2 2 2 x y z 2Ax 2By 2Cz D 0 vi 2 2 2A B C D 0 l phng

trnh mt cu tm I A;B;C , bn knh 2 2 2R A B C D . II. V tr tng i ca mt cu v mt phng:

Cho mt cu 2 22 2S : (x a) y b z c =R tm I a;b;c bn knh R v mt phng:

P : Ax By Cz D 0.

* Nu d I, P R th mt phng P v mt cu S khng c im chung.

* Nu d I, P R th mt phng P v mt cu S tip xc nhau ti ta tip im H. Ta c th tm ta tip m bng cch vit phng trnh ng thng i qua tm I ca mt cu v vung gc vi mp

P :

0

0

0

x x at

: y y bt H P

z z ct

.

* Nu d I, P R th mt phng P v mt cu S ct nhau theo giao tuyn l ng trn c phng trnh :

2 2 2 2x a y b z c R

Ax By Cz D 0

Bn knh ng trn 22r R d I, P .

Tm H ca ng trn l hnh chiu ca tm I mt cu S ln mt phng P .

III. V tr tng i ca mt cu v ng thng:

Cho mt cu 2 22 2S :(x a) y b z c =R tm I a;b;c bn knh R v ng thng

0

0

0

x x at

: y y bt t R

z z ct

.

* Nu d I, R th ng thng v mt cu S khng c im chung.

* Nu d I, P R th ng thng v mt cu S tip xc nhau ti ta tip im H. Ta c th tm ta tip m bng cch vit phng trnh mt phng i qua tm I ca mt cu v

vung gc vi ng thng : P : Ax By Cz D 0 H P .

* Nu d I, P R th ng thng v mt cu S ct nhau ti hai im phn bit, v ta 2 im im l A,B chnh l nghim ca h :

2 2 2 2

0

0

0

x a y b z c R

x x at

y y bt

z z ct

Cu VII.a(b) ( 1,0 im ).

Phng trnh bt phng trnh m v logarit. I. Cng thc s m v logarit cn nh.

0

a 1; a 0 o nguyen

ha

alog 1 0 o nguyen ha

1

a a

alog a 1

1aa

alog a

a . a a

a1

log a

aa

a

a alog b .log b; a,b 0,a 1

a . b a.b

aa1

log b .log b

a a; b 0

bb

aa

log a .log b

a a

a a alog b log c log b.c

a

a b log b

a a a

blog b log c log

c

.a a

a

b

1log b

log a

a a

ca

c

log blog b

log a

a.b a. b; a, b 0

a

log b b a

a a ; a 0;b 0b b

a alog b log b

e; ln a log a

a a

aloga 10

; lg a log a log a

.a a

a a

log b log c b c

a a ; a 1 a alog b log c b c; a 1

a a ; 0 a 1 a alog b log c b c; 0 a 1 II. Cc phng trnh - Bt phng trnh m v logarit thng gp. 1. Phng trnh Bt phng trnh m. a. a v cng c s.

* f x g xa a f x g x ri gii phng trnh tm nghim x.

* f x aa b f x log b x

* f x g xa a f x g x ; a 1

* f x g xa a f x g x ; 0 a 1 b. t n ph.

Dng 1: 2f x f xm.a n.a p 0 * t f xt a ( k: t 0 )

2* mt nt p 0 gii phng trnh tm t ri thay vo tm x. ( Bt phng trnh lm tng t )

Dng 2 : f x f xm.a n.b p 0 ** trong a.b 1 t f xt a ( k: t 0 ) f x1

bt

1

** mt n p 0t

gii phng trnh tm t ri thay vo tm x. (Bt phng trnh lm tng t )

Dng 3:

f x2f x 2f xm.a n. a.b p.b 0 * t f xt a ( k: t 0 )

2* mt nt p 0 gii phng trnh tm t ri thay vo tm x. (Bt phng trnh lm tng t ) 2. Phng trnh Bt phng trnh logarit.

alog f x c ngha f x 0

0 a 1

*

a a

f x 0,g x 0log f x log g x

f x g x

* balog f x b f x a

* a alog f x log g x *

Nu a 1 th

f x g x*

g x 0

Nu 0 a 1 th

f x g x*

f x 0

S phc. 1. nh ngha s phc. S phc l 1 biu din di dng z a bi ,ab R .Trong a l phn thc,b l phn o.

V ta qui c nh sau: 2 4m 4m 1 4m 2 3m 3i 1 ; i 1 ; i i ; i 1 ; i i m N . 2. S phc lin hp v mun ca n. Cho z a bi z a bi gi l s phc lin hp

mun s phc 2 2z z a b

3. Cc php ton trn tp hp s phc.

Cho hai s phc c dng 1 1 1 2 2 2

z a b i ; z a b i

Hai s phc bng nhau 1 21 2

1 2

a az z

b b

Php cng tr s phc 1 2 1 2 1 2z z a a b b i . Php nhn s phc

1 2 1 2 1 2 2 1 1 2z .z a .a a .b i a .b i b .b

Php chia s phc 1 1 2 21 1 2

2 22 2 2 2 2

a b i . a b iz z .z

z z .z a b

4. Cn bc hai v phng trnh s phc.

Cho a khi a 0

z a za i khi a 0

.

Cho 2 2x y a

z a bi z w m w x yi2xy b

gii tm x,y ri thay vo w.

* Cho phng trnh bc 2 : 2 2az bz c 0 a 0 .xt =b 4ac

khi 0 phng trnh c 2 nghim o phn bit : 1 2

b i b iz v z

2a 2a

.

khi 0 phng trnh c 1 nghim o kp 1 2b

z z2a

khi 0 phng trnh c 2 nghim thc phn bit :

1 2

b bz v z

2a 2a

5. Dng lng gic ca s phc.

Cho s phc z a bi gi r l modun, l acgumen ca z

2 2r a b

a r cos

b r sin

dng lng gic

z r cos isin

Cho hai s phc 1 1 1z r cos isin v 2 2 2z r cos isin

1 1 1 2 1 2 1 2 1 2 1 2 1 22 2

z rcos isin ; z .z r .r cos isin

z r

Cng thc Moa vr : Cho s phc

nn nz r cos isin z r cos isin r cosn isin n n N

T hp xc sut, nh thc Niu - tn.

I. T hp.

1. Hon v: nP n! n n 1 ! n n 1 n 2 ! ... n 1

2. Chnh hp:

knn!

A 1 k n .n k !

Tnh cht : nn nP A .

3. T hp:

knn!

C 0 k n .k! n k !

4. Cc tnh cht : n k kn n n nP A ; A C .k! ; k n k k 1 k K

n n n 1 n 1 nC C ; C C C 1 k n .

5. Nh thc Niu tn : n 0 n 1 n 1 1 2 n 2 2 n 2 2 n 2 n 1 1 n 1 n 0 n

n n n n n na b C a C a b C a b ... C a b C a b C a b .

6. H Qu:

* n 0 1 2 2 n n

n n n n1 x C xC x C ... x C .

* 0 1 n nn n nC C ... C 2

* n0 1 2 n

n n n nC C C ... 1 C 0

7. S hng tng qut trong khai trin n

a b l:

k n k k *k 1 nT C .a .b n N Hoc n i

k k n k *

n

n 0

C .a .b n N

.

II. Xc sut.

* Xc sut ca bin c A :

n A

P A . 0 P A 1n

Trong n A l s phn t ca bin c

A. n l s phn t ca khng gian mu .

* Tnh cht xc sut : P 0 ; P 1 ; P 0.

Nu A v B xung khc P A B P A P B cng thc cng xc sut.

A l bin c i ca A P A 1 P A .

A v B l bin c c lp P A.B P A .P B .

PHN II. THI I HC CAO NG CC NM.

S 1. THI TUYN SINH I HC KHI A NM 2011.

I. PHN CHUNG CHO TT C TH SINH ( 7,0 im ).

Cu I ( 2,0 im ). Cho hm s x 1

y C2x 1

.

1. Kho st s bin thin v v th C ca hm s cho.

2. Chng minh rng vi mi m ng thng y x m lun ct C ti hai im phn bit

A v B . Gi 1 2k ,k ln lt l h s gc ca cc tip tuyn vi C tiA v B . Tm m tng

1 2k k t gi tr ln nht.

Cu II (2,0 im ).

1. Gii phng trnh: 2

1 sin2x cos2x2 sin xsin2x.

1 cot x

2. Gii h phng trnh :

2 2 3

22 2

5x y 4xy 3y 2 x y 0 x, y .

xy x y 2 x y

Cu III ( 1,0 im ). Tnh tch phn 4

0

xsin x x 1 cosxI dx

xsin x cosx

Cu IV ( 1,0 im ). Cho hnh chp S.ABCc y ABC l tam gic vung cn ti B, AB BC 2a; hai mt phng SAB v SAC cng vung gc vi mt phng ABC . Gi M l trung im caAB ; mt phng qua SM v song song viBC , ct AC ti N. Bit gc gia hai mt phng SBC v ABC bng 060 . Tnh th tch khi chp S.BCNM v khong cch gia hai ng thng AB v SN theo a. Cu V ( 1 im ). Cho x,y,z l ba s thc thuc on 1;4 v x y, x z. Tnh gi tr nh nht

ca biu thc x y z

P .2x 3y y z z x

II.PHN RING ( 3,0 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ) A. Theo chng trnh chun Cu VI.a ( 2,0 im ). 1. Trong mt phng ta Oxy , cho ng thng : x y 2 0 v ng trn

2 2C : x y 4x 2y 0. Gi I l tm ca C ( A v B l cc tip im). Tm ta im M,

bit t gic MAIB c din tch bng 10. 2. Trong khng gian vi h ta Oxyz cho hai im A 2;0;1 ,B 0; 2;3 v mt phng

P : 2x y z 4 0. Tm ta im M thuc P sao cho MA MB 3.

Cu VII.a ( 1,0 im ). Tm tt c cc s phc z, bit: 22z z z.

B. Theo chng trnh nng cao. Cu VI.b ( 2,0 im ).

1. Trong mt phng ta Oxy , cho elip 2 2x y

E : 1.4 1 Tm ta cc im A v Bthuc

E , c honh dng sao cho tam gic OAB cn ti O v c din tch ln nht.

2. Trong khng gian vi h ta Oxyz , cho mt cu 2 2 2S : x y z 4x 4y 4z 0 v

im A 4;4;0 . Vit phng trnh mt phng OAB , bit im B thuc S v tam gic

OAB u. Cu VII.b ( 1,0 im ). Tnh mun ca s phc z, bit: 2z 1 1 i z 1 1 i 2 2i.

S 2. THI TUYN SINH I HC KHI B NM 2011.

I. PHN CHUNG CHO TT C TH SINH ( 7,0 im ). Cu I ( 2,0 im ). Cho hm s 4 2y x 2 m 1 x m 1 ,m l tham s.

1. Kho st s bin thin v v th hm s 1 khi m 1 .

2. Tm m th hm s 1 c ba im cc tr A,B,C sao cho OA BC; trong O l gc ta , A l im cc tr thuc trc tung, B v C l hai im cc tr cn li. Cu II (2,0 im ). 1. Gii phng trnh: sin2xcosx sinxcosx cos2x sinx cosx.

2. Gii phng trnh : 23 2 x 6 2 x 4 4 x 10 3x x .


2

0

1 x sin xI dx.

cos x

Cu IV ( 1,0 im ). Cho hnh lng tr 1 1 1 1ABCD.A B C D c y ABCD l hnh ch nht,AB a , AD a 3 . Hnh chiu vung gc ca im 1A trn mt phng ABCD trng vi giao im ca AC v BD . Gc gia hai mt phng 1 1ADD A v ABCD bng

060 . Tnh th tch khi lng tr cho v khong cch t im 1B n mt phng 1A BD theo a. Cu V ( 1 im ). Cho a v b l s thc dng tha mn 2 22 a b ab a b ab 2 .

Tm gi tr nh nht ca biu thc 3 3 2 2

3 3 2 2

a b a bP 4 9 .

b a b a

II. PHN RING ( 3,0 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ) A. Theo chng trnh chun Cu VI.a (2,0 im). 1. Trong mt phng ta Oxy , cho hai ng thng : x y 4 0 v d: 2x y 2 0. Tm ta

im N thuc ng thng d sao cho ng thng ON ct ng thng ti im M tha mn OM.ON 8.

2. Trong khng gian ta Oxyz . Cho ng thng x 2 y 1 z

:1 2 1

v mt phng

P : x y z 3 0. Gi I l giao im ca v P . Tm ta im M thuc P sao cho

MI vung gc vi v MI 4 14.

Cu VII.a ( 1,0 im ). Tm s phc z, bit: 5 i 3

z 1 0z


1. Trong mt phng ta Oxy , cho tam gic ABCc nh 1

B ;12

. ng trn ni tip tam gic

ABC tip xc vi cc cnh BC,CA,AB tng ng ti cc imD,E,F . Cho D 3;1 v ng thng EF c phng trnh y 3 0 . Tm ta nh A, bit A c tung dng.

2. Trong khng gian ta Oxyz , cho hai ng thng x 2 y 1 z 5

:1 3 2

v hai im

A 2;1;1 ,B 3; 1;2 . Tm ta im M thuc ng thng sao cho tam gic MAB c din

tch bng 3 5.

Cu VII.b ( 1 im ). Tm phn thc v phn o ca s phc

3

1 i 3z .

1 i

S 3. THI TUYN SINH I HC KHI D NM 2011.


Cu I ( 2,0 im ). Cho hm s 2x 1

y .x 1


2. Tm k ng thng y kx 2k 1 ct th C ti hai im phn bit A,B sao cho khong cch t A v Bn trc honh bng nhau. Cu II (2,0 im ).

1. Gii phng trnh: sin 2x 2cosx sin x 1

0.tan x 3

2. Gii phng trnh : 22 12

log 8 x log 1 x 1 x 2 0 x .


0

4x 1I dx.

2x 1 2

Cu IV ( 1,0 im ). Cho hnh chp S.ABC c y ABC l tam gic vung ti B,BA 3a,BC 4a; mt phng SBC vung gc vi mt phng ABC . Bit 0SB 2a 3 v SBC 30 . Tnh th tch khi chp S.ABCv khong cch t im B n mt phng SAC theo a.

Cu V ( 1,0 im ). Tm m h phng trnh sau c nghim:

3 2

2

2x y 2 x xy mx,y

x x y 1 2m

II.PHN RING ( 3 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ). A. Theo chng trnh chun Cu VI.a (2,0 im). 1. Trong mt phng ta Oxy , cho tam gic ABCc nh B 4;1 , trng tm l G 1;1 v ng thng cha phn gic trong ca gc A c phng trnh x y 1 0 . Tm ta cc nh A v

C.

2. Trong khng gian ta Oxyz . Cho im A 1;2;3 v ng thng x 1 y z 3

d : .2 1 2

Vit

phng trnh ng thng i qua im A, vung gc vi ng thng d v ct trcOx.

Cu VII.a ( 1,0 im ). Tm s phc z, bit z 2 3i z 1 9i. B. Theo chng trnh nng cao. Cu VI.b ( 2,0 im ). 1. Trong mt phng ta Oxy , cho im A 1;0 v ng trn 2 2C : x y 2x 4y 5 0.

Vit phng trnh ng thng ct C ti im M v N sao cho tam gic AMNvung cn ti A.

2. Trong khng gian ta Oxyz , cho ng thng x 1 y 3 z

:2 4 1

v mt phng

P : 2x y 2z 0. Vit phng trnh mt cu c tm thuc ng thng , bn knh bng 1 v

tip xc vi mt phng P .

Cu VII.b( 1,0 im ). Tm gi tr nh nht v gi tr ln nht ca hm s 22x 3x 3

yx 1

trn on

0;2 .

S 4. THI TUYN SINH CAO NG KHI A,B,D NM 2011.


Cu I ( 2,0 im ). Cho hm s 3 21

y x 2x 3x 1. C3

.


2. Vit phng trnh tip tuyn ca th C ti giao im ca C vi trc tung. Cu II ( 2,0 im ). 1. Gii phng trnh: 2cos4x 12sin x 1 0.

2. Gii bt phng trnh: 2 2x x x 2x 3 1 x 2x 34 3.2 4 0.

Cu III ( 1,0 im ). Tnh tch phn

2

1

2x 1I dx.

x x 1

Cu IV ( 1,0 im ). Cho hnh chp S.ABC c y ABC l tam gic vung cn ti B,AB a,SA vung gc vi mt phng ABC , gc gia hai mt phng SBC v ABC bng 030 . Gi M l trung im

ca cnhSC . Tnh th tch ca khi chp S.ABM theo a. Cu V ( 1,0 im ). Tm cc gi tr ca tham s thc m phng trnh sau c nghim:

6 x 2 4 x 2x 2 m 4 4 x 2x 2 x . II.PHN RING ( 3,0 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ). A. Theo chng trnh chun Cu VI.a ( 2,0 im ). 1. Trong mt phng vi h ta Oxy , cho ng thng d : x y 3 0. Vit phng trnh

ng thng i qua im A 2; 4 v to vi ng thng d mt gc bng 450.

2. Trong khng gian vi h ta Oxyz , hai im A 1;2;3 ,B 1;0; 5 v mt phng

P : 2x y 3z 4 0. Tm ta im M thuc P sao cho ba im A,B,M thng hng.

Cu VII.a ( 1,0 im ). Cho s phc z tha mn 2

1 2i z z 4i 20. Tm mun ca z.

B.Theo chng trnh nng cao. Cu VI.b ( 2,0 im ). 1. Trong mt phng vi h ta Oxy , cho tam gic ABCc phng trnh cc cnh l

AB: x 3y 7 0,BC: 4x 5y 7 0,CA : 3x 2y 7 0. Vit phng trnh ng cao k t nh A

ca tam gicABC .

2. Trong khng gian vi h ta Oxyz , cho ng thng x 1 y 1 z 1

d : .4 3 1

Vit phng

trnh mt cu c tm I 1;2; 3 v ct ng thng d ti hai im A,B sao cho AB 26.

Cu VII.b ( 1,0 im ). Cho s phc z tha mn 2z 2 1 i z 2i 0. Tm phn thc v phn o ca 1

.z


I.PHN CHUNG CHO TT C TH SINH ( 7,0 im ). Cu I ( 2,0 im ). Cho hm s 3 2y x 2x 1 m x m 1 , m l tham s thc. 1. Kho st s bin thin v v th ca hm s khim 1 .

2. Tm m th ca hm s 1 ct vi trc honh ti 3 im phn bit c honh 1 2 3x , x , x

tha mn iu kin 2 2 21 2 3x x x 4.

Cu II ( 2,0 im ).

1. Gii phng trnh: 1 sinx cos2x sin x

14cosx.

1 tan x 2

2. Gii bt phng trnh: 2

x x1.

1 2 x x 1

Cu III ( 1,0 im ). Tnh tch phn 1 2 x 2 x

x

0

x e 2x eI dx.

1 2e

Cu IV ( 1,0 im ). Cho hnh chp S.ABCD c y ABCD l hnh vung cnh a.Gi M v N ln lt l trung im ca cc cnh ABv AD; H l giao im ca CN viDM . Bit SH vung gc vi mt phng ABCD v SH a 3 . Tnh th tch khi chp S.CDNMv tnh khong cch gia hai ng thng DM v SC theo a.

Cu V ( 1,0 im ). Gii h phng trnh

2

2 2

4x 1 x y 3 5 2y 0x, y

4x y 2 3 4x 7

II. PHN RING ( 3,0 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ) A. Theo chng trnh chun. Cu VI.a ( 2,0 im ).

1. Trong mt phng ta Oxy , cho hai ng thng 1 2d : 3x y 0 v d : 3x y 0. Gi T l

ng trn tip xc vi 1d ti A, ct 2d ti hai im B v C sao cho tam gic ABCvung ti B. Vit

phng trnh ca T , bit tam gic ABCc din tch bng 3

2v im A c honh dng.

2. Trong khng gian ta Oxyz , cho ng thng x 1 y z 2

:2 1 1

v mt phng

P : x 2y z 0 . Gi C l giao im ca vi P , M l im thuc . Tnh khong cch t M n

P , bit MC 6.

Cu VII.a ( 1,0 im ). Tm phn o ca s phc z, bit rng 2

z 2 i 1 2i .

B. Theo chng trnh nng cao. Cu VI.b ( 2,0 im ). 1. Trong mt phng Oxy , cho tam gic ABCcn ti A c nh A 6;6 ;ng thng i qua trung im ca cc cnh AB v AC c phng trnh x y 4 0 . Tm ta cc nh B v C, bit im

E 1; 3 nm trn ng cao i qua nh C ca tam gic cho.

2. Trong khng gian ta Oxyz, cho im A 0;0; 2 v ng thng x 2 y 2 z 3

: .2 3 2

Tnh khong cch t A n .Vit phng trnh mt cu tm A, ct ti hai

im B v C sao choBC 8 .

Cu VII.b ( 1,0 im ). Cho s phc z tha mn

3

1 3iz

1 i

. Tm mdun ca s phc z iz.


I.PHN CHUNG CHO TT C TH SINH ( 7,0 im ).

Cu I ( 2,0 im ). Cho hm s 2x 1

y .x 1


2. Tm m ng thng y 2x m ct th C ti hai im phn bit A,B sao cho tam gic

OAB c din tch bng 3 ( O l gc ta ). Cu II (2,0 im ). 1. Gii phng trnh sin2x+cos2x cos x 2cos2x sinx 0.

2. Gii phng trnh 23x 1 6 x 3x 14x 8 0 x .


e

2

1

ln xI dx.

x 2 ln x

Cu IV ( 1,0 im ). Cho hnh lng tr tam gic u ' ' 'ABC.A BC c AB = a, gc gia hai mt phng

'A BC v ABC bng 060 . Gi G l trng tm tam gic 'A BC. Tnh th tch khi lng tr cho v

tnh bn knh mt cu ngoi tip t din GABC theo a. Cu V ( 1,0 im ). Cho cc s thc khng m a,b,c tha mn a b c 1. Tm gi tr nh nht ca

biu thc 2 2 2 2 2 2 2 2 2M 3 a b b c c a 3 ab bc ca 2 a b c . II. PHN RING ( 3,0 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ) A. Theo chng trnh chun. Cu VI.a ( 2,0 im ). 1. Trong mt phng ta Oxy , cho tam gic ABCvung ti A, c nh C 4;1 , phn gic trong gc A c phng trnh x y 5 0. Vit phng trnh ng thngBC , bit din tch tam gic

ABC bng 24 v c nh A c honh dng.

2. Trong khng gian ta Oxyz , cho cc im A 1;0;0 ,B 0;b;0 ,C 0;0;c , trong b,c

dng v mt phng P : y z 1 0. Xc nh b v c, bit mt phng ABC vung gc vi mt

phng P v khong cch t im O n mt phng ABC bng 1

3.

Cu VII.a ( 1,0 im ). Trong mt phng ta Oxy , tm tp hp im biu din cc s phc z tha

mn: z i 1 i z .


1. Trong mt phng ta Oxy , cho im A 2; 3 V elip 2 2x y

E : 1.3 2 Gi 1 2F v F l cc

tiu im ca E (F1 c honh m); M l giao im c tung dng ca ng thng 1AF vi

E ; N l im i xng ca 2F qua M. Vit phng trnh ng trn ngoi tip tam gic 2ANF .

2. Trong khng gian ta Oxyz , cho ng thng x y 1 z

: .2 1 2

Xc nh ta im M trn

trc honh sao cho khong cch t im M n bngOM .

Cu VII.b ( 1,0 im ). Gii h phng trnh:

2

x x 2

log 3y 1 xx,y .

4 2 3y


I.PHN CHUNG CHO TT C TH SINH ( 7,0 im ). Cu I ( 2,0 im ). Cho hm s 4 2y x x 6.


2. Vit phng trnh tip tuyn ca th C , bit tip tuyn vung gc vi ng thng 1

y x 1.6

Cu II (2,0 im ). 1. Gii phng trnh: sin 2x cos2x 3sin x cosx 1 0.

2. Gii phng trnh : 3 32x x 2 x 2 x 2 x 4x 44 2 4 2 x .

Cu III ( 1,0 im ). Tnh tch phn e

1

3I 2x ln xdx.

x

Cu IV ( 1,0 im ). Cho hnh chp S.ABCD c y ABCD l hnh vung cnh a, cnh bn SA a ;

hnh chiu vung gc t nh S trn mt phng ABCD l im H thuc cnh AC, AC

AH .4

Gi

CM l ng cao ca tam gic SAC . Chng minh M l trung im ca SA v tnh th tch khi t din SMBC theo a.

Cu V ( 1,0 im ). Tm gi tr nh nht ca hm s 2 2y x 4x 21 x 3x 10.

II.PHN RING ( 3,0 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ). A. Theo chng trnh chun Cu VI.a (2,0 im). 1. Trong mt phng ta Oxy , cho tam gic ABCc nh A 3; 7 , trc tm l H 3; 1 , tm

ng trn ngoi tip l I 2;0 . Xc nh ta nh C, bit C c honh dng.

2. Trong khng gian ta Oxyz . Cho hai mt phng P : x y z 3 0 v

Q : x y z 1 0. Vit phng trnh mt phng R vung gc vi P v Q sao cho khong

cch t O n R bng 2.

Cu VII.a ( 1,0 im ). Tm s phc z tha mn: 2z 2 v z l s thun o.

B. Theo chng trnh nng cao. Cu VI.b ( 2,0 im ). 1. Trong mt phng ta Oxy , cho im A 0;2 v l ng thng i qua O. Gi H l hnh chiu vung gc ca A trn . Vit phng trnh ng thng , bit khong cch t H n trc honh bngAH .

2. Trong khng gian ta Oxyz , cho hai ng thng 1 2

x 3 tx 2 y 1 z

: y t v : .2 1 2

z t

Xc

nh ta im M thuc 1 sao cho khong cch t im M n 2 bng 1.

Cu VII.a ( 1 im ). Gii h phng trnh

2

2 2

x 4x y 2 0x,y .

2log x 2 log y 0

S 8.

THI TUYN SINH CAO NG KHI A,B ,D NM 2010.

I.PHN CHUNG CHO TT C TH SINH ( 7,0 im ). Cu I ( 2,0 im ). 1. Kho st s bin thin v v th C ca hm s 3 2y x 3x 1

2. Vit phng trnh tip tuyn ca th C ti im c honh bng 1 Cu II (2,0 im ).

1. Gii phng trnh : 5x 3x

4cos cos 2 8sin x 1 cos x 52 2

2. Gii h phng trnh: 2 2

2 2x y 3 2x yx, y

x 2xy y 2


0

2x 1dx

x 1

Cu IV ( 1,0 im ). Cho hnh chp S.ABCD c y ABCD l hnh vung cnh a,mt phng SAB

vung gc vi mt phng y, SA SB , gc gia ng thng SC v mt phng y bng 045 .Tnh theo a th tch ca khi chp S.ABCD . Cu V ( 1,0 im ). Cho hai s thc dng thay i x,y tha mn iu kin 3x y 1 .Tm gi tr nh

nht ca biu thc 1 1

A .x xy

II.PHN RING ( 3,0 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ). A. Theo chng trnh chun.

Cu VI.a (2,0 im). Trong khng gian vi h ta Oxyz, cho hai im A 1; 2;3 ,B 1;0;1 v mt

phng P : x y z 4 0 .

1. Tm ta hnh chiu vung gc ca A trn P

2. Vit phng trnh mt cu S c bn knh bng AB

6,c tm thuc ng thng ABv S tip

xc vi P

Cu VII.a ( 1,0 im ). Cho s phc z tha mn iu kin 2

2 3i z 4 i z 1 3i .Tm phn thc

v phn o ca z. B. Theo chng trnh nng cao.

Cu VI.b ( 2,0 im ).Trong khng gian vi h ta Oxyz , cho ng thng x y 1 z

d :2 1 1

v

mt phng P : 2x y 2z 2 0.

1. Vit phng trnh mt phng cha d v vung gc vi P .

2. Tm ta im M thuc d sao cho M cch u gc ta O v mt phng P .

Cu VII.b ( 1,0 im ) Gii phng trnh 2z 1 i z 6 3i 0 trn tp hp cc s phc.



Cu I ( 2,0 im ). Cho hm s x 2

y 1 .2x 3

1. Kho st s bin thin v v th ca hm s 1 .

2. Vit phng trnh tip tuyn ca th hm s 1 , bit tip tuyn ct trc honh,trc tung ln lt ti hai im phn bit A,B v tam gic OAB cn ti gc ta O.

Cu II ( 2,0 im ).

1. Gii phng trnh

1 2sin x cos x3.

1 2sin x 1 sinx

2. Gii phng trnh 32 3x 2 3 6 5x 8 0 x .


3 2

0

I cos x 1 cos xdx.

Cu IV ( 1,0 im ). Cho hnh chp S.ABCD c y ABCD l hnh thang vung ti A v D; AB AD 2a,CD a; gc gia hai mt phng SBC v ABCD bng 060 .Gi I l trung im ca cnhAD .Bit hai mt phng SBI v SCI cng vung gc vi mt phng ABCD ,Tnh th thch khi chp S.ABCD theo a.

Cu V ( 1,0 im ). Chng minh rng vi mi s thc dng x,y,z tha mn x x y z 3yz, ta c:

3 3 3

x y x z 3 x y x z y z 5 y z .

II. PHN RING ( 3,0 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ). A. Theo chng trnh chun. Cu VI.a (2,0 im). 1. Trong mt phng vi h ta Oxy , cho hnh ch nht ABCDc im I 6;2 l giao im ca

hai ng cho AC v BD . im M 1;5 thuc ng thng ABv trung im E ca cnh CD thuc ng thng : x y 5 0 . Vit phng trnh ng thngAB .

2. Trong khng gian vi h ta Oxyz , cho mt phng P : 2x 2y z 4 0 v mt cu

2 2 2S : x y z 2x 4y 6z 11 0. Chng minh rng mt phng P ct mt cu S theo mt ng trn.Xc nh ta tm v bn knh ca ng trn .

Cu VII.a ( 1,0 im ). Gi 1 2z v z l hai nghim phc ca phng trnh 2z 2z 10 0. Tnh gi tr

ca biu thc 2 2

1 2A z z .

B. Theo chng trnh nng cao. Cu VI.b ( 2,0 im ). 1. Trong mt phng vi h ta Oxy ,cho ng trn 2 2C : x y 4x 4y 6 0 v ng

thng : x my 2m 3 0, vi m l tham s thc. Gi I lm tm ca ng trn C .Tm m

ct C ti hai im phn bit A v B sao cho din tch tam gic IAB ln nht.

2. Trong khng gian vi h ta Oxyz , cho mt phng P : x 2y 2z 1 0 v hai ng

thng 1 2x 1 y z 9 x 1 y 3 z 1

: ; : .1 1 6 2 1 2

Xc nh ta im M thuc ng thng 1

sao cho khong cch t M n ng thng 2 v khong cch t M n mt phng P bng nhau.

Cu VII.b ( 1,0 im ). Gii h phng trnh

2 2

2 2

2 2

x xy y

log x y 1 log xyx, y .

3 81


I. PHN CHUNG CHO TT C TH SINH ( 7,0 im ). Cu I ( 2,0 im ). Cho hm s 4 2y 2x 4x 1 .


2. Vi cc gi tr no ca m,phng trnh 2 2x x 2 m c 6 nghim thc phn bit ?

Cu II (2,0 im ).

1. Gii phng trnh 3sinx cos xsin 2x 3cos3x 2 cos4x sin x .

2. Gii h phng trnh 2 2 2

xy x 1 7yx, y .

x y xy 1 13y


3

2

1

3 ln xI dx.

x 1

Cu IV ( 1,0 im ). Cho hnh lng tr tam gic ' ' ' 'ABC.A BC c BB a, gc gia ng thng 'BB v mt phng ABC bng 060 ; tam gic ABCvung ti C v 0BAC 60 .Hnh chiu vung gc ca im

'B ln mt phng ABC trng vi trng tm ca tam gicABC . Tnh th tch khi t din 'A ABC theo a.

Cu V ( 1,0 im ). Cho cc s thc x,y thay i v tha mn 3

x y 4xy 2. Tm gi tr nh nht

ca biu thc: 4 4 2 2 2 2A 3 x y x y 2 x y 1. II. PHN RING ( 3,0 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ) A. Theo chng trnh chun Cu VI.a (2,0 im).

1. Trong mt phng vi h ta Oxy , cho ng trn 2 2 4C : x 2 y

5 v hai ng thng

1 2: x y 0, : x 7y 0. Xc nh ta tm K v tnh bn knh ca ng trn 1C ; bit ng

trn 1C tip xc vi cc ng thng 1 2, v tm K thuc ng trn C .

2. Trong khng gian vi h ta Oxyz , cho t din ABCDc cc nh A 1;2;1 ,B 2;1;3

,C 2; 1;1 v D 0;3;1 .Vit phng trnh mt phng P i qua im A,B sao cho khong cch t C

n P bng khong cch t D n P .

Cu VII.a ( 1,0 im ). Tm s phc z tha mn: z 2 i 10 v z.z = 25.

B.Theo chng trnh nng cao. Cu VI.b ( 2,0 im ). 1. Trong mt phng vi h ta Oxy , Cho tam gic ABCcn ti A c nh A 1,4 v cc nh B,C thuc ng thng : x y 4 0. Xc nh ta cc im B v C, bit din tch tam gic

ABCbng18 .

2. Trong khng gian vi h ta Oxyz , Cho mt phng P : x 2y 2z 5 0 v hai im

A 3;0;1 ,B 1; 1;3 . Trong cc ng thng i qua A v song song vi P ,hy vit phng trnh ng thng m khong cch t B n ng thng l nh nht. Cu VII.b ( 1,0 im ). Tm cc gi tr ca tham s m ng thng y x m ct th hm s

2x 1y

x

ti hai im phn bit A,B sao cho AB 4.


I.PHN CHUNG CHO TT C TH SINH ( 7,0 im ). Cu I ( 2,0 im ). Cho hm s 4 2y x 3m 2 x 3m c th l mC , m l tham s. 1. Kho st s bin thin v v th ca hm s cho khim 0 .

2. Tm m ng thng y 1 ct th mC , ti 4 im phn bit c honh nh hn 2. Cu II (2,0 im ).

1. Gii phng trnh 3cos5x 2sin3xcos2x sinx 0.

2. Gii h phng trnh

2

2

x x y 1 3 0

x, y .5x y 1 0

x


x

1

dxI .

e 1

Cu IV ( 1,0 im ). Cho hnh lng tr ng ' ' 'ABC.A BC c y ABC l tam gic vung ti B,

' 'AB a,AA 2a,AC 3a. Gi M l trung im ca on thng ' 'A C , I l trung im ca AM v 'A C . Tnh theo a th tch khi t din IABCv khong cch t im A n mt phng IBC . Cu V ( 1,0 im ). Cho cc s phc khng m x,y thay i v tha mn x y 1. Tm gi tr ln

nht v gi tr nh nht ca biu thc 2 2S 4x 3y 4y 3x 25xy. II.PHN RING ( 3,0 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ) A. Theo chng trnh chun Cu VI.a (2,0 im). 1. Trong mt phng vi h ta Oxy , cho tam gic ABC vi M 2;0 l trung im ca cnh AB. ng trung tuyn v ng cao qua nh A ln lt c phng trnh l 7x 2y 3 0 v

6x y 4 0. Vit phng trnh ng thngAC .

2. Trong khng gian vi h ta Oxyz , cho cc im A 2;1;0 ,B 1;2;2 ,C 1;1;0 v mt phng

P : x y z 20 0. Xc nh ta im D thuc ng thng ABsao cho ng thng CD song

song vi mt phng P . Cu VII.a ( 1,0 im ). Trong mt phng ta Oxy , tm tp hp im biu din cac s phc z tha

mn iu kin z 3 4i 2.


1. Trong mt phng vi h ta Oxy , cho ng trn 2 2C : x 1 y 1. Gi I l tm ca C .

Xc nh ta im M thuc C sao cho 0IMO 30 .

2. Trong khng gian vi h ta Oxyz , cho ng thng x 2 y 2 z

:1 1 1

v mt phng

P : x 2y 3z 4 0. Vit phng trnh ng thng d nm trong P sao cho d ct v vung gc vi ng thng . Cu VII.b ( 1,0 im ). Tm cc gi tr ca tham s m ng thng y 2x m ct th hm s

2x x 1y

x

ti hai im phn bit A,Bsao cho trung im ca on thng AB thuc trc tung.

S 12. THI TUYN SINH CAO NG KHI A,B,D NM 2009.

I. PHN CHUNG CHO TT C TH SINH ( 7,0 im ). Cu I ( 2,0 im ). Cho hm s 3 2y x 2m 1 x 2 m x 2 (1),vi m l tham s thc.

1. Kho st s bin thin v v th ca hm s 1 khi m 2

2. Tm cc gi tr ca m hm s 1 c cc i, cc tiu v cc im gi tr ca th hm s

1 c honh dng. Cu II (2,0 im ).

1. Gii phng trnh 2

1 2sin x cos x 1 sinx cos x.

2. Gii bt phng trnh x 1 2 x 2 5x 1 x


2x x

0

I e x e dx

Cu IV ( 1,0 im ). Cho hnh chp t gic u S.ABCD c AB a,SA a 2. Gi M, N v P ln lt l

trung im ca cc cnh SA,SB v CD.Chng minh rng ng thng MN vung gc vi ng thng SP.Tnh a th tch ca khi t dinAMNP . Cu V ( 1 im ). Cho a v b l hai s thc tha mn 0 a b 1 . Chng minh rng 2 2a ln b b lna lna ln b. II. PHN RING ( 3 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ) A. Theo chng trnh chun Cu VI.a (2,0 im). 1. Trong mt phng vi h ta Oxy , cho tam gic ABCc C 1; 2 ,ng trung tuyn k t A v ng cao k t B ln lt c phng trnh l 5x y 9 0 v x 3y 5 0, Tm ta cc nh

A v B.

2. Trong khng gian vi h ta Oxyz , cho mt phng 1P : x 2y 3z 4 0

2v P : 3x 2y z 1 0. Vit phng trnh mt phng P i qua im A 1;1;1 ,vung gc vi hai

mt phng 1 2P v P .

Cu VII.a ( 1,0 im ). Cho s phc z tha mn 2

1 i 2 i z 8 i 1 2i z. Tm phn thc v

phn o ca z. B. Theo chng trnh nng cao. Cu VI.b ( 2,0 im ). 1. Trong mt phng vi h ta Oxy , cho cc ng thng 1 : x 2y 3 0 v

2 : x y 1 0. Tm ta im M thuc ng thng 1 sao cho khong cch t im M n

ng thng 2 bng 1

2.

2. Trong khng gian vi h ta Oxyz , cho tam gic ABCc A 1;1;0 ,B 0;2;1 v trng tm

G 0;2; 1 . Vit phng trnh ng thng i qua im C v vung gc vi mt phng ABC .

Cu VII.b ( 1,0 im ). Gii phng trnh sau trn tp hp cc s phc : 4z 3 7i

z 2i.z i



Cu I ( 2,0 im ). Cho hm s

2mx 3m 2 x 2

y 1 ,x 3m

vi m l tham s thc.

1. Kho st s bin thin v v th ca hm s 1 khi m 1.

2. Tm cc gi tr ca m gc gia hai ng tim cn ca th hm s 1 bng 045 . Cu II (2,0 im ).

1. Gii phng trnh 1 1 7

4sin x .3sinx 4

sin x2

2. Gii h phng trnh

2 3 2

4 2

5x y x y xy xy

4x, y .

5x y xy 1 2x

4

Cu III ( 2,0 im ). Trong khng gian vi h ta Oxyz , cho im A 2;5;3 v ng thng

x 1 y z 2d : .

2 1 2

1. Tm ta hnh chiu vung gc ca im A trn ng thng d.

2. Vit phng trnh mt phng cha d sao cho khong cch t A n ln nht. Cu IV ( 2,0 im ).

1. Tnh tch phn 46

0

tan xI dx

cos2x

.

2. Tm cc gi tr ca tham s m phng trnh sau c ng hai nghim thc phn bit:

4 42x 2x 2 6 x 2 6 x m m . II. PHN RING ( 2 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ) A. Theo chng trnh chun Cu V.(2,0 im). 1. Trong mt phng vi h ta Oxy , hy vit phng trnh chnh tc ca Elip E bit rng

E c tm sai bng 5

3 v hnh ch nht c s ca E c chu vi bng 20 .

2. Cho khai trin n n

0 1 n1 2x a a x ... a x , trong *n N v cc h s 0 1 na ,a ,...,a tha mn

h thc 1 n0 na a

a ... 4096.2 2

Tm s ln nht trong cc s 0 1 na ,a ,...,a .

B. Theo chng trnh nng cao. Cu V. ( 2 im ).

1. Gii phng trnh 222x 1 x 1log 2x x 1 log 2x 1 4. 2. Cho lng tr ' ' 'ABC.A BC c di cnh bn bng2a , y ABC l tam gic vung ti A,

AB a,AC a 3 v hnh chiu vung gc vi nh 'A trn mt phng ABC l trung im ca

cnhBC . Tnh theo a th tch khi chp 'A .ABC v tnh cosin ca gc gia hai ng thng ' ' 'AA v BC .


I.PHN CHUNG CHO TT C TH SINH ( 8,0 im ). Cu I ( 2,0 im ). Cho hm s 3 2y 4x 6x 1 1 .


2. Vit phng trnh tip tuyn ca th hm s 1 , bit rng tip tuyn i qua im

M 1; 9 . Cu II (2,0 im ).

1. Gii phng trnh 3 3 2 2sin x 3cos x sin xcos x 3sin xcos x.

2. Gii h phng trnh 4 3 2 2

2

x 2x y x y 2x 9x, y .

x 2xy 6x 6

Cu III ( 2,0 im ). Trong khng gian vi h ta Oxyz, cho ba im A 0;1;2 ,B 2; 2;1 ,

C 2;0;1 . 1. Vit phng trnh mt phng i qua ba im A,B,C .

2. Tm ta ca im M thuc mt phng 2x 2y z 3 0 sao cho MA MB MC.

Cu IV ( 2,0 im ).

1. Tnh tch phn

4

0

sin x dx4

I .sin 2x 2 1 sinx cos x

2. Cho hai s thc x,y thay i tha mn h thc 2 2x y 1. Tm gi tr ln nht v gi tr nh

nht ca biu thc 2

2

2 x 6xyP .

1 2xy 2y

II. PHN RING ( 2,0 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ). A. Theo chng trnh chun Cu V. (2,0 im).

1. Chng minh rng k k 1 k

n 1 n 1 n

n 1 1 1 1

n 2 C C C

(n,k l cc s nguyn dng, knk n,C l s t hp

chp k ca n phn t). 2. Trong mt phng vi h ta Oxy , hy xc nh ta nh C ca tam gic ABC bit rng

hnh chiu vung gc ca C trn ng thng AB l im H 1; 1 , ng phn gic trong ca gc A c phng trnh x y 2 0 v ng cao k t B c phng trnh 4x 3y 1 0.

B. Theo chng trnh nng cao. Cu V. ( 2,0 im ).

1. Gii bt phng trnh: 2

0,7 6

x xlog log 0.

x 4

2. Cho hnh chp S.ABCD c y ABCD l hnh vung cnh 2a, SA a,SB a 3 v mt phng

SAB vung gc vi mt phng y. Gi M, N ln lt l trung im ca cc cnhAB,BC . Tnh theo a th tch ca hnh chp S.BMDN v tnh cosin ca gc gia hai ng thngSM,DN .


I. PHN CHUNG CHO TT C TH SINH ( 8,0 im ). Cu I ( 2,0 im ). Cho hm s 3 2y x 3x 4 1 .


2. Chng minh rng mi ng thng i qua im I 1;2 vi h s gc k k 3 u ct th

ca hm s 1 ti ba im phn bit I,A,Bng thi I l trung im ca on thng AB. Cu II (2,0 im ). 1. Gii phng trnh 2sin x 1 cos2x sin2x 1 2cosx.

2. Gii h phng trnh 2 2xy x y x 2y

x, y .x 2y y x 1 2x 2y

Cu III ( 2,0 im ). Trong khng gian vi h ta Oxyz, cho bn im

A 3;3;0 ,B 3;0;3 ,C 0;3;3 ,D 3;3;3 . 1. Vit phng trnh mt phng i qua bn imA,B,C,D .

2. Tm ta tm ng trn ngoi tip tam gicABC .

Cu IV ( 2,0 im ).

1. Tnh tch phn 2

3

1

ln xI dx.

x

2. Cho x,y l hai s thc khng m thay i. Tm gi tr ln nht v gi tr nh nht ca biu thc

2 2

x y 1 xyP

1 x 1 y

.

II. PHN RING ( 2 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ) A. Theo chng trnh chun Cu V.(2,0 im). 1. Tm s nguyn dng n tha mn h thc 1 3 2n 12n 2n 2nC C ... C 2048

( knC l s t hp chp ca

n phn t).

2. Trong mt phng vi h ta Oxy , cho parabol 2P : y 16x v im A 1;4 .Hai im phn

bit B,C (B v C khc A) di ng trn P sao cho gc 0BAC 90 . Chng minh rng ng thng BC lun i qua mt im c nh. B. Theo chng trnh nng cao. Cu V. ( 2 im ).

1. Gii bt phng trnh 2

1

2

x 3x 2log 0.

x

2. Cho lng tr ng ' ' 'ABC.A BC c y ABC l tam gic vung, AB BC a, cnh bn 'AA a 2. Gi M l trung im ca cnh BC. Tnh theo a th tch ca khi lng tr ' ' 'ABC.A BC v

khong cch gia hai ng thng 'AM v BC.

PHN III. TC GI BIN SON.

S 16. I. PHN CHUNG CHO TT C TH SINH ( 7,0 im ).

Cu I ( 2,0 im ). Cho hm s : 2x 1

y C .x 1

1. Kho st s bin thin v v th ca hm s C .

2. Gi Kd l ng thng i qua im A 2;2 v c h s gc l k. Tm k ng thng

Kd ct th hm s C ti hai im thuc hai nhnh ca th. Cu II (2,0 im ).

1. Gii phng trnh : sinx 3

tan x 2cosx 1 2

.

2. Gii phng trnh :

2log 100xlog 10x logx4 6 2.3 x .

Cu III ( 1,0 im ). Tnh tch phn sau : 2

0

x cos x2

I dxcos x 1

.

Cu IV ( 1,0 im ). Cho hnh chp S.ABCD c hai mt bn SAB , SAD cng vung gc vi mt

phng y,SA a,ABCD l hnh thoi cnh a v c gc 0A 120 .Tnh th tch hnh chp S.ABCD v

tnh khong cch t D n mt phng SBC .

Cu V ( 1,0 im ). Gii phng trnh : 2 23 3 3 32x 2x 1 x 1 x 2 x . II.PHN RING ( 3 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ) A.Theo chng trnh chun Cu VI.a (2,0 im) 1. Trong mt phng ta Oxy , cho A 2;1 v ng thng d : 2x 3y 4 0 .Vit phng

trnh ng thng qua A v to vi ng thng d mt gc bng 045 .

2. Trong khng gian ta Oxyz , cho M 1;2; 3 v ng thng x 1 y 1 z 3

d :3 2 5

. Vit

phng trnh ng thng qua im M, ct ng thng d v vung gc vi gi ca

vct a 6; 2; 3 .

Cu VII.a ( 1 im ). Tm tp hp nhng im M biu din s phc Z tha mn : z 3 4i 2.

B.Theo chng trnh nng cao Cu VI.b ( 2 im ) 1. Trong mt phng ta Oxy , cho hai ng trn 2 21C : x y 2x 4y 4 0;

2 22C : x y 2x 2y 14 0 vit phng trnh ng trn i qua giao im ca hai ng trn

trn v qua im M 0;1 .

2. Trong khng gian ta Oxyz , cho ng thng d :

x 1 2t

y 2 t t

z 3t

v mt phng

P : 2x y 2z 1 0 .Vit phng trnh mt cu S c tm thuc ng thng d sao cho khong

cch t tm mt cu S n mt phng P bng 1 R .

Cu VII.b ( 1 im ). Cho phng trnh 4 4log m 9 log m 3 3 .Hy tm phn thc, phn o ca

s phc m

z 1 i ,m N.

S 17. I.PHN CHUNG CHO TT C TH SINH ( 7,0 im ).

Cu I ( 2,0 im ). Cho hm s : 3 2y x 3 m 1 x 9x m C vi m l tham s thc.

1. Kho st s bin thin v v th hm s C ng vi m 1 .

2. Xc nh m hm s C t cc tr l 1 2x ,x sao cho 1 2x x 2 .

Cu II (2,0 im ).

1. Gii phng trnh: 1 sin 2x

cot x 2sin xsin x cosx 22

.

2. Gii phng trnh: 35 52log 3x 1 1 log 2x 1 .

Cu III ( 1,0 im ). Tnh tch phn sau : 5 2

1

x 1I dx

x 3x 1

Cu IV ( 1,0 im ). Cho hnh hp ' ' ' 'ABCDA BC D . Tnh th tch ca khi hnh ' ' ' 'ABCDA BC D bit

rng ' ' 'AA BD l t din u cnh bng a.

Cu V ( 1,0 im ). Cho cc s thc khng m x,y,z tho mn 2 2 2x y z 3 .

Tm gi tr ln nht ca biu thc : 5

A xy yz zxx y z

.

II.PHN RING ( 3,0 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ). A. Theo chng trnh chun. Cu VI.a (2,0 im)

1. Trong mt phng vi h to Oxy cho tam gic ABC c A 4;6 , phng trnh cc ng

thng cha ng cao v trung tuyn k t nh C ln lt l 2x y 13 0 v 6x 13y 29 0 .

Vit phng trnh ng trn ngoi tip tam gic ABC .

2. Trong khng gian vi h to Oxyz cho hnh vung ABCD c A 5;3; 1 ,C 2;3; 4 .Tm to

nh D bit rng nh B nm trong mt phng : x y z 6 0. Cu VII.a ( 1,0 im ). Cho s phc z sao cho z 10 v phn thc ca z bng 3 ln phn o.Tnh

z 1 .

B. Theo chng trnh nng cao. Cu VI.b ( 2,0 im ) 1. Trong mt phng to Oxy cho tam gic ABC c A 2;3 , trng tm G 2;0 . Hai nh

B v C ln lt nm trn hai ng thng 1 2d : x y 5 0v d : x 2y 7 0. Vit phng

trnh ng trn c tm C v tip xc vi ng thngBG .

2. Trong khng gian vi h to Oxyz cho cc im A 1;0;0 ,B 0;1;0 ,C 0;3;2 v mt phng

: x 2y 2 0. Tm to ca im M bit rng M cch u cc im A,B,Cv mt phng

.

Cu VII.b ( 1,0 im ). Cho s phc

n i

z n1 n n 2i

.Tm gi tr nh nht ca biu thc

A z 1 .


Cu I ( 2,0 im ). Cho hm s : 3 2 my x 3x mx 1 C ,( m l tham s).

1. Kho st s bin thin v v th hm s khim 3 .

2. Xc nh m mC ct ng thng y 1 ti ba im phn bit A 0;1 ,B,C sao cho cc tip

tuyn ca mC ti B v Cvung gc vi nhau.

Cu II (2,0 im ).

1. Gii phng trnh: 2 3

2

2

cos x cos x 1cos2x tan x

cos x

.

2. Gii h phng trnh: 2 2

2 2

x y xy 1 4y x,y .

y(x y) 2x 7y 2

Cu III ( 1,0 im ). Tnh tch phn sau : e 3

2

21

log xI dx

x 1 3ln x

Cu IV ( 1,0 im ). Cho hnh chp S.ABCD c y ABCD l hnh vung tm O cnh a 2 . Gc gia SD v mt y bng 045 . Gi M, N ln lt l trung im ca SA,SC . Mt phng BMN ct SO ti

I,SD ti K . Tnh th tch ca khi chp S.BMKN .

Cu V ( 1 im ). Cho a,b,c l cc s thc khng m tha mn a b c 1 .

Chng minh rng: 7

ab bc ca 2abc27

.

II. PHN RING ( 3 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ). A. Theo chng trnh chun. Cu VI.a (2,0 im) 1. Trong mt phng vi h ta Oxy ,cho tam gic ABC bit A 5;2 . Phng trnh ng

trung trc ca cnhBC , ng trung tuyn 'CC ln lt l d : x y 6 0v :2x y 3 0 .

Tm ta cc nh ca tam gic ABC . 2. Trong khng gian vi h ta Oxyz , hy xc nh to tm v bn knh ng trn ngoi

tip tam gic ABC , bit A 1;0;1 , B 1;2; 1 , C 1;2;3 .

Cu VII.a ( 1 im ). Cho 1 2z ,z l cc nghim phc ca phng trnh: 22z 4z 11 0 . Tnh gi tr

ca biu thc

2 21 2

21 2

z z

z z

.

B.Theo chng trnh nng cao. Cu VI.b ( 2 im ) 1. Trong mt phng vi h ta Oxy cho hai ng thng : x 3y 8 0, v

' : 3x 4y 10 0 im A 2 ;1 . Vit phng trnh ng trn c tm thuc ng thng , i qua im A v tip xc vi ng thng ' .

2. Trong khng gian vi h ta Oxyz , Cho ba im A 0;1;2 ,B 2; 2;1 ,C 2;0;1 .

Vit phng trnh mt phng ABC v tm im M thuc mt phng : 2x 2y z 3 0 sao

cho MA MB MC .

Cu VII.b (1 im) Gii h phng trnh: 2

1 x 2 y

1 x 2 y

2log ( xy 2x y 2) log (x 2x 1) 6 x,y

log (y 5) log (x 4)=1


Cu I ( 2,0 im ). Cho hm s : 2x 1

y Cx 1

1. Kho st s bin thin v v th hm s C .

2. Tm ta im M C sao cho khong cch t im I 1;2 ti tip tuyn ca C ti M l ln nht. Cu II (2,0 im ).

1. Gii phng trnh sau: 1 1

2cos3x 2sin3xsin x cosx

.

2. Gii phng trnh sau: 222x 1 x 1log 2x x 1 log 2x 1 4 .


2

0

x sin 2xI dx

cos x

Cu IV ( 1,0 im ). Cho hnh chp t gic u S.ABCD ,bit khong cch gia ABv mt phng

SCD bng 2. Gc gia mt bn v mt y bng 060 .Tnh th tch hnh chp S.ABCD .

Cu V ( 1,0 im ).Tm m phng trnh sau c 2 nghim phn bit:

2 210x 8x 4 m 2x 1 . x 1

II. PHN RING ( 3,0 im ). Th sinh ch c lm mt trong hai phn ( phn A hoc phn B ) A. Theo chng trnh chun. Cu VI.a (2,0 im) 1. Trong mt phng ta Oxy cho im I 2;0 v hai ng thng 1d : 2x y 5 0,

2d : x y 3 0 . Vit phng trnh ng thng i qua im I v ng thi ct 2 ng thng

1 2d v d Ti 2 im A,B sao cho IA 2IB .

2. Trong khng gian vi h ta Oxyz cho ng thng

x 2 t

d : y 2t

z 2 2t

. Gi l ng thng

qua im A 4;0; 1 song song vi d v I 2;0;2 l hnh chiu vung gc ca A trn d .Vit

phng trnh mt phng P cha ng thng , sao cho khong cch t mt phng P n

ng thng d l ln nht.

Cu VII.a ( 1,0 im ). Tm tp hp nhng im M biu din s phc z tha 2z 3 5i 2 .

B. Theo chng trnh nng cao Cu VI.b ( 2,0 im ) 1. Trong mt phng vi h ta Oxycho hai ng trn 2 2C : x y 2x 2y 1 0,

' 2 2C : x y 4x 5 0 cng i qua M 1;0 .Vit phng trnh ng thng qua im M ct hai ng trn 'C , C ln lt ti A,B sao cho MA 2MB.

2. Trong khng gian vi h ta Oxyz cho ng thng y 2

d : x z1

.Vit phng trnh mt

phng i qua ng thng d v to vi ng thng 'x 2 z 5

d : y 32 1

mt gc 030 .

Cu VII.b ( 1 im ). Cho phng trnh 2 2 2 2n 1 n 4 n 2 n 3C C 2 151 2 C C , tnh gi tr ca biu thc

4 3

n 1 nA 3ATn 1 !

. Cc s cho knn Z ,A l chnh hp chp k ca n phn t, knC l t hp chp k ca

n phn t.


Cu I ( 2,0 im ). Cho hm s 2 2

y | x | 1 . | x | 1 C .

1. Kho st s bin thin v v th hm s C .

2. Tm trn trc honh nhng im m t im k c ba tip tuyn phn bit n C . Cu II (2,0 im ).

1. Gii h phng trnh:

2 2x y 12

x y 2 x, y .

xy - x - y 1 x y - 2 6

2. Gii phng trnh: 2 2sin x tan x cos x cos2x 2 tan x

Cu III ( 1,0 im ). Tnh tch phn: 1

2 2

0

I x ln 1 x dx .

Cu IV ( 1,0 im ). Cho t din SABCc tam gic ABCvung cn nh B,AB a; cc cnh SA SB SC 3a a 0 .Trn cnh SA,SB ln lt ly im M, N sao cho SM BN a .Tnh th

tch khi chpC.ABNM theo a .

Cu V ( 1,0 im ). Vi mi s thc x,y tha iu kin 2 22 x y xy 1 . Tm gi tr ln nht v gi

tr nh nht ca biu thc 4 4x y

P .2xy 1

II. PHN RING ( 3 im ). Th sinh ch c lm mt trong hai p

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