cấu trúc Đề thi Đại mọc môn toán
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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.
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f(x)=x^ 3+3x^ 2-4
-3 -2 -1 1 2 3
-4
-3
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f(x)=-x^ 3+3x^ 2-4
-3 -2 -1 1 2 3
-4
-3
-2
-1
1
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x
y
O
f(x)=-x^ 3+3x^ 2-4x+2
-3 -2 -1 1 2 3
-4
-3
-2
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x
y
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f(x)=x^ 3+3x^ 2+4x+2
-3 -2 -1 1 2 3
-4
-3
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x
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f(x)=x^ 3-3x^ 2+3x+1
-3 -2 -1 1 2 3
-4
-3
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-1
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x
y
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f(x)=-x^ 3-3x^ 2-3x+1
-3 -2 -1 1 2 3
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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.
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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.
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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
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'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.
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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 .
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- 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.
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Trang 9
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:
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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.
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Trang 11
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 .
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Trang 12
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
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Trang 13
* 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
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Trang 14
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
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Trang 15
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 ...
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Trang 16
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
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Trang 17
* 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:
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Trang 18
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
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Trang 19
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
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Trang 20
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
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Trang 21
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
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Trang 22
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
du " " xy ra khi a b .
a,b,c 0 ta c 3
3a b c a b cabc abc3 3
du " " xy ra khi a b .
R
R
r
h
R
l
O
O
O . R
-
Trang 23
n s
a,b,c ... 0 ta c n s nn s
a b c ...
abc ...n
du " " xy ra khi a b .
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)
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Trang 24
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
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Trang 25
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.
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Trang 26
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 .
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Trang 27
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.
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Trang 28
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
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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
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Trang 30
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.
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* 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
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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.
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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.
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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
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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 .
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Trang 36
Cu III ( 1,0 im ). Tnh tch phn 3
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
B. Theo chng trnh nng cao. Cu VI.b ( 2,0 im ).
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.
I. PHN CHUNG CHO TT C TH SINH ( 7,0 im ).
Cu I ( 2,0 im ). Cho hm s 2x 1
y .x 1
1. Kho st s bin thin v v th C ca hm s cho.
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
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Trang 37
2. Gii phng trnh : 22 12
log 8 x log 1 x 1 x 2 0 x .
Cu III ( 1,0 im ). Tnh tch phn 4
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.
I. PHN CHUNG CHO TT C TH SINH ( 7,0 im ).
Cu I ( 2,0 im ). Cho hm s 3 21
y x 2x 3x 1. C3
.
1. Kho st s bin thin v v th C ca hm s cho.
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
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Trang 38
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
S 5. THI TUYN SINH I HC KHI A NM 2010.
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.
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Trang 39
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.
S 6. THI TUYN SINH I HC KHI B NM 2010.
I.PHN CHUNG CHO TT C TH SINH ( 7,0 im ).
Cu I ( 2,0 im ). Cho hm s 2x 1
y .x 1
1. Kho st s bin thin v v th C ca hm s cho.
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 .
Cu III ( 1,0 im ). Tnh tch phn
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
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Trang 40
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 .
B. Theo chng trnh nng cao. Cu VI.b ( 2,0 im ).
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
S 7. THI TUYN SINH I HC KHI D NM 2010.
I.PHN CHUNG CHO TT C TH SINH ( 7,0 im ). Cu I ( 2,0 im ). Cho hm s 4 2y x x 6.
1. Kho st s bin thin v v th C ca hm s cho.
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
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Trang 41
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
Cu III ( 1,0 im ). Tnh tch phn 1
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
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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.
S 9. THI TUYN SINH I HC KHI A NM 2009.
I.PHN CHUNG CHO TT C TH SINH ( 7,0 im ).
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 .
Cu III ( 1,0 im ). Tnh tch phn 2
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.
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Trang 43
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
S 10. THI TUYN SINH I HC KHI B NM 2009.
I. PHN CHUNG CHO TT C TH SINH ( 7,0 im ). Cu I ( 2,0 im ). Cho hm s 4 2y 2x 4x 1 .
1. Kho st s bin thin v v th ca hm s 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
Cu III ( 1,0 im ). Tnh tch phn
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
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Trang 44
'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.
S 11. THI TUYN SINH I HC KHI D NM 2009.
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
Cu III ( 1,0 im ). Tnh tch phn 3
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,
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Trang 45
' '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.
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 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
Cu III ( 1,0 im ). Tnh tch phn 1
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
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Trang 46
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
S 13. THI TUYN SINH I HC KHI A NM 2008.
I.PHN CHUNG CHO TT C TH SINH ( 8,0 im ).
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
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Trang 47
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 .
S 14. THI TUYN SINH I HC KHI B NM 2008.
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 .
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 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 .
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Trang 48
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 .
S 15. THI TUYN SINH I HC KHI D NM 2008.
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 .
1. Kho st s bin thin v v th ca hm s 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 .
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Trang 49
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
.
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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
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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 .
S 18. I.PHN CHUNG CHO TT C TH SINH ( 7,0 im ).
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
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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
S 19. I.PHN CHUNG CHO TT C TH SINH ( 7,0 im ).
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 .
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Cu III ( 1,0 im ). Tnh tch phn 3
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.
S 20. I.PHN CHUNG CHO TT C TH SINH ( 7,0 im ).
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 ).
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Trang 54
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