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510.76
B450D

JYEN VAN NHO - LE BAY - NGUYEN VAN THO

MOIi

BO DE THI
^Tii

LUAN

TOAN HOC
Danh cho thi sinh Icfp 12 on tap va thi Dai hoc, Cao dang
Bien soan theo ngi dung va cau true de thi cua Bg Glao due - Dao tao

0
Ha N O I

DVL.009154

NHA XUAT BAN DAI HQC QUOC GIA HA

NOI


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

BO

Df THI

T U LUA

TO^n HOC
^ Danh c h o thi sinh I6p 12 on tgp v a thi Dqi h o c - C a o d a n g
^ Bien soqn theo noi dung v a c d u true d4 thi c u a Bp GD&DT

NHA XUAT BAN DAI HOC QUOC GIA HA NOI


Jltue lue
Lifinoiddii

^

PHAN 1: T H I T U Y E N SINH D ^ I HQC - C A O DANG

3

6

C. De thi minh hoa

4

B. Mot so dieu can luTu y

3

A. Cau true de thi de thi tuyen sinh Dai hoc, Cao dang 2009

r

8

Dap an - thang diem

6

De thi tuyen sinh Dai hoc, Cao ding, khoi A

15

Dap an - thang diem

14

De thi tuyen sinh Dai hoc, Cao d^ng, khoi B, D

273

E. Mirdi de thi i\i luyen tap

21

D. Ba mi/di de thi CO IcJi giai

:

PHAN 2: T H I T O T N G H I E P T R U N G H Q C PHO THONG

289

291

B. De thi minh hoa

289

A. Cau true de thi Tot nghiep THPT 2009

314

D. De thi tham khao

297

C. De thi mau TNTHPT

292

Dap an - thang diem

318

De thi tot nghiep THPT phan ban 2008, Ian 2

316

De thi tot nghiep THPT phan ban 2008, Ian 1

315

De thi tot nghiep THPT phan ban 2007

314

De thi tot nghiep THPT phan ban 2006


Pfi^nl:

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

K. CMi TRUC DE THI TUYEN SINH DAI HOC, CAO DANG NAM 2 0 0 9
I. PHAN CHUNG CHO TAT CA THf SINH (7,0 di^m)
Cdu

I

A^^i dung kien thvlc
Diem
— Khdo sat, ve do thi ciia ham so.
— Cdc bdi loan lien quart den tint; dun^ ciia dao ham va do thi
ciia ham so: Chieu bie'n thien cua ham so. Cifc tri. Gia tri
Idn nha't va nho nhat cua ham so'. Tie'p tuyen, tiem can 2,0
(d\Jng va ngang) cua do thi ham so'. Tim tren do thi nhi^ng
diem CO tinh chat cho trtfdc; ti/dng giao giCfa hai do thi (mot
trong hai do thi la du"5ng thing);...

III II

IV
V

PhU(fn^ trinh, bat phu
Cong thiic lUcfng gidc, phU(fng trinh lUOng gidc.
Tim gidi han.
Tim nguyen ham, tinh tich phan.
Ifng dung ciia tich phan: Tinh dien tich hinh phing, the tich
khoi Iron xoay.
Hinh hoc khong gian (tong hcrp):Quan he song song, quan he
vuong goc cua di/dng thing, mSt phlng. Tinh dien tich xung
quanh cua hinh non tron xoay, hinh try tron xoay; tinh the tich
khoi lang tru, khoi chop, khoi n6n tron xoay, khoi tru tron
xoay; tinh dien tich mSt cau va the tich khoi cau.
Bdi todn tdng hap.

2,0
1,0

1.0
1,0

n.PHANRlfiNG (3,0diim)
Thi sinh chi difdc l^m mot trong hai phan (phan 1 hoac phan 2)
I. Theo chUtfng trinh Chudn:
Cm
NQI dung kien tMc

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

Diem
2,0
3


Cdu
-

Ngi dung kien thiic

^

Diem

Xac dinh toa do cua d i e m , vectd.

- Du'fJng tron, elip, mat cau.

T i n h gdc; tinh khoang each lijr d i e m den mat p h l n g . V i t r i

-

V i e t phu'dng trinh mat p h i n g , di/ftng t h i n g .

-

tifdng doi c u a - dUcJng thang, mat phang va mat cau.
-

So

pMc.

VII.a - To hap, xdc suci't, thonf> ke.
-

Bat ddn}> thvtc. Cuc tri ciia bleu thi'fc dgi

2. Theo chUOitg trinh Nang
Cdu

1,0
so.

cao:
NQi dung kien thi'tc

,

Diem

* PhUtfng phdp toa do trong mat phdni* vd trong khong

gian:

- Xac djnh toa dp c u a d i e m , vecttf.
- Dxiiing trdn, ha diTdng c o n i c , mat c a u .
Vl.b

2,0

- V i e t phiTPng trinh mat p h i n g , dtfcJng t h i n g .
- T i n h g o c ; tinh k h o a n g c d c h lit d i e m d e n difdng thang, mat
p h i n g ; k h o a n g e a c h - giffa hai di/dng l h a n g . V i tri tiTdng
d o i ciia dUtJng thang, m a t phang va mat c a u .
-

Sophi'/c.

, •, ^
, .
, , ,
/ ,
- Do thi ham phan thUc huu ti dang:

+ bx + c
v~

va

mot

ii'x + b'
Vll.b

so yen to lien quan.
- Su tiep xi'tc ciia hai dudng

1.0
cong.

ke.

- To h(/p, xdc sud't, thong

Idgarit.

- He phU(fng trinh mii vd

- Bat dang thUc. Cuc tri ciia hieu thi'fc dai
B. MOT S 6 D I E U C A N

so.

LUU Y:

DiTa vao cau true cua de thi va npi dung giijra hai bp sach theo chi/Png trinh
Chuan va chi/c^ng trinh Nang cao , chung ta can lifii y mot so van de nhU"sau :
I. P H A N C H U N G CHO T A T CA THf SINH

(7,0diem)

Cdu I: Trong phan nay chiing ta chi khao sat va ve do thj , cung nhif chi xet
cac bai toan l i e n quan d e n do thj cua ba loai ham so'.
4

,

• Hc^m bac 3 : y-cuc'+ hx^ + cjc + J , (« ^ O) .
• Ham bac 4 (dang trung phiTcfng): y - ax^ + hx^ + c , (a O) .
• Ham phan thuTc dang : y = ifilA ^ ^ o, at/ - be ^ O)
cx^-d
* Khi khao sat tinh chat cua ham so , tinh loi, 16m va viec tim diem uo'n ciaa
do thi CO the bo qua khong can xet (neu can thi chi can tim diem uon cua ham
bac 3 de suy ra tarn doi xiJng cua do thi , con ham bac 4 thi nen bo qua hoan
toan phan nay).
* Cac bai loan ve sir tiep xuc cua hai diTfJng cong cung se khong diTdc de cap
tdi trong phan chung nay .
* Cac bai toan ve tiem can cung chi de cap den liem can dtfng va tiem can
ngang

Cdu III:

* Viec uTng dung tich phan de tinh the tich cua khoi tron xoay chi c6 cac khoi
khi cho hinh phing quay quanh true Ox.
II. PHAN RIENG (3,0diem)
... u

1. Theo chittfitg trhih Chiidn:
Cdu Via:

* Trong phan nay doi vdi phiTdng phap tpa do trong mat phing ta chi can on
lai cac bai toan c6 lien quan den di/dng thang , du'dng tron va elip . cac bai toan
ve hypebol va parabol khong du'dc de cap tdi trong chi/dng trinh chuan , cac bai
toan ve tiep tuyen cua elip'cung du'dc bo qua .
* Doi vdi phi/dng phap toa dp trong khong gian phan khoang each tif diem
den dirdng thang va khoang each giffa hai diTdng thang cheo nhau cung diTdc bo

qua.

Cdu Vila:

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

2. Theo chUcfng trinh Ndng cao:
Cdu VIb:

Cac bai toan c6 lien quan den tie'p tuyen cua cac du'dng conic cung khong dc
cap tdi trong cau true de thi mdi nay . Nhff vay doi vdi cac dffdng conic chi can
on lai cac dang toan ve viet phifdng trinh ehinh tac , tim cac diem nlim trcn
conic thda tinh cha't nao do va cac bai toan ve mot so tinh chat dac trffng ci.i
tifng du'dng conic ^
5


C. DE THI MINH HOA
DUcfi day la hai de thi minh hoa va dap an - thang dientchi tiet cua Bg Gido
due va Dao tao, cdc ban cdn xem ki de Met dUctc nhUng yeu cdu cdn dat ditdc
khi lam bdi. TiT do rtit ra dUcfc each trinh bay Wi gidi mgt de thi cho ngdn gon
nhitng ddy du vd chinh xdc.
THI T U V ^ N SINH 961 HQC, CflO OANG -

K H 6 | fl

(Thcfi gian Idm bdi: 180 phut)

I. PHAN CHUNG CHO TAT CA THf SINH (7,0 diem)
Cau I (2,0 die'm)
Cho ham so y = -x^ + + 4 , trong do m la^tham so thiTc.
1. Khao sat sif bien thien va ve do thi cua ham so da cho, vdi m = 0.
,2. Tim tat ca cac gia tri cua tham so m de ham so da cho dong bien tren khoang
(0;+«)).
Cfiu II (2,0 di^'m)
1. Giai phU'dng trinh: V3(2cos^ A-+ cosj:-2J + (3-2cosx)sin-;r = 0
2. Giai phiTdng trinh: logj {x + 2)-3 + log^{x-5)^

+ log, 8 = 0
2

CfiuIII(l,0di6'm)
Tinh dien tich hinh phang gidi han bdi do thi ham so y = yje^ + 1 , true hoanh
hai duTcJng thing x = In3 , x = In 8.
CauIV(l,Odie'm)
Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh a, SA^SB = a,
mat phang (SAB) vuong goc vdi mat phing (ABCD). Tinh ban kinh mat cau
ngoai tie'p hinh chop S.ABCD.
CSu V (1,0 di^m)
'
Xet cac so'thufc difdng x,y,z thoa man dieu kien: x + y + z - l .
Tim gia tri nho nhaft cua bieu thiJc:
yz

zx

xy

II. PHAN RIENG (3,0 diem)
Thi sink chi diMc lam mot trong hai phdn (phdn 1 hoqc phdn 2)
1. Theo chUcfng trinh Chudn:

au VLa (2,0 diem)

-^

i

>

1. Trong mat phang toa do Oxy, cho di/dng tron ( C ) :

+

- bx + 5 = 0.

.

Tim diem M thuoc true tung sao cho qua M ke difdc hai tiep tuyen cua (C)
ma goc giSa hai tiep tuyen do bang 60".

2.

Trong khong gian vdi he toa do Oxyz, cho diem M ( 2 ; 1 ; 0) va
';c = l + 2r
{d):ly

= -l + t

. ,\

^

.

diTdng

thang: "

•

i

i

Viet phufcJng trinh tham so' cua dtfdng thang di qua diem M , cat va vuong goc
vdi dufdng thang fc?).

Cau VILa (1,0 diem)
Tim he so cua

j
trong khai trien thanh da thtfc ciia bieu thuTc:
P = lx^

+x-\f

2. Theo chitctng trinh ndngcao:

,.

CSu VI.b (2,0 diem)

'

•

1. Trong mat phang toa do Oxy, cho di^dng tron [C): x^ + y^ - 6x + 5 = 0.
Tim diem M thuoc true tung sao cho qua M ke di/dc hai tiep tuyen cua (C)
ma goc gii?a hai tiep tdyen do bang 60".
2. Trong khong gian vdi he toa dp Oxyz, cho d i l m M ( 2 ; 1; O) va di/dng thang
^ '

2

1

- I

Viet phufdng trinh chinh tic cua dtfcfng thang di qua diem M , cat va vuong
goc vcti dU'dng th^ng (d).

Cfiu VII.b (1,0 A\im)
Tim he so cua

trong khai trien thanh da thuTc ciia bieu thtfc:
P =

X^ + A-

\

7


Cdu
I
(2,0 diem)

\. 0,25

OAP A N - T H R N G

DI^M

Dap an

\Diem

diem)

V d i m = 0 , ta c6 ham so' y = -.v^ - 3JC" + 4
T a p xac djnh: D = x.
Sif bien thien:
• C h i e u bien thien: y ' = - 3 . V " - 6 . V . Ta c6:
y' =

0 «

'x = -2
x =0

r.v<-2

<=> v'<0<=>
•
[.r>0

0,50

<=> y ' > 0 < : = > - 2 < A < ( ) .
D o do:
+ H a m so nghjch bien tren m o i khoang ^-co; -2) va ((); +cc)
+ H a m so dong bien tren khoang ( - 2 ; ( ) ) .
CiTc t r i : H a m so dat ciTc tieu tai .v = 2 va
ycT = >'(~2) = 0 ; dat cifc dai tai A" = 0 va J C D = y(^0 = ^ •
Gic'Jihan:

iim y = +co ;

0,25

lim y =- o o .

B a n g bien thien:

0

y'

-2

X

y

0

+ 00

0

+

0,25

\

Xo
"

.

X-

CO

Dolhi:

+ D o thi c^t true tung tai d i e m
/\{0 ; 4 ) , citt true hoiinh tai
diem

0,25

; ()) va tiep xuc

v d i true hoanh tai C ( - 2 ; 0 ) .


2. (0,75diem)
Ham so da cho nghich bien tren khoang (0 ; +oo)
o

v' = -?>x^

«

3A-^

- 6A: + m < 0 , VJC> 0

+6x>m

,

VJC>0

(*)

0,25

Ta CO bang bien thien cua ham so v = 3.v"+6.r tren
(0;+a)):
s

I

0

Tir do, ta di/ffc: (*) o w < 0.
II
{2,0 diem)

1.(1,0 diem)
Phi/rtng trinh da cho tifcfng difdng vdi phi/dng trinh:
(2 sin jr - \/3) (V3 sin A-+ cos A-j = 0
<=>

sm X =
\/3 sin jr + cos A: = 0

x-[-\Y^

+ n7t, « e Z .

A- = - — + ^;r ,k&'L.

6
2. (1,0 diem)
Dieu kien: x > - 2
A-^5

0,50

0,50

(i

V(3i dieu kien do, ta c6:
Phifcfng trinh da cho tUdng dUtrng vcti phu'cJng trinh:
log2[(A- + 2 ) | A - 5 | ] - l o g 2 8

C>(A- f 2)|.v-5|-8
' A - - 3 A - - 1 8 ) ( . r - 3 v - 2 ) = ()

0,50


(1,0

Ill
diem)

L^-3A:-18 = 0

[A:^-3X + 2 = 0
31^17

<:=>j: = - 3 ; j t = 6 ; j : =

—

2

0,50

D o ' i c h i e ' u v d i d i e u k i e n (*), ta dufdc t a t ca c a c n g h i e m c u a
phtfctng t r i n h d a c h o l a : x = 6 v a x = ^

•

K i h i e u 5 la d i e n tich can tinh.
V i V e ' + l > 0 V ; c G [ l n 3 ; l n 8 ] n e n S = J yje'
In

B&t

yje'' +I=t,tac6

+l.d.x

0,25

3

dx^^^
/ ' - I

0,25

K h i x = l n 3 t h i / = 2 , k h i .v = l n 8 t h i t = 3

i t - i

i t - i j

U

0,50
2 r - l

2 '

= 2 +ln/-l
2

TV
(1,0

diem)

D o SA = S B =

+l

'

- l n / + l ^=2+l n - .

2
2

a) n e n

la tarn g i a c d e u .

G o i G va. I tiTcfng tfng l a t a r n c i i a tarn g i a c d e u SAB v a t a m
cua hinh v u o n g

ABCD.

0,50

G o i O la tarn c u a m a t c a u ngoai tiep hinh chop
ta CO OGl{SAB)

va

S.ABCD,

OIl{ABCD).

1

Suy ra:

M=p.V-\t°—

t r o n g d o H l a t r u n g d i e m c u a AB.

+ T a r n g i a c OCA v u o n g t a i G.

/„

'\''"'

b

•

/

0,25

c

10


K i hieu R la ban kinh cua mat cau ngoai tiep hinh chop
S.ABCD, ta c6:
0,25
R = OA = yJOG
V
(1,0 diem)

+ GA

= . — +
V 4

T a c o : P = — + — + ~ + —+
y
z
z
X
Nhan tha'y: x^

-xy>xy

=
9

— + —.
X
y

,

.
6
(*)

Vx,y&R
0,50

Do do: J:'^ + j " ^ > jry(jc + y ) , Vjc,y > 0
x^
v2
Hay — + — >x+v,\/x,v>0

y
X

• ;

v'
z^
TiTdng ti/, ta c6: — + — > V + z , V y , ^ > 0
z
y
'
—+—>z

+

. .

'• •••

'

xyx,z>o

z

X

Cong theo tufng ve'ba bat d i n g thuTc viifa nhan dUdc d tren,

0,50

ket hdp v d i (*), ta diTdc:
P > 2 ( j c + y + z) = 2 , \/x,y,z>0
Hdn nffa, ta lai CO F = 2 khi

va x + y + Z = 1

= V= e ="
3

V i vay, m i n P = 2 .
VI.a

I. (1,0

diem)

(2,0 diem)
V i e t l a i phuTdng trinh cua (C) dU'di dang: (A - 3)^ + y^ = 4

0,25

Tir do, (C) CO tarn / (3 ; 0) va ban kinh /? - 2 .
Suy ra triic tung khong c6 diem chung v d i difdng tron

(C).

Vi vay qua mot d i e m bat k i tren true tung luon ke dtfdc hai

0,25

tiep tuye'n den (C).
X e t d i e m M[0;m)

tuy y thuoc true tung.

Qua M ke cac tiep tuyen MA va MB cua (C) (A,B la cac tiep
diem). Ta c6: Goc giffa hai du'dng thang MA va MB bang
60"

o

/\A/B = 6 0 "
[AMB

= 120"

(1)
^
(2)

0,25

.

11


L

V i yV//la phan giac cua AMB ncn:
AMI

= 30"

A// =
sin

„ c^MI

30"

^
= 2R

o Vw" + 9 = 4<:=>w = ±\/7
(2)
^ ^

A.W -

60"

<=> M / -

Dc thay, khong c6

^'^ „ o A// sin 60"

3

/?

0,25

thoa (*).

Vay, C O tal ca hai diem can tim la: (o ;

va (o ;

.

2. (7,0 tfi^'w)
Goi H la hinh chicu vuong goc cua M Ircn J, ta c6 Af// la
0,25

dU'cJng ihtlng di qua M, cat vii vuong goc vc'li d.
Vi H ed

ncn toa do cua Hc6 diing: (l + 2r; - 1 + / ; - / ) .

Suyra

=(2r-l;-2 +f

VI MH 1 d \ (/ C O mot vecl« chi phi/dng la M" = (2 ; 1 ; - 1 ) ,
r

ncn 2 . ( 2 r ~ l ) + l . ( - 2 +

0,50

/) + (-!).(-/) = 0.

Tilf do, la dir«c / - - . V i the, MH =1 - :-~

3

l3

;-~

3

3J

Suy ra phu'ctng trinh tham so cua du'dng thang MH la:

1 - 4/

V =

2+/

A- =

0,25

:.^-2t

VILa
(1,0 diem)

Theo cong thuTc nhj thiJc Niu-ldn, ta c6:
P.c;:(.v-l)%ctr(.v-l)%... +

ctr'(.v-ir....

0,25

.Ctv'"(.v-1).QV^
Suy ra. khi khai trien P thanh da thu'c, A " chi xua't hien khi
khaitrien

C^'(A-lf

va

0,25
Q'.v-(A^l)^

He so'cua .v^ trong khai trien cua Cll[x-l)^'

la: C".C^.
0,25

He so ciia .v"^ trong khai trien cua C^.K^ {x ~ l)^ la: - Q ' . C " .
12

V i vay, he so cua

Vl.b

trong khai t r i c n P thanh da thi?c la:

0,25

1. (1,0 diem). X c m phan 1. Cau V l . a .

{1,0 diem)
2. {1,0

diem)

G o i H la hinh vuong goc ciia M i r c n J, la c6 A/// la difrJng
thang di qua M . cSl va vuong gc)c v d i d.

0,25

jc = 1 + 2/
D CO phUdng Irinh tham so la:

Vi H

y = - l+/

ncn tpa dp ciJa H c6 dang: (1 + 2/ ; - 1 + ? ; - / ) .

Suy ra MH = {2tV I MH Ld

\

^t \

0,50

\a d c6 mot veclO chi phu'dng la u = (2 ;1 ; - l ) ,

ncn 2 . ( 2 / - l ) + l . ( - 2 + /) + ( - l ) . ( - r ) = ( ) .
2
.
• (\
Tii do, la diWc / = - . V i the, MH =
V3

4

2^

3

3

Suy ra, phu'dng,lrinh chinh lac ciia duTJng ihflng MH la:
.y - 2 _ >• - 1 _

0,25

1

Vll.b
{1,0 diem)

Theo cong Ihifc nhj ihiYc Niu-Utn, la co:
P =--

(.V

-

1 ) % Ql.v^

( .

-

1)-%

- .

. - ^ ^ (.V -

if''

+ ••.

0,25

+ QU'"(X-I) + QV^
Suy ra, k h i khai trien P lhanh da ihifc. . v ' c h i xua't hien k h i

khai tricn C " ( A - 1 ) " va Q ' X " (A - l ) " \

0,25

He so' ciia .v' trong khai i r i c n ciia C^,' (A - l / ' la: ~ C " . Q ^ .
He so'ciia A' trong khai t r i c n ciia Q'.A" (A - 1)"^ la: + Q ' . c ] .
V i vay, he so ciia A ' trong khai t r i c n P lhanh da thuTc la:

-C^QSQ^.C^^ + IO.

0,25

0,25

13


Bi THI TUVdN SINH OR! HQC . CflO D^N^ - KH6| B.D

(Th&i gian lam bai: 180 phiit)
I. PHAN CHUNG CHO TAT cA THf SINH (7 di^m)
Cau I ( 2,0 di^m )
Cho ham so' y = ^"^^^ .
x-2

1. Khao sat sir bien thien va ve do thi (C) cua ham so da cho .
2. Tim tat ca cac gia tri cua tham so m de dtfdng thing y = 2x + m dt (C) tai hai
diem phan biet ma hai tiep tuyen cua (C) tai hai diem do song song vdi nhau
Cfiu II (2,0 die'm)
1. Giai phufdng trinh : (l + 2cos3jr)sinx + sin2jc = 2'sin^ 2x + —

4;

2. Giai phiTdng tnnh : log, - 2| + logj |JC + 5| + log, 8 = 0

'^

2

J-

• •.

Cfiu III (1,0 die'm )
Tinh dien tich hinh phSng gidi han bdi do thi ham so y =

x.\n^(x^ + \]

j

X +1

, true

tung, true hoanh va d i T d n g thing x = yle - \
Cfiu IV (1,0 di^m )
Cholangtru /\BC./4'fi'C'c6 day/\5Cla tam gidc deu canh a , AA'= 2a va
dirdng thing AA' tao vdi mat phing (ABC) mot g6c bang 60".Tinh the tich
khoi tu" dien y4C/i'B'theo a .
CSu V (1,0 di^m )
*

Tim ta't ca cac gia tri cua tham so a de bat p h i T d n g trinh j ; . ;
x^ + 'ix^ -\
CO nghi^m

.

* ,

11. PHAN RIENG (3,0 die'm )
Thisinh chi dii(fc lam mgt trong haiphdn (phdn 1 hodc phdn 2)
1. Theo chiMng trinh Chudn :
Cfiu VI.a ( 2,0 diem )
Trong khong gian vd'i he toa d p Oxyz , cho d i T d n g thing d c6 p h i T d n g trinh
14


jc-1

z-3

y-1

2

1

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

4

1. Tinh khoang each givi& duTdng t h i n g J va m a t p h i n g (P) .
2. K i hieu I la hlnh chieu vuong goc cua d tren (P) . V i e t phifdng trinh tham so'
cua dtfdng thang / .

j

i,

CSu V I L a ( 1 , 0 Aiim )
r i m cac so thi/c x,y
2. Theo chumg

thoa m a n dang thiJc : x{3 + 5i) + y{l-2if

trinh ndng

=9 + l4i .

cao:

Cfiu V L b ( 2,0 d i e m )

^ - ' 1

Trong khong gian v d i he toa do Oxyz, cho difdng thang d c6 phtfcfng trinh
—

—

=

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

1. Tinh khoang each giffa diTdng thang d \k mat phang (P).
2. K i hieu / la h i n h chieu vuong goc cua d tren (P).

|

•

?

J

V i e t phi/dng trinh chinh t^c

cua di/dng t h i n g / .

' }• I

|

Cau V I I . b ( 1 , 0 d i e m )

-|

Cho so p h u t z = l + V 3 / . H a y v i e t dang lifOng giac cua so phtfc 2^.

DAP

Cdu
I

AN-THANG
Bap

1.(1,25

f

DI^M

an

Diem

diem)

[2,0diem)
Tapxacdinh : D = R \ { 2 } .

'

Sir b i e n t h i e n :
n
•

C h i e u bien thien : y ' =

< 0 ,\/xeD

.

(x-lf
Suy ra, ham so nghich bien tren m o i khoang

0,50
{-°o;2)

va

(2 ; + 0 0 )
•

Ctfc trK H a m so khong c6 ctfc tri
15


•

Gidi

han

: lim y=

lim y = 2];

Urn >' = +co

vh.

lim >' = - co. Suy ra do thj hkm so c6 mpt ti$m can dtfng
0,25
la dirdng thing ;c = 2 ,

mpt ti$m can ngang Ik dtfdng

thing >' = 2 .
B5ng bien thicn :
X

-00

2

-

y"

-

0,25

2

y
-00

2

• Do t h i :
+ Do thi c i t true tung tai diem
, cat true hoanh tai
0,25

diem B

tr

+ Do thj nhan diem / ( 2 ; 2 ) (Ik
giao Ciia hai difcJng ti^m can) Ikm
tarn do'i xtfng .
2. (0,75 diem)
Di/dng thing y -2x + m c i t CC) tai hai diem ph§n biet mk
tiep tuyen tai 66 song song vdi nhau
2x + 3
<=> — -2x + m c6 hai nghipm phan biet

, jc^ thoa man
0,50

dieukien

v ' ( ; C | ) = 3''(jC2)

< : : > 2 J : ' + ( m - 6 ) j : - 2 m - 3 = 0 c6 hai nghiem phSn bi^t
,

khac 2 vk thoa man dieu kipn A:, +

=4

A = ( m - 6 ) +8(2m + 3 ) > 0
2.2^+(m-6).2-2m-3^0
6-m
—

o m = -2

0,25

.
- 4


II 1. (1,0 diem)
(2,0 diem)
PhiTdng trinh da cho ti/dng difdng vdi phiTdng trinh :
sin X + sm4x - sin 2x + sin 2 = 1 - cos 4x +
2)
<=> sin jc + sin 4.x: = 1 + sin 4 J: o sin x 1
c:> x = —TT + k2;r ,keZ
2
2. (1,0 diem)
Dieu kien : x:^2

0,50
0,25
0,25

Vdi dieu kiSn 66, PhiTdng trinh da cho tiTdng diTdng vdi
0,50
phifdng trinh:
log2 [\x + 2\\x- 5|] = log2 8

' y £, r -

x + 2 x-5 =8
A:^-3;C-18-0

x^-3x
<=>X =

0,50

+ 2-^0

-3;JC

=

6;A:

=

^^^Hl , th6a man di^u kiSn (*)

III Ki hieu S la dien tich can tinh .
(1,0

x.ln'^(x^+i\
0;
diem)
Vi
x'+l
'->oyxe
^jc.ln^fjc^+l)
nen 5 = |
'-.dx
xUl
0
Dat ln{^x^+l^ = t .taco dt = 2x.dx

771

Khi x = 0 thi r = 0,khi x = 4e^. thi r = l
Vivay : S - - } / ^ A = -/^ 1
THtJ VIENTINKBINHTHUAN

0,25

0,50
0,25
17


IV
(1,0
diem)

K i hieu A va V tiTcfng lirng la chieu cao va the ^ch cua kho'i
lang tru da cho , ta c6 :
V—h.S AAIIC
1
3

V
J

Goi H la hinh chieu vuong goc
cua A' tren mp(ABC),
A'H^h

ta c6

va A \ 4 ^ = 60"

Suyra: h =

A'A.sin60"=aj3

Do do,

A

V = h^^ifQ = av3.

V

=

Dieu kien : x > 1 . V d i dieu kien do bat phi/dng trinh da cho
ti/dng difcfng vdi bat phurdng trinh :

diem)

A;^+3JC^-1

Ta nhan thay , ham so: f{x)


= {^x^ +

-

3x^ - I ) ( V J C +

X'-

yfx^f

Dong bien tren [ l ; + o o ) .
Suy ra : / ( A : ) > / ( 1 ) = 3 V A : > 1 . V i the ton tai x > Ithoa
man (*) , hay baft phiTdng trinh da cho c6 nghiem klii va chi
khi a > 3.
VI.a

1. (1,0 diem)

(2,0 diem)
Ta CO : + A ( l ; 7 ; 3 ) e v a

M = (2 ; 1 ; 4) la mpt vec M chi

phifdng cua d .

'

+ n = (3 ; - 2 ; l ) la mot v6c td ph^p tuyen cua

(P).

18


Ma M . n = 2 . 3 - 1 . 2 - 4 . 1 = 0 v a / l « ( P )
0,25

(do 3 . 1 - 2 . 7 - 1 . 3 + 5 ^ 0 ) nen

d//{P).

Do do khoang each h giffa d va (P) chinh b^ng khoang each
0,25

tiif/ldenfPj.
^
V i v a y , h-

3.1-2.7-3 + 5
,
—
79 + 4 + 1

9714
14

.

0,25

2. (1,0 diem)
Ta CO dirdng thang d di qua d i e m v 4 ( l ; 7 ; 3 ) va c6 vee td chi
phiTdng M = (2 ; 1 ; 4 )
G o i d' la dtftJng th^ng di qua A va vuong goc v d i (P).
Do /i" = (3 ; - 2 ; 1) la mot vee td phap tuyen cua (P) nen n"

0,25

la mot vee td chi phi/dng cua d'. Suy ra , phi/cJng trinh cua d'
Ik:

x - l ^ y - 7 ^ z - 3
3

-2

-1

G o i /4' la giao dieV1 cua ^ ' v d i fPj , ta
x-\
la nghiem ciia he :

3

CO

A ' e / . Toa do A'

z-2>
-

2

-

1

3jc-2y-z + 5 = 0

-.-v

41
40
33
G i a i he tren ta difdc . x = — , v = — , z = — .
14 '
7
14

Hdn niya , v i

dll{P ) nen {d)ll{l).

0,50

V i vay ^ = (2 ; I ; 4 ) la

mot vee td chi phu'd ng cua / .
Suy ra phu'dng trinh tham so cua di/dng thang / 1 ^ :
41

^

14
40
v=— +/
•
7
33 ^
Z = — + 4/
14
VILa
(1,0

0,25

T a c o : Jc(3 + 5/) + y ( l - 2 / f = A:(3 + 5/) + > ' ( - 1 1 + 2/)
^

= ( 3 j t - l l y ) + (5;c + 2>')./

0,50

diem)
19


VI the cac so thifc x,y

thoa man de bai khi va chi khi

f3jc-lly = 9

0,25

|5A: + 2V = 14

2. (1,0 diem)

(1,0

Giai he tren, ta di/dc: x =
; v=—^ .
61 •
61
1. (1,0 diem). Xem phan 1. Cau Vl.a.

Vl.b
diem)

0,25

Ta CO di/c(ng thSng d di qua diem /4(1; 7 ; 3) va c6 vectd chi
phi/dng M' = (2 ; 1; 4)
Gpi d' la dU'cJng lhang di qua A va vuong goc vdi (P).
Do n = (3 ; - 2 ; l ) la mot vecliJ phap luycn cua (P) nen n

0,25

la mot vectd chi phu'dng cua d'. Suy ra/p^iiTdng Irinh cua d'
lala.

.-l_v-7_,--3
^ - _2 1 .

Goi A' la giao diem ciia J ' v('Ji (Pj, ta ccV A ' e / . Toa dp ^4' la
[.t-l _ v-7_z-3
nghiem cua he: -^3
- 2 - 1
[3A:-2v--f 5= 0

.
,
41
40
33
Giai he tren ta dU'dc: x- — , v = — , - = — .
14 •
7

14
Hcfn nSa, vi dll{P)

nen {d)ll{l).

^

0,50

V i vay « =(2 ; 1 : 4) la

mpl vecld chi phU'Ong ciia /.
Suy ra. phu"Png irinh chinh tac cua du'ting thdng / lii:

i.

1

41

40

X

Vll.b

V-

z^2

I

33

0,25

-

14
7 _
2
1
Dang lifPng giac cua r. la:

(hO
diem)

14
4 '
;
0,50

C O S — + /.sin —

3J

3

TCf do, theo cong thiJc Moa-vrP, la c6 dang lu'Png giac cua
-•^ la:

^
0,50

:

=32 cos— + /.sm— =32 cos — f/.sm —
I
3
3 j
L I 3J
I 3 JJ

20


D. HAI MUOI Dfe THI C6 idl

CIAI

1
I, PHAN C H U N G C H O T A T C A T H I SINH (7 di^'m )
Cfiu I ( 2,0 di^m )
Cho ham so >• = - ( w + m x ^

+ ( 3 W - 2 ) J : , trong do m la tham so thiTc.

1. Khao sat sir bie'n thien va ve do thi ciia ham so' da cho, vdi m = 2.
2. Tim tat ca cac gia tri cua tham so'm de ham so' da cho dong bie'n tren tap xac
dinh cua no .

CSu II (2,0di^m)

.

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

(1)

2. Giai phi/cfng Irmh:

ift3

| l o g , ( . . + 2)'-3 = l o g , ( 4 - x f + log,(^ + 6 f
4

4

(2)

'^'^

4

Cfiu m (1,0 di^m )
a>sx.dx

Tinh: / =

.

.

0 sin^ ji:-5sinx + 6

. ufiD

CSu IV (1,0 diem )
Cho lang tru di^ng ABC.A'B'C

c6 day la tarn giac deu. Mat phang y4'BC tao

vdi day mot goc 3 0 ° v a . t k m giac A'BC c6 dien tich bang 8. Tinh the tich
khoi lang tru .
Cau V (1,0 diem )
Gia suT X, y la hai so' difdng thay ddi thoa man dieu kien : x + y = — .
4
Tim gia tri nho nhat cua bieu thiJc : 5 = — + • ^
X 4y
II. PHAN RIENG (3,0 d i e m )
Thi sink chi dMc lam niQt trong haiphdn ( phdn 1 hoac phdn 2)
1. Theo chUcfitg trinh Chudn :
Cau VLa ( 2,0 diem )
1. Viet phiTdng trinh diTdng thang (A) di qua diem M{3; l ) v a cat true Ox, Oy
Ian lifdt tai B va C sao cho tam giac ABC can tai A vdi A (2 ; - 2 ) .
21


2. Trong khong gian vdi he tpa dp Oxyz, cho diem y4 ( 4 ; 0 ; O ) va diem
fi(jc„ ;>-() ; 0) ,{x() > 0 ,y^^ > 0) sao cho OB = 8 va goc AOB = 60". Xac dinh
tpa dp diem C tren true Oz de the tich tur dien OABC bang 8.

CSu V I L a (1, 0 diem )
TCr cac chu" so 0, 1, 2, 3, 4, 5 c6 the lap di/pc bao nhidu so tiT nhien mek moi
so CO 6 chff so khdc nhau va chff so' 2 diJng canh chi? so 3?
2. Theo chtMng trinh ndng cao :
CSu Vl.b ( 2,0 diem )
1. Vie't phU'Png trinh diTcfng thing di qua diem A/(4 ; l ) va c^t cdc tia Ox. Oy ian
li/Pt tai .4 v^ B sao cho gia tri cua tong OA + OB nho nhat.
2. Trong khong gian vdi he tpa dp Oxyz, cho tt? dien ABCD c6 ba dinh
A{2A

, B(3 ; 0 ; 1) , C(2 ; - 1 ; 3 ) , con dinh D nlm tren true Oy. Tim tpa

dp dinh D ne'u tdr dien c6 the tich V = 5.

'

CSu V I I . b (1, 0 di§m )
Tuf cac so 0; 1; 2; 3; 4; 5. Hoi c6 the thanh lap dj/Pc bao nhieu so c6 3 chff so
khong chia het cho 3 ma cac chff so trong moi so la khac nhau . -

MM aiai
Cfiu I .

\. Khi m = 2 iKi y =

Chieu bien thien :

•

Tap xac djnh :

•

Y^+2x^+Ax
D =K

o Gidi han cua ham so'tai v6 cffc : lim ^ = - c » ; hm y = + 00
j r - > + ao

o Bang bien thien :
• Taco : 3 ' ' - x ^ + 4 x + 4=:(x + 2)^ > 0 , V x e R
• Bang bien thien;

/

.

.

..^

-2

•oo

0

+

+

8

y

3

AJ


Ham so dong b i e n tren khoang (-00 ;+oo), h a m so k h o n g c6 ciTc t r i .
Do thi :
o

rV

Diem uon:
y" = 2\ 4

'

0o

37"

A; = - 2 => ^ ( - 2 ) =

--

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

k h i X d i qua d i e m x = - 2 , nen do thi cua
ham so c6 d i e m uon U
o

Cho

o

Do t h i :

x = 0=>

y = 0
•'
l a m t a m d o i xuTng

Do thi cua h a m so' nhan d i e m uon U

2. y = ^{m-l)x^
ycbtoy'

+mx^

+{3m-2)x

= {m-l)x'^+2mx

+ 3m-2>0

,\/xeR

•

K h i m = 1 =>>'' = 2jr + l > 0 < = > j : , suy ra m = 1 k h o n g thoa .

•

Khi m

9t

1

a = m-l>0
A' =
m >1
<=> i
-2m' +5/n-2<0

m^-(m-l)(3m-2)<0
m >1
m<—

1
2

haym>2

<=>m>2

Vay k h i m > 2 t h i h a m so l u o n l u o n dong big'n v d i m o i x .
CSu I I .
1. Giai phi/dng trinh
( 1 ) <=> 2 sin

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

<=> sin 2 + 1 + cos2X - (sin A: + cos J:) = I
< » s i n 2 x + COS2JC = sin

+ cosx

23


<=> 72
o

sin

sin 2x +

4)

7

4j
= sm x +

X +

—

4

V
Tt

= x + -- + A:2;r
4

=
—
4
x +
2. D i e u k i e n :

n -

2ji0

I

\ = /:2;r
n
, In
x=~+k—
6
3

+ /t2;r
4J

(k

-6
4-X>0<:>
x +

x^-2

6>0

( 2 ) o 3 1 o g j ^ | ; c + 2|-3 = 31og^ ( 4 - A : ) + 31og, {x + 6)
4

4

c:>logj^|j: + 2 | - l o g ,
4

-

'

4

4

= l o g , ( 4 - x ) + l o g , ( x + 6)

4^"*^

4

«

l o g , (4|x + 2|) = log I [ ( 4 - x){x

4

+ 6)]

4

o 4 | A : + 2| = ( 4 - x ) { . t + 6)
4 ( x + 2) = ( 4 - j r ) ( x + 6)
4{x + 2) = -{4-x){x
<=>

+ 6)

x^ + 6 J C - I 6 = 0
X

2

2 hay

X =

x^-i

x = l±V33

-2x-32 = 0

So vdi d i e u k i e n la c6 n g h i e m cua phtfcfng trinh : x = 2 hay x = 1 - V33 .
CSu I I I .
0
Dat r = sinx

<5ff

= cosx.t/x ; D o i can
0

dt
()t^-5t + 6
1^

=1
0

dl

=1

„(t-2)(t-3)

t-2

t-3

1

1

^

dt = ( l n | l - 3 | - l n | t - 2 | )

= ln

t-3
t-2

3

24