tỔng hỢp ĐỀ thi Đh mÔn toÁn khỐi a, b,d (2002-2010)

8/3/2019 TNG HP THI H MN TON KHI A, B,D (2002-2010)

1/27

b gio dc v o to K thi tuyn sinh i hc, cao nG nm 2002------------------------------ Mn thi : ton

chnh thc (Thi gian lm bi: 180 pht)_____________________________________________

Cu I (H : 2,5 im; C : 3,0 im)

Cho hm s : (1) ( l tham s).23223 )1(33 mmxmmxxy +++= m

1. Kho st s bin thin v v th hm s (1) khi .1=m 2. Tm k phng trnh: c ba nghim phn bit.033 2323 =++ kkxx3. Vit phng trnh ng thng i qua hai im cc tr ca th hm s (1).Cu II.(H : 1,5 im; C: 2,0 im)

Cho phng trnh : 0121loglog 232

3 =++ mxx (2) ( l tham s).m

1 Gii phng trnh (2) khi .2=m

2. Tm phng trnh (2) c t nht mt nghim thuc on [m 33;1 ].Cu III. (H : 2,0 im; C : 2,0 im )

1. Tm nghim thuc khong )2;0( ca phng trnh: .32cos

2sin21

3sin3cossin +=

+

++ x

x

xxx5

2. Tnh din tch hnh phng gii hn bi cc ng: .3,|34| 2 +=+= xyxxyCu IV.( H : 2,0 im; C : 3,0 im)1. Cho hnh chp tam gic u nh c di cnh y bng a. GiABCS. ,S v ln ltN

l cc trung im ca cc cnh v Tnh theo din tch tam gic , bit rngSB .SC a AMNmt phng ( vung gc vi mt phng .)AMN )(SBC

2. Trong khng gian vi h to cac vung gc Oxyzcho hai ng thng:

v .

=++

=+

0422

042:1

zyx

zyx

+=

+=

+=

tz

ty

tx

21

2

1

:2

a) Vit phng trnh mt phng cha ng thng)(P 1 v song song vi ng thng .2

b) Cho im . Tm to im)4;1;2(M Hthuc ng thng 2 sao cho on thng H c di nh nht.

Cu V.( H : 2,0 im)1. Trong mt phng vi h to cac vung gc Oxy , xt tam gic vung ti , ABC A

phng trnh ng thng lBC ,033 = yx cc nh vA B thuc trc honh v

bn knh ng trn ni tip bng 2. Tm ta trng tm ca tam gic .G ABC 2. Cho khai trin nh thc:

nx

n

n

nxx

n

n

xn

x

n

nx

n

nxx

CCCC

+

++

+

=

+

3

1

32

1

13

1

2

1

12

1

032

1

22222222 L

( n l s nguyn dng). Bit rng trong khai trin C v s hng th t13 5 nn C=

bng , tm vn20 n x .----------------------------------------Ht---------------------------------------------

Ghi ch: 1)Th sinhch thi cao ngkhng lm Cu V.

2) Cn b coi thi khng gii thch g thm.

H v tn th sinh:.................................................... S bo danh:.....................


2/27

b gio dc v o to k thi tuyn sinh i hc, cao ng nm 2002 chnh thc Mn thi : ton, Khi B.

(Thi gian lm bi : 180 pht)_____________________________________________Cu I. (H : 2,0 im; C : 2,5 im)

Cho hm s : ( ) 109 224 ++= xmmxy (1) (m l tham s).1. Kho st s bin thin v v th ca hm s (1) khi 1=m .2. Tm m hm s (1) c ba im cc tr.

Cu II. (H : 3,0 im; C : 3,0 im)1. Gii phng trnh: xxxx 6cos5sin4cos3sin 2222 = .

2. Gii bt phng trnh: ( ) 1)729(loglog 3 xx .

3. Gii h phng trnh:

++=+

=

.2

3

yxyx

yxyx

Cu III. ( H : 1,0 im; C : 1,5 im) Tnh din tch ca hnh phng gii hn bi cc ng :

44

2xy = v

24

2xy = .

Cu IV.(H : 3,0 im ; C : 3,0 im)1. Trong mt phng vi h ta cac vung gc Oxy cho hnh ch nht ABCD c tm

0;

2

1I , phng trnh ng thng AB l 022 =+ yx v ADAB 2= . Tm ta cc nh

DCBA ,,, bit rng nh A c honh m.2. Cho hnh lp phng 1111 DCBABCDA c cnh bng a .

a) Tnh theo a khong cch gia hai ng thng BA1 v DB1 .

b) Gi PNM ,, ln lt l cc trung im ca cc cnh CDBB ,1 , 11DA . Tnh gc gia

hai ng thng MP v NC1 .

Cu V. (H : 1,0 im)Cho a gic u nAAA 221 L ,2( n n nguyn ) ni tip ng trn ( )O . Bit rng stam gic c cc nh l 3 trong n2 im nAAA 221 ,,, L nhiu gp 20 ln s hnh ch nht

c cc nh l 4 trong n2 imn

AAA221

,,, L , tm n .

--------------------------------------Ht-------------------------------------------Ghi ch : 1)Th sinhch thicao ngkhng lm Cu IV 2. b) v Cu V.2) Cn b coi thi khng gii thch g thm.

H v tn th sinh:................................................................... S bo danh:...............................


3/27

B gio dc v o to K thi Tuyn sinh i hc ,cao ng nm 2002 chnh thc Mn thi : Ton, Khi D

(Thi gian lm bi : 180 pht)_________________________________________

CuI ( H : 3 im ; C : 4 im ).Cho hm s :

( )1x

mx1m2y

2

= (1) ( m l tham s ).

1. Kho st s bin thin v v th (C) ca hm s (1) ng vi m = -1.

2. Tnh din tch hnh phng gii hn bi ng cong (C) v hai trc ta .3. Tm m thca hm s (1) tip xc vi ng thng xy = .

Cu II ( H : 2 im ; C : 3 im ).1. Gii bt phng trnh : ( )x3x

2

. 02x3x22

.

2. Gii h phng trnh :

=+

+

=+

.y22

24

y4y52

x

1xx

2x3

Cu III ( H : 1 im ; C : 1 im ).Tm x thuc on [ 0 ; 14 ] nghim ng phng trnh :

04xcos3x2cos4x3cos =+ .

Cu IV ( H : 2 im ; C : 2 im ).1. Cho hnh t din ABCD c cnh AD vung gc vi mt phng (ABC); AC = AD = 4 cm ;

AB = 3 cm ; BC = 5 cm . Tnh khong cch t im A ti mt phng (BCD).2. Trong khng gian vi h ta cac vung gc Oxyz, cho mt phng (P) : 02yx2 =+

v ng thng md :( ) ( )

( )

=++++

=+++

02m4z1m2mx

01mym1x1m2( m l tham s ).

Xc nh m ng thng md song song vi mt phng (P).Cu V (H : 2 im ).1. Tm s nguyn dng n sao cho 243C2....C4C2C nn

n2n

1n

0n =++++ .

2. Trong mt phng vi h ta cac vung gc Oxy , cho elip (E) c phng trnh

19

y

16

x 22=+ . Xt im M chuyn ng trn tia Ox v im N chuyn ng trn tia Oy sao cho

ng thng MN lun tip xc vi (E). Xc nh ta ca M , N on MN c di nhnht . Tnh gi tr nh nht .

-------------------------Ht-------------------------

Ch :1. Th sinh ch thi cao ng khng lm cu V2. Cn b coi thi khng gii thch g thm.

H v tn th sinh : ................................................................ S bo danh.............................


4/27

B gio dc v o to k thi tuyn sinh i hc, cao ng nm 2003

-------------------------- Mn thi : ton khi A chnh thc Thi gian lm bi : 180 pht

___________________________________

Cu 1 (2 im). Cho hm s mx

mxmxy ((1)

1

2

++= l tham s).

1) Kho s t s bin thin v v th hm s (1) khi m = 1.2) Tm m th hm s (1) ct trc honh ti hai im phn bit v hai im c honh

dng.Cu 2 (2 im).

1) Gii phng trnh .2sin2

1sin

tg1

2cos1cotg 2 xx

x

xx +

+=

2) Gii h phng trnh

+=

=

.12

11

3xy

yy

xx

Cu 3 (3 im).1) Cho hnh lp phng . Tnh s o ca gc phng nh din [ ].. ' ' ' ' ABCD A B C D DCAB ,',2) Trong khng gian vi h ta cac vung gc Ox cho hnh hp ch nht

c trng vi gc ca h ta ,

yz

; 0; 0. ' ' ' ' ABCD A B C D A ( ), (0; ; 0), '(0; 0; ) B a D a A b

. Gi( 0, 0)a b> > l trung im cnh CC .'

a) Tnh th tch khi t din 'BDA M theo a v b .

b) Xc nh t sa

b hai mt phng v( ' ) A BD ( )BD vung gc vi nhau.

Cu 4 ( 2 im).

1) Tm h s ca s hng chax8

trong khai trin nh thc Niutn can

xx

+ 5

3

1 , bit rng

)3(73

14

+= +++ nCC

nn

nn

( n l s nguyn dng,x> 0, l s t hp chp k ca n phn t).knC

2) Tnh tch phn

+=

32

52 4xx

dxI .

Cu 5 (1 im).Cho x,y, z l ba s dng v x+y + z 1. Chng minh rng

.821

1

1

2

2

2

2

2

2 +++++z

zy

yx

x

HT

Ghi ch: Cn b coi thi khng gii thch g thm.

H v tn th sinh: .. . S bo danh: .


5/27

B gio dc v o to k thi tuyn sinh i hc, cao ng nm 2003----------------------- Mn thi : ton khi B

chnh thc Thi gian lm bi: 180 pht_______________________________________________

Cu 1 (2 im). Cho hm s ( l tham s).3 23 (1) y x x m= + m1) Tm th hm s (1) c hai im phn bit i xng vi nhau qua gc ta .m2) Kho st s bin thin v v th hm s (1) khi m =2.

Cu 2 (2 im).

1) Gii phng trnh2

otg tg 4sin 2sin2

x x xcx

+ = .

2) Gii h phng trnh

2

2

2

2

23

23 .

yy

x

xx

y

+=

+=

Cu 3 (3 im).1) Trong mt phng vi h ta cac vung gc Ox cho tam gic cy ABC

0, 90 . AB AC BAC = = Bit (1; 1)M l trung im cnh BC v2

; 03

G l trng

tm tam gic . Tm ta cc nh .

ABC , , A B C

2) Cho hnh lng tr ng c y l mt hnh thoi cnh ,

gc

. ' ' ' ' ABCD A B C D ABCD a

060BAD = . Gi l trung im cnh v l trung im cnh ' .Chng minh rng bn im

' N AA CC

', , ,B M D N

'

cng thuc mt mt phng. Hy tnh

di cnh ' theo a t gic AA B MDN l hnh vung.

3) Trong khng gian vi h ta cac vung gc Ox cho hai im

v im sao cho . Tnh khong cch t

trung im

yz

0)(2; 0; 0), (0; 0; 8)A B C (0; 6;AC

=

I ca BC n ng thng OA .Cu 4 (2 im).

1) Tm gi tr ln nht v nh nht ca hm s 24 . y x x= +

2) Tnh tch phn

4 2

0

1 2sin

1 sin 2 I dxx

= + .Cu 5 (1 im). Cho l s nguyn dng. Tnh tngn

2 3 10 1 22 1 2 1 2 1

2 3 1

nn

n n nC C Cn

+ + + + +

+ nC

( C l s t hp chp k ca phn t).kn n

----------------------------------Ht---------------------------------

Ghi ch: Cn b coi thi khng gii thch g thm.

H v tn th sinh.. S bo danh


6/27

B gio dc v o to k thi tuyn sinh i hc, cao ng nm 2003---------------------- Mn thi: ton Khi D chnh thc Thi gian lm bi: 180 pht

_______________________________________________

Cu 1 (2 im).1) Kho st s bin thin v v th ca hm s

2 2 4(1)

2

x xy

x

+=

.

2) Tm ng thng d ym : 2 2m mx m= + ct th ca hm s (1) ti hai im

phn bit.Cu 2 (2 im).

1) Gii phng trnh 2 2 2

sin tg cos 02 4 2

x xx

=

.

2) Gii phng trnh .2 222 2 x x x x + = 3

Cu 3 (3 im).1) Trong mt phng vi h ta cac vung gc cho ng trnOxy

4)2()1(:)( 22 =+ yxC v ng thng : 1 0d x y = .

Vit phng trnh ng trn ( i xng vi ng trn qua ng thng

Tm ta cc giao im ca v .

')C

(C

( )C .d

) ( ')C

2) Trong khng gian vi h ta cac vung gc Oxyz cho ng thng

3 2:

1 0.k

x ky z d

kx y z

0+ + =

+ + =

Tm ng thng vung gc vi mt phngk kd ( ) : 2 5 0 P x y z + = .

3) Cho hai mt phng v vung gc vi nhau, c giao tuyn l ng thng( )P ( )Q .Trn ly hai im vi , A B AB a= . Trong mt phng ly im , trong

mt phng ( ly im sao cho ,

( )P C

)Q D AC BD cng vung gc vi v

. Tnh bn knh mt cu ngoi tip t din v tnh khongcch t n mt phng AC BD

A

AB== ABCD( )BCD theo .a

Cu 4 ( 2 im).

1) Tm gi tr ln nht v gi tr nh nht ca hm s2

1

1

xy

x

+=

+trn on [ ]1; 2 .

2) Tnh tch phn2

2

0

I x x d = x .

Cu 5 (1 im).

Vi l s nguyn dng, gin 3 3na l h s ca3 3nx trong khai trin thnh a

thc ca ( 1 . Tm n 2 ) ( 2)nx x+ + n 3 3 26na n= .

------------------------------------------------ Ht ------------------------------------------------Ghi ch: Cn b coi thi khng gii thch g thm.

H v tn th sinh:.. . S bo danh:


7/27

B gio dc v o to thi tuyn sinh i hc, cao ng nm 2004------------------------------ Mn thi : Ton , Khi A

chnh thc Thi gian lm bi : 180 pht, khng k thi gian pht --------------------------------------------------------------

Cu I (2 im)

Cho hm s2x 3x 3

y2(x 1)

+ =

(1).

1) Kho st hm s (1).2) Tm m ng thng y = m ct th hm s (1) ti hai im A, B sao cho AB = 1.

Cu II (2 im)

1) Gii bt phng trnh 22(x 16) 7 xx 3 >x 3 x 3

+

.

2) Gii h phng trnh 1 442 2

1

log (y x) log 1y

x y 25.

=

+ =

Cu III (3 im)

1) Trong mt phng vi h ta Oxy cho hai im ( )A 0; 2 v ( )B 3; 1 . Tm ta trctm v ta tm ng trn ngoi tip ca tam gic OAB.

2) Trong khng gian vi h ta Oxyz cho hnh chp S.ABCD c y ABCD l hnh thoi,AC ct BD ti gc ta O. Bit A(2; 0; 0), B(0; 1; 0), S(0; 0; 2 2 ). Gi M l trung im

ca cnh SC.a) Tnh gc v khong cch gia hai ng thng SA, BM.

b) Gi s mt phng (ABM) ct ng thng SD ti im N. Tnh th tch khi chp S.ABMN.

Cu IV (2 im)

1) Tnh tch phn I =2

1

xdx

1 x 1+ .

2) Tm h s ca x8 trong khai trin thnh a thc ca8

21 x (1 x) + .

Cu V (1 im)

Cho tam gic ABC khng t, tha mn iu kin cos2A + 2 2 cosB + 2 2 cosC = 3.Tnh ba gc ca tam gic ABC.

------------------------------------------------------------------------------------------------------------------------

Cn b coi thi khng gii thch g thm.

H v tn th sinh............................................................................S bo danh.................................................


8/27

B gio dc v o to------------------------

chnh thc

thi tuyn sinh i hc, cao ng nm 2004Mn: Ton, Khi B

Thi gian lm bi: 180 pht, khng k thi gian pht -------------------------------------------

Cu I (2 im)

Cho hm s y = xxx 3231 23 + (1) c th (C).

1) Kho st hm s (1).2)Vit phng trnh tip tuyn ca (C) ti im un v chng minh rng l tip tuyn ca (C)

c h s gc nh nht.

Cu II (2 im)

1) Gii phng trnh xtgxx 2)sin1(32sin5 = .2) Tm gi tr ln nht v gi tr nh nht ca hm s

x

xy

2ln= trn on [1; 3e ].

Cu III (3 im)1) Trong mt phng vi h ta Oxy cho hai im A(1; 1), B(4; 3 ). Tm im C thuc ng

thng 012 = yx sao cho khong cch t C n ng thng AB bng 6.

2) Cho hnh chp t gic u S.ABCD c cnh y bng a, gc gia cnh bn v mt y bng ( o0 < < o90 ). Tnh tang ca gc gia hai mt phng (SAB) v (ABCD) theo . Tnh thtch khi chp S.ABCD theo a v .

3) Trong khng gian vi h ta Oxyz cho im A )4;2;4( v ng thng d:

+=

=

+=

.41

1

23

tz

ty

tx

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

Cu IV (2 im)

1) Tnh tch phn I = dxx

xxe

+

1

lnln31.

2) Trong mt mn hc, thy gio c 30 cu hi khc nhau gm 5 cu hi kh, 10 cu hi trungbnh, 15 cu hi d. T 30 cu hi c th lp c bao nhiu kim tra, mi gm 5 cu

hi khc nhau, sao cho trong mi nht thit phi c 3 loi cu hi (kh, trung bnh, d) vs cu hi d khng t hn 2 ?

Cu V (1 im)Xc nh m phng trnh sau c nghim

22422 1112211 xxxxxm ++=

++ .

------------------------------------------------------------------------------------------------------------------------


H v tn th sinh.................................................................................................S bo danh...........................


9/27

B gio dc v o to thi tuyn sinh i hc, cao ng nm 2004------------------------ Mn: Ton, Khi D chnh thc Thi gian lm bi: 180 pht, khng k thi gian pht

-------------------------------------------Cu I (2 im)

Cho hm s 3 2y x 3mx 9x 1= + + (1) vi m l tham s.1) Kho st hm s (1) khi m = 2.2) Tm m im un ca th hm s (1) thuc ng thng y = x + 1.

Cu II (2 im)1) Gii phng trnh .sin2sin)cossin2()1cos2( xxxxx =+

2) Tm m h phng trnh sau c nghim

=+

=+

.31

1

myyxx

yx

Cu III (3 im)1) Trong mt phng vi h ta Oxy cho tam gic ABC c cc nh );0();0;4();0;1( mCBA

vi 0m . Tm ta trng tm G ca tam gic ABC theo m. Xc nh m tam gic GABvung ti G.

2) Trong khng gian vi h ta Oxyz cho hnh lng tr ng 111. CBAABC . Bit ),0;0;(aA

0,0),;0;(),0;1;0(),0;0;( 1 >> babaBCaB .

a) Tnh khong cch gia hai ng thng CB1 v 1AC theo .,ba

b) Cho ba, thay i, nhng lun tha mn 4=+ ba . Tm ba, khong cch gia hai ng

thng CB1 v 1AC ln nht.

3) Trong khng gian vi h ta Oxyz cho ba im )1;1;1(),0;0;1(),1;0;2( CBA v mtphng (P): 02 =++ zyx . Vit phng trnh mt cu i qua ba im A, B, C v c tmthuc mt phng (P).

Cu IV (2 im)

1) Tnh tch phn I = 3

2

2 )ln( dxxx .

2) Tm cc s hng khng cha x trong khai trin nh thc Niutn ca

7

43 1

+

x

x vi x > 0.

Cu V (1 im)Chng minh rng phng trnh sau c ng mt nghim

01225 = xxx .

---------------------------------------------------------------------------------------------------------------------Cn b coi thi khng gii thch g thm.

H v tn th sinh.............................................................S bo danh........................................


10/27

B GIO DC V O TO-----------------------

CHNH THC

THI TUYN SINH I HC, CAO NG NM 2005Mn: TON, khi A

Thi gian lm bi: 180 pht, khng k thi gian pht ----------------------------------------

Cu I (2 im)

Gi m(C ) l th ca hm s1

y m xx

= + (*) ( m l tham s).

1) Kho st s bin thin v v th ca hm s (*) khi 1m .4

=

2) Tm m hm s (*) c cc tr v khong cch tim cc tiu ca m(C ) n timcn xin ca m(C ) bng

1.

2

Cu II (2 im)

1) Gii bt phng trnh 5x 1 x 1 2x 4. >

2) Gii phng trnh 2 2cos 3x cos 2x cos x 0. =

Cu III (3 im)1) Trong mt phng vi h ta Oxy cho hai ng thng

1d : x y 0 = v 2d : 2x y 1 0.+ =

Tm ta cc nh hnh vung ABCD bit rng nh A thuc 1d , nh C thuc 2d

v cc nh B, D thuc trc honh.

2) Trong khng gian vi h ta Oxyz cho ng thng x 1 y 3 z 3d :1 2 1

+ = =

v mt

phng (P) : 2x y 2z 9 0.+ + =

a)Tm ta im I thuc d sao cho khong cch t I n mt phng (P) bng 2. b)Tm ta giao im A ca ng thng d v mt phng (P). Vit phng trnh

tham s ca ng thng nm trong mt phng (P), bit i qua A v vung

gc vi d.

Cu IV (2 im)

1) Tnh tch phn 20

sin 2x sin xI dx.

1 3cos x

+=

+

2) Tm s nguyn dng n sao cho1 2 2 3 3 4 2n 2n 1

2n 1 2n 1 2n 1 2n 1 2n 1C 2.2C 3.2 C 4.2 C (2n 1).2 C 2005+

+ + + + + + + + + =L

( knC l s t hp chp k ca n phn t).

Cu V (1 im)

Cho x, y, z l cc s dng tha mn1 1 1

4.x y z

+ + = Chng minh rng

1 1 11.

2x y z x 2y z x y 2z+ +

+ + + + + +

------------------------------ Ht -----------------------------Cn b coi thi khng gii thch g thm.

H v tn th sinh................................................. s bo danh........................................

Mang Giao duc Edunet - http://www.edu.net.vn


11/27

B GIO DC V O TO-------------------------

CHNH THC

THI TUYN SINH I HC, CAO NG NM 2005Mn: TON, khi B

Thi gian lm bi: 180 pht, khng k thi gian pht --------------------------------------------------

Cu I (2 im)

Gim

(C ) l th ca hm s( )2x m 1 x m 1

y

x 1

+ + + +=

+

(*) ( m l tham s).

1) Kho st s bin thin v v th ca hm s (*) khi m 1.= 2) Chng minh rng vi mbt k, th

m(C ) lun lun c im cc i, im cc tiu

v khong cch gia hai im bng 20.

Cu II (2 im)

1) Gii h phng trnh( )2 39 3

x 1 2 y 1

3log 9x log y 3.

+ =

=

2) Gii phng trnh 1 sin x cos x sin 2x cos2x 0.+ + + + = Cu III (3 im)

1) Trong mt phng vi h ta Oxycho hai im A(2;0) v B(6;4) . Vit phng trnhng trn (C) tip xc vi trc honh ti im A v khong cch t tm ca (C) nim B bng 5.

2) Trong khng gian vi h ta Oxyz cho hnh lng tr ng 1 1 1ABC.A B C vi1

A(0; 3;0), B(4;0;0), C(0;3;0), B (4;0;4).

a)Tm ta cc nh 1 1A , C . Vit phng trnh mt cu c tm l A v tip xc vimt phng

1 1(BCC B ).

b)Gi M l trung im ca 1 1A B . Vit phng trnh mt phng (P) i qua hai imA, M v song song vi 1BC . Mt phng (P) ct ng thng 1 1A C ti im N .

Tnh di on MN.

Cu IV (2 im)

1) Tnh tch phn 20

sin2x cosxI dx

1 cosx

=+

.

2) Mt i thanh nin tnh nguyn c 15 ngi, gm 12 nam v 3 n. Hi c bao nhiucch phn cng i thanh nin tnh nguyn v gip 3 tnh min ni, sao cho mitnh c 4 nam v 1 n?

Cu V (1 im)

Chng minh rng vi mi x , ta c:x x x

x x x12 15 20 3 4 55 4 3

+ + + +

.

Khi no ng thc xy ra?

--------------------------------Ht--------------------------------


H v tn th sinh .................................................. S bo danh ...............................



12/27

B GIO DC V O TO-----------------------

CHNH THC

THI TUYN SINH I HC, CAO NG NM 2005Mn: TON, khi D

Thi gian lm bi: 180 pht, khng k thi gian pht -------------------------------------------

Cu I (2 im)Gi

m(C ) l th ca hm s 3 2

1 m 1y x x

3 2 3= + (*) ( m l tham s).

1) Kho st s bin thin v v th ca hm s (*) khi m 2.= 2) Gi M l im thucm

(C ) c honh bng 1. Tm m tip tuyn cam

(C ) ti

im M song song vi ng thng 5x y 0. =

Cu II (2 im)Gii cc phng trnh sau:

1) 2 x 2 2 x 1 x 1 4.+ + + + = 2) 4 4

3cos x sin x cos x sin 3x 0.

4 4 2

+ + =

Cu III (3 im)

1) Trong mt phng vi h ta Oxy cho im ( )C 2;0 v elp ( ) 2 2x yE : 1.4 1

+ = Tm

ta cc im A, B thuc ( )E , bit rng hai im A, B i xng vi nhau qua trc

honh v tam gic ABC l tam gic u.2) Trong khng gian vi h ta Oxyz cho hai ng thng

1

x 1 y 2 z 1d :

3 1 2

+ += =

v 2

x y z 2 0d :

x 3y 12 0.

+ =

+ =

a) Chng minh rng 1d v 2d song song vi nhau. Vit phng trnh mt phng(P) cha c hai ng thng 1d v 2d .

b) Mt phng ta Oxz ct hai ng thng 1 2d , d ln lt ti cc im A, B. Tnh

din tch tam gic OAB ( O l gc ta ).

Cu IV (2 im)

1) Tnh tch phn ( )2 sin x0

I e cos x cos xdx.

= +

2) Tnh gi tr ca biu thc ( )4 3

n 1 nA 3A

Mn 1 !+

+

=+

, bit rng2 2 2 2n 1 n 2 n 3 n 4C 2C 2C C 149+ + + ++ + + =

( n l s nguyn dng, kn

A l s chnh hp chp k ca n phn t v kn

C l s t hp

chp k ca n phn t).

Cu V (1 im)Cho cc s dng x, y, z tha mn xyz 1.= Chng minh rng

3 3 3 3 3 31 x y 1 y z 1 z x3 3.

xy yz zx

+ + + + + ++ +

Khi no ng thc xy ra?-------------------------------Ht--------------------------------Cn bcoi thi khng gii thch g thm.

H v tn th sinh.............................................. S bo danh..........................................



13/27

B GIO DC V O TO

CHNH THC

THI TUYN SINH I HC, CAO NG NM 2006

Mn thi: TON, khi AThi gian lm bi: 180 pht, khng k thi gian pht

PHN CHUNG CHO TT C CC TH SINHCu I (2 im)

1. Kho st s bin thin v v th ca hm s 3 2y 2x 9x 12x 4.= + 2. Tm m phng trnh sau c 6 nghim phn bit: 3 22 x 9x 12 x m. + =

Cu II (2 im)

1. Gii phng trnh: ( )6 62 cos x sin x sin x cos x

0.2 2sin x

+

=

2. Gii h phng trnh: ( )x y xy 3 x, y .x 1 y 1 4

+ =

+ + + =

Cu III (2 im)Trong khng gian vi h ta Oxyz , cho hnh l p phng ABCD.A'B'C'D' vi

( ) ( ) ( ) ( )A 0; 0; 0 , B 1; 0; 0 , D 0; 1; 0 , A ' 0; 0; 1 . Gi M v N ln lt l trung im ca AB

v CD .

1. Tnh khong cch gia hai ng thng A'C v MN.2. Vit phng trnh mt phng cha A 'C v to vi mt phng Oxy mt gc

bit1

cos .6

=

Cu IV (2 im)

1. Tnh tch phn: 22 20

sin2xI dx.

cos x 4sin x

=

+

2. Cho hai s thc x 0, y 0 thay i v tha mn iu kin: ( ) 2 2x y xy x y xy+ = + .Tm gi tr ln nht ca biu thc

3 3

1 1A .

x y= +

PHN TCHN: Th sinh chn cu V.a hoc cu V.bCu V.a. Theo chng trnh THPT khng phn ban (2 im)

1. Trong mt phng vi h ta Oxy, cho cc ng thng:1 2 3d : x y 3 0, d : x y 4 0, d : x 2y 0.+ + = = =

Tm ta im M nm trn ng thng 3d sao cho khong cch t M n ng thng

1d bng hai ln khong cch t M n ng thng 2d .

2. Tm h s ca s hng cha 26x trong khai trin nh thc Niutn ca n74

1x ,

x

+

bit

rng 1 2 n 202n 1 2n 1 2n 1C C ... C 2 1.+ + ++ + + =

(n nguyn dng, knC l s t hp chp k ca n phn t)

Cu V.b. Theo chng trnh THPT phn ban thim (2 im)1. Gii phng trnh: x x x x3.8 4.12 18 2.27 0.+ = 2. Cho hnh tr c cc y l hai hnh trn tm O v O ' , bn knh y bng chiu cao v

bng a. Trn ng trn y tm O ly im A, trn ng trn y tm O ' ly im Bsao cho AB 2a.= Tnh th tch ca khi t din OO'AB.

---------------------------------------Ht---------------------------------------Cn b coi thi khng gii thch g thm.H v tn th sinh: .......................................................... s bo danh: ..................................


14/27

B GIO DC V O TO

CHNH THC


Mn: TON, khi BThi gian lm bi: 180 pht, khng k thi gian pht

PHN CHUNG CHO TT C CC TH SINHCu I (2 im)

Cho hm s2x x 1y .x 2

+ =

+

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 tim cn xin

ca ( )C . Cu II (2 im)

1. Gii phng trnh: xcotgx sin x 1 tgxtg 4.2

+ + =

2. Tm m phng trnh sau c hai nghim thc phn bit:2

x mx 2 2x 1.+ + = + Cu III (2 im)

Trong khng gian vi h ta Oxyz, cho im A(0; 1; 2) v hai ng thng:

1 2

x 1 tx y 1 z 1

d : , d : y 1 2t2 1 1

z 2 t.

= + +

= = =

= +

1. Vit phng trnh mt phng (P) qua A, ng thi song song vi d1 v d2.2. Tm ta cc im M thuc d1, N thuc d2 sao cho ba im A, M, N thng hng.

Cu IV (2 im)

1. Tnh tch phn:ln5

x xln3

dxIe 2e 3

=+ .

2. Cho x, y l cc s thc thay i. Tm gi tr nh nht ca biu thc:( ) ( )

2 22 2A x 1 y x 1 y y 2 .= + + + + +

PHN TCHN: Th sinh chn cu V.a hoc cu V.bCu V.a. Theo chng trnh THPT khng phn ban (2 im)

1. Trong mt phng vi h ta Oxy, cho ng trn ( ) 2 2C : x y 2x 6y 6 0+ + = v im( )M 3; 1 . Gi 1T v 2T l cc tip im ca cc tip tuyn k t M n ( )C . Vit phng

trnh ng thng 1 2T T .2. Cho tp hp A gm n phn t ( )n 4 . Bit rng, s tp con gm 4 phn t ca A bng

20 ln s tp con gm 2 phn t ca A. Tm { }k 1, 2,..., n sao cho s tp con gm k phn

t ca A l ln nht.Cu V.b. Theo chng trnh THPT phn ban thim (2 im)

1. Gii bt phng trnh: ( ) ( )x x 25 5 5log 4 144 4log 2 1 log 2 1 .+ < + + 2. Cho hnh chp S.ABCD c y ABCD l hnh ch nht vi AB a, AD a 2= = , SA a= v

SA vung gc vi mt phng ( )ABCD . Gi M v N ln lt l trung im ca AD v SC;

I l giao im ca BM v AC. Chng minh rng mt phng (SAC) vung gc vi mtphng (SMB). Tnh th tch ca khi t din ANIB.

----------------------------- Ht -----------------------------Cn b coi thi khng gii thch g thm.H v tn th sinh.................................................................... s bo danh..............................................


15/27

B GIO DC V O TO

CHNH THC


Mn: TON, khi DThi gian lm bi: 180 pht, khng k thi gian pht

PHN CHUNG CHO TT C CC TH SINH

Cu I (2 im)

Cho hm s3

y x 3x 2= + .1. Kho st s bin thin v v th (C) ca hm s cho.2. Gi d l ng thng i qua im A(3; 20) v c h s gc l m. Tm m ng thng d

ct th (C) ti 3 im phn bit.

Cu II (2 im)1. Gii phng trnh: cos3x cos2x cosx 1 0.+ = 2. Gii phng trnh: ( )22x 1 x 3x 1 0 x . + + =

Cu III (2 im)Trong khng gian vi h ta Oxyz, cho im A(1;2;3) v hai ng thng:

1 2

x 2 y 2 z 3 x 1 y 1 z 1d : , d : .

2 1 1 1 2 1

+ += = = =

1. Tm ta im A' i xng vi im A qua ng thng d1.2. Vit phng trnh ng thng i qua A, vung gc vi d1 v ct d2.

Cu IV (2 im)

1. Tnh tch phn: ( )1 2x0

I x 2 e dx.=

2. Chng minh rng vi mi a 0> , h phng trnh sau c nghim duy nht:x ye e ln(1 x) ln(1 y)y x a.

= + +

=

PHN TCHN: Th sinh chn cu V.a hoc cu V.b

Cu V.a. Theo chng trnh THPT khng phn ban (2 im)1. Trong mt phng vi h ta Oxy, cho ng trn (C): 2 2x y 2x 2y 1 0+ + = v

ng thng d: x y 3 0. + = Tm ta im M nm trn d sao cho ng trn tm M, cbn knh gp i bn knh ng trn (C), tip xc ngoi vi ng trn (C).

2. i thanh nin xung kch ca mt trng ph thng c 12 hc sinh, gm 5 hc sinh lp A,4 hc sinh lp B v 3 hc sinh lp C. Cn chn 4 hc sinh i lm nhim v, sao cho 4hc sinh ny thuc khng qu 2 trong 3 lp trn. Hi c bao nhiu cch chn nh vy?

Cu V.b. Theo chng trnh THPT phn ban thim (2 im)

1. Gii phng trnh: 2 2x x x x 2x2 4.2 2 4 0.+ + = 2. Cho hnh chp tam gic S.ABC c y ABC l tam gic u cnh a, SA = 2a v SA vung

gc vi mt phng (ABC). Gi M v N ln lt l hnh chiu vung gc ca A trn ccng thng SB v SC. Tnh th tch ca khi chp A.BCNM.

----------------------------- Ht -----------------------------Cn b coi thi khng gii thch g thm.

H v tn th sinh............................................................. s bo danh.....................................................


16/27

B GIO DC V O TO

CHNH THC

THI TUYN SINH I HC, CAO NG NM 2007Mn thi: TON, khi A

Thi gian lm bi: 180 pht, khng k thi gian pht

PHN CHUNG CHO TT C CC TH SINH

Cu I (2 im)

Cho hm s2 2x 2(m 1)x m 4m

y (1),

x 2

+ + + +=

+ m l tham s.

1. Kho st s bin thin v v th ca hm s (1) khi m 1= .2. Tm m hm s (1) c cc i v cc tiu, ng thi cc im cc tr ca th cng vi gc ta

O to thnh mt tam gic vung ti O.

Cu II (2 im)

1. Gii phng trnh: ( ) ( )2 21 sin x cos x 1 cos x sin x 1 sin 2x.+ + + = + 2. Tm m phng trnh sau c nghim thc: 4 23 x 1 m x 1 2 x 1. + + =

Cu III (2 im)Trong khng gian vi h ta Oxyz, cho hai ng thng

1

x y 1 z 2d :

2 1 1

+= =

v 2

x 1 2t

d : y 1 t

z 3.

= +

= +

=

1. Chng minh rng 1d v 2d cho nhau.2. Vit phng trnh ng thng d vung gc vi mt phng ( )P : 7x y 4z 0+ = v ct hai ng

thng 1d , 2d .

Cu IV (2 im)

1. Tnh din tch hnh phng gii hn bi cc ng: ( )y e 1 x,= + ( )xy 1 e x.= + 2. Cho x, y, z l cc s thc dng thay i v tha mn iu kin xyz 1.= Tm gi tr nh nht ca

biu thc:2 2 2x (y z) y (z x) z (x y)

Py y 2z z z z 2x x x x 2y y

+ + += + +

+ + +

PHN TCHN: Th sinh chc chn lm cu V.a hoc cu V.b

Cu V.a. Theo chng trnh THPT khng phn ban (2 im)

1. Trong mt phng vi h ta Oxy, cho tam gic ABC c A(0; 2), B(2; 2) v C(4; 2). Gi H lchn ng cao k t B; M v N ln lt l trung im ca cc cnh AB v BC. Vit phng trnhng trn i qua cc im H, M, N.

2. Chng minh rng: 2n1 3 5 2n 12n 2n 2n 2n1 1 1 1 2 1C C C ... C2 4 6 2n 2n 1

+ + + + =

+

( n l s nguyn dng, knC l s t hp chp k ca n phn t).


1. Gii bt phng trnh: 3 13

2log (4x 3) log (2x 3) 2. + +

2.Cho hnh chp S.ABCD c y l hnh vung cnh a, mt bn SAD l tam gic u v nm trongmt phng vung gc vi y. Gi M, N, P ln lt l trung im ca cc cnh SB, BC, CD. Chngminh AM vung gc vi BP v tnh th tch ca khi t din CMNP.

---------------------------Ht---------------------------Cn b coi thi khng gii thch g thm.H v tn th sinh: ..s bo danh: .


17/27

B GIO DC V O TO

CHNH THC

THI TUYN SINH I HC, CAO NG NM 2007Mn thi: TON, khi B


PHN CHUNG CHO TT C TH SINH

Cu I. (2 im)Cho hm s: 3 2 2 2y x 3x 3(m 1)x 3m 1= + + (1), m l tham s.

1. Kho st s bin thin v v th ca hm s (1) khi m = 1.2. Tm m hm s (1) c cc i, cc tiu v cc im cc tr ca th hm s (1) cch u

gc ta O.

Cu II. (2 im)1. Gii phng trnh: 22sin 2x sin 7x 1 sin x.+ = 2. Chng minh rng vi mi gi tr dng ca tham s m, phng trnh sau c hai nghim thc

phn bit:

( )2x 2x 8 m x 2 .+ =

Cu III. (2 im)Trong khng gian vi h ta Oxyz, cho mt cu ( ) 2 2 2S : x y z 2x 4y 2z 3 0+ + + + = v

mt phng ( )P : 2x y 2z 14 0. + =

1. Vit phng trnh mt phng ( )Q cha trc Ox v ct ( )S theo mt ng trn c bn knhbng 3.

2. Tm ta im M thuc mt cu ( )S sao cho khong cch t M n mt phng ( )P ln nht.Cu IV. (2 im)

1.Cho hnh phng H gii hn bi cc ng: y x ln x, y 0, x e.

= = =

Tnh th tch ca khi trnxoay to thnh khi quay hnh H quanh trc Ox.

2. Cho x, y, z l ba s thc dng thay i. Tm gi tr nh nht ca biu thc:x 1 y 1 z 1

P x y z .2 yz 2 zx 2 xy

= + + + + +

PHN TCHN (Th sinh chc chn lm mt trong hai cu: V.a hoc V.b)


1. Tm h s ca s hng cha 10x trong khai trin nh thc Niutn ca n(2 x) ,+ bit:( )nn 0 n 1 1 n 2 2 n 3 3 nn n n n n3 C 3 C 3 C 3 C ... 1 C 2048 + + + =

(n l s nguyn dng, knC l s t hp ch p k ca n phn t).

2. Trong mt phng vi h ta Oxy, cho im ( )A 2; 2 v cc ng thng:d1: x + y 2 = 0, d2: x + y 8 = 0.

Tm ta cc im B v C ln lt thuc d1 v d2 sao cho tam gic ABC vung cn ti A.


1. Gii phng trnh: ( ) ( )x x2 1 2 1 2 2 0. + + = 2.

Cho hnh chp t gic u S.ABCD c y l hnh vung cnh a. Gi E l im i xng ca Dqua trung im ca SA, M l trung im ca AE, N l trung im ca BC. Chng minh MN vunggc vi BD v tnh (theo a) khong cch gia hai ng thng MN v AC.

---------------------------Ht---------------------------Cn b coi thi khng gii thch g thm.H v tn th sinh: ..S bo danh: .


18/27

B GIO DC V O TO

CHNH THC

THI TUYN SINH I HC, CAO NG NM 2007Mn thi: TON, khi D


PHN CHUNG CHO TT C TH SINH

Cu I. (2 im)

Cho hm s2x

y .x 1

=+

1.

Kho st s bin thin v v th ( )C ca hm s cho.2. Tm ta im M thuc (C), bit tip tuyn ca (C) ti M ct hai trc Ox, Oy ti A, B v tam gic

OAB c din tch bng1

.4

Cu II. (2 im)

1. Gii phng trnh: 2x xsin cos 3 cos x 2.2 2

+ + =

2. Tm gi tr ca tham s m h phng trnh sau c nghim thc:

3 3

3 3

1 1x y 5

x y

1 1x y 15m 10.

x y

+ + + =

+ + + =

Cu III. (2 im)Trong khng gian vi h ta Oxyz, cho hai im ( ) ( )A 1;4;2 ,B 1;2;4 v ng thng

x 1 y 2 z: .

1 1 2

+ = =

1. Vit phng trnh ng thng d i qua trng tm G ca tam gic OAB v vung gc vi mtphng ( )OAB .

2. Tm ta im M thuc ng thng sao cho 2 2MA MB+ nh nht.Cu IV. (2 im)

1. Tnh tch phn: e 3 21

I x ln xdx.=

2. Cho a b 0. > Chng minh rng: aa ba b

1 12 2 .

2 2

+ +

PHN TCHN (Th sinh chc chn lm mt trong hai cu: V.a hoc V.b)


1. Tm h s ca 5x trong khai trin thnh a thc ca: ( ) ( )5 102

x 1 2x x 1 3x . + +

2. Trong mt phng vi h ta Oxy, cho ng trn ( ) ( ) ( )2 2C : x 1 y 2 9 + + = v ng thngd : 3x 4y m 0. + =

Tm m trn d c duy nht mt im P m t c th kc hai tip tuyn PA,PB ti ( )C

(A, B l cc tip im) sao cho tam gic PAB u.


1. Gii phng trnh: ( )x x2 2 x1log 4 15.2 27 2log 0.4.2 3+ + + = 2.

Cho hnh chp S.ABCD c

y l hnh thang,

0

ABC BAD 90 ,= =

BA = BC = a, AD = 2a. Cnh

bn SA vung gc vi y v SA = a 2. Gi H l hnh chiu vung gc ca A trn SB. Chngminh tam gic SCD vung v tnh (theo a) khong cch t H n mt phng ( )SCD .

---------------------------Ht---------------------------Cn b coi thi khng gii thch g thm.H v tn th sinh: ..S bo danh: .


19/27

B GIO DC V O TO THI TUYN SINH I HC, CAO NG NM 2008Mn thi: TON, khi A

Thi gian lm bi 180 pht, khng k thi gian pht

PHN CHUNG CHO TT C TH SINHCu I (2 im)

Cho hm s2 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 o45 .

Cu II (2 im)

1. Gii phng trnh1 1 7

4sin x .3s inx 4

sin x2

+ =

2. Gii h phng trnh ( )

2 3 2

4 2

5x y x y xy xy

4 x, y .5

x y xy(1 2x)4

+ + + + =

+ + + =

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

x 1 y z 2d : .

2 1 2

= =

1. Tm ta hnh chiu vung gc ca im A trn ng thng d. 2. Vit phng trnh mt phng () cha d sao cho khong cch t A n () ln nht.

Cu IV (2 im)

1. Tnh tch phn

46

0

tg 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 ).

PHN RING __________ Th sinh chc lm 1 trong 2 cu: V.a hoc V.b __________Cu V.a. Theo chng trnh KHNG phn ban (2 im)

1. Trong mt phng vi h ta Oxy, hy vit phng trnh chnh tc ca elp (E) bit rng

(E) c tm sai bng5

3

v hnh ch nht csca (E) c chu vi bng 20.

2. Cho khai trin ( )n n

0 1 n1 2x a a x ... a x ,+ = + + + trong *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 .

Cu V.b. Theo chng trnh phn ban (2 im)1. Gii phng trnh 2 22x 1 x 1log (2x x 1) log (2x 1) 4. ++ + =

2. Cho lng tr ABC.A'B'C' c di cnh bn bng 2a, y ABC l tam gic vung ti A,

AB = a, AC = a 3 v hnh chiu vung gc ca nh A ' trn mt phng (ABC) l trungim ca cnh BC. Tnh theo a th tch khi chp A'.ABC v tnh cosin ca gc gia haing thng AA', B'C' .

...........................Ht...........................

Th sinh khng c sdng ti liu. Cn b coi thi khng gii thch g thm.H v tn th sinh:........................................................ S bo danh:...............................................

CHNH THC


20/27

B GIO DC V O TO THI TUYN SINH I HC, CAO NG NM 2008Mn thi: TON, khi B


PHN CHUNG CHO TT C TH SINHCu I (2 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 ti p tuyn ca th hm s (1), bit rng ti p tuyn i quaim ( )M 1; 9 .

Cu II (2 im)1. Gii phng trnh 3 3 2 2sin x 3cos x sinxcos x 3sin xcosx. =

2. Gii h phng trnh4 3 2 2

2

x 2x y x y 2x 9

x 2xy 6x 6

+ + = +

+ = + ( )x, y .

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

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

2. Tm ta ca im M thuc mt phng 2x 2y z 3 0+ + = sao cho MA MB MC.= = Cu IV (2 im)

1. Tnh tch phn4

0

sin x dx4

I .sin 2x 2(1 sin x cos x)

=

+ + +

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

tr nh nht ca biu thc2

2

2(x 6xy)P .

1 2xy 2y

+=

+ +

PHN RING Th sinh chc lm 1 trong 2 cu: V.a hoc V.bCu V.a. Theo chng trnh KHNG phn ban (2 im)

1. Chng minh rngk 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, k n, knC 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 gictrong ca gc A c phng trnh x y 2 0 + = v ng cao k t B c phng trnh

4x 3y 1 0.+ =

Cu V.b. Theo chng trnh phn ban (2 im)

1. Gii bt phng trnh2

0,7 6

x xlog log 0.

x 4

+


21/27

B GIO DC V O TO THI TUYN SINH I HC, CAO NG NM 2008Mn thi: TON, khi D


PHN CHUNG CHO TT C TH SINHCu I (2 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, B ng thi I l trung im ca on thng AB.Cu II (2 im)

1. Gii phng trnh 2sinx(1 cos2x) sin2x 1 2cosx.+ + = +

2. Gii h phng trnh

2 2xy x y x 2y

x 2y y x 1 2x 2y

+ + =

= (x, y ).

Cu III (2 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 cu i qua bn im A, B, C, D.2. Tm ta tm ng trn ngoi tip tam gic ABC.

Cu IV (2 im)

1. Tnh tch phn

2

31

lnxI 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

thc2 2

(x y)(1 xy)P .

(1 x) (1 y)

=

+ +

PHN RING Th sinh chc lm 1 trong 2 cu: V.a hoc V.bCu V.a. Theo chng trnh KHNG phn ban (2 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 k ca n phn t).2. Trong mt phng vi h ta Oxy, cho parabol (P) : 2y 16x= v im A(1;4). Hai im

phn bit B, C (B v C khc A) di ng trn (P) sao cho gc oBAC 90 .= Chng minh rng

ng thng BC lun i qua mt im cnh.Cu V.b. Theo chng trnh phn ban (2 im)

1. Gii bt phng trnh2

1

2

x 3x 2log 0.

x

+

2. Cho lng trng ABC.A'B'C' 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 trABC.A'B'C' v khong cch gia hai ng thng AM, B'C.

...........................Ht...........................

Th sinh khng c sdng ti liu. Cn b coi thi khng gii thch g thm.H v tn th sinh:........................................................ S bo danh:.............................................

CHNH THC


22/27

B GIO DC V O TO

CHNH THC

THI TUYN SINH I HC NM 2009

Mn thi: TON; Khi: A

Thi gian lm bi: 180 pht, khng kthi gian pht.

PHN CHUNG CHO TT C TH SINH (7,0 im):Cu I (2,0 im)

Cho hm s2

2 3

xy

x

+=

+(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 tip tuyn ct trc honh, trc tung ln lt ti

hai im phn bit A , B v tam gic OAB cn ti gc to .OCu II (2,0 im)

1. Gii phng trnh ( )( )( )

1 2sin cos3

1 2sin 1 sin

x x

x x

=

+ .

2. Gii phng trnh ( )32 3 2 3 6 5 8 0 .x x x + = Cu III (1,0 im)

Tnh tch phn ( )2

3 2

0

cos 1 cosI x

= xdx .

Cu IV (1,0 im)Cho hnh chp c y.S ABCD ABCD l hnh thang vung ti v ;D 2AB AD a= = , ;CD a= gc gia

hai mt phng v( )SBC ( )ABCD bng Gi l trung im ca cnh60 . I AD . Bit hai mt phng ( )SBI

v ( cng vung gc vi mt phng)SCI ( )ABCD , tnh th tch khi chp theo.S ABCD .aCu V (1,0 im)

Chng minh rng vi mi s thc dng , ,x y z tho mn ( ) 3 ,x x y z yz + + = ta c:

( ) ( ) ( )( )( ) ( )3 3

3 53.x y x z x y x z y z y z + + + + + + + +

PHN RING (3,0 im): Th sinh chc lm mt trong hai phn (phn A hoc B)A. Theo chng trnh Chun

Cu VI.a (2,0 im)1. Trong mt phng vi h to cho hnh ch nht,Oxy BCD c im l giao im ca hai ngcho

(6;2)I

C v BD . im ( )1;5M thuc ng thng AB v trung im E ca cnh thuc ngthng . Vit phng trnh ng thng

CD

: 5 0x y + = B .

2. Trong khng gian vi h to cho mt phng,Oxyz ( ) : 2 2 4 0P x y z = v mt cu( ) 2 2 2: 2 4 6 11 0.S x y z x y z + + = Chng minh rng mt phng ( )P ct mt cu ( )S theo mtng trn. Xc nh to tm v tnh bn knh ca ng trn .

Cu VII.a (1,0 im)

Gi v l hai nghim phc ca phng trnh1z 2z2 2 10z z 0+ + = . Tnh gi tr ca biu thc

2 2

1 2 .A z z = +

B. Theo chng trnh Nng cao

Cu VI.b (2,0 im)1. Trong mt phng vi h to cho ng trn,Oxy ( ) 2 2: 4 4 6C x y x y 0+ + + + = v ng thng

vi m l tham s thc. Gi l tm ca ng trn ( Tm : 2 3x my m + + = 0, I ).C m ct ( )C ti hai im phn bit v B sao cho din tch tam gic ln nht.IAB

2. Trong khng gian vi h to cho mt phng,Oxyz ( ) : 2 2 1P x y z 0 + = v hai ng thng1

1 9:

1 1 6

x y z + + = = , 2

1 3:

2 1

1

2

x y z + = =

. Xc nh toim M thuc ng thng 1 sao cho

khong cch t Mn 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 2log 1 log , .3 81x xy y

x y xy x y +

+ = + =

---------- Ht ----------Th sinh khng c sdng ti liu. Cn b coi thi khng gii thch g thm.

H v tn th sinh:.............................................; S bo danh................................


23/27

B GIO DC V O TO

CHNH THC

THI TUYN SINH I HC NM 2009Mn: TON; Khi: B

Thi gian lm bi: 180 pht, khng kthi gian pht

PHN CHUNG CHO TT C TH SINH (7,0 im)Cu I (2,0 im)

Cho hm s (1).42 4y x x= 2

1.

Kho st s bin thin v v th ca hm s (1).2. Vi cc gi tr no ca phng trnh,m 2 2| 2 |x x m = c ng 6 nghim thc phn bit ?Cu II (2,0 im)

1. Gii phng trnh 3sin cos sin 2 3 cos3 2(cos4 sin ).x x x x x+ + = + 2. Gii h phng trnh

2 2 2

1 7( , ).

1 13

xy x yx y

x y xy y

+ + =

+ + =

Cu III (1,0 im)

Tnh tch phn3

21

3 ln.

( 1)

xI d

x

+=

+x

Cu IV (1,0 im)

Cho hnh lng tr tam gic . ' ' 'ABC A B C c ' ,BB a= gc gia ng thng 'BB v mt phng bngtam gic

(ABC)

60 ; ABC vung ti vC BAC= 60 . Hnh chiu vung gc ca im 'B ln mt phng ( )ABC trng vi trng tm ca tam gic .BC Tnh th tch khi t din 'ABC theo .a

Cu V (1,0 im)Cho cc s thc , y thay i v tho mn ( )3 4 2.x y xy+ Tm gi tr nh nht ca biu thc+

4 4 2 2 2 23( ) 2( ) 1A x y x y x y= + + + + .

PHN RING (3,0 im)Th sinh chc lm mt trong hai phn (phn A hoc B)A. Theo chng trnh Chun

Cu VI.a (2,0 im)1. Trong mt phng vi h to cho ng trn,Oxy 2 2 4( ) : ( 2)

5C x y + = v hai ng thng 1 : 0x y , =

Xc nh to tm2 : 7 0x y = . K v tnh bn knh ca ng trn ( bit ng trn tip xc

vi cc ng thng v tm1);C 1( )C

1 2, K thuc ng trn ( ).C2. Trong khng gian vi h to cho t din,Oxyz ABCD c cc nh v

Vit phng trnh mt phng i qua sao cho khong cch t n bng khongcch t n (

(1;2;1), ( 2;1;3), (2; 1;1)A B C (0;3;1).D ( )P ,A B C ( )P

D ).PCu VII.a (1,0 im)

Tm s phc tho mn:z (2 ) 10z i + = v . 25.z z=

B. Theo chng trnh Nng caoCu VI.b (2,0 im)

1. Trong mt phng vi h to cho tam gic,Oxy BC cn ti A c nh v cc nh( 1; 4)A ,B C thucng thng Xc nh to cc im: 4x y = 0. B v bit din tch tam gic,C ABC bng 18.

2. Trong khng gian vi h to cho mt phng,Oxyz ( ) : 2 2 5 0P x y z + = v hai im ( 3;0;1),A Trong cc ng thng i qua(1; 1;3).B A v song song vi hy vit phng trnh ng thng m

khong cch t( ),P

B n ng thng l nh nht.Cu VII.b (1,0 im)

Tm cc gi tr ca tham s ng thngm y x m= + ct th hm s2 1x

yx

= ti hai im phn bit

sao cho,A B 4.AB = ---------- Ht ----------

Th sinh khng c sdng ti liu. Cn b coi thi khng gii thch g thm.H v tn th sinh:.............................................; S bo danh:................................


24/27

B GIO DC V O TO

CHNH THC

THI TUYN SINH I HC NM 2009Mn: TON; Khi: D


PHN CHUNG CHO TT C TH SINH (7,0 im)Cu I (2,0 im)

Cho hm s 4 2(3 2) 3y x m x= + + mm

C mc th l l tham s.( ),

1. Kho st s bin thin v v th ca hm s cho khi 0.m = 2. Tm m ng thng ct th ti 4 im phn bit u c honh nh hn 2.1y = (

mC )

Cu II (2,0 im)

1. Gii phng trnh 3cos5 2sin3 cos2 sin 0.x x x x = 2. Gii h phng trnh

22

( 1) 3 0( , ).5

( ) 1 0

x x y

x yx y

x

+ + =

+ + =

Cu III (1,0 im)

Tnh tch phn3

1

.1x

dxI

e=

Cu IV (1,0 im)Cho hnh lng trng c y l tam gic vung ti. ' ' 'ABC A B C ABC , , ' 2 , ' 3 .B AB a AA a A C a= = = Gi M l trung im ca on thng ' ',A C I l giao im ca v Tnh theo th tch khi t din vkhong cch tim n mt phng (

AM ' .A C a IABC A ).IBC

Cu V (1,0 im)Cho cc s thc khng m ,x y thay i v tho mn 1.x y+ = Tm gi tr ln nht v gi tr nh nht ca biu

thc 2 2(4 3 )(4 3 ) 25 .S x y y x xy= + + +

PHN RING (3,0 im)Th sinh chc lm mt trong hai phn (phn A hoc B)

A. Theo chng trnh ChunCu VI.a (2,0 im)

1. Trong mt phng vi h to ,Ox cho tam gic c l trung im ca cnh ng trungtuyn v ng cao qua nh ln lt c phng trnh l

y ABC (2;0)M .AB

A 7 2 3 0x y = v Vit phngtrnh ng thng

6 4 0.x y =.AC

2. Trong khng gian vi h to , cho cc im v mt phngXc nh toim

Oxyz (2;1;0), (1;2;2), (1;1;0)A B C ( ) : 20 0.P x y z + + = D thuc ng thng sao cho ng thng CD song songvi mt phng (

AB

).PCu VII.a (1,0 im)

Trong mt phng to ,Ox tm tp hp im biu din cc s phc tho mn iu kin |y z (3 4 ) | 2.z i =


1. Trong mt phng vi h to , cho ng trn .Oxy 2 2( ) : ( 1) 1C x y + = Gi l tm ca Xc nhtoim

I ( ).C

M thuc sao cho( )C IMO = 30 .

2. Trong khng gian vi h to , cho ng thngOxyz 2 2:1 1 1

x y+ = =

z

m

v mt phng

Vit phng trnh ng thng nm trong ( sao cho d ct v vung gc ving thng( ) : 2 3 4 0.P x y z + + = d )P

.Cu VII.b (1,0 im)

Tm cc gi tr ca tham s ng thngm 2y x= + ct th hm s2 1x x

yx

+ = ti hai im phn

bit sao cho trung im ca on thng thuc trc tung.,A B AB---------- Ht ----------

Th sinh khng c sdng ti liu. Cn b coi thi khng gii thch g thm.H v tn th sinh:.............................................; S bo danh:................................


25/27

B GIO DC V O TO

CHNH THC

THI TUYN SINH I HC NM 2010Mn: TON; Khi: A


I. PHN CHUNG CHO TT C TH SINH (7,0 im)Cu I (2,0 im)Cho hm sy=x3 2x2+ (1 m)x+m (1), m l tham s thc.

1. Kho st s bin thin v v th ca hm s khi m= 1.2. Tm m th ca hm s (1) ct trc honh ti 3 im phn bit c honh x1,x2,x3 tho mn iukin 2 2 21 2 3x x+ + < 4.

Cu II (2,0 im)

1. Gii phng trnh (1 sin cos 2 )sin 14 cos1 tan 2

x x x

x

+ + +

=

+.

2. Gii bt phng trnh21 2( 1

x x

x x

+ ) 1.

Cu III (1,0 im)Tnh tch phn I=1 2 2

0

2 d1 2

x x

x

x e x e

e

+ +

+ .

Cu IV (1,0 im)Cho hnh chp S.ABCD c y ABCD l hnh vung cnh a. Gi Mv N ln lt ltrung im ca cc cnh AB v AD; H l giao im ca CNvi DM. Bit SHvung gc vi mt phng

(ABCD) v SH=a 3 . Tnh th tch khi chp S.CDNMv tnh khong cch gia hai ng thngDMvSCtheo a.

Cu V (1,0 im)Gii h phng trnh2

2 2

(4 1) ( 3) 5 2 0

4 2 3 4 7

x x y y

x y x

+ + =

+ + =(x,yR).

II. PHN RING (3,0 im)Th sinh chc lm mt trong hai phn (phn A hoc B)A. Theo chng trnh ChunCu VI.a (2,0 im)1. Trong mt phng ta Oxy, cho hai ng thng d1: 3 0x y+ = v d2: 3x y = 0 . Gi (T) l

ng trn tip xc vi d1 ti A, ct d2 ti hai imB v Csao cho tam gic ABCvung ti B. Vit

phng trnh ca (T), bit tam gicABCc din tch bng3

2v imA c honh dng.

2. Trong khng gian toOxyz, cho ng thng : 12 1 1

x y z = =

2+v mt phng (P):x 2y+z= 0.

Gi Cl giao im ca vi (P), Ml im thuc . Tnh khong cch tMn (P), bit MC= 6 .Cu VII.a (1,0 im)Tm phn o ca s phcz, bit 2( 2 ) (1 2 )z i= + i .B. Theo chng trnh Nng caoCu VI.b (2,0 im)1. Trong mt phng toOxy, cho tam gic ABCcn ti A c nh A(6; 6); ng thng i qua trung

im ca cc cnhAB vACc phng trnh x+y 4 = 0. Tm to cc nhB v C, bit imE(1; 3)nm trn ng cao i qua nh Cca tam gic cho.

2. Trong khng gian toOxyz, cho im A(0; 0; 2) v ng thng : 2 22 3 2

3x y z + += = . Tnh

khong cch tAn . Vit phng trnh mt cu tmA, ct ti hai imB v Csao choBC= 8.

Cu VII.b (1,0 im)Cho s phcztha mn z=3(1 3 )

1i

i

. Tm mun ca s phc + i z.

----------- Ht ----------Th sinh khngc sdng ti liu. Cn b coi thi khng gii thch g thm.

H v tn th sinh:.............................................;S bo danh................................


26/27

B GIO DC V O TO

CHNH THC


Mn: TON; Khi: BThi gian lm bi: 180 pht, khng kthi gian pht

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

Cu I (2,0 im) Cho hm s 2 11

xyx

+=+

.

1. Kho st s bin thin v v th (C) ca hm s cho.2. Tm mng thngy=2x+m ct th (C) ti hai im phn bitA,B sao cho tam gic OAB

c din tch bng 3 (O l gc ta ).Cu II (2,0 im)1. Gii phng trnh (sin .2 cos 2 ) cos 2cos 2 sin 0x x x x x+ + =2. Gii phng trnh 23 1 6 3 14 8x x x x+ + = 0 (xR).Cu III (1,0 im) Tnh tch phn

( )21

lnd

2 ln

ex

I xx x

=

+.

Cu IV (1,0 im) Cho hnh lng tr tam gic u ' cAB=a, gc gia hai mt phng. ' 'ABC A B C ( ' )A BC v ( )ABC bng . Gi G l trng tm tam gic . Tnh th tch khi lng tr chov tnh bn knh mt cu ngoi tip t din GABCtheo a.

60o 'A BC

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 23( ) 3( ) 2a b b c c a ab bc ca a b c= + + + + + + + + .


Cu VI.a (2,0 im)1. Trong mt phng toOxy, cho tam gicABCvung tiA, c nh C( 4; 1), phn gic trong gcA cphng trnh x+y 5 = 0. Vit phng trnh ng thngBC, bit din tch tam gicABCbng 24 vnhA c honh dng.

2. Trong khng gian toOxyz, cho cc imA(1; 0; 0),B(0; b; 0), C(0; 0; c), trong b, c dngv mt phng (P):yz+ 1 = 0. Xc nh b v c, bit mt phng (ABC) vung gc vi mt phng

(P) v khong cch tim On mt phng (ABC) bng1

3.

Cu VII.a (1,0 im) Trong mt phng ta Oxy, tm tp hp im biu din cc s phcztha mn:(1 )z i i z = + .


1. Trong mt phng toOxy, cho imA(2; 3 ) v elip (E): 2 2 13 2

x y+ = . GiF1 vF2 l cc

tiu im ca (E) (F1 c honh m); Ml giao im c tung dng ca ng thngAF1 vi(E);Nl im i xng caF2 qua M. Vit phng trnh ng trn ngoi tip tam gicANF2.

2. Trong khng gian toOxyz, cho ng thng : 12 1 2

y z= = . Xc nh ta im Mtrn

trc honh sao cho khong cch tMn bng OM.

Cu VII.b (1,0 im) Gii h phng trnh2

2

log (3 1)

4 2 3x xy x

y

=

+ = (x,yR).

---------- Ht ----------Th sinh khng c sdng ti liu. Cn b coi thi khng gii thch g thm.

H v tn th sinh: .............................................; S bo danh: ...................................


27/27

B GIO DC V O TO

CHNH THC


Mn: TON; Khi: D


PHN CHUNG CHO TT C TH SINH (7,0 im)Cu I (2,0 im) Cho hm s .4 2 6y x x= +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 1

6y x= .

Cu II (2,0 im)1. Gii phng trnh s in 2 cos 2 3sin cos 1 0.x x x x + =2. Gii phng trnh 3 32 2 2 2 44 2 4 2 4x x x x x x+ + + + + + = + (xR).

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

3

2 ln

e

dI x xx

=

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

chiu vung gc ca nh Strn mt phng (ABCD) l im Hthuc onAC,AH=4

C. Gi CMl ng

cao ca tam gic SAC. Chng minh Ml trung im ca SA v tnh th tch khi t din SMBCtheo a.

Cu V (1,0 im) Tm gi tr nh nht ca hm s 2 24 21 3 1y x x x x= + + + + 0 .


Cu VI.a (2,0 im)1. Trong mt phng toOxy, cho tam gicABCc nhA(3; 7), trc tm lH(3; 1), tm ng trnngoi tip lI(2; 0). Xc nh ta nh C, bit Cc honh dng.

2. Trong khng gian toOxyz, cho hai mt phng (P):x+y+z 3 = 0 v (Q):xy+z 1 = 0. Vitphng trnh mt phng (R) vung gc vi (P) v (Q) sao cho khong cch tOn (R) bng 2.

Cu VII.a (1,0 im) Tm s phc z tha mn: |z| = 2 v z2 l s thun o.B. Theo chng trnh Nng cao

Cu VI.b (2,0 im)1. Trong mt phng ta Oxy, cho imA(0; 2) v l ng thng i qua O. GiHl hnh chiu

vung gc ca A trn . Vit phng trnh ng thng , bit khong cch tHn trc honh

bng AH.

2. Trong khng gian toOxyz, cho hai ng thng 1:3 t

y t

z t

= +

= =

v 2:2 1

2 1 2

y = =

z. Xc

nh ta im Mthuc 1 sao cho khong cch tMn 2 bng 1.

Cu VII.b (1,0 im) Gii h phng trnh2

2 2

4 2 0

2log ( 2) log 0

x x y

x

+ + =

y =(x,yR).

---------- Ht ----------

Th sinh khngc sdng ti liu. Cn b coi thi khng gii thch g thm.

H t th i h S b d h

tỔng hỢp ĐỀ thi Đh mÔn toÁn khỐi a, b,d (2002-2010)

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