Tuyen Tap Cac de Thi Thu Dai Hoc Cao Dang Tren Tap Chi Toan Hoc Tuoi Tre

8/2/2019 Tuyen Tap Cac de Thi Thu Dai Hoc Cao Dang Tren Tap Chi Toan Hoc Tuoi Tre

1/66

Tuyn tp cc thi thi hc, cao ng trn tp chTon hc v Tui trqua cc nm

Tp th lp 12T THPT Th x Cao Lnh Nin kho 2006-2009 GVCN: Thy Nguynnh Huy

TRNG TRUNG HC PH THNG TH X CAO LNH--------------

TP TH LP CHUYN TON NIN KHA 2006 2009

Nguyn c Tun -

TUYN TP CC THI TH

I HC , CAO NGTRN TP CH

QUA CQUA CQUA CQUA CCCCC NNNNMMMM

---- Thng 03-2009 ----


2/66



B GIO DC V O TOTP CH TON HC V TUI TR

NM2003

THI THI HC, CAO NG S 1Mn thi: TON

Thi gian lm bi: 180 phtCu I: (2 im)

Cho hm s: .424 mxmxxy ++=

1. Kho st v v th hm s khi .0=m 2. Tm cc gi tr ca m th hm s c ba im cc tr sao cho tam gic c nh l ba im cctr nhn gc ta lm trng tm.

Cu II: (2 im)1. Gii cc phng trnh :

( ) ( )( )xx xxxx = 2002loglogloglog 20022002

2. Tm tt c cc gi tr ca a tp xc nh ca hm s ( )xa

xaxf

+=

2

2cha tp gi tr ca hm

s ( ) .242 12 ++=

axxxg

Cu III: (2 im)1. Gii phng trnh :

( )xxxx 141488 sincos64sincos +=+ 2. Hai ng cao 11 ,BBAA ca tam gic nhn ABC ct nhau ti H. Gi R l bn knh ng trnngoi tip tam gic ABC.Chng minh rng din tch tam gic 11BHA bng CBACR cos.cos.cos.2sin.

2 .

Cu IV: (2 im)1. Cho t din OABCc: 0180 AOB BOC+ = gi l OD ng phn gic trong ca gc AOB

Hy tnh gc

BOD .2. Trong khng gian vi h ta cc vung gc Oxyz cho hai ng thng :

( )2 1 0

1 0

x y

x y z

+ + =

+ = ( )

3 3 0'

2 1 0

x y z

x y

+ + =

+ =

a. Chng minh rng hai ng thng ( ) v ( )' ct nhau.

b. Vit phng trnh chnh tc ca cp ng thng phn gic ca cc gc to bi ( ) v ( )' .

Cu V: (2 im)

1. Tnh tch phn :( )

24

4 2

4

sin

cos tan 2 tan 5

xdxI

x x x

= +

2. Trong hp ng 2n vin bi c n vin bi ging ht nhau v n vin bi xanh i mt khc nhauHi c bao nhiu cch khc nhau ly n vin bi t hp .

------------------ HT -------------------


3/66



P S HOC HNG DN GIIS 1-2003:

Cu I:

1. Cc bn t gii.2. p dng n l Vi-t bc ba. p s: : 6.m =

Cu II:

1.p s: 1001.x =

2.p s:3 17

.8

a+

>

Cu III:

1. Phng trnh v nghim. p dng BT Cauchy.

2. Cc bn t gii.

Cu IV:

1.p s: 090 .BOD =

2.

a. Chng minh h c nghim duy nht.

b. Dng vectn v.

p s:

1 32 2 ;

1 1 2 2 3 5

14 30 14 30 14 301 32 2 .

1 1 2 2 3 5

14 30 14 30 14 30

x zy

x zy

+

= =

+ + +

+

= =

Cu V:

1.t tant x= . p s: 32 ln 2 .8

I =

2.p s:0

2 .n

k n

n

k

C=

=

------------------ HT -------------------


4/66




NM2003


Thi gian lm bi: 180 pht

Cu I: (2 im)

Cho hm s : 423 += axxy 1. Kho st v v th hm s khi .3=a 2. Tm a phng trnh 423 ++ maxx = 0 lun c 3 nghim phn bit, vi mi gi tr ca mtha iu kin : .04


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Cu I:

1. Cc bn t gii.2. Lp bng bin thin.

p s: 3a .

Cu II:

1. p dng BT B.C.S. p s:1

2x y= =

2.p s:1

2.

Cu III:

1.t2 1 1

.3 10

xt t

x

+ =

p s:

1 2; .

3 4 5 4x

=

2. Cc bn t gii.

Cu IV:

1.p s:

25

.24V a h=

2. p s:6 6

.2 2 2 2

m

<

+ +

2. Tnh1

2 2

0

4 3I x x dx =

Cu IV: (2 im)1. Cho ng thng ( )d : 2 2 0x y = v hai im (0;1)A v (3;4)B . Hy tm to ca im

M trn ( )d sao cho 2 22MA MB + c gi tr nh nht.

2. Cho ng parabol c phng trnh 2 4y x= v gi s F l tiu im ca n. Chng minh rngnu mt ng thng i qua F v ct parabol ti hai im A, B th cc tip tuyn vi parabol ti A ,B vung gc vi nhau .

Cu V: (2 im)1. T cc ch s 0, 1, 2, 3, 4, 5, 6 ta c th vit bao nhiu ch s t nhin c 5 ch s khc nhau sao

cho trong nht thit c cc ch s 1 v 2 .2. Cho , ,x y z l cc s thc tho mn iu kin sau :04,01,01,0 >+>+>+=++ zyxzyx .

Hy tm gi tr ln nht ca biu thc:

1 1 4

x y zQ

x y z= + +

+ + +.

------------------ HT -------------------


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Cu I:

1. Cc bn c th t gii .

2. Hm s c cc i , cc tiu khi PT ' 0y = c hai nghim phn bit khc 1.

p s:4

13

m< < .

Cu II:

1.p s:f(0)=0.2. Ch s dng

3

3

4 sin 3 sin sin 3

4 cos 3 cos cos 3

x x x

x x x

=

= +

p s: ( )6

x k k Z

= + .

Cu III:

1. p s: 1 0x < <

2. C tht 3 2 sinx t=

p s: 2 1129 3

I = + .

Cu IV:

1. p s: (2;0)M 2. Cc bn t chng minh.

Cu V:

1. p s: 1056 s2. t 1, 1, 4a x b y c z = + = + = + , th , , 0, 6a b c a b c > + + = .

p s: max1

3Q = khi

1

2x y= = v 1z = .

------------------ HT -------------------


23/66




NM2005



1. Kho st v v th ca hm s :

2 2

3

x xy x

= .

2. Tnh din tch ca hnh phng c gii hn bi th ca hm s v trc honh .

Cu II: (2 im)1. Gi s , , ,a b c dl cc s thc tha mn ng thc : 2( ) ( )ab b c d c a b+ + + = + .Chng minh rngtrong ba bt phng trnh :

2 2 20 : 0 : 0x ax b x bx c x cx d + + + t nht mt bt phng trnh c nghim .

2. Vi nhng gi tr no ca a th h phng trnh :

2 2 2 2

1 1

x y a

a

x y

+ = + + =

c ng hai nghim?

Cu III: (2 im)1. Gii phng trnh lng gic:

1cos .cos2 .cos3 sin .sin2 .sin 3

2x x x x x x = .

2. Cho 3 4 4( ) (1 ) f x x x x = + + + .Sau khi khai trin v rt gn ta c :2 16

0 1 2 16( ) ...f x a a x a x a x = + + + + .Hy tnh gi tr ca h s 10a .

Cu IV: (3 im)1. Trong mt phng vi h trc to-cc vung gc Oxy cho Elip (E) c phng trnh l

2 2

2 21

x y

a b+ = (vi 0, 0a b> > ).Gi s ,A B l hai im thay i trn (E) sao cho OA vung gc

vi OB.

a. Tnh 2 21 1

OA OB + theo a v b .

b. Gi H l chn ng vung gc h tOxung AB .Tm tp hp cc im H khi ,A B thay i trn(E).

2. Cho hnh lp phng . ' ' ' 'ABCD A B C D vi cnh bng a . Hy tnh khong cch gia cnh'AA vi ng cho 'BD theo a

Cu V: (1 im)Cho , ,x y z al nhng s dng tha mn 1xyz = . Tm gi tr nh nht ca biu thc:

9 9 9 9 9 9

6 3 3 6 6 3 3 6 6 3 3 6

x y y z z x P

x x y y y y z z z z x x

+ + += + +

+ + + + + +.

------------------ HT -------------------


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Cu I:

1. Cc bn c th t gii.

2. p s:15

8ln22

S = .

Cu II:1. Cc bn t chng minh.

2. p s: 2 2a < < .

Cu III:

1.p s: ; ; ( ).8 2 12 3 4

x k x k x k k Z = + = + = +

2. Cc bn c th bin i4 4

3 3( ) 1 (1 ) (1 )(1 ) f x x x x x x = + + + = + +

p s: 10 22a = .

Cu IV:1.

a. p s:2 2

2 2 2 2

1 1 a b

OA OB a b

++ =

b. p s: Tp hp H l ng trn tm (O;2 2

2 2

a b

a b+)

2. p s:2

2

a.

Cu V:

Ch rng vi , 0a b > ta lun c:2 2

2 2

1

3

a ab b

a ab b

+

+ +

p s: min 2P = khi 1x y z= = = .

------------------ HT -------------------


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NM2005



Cu I: (2 im)

Cho hm s

3 2

( 3) (2 3 ) 2 .y x m x m x m = + + +

(1)1. Kho st v v th ca hm s vi

3

2m = .

2. Tm trn mt phng cc im cnh m thi hm s lun i qua vi mi m.3. Tm m th ca hm s ct trc honh ti ba im phn bit c honh lp thnh mt cps cng theo mt th t no .

Cu II: (2 im)1. Cho tam gic ABC c ba gc , ,A B C tho mn:

2 3tan tan

2 2 3

cos cos 1

A B

A B

+ =+ =

Chng minh rng tam gic ABC u.2. Gii bt phng trnh :

2

24

1 1

log (3 1)log ( 3 ) xx x .

2. V1

3x > nn 2 3 3 1.x x x+ > > T ta thy v tri ca phng trnh dng.

p s:2

13

x< < .

Cu III:

1.t x t= , sau khi th vo cc bn nhn thm lng lin hip.p s: 2lnI a= 2. p s: 6; 0a b= = .

Cu IV:

1. p s: (1;1;1)I ,( ) : 2 1 0Q x y =

2. p s:1 2

7 14 22

x y z+ = =

3. p s: 1; 1 3a b= < < .

Cu V:

2 2 2 2 2 2

2 2 2 2 2 2

5 cot 16 cot 27 cot (3 2) cot (12 4) cot (9 18) cot

(3 cot 12 cot ) (4 cot 9 cot ) (18 cot 2 cot ) 12

F A B C A B C

F A B B C C A

= + + = + + + + +

= + + + + +

p s: min 12F = khi1 1

cot 1, cot , cot2 3

A B C= = = .

------------------ HT -------------------


27/66




NM2005



Cu I: (2 im)

1. Kho st v v th ca hm s : 3 3 2y x x= + (C)2. Gi s , ,A B C l ba im thng hng phn bit thuc (C), tip tuyn vi (C) ti , ,A B C tng ngct li (C) ti ', ', 'A B C .Chng minh rng ', ', 'A B C thng hng .

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

2

2

1 1

1 3

x y

y x

+ = + =

2. Gii bt phng trnh :3 2

4 16

2

20 log 7 log 3 logx x x

x x x+ .

Cu III: (2 im)

1. Tam gic ABC c BC a= ;7

cos8

A = v din tch bng2 15

4

a.Gi , ,

a b ch h h ln lut l di

cc ng cao h t cc nh , ,A B C ca tam gic. Chng minh rng .a b ch h h= +

2. Tm gi tr ln nht ca hm s sin (1 6 cos )2 2

x xy = + .

Cu IV: (3 im)

1. Trn mt phng toOxy cho hai ng thng 1( ) : 2 1 0d x y + = v 2( ) : 2 7 0d x y+ = .

Lp phng trnh ng thng qua gc ta v to vi 1( )d , 2( )d tam gic cn c y thuc ng

thng . Tnh din tch tam gic cn nhn c .

2. Cho hnh lng tr tam gic 1 1 1.ABC AB C c cc mt bn l hnh vung cnh a .Gi D,E,F ln lt

l trung im cc on thng 1 1 1 1, , .BC A C C B Tnh khong cch gia DE v 1A F .

Cu V: (1 im)

Tnh2

0

1 sin

(1 cos ) xx

I dxx e

=

+

------------------ HT -------------------


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Cu I:

1. Cc bn c th t gii.

2. Cc bn t chng minh.

Cu II:

1. p s:1 3

( ; ) ( ; )2 2

x y =

2.t 2log x t= . p s:

5

1 1

44 84

10

161 2

x

x

x

x


29/66




NM2006



Cu I: (2 im)

Cho hm s 3 22 3 1 y x x= (C)1. Kho st s bin thin v v th ca hm s.2. Gi kd l ng thng i qua M (0;-1) v c h s gc k. Tm kng thng kd ct (C) ti baim phn bit.

Cu II: (3 im)

1. Trong mt phng vi h trc ta vung gc Oxy, cho tam gic ABC c A(1;0), hai ngthng tng ng cha ng cao k t B, C ca tam gic th t c phng trnh: 2 1 0x y + = v3 1 0x y+ = .

Vit phng trnh ng trn ngoi tip tam gic ABC.

2. Tm ta trc tm H ca tam gic ABC trong khng gian Oxyz vi A(3; 0; 0), B(0; 2; 0), C(0; 0;1).

3. Cho hnh chp tam gic u SABC, cnh y l a, cnh bn l b. Tnh khong cch t A n mtphng (SBC).

Cu III: (2 im)

1. Gii phng trnh: 2 25 1 54 12.2 8 0 x x x x + = .

2. Gii phng trnh:2cos4

cot tansin2

xx x

x= +

Cu IV: (2 im)

1. Tnh tch phn:1

20

4 5

3 2

x I dx

x x

+=

+ +

2. Mt trng THPT c 18 hc sinh gii ton din, trong c 7 hc sinh khi 12, 6 hc sinh khi11, 5 hc sinh khi 10. Hi c bao nhiu cch chn 8 hc sinh trong s 18 hc sinh trn i d tri hsao cho mi khi c t nht 1 hc sinh c chn?

Cu V: (1 im)Tm cc gc A, B, C ca tam gic ABC sao cho 2 2 2sin sin sinQ A B C = + t gi tr nh nht.

------------------ HT -------------------


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Cu I:

1. Cc bn t gii.

2. p s: 9 k 08

k v> .

Cu II:

1. p n: Phng trnh:( 2 236 10 43

07 7 7

x y x y+ + = )

2. p s: H12 18 36

; ;49 49 49

3. p s:

2 2

2 2

3( ; ) 4

a b ad A SBC

b a

=

.

Cu III:

1. p s: 3x = v9

4x =

2. p s:3

x k

= + ( )k Z .

Cu IV:

1. p s:27

ln4

I=

2. p s: ( )8 8 8 818 11 13 12 304351C C C C + + = .

Cu V:

p s: 30oA B= = v 120oC= .

------------------ HT -------------------


31/66




NM2006



Cu I: (2 im)

Cho hm s:2 3 3

1

x xy

x

+ +=

+(C)

1. Kho st s bin thin v v th ca hm s.2. Chng minh rng qua im M(-3;1) kc 2 tip tuyn ti th (C) sao cho hai tip tuyn vung gc vi nhau.

Cu II: (2 im)

1. Tm m bt phng trnh sau c nghim: 22 1 0. x m x+ + <

2. Tnh tch phn:1

3 1

0

x I e dx

+=

Cu III: (2 im)1. Gii phng trnh: 2log 23 1x x=

2. Gii phng trnh: ( )2 22 1

cos cos sin 13 3 2

x x x

+ + + = +

Cu IV: (2 im)1. Trong mt phng vi h trc ta vung gc Oxy cho parabol (P): 2y x= v im M(1;-1). Gis A, B l hai im phn bit, khc M, thay i trn (P) sao cho MA v MB lun vung gc vinhau. Chng minh rng ng thng AB lun i qua 1 im cnh.

2. Trong khng gian vi h ta vung gc Oxyz cho im A(1; -1; 1) v hai ng thng theo th

t c phng trnh: ( )1 : 1 2

3

x t

d y t

z t

=

= + =

; ( )23 3 0

:2 1 0

x y zd

x y

+ + =

+ =

Chng minh rng ( )1d , ( )2d v A cng nm trong mt mt phng.

Cu V: (2 im)1. C bao nhiu s t nhin chn gm 5 ch si mt khc nhau sao cho trong khng c mt

ch s 2.2. Tm gi tr nh nht ca biu thc

3 3 3

, x y z

Qy z x z x y

= + ++ + +

vix, y, z l cc s dng tha iu

kin: 6 x y z+ + .

------------------ HT -------------------


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Cu I:

Cc bn t gii.

Cu II:

1.p s: m > -1.

2.p s: I =22

3

e.

Cu III:

1.p s:x = 2.

2.p s:. 26

x k

= + ;5

26

x k

= + ; x k= ( )k Z .

Cu IV:

Cc bn t gii.

Cu V:

1.p s: 1680 + 4410 = 6090.

2. Hng dn: p dng BT Cauchy cho 3 s dng.

Qmin = 6, khix = y = z = 2.

------------------ HT -------------------


33/66




NM2006



Cu I: (2 im)

1. Kho st v v th hm s: 3 3 3 y x x= +

2. Tnh o hm cp n ca hm s:2

2004

5 6

xy

x x=

+.

Cu II: (2 im)1. Chng minh rng trong mi tam gic ABC ta lun c :

tan 3 tan 3 tan 33 3 3

A B C

4 tan tan tan 3

3 3 3

A B C = + +

.

2. Gii phng trnh:2 2

2 2

sin sin 22

sin 2 sin

x x

x x+ = .

Cu III: (2 im)

1. Tm gii hn:3

1 3lim .

1 1x x x

2. Tnh tch phn:1 2

20

.4

x dx

x +

Cu IV: (3 im)

1. Cho hai ng thng:1

2

2 4( ) : ,

1 1 18 6 10

( ) : ,2 1 1

x y zd

x y zd

+= =

+ = =

Trong h ta vung gc Oxyz. Lp phng trnh ng thng (d) ct 1( )d , ( )2d v (d) song song

vi trc Ox.

2. Cho t din OABC vi OA = a, OB = b, OC = c v OA, OB, OC i mt vung gc vi nhau.Tnh din tch tam gic ABC theo a, b, c. Gi , , l gc gia OA, OB, OC vi mt phng

(ABC). Chng minh rng: 2 2 2sin sin sin 1. + + = Cu V: (1 im)

Trong mt phng vi h trc ta vung gc Oxy cho parabol (P): 2y x= ta ly A(-1;1), B(3;9).Gi (D) l min phng gii hn bi on AB v (D).

Chng minh rng vi mi M bt k thuc cung nh AB ca (P) th34

ABM

D

S

S , DS l din

tch ca min (D), ABMS l din tch .ABM

------------------ HT -------------------


34/66




Cu I:

1. Cc bn t gii.2.p s: ( )

3 22004. 1 ! .

3 2nn

y nx x

=

Cu II:

1. Cc bn t gii.

2.p s:2

32

2 , ,3

x k

x k k Z

= +

= +

Cu III:

1. p s: 1.

2. p s:5 1 5

2ln2 2

I +

=

.

Cu IV:

1. p s:2 2 2 2 2 21 .

2ABCS a b b c c a= + +

2. Cc bn t gii.

Cu V:

Cc bn t gii.

------------------ HT -------------------


35/66




NM2006



Cu I: (2 im)

Cho hm s2 2 5

1

x kxy

x

+ =

(k l tham s).

1. Kho st v v th ca hm s (1) vi k=1.

2. Vi gi tr no ca tham s k th hm s c cc i, cc tiu v cc im cc i, cc tiu nm vhai pha ca ng thng (l): 2 0x y = .

Cu II: (2 im)

1. Gii phng trnh: ( )21

2 cos cos3

x x + + 28 1

s in2 3cos sin .3 2 3

x x x

= + + + +

2. Vi gi tr no ca tham skth hm s2

2

1lg 3

1

x kxy

x x

+=

+ + xc nh vi mix.

Cu III: (3 im)

1. Cho hnh chp t gic u S.ABCD c cnh dy bng ng cao v bng a. Tnh khong cchgia hai ng thng SC v AB.

2. Trong khng gian vi h ta Oxyz, cho ng thng ( ) c phng trnh :1 2

2 1 3

x y z = =

v mp(Q) i qua im M(1; 1; 1) v c vectphp tuyn ( )2; 1; 2n =

. Tm ta cc im

thuc( ) sao cho khong cch t mi im n mp(Q) bng 1.

Cu IV: (2 im)

1. Xc nh h s ca s hng cha 4a trong khai trin nh thc Newton2 2

n

aa

(vi 0a ),

bit rng tng cc h s ca 3 s hng u tin trong khai trin bng 97.

2. Tnh tch phn: 2

1

lnln .

1 ln

ex

I x dxx x

= +

+

Cu V: (1 im)Cho a thc: ( ) ( )2f x mx n p x m n p= + + + +

Vi m, n, p l ba s thc tha mn: ( )( ) 0m p m n p+ + + < .

Chng minh rng: ( )2 2 2 2 .n p m m n p np+ > + + +

------------------ HT -------------------


36/66




Cu I:

1. Cc bn t gii.2.p s 2 2 6 2 2 6k < < +

Cu II:

1. p s ( )22

x k k Z

= +

2. p s: 5 1.k < <

Cu III:

1.p s:2 5

5

ad=

2.p s: ( )1 9; 2;12A ; 2 ( 3;4; 6)A .

Cu IV:

1.p s: ( )

44

8 2 1120.C = 2.p s: ( )

21 2 .

3I e= + +

Cu V:

Cc bn t gii.

------------------ HT -------------------


37/66




NM2006



Cu I: (2 im)

Cho hm s2 1

1x xy

x

+=

(C)

1. Kho st v v th (C) ca hm s.

2. Vit phng trnh ng thng (d) i qua im3

0;2

A

v ct (C) ti hai im phn bit B, C

tha mn: 2 0 AB AC + =

.

Cu II: (2 im)

1. Gii h phng trnh:2 2

3 1 4 2

3

x xy y

x y

+ + = +

+ =

2. Gii bt phng trnh:( ) ( )

4

22

2 10.

log 2 25

xx

x x

+

Cu III: (2,5 im)Cho hnh hp ch nhtABCD.ABCD c th tch bng 1. GiI, J, Kln lt l trung im ca cc onthngAA, CD, AD.

1. Tnh th tch khi t dinBIJK.

2. Bit BK vung gc vi mt phng (ACD). Tnh di cc cnh ca hnh hp.3. Tm gi tr ln nht ca khong cch gia hai ng thng CIvAJ.

Cu IV: (2 im)

1. Tnh cc gc ca tam gic ABC, bit 2A = 3B v 2.a b=

2. Tnh32

4 20

cos.

cos 3cos 3

x I dx

x x

= +

Cu V: (1,5 im)

Trong mt trng hc c 5 em khi 12; 3 em khi 11 v 2 em khi 10 l cc hc sinh xut sc. Hic bao nhiu cch c 5 em hc sinh xut sc ca trng tham gia mt on i biu sao cho mikhi c t nht 1 em?

------------------ HT -------------------


38/66




Cu I:1. Cc bn t gii.

2. p s: ( )

3

.2:5 3

.4 2

yd

y x

=

=

Cu II:

1.p s ( ) ( ) ( ); : 1; 2 & 2;1 .x y

2. p s:

5;

4 0;0 3;

4 5 .

x

xx

x

<

<


39/66




NM2007



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

2

1x

y x= (C)2. Tm trn th (C) mt im c hanh ln hn 1 sao cho ti im ny tip tuyn ca (C) tovi 2 ng tim cn ca (C) to thnh 1 tam gic vi chu vi nh nht .

Cu II: (2 im)

Gii cc phng trnh sau :1. ( )2 2 3 3tan tan .sin 1 cos 0 x x x x =

2. ( )232 9 53

xx x

x

+ = +

Cu III: (2 im)

Cho2 2

0

sin2cos 3sin

x I dx

x x

=+

v2 2

0

cos2cos 3sin

x J dx

x x

=+

1. Tnh 9I - 4JvI+J.2. T suy ra kt qu ca I v J .

Cu IV: (2 im)

1. Trn mt phng vi h ta Descartes vung gc Oxy cho 3 ng thng

1 :3 4 4 0d x y = 2 : 6 0d x y+ = 3 : 3 0d x = Tm ta cc nh ca hnh vung ABCD bit rng A,C thuc d3, B thuc d1v D thuc d2 .

2. Cho 1, , ;33

a b c

. Chng minh rng :75

a b c

a b b c c a+ +

+ + +.

Cu V: (2 im)1. Gii phng trnh : 8.27 38.18 57.12 27 0 x x x + =

2. Cho hnh chp tam gic u S.ABC c ng cao SO=1 v y ABC c cnh bng 2 6 . Ccim M, N theo th t l trung im ca cnh AB, AC . Tnh th tch hnh chp S.AMN v bn knhmt cu ni tip hnh chp .

------------------ HT -------------------


40/66




Cu I:1. Bn t gii2. Gi im M thuc th tha , gi I l giao im 2 tim cn ng . A,B l giao im ca 2 timcn vi tip tuyn . Hy tnh din tch tam gic AIB chng minh tch IA.IB khng i . Sau dng nh l hm cos tnh di AB trong tam gic AB :

2 2 2 2. . .cos 2 . 8 AB IA IB IA IB AIB IA IB= +

p n : 44 4

1 11 ;2 2

2 2M

+ + +

.

Cu II:

1.a v phng trnh tch. p n :

2

4 2 1,cos

2 24

24

x k

x k

k Z

x k

x k

= = + =

= + +

= +

.

2.p n:3

11

x

x

=

=.

Cu III:

1.t tan2

xt= . p n :

9 4 1

1 13 1 13 3ln13 13 1 13 3

I J

I J

=

+ = + +

2. Gii phng trnh cu 1.

Cu IV:

1. Chng minh B v D i xng nhau qua d3 .Sau tm tm hnh vung ABCD l I , dn n hthc (a-2)2=1.p n : A(3;3), B(2;2), C(1;3), D(4;2); A(1;3), B(2;2), C(3;3), D(4;2).

2.p s: ( )1

; ; 3;1;3

a b c

=

v cc hon v.

Cu V:1. Xt hm s: ( ) 8.27 38.18 57.12 27 x x xf x = + . Hy chng minh hm s ny l ng bin.T suy ra phng trnh c nghim duy nht. p s:x=0.

2. p n:1 3 3

. ;3 2 4 2 2

SAMN AMN V SO S r = = =+

.

------------------ HT -------------------


41/66




NM2007



Cho hm s 3 2( 1) ( 1) 1y x m x m x= + + + .1.Kho st s bin v v th hm s khi 1m = .2.Chng t vi mi gi tr khc 0 ca m , th ca hm s ct trc honh ti 3 im phn bit A, BC trong B, C c honh ph thuc tham sm. Tm gi tr ca m cc tip tuyn ti B, C songsong vi nhau.

Cu II: (2 im)1. Gii phng trnh: 23 4sin 2 2cos 2 (1 2sin ) x x x = + .

2. Tm gi tr nh nht v gi tr ln nht ca hm s 3 2( ) 2 3 12 10 f x x x x= + trn [-3; 3].Cu III: (2 im)

Tam gic ABC c cc gc , ,A B C tha mn

sin

sin

sin

sin

2 4sin 1 4sin22

4sin 1 4sin2

A

B

B

C

A B

B C

+ = +

+ = +

.Chng minh tam gic ABC u.

Cu IV: (2 im)

Tnh tch phn3

2

4

tan

cos . 1 cos

x I dx

x x

=+

.

Cu V.A: (2 im) ( Dnh cho THPT khng phn ban)

Hnh chp t gic u SABCD c cnh y AB a= ; chiu cao 62

aSO = . Mt phng ( )P qua A

vung gc vi SC ct , ,SB SC SD ln lt ti ' ' ', , B C D .1. Tnh din tch thit din to thnh v tm t s th tch ca 2 phn hnh chp b ct bi mt phng( )P .

2. Tnh sin ca gc gia ng thng 'AC v mt phng ( )SAB Cu V.B: (2 im) ( Dnh cho THPT phn ban)

1. Tnh gi tr biu thc:5 7 2007

4 5 2008

...

...

i i iP

i i i

+ + +=

+ + +(trong 2 1i = )

2. Trong khng gian vi h trc ta Oxyz, cho tam gic ABC c ( 1; 3; 2)A ; ng cao BK

v trung tuyn CM ln lt nm trn cc ng thng:1

2

1 1 4( ) ;2 3 4

1 2 5( ) .

2 3 1

x y zd

x y zd

+ = =

+ = =

Lp phng trnh ng thng cha cc cnh ,AB ACca tam gic ABC.

------------------ HT -------------------


42/66




Cu I:1. Bn t gii2.Hanh giao im ca th vi trc honh l nghim pt : 3 2( 1) ( 1) 1 0 x m x m x + + + = T phng trnh trn => pcm .Cng t pt trn ta suy ra c cc h s gc ca tip tuyn ti B,C : ( 2) 1k m x m= + Vix1, x2 phn bit th, tip tuyn ti B v C song song nhau khi v ch khi kB=kcp s :m=2.

Cu II:

1.a v pt tch. p s : ( )

26

72

6

218 35 2

18 3

x k

x k

k Zkx

kx

= +

= +

= + = +

2. Dng o hm kho st .p s : Minf(x)=f(-1)=17; Maxf(x)=f(-3)=35.

Cu III :Hm s 2 4xy x= + ng bin cy(x)=1 x=0 . Ta c :

sin

sin

24sin 1 4sin sin sin2

A

B A B A B+ = + = .

Cu IV:

t 22 tant x= + th2 2

tan

cos 2 tan

xdxdt

x x=

+.

p s:5

3

5 3 I dt = = .

Cu V.A:Cc bn t gii.

Cu V.B:

1.p s:P=0.2. p n:

Phng trnh AB :1 3 2

2 9 7

x y z+ + = = . Phng trnh AC:

1 3 2

13 19 8

x y z+ + = =

.

------------------ HT -------------------


43/66




NM2007



Cu I: (2 im)Cho hm s 3 2 2 2(2 3) (2 9) 2 3 7( )my x m x m m x m m C = + + + +

1. Kho st hm s khi 0m = 2. Tm m ( )mC ct trc honh ti 3 im phn bit c honh 1 2 3, ,x x x khng nh hn 1.

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

1. 3 3 x x+ + = ;2. 2cos cos 2 cos3 5 7cos 2 . x x x x+ =

Cu III: (2 im)Trong khng gian vi h trc ta Oxyz cho mt phng ( )P c phng trnh 3 0 x y z+ + + = vcc im (3;1;1); (7;3;9); (2;2;2) A B C .1. Tnh khong cch t gc ta n mt phng ( )ABC .

2. Tm M thuc mt phng ( )P sao cho 2 3 MA MB MC + +

nh nht.

Cu IV: (2 im)

1. Tnh1 3

2 30 (1 )

x I dx

x=

+ .

2. Cho cc s dng , , x y z tha mn:

2 2

2 2

2 2

3 3 75

3 2716

x xy y

y z

z xz x

+ + =

+ =

+ + =

. Tnh 2 3P xy yz xz= + + .

Cu V.A: (2 im) ( Dnh cho THPT khng phn ban)

1. Trong mt phng ta Oxyz , hy lp phng trnh ng thng dcch im (1;1)A mkhong bng 2 v cch (2;3)B mt khong bng 4.

2. Cho dy s ( )nu c s hng tng qut3

5

195(1 )

16( 1)

nnn

n n

Cu C n

n

++

= +

. Tm cc s hng dng ca

dy.

Cu V.B: (2 im) ( Dnh cho THPT phn ban)1. Gii phng trnh trong tp hp s phc 2 0z z+ = .

2. Cho hnh chp t gic u .S ABCD c cnh y bng a v ASB = . Tm th tch hnh chp.S ABCD .

------------------ HT -------------------


44/66




Cu I:

1.Bn t

gii

2. PT hanh giao im c dng : ( ) ( )( )2 21 2 1 2 3 7 0 x x m x m m + + + = . Sau xt tip phng trnh bc 2 c 2 nghim phn bit ln hn 1.p s: 2 < m < 3.

Cu II:

1.t ( )3 3u x u= + . Ta a vc hi xng nh sau :3

3

u x

x u

=

= .

Gii h trn kt hp vi iu kin suy ra nghim .

p s: 7 132

x+

= .

2.a v phng trnh tch .p s: ( ) x k k Z = .

Cu III:

1. Vit phng trnh mt phng (ABC) . Sau dng cng thc khang cch tnh khang cch .

p s : ( )( )2 6

,3

d O mp ABC =

2. Chn im I sao cho: ( )2 3 0 4 IA IB IC + + =

Khi 2 3 MA MB MC + +

min khi MI

min hay M l hnh chiu ca I ln mp(P) .

T (4) ta tm c23 13 25

; ;6 6 6

I

.

Sau tm hnh giao im ca (d) qua I v vung gc mp(P)

p s :5 20 2

; ;9 9 9

M

Cu IV:

1.t ( )2tan tan 1 x dx d = = +

( )( )

34 43

220 0

tan . 1cos cos (cos )

161 tan

dJ d

= = + =

+

p s :1

16J = .


45/66



2. ( )

( )

2 2

2 2

2 2

3 3 75

3 27 5

16 6

x xy y

y z

z xy x

+ + =

+ =

+ + =

Bi ny kh r

i. Ta s

biu di

nx, y

quaz

.t x = kz . Ta c : ( ) ( ) ( )2 2 2 2 2 23 3 3 3 0 z xy x y z x xy y+ + + + + + =

2 22 1 z xz xy y zk

+ = = +

T (5) v (6) suy ra hng cp :

( )

22

2 2

21 3 27

1 16

zk

z k k

+ + =

+ + =

.

Gii ra gi tr ca kvz .p s: 24 3P = .

Cu V.A:Cc bn t gii.

Cu V.B:

1.t z = x+yi. p s: z1 = 0; z2 = i; z3 = -i.

2. Gi O l tm hnh vung ABCD . I l trung im AB. Tnh SO.

p s:3

.

. cos

6.sin2

S ABCD

aV

= .

------------------ HT -------------------


46/66




NM2007



Cho hm s( )2 2 3 6 1

2 x m x m

y x

+

= (1)1. Kho st v v th hm s khi (1) khi m=1 .2. Tm m hm s (1) c cc i , cc tiu ng thi 2 im cc i v cc tiu nm v 2 phaca ng thng y = -x + 7.

Cu II: (2 im)

1. Gii phng trnh : 3 3sin cos cos2 .tan .tan4 4

x x x x x

= +

2. Gii h phng trnh :( )

( )

3 2

3 2

1 2

1 2

x x x y

y y y x

+ = +

+ = +

.

Cu III: (2 im)

Trong khng gian Oxyz , cho 2 im A(1 ; -1; 2) , B(3; 1; 0) v mt phng (P) c phng trnh (P) cphng trnhx 2y - 4z +8 = 0 .1. Lp phng trnh ng thng (d) tha mn ng thi cc iu kin sau : (d) nm trong mtphng (P) , (d) vung gc vi AB v (d) i qua giao im ca ng thng AB vi mt phng (P)2. Tm ta C trong mt phng (P) sao cho CA=CB v mt phng (ABC) vung gc vi mt phng(P).

Cu IV: (2 im)

1. Tnh tch phn :1

2

0

3 6 1 I x x dx= + +

2. Chng minh rng:2 21 2 7 2 1 2 7 x xy y + + . Vix, y l cc s thc tha mn 2 2 3 x xy y + .

Cu V: (2 im)1. Gii phng trnh :

( ) ( )5 4log 3 3 1 log 3 1x x+ + = + .

2. Cho hnh chp tam gic u S.ABC c ng cao SH=h ,

ASB = .Tnh th tch hnh chp theo h v .

------------------ HT -------------------


47/66




Cu I:

1. Bn t gii

2 Hm s c cc i cc tiu khi( )

2

24 10 7'

2

x x my

x

+ =

= c 2 nghim phn bit .

Gi ta 2 im cc i , cc tiu l C(x1;y1) , CT(x2;y2) .

Dng iu kin khc pha xt du .

p s :41 3 229 41 3 229

4 4m

+< < .

Cu II:

1. Dng gc cung lin kt a v pt tch.

p s : ( )2 ; 22

x k x k k Z

= + = .

2.a v hm s ( ) 3 22 2 1 f t t t t = + + . Sau chng minh hm s lun ng bin .

Nux>y th 2y=f(x)>f(y)=2x . V l . Tng t=> x = y .

p s :1 5

1;2

x x

= = .

Cu III:

1.p n : ( )2 3

: 2 1 1

x y zd

+ = = .

2.p s : C(2 ;1 ;2).

Cu IV:

1. ( )( )1

2

0

4 3 1 I x dx=

t2

1 sin , ;

2 23

x t t

=

. Khi ( )0

3

21 cos 2

3

I t dt

= + .

p s :2 1

23 3I

= +

2. Cc bn t gii .


48/66



Cu V:

1.t ( ) ( )5 4log 3 3 1 log 3 1x x t+ + = + =

Ta c ( )3 1 4

3 2 5 33 3 1 5

x t

t tx t

+ = + = + + =

T (3) ta c1 2

3. 15 5

t t

+ =

. V tri nghch bin .

p s :x=1.

2. p s :3

2

3

3.cot 12

hV

=

.

------------------ HT -------------------


49/66




NM2008



Cho h th:

2 2 1( ) 1

x x mCm x

+ + = .

1. Kho st v v th hm sng vi m = 1.2. Tm m (Cm)c cc im cc i, im cc tiu v gc ta O lp thnh tam gic vung ti O

Cu II: (2 im)1. Gii phng trnh:

sin(3 ) sin 2 .sin( ).4 4

x x x

= +

2. Gii h phng trnh:

2 2 2

3 3 3

.log 3 log log

.log 12 log log

x y y x

x x y y

+ = +

+ = + .Cu III: (2 im)

1. Tnh cc tch phn sau:

2

31 . 1

dxI

x x=

+ ;

2

40

sin2

1 cos

x J dx

x

=+

.

2. Cho bn imA(5;1;3),B(1;6;2) , C(5;0;4) ,D(4;0;6). Chng minh rng hai ng thngAB vCD cho nhau. Tnh khong cch giaAB v CD v vit phng trnh ng vung gc chung cachng.

Cu IV: (2 im)

1. Gii phng trnh:32( 2)( 4 4 2 2) 3 1 x x x x + = .2. Cho a, b l cc s thc dng. Chng minh rng :

2 2

2 3

(1 )(1 )2 .

( 1)(1 )

a b b

a a b

+ +

+ +

Cu V.A: (2 im) ( Dnh cho THPT khng phn ban)1. Cho n l s nguyn dng vi n 2. Chng minh rng:

2 1 2 2 2 3 2 21 . 2 . 3 . ... . ( 1).2n nn n n nC C C n C n n

+ + + + = + 2. Cho tam gicABC. Xt tp hp gm nm ng thng song song viAB; su ng thng songsong viBCv by ng thng song song vi CA. Hi cc ng thng ny to ra bao nhiu hnh

bnh hnh, bao nhiu hnh thang?Cu V.B: (2 im) ( Dnh cho THPT phn ban)

Cho ng thng () c phng trnh 2 2 0x y + = v elip (E) c phng trnh2 2

18 4

x y+ = . Gi

sng thng () ct (E) ti hai imB v C.1. Tm im A thuc elip (E) tam gicABCcn ti A.2. Tm im A thuc elip (E) din tch tam gicABCt gi tr ln nht.

------------------ HT -------------------


50/66




Cu I:1. Cc bn t gii.2. (Cm) c im cc i M v im cc tiu N khi v ch khi m > - 2.

S dng iu kin ,0. =ONOM ta tm c5

7=m .

Cu II.

1. p s:24

kx = .

2. p s: ( ) .2log2;2log;3

4

3

4

=yx

Cu III.

1. p s:

+=

2

223ln

3

1I ;

4

=J .

2. p s:103

23693=IJ

Phng trnh ng vung gc chung:144

103

476

1221103

586

==

zy

x

.

Cu IV:1. p s:x = 32. Cc bn t chng minh.

Cu V.A:1. Cc bn t chng minh.2. p s: S hnh bnh hnh l: 675 ( hnh ). S hnh thang l: 1575 (hnh).

Cu V.B:

1. p s:

+

+

25

3921;

5

392,

25

3921;

5

39221 AA

2. p s: ( )2;2 A

------------------ HT -------------------


51/66




NM2008



Cho hm s: 4 2 2( ) : 2( ) 5 5Cm y x m x m m= + + + 1. Kho st s bin thin v v th (C1) ca hm s khi m = 1.2. Vi nhng gi tr no ca m th th (Cm) c im cc i v im cc tiu, ng thi cc imcc i v im cc tiu lp thnh mt tam gic u.

Cu II: (2 im)

1. Gii phng trnh:1

(1 cos )(1 cos 2 )(1 cos3 ) .2

x x x+ + + =

2. Gii h phng trnh:2

1 2

1 2

2log ( 2 2) log ( 2 1) 6

log ( 5) log ( 4) 1

x y

x y

xy x y x x

y x

+

+

+ + + + =

+ + =

Cu III: (2 im)

1. Tnh tch phn:

11 3 3

41

3

( ).

x x I dx

x

=

2. Cho cc s thc dng a, b, c tha mn ab + bc + cd= abc. Chng minh rng:4 4 4 4 4 4

3 3 3 3 3 31.

( ) ( ) ( )

a b b c c a

ab a b bc b c ca c a

+ + ++ +

+ + +

Cu IV: (2 im)Trong khng gian vi h ta Descartes Oxyz, cho mt phng (P) c phng trnh

2 1 0 x y z+ + = v ng thng (d) c phng trnh

2 2 0

2 2 0

x y

y z

=

+ + = .1. Tm to giao imA ca (d) v (P). Tnh so gc to bi (d) v (P).2. Vit phong trnh ng thng () i quaA, () nm trong mt phng (P) sao cho gc to bi haing thng () v (d) bng 450.

Cu V.A: (2 im) ( Dnh cho THPT khng phn ban)1. Vit phng trnh ng trn i qua hai imA(2;5) ,B(4;1) v tip xc vi ng thng cphng trnh 3 9 0.x y + = 2. Vi n l s nguyn dng, chng minh h thc sau:

1 2 2 2 22( ) 2( ) ... ( )

2

n n

n n n n

nC C n C C + + + =

Cu V.B: (2 im) ( Dnh cho THPT phn ban)

1.Gii phng trnh: 84 221 1

log ( 3) log ( 1) log 4 .2 4

x x x+ + =

2.Cho hnh chp t gic u S.ABCD c cnh y bng a, chiu cao cng bng a. GiE, K ln ltl trung im ca cc cnhAD vBC. Tnh bn knh ca mt cu ngoi tip hnh chp S.EBK.

------------------ HT -------------------


52/66




Cu I.1. Cc bn t gii.2.

k

(Cm) c ba

im c

c tr

l

m

< 2. Ccim c

c tr

ca (C

m

) l( ) ( ) ( ).1;2,1;2,55;0 2 mmCmmBmmA + p s: 3 32 =m .

Cu II.

1. p s:

23

2;

24mxkx +=+= .

2. p s: ( ) ( )1;2; =yx .

Cu III.

1. p s: 6=

I 2. Cc bn t gii.

Cu IV.1. p s: ( )1;0;1 A . Gc ( ) 030)(, =Pd .

2. p s: Hai ng thng tho mn l: ( )335

1

3132

1:1

+=

+=

+

zzx;

( )335

1

3132

1:2

+

+=

=

zyx

Cu V.A:

1. p s:2 2

1

2 22

( ) : ( 1) ( 2) 10( ) : ( 17) ( 10) 250C x y

C x y + =

+ =.

2. Cc bn t chng minh.

Cu V.B:

1. p s:3

3 2 3

x

x

=

= +.

2. p s:8

29aR =

------------------ HT -------------------


53/66




NM2008



Cu I: (2 im)

Cho hm s

2

( ) : 1

x

C y x

= 1. Kho st v v th (C) ca hm s.2. Tm cc gi tr ca tham sang thng (d) :y = a(x 3) ct th (C) ti hai im phnbit trong c t nht mt giao im c honh ln hn 1.

Cu II: (2 im)1. Gii phng trnh: 5 3 22sin 2sin .cos cos 2 sin 0 x x x x x+ + =

2. Gii h bt phng trnh:3 2

4 3 2

3 9 10 0

5 5 5 4 0

x x x

x x x x

+ +


54/66




Cu I.1. Cc bn t gii.2. p s: 0a .

Cu II.

1. p s:

kx

kx

+=

+=

4

22 .

2. p s: 14


55/66




NM2008



Cho hm s: ( )3 22 1 y x x m x m= + (1).

1. Kho st s bin thin v v th ca hm s khi m = 1.2. Trong trng hp hm s (1) ng bin trong tp s thc R, tm m din tch hnh phng giihn bi th ca hm s (1) v hai trc Ox, Oy c din tch bng 1.

Cu II: (2 im)1. Gii phng trnh nghim thc: 1 tan . tan 2 cos 3 . x x x =

2. Tm tt c cc gi tr ca tham sk phng trnh ( )1 4 2 1 2 x x xk k+ + = c nghim.

Cu III: (2 im)1. Trong mt phng vi h ta Oxy cho elip (E): 2 24 4.x y+ = Qua im M(1;2) k hai ngthng ln lt tip xc vi (E) ti A v B. Lp phng trnh ng thng i qua hai im A v B.

2. Cho tam gic ABC tha mn: ( )5

cos 2 3 cos 2 cos 2 0.2

A B C + + + =

Tnh ln ba gc ca tam gic .

Cu IV: (2 im)

1. Tnh tch phn sau:2

2 sin

0

2cos cos .2

xx I x x e dx

= +

2. Cho ba s thc dngx, y, z tha mn iu kin: 2 1. xy xz+ =

Tm gi tr nh nht ca biu thc:3 4 5

. yz zx xy

S x y z

= + +

Cu V.A: (2 im) ( Dnh cho THPT khng phn ban)1. Trong khng gian vi h trc ta Oxyz cho hai ng thng:

( )12 4 0

:3 0

x yd

z

+ =

= ( )2

0:

1 0

y zd

x

+ =

=

Lp phng trnh mt cu c bn knh nh nht tip xc vi c hai ng thng trn.2. C tt c bao nhiu s t nhin chn c 4 ch s, sao cho trong mi sng sau ln hn ch sng lin trc n?

Cu V.B: (2 im) ( Dnh cho THPT phn ban)1. Cho hnh chp t gic S.ABCD. y ABCD l hnh vung cnh bng a, SA vung gc vi mtphng (ABCD) v SA = a. Tnh din tch ca thit din to bi hnh chp vi mt phng qua Avung gc vi cnh SC.2. Gii bt phng trnh: 2 1log 3 log 2xx .

------------------ HT -------------------


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Cu I:1. Cc bn tgii.

2. p s: 316

=m .

Cu II:

1. Hng dn:

=

=

1cos

03cos

x

x.

2. p s:2

10


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NM2009



1. Kho st v v th ca hm s:

2

( ).1

x

y Cx

= 2. Chng minh rng vi mi gi tr thc ca m, ng thng y x m= + (d) lun ct th (C) tihai im phn bit A, B. Tm gi tr nh nht ca di on thng AB.

Cu II: (2 im)

1. Gii phng trnh2

2 13 .2 6x

x x = .

2. Gii phng trnh tan tan s in3 sin s in26 3

x x x x x

+ = +

.

Cu III: (1 im)Tnh th tch hnh chp S.ABCbit , , , 60 ,oSA a SB b SC c ASB= = = = 90 , 120 .o o BSC CSA= =

Cu IV: (1 im)

Tnh tch phn( )

2

30

sin .

sin 3.cos

x dxI

x x

=

+ .

Cu V: (1 im)

Tm gi tr nh nht ca biu thc 2 2 22 2 2log 1 log 1 log 4P x y z= + + + + + trong l cc s

dng , , x y z tha mn iu kin 8.xyz =

PHN DNH RING CHO CHNG TRNHCHUN.

Cu VIa: (2 im)1. Trong mt phng vi h trc ta vung gc Oxy, cho hai ng thng c phng trnh

11 0( ); x y d + + = 22 1 0( ). x y d = Lp phng trnh ng thng i qua im (1; 1)M ct 1( )d ,

2( )d tng ng ti A, B sao cho 2 0 MA MB+ =

.

2. Trong khng gian vi h trc ta vung gc Oxyz, cho mt phng (P) c phng trnh2 2 1 0 x y z+ + = v hai im (1: 7 : 1)A , (4;2;0)B . Lp phng trnh ng thng (d) l hnh

chiu vung gc ca ng thng AB trn mt phng (P).

Cu VIIa: (1 im)K hiu 1x , 2x l hai nghim phc ca phng trnh bc hai

22 2 1 0x x + = . Tnh gic tr cc s

phc21

1

x,

22

1.

x


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PHN DNH RING CHO CHNG TRNH NNG CAO.

Cu VIb: (2 im)1. Trong mt phng vi h trc ta vung gc Oxy, cho hyperbol (H) c phng trnh

2 2

19 4

x y

= . Gi s (d) l mt tip tuyn thay i v F l mt tring hai tiu im ca (H), k FMvung gc vi (d). Chng minh rng M lun nm trn mt ng trn cnh, vit phng trnhng trn .2. Trong khng gian vi h truc tc vung gc Oxyz, cho ba im (1;0;0)A , (0;2;0)B , (0;0;3)C .Tm ta trc tm ca tam gic ABC.

Cu VIIb: (1 im)Ngi ta s dng 5 cun sch Ton, 6 cun sch L, 7 cun sch Ha (cc cun sch cng loiging nhau) lm gii thng cho 9 hc sinh, mi hc sinh c hai cun sch khc loi. Trong s9 hc sinh trn c 2 bn Ngc v Tho. Tm xc sut hai bn Ngc v Tho c gii thng gingnhau.

------------------ HT -------------------


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Cu I:1. Cc bn t gii.

2. p s: min 2 2AB = 2m = .

Cu II:

1. Phng trnh c 3 nghim: p s:3

1

1 1 8log 6

4

x

x

= +

=

.

2. Phng trnh c 3 h nghim: p s:22

23

x k

x k

x k

=

=

= +

.

Cu III:

p s: Th tch l2

12V abc= .

Cu IV:

p s:3

6I= .

Cu V:

p s: min 5 2 2P x y z= = = = .

Cu VI.a:1. p s:ng thng:x=1.

2. Phng trnh ng thng:3 2

4 3 1

x y z = =

.

Cu VII.a:p s: Cc gi tr: (-2i; 2i).

Cu VI.b:Cc bn t gii.

Cu VII.b:Cc bn t gii.

------------------ HT -------------------


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NM2009



Cho hm s )(1

42Cx

x

y +

= .1. Kho st s bin v v th hm s(C).2. Tm trn th (C) hai im i xng nhau qua ng thng MN bit M (-3;0) v N (-1;-1).

Cu II: (2 im)

1. Gii phng trnh: .2

7

4

3cos4cos

2

12coscos4 4 =+

xxxx

2. Gii phng trnh: 1232.3 ++= xx xx .

Cu III: (1 im)

Tnh tch phn:

+

+=

2

0 cos1

sin1

dxex

xI

x .

Cu IV: (1 im)Cho hnh chp tam gic u S.ABC di cnh bn bng 1. Cc mt bn hp vi mt phng y mtgc . Tnh th tch hnh cu ni tip hnh chp S.ABC.

Cu V: (1 im)Trong khng gian vi h ta vung gc Oxyz cho ng thng l c phng trnh:

)(

24

2

32

Rt

tz

ty

tx

+=

=

+=

v hai im A(1;2;-1), B(7;-2;3).

Tm trn ng thng l nhng im sao cho tng khong cch t n A v B l nh nht.


Cu VIa: (2 im)1. Nm on thngc di 1 cm, 3 cm, 5 cm, 7 cm, 9cm. ly ngu nhin ba on thng trong nmon thng trn. Tm xc sut ba ton thng ly ra lp thnh mt tam gic.

2. Gii h phng trnh:

=

+=

5

8

yx

yyxyxx

.

Cu VIIa: (1 im)

Tm gi tr nh nht ca hm s:)sincos2(sin

cos2

xxx

xy

= , vi .

30

< x


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Cu VIb: (2 im)

1. Tm cc gi tr cax trong khai trin khai nh thc Newton:n

xx

+ 5 3log)2()310log( 22 bit rng

s hng th su ca khai trin bng 21 v 231 2 nnn CCC =+ .

2. Cho

+=

3

2sin

3

2cos3

i . Tm cc s phc sao cho =3 .

Cu VIIb: (1 im)Gi a, b, c l di ba cnh ca mt tam gic c chu vi bng 2. Chng minh rng:

2227

52 222


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Cu I:1. Cc bn t gii.2.p s: (0; 4); (2; 0)A B .

Cu II:1. p s: 8kx = .2. Phng trnh c 2 nghim: p s: 1;1 == xx .

Cu III:

p s: 2

eI = .

Cu IV:

p s:

3

2

sin tan

tan 4(1 sin )V

= + + .

Cu V:p s: (2;0;4)M .

Cu VI.a:

1. p s:35

3

C.

2. p s: )2;3();2;3();( =yx .

Cu VII.a:

p s: 13tan2

32min =

+= y .

Cu VI.b:1. p s:x=2.

2.

p s:

+

=

+

=

+

=

9

8sin

9

8cos3

9

2

sin9

2

cos3

9

4sin

9

4cos3

33

3

2

31

i

i

i

.


------------------ HT -------------------


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NM2009



Cu I: (2 im)

Cho hm s )1(1

)12( 2

= x

mxmy .

1. Kho st s bin v v th (C) ca hm s (1) ng vi m = -1.2. Tm m th ca hm s (1) tip xc vi ng thng y =x.

Cu II: (2 im)

1. Gii phng trnh: 09

log))9((log 22 =+

++x

xxx .

2. Gii h phng trnh:

=+

=+

++

yxyx

yx

xyyx

2

22 12

.

Cu III: (1 im)

1. Tm gii hn:3 22

2

0 1

)1ln(lim

2

xe

xL

xx +

+=

.

2. Tnh tch phn: +

=2

03)cos(sin

sin

dxxx

xdxI .

Cu IV: (1 im)

Cho hnh chp ct tam gic u ngoi tip mt hnh cu bn knh rcho trc. Tnh th tch hnhchp ct, bit rng cnh y ln gp i cnh y nh.

Cu V: (1 im)

Cho phng trnh: mxxx

x+=

12

12

13 2( vi m l tham s).

Tm m phng trnh cho c nghim duy nht.


Cu VIa: (2 im)1. Trong khng gian vi h ta Oxyz cho mt phng P c phng trnh:x+y+z+3=0; ng thng

dc phng trnh:12

2

1

1 zyx=

=

+v cc im A(3;1;1); B(7;3;9); C(2;2;2).

a. Vit phng trnh mt phng (Q) cha dv song song vi mt phng (P).

b. Tm ta ca im M thuc (P) sao cho MCMBMA 32 ++ nh nht.

2. Cho ng trn (C): 012622 =++ yxyx . Vit phng trnh ng thng di qua M(0;2) vct (C) theo mt dy cung c di l 4.


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Cu VIIa: (1 im)Chng minh rng vi mi s t nhin n ( vi n>2), ta c: )1(22 )1()2( > nnn nnn .


Cu VIb: (2 im)1. Trong khng gian h ta Oxyz cho mt phng ( ) c phng trnh: 3x+2y-z+4=0 v hai imA(4;0;0) v B(0;4;0). Gi I l trung im ca on thng AB. Tm ta giao im ca ng thngAB vi mt phng ( ) v xc nh ta im K sao cho KI vung gc vi mt phng ( ), ngthi K cch u gc ta O av2 mt phng ( ).

2. Cho elip (E) c phng trnh 125100

22

=+yx

. Tm cc im M thuc (E) nhn hai tiu im ca elip

(E) di mt gc 1200.

Cu VIIb: (1 im)

Chng minh rng vi mi s t nhin n ( vi n>2), ta c: )1ln()1ln(ln2

+> nnn .

------------------ HT -------------------


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Cu I:1. Cc bn t gii.2. p s: { }1\Rm .

Cu II:1. p s: 10=x .2. H phng trnh c 2 nghim: p s: )3;2();0;1();( =yx .

Cu III:

1. p s:3

7L

= .

2. p s:1

2I = .

Cu IV:Cc bn t gii.

Cu V:p s: Rm .

Cu VI.a:1.a. p s: 01 =++ zyx .

b. p s:

9

2;

9

20;

9

5M .

2. p s: 22;22

1+=+= xyxy .

Cu VII.a:Cc bn t gii.

Cu VI.b:

1. p s: Giao im: )0;16;12( ;

4

3;

2

1;

4

1K .

2. p s:

3150;

33101M ;

3150;

33102M ;

3150;

33103M v

3150;

33104M .


------------------ HT -------------------


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Thc hin:

Tp th Lp 12T

Trng THPT Th x Cao LnhNin kha: 2006 -2009

Gio vin ch nhim:

Thy Nguyn nh Huy

1. Tuyn tp cc nm 2003:Tp tht4. Chu trch nhim chnh: L Ngc c

2. Tuyn tp cc nm 2004:Tp tht5. Chu trch nhim chnh: Phm Ngc Trng

3. Tuyn tp cc nm 2005:Tp tht6. Chu trch nhim chnh: L Thanh Sang

4. Tuyn tp cc nm 2006:Tp tht3. Chu trch nhim chnh: Trnh Hong Anh

5. Tuyn tp cc nm 2007:Tp tht2. Chu trch nhim chnh: Nguyn Hong Vit Khnh

6. Tuyn tp cc nm 2008:Tp tht1. Chu trch nhim chnh: Nguyn Hng Hong

7. Tuyn tp cc nm 2009:Tp tht3. Chu trch nhim chnh: Nguyn c Tun

Nguyn c Tun

Tuyen Tap Cac de Thi Thu Dai Hoc Cao Dang Tren Tap Chi Toan Hoc Tuoi Tre

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Transcript of Tuyen Tap Cac de Thi Thu Dai Hoc Cao Dang Tren Tap Chi Toan Hoc Tuoi Tre