com - Dap an 40 de Thi Thu Mon Toan

7/31/2019 com - Dap an 40 de Thi Thu Mon Toan

1/260

B thi t lun- mn Ton Dnh cho hc sinh lp 12 n thi i hc- Cao ng

Vn Ph Quc- GV. Trng i hc Qung Nam D: 0982 333 443 ; 0934 825 925 1

1

A- PHN CHUNG (7,0 im)

Cu I (2,0 im) . Cho hm s: 4 22 3 y x x (C).

1. Kho st s bin thin v v th (C) ca hm s.2. Tm m ng thng y m ct th (C) ti bn im phn bit M, N, P, Q ( spth t t tri sang phi) sao cho di cc on thng MN, NP, PQ c gi s l d i 3cnh ca mt tam gic bt k.Cu II(2,0 im)

1. Gii phng trnh: 2sin .sin 4 2 2 cos 4 3 cos .sin .cos 26

x x x x x x

2. Gii h phng trnh:

2 22 3 8 1

, y8 3 13

x y y xx

x x y y.

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

21

1 4

x

x

x edx

x xe.

Cu IV(1,0 im). Tnh th tch khi t din ABCD bit AB = a, AC = b, AD = c

v 0BAC CAD DAB 60 .

Cu V (1,0 im). Chng minh phng trnh: 1 1xx

x x lun c nghim thc dng

duy nht.B- PHN RING (3,0 im). Th sinh ch c chn mt trong hai phnB.1. CHNG TRNH CHUNCu VI a (2,0 im)1.

Trong mt phng Oxy, cho ng thng : 1 0d x y v ng trn 2 2: 2 4 0C x y x y . Tm ta im M thuc ng thng d m qua k c hai

ng thng tip xc vi ng trn C ti A v B sao cho 060AMB .2. Trong khng gian Oxyz, cho 3 im ;0;0 , B 0; ;0 , C 0;0;A a b c vi a, b, c l cc

s dng thay i v tha mn 2 2 2 3a b c . Xc nh a, b, c sao cho khong cch tgc to O 0;0;0 n mt phng ABC t gi tr ln nht.

Cu VII a (1,0 im). Tm ,a b phng trnh 2 0z az b c nhn s phc1z i lm nghim.

B.2. CHNG TRNH NNG CAO

Cu VI b (2,0 im)1. Trong mt phng Oxy, cho prabol 2:P y x . Vit phng trnh ng thng d i quaM(1; 3) sao cho din tch hnh phng gii hn bi (P) v d t gi tr nh nht.2. Trong khng gian vi h to Oxyz, cho hai im A 1; 5; 0 , B 3; 3; 6 v ng

thng d:1 1

2 1 2

x y z

. Xc nh v tr ca im C trn ng thng d din tch tam

gic ABC t gi tr nh nht.

www.MATHVN.com

MATHVN.COM - Ton hc Vit Nam


2/260



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

2 32 2 4 2 4 2

4 1 2 22

1log 1 log 1 log 1 log 1

3x x x x x x x x .

BI GII

A- PHN CHUNGCu I1. Hc sinh t gii.2. Phng trnh honh giao im ca ng thngy = m v th (C):

4 2x 2x 3 m 4 2x 2x 3 m 0 1

t 2t x t 0 .Phng trnh trn thnh: 2t 2t 3 m 0 2

Gi 1 2 3 4x , x , x ,x ln ltl honh cc giao im M, N, P, Q.

Khi 1 2 3 4

x , x , x , x l nghim ca phng trnh (1).

Da vo th , ta thy vi iu kin: 4 m 3 th phng trnh (2) c hai nghiml: 1 2t 1 m 4 ; t 1 m 4 .

Suy ra 1 2 2 1 3 1 4 2x t , x t , x t , x t

Ta c 4 3 3 2 3MN PQ x x , NP = x x 2x

rng: iu kin ba s dng a, b, c l di 3 cnh ca mt l:

a b c

b c a

c a b

.

V MN PQ nn MN, NP, PQ l di 3 cnh ca mt tam gic bt k nn ta ch cn:

4 3 3 4 32 2 2 2MN PQ NP PQ NP x x x x x

hay 2 1 2 13 91

t 2 t t 4t 1 m 4 4 1 m 4 m 4 m5 25

Kt hp vi iu kin : 4 m 3 ta c:91

m 325

.

Cu II1. Ta c:

sin .sin 4 2 2 cos 3 cos .sin 46

x x x x x

sin 4 sin 3 cos 2 2 cos 6x x x x

sin 4 . sin sin cos cos 2 cos sin 4 2 cos 06 6 6 6

x x x x x x

www.MATHVN.com



3/260



cos 06

x

( v sin 4 2 0x )

2

6 2 3x k x k k

.

2. iu kin: 2 2x 3y 0 , y 8x 0 t 2 2u x 3y , v = y 8x ( u, v 0 )

H phng trnh thnh:

22 2 2 2 2

v 2u 12u v 1 v 2u 1

u v 13 u v 13 u 2u 1 13

2

v 2u 1

5u 4u 12 0

2 1

2236

5

v u

uu

vu

.

Khi :

2

2 2

2222

4 xy

3x 3y 2 x 3y 4

4 xy 8x 9y 8x 3 8x 93

222

24 2

1444 13

331 51 5 4 13 08 72 65 0

5 7

xxx yx

y yy

x xx x x xx x x

x y

Kt hp vi iu kin ta u ta thu c tp hp nghim ca h phng trnh l:

1;1 , 5; 7S .Cu III

Ta c:4 4x

2x2x x1 1

1 x e 1 1 1I dx dx

4x 4x exe xe

24

x1

1 1dx

e2 x

=4

x1

1 1dx

e2 x

=

4x

1x e

4

1 11

e e

Cu IV

Gi s a min a,b,c Trn cnh AC ly im E, AD ly im F sao choAB AE AF a .T din ABEF c bn mt l cc tam gic u bngnhau nn l t din u cnh bng a.

b

a

c

B

C

D

A

H

E

F

www.MATHVN.com



4/260



Ta d dng tnh c3

ABEF

a 2V

12 .

Gi H l chn ng cao h t B.

Ta c:

AEF

ACD

02

ABEF

0ABCD

1 1BH.S AEAFsin 60V a3 2

1 1V bcBH.S ACADsin 603 2

.

Suy ra ABCDabc 2

V12

.

Cu V

Ta c: xx 1x x 1 x 0 x 1 ln x x ln x 1 0

Xt hm s: f x x 1 ln x x ln x 1 x 0

1 1 1f x ln 1

x x x 1

Chng minh bt ng thc c bn ln 1 t t t > 0 ta suy ra1 1

ln 1x x

Do 1 1 1 1

f x 0 f xx x x 1 x 1

ng bin trn 0; (1)

Mt khc: do f x lin tc trn 0; v f 2 f 3 ln8 ln 9 ln81 ln 64 0 suy ra

tn ti 0 2;3 0x , sao cho 0f x 0 (2)T (1) v (2) iu phi chng minh.B- PHN RING

B.1. CHNG TRNH CHUNCu Via

1. Vit li C di dng 2 21 2 5x y

Vy C c tm 1;2I , bn knh 5R . ; 1M d M t t

Theo gi thit 60oAMB

Suy ra 30 2 2 2 5oAMI MI IA R

hay

2 22 23 3; 4

20 1 1 20 9

3 3;2

t MMI t t t

t M

2. Phng trnh mt phng (ABC):x y z

1a b c

2 2 2

1d d O; ABC

1 1 1a b c

B

A

M

I

www.MATHVN.com



5/260



2

2 2 22 2 2 2 2 2

1 1 1 1 1 1 1 1 19 a. b. c. a b c 3

a b c a b c a b c

.

Suy ra:2 2 2

1 1 13

a b c . Do :

1

3d .

Du bng xy ra 2 2 21 1 1 1 a b c 1a b c .

Vy1

Max d3

khi a b c 1 .

Cu VIIa

Theo , ta c: 2

1 i a 1 i b 0 21 2 0i i a ai b

a 2 i a b 0 a 2 0 a 2

a b 0 b 2

.

B.2. CHNG TRNH NNG CAOCu VIb1. Gi s d ct (P) ti hai im phn bit

2 2A a;a , B b;b ( b > a)PT ng thng d: y a b x ab Gi S l din tch hnh phng gii hn bi (P) v t d . Ta c:

b

2

a

S a b x ab x dx b

a

x a x b dx b

a

x a x b dx

=

b3

3 2

a1 a b 1x x abx b a3 2 6 .

Do M 1;3 d a b ab 3

Suy ra 3 32 2 22 1 1 1S b a a b 4ab ab 3 4ab

36 36 36

= 3321 8 128 8 2

ab 1 8 S36 36 9 3

8 2MinS ab 1 0 ab 1 a b 2

3 .

Vy ta lp c phng trnh ng thng d : y 2x 1 .2. Ta c: C d C 1 2t;1 t;2t

AB 2; 2;6 , AC 2t 2; t 4;2t

2 6 6 2 2 2

AB; AC ; ; 2t 24;8t 12;2t 12t 4 2t 2t 2t 2 2t 2 t 4

Gi S l din tch tam gic ABC.

www.MATHVN.com



6/260



Ta c: S = 2 2 21 1

AB;AC 2t 24 8t 12 2t 122 2

21 72t 144t 8642

= 21

72 t 1 792 3 222

.

Du = xy ra khi t 1 C 1;0; 2

Vy khi C 1;0;2 th MinS 3 22

Cu VIIbPhng trnh cho tng ng vi:

2 2 4 2 4 22 2 2 2log x x 1 log x x 1 log x x 1 log x x 1

2 2 4 2 4 22 2 2log x x 1 x x 1 log x x 1 log x x 1

4 2 4 2 4 22 2 2log x x 1 log x x 1 log x x 1

4 22log x x 1 0

4 2x x 1 1 4 2x 0

x x 0 x 1

x 1

Vy tp hp nghim ca phng trnh l: 1;0;1S

2

A- PHN CHUNG (7,0 im)Cu I (2,0 im) .

Cho hm s:

2 3

2

x

y x

(C).3. Kho st s bin thin v v th (C) ca hm s.4. Gi I l giao im ca hai tim cn. Tm im M thuc (C). Bit tip tuyn ca (C) tiM ct cc ng tim cn ti J v K sao cho ng trn ngoi tip tam gic IJK c dintch nh nht.Cu II(2,0 im)

1. Tm nghim 0;2

x

ca phng trnh sau y :

2 2 34sin 3 sin 2 1 2cos22 4

xx x

.

2. Gii h phng trnh: 3 32 2

8 27 18

4 6

x y y

x y x y

.

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

10 5 9

0

1 cos .sin .cosI x x xdx

.

www.MATHVN.com



7/260



Cu IV(1,0 im). Cho hnh chp S.ABC c y l tam gic ABC vung cn ti nh B,BA = BC = 2a, hnh chiu vung gc ca S trn mt phng y (ABC) l trung im E caAB v SE = 2a. Gi I, J ln lt l trung im ca EC, SC ; M l im di ng trn tia i

ca tia BA sao cho 00 90ECM v H l hnh chiu vung gc ca S trn MC.Tnh th tch ca khi t din EHIJ theo ,a v tm th tch ln nht.

Cu V (1,0 im). Chng minh rng: x 1

1 x 1 x2

x x x 0;1e

.

B- PHN RING (3,0 im)B.1. CHNG TRNH CHUNCu VI a (2,0 im)1. Trong mt phng Oxy, cho hnh thoi ABCD c phng trnh hai cnh AB, AD th tl: 2 2 0 ; 2x + y + 1= 0x y . Cnh BD cha im M 1;2 . Tm to cc nh cahnh thoi.

2. Trong khng gian Oxyz, cho ng thng 1 2:1 2 2

x y zd

. Vit phng trnh mt

phng (P) bit rng (P) cha ng thng d v to vi mt phng (xOy) mt gc nh nht.Cu VII a(1,0 im).

Tm tp hp im M m ta phc ca n tha mn iu kin: z 2 i 1 .

B.2. CHNG TRNH NNG CAOCu VI b (2,0 im)1. Trong mt phng Oxy, cho tam gic ABC cn ti BOx, phng trnh cnh AB c dng:

3 2 3 0x y ; tm ng trn ngoi tip tam gic l 0;2I . Tm to cc nh catam gic.2. Trong khng gian Oxyz, cho hai im A 2; 0; 0 v J 2; 0; 0 . Gi s l mt phngthay i, nhng lun i qua ng thng AJ v ct cc trc Oy, Oz ln lt ti cc im

B 0; b; 0 , C 0; 0; c vi b,c 0 . Chng minh rng:bc

b c2

v tm b, c sao cho din

tch tam gic ABC nh nht.Cu VII b(1,0 im).

Tnh0 0 1 1 2 2 3 3 2010 2010

2010 2010 2010 2010 20102 C 2 C 2 C 2 C 2 CP ...1.2 2.3 3.4 4.5 2011.2012

BI GII

A- PHN CHUNGCu I1. Hc sinh t gii.

www.MATHVN.com



8/260


9/260



3 33 3 318 3 18 27

13 3 3

u vu v u v uv u v u v

uvuv u v uv u v uv u v

. Suy ra u, v l

nghim ca phng trnh:2

3 5

23 1 0 3 5

2

t

t tt

.

Do :

3 5 3 53 5 3 52 2

4 42 2

3 3 5 3 3 53 3 5 3 3 5

2 2 2 2

x xx x

y yy y

.

Vy tp nghim ca h phng trnh cho l:

3 3 5 3 3 53 5 3 5; ; ;4 2 2 2S

.

Cu III

t 10 5 10 51 cos 1 cost x t x 9 4 4 910 5sin .cos sin .cos 2t dt x xdx x xdx t dt

Ta c: I 1 12

10 5 5 4 10 9 10 20

0 0 0

1 cos .cos .sin cos 2 1 2x x x xdx t t t dt t t dt

1

11 21

0

202

11 21 231

t t

.

Cu IVABC vung cn ti B nn

2 2 2 24 4 2 2AC AB BC a a a EBC vung cn ti B nn

2 2 2 24 5C BE BC a a a .Theo nh l ba ng vung gc ta c:EH HC . Suy ra: sinEH EC

1 5

2 2

a

EI EC .

1 1 5. sin 5 sin . cos2 2 2HEI

aS EH EI HEI a

= 25

sin28

a .

B A

C

S

M

E

I

J

H

www.MATHVN.com



10/260



Trong SCE , IJ l on trung bnh nn1

2IJ SE a v IJ HEI .

Do 3

21 1 5 5. sin 2 sin 2 .3 3 8 24EHIJ HEI

aV IJ S a a

EHIJV t gi tr ln nht khi sin 2 1 0 2

tc .4

Cu V

Xt hm s: x 1 x x x

1 x 1 x 1 x 1 x 1 xf x x x x x.x 1 x x x 0;1 .

x1 x

2

1 x x 1 xf x 2. ln x

1 x1 x

.

Xt: 1 x

g x 2. ln x1 x

2

21 xg x 0 g xx 1 x

ng bin trn 0;1

g x g 1 0 f x 0 x 0;1 f x nghch bin trn 0;1

11 x

11 1 x

x 1 x 1 x 1

1 2f x lim f x lim 1 x x .x 2lim 1 x 0;1

1 e1 x

.

B- PHN RINGB.1. CHNG TRNH CHUN

Cu VIa1. Ta c: A AB AD to im A l nghim ca h phng trnh:4

2 2 0 32 1 0 5

3

xx y

x yy

.

Do 4 5

;3 3

A

.

Gi ;N x y AC ( AC l tia phn gicBAD ). Hn na M v N nm cng pha i vi

ng thng AB nn ta c:

2 2 2 1

2 2 2 15 5

2 2 1 4 2 0 2 2 0

2 2 02 1 2 2 2 0

x y x y

x y x y

x y x y

x yx y

A

C

BD

M

N

www.MATHVN.com



11/260



2 2 2 1x y x y hay 3 0x y . Vy phng trnh ng thng cha cnh ACl: 3 0x y .

ng cho BD qua 1;2M nhn 1; 1BD ACu n

lm vect ch phng nn c

phng trnh l:1 2

3 0

1 1

x yx y

.

B AB BD to im B l nghim ca h phng trnh:2 2 0 4

3 0 1

x y x

x y y

. Do 4; 1B .

D AD BD to im D l nghim ca h phng trnh:2 1 0 4

3 0 7

x y x

x y y

. Do 4;7D .

Gi I AC BD to im I l nghim ca h phng trnh:3 0 0

3 0 3

x y x

x y y

. Do 0;3I .

V I l trung im AC nn4 13

;3 3

C

.

2. Gi vect php tuyn ca mt phng (P) l: 2 2 2; ; 0pn a b c a b c

.

vect php tuyn ca mt phng (xOy) l: 0;0;1xOyn

vect ch phng ca ng thng d l: 1;1;2du

. 0 2 0 2d PP u n a b c b a cd

.

Gi l gc gia hai mt phng (Q) v mt phng (xOy) 02

.

Ta c:

2 2 2 2 2 22 2cos

2 4 52

c c c

a b c a ac ca a c c

.

+ Nu 0c th cos 02

.

+ Nu 0c th ta chn 1c . Khi :

2 2

1 1 1cos

32 4 5 2 1 3a a a

.

Du = xy ra khi 1a . nh nht cos ln nht.

Do so snh hai trng hp ny ta c0;

2

1max cos3

khi 1 1a c ; b= .

Mt phng P qua 1; 2;0A d v 1; 1;1Pn

lm vect php tuyn nn c phng

trnh: 1. 1 1. 2 1. 0 0 3 0x y z x y z .Cu VIIaHai s phc lin hp c moun bng nhau, ta suy ra

www.MATHVN.com



12/260



z 2 i z 2 i (v z 2 i z 2 i z 2 i ).

T ta c: z 2 i 1 .

t ,z x iy x y .Suy ra :

2 2 2 2

2 1 2 1 1 2 1 1 2 1 1z i x y i x y x y

Tp hp cc im M l ng trn tm I 2;1 , bn knh R 1.

B.2. CHNG TRNH NNG CAOCu VI b1. Gi phng trnh ng thng AB i qua im

M(4;5) l: 2 24 5 0 0a x b y a b .Khi phng trnh ng thng BC i qua N(6;5) vvung gc vi AB l: 6 5 0b x a y .

Din tch ca hnh ch nht ABCD l:

2 22 2 2 2

4 33 4 4; . ; .

a b a ba b b ad P AB d Q BC

a ba b a b

.

Theo , ta c:

2 22 24 3

16 3 4a b a b

a b a b a ba b

.

Cho b = 1, ta c: 2 2 23 1 4 1 4 3 4 4a a a a a a

2 2 2

2 2 2

14 3 4 4 3 4 1 01

4 3 4 4 5 4 7 03

aa a a a a

aa a a a a

Vy phng trnh cnh AB l: 1 0 ; 3 11 0x y x y .2. mp qua A 2; 0; 0 , B 0; b; 0 , C 0; 0; c b,c > 0 nn c phng trnh l:

x y z1

2 b c

1 1 1 bc

J 1;1;1 1 b c 12 b c 2

AB 2;b;0 , AC 2;0;c

b 0 0 2 2 b

AB; AC ; ; bc;2c;2b0 c c 2 2 0

2 2 2 2 2 2 2 2ABC 1 1 1S AB;AC b c 4c 4b b c 4 b c2 2 2

.

p dng bt ng thc Bunhiacopski, ta c:

2 22 2 2 21.b 1.c 2 b c 4 b c 2 b c .

Do : 2 2 22 2

ABC

1 1 6S b c 2 b c 4 b c 2 b c b c

2 2 2

.

Du bng xy ra khi v ch khi b c.

C

A B

D P

Q N

M

www.MATHVN.com



13/260



T (1) suy ra2b

c 0 b 2.b 2

Suy ra ABC6 2b 6 4 6 4

S b b 2 b 2 42 b 2 2 b 2 2 b 2

p dng bt ng thc Cauchy cho 2 s dng: 4

b 2 4b 2 ABCS 4 6 .

Do ABCMinS 4 6 t c khib c 0

b c 4bcb c

2

.

Cu VIIbTa c:

k kk kk 2010

kk 1

k 12011

2 2010! 2 2010!2 C1

k 1 k! 2010 k ! k 1 k 1 ! 2010 k !

2 2011!1 12 C2011 k 1 ! 2011 k 1 ! 4022

3


Cu I (2,0 im) . Cho hm s: 3 21 5

4 43 2

y x mx mx (C).

5. Kho st s bin thin v v th (C) ca hm s khi m 0 .6. Tm m hm s t cc tr ti 1 2,x x sao cho biu thc :

222 1

2 21 2

5 12

5 12

x mx mmA

x mx m m

t gi tr nh nht.

Cu II(2,0 im)

3. Gii phng trnh: tan tan 2sin 1 6cos 3 sin 1 tan tan2

xx x x x x x

.

4. Gii h phng trnh:6 2 6

5 2

6 2 6

25

2

2 332

2 33

xyx x y

x x

xyy y x

x y

, yx .

1 2 20111 2 20112011 2011 2011

2011 0 02011

1P 2 C 2 C ... 2 C

40221 1

2 1 2 C .4022 2011

www.MATHVN.com



14/260



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

ln5

ln2

.10 1 1x x

dxI

e e

Cu IV (1,0 im). Cho hnh chp t gic S.ABCD c y ABCD l hnh vung tm O,

cnh bng a. Cnh bn SA vung gc vi y hnh chp v SA a 2 . Gi H v K ln lt

l hnh chiu ca A trn SB, SD. Chng minh SC AHK v tnh th tch O.AHK.Cu V (1,0 im). Tm m phng trnh sau c nghim:

4 3 3 3 4 1 1 0m x m x m B- PHN RING (3,0 im). Th sinh ch c chn mt trong hai phnB.1. CHNG TRNH CHUNCu VI a (2,0 im)3. Trong mt phng Oxy, cho hai ng trn:

2 22 2

1 2C : x y 9 ; C : x 1 y 1 25 . Gi A, B l cc giao im ca 1C

v 2C . Vit phng trnh ng thng AB. Hy chng minh rng nu K AB th

KI KJ vi I, J ln lt l tm ca 1C v 2C .

4. Trong khng gian Oxyz, cho im A 5;5;0 v ng thng x 1 y 1 z 7d :2 3 4

.

Tm to cc im B, C thuc d sao cho tam gic ABC vung cn ti A v

BC 2 17 .

Cu VII a(1,0 im). Gii phng trnh: 2z 2011 0 trn tp s phc .


Cu VI b (2,0 im)1. Trong mt phng Oxy, xc nh to cc im B v C ca tam gic u ABC bit

A 3; 5 v trng tm G 1;1 .

2. Trong khng gian Oxyz, cho hai im 0;0; 3 , N 2;0; 1M v mt phng

: 3 8 7 1 0x y z . Tm ta P nm trn mt phng sao cho tam gic MNPu.

Cu VII b(1,0 im). Gii h phng trnh:3 3log y log x

3 3

x 2y 27

log y log x 1

.

BI GIICu I1. Hc sinh t gii.2. TX: D =

2 5 4y x mx m .Hm s t cc tr ti 1 2,x x Phng trnh 0y c hai nghim phn bit 1 2,x x

www.MATHVN.com



15/260



2

1625 16 0 25

0

mm m

m

(1)

Theo nh l Viet, ta c: 1 2

1 2

5

4

x x m

x x m

V 1x l nghim ca phng trnh nn:2 21 1 1 15 4 0 5 4x mx m x mx m .

Do : 2 21 2 1 2 1 25 12 5 4 5 12 5 16 15 16 0x mx m mx m mx m m x x m m m .

Tng t, ta cng tnh c: 2 22 15 12 25 16 0x mx m m m .

Khi :22 2 22 1

2 2 2 21 2

5 12 25 162

5 12 25 16

x mx mm m m mA

x mx m m m m m

( Bt ng thc Cauchy cho 2 s dng).Du = xy ra

2 2 2

4 2 2 22 225 16

25 16 25 1625 16

m m mm m m m m m

m m m

20

24 16 0 2

3

m

m mm

. Kt hp vi (1) ta chn2

.3

m

Vy min 2A khi2

3m .

Cu II

1. iu kin: 2cos 0 cos 0 cos 0 cos 0

1 cos 0 cos 1cos 0 2cos 02 2

x x x xx x

x x

.

Ta c: tan tan 2sin 1 6cos 3 sin 1 tan tan2

xx x x x x x

sinsin 2tan tan 2sin 1 6cos 3 sin 1

cos cos2

x

xx x x x x

xx

cos cos sin sin2 2tan tan 2sin 1 6cos 3 sin

cos cos2

x xx x

x x x x xx

x

cos

2tan tan 2sin 1 6cos 3 sincos cos

2

x

x x x x xx

x

www.MATHVN.com



16/260



tan tan 2sin tan 6cos 3 tanx x x x x x

tan tan 2sin 6cos 3x x x x

2 2tan 1 2cos 3 1 2cos 1 2cos tan 3 0x x x x x

22 2 2 2 2

1

1 1 1 coscos cos cos 22 2 21

costan 3 sin 3 os 1 os 3cos4

xx x x

xx x c x c x x

2 1 1 2os os2 2 24 2 3 3

c x c x x k x k

k (tho iu kin)

2. Cng v theo v ca hai phng trnh trn ta c:

2 2

5 2 25

1 12

2 33 2 33xy x y

x x y y

2 2

2 25 51 121 32 1 32

xy x y

x y

(*)

Nhn xt: 2 2* 2 *VT xy x y VP .

0

* * 11

0

x yx y

VT VP x yx y

x y

.

Vy tp hp nghim ca h phng trnh cho l: 0;0 , 1;1S .Cu III

Ta c:

ln5 ln5

ln 2 ln 2

.10 1 1 10 1

x

x x x x

dx e dxI

e e e e

t 21 1 2x x xt e t e tdt e dx i cn: ln 2 1 ; ln 5 2x t x t

Khi :

22 2 2

21 1 1 1

2tdt dt 1 1 1 1 3 t 1 5I 2 dt ln ln

3 t 3 t 3 3 t 3 t 3 3 t 3 29 t t

.

Cu IV

+ Chng minh SC AHK

Ta c: BC AB

BC SAB BC AHBC SA

.

Mt khc AH SB nn AH SBC , do

AH SC . G

O

D

A B

C

S

H

K

I

M

www.MATHVN.com



17/260



Chng minh tng t ta cng c: AK SC .Vy SC AHK .+ Tnh th tch khi chp O.AHKGi G SO HK v I AG SC .V SC AHK nn SC AI .

Mt khc 2SA AC a nn tam gic SAC l tam gic vung cn ti A. Suy ra I l trungim SC v G l trng tm tam gic SAC.

Ta c: 2 2SC SA a nn1

2AI SC a v

2 2.

3 3

aAG AI

Ngoi ra: BD AC

BD SAC BD SCBD SA

.

T :

/ /

AHIK SC

BD SC BD AHIK

BD AHIK

.

Do : / / .AHK SBD HK BD

Ta c:2 2 2 2

3 3 3

HK SG aHK BD

BD SO .

Din tch ca tam gic AHK l:21 1 2 2 2 2 2

. . .2 2 3 3 9AHK

a a aS HKAG

Gi M l trung im ca AI th / /OM IC v1

.2 2

aOM IC

M SC AHK nn OM AHK .

Th tch hnh chp O.AHK l:2 31 1 2 2 2. . .

3 3 9 2 27AHKa a aV S OM

.

Cu Viu kin : 3 1.x

Khi : 3 3 4 1 1

4 3 3 3 4 1 1 04 3 3 1 1

x xm x m x m m

x x

(*)

V

2 2

x 3 1 x1

2 2

nn ta t x 3 2sin ; 1 x 2cos

t t tan , 0 , 0 t 12 2

.

Phng trinh (*) tr thnh:2

2

7 12 9

5 16 7

t tm

t t

(**)

Phng trnh cho c nghim khi v ch khi phng trnh (**) c nghim trn 0;1 .

Xt hm s: 2

2

7 12 9

5 16 7

t tf t

t t

www.MATHVN.com



18/260


19/260



Nu 0b th 2 2011 0a ( v l) . Do 0 0b a . Dn n: 2011b .

Vy s phc z cn tm l: 2011.i B.2. CHNG TRNH NNG CAOCu VI b (2,0 im)

1. Gi M l trung im BC, suy ra 3

0;42MA GA M

Trong tam ABM vung ti M ta c:

00

3 10cos30 2 30

cos30 32

AM AMAB

AB .

Suy ra:2 30

302

BM CM

Gi s 0 0;B x y . Khi 22 2

0 030 4 30MB x y (1)

Mt khc 0 0 0 0. 0 3 12 0 3 12AM BM AM BM x y x y

(2)

Thay (2) vo (1) ta c: 2 2 2

0 0 03 12 4 30 4 3y y y

0 0

0 0

7 9

1 9

y x

y x

Vy 9;7 , C 9;1B hoc 9;1 , C 9;7B .

2. ng thng MN qua M 0;0; 3 nhn MN 2;0; 2 lm vect ch phngnn c phng trnh:

x 2t

y 0 t

z 3 2t

.

Gi l mt phng trung trc ca on thng MN. Gi I l trung im MN I 1;0; 2 .

Mt phng qua I 1;0; 2 nhn 2;0;2MN

lm vect php tuyn nn c phng

trnh: 2. x 1 2. z 2 0 x z 1 0 .

P sao cho MNP u 2 2

P

MN NP

Gi s ta im N l 0 0 0; ;x y z , ta c:

0 0 0

0 0

22 20 0 0

3 8 7 1 0

1 0

3 8

x y z

x z

x y z

.

Gii h phng trnh ny ta tm c 2 2 1

P 2; 2; 3 , P ; ;3 3 3

.

Cu VIIbiu kin : , 0x y .

MB C

A

G

www.MATHVN.com



20/260



t 3

3

og 3

log 3

a

b

a l x x

b y y

HPT cho thnh: 23 93 2 3 27

1 11

b aaba b a b

a b a bb a

; ba l nghim ca phng trnh: 21

2 02

tt t

t

Khi :

31 1 9

2 2 12 2 91 1 1

3

x

a a y

b bx

a a

b by

( tho iu kin).

Vy tp hp nghim ca h phng trnh cho l: 1 1S 3;9 , ;9 3

.

4


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

1

xy

x

(C).

7. Kho st s bin thin v v th (C) ca hm s.8. Tm im M thuc (C) tng khong cch t M n hai trc to l nh nht.Cu II(2,0 im)1. Gii phng trnh: 2

2

cos 2 1tan 3tan

2 cos

xx x

x

.

2. Gii h phng trnh:33

2 3

1 3

82

y x

x y

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

2

3

4

tan tan xI x x e dx

.

Cu IV (1,0 im). Cho hnh chp S.ABC c SA ABC , tam gic ABC vung cn

nh C v SC a . Tnh gc gia hai mt phng SBC v ABC th tch khi chpln nht.Cu V (1,0 im). Cho a, b, c, d l cc s thc dng sao cho: 2 2 2 2 4a b c d .Chng minh: 3 3 3 3 8a b c d .B- PHN RING (3,0 im)

www.MATHVN.com



21/260



B.1. CHNG TRNH CHUNCu VI a (2,0 im)1. Trong mt phng vi h to Oxy, cho tam gic ABC vi 5,AB 1; 1 ,C ng

thng AB c phng trnh 2 3 0x y v trng tm G ca tam gic ABC thuc ngthng 2 0.x y Hy tm to cc im A v B.

2. Trong khng gian vi h to Oxyz, cho cc im 3;1;1 , B 7; 3; 9 , C 2; 2; 2A vmt phng (P) c phng trnh: 3 0x y z .

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

nh nht.

Cu VII a(1,0 im)

Gi A, B theo th t l cc im ca mt phng phc biu din s z khc 0 v1

2

iz z

.

Chng minh tam gic OAB vung cn.B.2. CHNG TRNH NNG CAOCu VI b (2,0 im)

1. Trong mt phng vi h to Oxy, cho ng thng : 2 1 2 0d x my vng trn 2 2: 2 4 4 0C x y x y . Gi I l tm ng trn (C). Tm m sao cho d ct(C) ti hai im phn bit A v B. Vi gi tr no ca m th din tch tam gic IAB ln nhtv tnh din tch .2. Trong khng gian Oxyz, cho tam gic ABC c A 1;2;5 v phng trnh hai ngtrung tuyn :

1 2

x 3 y 6 z 1 x 4 y 2 z 2d : ; d :

2 2 1 1 4 1

Vit phng trnh chnh tc cc cnh ca tam gic ABC.

Cu VII b(1,0 im).

Gii h phng trnh sau:

2 1

2 25 5

2 2 2

log 3 1 log 2 4 1

y x y x

x y y x y

.

BI GII

A- PHN CHUNGCu I1. Hc sinh t gii

2. Ta c: 000

1;1

xM x Cx

Gi Md l khong cch t M n hai trc ta , khi :

000

1;Ox ;

1Mx

d d M d M Oy xx

Chn 1;0 1MM H d . Do , tm MMind ta ch cn xt:

www.MATHVN.com



22/260



0

000

0 0

0

11 1

0 111 11

1

xx

xxx x

x

.

Vi 00 1x th

00 0 00 0 0

1 2 21 1 21 1 1M

xd x x x

x x x

00

22 1 2 2 2 2 2 2 1

1x

x

( Bt ng thc Cauchy).

Du bng xy ra0

00

0

0 1

2 121

1

x

xx

x

Vy 2 2 1Mmind khi 2 1;1 2M .

Cu II

1. iu kin:

cos 0cos 0

sin 2 0s 0 sin 0 2

2

xx

x x k kco x x

Phng trnh tng ng vi2

2 2 32

2sin 1cot 3tan tan 0 tan 1

cos tan

xx x x x

x x

tan 14

x x k k .

2. t 3,u 0 ; v = 1u x y . Suy ra:2 2 4

3 3 3 31 1x u x uy v y v

H cho tr thnh: 34

4 3 4 3

3 33 81

81 81

u v v uu u

u v u v

4 3 2 3 29 27 54 0 3 2 15 18 0u u u u u u u u .Do 0u nn 3 22 15 18 0u u u . Suy ra 3, v = 0u .

Khi :33

3 9

11 0

x x

yy

.

Vy 91

x

y

l nghim ca h phng trnh cho.

Cu III

2 x 2 x x 1 23 3 3

4 4 4

I tan x tan x e dx tan x.e dx t anx.e dx I I

S dng tch phn tng phn i vi 2I ta c:

www.MATHVN.com



23/260



232 1 1 24 3

4

t anx. 1 tan .x xI e x e dx e I I I I e

.

Cu IV

Ta c:

, 0;2

BC SBC ABC

SC BC SCA

CA BC

Trong SAC vung ti A, ta c: sin sinSA SC a .v cos cosAC SC a .

V ABC vung cn ti C nn:

2 2 21 1 1. cos2 2 2ABC

S AB AC AC a

.

Th tch hnh chp .S ABC l:

2

2 2 3 2 3. 1 1 1 1. sin . cos sin cos sin sin3 3 2 6 6S ABC ABC

aV SA S a a a

Xt hm s 3sin siny f , 0;2

.

t sint , 0; 0;12

t

.

Khi xt hm s 3 , t 0;1g t t t

21 3g t t

2

1

0;130 1 3 01

0;13

tg t t

t

Lp bng bin thin

Da vo bng bin thin ta suy ra

3 3 3 3

.0;1

0;2

2 3max max max .

6 6 6 273 3S ABC

t

a a a aV f g t

khi1

sin3

, 0;2

Cu V

A C

B

S

00

t

g t

g t

0 1

0

1

3

2

3 3

www.MATHVN.com



24/260



T 2 2 2 2 4a b c d suy ra 2 4 2.a a Khi 2 3 22 0 2a a a a .

Tng t ta cng chng minh c: 3 22b b , 3 22c c , 3 22d d .Cng v theo v ca bn bt ng thc trn ta c

3 3 3 3 2 2 2 22 8.a b c d a b c d

B- PHN RINGB.1. CHNG TRNH CHUNCu VI a

1. V G l trng tm tam gic ABC nn :1 3

1 3A B C A B G

A B C A B G

x x x x x x

y y y y y y

Suy ra 2 3A A B B G Gx y x y x y .

V G ng thng 2 0x y nn 2 0 2G G G G

x y x y .

Do : 2 6 8A A B B A A B B

x y x y x y x y

Li v A, B thuc ng thng 2 3 0x y nn: 2 3 0

12 3 0

A A

B B

x y

x y

.

2

8 3 2 3 2 610

A B

A A B B A A B B A B

A B

y yx y x y y y y y y y

x x

Mt khc 2 2

5 5A B A B

AB x x y y Nhng theo (1) 3 2 , x 3 2

A A B Bx y y nn 2A B B Ax x y y .

Suy ra 2 2 2

5 5A B A B A Bx x y y y y

Vy

1 11 1 2 2

2 2 3 32 2

A AA B A B

A B A BB B

y yy y y y

y y y yy y

Cui cng ta thu c:

1 34; , B 6;

2 2A

hoc3

6;2

A

,1

4;2

B

.

2. Gi s I l mt im tho 2 3 0IA IB IC

. Khi ta tnh c I23 13 25

; ;6 6 6

.

Ta c: 2 3 2 3 6MA MB MC MI IA MI IB MI IC MI

2 3MA MB MC

nh nht MI

nh nht M l hnh chiu vung gc ca I ln mtphng (P).

www.MATHVN.com



25/260



Gi l ng thng qua I nhn 1;1;1Pu n

lm vect ch phng nn c phng

trnh tham s :

23

613

625

6

x t

y t t

z t

.

Ta c M P to im M ng vi tham s t l nghim ca phng trnh :

23 13 25 79 5 20 23 0 ; ;

6 6 6 18 9 9 9t t t t M

.

Vy 2 3MA MB MC

nh nht khi5 20 2

; ;9 9 9

M

Cu VII a

Gi s z x yi th ta c A x; y . V z 0 nn2 2

x y 0 .Ta c

1 11 .

2 2 2 2

i x y x yz z i x yi i

Vy B c ta : ; .2 2

x y x yB

Ta li c:2 2 2 2

2 2 2 2; OB2 2 2

x y x y x yOA x y

.

2 2 2 2 2 22 .

2 2 2 2 2

x y x y x y y x x yAB x y

T , suy ra :2 2 2

OB AB .OA OB AB

Vy tam gic OAB vung cn ti B.

B.2. CHNG TRNH NNG CAOCu VI b1. ( )C c tm I (1; 2) v bn knh R = 3.d ct ( )C ti 2 im phn bit A, B

( , ) d I d R

22 2 1 2 3 2 m m 2 21 4 4 18 9m m m

25 4 17 0m m ( lun ng m ).

Ta c: 1 1 9

. sin .2 2 2

S IA IB AIB IA IBIAB

Vy: SIAB

ln nht l9

2khi 090AIB AB = 2 3 2R

3 2( , )

2d I d

d

B

A

I

www.MATHVN.com



26/260



3 2 21 2 2

2m m 2 216 16 4 36 18m m m

22 16 32 0m m 4 m 2. Thay ta A 1;2;5 vo hai phng trnh 1 2d , d thy khng tha. Do

1 2A d , A d . Gi s 1d l trung tuyn v t B, 2d l trung tuyn v t C 1 2, C dB d B 3 2b;6 2b;1 b C 4 c;2 4c;2 c .

1 2G d d Ta im G l nghim ca h :

3 6 1

2 2 1 .4 2 2

1 4 1

x y z

x y z

Gii h ny ta thu c x 3; y 6;z 1.

Do G 3;6;1

G l trng tm ca tam gic ABC nn

A B CG

A B CG

A B CG

x x x 1 3 2b 4 cx 33 3 2b c 1

y y y 2 6 2b 2 4cy 6 b 2c 4

3 3b c 5

z z z 5 1 b 2 c1z

33

b 2 B 7;2; 1

c 3 C 1;14; 1

T ta lp c phng trnh cc cnh ca tam gic ABC l:1 2 5

:1 0 1

7 2 1:

1 2 01 14 1

: .0 2 1

x y zAB

x y zBC

x y zCA

Cu VII b

2 1

2 25 5

2 2 2 1

log 3 1 log 2 4 1 2

y x y x

x y y x y

iu kin : 0y .

Chia c hai v ca (1) cho x2 0 ta c:

2 2 2 12 2 2 2 2 2 02 2 0

y x

y x y xy x y x

y x

www.MATHVN.com



27/260



Ta c: 2 1y x x y . Thay y x vo (2) ta c:

22 2

5 5 5

1log 3 1 log 2 4 1 log 3 1 2 1x x x x x x x

x

(3)

p dng bt ng thc Cauchy: 5 51

3 og 3 log 2 3 1VT l x

x

.

3 1VP .

Vy

2

1

3 3 1 1 1

1 0

xxVT VP x y

x

(tha iu kin y 0 )

Vy 1x y l nghim duy nht ca h phng trnh.

5


Cu I (2,0 im) . Cho hm s: 3 23 1 5 4 8y x m x m x mC

9. Kho st s bin thin v v th mC ca hm s khi m 0 .10.Tm m mC ct trc honh ti 3 im phn bit lp thnh mt cp s nhn.Cu II(2,0 im)

1. Gii phng trnh:3 1

8sincos sin

xx x

.

2. Gii phng trnh: 2 3 3 244 4 41 1 1 1 .x x x x x x x x

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

0

12

1 1

dxI

x x

.

Cu IV (1,0 im).Cho hnh chp S.ABCD c y ABCD l hnh thoi ; hai ng choAC = 2 3a , BD = 2a v ct nhau ti O; hai mt phng (SAC) v (SBD) cng vung gc

vi mt phng (ABCD). Bit khong cch t im O n mt phng (SAB) bng3

4

a.

Tnh th tch khi chp S.ABCD theo a.

Cu V (1,0 im). Tm m bt phng trnh sau v nghim:2

2

1 12 sinx sinx 7

sinx sinx2.

1 13 sinx sinx 12

sinx sinxm

B- PHN RING (3,0 im). Th sinh ch c chn mt trong hai phn

www.MATHVN.com



28/260



B.1. CHNG TRNH CHUNCu VI a (2,0 im)1. Trong mt phng vi h to Oxy, cho im 2;1A . Ly im B thuc trc Ox c

honh 0b v im C thuc trc Oy c tung 0c sao cho tam gic ABC vung tiA. Tm B, C sao cho din tch tam gic ABC ln nht.

2. Trong khng gian Oxyz cho cc im 2;0;0 , M 0; 3;6A . Vit phng trnh mtphng P cha A, M v ct cc trc ,Oy Oz ti cc im tng ng B, C sao cho

3.OABC

V Cu VII a(1,0 im).

Xt s phc: 1 2

i mz

m m i

. Tm m

1.

2z z .


Cu VI b (2,0 im)1. Trong mt phng vi h to Oxy, cho ng thng : 2 2 0x y v hai im

1;3 , B 3; 2A . Tm M trn sao cho MA MB t gi tr ln nht.

2. Trong khng gian Oxyz, cho hai im 2;3;0 , B 0; 2;0A v ng thng

: 0

2

x t

y

z t

. Tm C sao cho chu vi tam gic ABC nh nht.

Cu VII b(1,0 im).

Tm min xc nh ca hm s: 32 lg 2 lgln 8 4x xy

BI GIICu I

1. Hc sinh t gii

2. + iu kin cn : Gi s mC ct trc honh ti 3 im phn bit 1 2 3, ,x x x lp thnh

mt cp s nhn . Khi 3 23 1 5 4 8 0x m x m x c 3 nghim l 1 2 3, ,x x x

3 2

1 2 33 1 5 4 8 xx m x m x x x x x x x

3 23 1 5 4 8x m x m x 3 21 2 3 1 2 2 3 3 1 1 2 3x x x x x x x x x x x x x x x

Suy ra : 1 2 3 8x x x . V 1 2 3, ,x x x lp thnh mt cp s nhn nn22 1 3x x x . Do

32 28 2.x x Thay 2 2x vo phng trnh

3 23 1 5 4 8 0x m x m x ta thu c 4 2 0 2.m m

www.MATHVN.com



29/260



+iu kin : Vi 2m thay vo phng trnh 3 23 1 5 4 8 0x m x m x ta

thu c 3 2 1 2 37 14 8 0 1 2 4 0 1 , 2, 4x x x x x x x x x lpthnh mt cp s nhn .

Vy 2m l gi tr cn tm.

Cu II

1. iu kin: sin 0

sin 2 0 .cos 0 2

xx x k k

x

Phng trnh cho tng ng vi:28sin cos 3 sin cosx x x x

4 1 cos 2 cos 3 sin cosx x x x

4cos 4cos 2 cos 3 sin cosx x x x x

4cos 2 cos3 cos 3 sin cosx x x x x 1 3

cos sin cos3 cos cos32 2 3

x x x x x

3 2

3 6

3 23 12 2

x x k x k

k

x x k x k

tho iu kin ban u.

2. iu kin: 0 1x

t

4

44 4

0

01

1

uu x

vv x

u v

T phng trnh cho ta c :

2 2 3 2 2 2

2 2 0

u uv v v u u v

u v u v u uv v

1 0

1 0

u v u v u v

u v u v

( do u , v khng m v u , v khng ng thi bng 0 nn 0u v ).

Ta c cc h :

4

4 44

10 2)

112

uu v

au v

v

. Do

112 .

1 21

2

x

x

x

www.MATHVN.com



30/260



224 4 2 2

11)

1 2 2 1

u vu vb

u v u v uv u v

2 2

1 0

1 0 110

2 4 0 1 12

2 0

u v u

u v uv vu vuv

u v uv u v uuv

uv v

. Do

0

1 1 0

11

1 0

x

x x

xx

x

.

Vy tp hp nghim ca phng trnh cho l:1

0; ;12

S

.

Cu III

Ta c:

0 0

21 1

2 2

1 1 1 11

4 2

dx dxI

x xx

t1 1 1

sin costdt ; t ;2 2 2 2 2

x t dx

Lc :2 2 2

20 0 0

1cos cos 22 1 2 .

2 cos 2 cos 21 11 sin

4 4

tdtt

I dt dt Jt t

t

Cn tnh J2

0

dt

2 cos t

. t 2

2tan

2 1

x dtt dx

t

Ta c:1 12

2 20 0

2

21 2

1 32

1

dt

dttJt t

t

t 23 tan 3 1 tant u dt u du

26 6

20 0

3 1 tan13 3tan 3 6 3

u duJ du

u

.

Suy ra: 9 4 3

I 2.2 186 3

.

www.MATHVN.com



31/260



Cu IV

Trong ABO vung ti O ta c:

2 3tan 32

OA AC aABO

OB BD a

060ABO hay ABD u.

SAC ABCD

SBD ABCD SO ABCD

SO SAC SBD

.

Do ABD u nn vi H, K

ln lt l trung im ca AB, HB nn DH AB v DH = 3a ;

OK // DH v1 3

2 2

aOK DH OK AB AB (SOK)

Gi I l hnh chiu ca O ln SK ta c OI SK; AB OIOI (SAB) ,hay ;OI d O SAB

SOK vung ti O ta c:2 2 2

1 1 1

2

aSO

OI OK SO .

Din tch y: 24 2 . 2 3 2 3ABCD ABO

S S OA OB a a a .

Th tch khi chp S.ABCD:3

2.

1 1 3. 2 3

3 3 2 3S ABCD ABCDa a

V SO S a .

Cu VBt phng trnh cho tng ng vi:

2

2

1 12 sin sin 1

sin sin2

1 13 sin sin

sin sin

x xx x

x x mx x

.

t 1

sin , tsin

t x x k x

.

Bi ton tng ng vi vic tm m bt phng trnh sau ng vi mi t:2

2

2 12

3

t t

t t m

.

V mu thc xc nh t nn1

1 12 0 .12

t m Khi 23 0 tt t m .

V 22 1 0 tt t nn2

22

2 12 4 3 2 1 0

3

t tt t t m t

t t m

35

9 16 2 1 0 .12

m m

S

A

BK

H

C

O

ID

a

www.MATHVN.com



32/260



Vy35

12m l gi tr cn tm tha mn yu cu bi ton.

B- PHN RINGB.1. CHNG TRNH CHUNCu VI a

1. Ta c: ;0 , C 0;B b Ox c Oy vi , c 0b . 2; 1 , 2; 1AB b AC c

.

Do tam gic ABC vung ti A nn 0 2 2 1 0 1 2 2ABAC b c c b

.

2 21 1

2 1. 4 12 2ABC

S ABAC b c

2 2 21

2 1 4 4 2 2 1 12

b b b .

Do5

2 5 0 02

c b b .

Du = xy ra khi 2b , 1c .Vy min 1

ABCS

khi 2;0B , 0;1C .

2. Gi s 0; ;0 , C 0;0;B b c . Mt phng P qua A , B , C nn c phng trnh l :

1.2

x y z

b c

3 6

0; 3;6 1 6 3 .M P b c bcb c

Li c : 1 2 1. . . 3 9.3 3 2 3OABC OBC

bcV OA S bc bc

T ta cc cc h phng trnh :

(a) . 2

32 3

32 3 2 36 3 9 3

32 3 99 2 3 9 0 32

26

bc b

cc b c bb c b

b bbc b b bb

c

.

(ii ). 2

2 3 2 36 3 9

2 3 99 2 3 9 0

c b c bb c

b bbc b b

( H v nghim).

Vy phng trnh mt phng P cn tm l :

www.MATHVN.com



33/260



1 , 1.32 3 3 2 6

2

x y z x y z

Cu VII a

Ta c:

2

22 2 21 2

1 2 1 4m i m mii mz

m mi m m

2 2 2 2 2

2 22 2

1 2 1 2 1 1

1 1

m m m i m m m m i m

m m

=2 2

1

1 1

mi

m m

2 2

1

1 1

mz i

m m

Do

22

2 22

1 1 1 1 1. 1 2 1.

2 2 1 21

mz z m m

mm

B.2. CHNG TRNH NNG CAOCu VI b

1. t ; 2 2F x y x y . Do 1;3 . 3; 2 15 0F F nn A v B nm v hai pha

khc nhau i vi .Gi d l ng thng qua A vung gc vi . Khi d nhn 1;2du n

lm vect ch

phng nn c phng trnh:1 3

2 1 01 2

x yx y

.

Gi H d To im H l nghim ca h: 2 1 0 0

0;1

2 2 0 1

x y xH

x y y

.

Gi C l im i xng ca A qua nn H l trung im ca AC. Suy ra 1; 1C .

Ta c: 17.MA MB MC MB BC

Du = xy ra khi v ch khi .M BC ng thng BC qua 3; 2B nhn 4;1BC

lm vect ch phng nn c phng

trnh:3 2

4 5 04 1

x yx y

.

To im M cn tm l nghim ca h:9

2 2 0 79;7

4 5 0 22

xx y

Mx y y

.

Vy 17Max MA MB khi7

9;2

M

.

2. Ta c: ;0;2C C t t

Gi P l chu vi ca tam gic ABC, ta c:

P AB AC BC 2 215 6 2 2 8 17 2 4 6t t t y

www.MATHVN.com



34/260



2 22 215 6 2 2 2 3 2 1 2t t

Trong mt phng Oxy, t 2 2 ;3 , 2 1 ; 2u t v t

3 3AC BC u v u v

Du bng xy ra khi v ch khi u v v cng phng

2 2 3 72 52 1

tt

t

.

15 6 3 3 3MinP khi7 3

;0;5 5

C

.

Cu VII bHm s xc nh khi v ch khi

32 lg 2 lg

3 22 lg 2 lg32 lg 2 lg

008 4 0

8 48 4x x

x xx x

xx

9 2 lg 4 2 lg0 09 2 lg 4 2 lg2 2x xx x

x x

0 0100

lg 2 100

x xx

x x

.

Vy min xc nh ca hm s cho l: 100;D .

6


Cu I (2,0 im) . Cho hm s: 3 23 2y x x C

11.Kho st s bin thin v v th C ca hm s.12.Tm m C c im cc i v cc tiu nm v hai pha i vi ng trn

2 2 2: 2 4 5 1 0mC x y mx my m .Cu II(2,0 im)1. Tm nghim thuc khong 0; ca phng trnh:

sin 3 cos37 cos 4 cos 2

2sin 2 1

x xx x

x

.

2. Tm m h bt phng trnh sau c nghim:

5 1 5 1

2

7 7 2012 2012

2 2 3 0

x x x x

x m x m

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

2 4 211 3 1

dx

x x x x .

Cu IV (1,0 im). Cho hnh chp S.ABC c y l tam gic ABC vung ti C,, AB = 2a, SAAC a vung gc vi y. Gc gia hai mt phng (SAB) v (SAC) bng

www.MATHVN.com



35/260



060 . Gi H, K ln lt l hnh chiu ca A ln SB, SC. Chng minh AK HK v tnh thtch khi chp S.ABC.

Cu V (1,0 im). Cho , y, z 0,1x . Chng minh rng 1 1 1 1xyz x y z .

B- PHN RING (3,0 im). Th sinh ch c chn mt trong hai phnB.1. CHNG TRNH CHUNCu VI a (2,0 im)5. Trong mt phng Oxy, cho im 2;3M v ng thng

: 2 1 2 1 0m x m y m . Tm tham s thc m khong cch t M nng thng l ln nht.

6. Trong khng gian Oxyz, cho hai ng thng1

2 1:

2 1 2

x y zd

v 2

2 2

: 3

x t

d y t

z t

. Chng minh hai ng thng trn cho

nhau. Hy vit phng trnh mt cu (S) bit (S) c ng knh l on vung gc chung

ca 1 2,d d .Cu VII a(1,0 im). Cho M, N l hai im trong mt phng phc biu din theo th tcc s phc 1 2, zz khc 0 tha mn ng thc

2 21 2 1 2z z z z . Chng minh tam gic OMN

l tam gic u.B.2. CHNG TRNH NNG CAOCu VI b (2,0 im)1. Trong mt phng vi h ta Oxy, cho hnh ch nht ABCD c din tch bng 12, tmI l giao im ca hai ng thng 1 2: 3 0, d : 6 0d x y x y . Trung im M ca

cnh AD l giao im ca ng thng 1d vi trc Ox. Tm to cc nh ca hnh ch

nht.2. Trong khng gian vi h to Oxyz, cho mt phng v mt cu S ln lt c

phng trnh: 2 2 2

2 2 3 0 ; 1 2 4 25.x y z x y z Xt v tr tng i

gia mt cu S v mt phng . Vit phng trnh mt cu V i xng vi S qua

mt phng .

Cu VII b(1,0 im).Gii bt phng trnh: 2 2log 3 1 6 1 log 7 10x x .

BI GIICu I

1. Hc sinh t gii2. Hm s xc nh v lin tc trn .

Ta c: 23 6y x x ; 20

0 3 6 02

xy x x

x

Ta cc im cc tr: 0;2 , B 2; 2A .

Cch 1: (C) c im cc tr nm v hai pha i vi ng trn mC khi v ch khi

www.MATHVN.com



36/260



2 24 8 5 1 4 4 4 8 5 1 0m m m m m

2 2 25 8 3 5 4 7 0 5 8 3 0m m m m m m ( v 25 4 7 0 mm m ).3

15

m .

Cch 2: ng trn 2 2: 2 1mC x m y m c tm ; 2I m m , bn knh 1R .

Ta c:2

2 2 36 65 4 8 5 15 5 5

IB m m m R

im B nm pha ngoi

ng trn mC . Do im A nm pha trong ng trn mC , tc l:

2 2 31 5 8 4 1 5 8 3 0 15

IA R m m m m m .

Cu II

1.iu kin:

52

12sin 2 1

212

x k

x kx k

.

Phng trnh cho tng ng vi:3 33sin 4sin 4cos 3cos

7 cos 4 cos 22sin 2 1

x x x xx x

x

3 sin cos 4 sin cos 1 sin cos7 4 cos 2

2sin 2 1

x x x x x xx

x

sin cos 3 4 4sin cos7 cos 4 cos 2

2sin 2 1

x x x xx x

x

2

sin cos 2sin 2 17 cos 4 cos 2 7sin 4 1 2sin

2sin 2 1

x x xx x x x

x

2

1 2sin 62sin 7sin 3 0 2

5sin 3 2

6

x kx

x x

x x k

V 0;x nn ta c c hai nghim sau y:5

,6 6

x x

.

2. iu kin: 1x Ta c: 5 1 5 1 1 5 57 7 2012 2012 7 7 7 2012 1x x x x xx x (*)Nu 1x th * 0 *VP VT ( V l).

Bt phng trnh (*) lun ng khi 1 1x .H bt phng trnh cho c nghim Bt phng trnh 2 2 2 3 0x m x m c

nghim 1;1x Bt phng trnh: 22 2 3m x x x c nghim 1;1x

www.MATHVN.com



37/260



2 2 3

2

x xm

x

c nghim 1;1x

1;1min

xm f x

vi

2 2 3

2

x xf x

x

.

Hm s ny c vit li : 3

2f x x

x

, 1;1x

2

2 2

3 4 11 2 2

x xf x

x x

2 3 1;10

2 3 1;1

xf x

x

2 3 2 2 3f ; 1 1 2f f ;Do

1;1min 1 1 2

xf x f f

.

Vy 2m tho mn yu cu bi ton.Cu III

t x t dx dt .Khi :

1 1

2 4 2 2 4 21 11 3 1 1 3 1

dt dxI

t t t t x x x x

.

Suy ra: 1

1

2I g x g x dx

vi 2 4 21

1 3 1g x

x x x x

.

Ta c: 2 4 2 2 4 2

1 1

1 3 1 3g x g x

x x x x x x x x

=

2 4 2 2 4 2

2 4 2 2 4 2

1 3 1 1 3 1

1 3 1 1 3 1

x x x x x x x x

x x x x x x x x

2 4 2

2

2 4 2 2

2 1 3 1

1 3 1

x x x

x x x x

= 2 4 2

4 4 2 2 4 2 2 4 2 2

2 1 3 1

1 3 1 2 2 3 1 2 3 1

x x x

x x x x x x x x x x

2 4 24 2 4 2 2 4 2

2 1 3 1

2 2 4 2 3 1 2 3 1

x x x

x x x x x x x

2 4 2

4 2 4 2 2 4 2

1 3 1

1 2 3 1 3 1

x x x

x x x x x x x

2 4 2

22 4 2 2 4 2 4 2

1 3 1 1

11 3 1 3 1

x x x

xx x x x x x x x

.

Do :1 1 1

2 2 21 0 0

2 21 1 1

dx dx dxI I

x x x

t 2tan 1 tanx u dx u du

www.MATHVN.com



38/260



Vy 24 4

42 0

0 0

1 tan

1 tan 4

u duI du u

u

.

Cu IV..

Ta c: SA BC BC SAC BC AKAC BC

M AK SC nn AK SBC AK HK .

Hn na AK SBC AK SB

Suy ra: SB AHK SB HK

M SB SAB SBC nn 060AHK .

D dng tnh c2

ABC

a 3S

2

Trong tam vung AKH c0 3AK AHsin 60 AH

2

Cc tam gic SAB, SAC vung ti A nn2 2 2 2 2

1 1 1 1 1

AH SA AB SA 4a (1)

2 2 2 2 2 2 2 2 2

1 1 1 4 1 1 1 3 3

AK SA AC 3AH SA a AH 4SA 4a (2)

T (1) v (2) suy ra:2 2

1 1 a 2SA

4SA 2a 2 .

Vy

3

S.ABC

a 6V 12 .

Cu V

Cch 1: Do , , 0;1x y z nn

0;1

1 1 1 0;1

xyz

x y z

.

Khi : 3 31 1 1 1 1 1xyz x y z xyz x y z

p dng bt ng thc Cauchy ta c:

3 31 1

1 1 1 1

3 3

x y z x y zxyz x y z

Suy ra: 1 1 1 1xyz x y z .

Cch 2:

Ta c 0;1 1 0;1z z

Do : 1 1 1 1 1xyz x y z xy x y

p dng bt ng thc Bunhiacopski ta c:

A

C

B

S

H

K

www.MATHVN.com



39/260



2

1 1 1 1 1 1 1 1x y x y x x y y xy x y

Suy ra: 1 1 1 1xyz x y z .

B.1. CHNG TRNH CHUNCu VI a

1. Ta c:

2 2 2

2 2 3 1 2 1 7 8,

2 6 52 1

m m m md d M

m mm m

Suy ra:2

22

49 112 64

2 6 5

m md

m m

Cch 1:

Ta c:2

2 2 2 2 2 22

49 112 642 6 5 49 112 64

2 6 5

m md d m d m d m m

m m

hay 2 2 2 22 49 2 56 3 5 64 0d m d m d .

+ Nu 2 7 22 49 02

d d ( do 0d ) th 117 11735 02 70

m m .

+ Nu 27 2

2 49 02

d d th phng trnh bc hai n theo n m trn c nghim

khi v ch khi 22 2 256 3 2 49 5 64 0d d d

4 2 237 0 37 0d d d ( do 0d ) 2 37 37d d

Vi 37d ta c: 211

25 110 121 05

m m m .

Vy max 37d khi11

5m

Cch 2:

Xt hm s: 2

2

49 112 64

2 6 5

m mf m

m m

Tp xc nh D

2

22

70 234 176

2 6 5

m mf m

m m

28

70 70 234 176 0115

m

f m m m

m

49 49

lim , lim2 2x x

f m f m

.

www.MATHVN.com



40/260



Bng bin thin:

Da vo bng bin thin ta suy ra 11

max 375m

f m f

.

Vy max 37d khi11

5m .

2. ng thng 1d i qua 2;1;0M v c vect ch phng 1 1; 1; 2u

.

ng thng 2d i qua 2;3;0N v c vect ch phng 1 2;0;1u

.

Ta c: 1 2 1 2; 10 0 , d u u MN d cho nhau.

1 22 ;1 ;2 ; B 1 2 ;3;A a a a d b b d

AB l on vung gc chung ca 11 22

1. 0

, d 3. 0 0

AB u a

dAB u b

.

Do : 5 4 2

; ; , B 2;3;03 3 3

A

Mt cu (S) c tm11 13 1

; ;6 6 3

I , bn knh30

2 6

ABR

Vy phng trnh ca mt cu (S) l :

2 2 211 13 1 5

6 6 3 6 x y z .

Cu VII a

Ta c:

222 1 2 1 2 1 2 12 2

1 2 1 2 2 21 2 1 2 1 2 2 1

z z z z z z z zz z z z

z z z z z z z z

V 1 2, z 0z nn 1 2, 0z z .

T ta c:2 2

3 32 12 1 1 2 1 22 2

1 2

z zz z z z z z

z z

Do : 2 1 1 2z z z z M 1 2 2 1; ON = ; MN =OM z z z z .

Vy tam gic OMN u.B.2. CHNG TRNH NNG CAOCu VI b (2,0 im)

0t

f m

f m

8

711

50

49

2

492

37

0

www.MATHVN.com



41/260



1. Ta c : 1 2I d d to im I l nghim ca h :9

3 0 26 0 3

2

xx y

x yy

.

Do 9 3

;2 2I .

1 3;0M d Ox M .

Ta c : 2 I 3 2AB M

Gi thit12

12 . 12 2 23 2

ABCDS AB AD AD

V 1, M dI nn 1d AD .

ng thng AD qua 3;0M v nhn 1

1; 1AD du n

lm vect ch phng nn c

phng trnh:3 0

3 0

1 1

x yx y

.

To A, D c dng: ;3t t .Ta c:

2 22 22 2 3 3 2MA MD MA MD t t 2

26 8 0

4

tt t

t

.

Do : 2;1 , D 4; 1A hoc 4; 1 , D 2;1A .

+ Vi 2;1 , D 4; 1A suy ra 7;2C , 5;4B

+ Vi 4; 1 , D 2;1A suy ra 5;4C , 7;2B

2. + Mt cu S c tm 1; 2; 4I v bn knh 5R .

22 2

2.1 2 2.4 3; 3 5

2 1 2d I R

.

Vy mt phng ct mt cu S .

+ Gi J l im i xng ca I qua mt phng th V l mt cu c tm J, bn knh

5R .Gi d l ng thng i qua I v vung gc vi . Khi d nhn 2; 1; 2n

lm

vect ch phng nn c phng trnh tham s:1 2

2

4 2

x t

y t

z t

Gi H l trung im IJ. Khi H l hnh chiu vung gc ca I ln mt phng . To im H ng vi tham s t l nghim ca phng trnh:

2 1 2 2 2 4 2 3 0 1t t t t .

I

M

A

D C

B

www.MATHVN.com



42/260



Suy ra 1; 1; 2H . V H l trung im IJ nn d dng suy ra c J 3;0;0 .

Vy phng trnh mt cu (V) l: 2 2 23 25.x y z

Cu VII b (1,0 im)

iu kin:1

10.

3

x

Bt phng trnh cho tng ng vi:

2 23 1 6 3 1 6

log log 7 10 7 102 2

x xx x

3 1 2 10 8x x 3 1 4 3 1 10 4 10 64x x x x

2

4 3 1 10 23 16 3 1 10 23x x x x x x

2 36949 418 369 0 149

x x x

Kt hp vi iu kin ban u ta thu c tp hp nghim ca bt phng trnh trn l:

3691;49

S .

7


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

1

xy C

x

13.Kho st s bin thin v v th C ca hm s.2. Gi I l giao im hai tim cn. Vit phng trnh tip tuyn d ca th hm s bit d

ct tim cn ng v tim cn ngang ln lt ti A v B tho 5 26

cos26

BAI .

Cu II(2,0 im)

1. Gii phng trnh: )2

sin(2cossin

2sincot

2

1

x

xx

xx

2. Gii bt phng trnh sau: 2 2 23 2 4 3 2 5 4x x x x x x Cu III (1,0 im) Tnh din tch hnh phng gii hn bi ba ng sau:

Elip (E):2

2 1

4

xy , ng thng d: 2 3 4 0x y v trc honh.

Cu IV(1,0 im).Cho hnh chp SABCD c y ABCD l hnh thang vung ti A v D,2 , CD = a AB AD a , gc gia hai mt phng (SBC) l (ABCD) bng 060 . Gi I l

trung im ca cnh AD. Bit hai mt phng (SBI) v (SCI) cng vung gc vi mt phng(ABCD). Tnh th tch khi chp S.ABCD.

www.MATHVN.com



43/260



Cu V (1,0 im). Tm m phng trnh: 2 2cos 2mx x c ng 2 nghim thc phn

bit trong on 0;2

.

B- PHN RING (3,0 im). Th sinh ch c chn mt trong hai phnB.1. CHNG TRNH CHUN

Cu VI a (2,0 im)1. Trong mt phng vi h ta Oxy cho ABC c trng tm 2;0 .G Bit phng

trnh cc cnh AB,AC theo th t l 4 14 0x y , 2 5 2 0.x y Tm ta cc nhA,B,C .2. Trong khng gian Oxy cho cc im 3;5; 5 , B 5; 3;7A v mt phng

: 0P x y z . Tm im M P sao cho 2 2MA MB nh nht .Cu VII a(1,0 im)

Trong khai trin sau y c bao nhiu s hng hu t 43 5n

bit n tha mn

1 2 3 2 4964 1 4 1 4 1 4 1... 2 1

n

n n n nC C C C .B.2. CHNG TRNH NNG CAOCu VI b (2,0 im)1. Cho parabol 2.y x Mt gc vung nh O ct Parabol ti 1A v 2A . Hnh chiu ca

1 2, AA ln Ox l 1 2, BB . Chng minh rng: 1 2.OB OB const .

2. Cho mt cu: 2 2 2: 2 2 2 0S x y z x z v cc im 0;1;1 ,A B 1; 2; 3

C 1;0; 3 . Tm im D thuc mt cu (S) sao cho th tch t din ABCD ln nht.

Cu VII b(1,0 im)Tm s nguyn dng n b nht 31

n

i

il s thc

BI GII

A- PHN CHUNGCu I1. Hc sinh t gii

2. Gi 00 00

3 2; 1

1

xM x C x

x

Tip tuyn d vi (C) ti M c phng trnh:

0 020 0

3 2 5

1 1

xy x x

x x

.

Do d ct tim cn ng, tim cn ngang ln lt ti A, B v IAB c 5 26

cos26

BAI nn

tan 5BAI .

www.MATHVN.com



44/260



Li c tan BAI l h s gc ca tip tuyn d m

0 2

0

50

1y x

x

nn

2 002

00

055 1 1

21

xx

xx

.

Vy c hai tip tuyn tho mn yu cu bi ton: 5 2 ; 5 2y x y x .Cu II1. iu kin: .0cossin,0sin xxx Ta c:

1 sin 2cot 2sin( )

sin cos 22

xx x

x x

0cos2

cossin

cossin2

sin2

cos

x

xx

xx

x

x

2cos 2cos0

sin cos2sin

x x

x xx

cos 0

cos sin( ) sin 2 0sin sin 24

4

x

x x xx x

222

242 4 3

4 3

x k

x k

x k kk

xk

x

( tho iu kin ban u)

Vy phng trnh cho c hai h nghim: 2 2

4 3

x k

kk

x

2. iu kin:

2

2

2

3 2 0 1 2

4 3 0 1 3 1 4

1 45 4 0

x x x x

x x x x x x

x xx x

.

2 2 23 2 4 3 2 5 4x x x x x x

2 2 2 23 2 5 4 4 3 5 4 0x x x x x x x x

2 2 2 2

2 1 10

3 2 5 4 4 3 5 4

x x

x x x x x x x x

www.MATHVN.com



45/260



2 2 2 2

2 11 0

3 2 5 4 4 3 5 4

xx

x x x x x x x x

1 0 1x x .

Kt hp vi iu kin ban u, ta c: 1 4x x .

Cu III

+ Giao im ca d v (E) l3

A 1;2

+ d ct trc honh ti B 4;0 .Gi A l chn ng vung gc h t Axung Ox. Khi din tch hnh phng cn tm

l phn gch xin nh hnh v.Ta c: AKB AKCS S S .

AKB1 1 3 3 3S AK.KB .32 2 2 4

Nhng im thuc hnh phng c tung y 0 nn t phng trnh (E) suy ra:

21y 4 x2

.

Do :2

2AKC

1

1S 4 x dx

2

t 2sin 2cosx t dx tdt Khi :

2 2 2 2

2 2

66 6 6

1 1 34 4sin .2cos 2 cos 1 cos 2 sin 2

2 2 3 4AKCS t tdt tdt t dt t t

Vy3 3 3

S 34 3 4 3

(

vdt).Cu IVTa c:

SBI ABCD

SCI ABCD SI ABCD

SI SBI SCI

.

Gi I l hnh chiu ca I ln BC.Theo nh l 3 ng vung gcsuy SH BC .

M BC SBC ABCD nn 060SHI

y

A

K CBO x

I

A

D C

B

S

E

H

www.MATHVN.com



46/260



l gc gia hai mt phng

SBC v (ABCD).

Ta c : 21 1

2 2 3 .2 2ABCD

S AD AB CD a a a a

22 2 2 21 1 1 3

3 . . 32 2 2 2IBC ABCD ICD IABa

S S S S a ID CD IA AB a a a .

K CE AB , ta c:

2

2

2 2 2 2

32.2 3 3 52

54IBC

a

S a aIH

BC CE BE a a

Trong tam gic SIH vung ti I, ta c: 03 5 3 15

tan 60 . 35 5

a aSI IH .

Do :3

21 1 3 15 3 15. . .33 3 5 5ABCD

a aV SI S a ( vtt).

Cu V

+ R rng: 0x l mt nghim ca PT.

+ Vi 0;2

x ta c:

2

2

4sin2

x

mx

. t2

xt , 0; 0;

2 4x t

Ta c phng trnh sau:2

sin, t 0;

4

tm

t.

Xt hm s: 2

sin, t 0;

4

tf t

t.

3sin2

1 tan 0 t 0;4

tf t t

tsuy ra f(t) nghch bin trn 0;

4

.

0

lim 1t

f t

.

Phng trnh trn c 2 nghim thc phn bit tong on 0;2

PT f t m c ng 1

nghim thc trong 0;4

.

Bng bin thin :

Da vo bng bin thin ta tm c :2

81m

.

B- PHN RING (3,0 im). Th sinh ch c chn mt trong hai phnB.1. CHNG TRNH CHUN

0

t

f t

f t

4

1

2

8

www.MATHVN.com



47/260



Cu VI a (2,0 im)

1. Ta c :A AB AC Ta im A l nghim ca h4 14 0 4

2 5 2 0 2

x y x

x y y

hay A 4;2

; 4 14B AB B b b ; 2 1; 5 5C AC C c c

V G l trng tm ABC nn

4 6 3 3; 22 1

2 4 14 0 1 1;05 5

b cb B

b c c C

Vy A 4; 2 , 3; 2B , 1;0C .2. Gi H l trung im ca on thng AB. Theo cng thc di trung tuyn MH ca tam

gic MAB ta c:2

2 2 222

ABMA MB MH .

2 2

MA MB nh nht MH nh nht.Ta c: 1;1;1H c nh. Suy ra MH nh nht M l chn ca ng vung gc h t Hn (P).

rng 1;1;1OH

l vect php tuyn ca mp (P) v O P nn M O 0;0;0 .

Vy 2 2MA MB nh nht khi v ch khi M trng vi gc ta v

2 2 2 2 142Min MA MB Min OA OB .Cu VII a (1,0 im)

Ta c: 4n 1 0 1 2 2 3 3 4n 1 4n 1

4n 1 4n 1 4n 1 4n 1 4n 11 x C C x C x C x ... C x

Chn x 1 4n 1 0 1 2 3 4n 14n 1 4n 1 4n 1 4n 1 4n 12 C C C C ... C

0 1 2 3 2n4n 1 4n 1 4n 1 4n 1 4n 12 C C C C ... C Suy ra 4n2 0 1 2 3 2n4n 1 4n 1 4n 1 4n 1 4n 1C C C C ... C

Hay 4n 4962 2 4n 496 n 124.

124 k k124 124124 124 k k

k k4 4 2 4124 124

k 0 k 0

3 5 C 3 5 C 3 5

.

Trong khai trin c s hng hu t

124 k 2

k 4

0 k 124

k 4 k 4t 0 t 310 k 124 0 4t 124

C 32 gi tr ca t suy ra c 32 gi tr ca k. Vy trong khai trin trn c 32 s hng hu t.

www.MATHVN.com



48/260



B.2. CHNG TRNH NNG CAOCu VI b (2,0 im)

1. Gi s 21 0 0;A x x P .Khi :

1B l hnh chiu ca 1A ln Ox nn c ta

l 1 0 1 0;0B x OB x .

Phng trnh ng thng 1 0:OA y xx .

Do 2 1OA OA nn phng trnh ng thng 20

1:OA y x

x .

Ta 2

A l nghim ca h phng trnh:

2

2 20 00

1 1;1

y x

Ay x x x

x

.2

B l hnh chiu

ca 2A ln Ox nn c ta l 2 20 0

1 1;0B OB

x x

. V th 1 2. 1OA OB .

2. S c tm 1; 0; 1I , bn knh 2R .

1; 3; 4 , 1; 1; 4AB AB

.

Gi l mt phng cha 3 im A, B, C nhn , 8; 8; 4n AB AC

lm vect phptuyn nn c phng trnh: 8 1 8 2 4 3 0 2 2 1 0x y x x y z .

22 2

2 0 1 1 2, 2

32 2 1d I R

S .

Ta c:1

.3ABCD D ABC

V h S

nnABCD

V ln nhtD

h ln nht.

Gi1 2

D D l ng knh ca S vung gc vi mt phng

.

V D l im bt k thuc S nn 1 2, max , , ,d D d D d D .

Du bng xy ra khi v ch khi D trng vi mt trong hai im 1D hoc 2D .

2A

1A

1B2BO

www.MATHVN.com



49/260



1 2D D qua I nhn vect php tuyn ca lm vect ch phng nn c phng trnh

tham s: 1 2

1 2

: 2 t

1

x t

D D y t

z t

.

Gi 0 0 0 1 21 2 ; 2 ; 1D d d d D D l im cn tm. Khi D l nghim ca phng

trnh: 2 22

0 0 0 0 0 0

21 2 4 1 2 1 2 2 1 2 0

3d d d d d d .

Ta c: 09 2

,3

dd D

. V

22 9. 29. 233

3 3

nn D phi ng vi 0

2

3d .

Vy 7 4 1; ;3 3 3

D

l im cn tm.

Cu VIIb (1,0 im)

Ta c: 3 2 os sin ; 1 = 2 os i sin .6 6 4 4

i c i i c

3 5 52 os i sin .

1 12 12

ic

i

Do :23 1 5 5

2 os isin .1 12 12

nn

n n

ci

S l thc khi v ch khi 5 5 5

sin 0 .12 12 12

n n nk k k

S nguyn dng b nht cn tm l: 12n .

8


Cu I (2,0 im) . Cho hm s: 3 21 8

33 3

y x x x (C).

14.Kho st s bin thin v v th C ca hm s.2. Lp phng trnh ng thng d song song vi trc honh v ct th (C) ti hai imphn bit A, B sao cho tam gic OAB cn ti O ( O l gc to ).Cu II(2,0 im)

1. Gii phng trnh: 21

1 4sin sin 32

x x .

www.MATHVN.com



50/260



2. Gii phng trnh : 4 2 21 1 2x x x x .

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

0 1 sin cos

dxI

x x

Cu IV(1,0 im).

Cho lng tr tam gic .ABC A B C c y ABC l tam gic u cnh av nh A cch u cc nh A, B, C. Cnh AA to vi y gc 060 . Tnh th tch khi

lng tr.

Cu V (1,0 im). Cho cc s thc x, y, z tha:2 2

2 2

3

16

x xy y

y yz z

.

Chng minh rng: 8xy yz zx .

B- PHN RING (3,0 im). Th sinh ch c chn mt trong hai phnB.1. CHNG TRNH CHUNCu VI a (2,0 im)

1.Trong mt phng Oxy, cho im 7;8P v hai ng thng: 1 : 2 5 3 0,d x y 2d :5 2 7 0x y ct nhau ti A. Vit phng trnh ng thng d i qua P v to vi

1 2,d d thnh tam gic cn ti A v c din tch bng29

2.

2. Trong khng gian Oxyz, cho im 4;5;6 .H Vit phng trnh mt phng (P) qua H,ct cc trc to Ox, Oy, Oz ln lt ti A, B, C sao cho H l trc tm ca tam gic ABC.Cu VII a(1,0 im) . Tnh ni vi n .B.2. CHNG TRNH NNG CAOCu VI b (2,0 im)1. Trong mt phng Oxy, cho Parabol

2: 64P y x v ng thng : 4 3 46 0x y .

Tm A thuc (P) sao cho khong cch t A n nh nht. Tnh khong cch nh nht . 2. Trong khng gian Oxyz, cho mt phng (P) ct Ox, Oy, Oz ln lt ti

A a;0;0 , B 0; b;0 , C 0;0;c Gi , , ln lt l cc gc ca cc mt phng (OAB), (OBC) , (OCA) vi mt phng

(ABC). Chng minh rng: 2 2 2os os os 1.c c c Cu VII b(1,0 im)

Gii h phng trnh:

1)4(log)5(log

6)12(log)22(log2

21

221

xy

xxyxxy

yx

yx

BI GIIA- PHN CHUNGCu I1. Hc sinh t gii2. V d song song vi trc honh nn phng trnh ca d l: 0y m m .Phng trnh honh giao im ca (C) v d:

www.MATHVN.com



51/260



3 2 3 21 8

3 3 9 8 3 0 13 3x x x m x x x m

d ct (C) ti hai im phn bit A, B sao cho tam gic OAB cn th phng trnh (1)phi c cc nghim: 1 1 2, ,x x x ( vi 1 1,x x l honh ca , BA ).

Khi 1 2,x x l nghim ca phng trnh:

2 2 3 2 2 21 2 2 1 1 20 0x x x x x x x x x x x (2)

ng nht (1) v (2) ta c:

2

21

21 2

3

9 9.3 8 3

8 3

x

x m

x x m

hay19

3m .

Vy19

:3

d y .

Cu II1. Nhn xt: cos 0x khng phi l nghim ca phng trnh . Do , nhn c hai v caphng trnh cho cos 0x ta c:

2 21 1

cos 4cos sin sin 3 cos cos 4cos 1 cos sin 3 cos2 2

x x x x x x x x x x

32sin 3 4cos 3cos cos 2sin 3 cos3 cos sin 6 cosx x x x x x x x x

2

14 7sin 6 sin22

10 5

x k

x x k

x k

.

15.iu kin: 22

1 01

1

xx

x x

.

Khi : 42 2 21 1 1x x x x x x ( do 1x ).

Suy ra: 4 4 42 2 2 21 1 1 1x x x x x x x x p dng bt ng thc Cauchy ta c:

4 4 4 42 2 2 21 1 2 1. 1 2x x x x x x x x

Do : 4 2 21 1 1x x x x . Vy phng trnh cho v nghim.Cu III

t2

2tan

2 1

x dtt dx

t

10 0 ; x =

3 3x t t

1 1

3 3 1

32 0

20 02 2

2 1ln 1 ln 1

12 1 31 1

1 1

dt dt I t

tt tt

t t

.

www.MATHVN.com



52/260



Cu IVGi H l hnh chiu ca A ln mp (ABC)v M l trung im ca BC.

Do ABC l tam gic u vA A A B A C nn H l trng tm tamgic ABC.

Ta c:2 2 3 3

3 3 2 3

a aAH AM .

Mt khc AH l hnh chiu ca AA ln

mt phng (ABC) nn 060A AH l gchp bi cnh bn AA v ABC .Trong tam gic A HA vung ti H, ta c :

0 3

tan 60 . 33

a

A H AH a

.2 3

4ABCa

S

.

Vy2

3.

3 3. .

4 4ABC A B C ABCa

V A H S a a .

Cu Vp dng bt ng thc Bunhiacopski, ta c:

2

23 3 3

4 2 2 2 2

x zxy yz zx y z x y

2 2

2 2 2 2 2 23 3 482 4 4 2

x zy x z y x xy y y yz z

2

64 8xy yz zx xy yz zx ( pcm).

B- PHN RINGB.1. CHNG TRNH CHUNCu VI a1. Ta c: 1 2A d d to im A l nghimca h :

2 5 3 0 15 2 7 0 1x y x

x y y

hay 1; 1A .

Phng trnh cc ng phn ca gc to bi

1 2,d d l: 1 2: 7 3 4 0 , :3 7 10 0x y x y

V d to vi 1 2,d d mt tam gic vung cn nn

M

A

B

C

A'

C'

B'

H

d2

dCB

A

d1

HP

www.MATHVN.com



53/260



1 1

2 2

3 7 0

7 3 0

d x y C

d x y C

Mt khc 7;8P d nn 1 277, C 25C .

Suy ra:: 3 7 77 0

: 7 3 25 0

d x y

d x y

Gi 1 2, CB d d d d

V tam gic ABC vung cn ti A nn 21 1 29

292 2 2ABC

S ABAC AB AB

.

v 2 58BC AB

Suy ra:

292.2 582

258ABCSAH

BC .

Vi :3 7 77 0d x y , ta c:

22

3.1 7. 1 77 87 58;

2583 7d A d AH

( loi).

Vi : 7 3 25 0d x y , ta c : 2 2

7.1 3. 1 25 29 58;

2587 3d A d AH

( tho).

Vy : 7 3 25 0d x y .

2. Gi s : ; 0; 0 , 0; ;0 , 0; 0;P Ox A a P Oy B b P Oz C c Khi (P) c phng trnh : 1

x y z

a b c .

Ta c : 4 5 6

4;5;6 1H P

a b c

4 ;5;6 , 4;5 ;6 , 0; ; , ;0;AH a BH b BC b c AC a c

V H l trc tm tam gic ABC nn. 0 5 6 0

4 6 0. 0

AH BC b c

b cBH AC

Gii h phng trnh :

774 5 6

1 477

5 6 05

4 6 077

6

a

a b c

b c b

b c

c

.

Vy phng trnh mt phng (P) l : 177 77 77

4 5 6

x y z .

Cu VIIa

+ Nu 4n k k th 4 4 1k

n ki i i

www.MATHVN.com



54/260



+ Nu 4 1n k k th 4 1.n ki i i i i

+ Nu 4 2n k k th 4 2 1. 1 1n ki i i

+ Nu 4 3n k k th 4 3 1n ki i i i i .


Cu VI b

1. Ta c: 2

2: 64 ;64

aA P y x A a

2

22

2 2

4. 3 4664 1 1

, 48 736 24 16080 804 3

aa

d A a a a

21 160

24 160 280 80

a .

Du bng xy ra khi ch khi 24 0 24a a .

Lc Mind A, 2 khi 9; 24A .2.

: 1 0x y z

P ABC a b c

c phng vect php tuyn1 1 1

, ,na b c

mp OAB c vect php tuyn 1 0,0,n OC c

( )mp OBC c vect php tuyn 2 ,0,0n OA a

mp OAC c vect php tuyn 3 (0, , 0)n OB b

Gi , , ln lt l gc gia cc mt phng , ,OAB OBC OCA vi mp ABC .

Do :

2 2 22 2 2 2 2 2

1 1 1 10 0os

1 1 1 1 1 10 0

ca b c cc

ca b c a b c

(1)

2 2 22 2 2 2 2 2

1 1 1 10 0os

1 1 1 1 1 10 0

aa b c ac

aa b c a b c

(2)

2 2 22 2 2 2 2 2

1 1 1 10 0os

1 1 1 1 1 10 0

ba b c bc

ba b c a b c

(3)

T (1), (2) v (3) suy ra: 2 2 2os os os 1.c c c

www.MATHVN.com



55/260



Cu VIIb

iu kin:

1;2

0,14

yy

xx

21 2

1 2

21 21 2

1 21 2

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

log ( 5) log ( 4) 12 2 log 2 2 log 1 62 log 1 2 log 1 6

log 5 log 4 1log 5 log 4 1

x y

x y

x yx y

x yx y

xy x y x x

y x

y xx y x

y xy x

21 2 1 1

1 2 1 2

log 2 log 1 2 log 2 2 log 2 1 0

log 5 log 4 1 log 5 log 4 1

x y x x

x y x y

y x y y

y x y x

1

1 11 2

log 2 1 1

log 4 log 4 1log 5 log 4 1

x

x xx y

y y x

x xy x

2 21

1 1 1 1 24 4

1log 1 1 4 3 4 2 04 4x

y x y x y x y x xx x

yx x x x x xx x

.

Vy2

1

x

y

l nghim ca h phng trnh cho.

9


Cu I (2,0 im) . Cho hm s: 3 22 3 4y x mx m x c th mC

16.Kho st s bin thin v v th C ca hm s khi 1m .2. Cho ng thng : 4d y x v im 1;3E . Tm tt c cc gi tr ca tham s m sao

cho d ct mC ti ba im phn bit 0;4 , ,A B C sao cho tam gic EBC c din tch bng4 .Cu II(2,0 im)

1. Gii phng trnh: 3 32 3 2

cos3 cos sin 3 sin8

x x x x

.

2. Gii h phng trnh:

2

21 4 ,1 2

x y y x y x yx y x y

.

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

4

2

ln 9

ln 9 ln 3

xI dx

x x

www.MATHVN.com



56/260



Cu IV(1,0 im).Cho lng tr ng t gic u .ABCD A B C D c chiu cao bng h.

Gc gia hai ng cho ca hai mt bn k nhau k t mt nh bng 0 00 90 .Tnh th tch khi lng tr cho.Cu V (1,0 im). Gii phng trnh:

2 2

2 2 2

3 2 2 2 3 1033 3 4 4 3

x x x x x x x x x x

x x x x x x x x x x x x x x x x

B- PHN RING (3,0 im). Th sinh ch c chn mt trong hai phnB.1. CHNG TRNH CHUNCu VI a (2,0 im)1. Trong mt phng Oxy, cho hnh bnh hnh ABCD c din tch bng 4. Bit to ccnh 2;0 , B 3;0A v I l giao im ca hai ng cho AC v BD, I nm trn ng

thng y x . Xc nh to cc im C, D.

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

1 1 2

x y zd v 2

1 1:

2 1 1

x y zd

.

Chng minh 1 2,d d cho nhau. Tm 1 2,A d B d sao cho ng thng AB song song vi

mt phng : 0P x y z v di 2AB .

Cu VII a(1,0 im)Trn cc cnh AB, BC, CD, DA ca hnh vung ABCD ln lt cho 1, 2, 3 v n im phnbit khc A, B, C, D. Tm n s tam gic c 3 nh ly t 6n im cho l 439.B.2. CHNG TRNH NNG CAOCu VI b (2,0 im)1. Trong mt phng Oxy, lp phng trnh ng trn C qua M 2; 4 v tip xc vi

hai trc ta .

2. Trong khng gian vi h ta Oxyz, cho ba im 1;0; 1 , 2;3; 1 , 1;3;1A B C v

ng thng1 3

:1 1 2

x y zd

. Tm ta im D thuc ng thng d sao cho th tch

khi t din ABCD bng 1. Vit phng trnh tham s ca ng thng qua trc tm Hca tam gic ABC v vung gc vi mt phng (ABC).Cu VII b(1,0 im)Gii phng trnh: 2 0z z .

BI GIIA- PHN CHUNG

Cu I1. Hc sinh t gii2. Phng trnh honh giao im ca mC v d:

3 2 3 22

02 3 4 4 2 2 0

2 2 0 *

xx mx m x x x mx m x

x mx m

www.MATHVN.com



57/260



d ct mC ti ba im phn bit 0;4 , B,CA Phng trnh (*) c hai nghim phn

bit khc 02 1 22 0

22 0

m mm m

mm

.

Gi 1 2,x x theo th t l honh ca cc im ,B C. Do 1 2,x x l hai nghim phn bit

ca phng trnh (*). Theo nh l Viet, ta c: 1 2

1 2

22

x x m

x x m

.

21 1

4 , . 4 2. 4 4 2 322 2EBC

S d E d BC BC BC BC

22 2

2 1 2 1 2 24 4 32 2 32x x x x x x

2 2 2

2 1 1 2 1 216 4 16 4 4 2 16x x x x x x m m

2 26 03

mm m

m

. So vi iu kin ta u ta tm c 3m

Vy 3m l gi tr cn tm tho mn yu cu bi ton.Cu II

1. Ta c: 3 32 3 2

cos3 cos sin 3 sin8

x x x x

cos3 3cos 3sin sin 3 2 3 2cos3 . sin 3 .

4 4 8

x x x xx x

2 22cos 3 6cos3 cos 6sin 3 sin 2sin 3 2 3 2x x x x x x

2 22 cos 3 sin 3 6 cos3 cos sin 3 sin 2 3 2x x x x x x

2 16 2cos42

16 2

x k

x k

x k

.

2. + Vi 0y ta c:

2

2

1 0

1 2 0

x

x x

( H v nghim).

+ Vi 0y , ta c :

2

2

2 2

12 21 4

1 2 1

2 1

xy x

x y y x y y

x y x y xy x

y

Suy ra:2 1

Documents

com - Dap an 40 de Thi Thu Mon Toan