10 de thi thu 2015

Tuyn tp 10 thi th nm 2015

GV: Nguyn Tt Thu (0942444556) 1

S 1 Cu 1. Cho hm s 3 23 1 3 2y x m x m x m (1) vi m l tham s. 1) Kho st s bin thin v v th ca hm s (1) khi 2m 2) Tm m th hm s (1) t cc i, cc tiu ti nhng im c honh dng. Cu 2. 1) Cho tan 2x . Tnh gi tr ca biu thc 2 2sin 2sin 2 3cosP x x x .

2) Cho s phc z tha mn 3 3iiz iz

. Tnh m un ca s phc 10z

, bit z c phn thc ln

hn 1 .

Cu 3. Gii phng trnh 21 2 18 . 2 4xx x x .

Cu 4. Gii h phng trnh 5 4 10 6

32 2 21 2 1 4

x xy y y

y x y x

.

Cu 5. Tnh tch phn 2

2

0sin cos

2xI x x dx

.

Cu 6. Cho hnh chp .S ABC c ABC vung ti A , , 3AB a AC a . Hnh chiu ca S ln mt phng ABC l im H nm trn cnh BC sao cho 3BH CH . Gc gia cnh bn SA v mt phng ABC bng 060 . Tnh th tch khi chp .S ABC v khong cch gia hai ng thng SB v AC . Cu 7. Trong mt phng vi h ta Oxy , cho tam gic ABC c chn ng cao h t nh A l

17 1;5 5

H

, chn ng phn gic trong ca gc A l 5; 3D v trung im ca cnh AB l

0;1M . Tm ta nh C .

Cu 8. Trong khng gian Oxyz cho ng thng 1 1d :1 1 2

x y z v mt phng a c phng

trnh 2 2 5 0x y z . Chng minh rng ng thng d ct mt phng ti I . Vit phng trnh mt cu S c tm I v ct mt phng theo mt ng trn c din tch bng 16 . Cu 9. Mt chic hp c 10 qu cu c kch thc nh nhau c nh s t 1 n 10 . T chic hp ny ly ngu nhin ra 3 qu cu. Tm xc sut cc s trn ba qu cu ly c l di ba cnh ca mt tam gic vung. Cu 10. Cho cc s thc khng m , ,x y z tha mn 2x y z . Tm gi tr nh nht ca biu thc

2 2 2 2 2 21 1 1P

x y y z z x

.


GV: Nguyn Tt Thu (0942444556) 2

S 2

Cu 1. Cho hm s 11

xyx

c th C . 1) Kho st s bin thin v v th C 2) Tnh din tch hnh phng gii hn bi th C v ng thng : 3 1d y x . Cu 2.

1) Cho cc s thc ,a b tha , 02 2

a a b v 1 1sin , cos3 5

a a b . Tnh sin b .

2) Tm s phc z bit 25 8 6z iz

.

Cu 3. Gii phng trnh 29 13

4 log 3 log 2 0x x .

Cu 4. Gii phng trnh: 22 3612

2 1 8 7x x x

x x

.

Cu 5. Tnh tch phn 2

0

sin 23 4 sin cos2

xdxIx x

.

Cu 6. Cho hnh chp .S ABCD c y ABCD l hnh thoi, 2 3AC a , 2BD a v O l giao im ca hai ng cho, hai mt phng SAC v SBD cng vung gc vi mt phng ABCD .

Bit khong cch t im O n mt phng SAB bng 34a . Tnh th tch khi chp .S ABCD

theo a.

Cu 7. Trong mt phng ta Oxy cho tam gic ABC vung cn ti A , 11 17;3 3

D

l im nm

trn cnh BC sao cho 2BD CD . ng trung tuyn BM ca tam gic ABC c phng trnh 4 7 37 0x y . Xc nh ta cc nh ca tam gic ABC , bit 0Bx .

Cu 8. Trong khng gian vi h trc ta Oxyz , cho ng thng 1 3:1 1 4

x y z v im

0; 2;0M . Vit phng trnh mt phng P i qua im M song song vi ng thng ng thi khong cch gia ng thng v mt phng P bng 4 .

Cu 9. Tm h s ca s hng cha 5x trong khai trin nh thc Newton ca 21 3 nx bit n l s nguyn dng tha 3 22 100n nA A .

Cu 10. Cho , , 0a b c tha 2 2 2 34

a b c . Tm gi tr nh nht ca biu thc

1 1 1P a b c .


GV: Nguyn Tt Thu (0942444556) 3

S 3 Cu 1. Cho hm s 3 23 1 2 2y x m x mx m (1) 1) Kho st s bin thin v v th hm s (1) khi 0m 2) Tm m hm s (1) t cc tiu ti 2x . Cu 2.

1) Gii phng trnh 21 sin 2 3sin1 tan 2

x xx

.

2) Trong mt phng Oxy , hy tm tp hp im M biu din s phc z tha mn

1z i iz .

Cu 3. Gii phng trnh 2 4log 1 2 log 3 2 2 0x x . Cu 4. Tm m h phng trnh sau c nghim

3 2 3 23 10 5 11

(4 3) 1 (3 4) 1 1 0

y y y x x x

m y m x m

.

Cu 5. Tnh tch phn 4

6

1 cos2sin 2

xI dxx

.

Cu 6. Cho hnh chp .S ABCD c y ABCD l hnh thoi c tam gic ABC u cnh a . Hnh chiu vung gc ca S ln mt phng ABCD l trung im H ca cnh AB . Gc gia cnh bn SC v mt phng ABCD bng 060 . Tnh theo a , th tch ca khi chp .S ABCD v khong cch giao im O ca AC vi BD n mt phng SCD . Cu 7. Trong mt phng Oxy cho tam gic ABC c 7; 1H l chn ng vung gc h t A xung BC v 4; 4I l tm ng trn ngoi tip tam gic ABC . Tm ta cc nh ABC , bit phn gic trong gc A c phng trnh 3 11 0x y .

Cu 8. Trong khng gian Oxyz cho ng thng 1 1:2 1 1

x y zd

v im 1; 2; 3A . Tm ta hnh chiu vung gc ca A ln ng thng d . Vit phng trnh mt cu tm A v tip xc vi ng thng . Cu 9. Trong mt mn hc, thy gio c 30 cu hi khc nhau gm 5 cu hi kh, 10 cu hi trung bnh, 15 cu hi d. T 30 cu hi c th lp c bao nhiu kim tra, mi gm 5 cu hi khc nhau, nht thit phi c 3 loi cu hi v s cu hi d khng t hn 2.

Cu 10. Cho cc s thc khng m ,x y tha mn 3 2 52 1 3x yx y

. Tm gi tr ln nht ca biu thc

5 4P x y .


GV: Nguyn Tt Thu (0942444556) 4

S 4 Cu 1. Cho hm s 4 22 2y x x c th C . 1) Kho st s bin thin v v th C 2) Vit phng trnh tip tuyn ca th C , bit tung tip im bng 6 . Cu 2.

1) Cho cot 3x . Tnh gi tr ca biu thc 3

3sin cos

2 cos sin

x xPx x

.

2) Cho s phc z tha mn 2 3 6z z i . Tnh gi tr biu thc 2 3z z z

Cu 3. Gii phng trnh: 14 29.10 25 0x x x .

Cu 4. Gii bt phng trnh: 223 2 6 2 2 2 2x x x x x x x .

Cu 5. Tnh tch phn 1 3

20 4

xI dxx

.

Cu 6. Cho lng tr ng . ' ' 'ABC A B C c y ABC l tam gic vung ti A , , 3AB a AC a . Gc gia hai mt phng ABC v 'A BC bng 060 . Gi M l trung im cnh BC . Tnh th tch khi lng tr . ' ' 'ABC A B C v c sin ca gc gia hai ng thng AB v 'B M . Cu 7. Trong mt phng Oxy cho tam gic ABC cn ti B c trung im M ca on AB nm trn

ng thng 0x y v 3 1; , 1; 35 5D E

theo th t l chn ng vung gc h t A v B .

Tm ta cc nh , ,A B C . Cu 8. Trong khng gian Oxyz cho mt phng : 2 2 1 0x y z v ng thng

2 1:1 1 2x y z

. Vit phng trnh ng thng d nm trong mt phng , ng thi d ct v vung gc vi ng thng . Cu 9. T 10 ch s 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 c th lp c bao nhiu s gm 6 ch s khc nhau sao cho s phi c mt hai ch s 0 v 1. Cu 10. Cho , , 0a b c tha mn 3 3 3 3a b c . Tm gtln ca biu thc

3P ab bc ca abc .


GV: Nguyn Tt Thu (0942444556) 5

S 5

Cu 1. Cho hm s

2 11

xyx

c th l C . 1) Kho st s bin thin ca hm s v v th C 2) Vit phng trnh tip tuyn ca C , bit tip tuyn song song vi ng thng 3 1y x . Cu 2.

1) Cho tan 2x . Tnh gi tr ca biu thc sin sin 2 sin 3cos cos2 cos3

x x xPx x x

.

2) Cho s phc z tha z 2 i z 3 5i . Tm phn o ca s phc z . Cu 3. Gii bt phng trnh 29 3log ( 1) 2 log ( 1) 3 0x x .

Cu 4. Gii h phng trnh 2 2 2

5 2

2 3 2

2 2 0

xy x y x y

y x y xy x

.

Cu 5. Tnh tch phn 2

1

ln

(2 ln )

e xI dxx x

.

Cu 6. Cho lng tr . ' ' 'ABC A B C c tam gic ABC vung ti A , , 3AB a AC a . Hnh chiu vung gc ca 'A ln mt phng ABC l im H thuc cnh BC sao cho 3CH BH . Gc gia ng thng 'AA v mt phng ABC bng 060 . Gi M l trung im 'BB . Tnh theo a th tch ca khi t din 'ABCB v khong cch gia hai ng thng AM v 'CC . Cu 7. Trong mt phng Oxy cho tam gic ABC ni tip ng trn C c phng trnh 2 2x 1 y 1 25 . Bit D 0;3 , E 5;3 l chn ng cao h t nh A v B . Xc nh ta cc nh ca tam gic ABC , bit C c tung dng.

Cu 8. Trong khng gian Oxyz cho ng thng 2 3:1 1 2x y zd

v hai mt phng

: 2 2 1 0x y z , : 2 2 7 0x y z . Vit phng trnh mt cu (S) c tm nm trn ng thng d v (S) tip xc vi hai mt phng v . Cu 9. Trong chic hp c 21 vin bi c kch thc ging nhau gm 5 vin bi xanh, 7 vin bi v 9 vin bi vng. Ly ngu nhin ra 4 vin bi. Tm xc sut 4 vin bi ly ra c ng 3 mu. Cu 10. Cho hnh hp lp phng c cnh bng 1 . Trong khi lp phng cho ba im , ,M N P

khng thng hng. Chng minh rng 32MNP

S .


GV: Nguyn Tt Thu (0942444556) 6

S 6 Cu 1. Cho hm s 3 2y 2x 3x 1 c th l (C). 1) Kho st s bin thin v v th (C) 2) Vit phng trnh tip tuyn ca (C), bit tip tuyn vung gc vi ng thng x 12y 1 0 . Cu 2. 1) Chng minh rng biu thc sau khng ph thuc vo x

1 sin 2 1 cos 21 tan 42

xP xx

.

2) Cho s phc z tha iz 2i 1 z 1 13i . Tnh z . Cu 3. Gii bt phng trnh 2 3log log 2 0x x .

Cu 4. Gii h phng trnh

2 2

3 2 2 3

3 2 9 6

3 3 2 8

y x x x y xy

x x y y x y

.

Cu 5. Tnh tch phn

0 3 2

221

x 3x x 3I dxx 2x 3

.

Cu 6. Cho hnh chp S.ABCD c y ABCD l hnh thoi cnh a v 0BAD 60 . Hnh chiu ca S ln mt phng ABCD l trng tm tam gic ABC . Gc gia mt phng ABCD v SAB bng 060 . Tnh th tch khi chp S.ABCD v khong cch t B n mt phng SCD . Cu 7. Trong mt phng Oxy cho hnh ch nht ABCD c AB 2AD , ta nh D 1;1 v

M 5;5 l im nm trn cnh AB sao cho AM 3MB . Tm ta cc nh A , B v C , bit im A c honh m.

Cu 8. Trong khng gian Oxyz cho ng thng x 5 y 2 z 5d :2 1 3

, mt phng

: 2x y 2z 2 0 v hai im A 0; 1; 2 , B 2;3;1 . Vit phng trnh mt phng i qua hai im A,B v ct ng thng d ti C sao cho C cch mt phng mt khong bng 2 . Cu 9. Mt hp cha 12 vin bi kch thc nh nhau, trong c 5 bi mu xanh v c nh s t 1 n 5 , 4 vin bi mu c nh s t 1 n 4 v 3 vin bi mu vng c nh s t 1 n 3 . Ly ngu nhin 2 vin bi t hp . Tm xc sut 2 vin bi ly ra va khc mu va khc s. Cu 10. Cho cc s thc dng a, b, c tha:

2 2 2 2 2 2 7 abca b b c c a2

.

Tm gi tr nh nht ca biu thc 2 2 2a b cP

b c c a a b

.


GV: Nguyn Tt Thu (0942444556) 7

S 7 Cu 1. Cho hm s 3 23 1y x x (1). 1) Kho st s bin thin v v th hm s (1) 2) Vit phng trnh tip tuyn ca th hm s (1), bit tip tuyn song song vi ng thng

9 4y x . Cu 2. 1) Gii phng trnh 3 cos3 2sin 2 sin 3x x x 2) Cho s phc z tha mn 1 2 4 3i z i iz i . Tm phn o ca s phc z . Cu 3. Gii phng trnh 22 1

2

4 log 3 log 1 0x x .

Cu 4. Gii h phng trnh 3 3 2

23 8 0

2 4 0

x y x y

y xy y

( ,x y ).

Cu 5. Tnh tch phn 1

0 3 1xdxIx

.

Cu 6. Cho hnh chp .S ABCD c y ABCD l hnh thang vung ti A v B , AB BC a , 2AD a . Cnh bn SA vung gc vi y v gc gia ng thng SC vi mt phng ABCD

bng 060 . Tnh th tch ca khi chp .S ABCD v khong cch t im D n mt phng SBC . Cu 7. Trong mt phng Oxy , cho hnh vung ABCD c 1;4A v hai im ,M N ln lt nm trn hai cnh , BC CD . Bit rng hnh chiu vung gc ca ,M N trn ,AN AM tng ng l

1;0 , 4;1P Q . Tm ta nh C .

Cu 8. Trong khng gian Oxyz cho ng thng 1 1:1 2 3

x y zd

v mt phng c phng trnh 2 2 3 0x y z . Chng minh rng ng thng d ct mt phng ti I . Vit phng trnh mt cu tm I v ct mt phng theo giao tuyn l mt ng trn c bn knh bng 3 . Cu 9. Trong mt hp c 9 chic th kch thc nh nhau v c nh s t 1 n 9 . Rt ngu nhin ra 4 chic th t hp . Tnh xc sut tch cc s trn 4 th c rt ra l s chn. Cu 10. Cho cc s thc , , 0;2x y z . Tm gi tr nh nht ca biu thc

2 2 23 3 3

96102

P x y z xy yz zxx y z

.


GV: Nguyn Tt Thu (0942444556) 8

8

Cu 1. Cho hm s 11

xyx

.

1. Kho st s bin thin v v th (C) ca hm s cho 2. Tm m th C ct ng thng : 2 1d y x m ti hai im phn bit c honh dng. Cu 2.

1) Cho 1sin3

x . Tnh gi tr ca biu thc 2cos tan2 tan cos

x xPx x

.

2) Trong mt phng Oxy , hy tm tp hp im M biu din s phc z tha mn

3 4z i iz .

Cu 3. Gii bt phng trnh 116 25.12 9 0x x x .

Cu 4. Gii bt phng trnh 33 2 2 3. 2 1 22

x xx

.

Cu 5. Tnh tch phn 1

0 1 3 1xdxI

x

.

Cu 6. Cho hnh lng tr ng ABC.A 'B 'C ' c tam gic ABC vung ti A , AB a, BC 2a v AA ' 2a . Gi M l trung im ca cnh BB ' . Tnh th tch khi chp BMCA ' v c sin ca gc gia hai ng thng A 'M v BC . Cu 7. Trong mt phng vi h trc to Oxy cho tam gic ABC cn ti A , D l trung im ca

cnh AC . Bit 1; 0K , 1 ;43E

ln lt l tm ng trn ngoi tip tam gic ABC , trng tm

tam gic ABD . Cc im 1;6 , 9;2P Q ln lt thuc cc ng thng , AC BD . Tm to cc nh , , A B C bit D c honh dng. Cu 8. Trong khng gian Oxyz cho mt phng : 2 2 5 0x y x v im 1;1; 1A . Vit phng trnh mt cu tm A v tip xc vi mt phng v tm ta tip im. Cu 9. Mt bi thi trc nghim c 50 cu hi. Mi cu hi c 4 phng n tr li trong ch c mt cu tr li ng. Nu tr li ng th c 0,2 im, nu tr li sai th khng c im. Bn Nam khng hc bi nn lm bi bng cch nh ngu nhin. Tnh xc sut bn Nam c 5 im.

Cu 10. Cho cc s thc , (0;1)a b tha 2 2 2 21 1a b a b b a . Tm gi tr nh nht ca biu thc sau:

8(1 ) 191 1

a bPa b

.


GV: Nguyn Tt Thu (0942444556) 9

S 9 Cu 1. Cho hm s 3 26x 9x 1y x (1). 1) Kho st s bin thin v v th (C) ca hm s (1). 2) Tm m phng trnh 3 22 12 18 0x x x m c 3 nghim phn bit. Cu 2. 1) Gii phng trnh 3 cos5 2sin 4 .cos sin 3x x x x . 2) Tm s phc z tha mn (1 ) 3 4 1 2i z i i . Cu 3. Gii bt phng trnh 0,2 0,2 0,2log log ( 1) log ( 2)x x x .

Cu 4. Gii h phng trnh 2 23

2 24 2 (3 1) 8 ( 1) .2 1 2( 1) 3 ( 1) 1

x x x y

x y x y x

Cu 5. Tnh din tch hnh phng gii hn bi thi hai hm s y 3 x v y x 2 . Cu 6. Cho hnh chp S.ABC c tam gic ABC vung ti A , AB a, AC a 3 . Hnh chiu ca S ln mt phng ABC l im H nm trn cnh AC sao cho AH 2CH . Gc gia ng thng SB v mt phng (ABC) bng 060 . Tnh th tch khi chp S.ABC v khong cch gia SC v AB . Cu 7. Trong mt phng ta Oxy , cho hnh thang ABCD vung ti A v D , c 2CD AB .

Hnh chiu vung gc ca D ln ng thng AC l 11 7;5 5

H

. Gi N l trung im ca HC .

Tm ta cc nh , ,A C D bit 4;2B v ng thng DN c phng trnh 3 4 0x y .

Cu 8. Trong khng gian Oxyz cho im A 1;0; 1 v ng thng d: x 1 y 1 z2 2 1

. Vit

phng trnh mp qua A v vung gc vi d. Tm ta hnh chiu vung gc ca A trn d. Cu 9. Trong chic hp c 5 vin bi xanh, 7 vin bi v 9 vin bi vng. Ly ngu nhin ra 4 vin bi. Tm xc sut 4 vin bi ly ra c ng 3 mu. Cu 10. Cho cc s thc , ,a b c tha 0abc v 0a b c . Tm gi tr nh nht ca biu thc:

1 1 1 12 81 abcP ab bc caa b c ab bc ca


GV: Nguyn Tt Thu (0942444556) 10

S 10 Cu 1. Cho hm s 3 23 1 3 1y x m x mx m (1). 1) Kho st s bin thin v v th hm s (1) khi 0m 2) Tm m hm s (1) ng bin trn . Cu 2.

1) Gii phng trnh 2 x2sin x sin 2x 2cos2

.

2) Cho s phc z tha z 2z 2i 1 3z 38i 11 . Tm phn thc, phn o ca s phc 2w 1 2z 3z .

Cu 3. Tm cc s thc a, b tha mn ng thi hai iu kin sau: 5a 8b 37 v 2 log(a 3b) log 4 log a log b .

Cu 4. Gii h phng trnh :

2 2

22 2

x 1 x y y 1 1

x 1 y 1 8 y x 3 17

.

Cu 5. Tnh th tch ca khi trn xoay to nn do ta quay hnh D quanh trc Ox, vi D l hnh gii

hn bi cc ng: 2cos sin , 0, 0,2

y x x x y x x .

Cu 6. Cho khi lng tr ABC.A B C c di cnh bn bng 2a , y ABC l tam gic vung ti A, AB a, AC a 3 v hnh chiu vung gc ca nh A ' trn mt phng (ABC) l trung im ca cnh BC . Tnh theo a th tch khi chp A .ABC v tnh cosin ca gc gia hai ng thng AA ' v B 'C ' . Cu 7. Trong mt phng Oxy cho hnh ch nht ABCD c AB 2AD . ng thng AC c phng trnh 3x y 12 0 . im M 4; 2 nm trn cnh CD sao cho DM 3CM . Tm ta cc nh ca hnh ch nht ABCD bit Cy 1 . Cu 8. Trong khng gian Oxyz cho 3 im A 1;0; 1 ,B 2;3; 1 , C 1;3;1 v ng thng

x 1 y z 5d :1 1 2

. Vit phng trnh tham s ca ng thng i qua trc tm H ca tam gic

ABC v vung gc vi mt phng ABC . Cu 9. Cho khai trin n n0 1 n(1 2x) a a x ... a x , trong n * . Tm s ln nht trong cc s 0 1 na , a , ..., a , bit cc h s 0 1 na , a , ..., a tha mn h thc

1 n0 n

a aa ... 4096

2 2 .

Cu 10. Cho , , 0a b c ; a c . Tm gi tr ln nht ca biu thc

2 2 2 2

a b cPc aa b b c

.


GV: Nguyn Tt Thu (0942444556) 11

Hng dn gii.

S 1 Cu 1. 1) Bn c t lm 2) Ta c 2' 3 2 1 2y x m x m

Suy ra 2' 0 2 1 2 0y x m x m (*). Yu cu bi ton tha mn khi v ch khi (*) c hai nghim dng phn bit

Hay 2' 3 3 0

2 1 0 2

2 0

m mS m m

P m

.

Vy 2m l nhng gi tr cn tm. Cu 2.

1) Ta c tan 2 sin 2cosx x x v 22

1 1cos51 tan

xx

.

Suy ra 2 2 2 2 74 cos 8cos 3cos 7 cos5

P x x x x .

2) t z x yi z x yi .

Ta c 2 23 3 . 3 3 3 3 3iiz i iz z i iz i x y i xi yz

2 2

21 1 21 3

1 13 3 3 2 0

y x xx y xy yy x x

.

Do z c phn thc ln hn 1 nn ta c 10 102 2 2 4 22

z i i iz i

.

Vy 2 210 4 2 2 5z

.

Cu 3. Phng trnh cho tng ng vi

2 21 12 2 1 33 2 4 22 22 .2 2 2 2x xx x xx x x

2 21 53 2 4 2 4 5 0 1,

2 4xx x x x x x x .

Vy nghim ca phng trnh cho l 51,4

x x .

Cu 4. Ta thy 0y khng l nghim ca h, nn phng trnh th nht ca h tng ng vi

55 2 0x x xy y y x y x

y y y

.

Thay vo phng trnh th hai ca h ta c 3 21 2 1 4x x x x (*).


GV: Nguyn Tt Thu (0942444556) 12

T (*) ta suy ra

2

23 23

2 5 2 52 5 2 51 4 01 4 1 03 5 1 01 2

xxx xx x xx x xx x

2 5 2 5

2 51;2 5 2 5;

xx

x

.

Th li ta c nghim ca h l: ; 2 5; 2 5x y

.

Cu 5. Ta c 2 2

0 0

1 1sin 1 cos 2sin sin 22 4

I x x x dx x x x dx

.

t 12sin sin 2 2 cos cos22

du dxu xdv x x dx v x x

.

Suy ra 22

0 0

1 1 1 12cos cos 2 2 cos cos24 2 4 2

I x x x x x dx

2

0

1 1 82sin sin 2 .16 4 4 16

x x

Cu 6.

E D

H

A C

B

S

K

Ta c 0, ( ) 60SAH SA ABC v 2BC a . Suy ra 2aCH v 3cos

2C .

p dng nh l C sin cho tam gic AHC ta c 2 2

2 2 2 2 3 7 72 . . cosC 3a 2. 3. .4 2 2 4 2a a a aAH AC CH AC CH a AH .

Do 021. tan 602

aSH AH . Li c 21 3.

2 2ABCaS AB AC .

Th tch khi chp .S ABC l: 21 7.

3 4ABCaV SH S .

Dng hnh bnh hnh ABDC , ta c AC ABD nn


GV: Nguyn Tt Thu (0942444556) 13

3, , ( ) , ( ) , ( )2

d SB AC d AC SBD d C SBD d H SBD .

K , HE BD HK SE ta c 2 2

., ( ) HE HSd H SBD HKHE HS

.

M 2 2 23 3 3

aHE CD AB nn 212205

HK a .

Vy 21, 3 205d SB AC a . Cu 7.

M

DHB C

A

Ta c 8 16; 2; 1 , 1;25 5 BC AHHD n n

.

Suy ra phng trnh ca : 2 7 0BC x y v : 2 3 0HA x y . Do 3 2 ; , ;2 7A a a B b b .

Ta c 3 2 0 3 3;3 , 3; 12 7 2 3a b a A B

a b b

.

Gi 'M i xng vi M qua AD , ta c 'M AC . Ta c 8; 0 ,AD

suy ra phng trnh : 3 0, ' : 0AD y MM x .

Suy ra 'MM ct AD ti 0;3E , suy ra ' 0;5M . Phng trnh : 2 3 15 0AC x y .

Ta 92 7 0 9: ;22 22 3 15 0 2

x y xC Cx y y

.

Cu 8.

Xt h phng trnh 1 0 21 1

2 1 0 11 1 22 2 5 0 2 2 5 0 1

x y xx y zy z y

x y z x y z z

.

Vy d ct ti 2;1;1I . Gi ,R r ln lt l bn knh mt cu cn vit v bn knh ca ng trn c din tch bng 16 . Ta c 2 16 4r r . V I nn ta suy ra 4R r .

Phng trnh mt cu 2 2 2: 2 1 1 16S x y z . Cu 9.


GV: Nguyn Tt Thu (0942444556) 14

Ta c khng gian mu 310 120n C . Gi A l bin c cn tnh xc sut. Gi a b c l ba s ghi trn ba qu cu chn c v ba s lp thnh ba cnh ca mt tam gic vung. Ta c 2 2 2a b c . Ta c cc b , ,a b c l 3,4,5 , 6,8,10 .

Do 2n A . Vy 2 1

120 60n A

P An

.

Cu 10.

Gi s min , ,z x y z . t , 2 2z za x b y , suy ra 2a b x y z . Ta c

222 2 2 2 2

4 2z zx z x xz x xz x a

Tng t: 2

2 2 22zy z y b

V 2 2

2 2 2 22 2z zx y x y a b

.

Do 2 2 2 2 2 2 2 2 2 2

1 1 1 1 1 1 1 3 1 14 4

Pa b a b a b a b a b

2 2 2 2

1 1 3 1 12 4aba b a b

.

Mt khc vi mi , 0u v ta c 1 1 4u v u v

v

2 2 21 1 2 8

uvu v u v

.

Nn ta c

2 2 2 2 2 2 2 21 1 3 1 1 4 3 8 10 5

2 4 4 22aba b a b a b ab a b a b

Suy ra 52

P . ng thc xy ra khi 0, 1z x y .

Vy 5min2

P .

S 2

Cu 1. 1) Bn c t lm. 2) Phng trnh honh giao im ca C v d

21 13 1 3 0 , 01 3

x x x x x xx

.

Din tch hnh phng cn tnh l

0 0

1 13 3

1 23 1 2 31 1

xS x dx x dxx x


GV: Nguyn Tt Thu (0942444556) 15

02

13

3 5 22 2 ln 1 2 ln2 6 3

x x x

(vdt).

Cu 2.

1) Ta c 2 2 2cos 1 sin3

a a v 2 2 6sin 1 cos ( ) 5a b a b . Suy ra sin sin sin cos cos sinb a b a a b a a b a

2 6 2 2 1 1 8 3 1. .5 3 5 3 15

.

2) t z x yi z x yi .

Ta c 2 225 8 6 . 25 8 6 25 8 6z i z z i z x y i x yiz

2 2 25 8 6 8 6x y x y y x i

2 2 2 2

4 48 6 0 43 325 50 325 8 6 25 6 9 09 3

x yy x x y xyx y x y y y y y

.

Vy 4 3z i . Cu 3. Phng trnh cho tng ng vi

22

3 3 3 314 log 3 log 2 0 log 3log 2 02

x x x x

33

log 1 3log 2 9

x xx x

.

Vy phng trnh cho c nghim l 3, 9x x .

Cu 4. T phng trnh ta suy ra 4964

x .

Ta c 236 36 2 3612 2 .

2 1 2 1 2 1 2 1x x xx x

x x x x

(1).

ng thc xy ra khi 236 2 36 0 42 1

x x x xx

.

Mt khc 8 7 4 4 7 2 4 7 2 1x x x x

Nn 2 22 36 2 36

2 18 7x x x x

xx

(2).

ng thc xy ra khi 4x . T (1) v (2) ta suy ra nghim ca phng trnh cho l 4x . Cu 5.

Ta c 2 2

sin 2 2sin cos sin cos3 4sin cos2 2sin 4sin 2 1 sin

x x x x xx x x x x

.

t 1 sin cost x dt xdx . i cn 0 1, 22

x t x t .


GV: Nguyn Tt Thu (0942444556) 16

Suy ra 22 2

2 211 1

1 1 1 1 1ln ln 22

tI dt dt tt tt t

.

Cu 6.

E OB

AD

C

S

K

K ,OE AB OK SE ta c 3, ( ) 4aOK d O SAB .

Ta c 2 2 2 2

. . 3 323

OA OB a a aOEOA OB a a

.

Suy ra 2 2 2 2 2 2

1 1 1 16 4 423 3

aSOOS OK OE a a a

. Li c 21 . 2 32ABCD

S AC BD a .

Th tch khi chp .S ABCD l 3

21 1 3. .2 33 3 2 3ABCD

a aV SO S a .

Cu 7.

Ta c 2 2 1 23 3 3 3AD AB BC AB AC AB AB AC

v 12

BM AB AC

Suy ra 2 21 1. 03 3

AD BM AB AC AD BM

.

Phng trnh : 7 4 3 0AD x y . Gi I l giao im ca AD v BM , suy ra ta ca I l nghim ca h

134 7 37 0 13 195 ;

5 57 4 3 0 195

xx y Ix y y

.

t AB a , ta c 2 2

2 1 2 5 53 3 9 3

a aAD AB AC AD

2 2

. 5 35 5

AM AB a AIAIADAM AB

Do

13 3 1615 2 153 1;1

2 119 3 285 2 15

AA

AA

xx

IA DI Ay

y

.


GV: Nguyn Tt Thu (0942444556) 17

Ta c 2 65 2 13 2 135

AI a AB .

M 37 4 30 4; 1;7 7

b bB b AB b

nn

2

2 230 4 412 13 1 52 5 26 123 0 , 37 5

bAB b b b b b

.

Do 0Bx nn 3 3;7b B .

Ta c 3 7;52BC BD C

.

I

D

M

C

A B

Cu 8. ng thng c 1;1;4u

l VTCP v i qua 1;3;0A .

Gi ; ;n a b c

vi 2 2 2 0a b c l VTPT ca mt phng P . Phng trnh P c dng :

2 0ax by cz b . V P nn . 0 4 0 4n u a b c a b c

.

Mt khc , ( ) 4 , ( ) 4d P d A P

Hay 2 2 22 2 2

3 24 4

a b bb c b c b c

a b c

2 2 2 2 2 22 2 8 17 10 16 0 2 , 8b bc c b bc c b bc c b c b c . +) 2b c ta chn 1 2, 2c b a . Phng trnh : 2 2 4 0P x y z . +) 8b c ta chn 1 8, 4c b a . Phng trnh : 4 8 16 0P x y z . Cu 9.

iu kin 3

nn

.

Ta c

3 ! 1 23 !n

nA n n nn

v

2 ! 12 !n

nA n nn

.

Do 3 2 3 22 100 1 2 2 1 100 100 0 5n nA A n n n n n n n n .

Khi 102 10

100

1 3 1 3 3n kk

ix x C x

.


GV: Nguyn Tt Thu (0942444556) 18

H s cha 5x l 5 510.3C .

Cu 10. Khng mt tnh tng qut, ta gi s 1min , , 2c a b c c .

Ta c 2 2 32 2 41 1 1

2

a b a b ca b a b ab

2 2 211 4 14

2 8

a b c c .

Suy ra 24 1 1

8

c cP

.

Xt hm s 2 3 24 1 1 4 4 1f c c c c c c vi 10; 2c

Ta c 2 1 1'( ) 12 8 1 '( ) 0 ,6 2

f c c c f c c c .

V 0

1 1 25lim ( ) 1,2 6 27c

f c f f

nn suy ra 25 1 0;27 2f c c

.

Suy ra 25216

P .

ng thc xy ra khi 2 2 2

51 1 6

3 5 14 36 6

1 1 16 6 6

aa b a b

a b c ab b

c c c

v cch hon v.

Vy 25min216

P .

S 3

Cu 1. 1) Bn c t lm 2) Ta c 2' 3 6 1 2y x m x m , '' 6 6(1 )y x m . Suy ra '(2) 10 , ''(2) 6 1y m y m . V hm s t cc tiu ti 2x nn ta c '(2) 0 0y m . Khi ''(2) 6 0y nn hm s t cc tiu ti 2x . Vy 0m l gi tr cn tm. Cu 2.

1) iu kin cos 0tan 1

xx

.

Ta c 2 sin cos1 sin 2 sin cos , 1 tancosx xx x x x

x

Nn phng trnh cho tng ng vi

2 3cos sin cos sin sin 2 12 4

x x x x x x k .


GV: Nguyn Tt Thu (0942444556) 19

Vy nghim ca phng trnh l ,4

x k k .

2) Gi ;M x y , ta c z x yi . Suy ra 1 1 1 , z i x y i iz y ix .

Do 2 2 2 21 1 1z i iz x y y x

2 2 2 22 1 2 1 1 0x x y y x y x y . Vy tp hp im M l ng thng 1 0x y . Cu 3. iu kin 1x . Bt phng trnh 2 2log 1 log 3 2 2 0x x 2 2log 4 1 log 3 2 4 4 3 2 2x x x x x . Vy nghim ca bt phng trnh l 1 2x . Cu 4. Ta c 3 2 3 23 10 5 11y y y x x x

3 23 2 3 2 2 3 2y y y x x x 2y x

(Do hm s 3 2( ) 3f t t t t l hm lin tc v ng bin trn ). Thay vo phng trnh th hai ta c

(4 3) 3 (3 4) 1 1 0m x m x m (1). H c nghim khi v ch khi (1) c nghim. iu kin : 3 1x . Phng trnh (1) (4 3 3 1 1) 3 3 4 1 1m x x x x

3 3 4 1 14 3 3 1 1

x x mx x

(2).

V 2 23 1 4x x nn ta c th t: 2

2

2

23 2111 21

txttxt

, vi 0 1t .

Khi (2) tr thnh: 2 2 2

2 2 212 8(1 ) 1 7 12 9 ( )16 6(1 ) 1 5 16 7

t t t t tm f tt t t t t

(3).

(1) c nghim x (3) c nghim [0;1]t . Xt hm s ( )f t vi [0;1]t . Ta thy f lin tc trn 0;1 v c

2

2 252 8 60'( ) 0 [0;1](5 16 7)

t tf t tt t

.

Suy ra 7 9(1) ( ) (0) [0;1]9 7

f f t f t .

Do phng trnh (1) c nghim 7 99 7

m .

Vy 7 99 7

m l nhng gi tr cn tm.


GV: Nguyn Tt Thu (0942444556) 20

Cu 5.

Ta c 21 cos2 2cos cos

sin 2 2sin cos sinx x x

x x x x

.

Suy ra 4

4

66

cos ln sin ln 2sin

xdxI xx

.

Cu 6.

Ta c tam gic ABC u cnh a v H l trung im AB nn 32

aHC .

V HC l hnh chiu ca SC ln mt phng ABCD nn 060SCH .

Suy ra 0 3. tan 602aSH HC . Li c

2 322ABCD ABC

aS S .

Th tch khi chp .S ABCD l: 2 31 3 3 3. .

3 2 2 4a a aV .

Gi M l giao im ca HO vi CD ta c O l trung im ca HM .

Suy ra 1, ( ) , ( )2

d O SCD d H SCD . K HK SC .

Ta c ( )CH CD CD SCH , suy ra ( )HK CD HK SCD .

Hay 2 2

. 3, ( )4

HC HS ad H SCD HKHC HS

.

Vy 3, ( )2ad O SCD .

MOH

B

AD

C

S

K

Cu 7. Gi D l giao im ca phn gic trong gc A vi ng trn I v 'I i xng vi I qua AD . Ta c phng trnh ' : 3 8 0II x y .

Gi 'K II AD , ta c

73 11 0 7 52: ;

2 23 8 0 52

xx yK Kx y y

.

V K l trung im 'II nn ' 3; 1I . Ta c 'AKI DKI nn 'AI ID , m AH ID nn 'I AH .


GV: Nguyn Tt Thu (0942444556) 21

' 4;0 0;1HI n

l VTPT ca AH . Phng trnh : 1 0AH y .

Ta im A l nghim ca h 3 11 0 81 0 1

x y xy y

, suy ra 8; 1A .

Phng trnh : 7 0BC x .

Phng trnh ng trn 2 2: 4 4 25ABC x y .

Ta ,B C l nghim ca h 2 2

7 0 70, 84 4 25

x xy yx y

.

Vy 8; 1A , 7; 8B , 7; 0C hoc 8; 1 , 7; 0 , 7; 8A B C .

KI'

D

H

I

A

B

C

Cu 8. Gi H l hnh chiu ca A ln d , suy ra 1 2 ; 1 ;H t t t . Ta c 2 ; 3; 3AH t t t

v 2;1; 1u

l VTCP ca d .

V AH d nn ta c . 0AH u

hay 4 3 3 0 1 3;0; 1t t t t H .

V mt cu tm A tip xc vi d nn ta c 2 3R AH . Phng trnh mt cu l:

2 2 21 2 3 12x y z . Cu 9. Mi kim tra c s cu d l 2 hoc 3, nn c cc trng hp sau: c 2 cu d, 2 cu trung bnh, 1 cu kh suy ra c 2 2 115 10 5. .C C C .

c 2 cu d, 1 cu trung bnh, 2 cu kh suy ra c 2 1 215 10 5. .C C C .

c 3 cu d, 1 cu trung bnh, 1 cu kh suy ra c 3 1 115 10 5. .C C C .

Vy tt c c: 2 2 115 10 5. .C C C + 2 1 215 10 5. .C C C +

3 1 115 10 5. .C C C = 23625 + 10500 + 22750 = 56875 .

Cu 10. p dng bt ng thc C si ta c 1 1 5 4 95 4

2 2 2x y x yP .

ng thc xy ra khi 1x y .


GV: Nguyn Tt Thu (0942444556) 22

Vi ;M x y ta t 5 4f M x y .

Ta thy min nghim ca h 3 2 52 1 3x yx y

l min c gii hn bi t gic li AOBC vi 10;

3A

,

1;1B v 5 ; 03

C

.

Ta c 4 25, 9, 0,3 3

f A f B f O f C . T y, suy ra 5 4 9x y .

Do 9 9 9

2P . Du = xy ra khi 1x y . Vy max 1P .

S 4 Cu 1. 1) Bn c t lm. 2) Ta c 3' 4 4y x x . Gi 0 0;M x y l tip im. Ta c 0 6y Hay 4 2 4 20 0 0 0 02 2 6 2 8 0 2x x x x x .

+) 0 02 '( ) 24x y x . Phng trnh tip tuyn 24 2 6 24 42y x x +) 0 02 ' 24x y x . Phng trnh tip tuyn 24 2 6 24 42y x x . Cu 2. 1) Chia c t v mu cho 3sin x ta c

2 3

21 cot cot 1 9 27 37

2.3.10 1 612cot 1 cot 1

x xPx x

.

2) Gi s z x yi 5 34

i

T gi thit, ta c: 2 2

2 2 255 53 2 . 3 44 4 2 3

x yz i x y xy i ixy

2 2 4 22 2

3 3 3 12 2 2 39 5 9 54 5 9 0 24 44 4

y y xyx x xyx x x x

x x

222 39 2 3 ,9 2 3 4 323 3

x x xx x z iy y

Vy, 2 3 5 25 125 155z z z Cu 3. Phng trnh x x x10.25 29.10 10.4 0


GV: Nguyn Tt Thu (0942444556) 23

2x x5 510 29 10 02 2

x

x

5 52 2

x 15 22 5

.

Cu 4. iu kin 2x . Bt phng trnh tng ng vi

2 2 22 1 2 2 2 2 2x x x x x x x (1). Ta c (1) 2 2 1 2 2VT x x x x . Mt khc

21 2 2 2 2 ( 1) 2 2 2 2 2 2x x x x x x x x x x x Suy ra

2(1) 2 2 2VT x x x . Du = xy ra khi 2x . Vy bt phng trnh cho c nghim 2x . Cu 5.

Ta c 11

2 22

00

4 1 4 12 ln 4 2 ln2 3 24

xI x dx x xx

Cu 6.

K AH BC , ta c 0' 60A HA v 2 2

. 32

AB AC aAHAB AC

.

Suy ra 0 3 3' . tan 60 . 32 2

a aAA AH . 21 3.

2 2ABCaS AB AC .

Th tch khi lng tr 33 3'.4ABC

aV AA S .

Gi N l trung im AC , ta c MN AB . Do , ' , 'AB B M MN B M .

Ta c 2

2 2 21 9 13, ' '2 2 4 2

a a aMN AB B M B B BM a .

22 2 2 2 23 7 ' ' 2

4 2a aBN AB AN a B N B B BN a .

p dng nh l c sin cho tam gic 'B MN ta c

2 2

22 2 213 4' ' 14 4cos '

2 . ' 13 132 .2 2

a a aMN B M B NNMBMN MB a a

.

Vy 1cos ' ,13

B M AB .


GV: Nguyn Tt Thu (0942444556) 24

NM

H

NM

C'

B'

AC

B

A'

A B

H

C

Cu 7.

D

EM

B C

A

Gi ;M m m . Ta c t gic ABDE ni tip ng trn tm M nn ta c MD ME

Hay 2 2

2 23 1 1 3 2 2; 25 5

m m m m m M

.

V tam gic ABC cn ti B nn E l trung im cnh AC . Ta c 3;1ME

, suy ra 1; 3n

l VTPT ca ng thng BC .

Phng trnh : 3 0BC x y Phng trnh : 3 2 0AD x y . Gi ; 2 3 2 ;4 3A a a C a a M C BC nn 2 3 4 3 0 1a a a . Suy ra 1;5 , 3;1 , 3; 1A C B . Cu 8. ng thng c 1; 1;2u

l VTCP

Mt phng c 2; 2;1n

l VTPT.

V d nm trong v vung gc vi nn d nhn vect 1 3;3; 0u u n

lm VTCP.

Gi ; 2 ; 1 2M d M M t t t .


GV: Nguyn Tt Thu (0942444556) 25

Mt khc M nn ta c 2 2 2 1 2 1 0 1 1;1;1t t t t M .

Phng trnh 1

: 11

x td y t

z

, t .

Cu 9. Gi s c 6 ch s l 1 2 3 4 5 6a a a a a a , s to thnh c 6 ch s 6 v tr. C 5 cch chn v tr cho ch s 0 (tr v tr u tin) C 5 cch chn v tr cho ch s 1 (tr v tr chn cho ch s 0) Ly 4 ch s trong 8 ch s cn li t vo 4 v tr cn li c 48A cch

Vy c 485.5.A 42000 s tha mn yu cu bi ton. Cu 10. Ta c 3 3 3 2 2 23a b c abc a b c a b c ab bc ca t 3t a b c , ta c

33 3 3abc t t ab bc ca

Suy ra 2 3 33 abcab bc ca t t t

Do 22 2 2 33 3 3 3 3 ( )27

t tabc tP t abc t abc t f tt t t t t

.

Kho st hm s ( )f t ta c 3 8f t f , suy ra 8P . ng thc xy ra khi 1a b c .

S 5 Cu 1. 1) Bn c t lm.

2) Ta c 2

3'1

yx

.

Gi 0 0;M x y l tip im. V tip tuyn song song vi ng thng 3 1y x nn ta c

2 00 0200

03' 3 3 1 121

xy x x

xx

.

+) 0 00 1x y . Phng trnh tip tuyn 3 1y x (loi)

+) 0 02 5x y . Phng trnh tip tuyn 3 2 5 3 11y x x . Vy phng trnh tip tuyn cn tm l 3 11y x . Cu 2.

1) Ta c sin sin 3 sin2cos cos3 cos2

x x xPx x x

sin 2 2 cos 12sin 2 cos sin 22cos2 cos cos2 cos2 2cos 1

x xx x xx x x x x

2 2

2 tan 2.3 3tan 241 tan 1 3

xxx

.


GV: Nguyn Tt Thu (0942444556) 26

2) t z x yi z x yi z 2 i x yi 2 i 2x y x 2y i Suy ra z 2 i z 3x y x y i . Nn t bi ta c 3x y 3 x 2

x y 5 y 3

.

Vy phn o ca z bng 3 . Cu 3. iu kin: x 1

Bt phng trnh 23 31 log (x 1) 2 log (x 1) 3 04

23 3 3log (x 1) 8 log (x 1) 12 0 2 log x 1 6 10 x 730 . Vy nghim ca phng trnh l 10 x 730 . Cu 4. iu kin 0x . Ta thy 0x suy ra 0y l nghim ca h. Xt 0xy .

H

2 22 2

4 4

2 2 2 23 3

2 22 1 0 2 1

y yx xy yx x

y yx xx y x y

.

t 22 , 0ya x b

y x ta c h

22

222

1 2 (1)2 3

1 2 2 3 (22 1

a ba b

b bb a

Ta c 4 2 3 2(2) 2 2 1 0 1 2 2 1 0 1b b b b b b b (do 0b ) suy ra 1a .

Do vy ta c

24

5

1

2 2 01

yx yxy yx

y

(*).

Xt hm s 5 2f y y y c 4'( ) 4 1 0f y y vi mi y nn hm s ( )f y ng bin

trn v 1 0f nn 1*1

yx

l nghim ca h cho.

Cu 5.

t 1lnu x du dxx

Ta c

1 1

2 20 0

1 222 2

uI du duuu u


GV: Nguyn Tt Thu (0942444556) 27

1

0

2ln 22

uu

2 3 1ln 3 ln 2 1 ln3 2 3

.

Cu 6.

Ta c 2 2 22aBC AB AC a BH .

V AH l hnh chiu vung gc ca 'AA ln mt phng ABC nn 'A AH l gc gia 'AA v mt phng ABC , hay 0' 60A AH . p dng nh l c sin cho tam gic ABH ta c

2 2 2 2 . .cosAH AB BH AB BH ABH 2 2

2 1 32. . .4 2 2 4a a aa a

Suy ra 32

aAH . Do 0 3' . tan 602aA H AH .

Din tch tam gic ABC l 21 3.

2 2ABCaS AB AC .

Th tch khi t din 'ABCB l 2 31 1 3 3 3' . . .

3 3 2 2 4ABCa a aV A H S .

Ta c ' ' 'AM ABB A CC nn , ' , ' ' 4 , ( ' ')d AM CC d C ABB A d H ABB A .

K , 'HE AB HK A E , ta c 1 34 4

aHE AC v

2 2 2 2

3 3.. ' 34 2, ( ' ')2 13' 3 9

16 4

a aHE HA ad H ABB A HK

EH A H a a

.

Vy 6, '13ad AM CC .

M

E

C'

B'

H

A C

B

A'

K

Cu 7.


GV: Nguyn Tt Thu (0942444556) 28

x

E

D

I

A

B

C

K tip tuyn Cx ca ng trn (C).

Ta c: 12

xCB BAC sBC .

Mt khc, t gic ABDE ni tip nn BAE EDC . Suy ra xCB EDC DE Cx M Cx IC DE CI . V 5; 0DE

v 1;1I nn ta c phng trnh : x 1 0CI . Suy ra 1;C c , thay ta ca C vo phng trnh (C) ta c:

21 25 4, 6c c c . Do C c tung dng nn 1; 6C . Ta c 1; 3DC

phng trnh : 3 3 0 ; 3 3DC x y B b b

B C nn 2 2 21 3 2 25 2 0 1, 2b b b b b b . Suy ra 2; 3B .

6;3EC

nn phng trnh : 2 11 0 2 11;EC x y A a a .

2 2 22 12 1 25 10 24 0 6, 4A C a a a a a a . Suy ra 3;4A . Vy 3;4A , 2; 3B , 1;6C . Cu 8. Gi I l tm ca mt cu (S), I d nn ; 2 ; 3 2I t t t . V (S) tip xc vi hai mt phng v nn ( , ( )) ( , ( ))d I d I

5 11 7 15 11 7 1 5, 1

3 3t t

t t t t

+) 1 1;1;1 , 2t I R . Phng trnh mt cu (S)

2 2 21 1 1 4x y z . +) 5 5;7;13 , 12t I R . Phng trnh mt cu (S)

2 2 25 7 13 144x y z . Cu 9. Ta c khng gian mu 421n C . Gi A l bin c ly c 4 vin bi c 3 mu. Ta c cc trng hp sau:


GV: Nguyn Tt Thu (0942444556) 29

+) 4 vin bi ly c c 2 xanh, 1 bi v 1 bi vng. S cch ly trong trng hp ny l: 2 1 15 7 9. .C C C . +) 4 vin bi ly c c 1 xanh, 2 bi v 1 bi vng. S cch ly trong trng hp ny l: 1 2 15 7 9. .C C C . +) 4 vin bi ly c c 1 xanh, 1 bi v 2 bi vng. S cch ly trong trng hp ny l: 1 1 25 7 9. .C C C .

Suy ra 2 1 1 1 2 1 1 1 25 7 9 5 7 9 5 7 9. . . . . . 2835n A C C C C C C C C C .

Vy n( ) 9( )( ) 19AP A

n

.

Cu 10. Xt hnh hp . ' ' ' 'ABCD A B C D v ba im , ,M N P nm trong khi lp phng c din tch S . Gi

1 2 3, ,S S S l din tch hnh chiu ca MNP ln cc mt ABCD , ' 'AA B B v ' 'ADD A v , , l gc gia mt MNP vi cc mt phng trn.

Gi n

l VTPT ca MNP , ta c 'n xAB yAD zAA

.

T , ta chng minh c 2 2 2cos cos cos 1

Suy ra 2 2 2 2 2 2 2 21 2 3cos cos cosS S S S S . Mt khc vi ba im nm trong mt hnh vung n v th din tch tam gic to bi ba im

khng vt qu 12

.

Do 21 1 1 3 34 4 4 4 2

S S .

S 6 Cu 1. 1) Bn c t lm 2) Ta c 2y ' 6x 6x .

Gi 0 0M x ;y l tip im. Do tip tuyn vung gc vi ng thng 1 1x 12y 1 0 y x12 12

nn ta c 20 0 0 0 0 01y ' x . 1 y ' x 12 x x 2 0 x 1, x 212

.

+) 0 0x 1 y 4 . Phng trnh tip tuyn l: y 12 x 1 4 12x 8 . +) 0 0x 2 y 5 . Phng trnh tip tuyn l: y 12 x 2 5 12x 19 . Cu 2.

1) Ta c 2

cos sin cos 1 sin 2 cos2sin cos 2x x x

P x xx x

21 1cos sin cos 2sin cos 2 cos 12 2x x x x x x . Vy P khng ph thuc vo x . 2) t z x yi , ta c iz y xi, 2i 1 z 2i 1 x yi 2y x 2x y i .


GV: Nguyn Tt Thu (0942444556) 30

Suy ra iz 2i 1 z y x 3x y i . Theo bi iz 2i 1 z 1 13i nn ta c

y x 1 x 3 z 3 4i z 53x y 13 y 4

.

Cu 3. Bt phng trnh 2log 3 log 2 0x x log 2log 1

xx

1000 10x

x

l nghim ca bt phng trnh cho.

Cu 4.

iu kin 0 x 2y 3

.

Nu y 0 th phng trnh th nht ca h v nghim, suy ra y 0 v do y 3 . Khi , phng trnh th hai ca h tng ng vi

3 2 3 23 3 3 6 8x x x y y y (*).

V hm s 3 2( ) 3 3f x x x x c 2'( ) 3 6 3 0 f x x x x nn ta suy ra 2 26 0;2f x f x (1).

Hm s 3 23 6 8g y y y y c 2' 3 6 6 0 g y y y y nn suy ra 3 26 3g y g y (2).

T (1) v (2), suy ra 2*3

xy

.

Vy h c nghim 23

xy

.

Cu 5. Ta c 3 2 2x 3x x 3 x 1 x 2x 3 t 2 1t x 2x 3 dt x 1 dx

2 .

i cn x 1 t 2, x 0 t 3 .

Khi 33 3

2 222 2

1 t 6 1 1 6 1 6 1 3I dt dt ln t ln 12 2 t 2 t 2 2t t

. Cu 6.

NH

A

B C

D

S

M

K

Gi H l trng tm tam gic ABC, suy ra SH (ABCD) . K MH vung gc vi AB, M thuc AB.


GV: Nguyn Tt Thu (0942444556) 31

Ta c SMH l gc gia hai mt phng SAB v ABCD , do 0SMH 60 . V HB 1

DB 3 nn 1 1 a 3 a 3MH d D,AB

3 3 2 6 , suy ra 0 aSH MH.tan60

2 .

Mt khc tam gic ABD u cnh a nn 2 2

ABCD ABDa 3 a 3S 2S 2.

4 2 .

Th tch khi chp S.ABCD l 2 3

ABCD1 1 a a 3 a 3V SH.S . .3 3 2 2 12

.

Ta c 3d B, (SCD) d H,(SCD)2

.

Gi N, K theo th t l hnh chiu ca H ln CD v SN, khi d H,(SCD) HK . V 2 2 a 3 a 3HN d B,CD

3 3 2 3 nn

2 2

SH.HN a 7HK7SH HN

.

Vy 3a 7d AB,SC14

.

Cu 7.

M

CD

A B

t 32 ,2xAD x AB x AM .

Ta c 2 2 2AD AM MD 2

2 2 2 29 6 4 16 44xx x x .

Gi ;A x y , ta c 2 2

1 5 1 5 0. 04 1 1 16

x x y yAD AMAD x y

2 2 2 2

2 24 6 0 4 6 0

3 2 7 02 2 14 0

x y x y x y x yx yx y x y

2

7 31 1;52

513 22 35 0

xy x Ay

x x

.

Li c 3AM MB

, suy ra 7;5B . Gi I l trung im BD , suy ra 3; 3I . Do I l trung im AC nn ta c 7;1C . Vy 1;5 , 7;5 , 7;1A B C . Cu 8. V C d nn C 5 2t; 2 t;5 3t . Theo bi 3td C, ( ) 2 2 t 2

3

+) t 2 C 9; 4;11 AB 2;4;3 , AC 9; 3;14 AB AC 65; 1; 42


GV: Nguyn Tt Thu (0942444556) 32

Suy ra n 65; 1; 42

l VTPT ca . Phng trnh : 65x y 42z 85 0 . +) t 2 C 1;0; 1 AB 2;4; 3 ,AC 1;1;1 AB AC 1;1; 2

Suy ra n 1;1; 2

l VTPT ca . Phng trnh : x y 2z 3 0 . Cu 9. S cch ly hai vin bi t hp l 212 66C S cch ly ra hai vin bi gm 1 vin mu xanh, 1 vin mu v khc s l 4.4 16 . S cch ly ra hai vin bi gm 1 vin mu xanh, 1 vin mu vng v khc s l 3.4 12 . S cch ly ra hai vin bi gm 1 vin mu , 1 vin mu vng v khc s l 3.3 9 . Nh vy s cch ly ra 2 vin bi t hp va khc mu va khc s l 16 12 9 37 .

Suy ra xc sut cn tnh l 37 0,5666

P

Cu 10.

Ta c: 2 2b c 2(b c )

Suy ra 2 2 2 2

2 22 2

a a b c 1 b cb c 22(b c )

Tng t: 2 2 2 2

2 22 2

b a b c 1 c ac a 22(c a )

; 2 2 2 2

2 22 2

c a b c 1 a ba b 22(a b )

Suy ra 2 2 2

2 2 2 2 2 2

2 2 2 2 2 2

a b c 1 1 1P2 a b b c c a

1 a b b c c a2

Ta c: 2 2 2 2 2 2 2 2 2 2 2 2

1 1 1 9

a b b c c a a b b c c a

V 2

2 2 2 2 2 2 2 2 21a b c a b b c c a6

Suy ra 2 2 2 2 2 21P a b b c c a2 2

t 32 2 2 2 2 2t a b b c c a 2 ab bc ca 3 2 abc Suy ra 31 t abc

54 2 nn t gi thit ta suy ra

2 2 2 2 2 2 37 abc 7 1t a b b c c a t1082 2

3 2t 108t 378 2 0 (t 3 2)(t 3 2t 126) 0 t 3 2 .

Suy ra 3 2 3P22 2

.


GV: Nguyn Tt Thu (0942444556) 33

ng thc xy ra khi v ch khi 3 3

a b ca b c

a b c 17 a3 2a a 6a 7 02

.

Vy 3minP2

t c khi v ch khi a b c 1 .

S 7

Cu 1. 1) Bn c t lm. 2) Gi 0 0;M x y l tip im. V tip tuyn song song vi ng thng 9 4y x nn ta c

0' 9y x . Hay 2 20 0 0 0 0 03 6 9 2 3 0 1, 3x x x x x x .

+) 0 01 5x y . Phng trnh tip tuyn 9 1 5 9 4y x x . +) 0 03 1x y . Phng trnh tip tuyn l : 9 3 1 9 28y x x . Vy phng trnh tip tuyn cn tm l 9 28y x . Cu 2. 1) Phng trnh tng ng vi

3 cos3 sin 3 2sin 2 sin 3 sin 23

x x x x x

23 2 23 15 5 ,

23 2 2 23 3

kx x k xk

x x k x k

l nghim ca phng trnh .

2) Tm s phc z t z x yi , ta c

1 2 1 2 1i z i iz i x y i i x yi 3 2 1x y x y i

Do 3 2 4 11 2 4 31 3 1

x y xi z i iz ix y y

.

Vy phn o ca s phc z bng 1 . Cu 3. Phng trnh 22 24 log 3log 1 0x x

2

2 4

2log 111log

4 2

xx

xx

l nghim ca phng trnh cho.

Cu 4. Ta thy 0y khng l nghim ca h. Do , h cho tng ng vi 3 3 2

3 2 23 8 0

3 3 6 12 0

x y x y

y xy y y

.

Tr hai phng trnh ca h ta c: 3 3 2 2 22 3 3 6 12 8 0x y x y xy y y

3 32 2 2 2x y y x y y x y . Thay vo phng trnh th hai ca h ta c: 2 4 2y y .


GV: Nguyn Tt Thu (0942444556) 34

Vy nghim ca h l ; 2;2 , 6; 2x y .

Cu 5. t 2 1 23 13 3

tt x x dx tdt .

i cn: 0 1, 1 2x t x t .

Suy ra 2

2 22

1 1

1 2. 23 3 19

t tdtI t dt

t

2

3

1

2 1 89 3 27

t t

.

Cu 6. Ta c 23

2 2ABCDAB AD BC aS

.

V AC l hnh chiu vung gc ca SC ln mt phng ABCD nn gc SCA l gc gia SC v mt phng ABCD , hay l 060SCA .

Li c 2 2 2AC AB BC a nn 0. tan 60 6SA AC a .

Th tch khi chp .S ABCD l: 31 6.

3 2ABCDaV SA S .

Ta c , ( ) , ( )AD BC AD ABC d D SBC d A SBC . K AH SB , do BC SAB BC AH AH SBC .

Suy ra 2 2

. 6, ( )7

AS AB ad A SBC AHAS AB

. Vy 6, ( ) 7d D SBC a .

AD

B

S

C

H

Cu 7.


GV: Nguyn Tt Thu (0942444556) 35

P

Q

NI

B C

A D

M

Ta c (0; 4); (3; 3); 4; 3 2AP AQ AP AQ

.

0. 1cos cos( , ) 45. 2

AP AQPAQ AP AQ PAQAP AQ

.

Gi P l giao im ca ng thng AM vi ng cho BD. T gic APND c 0' ' 45P AN P DN t gic APND ni tip. Do 0' 90AP N hay ' 'AM P N P P hay P thuc ng cho BD. Chng minh tng t ta cng c Q thuc ng cho BD.

(3;1)PQ

PT ng thng BD l 3 1 0x y PT ng thng AC l 3 7 0x y .

Gi I l tm hnh vung ta c ta I tha mn h 3 7 03 1 0

x yx y

. Suy ra 11 2;

5 5I

.

Do I l trung im ca AC nn 17 16;5 5

C

.

Cu 8. Ta c 1; 2; 3u

l VTCP ca d v 2; 1;2n

l VTPT ca . V . 6 0u n

nn d v ct nhau ti I . Ta c 1 ; 1 2 ; 3I t t t , li c I nn 2 1 1 2 6 3 0 1 2;1; 3t t t t I .

V I nn mt cu S ct theo giao tuyn l ng trn ln. Do , bn knh mt cu S l 3R .

Vy phng trnh mt cu S l: 2 2 22 1 3 9x y z . Cu 9. Ly bn th trong 9 th suy ra 49 126n C Gi A l bin c " Tch cc s trn th c rt ra l s chn " Suy ra A l bin c " Tch cc s trn th c rt ra l s l " Tch cc s trn th c rt ra l s l th ba th c rt ra phi c s u l.

Do 45 55 126n A C P A Vy ta c 5 1211 1 126 126P A P A . Cu 10.


GV: Nguyn Tt Thu (0942444556) 36

Ta c 2x 0;2 x x 2 0 x 2x . Tng t 2 2y 2y, z 2z . Suy ra 2 2 2x y z 2 x y z . Mt khc 3 33 3x y x y 3xy x y x y v z 2z nn 3 3 32 x y z 2 x y z . t t x y z 0 t 6 . Ta c 2 2 2 2

3 3 3

96P 5 x y z 4 x y z2 x y z

22 48 4 45t 8t 5 t 2 12 t 20 12.2 t. 20 28t t t

.

ng thc xy ra chng hn khi x 2y z 0

.

S 8

Cu 1. 1) Bn c t lm. 2) Phng trnh honh giao im ca C v d

1 2 1 2 1 1 11

x x m x m x xx

( 1x )

22 4 0x m x m (1). Yu cu bi ton c tha mn khi v ch khi (1) c hai nghim dng khc 1

Hay l

2 4 02 4 0

44 00 02

02

mm m

mm mS mmP

.

Vy 0m l nhng gi tr cn tm. Cu 2.

1) Ta c 2 2

2 22 cos sin 2sin sin 2 19

142sin cos sin 2sin 1

x x x xPx x x x

.

2) Gi ;M x y , suy ra z x yi v 3 4 3 4 , z i x y i iz y ix .

Do 2 2 2 23 4 3 4z i iz x y x y 6 8 25 0x y . Vy tp hp im M l ng thng 6 8 25 0x y .

Cu 3. Chi hai v bt phng trnh cho 9x ta c 24 25 4 3 4 4. 1 0 1 1

3 12 3 4 3 3

x x xx

.

Cu 4. iu kin: 2x .

Phng trnh 32 1 2 2 3. 2 12x x x

x

.

t 32 , 2 1a x b x ta c phng trnh


GV: Nguyn Tt Thu (0942444556) 37

33 2 3

22 3 2 3 0b a b a a b b

a

2 2 0 , 2a b a b a b b a . Mt khc 2 32 3a b . +) a b , ta c 2 32 3 1 2 1 1a a a x x . +) 2b a , ta c 2 3 3 22 8 3 8 2 3 0a a a a (1). V 0a nn p dng bt ng thc C si ta c

3 3 2 3 21 3 2 1 2a a a a a . Do , ta suy ra c (1) v nghim. Vy phng trnh cho c nghim duy nht 1x .

Cu 5. t 2 1 23 13 3

tt x x dx tdt .

i cn: 0 1, 1 2x t x t .

Ta c

222 2 3 2

1 1 1

1 2. 2 2 53 3 11 9 9 3 2 27

t tdt t tI t t dtt

.

Cu 6.

N

M

B'

C'A'

A C

B

H

V ng cao A 'H ca tam gic A 'B'C ' , ta c A 'H BMC v A 'B '.A 'C ' a 3A 'HB 'C ' 2

.

M 2MBC1 1S BC.BM .2a.a a2 2

.

Th tch khi t din BMCA ' l 3

2MBC

1 1 a 3 a 3V A 'H.S . .a3 3 2 6

.

Gi N l trung im CC, ta c MN BC nn gc gia hai ng thng AM v BC bng gc gia hai ng thng AM v MN. Ta c 2 2 2 2A 'M A 'B ' B 'M 2a , 2 2 2 2 2 2A 'N C'N A 'C ' a 3a 4a

Suy ra 2 2 2A 'N A 'M MN 2cosMA 'N2A 'N.A 'M 4 .


GV: Nguyn Tt Thu (0942444556) 38

Vy c sin ca gc gia hai ng thng A 'M v BC bng 24

.

Cu 7. Gi G l trng tm tam gic ABC v F l trung im AB , ta c

13

FE FG EG DCFD FC

M DC DK EG DK . Li c KG ED nn ta c G l trc tm tam gic EKD EK DG . M Q thuc BD nn phng trnh ca :BD 6 21 0.x y

Suy ra 21; ( 0).6

tD t t

.

Ta c 15 211; , 1; .6 6

t tDP t DK t

V DP DK

nn 315 211 1 0 1176 637

tt tt tt

3t (v

0t ) (3;4).D ng thng AC i qua D v P nn c phng trnh 2 11 0.x y ng thng AK i qua K v vung gc vi DE nn : 1 0.KA x Suy ra A(1; 5), kt hp vi D l trung im ca AC suy ra C(5; 3). ng thng BC i qua C v vung gc vi AK nn : 3 0.BC y V ( 3; 3).B DB BC B Vy (1;5), ( 3; 3), (5;3).A B C

G

EF

K

D

MB C

A

QP

Cu 8. V mt cu c tm A v tip xc vi mt phng nn mt cu c bn knh l

1 2 2 5 4R d A,( )

31 4 4

.

Vy phng trnh mt cu cn lp l: 2 2 2 16x 1 y 1 z 1 9 .

Gi M l tip im, suy ra IM nn phng trnh ca x 1 t

IM : y 1 2tz 1 2t

.

Suy ra M 1 t;1 2t; 1 2t .


GV: Nguyn Tt Thu (0942444556) 39

M M nn 41 t 2 1 2t 2 1 2t 5 0 t 9 .

Vy 13 17 1M ; ;9 9 9

.

Cu 9. C 50 cu v mi cu c 4 phng n tr li nn 50n 4 Gi A l bin c "Bn Nam c 5 im" Bn Nam c 5 im tc l tr li ng 25 cu v tr li sai 25 cu Chn 25 cu ng trong 50 cu c 2550C cch(v mt cu ng ch c mt phng n) Cn 25 cu cn li sai. Mi cu sai c ba phng n la chn nn s cch chn cho 25 cu cn li u sai l 253 , Theo quy tc nhn ta c 25 2550n A C .3

Vy ta c

25 25550

50

n A C .3P(A) 8,45.10

n 4.

Cu 10.

Ta c 2 2 2 21 1a b a b b a

2 21 1 0a a b b b b

2 2 2 2

2 2

1 10

1 1

a a b b a b

a b b a

2 2 2 21 0 1a b a b Do tn ti hai gc nhn ,x y sao cho cos , sina x b x .

Suy ra 2 29(1 tan )

28 tan 9 tan 8 tan2 4 2 2 1 tan

2

xx x xP

x

.

t tan 0;12xt t , ta c

2 9 18 ( )1

tP t f t

t

.

Xt hm s ( )f t vi 0;1t ta c:

3 2 2

2 2 2

2 8 16 8 9 2 2 1 4 10 918'( ) 161 1 1

t t t t t tf t t

t t t

.

Suy ra 1'( ) 02

f t t . Lp bng bin thin ta c 1( ) 5 52

f t f P

.

ng thc xy ra khi 2

2

1 tan1 3 42tan cos , cos sin2 2 5 51 tan

2

xx a x b y x

x

.

Vy min 5P .

S 9


GV: Nguyn Tt Thu (0942444556) 40

Cu 1. 1) Bn c t lm. 2) Phng trnh cho tng ng vi

3 26 9 1 12mx x x .

Phng trnh c ba nghim phn bit khi v ch khi ng thng 12my ct (C) ti 3 im phn

bit. Hay l:

1 1 3 0 82m m .

Cu 2. 1) Phng trnh 3 cos5x sin5x sin 3x sin 3x 3 cos5x sin 5x 2sin 3x

5x 3x k2

3sin 5x sin 3x3 5x 3x k2

3

kx

24 4 , k2x k3

.

2) Ta c 4 2(1 ) 3 4 1 21

ii z i i zi

(4 2 )(1 ) 2 6 1 3(1 )(1 ) 2

i i iz z z ii i

.

Cu 3. iu kin: 0x . Bt phng trnh cho tng ng vi

2 20,2 0,2log ( ) log ( 2) 2 2x x x x x x x (do 0x ).

Vy nghim ca bt phng trnh cho l 2x . Cu 4. iu kin 1x . H cho tng ng vi

2 23

2 212 2 (3 1) 24 3( 1)

4 2 4( 1) 6 2( 1) 1

x x x y

x y x y x

.

Tr hai phng trnh ca h theo v ta c

22312 2 3 1 18 3 3 1 1x x x x y x (*). Ta c 21x x x , do :

2 2(*) 11 3 3 7 11 3 10 8 (6 2) 8VT x x x x x x x

3 33. 8 . 6 2 .8 12. 2 3 1x x x x .

Suy ra 11 1 0(*)18 6 2 8

xy xyx x

.

Th li, ta thy nghim ca h l 11

xy

.


GV: Nguyn Tt Thu (0942444556) 41

Cu 5. Phng trnh honh giao im ca hai th

2x 2

3 x x 2 x 1, x 4x 5x 4 0

.

Vi 1 x 4 ta thy 3 x x 2 nn din tch hnh phng cn tnh l:

44 2

1 1

x 1 7S 3 x x 2 dx 2x x x 42 2 2

(vdt).

Cu 6.

HA C

B

S

K

Ta c 2 22 2a 3 a 21AH AC BH AH AB3 3 3

.

V HB l hnh chiu ca SB ln mt phng ABC nn 0SBH SB, (ABC) 60 .

Suy ra 0SH HB.tan60 a 7 , 2

ABCa 3S

2 .

Vy th tch khi chp S.ABC l: 2 3

ABC1 1 a 3 a 21V SH.S a 7.3 3 2 6

.

Ta c CA 3d C, SAB d H,(SAB) d H,(SAB)HA 2 . K HK SA , ta c AB AC, AB SH AB (SAC) HK AB .

M HK SA nn HK (SAB) hay 2 2

SH.AH 2a 7d H, (SAB) HK5SH AH

.

Vy 3a 7d C, (SAB) 5 . Cu 7. Gi K l trung im ca DH , khi NK l ng trung bnh ca tam gic DHC

1/ /2

NK CD

M 1/ /2

AB CD / /NK AB ABNK l hnh bnh hnh.

Ta c DK AN KNK AD

l trc tm ca tam gic ADN

/ /DN AK BN DNBN AK


GV: Nguyn Tt Thu (0942444556) 42

ng thng BN i qua 4; 2B v vung gc vi DN nn c phng trnh 3 14 0x y

Ta im N l nghim ca h

233 4 0 23 15 ;

5 53 14 0 15

xx y Nx y y

V N l trung im ca HC nn 7; 1C ng thng AC i qua C v H nn c pt : 2 5 0x y DH i qua H v vung gc vi AC nn c pt 2 3 0x y

Ta im D l nghim ca h 2 3 0 1 1; 13 4 0 1

x y x Dx y y

ng thng AD qua D v c 6;0DC

l VTPT nn c phng trnh:

6 1 0 1 0 1 0x y x Ta im A l nghim ca h 2 5 0 1 1;2

1 0 2x y x Ax y

.

Vy 1;2 , 7; 1 , 1; 1A C D . Cu 8. Gi () l mt phng qua A 1;0; 1 v () d. Ta c : da n 2; 2; 1

.

Suy ra phng trnh : 2x 2y z 3 0 . Hnh chiu A ln d l giao im I ca () v d.

A d A 1 2t; 1 2t; t

1A 2 2t 1 2 2t 1 t 3 0 t 3 . Vy 5 1 1I ; ;3 3 3

.

Cu 9. Ta c khng gian mu 421n C . Gi A l bin c ly c 4 vin bi c 3 mu. Ta c cc trng hp sau: +) 4 vin bi ly c c 2 xanh, 1 bi v 1 bi vng. S cch ly trong trng hp ny l: 2 1 15 7 9C .C .C . +) 4 vin bi ly c c 1 xanh, 2 bi v 1 bi vng. S cch ly trong trng hp ny l: 1 2 15 7 9C .C .C . +) 4 vin bi ly c c 1 xanh, 1 bi v 2 bi vng. S cch ly trong trng hp ny l: 1 1 25 7 9C .C .C .

Suy ra 2 1 1 1 2 1 1 1 25 7 9 5 7 9 5 7 9n A C .C .C C .C .C C .C .C 2835 . Vy n(A) 9P(A)

n( ) 19

.

Cu 10. t ,m ab bc ca n abc Do 2 2 20 2( ) 0 , 0a b c ab bc ca a b c m n Khi : (1 ) 12 8m m nP

n m

p dng bt ng thc C si ta c: 3 28 8 12m n n mn


GV: Nguyn Tt Thu (0942444556) 43

2 24 4m n mn Suy ra 3 2 212 8 16m m n n mn

Do : (1 ) 12 8 16m m nPn m

ng thc xy ra khi v ch khi 2, 1m n , tc l , ,a b c l ba nghim ca phng trnh

3 2 1 52 1 0 ( 1)( 1) 0 1,2

x x x x x x x

Vy min 16P t c khi , ,a b c l hon v ca b 1 5 1 51; ;2 2

.

S 10 Cu 1. 1) Bn c t lm. 2) Ta c 2' 3 2 1y x m x m . Hm s ng bin trn khi v ch khi ' 0 y x Hay 2 2 1 0 x m x m x

2 3 5' 3 1 02

m m m hoc 3 52

m .

Cu 2. 1) Phng trnh 2sin x 2sin x cos x 1 cos x 2sin x 1 cos x 1 cos x 0 2sin x 1 1 cos x 0

51 x k2 , x k2sin x , k6 62

cos x 1 x k2

.

2) t z x yi z x yi z 2z 3x yi Nn z 2z 2i 1 3x yi 2i 1 3x 2y 6x y i Suy ra z 2z 2i 1 4z x 2y 6x 5y i . M z 2z 2i 1 3z 38i 11 nn ta c x 2y 11 x 3 z 3 4i6x 5y 38 y 4

.

Suy ra 2w 1 2 3 4i 3 3 4i 14 80i w 2 1649 . Cu 3. iu kin: a 3b 0 Ta c: 2 log(a 3b) log 4 log a log b 2 2log(a 3b) log(4ab) (a 3b) 4ab

Do , ta c h : 2 25a 8b 37 5a 8b 37

(a b)(a 9b) 0a 10ab 9b 0

5a 8b 37 a 9

a 9b 0 b 1

.

Cu 4. iu kin y 1 .

Ta c 2 2 2 2x 1 x y y 1 1 x 1 x y y 1

2 2y x x 1 y 1 (1)


GV: Nguyn Tt Thu (0942444556) 44

22 2 2 2 2y x x 1 y 1 xy x 1 y 1 2 2 2 2 2 2

xy 0 xy 0x y x 1 y 1 y x 1

(2).

T (1) ta c 2 2

2 2 2 2

x y 2 1y xx 1 y 1 x 1 y 1

Suy ra 2 2 1x 1 y 1y x

. T y ta c 0 y x 1 .

Thay vo phng trnh th hai ta c: 2

1 8 y x 3 17y x

.

t t y x, t 0;1 , ta c phng trnh

21 8 t 3 17t

(3).

Xt hm s 21f (t) 8 t 3, t 0;1t

c

33 3

2 2t t 32 4f '(t)t 3t t t 3

.

Ta c 3 6f '(t) 0 2t t 3 0 4t t 3 0 5 4 3 2t 1 4t 4t 4t 4t 4t 3 0 t 1 . Suy ra f (t) f (1) 17 t 0;1 . Do (3) c nghim duy nht t 1 . Vy ta c

2 2y x 1 x 0

y 1y x 1

. Th li ta thy cp nghim ny tha h cho.

Vy nghim ca h l x 0y 1

.

Cu 5. Ta c th tch khi trn xoay cn tnh l:

2 2 2 22 2 2

0 0 0 0

V y dx (x cos x sin x)dx x cos xdx sin xdx

Ta c: 2 2 22

00 0

1 1 1sin xdx (1 cos2x)dx x sin 2x 2 2 2 4

.

t u x du dxdv cos xdx v sin x

2 220

0 0

x cos xdx x sin x sin xdx 12

Vy (3 4)V ( 1)2 4 4 (vtt).

Cu 6.


GV: Nguyn Tt Thu (0942444556) 45

C'

B'

H

A

B

C

A'

Tnh th tch khi chp A .ABC . Gi H l trung im BC.

Theo bi ra ta c A H (ABC) v 2 2 2 21 1 1AH BC AC AB a 3a a.2 2 2

Do 2 2 2 2A H A A AH 3a A H a 3.

Vy th tch ca khi chp A .ABC l 3

A .ABC ABC1 aV A H.S3 2

.

Tnh gc gia hai ng thng AA ' v B 'C ' . Do AA / /BB ,B C / /BC nn gc gia hai ng thng AA ' v B'C ' cng l gc gia hai ng thng BB v BC. p dng Pitago cho tam gic A B H ta c 2 2HB A B A H 2a. Suy ra tam gic B BH cn ti B ' . Do B BH l gc gia hai ng thng AA v B C .

Vy BH a 1cos .BB 2.2a 4

Cu 7. Gi I l giao im ca AC vi BM . Xt hai tam gic vung ADC v MCB c 0ADC MCB 90 v AD MC 1

DC CB 2 nn suy ra ADC MCB .

Dn n 0DCA CBM KMC MCI BMC CBM 90 hay BM AC . Phng trnh BM : x 3y 10 0 . Gi C c;12 3c MC c 4;14 3c

. ng thng AC c u 1; 3

l VTCP.

Ta c MC.uDC 2 2cos MC,u cosMCA AC 5 5MC. u

.

22

10c 46 2 5c 23 2 10c 92c 212510c 92c 212. 10

2 2 2 215c 23 20c 184c 424 5c 46c 105c 0 c 5, c5

.

V Cy 1 nn ta c c 5 C 5; 3 , MC 1; 1

.


GV: Nguyn Tt Thu (0942444556) 46

Phng trnh BC : x y 8 0 .

Ta im x y 8 0 x 7B : B 7; 1x 3y 10 0 y 1

.

Ta c D DD D

4 x 3 x 1DM 3MC D 1;1

2 y 3 y 1

.

Trung im J ca BD l J 4;0 . Do A i xng vi C qua J nn A 3;3 . Vy A 3;3 , B 7; 1 , C 5; 3 , D 1;1 .

I

M

B

CD

A

Cu 8. Ta c AB, AC (6; 2;3)

l mt VTPT ca mt phng (ABC)

Phng trnh ca (ABC) : 6x 2y 3z 3 0 . Gi H(x; y;z) l trc tm tam gic ABC

AH.BC 0

BH.AC 0H (ABC)

85x49x 2z 3 01353y 2z 7 0 y49

6x 2y 3z 3 0 31z49

85 135 31H ; ;49 49 49

.

V i qua H v vung gc vi (ABC) nn c phng trnh : 85 135 31x y z49 49 49

6 2 3

.

Cu 9. t n n0 1 nf (x) (1 2x) a a x ... a x

Suy ra n1 n0 na a 1a ... f 22 22

nn ta c n2 4096 hay n 12 .

Vi mi k 0,1, 2, ...,11 ta c k k k 1 k 1k 12 k 1 12a 2 C , a 2 C , nn kk 1

a k 1a 2(12 k)

Do kk 1

a1

a khi k 1 1

2(12 k)

hay 23k3

.

M k Z nn suy ra k 7 . Dn ti, 0 1 8a a ... a

Tng t k 8 9 12k 1

a1 k 7 a a ... a

a

S ln nht trong cc s 0 1 12a , a , ..., a l8 8

8 12a 2 C 126720 . Cu 10.

t , , 1, 1b c ax y z xyz za b c

. Suy ra 1xy .


GV: Nguyn Tt Thu (0942444556) 47

Khi 2 2

1 1 111 1

Pzx y

.

Vi 1xy ta chng minh c

2 2

1 1 211 1 xyx y

(*).

Tht vy

2 2 2 2

1 1 2 4(*)11 1 1 1 xyx y x y

(**).

Ta c

2

2 2 2 2

11 1 2 011 1 1 1 1

x y xyxyx y x y xy

do 1xy .

22 2 2 2 2 2 2 21 1 1 2 1 1x y x y x y x y xy xy Suy ra

2 2

2 211 1 xyx y

.

Do (**) ng.

Vy

2 1 2 1 ( )1 1 1

zP f zxy z z

Xt hm s ( )f z vi 1z ta c

3

1 2 122 1'(z)

12 1

z zz zzf

zz z

Suy ra '( ) 0 z 4f z .

Lp bng bin thin ta c 4 5f z f hay 5P .

ng thc xy ra khi

4

1 12

z

x yz

.

Vy max 5P .


GV: Nguyn Tt Thu (0942444556) 48

Documents

10 de thi thu 2015