TRNG I HC VINH TRNG THPT CHUYN

KHO ST CHT LNG LP 12, LN 3 - NM 2013 Mn: TON; Khi: A v A1; Thi gian lm bi: 180 pht

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

Cu 1 (2,0 im). Cho hm s 2231 24 += mxxy (1), vi m l tham s.

a) Kho st s bin thin v v th ca hm s (1) khi .34=m

b) Tm m th ca hm s (1) c ba im cc tr to thnh mt tam gic c tm ng trn ngoi tip trng gc ta O. Cu 2 (1,0 im). Gii phng trnh .2sin)cos2(sin2cos)cos1(3sin xxxxxx +=+ Cu 3 (1,0 im). Gii phng trnh ).5(212)2(3)12(4 322 xxxxxx +=++ Cu 4 (1,0 im). Tnh th tch khi trn xoay to thnh khi quay hnh phng gii hn bi cc ng

3,0,3

12

==++= xy

xxy xung quanh trc honh.

Cu 5 (1,0 im). Cho hnh lng tr '''. CBAABC c cc y l tam gic u cnh 3a. Hnh chiu vung gc ca 'C ln mt phng )(ABC l im D tha mn iu kin DBDC 2= . Gc gia ng thng 'AC v mt phng )'''( CBA bng .450 Tnh theo a th tch khi lng tr '''. CBAABC v tnh csin gc gia hai ng thng 'BB v AD. Cu 6 (1,0 im). Cho cc s thc khng m x, y, z tha mn 521211 2 =+++++ zyx . Tm gi tr ln nht ca biu thc .2 333 zyxP ++=

II. PHN RING (3,0 im) Th sinh ch c lm mt trong hai phn (phn a hoc phn b) a. Theo chng trnh Chun Cu 7.a (1,0 im). Trong mt phng vi h ta ,Oxy cho tam gic ABC c ),5;4( B phng trnh cc ng thng cha ng cao k t A v trung tuyn k t B ln lt l 073 = yx v .01 =++ yx Tm ta cc im A v C bit din tch tam gic ABC bng 16.

Cu 8.a (1,0 im). Trong khng gian vi h ta ,Oxyz cho cc ng thng ;2

311

1:1== zyxd

;1

12

21

:2== zyxd .

112

23:3

zyxd =+= Tm ta im P thuc 1d v im Q thuc 2d sao cho ng

thng PQ vung gc vi 3d v di PQ nh nht. Cu 9.a (1,0 im). Trong mt phng ta Oxy, tm tp hp im biu din cc s phc z tha mn

11 ++++

+z

izz

iz l s thun o.

b. Theo chng trnh Nng cao

Cu 7.b (1,0 im). Trong mt phng vi h ta ,Oxy cho im ).2;32(M Vit phng trnh chnh tc ca elp (E) i qua M bit rng M nhn hai tiu im ca (E) di mt gc vung.

Cu 8.b (1,0 im). Trong khng gian vi h ta ,Oxyz cho im )2;3;1(K v mt phng .03:)( =++ zyxP Vit phng trnh ng thng d i qua K, song song vi mt phng )(Oyz v to vi

(P) mt gc c .2tan = Cu 9.b (1,0 im). Gii h phng trnh ).,(

)log1(2)22(log)1(log)36(333.2

222

2

R

+=+++= +

yxyxyx

yxxyx

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

Ghi ch: 1. BTC s tr bi vo cc ngy 18, 19/5/2013. nhn c bi thi, th sinh phi np li phiu d thi cho BTC.

2. K kho st cht lng ln cui nm 2013 s c t chc vo chiu ngy 15 v ngy 16/6/2013. ng k d thi ti vn phng trng THPT Chuyn t ngy 18/5/2013.

1

TRNG I HC VINH TRNG THPT CHUYN

P N KHO ST CHT LNG LP 12, LN 3 - NM 2013 Mn: TON Khi A, A1; Thi gian lm bi: 180 pht

Cu p n im

a) (1,0 im)

Khi 34=m hm s tr thnh .2

38

31 24 += xxy

1o. Tp xc nh: R=D , y l hm s chn. 2o. S bin thin:

* Gii hn ti v cc: .)238

31(limlim 42

4 +=+= xxxy xx

* Chiu bin thin: Ta c .,3

1634' 3 R= xxxy

===

20

0'xx

y ; >

m Khi th hm s c 3 im cc tr l )32;3(),2;0( 2mmBA v ).32;3( 2mmC

0,5

Cu 1.

(2,0 im)

Tam gic ABC c tm ng trn ngoi tip trng gc ta O khi v ch khi OCOBOA ==

0)133)(1()32(32 222 =++= mmmmmm

===

.

6213

0,1

m

mm

Kt hp iu kin 0>m ta c gi tr ca m l .6

213,1 +== mm

0,5

Phng trnh cho tng ng vi xxxxxxxx cos2sin2sin2sin2coscos2cos3sin +=+

.0)1sin)(cossin(cos

sincos2cossin3sin)sin2sincos2(cos2cos3sin

=++=

+++=+

xxxxxxx

xxxxxxxx

0,5

Cu 2.

(1,0 im)

* .4

1tan0sincos kxxxx +===+

xO 2

y

2

310

2

x +0 0 0 0 +'y

y 3

10

+

2

2

2+

+

310

2

*

+=

=

+=+

+=+=

+=

.22

2

244

244

21

4cos1sincos

kx

kx

kx

kxxxx

Vy nghim ca phng trnh l kx +=4

, .,22

,2 Z+== kkxkx

0,5

iu kin: .21x

Phng trnh cho tng ng vi )254(212)2(3 23 += xxxxxx v

+==

+=

)1()12(2123

2)12)(2(212)2(3

2

2

xxxx

xxxxxxx

0,5

Cu 3.

(1,0 im)

Phng trnh (1) tng ng vi

02123)12(2 2 =+ xxxx .0212.312.2 2 =+ xx

xx (2)

t .0,12 = txxt Khi phng trnh (2) tr thnh

,210)2)(12(0232 2 ==+=+ ttttt .0t

Suy ra ,3240482 ==+ xxx tha mn iu kin. Vy nghim ca phng trnh l .324,324,2 =+== xxx

0,5

Ta c .103

12

==++ x

xx

Th tch V cn tnh l th tch khi trn xoay to thnh khi quay hnh thang cong gii hn bi

cc ng 3,1,0,3

12

===++= xxy

xxy xung quanh Ox.

Suy ra

+++=+

+=3

122

3

12

2

d3

23

21d3)1( x

xxxx

xxV

( ) .3

d2)3ln4(3

d2|3|ln3

12

3

12

1

32

++=+++= x

xx

xxx (1)

0,5

Cu 4.

(1,0 im)

Tnh .3

d3

12

+=

xxI t .tan3 tx = Khi 1=x th ,

6=t khi 3=x th

3=t v

.cos

d3d 2 ttx = Suy ra .

63d

31

cosd3.

)tan1(31

3d 3

6

3

6

22

3

12

==+=+=

t

tt

txxI

Thay vo (1) ta c .33)3ln4(

2 +=V

0,5

T gi thit ).(' ABCDC V )'''//()( CBAABC nn .45'))'''(,'())(,'( 0=== ADCCBAACABCAC

S dng nh l cosin cho 7aADABD = 745tan' 0 aADDC == .

Suy ra th tch lng tr

.4219

43)3(.7.' 3

2

aaaSDCV ABC ===

Cu 5.

(1,0 im)

0,5

'B

A C

B

D

'C

a3 045

'A

3

V '//' BBAA nn ADAADAAADBB '),'(),'( == (1) V )(' ABCDC nn ,16''''''')'''(' 2222 aACDCDAACDCCBADC =+=

.11''' 22 aDCDCCCAA =+== Suy ra .771

'.2'''cos222

=+=ADAA

DAADAAADA (2)

T (1) v (2) suy ra .771|'cos|),'cos( == ADAADBB

0,5

Vi hai s khng m a, b ta c .1111 baba ++++++ (1) Tht vy, babababa ++++++++++ 122)1)(1(22)1( ,11 baabba +++++ lun ng. Du ng thc xy ra khi 0=a hoc .0=b p dng (1) ta c zyxzyx 21211212115 22 ++++++++++= .2212 2 zyx ++++ Suy ra ,8222 ++ zyx hay .

24

2xzy + (2)

0,5

Cu 6.

(1,0 im)

Khi .2

42)(232

333

+++ xxzyxP (3)

Ch rng, t (2) v x, y, z khng m ta c .220 x Xt hm s

323

242)(

+= xxxf trn ].22;0[ Ta c

[ ].)16(2)12()2(43

2436)(' 22

222 xxxxxxxxxf +=

=

Vi ]22;0[x ta c ===

.20

0)('xx

xf

T 232)22(,24)2(,64)0( === fff suy ra ].22;0[,64)( xxf (4) T (3) v (4) ta c ,64P du ng thc xy ra khi 4,0 === zyx hoc .4,0 === yzx Vy gi tr ln nht ca P l 64, t c khi 4,0 === zyx hoc .4,0 === yzx

0,5

.073:pt =+ yxBCAHBC ),37;( ccCBCC ).;73( aaAAHA +

Suy ra trung im AC l .2

73;2

73

+++ cacaM

Do .082012

732

7301: =+=+++++=++ cacacayxBMM (1)

0,5

Cu 7.a (1,0 im)

Ta c10

|1410|),(,410)312()4( 22 +===+= aBCAdAHcccBC

.16|)75)(4(|.21 =+== acAHBCSABC (2)

T (1) v (2) suy ra

====

.573;

536,

52;

529

)1;2(),3;2(

536,

52

2,3

CA

CA

ca

ca

0,5

Cu 8.a (1,0 im)

)1;22;(),32,,1( 21 ++++ qqqQdQpppPdP . Suy ra ).22;22;1( ++++= qpqpqpPQ

0)22(1)22(1)1(20.33 =++++++= qpqpqpPQudPQ 02 =++ qp hay .2+= qp

0,5

A

B H C

M

4

Suy ra .2727)3(245122)6(9 22222 ++=++=+++= qqqqqPQ Suy ra 33min =PQ khi 3,1 == qp hay ).2;4;3(),1;1;0( QP 0,5 Gi s yixz += v im biu din s phc z l ).;( yxM Ta c .

)1()1(22)(2

1||2)(||2

11 2222

2

2

yxixxyx

zzzizzizzz

ziz

ziz

++++++=+++

+++++=++++

+ 0,5

Cu 9.a (1,0 im)

11 ++++

+z

izz

iz l s thun o

=+

+

++=++

).0;1();(41

21

0)1(

02)(2 22

22

22

yx

yxyx

xyx

Vy tp hp im M l ng trn 41

21 2

2

=+

+ yx b i im ).0;1( 0,5

Phng trnh chnh tc ca ).0(1:)( 22

2

2

>>=+ baby

axE

.1412)( 22 =+ baEM (1)

.1642190 2221

021 ===== baccFFMOMFF (2)

0,5

Cu 7.b (1,0 im)

T (1) v (2) suy ra .1824

:)(8

24 222

2

=+ =

= yxEba

0,5

Phng trnh .0:)( =xOyz Gi s d c vtcp ).0(),;;( 222 ++ cbacbaud Ta c 0.1

0.

)()2;3;1()//( =

= a

nu

OyzKOyzd

Oyzd

hay .0=a Suy ra );;0( cbud v ).1;1;1(Pn 0,5

Cu 8.b (1,0 im)

Do 00 900 nn t .32sin2tan == M

22.3||))(,sin(cb

cbPd+

+=

.00)(32

.3|| 2

22===

++ cbcb

cbcb

Chn ).1;1;0(1 === ducb Suy ra phng trnh

+=+=

=

.231

:tzty

xd

0,5

iu kin:

>+>>.01

0,1xy

yx

Phng trnh th nht ca h tng ng vi xxyyx 6.3333.2 21 +=+ ++ .1333)32(3)32( 11 +==+=+ ++ xyxyxxyx

Phng trnh th hai ca h tng ng vi .2)1)(1(2log)1)(1(log 2222 yxyxyxyx =++=++

0,5

Cu 9.b (1,0 im)

T ta c

=>+=

=+>+=

=++>+=

01

0121

012)1)(1(

0122 xx

xyyxy

xyyxyx

xy

==

+=+=

.2

53,2

512

53,2

51

yx

yx

0,5

TRNG I HC VINH TRNG THPT CHUYN

KHO ST CHT LNG LP 12, LN 3 - NM 2013 Mn: TON; Khi: D; Thi gian lm bi: 180 pht

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

Cu 1 (2,0 im). Cho hm s 2231 24 += mxxy (1), vi m l tham s.

a) Kho st s bin thin v v th ca hm s (1) khi 34=m .

b) Tm m th ca hm s (1) c ba im cc tr to thnh mt tam gic c tm ng trn ngoi tip trng gc ta O. Cu 2 (1,0 im). Gii phng trnh .2sin)cos2(sin2cos)cos1(3sin xxxxxx +=+ Cu 3 (1,0 im). Gii bt phng trnh ( ) ( )422 12323 xxxx

1

TRNG I HC VINH TRNG THPT CHUYN

P N KHO ST CHT LNG LP 12, LN 3 - NM 2013 Mn: TON Khi D; Thi gian lm bi: 180 pht

Cu p n im

a) (1,0 im)

Khi 34=m hm s tr thnh .2

38

31 24 += xxy

1o. Tp xc nh: R=D , y l hm s chn. 2o. S bin thin:

* Gii hn ti v cc: .)238

31(limlim 42

4 +=+= xxxy xx

* Chiu bin thin: Ta c .,3

1634' 3 R= xxxy

===

20

0'xx

y ; >

m Khi th hm s c 3 im cc tr l )32;3(),2;0( 2mmBA v ).32;3( 2mmC

0,5

Cu 1.

(2,0 im)

Tam gic ABC c tm ng trn ngoi tip trng gc ta O khi v ch khi OCOBOA ==

0)133)(1()32(32 222 =++= mmmmmm

===

.

6213

0,1

m

mm

Kt hp iu kin 0>m ta c gi tr ca m l .6

213,1 +== mm

0,5

Phng trnh cho tng ng vi xxxxxxxx cos2sin2sin2sin2coscos2cos3sin +=+

.0)1sin)(cossin(cos

sincos2cossin3sin)sin2sincos2(cos2cos3sin

=++=

+++=+

xxxxxxx

xxxxxxxx

0,5

Cu 2.

(1,0 im)

* .4

1tan0sincos kxxxx +===+

xO 2

y

2

310

2

x +0 0 0 0 +'y

y 3

10

+

2

2

2 +

+

310

2

*

+=

=

+=+

+=+=

+=

.22

2

244

244

21

4cos1sincos

kx

kx

kx

kxxxx

Vy nghim ca phng trnh l kx +=4

, .,22

,2 Z+== kkxkx

0,5

Bt phng trnh cho tng ng vi .0233)3(2 224

3

V '//' BBCC nn '.),'(),'( ACCACCCACBB == (1) Ta c 222222 14'',11'' aADDCACaDCDCCC =+==+=

.111

'.2'''cos222

=+=CCCA

ACCCCAACC (2)

T (1) v (2) suy ra .111'cos),'cos( == ACCACBB

0,5

T gi thit ta c )21)(1(22216 22 yxyx +++++= .21222221222 22222 yxyxyxyxyx +++++++++++= T ta c ,321 2 ++ yx hay .822 + yx Suy ra .220,

24

2

xxy

Khi 2++ x

yxP .2

22

12

22

222

42

+++=+++=+

+

xxx

xx

x

x

x (1)

0,5

Cu 6.

(1,0 im)

Xt hm s 2

22

1)( +++= xxxf trn ].22;0[ Ta c

].22;0[,0)2(2

4)(' 22

++= x

xxxxf

Suy ra .22)22()( = fxf (2) T (1) v (2) ta c ,22P du ng thc xy ra khi .0,22 == yx Vy gi tr ln nht ca P l ,22 t khi .0,22 == yx

0,5

.073:pt =+ yxBCAHBC ),37;( ccCBCC ).;73( aaAAHA +

Suy ra trung im AC l .2

73;2

73

+++ cacaM

Do .082012

732

7301: =+=+++++=++ cacacayxBMM (1)

0,5

Cu 7.a (1,0 im)

Ta c 10

|1410|),(,410)312()4( 22 +===+= aBCAdAHcccBC

.16|)75)(4(|.21 =+== acAHBCSABC (2)

T (1) v (2) suy ra

====

.573;

536,

52;

529

)1;2(),3;2(

536,

52

2,3

CA

CA

ca

ca

0,5

)1;22;(),32,,1( 21 ++++ qqqQdQpppPdP . Suy ra ).22;22;1( ++++= qpqpqpPQ

0)22(1)22(1)1(20.33 =++++++= qpqpqpPQudPQ 02 =++ qp hay .2+= qp

0,5

Cu 8.a (1,0 im)

Suy ra .2727)3(245122)6(9 22222 ++=++=+++= qqqqqPQ Suy ra 33min =PQ khi 3,1 == qp hay ).2;4;3(),1;1;0( QP

0,5

Cu 9.a

Gi s yixz += v im biu din s phc z l ).;( yxM Ta c .

)1()1(22)(2

1||2)(||2

11 2222

2

2

yxixxyx

zzzizzizzz

ziz

ziz

++++++=+++

+++++=++++

+ 0,5

A

B H C

M

4

(1,0 im)

11 ++++

+z

izz

iz l s thun o

=+

+

++=++

).0;1();(41

21

0)1(

02)(2 22

22

22

yx

yxyx

xyx

Vy tp hp im M l ng trn 41

21 2

2

=+

+ yx b i im ).0;1( 0,5

Phng trnh chnh tc ca ).0(1:)( 22

2

2

>>=+ baby

axE

.1412)( 22 =+ baEM (1)

.1642190 2221

021 ===== baccFFMOMFF (2)

0,5

Cu 7.b (1,0 im)

T (1) v (2) suy ra .1824

:)(824 22

2

2

=+ =

= yxEba

0,5

Phng trnh .0:)( =xOyz Gi s d c vtcp ).0(),;;( 222 ++ cbacbaud Ta c 0.1

0.

)()2;3;1()//( =

= a

nu

OyzKOyzd

Oyzd

hay .0=a Suy ra );;0( cbud v ).1;1;1(Pn 0,5

Cu 8.b (1,0 im)

Do 00 900 nn t .32sin2tan == M

22.3||))(,sin(cb

cbPd ++=

.00)(32

.3|| 2

22===+

+ cbcbcb

cb

Chn ).1;1;0(1 === ducb Suy ra phng trnh

+=+=

=

.231

:tzty

xd

0,5

iu kin:

>+>>.01

0,1xy

yx

Phng trnh th nht ca h tng ng vi xxyyx 6.3333.2 21 +=+ ++ .1333)32(3)32( 11 +==+=+ ++ xyxyxxyx

Phng trnh th hai ca h tng ng vi .2)1)(1(2log)1)(1(log 2222 yxyxyxyx =++=++

0,5

Cu 9.b (1,0 im)

T ta c

=>+=

=+>+=

=++>+=

01

0121

012)1)(1(

0122 xx

xyyxy

xyyxyx

xy

==

+=+=

.2

53,2

512

53,2

51

yx

yx

0,5