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SGD & T KHNH HOTRNG THPT L TTRNG

THI THI HC LN I - NM HC 2014-2015Mn: TON

Thi gian lm bi: 180 pht khng kthi gian giao

Cu I: (2) Cho hm s2 1

( )1

xy f x

x

= =

1. Kho st sbin thin v vth(C) ca hm s.2. Gi I l giao im ca hai ng tim cn ca th(C) , hy tm trn th(C) im M c

honh dng sao cho tip tuyn vi th(C) ti im M ct ng tim cn ng , tim cnngang ln lt ti A v B tha mn: 2IA2+ IB2= 12 .

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

1.(1 s inx)(2sin 2 6 cosx 2sinx 3)

22cos 1

x

x

+ + +=

+

2. 3 227 3 31log log ( 4) log ( 2)4x x x+ + =

Cu III: (1) Tnh tch phn :1 2

0

( )

2

x

x

x x eI dx

x e

+=

+

Cu IV: (1) Trong mt phng vi hta Oxy ,cho hai im A(1;2); B(4;1) v ng thngd: 3x-4y+5=0. Vit phng trnh ng trn (C) i qua A,B v ct d ti C, D sao cho CD = 6.

Cu V: (1) Trong mt chic hp c cha 6 vin bi , 5 vin bi vng v 4 vin bi trng. Lyngu nhin trong hp ra 4 vin bi. Tnh xc sut trong 4 vin bi ly ra khng c c3 mu.

Cu VI: (1) Cho hnh chp u S.ABCD c di cnh y bng a, mt bn ca hnh chp tovi mt y mt gc 600. Mp(P) cha AB v i qua trng tm G ca SAC ct SC , SD ln ltti M,N . Tnh thtch khi chp S.ABMN theo a

Cu VII: (1) Gii hphng trnh :3 2 3

3 2

6 13 10

2 5 3 3 10 6

x x x y y

x y x y x x y

+ = + +

+ + = +

Cu VIII: (1) Cho 0,, zyx v 3=++ zyx .Tm gi trnhnht ca biu thc:

xzzyyxP

+++

+++

++=

)1ln(24

1

)1ln(24

1

)1ln(24

1

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

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Trang 1

P N MN TON THI THLN I NM HC 2014-2015 - KHI 12Cu p n im

1. Kho st sbin thin v vth(C) : 2 1( )1

xy f x

x

= =

: (1)

+ Tp xc nh : D = R \ {1}

+( )

2

1' 0;

1y x D

x

= <

: Hm snghch bin trn cc khong ( ;1)

v (1; )+

0.25

+1 1

lim ; limx x

y y +

= = + : TC x = 1

+ lim 2x

y+

= : TCN y = 2 0.25

+ Bng bin thin:x - 1 + y' - -y 2 +

- 2

0.25

+ im c bit : (0;1) ;1

;02

+ th:

2

1

I

1

0.25

2. (1)

+ I(1;2) . Gi M(x0; 0

0

2 1

1

x

x

) (C) , x0> 0 ; 0 1x

+ Pttt vi (C) ti M : 002

0 0

2 11: ( )

( 1) 1

xd y x x

x x

= +

0.25

+ A l giao im ca d v TC A(1; 00

2)

1

x

x

+ B l giao im ca d v TCN B(2x0-1; 2)0.25

+ Tnh c IA2=

( )

2

0

4

1x

; IB2= 4(x0 1)2

+ 2IA2+ IB2= 12 2 4 20 0 02

0

2( 1) 3 ( 1) 3( 1) 2 0

( 1) x x x

x+ = + =

0 0

20 00

20 00

0 0

1 1 2

1 1 0 (loai)( 1) 1

1 2 1 2( 1) 2

1 2 1 2 (loai)

x x

x xx

x xx

x x

= =

= = = = = + =

= =

0.25

Cu I(2)

+ KL: Vy c 2 im cn tm: M1(2;3) ; M2(1+ 2 ; 2+2

2) 0.25

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Trang 2

1. Gii phng trnh: (1 s inx)(2sin 2 6cosx 2sinx 3) 22cos 1

x

x

+ + +=

+(1)

+ iu kin: 1 2cos 2 ;2 3

x x k k Z

+

+

0.25

(1) (1 s inx)(4sin cos 6cos 2sin 3) 22cos 1

x x x x

x

+ + + =

+

(1 s inx)(2 sin 3)(2 cosx 1)2

2cos 1

x

x

+ + =

+ 0.25

(1 s inx)(2sin 3) 2x + = 22sin sinx 1 0x + = 0.25

22

sinx 121

6sinx2 5

26

x k

x k

x k

= +=

= + = = +

tha mn iu kin

0.25

2. Gii phng trnh: 3 227 3 31

log log ( 4) log ( 2)4

x x x+ + = (1)

+ K : 0 2x< Vi K trn, (1) 3 3 3log log ( 4) log 2x x x + + =

[ ]3 3log ( 4) log 2x x x + = ( 4) 2x x x + =

0.25

2

( 4) 2

2

( 4) 2

x

x x x

x

x x x

>

+ =

+ + =

= = =

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Trang 3

C

D

D

O

I

B

I

Nhn xt A thuc d nn A trng vi C hay D . (GisA trng C)Gi I(a;b) l tm ng trn (C), bn knh R>0.(C) i qua A,B nn IA=IB=R

2 2 2 2(1 ) (2 ) (4 ) (1 )a b a b R + = + = 3 6b a =

0.25

Suy ra I(a;3a-6) v R = 210 50 65a a + (1)

Gi H l trung im CD IH CD v IH = d(I;d) =9 29

5

a +

R=IC= ( )2

2 2 9 29925

aCH IH

+ = + (2)

0.25

T(1) v (2) , c:2

10 50 65a a + =

( )2

9 29

9 25

a

+

2

113 56 43 0 43

13

a

a aa

= + = =

0.25

Cu IV(1)

+ a=1 (1; 3); 5I R = . Pt ng trn (C): (x-1)2+(y+3)2=25

+43

13a = 43 51 5 61( ; );

13 13 13I R = .

Pt ng trn (C):2 2

43 51 1525

13 13 169x y

+ =

0.25

Scch chn 4 vin bi bt ktrong hp : 415 1365C = cch 0.25

+ Chn 2 bi , 1 bi trng , 1 bi vng: 2 1 16 5 4. .C C C

+ Chn 1 bi , 2 bi trng , 1 bi vng: 1 2 16 5 4. .C C C

+Chn 1 bi , 1 bi trng , 2 bi vng: 1 1 26 5 4. .C C C

Scch chn 4 vin bi c 3 mu :2 1 16 5 4. .C C C +

1 2 16 5 4. .C C C +

1 1 26 5 4. .C C C = 720 cch

0.25

Scch chn 4 vin bi khng c3 mu : 1365-720 = 645 cch 0.25

Cu V(1)

Xc sut cn tm : P =645 43

1365 91=

0.25

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Trang 4

JI

N

MG

O

C

AD

B

S

Gi O l giao im ca AC v BDS.ABCD l hnh chp tgic u nn SO (ABCD)Gi I, J ln lt l trung im ca AB v CD , xc nh c gc gia mt bn

(SCD) v mt y (ABCD) l 060SJI = .

0.25

Nhn xt SIJ u ; SO =3

2

a; VS.ABCD =

31 3.

3 6ABCDa

SO S = (vtt) 0.25

Trong (SAC) , AG ct SC ti M , M l trung im ca SCC/minh c MN// AB v N l trung im ca SD

.

1 1

2 4

SABMSABM S ABCD

SABC

V SMV V

V SC

= = =

.

1 1

4 8SAMN

SAMN S ABCD

SACD

V SM SN V V

V SC SD= = =

0.25

Cu VI(1)

3

. . . .

3 3

8 16S ABMN S ABM S AMN S ABCDa

V V V V = + = = (vtt)0.25

Gii hphng trnh :3 2 3

3 2

6 13 10 (1)

2 5 3 3 10 6 (2)

x x x y y

x y x y x x y

+ = + +

+ + = +

(1)

3 3(1) ( 2) ( 2)x x y y + = + Xt hm sf(t) = t3+t , t R c f (t) = 3t2+1>0, t R f(t) ng bin trn R v (1) 2x y = (3)

0.25

Thay (3) vo (2): 3 2 53 3 5 2 3 10 26 (4); 12

x x x x x x+ = + 0.25

+ Chng minh g(x) = 3 3 5 2x x+ ng bin trn on5

1;2

+ Chng minh h(x) = 3 23 10 26x x x + nghch bin trn on5

1;2

g(2) = h(2) = 2 x=2 l nghim duy nht ca pt (4)

0.25

Cu VII(1)

p s ( ) ( ); 2;0x y = 0.25

Cho 0,, zyx v 3=++ zyx .Tm gi trnhnht ca biu thc:

xzzyyxP ++++++++= )1ln(241

)1ln(241

)1ln(241

Cu VIII(1)

Vi cba ,, >0,p dng bt ng thc Csi ta c :1 1 1 1 1 1 9

( ) 9a b ca b c a b c a b c

+ + + + + +

+ + (1)

Du = xy ra cba ==

p dng (1) ta czzyyxx

P++++++

)1ln(2)1ln(2)1ln(212

9

0.25

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Trang 5

Xt [ ]3;0,)1ln(2)( += ttttf

t

t

ttf

+

=

+=

1

11

1

2)(' ; 10)(' == ttf

32ln4)3(,14ln)1(,0)0( === fff 14ln)(32ln4 tf

0.25

12ln 2 9 ( ) ( ) ( ) 3ln 4 3

12ln 2 3 ( ) ( ) ( ) 12 9 3ln 4

f x f y f z

f x f y f z

+ +

+ + + + +

9 9 3

12 ( ) ( ) ( ) 9 3ln 4 3 ln 4P

f x f y f z =

+ + + + +

Vy 3 13 ln 4

MinP x y z= = = =+

0.25

Mi cch gii khc ng ca hs u cho im tng ng vi mi phn ca cu