Trng THPT Chuyn H Tnh

Nm hc 2013-2014 THI TH I HC LN 3 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: 1) Kho st v v th (C) ca hm s: y=x4-2x2-3. 2) Tm m phng trnh sau c 4 nghim phn bit: 2x2-x4+log2(m

2-m+1)=0

Cu 2: Gii phng trnh: xtan2 =x

x

cos

)4

(sin2 2

Cu 3: Gii h phng trnh:

2)(

1

31

2

2

22 xyyx

yx

yyx

x

Cu 4:Tnh tch phn: I= 2

022cossin3

2sin2cos3

dxxx

xx

Cu 5: Cho hnh lng tr ABC.A'B'C' c y ABC cn ti A, AB=a v gc BAC=120o, hnh chiu vung gc ca B' ln (ABC) l H trung im ca cnh BC. Bit gc gia AB' v mt phng (A'B'C') l 60o. Tnh th tch khi lng tr cho, gc gia B'C v mt phng (ABA'). Cu 6: Cho cba ,, phn bit tha mn: 1 cba v 0 cabcab

Tm gi tr nh nht ca biu thc sau: cabcabaccbba

P

5222

II. PHN RING(3,0 im):Th sinh ch c lm phn A hoc phn B. A. Theo chng trnh Chun: Cu 7a:Trong mt phng vi h ta Oxy cho hai ng trn: x2 + y2 - 4x - 4y +4 = 0(C) v x2 + y2 - 16x +8y + 28 = 0(C'). Vit phng trnh ng thng qua A(4;2) ct cc ng trn trn theo cc dy cung c di bng nhau. Cu 8a:Trong khng gian Oxyz hy lp phng trnh mt phng (P) song song vi mt phng

(Q): x +2y + z - 1= 0, ct mt cu (S): x2+y2+z2-2x-2y-2z-6=0 theo ng trn c chu vi l 30 .

Cu 9a: Cho s phc z c phn thc dng tha mn: 5z v 432 iz . Tnh: A=2

113

z

z

B. Theo chng trnh Nng cao: Cu 7b: Trong mt phng vi h ta Oxy cho ng trn: x2 + y2 - 2x - 2y -2= 0(C) v ng thng : x+y+4=0. M thay i trn , MA v MB l cc tip tuyn k t M n (C) (A, B l cc tip im). Tm C c nh m ng thng AB lun i qua khi M thay i trn .

Cu 8b: Trong khng gian Oxyz cho : 43

1

2

1 zyx

v A(2;3;1). Vit phng trnh ng thng d qua A,

ct ti B c ta l nhng s nguyn v to vi gc c c-sin bng 58

6.

Cu 9b: Tm m ng thng y=2x+m ( ) ct th y=1

2

x

x(C) ti A, B phn bit sao cho AB c di ngn

nht. Ch :Mi cu nh 1 im, th sinh khng dng ti liu v gim th khng gii thch g thm.

TRNG THPT CHUYN

H TNH

K THI TH I HC LN 3 NM 2014 HNG DN CHM

Mn: TON; Khi: A&A1 Cu Ni dung

* Tp xc nh: R * S bin thin: +) y'=4x3-4x=0 x=0 hoc x= 1 Hm s b trn: (-1;0), (1;+ ). Hm s nb trn: (- ;-1), (0;1) im c: (0;-3). im ct: ( 1;-4) +) Gii hn:

x

Limy , x

Limy .

+) Bng bin thin:

0,5

0,25

0,25

Cu1.1

* th: im ct Ox: ( 3 ;0).

im un: ( )9

32;

3

1

8

6

4

2

2

4

6

15 10 5 5 10 15

f x( ) = x4 2x2 3

+) Pt tng ng: x4-2x2-3=log2(m2-m+1)-3. Yu cu bi ton tha mn (C) ct t y=

log2(m2-m+1)-3 ti 4 im phn bit

0,25

+) Cn c (C) ta phi c: -4

+)Khi pt cho (sinx+1)(2cosx-1)=0

23

22

kx

kx

0,5

+)i chiu vi /k ta c nghim:

23

kx . 0,25

+) iu kin: x+y 0. H

3)1

)(()1

()(

3)1

()(

22

yxxyx

yxxyx

yxxyx

0,25

+) t : a=x-y, b=x+yx

1 . Gii h:

3

322 abba

ba

Ta c: a=1, b=2 hoc a=2, b=1. 0,25

+) Vi a=1, b=2: Gii ra c cc nghim: (1;0), (2

1;

2

3) 0,25

Cu 3

+) Vi a=2, b=1: H v nghim.

Vy h c 2 nghim (x ;y): (1;0), (2

1;

2

3). 0,25

Cu 4 +) t t= sinx, a c v: I=

1

0

2 132

43dt

tt

t 0,5

+) I =

1

01

1dt

t +

1

012

2dt

t = ln6 0,5

N

C'

B'

A'

H

C

B

A

+) Gc gia AB' v (A'B'C') l B'AH (do (ABC) // (A'B'C') ).

AH = 0,5a suy ra: B'H = 2

3a.

0.25

Cu 5

+) dt(ABC) = 4

32a, VABC.A'B'C' =B'H.dt(ABC) =

8

3 3a 0,25

+) Gi E l hnh chiu C ln (ABA') suy ra: =CB'E l gc cn tm.

Ta c: sin =CE/CB' , CE=d(C;(ABA'))=2.d(H;(ABA')), do H trung im ca BC.

0.25

+) K HN AB, HK NB', cm c: KH= d(H;(ABA')).

Tnh c: sin =5

2 . Vy =arcsin

5

2

0,25

+) Khng mt tq gi s a>b>c ta c: P=cabcabcacbba

5222.

Ta c: m, n>0 th 22

22411

nmnmnm

(1), du "=" c khi v ch khi m=n.

p dng (1): P= cabcabcacabcabcacbba

2

101052)

11(2

)31)(1(

220

)4)((

220

bbbcaca

(2)

0.5

Cu 6

+) Mt khc: 3(1-b)(1+3b)=(3-3b)(1+3b) 4)3133(4

1 2 bb suy ra:

3

2)31)(1( bb . Kt hp (2) ta c: P 610 . Du "=" c khi v ch khi:

)(1

3133

2

cbacba

bb

cabcabca

cbba

6

62

3

1

6

62

c

b

a

. Vy: minP = 610 .

0,5

+) I(2;2) tm (C), R=2. I'(8;-4) l tm (C'), R'=2 13 A(4;2), B(2;0) l cc giao im ca (C) v (C'). ng thng qua A, B c pt: x-y-2=0 l mt trng hp cn tm.

0,5

Cu7a

K

F

E

N

M

B

A

I'I

+) Gi s qua A, ct (C) v (C') ti M, N (khc B) v AM=AN. Gi K,E,F l trung im II' , AM, AN suy ra qua A v vung gc KA. Pt l: x-3y+2=0. Kt lun: C 2 ng thng tha mn YCBT l: x-y-2=0, x-3y+2=0.

0,5

+) (S) c tm I(1;1;1) v bn knh R=3. ng trn do (P)ct (S) c chu vi

30 c bn knh r=2

15.

0,25

+) Khong cch t I n (P) l: d=2

322 rR . 0,25

Cu8a

+) Mt phng (P) do song song (Q) nn c dng: x+2y+z+m=0 (P).

d=2

3

6

4

m

7

1

m

m

0,25

+) Gi tr m=-1 b loi do (P) trng (Q). Vy phng trnh mt phng cn tm l: x+2y+z-7 = 0.

0,25

+) Gi z=a+bi, a v b l cc s thc. Theo gt ta c:

16)3()2(

522

22

ba

ba 0,25

+) Gii h trn c:

1

2

b

a(tha mn) ,

13

19

13

22

b

a

(loi) 0,25

Cu9a

+) Vy z=2+i suy ra: A= i2713 = 898 . 0,5

+) I(1;1), R=2, d(I; )= 2

6R suy ra v (C) khng c im chung. 0,25

+)Gi M(m;-m-4) . Tnh c: MA2=MB2=2m2+8m+22. ng trn (C') tm M bn knh MA c pt: (x-m)2+(y+m+4)2=2m2+8m+22. 0,25

+)Ly pt (C) v pt (C') tr v theo v c pt ng thng qua AB l: (m-1)x-(m+5)y+2=0 (AB). 0,25

Cu7b

I

B

A

M

+) Gi C(x0;y0) l im m AB lun i qua, ta c: (m-1)x0-(m+5)y0+2=0 vi

mi m. Suy ra: C(3

1;

3

1)

0,25

+) Gi B(1+2t;-1+3t;4t) . Ta c:

u =(2;3;4) l VTVP ca v

AB =(2t-1;3t-4;4t-1) . 0.25

+) Gi l gc gia d v , theo gt ta c: cos =183629.29

1829

58

62

tt

t.

Gii ra ta ch nhn t=0 do ta B nguyn. Vy B(1;-1;0) 0,5

Cu8b

+) ng thng cn tm qua A, B nn c phng trnh:

1

1

4

3

1

2

zyx (d) 0.25

+) ct (C) ti A, B phn bit khi v ch khi pt: x2+(m-2)x-m=0 c 2 nghim phn bit khc 1 mR

0.5

Cu9b

+) Gi A(x1;y1), B(x2;y2). Suy ra: AB2=5m2+20 20. Du "=" c khi v ch khi

m=0. Kt lun: m=0. 0.5