2004年高考理科数学(全国二卷)真题
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2004 ()
12 5 60
1 M{x|x24} N{x|x22x30} MN
A{x|x2}
B{x|x3}
C{x|1x2}
D{x|2x3}
2 lim x 2 x 2
n1 x 2 4x 5
A
1 2
B1
C
2 5
D
1 4
3
1 2
32i 1
A�
B 2
C 1
D 1 2
4 C (x1)2y21 yx C
5
5
()tanA2tanB
() AB3 AB
- 3-
18( 12 ) 8 3 8 AB
4 ()AB ()A
19( 12 ) {an} n Sn a11an1 n 2 Sn n123... n
(){ Sn } n ()Sn14an
- 4-
20( 12 ) ABC-A1B1C1 ACB90oAC1CB 2 AA11 AA1B1B DB1C1 M
n
Sn 1
n 1 Sn
2
(n=1,2,3...).{
Sn n
}
1
2
n
III Sn1 4 Sn1 (n 2) Sn+1=4(n+1)� Sn1 =4an(n 2 )
n 1 n1
n1
a2=3S1=3, S2=a1+a2=4=4a1, n1 Sn+1=4an.
20(I) CA1AC1CM CA1= 2 CB=CA1= 2 CBA1
2
C D2
12 12345 5 23145
43521
A56
B57
C58
D60
4 4 16 13 3 2 2
0
1
2
P
14 xy
- 2-
x 0 x y
2x y 1
z3x2y
15 2x22y21
16
()
6 74
17 ( 12 )
ABC sin(AB) 3 sin(AB) 1
A
A
D
A' A'
D A1B CDA1B A1C1=1C1B1= 2 A1B1= 3 BB1=1A1B=2
C
B
D
C
B
C'
M B' C' M B'
A1CB D A1B CD= 1 A1B=1CD=CC1
2
DM= 1 AC1= 2 DM=C1MCDNCC1MCDM=CC1M=90�,
2
2
CDDM
B1F2=B1B2+BF2=1+( 2 )2= 3 .
2
2
cosB1GF=
B G2 1
FG2
B F2 1
( 3 )2 ( )12
2
2
3 2
3
2B1G FG
2 3 1
3
22
-arccos 3
3
C
(I):B( 2 ,0,0),B1( 2 ,1,0),A1(0,1,1),D( 2 , 1 , 1 ),
2 tan 2 B 4 tan B 1 0
tan B 2 6 , B tan B 2 6 , tan A 2 tan B =2+ 6
2
2
AB CD AB=AD+DB= CD CD 3CD AB=3 tan A tan B 2 6
CD=2+ 6
- 7-
AB 2+ 6
3 5 1 5
sin Acos cos Asin
B B
2
5 1
5
tan A tan B
2 tan A 2 tan B .
(II) <A+B< , sin( A B) 3 , cos( A B) 4 , tan( A B) 3
2
5
5
4
tan A tan B 3 , tan A 2 tan B 1 tan A tan B 4
D 6 3
8 A(12) 1 B(31) 2
A1
B2
C3
D4
9
l
e
(
4
,
3)
O(0,0)
A(1,-2)
l
55
O1 A1 O1A1 e
A 11 5
B 11 5
C2
D2
10 yxcosxsinx
A( 3) B(2)
22
C( 35)
22
D(23)
11 ysin4xcos2x
A
4
B
()CD BDM () B1BD CBD
21( 12 ) Cy24xF C F l C AB
() l 1 OA OB () FB AF [49] l y
- 5-
(22)( 14 ) f(x)ln(1x)xg(x)xlnx (1) f(x) (2) 0ab0g(a)g(b)2g( a b )(ba)ln2 2
18(I) 2
C2C 2 3 C84 5
6 7
(II)A
C2 C 2 35 C84
3C5
3C1
C84
1 2
19I a1=1,an+1= n 2 Sn(n=1,2,3...)
n
S2
a2= 2 1 S1=3a1, S2 4a1 2 , S1 1,
1
22
1
22 S1
1
an+1=Sn+1-Sn(n=1,2,3...), Sn+1-Sn= n 2 Sn(n=1,2,3...)nSn+1=2(n+1)Sn,
A(x1)2y21
Bx2y21
Cx2(y1)21
Dx2(y1)21
5 ytan(2x)( ,0)
12
A
B
C
6
6
12
D
12
6 yex
A. yex y B. yex
- 1-
C. yex y
D. yex
7 O 1ABC
O ABC
2
A 1 3
B 3 3
C 2 3
A1BDM BDM CD BDM
(II) FG BCBD B1GFGB1F
- 8-
FGCDFG= 1 CD FG= 1 FGBD.
2
2
A
A'
BB1A1A D,
BD=B1D= 1 A1B=1
2
C
D
FG
C' M
BB1D 1 B1GBD B
B'
B1G=
3
2
B1GF
2 , 1 , 1 )
- 6-
2004 ( )
来自百度文库2 5 60
1C
2A
3C
4C
5A
D
7B
8B
9D
10B
11B
C
6 12
4 4 16
130.10.60.3
145
15 1 x2y21 2
17(I)sin(A+B)= 3 ,sin(A-B)= 1
5
5
16
sin
Acos
B
cos
Asin
B
sin Acos B cos Asin B
2 22
M(
2 ,1,0),
2
CD
(
2
,
1 2
,
1 ), A
22
B
1(
2
,-1,-1),
DM
(0,
1 ,-
2
1 ),
2
CD
A1B
0,CD
DM
0,
CDA1B,CDDM. A1BDM BDM
CD BDM
(II): BD G B1G ( 3 2 1, ,1), BD (-
G
4 44
12 5 60
1 M{x|x24} N{x|x22x30} MN
A{x|x2}
B{x|x3}
C{x|1x2}
D{x|2x3}
2 lim x 2 x 2
n1 x 2 4x 5
A
1 2
B1
C
2 5
D
1 4
3
1 2
32i 1
A�
B 2
C 1
D 1 2
4 C (x1)2y21 yx C
5
5
()tanA2tanB
() AB3 AB
- 3-
18( 12 ) 8 3 8 AB
4 ()AB ()A
19( 12 ) {an} n Sn a11an1 n 2 Sn n123... n
(){ Sn } n ()Sn14an
- 4-
20( 12 ) ABC-A1B1C1 ACB90oAC1CB 2 AA11 AA1B1B DB1C1 M
n
Sn 1
n 1 Sn
2
(n=1,2,3...).{
Sn n
}
1
2
n
III Sn1 4 Sn1 (n 2) Sn+1=4(n+1)� Sn1 =4an(n 2 )
n 1 n1
n1
a2=3S1=3, S2=a1+a2=4=4a1, n1 Sn+1=4an.
20(I) CA1AC1CM CA1= 2 CB=CA1= 2 CBA1
2
C D2
12 12345 5 23145
43521
A56
B57
C58
D60
4 4 16 13 3 2 2
0
1
2
P
14 xy
- 2-
x 0 x y
2x y 1
z3x2y
15 2x22y21
16
()
6 74
17 ( 12 )
ABC sin(AB) 3 sin(AB) 1
A
A
D
A' A'
D A1B CDA1B A1C1=1C1B1= 2 A1B1= 3 BB1=1A1B=2
C
B
D
C
B
C'
M B' C' M B'
A1CB D A1B CD= 1 A1B=1CD=CC1
2
DM= 1 AC1= 2 DM=C1MCDNCC1MCDM=CC1M=90�,
2
2
CDDM
B1F2=B1B2+BF2=1+( 2 )2= 3 .
2
2
cosB1GF=
B G2 1
FG2
B F2 1
( 3 )2 ( )12
2
2
3 2
3
2B1G FG
2 3 1
3
22
-arccos 3
3
C
(I):B( 2 ,0,0),B1( 2 ,1,0),A1(0,1,1),D( 2 , 1 , 1 ),
2 tan 2 B 4 tan B 1 0
tan B 2 6 , B tan B 2 6 , tan A 2 tan B =2+ 6
2
2
AB CD AB=AD+DB= CD CD 3CD AB=3 tan A tan B 2 6
CD=2+ 6
- 7-
AB 2+ 6
3 5 1 5
sin Acos cos Asin
B B
2
5 1
5
tan A tan B
2 tan A 2 tan B .
(II) <A+B< , sin( A B) 3 , cos( A B) 4 , tan( A B) 3
2
5
5
4
tan A tan B 3 , tan A 2 tan B 1 tan A tan B 4
D 6 3
8 A(12) 1 B(31) 2
A1
B2
C3
D4
9
l
e
(
4
,
3)
O(0,0)
A(1,-2)
l
55
O1 A1 O1A1 e
A 11 5
B 11 5
C2
D2
10 yxcosxsinx
A( 3) B(2)
22
C( 35)
22
D(23)
11 ysin4xcos2x
A
4
B
()CD BDM () B1BD CBD
21( 12 ) Cy24xF C F l C AB
() l 1 OA OB () FB AF [49] l y
- 5-
(22)( 14 ) f(x)ln(1x)xg(x)xlnx (1) f(x) (2) 0ab0g(a)g(b)2g( a b )(ba)ln2 2
18(I) 2
C2C 2 3 C84 5
6 7
(II)A
C2 C 2 35 C84
3C5
3C1
C84
1 2
19I a1=1,an+1= n 2 Sn(n=1,2,3...)
n
S2
a2= 2 1 S1=3a1, S2 4a1 2 , S1 1,
1
22
1
22 S1
1
an+1=Sn+1-Sn(n=1,2,3...), Sn+1-Sn= n 2 Sn(n=1,2,3...)nSn+1=2(n+1)Sn,
A(x1)2y21
Bx2y21
Cx2(y1)21
Dx2(y1)21
5 ytan(2x)( ,0)
12
A
B
C
6
6
12
D
12
6 yex
A. yex y B. yex
- 1-
C. yex y
D. yex
7 O 1ABC
O ABC
2
A 1 3
B 3 3
C 2 3
A1BDM BDM CD BDM
(II) FG BCBD B1GFGB1F
- 8-
FGCDFG= 1 CD FG= 1 FGBD.
2
2
A
A'
BB1A1A D,
BD=B1D= 1 A1B=1
2
C
D
FG
C' M
BB1D 1 B1GBD B
B'
B1G=
3
2
B1GF
2 , 1 , 1 )
- 6-
2004 ( )
来自百度文库2 5 60
1C
2A
3C
4C
5A
D
7B
8B
9D
10B
11B
C
6 12
4 4 16
130.10.60.3
145
15 1 x2y21 2
17(I)sin(A+B)= 3 ,sin(A-B)= 1
5
5
16
sin
Acos
B
cos
Asin
B
sin Acos B cos Asin B
2 22
M(
2 ,1,0),
2
CD
(
2
,
1 2
,
1 ), A
22
B
1(
2
,-1,-1),
DM
(0,
1 ,-
2
1 ),
2
CD
A1B
0,CD
DM
0,
CDA1B,CDDM. A1BDM BDM
CD BDM
(II): BD G B1G ( 3 2 1, ,1), BD (-
G
4 44