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湖北省孝感高级中学2012-2013学年高一下学期期末考试数学试卷


一、选择题(本大题共 10 个小题,每小题 5 分,共 50 分,在每小题给出的四个选项中,只 有一项是符合题目要求的)
1. 直线 3x ? y ? 2 ? 0 的倾斜角为( )

A.150

B.120

C. 60

D. 30

2. 已知关于 x 的不等式 ax 2 ? bx ? 2 ? 0 的解集是 ? ?1, 2? ,则 a ? b 的值为( )

A.1

B.-1

C.0

D. -2

3.在 ?ABC 中, a ? 4, b ? 4 3, C ? 60 ,则 ?ABC 的面积为( )

A.24

B.12

C.8 3

D.16 3

4. 某几何体的三视图如图所示,则该几何体的表面积等于( )

A. 12?

B. 10?

C. 13 ? 3

D. 28 ? 3

5. 点 E、F、G、H 分别是四面体 ABCD 的边 AB、BC、CD、DA

的中点,则四面体的六条棱中与平面 EFGH 平行的条数是( )

2
3 2 正视图

2 侧视图

A.0

B.1

C.2

D.3

6. 在等比数列{an}中,已知 a1a3a11 ? 8 ,则 a2a8 等于( )

A.16

B.12

C.6

D.4

7. 已知点 A?1,3? , B ?3,1? ,C ?0,0? ,则 ?ABC 的面积为( )

2 俯视图

A.4

B.5

C.6

D.7

8. 四棱锥V ? ABCD 中,底面 ABCD 是边长为 2 的正方形,其他四个侧面都是侧棱长

为 5 的等腰三角形,则二面角V ? AB ? C 的大小为( )

A. 30

B. 45

C. 60

9. 如右图是正方体的平面展开图,则在这个正方体中:

①BM 与 ED 平行

②CN 与 BE 是异面直线

③CN 与 BM 成 60o 角

④DM 与 BN 是异面直线

D.120
N ED
E
EA
EE

CM
EE
B
EF
E

以上四个命题中,正确命题的序号是( )

A.①②③

B.②④ C.③④ D.②③④

10.由曲线 x2 ? y2 ? 1 x ? 1 y 围成的图形的面积为( ) 22

A. 1 2

B.1

C. ? 2

D. ? + 1 42

13.已知 a、b 是不重合的两条直线,?、?、? 是不重合的三个平面,给出下列结论,其

中正确的结论的序号是
①若? // ? , a ? ? ,则 a // ? ;
③若? ⊥ ? , ? ⊥? ,则? ∥? ;

.
②若 a 、 b 与? 所成角相等,则 a // b ;
④若 a ? ? , a ? ? ,则? // ? .

14 . 已 知 圆 O : x2 ? y2 ? 4 内 有 一 点 P(0,1), 过 点 P 的 直 线 l 交 圆 O 于 A, B 两 点 。 若

AB ? 14 ,则直线 l 的方程为

x2 ?1

15.若方程

? kx 有两个实数根,则实数 k 的取值范围是

x ?1

三、解答题(本大题共 6 个小题,共 75 分,解答题必须要写演算步骤,证明过程,文字说 明)
16. (本小题满分 12 分)在锐角三角形△ ABC 中, a, b, c 分别是角 A, B,C 所对的边,且
3a ? 2c sin A . (1)求角 C 的大小; (2)若 c ? 7 , ?ABC 的面积为 3 3 ,求 a ? b 的值.
2

17.(本小题满分 12 分)已知直线 l1 经过点 A??3, 0? , B(3, 2) ,直线 l2 经过点 B ,且 l1 ? l2 .
(1)求经过点 B 且在两坐标轴上的截距相等的直线的方程;
(2)设直线 l2 与直线 y ? 8x 的交点为 C ,求 ?ABC 外接圆的方程.

19. (本小题满分 12 分)一个化肥厂生产甲、乙两种混合肥料,生产 1 车皮甲种肥料的主 要原料是磷酸盐 4 吨、硝酸盐 18 吨;生产 1 车皮乙种肥料需要的主要原料是磷酸盐 1 吨、硝酸盐 15 吨。现库存磷酸盐 10 吨、硝酸盐 66 吨,在此基础上生产这两种混合肥 料,若生产 1 车皮甲种肥料,产生的利润为 10000 元;生产 1 车皮乙种肥料,产生的利 润为 5000 元,那么分别生产甲、乙两种肥料各多少车皮,能够产生最大利润?

? ? 20. (本小题满分 13 分)已知等差数列 an 中,公差 d ? 0 ,其前 n 项和为 Sn ,且满足

a2 ? a4 ? 45, a1 ? a5 ? 14 。

? ? (1) 求数列 an 的通项公式及其前 n 项和 Sn ;

? ? ? ? ? ? (2)



bn

=

1 an2 ?1

n? N*

,若数列 cn

满足

c1 =

?

1 4

,cn?1

?

cn

=bn

,求数列

cn

的通项公式 cn ;
? ? (3) 求 f ? n? ? n ? bn n ? N * 的最小值。 9 cn
21. (本小题满分 14 分)已知定点 O ?0,0? , A?3,0? ,动点 P 到定点 O 距离与到定点 A 的
距离的比值是 1 . ?
(1) 求动点 P 的轨迹方程,并说明方程表示的曲线; (2) 当 ? ? 4 时,记动点 P 的轨迹为曲线 D .
①若 M 是圆 E : ? x ? 2?2 ? ? y ? 4?2 ? 64 上任意一点,过 M 作曲线 D 的切线,
切点是 N ,求 MN 的取值范围;
②已知 F ,G 是曲线 D 上不同的两点,对于定点 Q(?3,0) ,有 QF ? QG ? 4 . 试问无论 F 、 G 两点的位置怎样,直线 FG 能恒与一个定圆相切吗?若能, 求出这个定圆的方程;若不能,请说明理由.

孝感高中 2012-2013 学年度高一下学期期末考试 高一数学参考答案

二、填空题
11、 ? 1 2
14、 y ? x ? 1或 y ? ?x ? 1
三、解答题

12、 ?

x+2?2

?

? ??

y

?

3 2

?2 ??

?

5 4

15、 0 ? k ? 1或1 ? k ? 2 .

13、①④

16.解(1)由正弦定理得 3a ? 2c sin A ? 3 sin A ? 2sin C sin A ? sin C ? 3 2

?ABC 是锐角三角形,故 C =60?

……………………6 分

(2)

S?ABC

=

1 2

ab sin C

?

33 2

?

ab

?

6

将 c ? 7 代入 cos C ? a2 ? b2 ? c2 ? 1 得到 a2 ? b2 ? 13

2ab

2

则 ?a ? b?2 ? a2 ? b2 ? 2ab ? 25 ,即 a ? b ? 5 ……………………12 分

17.解:(1)若直线过原点,则方程为 y ? 2 x 3

……………………3 分

若直线不过原点,则方程为 y ? ? x ? 5 ……………………6 分

(2)直线

l1

经过点

A

?

?3,

0

?

,

B(3,

2)

,则

l1

的斜率为

1 3

设直线 l2 的方程 3x ? y ? c ? 0

18.(1)证明:∵底面 ABCD 是正方形∴AC⊥BD ∵PD⊥底面 ABCD∴PD⊥AC

AC AC

? ?

BD? PD ??

?

AC

?

面PBD?? ?

?

面AEC

?

面PDB

AC ? 面AEC

? ?

…………………6 分

(2)设 AC 与 BD 交于 O 点,连接 EO

则易得∠AEO 为 AE 与面 PDB 所成的角

…………………………9 分

在 Rt△AEO 中,OE= 1 PD= 2 AB,

2

4

AO= 2 AB 4
故 AE 与面 PDB 所成角的正切值为 2

O …………………………12 分

19.解:设生产甲种肥料 x 车皮、乙种肥料 y 车皮,能够产生利润 z 元,目标函 数为 z=10000x+5000y.

?4x ? y ? 10,

由题意满足以下条件:

??18x ??x ?

? 15 0,

y

?

66,

?? y ? 0.

可行域如图.

由图可以看出,当直线 y=

-2x+2z 经过可行域上的点 M

时,截距最大,即 z 最大.

…………………………6 分









?18x ? 15y ? 66, ??4x ? y ? 10,

得 M 的坐标为 x=2,y=2.所以

zmax=10000x+5000y=30000. 故生产甲 2 车皮,乙 2 车皮能够产生最大的利润。……………………12 分

20.解:(1)因为数列{a n }是等差数列,所以 a1 +a 5 =

a +a =14.

2

4

因为

d>0,所以解方程组

???aa22

? a4

a4 ?

? 14,得 45.

a

2

=5,a

4

=9.

故 a1 =3,d=2,a n =2n+1,S n =n2+2n …………………………4 分

(2)因为

b

n



1 an2 ?

1

(n∈N*),a

n

=2n+1,



b

n



1 4n(n

?

1)

.

数列{c

n

}满足

c

1

=-

1 4



Cn?1

?

Cn

=

1 4n(n+1)

=

1 4

? ??

1 n

?

n

1 ?

1

? ??





Cn

?

Cn-1 =

1 4

? ??

1 n-1

?

1 n

? ??

…………

C2

?

C1 =

1 4

?1 ?? 1

?

1 2

? ??

以上各式相加得:

Cn

?

C1 =

1 4

?1 ?? 1

?

1 n

? ??

?

n

?

2?

因为 C1 =

1 4

,所以 cn

?

?

1 4n

?n

?

2?

? ? 验证 C1=

1 4

也满足,故

cn

?

?

1 4n

n? N*

…………………………9 分

21.解(1)设动点 P 的坐标为 ? x, y? ,则由 ? PO ? PA ,得 ?(x2 ? y2 ) ? (x ? 3)2 ? y2 , 整理得: ?? ?1? x2 ? ?? ?1? y2 ? 6x ? 9 ? 0 .

由条件知 ? ? 0 , 当 ? ? 1 时,方程可化为: 2x ? 3 ? 0 ,
故方程表示的曲线是线段 OA 的垂直平分线

…………………………2 分



?

?

1

时,则方程可化为

? ??

x

?

3 ?2 ? ?1??

?

y2

?

?3

? ??

?

?

? ?2

?

1?

? ??



故方程表示的曲线是以

? ??

?

?

3 ?1

,0

? ??

为圆心,

3? ? ?1

为半径的圆.

……………4 分

(2)当 ? ? 4 时,曲线 D 的方程是 x2 ? y2 ? 2x ? 3 ? 0 ,

则曲线 D 表示圆,圆心是 D??1,0? ,半径是 2 .

①由 DE ? ?2 ?1?2 ? ?4 ? 0?2 ? 5 ,及 5 < 8 - 2 知:两圆内含,且圆 D 在圆 E 内部.

由 MN 2 ? MD 2 ? DN 2 有: MN 2 ? MD 2 ? 4 ,故求 MN 的取值范围就是求 MD 的

取值范围.而 D 是定点, M 是圆上的动点,故过 D 作

圆 E 的直径,得 MD ? 8 ? 5 ? 3 , MD ? 8 ? 5 ? 13 ,

min

max

故 5 ≤ MN 2 ≤165 ,即 5 ≤ MN ≤ 165 .

………………………9 分

②解法一:设点 Q 到直线 FG 的距离为 d , ?FQG ? ? ,

则由面积相等得到 QF ? QG sin? ? d FG ,且圆的半径 r ? 2 .

即 d ? 4sin? ? 4sin? ? 1.于是顶点 Q 到动直线 FG 的距离为定值, FG 2r sin?

故动直线 FG 与定圆 (x ? 3)2 ? y2 ? 1恒相切.

解法二:设 F ,G 两点的坐标分别为 F ? x1, y1 ? ,G ? x2 , y2 ? ,则由 QF ? QG ? 4 有:
(x1 ? 3)2 ? y12 ? (x2 ? 3)2 ? y22 ? 4 ,结合 x12 ? y12 ? 2x1 ? 3 ? 0, x22 ? y22 ? 2x2 ? 3 ? 0 有: 4x1 ? 12 ? 4x2 ? 12 ? 4 ? x1x2 ? 3(x1? x2 ) ? 8 ? 0 。
若经过 F 、 G 两点的直线的斜率存在,设直线 FG 的方程为 y ? mx ? n ,由

? ? ?
? ?

y ? mx x2 ? y2

? ?

n 2

x

?

3

?

0

,消去

y 有:

1? m2

x2 ? ?2mn ? 2? x ? n2 ? 3 ? 0 ,

则 x1? x2

?

2mn ? 2 ? 1? m2

, x1x2

? n2 -3 1? m2



所以 x1x2

n2 ? 3 ? 3(x1?x2 ) ? 8 ? 1 ? m2

?

?6mn ? 6 1? m2

?

8 ? 8m2 1? m2

? 0 ,由此可得 8m2

? 6mn ? n2

?1,

也即 (3m ? n)2 ? 1? m2 ? 3m ? n ? 1 ( ※ )
1? m2

假设存在定圆 ? x ? a?2 ? ? y ? b?2 ? r2 ,总与直线 FG 相切,则

d ? ma ? b ? n 是定值 r ,即 d 与 m, n 无关。
1? m2


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