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四川省宜宾县第一中学校2018_2019学年高二数学上学期期中试题理


2018 年秋四川省宜宾县一中高二期中考试 数学(理)试卷
一.选择题(本大题共 12 个小题,每小题 5 分,共 60 分) 1. 复数 z ? A. i

5i ? 2?i
B. 1 ? i C. 1 ? 2i D. 1 ? i

2. 过点 ?1, ?1? 且与直线 x ? 2 y ? 1 ? 0 平行的直线方程为 A. x ? 2 y ? 1 ? 0 C. x ? 2 y ? 3 ? 0 3.与直线 l : 3x ? 5 y ? 4 ? 0 关于 x 轴对称的直线的方程为 A. 5x ? 3 y ? 4 ? 0 C. 3x ? 5 y ? 4 ? 0 B. 3x ? 5 y ? 4 ? 0 D. 5x ? 3 y ? 4 ? 0 B. x ? 2 y ? 1 ? 0 D. 2 x ? y ? 1 ? 0

4. “ a ? b ? 2 ab 是 a ? ? 0, ??? , b ? ? 0, ??? ”的 A.充分不必要条件 C.充要条件
2 2

B.必 要不充分条件 D.既不充分也不必要条件

5.过点 ? 2,1? 的直线中,被圆 x ? y ? 2 x ? 4 y ? 0 截得的弦长最大的直线方程是 A. 3x ? y ? 5 ? 0 C. x ? 3 y ? 5 ? 0 B. 3x ? y ? 7 ? 0 D. x ? 3 y ? 5 ? 0

6.过椭圆 周长为 A.

x2 ? y 2 ? 1 的左焦点 F1 作直线 交椭圆于 A, B 两点, F2 是椭圆右焦点,则 ?ABF2 的 2

B.

C.

D.

7. ? , ? 是两个平面, m, n 是两条直线,有下列四个命题: (1)如果 m ? n, m ? ? , n / / ? ,那么 ? ? ? .(2)如果 m ? ? , n / /? ,那么 m ? n . (3)如果 ? / / ? , m ? ? ,那么 m / / ? .(4)如果 m / / n, ? / / ? ,那么 m 与 ? 所成的角和 n 与 ? 所成的角相等. 其中正确的命题个数为

-1-

A. 1 B. 2 C. 3 D. 4 8. 已知圆柱的高为 1 ,它的两个底面 的圆周在直径为 2 的同一个球的球面上,则该圆柱的体积 为 A.

?

B.

3? 4

C.

? 2

D.

? 4

9. 已知直线 l1 : 4 x ? 3 y ? 6 ? 0 和直线 l2 : x ? ?1 ,抛物线 y 2 ? 4 x 上一动点 P 到直线 l1 和直 线 l2 的距离之和的最小值是 A.

37 16

B. 3

C.

11 5

D. 2

10.设点 A? ?2,3? , B ?3,2 ? ,若直线 ax ? y ? 2 ? 0 与线段 AB 没有交点,则 a 的取值范围是 A. ? ??, ? ? ? ? , ?? ? 2 3

? ?

5? ?

?4 ?

? ?

B. ? ? , ? ? 2 3? D. ? ??, ? ? ? ? , ?? ? 3 2

? 5 4?

C. ? ?

? 4 5? , ? ? 3 2?
2

? ?

4? ?

?5 ?

? ?

11. 已知二次函数 f ? x ? ? ax ? 2x ? c ? x ? R ? 的值域为 ?0,??? ,则 A.8 B. 4 2 C.4

a ?1 c ?1 ? 的最小值为 c a
D. 8 2

12.椭圆

x2 y 2 ? ? 1 ( a ? b ? 0 ) 的左,右顶点分别是 a 2 b2
成等比数列,则此椭圆的离心率为

,左,右焦点分别是

,若

A.

1 4

B.

5 5

C.

1 2

D.

二.填空题(本大题共 4 个小题,每小题 5 分,共 20 分)

x? y ?0
13.若 x, y 满足约束条件 {x ? y ? 2 ? 0 ,则 z ? 3x ? 4 y 的最小值为__________.

y?0
14.动圆过点 ?1,0 ? ,且与直线 x ? ?1 相切,则动圆的圆心的轨迹方程为__________. 15.在四面体 ABCD 中, AB ? CD ? 5 , AC ? BD ? 13 , AD ? BC ? 10 ,则该四面体 外接球的表面积为__________.

-2-

16.已知 A(1, 2), B( ?1, 2) ,动点 P 满足 AP ? BP ,若双曲线

x2 y 2 ? ? 1(a ? 0, b ? 0) 的渐近 a 2 b2

线与动点 P 的轨迹没有公共点,则双曲线离心率的取值范围是__________

三.解答题(本大题共 6 个小题,共 70 分) 17.(本大题满分 10 分)
2 2 已知命题 P :不等式 x ?1 ? a ? a ? 2 的解集为 R ;命题 Q :圆 ? x ? 1? ? y ? 4 上至少有 2

三个点到直线 ax ? y ? 1 ? 0 的距离为 1 .若命题 P 和 Q 中有 且只有一个为真,求实数 a 的取值 范围.

18.(本大题满分 12 分) 直线 l 经过两直线 l1 : 2x ? y ? 4 ? 0 与 l2 : x ? y ? 5 ? 0 的交点,且与直线 x ? 2 y ? 6 ? 0 垂直. (1) 求直线 l 的方程; (2)若点 P(a,1) 到直线 l 的距离为 5 ,求实数 a 的值.

19.(本大题满分 12 分) 如图,在四棱锥 P﹣ABCD 中,PC⊥底面 ABCD,ABCD 是直角梯形,AB⊥AD,AB∥CD, AB=2AD=2CD=PC=2.E 是 PB 的中点. (1)求证:平面 EAC⊥平面 PBC; (2)求二面角 P—AC—E 的余弦值; (3)求直线 PA 与平面 EAC 所成角的正弦值.

-3-

20.(本大题满分 12 分) 已知圆心在直线 y ? 4 x 上,且与直线 l : x ? y ? 2 ? 0 相切于点 P ?1,1? (1)求圆的方程 (2)直线 kx ? y ? 3 ? 0 与该圆相交于 A, B 两点,若点 M 在圆上,且有向量

OM ? OA ? OB ( O 为坐标原点),求实数 k .

21.(本大题满分 12 分)

x2 y 2 3 已知椭圆 C : 2 ? 2 ? 1(a ? b ? 0) 的离心率为 ,点 M ? 2,1? 在椭圆 C 上 a b 2
(1)求椭圆 C 的方程 (2)直线 l 平行于 OM (O 为坐标原点 ), 且与椭圆 C 交于 A, B ? 两个不同的点,若 ?AOB 为钝角, 求直线 l 在 y 轴上的截距 m 的取值范围

22.(本大题满分 12 分) 如图, O 为坐标原点,椭圆 C1 :

x2 y 2 ? ? 1(a ? b ? 0) 的左、 右焦点分别为 F1 , F2 ,离心 率为 e1 ; a 2 b2

双曲线 C2 :

x2 y 2 3 ? 2 ? 1 的左、右焦点分别为 F3 , F4 ,离心率为 e2 .已知 e1e2 ? ,且 2 a b 2

F2 F4 ? 3 ? 1

-4-

(1)求 C1 , C2 的方程; (2)过 F1 作 C1 的不垂直于 y 轴的弦 AB , M 为 AB 的中点,当直线 OM 与 C2 交于 P , Q 两点 时,求四边形 APBQ 面积的最小值.

-5-

2018 年秋四川省宜宾县一中高二期中考试 数学(理)试卷答案 一.选择题 1.C 2.C 3.B 11.C 12.B 二.填空题[:] 13. ?1 14. y 2 ? 4 x 15. 14? 16. ?1, 2 ? 4.B 5.A 6.B 7.C 8.B 9.C 10.C

三.解答题 17.命题 P : a 2 ? a ? 2 ? 0 ? ?1 ? a ? 2 命题 Q :

a ?1 a2 ? 1

?1? a ? 0

若 P 真 Q 假,则有: ?

??1 ? a ? 2 ? ?1 ? a ? 0 , ?a ? 0

若 P 假 Q 真,则有:

?a?2

综上可得:实数 a 的取值范围为 ? ?1,0? ??2, ??? .

18.解:(1)有题得: { 即交点为 (1, 6)

2x ? y ? 4 ? 0 x ?1 ?{ x? y ?5 ? 0 y?6

∵ l 与 x ? 2 y ? 6 ? 0 垂直,则 kl ? ?

1 ? ?2 1 2

∴ l : y ? 6 ? ?2( x ? 1) 即 2x ? y ? 8 ? 0 (2)点 P(a,1) 到直线 l 的距离为 5 ,则

12a ? 71 ? 5 ? 12a ? 71 ? 5 ? a ? 6 或 1 5
19.(1)解:解析:(1)证明:∵PC⊥平面 ABCD,AC? 平面 ABCD,∴AC⊥PC, ∵AB=2,AD=CD=1, ∴AC=BC= , ∴AC +BC =AB ,∴AC⊥BC,
2 2 2

-6-

又 BC∩PC=C,∴AC⊥平面 PBC, ∵AC? 平面 EAC,∴平面 EAC⊥平面 PBC. (2)由(1)知 AC⊥平面 PBC 即为二面角 P—AC—E 的平面角.

∴在

, 又 E 为中点,可得



从而二面角 P—AC—E 的余弦值为 (3)作 ,F 为垂足 平面 PBC=CE,

由(Ⅰ)知平面 EAC⊥平面 PBC,又∵平面 EAC ∴ ,连接 AF,则

就是直线 PA 与平面 EAC 所成的角.

由(Ⅱ)知

,由等面积法可知,







中,



即直线 PA 与平面 EAC 所成角的正弦值为



20.(1)设圆的方程为 ( x ? a) ? ( y ? 4a) ? r 因为直线相切,
2 2 2

圆心到直线的距离 d ?

a ? 4a ? 2 2

?r,

且圆心与切点连线与直线 l 垂直

4a ? 1 (?1) ? ?1 a ?1

-7-

可得 a ? 0, r ? 2 ,所以圆的方程为: x2 ? y 2 ? 2 (2)直线与圆联立: {

kx ? y ? 3 ? 0 x2 ? y 2 ? 2

,

得: (1 ? k 2 ) x2 ? 6kx ? 7 ? 0 , ? ? 8k 2 ? 28 ? 0 解得 k ?

7 7 或k ? ? . 2 2
6k 7 6 , x1 x2 ? y1 ? y2 ? 2 2 1? k 1? k 1? k 2

设 A? x1, y1 ? B ? x2 , y2 ? , x1 ? x2 ? ?

M ( x1 ? x2 , y1 ? y2 ) 代入圆方程 ( x1 ? x2 )2 ? ( y1 ? y2 )2 ? 2 ,
求得 k ? ? 17

c 3 ? a 2 4 1 3 21.(1)因为椭圆的离心率为 ,点 M (2,1) 在椭圆 C 上所以 { 2 ? 2 ? 1 ,解得 a b 2 a 2 ? b2 ? c2 e?

a ? 2 2, b ? 2, c ? 6 故椭圆 C 的标准方程为

x2 y 2 ? ?1 8 2
1 ,又 l 在 y 轴上的截距 m ,故 l 的方程 2

(2)由直线 l 平行于 OM 得直线 l 的斜率为 k ? kOM ?

1 x?m 1 2 2 2 为 y ? x ? m 由{ 2 得 x ? 2mx ? 2m ? 4 ? 0 ,又直线与椭圆 C 交于 A, B ? 两个不 2 2 x y ? ?1 8 2 y?
同的点,设 A ? x1 , y1 ? , B ? x2 , y2 ? ,则 x1 ? x2 ? ?2m, x1x2 ? 2m2 ? 4 .所以

? ? (2m)2 ? 4(2m2 ? 4) ? 0 ,于是 ?2 ? m ? 2 , ?AOB 为钝角等价于 OA ? OB ? 0 ,且 m ? 0
则 OA ? OB ? x1 x2 ? y1 y2 ? x1 x2 ? ?

m ?1 ?? 1 ? 5 x1 ? m ?? x2 ? m ? ? x1 x2 ? ? x1 ? x2 ? ? m 2 ? 0 即 2 ?2 ?? 2 ? 4

m2 ? 2 ,又 m ? 0 ,所以 m 的取值范围为 ? 2, 0 ? 0, 2

?

? ?

?

-8-

22.(1)因为 e1e2 ?
4 4 即a ?b ?

a 2 ? b2 a 2 ? b2 3 3 ,所以 , ? ? 2 a a 2

3 4 a ,因此 a 2 ? 2b2 ,从而 F2 (b,0) , F4 4

?

3b, 0 ,

?

于是 3b ? b ? F2 F4 ? 3 ? 1 ,所以 b ? 1 ,所 以 a 2 ? 2 , 故 C1 , C2 的方程分别为

x2 x2 ? y 2 ? 1, ? y 2 ? 1. 2 2

(2)因 AB 不垂直于 y 轴,且过点 F1 (?1,0) , 故可设直线 AB 的方程为 x ? my ? 1 .

x ? my ? 1,
由{ x2

2

? y2 ? 1

得 (m2 ? 2) y 2 ? 2my ?1 ? 0 .

易知此方程的判别式大于 0.设 A( x1 , y1 ) , B( x2 , y2 ) ,则 y1 , y2 是上述方程的两个实根, 所以 y1 ? y2 ?

2m ?1 4 , y1 y2 ? 2 ,因此 x1 ? x2 ? m( y1 ? y2 ) ? 2 ? ? 2 , 2 m ?2 m ?2 m ?2

于是 AB 的中点为 M ? ?

2 m ? ? , 2 ?, 2 ? m ?2 m ?2?
m m , P、Q 的方程为 y ? ? x ,即 mx ? 2 y ? 0 . 2 2

故直线 P、Q 的斜率为 ?

y??
由{
2

m x, 2

x ? y2 ? 1 2

得 (2 ? m ) x ? 4 ,
2 2

2 2 所以 2 ? m ? 0 ,且 x ?

m2 4 2 y ? , , 2 ? m2 2 ? m2

从而 PQ ? 2 x ? y ? 2
2 2

m2 ? 4 .如图,设点 A 到直线 P、Q 的距离为 d , 2 ? m2 mx1 ? 2 y1 ? mx2 ? 2 y2 m2 ? 4
.

则点 B 到直线 P、Q 的距离也为 d ,所以 2d ?

-9-

因为点 A , B 在直线 mx ? 2 y ? 0 的异侧,所以 (mx1 ? 2 y1 ) ? (mx2 ? 2 y2 ) ? 0 , 于是 mx1 ? 2 y1 ? mx2 ? 2 y2 ? mx1 ? 2 y1 ? mx2 ? 2 y2 ,

从而 2d ?

(m2 ? 2) y1 ? y2 m2 ? 4
2 2 ? 1 ? m2 m2 ? 4

.又因为 y1 ? y2 ? ( y1 ? y2 ) ? 4 y1 y2 ?
2

2 2 ? 1 ? m2 , m2 ? 2

所以 2d ?

.故四边形 APBQ 的面积

S?

1 2 2 ? 1 ? m2 3 PQ ? 2d ? ? 2 2 ? ? 1 ? . 2 2 ? m2 2 ? m2

2 而 0 ? 2 ? m ? 2 ,故当 m ? 0 时, S 取得最小值 2 .

综上所述,四边形 APBQ 面积的最小值为 2 .

- 10 -


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