2017-2018年上海市七宝中学高三下3月月考

2017-2018年七宝中学高三下3月月考

2018.03

一. 填空题

1. 已知集合{|3}{|1}A x x x x =≥<-，则C A =R

2. 复数1i

z i

-（其中i 为虚数单位）的模为 3. 某学校有8个社团，甲、乙两位同学各自参加其中一个社团，且他俩参加各个社团的可能性相同，则这两位同学参加同一个社团的概率为

4. 从编号0，1，2，???，79的80件产品中，采用系统抽样的方法抽取容量是5的样本，若编号为28的产品在样本中，则该样本中产品的最大编号为

5. 设奇函数()f x 的定义域为[5,5]-，若当[0,5]x ∈时，()f x 的图像如图，则不等式()0f x <的解是

6. 若12

log 11

a <-，则a 的取值范围是 7. 在平面直角坐标系xOy 中，设A 是半圆2

:2O x y +=（0x ≥）上一点，直线OA 的倾斜角为

45°，过点A 作x 轴的垂线，垂足为H ，过H 作OA 的平行线交半圆于点B ，则直线AB 的方程是 8. 在ABC ?中，D 是BC 的中点，8AD =，20BC =，则AB AC ?的值为 9. 设x 、y 、z 是实数，9x 、12y 、15z 成等比数列，且1x 、1y 、1z 成等差数列，则x z z x

+ 的值是

10. 设实数a 、b 、c 满足221a b c +≤≤，则a b c ++的最小值为

11. 若不等式2

(1)[3(1)1]0mx m x m --+-≥对任意(0,)m ∈+∞恒成立，则实数x 的值为

12. 设等差数列{}n a 的公差为d ，前n 项和为n S ，且11a ≥，2424a ≥，12168S ≤，则2

9a d -的取值范

围是

二. 选择题

13. 若函数2()21

x f x =-+，则该函数在(,)-∞+∞上是（）

A. 单调递减无最小值

B. 单调递减有最小值

C. 单调递增无最大值

D. 单调递增有最大值

14. “22a -≤≤”是“实系数一元二次方程210x ax ++=有虚根”的（）条件

A. 必要不充分

B. 充分不必要

C. 充要

D. 既不充分也不必要

15. 已知a 、b 是单位向量，0a b ?=，若向量c 满足||1c a b --=，则||c 的取值范围是（）

A. 1]

B. 2]

C. 1]

D. 2]+

16. 将半径都为1的4个钢球完全装入形状为正四面体的容器里，这个正四面体的高的最小值为（）

A. 3

B. 23+

C. 43

+ D. 3

三. 解答题

17. 在ABC ?中，已知9AB AC ?=，16AB BC ?=-，求：（1）AB 的值；（2）sin()

sin A B C

-的值.

18. 如图，在直棱柱1111ABCD A B C D -中，AD ∥BC ，90BAD ∠=?，AC BD ⊥，1BC =，

13AD AA ==.

（1）证明：1AC B D ⊥；

（2）求直线11B C 与平面1ACD 所成角的正弦值.

19. 将52名志愿者分成A 、B 两组参加义务植树活动，A 组种植150捆白杨树苗，B 组种植 200捆沙棘树苗，假定A 、B 两组同时开始种植. （1）根据历年统计，每名志愿者种植一捆白杨树苗用时

25小时，种植一捆沙棘树苗用时12

小时，应如何分配A 、B 两组的人数，使植树活动持续时间最短？

（2）在按（1）分配的人数种植1小时后发现，每名志愿者种植一捆白杨树苗用时仍为

小时，而每名志愿者种植一捆沙棘树苗实际用时2

小时，于是从A 组抽调6名志愿者加入B 组继续种植，求植树活动所持续的时间.

20. 已知左焦点为(1,0)F -的椭圆过点(1,

E ，过点(1,1)P 分别作斜率为1k 、2k 的椭圆的动弦AB 、CD ，设M 、N 分别为线段AB 、CD 的中点. （1）求椭圆的标准方程；

（2）若P 为线段AB 的中点，求1k ；

（3）若121k k +=，求证：直线MN 恒过定点，并求出定点坐标.

21. 已知数列{}n a 满足2

12n a a a n ++???+=（n ∈*N ）.

（1）求数列{}n a 的通项公式；

（2）对任意给定的k ∈*N ，是否存在,p r ∈*

N （k p r <<）使

a 、1p a 、1r a 成等差数列？若存在，用k 分别表示p 和r （只要写出一组），若不存在，请说明理由；

（3）证明：存在无穷多个三边成等比数列且互不相似的三角形，其边长为1n a 、2n a 、3n a .

参考答案

1、[)1,3-

2、

2 3、1

4、76

5、()(]2,02,5-

6、4a >

7、1y =+ 8、36- 9、

3415 10、12- 11、1x = 12、2498,16??

????

13-16、AAAC 17、（1）5；（2）

18、（1）证明略;（2 19、（1）A 组20人，B 组32人；（2）

小时 20、（1）22132x y +=；（2）23-；（3）20,3?

?- ??

21、（1）21n a n =-；（2）21p k =-，2452r k k =-+；（3）证明略

2018-2019学年上海中学高三上英语期中英语试卷

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