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2017-2018-上海市大同中学高三下3月月考发

2017-2018-上海市大同中学高三下3月月考发
2017-2018-上海市大同中学高三下3月月考发

2017学年第二学期3月考试试卷 2018.3

高三数学

一、填空题

1. 已知集合U R =,集合{}

|2,x M y y x R ==∈,集合

(){}

|lg 3N x y x ==-,则

()U C M N ?=

____________ 2. 已知幂函数

()

f x

过点

(,则()f x 的反函数为____________

3. 直线23x y +=的倾斜角是____________(用反三角表示)

4. 三阶行列式

4235

4

112

k ---第2行第1列元素的代数余子式为10-,则k =____________

5. 将一个半圆面围成圆锥的侧面,则其任意两条母线间夹角的最大值为____________

6. 某个不透明的袋中装有除颜色外其它特征完全相同的7个乒乓球(袋中仅有白色和黄色两种颜色的

球),若从袋中随机摸一个乒乓球,得到的球是白色乒乓球的概率是2

7,则从袋中一次随机摸两个球,

得到一个白色乒乓球和一个黄色乒乓球的概率是____________

7. 已知

{}n a 是首项为a ,公差为1的等差数列,

1n

n n a b a +=

,若对任意的*n N ∈,都有8n b b ≥成立,则

实数a 的取值范围是____________

8. 若双曲线()222

210,b 0x y a a b -=>>上存在四个不同的点A 、B 、C 、D ,使四边形ABCD 为菱形,则b a

的取值范围为____________

9. 动点(),P a b 在不等式组20

00x y x y y +-≤??

-≥??≥?表示的平面区域内部及其边界上运动,则

3

1a b w a +-=-的取值范围是____________

10. 在面积为2的ABC 中,E 、F 分别是AB 、AC 的中点,点P 在直线EF 上,则2

PC PB BC ?+的最小值是____________

11. 函数()()2sin 206f x x πωω?

?=+> ???图像上有两点(),A s t ,()()2,22B s t t π+-<<,若对任意

s R ∈,线段AB 与函数图像都有五个不同交点,若()f x 在[]12,x x 和[]34,x x 上单调递增,在

[]23,x x 上单调递减,且

()4321322

3x x x x x x -=-=

-,则1x 的所有可能值是____________

12. 设*

n N ∈,圆

()

222:0n n n C x y R R +=>与y 轴正半轴的交点为n P

,与曲线y =

的交点为

()

,n n n Q x y ,直线

n n

P Q 与x 轴的交点为

()

,0n A a ,若数列

{}n x 满足:13x =,143n n x x +=+,要使

数列

{}1n n a Pa +-成等比数列,则常数P=____________

二、选择题

13. 下列命题中,错误的是( )

A. 过平面α外一点可以作无数条直线与平面α平行

B. 与同一个平面所成的角相等的两条直线必平行

C. 若直线l 垂直平面α内的两条相交直线,则直线l 必垂直平面α

D. 垂直于同一个平面的两条直线平行

14. 已知z C ∈,i 是虚数单位,z 是z 的共轭复数,则下列说法与“z 为纯虚数”不等价的是( )

A. 2

0z <

B. 0z z +=

C. Re 0z =且Im 0z ≠

D.

i

z z =或

i

z z =-,且

z ≠

15. 若两个函数的图象经过若干次平移后能够重合,则称这两个函数为“同形”函数,给出下列四个函数:

()122log f x x

=,

()()

22log 2f x x =+,

232log f x

=,

()

42log 2f x =则“同形”函数是( )

A.

()

1f x 与

()

2f x

B.

()

2f x 与

()3f x

C.

()

2f x 与

()4f x D.

()

1f x 与

()

4f x 16. 正方形

1S 和

2

S 内接于同一个直角三角形ABC 中,如图所示,设A α∠=,若

1441

S =,

2440

S =,

则sin 2α的值为( )

A. 111

B. 110

C. 121

D. 19

三、解答题

17. 如图,等腰Rt AOB ,OA=OB=2,点C 是OB 的中点,AOB 绕BO 所在的边逆时针 旋转一周.

(1)求ABC 旋转一周所得旋转体的体积V 和表面积S ;

(2)和OA 逆时针旋转至OD ,旋转角为θ,[)

0,2θπ∈,且满足AC ⊥BD ,求θ.

18. 已知函数()()23

3sin sin cos f x x x x x R =+-

∈.

(1)求函数

()

f x 的最小正周期T 与单调递增区间;

(2)在ABC 中,若

()()1

2f A f B ==

,求角C 的值.

19. 如图1,OA 、OB 是某地一个湖泊的两条互相垂直的湖堤,线段CD 和曲线段EF 分别是湖泊中的一

座栈桥和一条防波堤,为观光旅游的需要,拟过栈桥CD 上某点M 分别修建与OA 、OB 平行的栈桥MG 、MK ,且以MG 、MK 为边建一个跨越水面的三角形观光平台MGK ,建立如图2所示的坐标系,

测得线段CD 的方程是()

220020x y x +=≤≤,曲线段EF 的方程是

()

200540xy x =≤≤,设点

M 的坐标为

(),s t ,记z s t =?(题中所涉及

的长度单

位均为米,栈桥和防波堤都不计宽度). (1)求z 的取值范围;

(2)试写出三角形观光平台MGK 面积

MGK

S

关于z 的函数解析式,并求出该面

积的最小值.

20. 各项均为正数的数列{}n a 的前n 项和为n S ,满足()()2*21n n n S a a n N +=+∈.

(1)求数列

{}n a 的通项公式;

(2)求数列{}n b 满足()*112,2n n b b b n N +==∈,数列{}n c 满足()*

,21,2n n

n

a n k c k N

b n k =-?=∈?=?,数列

{}n c 的前n 项和为n T ,求n T ;

(3)若数列()2

*244n n P n n N =+∈,甲同学利用第(2)问中的n T ,试图确定

()*22k k T P k N -∈的值是否可以等于2018?为此,他设计了一个程序(如图),但乙同学认为这个程序如果被执行会是

一个“死循环”(即程序会永远循环下去,而无法结束),你是否同意乙同学的观点?请说明理由.

21. 如图,直线:l y kx b =+与抛物线2

2x py =(常数0p >)相交于不同的两点()11,A x y 、()22,B x y ,

21x x h

-=(h 为定值),线段AB 的中点为D ,与直线:l y kx b =+平行的切线的切点为C (不

与抛物线对称轴平行或重合且与抛物线只有一个公共点的直线称为抛物线的切线,这个公共点为切

点).

(1)用k 、b 表示出C 点、D 点的坐标,并证明CD 垂直于x 轴; (2)求ABC 的面积,证明ABC 的面积与k 、b 无关,只与h 有关;

(3)小张所在的兴趣小组完成上面两个小题后,小张连AC 、BC ,再作与AC 、BC

平行的切线,切点分

别为E 、F ,小张马上写出了ACE 、BCF 的面积,由此小张求出了直线l 与抛物线围成的面积,你认为小张能做到吗?请你说明理由.

参考答案

1、(],0-∞

2、()()12,0f x x x -=≥

3、arctan 2π-

4、14-

5、60

6、10

21 7、()8,7-- 8、()1,+∞ 9、(

][),13,-∞-+∞ 10、2

3

11、,6

k k Z

π

π-

+∈ 12、2或4

13-16、BBCB

17、(1)43V π=,()

4225S π

=+;(2)23π或43π

18、(1)T π=,5,,1212k k k Z ππππ??-+∈????;(2)6π

19、(1)

75,502z ??∈??

??;(2)1400004002S z z ??

=+- ???,最小值为225平方米. 20、(1)n 1n a =+;(2)()221

2421,243,4324,21

433n

n n n n n k T k N n n n k *

+?++-=??=∈?++?+-=+??;(3)乙同学观点正确

21、(1)2,2pk C pk ?? ?

??,()2,D pk pk b +,,C D 的横坐

标相同,所以CD x ⊥轴;

(2)316h p ;(3)能,3

12h p

2018-2019学年上海中学高三上英语期中英语试卷

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