2015西城区高三一模数学理试题及答案word版

北京市西城区2015年高三一模试卷

数

第I 卷一、选择题：本大题共 8小题，每小题符合题目要求的一项.

1 .设集合 A {0,1}，集合 B {x|x a}，若 AI B

(A ) a <1

(B ) a >1

学(理科)

2015.4

(选择题共40分)

5分，共40分.在每小题列出的四个选项中，选出

，则实数a 的取值范围是(

)

2 ?复数z 满足z i

3 i ，则在复平面内，复数

(A) 第一象限 (C )第三象限 3.在极坐标系中，曲线

p 二2cos 9是(

(A )过极点的直线

(C )关于极点对称的图形

(D ) a w 0

z 对应的点位于 (

)

(B) 第二象限 (D) 第四象限

)

(B )半径为2的圆 (D )关于极轴对称的图形

4?执行如图所示的程序框图，若输入的

x 的值为3,

则输出的n 的值为( )

(A) 4 (B) 5 (C) 6 (D)

开始 /输入X /

5?若函数f(x)的定义域为R ，则“ x R , f(x 1) f(x) ”是“函数f(x)为增函数”的()

于20元，那么2枝玫瑰和3枝康乃馨的价格的比较结果是( )

1 2 1 2

8.已知抛物线y 二 x 和y 二- x + 5所围成的封闭

4 16

曲线如图所示，给定点 A(0, a)，若在此封闭曲线上三对不同的点，满足每一对点关于点 A 对称，则实数a 的取值范围是(

)

(A) (1,3)

，3)

(A )充分而不必要条件 (C )充分必要条件

(B )必要而不充分条件 (D )既不充分也不必要条件

6. —个几何体的三视图如图所示，则该几何体的体积的是(

)

(A)

6 (B) 23

1 (D) 7

7.已知6枝玫瑰与3枝康乃馨的价格之和大于 24元，而4枝玫瑰与 4枝康乃馨的价格之和小

(A ) 2枝玫瑰的价格高 (C) 价格相同

(B ) 3枝康乃馨的价格高 (D) 不确定

(B) (2,4)

(D)(二，

正(主)视图侧(左)视图

俯视图

第n卷(非选择题共no分)

二、填空题：本大题共6小题，每小题5分，共30分.

9. 已知平面向量a, b满足a (1, 1), (a b) (a b)，那么|b|= _____________ .

2 2

10. 已知双曲线C：冷每1(a 0,b 0)的一个焦点是抛物线y2 8x的焦点，且双曲线

a b

C的离心率为2，那么双曲线C的方程为_______ .

11?在ABC中，角A, B, C所对的边分别为a, b, c.若A - , cos B 空7, b 2 , 3 7则a ____ .

12?若数列｛久｝满足a 2，且对于任意的m,n N*，都有a m n a m a n，则a3 —；数列｛a n｝前10项的和S o ______________ .

13. 某种产品的加工需要A, B, C, D, E五道工艺，其中A必须在D的前面完成(不一定

相邻)，其它工艺的顺序可以改变，但不能同时进行，为了节省加工时间，B与C必须相令邻，那么完成加工该产品的不同工艺的排列顺序有______ 种?(用数字作答)

14. 如图，四面体ABCD的一条棱长为x，其余棱长均为1,

记四面体ABCD的体积为F(x)，则函数F(x)的单

调增区间是 ____ ;最大值为_____ .

三、解答题：本大题共 6小题，共80分?解答应写出必要的文字说明、证明过程或演算步骤.

15. （本小题满分13分）

设函数

f (x) 4cosxsi n(x )

.,3 , x R .

（I ）当x [0,—]时，求函数f （x ）的值域；

（H ）已知函数 y f （x ）的图象与直线y 1有交点，求相邻两个交点间的最短距离

16. （本小题满分13分）

2014年12月28日开始，北京市公共电汽车和地铁按照里程分段计价 ?具体如下表?（不

考虑公交卡折扣情况）

乘公共电汽车

方案

10公里（含）内2元；

10公里以上部分，每增加 1元可乘坐5公里（含）. 乘坐地铁方案（不含机场线）

6公里（含）内3元；

6公里至12公里（含）4元； 12公里至22公里（含）5元； 22公里至32公里（含）6元；

32公里以上部分，每增加 1元可乘坐20公里（含）.

已知在北京地铁四号线上，任意一站到陶然亭站的票价不超过 5元，现从那些只乘坐四

号线地铁，且在陶然亭站出站的乘客中随机选出

120

人

（川）小李乘坐地铁从 A 地到陶然亭的票价是

车所花交通费也是 5元，假设小李往返过程中乘坐地铁和公共电汽车的路程均为写出s

（I ）如果从那些只乘坐四号线地铁，且在

陶然亭站出站的乘客中任选

1人，试估计此人乘

坐地铁的票价小于 5元的概率;

（n ）从那些只乘坐四号线地铁，且在陶然

亭站出站的乘客中随机选 2人，记X 为这2人乘坐地铁的票价和，根据统计图，并以频率作为概率，求X 的分布列和数学期望; s 公里，试

5元，返程时，小李乘坐某路公共电汽

的取值范围.（只需写出结论）

17. (本小题满分14分)

如图，在五面体 ABCDEF 中，四边形 ABCD 是边长为4的正方形，EF//AD , 平面ADEF 平面ABCD ，且BC 2EF , AE AF ，点G 是EF 的中点.

(I)证明：AG 平面ABCD ;

(n)若直线BF 与平面ACE 所成角的正弦值为

，求AG 的长;

19.(本小题满分14分)

2 2

设F 1, F 2分别为椭圆E:务每 1(a

a b

且点P 和F 1关于点C(0,-)对称.

(I)求椭圆E 的方程;

(n)过右焦点F 2的直线I 与椭圆相交于 A , B 两点，过点P 且平行于AB 的直线

与椭

圆交于另一点Q ，问是否存在直线I ，使得四边形PABQ 的对角线互相平分？若存在，求出

I 的方程；若不存在，说明理由

(川)判断线段AC 上是否存在一点

若不存在，说明理由. 18. (本小题满分13分) 使MG //平面ABF ?若存在，求出列的值;

e n

f (x) 1零点个数,

In 二，函数g(x)

(I)当n 1时，写出函数y 设n N ,函数f(x)

(°,).

并说明理由;

(n)若曲线y f(x)与曲线

g(x)分别位于直线I : y 1的两侧，

求n 的所有可能取

b 0)的左、右焦点，点

P(1,—)在椭圆E 上,

20．(本小题满分13 分)

已知点列T:P1(x1,y1),P2(x2,y2),L ,P k(x k,y k)(k N

x i y i x i 1 1,与x i y i 1 y i x i 1,

y i 1 1

(i 2,3,L ,k) 中有且仅有一个成立．

(I)写出满足k 4且巳(3,2)的所有点列;

(H)证明：对于任意给定的k ( k N*, k>2 ),不存在点列T ,使得X i1y

(川)当k 2n 1 且P an i(n,n) (n N , n>2 )时，求X i

y i的最大

值．

2015年北京市西城区初三一模考试数学试题及答案word解读

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