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2005年北京市海淀区数学中考试卷

2005年北京市海淀区数学中考试卷
2005年北京市海淀区数学中考试卷

(海淀卷)数学试卷第1页(共4页)

北京市2005年高级中等学校招生统一考试(海淀卷)

数 学 试 卷

学校 姓名 准考证号

一、选择题:(本题共24分,每小题4分)

在下列各题的四个备选答案中,只有一个是正确的.

1.一个数的相反数是3,则这个数是

A .1

3

- B .13

C .3-

D .3

2.同时抛掷两枚质地均匀的正方体骰子,骰子的六个面上分别刻有1到6 的点数, 下列事件中是不可能事件的是 A .点数之和为12

B .点数之和小于3

C .点数之和大于4且小于8

D .点数之和为13 3.已知2(1)|2|0m n -++=, 则m+n 的值为

A .-1

B .-3

C .3

D .不确定

4.如图,C 是⊙O 上一点,O 是圆心. 若∠C =35°,则∠AOB

的度数为 A .35° B .70° C .105°

D .150°

5.如图,电线杆AB 的中点C 处有一标志物,在地面D 点处测得标志物的仰角为45°,若点D 到电线杆底部点B 的距离为 a ,则电线杆AB 的长可表示为 A .a B .2a C .3

2

a D .52

a

C

A

B

O

D

C

B

A

6.用一块等边三角形的硬纸片(如图1)做一个底面为等边三角形且高相等的无盖的盒子(边缝忽略不计,如图2),在△ABC的每个顶点处各需剪掉一个四边形,其中四边形AMDN中,∠MDN的度数为

A.100°B.110°

C.120°

D.130°

二、填空题:(本题共24分,每小题

4分)

7.103000用科学记数法可表示为

.

8

.函数y=

中,自变量x的取值范围是

.

9.某校初三(2)班想举办班徽设计比赛,全班50

名同学,计划每位同学交设计方案一份,拟评选出10份为一等奖,那么该班某位同学获一等奖的概率为

.

10

、a,b,都有=

a和= b .

例如,,,则()= .

11.已知圆柱的底面半径为2cm,母线长为3cm,则该圆柱的侧面展开图的面积为

cm2.

12.把编号为1,2,3,4,…的若干盆花按右图所示摆放,

花盆中的花按红、黄、蓝、紫的颜色依次循环排列,则

第8行从左边数第6盆花的颜色为色.

三、解答题:(本题共30分,每小题5分)

13.计算:310

22(tan30cos45)

-

-??-?.

14.先化简,再求值:

2

62

393

m

m m m

+--

,其中2

m=-.

15.解方程组

41,

216.

x y

x y

-=-

?

?

+=

?

16.解不等式21

x-≥

101

.

6

x+

17.如图,矩形ABCD中,AC与BD交于O点,BE⊥AC

于E,CF⊥BD于F.

求证BE = CF.

图1 图2

B

A

D

M N

A

C

B

D

O

F

E

(海淀卷)数学试卷第2页(共4页)

(海淀卷)数学试卷第3页(共4页)

18.如图,梯形ABCD 中,AB // DC ,∠B = 90°,E 为BC 上

一点,且AE ⊥ED .若BC = 12,DC = 7,BE ∶EC = 1∶2,求AB 的长.

四、解答题:(本题共20分,第19、21题各5分,第20题4分,第22题6分)

19.已知反比例函数k y x =

的图象经过点1

(4,),2

若一次函数1y x =+的图象平移后经过该反比例函数图象上的点B (2,m ),求平移后的一次函数图象与x 轴

的交点坐标.

20.甲、乙两名运动员在6次百米跑训练中的成绩如下.(单位:秒)

请你比较这两组数据中的众数、平均数、中位数,谈谈你的看法.

21.如图,△ABO 中,OA = OB ,以O 为圆心的圆经过AB 中点C ,且分别交OA 、

OB 于点E 、F . (1)求证 AB 是⊙O 的切线;

(2)若△ABO 腰上的高等于底边的一半,且

AB

= 求 ECF

的长.

22.印制一本书,为了使装订成书后页码恰好为连续的自然数,可按如下方法操作:

先将一张整版的纸,对折一次为4页,再对折一次为8页,连续对折三次为16页,…… ;然后再排页码.如果想设计一本16页的毕业纪念册,请你按 图1、图2、图3(图中的1,16表示页码)的方法折叠,在图4中填上按这种折叠方法得到的各页在该面相应位置上的页码.

16

1

图4

图1 图3 图2 116

1

161

E D

C

B

A

F

E

C

B

O

A

(海淀卷)数学试卷第4页(共4页)

五、解答题:(本题共22分,第23题6分,第24题8分,第25题8分)

23.如图所示,一根长2a 的木棍(AB ),斜靠在与

地面(OM )垂直的墙(ON )上,设木棍的中点为P . 若木棍A 端沿墙下滑,且B 端沿地面向右滑行.

(1)请判断木棍滑动的过程中,点P 到点O 的

距离是否变化,并简述理由.

(2)在木棍滑动的过程中,当滑动到什么位置

时,△AOB 的面积最大?简述理由,并求出面积的最大值.

24.已知抛物线22y x mx m =-+-.

(1)求证此抛物线与x 轴有两个不同的交点;

(2)若m 是整数,抛物线22y x mx m =-+-与x 轴交于整数点,求m 的值; (3)在(2)的条件下, 设抛物线顶点为A ,抛物线与x 轴的两个交点中右侧交

点为B .若M 为坐标轴上一点,且MA = MB ,求点M 的坐标.

25.已知△ABC ,分别以AB 、BC 、CA 为边向形外作等边三角形ABD 、等边三角形

BCE 、等边三角形ACF .

(1)如图1,当△ABC 是等边三角形时,请你写出

满足图中条件,四个成立的结论;

(2)如图2,当△ABC 中只有∠ACB = 60° 时,请你

证明ABC S 与ABD S 的和等于BCE S 与ACF S 的和.

图1

F

E D

C

B A

图2

A

B D

F

C

E

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