初中毕业生学业考试数学试卷
说明：全卷共8页，考试时间90分钟，满分120分．
| 题号 | 一 | 二 | 三 | 总分 | |||||||||
| 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | ||||
| 得分 | |||||||||||||
一、选择题：每小题3分，共15分；每小题给出四个答案，其中有一个是正确的，把所选答案的编号填写在题目后面的括号内．
1．
等于（ ）
Ａ．2 Ｂ．
Ｃ．
Ｄ．
2．小明在镜中看到身后墙上的时钟，实际时间最接近8时的是下图中的（ ）
3．我市大部分地区今年5月中、下旬的天气情况是：前5天小雨，后5天暴雨．那么能反映我市主要河流水位变化情况的图象大致是（ ）
4．如图1，把矩形
沿
对折，若
，则
等于（ ）
Ａ．
Ｂ．
Ｃ．
Ｄ．
5．在同一平面直角坐标系中，直线
与双曲线
的交点个数为（ ）
Ａ．0个 Ｂ．1个 Ｃ．2个 Ｄ．无法确定
二、填空题：每小题3分，共30分；答案填写在该题的横线上．
6．我市约有495万人口，用科学记数法表示为 人．
7．如果一个几何体的主视图是等腰三角形，那么这个几何体可以是 ．（填上满足条件的一个几何体即可）
8．一个袋中装有6个红球、4个黑球、2个白球，每个球除颜色外完全相同，从袋中任意摸出一个球，那么摸出 球的可能性最大．
9．计算：
．
10．计算：
．
11．将抛物
向左平移1个单位后，得到的抛物线的解析式是 ．
12．当
时，分式
的值为零．
13．能使平行四边形为正方形的条件是
．（填上一个符合题目要求的条件即可）
14．如图2，两个半圆中，小圆的圆心
在大
的直径
上，
长为4的弦
与直径平行且与小半圆相切，那么圆中阴影部
分面积等于 ．
15．如图3，已知
的周长为
，分别连接
的中点
得
，再连接
的中点
得
，再连接
的中点
得
，这样延续下去，最后得
．
设的周长为
，的周长为
，
的周长为
， 的周长为
，则
．
三、解答题：本大题有10小题，共75分．
16．本小题满分6分．
因式分解：
．
17．本小题满分6分．
解不等式组：
18．本小题满分6分．
如图4是某文具店在2005年卖出供学生使用的甲、乙、丙三种品牌科学计算器个数的条形统计图，试解答下面问题：
（1）求卖出甲、乙、丙三种科学计算器的个数的频率；
（2）根据以上统计结果，请你为该文具店进货提出一条合理化建议．
 |
19．本小题满分6分．
如图5，已知的顶点
的坐标分别是
．
（1）作出关于原点
中心对称的图形；
（2）将绕原点按顺时针方向旋转
后得到，画出，并写出点
的坐标．
20．本小题满分7分．
小明与小华在玩一个掷飞镖游戏，如图
甲是一个把两个同心圆平均分成8份的靶，当飞镖掷中阴影部分时，小明胜，否则小华胜（没有掷中靶或掷到边界线时重掷）．
（1）不考虑其他因素，你认为这个游戏公平吗？说明理由．
（2）请你在图乙中，设计一个不同于图甲的方案，使游戏双方公平．
 |
21．本小题满分7分．
梅华中学九年级数学课外学习小组某下午实践活动课时，测量朝西教学楼前的旗杆的高度．如图7，当阳光从正西方向照射过来时，旗杆的顶端
的影子落在教学楼前的坪地
处，测得影长
与地面的夹角
．在同一时刻，测得一根长为1m的直立竹竿的影长恰为4m．根据这些数据求旗杆的高度．（可能用到的数据：
，结果保留两个有效数字）
D
22．本小题满分8分．
某公司开发生产的1200件新产品需要精加工后才能投放市场，现有甲、乙两个工厂都想加工这批产品．公司派出相关人员分别到这两间工厂了解生产情况，获得如下信息：
信息一：甲工厂单独加工完成这批产品比乙工厂单独加工完成这批产品多用10天；
信息二：乙工厂每天比甲工厂多加工20件．
根据以上信息，求甲、乙两个工厂每天分别能加工多少件新产品？
23．本小题满分8分．
用两个全等的正方形和
拼成一个矩形
，把一个足够大的直角三角尺的直角顶点与这个矩形的边
的中点
重合，且将直角三角尺绕点按逆时针方向旋转．
（1）当直角三角尺的两直角边分别与矩形的两边
相交于点
时，如图
甲，通过观察或测量
与
的长度，你能得到什么结论？并证明你的结论．
（2）当直角三角尺的两直角边分别与
的延长线，的延长线相交于点时（如图乙），你在图甲中得到的结论还成立吗？简要说明理由．
24．本小题满分10分．
如图9，直线
的解析式为
与
轴，
轴分别交于点
．
（1）求原点到直线的距离；
（2）有一个半径为1的
从坐标原点出发，以每秒1个单位长的速度沿轴正方向运动，设运动时间为
（秒）．当与直线相切时，求的值．
25．本小题满分11分．
如图10，点在抛物线
上，过点作与轴平行的直线交抛物线于点
，延长
分别与抛物线
相交于点
，连接
，设点的横坐标为，且
．
（1）当
时，求点
的坐标；
（2）当为何值时，四边形的两条对角线互相垂直；
（3）猜想线段与之间的数量关系，并证明你的结论．
 |
初中毕业生学业考试
数学试卷参考答案及评分意见
一、选择题：每小题3分，共15分
1．D 2．B 3．B 4．A 5．C
二、填空题：每小题3分，共30分
6．
； 7．圆锥或正三棱锥或正四棱锥； 8．红； 9．3 10．
；
11．
； 12．
； 13．
且
或
且
等；
14．
； 15．
．
三、解答下列各题：共75分
16．解：原式
············································································· 2分
····················································································· 4分
············································································· 6分
17．解：由
，得
，······································································· 2分
由
，得
，·············································································· 4分
原不等式组的解集是：
．····························································· 6分
18．解：（1）卖出甲计算器个数的频率：
········································ 2分
卖出乙计算器个数的频率：
········································· 3分
卖出丙计算器个数的频率：
．····································· 4分
（2）
，·············································································· 5分
该文具店进甲、乙、丙三种科学计算器时，按
的比例进货．············· 6分
（或该文具店进货时，丙科学计算器进多一些，而甲、乙科学计算器进少一些．类 似这样的合理答案5分）
19．（1）正确画出图形·········································································· 3分
（2）正确画出图形·········································································· 5分
························································································ 6分
20．解：（1）这个游戏公平．······························································· 2分
根据图
甲的对称性，阴影部分的面积等于圆面积的一半，
这个游戏公平．·································································································· 4分
（2）把图乙中的同心圆平均分成偶数等分，再把其中的一半作为阴影部分即可．（图略） 7分
21．解：如图，过点
分别作
于点
的延长线于
．······· 1分
在
中，
，··································································· 2分
m··········································· 3分
又
········································································································· 5分
．···································································· 6分
．··············································································· 7分
22．解：设甲工厂每天能加工件新产品，··································································· 1分
则乙工厂每天能加工
件新产品．······························································· 2分
依题意得方程 
．····································································· 4分
解得
或
（不合题意舍去），····························································· 6分
经检验是所列方程的解，
．····································································································· 7分
答：甲工厂每天能加工40件新产品，乙工厂每天能加工60件新产品．·················· 8分
23．解：（1）
．···························································································· 2分
四边形和都是正方形，
，
，·················· 3分
，
，······································································································· 4分
．··················································································· 5分
（2）结论仍然成立．··················································································· 6分
同理可证
，
．······························································ 8分
24．解：（1）在
中，令
，得
，得
．
令
，得
，得
，
．····················································· 2分
设点到直线的距离为
，
，
．·························································································· 4分
（其它解法参照给分）
（2）如图，设与直线相切于点，连，则
，······························· 5分
，
······································································· 6分
由（1）得
，
，
（秒）．···························································································· 8分
根据对称性得
，
（秒）．························································· 9分
当与直线相切时，
秒或
秒．······················································· 10分
25．解：（1）点在抛物线上，且
，
，······················· 1分
点与点关于轴对称，
．························································ 2分
设直线
的解析式为
，
．······················································································· 3分
解方程组
，得
．································································· 4分
（2）当四边形的两对角线互相垂直时，由对称性得直线
与轴的夹角等于
所以点的纵、横坐标相等，···························································································································· 5分
这时，设
，代入，得
，
．
即当
时，四边形的两条对角线互相垂直．········································· 7分
（3）线段
．································································································ 8分
点在抛物线，且
，
得直线的解析式为
，
解方程组
，得点
······················································· 9分
由对称性得点
，················································· 10分
，
．···································································································· 11分