高中阶段学校招生考试数学试卷课改实验区
说明：全卷共8页，考试时间90分，满分150分．
一、选择题（每小题3分，共36分，每小题给出4个答案，其中只有一个正确，把所选答案的编号写在题目后面的括号内）
1．
的相反数是（ ）
A． B．
C．
D．
2．今年我市参加中考的人数约是
，数据用科学记数法表示为（ ）
A．
B．
C．
D．
3．在下列长度的四根木棒中，能与3cm，7cm两根木棒围成一个三角形的是（ ）
A．7cm B．4cm C．3cm D．10cm
4．下列运算正确的是（ ）
A．
B．
C．
D．
5．点
关于
轴对称的点的坐标是（ ）
A．
B．
C．
D．
6．下图中所示的几何体的主视图是（ ）
 |
7．下列事件是必然事件的是（ ）
A．今年10月1日湛江的天气一定是晴天
B．2008年奥运会刘翔一定能夺得110米跨栏冠军
C．当室外温度低于
℃时，将一碗清水放在室外会结冰
D．打开电视，正在播广告
8．图1是
两国2005年财政经费支出情况的扇形统计图．根据统计图，下面对两国全年教育经费支出判断正确的是（ ）
A．
国比
国多
B．国比国多
C．国与国一样多
D．无法确定哪国多
9．数据12，10，13，8，17，10，21的中位数是（ ）
A．8 B．10 C．13 D．12
10．在一个不透明的口袋中，有大小、形状完全相同，颜色不同的球15个，从中摸出红球的概率为
，则袋中红球的个数为（ ）
A．10 B．15 C．5 D．3
11．小颖从家出发，直走了20分钟，到一个离家1000米的图书室，看了40分钟的书后，用15分钟返回到家，下图中表示小颖离家时间与距离之间的关系的是（ ）
12．如图2，
的半径为5，弦
的长为8，点
在线段（包括端点
）上移动，则
的取值范围是（ ）
A．
B．
C．
D．
二、填空题（每小题3分，共24分，请把答案填在横线上）
13．分解因式：
．
14．请写出一个图象位于第二、四象限的反比例函数： ．
15．数据100，99，99，100，102，100的方差
．
16．如图3，已知直线
，
，
，则
度．
17．图4是平面镜里看到背向墙壁的电子钟示数，这时的实际时间应该是 ．
 | |||
 | |||
18．下面是一个简单的数值运算程序，当输入的值为2时，输出的数值是 ．
 |
19．如果一个扇形的圆心角为
，半径为
，那么该扇形的弧长是 ．
20．观察下列顺序排列的等式：
，…．试猜想第
个等式（为正整数）：
．
三、解答题（每小题6分，共30分）
21．计算：
．
22．先化简，再求值：
，其中
．
23．如图5，请你画出方格纸中的图形关于点
的中心对称图形，并写出整个图形的对称轴的条数．
 |
24．近年来，我市开展以“四通五改六进村”为载体，以生态文明为主要特色的新农村建设活动取得了明显成效．下面是市委领导和市民的一段对话，请你根据对话内容，替市领导回答市民提出的问题（结果精确到0.1%）．
 |
领导
市民
25．如图6，点
分别为四边形
的边
的中点，试判断四边形
的形状，并证明你的结论．
 |
四、解答题（每小题9分，共36分）
26．小刘同学为了测量雷州市三元塔的高度，如图7，她先在
处测得塔顶
的仰角为
，再向塔的方向直行35米到达
处，又测得塔顶的仰角为
，请你帮助小刘计算出三元塔的高度（小刘的身高忽略不计，结果精确到1米）．
 |
27．为了让学生了解安全知识，增强安全意识，我市某中学举行了一次“安全知识竞赛”．为了了解这次竞赛成绩情况，从中抽取了部分学生的成绩（得分取整数，满分为100分）为样本，绘制成绩统计图，如图8所示，请结合统计图回答下列问题：
（1）本次测试的样本容量是多少？
（2）分数在80.5~90.5这一组的频率是多少？
（3）若这次测试成绩80分以上（含80分）为优秀，则优秀人数不少于多少人？
 |
28．某工厂现有甲种原料280kg，乙种原料190kg，计划用这两种原料生产两种产品50件，已知生产一件产品需甲种原料7kg、乙种原料3kg，可获利400元；生产一件产品需甲种原料3kg，乙种原料 5kg，可获利350元．
（1）请问工厂有哪几种生产方案？
（2）选择哪种方案可获利最大，最大利润是多少？
29．如图9，是的直径，
平分
，交于点
，过点作直线
，交
的延长线于点
，交的延长线于点．
（1）求证：
是的切线；
（2）若
，
，求的长．
五、解答题（每小题12分，共24分）
30．如图10，在
中，
，
，把边长分别为
的个正方形依次放入
中，请回答下列问题：
（1）按要求填表
|
| 1 | 2 | 3 |
|
|
（2）第个正方形的边长
；
（3）若
是正整数，且
，试判断的关系．
 |
31．已知抛物线
与轴相交于点
，
，且
是方程
的两个实数根，点为抛物线与
轴的交点．
（1）求
的值；
（2）分别求出直线
和
的解析式；
（3）若动直线
与线段
分别相交于
两点，则在轴上是否存在点，使得
为等腰直角三角形？若存在，求出点的坐标；若不存在，说明理由．
 |
高中阶段学校招生考试数学试题课改实验区
参考答案及评分标准
一、选择题（每小题3分，共36分）
| 题号 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| 答案 | B | C | A | B | D | D | C | D | D | C | A | A |
二、填空题（每小题3分，共24分）
13．
14．
等 15．1 16．80 17．
18．1 19．
20．
三、解答题（每小题6分，共30分）
21．解：原式
··························································································· 4分
．······································································································ 6分
22．解：原式
························································································· 2分
······························································································ 3分
··································································································· 4分
当时，原式
························································································ 5分
························································································· 6分
23．解：如图1，·········································································································· 4分
共有4条对称轴．········································································································· 6分
 |
24．解：设平均每年生态文明村增长率是，根据题意，得·········································· 1分
····················································································· 3分
解得：
（不合题意，舍去）·················································· 5分
答：平均每年生态文明村增长率约是18.1%．································································ 6分
25．解：四边形是平行四边形··········································································· 1分
证明：连结，如图2．
分别是
的中点，
是的中位线，……………………2分
，且
．………………3分
同理：
，且
，…………4分
．……………………………………5分
四边形是平行四边形．·················································································· 6分
26．解：在
中，
．············································································································ 2分
在
中，
．············································································································ 4分
，
，··························································································· 6分
（米）········································································ 8分
答：三元塔的高度约是34米．······················································································ 9分
27．解：（1）
，
本次测试的样本容量是100．····················································································· 3分
（2）
．
分数在80.5~90.5这一组的频率是0.52．···································································· 6分
（3）
，
优秀人数不少于75人．····························································································· 9分
28．解：（1）设生产产品件，生产产品
件，则······································· 1分
································································································· 2分
解得：
．······························································································· 3分
为正整数，可取30，31，32．
当
时，
，
当
时，
，
当
时，
，·························································································· 4分
所以工厂可有三种生产方案，分别为：
方案一：生产产品30件，生产产品20件；
方案二：生产产品31件，生产产品19件；
方案三：生产产品32件，生产产品18件；··························································· 5分
（2）方案一的利润为：
元；
方案二的利润为：
元；
方案三的利润为：
元．······················································ 8分
因此选择方案三可获利最多，最大利润为19100元．····················································· 9分
29．（1）证明：连结
，如图3．
平分，
．……………………1分
，
，……………………2分
，
．……………………………3分
，
，
是的切线．·································································································· 4分
（2）设
是的半径，
在
中，
············································································ 5分
即
．
解得
．·················································································································· 6分
，
，
．·································································································· 7分
即
．
解得
．·························································································· 8分
．························································································ 9分
30．（1）
········································································································ 6分
（2）
．················································································································ 8分
（3）
······················································································· 10分
．································································································· 11分
．······································································································· 12分
31．解：（1）由，得
．
，··································································································· 1分
把两点的坐标分别代入联立求解，得
．······································································································ 2分
（2）由（1）可得
，
当
时，
，
．
设
，把
两点坐标分别代入
，联立求得
．直线的解析式为
．····················································· 3分
同理可求得直线的解析式是
．···························································· 4分
（3）假设存在满足条件的点，并设直线
与轴的交点为
．
①当
为腰时，分别过点作
轴于
，作
轴于
，如图4，则
和
都是等腰直角三角形，
，
．
，
，
，即
．
解得
．················································································································ 6分
点的纵坐标是
，点在直线上，
，解得
，
．
，同理可求
．··············································································· 8分
②当为底边时，
过的中点
作
轴于点
，如图5，
则
，
由
，
得
，即
，
解得
．…………………………………………9分
同1方法．求得
，
，
．····························································· 11分
结合图形可知，
，
，
是
，也满足条件．
综上所述，满足条件的点共有3个，即
．················ 12分
说明：以上各题如有其他解（证）法，请酌情给分．