初中毕业升学数学考试
（课改）
一、填空题：本大题共12小题，每小题2分，共24分．
1．比较大小：
．
2．计算：
．
3．已知
，则
的余角等于 ．
4．投一枚均匀的正方体骰子，面朝上的点数是5的概率是 ．
5．反比例函数
的图象的一个分支如图1所示，
则另一个分支在第 象限．
6．方程
的解是 ．
7．正
边形的一个外角等于
，则
．
8．已知不等式组
无解，则
的取值范围是 ．
9．如图2的围棋盘放在平面直角坐标系内，黑棋
的坐标为
，
那么白棋
的坐标是 ．
10．如图3，
中，
，
，
，则
．
11．若
，
，且
，则
．
12．如图4，
，
为
的中点，
都是半径为3的
的切线，
为切点，则
的长
为 ．
二、选择题：本大题共8小题，每小题3分，共24分．在每小题给出的四个选项中，只有一项是符合题意的，请将你认为正确答案的序号填在题后的括号内．
13．方程
的解是（ ）
Ａ．
Ｂ．
Ｃ．或
Ｄ．或
14．图5是一个正方体的展开图，将它折叠成正方体后，
“建”字的对面是（ ）
Ａ．和 Ｂ．谐 Ｃ．社 Ｄ．会
15．为了做一个试管架，在长为
的木板上钻3个小孔（如图6），每个小孔的直径为2cm，则
等于（ ）
Ａ．
cm Ｂ．
cm Ｃ．
cm Ｄ．
cm
16．哥哥身高1.68米，在地面上的影子长是2.1米，
同一时间测得弟弟的影子长1.8米，则弟弟身高是（ ）
Ａ．1.44米 Ｂ．1.52米
Ｃ．1.96米 Ｄ．2.25米
17．某超市购进了一批不同价格的运动鞋，根据近几年统计的平均数据，运动鞋单价为40元，35元，30元，25元的销售百分率分别为
，
，
，
．要使该超市销售运动鞋收入最大，该超市应多购单价为（ ）的运动鞋．
Ａ．40元 Ｂ．35元 Ｃ．30元 Ｄ．25元
18．下列等式必定成立的是（ ）
Ａ．
Ｂ．
Ｃ．
Ｄ．
19．如图7，是菱形
的对角线的交点，
分别是
的中点．下列结论：①
；②四边形
是中心对称图形；③
是轴对称图形；④
．其中错误的结论有 ．
Ａ．1个 Ｂ．2个 Ｃ．3个 Ｄ．4个
20．如图8，直线
相交于点，
，半径为1cm的
的圆心在射线
上，且与点的距离为6cm．如果以1cm/s的速度沿由向的方向移动，那么（ ）秒种后与直线
相切．
Ａ．4 Ｂ．8
Ｃ．4或6 Ｄ．4或8
三、解答题：本大题共8小题，满分共72分．解答应写出文字说明，证明过程或演算步骤．
21．（本题满分7分）
计算：
．
22．（本题满分7分）
化简：
．
23．（本题满分8分）
如图9，相交于
，现给出如下三个论断：
①
；②
；③
．
请你选择其中两个论断为条件，另外一个论断为结论，构造一个命题．
（1）在构成的所有命题中，真命题有 个．
（2）在构成的真命题中，请你选择一个加以证明．
你选择的真命题是：
（用序号表示）．
24．（本题满分8分）
|
| 14岁 | 15岁 | 16岁 |
| 频数记录 | 正正正 | 正正正正正 | 正正正 |
| 频数 | 15 | 10 | |
| 频率 | 0.5 |
右表是某班学生年龄统计表．
（1）请你把表中未填的项目补充完整；
（2）从表中可以看出，众数是 ，
中位数是 ，平均数是 ；
（3）请你根据统计表，在图10中画出该
班学生年龄统计直方图（要求标出数字）．
25．（本题满分8分）
如图11，梯形中，
，
是中位线，
于
，
于
，梯形的高
．沿着
分别把
，
剪开，然后按图中箭头所指方向，分别绕着点旋转
，将会得到一个什么样的四边形？简述理由．
26．（本题满分10分）
福林制衣厂现有24名制作服装工人，每天都制作某种品牌衬衫和裤子，每人每天可制作衬衫3件或裤子5条．
（1）若该厂要求每天制作的衬衫和裤子数量相等，则应安排制作衬衫和裤子各多少人？
（2）已知制作一件衬衫可获得利润30元，制作一条裤子可获得利润16元，若该厂要求每天获得利润不少于2100元，则至少需要安排多少名工人制作衬衫？
27．（本题满分12分）
已知抛物线
与直线
相交于点
．
（1）求抛物线的解析式；
（2）请问（1）中的抛物线经过怎样的平移就可以得到
的图象？
（3）设抛物线上依次有点
，其中横坐标依次是
，纵坐标依次为
，试求
的值．
28．（本题满分12分）
如图12，是的内接三角形，直径
，交
于
，
，
的延长线相交于．
（1）求证：
；
（2）若
，试求的半径；
（3）当
是什么类型的弧时，
的外心在的外部、内部、一边上？（只写结论，不用证明）
初中毕业数学升学考试
课改 参考答案
一、填空题
1．＞ 2．
3．
4．
5．四 6． 7．18 8．
9．
10．
11．
12．
二、选择题
13．Ｄ 14．Ｃ 15．Ｃ 16．Ａ 17．Ｂ 18．Ｃ
19．Ａ 20．Ｄ
三、解答题
21．解：原式
····················································································· 5分
．······································································································ 7分
22．解法一：原式
···································································· 2分
························································································· 4分
．···························································································· 7分
解法二：原式
········································································ 2分
·············································································· 4分
．······················································································· 7分
解法三：原式
················································································ 4分
．··································································································· 7分
23．（1）2····················································································································· 2分
（2）你选择的真命题是：
······································································· 4分
证明：在
和
中，
，
．·································································································· 6分
．··············································································································· 8分
选择命题二：
······························································································· 4分
证明：在
和
中，
，
，
．······································································································ 6分
．················································································································· 8分
24．解：（1）
|
| 14岁 | 15岁 | 16岁 |
| 频数记录 | 正正正 | 正正正 正正 | 正正 |
| 频数 | 15 | 25 | 10 |
| 频率 | 0.3 | 0.5 | 0.2 |
（2）15岁，15岁，14.9岁；
（3）
25．解：将会得到一个正方形，理由如下：··································································· 1分
，
．············································································································· 2分
是梯形的中位线，
，
．·············································································································· 4分
梯形的高
，
梯形的高
．··································································································· 5分
设绕点旋转后点落在
处，绕点
旋转后，点落在
处则
，
在
所在的直线上．
是梯形的高．························································································· 6分
，
．······················································ 7分
四边形
是正方形．······················································································ 8分
26．解：（1）设应安排名工人制作衬衫，依题意，得················································· 1分
．·········································································································· 3分
解之，得
．········································································································· 4分
．
答：应安排15名工人制作衬衫，9名工人制作裤子．···················································· 5分
（2）设应安排
名工人制作衬衫，依题意，得····························································· 6分
．··············································································· 8分
解之，得
．······································································································· 9分
答：至少应安排18名工人制作衬衫．·········································································· 10分
27．解：（1）点在直线上，
．····································································································· １分
把
代入，
得
．求得
．·················································································· 3分
抛物线的解析式是
．······································································· 4分
（2）
．
顶点坐标为
．···································································································· 6分
把抛物线向左平移3个单位长度得到
的图象，再把的图象向下平移1个单位长度得到
的图象．······························································································· 7分
（3）由题意知，
的横坐标是连续偶数，所以
的横坐标是
，纵坐标为
所对应的纵坐标依次是
．············································································································· 9分
····················································································· 10分
．········································································· 12分
28．（1）证明：连结
．··························································································· 1分
，
．
．················································································· 2分
，
．
．··································································································· 3分
又
，
．········································································································ 4分
（2）解：连结
，则
．·································································· 5分
，
．········································································································ 6分
，
．································································································· 7分
．············································································································· 8分
．
，即的半径为2．··················································································· 9分
（3）解：当是劣弧时，的外心在的外部．······························· 10分
当是半圆时，的外心在的边上．··············································· 11分
当是优弧时，的外心在的内部．··············································· 12分