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数 学
考生注意：本学科试卷共三道大题25小题，满分120分，考试时量120分钟．
一、选择题（本大题共8小题，每小题3分，满分24分）
1．在实数0， ， ，0.74， 中，无理数有（ ）
A．1个 B．2个 C．3个 D．4个
2．用计算器求 值时，需相继按“2”，“ ”，“3”，“ ”键，若小红相继按“ ”，“2”，“ ”，“4”，“ ”键，则输出结果是（ ）
A．4 B．5 C．6 D．16
3．下图所示的几何体的主视图是（ ）
4．不等式组 的解集在数轴上表示为（ ）
5．下列运算正确的是（ ）
A． B．
C． D．
6．下列不是必然事件的是（ ）
A．两直线相交，对顶角相等 B．三角形的外心到三个顶点的距离相等
C．三角形任意两边之和大于第三边 D．两相似多边形面积的比等于周长的比
7．如图， ，且 ， ，
则 的度数是（ ）
A． B．
C． D．
8．为了预防“HINI”流感，某校对教室进行药熏消毒，药品燃烧时，室内每立方米的含药量与时间成正比；燃烧后，室内每立方米含药量与时间成反比，则消毒过程中室内每立方米含药量 与时间 的函数关系图象大致为（ ）
二、填空题（本大题共8小题，每小题3分，满分24分）
9． 的绝对值为 ．
10．如图， 是 的内切圆，与边
的切点分别为 ，若 ，则 ．
11．张家界国际乡村音乐周活动中，来自中、日、美的三名音乐家准备在同一节目中依次演奏本国的民族音乐，若他们出场先后的机会是均等的，则按“美—日—中”顺序演奏的概率是 ．
12．将函数 的图象向上平移2个单位，得到函数 的图象．
13．分解因式 ．
14．我市甲、乙两景点今年5月上旬每天接待游客的人数如图所示，甲、乙两景点日接待游客人数的方差大小关系为： ．
15．对于正实数 作新定义： ，在此定义下，若 ，则 的值为 ．
16．如图，等腰梯形 中， ，且 ， 为 上一点， 与 交于点 ，若 ，则 ．
三、解答题（本题共9小题，满分72分）
17．（本小题6分）
计算
18．（本小题6分）
小明将一幅三角板如图所示摆放在一起，发现只要知道其中一边的长就可以求出其它各边的长，若已知 ，求 的长．
19．先化简，后求值（本小题6分）
其中
20．（本小题6分）
在建立平面直角坐标系的方格纸中，每个小方格都是边长为1的小正方形， 的顶点均在格点上，点 的坐标为 ，请按要求画图与作答
（1） 把 绕点 旋转 得 ．
（2）把 向右平移7个单位得 ．
（3） 与 是否成中心对称，若是，找出对称中心 ，并写出其坐标．
21．列方程解应用题（本小题9分）
“阳黄公路”开通后，从长沙到武陵源增加了一条新线路，新线路里程在原线路长360Km的基础上缩短了50Km，今有一旅游客车和小车同时从长沙出发前往武陵源，旅游客车走新线路，小车因故走原线路，中途停留6分钟．若小车速度是旅游客车速度的1.2倍，且两车同时到达武陵源，求两车的速度各是多少？
22．（本小题9分）
如图，有两个动点 分别从正方形 的两个顶点 同时出发，以相同速度分别沿边 和 移动，问：
（1）在 移动过程中， 与 的位置和大小有何关系？并给予证明．
（2）若 和 相交点 ，图中有多少对相似三角形？请把它们写出来．
23．（本小题9分）
我市今年初三体育考试结束后，从某县3000名参考学生中抽取了100名考生成绩进行统计分析（满分100分，记分均为整数），得到如图所示的频数分布直方图，请你根据图形完成下列问题：
（1）本次抽样的样本容量是 ．
（2）请补全频数分布直方图．
（3）若80分以上（含80分）为优秀，请你据此．估算该县本次考试的优秀人数．
24．（本小题9分）
有若干个数，第1个数记为 ，第2个数记为 ，第3个数记为 ， 第 个数记为 ，若 ，从第二个数起，每个数都等于1与前面那个数的差的倒数．
（1）分别求出 的值．
（2）计算 的值．
25．（本小题12分）
在平面直角坐标系中，已知 ， ，且以 为直径的圆交 轴的正半轴于点 ，过点 作圆的切线交 轴于点 ．
（1）求过 三点的抛物线的解析式
（2）求点 的坐标
（3）设平行于 轴的直线交抛物线于 两点，问：是否存在以线段 为直径的圆，恰好与 轴相切？若存在，求出该圆的半径，若不存在，请说明理由？
张家界市2009年初中毕业学业考试数学试卷答案
一、选择题
1．B 2．A 3．B 4．A 5．C 6．D 7．C 8．A
二、填空题
9． 10． 11． 12．
13． 14． 15．16 16．
三、解答题
17．原式 ·································································· 3分
································································································· 4分
···································································································· 5分
······························································································································· 6分
18．解：
······························································································ 2分
，则
·································································································· 4分
··················································································································· 5分
······································································································· 6分
19．解：原式
························································· 2分
································································································ 3分
························································································································ 4分
当 时
········································································································ 6分
20．注：每问2分
（3）
21．解：设旅游客车速度为 Km/h，则小车为 Km/h··············································· 1分
·········································································································· 3分
解方程得 ·········································································································· 7分
经检验 是方程的根，且合题意 Km/时········································· 8分
答：小车的平均速度为120Km/时··················································································· 9分
22．解：（1）在正方形 中， ，
·················································································································· 1分
（SAS）························································································ 2分
······································································································· 3分
······························································································ 4分
在 中，
················································································································· 6分
（2）有5对相似三角形································································································· 7分
··································································· 9分
23．（1）100·················································································································· 2分
（2）···························································································································· 5分
（3）
该县优秀人数约为1800人····························································································· 9分
24．解：（1） ··········································································· 2分
······································································································· 4分
············································································································ 6分
（2） ······················································ 9分
25．解：（1）令二次函数 ，则
········································································································· 1分
··················································································································· 2分
过 三点的抛物线的解析式为 ········································ 4分
（2）以 为直径的圆圆心坐标为
·································································································· 5分
为圆 切线 ················································································ 6分
······························································ 8分
坐标为 ········································································································ 9分
（3）存在···················································································································· 10分
抛物线对称轴为
设满足条件的圆的半径为 ，则 的坐标为 或
而 点在抛物线 上
故在以 为直径的圆，恰好与 轴相切，该圆的半径为 ， ··········· 12分
注：解答题只要方法合理均可酌情给分
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