2022年山东省菏泽市单县中考三模数学试题(word版含答案)

2022年初中学业水平模拟测试

数学试题（三）

注意事项：

1.本试题共24个题，满分120分，考试时间120分钟。

2.请把答案写在答题卡上，选择题用2B铅笔填涂，非选择题用0.5毫米的黑色墨水签字笔书写在答题卡的指定区域内，写在其它区域不得分。

一、选择题（本大题共8个小题，每小题3分，共24分，在每小题给出的四个选项中，只有一个选项是正确的，请把正确选项的序号涂在答题卡的相应位置.）

1.实数a，b在数轴上的位置如图所示，则下列式子正确的是（    ）

A. B. C. D.

2.一列数4，5，6，4，4，7，x的平均数是5，则中位数和众数分别是（    ）

A.4，4 B.5，4 C.5，6 D.6，7

3.下列运算正确的是（    ）

A. B. C. D.

4.已知二元一次方程组，则的值为（    ）

A.2 B.6 C.-2 D.-6

5.已知，，则的值是（    ）

A.2 B. C.3 D.

6.如图，在中，，，，将绕点A逆时针旋转得到，使点落在边上，连结，则的值为（    ）

A. B. C. D.

7.如图，是的外接圆，交于点E，垂足为点D，，的延长线交于点F.若，，则的长是（    ）

A.10 B.8 C.6 D.4

8.如图，二次函数的图象经过点，，与y轴交于点C.下列结论：

①         ②当时，y随x的增大而增大  ③  ④.

其中正确的个数有（    ）

A.1个 B.2个 C.3个 D.4个

二、填空题（本大题共6个小题，每小题3分，共18分.把结果填写在答题卡相应区域内）

9.若一个扇形的圆心角为60°，面积为，则这个扇形的弧长为_________（结果保留）.

10.已知是一元二次方程的一个根，则m的值为_________.

11.关于x的不等式组恰好有2个整数解，则实数a的取值范围是_________.

12.如图，在平面直角坐标系中，的顶点A、B的坐标分别为、.把沿x轴向右平移得到，如果点D的坐标为，则点E的坐标为_________.

13.如图所示，反比例函数的图象经过矩形的边的中点D，则矩形的面积为________.

14.如图，在平面直角坐标系中，将边长为1的正方形绕点O顺时针旋转45°后得到正方形，依此方式，绕点O连续旋转2019次得到正方形，那么点的坐标是_________.

三、解答题（本大题共78分，把必要的证明过程或演算步骤写在答题卡的相应区域内）

15.（6分）计算：.

16.（6分）先化简，再求值：，其中a，b满足.

17.（6分）如图，在和中，，.

（1）求证：；

（2）若，，求的长.

18.（6分）由我国完全自主设计，自主建造的首艘国产航母于2018年5月成功完成首次海上试验任务.如图，航母由西向东航行，到达B处时，测得小岛A在北偏东60°方向上，航行20海里到达C点，这时测得小岛A在北偏东30°方向上，小岛A周围10海里内有暗礁，如果航母不改变航线继续向东航行，有没有触礁危险？请说明理由.

19.（7分）如图，已知反比例函数（）的图象和一次函数的图象都过点，过点P作y轴的垂线，垂足为A，O为坐标原点，的面积为1.

（1）求反比例函数和一次函数的表达式；

（2）设反比例函数图象与一次函数图象的另一交点为M，过M作x轴的垂线，垂足为B，求五边形的面积.

20.（7分）某电商在抖音上对一款成本价为40元的小商品进行直播销售，如果按每件60元销售，每天可卖出20件.通过市场调查发现，每件小商品售价每降低5元，日销售量增加10件.若日利润保持不变，商家想尽快销售完该款商品，每件售价应定为多少元？

21.（10分）为加快推进生活垃圾分类工作，其中，可回收物用蓝色收集桶，有害垃圾用红色收集桶，厨余垃圾用绿色收集桶，其他垃圾用灰色收集桶.为了解学生对垃圾分类知识的掌握情况，某校宜传小组就“用过的餐巾纸应投放到哪种颜色的收集桶”在全校随机采访了部分学生，根据调查结果，绘制了如图所示的两幅不完整的统计图.根据图中信息，解答下列问题：

（1）此次调查一共随机采访了_________名学生，在扇形统计图中，“灰”所在扇形的圆心角的度数为_________度；

（2）补全条形统计图（要求在条形图上方注明人数）；

（3）若该校有3600名学生，估计该校学生将用过的餐巾纸投放到红色收集桶的人数；

（4）李老师计划从A，B，C，D四位学生中随机抽取两人参加学校的垃圾分类知识抢答赛，请用树状图法或列表法求出恰好抽中A，B两人的概率.

22.（10分）如图，已知是的直径，C为上一点，的角平分线交于点D，F在直线上，且，垂足为E，连接、.

（1）求证：是的切线；

（2）若，的半径为3，求的长.

23.（10分）如图1，在中，，，点D、E分别在边、上，，连接，点M，P，N分别为、、的中点.

（1）观察猜想；图1中，线段与的数量关系是___________，位置关系是___________.

（2）探究证明：把绕点A逆时针方向旋转到图2的位置，连接，，，判断的形状，并说明理由；

（3）拓展延伸：把绕点A在平面内自由旋转，若，，请直接写出面积的最大值.

24.（10分）如图，抛物线与x轴交于A、B两点，与y轴交于C点，，，连接和.

（1）求抛物线的表达式；

（2）点Ｄ在抛物线的对称轴上，当的周长最小时，点D的坐标为___________.

（3）点E是第四象限内拋物线上的动点，连接和.求面积的最大值及此时点E的坐标；

（4）若点M是y轴上的动点，在坐标平面内是否存在点N，使以点A、C、M、N为顶点的四边形是菱形？若存在，请直接写出点N的坐标；若不存在，请说明理由.

2022年九年级数学模拟三参考答案

一、选择题（本大题共8个小题，每小题3分，共24分.）

二、填空题（本大题共6个小题，每小题3分，共18分.）

9.    10.-1    11.    12.    13.4    14.

三、解答题：（本大题共10个小题，共78分，解答应写出必要的文字说明、证明过程或演算步骤）

15.解：原式········································································（2分）

·················································································（4分）

.·················································································（6分）

16解：原式·········································································（2分）

·················································································（3分）

，················································································（4分）

∵，

∴，，

，，··············································································（5分）

原式.（6分）

17证明：（1）∵.∴，

∴，··············································································（2分）

又∵，∴；·········································································（4分）

（2）∵；

∴，··············································································（5分）

又∵，∴.··········································································（6分）

18解:如果渔船不改变航线继续向东航行，没有触礁的危险，

理由如下：过点A作，垂足为D，························································（1分）

根据题意可知，，···································································（2分）

∵，

∴，

∴，··············································································（3分）

在中，

，，，

∴，··············································································（4分）

∴，··············································································（5分）

∴渔船不改变航线继续向东航行，没有触礁的危险.···········································（6分）

19:解（1）∵过点P作y轴的垂线，垂足为A，O为坐标原点，的面积为1.

∴，∴，··········································································（1分）

∵在第一象限，

∴，∴反比例函数的表达式为；·························································（2分）

∵反比例函数（）的图象过点，

∴，∴，··········································································（3分）

∵一次函数的图象过点，

∴，解得，

∴一次函数的表达式为；······························································（4分）

（2）设直线交x轴、y轴于C、D两点，

∴，，

解得或，

∴，，············································································（5分）

∴，，，，·········································································（6分）

∴五边形的面积为

.·················································································（7分）

20解：设售价应定为x元，则每件的利润为元，日销售量件，···································（1分）

依题意，得：，·····································································（3分）

整理，得：，·······································································（5分）

解得：，（舍去）.···································································（6分）

答：售价应定为50元；································································（7分）

21.解：（1）此次调查一共随机采访学生（名），············································（1分）

在扇形统计图中，“灰”所在扇形的圆心角的度数为，········································（2分）

故答案为：200，198；

（2）绿色部分的人数为（人），························································（3分）

补全图形如下：·····································································（4分）

（3）估计该校学生将用过的餐巾纸投放到红色收集桶的人数（人）；····························（5分）

（4）列表如下：····································································（8分）

由表格知，共有12种等可能结果，其中恰好抽中A，B两人的有2种结果，⋯（9分）

所以恰好抽中A，B两人的概率为.························································（10分）

22.解：（1）如图，连接，·····························································（1分）

∵，

∴，

∵平分，

∴，

∴，··············································································（2分）

∴，

∴， （3分）

∵，

∴，

∴，即，

∴是的切线；·······································································（4分）

（2）∵是的直径，

∴，

∴，则，··········································································（5分）

在中，，，

∴，即，··········································································（6分）

解得，············································································（7分）

由（1）知是的切线，

∴，

∵，

∴，

∴，则，··········································································（8分）

在中，，

由勾股定理可得，，即，

解得，则，·········································································（9分）

由（1）知，

∴，即，解得.·······································································（10分）

23.解：（1），，

∵点P，N是，的中点，

∴，，

∵点P，M是，的中点，

∴，，············································································（1分）

∵，，

∴，

∴，··············································································（2分）

∵，

∴，

∵，

∴，

∵，

∴，··············································································（3分）

∴，

∴，··············································································（4分）

故答案为：，，

（2）由旋转知，，

∵，，

∴，

∴，，············································································（5分）

同（1）的方法，利用三角形的中位线得，，，

∴，

∴是等腰三角形，.（6分）

同（1）的方法得，，

∴，

同（1）的方法得，，

∴，

∵，

∴

，················································································（7分）

∵，

∴，

∴，

∴是等腰直角三角形，································································（8分）

（3）如图2，同（2）的方法得，是等腰直角三角形，

∴最大时，的面积最大，

∴且在顶点A上面，

∴最大，··········································································（9分）

连接，，

在中，，，

∴，

在中，，，

∴，

∴.···············································································（10分）

24.解：（1）∵，，∴，，

∵抛物线过点A、C，

∴················································································（1分）

解得，

∴抛物线解析式为；··································································（2分）

（2）

∵当时，，解得，，，∴，

抛物线对称轴为直线，

∵点D在直线上，点A、B关于直线对称，

∴，，

∴当点B、D、C在同一直线上时，

最小，············································································（3分）

设直线的表达式为，∴，

解得，，∴直线：，

∴，∴，

故答案为；·········································································（4分）

（3）过点E作轴于点G，交直线与点F，

设（），则，

∴，··············································································（5分）

∴

，················································································（6分）

∴当时，面积最大，

∴，

∴点E坐标为时，面积最大，最大值为.····················································（7分）

（4）存在点N，使以点A、C、M、N为顶点的四边形是菱形.

∵，，∴，·········································································（8分）

①若为菱形的边长，如图3，

则且，，

∴，，；··········································································（9分）

②若为菱形的对角线，如图4，则，，

设，

∴，

解得，，

∴，··············································································（10分）

综上所述，点N坐标为，，，.

（以上各题如有其它解法，酌情给分）