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【330338】2020年内蒙古通辽市中考数学试题

时间:2025-02-09 10:58:26 作者: 字数:25078字

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 内蒙古通辽市2020年中考数学试题

注意事项:

 1.本试卷共6页,26小题,满分为120分,考试时间为120分钟.

 2.根据网上阅卷需要,本试卷中的所有试题均按要求在答题卡上作答,答在本试卷上的答案无效.

 3.考试结束后,将本试卷与答题卡分别封装一并上交.

一、选择题(本题包括10小题,每小题3分,共30分,每小题只有一个正确答案,请在答题卡上将代表正确答案的字母用2B铅笔涂黑)

1.2020年我市初三毕业生超过30000人,将30000用科学记数法表示正确的是(   )

A.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> B.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> C.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> D. 3

【答案】B

【解析】

【分析】

科学记数法的表示形式为a×10n的形式,其中1≤|a|10n为整数.确定n的值时,要看把原数变成a时,小数点移动了多少位,n的绝对值与小数点移动的位数相同.当原数绝对值>1时,n是正数;当原数的绝对值<1时,n是负数.

【详解】解:将30000用科学记数法表示为3×104
故选:B

【点睛】此题主要考查了科学记数法的表示方法.科学记数法的表示形式为a×10n的形式,其中1≤|a|10n为整数,表示时关键要正确确定a的值以及n的值.

2.下列说法不正确的是(   )

A.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 2个数a的和 B.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 2和数a的积

C.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 是单项式 D.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 是偶数

【答案】D

【解析】

【分析】

根据2a的意义,分别判断各项即可.

【详解】解:A <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> =a+a,是2个数a的和,故选项正确;

B <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> =2×a,是2和数a的积,故选项正确;

C <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 是单项式,故选项正确;

D、当a为无理数时, <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 是无理数,不是偶数,故选项错误;

故选D.

【点睛】本题考查了代数式的意义,注意a不一定为整数是解题的关键.

3.下列事件中是不可能事件的是(   )

A. 守株待兔 B. 瓮中捉鳖 C. 水中捞月 D. 百步穿杨

【答案】C

【解析】

【分析】

不可能事件是一定不会发生的事件,依据定义即可判断.

【详解】解:A、守株待兔,不一定就能达到,是随机事件,故选项不符合;

B、瓮中捉鳖是必然事件,故选项不符合;

C、水中捞月,一定不能达到,是不可能事件,选项不符合;

D、百步穿杨,未必达到,是随机事件,故选项不符合;

故选C.

【点睛】本题考查了随机事件,解决本题需要正确理解必然事件、不可能事件、随机事件的概念.必然事件指在一定条件下一定发生的事件.不可能事件是指在一定条件下,一定不发生的事件.不确定事件即随机事件是指在一定条件下,可能发生也可能不发生的事件.

4.如图,将一副三角尺按下列位置摆放,使 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 互余的摆放方式是(   )

A.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> B.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

C <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> D.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【答案】A

【解析】

【分析】

根据图形,结合互余的定义判断即可.

【详解】解:A、∠α与∠β互余,故本选项正确;

B、∠α+∠β90°,即不互余,故本选项错误;

C、∠α+∠β=270°,即不互余,故本选项错误;

D、∠α+∠β=180°,即互补,故本选项错误;

故选A.

【点睛】本题考查了对余角和补角的应用,主要考查学生的观察图形的能力和理解能力.

5.若关于x的方程kx2﹣6x+9=0有实数根,则k的取值范围是(  )

A. k1 B. k≤1 C. k1k≠0 D. k≤1k≠0

【答案】B

【解析】

【详解】解:(1)当k=0时,-6x+9=0,解得x= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>
2)当k≠0时,此方程是一元二次方程,
关于x的方程kx
2-6x+9=0有实数根,
∴△=
-6
2-4k×9≥0,解得k≤1
由(1)、(2)得,k的取值范围是k≤1
故选B

6.根据圆规作图的痕迹,可用直尺成功地找到三角形内心的是(   )

A.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> B.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

C.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> D.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【答案】B

【解析】

【分析】

根据三角形内心的定义,三角形内心为三边的垂直平分线的交点,然后利用基本作图和选项进行判断.

【详解】解:三角形内心为三个角的角平分线的交点,

由基本作图得到B选项作了两个角的角平分线,

而三角形三条角平分线交于一点,从而可用直尺成功找到三角形内心.
故选:B

【点睛】本题考查了作图-基本作图:熟练掌握基本作图(作一条线段等于已知线段;作一个角等于已知角;作已知线段的垂直平分线;作已知角的角平分线;过一点作已知直线的垂线).也考查了三角形的内心.

7.如图, <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 分别与 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 相切于 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 两点, <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,则 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> (   )

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

A.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> B.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> C.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> D.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【答案】C

【解析】

【分析】

连接OAOB,根据切线的性质定理,结合四边形AOBP的内角和为360°,即可推出∠AOB的度数,然后根据圆周角定理,即可推出∠C的度数.

【详解】解:连接OAOB
直线PAPB分别与⊙O相切于点AB
∴OA⊥PA
OB⊥PB
∵∠P=72°

∴∠AOB=108°

∵C
是⊙O上一点,
∴∠ACB=54°

故选:C

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【点睛】本题主要考查切线的性质、四边形的内角和、圆周角定理,关键在于熟练运用切线的性质,通过作辅助线构建四边形,最后通过圆周角定理即可推出结果.

8.如图, <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的中线,四边形 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 是平行四边形,增加下列条件,能判断 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 是菱形的是(   )

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

A.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> B.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> C.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> D.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【答案】A

【解析】

【分析】

根据菱形的判定方法逐一分析即可.

【详解】解:A、若 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,则AD=BD=CD=AE,∵四边形ADCE是平行四边形,则此时四边形ADCE为菱形,故选项正确;

B、若 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,则四边形ADCE是矩形,故选项错误;

C、若 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,则∠ADC=90°,则四边形ADCE是矩形,故选项错误;

D、若 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,而ABAD,则AE≠AD,无法判断四边形ADCE为菱形,故选项错误.

故选A.

【点睛】本题考查了菱形的判定,还涉及到平行四边形的性质,矩形的判定,等腰三角形的性质,解题的关键是掌握判定定理.

9.如图, <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 交双曲线 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 于点A,且 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,若矩形 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的面积是8,且 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 轴,则k的值是(   )

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

A. 18 B. 50 C. 12 D.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【答案】A

【解析】

【分析】

过点A和点C分别作x轴的垂线,垂足为EF,得到△OAE∽△OCF,设点Amn),求出ABBC,利用矩形ABCD的面积为8求出mn,即k.

【详解】解:过点A和点C分别作x轴的垂线,垂足为EF

AE∥CF

∴△OAE∽△OCF

OCOA=53

OFOE=CFAE=53

设点Amn),则mn=k

OE=mAE=n

OF= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> CF= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

AB=OF-OE= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> BC=CF-AE= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

矩形ABCD的面积为8

AB·BC= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> × <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> =8

mn=18=k

故选A.

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【点睛】本题考查了相似三角形的判定和性质,反比例函数表达式,矩形的性质,解题的关键是利用相似三角形的性质表示出线段的长.

10.从下列命题中,随机抽取一个是真命题的概率是(   )

1)无理数都是无限小数;

2)因式分解 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

3)棱长是 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的正方体的表面展开图的周长一定是 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

4)弧长是 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,面积是 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的扇形的圆心角是 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

A.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> B.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> C.  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> D. 1

【答案】C

【解析】

【分析】

分别判断各命题的真假,再利用概率公式求解.

【详解】解:(1)无理数都是无限小数,是真命题,

2)因式分解 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,是真命题,

3)棱长是 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的正方体的表面展开图的周长一定是 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,是真命题,

4)设扇形半径为r,圆心角为n

弧长是 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,则 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> = <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,则 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

面积是 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,则 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> = <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,则 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 360×240

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,则n=3600÷24=150°

故扇形的圆心角是 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,是假命题,

则随机抽取一个是真命题的概率是 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

故选C.

【点睛】本题考查了命题的真假,概率,扇形的弧长和面积,无理数,因式分解,正方体展开图,知识点较多,难度一般,解题的关键是运用所学知识判断各个命题的真假.

二、填空题(本题包括7小题,每小题3分,共21分,将答案直接填在答题卡对应题的横线上)

11.计算:

1 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ______;(2 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ______;(3 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ______

【答案】 (1). 1 (2).  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> (3). -1

【解析】

【分析】

根据零指数幂,特殊角的三角函数值,乘方运算法则分别计算即可.

【详解】解: <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 1

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> = <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> -1

故答案为:1 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> -1.

【点睛】本题考查了零指数幂,特殊角的三角函数值,乘方运算,掌握运算法则是关键.

12.若数据3a353的平均数是3,则这组数据中(1)众数是______;(2a的值是______;(3)方差是______

【答案】 (1). 3 (2). 1 (3). 1.6

【解析】

【分析】

根据平均数的定义先求出a的值,再根据众数的定义、以及方差公式进行计算即可得出答案.

【详解】解:根据题意得,

3+a+3+5+3=3×5

解得:a=1

则一组数据13335的众数为3

方差为: <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> = <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> =1.6

故答案为:(13;(21;(31.6

【点睛】此题考查了众数、平均数和方差,用到的知识点是众数、平均数和方差的求法,注意计算不要出错.

13.如图,点O在直线 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 上, <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,则 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的度数是______

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【答案】 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【解析】

【分析】

根据补角的定义,进行计算即可.

【详解】解:由图可知:∠AOC和∠BOC互补,

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

∴∠BOC=180°- <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> = <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

故答案为: <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> .

【点睛】本题考查了补角的定义,和角的计算,关键是掌握角的运算方法.

14.如图,用大小相同的小正方形拼大正方形,拼第1个正方形需要4个小正方形,拼第2个正方形需要9个小正方形……,按这样的方法拼成的第 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 个正方形比第n个正方形多_____个小正方形.

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【答案】2n+3

【解析】

【分析】

首先根据图形中小正方形的个数规律得出变化规律,进而得出答案.

【详解】解:∵第一个图形有22=4个正方形组成,
第二个图形有3
2=9个正方形组成,
第三个图形有4
2=16个正方形组成,
n个图形有(n+1
2个正方形组成,第n+1个图形有(n+22个正方形组成

n+22-n+12

=2n+3

故答案为:2n+3

【点睛】此题主要考查了图形的变化类,根据图形得出小正方形的变化规律是解题关键.

15.有一个人患了新冠肺炎,经过两轮传染后共有169人患了新冠肺炎,每轮传染中平均一个人传染了______个人.

【答案】12

【解析】

【分析】

设平均一人传染了x人,根据有一人患了流感,经过两轮传染后共有169人患了流感,列方程求解

【详解】解:设平均一人传染了x人,
x+1+
x+1x=169
解得:x=12x=-14(舍去).
平均一人传染12人.
故答案为:12

【点睛】本题考查理解题意的能力,关键是看到两轮传染,从而可列方程求解.

16.如图,在 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 中, <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,点P在斜边 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 上,以 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 为直角边作等腰直角三角形 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,则 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 三者之间的数量关系是_____

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【答案】PA2+PB2=PQ2

【解析】

【分析】

AP2PB2都用PCCD表示出来,结合Rt△PCD中,可找到PCPDCD的关系,从而可找到PA2PB2PQ2三者之间的数量关系;

【详解】解:过点CCD⊥AB,交AB于点D

∵△ACB为等腰直角三角形,CD⊥AB

CD=AD=DB
∵PA
2=AD-PD2=CD-PD2=CD2-2CD•PD+PD2
PB
2=BD+PD2=CD+PD2=CD2-2CD•PD+PD2
∴PA
2+PB2=2CD2+2PD2=2CD2+PD2),
Rt△PCD中,由勾股定理可得PC
2=CD2+PD2
∴PA
2+PB2=2PC2
∵△CPQ
为等腰直角三角形,且∠PCQ=90°
∴2PC
2=PQ2
∴PA
2+PB2=PQ2
故答案为PA
2+PB2=PQ2.

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【点睛】本题考查了等腰直角三角形的性质,勾股定理的应用,关键是作出辅助线,利用三线合一进行论证.

17.如图①,在 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 中, <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,点E是边 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的中点,点P是边 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 上一动点,设 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> .图②是y关于x的函数图象,其中H是图象上的最低点..那么 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的值为_______

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【答案】7

【解析】

【分析】

BAC的平行线,过CAB的平行线,交于点D,证明四边形ABCD为菱形,得到点A和点D关于BC对称,从而得到PA+PE=PD+PE,推出当PDE共线时,PA+PE最小,即DE的长,观察图像可知:当点P与点B重合时,PD+PE= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,分别求出PA+PE的最小值为3PC的长,即可得到结果.

【详解】解:如图,过BAC的平行线,过CAB的平行线,交于点D

可得四边形ABCD为平行四边形,又AB=AC

四边形ABCD为菱形,点A和点D关于BC对称,

PA+PE=PD+PE

PDE共线时,PA+PE最小,即DE的长,

观察图像可知:当点P与点B重合时,PD+PE= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

EAB中点,

BE+BD=3BE= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

BE= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> AB=BD= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

∵∠BAC=120°

∴∠ABD=180°-120°÷2×2=60°

∴△ABD为等边三角形,

DE⊥AB,∠BDE=30°

DE=3,即PA+PE的最小值为3

即点H的纵坐标为a=3

当点PDEBC交点时,

AB∥CD

∴△PBE∽△PCD

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

菱形ABCD中,AD⊥BC

BC=2× <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> =6

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

解得:PC=4

即点H的横坐标为b=4

a+b=3+4=7

故答案为:7.

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【点睛】本题考查动点问题的函数图象,解答本题的关键是明确题意,找出所求问题需要的条件,利用数形结合的思想解答.

三、解答题(本题包括9小题,共69分,每小题分值均在各题号后面标出,请在答题卡上写出各题解答的文字说明、证明过程或计算步骤)

18.解方程: <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> .

【答案】 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> .

【解析】

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 分析】

首先去掉分母,观察可得最简公分母是xx﹣2),方程两边乘最简公分母,可以把分式方程转化为整式方程求解,然后解一元一次方程,最后检验即可求解.

【详解】去分母,得 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

去括号,得 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

移项,合并同类项,得 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

x的系数为1,得 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

经检验, <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 是原方程的根,

原方程的解为 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【点睛】本题考查了解分式方程,熟练掌握解分式方程的步骤以及注意事项是解题的关键.


19.A处看一栋楼顶部的仰角为 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,看这栋楼底部的俯角为 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> A处与楼的水平距离 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,若 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,求这栋楼高.

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【答案】270

【解析】

【分析】

根据正切的定义分别求出BDDC的长,求和即可.

【详解】解:在Rt△ABD中,tanα= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>
BD=AD•tanα=90×0.27=24.3
Rt△ACD中,tanβ=
 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>
CD=AD•tanβ=90×2.73=245.7
∴BC=BD+CD=24.3+245.7=270

答:这栋楼高约为270米.

【点睛】本题考查的是解直角三角形的应用-仰角俯角问题,正切理解仰角俯角的概念、熟记锐角三角函数的定义是解题的关键.

20.用※定义一种新运算:对于任意实数mn,规定 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,如: <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

1)求 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

2)若 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,求m的取值范围,并在所给的数轴上表示出解集.

【答案】(1) <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> (2) <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,图见解析

【解析】

【分析】

1)根据新定义规定的运算法则列式,再由有理数的运算法则计算可得;
2)根据新定义列出关于x的不等式,解不等式即可得.

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 详解】解:(1 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> = <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

2)∵ <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>
 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

解得: <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

将解集表示在数轴上如下:

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【点睛】本题主要考查解一元一次不等式和二次根式的混合运算,解题的关键是根据新定义列出算式和一元一次不等式及解一元一次不等式的步骤

21.甲口袋中装有2个相同小球,它们分别写有数字12;乙口袋中装有3个相同小球,它们分别写有数字345;丙口袋中装有2个相同小球,它们分别写有数字67.从三个口袋各随机取出1个小球.用画树状图或列表法求:

1)取出的3个小球上恰好有一个偶数的概率;

2)取出的3个小球上全是奇数的概率.

【答案】1 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ;(2 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【解析】

【分析】

1)画出树状图,由树状图求得所有等可能的结果,然后利用概率公式求解即可求得答案;

2)画出树状图,由树状图求得所有等可能的结果,然后利用概率公式求解即可求得答案;

【详解】解:画树状图得:

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

1)∵共有12种等可能的结果,取出的3个小球上恰好有1个偶数数字的有5种情况,
取出的3个小球上只有1个偶数数字的概率是 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

2)∵共有12种等可能的结果,取出的3个小球上全是奇数数字的有2种情况,
取出的3个小球上全是奇数数字的概率是 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>
.

【点睛】此题考查的是用树状图法求概率.树状图法可以不重复不遗漏的列出所有可能的结果,适合两步或两步以上完成的事件;注意概率=所求情况数与总情况数之比.

22.如图, <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的直径 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 交弦(不是直径) <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 于点P,且 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> .求证: <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【答案】见解析

【解析】

【分析】

连接ACBD,证明△PAC∽△PDB,得到 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,再根据 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 得到 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,从而得到PC=PD,根据垂径定理得出结果.

【详解】解:连接ACBD

在△PAC和△PBD中,

A=∠D,∠C=∠B

∴△PAC∽△PDB

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

PC=PD

AB为直径,

AB⊥CD.

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【点睛】本题考查了圆周角定理,相似三角形的判定和性质,垂径定理,解题的关键是证明△PAC∽△PDB,得到 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> .

23.某校研究学生的课余爱好情况,采取抽样调查的方法,从阅读、运动、娱乐、上网四个方面调查了若干名学生的兴趣爱好,并将调查结果绘制成如图所示的两幅不完整的统计图,请你根据图中提供的信息解答下列问题:

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

1)在这次调查中,共调查了多少名学生;

2)补全条形统计图;

3)若该校爱好运动的学生共有800名,则该校学生总数大约有多少名.

【答案】1100;(2)见解析;(32000

【解析】

【分析】

1)根据爱好运动人数的百分比,以及运动人数即可求出共调查的人数;
2)根据两幅统计图即可求出阅读的人数以及上网的人数,从而可补全图形.
3)利用样本估计总体即可估计爱好运动的学生人数.

【详解】解:(1)爱好运动的人数为40,所占百分比为40%
共调查人数为:40÷40%=100
2)爱好上网的人数所占百分比为10%
爱好上网人数为:100×10%=10
爱好阅读人数为:100-40-20-10=30
补全条形统计图,如图所示,

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>
3)爱好运动的学生人数所占的百分比为40%
该校共有学生大约有:800÷40%=2000人;

【点睛】本题考查统计,解题的关键是正确利用两幅统计图的信息,本题属于中等题型.

24.某服装专卖店计划购进 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 两种型号的精品服装.已知2A型服装和3B型服装共需4600元;1A型服装和2B型服装共需2800元.

1)求 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 型服装的单价;

2)专卖店要购进 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 两种型号服装60件,其中A型件数不少于B型件数的2倍,如果B型打七五折,那么该专卖店至少需要准备多少货款?

【答案】1A型女装的单价是800元,B型女装的单价是1000元;(247000

【解析】

【分析】

1)设A型女装的单价是x元,B型女装的单价是y元.根据“2A型女装和3B型女装共需4600元;1A型女装和2B型女装共需2800元”列出方程组并解答;
2)设购进A型女装m件,则购进B型女装(60-m)件,依据“A型的件数不少于B型件数的2倍”求得m的取值范围,然后根据购买方案求得需要准备的总费用.

【详解】解:(1)设A型女装的单价是x元,B型女装的单价是y元,

依题意得: <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

解得: <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

答:A型女装的单价是800元,B型女装的单价是1000元;

2)设购进A型女装m件,则购进B型女装(60-m)件,
根据题意,得m≥260-m),
∴m≥40

设购买AB两种型号的女装的总费用为w元,
w=800m+1000×0.75×
60-m=50m+45000
∴w
m的增大而增大,
m=40时,w
最小=50×40+45000=47000
答:该专卖店至少需要准备47000元的贷款.

【点睛】本题考查了一元一次不等式的应用,二元一次方程组的应用.解决问题的关键是读懂题意,找到关键描述语,找到所求的量的等量关系.

25.中心为O的正六边形 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的半径为 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> .点 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 同时分别从 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 两点出发,以 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的速度沿 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 向终点 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 运动,连接 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,设运动时间为 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

1)求证:四边形 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 为平行四边形;

2)求矩形 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的面积与正六边形 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的面积之比.

【答案】1)见解析;(223

【解析】

【分析】

1)只要证明△ABP≌△DEQSAS),可得BP=EQ,同理PE=BQ,由此即可证明;

2)过点B,点EBN⊥CDEM⊥CD,连接OCOD,过点OOH⊥CD分别求出矩形 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的面积和正六边形 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的面积,从而得到结果.

【详解】解:(1)证明:∵中心为O的正六边形ABCDEF的半径为6cm
∴AB=BC=CD=DE=EF=FA
,∠A=∠ABC=∠C=∠D=∠DEF=∠F
PQ同时分别从AD两点出发,以1cm/s速度沿AFDC向终点FC运动,
∴AP=DQ=t
PF=QC=6-t
在△ABP和△DEQ中,

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>
∴△ABP≌△DEQ
SAS),
∴BP=EQ
,同理可证PE=QB
四边形PEQB是平行四边形;

2)由(1)可知四边形PEQB是平行四边形

当∠BQE=90°时,四边形PEQB是矩形

过点B,点EBN⊥CDEM⊥CD,连接OCOD,过点OOH⊥CD

∴∠BNQ=∠QME=90°

∴∠BQN+∠NBQ=90°,∠BQN+∠EQM=90°

∴∠NBQ=∠EQM

∴△NBQ∽△MQE

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

又∵正六边形ABCDEF的半径为6

正六边形ABCDEF的各边为6,∠BCQ=∠EDQ=120°

Rt△BNCRt△EDM中,∠NBC=∠DEM=30°

NC=DM= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> BN=EM= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,解得:

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> (舍去)

即当PF重合,QC重合时,四边形PEQB是矩形

此时矩形PEQB的面积为 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

在正六边形ABCDEF中,∠COD=60°OC=OD

∴△OCD是等边三角形,OC=OD=CD=6OH= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

S六边形ABCDEF= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

S矩形PBQES六边形ABCDEF= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> =23
 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

【点睛】本题考查正多边形、平行四边形的判定和性质、矩形的性质与判定,解题的关键是灵活运用所学知识解决问题,属于中考常考题型.

26.如图,在平面直角坐标系中,抛物线 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> x轴交于点 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,与y轴交于点C,且直线 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 过点B,与y轴交于点D,点C与点D关于x轴对称.点P是线段 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 上一动点,过点Px轴的垂线交抛物线于点M,交直线 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 于点N

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

1)求抛物线的函数解析式;

2)当 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 的面积最大时,求点P的坐标;

3)在(2 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 条件下,在y轴上是否存在点Q,使得以 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 三点为顶点的三角形是直角三角形,若存在,直接写出点Q的坐标;若不存在,说明理由.

【答案】1 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ;(2)(20);(3)存在,(012)或(0-4)或(0 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> )或(0 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> .

【解析】

【分析】

1)根据直线 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 求出点B和点D坐标,再根据CD之间的关系求出点C坐标,最后运用待定系数法求出抛物线表达式;

2)设点P坐标为(m0),表示出MN的坐标,再利用三角形面积求法得出S△BMD= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,再求最值即可;

3)分当∠QMN=90°时,当∠QNM=90°时,当∠MQN=90°时,三种情况,结合相似三角形的判定和性质,分别求解即可.

【详解】解:(1)∵直线 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 过点B,点Bx轴上,

y=0,解得x=6,令x=0,解得y=-6

B60),D0-6),

C和点D关于x轴对称,

C06),

抛物线 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 经过点B和点C,代入,

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,解得: <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

抛物线的表达式为: <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

2)设点P坐标为(m0),

则点M坐标为(m <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ),点N坐标为(mm-6),

MN= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> -m+6= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

S△BMD=S△MNB+S△MND

= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

=-3m-22+48

m=2时,S△BMD最大=48

此时点P的坐标为(20);

3)存在,

由(2)可得:M212),N2-4),

设点Q的坐标为(0n),

当∠QMN=90°时,即QM⊥MN,如图,

可得,此时点Q和点M的纵坐标相等,

Q012);

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

当∠QNM=90°时,即QN⊥MN,如图,

可得,此时点Q和点N的纵坐标相等,

Q0-4);

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

当∠MQN=90°时,MQ⊥NQ,如图,

分别过点MNy轴的垂线,垂足为EF

∵∠MQN=90°

∴∠MQE+∠NQF=90°,又∠MQE+∠QME=90°

∴∠NQF=∠QME

∴△MEQ∽△QFN

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ,即 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

解得:n= <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>  <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

Q0 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> )或(0 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> ),

 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a>

综上:点Q的坐标为(012)或(0-4)或(0 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> )或(0 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> .

【点睛】本题是二次函数综合题,考查了二次函数 <a href="/tags/51/" title="试题" class="c1" target="_blank">试题</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/1060/" title="内蒙古" class="c1" target="_blank">内蒙古</a> <a href="/tags/1064/" title="通辽" class="c1" target="_blank">通辽</a> 表达式,相似三角形的判定和性质,直角三角形的性质,二次函数的最值,解一元二次方程,解题时要注意数形结合,分类讨论思想的运用.