![](https://wkrtcs.bdimg.com/rtcs/image?w=554&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"554","h":"793","dataType":"png","c":"word/media/image1.png","imgOriW":707,"imgOriH":1013})
2019届石家庄市高三一模考试
数学（理）参考答案
选择题
A卷答案：1-5 CDACB 6-10BCCBD 11-12DA
B卷答案：1-5 CDBCA 6-10ACCAD 11-12DB
二、填空题
13. 1 14. ![](https://wkrtcs.bdimg.com/rtcs/image?w=84&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"84","h":"41","dataType":"gif","c":"word/media/image5.png"})
或
15. ![](https://wkrtcs.bdimg.com/rtcs/image?w=25&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"25","h":"23","dataType":"gif","c":"word/media/image7.png"})
16. 10
三、解答题
17. 解: （1） ∵△ABC三内角A、B、C依次成等差数列，∴B=60°
设A、B、C所对的边分别为![](https://wkrtcs.bdimg.com/rtcs/image?w=13&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"13","h":"15","dataType":"gif","c":"word/media/image8.png"})
、、,由=可得.……2分
∵![](https://wkrtcs.bdimg.com/rtcs/image?w=93&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"93","h":"19","dataType":"gif","c":"word/media/image14.png"})
，由正弦定理知，∴. ……4分
△ABC中，由余弦定理可得![](https://wkrtcs.bdimg.com/rtcs/image?w=184&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"184","h":"21","dataType":"gif","c":"word/media/image17.png"})
，∴b=.
即![](https://wkrtcs.bdimg.com/rtcs/image?w=28&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"28","h":"19","dataType":"gif","c":"word/media/image19.png"})
的长为 ……6分
（2）∵BD是AC边上的中线，∴![](https://wkrtcs.bdimg.com/rtcs/image?w=123&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"123","h":"41","dataType":"gif","c":"word/media/image20.png"})
……8分
∴![](https://wkrtcs.bdimg.com/rtcs/image?w=213&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"213","h":"41","dataType":"gif","c":"word/media/image21.png"})
==
![](https://wkrtcs.bdimg.com/rtcs/image?w=117&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"117","h":"41","dataType":"gif","c":"word/media/image24.png"})
，当且仅当时取“=” ……10分
∴![](https://wkrtcs.bdimg.com/rtcs/image?w=55&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"55","h":"32","dataType":"gif","c":"word/media/image26.png"})
,即BD长的最小值为3. ……12分
18. 解：（1）证明：在![](https://wkrtcs.bdimg.com/rtcs/image?w=45&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"45","h":"19","dataType":"gif","c":"word/media/image27.png"})
中，，，，由余弦定理可得，
![](https://wkrtcs.bdimg.com/rtcs/image?w=132&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"132","h":"23","dataType":"gif","c":"word/media/image32.png"})
，，…………2分
![](https://wkrtcs.bdimg.com/rtcs/image?w=192&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"192","h":"21","dataType":"gif","c":"word/media/image34.png"})
，
![](https://wkrtcs.bdimg.com/rtcs/image?w=120&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"120","h":"23","dataType":"gif","c":"word/media/image35.png"})
，，.…………4分
（2）法1：在平面![](https://wkrtcs.bdimg.com/rtcs/image?w=39&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"39","h":"19","dataType":"gif","c":"word/media/image38.png"})
中，过点作，以所在的直线分别为轴建立空间直角坐标系如图所示：
![](https://wkrtcs.bdimg.com/rtcs/image?w=276&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"276","h":"27","dataType":"gif","c":"word/media/image44.png"})
…………6分
![](https://wkrtcs.bdimg.com/rtcs/image?w=182.66666666667&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"182.66666666667","h":"234.66666666667","dataType":"gif","c":"word/media/image46.png"})
设平面的一个法向量为
则![](https://wkrtcs.bdimg.com/rtcs/image?w=159&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"159","h":"56","dataType":"gif","c":"word/media/image49.png"})
解得，，
即![](https://wkrtcs.bdimg.com/rtcs/image?w=84&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"84","h":"27","dataType":"gif","c":"word/media/image53.png"})
…………8分
设平面![](https://wkrtcs.bdimg.com/rtcs/image?w=37&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"37","h":"19","dataType":"gif","c":"word/media/image54.png"})
的一个法向量为
则![](https://wkrtcs.bdimg.com/rtcs/image?w=156&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"156","h":"56","dataType":"gif","c":"word/media/image56.png"})
解得![](https://wkrtcs.bdimg.com/rtcs/image?w=55&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"55","h":"27","dataType":"gif","c":"word/media/image57.png"})
，，即…………10分
![](https://wkrtcs.bdimg.com/rtcs/image?w=342&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"342","h":"56","dataType":"gif","c":"word/media/image61.png"})
由图可知二面角![](https://wkrtcs.bdimg.com/rtcs/image?w=77&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"77","h":"19","dataType":"gif","c":"word/media/image62.png"})
为锐角，所以二面角的余弦值为。……12分
法2：由（1）可知平面![](https://wkrtcs.bdimg.com/rtcs/image?w=51&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"51","h":"19","dataType":"gif","c":"word/media/image64.png"})
平面，
所以二面角的余弦值就是二面角的正弦值，…………6分
作![](https://wkrtcs.bdimg.com/rtcs/image?w=71&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"71","h":"19","dataType":"gif","c":"word/media/image67.png"})
于点，则平面，
![](https://wkrtcs.bdimg.com/rtcs/image?w=148&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"148","h":"200","dataType":"gif","c":"word/media/image70.png"})
作于点，连接，则
![](https://wkrtcs.bdimg.com/rtcs/image?w=15&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"15","h":"13","dataType":"gif","c":"word/media/image75.png"})
为二面角的平面角；…………8分
![](https://wkrtcs.bdimg.com/rtcs/image?w=15&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"15","h":"13","dataType":"gif","c":"word/media/image77.png"})
点为中点，点为中点，
在![](https://wkrtcs.bdimg.com/rtcs/image?w=64&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"64","h":"19","dataType":"gif","c":"word/media/image80.png"})
中，，，
![](https://wkrtcs.bdimg.com/rtcs/image?w=84&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"84","h":"45","dataType":"gif","c":"word/media/image83.png"})
…………10分
![](https://wkrtcs.bdimg.com/rtcs/image?w=179&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"179","h":"45","dataType":"gif","c":"word/media/image84.png"})
,所以二面角的余弦值为。…………12分
19. 解答：根据题意可得
![](https://wkrtcs.bdimg.com/rtcs/image?w=256&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"256","h":"297","dataType":"gif","c":"word/media/image85.png"})
……..部分对给2分，全对给4分
![](https://wkrtcs.bdimg.com/rtcs/image?w=15&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"15","h":"21","dataType":"gif","c":"word/media/image86.png"})
的分布列如下：
| 30 | 31 | 32 | 33 | 34 | 35 | 36 |
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本文来源：https://www.2haoxitong.net/k/doc/6cded2fda517866fb84ae45c3b3567ec112ddc2d.html
文档为doc格式
![](https://wkrtcs.bdimg.com/rtcs/image?w=23&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"23","h":"41","dataType":"gif","c":"word/media/image87.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=23&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"23","h":"41","dataType":"gif","c":"word/media/image88.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=16&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"16","h":"41","dataType":"gif","c":"word/media/image89.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=23&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"23","h":"41","dataType":"gif","c":"word/media/image90.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=23&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"23","h":"41","dataType":"gif","c":"word/media/image91.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=23&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"23","h":"41","dataType":"gif","c":"word/media/image92.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=31&md5sum=a9a970c02eb66719a7dfc13496079d70&sign=9d6b7c7979&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=170&ipr={"t":"img","w":"31","h":"41","dataType":"gif","c":"word/media/image93.png"})