2019年湖南省普通高中学业水平考试
数 学
1、选择题：本大题共10小题，每小题4分，共40分，在每小题给出的四个选项中，只有一项是符合题目要求的.
1、下列几何体中为圆锥的是
![](https://wkrtcs.bdimg.com/rtcs/image?w=531.00&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_7","w":"531.00","h":"123.00","dataType":"png","c":"word/media/image1.png","imgOriW":531,"imgOriH":123})
2、已知集合![](https://wkrtcs.bdimg.com/rtcs/image?w=192.06666666667&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_8","w":"192.06666666667","h":"24.933333333333","dataType":"png","c":"word/media/image2_1.png","imgOriW":3700,"imgOriH":480})
![](https://wkrtcs.bdimg.com/rtcs/image?w=176&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_9","w":"176","h":"46.666666666667","dataType":"png","c":"word/media/image3_1.png","imgOriW":2653,"imgOriH":703})
3、![](https://wkrtcs.bdimg.com/rtcs/image?w=184.00&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_6","w":"184.00","h":"294.00","dataType":"png","c":"word/media/image4.png","imgOriW":184,"imgOriH":294})
函数的最小正周期是
![](https://wkrtcs.bdimg.com/rtcs/image?w=196&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_9","w":"196","h":"65.333333333333","dataType":"png","c":"word/media/image6_1.png","imgOriW":2393,"imgOriH":797})
4、执行如图1所示的程序框图，若输入![](https://wkrtcs.bdimg.com/rtcs/image?w=14.666666666667&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_8","w":"14.666666666667","h":"14.666666666667","dataType":"png","c":"word/media/image7_1.png","imgOriW":388,"imgOriH":388})
的值为2，则输出
![](https://wkrtcs.bdimg.com/rtcs/image?w=14.666666666667&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_8","w":"14.666666666667","h":"17.333333333333","dataType":"png","c":"word/media/image8_1.png","imgOriW":388,"imgOriH":458})
的值为
![](https://wkrtcs.bdimg.com/rtcs/image?w=176&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_9","w":"176","h":"46.666666666667","dataType":"png","c":"word/media/image9_1.png","imgOriW":2653,"imgOriH":703})
5、某班有男生30人，女生20人，现用分层抽样的方法从中抽取10人参加一项活动，则抽取的男生的人数为
![](https://wkrtcs.bdimg.com/rtcs/image?w=176&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_9","w":"176","h":"46.666666666667","dataType":"png","c":"word/media/image10_1.png","imgOriW":2653,"imgOriH":703})
6、已知向量![](https://wkrtcs.bdimg.com/rtcs/image?w=182.66666666667&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_8","w":"182.66666666667","h":"22.666666666667","dataType":"png","c":"word/media/image11_1.png","imgOriW":3791,"imgOriH":470})
7、已知![](https://wkrtcs.bdimg.com/rtcs/image?w=85.333333333333&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_8","w":"85.333333333333","h":"21.333333333333","dataType":"png","c":"word/media/image12_1.png","imgOriW":2257,"imgOriH":564})
则下列不等式恒成立的是
![](https://wkrtcs.bdimg.com/rtcs/image?w=288&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_9","w":"288","h":"48","dataType":"png","c":"word/media/image13_1.png","imgOriW":3389,"imgOriH":565})
8、函数![](https://wkrtcs.bdimg.com/rtcs/image?w=70.666666666667&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_8","w":"70.666666666667","h":"24","dataType":"png","c":"word/media/image14_1.png","imgOriW":1869,"imgOriH":635})
的图像大致为
![](https://wkrtcs.bdimg.com/rtcs/image?w=517.00&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_7","w":"517.00","h":"140.00","dataType":"png","c":"word/media/image15.png","imgOriW":517,"imgOriH":140})
9、设![](https://wkrtcs.bdimg.com/rtcs/image?w=49.333333333333&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_8","w":"49.333333333333","h":"18.6","dataType":"png","c":"word/media/image16_1.png","imgOriW":1305,"imgOriH":493})
的内角所对的边分别为.若
![](https://wkrtcs.bdimg.com/rtcs/image?w=196&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_9","w":"196","h":"53.333333333333","dataType":"png","c":"word/media/image20_1.png","imgOriW":2635,"imgOriH":717})
10、已知直线![](https://wkrtcs.bdimg.com/rtcs/image?w=108&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_8","w":"108","h":"25.333333333333","dataType":"png","c":"word/media/image21_1.png","imgOriW":2857,"imgOriH":670})
与圆心在的圆相切，则圆的方程为
![](https://wkrtcs.bdimg.com/rtcs/image?w=384&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_9","w":"384","h":"50.666666666667","dataType":"png","c":"word/media/image24_1.png","imgOriW":3837,"imgOriH":506})
2、填空题：本大题共5小题，每题4分，共20分
11、已知直线![](https://wkrtcs.bdimg.com/rtcs/image?w=197.33333333333&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_8","w":"197.33333333333","h":"22.666666666667","dataType":"png","c":"word/media/image25_1.png","imgOriW":3952,"imgOriH":453})
若，则______
12、![](https://wkrtcs.bdimg.com/rtcs/image?w=141.00&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_5","w":"141.00","h":"176.00","dataType":"png","c":"word/media/image28.png","imgOriW":141,"imgOriH":176})
若则______
13、已知![](https://wkrtcs.bdimg.com/rtcs/image?w=178.66666666667&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_8","w":"178.66666666667","h":"22.666666666667","dataType":"png","c":"word/media/image31_1.png","imgOriW":3746,"imgOriH":475})
____________
14、如图2所示，将一个圆八等分，假设在该圆内随机撒一粒豆子，则豆子落在阴影部分的概率为_________.
15、若关于![](https://wkrtcs.bdimg.com/rtcs/image?w=14.666666666667&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_8","w":"14.666666666667","h":"14.666666666667","dataType":"png","c":"word/media/image32_1.png","imgOriW":388,"imgOriH":388})
的不等式的解集为，
则![](https://wkrtcs.bdimg.com/rtcs/image?w=56&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_2","w":"56","h":"16","dataType":"png","c":"word/media/image35_1.png","imgOriW":1481,"imgOriH":423})
_________.
3、解答题：本大题共5小题，共40分，解答应写出文字说明、证明过程或演算步骤.
16. （本小题满分6分）
已知![](https://wkrtcs.bdimg.com/rtcs/image?w=16&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_2","w":"16","h":"14.666666666667","dataType":"png","c":"word/media/image36_1.png","imgOriW":423,"imgOriH":388})
为锐角，且.
（1）求![](https://wkrtcs.bdimg.com/rtcs/image?w=38.666666666667&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_2","w":"38.666666666667","h":"14.666666666667","dataType":"png","c":"word/media/image38_1.png","imgOriW":1023,"imgOriH":388})
的值；
（2）求![](https://wkrtcs.bdimg.com/rtcs/image?w=45.333333333333&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_2","w":"45.333333333333","h":"18.6","dataType":"png","c":"word/media/image39_1.png","imgOriW":1199,"imgOriH":493})
的值.
17、（本小题满分8分）
甲、乙两个学习小组各有7名同学，在某次数学测试中，测试成绩的茎叶图如图3所示，（1）求甲组同学成绩的中位数和乙组同学成绩的众数；
![](https://wkrtcs.bdimg.com/rtcs/image?w=231.00&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_5","w":"231.00","h":"192.00","dataType":"png","c":"word/media/image40.png","imgOriW":231,"imgOriH":192})
（2）从这次测试成绩在90分以上的学生中，随机抽取1名学生，求抽到的这名学生来自甲组的概率.
18、（本小题满分8分）
如图4所示，三棱柱![](https://wkrtcs.bdimg.com/rtcs/image?w=104&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_2","w":"104","h":"22.666666666667","dataType":"png","c":"word/media/image41_1.png","imgOriW":2751,"imgOriH":599})
中，
（1）![](https://wkrtcs.bdimg.com/rtcs/image?w=266.00&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_5","w":"266.00","h":"232.00","dataType":"png","c":"word/media/image43.png","imgOriW":266,"imgOriH":232})
求证：
（2）已知![](https://wkrtcs.bdimg.com/rtcs/image?w=113.33333333333&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_2","w":"113.33333333333","h":"21.333333333333","dataType":"png","c":"word/media/image45_1.png","imgOriW":2998,"imgOriH":564})
且异面直线所成的角为，求三棱柱的体积.
19、（本小题满分8分）
已知数列![](https://wkrtcs.bdimg.com/rtcs/image?w=32&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_2","w":"32","h":"24","dataType":"png","c":"word/media/image48_1.png","imgOriW":846,"imgOriH":635})
满足
（1）求数列的通项公式；
（2）已知数列![](https://wkrtcs.bdimg.com/rtcs/image?w=30.666666666667&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_2","w":"30.666666666667","h":"24","dataType":"png","c":"word/media/image50_1.png","imgOriW":811,"imgOriH":635})
满足设，若数列为等比数列，求数列的前项和.
20、（本小题满分10分）
已知二次函数![](https://wkrtcs.bdimg.com/rtcs/image?w=182.66666666667&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_2","w":"182.66666666667","h":"24","dataType":"png","c":"word/media/image56_1.png","imgOriW":3690,"imgOriH":485})
（1）若![](https://wkrtcs.bdimg.com/rtcs/image?w=38.666666666667&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_2","w":"38.666666666667","h":"21.333333333333","dataType":"png","c":"word/media/image57_1.png","imgOriW":1023,"imgOriH":564})
为偶函数，求的值；
（2）判定函数在区间内是否有零点，请说明理由；
（3）已知函数![](https://wkrtcs.bdimg.com/rtcs/image?w=341.33333333333&md5sum=4187591295fcf040e417b98e62f5a17f&sign=44daf80914&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=279&ipr={"t":"img","r":"r_2","w":"341.33333333333","h":"42.6","dataType":"png","c":"word/media/image60_1.png","imgOriW":3920,"imgOriH":490})
的最大值.
本文来源：https://www.2haoxitong.net/k/doc/93f59f0f81eb6294dd88d0d233d4b14e84243e95.html
文档为doc格式