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1 . 潮汐现象是地球上的海水受月球和太阳的万有引力作用而引起的周期性涨落现象.某观测站通过长时间观察,发现某港口的潮汐涨落规律为
(其中
,
),其中y(单位:
)为港口水深,x(单位:
)为时间
,该观测站观察到水位最高点和最低点的时间间隔最少为
,且中午12点的水深为
,为保证安全,当水深超过
时,应限制船只出入,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96ada8345cc6de95a2e3d9f550faeef2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13378be06b6b01bcad1d261ff14e87cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4456675a5dbe545462a22cef9aca8fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15e00f40396e914d1d9955bd7785f1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1933311c0c090e1138e4dd388b7adf8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2727b98332d0166f96ab098a5381e5bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c427170540df972d1cee3ce4baf25bc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00be2f5a88cf57caaaa92369367d210e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00be2f5a88cf57caaaa92369367d210e.png)
A.![]() |
B.最高水位为12![]() |
C.该港口从上午8点开始首次限制船只出入 |
D.一天内限制船只出入的时长为![]() |
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名校
2 . 在
的展开式中,
的一次项的系数为___________ (用数字作答).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3b6f6a29396d262ed67cdf31e1e1dc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
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4卷引用:湖南省湘潭市2024届高三下学期3月质量检测数学试题
3 . 已知函数
,
.
(1)当
时,求
的单调区间;
(2)当
时,记
的极小值点为
,证明:
存在唯一零点
,且
.(参考数据:
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f385b23c5ed85f350ffa395cd860f58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22dd8b3dc4c609bab82d356a5cc2208d.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e10e1c43b86a8cd4360ca9b57232164.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2ffc0733cb65fb25e9096618fff3348.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/697572b42c40f498ed398099c659df1f.png)
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2卷引用:湖南省湘潭市2024届高三下学期3月质量检测数学试题
名校
4 . 设等差数列
的前
项和为
,若
,
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22c5c4ac959eb2c4b74afabc9cdd3a6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41be41a5a4965ebd346e7ee74d21f0f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/614306cc3f34bdee4d5d885b79667645.png)
A.34 | B.35 | C.36 | D.38 |
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3卷引用:湖南省湘潭市2024届高三下学期3月质量检测数学试题
解题方法
5 . 已知圆锥
的侧面展开图为一个半圆,
为底面圆
的一条直径,
,B为圆O上的一个动点(不与A,C重合),记二面角
为
,
为
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e5ef91fb27dd684a27ae7f1993cfba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb5b12692517a39c320f99a479eb055.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10c12179820a82983959cb3885070e77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e15bd20c69abb74364f3ed5f2bf7781.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
A.圆锥![]() ![]() |
B.三棱锥![]() ![]() |
C.若![]() ![]() ![]() |
D.若![]() ![]() |
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2卷引用:湖南省湘潭市2024届高三下学期3月质量检测数学试题
解题方法
6 . 已知椭圆
:
的离心率为
,
为坐标原点,
,
为椭圆C的左、右焦点,点P在椭圆C上(不包括端点),当
时,
的面积为
,
(1)求椭圆C的标准方程;
(2)设点
,
,直线
,
分别与椭圆C交于异于点P的M、N两点,记直线
,
的斜率分别为
,
,求
的值,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2dfb22c6f1c155747100e7536cd1abd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33d776753746914c2410a3946c357f35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edb57d84f9bbcb3e30d4ce7e2e1e8604.png)
(1)求椭圆C的标准方程;
(2)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/383f12cb70ca55eba4ff012771dbfa9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3544997cc034ed882c0d0a3bdbf5f957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef49a3ca580a144cc65a609c167facc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4757181824e15e0f21e5bdd55448783.png)
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2卷引用:湖南省湘潭市2024届高三下学期3月质量检测数学试题
解题方法
7 . 已知函数
,记函数
,
的值域分别为
,若
,则
的取值范围是___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dd1a089301386fa8ea80217d6761b7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/276b142a9d9f0a87425a668dd6501f15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2卷引用:湖南省湘潭市2024届高三下学期3月质量检测数学试题
解题方法
8 . 已知
,
为双曲线
:
的左、右焦点,点
满足
,N为双曲线C的右支上的一个动点,O为坐标原点,则()
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ea7fdd0a77164d3dc0c032e6cb90b30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e8e445500f8a9de2c8ead8b2f24b1fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a35b148c043ddff87cc37c8891138187.png)
A.双曲线C的焦距为4 |
B.直线![]() |
C.![]() |
D.![]() |
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2卷引用:湖南省湘潭市2024届高三下学期3月质量检测数学试题
解题方法
9 . 已知F为抛物线C:
的焦点,O为坐标原点,过F且斜率为1的直线交抛物线C于A,B两点,直线
,
分别交抛物线C的准线于P,Q两点,若
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96724b211bf3e56d588bd430aa3f2894.png)
___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7089148c36cb3c39af71de653756396a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2c3d2cba96f6f03520c0b3f6e4da03e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf1438142deeac876fc7dc50552e552.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8ac3ffc4fa0746f7dd532a5b7ed7e1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47f449e4a47313fce67e5a7f05e86195.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96724b211bf3e56d588bd430aa3f2894.png)
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2卷引用:湖南省湘潭市2024届高三下学期3月质量检测数学试题
10 . 在三棱台
中,
为等边三角形,
,
平面
,
分别为
,
的中点,
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/24/7dffb745-2037-42ce-bbd9-a9f7eba59151.png?resizew=193)
(1)证明:平面
平面
;
(2)若
,设
为线段
上的动点,求
与平面
所成的角的正弦值的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73a9ad711b25c36dae0c2a2cedff9954.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2a61472439de1ba85cfe33840b775f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5845ccc0d735dc14c92a8926d9b1def6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/24/7dffb745-2037-42ce-bbd9-a9f7eba59151.png?resizew=193)
(1)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca81233fe44d2853df1a484580b6008.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e74395b07e8153a0ef0bbcb5881013f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f88e7df45acca3fc3d3da3370f0c32bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10d8eb4a9f462ca0c1d49c3fe91e720d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
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