1 . 已知
的两个顶点
,
,点G为
的重心,边
上的两条中线的长度之和为6,记点G的轨迹为曲线E.
(1)求曲线E的方程;
(2)若点P是曲线E上的任意一点,
,
,
,
,直线PC,PD与x轴分别交于点M,N.
①求
的最大值;
②判断
是否为定值.若为定值,求出该定值;若不为定值,求出它的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea8dfd0ebf16476994cf04dd0f7fafca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17618d8d22ebb3fd6835a7eb139b4f95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13683e2ecf2164a0adbfdb9923d210a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea8dfd0ebf16476994cf04dd0f7fafca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6bf421591c540d317587bdac885e9c.png)
(1)求曲线E的方程;
(2)若点P是曲线E上的任意一点,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/913f78382630e50543e5f7192cae3ed3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c850811ba59a05e945a665196539a048.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dbc9361f0be545e55976a052e95e960.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a375e47ac52f067ac1f7b3f73dbc41ee.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/084cf5ffced059f5653ee2a1023518b7.png)
②判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9467e420cfaea81eb1e7245e3f14c503.png)
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解题方法
2 . 2024年初,冰城哈尔滨充分利用得天独厚的冰雪资源,成为2024年第一个“火出圈”的网红城市,冰城通过创新营销展示了丰富的文化活动,成功提升了吸引力和知名度,为其他旅游城市提供了宝贵经验,从2024年1月1日至5日,哈尔滨太平国际机场接待外地游客数量如下:
(1)计算
的相关系数
(计算结果精确到0.01),并判断是否可以认为日期与游客人数的相关性很强;
(2)请根据上表提供的数据,用最小二乘法求出
关于
的线性回归方程;
(3)为了吸引游客,在冰雪大世界售票处针对各个旅游团进行了现场抽奖的活动,具体抽奖规则为:从该旅游团中随机同时抽取两名游客,两名游客性别不同则为中奖.已知某个旅游团中有5个男游客和
个女游客,设重复进行三次抽奖中恰有一次中奖的概率为
,当
取多少时,
最大?
参考公式:
,
,
,
参考数据:
.
![]() | 1 | 2 | 3 | 4 | 5 |
![]() | 45 | 50 | 60 | 65 | 80 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d708cb763716467219215cdc0782c0a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
(2)请根据上表提供的数据,用最小二乘法求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(3)为了吸引游客,在冰雪大世界售票处针对各个旅游团进行了现场抽奖的活动,具体抽奖规则为:从该旅游团中随机同时抽取两名游客,两名游客性别不同则为中奖.已知某个旅游团中有5个男游客和
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84ea3a20cdf5915c2b51d0c401b9e834.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
参考公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/609343476099195950454b8809930066.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ebff20f21ae41fd8d1f1e3145895842.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d62e7e496bab282e2475829358054202.png)
参考数据:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47bb3f35e3db7c1f3a3dd3eb20151b5f.png)
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3 . 曲率是曲线的重要性质,表征了曲线的“弯曲程度”,曲线曲率解释为曲线某点切线方向对弧长的转动率,设曲线
具有连续转动的切线,在点
处的曲率
,其中
为
的导函数,
为
的导函数,已知
.
(1)
时,求
在极值点处的曲率;
(2)
时,
是否存在极值点,如存在,求出其极值点处的曲率;
(3)
,
,当
,
曲率均为0时,自变量最小值分别为
,
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd817a1014876a72ad1971548ed6f52c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe7522a3f232bd0b7a7850ae674db43f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bad7aa241de8ac2738629f7361a7c8bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10acd6d864583617dd3e71240bf0c857.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea058d082b5f7517c3b6a6359dbcb44a.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4d795709b0abcf47bceec2250f2f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0c51e20ceeca65fe6821130d94b794c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3387f1c69de6c2407212536b35150e5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fb22f6880c74b35a8285cbb51a50fb1.png)
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4 . 下列说法正确的是( ).
A.函数![]() ![]() ![]() |
B.函数![]() ![]() |
C.已知函数![]() ![]() ![]() ![]() ![]() |
D.若![]() ![]() ![]() |
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5 . 关于函数
有下列结论:
(1)函数
存在最小值但没有最大值;
(2)函数
存在两个零点,且两个零点的和小于1;
(3)函数
存在唯一的极小值点
,且
;
(4)函数
存在唯一的极大值点
,且
.
其中正确的是__________ .(填写所有正确结论的编号)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2db29e99cb06dfd0d9dbdd3a0d387fd.png)
(1)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)函数
![](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/caae75ee744fa4662f1dc8fbe3995789.png)
(4)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f803d4aab3143f9e43d7a024aab305e9.png)
其中正确的是
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6 . 已知正方体
棱长为2,
为棱
的中点,
为正方体表面上一动点,下列说法中正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
A.点![]() ![]() ![]() ![]() ![]() |
B.点![]() ![]() ![]() ![]() ![]() |
C.当点![]() ![]() ![]() ![]() |
D.若![]() ![]() ![]() |
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7 . 设复数
,其在复平面内对应点为
,且
,复数
,其在复平面内对应点为
,且
,若存在
的轨迹上的两点
、
,使
,则
的取值范围为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8f41acaa456df1ac6e5af4fc056f780.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c69a18ac82d772e7c7707efe8f44eb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d72aeab648917f787ddb15d9039b8856.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e380c51ad6d645900156ff061992799b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a86380a6d6501f6504dcb4aa5e3099f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eae863e7a1f1fed09f1075de4a817c63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f42143191548a417aa602e2f884df244.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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8 . 已知
,
,平面内动点P满足
.
(1)求动点P的轨迹C的方程;
(2)动直线
交C于A、B两点,O为坐标原点,直线
和
的倾斜角分别为
和
,若
,求证直线
过定点,并求出该定点坐标;
(3)设(2)中定点为Q,记
与
的面积分别为
和
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48d195911a91d12edd5685f6cd963fa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/743eac5ef7cd9452d9678d797da748ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb18941b1e55e62e6c3f54a35ccb214.png)
(1)求动点P的轨迹C的方程;
(2)动直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46f478abeeb4da23121b652cf907972d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(3)设(2)中定点为Q,记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/592f84fbbd939b954f52dc6b8c009b53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a51aab5c5e99207337fb64603887579c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/884d40a97fd767e95f34f3b91ab8d84c.png)
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9 . 正三棱柱
内切球(球与上下底面和侧面都相切)的半径是
为棱
上一点,若二面角
为
,则平面
截内切球所得截面面积为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a266ae7eacf6133d6e6dcc7c7eeb7f8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/325fbf7c39864c58789bc8ebe853dbe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ac09dc1ca2cdd7aef28c218763d3e4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/500df0e782bb081e608f4bc1d576afcf.png)
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10 . 如图抛物线
,过
有两条直线
与抛物线交于
与抛物线交于
,
斜率为1,求
;
(2)是否存在抛物线
上定点
,使得
,若存在,求出
点坐标并证明,若不存在,请说明理由;
(3)直线
与直线
相交于
两点,证明:
为
中点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/159a2d4945049c5a8f525f18763eba91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52b385adf402cd17bddace02fdb32030.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/254d90ef7eba319615e1fd6e01f6abd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c11bcc7389ee99011b2d77ee0bb4018.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e1e4115d78e625e9e0f47cdade3286.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2401bc9c26cc3b0b8384c7139bd58fff.png)
(2)是否存在抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63b0b8816e5284e07f4377c88fa7980e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
(3)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d585d2d6643471640905d234d9538c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6b0a3fa475b24f57ecd79c681259561.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
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