1 . 已知椭圆
的左、右焦点分别为
,
,离心率为
,点
,且
为等腰直角三角形.
(1)求椭圆
的标准方程;
(2)设点
为
上的一个动点,求
面积的最大值;
(3)若直线
与
交于
两点,且
,证明:直线
过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f20a7cb8494b71121a4d176f923d4318.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/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1b6f209d1a805437046ca6ef79dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33d776753746914c2410a3946c357f35.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50667ce76318b23b69ac6b8e16dc80d0.png)
(3)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c27ca17526003f779a0b68993a26180.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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5卷引用:山西省临汾市部分学校2023-2024学年高二下学期5月质量检测数学试题
山西省临汾市部分学校2023-2024学年高二下学期5月质量检测数学试题河南省部分重点高中(金科未来)2023-2024学年高二下学期5月大联考数学试题河南省部分重点高中2023-2024学年高二下学期5月质量检测数学试题(已下线)湖南省岳阳市第一中学等多校2023-2024学年高二下学期5月月考数学试题(已下线)模型6 非对称结构和齐次化处理问题模型
名校
2 . 已知函数
.
(1)当
时,求曲线
在点
处的切线方程;
(2)已知
有两个极值点.
(ⅰ)求
的取值范围;
(ⅱ)若
的极小值小于
,求
的极大值的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e589b525104e0f2f599ed6ecf27701fd.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5828873f8369183faf71181cda5b61d2.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(ⅰ)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09a691a1db38dabca5af44a4c6817d21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
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5卷引用:山西省临汾市部分学校2023-2024学年高二下学期5月质量检测数学试题
山西省临汾市部分学校2023-2024学年高二下学期5月质量检测数学试题河南省部分重点高中(金科未来)2023-2024学年高二下学期5月大联考数学试题河南省名校2023-2024学年高二下学期5月质量检测数学试题河南省部分重点高中2023-2024学年高二下学期5月质量检测数学试题(已下线)专题08 导数的运算、几何意义及极值最值常考题型归类--高二期末考点大串讲(人教B版2019选择性必修第三册)
名校
3 . 近年来,我国青少年近视问题呈现高发性、低龄化、重度化趋势. 已知某校有学生200人,其中40人每天体育运动时长小于1小时,160人每天体育运动时长大于或等于1小时,为研究体育运动时长与青少年近视的相关性,研究人员采用分层随机抽样的方法从学生中抽取50人进行调查,得到以下数据:
(1)请完成上表,并依据小概率值
的独立性检验,能否认为学生是否近视与体育运动时长有关?
(2)为进一步了解近视学生的具体情况,现从调查的近视学生中随机抽取3人进行进一步的检测,设随机变量
为体育运动时长小于1小时的人数,求
的分布列和数学期望.
附:
参考公式:
,其中
.
体育运动时长小于1小时 | 体育运动时长大于或等于1小时 | 合计 | |
近视 | 4 | ||
无近视 | 2 | ||
合计 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83caa0ad94044a1e206b1cc0b3f85080.png)
(2)为进一步了解近视学生的具体情况,现从调查的近视学生中随机抽取3人进行进一步的检测,设随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
附:
0.10 | 0.05 | 0.010 | 0.005 | 0.001 | |
2.706 | 3.841 | 6.635 | 7.879 | 10.828 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2187714e660234f0b72f2b47d3ea685a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/356b05e46b10ee51c3e43546d73ec96c.png)
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5卷引用:山西省临汾市部分学校2023-2024学年高二下学期5月质量检测数学试题
山西省临汾市部分学校2023-2024学年高二下学期5月质量检测数学试题河南省部分重点高中(金科未来)2023-2024学年高二下学期5月大联考数学试题河南省部分重点高中2023-2024学年高二下学期5月质量检测数学试题(已下线)湖南省岳阳市第一中学等多校2023-2024学年高二下学期5月月考数学试题(已下线)专题07 回归方程与独立性检验--高二期末考点大串讲(苏教版2019选择性必修第二册)
名校
4 . 如图,在直三棱柱
中,
,
,
,
为
的中点.
;
(2)设
为
的中点,
在棱
上,满足
平面
,求
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c3e9ef3e849788645552cfb0735d987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ef8866ccf160ddc441bf69c5d3a3d5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36c4559d27e3905980d1a4f1856f07de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1f229274a6e17977cc047814212589.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cff8e4e75b207f6eb4f0d1052ce250c.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db807b09cc550f476b3f8fa0c6a14425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/780d3f5f4c4419913c1232b7aae03ade.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/780d3f5f4c4419913c1232b7aae03ade.png)
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4卷引用:山西省临汾市部分学校2023-2024学年高二下学期5月质量检测数学试题
5 . 已知数列
满足
,
.
(1)证明:数列
为等比数列;
(2)在
与
之间插入
个数,使得这
个数组成公差为
的等差数列,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57bfc1f8772f31748bfdc280d0712fc0.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e2de706dc5f0439b989273a5367f63a.png)
(2)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f255d0395fba51ca2d44293cca42e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/217b927efe12a98e1082ecd7f035b921.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4792fd59c4ca11ff03dc32e367c3983f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2d1ae060458f733025fc82f7c7b14f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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4卷引用:山西省临汾市部分学校2023-2024学年高二下学期5月质量检测数学试题
6 . 如图,在平面直角坐标系中,
和
是
轴上关于原点对称的两个点,过点
倾斜角为
的直线
与抛物线
交于
两点,且
.
为
的焦点,求证:
;
(2)过点
作
轴的垂线,垂足为
,若
,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bb4dd4670828f75bc573b52cdd02e1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e08136e6bc876f29a13d1204d9d621db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fea0b6471289760543596f5f45aa43ae.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14cc1572ca21da2e3271484f127a5094.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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3卷引用:山西省临汾市2024届高三下学期考前适应性训练(三)数学试题
7 . 已知函数
的图象可由函数
的图象平移得到,且关于直线
对称.
(1)求
的值;
(2)求函数
的单调递增区间.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2afc3ff03a15347e4b1e270d34825953.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42b7303c20aa276076074282802c004a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9895bf4192f5c55c16f8270d53c49b13.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60edd6f22af315c5a5d77e741149e63f.png)
(2)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e41ead8c199f0b65d4444e646740054e.png)
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解题方法
8 . 已知函数
.
(1)求
在
处的切线方程;
(2)若曲线
与直线
有且仅有一个交点,求
的取值范围;
(3)若曲线
在
处的切线与曲线
交于另外一点
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c2e33e637f3ce3f7163031be3bedfd0.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
(2)若曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1447dbe580ac5c825776995118e75acf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e209a60f1031bc4978c0cfce9bc1358.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2c9a4bbe272cdfd820fbbcfeceea4fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bc1ae6f6b30f3f422b6d4eb22c2d216.png)
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解题方法
9 . 某导弹试验基地,对新研制的
型导弹进行最后确定试验.
(1)据以往多次试验,
型导弹每次击中空中目标的概率为
.用该导弹对目标进行连续射击,若击中2次,则目标被击落,射击停止;若射击达到5次,不管目标击落与否,则结束试验.求射击次数
的分布列并计算其期望;
(2)据以往多次试验,
型导弹每次击中空中目标的概率为
.用该导弹对目标进行连续射击,若击中1次,则目标被击落,射击停止.请完成以下关于射击次数
的分布列,并证明:
.
(参考公式:若
,则
.)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
(1)据以往多次试验,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(2)据以往多次试验,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c624cacd18c986d5268935b1faab38ba.png)
![]() | 1 | 2 | 3 | … | ![]() | … |
![]() | … | … |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b70d4a3fc3e01b5a6358cf4e57578e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fc9b039c2507ad25dc25c6b039da7c1.png)
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10 . 在计算机科学中,
维数组
是一种基础而重要的数据结构,它在各种编程语言中被广泛使用.对于
维数组
,
,定义
与
的差为
与
之间的距离为
.
(1)若
维数组
,证明:
;
(2)证明:对任意的数组A,B,C,有
;
(3)设集合
中有
个
维数组,记
中所有两元素间的距离的平均值为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6b93450a25edf994283872816f0a175.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47d3f2ae3aa8322ec0e2c36471bfc7f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb963b1d6c794e8d3331214473f44d61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90154ab818dc6fd319683db8b642c3f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/447cf234057b2b23e80cc0ec6770de2c.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/288e149a1d5633a31b2fb684564e1547.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebde3a4de6911a9cf5dcc9641c76f5dd.png)
(2)证明:对任意的数组A,B,C,有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8513f18376e4e456b939d0f1cdb6e602.png)
(3)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/176ff134abb231aa2b562ca6fe240a44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423cc16756424271a003917fbca775b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3392856fc8ba3250d76f281e9f04129.png)
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