名校
解题方法
1 . 已知函数
.
(1)当
时,
恒成立,求实数
的取值范围;
(2)已知直线
是曲线
的两条切线,且直线
的斜率之积为1.
(i)记
为直线
交点的横坐标,求证:
;
(ii)若
也与曲线
相切,求
的关系式并求出
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6453c2284ab370e0c3817f5e14bafa7d.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a3c442579603164f3fc19458677d307.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43920f5171ed31db2520ef00e4c5fc24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)已知直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50297ad9f7256b4d2efc3462289f18b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9849df491461fb04b28fd5fe6017753e.png)
(i)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9849df491461fb04b28fd5fe6017753e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f90560052fe43871fd3d594c771723c.png)
(ii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9849df491461fb04b28fd5fe6017753e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
您最近一年使用:0次
解题方法
2 . 设Sn是数列
的前n项和,定义等斜率数列
且
等式
恒成立.
(1)若
是首项为1,公比为3的等比数列,请判断
是否为等斜率数列,并说明理由;
(2)已知
是等斜率数列,证明:
是等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f0e0373a4e95709a67c312cdc054466.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d48bb696708fd77448c1427b6e769fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2d0c1a0bddea64281c61f2851b37634.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
您最近一年使用:0次
3 . 设椭圆C:
的左、右顶点和椭圆
的左、右焦点均为E,F.P是C上的一个动点(异于E,F),已知直线EP交直线
于点A,直线FP交直线
于点B.直线AB与椭圆
交于点M,N,O为坐标原点.
(1)若b为定值,证明:
为定值;
(2)若直线OM,ON的斜率之积恒为
,求b.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd8c4b0e11e5894346b27e4656336c67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9e7a9b887d3dca1953a4f271d7b60a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f09e84f509537abd3dfaa20d7915675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee023dde0cd66b17a3ec5775d4b7d714.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(1)若b为定值,证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2b0ba14e41e306e5633ad4bf1cdedd8.png)
(2)若直线OM,ON的斜率之积恒为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3389f53711264b0acba3ba6019f8b908.png)
您最近一年使用:0次
名校
解题方法
4 . 在初等数论中,对于大于1的自然数,除了1和它自身外,不能被其它自然数整除的数叫做素数,对非零整数a和整数b,若存在整数k使得
,则称a整除b.已知p,q为不同的两个素数,数列
是公差为p的等差整数数列,
为q除
所得的余数,
为数列
的前n项和.
(1)若
,
,
,求
;
(2)若某素数整除两个整数的乘积,则该素数至少能整除其中一个整数,证明:数列
的前q项中任意两项均不相同;
(3)证明:
为完全平方数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de8b048de4625c67d7f74a4eda94877a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/451b1923f70f26e57c557ffe606f7016.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26d9b6c86435e0ceff94d8ad1cd03737.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/addee6ce5163a2580888ce2da22714af.png)
(2)若某素数整除两个整数的乘积,则该素数至少能整除其中一个整数,证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ab4a56d5549a49adae4f1f17926dd8b.png)
您最近一年使用:0次
名校
解题方法
5 . 已知
,其中
.
(1)当
,
时,
①任意写出
的一条对称轴;
②求证:
;
(2)若对任意
,
,求
所能取到的最小值和最大值,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aba354888ba7e2065e85656c20f31005.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/360ff131c51a4ef6745538c18cec92c2.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/191d9381c4f252fbb5553ba72462d0aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5805d32dc3582d0a706c015875c15eb9.png)
①任意写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
②求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e9c599e8d420006448905acec2b8234.png)
(2)若对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e9c599e8d420006448905acec2b8234.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20d6fc9b90f370fbb27552876b650f8f.png)
您最近一年使用:0次
2024·全国·模拟预测
名校
解题方法
6 . 已知直线
和椭圆
.
(1)证明:
与
恒有两个交点;
(2)若
为
与
的两个交点,过原点且垂直于
的直线交
于
两点,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b648c913b1b66abb9cd526dd8a7b2ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf0b0ae6048bf94ccd5a3b4a8aafd81e.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ce9db8bf6dff7b5b1feb849f5532f22.png)
您最近一年使用:0次
解题方法
7 . 西附高中为了解“方洲路”,“普惠路”两个校区高二学生的数学水平,随机抽取200名学生进行调查统计,得到如下
列联表:
(1)依据小概率值
的
独立性检验,判断两校区学生的数学成绩优秀率是否有差异?
(2)西附高中不仅关注学生的学习成绩,更加注重学生的身心健康,德智体美劳全面发展.
①从上述参与调查的200人中按分层抽样从两校区抽出10人,再从10人中随机抽取3人参加“书记有约”活动,设其中来自“方洲路”的学生数为随机变量X,求随机变量X的分布列;
②为更好了解上述身体状况,将这200名同学排在一起逐个依次体检,己知每位同学体检所需时间为1分钟,求证:数学优秀同学体检全部结束所需时间
的期望
.
附:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7dc70b5e1ba847b9918a50f67bfbe8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b72fcdc709e77910cd36a26369648b3.png)
优秀 | 不优秀 | 合计 | |
方洲路 | 30 | 90 | 120 |
普惠路 | 25 | 55 | 80 |
合计 | 55 | 145 | 200 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0298d106f2b72aadf3cffce041a25da6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/953e89f7e797bf8fe9ff31e0d2f66728.png)
(2)西附高中不仅关注学生的学习成绩,更加注重学生的身心健康,德智体美劳全面发展.
①从上述参与调查的200人中按分层抽样从两校区抽出10人,再从10人中随机抽取3人参加“书记有约”活动,设其中来自“方洲路”的学生数为随机变量X,求随机变量X的分布列;
②为更好了解上述身体状况,将这200名同学排在一起逐个依次体检,己知每位同学体检所需时间为1分钟,求证:数学优秀同学体检全部结束所需时间
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a4a9223f80c2d41ae0f933a20e00180.png)
附:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7dc70b5e1ba847b9918a50f67bfbe8f.png)
0.1 | 0.050 | 0.010 | 0.001 | |
2.706 | 3.841 | 6.635 | 10.828 |
您最近一年使用:0次
8 . 设函数
为定义在区间
上的可导函数,记
的导函数为
,若对
,都有
或
恒成立,则称
为区间
上的“原导同号函数”.
(1)证明:
为
上的“原导同号函数”;
(2)是否存在实数
,使
为
上的“原导同号函数”,若存在,求出
的取值范围;若不存在,请说明理由;
(3)若
为
上的“原导同号函数”,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc1da2db85b44ae9ced8c09cd19593e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/126f7f1e8cdd38225803c6ec59968660.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22a935456da83aed9c3f485152e541f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36489856ced75bf35dea7b12c2b6bcd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5d6243e93c41978871cb23d8e66148d.png)
(2)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b14ce24da9de20311832866834d78a05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5d6243e93c41978871cb23d8e66148d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6146c34c9aad4d49938e086d3b18c774.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8845c0d06613fabb0358d5392cca38b3.png)
您最近一年使用:0次
名校
解题方法
9 . 已知圆的方程
,
,
,抛物线过
两点,且以圆的切线为准线.
(1)求抛物线焦点的轨迹C的方程;
(2)已知
, 设x轴上一定点
, 过T的直线交轨迹C于
两点(直线
与
轴不重合),求证:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54329a84abb204cecb237b2bf2ff2bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cd99c5000629d7f49499d666e68f40d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852b303689c31189cd47bb4a3220f9fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
(1)求抛物线焦点的轨迹C的方程;
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c153027427477bcd0a7228b14ce96cc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eaf5077ae09443a75fb8301841116a9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b800af3d8614f3f30eefa7de8d0b581f.png)
您最近一年使用:0次
2024-02-03更新
|
899次组卷
|
3卷引用:四川省成都市2023-2024学年高二上学期期末校级调研联考数学试题
解题方法
10 . 如图,
为坐标原点,
为抛物线
的焦点,过
的直线交抛物线于
两点,直线
交抛物线的准线于点
,设抛物线在
点处的切线为
.
与
轴的交点为
,求证:
;
(2)过点
作
的垂线与直线
交于点
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b072ff6d1b83232bebd7d4709ffba4ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2c3d2cba96f6f03520c0b3f6e4da03e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2e095298cfb23d9f47811556fc9f9a.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2c3d2cba96f6f03520c0b3f6e4da03e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa3eae6aa7672b9eef122fb7a1dab14e.png)
您最近一年使用:0次
2024-03-13更新
|
1608次组卷
|
5卷引用:湖北省七市州2024届高三下学期3月联合统一调研测试数学试题
湖北省七市州2024届高三下学期3月联合统一调研测试数学试题山东省潍坊市昌乐北大公学学校2024届高三下学期3月监测数学试题四川省成都市教育科学研究院附属中学2023-2024学年高三下学期4月综合测试数学(理科)试题(已下线)湖北省七市州2024届高三下学期3月联合统一调研测试数学试题变式题16-19甘肃省兰州市2024届高三下学期三模数学试题