名校
1 . (1)求证:已知
,
,
,
,
,并指出等号成立的条件;
(2)求证:对任意的
,关于
的两个方程
与
至少有一个方程有实数根(反证法证明);
(3)求证:使得不等式
对一切实数
,
,
都成立的充要条件是
,
,
且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/941b8c37cb9b036a5d7faa7eac01fa6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/878f834c03d26711f64bb3abe20e5488.png)
(2)求证:对任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb63478132d4c1fef3c17e591919da83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ace8f8a779c8f039407b7cae737d7212.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/751ee06608e9b40cd42cc4b48165e37c.png)
(3)求证:使得不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e774028355336f9a47e4e5194f3e7b06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.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/fff8a8a07e9fab2efc5be33f1339112f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0ad8d91c1ce139fbf2382a6e8a8f674.png)
您最近一年使用:0次
2 . (1)求证:
;
(2)求证:
;
(3)若m、n、r均为正整数,试证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/588855663a97d8fc98e41368c9f0c887.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724a6dd2bb85b676a9ddbcb4d8ede156.png)
(3)若m、n、r均为正整数,试证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6ca3166112603878ea3d79170b7632d.png)
您最近一年使用:0次
名校
3 . 给定不共面的4点,作过其中3个点的平面,所有4个这样的平面围成的几何体称为四面体(如图所示),预先给定的4个点称为四面体的顶点,2个顶点的连线称为四面体的棱,3个顶点所确定的三角形称为四面体的面.求证:四面体中任何一对不共顶点的棱所在的直线一定是异面直线.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/1/7591e2f1-42ef-474b-ae38-6e946dfe7429.png?resizew=151)
(1)请你用异面直线判定定理证明该结论;
(2)请你用反证法证明该结论.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/1/7591e2f1-42ef-474b-ae38-6e946dfe7429.png?resizew=151)
(1)请你用异面直线判定定理证明该结论;
(2)请你用反证法证明该结论.
您最近一年使用:0次
名校
4 . 定理(三角不等式),对于任意的
、
,恒有
.定义:已知
且
,对于有序数组
、
、
、
,称
为有序数组
、
、
、
的波动距离,记作
,即
,请根据上述俼息解决以下几个问题:
(1)求函数
的最小值,并指出函数取到最小值时
的取值范围;
(2)①求有序数组
、
、
、
的波动距离
;
②求证:若
、
、
、
且
,则
;题(注:该命题无需证明,需要时可直接使用).设两两不相等的四个实数
、
、
、
,求有序数组
、
、
、
的波动距离
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dd0914dc4d4c7f75710ff460a286fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49506c61cf5c61605f1cf90a440348cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd4613271f782a90ab580131d09d03d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dec475a4298eab592d6589aab8915276.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ef141315bf951ddcd300f0743a16897.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7cfd590897d8d908066c781c63a812d.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d3b887215cd1514d3e2e79063729a4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)①求有序数组
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91e07104b699c4012be2d26160976a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7330e52932883877de428cfe91962b96.png)
②求证:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a876ecb804eb0553c246e5fcc40b708.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b73abfe4bc26b1ded680d7abb1a2cac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/effb89a4bffb74028211ecfe671b79d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d46944e1594eec140cacd7b454342561.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daf464629fa321a6ff7401ab79f07083.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cc0ce632fa217dc77f6c92afd311815.png)
您最近一年使用:0次
2022-08-22更新
|
417次组卷
|
7卷引用:第二章 等式与不等式(压轴题专练)-速记·巧练(沪教版2020必修第一册)
(已下线)第二章 等式与不等式(压轴题专练)-速记·巧练(沪教版2020必修第一册)上海市吴淞中学2023-2024学年高一上学期期中数学试题上海市控江中学2021-2022学年高一上学期期中数学试题(已下线)专题02 等式与不等式(练习)-2上海市高桥中学2022-2023学年高一上学期期中数学试题(已下线)期中模拟预测卷03(测试范围:前三章)-2022-2023学年高一数学上学期期中期末考点大串讲(沪教版2020必修第一册)(已下线)上海高一上学期期中【压轴42题专练】(2)
名校
5 . 若数列
的子列
均为等差数列,则称
为k阶等差数列.
(1)若
,数列
的前15项与
的前15项中相同的项构成数列
,写出
的各项,并求
的各项和;
(2)若数列
既是3阶也是4阶等差数列,设
的公差分别为
.
(ⅰ)判断
的大小关系并证明;
(ⅱ)求证:数列
是等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6445438ad302d53a1a94d36d1348f9b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b80c1ed7b10ac7ca1cd81cdd39a8fcc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c15034b5b5cb8ca7ef64bb7517a19a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b5708191e2a4453461ea398c4e16ab6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec7b0cec593a0ea2980b968d3aa826e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367c96a0ff95b92877eda2a7c98871e1.png)
(ⅰ)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367c96a0ff95b92877eda2a7c98871e1.png)
(ⅱ)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
您最近一年使用:0次
2022-11-02更新
|
460次组卷
|
3卷引用:北京市第六十六中学2024届高三上学期期中质量检测数学试题
名校
解题方法
6 . 如图,已知
为二次函数
的图像上异于顶点的两个点,曲线
在点
处的切线相交于点
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/14/b1c9ee20-7af4-4018-a49a-6703b2da8013.png?resizew=203)
(1)利用抛物线的定义证明:曲线
上的每一个点都在一条抛物线上,并指出这条抛物线的焦点坐标和准线方程;
(2)求证:
成等差数列,
成等比数列;
(3)设抛物线
焦点为
,过
作
垂直准线
,垂足为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/031da5d48fbe63745429b1add253344f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c60b6eee6448a408616e1b61bd793f0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf210c8c9e83e70f2d3ede1e18a5f3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/031da5d48fbe63745429b1add253344f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7775aa57ca0e62216f3039ed88dceed0.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/14/b1c9ee20-7af4-4018-a49a-6703b2da8013.png?resizew=203)
(1)利用抛物线的定义证明:曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf210c8c9e83e70f2d3ede1e18a5f3d.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/297426b8f7938c8d14f42a481a19c3a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a2b440f7aac4b432fef8f4c9f8e3f76.png)
(3)设抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf210c8c9e83e70f2d3ede1e18a5f3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35d58f9019097bd05037aefd5c322916.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31b7a8d232e9a11f5d471f47a1294cd4.png)
您最近一年使用:0次
解题方法
7 . 如图,在直角梯形
中,
,
,
,并将直角梯形
绕AB边旋转至ABEF.
平面ADF;
(2)求证:直线
平面ADF;
(3)当平面
平面ABEF时,再从条件①、条件②、条件③这三个条件中选择一个作为已知,使平面ADE与平面BCE垂直.并证明你的结论.
条件①:
;
条件②:
;
条件③:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10df84d553a8826a7ce9bff4bf0d95b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1134c8e3440abb6cd385af2c169037fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6045266f6db39e41b7abde762d9e9a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21f9157fce2a8339d281178c7c0bccbe.png)
(2)求证:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11d27ff0b39832f094ec51e28721d739.png)
(3)当平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf9a6db3571fa57bfa2d5e4d44c51b3.png)
条件①:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6166b9a5437671bcba31e17c375eb39.png)
条件②:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4aca5534bce25acaeb7379deed8f8f.png)
条件③:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c182a9d9fd0a7023b710cd671d9468e7.png)
您最近一年使用:0次
2022-07-08更新
|
1295次组卷
|
11卷引用:7.2 空间几何中的垂直(精练)
(已下线)7.2 空间几何中的垂直(精练)(已下线)7.1 空间几何中的平行与垂直(精讲)(已下线)高考新题型-立体几何初步(已下线)8.6.1 空间直线、平面的垂直(精练)-2022-2023学年高一数学一隅三反系列(人教A版2019必修第二册)(已下线)模块三 专题9(劣构题)拔高能力练(北师大版)(已下线)模块三 专题9(劣构题)基础夯实练(人教B)(已下线)模块三 专题9(劣构题)拔高能力练人教A版)(已下线)2023年高考全国乙卷数学(理)真题变式题16-20(已下线)模块三 专题10(劣构题)拔高能力练(苏教版)北京市丰台区2021-2022学年高一下学期期末练习数学试题(已下线)专题06 空间中点线面的位置关系6种常考题型归类(1)-期期末真题分类汇编(北京专用)
解题方法
8 . 已知抛物线
,
,
是C上两个不同的点.
(1)求证:直线
与C相切;
(2)若O为坐标原点,
,C在A,B处的切线交于点P,证明:点P在定直线上.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fa2c731aaa4005382d5b4324e29fbb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9fb1a589404b101361fab4a264af920.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d4adb1a0c5fbcaa7cb61b2febdb7db3.png)
(1)求证:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d031516b8b9572a1973e44004a30493a.png)
(2)若O为坐标原点,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9eb512456bcc994ea2354e9525d3f282.png)
您最近一年使用:0次
2022-07-25更新
|
1235次组卷
|
6卷引用:专题3-6 抛物线综合大题归类(讲+练)-【巅峰课堂】2023-2024学年高二数学热点题型归纳与培优练(人教A版2019选择性必修第一册)
(已下线)专题3-6 抛物线综合大题归类(讲+练)-【巅峰课堂】2023-2024学年高二数学热点题型归纳与培优练(人教A版2019选择性必修第一册)江西省名校联考2023届高三7月第一次摸底测试数学(理)试题抛物线的综合问题(已下线)专题6 判断位置关系的运算(基础版)(已下线)专题3.14 直线与抛物线的位置关系-重难点题型检测-2022-2023学年高二数学举一反三系列(人教A版2019选择性必修第一册)(已下线)专题05 抛物线8种常见考法归类(2)
名校
解题方法
9 . 设数列
的前
项和为
,且
,数列
满足
,其中
.
(1)证明
为等差数列,求数列
的通项公式;
(2)求使不等式
对任意正整数
都成立的最大实数
的值;
(3)当
时,求证:
.
![](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/851afb5fa82c3e4448ac7b674d143cdf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/661fb8eb9ebf28433198329f10dbafc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
(1)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25cbe66fe4e84b4022721122baab4a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)求使不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47270ec036e4354fd32318aa37e16221.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c902b3e253f48c784aabb9c8f041458b.png)
您最近一年使用:0次
名校
10 . 汽车前灯反射镜曲面设计为抛物曲面(即由抛物绕其轴线旋转一周而成的曲面).其设计的光学原理是:由放置在焦点处的点光源发射的光线经抛物镜面反射,光线均沿与轴线平行方向路径反射,而抛物镜曲面的每个反射点的反射镜面就是曲面(线)在该点处的切面(线).定义:经光滑曲线上一点,且与曲线在该点处切线垂直的直线称为曲线在该点处的法线.设计一款汽车前灯,已知灯口直径为20cm,灯深25cm(如图1).设抛物镜面的一个轴截面为抛物线C,以该抛物线顶点为原点,以其对称轴为x轴建立平面直角坐标系(如图2)抛物线上点P到焦点距离为5cm,且在x轴上方.研究以下问题:
(2)求P点坐标.
(3)求抛物线在点P处法线方程.
(4)为证明(检验)车灯的光学原理,求证:由在抛物线焦点F处的点光源发射的光线经点P反射,反射光线所在的直线平行于抛物线对称轴.
(2)求P点坐标.
(3)求抛物线在点P处法线方程.
(4)为证明(检验)车灯的光学原理,求证:由在抛物线焦点F处的点光源发射的光线经点P反射,反射光线所在的直线平行于抛物线对称轴.
您最近一年使用:0次