真题
解题方法
1 . 如图,椭圆的长轴
与x轴平行,短轴
在y轴上,中心为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/12/3ed9fd99-c37e-4477-bfd6-d8eb42e17367.png?resizew=293)
(1)写出椭圆的方程,求椭圆的焦点坐标及离心率;
(2)直线
交椭圆于两点
;直线
交椭圆于两点
,
.求证:
;
(3)对于(2)中的中的在
,
,
,
,设
交
轴于
点,
交
轴于
点,求证:
(证明过程不考虑
或
垂直于
轴的情形)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/473913c0887bb64d386f4c02f1853452.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bc9076974ebd6331d67055302be8167.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e395571ff5d1ea9ea8ceb06522211f89.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/12/3ed9fd99-c37e-4477-bfd6-d8eb42e17367.png?resizew=293)
(1)写出椭圆的方程,求椭圆的焦点坐标及离心率;
(2)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/766bc42b7ead98238a339bb4dc42bb51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48e9c4ea393bbf064453e91f4800f967.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87f8af9ce5d927e6f422de42ead6ffb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a060ffc86c94a526d4d1086e5590a4f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eea915b7c0562b239ea553b9ed2f9897.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6318191342aedeaeeddb0f259ed759b3.png)
(3)对于(2)中的中的在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6655e2fa64a32cd12fe0279afd65d73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce8f887360a533f0a25b0b34fb11f0a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15abfafc59b6f9f01f3be4db4df797d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6655e2fa64a32cd12fe0279afd65d73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce8f887360a533f0a25b0b34fb11f0a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
您最近一年使用:0次
名校
2 . 《见微知著》谈到:从一个简单的经典问题出发,从特殊到一般,由简单到复杂:从部分到整体,由低维到高维,知识与方法上的类比是探索发展的重要途径,是思想阀门发现新问题、新结论的重要方法.
阅读材料一:利用整体思想解题,运用代数式的恒等变形,使不少依照常规思路难以解决的问题找到简便解决方法,常用的途径有:(1)整体观察;(2)整体设元;(3)整体代入;(4)整体求和等.
例如,
,求证:
.
证明:原式
.
波利亚在《怎样解题》中指出:“当你找到第一个藤菇或作出第一个发现后,再四处看看,他们总是成群生长”类似问题,我们有更多的式子满足以上特征.
阅读材料二:基本不等式
,当且仅当
时等号成立,它是解决最值问题的有力工具.
例如:在
的条件下,当x为何值时,
有最小值,最小值是多少?
解:∵
,∴
,即
,∴
,
当且仅当
,即
时,
有最小值,最小值为2.
请根据阅读材料解答下列问题
(1)已知如
,求下列各式的值:
①
___________.
②
___________.
(2)若
,解方程
.
(3)若正数a、b满足
,求
的最小值.
阅读材料一:利用整体思想解题,运用代数式的恒等变形,使不少依照常规思路难以解决的问题找到简便解决方法,常用的途径有:(1)整体观察;(2)整体设元;(3)整体代入;(4)整体求和等.
例如,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca27cc54ca0332245f5167488daa3408.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e2764ccd2cfe6de0c53dce98e45b120.png)
证明:原式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87898da3367d13667477a10c9cc47ac2.png)
波利亚在《怎样解题》中指出:“当你找到第一个藤菇或作出第一个发现后,再四处看看,他们总是成群生长”类似问题,我们有更多的式子满足以上特征.
阅读材料二:基本不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a28514741f365301978e922fdca0fcc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f22fec5a381ae8aca93d876e54c79de.png)
例如:在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13f40c24c64bbb0645fcf585f4e66872.png)
解:∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c42b50f6f9e56ea5f222b0a40cb4a3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91bb4a7110c19cd10cb915e55438314b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d32ba3941cef6b1d549f050f0d314e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63af71b9e6f71cd26e6e97541154cd8c.png)
当且仅当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b6a593ef3641dbd11e324dbe78b4dc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13f40c24c64bbb0645fcf585f4e66872.png)
请根据阅读材料解答下列问题
(1)已知如
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca27cc54ca0332245f5167488daa3408.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f0dd92f322200ecabfb74ffd7cf3f4a.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af71e37295978173629004816b65791a.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56667aabbe787eb1c3189d487d203e22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d9093a255130a938a4d84595c0c56ce.png)
(3)若正数a、b满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca27cc54ca0332245f5167488daa3408.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ab1cbf887eca130c254f6e0cf3fdb2f.png)
您最近一年使用:0次
2021-10-29更新
|
530次组卷
|
3卷引用:江苏省南通中学2020-2021学年高一上学期开学考试数学试题
江苏省南通中学2020-2021学年高一上学期开学考试数学试题江西省南昌市第二中学2023-2024学年高一上学期月考数学试题(一)(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)
3 . 设直线
,曲线
.若直线
与曲线
同时满足下列两个条件:①直线
与曲线
相切且至少有两个切点;②对任意
都有
.则称直线
为曲线
的“上夹线”.
(1)已知函数
.求证:
为曲线
的“上夹线”;
(2)观察下图:
的“上夹线”的方程,并给出证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70087bf78bee970f6ecf583ca1fccc42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0016d106579d6b26cf2960cf744f317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d9dc155203792c9983b2118b7730088.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
(1)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c043c3bf7b638f8bb635ee098130560.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c31c4f39399ec245a67db2933ed639f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)观察下图:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d08fe48eafb7a58cb673cc4bce2aa0e7.png)
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4 . 过曲线
:
上的点
作曲线
的切线
与曲线
交于
,过点
作曲线
的切线
与曲线
交于点
,依此类推,可得到点列:
,已知
.
(1)求点
,
的坐标;
(2)求数列
的通项公式;
(3)记点
到直线
(即直线
)的距离为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c904567c3b3734e1eca8d042ef7a7b2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b225d772013d021cf1bfe7b9421fa5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6b7e35faab6d74fa0c36599c39d1698.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9cb8e6ff801523b0304576cd69fd2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b51443d97c22cfa55a47270bfdd7b37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f724c379a00959905b87eedbe6d61fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87a60302649eb940748da818199e55da.png)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9cb8e6ff801523b0304576cd69fd2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/797e67927616b141ed7c6b83f8b6f4fb.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
(3)记点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29a5b0f908cdae073db61be5b42fbcf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c93f04dcabdafec74f98f4a1f4faa3fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82e260b088f071983f254ce8f5163fcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91321f9e0b712ccef47c4b9e0baca333.png)
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5 . 已知
、
是椭圆
上两动点,
为原点,定点
,向量
,
在向量
方向上的投影分别为
,
,且
,动点
满足
.
(1)求点
的轨迹
的方程;
(2)记点
,
,求证:无论动点
在轨迹
上如何运动,
恒为一个常数.
![](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/f578c87d528cc32b2acb0e913391c26e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33a62c9695f5a1691c5fe8724fa764b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4f605ec0729ce6d72237ad662a06862.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fc9656d8286c4d6fa309d6ae347c89e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21909dd065ccc349a2cbfd4c3cf4976b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca70897450a4208d95018c8fac6138ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98094053649f93909ac555de3694ad52.png)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
(2)记点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae140e4db2c5563e5f902fcbebaac262.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c95963e8e4dcc511f0d86b1853ddcdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b894af0328f96959e0ef1c19ff50cdd.png)
您最近一年使用:0次
解题方法
6 . 设函数为
上的增函数,令
.
(1)判断并证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46be55c8f2760d6db125f46691a3de48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5226c58ca852742dca2b380d1fd4042e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/450398974b1561ca801e102e16df6789.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85a4d51c0394ade8f125957afaa6e994.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f93e1aa555d1670dfed9c8861bed364d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
您最近一年使用:0次
7 . 已知函数
,其中
,
为实数且
.
(1)当
时,根据定义证明
在
单调递增;
(2)求集合
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9021fa0cc7427f534fbf1ec1db79c4b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2c80c26a794a844127aae7dee87c93b.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f0d68648b10fce54dfc19c5ee60086d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c0070f5fd205c3c48a41bd865a7049b.png)
(2)求集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac3810dfe2b1eb718bae23bf0bab35e.png)
您最近一年使用:0次
解题方法
8 . 已知
是整系数方程
的一个无理根,求证:存在常数
,使得对任意互质的正整数p,q,均有
,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1146110d4382c714c10de00dd1273b7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75aed2b995e34438a29a4170cf535914.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08a12ee40200dc7bfcc26beea31e2328.png)
您最近一年使用:0次
9 . 如图1,设是一个锐角三角形,且
,
为其外接圆,
分别为其外心和垂心,
为圆
直径,
为线段
上一动点且满足
.
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/186354626d5a1d13b88dfb4df916947b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
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