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
真题
2 . 如图,
为椭圆的两个顶点,
为椭圆的两个焦点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/18/81b84268-931b-499e-914d-cc9003bd1086.png?resizew=397)
(1)写出椭圆的方程及准线方程;
(2)过线段
上异于O,A的任一点K作
的垂线,交椭圆于P,
两点,直线
与
交于点M.求证:点M在双曲线
上.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32622697953ab5e4b4e35cd62d1b1c64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/18/81b84268-931b-499e-914d-cc9003bd1086.png?resizew=397)
(1)写出椭圆的方程及准线方程;
(2)过线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2708fa6298e52f617383efc175b71ddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d0ff310aabd2282b539537ebed3f788.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aba947934b05caae8b0c1fb1a522a0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6d49e09b17b19d7c787e95bcf1e9731.png)
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真题
解题方法
3 . 已知
,
,其中
,设
,
.
(1)写出
;
(2)证明:对任意的
,恒有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0201063518911954b565c33f4e6922b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ef9a5c965598ea0f492ade8bf01f85c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9dbc206aad9e1a0edfb2504e513d3a9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1497c9cb334ca9a1d7b817abb8034735.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66f0ca536621ec8db02707ba65917029.png)
(1)写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6645a5979b3436efdf7d76210d060b7.png)
(2)证明:对任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05adfa1f46f8d2eb486991e61b727f27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9653a00340ce6cfb8d273cc36b1c01d8.png)
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4 . 设函数
.
(1)证明:当
,且
时,
;
(2)点
在曲线
上,求曲线在点P处的切线与x轴和y轴的正向所围成的三角形面积表达式(用
表达).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fb1c53efea67064ff2ddac3ceea3378.png)
(1)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8e14206c7d228a7c2259a7b27da8813.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f94345694d4215284c41f87146795ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dea134f599285e3d32d2ab3e7186990.png)
(2)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab66ffc45b32fa9189415b1edfab9b9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
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真题
5 . 设函数
(
,且
,
)
(1)当
时,求
的展开式中二项式系数最大的项;
(2)对任意的实数
,证明
(
是
的导函数);
(3)是否存在
,使得
恒成立?若存在,试证明你的结论并求出
的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f69438543e3a9928c1f2295262f6579c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10e468312d09c6563c9094b710a35a65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54c50fb5615e36df436d747356b00d78.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39cc033406da2cdd342308972c6701f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f71935797074e889f15a62ac370a9c1.png)
(2)对任意的实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29a0ffe57e983469b590914f5ce06d88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)是否存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c38f3baf9a34265fbdb5c65dd1664d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2b5fb0bc5207a915464ef8a5f40f5e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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6 . 若A、B是抛物线
上的不同两点,弦
(不平行于y轴)的垂直平分线与x轴相交于点P,则称弦
是点P的一条“相关弦”.已知当
时,点
存在无穷多条“相关弦”.给定
.
(1)证明:点
的所有“相关弦”的中点的横坐标相同;
(2)试问:点
的“相关弦”的弦长中是否存在最大值?若存在,求其最大值(用
表示);若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745de5ef1fd897d16e37464172d5e8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c0aa2ef928b6e3341d0a0dc6d8055b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/924886b76c218169b2f4d00bb7e9563c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b042740af8a73f8add3ec9c586b4e540.png)
(1)证明:点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9c9b882323edba16b3625458239b6f3.png)
(2)试问:点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9c9b882323edba16b3625458239b6f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
您最近一年使用:0次
真题
解题方法
7 . 已知函数
,且存在
,使
.
(1)证明:
是
上的单调增函数;
(2)设
,
,
,
,其中
.证明:
;
(3)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c3cfd92b7157867ed0bbf56b6ea2c9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b95189511359447a21cc4e22b3ac972.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66f66a2b3d90f0d935d6c8ebaf675349.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62b6ab454199d2738ea1b5cefb133d50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a0cf06beb7cfde2c2ce4796bfe6d7c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f347a1bd45e8fe728bef4952ff2e6f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14c5a726124806fc0936968107e106e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7247567230a3bebb8fa497c2b22bb02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8c6648cdc6f9ffd069014c2d642400e.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa1febbc0b83f99c6a1a9814c5f6a1c0.png)
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真题
8 . 设点
和抛物线
,其中
,
由以下方法得到:
,点
在抛物线
上,点
到
的距离是
到
上点的最短距离,……,点
在抛物线
上,点
到
的距离是
到
上点的最短距离.
(1)求
及
的方程.
(2)证明
是等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ddf358a5bd45285693963081eb45534.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4165af644315776a4ca477ff04e1537a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eb213494d7551a41dc5153de3bfc850.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87a60302649eb940748da818199e55da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec71037a96b7a4bb1653de301369cbd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4921fcafcfe1ab0a10af0ed484afa5d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa6b09f39af8d61f60a430cbcadc6027.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9cb8e6ff801523b0304576cd69fd2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f536611d68bd7e72f580602902ebdd40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1ec8d169ca3965aa240cdaad351482a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb07a9ee7fa07fcec6a679c9bee53a01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c15016fc7de1cd5971b7d38c70071e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcc31dcdb99754fc452ff2b92a2fb8c9.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
您最近一年使用:0次
真题
解题方法
9 . 如图,以椭圆
的中心O为圆心,分别以a和b为半径作大圆和小圆.过椭圆右焦点
作垂直于x轴的直线交大圆于第一象限内的点A.连结
交小圆于点B.设直线
是小圆的切线.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/16/de47bcec-2815-4e0e-9a2b-92ce47ec6e1c.png?resizew=190)
(1)证明
,并求直线
与y轴的交点M的坐标;
(2)设直线
交椭圆于P、Q两点,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98a9a660910d4efa74ccdcf120273993.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/16/de47bcec-2815-4e0e-9a2b-92ce47ec6e1c.png?resizew=190)
(1)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/502491f4e48e1d74ca8cc709840c30b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
(2)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e194a24c02cf7c0b8a92b56731188f6.png)
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10 . 在研究并行计算的基本算法时,有以下简单模型问题:用计算机求n个不同的数
的和
,计算开始前,n个数存贮在n台由网络连接的计算机中,每台机器存一个数.计算开始后,在一个单位时间内,每台机器至多到一台其他机器中读数据,并与自己原有数据相加得到新的数据,各台机器可同时完成上述工作.为了用尽可能少的单位时间,即可完成计算,方法可用下表表示:
(1)当
时,至少需要多少个单位时间可完成计算?把你设计的方法填入下表:
(2)当
时,要使所有机器都得到
,至少需要多少个单位时间可完成计算?(结论不要求证明)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c282c52b28800637f2a309427aa952.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/054e88e93017889ec167b416af64d9cf.png)
机器号 | 初始时 | 第一单位时间 | 第二单位时间 | 第三单位时间 | |||
被读机号 | 结果 | 被读机号 | 结果 | 被读机号 | 结果 | ||
1 | ![]() | 2 | ![]() | ||||
2 | ![]() | 1 | ![]() |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fac3649308b528fd56545ba102dc42d5.png)
机器号 | 初始时 | 第一单位时间 | 第二单位时间 | 第三单位时间 | |||
被读机号 | 结果 | 被读机号 | 结果 | 被读机号 | 结果 | ||
1 | ![]() | ||||||
2 | ![]() | ||||||
3 | ![]() | ||||||
4 | ![]() |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a66fd124399b79d05102348954f528c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6ebe04c3c961eee27d89170f64af70c.png)
您最近一年使用:0次
2022-11-09更新
|
134次组卷
|
2卷引用:2002年普通高等学校招生考试数学(理)试题(北京卷)