2020高三·全国·专题练习
名校
1 . 易知椭圆
,其短轴为4,离心率为e1.双曲线
的渐近线为
,离心率为e2,且
.
(1)求椭圆E的方程;
(2)设椭圆E的右焦点为F,过点G(4,0)斜率不为0的直线交椭圆E于M、N两点设直线FM和FN的斜率为
,试判断
是否为定值,若是定值,求出该定值;若不是定值,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a200ca2c4af794f4d1c6a5443830b5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a90adb1f0e193b178f7e387b21a174d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d3051f43ac48c0a730a791b8a93ad37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7bcaf273474069ffe4d6e0db22e2cbf.png)
(1)求椭圆E的方程;
(2)设椭圆E的右焦点为F,过点G(4,0)斜率不为0的直线交椭圆E于M、N两点设直线FM和FN的斜率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90963760acac7bfad3ae03088c6c80b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b69e3f7ddd51215d00661c09cd900d60.png)
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6卷引用:2019年全国高中数学联赛甘肃省预赛
2019年全国高中数学联赛甘肃省预赛甘肃省高台县第一中学2021-2022学年高三上学期期中考试数学(理科)试题(已下线)秒杀题型11 圆锥曲线中的定值与定点-2020年高考数学试题调研之秒杀圆锥曲线压轴题宁夏自治区银川市银川九中、石嘴山三中、平罗中学三校2020届高三下学期联考数学(文)试题西藏日喀则市2021届高三学业水平考试数学(文)试题吉林省吉林市第一中学2022-2023学年高二上学期期末数学试题
2 . 已知x>0,y>0,且
,则x+2y的最小值为____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51376dbee9c1fcebf4c93e4e55b1e6a1.png)
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名校
3 . 已知椭圆
过点
,且右焦点为
.
(1)求椭圆
的方程;
(2)过点
的直线
与椭圆
交于
两点,交
轴于点
.若
,求证:
为定值;
(3)在(2)的条件下,若点
不在椭圆
的内部,点
是点
关于原点
的对称点,试求三角形
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7d72a07a4e5acfc140a3cea1f26b951.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15ed90ebf0061c8a79beed307fc1719a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be92f0e0012a7696c78e3e00513edefd.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](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/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a898cb989ec6bf8f509aab49c043dd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fd1f0ace9ca0b79929e73af6c201c2e.png)
(3)在(2)的条件下,若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c606f78391198b6648ba0b92b60f8cf.png)
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5卷引用:2018年全国高中数学联赛甘肃省预赛
2018年全国高中数学联赛甘肃省预赛上海市南洋模范中学2019-2020学年高三上学期期中数学试题浙江省名校协作体2018-2019学年高二下学期开学联考数学试题上海市南洋模范中学2023届高三上学期开学考数学试题(已下线)第五篇 向量与几何 专题5 调和点列 微点3 调和点列(三)
4 . 设函数
(
).
(1)讨论
的单调性;
(2)如果
有两个极值点
和
,我们记过点
的直线斜率为
.问:是否存在
,使得
?若存在,求出
的值,若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cff4255bff210170362868e710db87ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/450669619f798d4e05e3f67f72f34282.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0de3129f0c484d2542e4d0810cbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b680c9cf0ac2e7aed2f445597eb47446.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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名校
5 . .
已知点
是椭圆
右焦点,点
、
分别是x轴、 y上的动点,且满足
,若点
满足
.
(1)求
点的轨迹
的方程;
(2)设过点
任作一直线与点
的轨迹
交于
、
两点,直线
、
与直线
分别交于点
、
(其中
为坐标原点),试判断
是否为定值?若是,求出这个定值;若不是,请说明理由.
已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a84a8b67c0f5113f0b38ef3d184c2d13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8649ce18c628d0e03e72cef541f8284f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fbb6cae4ac201f350e9856544320303.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8369711615682d008a80ef16d2b2d51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c55653fdc089396e3234de1b5858de7.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设过点
![](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/c5db41a1f31d6baee7c69990811edb9f.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/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31675a03001a15f0e810571f6a517fe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://img.xkw.com/dksih/QBM/2012/2/24/1570771606020096/1570771611451392/STEM/d49b17e185e64167b350448541edce19.png)
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3卷引用:2016年全国高中数学联赛甘肃赛区预赛试题
6 . 已知函数
.
(1)设实数k使得
恒成立,求k的取值范围.
(2)设
.若函数
在区间
内有两个零点,求k的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f53f81bca037a4383c1fab122a3cd3d.png)
(1)设实数k使得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5df785db9176182d55ffac97bfa3f29.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f094456ccfcfba13838e3a97f77b319.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08d9a096b1253a0d451a949027e65e77.png)
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3卷引用:2016年全国高中数学联赛甘肃赛区预赛试题
7 . 设数列
的前n项和为
,点
在
的图像上.
(1)求数列
的通项公式;
(2)求
,且对任意的正整数n,均有
.证明:对任意
,总有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7119c1ea2321accb7e8bb391fa9c8a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16418488e7184ec275286d55e709a48d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec8c7aac435710854216e3bca16176d7.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff58aa3ab29807fda113b73463e3e305.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f8a0c487e4c80517667c359ee396b44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7486e37af3a2c7779b31ccc1523da0bf.png)
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2卷引用:2016年全国高中数学联赛甘肃赛区预赛试题
8 . 设f(x)为定义在整数集上的函数,满足条件
(1)
,
;
(2)对任意的
、
均有
则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba2d9db9bb4ada7ba542b14627d5bfc3.png)
______ .
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/249a976e88133f3b3733f09137cf5c42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dec65a2bec3d4296c613a80b3ae41d5e.png)
(2)对任意的
![](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/acdd92c0f57dbca21d7dac2252dfae08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba2d9db9bb4ada7ba542b14627d5bfc3.png)
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9 . 已知椭圆
:
的左、右焦点为F1、F2,设点F1、F2与椭圆短轴的一个端点构成边长为4的正三角形.
(1)求椭圆
的标准方程;
(2)过椭圆
上任意一点
作椭圆
的切线与直线
的垂线
交于点M,求点M的轨迹方程;
(3)若切线
与直线
交于点,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da01a3abe1c9dc4e6283afa0dc1a0d39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d8ad9e94d07405a6be585f81a0d623b.png)
(3)若切线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/438f34bc8b04e8c494b91306ac6fe352.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/639c3d2ff5ee566fcc1b69c65712a661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df51575d9a6b6c449c50d553c68ebb1.png)
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10 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51866bf4c9401ec9529d6ff9643a2a28.png)
(1)若函数
在区间
上为单调增函数,求
的取值范围;
(2)设
,证明:
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51866bf4c9401ec9529d6ff9643a2a28.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ab5e0524def52baf53480b8726784ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66eba129d92ede31b728e2590c4db2a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79c425d7c4fd26e00592c3fa09f4d4f8.png)
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2018-12-06更新
|
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2卷引用:2015年全国高中数学联赛甘肃赛区预赛试题