2023高二上·江苏·专题练习
1 . 利用数学归纳法证明“
”时,由
到
时,左边应添加因式__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70f716858a6bb27117518863575c4bbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b00f4eb7f1bd2ccefbabf0c1dfa8f69.png)
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2 . 阿波罗尼斯(古希腊数学家),证明过这样的一个命题:平面内与两定点距离之比为常数
(
且
)的点的轨迹是圆,后人将这个圆称为阿波罗尼斯圆.在
中,
,
,当
面积最大时,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a70fd837c70b944a66450bbcf6946bc.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f0d68648b10fce54dfc19c5ee60086d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c525393775354325cbf7839366ca50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ce12d0462e68450e7a47bfb43621742.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0907a673d52825cd7df84b400972d4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a70fd837c70b944a66450bbcf6946bc.png)
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3 . 用数学归纳法证明
(
且
),第一步要证明的不等式是______ ,从
到
时,左端增加了________ 项.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6846743e55bb6f2ee46b2d03ba626461.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a70b95c53fb6655721e2a8c61f5c2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10e468312d09c6563c9094b710a35a65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ef7ca2b3e8061384501f668e59696a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
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4 . 阿波罗尼斯(约公元前262-190年)证明过这样一个命题:平面内到两定点距离之比为常数
的点的轨迹为圆,已知
分别是圆
与直线
上的点,O 是坐标原点,则
的最小值为_______
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b991d4173297923de7c4c1fa48bfae61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c73a64354d8312e10af030d32afe069.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2924070a023138483fbbbf0ccaa73b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/734f00633600b5349c5c83c782332614.png)
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名校
解题方法
5 . 古希腊数学家阿波罗尼斯的著作《圆锥曲线论》是古代世界光辉的科学成果,它将圆锥曲线的性质网罗殆尽,几乎使后人没有插足的余地.他证明过这样一个命题:平面内与两定点距离的比为常数
(
且
)的点的轨迹是圆,后人将之称为阿波罗尼斯圆.现有椭圆
,
,
为椭圆
长轴的端点,
,
为椭圆
短轴的端点,
,
分别为椭圆
的左右焦点,动点
满足
,
面积的最大值为
,
面积的最小值为
,则椭圆
的离心率为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f0d68648b10fce54dfc19c5ee60086d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c525393775354325cbf7839366ca50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3b76e364a93cd78537c6c97b88021f1.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/0b68df477b3ee45ac0f725db00d465a1.png)
![](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/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9eb03004d88965988819597132637b8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a11cb104b04c4e6a1be700e81da279a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ee05b3210c8964deef8ff771173d288.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/781e6927e3bc512359dc8b0c11e195d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
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6 . 《几何原本》卷2的几何代数法(以几何方法研究代数问题)成了后世西方数学家处理问题的重要依据.通过这一原理,很多代数的公理或定理都能够通过图形实现证明,也称之为无字证明.现有如图所示的图形,点
在以
为直径的半圆上,
为圆心,点
在半径
上(不与
点重合),且
.设
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1bc69e886c7afedf1c9233e9a2a6870.png)
__________ (用
表示),由
可以得出的关于
的不等式为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ebef5bab02280cdc99cc7f689135cd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294726f8e596ce099d050ebcd538e421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1bc69e886c7afedf1c9233e9a2a6870.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76edbc800f52f6f8f710b1d7179fb31f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/26/82b41fab-4a23-4fdb-8191-a5a97d6b0134.png?resizew=160)
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7 . 阿波罗尼斯(古希腊数学家,约公元前262~190年)的著作《圆锥曲线论》是古代世界光辉的科学成果,它将圆锥曲线的性质网罗殆尽,几乎使后人没有插足的余地.他证明过这样一个命题:平面内与两定点距离的比为常数
且
的点的轨迹是圆,后人将这个圆称为阿氏圆.现有
,
,求点
的轨迹方程为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fbf56f44f995858afc4f6ae1306bdbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15bb8775b827a649b07b6c2f8c3ea284.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbec23b2248bd42cce868b83be46e5f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/928d95e1c2569436581af40ee38ac1ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
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名校
解题方法
8 . 《几何原本》中的几何代数法是以几何方法研究代数问题,这种方法是后西方数学家处理问题的重要依据,通过这一原理很多的代数公理或定理都能够通过图形实现证明,也称之为无字证明,现有图形如图所示,C为线段
上的点,且
,
,O为
的中点,以
为直径作半圆,过点C作
的垂线交半圆于D,连接
,
,
,过点C作
的垂线,垂足为E,若不添加辅助线,则该图形可以完成的所有无字证明为__________ .(填写序号)
①
;②
;
③
;④
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a3d296e0d7154a170cb7d3ae42989b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4a88b719166fcc1431f876bc8c5656c.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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683c590673eece14fea3319c4fd5eb55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683c590673eece14fea3319c4fd5eb55.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/1/3829f1ef-4d36-4411-b991-19beda49ad27.png?resizew=169)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb90c316d8a99694396de80ed0b0cf25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09adf1ac3103c5ddfe7a92856cb2afbe.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf323321cd5f80e909c243941c33f7db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a305a51783a797bdda25197e090feb05.png)
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名校
解题方法
9 . 我国后汉时期的数学家赵爽通过弦图利用出入相补法证明了勾股定理,在我国历史上还有多人通过出入相补法证明过勾股定理,如下图为我国清末数学家华蘅芳证明勾股定理时构造的图形,在该图中
是以
为斜边的直角三角形,分别以
为边作3个正方形,点
在直线
上,
,记
的周长与面积分别为
,则
的最大值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0e08a39c6619123557148d195abfbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad9c4e14fcf121dca94b9a5549a93288.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3de8fa2fa9271308932731a418f21f95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f484be7264a86c5dcfe4e25d6459ac43.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/23/670c1694-0cdb-4130-9fd8-cc9450265163.png?resizew=168)
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2023高三·全国·专题练习
10 . 阅读下面题目及其解答过程.
如图,在直三棱柱
中,
,
,
分别为
,
的中点.
![](https://img.xkw.com/dksih/QBM/2023/12/1/3379858988621824/3380140028157952/STEM/925fc86dad1c48f09cf7adf52bb8d990.png?resizew=138)
(1)求证:
;
(2)求证:
.
解:(1)取
的中点
,连接
,
,如图所示.
![](https://img.xkw.com/dksih/QBM/2023/12/1/3379858988621824/3380140028157952/EXPLANATION/d8115a60a30b48ac99ac18fb6deb2ed9.png?resizew=139)
在
中,
,
分别为
,
的中点,
,
.
由题意知,四边形
为_ .
为
的中点,
,
.
,
.
四边形
为平行四边形,
.又_ ,
平面
,
.
(2)
为直三棱柱,
平面
.
又
平面
,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
_ .
,且
,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
_ .
又
平面
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
_ ,
.
以上题目的解答过程中,设置了①~⑤五个空格,如下的表格中为每个空格给出了两个选项,其中只有一个符合逻辑推理.请选出符合逻辑推理的选项(只需填写“A”或“B”).
如图,在直三棱柱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36c4559d27e3905980d1a4f1856f07de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://img.xkw.com/dksih/QBM/2023/12/1/3379858988621824/3380140028157952/STEM/925fc86dad1c48f09cf7adf52bb8d990.png?resizew=138)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33cce492aefef0c3a24fffcae3a3ccba.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9874eca4abea481fa84eb772a920f9c7.png)
解:(1)取
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1f229274a6e17977cc047814212589.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/735056c174e8dd7906257a2a50a962a7.png)
![](https://img.xkw.com/dksih/QBM/2023/12/1/3379858988621824/3380140028157952/EXPLANATION/d8115a60a30b48ac99ac18fb6deb2ed9.png?resizew=139)
在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4310db23fc79936c7182361e652bab1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1f229274a6e17977cc047814212589.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb1038fe742b2121709231eed48fcb11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcfbf154e19cbd0580d58ccc9bac077c.png)
由题意知,四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f96c673a2381f118ea2d3efc0bca1f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c56afd59592dbb194c87cdd725b7dc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adec13cc3d4b82438803ac7bfa18d61b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38d0463b6e3d27b5cfc1df0e6c14fbef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/758d6c28ad9f09ae4c5dbe5649cdf9f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeeadcae4a2964c73187962918724ae7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18d5a164bf56f8fb92527ad78bc10ccf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b28e425314ef91a4b7d9522ac79fbed6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e3ffd599e4fb57893b141bad96c66b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e53b212640dadf751ef7f65a78a209.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/625899f6b0246330b5ac95b6538f5ca6.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/215eb19188ab59c8ec06776d0aee2085.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a753598c7dafac4e9f2841b8b9a7132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
又
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be509ef5101aae24609ff9941cb246fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bab896c46e21eade473ddabf245263d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83499936f532ddce9068dd1ff8eb2b01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
又
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e3ffd599e4fb57893b141bad96c66b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e53b212640dadf751ef7f65a78a209.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0375c6c592f61ee820127b9261e96d5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/691da5f9c37b146ea9abbc50b8560c51.png)
以上题目的解答过程中,设置了①~⑤五个空格,如下的表格中为每个空格给出了两个选项,其中只有一个符合逻辑推理.请选出符合逻辑推理的选项(只需填写“A”或“B”).
空格序号 | 选项 |
① | A.矩形 B.梯形 |
② | A.![]() ![]() ![]() ![]() |
③ | A.![]() ![]() |
④ | A.![]() ![]() ![]() ![]() |
⑤ | A.![]() ![]() |
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