1 . 已知椭圆
,点
在椭圆上,如图,用
表示椭圆在点
处切线的单位向量.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/30/198f7b15-d256-45b8-8ad4-3cc9057793bf.png?resizew=120)
(1)设
,求
的最大值;
(2)是否存在定圆
,使得圆
的任一切线与
的交点
满足
,若存在,求出圆
方程,若不存在,请说明理由
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4a6e483672a226118dff5a39aa28449.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f4fbb5a4568dd3e4baec9f8358552b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58e8744362c6e224146461b97faf9821.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/30/198f7b15-d256-45b8-8ad4-3cc9057793bf.png?resizew=120)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75874285daf265905257368573ded035.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)是否存在定圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0a21698cc2ceabd28d995692ab2bfc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.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/3825ccc273ef9a672a606432d165b866.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
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名校
2 . 如图,给定外心为
的锐角
,令
分别为
到对边的垂足.
为
的外接圆在
和
处的切线的交点.一条经过
且垂直于
的直线交直线
于
为
在
上的投影.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80273dbcfcd2fda629a42d425ff25199.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e30c0a5c92f50dce1f7624709950ff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](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/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8ca00309261a540934d9b3ed9ba05b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95381ea2e4234a389f04150ff11660ef.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/1/73eec771-3787-4927-b668-78515fb9d732.png?resizew=198)
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3 . 甲、乙两人分别进行投硬币和掷图钉试验,每人各进行100次试验.设
为前k次试验中硬币正面向上的次数,
为前k次试验中图钉针尖朝下的次数,记
.
(1)若
,问是否存在常数P,不论试验过程中
如何变化,均存在某个
,使得
?若存在,求出所有P的可能值;若不存在,请说明理由;
(2)若
,问是否存在常数Q,不论试验过程中
如何变化,均存在某个
,使得
?若存在,求出所有Q的可能值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f255d0395fba51ca2d44293cca42e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/233427826eb2233641fc3a9805f6d206.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/115e6f9387ea866bc7a9065bb256f855.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae727a3da730ebf0c3e2d6673d7fdb91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbe0f61ec213734b7ae080719fa5cdd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afe2b62374f95f9d569a6d5739dda128.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9f668c772ae36645251661e083ce7f8.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98c5596c8546e9600062fea9f799f251.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9778f679a2495d92a52b36e5e86d4b31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afe2b62374f95f9d569a6d5739dda128.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46f6d82d629fe8d24b2d8a14a71a8bbf.png)
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4 . 设点
,过点F作斜率为k的直线l交椭圆
于C,D两点.
(1)记直线
的斜率分别为
.从下列①②③三个式子中任选其一,当k变化时,判断该式子是否为定值,若是,求出定值;若不是,请说明理由.
①
;②
;③
.
(2)当直线
分别交双曲线
的下支于P,Q两点(异于点B)时,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20421239f70724807fc2e42f74b1f02d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f182db8ffa0d36588407eae8aeb7aa92.png)
(1)记直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5913ac8f576f9389802005ae712205d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c65902e35640cf2c8d4111c36b40145.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b881044b5c73db6fcce110525741b02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89c20aa9e89d8cc15be390051124ee5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0601dc0a83ff1c3b01dce7cdaa18cabb.png)
(2)当直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2013303974d530fce9a9930938f4c2b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2aea802c49af59b44b8de07f6ef56d90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411e6e928c39bb54f031aa3e5d492852.png)
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解题方法
5 . 设函数
的定义域为I,区间
,如果对于任意的常数
,都存在实数
,满足
,且
,那么称
是区间
上的“绝对差发散函数”.则下列函数是区间
上的“绝对差发散函数”的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/306621a2b3d220bbe34027c1aa503b28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2480f87a11c4cd450bc9454ea7276722.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fe1c31a81f198c443e71b83ca662939.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbe1b0b3f9714e1f770d4ca0dca58649.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85e86867a0285838dfe2401388d2900d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30277e0be448b4955903e81e8795e45d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab1242ec96ac54e2fd418988d5190a88.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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6 . 已知
,求最大的实数
,使得对任意大于2022的正整数
及实数
,存在集合
的一个子集
满足
对所有
恒成立且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3c5cb4d381a7d02ed31e3d92ce4306e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce1132157a33c82610c2d5035493d024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a425978da20cebf8c4c63953579e7b35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6c6971463b9839ecc1c2352ae0afb7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0478ec4c7dcce9fa624789a6223eb53d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44c4dce884147e801b50675b9c0714e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e64c35495f9f68ad197ba30902915c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d81314852212a08f2e6d36338d0368a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0a9f3aaedf6f256f10159eaf262a5b.png)
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7 . 设
为整数,数列
定义为
,
,且对任意
都有
.试求所有的
,使得这个数列的每一项都是完全平方数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c7bf0d4b73924e85ee5be8d6b170ee7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e18aa1cccf2f341a7886ca7a22d63513.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01fc0bf43fa7dcffcf1daebc1b0830c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97bccd30b914efbdf1bcedfb60d63b1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5575ee73675d9fb24280bb3d8d33361.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db0b397225aafbf5906a2b928c8f2947.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ab4717e4827480f0f6f4ded85e52eab.png)
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8 . 给定正整数n,
,记
从
的一一映射f称为是可
划分的:若X可划分为k个非空子集
,
,…,
,且
(
,2,…,k)(即
,且
,
,…,
两两的交集为空集,
).已知f是一个X的
划分的一一映射,
,
,…,
是1,2,…,n的一个排列,则
的最小值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7863f720c810be5158d84a0e9b5e5be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b056db510d3e75cc12adc855015ad448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31556cfdfe2ebbf0dacfbb9e28f968ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/307e85746fa628650484d89e3b9809ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9721059d158853671eaf19e39769b13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd8db4b168ddbcba90ac9b31d36a0432.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7f2c72ab559a0615db4c51327b78d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd2781fa12b524b3b01f02d7b9d8f68a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a88501f72db91e5cf1a42114b4300ad6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb67fb9b5a8e475087ecc2a0aaca869a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9721059d158853671eaf19e39769b13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd8db4b168ddbcba90ac9b31d36a0432.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7f2c72ab559a0615db4c51327b78d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7187f7b32e990b39cf9df84428660a42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/307e85746fa628650484d89e3b9809ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d04a8b7a7595251251b8e0b7e665e8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5453ec6a9e8b96357c888ea863ddcf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a10902a745902e85100175a0d7788fe.png)
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9 . 四面体,又叫三棱锥,是一种简单多面体.指空间两两不相交且不共线的四个平面在空间割出的封闭多面体.它有
个面、
个点、
条棱、
个二面角.若一个四面体的四个顶点
,
,
,
.则可记为四面体
.对下列特殊的四面体,请选择正确得选项( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8c4c029e552954bd493b49aeab82d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8c4c029e552954bd493b49aeab82d5.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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
A.若四面体![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
B.若四面体![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
C.各面均为直接三角形且有至少三条棱长为![]() ![]() |
D.若一个平面与正四面体相交得到一个钝角三角形,则该钝角总小于![]() |
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10 . 设
,
,且
.称
为好数,如果
使上述所定义的
满足
且
.求全体好数在数轴上所对应的所有区间的长度之和.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047056c99b39c70fa40d3c8178e5b631.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/719c69e9c14fc76e23728a24157a1539.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c80c869a76a4b0cb8633719d7f5603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40d4ebc2dd452f165ed53fc594e54e34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44be578a262959b5b2d97a1ce25f7b42.png)
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