1 . 已知两条抛物线
,
.
(1)求
与
在第一象限的交点的坐标.
(2)已知点A,B,C都在曲线
上,直线AB和AC均与
相切.
(ⅰ)求证:直线BC也与
相切.
(ⅱ)设直线AB,AC,BC分别与曲线
相切于D,E,F三点,记
的面积为
,
的面积为
.试判断
是否为定值,若是,求出该定值;若不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac8a3bffe545af2299cf999d44767206.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d1138c04fc3a0e1c217db0d432e4aff.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(2)已知点A,B,C都在曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(ⅰ)求证:直线BC也与
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(ⅱ)设直线AB,AC,BC分别与曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235f0a6fb218d28383e6f27f2df1f50f.png)
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2 . 对于平面向量
,定义“
变换”:
,其中
表示
中较大的一个数,
表示
中较小的一个数.若
,则
.记
.
(1)若
,求
及
;
(2)已知
,将
经过
次
变换后,
最小,求
的最小值;
(3)证明:对任意
,经过若干次
变换后,必存在
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4fde5542ad04744c14f912648f3aa0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41bd66e602e9c043218806708e943c2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb00071815c94c090a4095b4964fefb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f7bc9573b3a8758511c63731db18183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96340894e8fb63c00d778b4d654d0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f7bc9573b3a8758511c63731db18183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/701ab98a2bf1135cd989822b0738e11d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/484c1b7bc2fc5677406e20180f667200.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0624499e16b73afec432dd1afd6153d.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b162d1d5bfaa7760678ea3d624beb171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5c19921380da55f5f1a00809a34503.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35234a3829d238ea479fef9cec166468.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8aceb3666a9d49ef40c39eac116ccd5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20cfb8c8707c3960bf1fd46b805e481d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a887552671e6d4df390320ee9a36150.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)证明:对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f389ec068eb1d1aa586b79097d70a7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dcd78ec8777a8e6e5b32222cdb15c05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06296b9023c1dca6f44b8297842bef7c.png)
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名校
解题方法
3 . 已知数列
满足:
,
,其中
为数列
的前n项和.
(1)求数列
的通项公式;
(2)设m为正整数,若存在首项为1且公比为正数的等比数列
(
),对任意正整数k,当
时,都有
成立,求m的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59dd6c97d2ee3e74ba5730f1cbcc1d43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b11b53600b5f8795438b5071251fe32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
(2)设m为正整数,若存在首项为1且公比为正数的等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ef4c4439b36c2847b0056a116d56d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/930bc56406e69b785b37a83d48e36724.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d3d686e972baa72df81389555897acd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41075383e3b80a1ac8faf5edec08f620.png)
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4 . 已知定点
,动点N在直线
上,过点N作l的垂线,该垂线与NF的垂直平分线交于点T,记点T的轨迹为曲线C.
(1)求曲线C的方程;
(2)已知点P、A、B是曲线C上的点,且
.
(i)若点P的坐标为
,则动直线AB是否过定点?如果过定点,请求出定点坐标,反之,请说明理由;
(ii)若
,求
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5421a28dc3675ae20190d6090793246e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d42502a5730e1930d77d7100d1e34707.png)
(1)求曲线C的方程;
(2)已知点P、A、B是曲线C上的点,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83640592853a53872d7af69c0cffc1bb.png)
(i)若点P的坐标为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc7334402b0f5f606a82d027208e3e2c.png)
(ii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d81529452eb2cb8578fd164db9e957ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
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5 . 若有穷数列
满足:
,则称此数列具有性质
.
(1)若数列
具有性质
,求
的值;
(2)设数列
具有性质
,且
为奇数,当
时,存在正整数
,使得
,求证:数列
为等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef94592b70bea840c747393959c71b39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c735a110f4cf68dea9133c78e205b43f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf404363fe0f057007b8e8d90a775d90.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/030ef2d631bb39945bb752932146364b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf404363fe0f057007b8e8d90a775d90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc9927f218d1b9cd9d7a8b979da6c669.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf9f50605db5d5f8f3a01ee8e474a112.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88e6153f9e3bfe84d3a61f388c7fa2b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd00e20f967cb2bdce939165abd38440.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1c87acdb6ce8286ea7d256b96801507.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
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解题方法
6 . 已知集合
,
,
,若
,
,
或
,则称集合A具有“包容”性.
(1)判断集合
和集合
是否具有“包容”性;
(2)若集合
具有“包容”性,求
的值;
(3)若集合C具有“包容”性,且集合C的子集有64个,
,试确定集合C.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe38d1b1e18720a878bd7442c8f094de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ed006b944ea64f970fee46e2f558467.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e63c91626ffa91e590925e6f206c3c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/957d41dbe52b49c3a7339e3519a3fe84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c46de01c5104b9112a688df37eadb000.png)
(1)判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bac46c6c52fc8b8e8f76084352e1893f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b3ed03b0f8fb8b88d7edf6165345c6f.png)
(2)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6acf45d9d64697db902fe8faa15394d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c925be255ca736a53b24d13ddede1a86.png)
(3)若集合C具有“包容”性,且集合C的子集有64个,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f65f44d4f1341a7d80897a54a6778fae.png)
您最近一年使用:0次
7 . 若函数
在
上有定义,且对于任意不同的
,都有
,则称
为
上的“k类函数”.已知函数
.
(1)求曲线
在点
处的切线方程;
(2)求
的单调区间;
(3)若
为
上的“3类函数”,求实数a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4776c85b79df196f606d3ebf3697fbc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9d6fe21d6ed78bfc1d2b9cc41a766c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeaea651d833a0fd5a2a7f0cbaeb4ae0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4776c85b79df196f606d3ebf3697fbc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/485825faa883e045117e914fc4094908.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bea9227dd0104da58e0c40952cc87ed.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da34ce730f711c09909d53806fe2330a.png)
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8 . 已知
.
(1)讨论
的单调性;
(2)若
且
有2个极值点
,
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cb9ba8eaf2f8800b54661ab87e65978.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](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/ffd888afdcfdb3e91a157d50f65e915e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb2466125d51567743b63ab7d17739f3.png)
您最近一年使用:0次
名校
解题方法
9 . 若函数
在
上有定义,且对于任意不同的
,都有
,则称
为
上的“
类函数”.
(1)若
,判断
是否为
上的“2类函数”;
(2)若
,为
上的“2类函数”,求实数a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32f0e1ac411fd3a260a5c71df178bd7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b62f64bce0222f01a519ab1b26236bcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee4613910bb8aa030db2fc5d2768e533.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12f248318141e0016d38f9f5a692797f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a3616e4a7268ef41b750fe22afbcd74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12f248318141e0016d38f9f5a692797f.png)
您最近一年使用:0次
2024-05-08更新
|
259次组卷
|
3卷引用:广西南宁市第二中学·柳州高级中学2023-2024学年高二下学期5月联考数学试题
10 . 设
,用
表示不超过x的最大整数,则
称为取整函数,取整函数是德国数学家高斯最先使用,也称高斯函数.该函数具有以下性质:
①
的定义域为R,值域为Z;
②任意实数都能表示成整数部分和纯小数部分之和,即
,其中
为x的整数部分,
为x的小数部分;
③
;
④若整数a,b满足
,则
.
(1)解方程
;
(2)已知实数r满足
,求
的值;
(3)证明:对于任意的大于等于3的正整数n,均有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb63478132d4c1fef3c17e591919da83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f161c2a3717f1b6c62d0d7dae0b606.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5b7f26fe1977bda9de200debe99f020.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5b7f26fe1977bda9de200debe99f020.png)
②任意实数都能表示成整数部分和纯小数部分之和,即
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9643772929ed7ee674ae68adb5381265.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f161c2a3717f1b6c62d0d7dae0b606.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/216921512381b9ebbb9cc59ecc9eb427.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5f7e2f76a9643572acc81394e9b965a.png)
④若整数a,b满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4513fc3f11c7030d7c83294335de57f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1e38f0d07ed41a7e373b3f8a281eef.png)
(1)解方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ab4334cd34a187b787278e1b2cb214b.png)
(2)已知实数r满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ed5c396204fbca3ef755668b277f6a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/904778775ee8cf551428f21b5b0ca915.png)
(3)证明:对于任意的大于等于3的正整数n,均有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39219116316e31189df7d04d6b9f428b.png)
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