名校
解题方法
1 . 对给定的实数a,b,q,其中
,
.如果函数
,
:满足(1)对任意的
,
且
;(2)对任意的
,
.则称
为在区间
上的一个“q-压缩函数”.区间
上所有“q-压缩函数”构成的集合记作
.
(1)判断下列函数,是否属于集合
?(直接写出结论)
①
②
③![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f80e8d2d66b432fd2b8a2830ba21e5dd.png)
(2)设
,若
求实数a的取值范围.
(3)设
.若对任意的
,
,均有
,求M的最小值,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6a46e678bf9d2df5ad4c782b3dc22f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ae99e050d0f1cfc0447304f06424d17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e02cab1add26335b3cb43d5b54c7c853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e207cf62e3a7e282eac4c4a3455bbf9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e02cab1add26335b3cb43d5b54c7c853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48318c1b3f9dffc48e3ce0626b277e4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/664c92988d61280ed12488f270151940.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a8b4f05d2ba1a5514ebf390dbbd325.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a293e0b4f751f4102c795b567a633194.png)
(1)判断下列函数,是否属于集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3c564b7a711c6ac5945a0d41b2564c8.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a740cdaca0afce73f68c890f86acc6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cef72cf8297d91045117d50b4e9d3476.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f80e8d2d66b432fd2b8a2830ba21e5dd.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db0e64d809acabedccf8a8516ff8b6a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c109bacd687055d3a6a79978dfa7c0e.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e39b23fb1290ed449f1700c42c3efc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfcd04b625189228b6d697edf095f7c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c97169936ca9785dfca39ca368c65c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b0aa28b363cb79e6f8988b9b2bad7dc.png)
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2 . 已知函数
(
).
(1)讨论函数
的单调性;
(2)当
时,
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83f1370a72ac721431abc2798ee4dba0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10bbdef421c976962a270a2beabbad91.png)
(1)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/636289ad84b4a3a51095dd32ca201f94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88e9ab25a1f62a3a17514b2903345e9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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3 . 如图所示数阵,第
行共有
个数,第
行的第1个数为
,第2个数为
,第
个数为
.规定:
.
(2)从第1行起,每一行最后一个数依次构成数列
,设数列
的前
项和为
,是否存在正整数
,使得对任意正整数
,
恒成立?如存在,请求出
的最大值;如不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5726ebd95b22e5f7971028479790df77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0623207595425920f16e76a7f8f268b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a29a285201fd7e0ad70fa7431cb89a79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df0749c4129afc0c704155f522290b25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee754f9af67705c16c167733b5ef75db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5222268dda9dcb9b660f3cbedbb37757.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9ef9ec4340eabb42722042c65cc60d8.png)
(2)从第1行起,每一行最后一个数依次构成数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23e8660fb54ba32b037b392b75316087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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4 . 已知函数
.
(1)当a=1时,求曲线
在点
处的切线方程;
(2)讨论函数
的单调性;
(3)若
的有三个零点,求a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba17f18c17d3d9f4145022baeb5c72bb.png)
(1)当a=1时,求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ea9824af71c9da5db5a00ec06063024.png)
(2)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
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名校
5 . 已知函数
.
(1)讨论函数
的单调性;
(2)设
,对任意
,且
,使
恒成立,求正实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5af60d06658c16bb2bb2dae7007c9da.png)
(1)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a84cb24c6b9d85d81f57999b275c86e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e694ba0d3ee2b92073f41ef5aad4eecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/684bcf84f0a266515bfafde0da903050.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a69c9837c55274b9aa0b311fbc18fbc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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名校
6 . 射影几何学中,中心投影是指光从一点向四周散射而形成的投影,如图,光从
点出发,平面内四个点
经过中心投影之后的投影点分别为
.对于四个有序点
,若
,
,定义比值
叫做这四个有序点的交比,记作
.
时,称
为调和点列,若
,求
的值;
(2)①证明:
;
②已知
,点
为线段
的中点,
,
,求
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42c2d86d8daea5e652d99fe1c6bc3f9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34fc2a215a63f1846cdc94cc0260d4ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcda6a2a013e61f30eac744d57ab86fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9440bcb5362e00e5a6b4af27940b3007.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a881d00bcb6fcdc1029c55898c464d3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e1c84057882768f20a01365c81b6760.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09e839a2f596ac7266b6ff41a35c4a94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274f162e5e5a9d358342ddbe2b6c1519.png)
②已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73966616bd0b56416b4089a6dc884347.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c5ed371ae0038e0d5d2717418869b38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70bcf4326b5da2c4cf1caf567b55d1a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bb0c703f6effcbcf1770569971b3cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d34da8e5ecc3d124fd1455c8a18bd45a.png)
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7 . 已知椭圆
短轴长为2,椭圆
上一点
到
距离的最大值为3.
![](https://img.xkw.com/dksih/QBM/2024/5/26/3504894981390336/3520242638839808/STEM/32c129353263403684caca44141869ef.png?resizew=151)
(1)求
的取值范围;
(2)当椭圆
的离心率达到最大时,过原点
斜率为
的直线
与
交于
两点,
分别与椭圆
的另一个交点为
.
①是否存在实数
,使得
的斜率
等于
?若存在,求出
的值;若不存在,说明理由;
②记
与
交于点
,求线段
长度的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7e5578ca83f5bd5c285994061b9c015.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9059420851d219dba2189975f8eb5178.png)
![](https://img.xkw.com/dksih/QBM/2024/5/26/3504894981390336/3520242638839808/STEM/32c129353263403684caca44141869ef.png?resizew=151)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)当椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7708c156249bb475564027f73313fa8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f2ec2e454afe8452a6f1714a986aa38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235557c7e459c7100787e436826c198e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9f963b16dcebd808b41842febc329a6.png)
①是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7e08d5c04f0431fb57b33a01717b599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e4b2e1a496f0df7918efed76a51f34c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
②记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
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2卷引用:辽宁省大连市部分学校2024届高三下学期联合模拟考试数学试题
8 . 已知数列
的奇数项是首项为1的等差数列,偶数项是首项为2的等比数列.数列
前
项和为
,且满足![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53c1afb2ab393153baf1dbb0b7457254.png)
(1)求
;
(2)求数列
的通项公式及数列
的前2k项和
;
(3)在数列
中,是否存在连续的三项
,按原来的顺序成等差数列?若存在,求出所有满足条件的正整数
的值;若不存在,说明理由
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53c1afb2ab393153baf1dbb0b7457254.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fec70234b2136c08abd7a59726cc0ea0.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1254067b56565394384de5be7b8f3ec1.png)
(3)在数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a91239c38be30570f5905f56d03b0ecb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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2卷引用:上海市吴淞中学2023-2024学年高二下学期第二次调研(5月)数学试卷
名校
9 . 如图所示,在半径为1的球
的内接八面体
中,顶点
分别在平面
两侧,且四棱锥
与
都是正四棱锥.设二面角
的平面角的大小为
.
体积的最大值;
(2)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73cbd9eb22f75ad5304d8491b314a9a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55c6caa0455442437177ab9b995df37b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f96627abd793ca157d4dd1587f584d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73cbd9eb22f75ad5304d8491b314a9a9.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc24d605ad707ad0e76059d8a31f50d3.png)
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10 . 已知函数
,
.
(1)讨论
的单调性;
(2)若
有两个零点,求实数
的取值范围;
(3)若
对任意的
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5eaa6d55cdb24cff59f22f8a09b27160.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/026d50aeb347823e800aa11442b80331.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/780876acd6f251de9b8510f4def91b5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2fb40a36a293471742ce75f6b9635b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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4卷引用:山东省菏泽第一中学八一路校区2024届高三5月月考数学试题