名校
解题方法
1 . 若函数
与
在区间
上恒有
,则称函数
为
和
在区间
上的隔离函数.
(1)若
,判断
是否为
和
在区间
上的隔离函数,并说明理由;
(2)若
,且
在
上恒成立,求
的值;
(3)若
,证明:
是
为
和
在
上的隔离函数的必要条件.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8fd1e808e015f4cb43d2e3a0529ac6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/212983432fdb9bb12719fc9be4b410d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d96ebc8df4c7e77bc256961f29a7a2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33a546f4187414229e8e7f4f95487e17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b90551ed750a9e91f39d9b5079d9fef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6f1fbec359432e5754bb5db50ba22b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/480db4ea21e9f26ba5e527716477d0c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
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|
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2卷引用:河北省沧州市部分高中2024届高三下学期二模考试数学试题
名校
2 . 已知函数
.
(1)当
时,求曲线
在点
处的切线与两坐标轴围成的三角形的面积;
(2)若
有两个极值点
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/308f5206245a0c74155f47405dc07c03.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5828873f8369183faf71181cda5b61d2.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b1ab2e5e3dd3a1c768a88eb182b44d9.png)
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解题方法
3 . 已知点
分别为椭圆
的左、右焦点,过
的直线l(斜率不为0)交椭圆C于P,Q两点,当直线l的斜率不存在时,
.
(1)求椭圆C的离心率;
(2)若点A,B分别为椭圆C的左、右顶点,且
面积的最大值为
,直线
与直线
相交于点M,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6dc791ae552024ea0df7905bf190f30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/320c0dd8f44d5c72757b1ad1318800d6.png)
(1)求椭圆C的离心率;
(2)若点A,B分别为椭圆C的左、右顶点,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38387ba1cadfd3dfc4dea4ca9f613cea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf702adb116c1e46569eb7050d029f49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c3d833dad7bda8887fdfcaf00bb4c49.png)
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名校
4 . 设
,
,
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adc199f6ba069286a6ed3b38215ee972.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0eaf6f945de3707e5fccbfed7915d0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4580cc037c0c760c728cdbb74a8154c6.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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|
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2卷引用:河北省石家庄市第二中学教育集团2023-2024学年高二下学期期中考试数学试题
名校
解题方法
5 . 柯西是一位伟大的法国数学家,许多数学定理和结论都以他的名字命名,柯西不等式就是其中之一,它在数学的众多分支中有精彩应用,柯西不等式的一般形式为:设
,则
当且仅当
或存在一个数
,使得
时,等号成立.
(1)请你写出柯西不等式的二元形式;
(2)设P是棱长为
的正四面体
内的任意一点,点
到四个面的距离分别为
、
、
、
,求
的最小值;
(3)已知无穷正数数列
满足:①存在
,使得
;②对任意正整数
,均有
.求证:对任意
,
,恒有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81a8a1b208f491296432e9e6bf0e91c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0653d6a0e8778ad47b06d5f6b88cffa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/419c991c4022ef12d4801e119018b587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f31a068fb311eff550b3088a212fb2f0.png)
(1)请你写出柯西不等式的二元形式;
(2)设P是棱长为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5edf900c810371fb21297c15f86d8743.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b31ac1def558351e2e3ed1235c570530.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/342d0252c1b2f7d2a84b5c985d19d547.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d31659f106fba3c9750661eb0e3c3eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dde93376f5d29f8f7d501122759b0ab.png)
(3)已知无穷正数数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c24ecf9e59082e563372b12981d03fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ee33826e02eda7aa6221649355a5709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9db6b0bf3d360830fff618193c595b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a33ac34aa03dc7f0a5faad6dc664ec6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cca1d86c9f078347773f700fee49d1d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d191d6de821fbb06a51b5a20112db6de.png)
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|
337次组卷
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2卷引用:河北省邯郸市2024届高三下学期高考保温数学试题
名校
解题方法
6 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00122222a860a62964ae33dae78b10a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e5ff2705eb737adef9a6dc70559d79.png)
(1)若
,求
的取值范围;
(2)若
既存在极大值,又存在极小值.
①求a的取值范围;
②当
时,
分别为
的极大值点和极小值点,且
,求实数k的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00122222a860a62964ae33dae78b10a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e5ff2705eb737adef9a6dc70559d79.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b108ab31cc093f03cf48ad65429889e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
①求a的取值范围;
②当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7326ea56be82bd616fec7e6aa3c884c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f386803debe019dfca91cb18a09c1b1.png)
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2024-06-03更新
|
170次组卷
|
2卷引用:河北省邢台市第一中学2023-2024学年高二下学期期中测试数学试题
7 . 已知函数
.
(1)求函数
的单调性;
(2)若
有两个不相等的零点
,且
.
①证明:
随
的增大而增大;
②证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2171c919d54a0168197a4ae8222792c8.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1afa638b397608f0ad39f1beab9b2243.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26d8dafc71b106f39f4e15442220897b.png)
①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8dc531505ec45b8eb8ae4fad88d69e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
②证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7f7298bf3df9eef84d25c1ccaedf3e6.png)
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解题方法
8 . 对于函数
和
,设
,若存在
使得
,则称
和
互为“零点相邻函数”.设
,
,且
和
互为“零点相邻函数”.
(1)求
的取值范围;
(2)令
(
为
的导函数),分析
与
是否互为“零点相邻函数”;
(3)若
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed1f6dc102cde1ad73261dd011fe2d91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e288596fa3811dd2c17bded60e82e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7c5d913c7b5aaf5a3ed0054e6b4647c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b22ec92162aa1b78e8768e5fa0294f41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6caa47b05ebbf9816c4e6f159c740f0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b4b7fddfba71bb09e7e5ab7f1a2f213.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22add663bd26e87d972a10dc5fd9ada1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34937ab7546361c8bb4873a164ced32b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28789ca506fc253b4019f92998e14094.png)
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名校
9 . 若数列
若满足递推关系
其中
为常数,我们称该数列为k阶常系数齐次线性递推数列,并称方程
为递推关系式(*)的特征方程,该方程的根称为数列
的特征根.我们有以下结论:对于k阶常系数齐次线性递推数列,若其不同的特征根为
,
,…,
,且特征根
的重数为
,则数列
的通项公式为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a573c3e3b4c5b02a85d309aee9ffbc2.png)
其中
,
,这里
都是常数,它们由数列初始值可以确定.
(1)若数列
满足
,且
,
,
,求数列
的通项公式;
(2)若数列
满足对于所有非负整数m,n(
),
都成立,且
,求数列
的通项公式;
(3)设边长为1的正六边形ABCDEF,O是六边形的中心,除了六边形的每一条边,我们还从点O到每个顶点连一条线段,共得到12条长度为1的线段,一条路径是指动点沿着上述线段(全部或部分)移动,始点终点均为点O的一条移动路线.求长度为2024的路径共有多少条?(注:根的重数就是方程中同样根的数量)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22ab67ba8b0719104e78cfa6ce029290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11e26bb035fe18631ca09dd61ba446d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0c87ab1d7f0eaf58fb90e7087ad7e71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.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/8256967311eda335e21bb88f6e726fb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec6ba141730fd5aae78ada1a8eb17d21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d68e94f023b09352f46cf2ff3afb291c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a573c3e3b4c5b02a85d309aee9ffbc2.png)
其中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed9176aeda3df453783774182340e074.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f9b96ef08d0169c0c8ff9a06eb0c5d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1c5521a39235f0b9cdf432d5903aa83.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac06043337b08fece3c5762766fdb2a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/329785900390130a04a57d0b55aaa569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ddb84ee3769b8977d138638120ed820.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bd68dad20a530c17474ad6c73be07e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b67fc21d26aead8dcbfb36d7df8aa895.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)设边长为1的正六边形ABCDEF,O是六边形的中心,除了六边形的每一条边,我们还从点O到每个顶点连一条线段,共得到12条长度为1的线段,一条路径是指动点沿着上述线段(全部或部分)移动,始点终点均为点O的一条移动路线.求长度为2024的路径共有多少条?(注:根的重数就是方程中同样根的数量)
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10 . 设函数
,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e5f844201df5c98dd55b9f383377634.png)
A.n为奇数时,![]() ![]() |
B.![]() ![]() ![]() |
C.![]() ![]() ![]() |
D.![]() ![]() |
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