名校
解题方法
1 . 数列
满足
则称数列
为下凸数列.
(1)证明:任意一个正项等比数列均为下凸数列;
(2)设
,其中
,
分别是公比为
,
的两个正项等比数列,且
,证明:
是下凸数列且不是等比数列;
(3)若正项下凸数列的前
项和为
,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0bee75d4d83c0b76421fd87113e4dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)证明:任意一个正项等比数列均为下凸数列;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f67fc95a626251da11649acb5e1706f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c340d7d093dd4a275ffea4b87cd26827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6268630d5e5288048d32f4aa5c8bc02d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c171ff5c2728e7cf00a88f88de14f308.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3755d7aa870e2f199d6c12264fc9be86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
(3)若正项下凸数列的前
![](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/0002f427eded1721f43d60dd0fd3ffe0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd419dc0a6580ab97777b2cb8fd7cded.png)
您最近一年使用:0次
2024-06-12更新
|
1142次组卷
|
5卷引用:2024届广东省三模数学试题
2 . 已知
为坐标原点,
,
.
(1)判断
的形状,并给予证明;
(2)若
,求证:
、
、
三点共线;
(3)若
是线段
上靠近点
的四等分点,求
的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51116e96f4c35d90677e91e0aa914111.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9612a17c77d5d6ded6123e12f9c8914.png)
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4bdb30cad5418d2b634e697d2d8e46e.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)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
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解题方法
3 . 在数学中,不给出具体解析式,只给出函数满足的特殊条件或特征的函数称为“抽象函数”.我们需要研究抽象函数的定义域、单调性、奇偶性等性质.对于抽象函数
,当
时,
,且满足:
,均有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367936b458618efb6b2eadc843e5d6ba.png)
(1)证明:
在
上单调递增;
(2)若函数
满足上述函数的特征,求实数
的取值范围;
(3)若
,求证:对任意
,都有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c73a98c1b3504e09bfbe0db849b0d24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6571b33b56c6cd88f2f6e091031bcf40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367936b458618efb6b2eadc843e5d6ba.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/639c5f8b7a1a268c904d04356f0d1b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/249a976e88133f3b3733f09137cf5c42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3be9b79f42bbf0de1851607050c3e8d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/219598f1289ddb370d632ea141731d52.png)
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名校
4 . (1)证明“直线与平面垂直的判定定理”:如果一条直线与一个平面内的两条相交直线垂直,则该直线与此平面垂直.
已知:如图,
,
,
,
.求证:
;
![](https://img.xkw.com/dksih/QBM/2023/11/17/3369796464435200/3370169716801536/STEM/653a2bc095e040b2a0c772ff8704c289.png?resizew=130)
(2)证明:平行四边形两条对角线的平方和等于两条邻边的平方和的两倍.
如图,四边形
是平行四边形.求证:
.
已知:如图,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6182bd53bccdad13334835221362a4d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60750b5eab6344496e925eb603cab46a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff290c28b42c8380283f6259daaec5c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac16b6d9ffc65507c5cd4083a1363937.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e380108ba2cf04e68a5a9393d2b921c.png)
![](https://img.xkw.com/dksih/QBM/2023/11/17/3369796464435200/3370169716801536/STEM/653a2bc095e040b2a0c772ff8704c289.png?resizew=130)
(2)证明:平行四边形两条对角线的平方和等于两条邻边的平方和的两倍.
如图,四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7105465941e9c130703b15790c6c1ecf.png)
![](https://img.xkw.com/dksih/QBM/2023/11/17/3369796464435200/3370169716801536/STEM/35d2213ed5264d45abd83c78d2631c9a.png?resizew=141)
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名校
5 . 如图1,AB为⊙O的直径,点P是直径AB上任意一点,过点P作弦
,垂足为P,过点B的直线与线段AD的延长线交于点F,且∠F=∠ABC.
![](https://img.xkw.com/dksih/QBM/2022/8/3/3036407790297088/3042341699436544/STEM/1f38f551e5a54e12bfa08392d396732b.png?resizew=302)
(1)若CD=
,BP=4,求⊙O的半径;
(2)求证:直线BF是⊙O的切线;
(3)当点P与点O重合时,过点A作⊙O的切线交线段BC的延长线于点E,在其它条件不变的情况下,判断四边形AEBF是什么特殊的四边形?请在图2中补全图象并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b757f0c42ae5c9a2d6a4b19e5877b27.png)
![](https://img.xkw.com/dksih/QBM/2022/8/3/3036407790297088/3042341699436544/STEM/1f38f551e5a54e12bfa08392d396732b.png?resizew=302)
(1)若CD=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38387ba1cadfd3dfc4dea4ca9f613cea.png)
(2)求证:直线BF是⊙O的切线;
(3)当点P与点O重合时,过点A作⊙O的切线交线段BC的延长线于点E,在其它条件不变的情况下,判断四边形AEBF是什么特殊的四边形?请在图2中补全图象并证明你的结论.
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解题方法
6 . (1)求函数
的单调区间.
(2)用向量方法证明:已知直线l,a和平面
,
,
,
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9b6a91900d0dfa6296cdee22fdd6fe6.png)
(2)用向量方法证明:已知直线l,a和平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c29f79e8e51e7c35213df9ebe697bcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72d2a947e3fdc214d40a7d3f54679a73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c9b2c3117321788078867bd0701743b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ad25ad7785af488a004cae4436019ff.png)
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7 . 若函数
的图象上的两个不同点处的切线互相重合,则称该切线为函数
的图象的“自公切线”,称这两点为函数
的图象的一对“同切点”.
(1)分别判断函数
与
的图象是否存在“自公切线”,并说明理由;
(2)若
,求证:函数
有唯一零点且该函数的图象不存在“自公切线”;
(3)设
,
的零点为
,
,求证:“存在
,使得点
与
是函数
的图象的一对‘同切点’”的充要条件是“
是数列
中的项”.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbf5beca5f1a475dbf003bb2e27d51dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbf5beca5f1a475dbf003bb2e27d51dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbf5beca5f1a475dbf003bb2e27d51dd.png)
(1)分别判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2750eb2ffdae5d0be38bda2ebb51875b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43c6e387dd234bb49f53df1668d5e63e.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10bbdef421c976962a270a2beabbad91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4157d7a3d18b13df5428790499406f7d.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039124ad765f2a9d8d3382bdc60a3d6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca1551e58c685b32149bffcb9329e710.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdab40c21646025ac21019cf6e883c54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a943c3df48c0961838d083e1c34fdbdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4fa720a5bafa2bb6ec5c60197e74a5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2b9643da0c0fea4f099f9a9133d6076.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3d2fad3eba14b645100f279cf2af2ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086e9b14c35ef3c57b20f5e952ebf9c8.png)
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8 . 定义两个
维向量
,
的数量积
,
,记
为
的第k个分量(
且
).如三维向量
,其中
的第2分量
.若由
维向量组成的集合A满足以下三个条件:①集合中含有n个n维向量作为元素;②集合中每个元素的所有分量取0或1;③集合中任意两个元素
,
,满足
(T为常数)且
.则称A为T的完美n维向量集.
(1)求2的完美3维向量集;
(2)判断是否存在完美4维向量集,并说明理由;
(3)若存在A为T的完美n维向量集,求证:A的所有元素的第k分量和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c3e014b5001732bc4b37be2b03c4033.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f00efefb7f52ad5c9dbdb180e577ee54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028076d0553b70f0fdae6beff69a10ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8c5d928c389d3abb01ca33fedf17efb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00da2c261a6ecd7533ffb8e153eaa506.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4d9e604bcc449034230149a89d746a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53b4b3879d1c6debf0333008f686634e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea687406a05d37d0761cd1a3455c804f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/773ba4a7e65c27fb359ba7aadd49f797.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d344174267f996c7cefecfd6985d380.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75cbe607b41f76db6418ce01831a1d42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4d9e604bcc449034230149a89d746a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fdb50eea11f40d9f3c37052c45894a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57aabc9b21bc15ae35720679a7b6d1ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10772b376bc43eb5c33cfd7ba9771657.png)
(1)求2的完美3维向量集;
(2)判断是否存在完美4维向量集,并说明理由;
(3)若存在A为T的完美n维向量集,求证:A的所有元素的第k分量和
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/912092af5b5301c659e6e86a7e858f38.png)
您最近一年使用:0次
2024-04-23更新
|
659次组卷
|
2卷引用:广东省汕头市潮阳区河溪中学2023-2024学年高三下学期第二学月质检数学试题
解题方法
9 . 已知抛物线
:
,直线
与抛物线
交于
,
两点,
为坐标原点.
(1)若直线
过
的焦点
.
(
,
)与
交于四点
,
,
,
,记弦
,
的中点分别为
,
,求证:线段
被定点平分,并求定点坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745de5ef1fd897d16e37464172d5e8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/1dde8112e8eb968fd042418dd632759e.png)
(1)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
(i)当的面积最小时,求直线
的方程;
(ii)当,记
的外接圆
与
的另一个交点为
,求
;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93a01cad957bd5a9e4f0c931cf5510a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dd0914dc4d4c7f75710ff460a286fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45ba6b6aa6c3f9faba6b03bc193a6e61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f0009063fe00277645aff1be6e32471.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
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名校
10 . 定义函数
的“源向量”为
,非零向量
的“伴随函数”为
,其中
为坐标原点.
的“伴随函数”为
,求
在
的值域;
(2)若函数
的“源向量”为
,且以
为圆心,
为半径的圆内切于正
(顶点
恰好在
轴的正半轴上),求证:
为定值;
(3)在
中,角
的对边分别为
,若函数
的“源向量”为
,且已知
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9153386601e89709ded16e6e56cc86b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e80896903107cb0ec517fedffa3f735.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e80896903107cb0ec517fedffa3f735.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9153386601e89709ded16e6e56cc86b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ead0f45df9fc9e5a6a90a048daf15ce0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9b0339e96e32d6fa1a092824850ef8d.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f6183bf0dcb6c744b27f6963007bda5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e723e57753f0a4fe1ef8ca1aee0e2117.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40589f60d5b9e76464c084d80fe92c0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeca565ad5dfdba18cf431dd3b84c57e.png)
(3)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18c49ca8562b98657ca9c499093f7233.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/896785f1902334350af510775d152f98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d76137ec77bd3221aa3842cabebe4910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3941f79eb3ae64e0f735ae45308e5b19.png)
您最近一年使用:0次
2024-04-07更新
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734次组卷
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2卷引用:广东省中山市迪茵公学2023-2024学年高一下学期第一次月考数学试题