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
7日内更新
|
1112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)
您最近一年使用:0次
名校
3 . (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)
您最近一年使用:0次
解题方法
4 . 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)
您最近一年使用:0次
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.
您最近一年使用:0次
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)
您最近一年使用:0次
解题方法
7 . 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)
您最近一年使用:0次
名校
解题方法
8 . 如图,在圆台
中,
为轴截面,
为下底面圆周上一点,
为下底面圆
内一点,
垂直下底面圆
于点
.
平面
;
(2)若
为等边三角形,求平面
和平面
的交线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae0a4f38420bb9215dbc9c875b755838.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9b7b7793d29d66dfdd89e7a6564a35c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e64beb125bd45dde1a2b17cdd74001ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce1b066f8869d0ff4513f7a99745125.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6d0f9440606475f093d453bfa4d08e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df1e2381971c4dbd3d53dea8ce33e086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a8035fc825a001d7d9a3dacd8271662.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbd1c4e883518a7ac5a7517615e47e86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a8035fc825a001d7d9a3dacd8271662.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1eddaf3f33bd9a99162c061c9dd99aee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e61dc0fec2de4694075281e882d3c5ac.png)
您最近一年使用:0次
2024-05-01更新
|
884次组卷
|
3卷引用:数学(广东专用03,新题型结构)
9 . 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)
您最近一年使用:0次
名校
解题方法
10 . 基本不等式可以推广到一般的情形:对于
个正数
,它们的算术平均不小于它们的几何平均,即
,当且仅当
时,等号成立.若无穷正项数列
同时满足下列两个性质:①
;②
为单调数列,则称数列
具有性质
.
(1)若
,求数列
的最小项;
(2)若
,记
,判断数列
是否具有性质
,并说明理由;
(3)若
,求证:数列
具有性质
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efff8ec14cb242e793afab4468bf2e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2617515e5ce81b3f5d9f4e806b21b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6879960be91ea52297d587e9a014f54a.png)
![](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/dad2a36927223bd70f426ba06aea4b45.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bce59ae5baacab766b0915722377a746.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1bc99b9545c8c838e99b7be9c6d1046.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f20e03ee7d9307a0a4d242fffda381d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d013861990cf331c82eb453416ae31bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4247739746b8ddf1403541047e8b5580.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
您最近一年使用:0次
2024-02-21更新
|
3141次组卷
|
7卷引用:黄金卷04(2024新题型)
(已下线)黄金卷04(2024新题型)广东省广州市西关外国语学校2023-2024学年高二下学期期中数学试题安徽省部分省示范高中2024届高三开学联考数学试卷湖南省2024年高三数学新改革提高训练三(九省联考题型)湖北省荆州市沙市中学2024届高三下学期3月月考数学试题(已下线)压轴题03不等式压轴题13题型汇总-2辽宁省朝阳市建平县实验中学2024届高三第五次模拟考试数学试题