名校
解题方法
1 . (1)证明:当
时,
;
(2)已知正项数列
满足
.
(i)证明:数列
为递增数列;
(ii)证明:若
,则对任意正整数
,都有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca947e8ea00b7a485097ecafd2dfcae9.png)
(2)已知正项数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f91f8b7476e67db488d85c3a14ffa6d.png)
(i)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0496f142d8ae5acb06e83526eaa3ef87.png)
(ii)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90f2db682457da2d4abd0e7cca1bdf40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38d8a310323506a3c2f3626dec8d781f.png)
您最近一年使用:0次
解题方法
2 . 已知
且
,设
是空间中
个不同的点构成的集合,其中任意四点不在同一个平面上,
表示点
,
间的距离,记集合![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58a65680a7f5b5b93239c7dbdc1edd22.png)
(1)若四面体
满足:
,
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce519312a849963b376c202c3f9d7cf7.png)
①求二面角
的余弦值:
②若
,求![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18c84afeae87337f9b22fa12902222d1.png)
(2)证明:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e1ef3399691fa63838aa0474d25b9dc.png)
参考公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5ebfda261c4a27e1fa2ee5fc6d4bdfb.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/58a65680a7f5b5b93239c7dbdc1edd22.png)
(1)若四面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95c12f98844971f91baaeed4775a72e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bd6a2b112facda441f4e34bf5c145fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce519312a849963b376c202c3f9d7cf7.png)
①求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c2898853a3396f0878af9eac934416d.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e10e0b10442a269fe929eb8e592cb1ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18c84afeae87337f9b22fa12902222d1.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e1ef3399691fa63838aa0474d25b9dc.png)
参考公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c27d71b7260e008ebefdb79da3a2f3e4.png)
您最近一年使用:0次
名校
解题方法
3 . 已知
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4d7c90daa21eebaa46dd133d8e2f903.png)
(1)证明:当
时,
;
(2)令
,
(i)证明:当
时,
;
(ii)是否存在正实数
,使得
恒成立,若存在,求
的最小值,若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40f4e1236d7dc0366d9523d0cbb426be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4d7c90daa21eebaa46dd133d8e2f903.png)
(1)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e636cb16fed46289f92b91910986cdf6.png)
(2)令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbf6f1bd3b60dd7cf0d288ecb38922c5.png)
(i)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03488728d98c4c61c0b1998ccbbb535c.png)
(ii)是否存在正实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2b7ea47ee2b64ebfc1f06da577f07d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
4 . 高斯是德国著名的数学家,近代数学的奠基者之一,享有“数学王子”的称号,用其名字命名的“高斯函数”定义为:对于任意实数x,记
表示不超过x的最大整数,则
称为“高斯函数”.例如:
,
.
(1)设
,
,求证:
是
的一个周期,且
恒成立;
(2)已知数列
的通项公式为
,设
.
①求证:
;
②求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7179c645736d68c90023f83d7f11ed01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a84bb2c73d7560f8543ee90fd3cfd87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ce33c9b4713d3027ffcc1321800bfdc.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc886ed87255efa6007b3e5d1df429a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b92b70365c63607daecdc8deb73ecf.png)
(2)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69cde4038ab5a4ede107d02d41861fba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/058f67ed1a4372f6a807c14d4c8fa3a4.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbb605f0ddd311d1a092c5be5ae29260.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21f275ffdb11a31a07fc8569ddda7a6a.png)
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名校
5 . 在平面直角坐标系中,点在抛物线
上.
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8c63427d85384dea74549213d705149.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b66a5b7813e902306477f91f9f4084cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e5c62f22d7afc5627fcb86599faa8e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
①若,求证:
;
②设圆在
,
处的切线的交点为
,求证:直线
过定点.
附:若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb6a4781b020b879519321e05c299f6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e885d3743ae3d9f0fbd740b75f900083.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb6a4781b020b879519321e05c299f6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/972da921452cc4d0332afe28b11098b4.png)
您最近一年使用:0次
名校
6 . 抛物线
与椭圆
有相同的焦点,
分别是椭圆的上、下焦点,P是椭圆上的任一点,I是
的内心,
交y轴于M,且
,点
是抛物线上在第一象限的点,且在该点处的切线与x轴的交点为
,若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16231e127f8a00b343c1986f65f0ab56.png)
____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8516f71467b419293fa27df70bdaed74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/332f0dc7afc21adbab48acae2eaf875b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33d776753746914c2410a3946c357f35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ede935419d69a161bb22fd513647da06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3900190c901795456d20b1939916dafd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcc675d3c35af2425ec134743250ceae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c153922d3e1fec7dcb99c1713459547.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57aa9c9b0ab417c0b952809669f6161b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16231e127f8a00b343c1986f65f0ab56.png)
您最近一年使用:0次
2024-02-27更新
|
908次组卷
|
5卷引用:重庆市开州中学2024届高三下学期高考模拟考试(二)数学试题
解题方法
7 . 设
为抛物线
的焦点,
是抛物线
的准线与
轴的交点,
是抛物线
上一点,当
轴时,
.
(1)求抛物线
的方程.
(2)
的延长线与
的交点为
,
的延长线与
的交点为
,点
在
与
之间.
(i)证明:
,
两点关于
轴对称.
(ii)记
的面积为
,
的面积为
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3baa6d3cf25af55055fb8e1c4dcccd91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/323232ab36943d1d5d2831d70ffcff87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a780261d7c91ccf6b5dc0f580146b1a4.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(i)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(ii)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6a0c85deb80d8e63bc60127e803f7ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72d4c08ed526b54460c4d6fda1c11b10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74d7df4d3cf520d9d9355394c0a884bb.png)
您最近一年使用:0次
2024-02-05更新
|
525次组卷
|
3卷引用:重庆市缙云教育联盟2024届高三下学期3月月度质量检测数学试题
8 . 正多面体又称为柏拉图立体,是指一个多面体的所有面都是全等的正三角形或正多边形,每个顶点聚集的棱的条数都相等,这样的多面体就叫做正多面体.可以验证一共只有五种多面体.令
(
均为正整数),我们发现有时候某正多面体的所有顶点都可以和另一个正多面体的一些顶点重合,例如正
面体的所有顶点可以与正
面体的某些顶点重合,正
面体的所有顶点可以与正
面体的所有顶点重合,等等.
(1)当正
面体的所有顶点可以与正
面体的某些顶点重合时,求正
面体的棱与正
面体的面所成线面角的最大值;
(2)当正
面体在棱长为
的正
面体内,且正
面体的所有顶点均为正
面体各面的中心时,求正
面体某一面所在平面截正
面体所得截面面积;
(3)已知正
面体的每个面均为正五边形,正
面体的每个面均为正三角形.考生可在以下2问中选做1问.
(第一问答对得2分,第二问满分8分,两题均作答,以第一问结果给分)
第一问:求棱长为
的正
面体的表面积;
第二问:求棱长为
的正
面体的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/785869573d25ad8fe2cffd37dfcab4fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d8fa6d22b58fbd61c43ee524cb30394.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
(1)当正
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
(2)当正
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
(3)已知正
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
(第一问答对得2分,第二问满分8分,两题均作答,以第一问结果给分)
第一问:求棱长为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
第二问:求棱长为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
您最近一年使用:0次
2023-11-10更新
|
562次组卷
|
3卷引用:重庆市乌江新高考协作体2024届高三上学期高考第一次联合调研抽测数学试题
重庆市乌江新高考协作体2024届高三上学期高考第一次联合调研抽测数学试题(已下线)专题22 新高考新题型第19题新定义压轴解答题归纳(9大核心考点)(讲义)上海师范大学附属中学闵行分校2023-2024学年高二上学期期中数学试题
23-24高三上·重庆·开学考试
9 . 已知抛物线
,过
且斜率为相反数的直线
,
交抛物线于A,B两点(异于点P),点H为
的垂心.
(1)证明:点H在定直线上;
(2)若有且仅有2个不同的
面积为S,求S的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e8953ded144195804384dcb494d5e2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3c9708ef0dc6d6f5dcf6596d3e4f6e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
(1)证明:点H在定直线上;
(2)若有且仅有2个不同的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f584dfa75ec20e4cba4216998b454dd.png)
您最近一年使用:0次
名校
10 . 单位向量
,
,
的两两夹角为
,若实数
,
,
满足
,则下列结论中正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0411792b587ddd3e04440392f011c224.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b95d660852c5226ff65a21cfb36b8b39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17ccfda9afde4441cfd1d4df5fe9622d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d88591679796c52024d11c4de641bdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/145ec8e9ef187afb4bf6d27a8ab8be22.png)
A.![]() ![]() | B.![]() ![]() |
C.![]() ![]() | D.![]() ![]() |
您最近一年使用:0次
2023-07-27更新
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745次组卷
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3卷引用:重庆市北碚区西南大学附属中学校2024届高三上学期11月期中数学试题