1 . 若当且仅当
时,等差数列
的前
项和
取得最大值,则数列
的通项公式可以是________ .(写出满足题意的一个通项公式即可)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08ec5d76db9bd05547932966c9913dc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.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/63d471926f7b27322d90c82b9ce21d3d.png)
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名校
2 . 关于任意平面向量可实施以下6种变换,包括2种
变换和4种
变换
模变为原来的
倍,同时逆时针旋转
;
模变为原来的
倍,同时顺时针旋转
;
模变为原来的
倍,同时逆时针旋转
;
:模变为原来的
倍,同时顺时针旋转
;
模变为原来的
倍,同时逆时针旋转
;
模变为原来的
倍,同时顺时针旋转![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd4c5016bd1fc6582078299b9cf8b392.png)
记集合
,若每次从集合
中随机抽取一种变换,每次抽取彼此相互独立,经过
次抽取,依次将第
次抽取的变换记为
,即可得到一个
维有序变换序列,记为
,则以下判断中正确的序号是__________ .
①单位向量
经过奇数次
变换后所得向量与向量
同向的概率为
;
②单位向量
经过偶数次
变换后所得向量与向量
同向的概率为
;
③若单位向量
经过
变换后得到向量
,则
中有且只有2个
变换;
④单位向量
经过
变换后得到向量
的概率为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc13a607ac0c7f76d252d7cb1bb040fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/595044a7750ab4f84519041979c3d780.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aabf0b3987b45fc69f88f474e1190619.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2ff7caec4fdd8fb54a3ffbff9692414.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c35f6481d876bc632b8e05c6d87b6214.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2ff7caec4fdd8fb54a3ffbff9692414.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b342300de8aeeefd2d5ee25f395df8c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1e5fa72f2878b476bc57f0df12d6555.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2a14b56f7f7bb23cd4bf9041e440a46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1e5fa72f2878b476bc57f0df12d6555.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/979301d0cd793fb5282c76d6e0e2e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd4c5016bd1fc6582078299b9cf8b392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea59db613a395984867113370a045436.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd4c5016bd1fc6582078299b9cf8b392.png)
记集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02ef6e24288a68823a7d3143eb547a72.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/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4da49c5d7719cf1a92e825f425867ece.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1d0009ab474b946a2abb29803250a1e.png)
①单位向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deab0dd26dbb0badb5f6006722f8514a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc13a607ac0c7f76d252d7cb1bb040fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/233e17f629c5cb175e4ec1fadc872f19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
②单位向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deab0dd26dbb0badb5f6006722f8514a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/595044a7750ab4f84519041979c3d780.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/468322cef1887fec96c304371e580bc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
③若单位向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deab0dd26dbb0badb5f6006722f8514a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24cb1319a55351f60be7045aa860eb38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68ff72b110ee53982970799db042867c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24cb1319a55351f60be7045aa860eb38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc13a607ac0c7f76d252d7cb1bb040fd.png)
④单位向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deab0dd26dbb0badb5f6006722f8514a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24cb1319a55351f60be7045aa860eb38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68ff72b110ee53982970799db042867c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bdb68ec34b2c959767c2ae8cdab03c2.png)
您最近一年使用:0次
2021-06-04更新
|
672次组卷
|
5卷引用:北京市2021届高三高考模拟数学试题
北京市2021届高三高考模拟数学试题北京市第二中学2022届高三上学期期中考试数学试题(已下线)课时27 平面向量的分解定理及应用-2022年高考数学一轮复习小题多维练(上海专用)(已下线)专题09 平面几何与向量-备战2022年高考数学(文)母题题源解密(全国乙卷)(已下线)第11讲 平面向量-3
名校
3 . 对集合
,定义![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aca578919b59d25a1bb2b5ea8e33da46.png)
①若
的元素个数为4,则
可以为:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f1e5d29de6e4d72bfed62d9c14dde5.png)
________ ,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f9fabbbe61a759e52ec975215e2e7c.png)
________ (写出一组即可)
②若集合
满足:存在
的子集
,使得
的元素个数不小于100,且对任意
,均有
,则集合
的元素个数的最小值是________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aca578919b59d25a1bb2b5ea8e33da46.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81e37e656d05244fe3a5769cd1446725.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f1e5d29de6e4d72bfed62d9c14dde5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f9fabbbe61a759e52ec975215e2e7c.png)
②若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81e37e656d05244fe3a5769cd1446725.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57a5235dcfd51f1ba4c1ee0037e167c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dfa843f0c5458e8b9ad3244680f7679.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
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名校
4 . 从本质上来讲,声音实际上是一种简谐振动产生的机械波,也称声波.声音两个最主要的要素:响度和音调,分别由振动的振幅和频率刻画.其中最基本的声波就是简谐振动所产生的正弦波.纯音是以某个固定频率进行简谐振动所产生的声波,且纯音的函数可以表示为:
,其中
,
,则这个函数的频率
为___________ (写出表达式即可)(注:频率是周期的倒数)一般说的
,
,
,
,
,
,
又是什么呢?这些唱名是音调的一种记法,音调
与频率
之间的关系为
.已知标准音
(也是纯音)的音调为
,那么标准音
对应的函数中![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9448bf1d6136b3a108e7ea60506ad39.png)
___________ .已知标准音
和标准音
的频率比为
,那么标准音
的音调为___________ .(取
,
,结果精确到小数点后两位).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e7c7212e16f10017a19204af1fb09e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13378be06b6b01bcad1d261ff14e87cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4456675a5dbe545462a22cef9aca8fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dd046d79dd0da3d97db1084d9e5a8dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adf06fea4c46faf2af9fb80e83929ce3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5939c344ea0a6705f2fb114d8fcc8954.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bbf7540b99474d0aa6e6d8556a4b399.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/812d71a0a42fa3007542c8a68a654188.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df9841d7e004d702fd65cadbb10e4220.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8be8824febd2a5f71b3921b1879b933b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/171102a883b22fe6ca578efc8926f5b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a84b95bf095fdba387cdf259105369.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df9841d7e004d702fd65cadbb10e4220.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ef8f307eb43bf38ddc6b1b33b94b82f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df9841d7e004d702fd65cadbb10e4220.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9448bf1d6136b3a108e7ea60506ad39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dd046d79dd0da3d97db1084d9e5a8dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df9841d7e004d702fd65cadbb10e4220.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf1484ddc948caf40e4c3243bc959584.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dd046d79dd0da3d97db1084d9e5a8dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49f8340ff8ef6994abddab919418423b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf0227933292a639a0e4e719d33f93a0.png)
您最近一年使用:0次
5 . 有一张矩形纸片
,按下面步骤进行折叠:
第一步:如图①,将矩形纸片
折叠,使点
,
重合,点
落在点
处,得折痕
;
第二步:如图②,将五边形
折叠,使
,
重合,得折痕
.再打开;
第三步:如图③,进一步折叠,使
,
均落在
上,点
,
落在点
处,点
,
落在点
处,得折痕
,
.
这样,就可以折出一个五边形
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/16/f542cf2b-0c93-4fa8-ac31-578892903940.png?resizew=141)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/16/669fd515-afbb-46a9-8023-86dc67d158c2.png?resizew=141)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/16/6c2b59a9-a749-4b81-9264-bd4b80f19283.png?resizew=141)
(1)适当添加辅助线,请写出图①中三组全等三角形______,______,______;(写出不同的三组即可)
(2)若这样折出的五边形
(如图③)恰好是一个正五边形,当
,
,
①请写出一个
与
的关系式,并加以证明;
②设正五边形的边长
,请求出边长
(用
或
表示
).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
第一步:如图①,将矩形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
第二步:如图②,将五边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8149f05bbaf25926ad9ac243a04b482.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae72f1da4f5d540dc1e6f42ea08f952e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e72b2e1ff83e95df048745322982451.png)
第三步:如图③,进一步折叠,使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae72f1da4f5d540dc1e6f42ea08f952e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e72b2e1ff83e95df048745322982451.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3b1a5427d8ff23df0f3ec194756c84c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f6927cc2a930203ac34366383e76ef.png)
这样,就可以折出一个五边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db24888830edd7df393bf95fde1281e8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/16/f542cf2b-0c93-4fa8-ac31-578892903940.png?resizew=141)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/16/669fd515-afbb-46a9-8023-86dc67d158c2.png?resizew=141)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/16/6c2b59a9-a749-4b81-9264-bd4b80f19283.png?resizew=141)
(1)适当添加辅助线,请写出图①中三组全等三角形______,______,______;(写出不同的三组即可)
(2)若这样折出的五边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db24888830edd7df393bf95fde1281e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d42e97eee705d164e6ac6de9ecd6d1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c52b857c7ca55dfa6da108df1d3cee2.png)
①请写出一个
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
②设正五边形的边长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c21350a49d9229ef239253f7c3d366ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.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/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
6 . 下列说法错误的是( )
A.使得![]() ![]() |
B.充分条件就是“有之即可,无之未必不行” |
C.必要条件就是“有之未必行,无之必不行” |
D.没有证明的猜想不是命题 |
您最近一年使用:0次
7 . 在直角坐标系
中,经过点
,且关于
轴对称的曲线的方程是__________ .(填上正确的一个方程即可,不必考虑所有的情形)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/448c0a5ee776d19ce8e42ac9a5fd27c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
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解题方法
8 . 正四棱锥
的展开图如图所示,侧棱
长为1,记
,其表面积记为
,体积记为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/e2bf084b-f7e5-47d8-add0-6ed4bfada543.png?resizew=202)
(1)求
的解析式,并直接写出
的取值范围;
(2)求
,并将其化简为
的形式,其中
为常数;
(3)试判断
是否存在最大值,最小值?(写出结论即可)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a6e2867f32d3f1c3cd36cd3a11a8580.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3c55c1c441f921d874702a4f19ed17f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/794f2c6bd63355105d179d11306a9cae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c9d76fb48eb30e7946cb96047e08206.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/e2bf084b-f7e5-47d8-add0-6ed4bfada543.png?resizew=202)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/794f2c6bd63355105d179d11306a9cae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29d0adafeb8e5d088e974f1246880055.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a296bb758c36b50b102a4ceb2dea42bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
(3)试判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29d0adafeb8e5d088e974f1246880055.png)
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2022-07-05更新
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7卷引用:北京一零一中学2021-2022 学年高一下学期期末考试数学模拟试题(一)
北京一零一中学2021-2022 学年高一下学期期末考试数学模拟试题(一)上海市洋泾中学2022-2023学年高二上学期期中数学试题湖北省郧阳中学、恩施高中、沙市中学、随州二中、襄阳三中2022-2023学年高二上学期10月联考数学试题湖北省五校(郧阳中学、恩施高中、沙市中学、随州二中、襄阳三中)2022-2023学年高二上学期10月月考数学试题湖北省黄石市第二中学2023-2024学年高二上学期9月月考数学试题(已下线)湖南省长沙市雅礼中学2024届高三上学期月考(二)数学试题变式题19-22(已下线)期中测试卷01(测试范围:第10-11章)-2023-2024学年高二数学单元速记·巧练(沪教版2020必修第三册)
21-22高二上·北京西城·期中
名校
9 . 对于给定的正整数n,记集合
,其中元素
称为一个n维向量.特别地,
称为零向量.设
,
,
,定义加法和数乘:
,
.对一组向量
,
,…,
(
,
),若存在一组不全为零 的实数
,
,…,
,使得
,则称这组向量线性相关.否则,称为线性无关.
(1)对
,判断下列各组向量是线性相关还是线性无关,并说明理由.
①
,
;②
,
,
;③
,
,
,
.
(2)已知向量
,
,
线性无关,判断向量
,
,
是线性相关还是线性无关,并说明理由.
(3)已知
个向量
,
,…,
线性相关,但其中任意
个都线性无关,证明下列结论:
①如果存在等式
(
,
),则这些系数
,
,…,
或者全为零,或者全不为零;
②如果两个等式
,
(
,
,
)同时成立,其中
,则
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d916961d15be6e499999e2fbe51c456.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac852f327effde190b9ebf3dd08e037c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58a13d67dfa8d6f81c9aa06ca42c9e68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aff8d9b6533ff319420cdc5e8740b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2668c56f2e7bb75917c8b7d7b09c8926.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b18d4591126e3801e10b93dadf2df3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14dfee189ba5596b1fe73fef044b4687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4146baa4c775fef17a3eb033b2c1691.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a40932863dcc8cb8356bbd487eaf196.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d43fd777be9b2e4fdfe3fbe9a7cdde79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15e74e0eba2b8dbf3d09f6e6382f4c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37c567e4139b2be7a0ac01f3430d2d54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1445d8e79cc2309e20001952f2b742d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d2dc187e5bae6ab191f2442948763ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5259301dfd6e050de40be969068229b4.png)
(1)对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74044239e1bc20b87b6be3f40b282675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be93e52fa914240b43743270b76a8fb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74044239e1bc20b87b6be3f40b282675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be93e52fa914240b43743270b76a8fb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d159653ffeca0f732a520f9d52eaf5fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cb112690d7eb559b09a4f7666ff10ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7308ce4466feaf0c214ac02503c83029.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/250be63dcc2692aceeed802b008e767b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca2e34d347c50b4de2ac8a306ead0e9c.png)
(2)已知向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac852f327effde190b9ebf3dd08e037c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e257e6c38a24ced4f9423f7f51b0e92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f168ab643bde113e92134b3fa8139665.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85f7e488464e41e1a1e1eed427154aa6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81c769ca2d340eb5ad4d16cd54852990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e6faf15e812aa5e0f9272c8dbaa3591.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b2ea5126ca30d68cc8b70f03860ba2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a40932863dcc8cb8356bbd487eaf196.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d43fd777be9b2e4fdfe3fbe9a7cdde79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c0c36e75691a88bd6e38e6269e69fef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f1e94f660b7d05de4be4b5fbd9041f4.png)
①如果存在等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ad741efadd0b14577a80df4c243e3b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7368220470ebd17a0fd2bcf1f9fd495.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/299d6864de87fd34448379b795776334.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27628b047da341c79074ea4aa938ddc8.png)
②如果两个等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ad741efadd0b14577a80df4c243e3b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/898ac87221725132b69725b4042bb441.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7368220470ebd17a0fd2bcf1f9fd495.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d66379e8d6cb9d00a3287d434fa3c9f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/299d6864de87fd34448379b795776334.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f165fa05c199307b89a91027dd6d696.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecd9175a6d8d1d75238a4b60db254846.png)
您最近一年使用:0次
2021-11-19更新
|
2688次组卷
|
13卷引用:北京市第四中学2021-2022学年高二上学期期中考试数学试题
(已下线)北京市第四中学2021-2022学年高二上学期期中考试数学试题北京市昌平区第二中学2022-2023学年高二上学期10月月考数学试题江苏省南通市如皋市2024届高三上学期1月诊断测试数学试题湖北省孝感市高级中学2024届高三上学期期末数学试题广东省深圳市南山区华侨城中学2024届高三下学期一模适应性考试数学试题重庆市九龙坡区育才中学2023-2024学年高一下学期寒假检测定时训练数学试题(已下线)最新模拟重组精华卷1-模块一 各地期末考试精选汇编(已下线)新题型01 新高考新结构二十一大考点汇总-3(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)(已下线)黄金卷03(2024新题型)福建省福州外国语学校2023-2024学年高一下学期3月月考数学试卷辽宁省东北育才学校双语校区2023-2024学年高一下学期期中考试数学试题湖南省衡阳市第八中学2024届高三下学期适应考试数学试题
名校
解题方法
10 . 《瀑布》(图1)是埃舍尔最为人所知的作品之一,图中的瀑布会源源不断地落下,落下的水又逆流而上,荒唐至极,但又会让你百看不腻.画面下方还有一位饶有兴致的观察者,似乎他没发现什么不对劲.此时,他既是画外的观看者,也是埃舍尔自己.画面两座高塔各有一个几何体,左塔上方是著名的“三立方体合体”由三个正方体构成,右塔上的几何体是首次出现,后称“埃舍尔多面体”(图2)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/16/68d54bad-137e-48be-86d4-e3a12933ebf6.png?resizew=315)
埃舍尔多面体可以用两两垂直且中心重合的三个正方形构造,设边长均为2,定义正方形
的顶点为“框架点”,定义两正方形交线为“极轴”,其端点为“极点”,记为
,将极点
,分别与正方形
的顶点连线,取其中点记为
,如(图3).埃舍尔多面体可视部分是由12个四棱锥构成,这些四棱锥顶点均为“框架点”,底面四边形由两个“极点”与两个“中点”构成,为了便于理解,图4我们构造了其中两个四棱锥
与
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/17/aadac78f-add7-45ab-b5bc-c5856d61f0bd.png?resizew=219)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/17/4dc361d0-a928-475e-a5ff-08809066b709.png?resizew=219)
(1)求异面直线
与
成角余弦值
(2)求平面
与平面
的夹角余弦值
(3)求埃舍尔体的表面积与体积(直接写出答案)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/16/68d54bad-137e-48be-86d4-e3a12933ebf6.png?resizew=315)
埃舍尔多面体可以用两两垂直且中心重合的三个正方形构造,设边长均为2,定义正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65e8d40a892330cb0462f5e1eb388933.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee1c51f15c934050099b460b19a04f4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9195bc5917cc0dcef221f17561d1cdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/531269bd0f80e68bdc3982e864c254e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c463077e1b30d448275ecb3db350204.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76269a5843b60ca3f361ca5510f1b9ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3515ff4df04d24912acbf35d327e1f8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/17/aadac78f-add7-45ab-b5bc-c5856d61f0bd.png?resizew=219)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/17/4dc361d0-a928-475e-a5ff-08809066b709.png?resizew=219)
(1)求异面直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ff64e3c1ca2c71aa14f1786c72993ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41264a5ce05a6cf424fb63ac6ccf42e1.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/948272ac8389de36ff0a1bed7b76ac5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ee63e2e78d42068eda47e947612829c.png)
(3)求埃舍尔体的表面积与体积(直接写出答案)
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